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Thèse

Case of slabs subjected to impact

Présentée devant

L’Institut National des Sciences Appliquées de Lyon

Pour obtenir

Le grade de docteur

Formation doctorale :

Génie Civil

École doctorale : Mécanique, Energétique, Génie Civil, Acoustique (MEGA)

Par

Fidaa KASSEM

(Ingénieur)

Jury

Rapporteur Fabrice GATUINGT Professeur (CNRS) (Université Paris Saclay)

Rapporteur Abdellatif KHAMLICHI Professeur (Faculté des Sciences de Tétouan)

Examinateur Elias BOU-SAID Ingénieur (R&D EGIS France, IOSIS Industries)

Co-directeur David BERTRAND MCF (INSA de Lyon)

Directeur Ali LIMAM Professeur (INSA de Lyon)

© [F. Kassem], [2015], INSA Lyon, tous droits réservés

INSA Direction de la Recherche - Ecoles Doctorales – Quinquennal 2011-2015

CHIMIE http://www.edchimie-lyon.fr Université de Lyon – Collège Doctoral

Sec : Renée EL MELHEM Bât ESCPE

Bat Blaise Pascal 3e etage 43 bd du 11 novembre 1918

04 72 43 80 46 69622 VILLEURBANNE Cedex

Insa : R. GOURDON Tél : 04.72.43 13 95

secretariat@edchimie-lyon.fr directeur@edchimie-lyon.fr

E.E.A. ELECTROTECHNIQUE, AUTOMATIQUE Ecole Centrale de Lyon

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69134 ECULLY

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E2M2 MICROBIOLOGIE, MODELISATION Laboratoire de Géologie de Lyon

http://e2m2.universite-lyon.fr Université Claude Bernard Lyon 1

Bât Géode – Bureau 225

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Bat Atrium- UCB Lyon 1 69622 VILLEURBANNE Cédex

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Insa : S. REVERCHON Sylvie.reverchon-pescheux@insa-lyon.fr

Safia.ait-chalal@univ-lyon1.fr fabrice.cordey@ univ-lyon1.fr

EDISS SANTE INSERM U1060, CarMeN lab, Univ. Lyon 1

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Sec : Safia AIT CHALAL 11 avenue Jean Capelle INSA de Lyon

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Insa : Emmanuelle.canet@univ-lyon1.fr

Safia.ait-chalal@univ-lyon1.fr

INFOMATHS MATHEMATIQUES LIRIS – INSA de Lyon

http://infomaths.univ-lyon1.fr Bat Blaise Pascal

7 avenue Jean Capelle

Sec :Renée EL MELHEM 69622 VILLEURBANNE Cedex

Bat Blaise Pascal Tél : 04.72. 43. 80. 46 Fax 04 72 43 16 87

3e etage Sylvie.calabretto@insa-lyon.fr

infomaths@univ-lyon1.fr

http://ed34.universite-lyon.fr INSA de Lyon

Matériaux

MATEIS

Sec : M. LABOUNE Bâtiment Saint Exupéry

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Ed.materiaux@insa-lyon.fr

MEGA CIVIL, ACOUSTIQUE INSA de Lyon

http://mega.universite-lyon.fr Laboratoire LAMCOS

Bâtiment Jacquard

Sec : M. LABOUNE 25 bis avenue Jean Capelle

PM : 71.70 –Fax : 87.12 69621 VILLEURBANNE Cedex

Bat. Direction 1er et. Tél : 04.72 .43.71.70 Fax : 04 72 43 72 37

mega@insa-lyon.fr Philippe.boisse@insa-lyon.fr

ScSo http://recherche.univ-lyon2.fr/scso/ Université Lyon 2

86 rue Pasteur

Sec : Viviane POLSINELLI 69365 LYON Cedex 07

Brigitte DUBOIS Tél : 04.78.77.23.86 Fax : 04.37.28.04.48

Insa : J.Y. TOUSSAINT isavonb@dbmail.com

viviane.polsinelli@univ-lyon2.fr

*ScSo : Histoire, Géographie, Aménagement, Urbanisme, Archéologie, Science politique, Sociologie, Anthropologie

© [F. Kassem], [2015], INSA Lyon, tous droits réservés

Dedication

ii

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© [F. Kassem], [2015], INSA Lyon, tous droits réservés

Abstract

Reinforced concrete (RC) structures are subjected to several sources of uncertainties that highly

aect their response. These uncertainties are related to the structure geometry, material properties

and the loads applied. The lack of knowledge on the potential load, as well as the uncertainties

related to the structure features shows that the design of RC structures could be made in a reliabil-

ity framework. This latter allows propagating uncertainties in the deterministic analysis. However,

in order to compute failure probability according to one or several failure criteria mechanical and

stochastic models have to be coupled, which can be very time consuming and in some cases im-

possible. Indeed, either the complexity of the deterministic model considered implies important

computing time (from minutes to hours) or reliability methods evaluating failure probability require

a too large number of simulations of the deterministic model.

The platform OpenTURNS is used to perform the reliability analysis of three dierent structures

considered in the present study and propagate uncertainties in their physical models. OpenTURNS

can be linked to any external software using the generic wrapper. Therefore, OpenTURNS is coupled

to CASTEM to study the reliability of a RC multiber cantilever beam subjected to a concentrated

load at the free end, to Abaqus to study the reliability of RC slabs which are subjected to accidental

impacts, and to ASTER to study the reliability of a prestressed concrete containment building.

Among structures considered in this study, only the physical problem of reinforced concrete (RC)

slabs subjected to impact is investigated in detail. The design of such type of structures is generally

carried out under static or pseudo-static loading. In the case of dynamic loads such as impacts, the

force applied on the structural member is often assessed from energetic approaches and then, for

the sake of simplicity, an equivalent pseudo-static loading is considered for its design. However, in

some particular cases, the dynamic response of the structure cannot be simplied to a quasi-static

response. The transient dynamic analysis has to be performed accounting for the main physical

processes involved. Thus accurate models are needed to describe and to predict the capacity of the

RC member and in particular the impact force.

This study focuses on RC slabs impacted by adropped object impact during handling operations

within nuclear plant buildings. When a dropped object impacts a RC slab, damage can arise at the

impact zone depending on the impact energy and the relative masses of the colliding bodies. An

optimal design requires taking into account the potential development of nonlinearities due to the

material damage (concrete cracking, steel yielding, etc.).

The aim of this study is to address the issue of reliability computational eort. Two strategies

are proposed for the application of impacted RC slabs. The rst one consists in using deterministic

analytical models which predict accurately the response of the slab. In general, the analytical model

assumptions are constraining and only few slab congurations can be studied. The objective is

to reduce the computational cost to the minimum and to explore as far as possible how to use

these types of analytical models. In the opposite case, when nite element models are needed, the

second strategy consists in reducing the number of simulations needed to assess failure probability

or substituting the nite element model by a meta-model less expensive in computational time.

The rst part of this study describes the behavior of reinforced concrete slabs subjected to impact

and determines the failure modes of slabs according to their characteristics and impact conditions. In

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© [F. Kassem], [2015], INSA Lyon, tous droits réservés

this study only low velocities are considered. Then two deterministic models are used and evaluated:

a 3D nite element model simulated with the commercial code Abaqus/Explicit and an analytical

mass-spring model. For the simulation based on nite element method (FEM), the RC slab and the

impactor are fully modeled. The steel is modeled as an elasto-plastic material with hardening and,

the concrete behavior is described by the damaged plasticity model. The model is validated with

experiments carried out on several RC slabs under drop-weight loads at Heriot-Watt University by

Chen and May [34]. As an alternative modeling, a simplied analytical model of the slab is used.

The model consists in a two degrees of freedom mass-spring system which accounts for potential

viscous damping. The rst (resp. the second) spring represents the stiness of the slab (resp. the

contact). A frequency decrease approach is used to describe the degradation of RC slabs stiness.

The natural frequency of the slab is updated as a function of the maximal displacement reached

during the loading.

Next, a reliability analysis of reinforced concrete slabs under low velocity impact is performed.

Firstly, the choice of random variable inputs and their distributions, failure criteria and probabilistic

methods are discussed. In the case where the nonlinear behavior of materials is considered in the

model, the failure criterion is dened according to the maximum displacement of the slab at the

impact point. Finally, the probabilistic method dedicated to assess failure probability is chosen. This

latter has to be adapted in order to reduce the computational cost and keeping a good accuracy.

The Monte Carlo and importance sampling methods are coupled with the mass-spring model, while

FORM is used with the nite element model.

The two strategies used to reduce the computational cost of a reliability analysis are compared

in order to verify their eciency to calculate the probability of failure. Finally, a parametric study

is performed to identify the inuence of deterministic model parameters on the calculation of failure

probability (dimensions of slabs, impact velocity and mass, boundary conditions, impact point,

reinforcement density). This study helps to locate the most critical impact points for the slab

design, and illustrate how to optimize the design of reinforced concrete slabs under impact in terms

of material choice, concrete properties and ratio of reinforcement.

Keywords: RC structures, RC slabs, impact at low velocity, dynamic, material nonlinear behav-

iors, deterministic approaches, reliability approaches, failure probability.

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© [F. Kassem], [2015], INSA Lyon, tous droits réservés

Résumé

Dans le domaine du génie civil, le dimensionnement des structures en béton armé est essentiellement

basé sur des démarches déterministes. Cependant, les informations fournies par des analyses déter-

ministes sont insusantes pour étudier la variabilité de la réponse de la dalle. En outre, la réponse

des structures en béton armé sont fortement inuencés par de nombreuses sources d'incertitudes. Le

manque de connaissance des charges appliquées ainsi que les incertitudes liées à la géométrie de la

dalle et les caractéristiques des matériaux nécessitent donc l'utilisation d'une approche abiliste qui

permet la propagation de ces incertitudes dans les analyses déterministes. De nombreuses méthodes

basées sur la théorie des probabilités ont été développées, elles permettent d'une part de calculer

la probabilité de défaillance des structures et, d'autre part, d'étudier l'inuence de la variabilité

des variables de conception sur le comportement de la structure étudiée. L'approche abiliste est

basée sur le principe de couplage mécano-abiliste qui consiste à coupler un modèle stochastique,

contenant les caractéristiques probabilistes des variables aléatoires d'entrée, et un modèle détermin-

iste, permettant d'obtenir une estimation quantitative des variables de sortie ainsi que d'évaluer

les fonctions de performance qui décrivent les états défaillants de la structure étudiée. Cependant

un couplage mécano-abiliste peut être très exigeant en temps de calcul, soit à cause des modèles

déterministes dont un seul lancement peut être de l'ordre de quelques minutes à quelques heures

en fonction de son degré de complexité, soit à cause des méthodes d'évaluation de la probabilité de

défaillance qui exigent parfois un nombre d'appels très important au modèle déterministe.

Dans cette étude, il est proposé de s'appuyer sur la plateforme généraliste de traitements des

incertitudes OpenTURNS, développé par EDF R&D, EADS et la société PHIMECA pour eectuer

les analyses abilistes. Cette plateforme open source, permet de nombreuses méthodes de traitement

des incertitudes, Monte Carlo, Cumul Quadratique, méthodes de abilité FORM/SORM, tirage

d'importance, ..., qui méritent d'être testées dans le cadre d'applications du génie civil. Cette

plateforme possède une interface qui facilite le couplage avec n'importe quel logiciel externe. Dans

le cadre de cette thèse, la méthodologie propre aux problématiques des ouvrages du génie civil est

développée et validée tout d'abord sur un cas simple de structures en béton armé. Le cas d'une

poutre encastrée en béton armée est proposé. Le système est modélisé sous CASTEM par une

approche aux éléments nis de type multibre. Puis la abilité d'une dalle en béton armé impactée

par une masse rigide à faible vitesse est étudiée en couplant OpenTURNS à Abaqus. Enn, une

enceinte de connement en béton précontrainte modélisée sous ASTER est étudiée d'un point de

vue probabiliste.

Parmi les structures considérées dans cette étude, seul le problème physique des dalles en béton

armé soumises à impact est examiné en détail. Le problème d'impact fait partie des problèmes

dynamiques nonlinéaires les plus diciles parce qu'il pose le problème du contact accompagné par

une variation temporelle des vitesses. Le comportement d'une dalle en béton armé soumise à impact

est très complexe. Sa réponse peut s'exprimer en deux termes: la réponse globale et la réponse locale.

Le cas où l'impacteur a une vitesse initiale importante, la dalle a une réponse locale caractérisée

par un endommagement qui se localise près de la zone d'impact où la pénétration, la perforation et

l'écaillage ont lieu. Avec une vitesse initiale faible, la réponse globale se caractérise par la ssuration

progressive des faces comprimée et tendue, l'endommagement est lié à une réponse en mode de

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© [F. Kassem], [2015], INSA Lyon, tous droits réservés

cisaillement ou en exion. La conception de ce type de structures est généralement eectuée sous

des charges statiques ou pseudo-statiques. Dans le cas de charges dynamiques tels que les impacts, la

force appliquée à la structure est souvent évaluée à partir des approches énergétiques et une charge

statique équivalente est considérée dans sa conception pour simplier le problème. Cependant dans

certains cas spéciques, la réponse dynamique de la structure ne peut pas être simpliée à une

réponse quasi-statique. Une analyse dynamique doit être eectuée en tenant compte des principaux

phénomènes physiques impliqués. Ainsi des modèles précis sont nécessaires pour décrire et prédire la

performance d'une structure en béton armé, notamment lorsqu'elle est soumise à une force d'impact.

Cette étude porte sur des dalles en béton armé qui peuvent être soumises à une chute de colis

dans les centrales nucléaires, ce qui peut conduire à un endommagement dans la zone d'impact.

Lorsqu'une dalle en béton armé est soumise à un impact, l'endommagement peut se localiser à la

zone d'impact et varie en fonction de l'énergie cinétique d'impact et des masses relatives des corps

entrant en collision. Dans ce cas, un dimensionnement optimal nécessite la prise en compte de

la charge d'impact lors de la conception, ainsi que le développement des nonlinéarités dues aux

comportements des matériaux (ssurations du béton, plastication des aciers, etc.). L'objectif de

cette étude est de proposer des solutions pour diminuer le temps de calcul d'une analyse abiliste

en utilisant deux stratégies dans le cas des dalles impactées. La première stratégie consiste à utiliser

des modèles analytiques qui permettent de prédire avec précision la réponse mécanique de la dalle et

qui sont moins coûteux en temps de calcul. L'idée directrice est de créer des modèles économiques

en temps de calcul permettant de tester facilement plusieurs congurations de dalle. La deuxième

stratégie consiste à réduire le nombre d'appels au modèle déterministe, surtout dans le cas des

modèles par éléments nis, en utilisant des méthodes probabilistes d'approximation ou des méthodes

qui permettent de substituer le modèle par éléments nis par un modèle approximé qu'on appelle

méta-modèle.

La première partie de cette étude présente le comportement des dalles en béton armé lorsqu'elles

sont soumises à impact et détermine les modes de rupture des dalles en fonction de leurs caractéris-

tiques et des conditions d'impact. Ensuite deux modèles déterministes sont utilisés et évalués an

d'étudier les phénomènes dynamiques appliqués aux dalles en béton armé sous impact: un modèle

par éléments nis en 3D modélisé sous Abaqus qui permet une grande souplesse en terme de mod-

élisation et un modèle simplié de type masse-ressort amorti à deux degrés de liberté qui permet

de prendre en compte les principaux phénomènes physiques impliqués. Pour le modèle par éléments

nis, la dalle avec son ferraillage ainsi que l'impacteur sont entièrement modélisés. L'acier est mod-

élisé à l'aide d'une loi de comportement élasto-plastique avec écrouissage, alors que le comportement

du béton est décrit par le modèle élasto-plastique endommageable (concrete damage plasticity) qui

est intégré dans Abaqus. La complexité du modèle par éléments nis se situe au niveau des non-

linéarités liées d'une part à la gestion du contact et d'autre part aux comportements des matériaux,

ce qui entraîne une augmentation forte en terme du temps de calcul. Ce modèle est validé avec

des essais expérimentaux réalisés à l'université Heriot-Watt par Chen et May sur plusieurs dalles

en béton armé soumises à une chute d'une masse rigide [34]. Une alternative intéressante est pro-

posée dans cette étude en considérant un modèle simplié basé sur la modélisation de la structure

réelle par un système masse-ressort amorti à deux degrés de liberté. Ce modèle simplié prend en

compte les nonlinéarités du béton armé par une approche de chute en fréquence (i.e. la rigidité

structurelle). Ce système décrit l'interaction entre la dalle et l'impacteur dont leurs masses inter-

agissent par l'intermédiaire de deux ressorts et prend en compte l'amortissement visqueux lors de la

résolution par l'intermédiaire de deux amortisseurs en parallèle avec les ressorts.

L'étape suivante consiste à étudier la abilité des dalles en béton armé soumises à l'impact d'une

masse à faible vitesse en évaluant la probabilité de défaillance. Le calcul de cette probabilité s'appuie

tout d'abord sur le choix des variables d'entrée, des critères de défaillance ainsi que des méthodes

probabilistes à coupler avec les modèles déterministes. Les variables d'entrée sont modélisées en tant

que des variables aléatoires dont il faut dénir leurs lois de distributions. Le choix des critères de

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© [F. Kassem], [2015], INSA Lyon, tous droits réservés

défaillance est aussi essentiel dans une étude abiliste, un seul critère de défaillance est étudié dans

le cas où le comportement nonlinéaire des matériaux est considéré dans le modèle. Le critère est

lié au déplacement maximal que peut subir la dalle au point d'impact. Le seuil de déplacement est

choisi de façon à ne pas avoir de plastication dans les armatures ou au moment de l'apparition des

ssures dans le béton. La dernière étape à considérer dans une étude abiliste nécessite à bien choisir

la méthode probabiliste à utiliser pour calculer la probabilité de défaillance, elle doit être adaptée

au type du modèle déterministe et pour des temps de calcul optimisés. Dans cette étude, nous avons

couplé les méthodes Monte Carlo et simulation d'importance avec le modèle de type masse-ressort.

FORM est utilisée avec le modèle par éléments nis, ce qui permet un nombre d'appels réduit au

modèle par éléments nis.

Les deux stratégies utilisées pour réduire le temps de calcul d'une étude abiliste sont comparées

an de vérier l'ecacité de chacune pour calculer la probabilité de défaillance. Enn, une étude

paramétrique est réalisée an d'étudier l'eet des paramètres d'entrées des modèles déterministes

sur le calcul de la probabilité de défaillance. Nous avons choisi dans cette étude de considérer les

paramètres liés aux dimensions de la dalle, la vitesse d'impact, les conditions aux limites de la dalle,

la position du point d'impact et le taux de ferraillage. Cette étude permet de localiser les points

d'impact les plus défavorables pour la réponse de la structure, et d'illustrer les moyens d'optimiser

le dimensionnement des dalles en béton armé sous impact en termes de choix des matériaux, carac-

téristiques du béton et pourcentage d'armatures.

Mots-clès: Structures en béton armé, dalles en béton armé, impact à faible vitesse, dynamique,

comportement nonlinéaire des matériaux, approches déterministes, approches abilistes, probabilité

de défaillance.

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© [F. Kassem], [2015], INSA Lyon, tous droits réservés

Contents

Dedication ii

Contents viii

Nomenclature xiii

List of Figures xvi

List of Tables xxii

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Research motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Study scope and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Thesis contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Reinforced concrete slabs behavior under impact: Dynamic response and lit-

erature review 8

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Reinforced concrete slabs behavior under impact . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Impact dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Types of impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Failure modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.4 Energy considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Methods to determine the response of RC slabs under impact . . . . . . . . . . . . . 20

2.3.1 Experimental approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1.1 Zineddin and Krauthammer tests . . . . . . . . . . . . . . . . . . . 20

2.3.1.2 Chen and May tests . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1.3 Hrynyk tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.2 Analytical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.2.1 Tonello model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2.2 CEB model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2.3 Abrate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2.4 Delhomme et al. model . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.2.5 Yigit and Christoforou model . . . . . . . . . . . . . . . . . . . . . 31

2.3.3 Finite element approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.3.1 Berthet-Rambaud et al. model . . . . . . . . . . . . . . . . . . . . . 33

2.3.3.2 Mokhatar and Abdullah model . . . . . . . . . . . . . . . . . . . . . 35

2.3.3.3 Trivedi and Singh model . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Mechanical behavior of concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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2.4.1 Uniaxial compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4.2 Uniaxial tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4.2.1 Uniaxial tensile stress-strain curve . . . . . . . . . . . . . . . . . . . 43

2.4.2.2 Uniaxial stress-crack opening curve . . . . . . . . . . . . . . . . . . 46

2.5 Mechanical behavior of steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.5.1 Uniaxial tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.5.2 Idealizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.5.3 Steel-concrete bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Finite element analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Choice of commercial FE software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Components of an Abaqus analysis model . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 Conguring analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.6 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.6.1 Concrete constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.6.2 Concrete damage plasticity model (CDP) . . . . . . . . . . . . . . . . . . . . 59

3.6.2.1 Plastic ow and yield surface . . . . . . . . . . . . . . . . . . . . . 60

3.6.2.2 Stress-strain curve for uniaxial compression . . . . . . . . . . . . . . 61

3.6.2.3 Stress-strain curve for uniaxial tension . . . . . . . . . . . . . . . . 62

3.6.2.4 Damages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.6.3 Constitutive model of reinforcing steel . . . . . . . . . . . . . . . . . . . . . . 65

3.7 Finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.8 Contact modeling between slab and impactor . . . . . . . . . . . . . . . . . . . . . . 66

3.9 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.9.1 Approaches to represent steel in RC numerical analysis . . . . . . . . . . . . . 67

3.9.2 Tie constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.10 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1 Introduction to reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Principles of structural reliability analysis . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Uncertainty sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.1 Physical uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.2 Measurement uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.3 Statistical uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.4 Model uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.5 Other uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4 Reliability approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4.1 Modeling of uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4.1.1 Joint probability density function . . . . . . . . . . . . . . . . . . . 77

4.4.1.2 Covariance and correlation . . . . . . . . . . . . . . . . . . . . . . . 77

4.4.2 Variable of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4.2.1 Dispersion analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4.2.2 Distribution analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4.3 Probability of failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4.3.1 Limit states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4.3.2 Approximation methods . . . . . . . . . . . . . . . . . . . . . . . . . 81

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4.4.3.3 Simulation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4.3.4 Response surface methods . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.3.5 Comparison of methods according to some reviews in the literature 89

4.5 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.6 Statistical descriptions of random variables . . . . . . . . . . . . . . . . . . . . . . . 91

4.6.1 Concrete properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.6.1.1 Compressive strength of concrete . . . . . . . . . . . . . . . . . . . . 91

4.6.1.2 Tensile strength of concrete . . . . . . . . . . . . . . . . . . . . . . . 93

4.6.1.3 Young's Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.6.2 Steel properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.6.2.1 Yield strength of steel . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.6.2.2 Modulus of elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.6.3 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2 FEM of Chen and May slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2.1 Creating model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2.2 Creating parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2.2.1 Part of slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2.2.2 Part of reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2.2.3 Part of impactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2.2.4 Part of support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2.3 Creating materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2.3.1 Identication of constitutive parameters for CDP model . . . . . . 105

5.2.3.2 Identication of parameters for reinforcing steel stress-strain curve . 107

5.2.4 Dening and assigning section properties . . . . . . . . . . . . . . . . . . . . . 108

5.2.5 Dening the assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.2.6 Meshing the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2.6.1 Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.2.6.2 Impactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.2.7 Conguring analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.2.8 Applying loads, boundary and initial conditions to the model . . . . . . . . . 116

5.2.9 Creating interaction and constraints . . . . . . . . . . . . . . . . . . . . . . . 116

5.2.10 Output requests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2.11 Creating an analysis job . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.3 Discussion of results of Chen and May slabs FEM . . . . . . . . . . . . . . . . . . . . 118

5.3.1 Comparison of mesh types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.3.2 Comparison of contact algorithms . . . . . . . . . . . . . . . . . . . . . . . . 119

5.3.3 Mesh sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.3.4 Choice of materials behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.3.5 Validation of model for other slabs . . . . . . . . . . . . . . . . . . . . . . . . 122

5.4 Deterministic models of the slab in nuclear plant . . . . . . . . . . . . . . . . . . . . 129

5.4.1 Finite element model (FEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.4.2 Mass-spring models (MSM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.4.2.1 Mass-spring model without damping . . . . . . . . . . . . . . . . . . 134

5.4.2.2 Mass-spring model with damping . . . . . . . . . . . . . . . . . . . . 134

5.4.2.3 Identication of MSM parameters . . . . . . . . . . . . . . . . . . . 135

5.5 Discussion of results of deterministic models of the slab in nuclear plant . . . . . . . 138

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5.5.1 Comparison of FEM and MSM results . . . . . . . . . . . . . . . . . . . . . . 138

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.2 Application to RC beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2.1 Deterministic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2.2 Probabilistic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2.3 Failure criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.2.4 Discussion of results of beam reliability . . . . . . . . . . . . . . . . . . . . . 146

6.2.4.1 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.2.4.2 Distribution analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.2.4.3 Comparison of probabilistic methods . . . . . . . . . . . . . . . . . . 149

6.2.4.4 Metamodeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.2.4.5 Conclusion remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.3 Application to RC slab subjected to impact - Elastic behavior . . . . . . . . . . . . . 165

6.3.1 Deterministic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.3.1.1 Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.3.1.2 Model based on plate theory (PT) . . . . . . . . . . . . . . . . . . . 166

6.3.1.3 Mass-spring model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.3.2 Probabilistic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.3.3 Failure criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.3.4 Discussion of results of slab reliability (elastic behavior) . . . . . . . . . . . . 169

6.3.4.1 Comparison of deterministic models . . . . . . . . . . . . . . . . . . 169

6.3.4.2 Comparison of probabilistic methods . . . . . . . . . . . . . . . . . . 170

6.3.4.3 Comparison of failure criteria . . . . . . . . . . . . . . . . . . . . . . 171

6.3.4.4 Parametric study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.3.5 Conclusion remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.4 Application to RC slab subjected to impact - Nonlinear behavior . . . . . . . . . . . 176

6.4.1 Deterministic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.4.2 Probabilistic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.4.2.1 Concrete properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.4.2.2 Steel properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.4.2.3 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.4.3 Failure criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.4.4 Discussion of results of slab reliability (Nonlinear behavior) . . . . . . . . . . 179

6.4.4.1 Comparison of strategies . . . . . . . . . . . . . . . . . . . . . . . . 179

6.4.4.2 Parametric study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.4.5 Conclusion remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.5 Application to containment building of nuclear power plant . . . . . . . . . . . . . . 187

6.5.1 Deterministic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.5.1.1 FE software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

6.5.1.2 Material properties and behaviors . . . . . . . . . . . . . . . . . . . 191

6.5.1.3 Boundary conditions and loading . . . . . . . . . . . . . . . . . . . . 191

6.5.1.4 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

6.5.1.5 Delayed strains in concrete . . . . . . . . . . . . . . . . . . . . . . . 192

6.5.2 Probabilistic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.5.3 Failure criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.5.4 Discussion of results of containment building reliability . . . . . . . . . . . . . 195

6.5.4.1 Deterministic output variable . . . . . . . . . . . . . . . . . . . . . . 195

6.5.4.2 Dispersion analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

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6.5.4.3 Distribution analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

6.5.4.4 Failure probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

6.5.4.5 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

6.5.4.6 Convergence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.5.4.7 Conclusion remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

7.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Bibliography 208

A Probability distributions in OpenTRUNS 13.2 219

B MSM results - comparison with FEM results 223

B.1 Impact kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

B.2 Impactor velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

B.3 Impactor mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

B.4 Steel diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

B.5 Concrete density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

B.6 Slab thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

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Nomenclature

σs Steel stress

elastic phase

εin

c Crushing (inealstic) strain of concrete

εpl

c Plastic strain of concrete in compression

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εte Tensile elastic strain of concrete corresponding to the end of linear elastic

phase

εck

t Cracking strain of concrete

εpl

t Plastic strain of concrete in tension

e RC slab thickness

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Kc Ratio of the second invariant on the tensile meridian to that on the com-

pressive meridian at any given value of the pressure invariant (parameter

of CDP model)

Pf Failure probability

X A random variable

Y Variable of interest

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List of Figures

2.1 Some accidental impact loading cases in civil engineering eld: a) Rockfall on protection

galleries, b) Aircraft impact on nuclear containments, c) Dropped objects impact during

handling operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Accidental dropped object impact during handling operations within nuclear plant buildings 10

2.3 Response types during impact on plates according to [149] . . . . . . . . . . . . . . . . 11

2.4 Structure and impactor deformations according to the impact type [45] : a) Initial con-

ditions (t=0), b) Hard impact (t>0), c) Soft impact (t>0) . . . . . . . . . . . . . . . . . 13

2.5 Mass-spring model proposed by Eibl [55] to simulate an impact [45] . . . . . . . . . . . 14

2.6 Classication of low velocity impacts according to Koechlin and Potapov [95] . . . . . . 15

2.7 Overall response failure modes of impacted RC slabs [119] : a) Flexural failure, b) Punch-

ing shear failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.8 Local damage failure modes of impacted RC slabs [119] : a) Surface failure, b) spalling,

c) scabbing, d) perforation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.9 Energy transformation process during an impact event [130] . . . . . . . . . . . . . . . 19

2.10 Impact test system used by Zineddin and Krauthammer [189] . . . . . . . . . . . . . . . 21

2.11 Load-time history for slabs impacted with a drop height of 152 mm [189] . . . . . . . . . 22

2.12 Failure mode of slabs impacted with a drop height of 610 mm and reinforced with rein-

forcement type: (a) Reinf. 3, (b) Reinf. 1, (c) Reinf. 1-Failure of one steel bar [189] . . . 22

2.13 Cracks patterns on top surface of slabs reinforced with reinforcement type Reinf. 3 and

impacted with a drop height of : (a) 305 mm, (b) 610 mm[189] . . . . . . . . . . . . . . 23

2.14 Details of slabs (dimensions in mm): a) 760 mm square slabs, b) 2320 mm square slabs,

c) Boundary conditions [34] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.15 Details of impactors (dimensions in mm): a) Hemispherical impactor, b) Cylindrical

impactor [34] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.16 Transient impact load of slabs 2-6 [34] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.17 Local damage of slabs bottom faces after impact: (a) Slab 2, (b) Slab 3, (c) Slab 4, (d)

Slab 5, (e) Slab 6 [34] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.18 Details of slabs (dimensions in mm): a) Slab with additional links at the impact region,

b) Boundary conditions [84] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.19 Cracking patterns on the bottom surface of a slab with a longitudinal reinforcement ratio

of 0.42%: a) Impact #1, b) Impact #2, c) Impact #3 [84] . . . . . . . . . . . . . . . . . 27

2.20 Final cracking patterns on the bottom surface of slabs with a longitudinal reinforcement

ratio of: a) 0.273%, b) 0.42%, c) 0.592% [84] . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.21 One degree of freedom mass-spring model used in Tonello IC design oce [174] . . . . . 29

2.22 Simplied model of CEB for a hard impact [29] . . . . . . . . . . . . . . . . . . . . . . 30

2.23 Nonlinear force-displacement relationship for spring: (a) R1 , (b) R2 [29] . . . . . . . . . 30

2.24 Mass-spring models of impact on composite structures according to [10] . . . . . . . . . 31

2.25 The two mass-spring-damper models used by [47] to analyze SDR response: (a) contact:

model, (b) post-impact model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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2.26 Models for low velocity impact response in structures according to [184] . . . . . . . . . 32

2.27 3D FE model of SDR protection galleries proposed by Berthet-Rambaud et al. [19] . . . 33

2.28 Damage distribution in and around the repaired zone [19] . . . . . . . . . . . . . . . . . 34

2.29 Comparison of numerical and experimental results: (a) Calibration on friction parameter,

(b) Vertical displacement at a point near the impact zone [19] . . . . . . . . . . . . . . . 34

2.30 Details of nite element model of Mokhatar and Abdullah [137] . . . . . . . . . . . . . . 35

2.31 Comparison of numerical and experimental transient impact force with dierent: (a)

Mesh densities, (b) Constitutive concrete models [137] . . . . . . . . . . . . . . . . . . . 36

2.32 Final crack pattern of Slab3 bottom face: (a) Experimental result, (b) Numerical result

[137] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.33 Details of Trivedi and Singh FE model: (a) One fourth of a slab with 2 meshes of

longitudinal and transversal steel bars, (b) Time-amplitude curve in case of a hammer

dropped with a height of 610 mm [175] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.34 Identication of slabs failure mode by inection point criterion: (a) Transition from

exure to punching shear, (b) Flexural mode [175] . . . . . . . . . . . . . . . . . . . . . 38

2.35 Identication of a slab exural failure mode by strain based failure criterion: (a) Top

elements, (b) Bottom elements [175] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.36 Uniaxial compressive stress-strain curve for concrete [4] . . . . . . . . . . . . . . . . . . 40

2.37 Uniaxial tensile stress-strain curve for concrete [156] . . . . . . . . . . . . . . . . . . . . 44

2.38 Concrete stress-crack opening curve with: (a) Linear softening branch [7], (b) Bi-linear

softening branch [78], (c) Tri-linear softening branch [170] . . . . . . . . . . . . . . . . . 46

2.39 Experimental tensile stress-strain curve for reinforcing steel [112] . . . . . . . . . . . . . 49

2.40 Average tensile stress-strain curve for reinforcing steel embedded in concrete [16] . . . . 51

3.1 Unloading response of: (a) elastic plastic, (b) elastic damage, (c) elastic plastic damage

models [88] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Drucker-Prager hyperbolic function of CDP ow potential and its asymptote in the merid-

ian plane [94] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3 Yield surface for the CDP model in: (a) plane stress, (b) the deviatoric plane correspond-

ing to dierent values of Kc [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 Response of concrete to uniaxial loading in compression for CDP model [6] . . . . . . . 63

3.5 Response of concrete to uniaxial loading in tension for CDP model [6] . . . . . . . . . . 64

3.6 Finite elements used for the problem involving contact-impact of reinforced concrete [5] 66

3.7 Constraints with the contact pair algorithm: a) Kinematic contact, b) Penalty contact [5] 67

3.8 Contact interaction properties: a) Hard contact model, b) Coulomb friction model [5] . 68

3.9 Approaches to represent steel in RC numerical analysis: a) smeared approach, b) embed-

ded approach, c) discrete approach [138] . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.10 Tie constraint to tie two surfaces together in Abaqus [5] . . . . . . . . . . . . . . . . . . 69

4.2 PDF of Gauss, Lognormal, Uniform and Weibull distributions with same mean and stan-

dard deviation [107] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3 CDF of Gauss, Lognormal, Uniform and Weibull distributions with same mean and

standard deviation [107] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.4 Correlation of two random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5 Representation of a sample of an output variable by a histogram [107] . . . . . . . . . . 80

4.6 Reliability index in standard space [44] . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.7 Approximation of limit state surface by FORM [75] . . . . . . . . . . . . . . . . . . . . 83

4.8 Approximation of limit state surface by SORM [75] . . . . . . . . . . . . . . . . . . . . 84

4.9 Principle of Monte Carlo method, illustration in the standard space [46] . . . . . . . . . 85

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4.10 Principle of importance sampling method, illustration in the standard space [46] . . . . 86

4.11 Principle of directional simulation method, illustration in the standard space [46] . . . . 87

4.12 Principle of Latin hypercube sampling method, illustration in the standard space [46] . 88

4.13 Linear stepped model for tension stiening [35] . . . . . . . . . . . . . . . . . . . . . . . 94

5.2 Part of reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.3 Part of impactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.4 Part of support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.5 Reinforcement in the model assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.6 Assembly of the model after creating instances . . . . . . . . . . . . . . . . . . . . . . . 110

5.7 Mesh of cylindrical support instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.8 Partitions used to create Mesh1 and Mesh2 . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.9 Mesh1 and Mesh2 of slab in xy-plane and in the direction of thickness . . . . . . . . . . 111

5.10 Partitions used to create Mesh3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.11 Mesh3 of slab in xy-plane and in the direction of thickness . . . . . . . . . . . . . . . . 112

5.12 Partitions used to create Mesh4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.13 Mesh4 of slab in xy-plane and in the direction of thickness . . . . . . . . . . . . . . . . 113

5.14 Mesh4 with dierent sizes of impact region . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.15 Mesh4 for dierent impact positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.16 The impactor mesh used with Mesh1, Mesh2 and Mesh3 of slab . . . . . . . . . . . . . 115

5.17 The impactor mesh used with Mesh4 of slab . . . . . . . . . . . . . . . . . . . . . . . . 115

5.18 Comparison of dierent mesh types proposed for Slab2 . . . . . . . . . . . . . . . . . . 119

5.19 Comparison of contact algorithms available in Abaqus for Slab2 meshed with Mesh4 . . 120

5.20 Sensitivity of mesh in term of the number of elements in the thickness direction and the

size of elements in the impact zone for Slab2 meshed with Mesh4 . . . . . . . . . . . . . 121

5.21 Sensitivity of mesh in term of the size of elements in the slab and the size of elements in

the reinforcement for Slab2 meshed with Mesh4 . . . . . . . . . . . . . . . . . . . . . . 121

5.22 Comparison of concrete compressive stress-strain curves for Slab2 . . . . . . . . . . . . 123

5.23 Comparison of concrete compressive stress-strain curves for Slab2 . . . . . . . . . . . . 124

5.24 Comparison of concrete compressive stress-strain curves for Slab2 . . . . . . . . . . . . 125

5.25 Comparison of concrete tensile stress-strain curves for Slab2 . . . . . . . . . . . . . . . 126

5.26 Comparison of concrete tensile stress-strain curves for Slab2 . . . . . . . . . . . . . . . 127

5.27 Comparison of concrete tensile stress-cracking displacement curves for Slab2 . . . . . . 128

5.28 Comparison of fracture energy values for Slab2 . . . . . . . . . . . . . . . . . . . . . . . 129

5.29 Comparison of steel stress-strain curves for Slab2 . . . . . . . . . . . . . . . . . . . . . . 130

5.30 Comparison of steel stress-strain curves for Slab2 . . . . . . . . . . . . . . . . . . . . . . 131

5.31 Comparison of C6T1S1 and C6D3S1 simulations for Slab3 and Slab4 . . . . . . . . . . 132

5.32 Comparison of C6T1S1 and C6D3S1 simulations for Slab5 and Slab6 . . . . . . . . . . 132

5.33 Finite element model of slab: a) meshing, b) reinforcement . . . . . . . . . . . . . . . . 134

5.34 Mass-spring model without damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.35 Mass-spring model with damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.36 Slab stiness decrease in term of displacement for several values of thickness and the

variation of the stiness initial value in term of the thickness . . . . . . . . . . . . . . . 137

5.37 Comparison of mass-spring models with and without damping . . . . . . . . . . . . . . 139

5.38 Comparison of FEM and MSM for the same kinetic energy . . . . . . . . . . . . . . . . 141

6.1 RC beam tested experimentally in the framework of the LESSLOSS project [108] . . . . 144

6.2 Discretization of multiber beam into elements, nodes and degrees of freedom [76] . . . 144

6.3 Geometrical properties of the multiber beam . . . . . . . . . . . . . . . . . . . . . . . 145

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6.4 Beam displacement distribution estimated with Kernel Smoothing method . . . . . . . 150

6.5 Comparison of the beam displacement sample to several parametric distributions . . . . 150

6.6 QQ-plot test to graphically compare the beam displacement sample to several parametric

distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.7 Number of calls of beam FEM in term of failure probability magnitude for dierent

probabilistic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.8 Eect of the number of random variables on the accuracy and precision of dierent

probabilistic methods in comparison to MC . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.9 Variation of failure probability in terms of the force applied at the free end of the beam

and the number of random variables considered . . . . . . . . . . . . . . . . . . . . . . 157

6.10 Eect of the distribution type of random variables on the accuracy and precision of

dierent probabilistic methods in comparison to MC . . . . . . . . . . . . . . . . . . . . 158

6.11 Eect of the type of random variables distribution on the estimation of failure probability

for dierent probabilistic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.12 Eect of the COV of random variables on the accuracy and precision of dierent proba-

bilistic methods in comparison to MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.13 Variation of failure probability in term of the mean of the force for dierent values of COV161

6.14 Variation of failure probability in term of the COV for dierent values of the mean of the

force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.15 Comparison of the beam displacement distributions estimated with the initial model and

Taylor expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.16 Comparison of the beam displacement distributions estimated with the initial model and

Least Square method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.17 Comparison of the beam displacement distributions estimated with the initial model and

polynomial chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.18 Comparison of failure probability estimated with the initial model and several response

surface types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.19 Finite element model of slab: a) meshing, b) reinforcement . . . . . . . . . . . . . . . . 166

6.20 Failure probability with dierent deterministic models and displacement criterion . . . 170

6.21 Comparison of dierent failure criteria in term of impact velocity mean . . . . . . . . . 172

6.22 Eect of the mean of slab length for the displacement criterion . . . . . . . . . . . . . . 172

6.23 Eect of the mean of slab length for the displacement criterion . . . . . . . . . . . . . . 173

6.24 Eect of the mean of concrete density for the displacement criterion . . . . . . . . . . . 174

6.25 Eect of the mean of steel bars diameter for the displacement criterion . . . . . . . . . 174

6.26 Eect of the mean of impactor mass for the displacement criterion . . . . . . . . . . . . 175

6.27 Eect of the mean of slab stiness for the displacement criterion . . . . . . . . . . . . . 175

6.28 Eect of the mean of impactor velocity for the displacement criterion (m=3600 kg) . . 180

6.29 Eect of the mean of impactor mass for the displacement criterion . . . . . . . . . . . . 181

6.30 Eect of the mean of slab thickness for the displacement criterion . . . . . . . . . . . . 182

6.31 Eect of the mean of concrete compressive strength for the displacement criterion . . . 183

6.32 Eect of the mean of steel yield strength for the displacement criterion . . . . . . . . . 184

6.33 Eect of the mean of steel diameter for the displacement criterion . . . . . . . . . . . . 185

6.34 Eect of numbers of transversal and longitudinal bars for the displacement criterion . . 185

6.35 Eect of stirrups for the displacement criterion . . . . . . . . . . . . . . . . . . . . . . . 186

6.36 Eect of boundary conditions for the displacement criterion . . . . . . . . . . . . . . . . 187

6.37 Eect of impact position for the displacement criterion . . . . . . . . . . . . . . . . . . 188

6.38 Containment building [161] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.39 Geometry of the containment building of the Flamanville nuclear power plant . . . . . 190

6.40 The selected zone of the containment building to be modeled . . . . . . . . . . . . . . . 190

6.41 Mesh of the selected zone of study of the containment building . . . . . . . . . . . . . . 192

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6.42 Horizontal and vertical cables considered in the reliability analysis of containment build-

ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.43 Variation of the average tensile stress in term of height . . . . . . . . . . . . . . . . . . 196

6.44 Mean and COV of height in term of the number of simulations . . . . . . . . . . . . . . 197

6.45 Histogram of tension zone height samples obtained with initial model and meta-model

for 1000 simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.46 CDF of tension zone height samples obtained with initial model and meta-model for 1000

simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

6.47 PDF of tension zone height samples obtained with initial model and meta-model for 1000

simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.48 Comparison of initial model and meta-model output samples using QQ-plot . . . . . . . 199

6.49 Failure probability and reliability index estimated with dierent probabilistic methods

depending on the number of points of the design of experiments . . . . . . . . . . . . . 200

6.50 Convergence at level 0.95 of the estimate of failure probability for dierent probabilistic

simulation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

6.51 Importance factors of containment random variables using quadratic combination method

and FORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.52 Importance factor of concrete shrinkage coecient in term of number of points in the

design of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.53 Comparison of initial model and meta-model for dierent number of points in the design

of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

B.1 Displacement and velocity of slab and impactor: v = 2.5 m/s, m = 34151 kg . . . . . . . 223

B.2 Displacement and velocity of slab and impactor: v = 5 m/s, m = 8537 kg . . . . . . . . 224

B.3 Displacement and velocity of slab and impactor: v = 7.7 m/s, m = 3600 kg . . . . . . . 225

B.4 Displacement and velocity of slab and impactor: v = 10.5 m/s, m = 1936 kg . . . . . . . 226

B.5 Displacement and velocity of slab and impactor: v = 13.2 m/s, m = 1225 kg . . . . . . . 227

B.6 Displacement and velocity of slab and impactor: v = 15.5 m/s, m = 888 kg . . . . . . . 228

B.7 Displacement and velocity of slab and impactor: v = 19.7 m/s, m = 550 kg . . . . . . . 229

B.8 Displacement and velocity of slab and impactor: v = 2.5 m/s . . . . . . . . . . . . . . . 230

B.9 Displacement and velocity of slab and impactor: v = 5 m/s . . . . . . . . . . . . . . . . 231

B.10 Displacement and velocity of slab and impactor: v = 7.7 m/s . . . . . . . . . . . . . . . 232

B.11 Displacement and velocity of slab and impactor: v = 10.5 m/s . . . . . . . . . . . . . . . 233

B.12 Displacement and velocity of slab and impactor: v = 13.2 m/s . . . . . . . . . . . . . . . 234

B.13 Displacement and velocity of slab and impactor: v = 15.5 m/s . . . . . . . . . . . . . . . 235

B.14 Displacement and velocity of slab and impactor: v = 19.7 m/s . . . . . . . . . . . . . . . 236

B.15 Displacement and velocity of slab and impactor: m = 750 kg . . . . . . . . . . . . . . . . 237

B.16 Displacement and velocity of slab and impactor: m = 1000 kg . . . . . . . . . . . . . . . 238

B.17 Displacement and velocity of slab and impactor: m = 2000 kg . . . . . . . . . . . . . . . 239

B.18 Displacement and velocity of slab and impactor: m = 3000 kg . . . . . . . . . . . . . . . 240

B.19 Displacement and velocity of slab and impactor: m = 3600 kg . . . . . . . . . . . . . . . 241

B.20 Displacement and velocity of slab and impactor: m = 5000 kg . . . . . . . . . . . . . . . 242

B.21 Displacement and velocity of slab and impactor: m = 6000 kg . . . . . . . . . . . . . . . 243

B.22 Displacement and velocity of slab and impactor: dA = 0.006 m . . . . . . . . . . . . . . 244

B.23 Displacement and velocity of slab and impactor: dA = 0.008 m . . . . . . . . . . . . . . 245

B.24 Displacement and velocity of slab and impactor: dA = 0.01 m . . . . . . . . . . . . . . . 246

B.25 Displacement and velocity of slab and impactor: dA = 0.012 m . . . . . . . . . . . . . . 247

B.26 Displacement and velocity of slab and impactor: dA = 0.014 m . . . . . . . . . . . . . . 248

B.27 Displacement and velocity of slab and impactor: dA = 0.016 m . . . . . . . . . . . . . . 249

B.28 Displacement and velocity of slab and impactor: dA = 0.02 m . . . . . . . . . . . . . . . 250

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B.29 Displacement and velocity of slab and impactor: dA = 0.03 m . . . . . . . . . . . . . . . 251

B.30 Displacement and velocity of slab and impactor: ρc = 1200 kg/m3 . . . . . . . . . . . . . 252

B.31 Displacement and velocity of slab and impactor: ρc = 1600 kg/m3 . . . . . . . . . . . . . 253

B.32 Displacement and velocity of slab and impactor: ρc = 2000 kg/m3 . . . . . . . . . . . . . 254

B.33 Displacement and velocity of slab and impactor: ρc = 2300 kg/m3 . . . . . . . . . . . . . 255

B.34 Displacement and velocity of slab and impactor: ρc = 2500 kg/m3 . . . . . . . . . . . . . 256

B.35 Displacement and velocity of slab and impactor: ρc = 3000 kg/m3 . . . . . . . . . . . . . 257

B.36 Displacement and velocity of slab and impactor: ρc = 3500 kg/m3 . . . . . . . . . . . . . 258

B.37 Displacement and velocity of slab and impactor: ρc = 4000 kg/m3 . . . . . . . . . . . . . 259

B.38 Displacement and velocity of slab and impactor: ρc = 4500 kg/m3 . . . . . . . . . . . . . 260

B.39 Displacement and velocity of slab and impactor: e = 0.3 m . . . . . . . . . . . . . . . . . 261

B.40 Displacement and velocity of slab and impactor: e = 0.4 m . . . . . . . . . . . . . . . . . 262

B.41 Displacement and velocity of slab and impactor: e = 0.5 m . . . . . . . . . . . . . . . . . 263

B.42 Displacement and velocity of slab and impactor: e = 0.6 m . . . . . . . . . . . . . . . . . 264

B.43 Displacement and velocity of slab and impactor: e = 0.7 m . . . . . . . . . . . . . . . . . 265

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List of Tables

2.2 Impact loading protocol [84] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Values of εc0 parameter for high strength concrete [54] . . . . . . . . . . . . . . . . . . . 41

4.2 Summary on compressive strength of concrete according to various references . . . . . . 93

4.3 Summary on tensile strength of concrete according to various references . . . . . . . . . 95

4.4 Summary on Young's modulus of concrete according to various references . . . . . . . . 96

4.5 Summary on yield and ultimate strengths of steel according to various references . . . . 97

4.6 Summary on elasticity modulus of steel according to various references . . . . . . . . . . 98

4.7 Summary on dimensions of RC members according to various references . . . . . . . . . 99

5.1 Linear mechanical properties of concrete and steel materials for Chen and May slabs . . 104

5.2 Nonlinear mechanical properties of concrete and steel materials for Chen and May slabs 105

5.3 Parameters of the damaged plasticity model . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4 Compressive concrete behaviors for CDP model . . . . . . . . . . . . . . . . . . . . . . . 106

5.5 Tensile concrete behaviors for CDP model . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.6 Idealized stress-strain curves for reinforcing steel . . . . . . . . . . . . . . . . . . . . . . 108

5.7 Linear mechanical properties of concrete and steel materials for the slab in nuclear plant 132

5.8 Nonlinear mechanical properties of concrete and steel materials for the slab in nuclear

plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.1 Random variables of the multiber beam and their descriptions . . . . . . . . . . . . . 145

6.2 Importance factors of beam random variables . . . . . . . . . . . . . . . . . . . . . . . . 146

6.3 Stress and resistance variables of the cantilvever beam and their variability inuence . . 147

6.4 Omission factors of the cantilvever beam variables and their inuence on Pf . . . . . . . 147

6.5 Variability of the cantilever beam variables according to their mean and standard devia-

tion values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.6 Summary of sensitivity studies for the cantilever beam . . . . . . . . . . . . . . . . . . . 148

6.7 Results of Kolmogorov-Smirnov test to verify beam displacement distribution . . . . . . 149

6.8 Failure probability estimated with dierent methods for all the cases studied of the can-

tilever beam (number of random variables=2) . . . . . . . . . . . . . . . . . . . . . . . . 152

6.9 Failure probability estimated with dierent methods for all the cases studied of the can-

tilever beam (number of random variables=3) . . . . . . . . . . . . . . . . . . . . . . . . 152

6.10 Failure probability estimated with dierent methods for all the cases studied of the can-

tilever beam (number of random variables=7) . . . . . . . . . . . . . . . . . . . . . . . . 153

6.11 Failure probability estimated with dierent methods for dierent types of probability

distribution (number of random variables=2) . . . . . . . . . . . . . . . . . . . . . . . . 159

6.12 Random variables of the RC slab and their statistical descriptions (elastic behavior) . . 168

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6.13 Comparison of displacement of slab at the impact point for dierent deterministic models

and dierent values of velocity (elastic behavior) . . . . . . . . . . . . . . . . . . . . . . 170

6.14 Comparison of probabilistic methods combined with the MSM for the displacement criterion171

6.15 Deterministic variables, random variables and their statistical descriptions . . . . . . . 178

6.16 Factorial experiment factors and mass-spring model parameters . . . . . . . . . . . . . . 179

6.17 Failure probability with FE and mass-spring models and displacement criterion . . . . . 179

6.18 Random variables of the containment building and their descriptions . . . . . . . . . . . 194

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Chapter 1

Introduction

1.1 Background

In civil engineering eld, the main problem consists in nding an optimal design that ensures to

maximum the continuous safety of structures throughout its service life. The design should also en-

sure a satisfactory structural performance with an economical cost. Reinforced concrete structures

are subjected to several sources of uncertainties that highly aect their response. These uncertain-

ties are related to the structure geometry, material properties and the loads applied. Engineers

have always used traditional and deterministic approaches to simplify the problem and account for

these uncertainties through the application of safety factors according to limit states. However,

these factors are based on engineering judgment, they are calibrated to a certain level of security

and presented as xed values. In addition, they do not provide sucient information on the eect

of dierent uncertain variables on the structural safety. Consequently, for an optimal and robust

design that guarantees a real prediction of the behavior of structures, uncertainties should be taken

into account and propagated in the structural deterministic analysis. To address this issue, proba-

bilistic approaches were developed to correctly model the variation of input variables and to ensure

a higher reliability of representative mechanical models used for the design of new structures or the

monitoring of performance of existing structures. These approaches allow assessing failure proba-

bility of structures according to one or several criteria, but also studying the inuence of uncertain

variables on the structure response. Sensitivity analysis can be performed to analyze the variation

of structure resistance during the design phase or the surveillance testing phase with respect to

conventional parameters, such as loads and material properties, but also with respect to nonlinear

inaccurately known parameters such as the creep in concrete or its damage due to cracking. Thus,

structural reliability analysis based on the principle of combining a stochastic model with a deter-

ministic model enables to deduce a more realistic estimate of the safety margin in terms of the

variation of dierent variables. Stochastic models should include the probabilistic characteristics

of random input variables including a suitable probability distribution and appropriate values of

their mean and coecient of variation. Deterministic models allow predicting structural response

and evaluating the variables of interest, they can be based on analytical, empirical or numerical

deterministic approaches.

Therefore, the physical problem and mechanical phenomena applied should be evaluated be-

fore performing reliability analysis. Among structures considered in this study, only the physical

problem of reinforced concrete (RC) slabs subjected to impact is investigated in detail. Such types

of problem is classied as low velocity impact event and considered as one of the most dicult

nonlinear dynamic problems in civil engineering because it involves several factors, including the

nonlinear behavior of concrete, the interaction eects between the concrete and reinforcement, and

contact mechanics between the slab and the impactor. The complexity of impact problems also lies

in the dynamic response of RC slabs and the time-dependent evolution of impact velocity. Design

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of RC slabs to resist impact loads represents an area of research that is increasingly gaining impor-

tance. Their design is generally based on static method which is often not suciently accurate to

describe an impact phenomenon. An optimal design requires taking into account the main physical

processes involved and the potential development of nonlinearities due to material damage. Numer-

ous studies were carried out on the transient behavior of RC slabs under low velocity impact and

three approaches were usually used. Many researchers studied the actual response of impacted slab

with the development of crack patterns and resulting failure modes through full-scale impact tests

[34, 189]. Advanced numerical approaches, which take into account materials constitutive models

and contact algorithms, have proven to be reliable to analyze the failure modes of RC slabs subjected

to impact [19, 119, 137]. Simplied analytical methods were also developed and used as an initial

approximation of the impact behavior of slabs [10, 29, 47], they are generally based on simplifying

assumptions and account for the complexity of the studied phenomenon. However, these approaches

are deterministic and do not consider the randomness of the basic variables of an impact problem.

These variables aect the response of the slab which has also a random character, thus reliability

analysis seems to be the more appropriate to study the behavior of slabs under impact. Based on

this approach, a number of studies were carried out to estimate the reliability of dierent types of

RC structures. Low and Hao [111] performed a reliability analysis of RC slabs subjected to blast

loading using two loosely coupled single degree of freedom system and Val et al. [177] evaluated the

reliability of plane frame structures with ultimate limit state functions. However, reliability analyses

that consider the eect of uncertainties in a low velocity impact phenomenon are scarce.

The lack of knowledge on the potential load applied to a RC structure, as well as the uncertainties

related to its features (geometry, mechanical properties) shows that the design of RC structures could

be made in a reliability framework. Structural reliability provides the tools necessary to account for

these uncertainties and evaluate an appropriate degree of safety. This research discusses the use of

reliability analysis for three dierent civil engineering applications of dierent degrees of complexity.

The platform OpenTURNS is used to perform the reliability analysis of the RC structures considered

in the present study and propagate uncertainties in their physical models. OpenTURNS is used

because it oers a large set of methods that enable the quantication of the uncertainty sources,

the uncertainty propagation and the ranking of the sources of uncertainty. However, OpenTURNS

should be linked to the software used to perform deterministic analysis. The link of any external

software to OpenTURNS can be made using the generic wrapper and is called wrapping. Wrapping

allows an easy connection to the deterministic simulation as well as an easy evaluation of its response

due to several features provided in OpenTURNS. A wrapper is a Python based programming and

necessitates the description of input and output variables using regular expressions that enable the

substitution of the values of these variables with random values in the initial deterministic model.

Thus, the input le of the deterministic code and the output le corresponding to output variable

values should be specied, since the rst will be modied according to the random values generated

and the second will be read by OpenTURNS. The command to submit the deterministic model

should be also provided in the wrapper, it allows running the external software using the specic

random values generated for each call of the wrapper. A wrapper works as follows:

The wrapper uses the input le of the deterministic code in order to create a new le corre-

sponding to the random values generated. The new le will be placed in a temporary directory.

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Once the submission command of the deterministic model is accomplished, the wrapper reads

the results le and send the mechanical response obtained to OpenTURNS in order to perform

a reliability analysis for the structure considered.

The capacity of OpenTURNS to be linked to dierent nite element software is tested through

the three applications considered in this study. Therefore, OpenTURNS is linked to CASTEM to

study the reliability of a RC multiber cantilever beam subjected to a concentrated load at the free

end. It is also coupled to Abaqus in order to study the reliability of RC slabs which are subjected

to accidental dropped object impact during handling operations within nuclear plant buildings.

Furthermore, a combination of ASTER and OpenTURNS is considered in the aim of studying the

reliability of a prestressed concrete containment building. In the case where a structure is presented

with an analytical model, OpenTURNS is combined with Matlab.

The combination of mechanical and stochastic models is necessary in reliability analysis in order

to compute failure probability for one or several failure modes, which can be very time consuming

and in some cases impossible. Indeed, either the complexity of the deterministic model considered

implies important computing time (from minutes to hours) or reliability methods evaluating failure

probability require a too large number of simulations of the deterministic model.

There are many objectives behind this research. One of these objectives is to discuss the principles

of structural reliability analysis, as well as the types of uncertainty related to the structure and the

deterministic model used. This study describes the basic statistical concepts required to perform

a reliability analysis in structural engineering, as well as the probabilistic methods available in

OpenTURNS to assess failure probability of existing or new structures according to a specic limit

state function.

Another main objective of the current research is to address the issue of computational eort of

reliability analysis. Thus, two strategies are proposed to accurately assess failure probability with

the minimum computational cost:

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The rst strategy consists in using deterministic analytical models that are as simple as pos-

sible, but that are able to predict the structural response appropriately. In general, the con-

straining assumptions of analytical models allow studying only a few congurations of the

structure. Thus, the objective is to reduce the computational cost of the deterministic anal-

ysis to the minimum, while also explore as far as possible how to use this type of simplied

models. Due to their low computational time, these models can be coupled with simulation

probabilistic methods that need to carry out several deterministic analyses.

The second strategy is used when the structure is simulated with a nite element model.

It consists in choosing an appropriate probabilistic method where failure probability can be

assessed from a small number of simulations of the numerical model, or substituting the nite

element model by a meta-model less expensive in computational time.

A simple application to a RC beam is rstly used as a preliminary example in the aim of mas-

tering the basics of OpenTURNS and examining the eciency of several probabilistic methods

available in OpenTURNS. This application enables to compare probabilistic methods with

an example necessitating low computational eort in order to initially select an appropriate

method to use in the case of more complex structures depending on the deterministic model

considered and the computational eort constraints required.

The second application focuses on RC slabs which are subjected to accidental dropped object

impact during handling operations within nuclear plant buildings. In this case, dropped objects

are characterized by small impact velocities and damage can arise at the impact zone depending

on the impact energy and the relative masses of the colliding bodies. The following example

is used to compare the strategies proposed to reduce the computational eort of reliability

analysis, as well as to perform parametric studies in order to locate the most critical impact

points for the slab design and illustrate how to optimize the design of reinforced concrete

slabs under impact in terms of dimensions, material properties and reinforcement ratio. The

reliability of slabs subjected to impact assuming an elastic linear behavior of materials is

examined rst followed by the reliability of slabs when nonlinear behaviors of concrete and

steel are considered in deterministic models.

The third application consists of a prestressed containment building in order to study the

stress evolution with time in the containment during periodic surveillance testing carried out

20 years after its implementation. The problem is examined under aging phenomena, including

relaxation of reinforcement, creep and shrinkage of concrete. The aim of this application is to

investigate the eciency of meta-modeling strategy in reliability analysis of very complicated

structures. In this case, the deterministic model is approximated by a polynomial function and

the reliability analysis is performed for this approximation. The computational eort of this

strategy depends on the number of simulations needed to obtain an accurate approximated

model.

As previously mentioned, only the physical problem of reinforced concrete (RC) slabs subjected to

unplanned impact events due to dropped objects is investigated in detail. The aim of this application

is to provide an ecient procedure to predict RC slabs response under low velocity impact. Numerical

and analytical approaches are used and models are validated using physical experimental tests from

the literature which provide high quality input data and a wide range of output results. As a part

of the fundamental objectives of this research, the following investigations are carried out in order

to be able to examine the structure performance and carry out designs:

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Study the complex phenomenon of impact and the structural behavior associated to impact

by:

classifying the impact response types of RC slabs according to the impactor mass and

velocity,

dening the impact event types according to the impactor deformability, material prop-

erties and impact velocity,

describing the failure modes of RC slabs under impact according to their characteristics

and impact conditions,

presenting the energy transformation process of the impact energy to slabs as dierent

forms of energy.

posed in the literature,

choosing suitable material constitutive models to use in nite element analysis and iden-

tifying properly their parameters,

ability analysis of impacted RC slabs.

presenting the dierent steps necessary to create a nite element model using the software

selected,

describing the principles of nite element components implemented in the software and

adopted for the model.

explicit conguration, which allows taking into account the nonlinear behavior of concrete

and modeling the contact between the impactor and the slab. The model should also allow a

better representation of the actual structure geometry and boundary conditions. The model is

validated with experiments carried out on several RC slabs under drop-weight loads at Heriot-

Watt University by Chen and May [34]. Then, the model adopted is used to model RC slabs

which are subjected to accidental dropped object impact during handling operations within

nuclear plant buildings.

Develop a simplied analytical model that consists of two degrees of freedom mass-spring

system which accounts for potential viscous damping, and describes the degradation of slab

stiness using a frequency decrease approach

The modeling procedure adopted in this study simulates the impacted slabs as close as possible

to the actual structure without any geometrical or material simplications. The full geometry of

slabs is considered in the model and no symmetry boundary conditions are modeled in order to be

able to study several congurations of impact position. The model is parametrized through a code

developed using a Python script in Abaqus Python edition. It should be noted that strain rate eect

is not taken into account in this research as only low velocity impacts are studied.

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1.4 Thesis contents

Chapter 2 This chapter provides a classication of impact response types of RC slabs according

to the impactor mass and velocity. Then a denition of impact event types is introduced and the

behavior of RC slabs subjected to a transient dynamic load is described by determining their failure

modes according to their characteristics and impact conditions. An overview of approaches used

to determine slabs response is also given and several studies from the literature performed on their

transient behavior under low velocity impact are detailed. Next, the nonlinear static and dynamic

mechanical material behaviors are presented and many uniaxial stress-strain relationships for steel

and concrete in compression and tension proposed in the literature are provided.

Chapter 3 This chapter discusses the choice of deterministic software used in the present study

to simulate the problem of RC slabs which are subjected to accidental dropped object impact during

handling operations within nuclear plant buildings. First, the basic phases of nite element analysis

are described and the dierent steps which are necessary to create a nite element model in Abaqus

are presented. Furthermore, plasticity constitutive models used to represent concrete and steel in

numerical simulations are discussed by describing and identifying their fundamental parameters,

and contact algorithms used for modeling the interaction between two bodies are presented. Other

numerical features which are necessary for simulating impact analysis are also discussed.

Chapter 4 In this chapter, the principles of structural reliability analysis, as well as the types

of uncertainty related to the structure and the deterministic model used, are presented. Next, the

steps of a reliability analysis are detailed and the methods used in OpenTURNS for uncertainty

quantication, uncertainty propagation and sensitivity analysis are described. Finally, statistical

descriptions of random variables intervening in RC structures are examined according to several

studies in the literature.

Chapter 5 In this chapter, two deterministic models are used and evaluated for the problem of im-

pacted slabs. The rst model consists of a 3D nite element model simulated with Abaqus/Explicit.

The steel is modeled as an elasto-plastic material with hardening and, the behavior of concrete is

described by the damage plasticity model. A detailed step-by-step procedure for creating FE models

of impacted slabs with Abaqus is described. This model is used for Chen and May RC slabs [34]

and validated with experimental results. The model adopted is used to simulate RC slabs which

are subjected to accidental dropped object impact during handling operations within nuclear plant

buildings. Then, a simplied analytical model is also used for these slabs. It consists of a two degrees

of freedom mass-spring system which accounts for potential viscous damping. A frequency decrease

approach is used to describe the degradation of the slab.

Chapter 6 In this chapter, 3 case studies are used in order to address the issue of solving relia-

bility problems in the domain of civil engineering. The rst application consists of a RC multiber

cantilever beam and is used in the aim of mastering the basics of OpenTURNS and examining the

probabilistic methods proposed in OpenTURNS to estimate failure probability in structural reliabil-

ity analysis in terms of their accuracy, precision and computational eort. The second application

is the problem of RC slabs subjected to impact in the aim of addressing the issue of computational

cost of reliability analysis and proposing computational strategies allowing the accurate assessment

of the failure probability for minimum computational time. Firstly, the problem of impacted slab is

studied assuming a exural mode of failure and an elastic behavior for steel and concrete. Following

this, the problem of impacted slabs is studied using deterministic models that take into account the

nonlinear material properties. In both cases, elastic and nonlinear behaviors, a parametric study is

performed to identify the inuence of deterministic model parameters on the reliability of RC slabs

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under low velocity impact. The third application consists of the prestressed concrete containment

building of the Flamanville nuclear power plant and is considered in the aim of presenting a proce-

dure to be followed to study the response of very complicated structures in a reliability framework.

Polynomial chaos expansion is used to simplify the physical model and study the reliability of the

containment.

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Chapter 2

impact: Dynamic response and literature

review

2.1 Introduction

In civil engineering eld, reinforced concrete (RC) structures are often subjected to some extreme

dynamic loadings due to accidental impacts of rigid bodies that may occur during their service life

with a very low probability of occurrence. The failure resulting consequences of structures subjected

to such extremely severe loadings might be extremely high, which makes their analysis and design

very complex especially when working with nonelastic materials. Impact loadings are characterized

by a force of considerable magnitude applied within a short duration, they may be caused by falling

rock impact on protection galleries, missile and aircraft impact on nuclear containments, vehicles

collision with buildings or bridges, ships or ice crash impact with marine and oshore structures,

ying objects due to natural forces such as tornados and volcanos, fragments generated due to

military or accidental explosions on civil structures, or by dropped objects impact on industrial or

nuclear plant oors as a consequence of handling operations. Figure 2.1 shows some examples of

impact loading cases in civil engineering eld. For all these applications, the problem of impact has

to be studied with convenient considerations since structures undergo, depending on their stiness,

dierent failure modes such as exure and punching shear failure, in addition to dierent types

of local damage such as crushing, cracking, scabbing, spalling and perforation. Furthermore, the

velocity, angle and position of impact, as well as the shape, mass and rigidity of impactors vary

broadly from one application to another.

Design of RC slabs to resist impact loads represents an area of research that is increasingly gaining

importance and numerous studies were carried out by many researchers on the behavior of RC slabs

subjected to dynamic impact loading, especially under high velocity regimes [17, 52, 80, 86, 92].

However, when slabs are exposed to unplanned impact events such as accidental collisions with

dropped objects, the problem is classied as low velocity impact event. Several studies that will be

detailed in section 2.3 were carried out on the transient behavior of RC slabs under low velocity

impact, but there is a growing need to more investigate the behavior of slabs for low velocity

situations.

In the available design codes of civil engineering, the design of reinforced concrete (RC) structures

subjected to impact is generally based on approximate static method. The dynamic forces during

impact are usually considered by applying an equivalent dynamic impact factor and converted into

a static force of equal magnitude. Such static design is often not suciently accurate to describe an

impact phenomenon, as not only the maximum value of the impact force is crucial to predict the

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Figure 2.1: Some accidental impact loading cases in civil engineering eld: a) Rockfall on protection

galleries, b) Aircraft impact on nuclear containments, c) Dropped objects impact during handling

operations

structural dynamic behavior, but also the loading duration and rate must be taken into account.

The use of approximate static methods results in either an under- or over-dimensioned design for

structures and, in some particular cases, the dynamic response of structures cannot be simplied

to a quasi-static response. Moreover, an equivalent dynamic force can be just used in the case

when structures withstand exural mode. Nevertheless, during impacts, structures do not only fail

under exural mode but they may fail under punching shear or local damage, and higher modes of

vibration can be expected. Thus the transient dynamic analysis has to be performed accounting

for the main physical processes involved and accurate models are needed to describe and predict

the structural behavior of RC members when subjected to impact loading. An optimal design that

guarantees the structure resistance with an economical cost requires taking into account the potential

development of nonlinearities due to the material damage (concrete cracking, steel yielding, etc.).

Abrate [9] indicated that the study of impact on composite structures such as RC slabs involves

many dierent topics (contact mechanics, structural dynamics, strength, stability, fatigue, damage

mechanics, micromechanics) and that the prediction of dynamic response of composite structures to

impact can be made by using analytical or nite element models which must appropriately account

for the motion of the impactor, the overall motion of the structure, and the local deformation in

the contact zone and in the area surrounding the impact point. Abrate also indicated that the

parameters which aect the impact resistance of composite structures are the properties of the

matrix, the reinforcing bers, the ber-matrix interfaces, the size, the boundary conditions, and the

shape, mass, and velocity of the impactor.

The present study focuses on RC slabs which are subjected to accidental dropped object impact

during handling operations within nuclear plant buildings (Figure 2.2). In this case, dropped objects

are characterized by small impact velocities and damage can arise at the impact zone depending on

the impact energy and the relative masses of the colliding bodies. The current chapter rst provides a

classication of impact response types of RC slabs according to the impactor mass and velocity. Then

a denition of impact event types is introduced and the behavior of RC slabs subjected to a transient

dynamic load is described by determining their failure modes according to their characteristics and

impact conditions. Afterwards, an overview of approaches used to determine slabs response is given

and several studies from the literature performed on their transient behavior under low velocity

impact are detailed. Next, the nonlinear static and dynamic mechanical material behaviors are

presented and many uniaxial stress-strain relationships for steel and concrete in compression and

tension proposed in the literature are provided.

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Figure 2.2: Accidental dropped object impact during handling operations within nuclear plant build-

ings

2.2.1 Impact dynamics

During handling and maintenance operations within nuclear plant buildings, impacts on RC slabs

by foreign objects as tools and nuclear material packages can be expected to occur. Impacts create

internal damage that depends on the load (velocity and mass of the impactor) and may causes severe

reductions in strength [9]. The internal damage is often undetectable by visual inspection, hence the

eects of dropped object impacts on RC slabs must be understood and taken into account in the

design process. According to Abrate [10], composite structures subjected to impact have dierent

types of behavior which must be properly evaluated in order to interpret experimental results and

select convenient analytical or numerical models. Abrate studied and classied the currently available

models used to predict the dynamic of impacts between a foreign object and a composite structure

in the aim of presenting an approach to select an appropriate model for each particular case study.

He showed that developing a model to analyze the overall response of the structure and the contact

phenomenon requires understanding the dierent types of structural behavior, as well as the eects

of several parameters on the impact dynamics.

According to [9], when the impactor enters in contact with the structure, shear and exural waves

propagate away from the impact point and reect back when they reach the back face. These waves

propagate at dierent speeds and decline progressively in response to materials damping, energy

diusion and several geometrical eects [151]. Thus, the impact response can be controlled by wave

propagation and strongly inuenced by impact duration [149]. The impact duration inuence is

illustrated in Figure 2.3, which shows three dierent types of response for very short, short and long

impact times.

The response dened for very short impact times is dominated by three-dimensional wave prop-

agation through the thickness direction and the impact time is in the order of waves transition time.

This response is unaected of structure size and boundary conditions and is generally associated

with ballistic impacts (Figure 2.3.a). A denition of ballistic impacts was given by [9] as impacts

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Figure 2.3: Response types during impact on plates according to [149]

resulting in complete perforation of the structure with a ballistic limit equal to the lowest initial

velocity of the impactor causing this perforation. The ballistic limit increases with the structure

thickness and is aected by material densities and mechanical properties. For short impact times,

the response is governed by exural waves and shear waves and is typical for impact by hail and

runway debris (Figure 2.3.b). In this case, the response remains independent of boundary condi-

tions as long as main waves have not reached a boundary. For long impact times, the response is

quasi-static inuenced by structure dimensions and boundary conditions. A typical example of

long impact times is dropping of heavy tools. In this case, the time needed by waves to reach struc-

ture boundaries is much less than impact duration and the quasi-static response is in the sense

that deection and load have the same relation as in a static case (Figure 2.3.c). The two latter

response types can be associated with the case of RC slabs subjected to foreign object impact.

A criterion related to the three-dimensional stress distribution under the impactor was proposed

by Abrate [9] to classify impact dynamics by distinguishing between high-velocity and low-velocity

impacts. His criterion consisted of dening low-velocity impacts in cases where stress wave prop-

agation through the thickness has no signicant role. In this case, damage is not introduced in

the early stages of impact, but it is initiated after the overall structure bending behavior is estab-

lished. For low-velocity impacts, stress levels remain low due to several wave reections through the

thickness and the structure dynamic properties are usually not aected by the presence of damage.

Conversely, Abrate indicated that damage for high-velocity impacts is introduced during the rst

few compressive wave reections through the thickness, while overall structure behavior is not yet

initiated. Abrate's criterion dened high-velocity impacts as cases related to the ratio of impact

velocity versus the speed of propagation of compressive waves through the thickness direction, so

that this ratio must be larger than the maximum failure strain. During high velocity impacts, fail-

ure near the structure back face may occur due to tensile stresses generated by the compressive

wave after reection. A velocity of 20 m/s was suggested by Olsson [151] as the upper limit of low

velocity impacts, while the suggested limit of medium velocity ranges from 20 to 100 m/s. Very

high velocities were considered to be greater than 2 km/s [9]. Classifying impact events by using

velocity is the most common, however this classication is highly relative. For example in the case

of impactors dropped at low velocity (a few m/s) on large plates, the response was demonstrated to

be associated with high-velocity impact since it is governed by wave propagation [148].

Further, the initial kinetic energy of the impactor is an important parameter that aects the

structure response [9, 149]. In consequence, although the kinetic energy is the same, a structure

subjected to an impact of a large mass with a low initial velocity behaves dierently than in case of

a smaller mass with higher velocity. For the rst problem, the indentation eect is negligible and

the impact may involve an overall response of the structure. In the other case, damage initiates at

earlier stages and the response can be introduced by deformations in a small zone surrounding the

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impact point. Impactors with small mass cause more localized deection and higher impact loads

with a signicant eect of the indentation on absorbing the impact energy that can be restituted

to the impactor or transferred to the structure. The inuence of impactor mass on the response

behavior can be associated to Figure 2.3 as mentioned in [150]. A ballistic response occurs for very

small masses, moderately small masses cause a response dominated by shear and exural waves and

large masses cause a quasi-static response.

However, Olsson [149] showed that response and damage are not inuenced only by impact

velocity, duration or initial kinetic energy, but also by the impactor vs. structure mass ratio. Olsson

derived a simple criterion for impact plates response controlled by exural waves and distinguished

between small-mass, intermediate-mass and large-mass impacts. A sucient condition for small-

mass impact response is that dominating exural waves does not reach the boundaries during the

impact duration. For central impact on a quasi-isotropic plate, small-mass criterion corresponds to

impactor masses less than one fth of the mass of the plate area aected by the impact:

The response with intermediate-mass impactors is more complex and the mass criterion becomes:

For large-mass impactors, the exural wave reaches boundary conditions and a quasi-static

impact response occurs for:

where M is the impactor mass and Mp is the mass of the impacted plate. For non-central

impacts, Mp is calculated as the mass of a square zone centered at the impact with a side that

coincides with the closest plate edge [151]. For square plates, the condition of small-mass impact

response corresponds to a mass ratio less than 1/4 (M/Mp ≤ 1/4). It should be noted that this mass

criterion becomes irrelevant for impact velocity resulting in penetration [151].

An existing denition allows normally classifying impact events into either hard or soft according

to the impactor deformability. For hard impact, impactor is generally considered as rigid and its

deformation is negligible compared to the structure deformation (Figure 2.4.b). In this case, the

impact kinetic energy is to a great extent absorbed by the structure deformation and the impactor

remains undamaged or barely deformed during impact. Hard impact results in both local and overall

dynamic response of the structure and failure may occur due to complicated stress waves. According

to [132], the shape and dimensions of impactor, as well as impact velocity are the essential parameters

that may aect the classication of an impact as hard.

However, in the case of soft impact, the structure deformation is very small compared to the

impactor. In other words, the structure that resists to impact is assumed to remain undeformed,

while the impactor is strongly damaged and the impact kinetic energy is fully transferred into

deformation energy of the impactor (Figure 2.4.c). The propagation of stress waves is considered as

negligible and the structure failure response can be compared to that under static loading with no

rebound of the impactor. An impact can be considered as soft owing to the failure region and the

ratio between the impacted structure and the impactor masses [132]. The case of soft impact can

be illustrated by a crash of a vehicle against a rigid wall or a crash of an airplane against a thick

concrete containment wall of a nuclear power plant [29].

This basic and well-known denition of soft and hard impacts was proposed by Eibl [55] in order

to consider the dissymmetry of an impact event along with the dierence between structure and

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Figure 2.4: Structure and impactor deformations according to the impact type [45] : a) Initial

conditions (t=0), b) Hard impact (t>0), c) Soft impact (t>0)

impactor responses. The dissymmetry of an impact event is due to the fact that the structure is

stationary before impact, while the impactor moves with an initial velocity V (0). As mentioned

earlier, one of the two bodies in collision shows small deformations compared to the other one

during soft and hard impact, which involves a dissymmetry in the behaviors of the structure and

the impactor.

The classication of Eibl is based on a mass-spring model (Figure 2.5) that consists of the

structure (m2 ) and the impactor (m1 ) masses, a contact spring associated with a stiness k1 and a

second spring with a stiness k2 . The contact spring simulates the force that arises when the two

bodies are in contact, while the second represents the structure behavior and its resisting force to

impact. The mass-spring model is governed by two dierential equations that describe the motion

of the two bodies and can be expressed as follows:

(

m1 ẍ1 (t) + k1 [x1 (t) − x2 (t)] =0

(2.4)

m2 ẍ2 (t) − k1 [x1 (t) − x2 (t)] + k2 x2 (t) = 0

In the case of a soft impact (x1 (t) x2 (t)), the contact force F (t) can always be deduced only

from the impactor deformation by solving the rst independent equation of the following system:

(

m1 ẍ1 (t) + k1 x1 (t) = 0

(2.5)

m2 ẍ2 (t) + k2 x2 (t) = F (t)

with F (t) = k1 x1 (t). The second equation allows determining the structure response under an

independently acting force F (t).

Hard impacts require to consider the local as well as the overall deformations of the structure,

thus the two equations of system (2.4) cannot be independent and spring stinesses must be de-

termined. Section 2.3.2.2 shows how to calculate k1 and k2 for hard impacts as indicated by the

Euro-International Concrete Comity (CEB) [29] that assumes the same characterization of soft and

hard impact as Eibl.

Nevertheless, Koechlin and Potapov [95] found that the denition of Eibl is qualitative and

not precise enough to make a distinction between soft and hard impacts for low velocity impact

events. Alternatively, they proposed a more precise quantitative classication which is based on the

material properties and the impactor velocity, and permits to predict the type of impact that is

expected even before the observation of the structural response during impact. They indicated that

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Figure 2.5: Mass-spring model proposed by Eibl [55] to simulate an impact [45]

considering the impactor deformation as described previously as the essential property of the impact

type specication is not completely true since the terms hardness and softness are relative and

the designation of a rigid impactor or a rigid structure is not identical for all studies that examine the

eect of impact on RC structures. Consequently, they showed that the characterization of soft and

hard impacts is related to the structure failure mode and to the fact that the impactor penetrates

or not the structure. They indicated that in the case of soft impacts, failure occurs due to a shear

cone failure mode, the structure is not damaged and the impactor is crushed. On the contrary,

perforation is a consequence of a local failure and the impactor penetrates the structure for hard

impacts.

In order to dene an impact type, the new criterion of Koechlin and Potapov consists in com-

paring the structure compressive strength σs with the stress applied by the impactor during impact.

This latter stress represents the sum of the impactor limit strength σi and a second component that

depends on the impactor density ρi and velocity vi . This criterion allows verifying if the structure

withstands the impact and is expressed by:

σs = σi + ρi vi2 (2.6)

Equation (2.6) can be written in terms of non-dimensional parameters, which permits to compare

impacts at dierent scales and gives the boundary between hard and soft impacts (Figure 2.6):

σi ρi vi2

+ =1 (2.7)

σs σs

This boundary provides the type of failure that could occur during impact and species whether

the structure fails due to a direct perforation or a punching shear. In other words, it represents the

boundary between the potential global and local failure modes.

Numerous experimental studies were carried out to assess the eect of soft and hard impact

on the concrete behavior and the response of RC structures. Koechlin and Potapov presented and

classied these tests using their new criterion that is based on an explicit graphic representation

as shown above. Few experimental tests that were considered by their authors as soft impact

are available in the literature, namely EDF tests [51] and Meppen tests [90, 140, 141, 162]. The

classication of these experiments as soft impact is in accordance with Koechling and Potapov new

classication. The EDF tests constituted a part of a research program that was initiated by EDF

(Eléctricité de France/ French Electric Power Company) in order to analyze the nonlinear response

of RC slabs subjected to short time accidental loads due to soft missiles impact. The main purpose

of EDF tests was to deduce some practical rules for plastic design and reduce the reinforcement

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Figure 2.6: Classication of low velocity impacts according to Koechlin and Potapov [95]

ratio with ensuring that no failure or even large deection would occur during impact. The full

scaled Meppen tests consisted in a series of 21 RC slabs impacted by highly deformable missiles

and aimed to assess the safety of containment buildings of German nuclear power plants against

potential aircraft impacts. Meppen tests examined the eect of some parameters such as bending

and shear reinforcement ratios, the impactor velocity and deformation, and the slab thickness on

the bending and shear bearing capacities of slabs. They proposed a validation of methods that are

usually applied to analyze aircraft impact loads with regard to the structural design.

Other experiments with soft missiles [97, 147, 166, 167] were incorrectly referred as soft impact

due to the reason that the impactor is deformable. However, Koechlin and Potapov found that,

according to their criterion, these tests must be classied as hard impact since the failure mode of

slabs is similar to the hard impact one. Kojima tests [97] were performed in the aim of investigating

the local behavior of RC slabs subjected to missile impact with varying the missile and slabs prop-

erties. Soft-nosed and hard-nosed missiles were used to study the degree of slab damages in both

cases, while critical reinforcement ratio and slab thickness were evaluated in order to analyze slabs

resistance to impact and prevent perforation. Ohno et al. [147] investigated experimentally the

eect of missile nose shape on the local damage of RC slabs when impacted by deformable missiles.

Five shapes of missiles were used for the impact tests and empirical formulas for perforation and

scabbing were developed. The investigations carried out by Sugano et al. [166, 167] are among

other experimental studies that examined local damage caused by deformable missiles impact on

RC structures. Sugano et al. tests showed that reduced and full scale tests give similar results and

that reinforcement ratio has no eect on local damage.

Moreover, it should be noted that tests performed by Kojima [97] and Sugano et al. [166,

167] with rigid missiles were also classied as hard impact by Koechlin and Potapov, which is in

accordance with Eibl classication. CEA-EDF experiments [61, 71, 77] were also considered as

hard impact for both criteria. They were performed by CEA (Commissariat à l'énergie atomique

et aux énergies alternatives/ French Alternative Energies and Atomic Energy Commission ) and

EDF to predict the perforation limit of RC slabs subjected to rigid missile impacts with varying

slabs thickness, concrete strength, reinforcement ratio and impact velocity. Likewise, Koechlin and

Potapov described impacts with high velocity as hard impacts, which is in agreement with the usual

observations of Bischo and Perry [22].

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2.2.3 Failure modes

Structures subjected to impact loading show dierent behavior compared to that under static load, as

a result of the transient and localized properties of impact loading. Furthermore, dynamic properties

of materials are dierent than those under static loading and may be aected by the strain rate or

load rate. Therefore, a solid understanding of this type of structures behavior under an impact

event is essential to develop a satisfactory design and prevent their collapse. The prediction of the

behavior of RC slabs when they are subjected to impact loading is of considerable complexity since

impact event involves several phenomena and slabs have been observed to react both locally and

globally. Failure modes of slabs during impact change with time and are inuenced by stress waves

and inertial forces. The impact behavior of concrete is also considered as quite complicated due to

heterogeneity of material. Several parameters may signicantly inuence the impact response of RC

slabs:

Slab dimensions and boundary conditions control the stiness of the slab.

Material properties have a signicant inuence on the slab transient response by aecting the

contact and overall slab stinesses.

Impactor characteristics including impact velocity, shape, position, mass and rigidity inuence

the impact dynamics.

Changes in slab stiness arise from cracking or crushing of concrete, yielding of reinforcement, or

other mechanisms such as local unloading behaviors and post-cracking dilation [84].

A RC slab subjected to a dropped object will be exposed to a transient dynamic load. As a result,

both local and global dynamic response of the slab can be induced. The identication of the various

modes of failure that occur during impact is important to correctly evaluate the dynamic response.

The elastic-plastic response of RC slabs may cause two main overall response failure modes: exural

failure or punching shear failure [119] (Figure 2.7). At exural failure mode, the slab bends strongly

and fails due to excessive tension stresses which lead to the formation of cracks through the RC slab

thickness and yielding in tension reinforcement. At the punching shear failure mode, a shear cone

that engenders tensile stresses within the transverse steel is created and the slab fails due to excessive

shear stresses. The kinematic mechanism of punching was described by [173] for a static load: the

application of an increasing load to a RC slab presents a roughly circular crack that appears around

the loading zone on the slab tension surface and propagates subsequently into the compressed zone

of concrete. Then new exural cracks arise and inclined shear cracks are observed. The further

increase in load develops curved shear cracks which aect steel in tension and compression zone.

Punching shear mode is characterized by brittle response with minimal tension cracking damage and

little or no yielding of reinforcement [130]. However, an intermediate failure mode between exural

and punching shear failure modes can also be expected [8]. A combined exure-shear failure mode

is characterized by a exural failure at the early stages of loading followed by a transition into the

punching shear failure mode.

The local damage of impacted slabs can be divided into four local failure modes: surface crushing,

spalling of concrete and formation of a crater on the impacted surface, scabbing of the rear face and

perforation [138] (Figure 2.8). The severity of the local damage is primarily dependent upon the

velocity of the impactor [93]. According to [92, 138] when the impactor strikes the slab for low impact

velocities, a failure surface mode occurs. At failure surface, concrete is fallen o the impacted face

of the slab at and around the impact zone. The impactor penetrates the slab to low depth and then

bounces back. The further increase in velocity causes ejection of concrete from the upper surface and

spalling occurs producing a spall crater in the surrounding zone of impact. The impactor penetrates

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Figure 2.7: Overall response failure modes of impacted RC slabs [119] : a) Flexural failure, b)

Punching shear failure

deeper into the RC slab without causing any eect on rear face [93] and sticks to the slab rather

than rebounding. Spall crater surface is generally greater than the impactor cross-sectional surface

and depends upon the mass, velocity and shape of the impactor. Normally, impactors with low

velocity cause more spalling with a bigger spall crater than for high velocity cases [109]. As the

impactor velocity increases further, scabbing arises on the rear face and pieces of concrete will be

spalled o the back surface of the slab. Since concrete is very weak in tension, scabbing takes place

when the tensile stresses generated by the tensile reected wave produced during impact become

equal or higher than the concrete tensile strength. Generally, scabbing shows a wider zone and

higher depth than spalling and indicates that concrete has no further strength remained to resist

more local impact eects [109]. If the impactor velocity is high enough the impactor will perforate

the slab and exit from its rear face with residual velocity. Perforation is the last process of damage

due to impact event and is caused by extending penetration hole through scabbing crater. The local

damage failure modes are caused by stress wave response and usually occur in conjunction with the

two overall response failure modes.

The evaluation of each of the above failure modes is completely dicult either experimentally

or numerically. For example, complete perforation usually takes place under high-velocity impact

when slab deformations are localized in the impact zone, but it can also be achieved under low-

velocity impacts after the slab reaches its overall deections [9]. Spalling and scabbing generally

occur during bending failure, while they cannot be observed during a full penetration [189]. These

failure modes must be taken into account when designing RC slabs under impact: it is obvious that

exural failure mode represents the critical parameter to design longitudinal reinforcement, punching

shear is important to determine shear reinforcement, while spalling and scabbing helps to evaluate

the appropriate concrete properties [19].

The onset and growth of damage in slabs represent another aspect that should be treated as

well to a better understanding of slabs dynamic behavior under impact, since the main orientation

of major cracks can be helpful to distinguish the corresponding failure mode during impact event.

According to Abrate [9] that studied the impact on composite structures, during a low-velocity

impact event, two types of cracks are observed, namely tensile cracks and shear cracks. Tensile

cracks are created by tensile exural stresses and appear when in plane normal stresses exceed

the concrete tensile strength. Shear cracks are inclined relative to the normal to the midplane

and indicate a signicant role of shear stresses. Cracks initiate the damage process and follow a

complicated and irregular path in concrete. For thick slabs, cracks are rst induced on the upper

surface subjected to impact as a result of high and localized contact stresses, then they progress

from the top downwards. For thin slabs, cracks are induced on the bottom surface due to bending

stresses, which leads to a reversed shear cracks pattern. Therefore, a detailed prediction of nal

cracks pattern is very dicult, but it is not necessary since its shape has no important contribution

on the reduction in slab residual properties.

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Figure 2.8: Local damage failure modes of impacted RC slabs [119] : a) Surface failure, b) spalling,

c) scabbing, d) perforation

Zineddin and krauthammer [189] performed an experimental study in the aim of understanding

the dynamic behavior of structural concrete slabs under impact loading. It was shown that the

failure modes of slabs are dependent on the rate of the applied load, i.e. static or quasi-static

loading favors a global bending mode while for harder and shorter impact the local response could

dominate the slab behavior with a likelihood of punching mode. In case of soft impact, a direct

shear failure could take place at the boundary conditions due to high stresses. These results are in

agreement with Miyamoto and King [130] study that also considered the eect of loading rate on slab

failure mode. Miyamoto and King proposed a dynamic design procedure for RC slabs subjected to

soft impact loads and found that failure modes are aected by loading rates, slabs fail under exural

mode for low loading rates while the punching shear failure is dominant for higher loading rates. The

experiments of Zineddin and Krauthammer showed also that the response of a slab is aected by the

amount of steel reinforcement and the drop height (i.e. the impact velocity), RC slabs may change

their failure mode from a exural failure at low drops to punching one under higher drops. As well,

Martin [119] indicated that the overall response failure of a RC slab depends upon the impactor

velocity and reinforcement strength: for lower impact velocities exural failure is more likely while

for high impact velocities punching shear failure occurs; for a strong reinforcement exural failure

is more likely and for a weaker one punching shear failure takes place. However, reinforcement ratio

shows an adverse eect on RC slabs behavior under impact, as indicated by Abbasi et al. [8]. Abbasi

et al. mentioned that for weakly reinforced slabs failure is identied by a global and pure exural

mode with a measure of ductility due to reinforcement plasticity, whereas increasing reinforcement

ratio enhances the punching capacity and slabs may fail under combined exural-shear mode for

medium levels of reinforcement and under punching shear mode for high reinforcement ratio.

An impact event can be dened as an interaction of two bodies where mechanical energies are

transformed. Miyamoto and King [130] mentioned that the energy criterion would be the most

ecient method of designing concrete structures under impact loads, especially for a exural failure

mode. They show that structural failure of a concrete structure subjected to impact likely occurs if

the structure is not capable of absorbing all of the energy transmitted during impact. Therefore it

is necessary to examine the energy balance, especially in case of evaluating the results of numerical

impact analyses. As illustrated in Figure 2.9, before the impact there is only the impactor kinetic

energy of which the main part is then transmitted to the RC slab when the impactor hits the slab.

After impact a small part of the impactor kinetic energy is transferred to slab as kinetic energy

due to slab vibrations that occur as a result of impact, while a considerable part of this energy is

converted into the energy absorbed by the structure. The energy not transmitted into the structure

is progressively converted into elastic and plastic strain energy in the impactor as a result of its elastic

and plastic deformation. The energy absorbed by the RC slab involves irrecoverable (plastic) energy

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Figure 2.9: Energy transformation process during an impact event [130]

and recoverable strain (elastic) energy. These energies are absorbed by the mechanisms of concrete

and reinforcement [92], as well as elastic and plastic parts of strain energy in slab are associated

with its deformation after impact [119]. A part of the recoverable strain energy is subsequently

re-transmitted to slab as kinetic energy in response to slab vibrations, as indicated in Figure 2.9

by a dashed line. In addition to plastic deformation, the irrecoverable energy in RC slabs results

from the formation of cracks, friction and damping. Thus, a part of the initial kinetic energy of

the impactor is dissipated as damage energy due to concrete damage and as viscous energy due to

viscous damping inside the RC slab.

Martin [119] indicated that an energy balance of an impact on a RC slab can be written as

follows:

I I I S S S

Ekin0 = Ekin1 + Estr1 + Ekin1 + Estr1 + Edam + Evis (2.8)

with

I

Ekin0 = kinetic energy of impactor before impact

I

Ekin1 = kinetic energy of impactor after impact

I

Estr1 = strain energy of impactor after impact

S

Ekin1 = kinetic energy of concrete slab after impact

S

Estr1 = strain energy of concrete slab after impact

S

Edam = damage energy due to damage of concrete

Evis = viscous energy due to viscous damping

Miyamoto et al. [133] found that the kinetic energy transmitted to a RC slab structure by an

impacting body as well as the energy absorbed by the structure are aected by the nal structural

failure mode and RC slabs may respond in several ways as indicated by [84]. For overall response

failure modes the energy absorbed is due to global structural bending or shear deformations of

the slab, while the main part of impact energy is dissipated through local damage mechanisms for

local response failure modes. In case where both local and global dynamic response of the slab

are involved, the initial kinetic energy of impactor is dissipated through a combination of structure

deformations and local damage mechanisms. Moreover, Miyamoto et al. indicated that the energy

absorbed by slabs under exural failure mode is higher than in case of punching shear failure mode

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which results in less energy being dissipated. The process of dissipating energy is also aected by

mechanical properties of the RC structure and the impactor, it can vary signicantly depending

upon the type of impact event (hard or soft) [84, 119]. Therefore, the energy transformation process

of the impactor kinetic energy to dierent forms of energy mentioned in equation 2.8 is inuenced

by the impactor velocity, the slab rigidity and the reinforcement of the concrete slab; e.g. in case of

a strongly reinforced slab subjected to a soft impact with a high initial velocity, the majority of the

impact energy is converted into strain energy of the impactor [119]. The impactor shape and size

have also a considerable eect on the energy absorption mechanism since failure modes can be quite

dierent, consequently impactors with nose shape dissipate more energy during penetration than

at impactors and the energy absorbed during a perforation failure mode increases with impactor

diameter [9].

This section presents the three approaches that are usually used to determine the global and local

dynamic responses of RC slabs subjected to impact loading. Several studies from the literature that

are considered as important investigations in the context of impact phenomenon eect on RC slabs

are also presented for each approach. These three approaches can be classied as follows:

Experimental methods based on either laboratory or eld studies and leading to empirical

formula;

This approach, based on either laboratory or eld studies, is considered to be fundamental for

extending the understanding of the dynamic behavior of RC slabs under the action of impact loading.

The main aim of this approach is to study the actual response of the slab including the development

of crack patterns and the resulting failure modes. Various data can be gathered from tests such

as the variation of the impact load with respect to the slab deection, the impactor acceleration,

the reactions at supports and the reinforcing steel strains. Experimental data are important for

validating analytical and numerical models. The accuracy of the measurement of physical parameters

depends on observations and the equipment used to store data from the test. Empirical expressions

which are derived from experimental data are especially important due to their simplicity to represent

the complex impact phenomena and can be useful to get a rst order approximation of impact eects.

However, experimental approach is not a cost-eective practicable solution since it requires a

series of full-scale tests in order to perform a reliable estimation of RC structures response. This

approach can be an excessively time-consuming and costly procedure, especially in terms of providing

the necessary test materials and equipment.

Zineddin and Krauthammer [189] performed an experimental study in the aim of understanding the

dynamic behavior of structural concrete slabs under impact loading, and investigating the eect of

reinforcement type and the applied load on the dynamic response of RC slabs.

9 slabs of 3.353x1.524 m span and 90 mm thickness were tested and reinforced with three dierent

types of reinforcement placed along the slabs width and length at spacing of 152 mm, as follows:

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Figure 2.10: Impact test system used by Zineddin and Krauthammer [189]

1 single mesh of longitudinal and transversal steel bars of 9.5 mm diameter, located at slabs

center with a cover of 45 mm (Reinf. 1).

mm from the top and bottom of slab faces (Reinf. 2).

2 meshes of longitudinal and transversal steel bars of 9.5 mm diameter, placed at a cover

distance of 25 mm from the top and bottom of slab faces (Reinf. 3).

Slabs were impacted with a cylindrical drop hammer with a 2608 kg mass and 250 mm diameter.

The impact mass was dropped from dierent heights of 152 mm, 305 mm or 610 mm at the center

of slabs in order to examine the variation of slab failure modes under low and high drop heights.

Slabs were supported on a sti steel frame of 305 mm width and bolted on all four sides of the top

face with steel channels of the same width (Figure 2.10). Zineddin and Krauthammer mentioned

that boundary conditions could be described as somewhere between simply supported and xed.

Output that were gathered from these impact tests included impact force-time history, slab

deection, reinforcement strains, as well as accelerations that were measured for the slab, the steel

frame and the impact mass. Cracks propagation was also examined by means of high-speed videos.

Zineddin and Krauthammer studied the transition of slab behavior modes in term of force-time

history curves and indicated that, according to these curves, four stages could be distinguished:

1. Slab resistance to the impact load as an elastic plate, followed by concrete failure at the impact

zone.

2. Reinforcement resistance with severe cracking of concrete around the perimeter of the impacted

zone.

3. Steel yielding that started in bars under the impact mass and propagated radially in the

non-impacted zone.

4. Increase in the slab deection at the center, which may lead to a concrete ejection or slab

perforation.

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Figure 2.11: Load-time history for slabs impacted with a drop height of 152 mm [189]

Figure 2.12: Failure mode of slabs impacted with a drop height of 610 mm and reinforced with

reinforcement type: (a) Reinf. 3, (b) Reinf. 1, (c) Reinf. 1-Failure of one steel bar [189]

Figure 2.11 shows these four stages for 3 slabs with 3 dierent types of reinforcement and impacted

with a drop height of 152 mm. These curves indicate that slab response is aected by reinforcement

type and, particularly by reinforcement ratio. Although all curves had a similar shape, the peak

load for slab with lower reinforcement ratio (Reinf. 1) was not signicant as the peak for other slabs

and was reached at earlier time. Reinforcement ratio had also an important inuence on slab failure

modes, as can been seen in Figure 2.12 which presents the failure modes of two slabs impacted

with a drop height of 610 mm but with dierent reinforcement ratio. In fact, slabs with more steel

reinforcement showed a localized punching shear failure (Figure 2.12.a), whereas slabs with less

steel reinforcement failed by brittle failure of concrete with a considerable amount of debris ejected

from the bottom surface (Figure 2.12.b). Reinforcement bars were yielded for both slabs, but one

longitudinal steel bar was cut in the case of slab with lower reinforcement ratio (Reinf. 1) due to

the excessive shear failure that occurs at the impact zone and to its location at the mid-thickness

of the slab (Figure 2.12.c).

Zineddin and Krauthammer found that slabs response was also aected by drop height of the

impact mass, hence RC slabs may change their failure mode from a exural at low drops to punching

one under higher drops. Slabs impacted with the lower drop height of 152 mm were designed in the

objective to fail by exural failure mode regardless of their reinforcements. Nevertheless, only slab

reinforced with Reinf. 3 type underwent exural failure with large cracks on the bottom surface,

and the two slabs reinforced with Reinf. 1 and Reinf. 2 types failed in a brittle manner since the

change in the failure mode was compensated by the strength increase due to the loading rate. For

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Figure 2.13: Cracks patterns on top surface of slabs reinforced with reinforcement type Reinf. 3 and

impacted with a drop height of : (a) 305 mm, (b) 610 mm[189]

2 98.7 Hemispherical 6.5

3 98.7 Flat 6.5

4 98.7 Hemispherical 8.0

5 196.7 Hemispherical 8.7

6 382.0 Hemispherical 8.3

higher drops, a local failure mode dominated the behavior of slabs. Consequently, tests performed

with 305 mm and 610 mm drops showed a punching shear failure with a smaller punching shear hole

and more cracks at the top slab surface for higher drops (Figure 2.13)

A series of experiments were carried out on several RC slabs under drop-weight loads at Heriot-Watt

University by Chen and May [34]. The aim of these impact tests was to investigate the behavior of

RC slabs subjected to high-mass, low-velocity impact and to provide sucient input data with high

quality for the validation of numerical procedures.

Six square slabs were tested under impact loads using a drop-weight impact system. Four 760

mm square slabs with a depth of 76 mm (Slabs 1-4) were reinforced with 6 or 8 mm diameter high

yield steel bars at the bottom and the top with a concrete cover of 12 mm. Slab 5 and Slab 6 were

2320 mm square and 150 mm thick and reinforced with 12 mm diameter high yield steel bars at the

bottom and the top with a concrete cover of 15 mm. The slabs, shown in Figure 2.14, consisted of

two parts: a reinforced concrete region surrounded by a steel support of 17.5 mm width, where the

support was clamped at the four corners to restrain horizontal and vertical displacements (Figure

2.14.c).

The striker that struck the slabs was released with a velocity of up to 8.7 m/s and consisted of

a mass, a load cell and an impactor. The tests were performed with a striker mass of up to 380

kg and a stainless steel hemispherical impactor of a 90 mm diameter dropped at the center of slabs

(Figure 2.15a), except for Slab 3 which was impacted by a mild steel cylindrical impactor of a 100

mm diameter (Figure 2.15b). Although the vertical movement of the impactor was not restrained

and a second impact might occur after a rebound, only the rst impact of duration of 20-30 ms was

considered to study the slabs behavior. Table 2.1 summarizes the details of impact test for each

slab.

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Figure 2.14: Details of slabs (dimensions in mm): a) 760 mm square slabs, b) 2320 mm square slabs,

c) Boundary conditions [34]

impactor [34]

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Figure 2.16: Transient impact load of slabs 2-6 [34]

These experimental studies, considered as well monitored, allowed gaining a better understanding

of the impact behavior of RC slabs. The transient impact force-time histories were determined

using a load cell placed between the mass and the impactor. Strain gauges placed inside bars and

accelerometers were used to measure transient reinforcement strains and accelerations, respectively.

Experimental output were sampled at a rate of 500 kHz and ltered with a Butterworth lter using

a cuto frequency of 2 kHz. The local impact response of slabs was recorded with a high-speed

camera to show the development of failure modes in terms of cracking, scabbing and spalling.

Impact force-time histories show a plateau for slabs 2, 3, 4 and 6 after reaching the peak load

(Figure 2.16). The presence of this plateau is related to the fact that local failures due to penetration

and scabbing, accompanied by reinforcement yielding, were observed. As can be seen in Figures

2.17.d and 2.17.e, scabbing was not fully developed for slab 5 while more scabbing was observed on

slab 6 which have the same geometric and material properties as slab 5, but impacted by a higher

mass. Moreover, these two slabs of 150 mm thick show a less amount of concrete debris ejected due

to scabbing than the 76 mm thick slabs. For slab 4 which is similar to slab 2 but impacted with a

higher velocity, a signicant amount of penetration of the impactor in the slab occurred, indicating

that the thickness used was not sucient to prevent penetration for velocities higher than 6.5 m/s

(Figure 2.17.c). As illustrated in Figures 2.17.a and 2.17.b, the size and shape of scabbing zone could

be aected by the impactor shape since the stress wave caused by the progressive contact between

the impactor and the slab could propagate more uniformly in the case of a hemispherical impactor.

Thus, the scabbing zone on slab 2 bottom face caused by the hemispherical impactor had a more

circular shape than that created on slab 3 with a at impactor.

From these experimental investigation results, it can be seen that several parameters such as the

slab thickness, the impactor mass, velocity and shape have a signicant eect on RC slabs behavior

under impact.

The research program of Hrynyk [84] consisted of several tests of RC and steel ber reinforced

concrete (SFRC) slabs subjected to dynamic drop-weight impact, in the aim of verifying the accuracy

of his nite element software program (VecTor4) developed on the basis of capturing shear-critical

behavior for the analysis of RC structures under impact. The experimental program consisted of a

series of tests of eight intermediate-scale simply supported slabs subjected to high-mass low-velocity

impact. The experiments, considered as well-instrumented, investigated the behavior of RC and

SFRC slabs under impact and addressed the lack of high quality data issue in the research eld

of RC structures under dynamic loading where limited data are available in the literature. Thus,

these experiments allow obtaining a well-documented data set that might be useful to develop and

evaluate any further numerical analysis of slabs under impact.

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Figure 2.17: Local damage of slabs bottom faces after impact: (a) Slab 2, (b) Slab 3, (c) Slab 4, (d)

Slab 5, (e) Slab 6 [34]

Impact # 1 2 3 4 5 6 7 8 9 10

Drop-weight mass (kg) 150 180 210 240 240 270 270 300 300 300

Amongst the eight slabs, only impact tests of the four RC square slabs of 1.80 m width and

130 mm thickness are presented in the following section. The slabs were reinforced with top and

bottom layers with equal amount of reinforcement in x and y directions (ρx = ρy ). Upper and lower

reinforcing layers were joined with steel links in the corners and the center regions of slabs to prevent

reinforcement movement during casting. Figure 2.18.a presents the geometry and reinforcement

layers of a slab with additional links at the center that were used in the aim of stiening the impact

region. In order to reduce axial connement eects with ensuring an overall stability boundary

conditions during tests, the four corner supports were restrained to translate vertically and free

to rotate in all directions, while lateral restraint conditions varied with a simple pinned support,

two supports restrained to translate laterally in only one direction, and a support free to translate

laterally in all directions (Figure 2.18.a). Slabs were impacted sequentially with a drop-weight of a

constant velocity of 8.0 m/s but an increasing mass ranging from 120 kg to 300 kg. The contact

surface of the impactor with slabs had a at square shape of 300x300 mm and the impact was dened

as hard. For all except one of the experiments, slabs were impacted with an initial mass of 150 kg

which was increased for the subsequent impacts as indicated in Table 2.2. Hrynyk considered that

the impact testing of slabs could be terminated in case of a completion of the impact loading protocol

presented in Table 2.2, a signicant decrease of the measured support reaction forces between two

successive impacts, or a potential damage of the instrumentation under an additional impact.

Experimental results were obtained using detailed instrumentation and data measurement tech-

niques such as load cells to measure reaction forces at the four corner supports, accelerometers to

estimate the transverse acceleration behavior of slabs and the applied impact force, potentiometers

to determine transverse and lateral slab displacements, and strain gauges to evaluate the magnitude

and rate of strain in reinforcing bars. Furthermore, a high-speed video camera was used to observe

cracking patterns, mass penetration and concrete scabbing.

Figure 2.19 illustrates the typical damage behavior and cracking patterns of RC slabs that were

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Figure 2.18: Details of slabs (dimensions in mm): a) Slab with additional links at the impact region,

b) Boundary conditions [84]

Figure 2.19: Cracking patterns on the bottom surface of a slab with a longitudinal reinforcement

ratio of 0.42%: a) Impact #1, b) Impact #2, c) Impact #3 [84]

observed during three consecutive impacts with masses ranging from 150 kg to 210 kg. Cracking

patterns were marked after each impact event. As can be seen, under the rst impact, cracks

developed circumferentially along the reinforcing bars conguration and more densely in the impact

region, while no scabbing or penetration was apparent. Then a punching shear failure occurred at

the location of circumferential cracks as a result of subsequent impacts, associated with extensive

concrete scabbing and mass penetration.

Hrynyk varied the reinforcement ratio parameter in order to study its inuence on the response

of slabs. It was found that slabs stiness increased with the reinforcement ratio while it had a limited

inuence on slabs capacity to absorb impact energy. The nal cracking patterns of three RC slabs

with dierent reinforcement ratio presented, for all cases, an extensive damage at the impact region

with a concrete scabbing occurring on the bottom surface (Figure 2.20). The bottom reinforcement

layer bars were more exposed in the case where the slab was less reinforced.

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Figure 2.20: Final cracking patterns on the bottom surface of slabs with a longitudinal reinforcement

ratio of: a) 0.273%, b) 0.42%, c) 0.592% [84]

It should be noted that the experimental program of Hrynyk provided all data necessary for a

numerical analysis since it also included a series of tensile coupon tests to evaluate the stress-strain

curve of reinforcing steel, as well as a series of concrete cylinder and bending prism tests which

were performed to characterize the properties of concrete and investigate its compressive and tensile

behaviors. Furthermore, in comparison to the RC slabs response, SFRC slabs showed a dierent

behavior as the addition of steel bers resulted in an important increase in slabs strength and post-

cracking stiness. Slabs with higher ber volume fraction led to reduce the development of the

localized punching behavior and prevented a fully scabbing of concrete from the bottom surfaces of

slabs. The ber volume fraction had a signicant inuence on the midpoint displacement responses,

the support reaction magnitudes and the increasing of impact energy capacities of slabs that tended

to fail more toward exural mode.

In order to be able to choose an appropriate and accurate simplied model, the eect of impact

on composites has been extensively studied by several researchers [10, 39, 40, 42, 41, 149]. They

indicated that understanding the dierent types of response and the knowledge of parameters gov-

erning the response type are necessary for the selection of an appropriate analytical model. Abrate

[9] specied that several analytical models may be used to predict the impact dynamics, namely

spring-mass models, energy-balance models, innite plate models and models based on plate theory.

In this section, analytical mass-spring models developed in various research studies to predict the

dynamic response of RC slabs subjected to low impact velocity are presented.

The analytical approach, which is commonly regarded to be the most representative simplied

method to simulate an impact event between a slab and an impactor, consists in a one or more degrees

of freedom mass-spring system. The slab and the impactor are modeled as two colliding masses and

springs are used to represent the local and global behaviors of the slab. In the case of hard impact

(the impactor is considered to be rigid), the analytical model is based on the assumption that the

deformation and failure of the impactor are negligible. The local slab response is normally neglected

and the approach focuses on the global response which plays an important role for low impact

velocities. Analytical approaches have limited applicability since they are based on assumptions

which separate the local and global response of the slab, therefore they can only be used as an initial

approximation of the slab impact behavior.

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Figure 2.21: One degree of freedom mass-spring model used in Tonello IC design oce [174]

Tonello IC design oce used a mass-spring model with one degree of freedom to represent impact on

RC slabs [174]. Their approach consisted in estimating the maximum dynamic load reached during

an impact on the structure and took into account the elasto-plastic behavior of materials. Several

assumptions were adopted for this model: the impact is considered as soft, the impactor stays in

contact with the slab after impact (no rebound or penetration are allowed), and the slab equivalent

mass is calculated based on a static bending deformation (Figure 2.21).

The Bulletin of Information initiated by CEB [29] provides simplied models in the aim of helping

and guiding civil engineers in designing RC structures subjected to impact. This bulletin shows that,

in the case of a hard impact where the kinetic energy of the impactor is absorbed by the structure

deformation, an impact problem on RC slabs can be reduced to a mass-spring model with two

degrees of freedom as shown in Figure 2.22. In this case, the slab local behavior must be considered

as well as its overall deformation and the two dierential equations of equilibrium for the two masses

can be written as:

(

m2 ü2 + R2 (u2 − u1 ) =0

(2.9)

m1 ü1 + R1 u1 − R2 (u2 − u1 ) = 0

m1 and m2 represent the two colliding masses. The spring R1 represents the overall behavior

of the slab and has a nonlinear force-displacement relationship which can be considered similar to

a static behavior (Figure 2.23.a), while R2 represents the local behavior and simulates the contact

force between the impactor and the structure. The nonlinear relationship of R2 is more dicult to

determine, but can be obtained by using a nonlinear nite element code. CEB indicated that the

general force-displacement relationship of R2 in the contact zone of a solid, e.g. concrete, shows

an elastic compression phase in the range 0 < 4u < 4u1 , followed by an elastic-plastic phase for

4u1 < 4u < 4u2 where irreversible internal damage occurs (Figure 2.23.b).

Abrate [10] has presented an approach for selecting an appropriate model for analyzing the impact

dynamics for each particular case. He classied models according to how the structure is modeled:

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Figure 2.22: Simplied model of CEB for a hard impact [29]

Figure 2.23: Nonlinear force-displacement relationship for spring: (a) R1 , (b) R2 [29]

spring-mass models, energy balance models, complete models, and a model for impact on innite

plates.

The complete model consists of the impactor mass M1 , the eective mass of the structure M2 ,

the nonlinear contact stiness K, the shear stiness Ks , the bending stiness Kb and the nonlinear

membrane stiness Km (Figure 2.24.a). This model presents two situations for which a nonlinear

SDOF can provide accurate predictions of the contact force history. The rst situation is when the

overall deection of the structure is negligible compared to the local indentation. In this case, the

response of the structure can be modeled by the spring in Figure 2.24.b and the equation of motion

is:

3/2

M1 ẍ1 + kx1 =0 (2.10)

The second situation is when the membrane stiening is signicant and the deections of the

structure are large. In this case, the local indentation is negligible and the equation of motion of

the nonlinear SDOF model is:

M ẍ + kb x + km x3 = 0 (2.11)

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Figure 2.24: Mass-spring models of impact on composite structures according to [10]

Delhomme et al. [47] performed their study in the aim of analyzing the structural response of Struc-

turally Dissipating Rock-shed (SDR) protective system against rockfalls by a mass-spring model.

The principle of the model consisted in separating the contact phase between the block and the

slab from the slab free vibration phase after impact. Thus, two mass-spring-damper models were

generated: the contact model with two degrees of freedom that simulates the contact phase during

the rst milliseconds (Figure 2.25.a) and the post-impact model with one degree of freedom that

simulates the free vibration phase for several seconds (Figure 2.25.b).

Delhomme et al. [47] indicated that M∗ and ks which represent the equivalent mass and the

static stiness of the slab, respectively, are not constant during the impact phenomenon since they

depend on the slab surface in bending. Thus, M1∗ is lower than M2∗ and ks1 is higher than ks2 , since

in the contact phase only a reduced surface is considered in bending, while the equivalent mass and

the stiness are calculated for the total slab surface for the post-impact model.

The contact was considered as an elasto-plastic in order to take into account the cracking of

concrete and the Hertz contact law was used to estimate the contact force. Therefore, the equations

of the contact model can be written as:

(

mb (üb + g) + kc (ub − us )3/2 =0

(2.12)

M1∗ üs − kc (ub − us )3/2 + ks1 (us ) + c1 u̇s =0

where mb , ub are respectively the mass and the displacement of the block, us is the slab dis-

placement at the impact point. The contact stiness kc and the impedances c were derived from

experimental results.

The model shown in Figure 2.26.a was proposed by [184] to model low velocity impact response of

composite plates. In this model, ms represents the modal mass and the impactor has a mass of

mi and an initial velocity of v0 . The plate stiness Kst and the linear contact stiness Ky account

for the structural contribution and the local contact behavior, respectively. The amount of energy

transferred to the structure during impact is presented by the impedance c. This full model is used

for impacts that do not correspond to small or large mass impacts.

According to [42], if the size of the structure is very large and the impactor mass is very small,

the impact is dened as an innite structure impact which can be modeled as a SDOF (Single Degree

Of Freedom) system, where the mass of the impactor is supported by the local contact stiness and

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Figure 2.25: The two mass-spring-damper models used by [47] to analyze SDR response: (a)

contact: model, (b) post-impact model

Figure 2.26: Models for low velocity impact response in structures according to [184]

the structure acts as a damping mechanism (Figure 2.26.b). During this kind of impacts, the waves

are not reected back from the boundaries.

On the other hand, if the impactor mass is very large, the mass of the structure can be neglected

and the impact response can be modeled with a SDOF system, where the mass of the impactor

is supported by the local contact stiness and the static stiness of the structure in series (Figure

2.26.c). This type of impact is dened as a quasi-static impact and its duration is relatively long.

In order to solve a complete impact problem and to gain knowledge of the physical behavior of slabs,

numerical approaches such as nite element (FE) method can be considered to be more appropriate to

determine the behavior of RC slabs subjected to impact loading. A three-dimensional nite element

model allows modeling, as close as possible to the real case, the various aspects of any complex

problem including the boundary conditions and the loads applied to the structure. In the case of

impacted slabs, the RC slab and the impactor can be fully modeled and nonlinearities due to material

behaviors can be taken into account. The numerical simulation of RC slabs subjected to impact

loads is a complex phenomenon. Therefore an accurate simulation requires using an appropriate

model for concrete and a contact algorithm which can be used in nonlinear explicit nite element

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Figure 2.27: 3D FE model of SDR protection galleries proposed by Berthet-Rambaud et al. [19]

analysis. Once the FE model eectiveness is veried with precision, numerical simulations can

replace costly full-scale tests by providing more details of the dynamic response and an expansion

of experimental measurements which would be inaccessible during tests. They also provide a large

output data including the stress and deformation elds and make possible to perform sensitivity

analyses and parametric studies to determine the inuence of one parameter or another. Despite all

these advantages that can oer FE approaches, the main diculty consists in choosing appropriately

the set of numerical tools in order to accurately simulate the impact phenomenon and to identify

the induced failure modes due to transient dynamic loading.

Berthet-Rambaud et al. [19] investigated numerically the response of SDR protection galleries in

order to improve their design and reduce costs in response to the increasing request of using such type

of structures to provide protection in mountainous regions against rockfalls risk. These galleries are

composed of a RC slab which is directly subjected to falling rock impacts, and steel fuse supports that

allow transmitting slab reactions to the sub-structures and dissipating energy. Berthet-Rambaud

et al. used ABAQUS to propose a 3D model of the slab and supports (Figure 2.27) and compared

their numerical results to experimental tests that were conducted by Delhomme et al. [47] on a 1/3

reduced scale system model. The slab of 4.8 m wide and 12 m length was modeled and meshed

with C3D8R solid elements with four layers in the thickness of 0.28 m. The slab was impacted, at

various positions, by a cubic RC block of 450 kg mass and a velocity varying from 17.2 to 24.2 m/s.

The block was introduced in the FE model respecting the actual block geometry of the experiments

and meshed with C3D4 tetrahedral elements. The slab is supported by two lines, each with 11

steel fuse supports spaced at a distance of 1.14 m. The supports were considered with all their

components in the model, but their simulation will not be detailed in the present study (for more

details about supports simulation, see [19]). The slab and block reinforcements were represented

by bar elements which were embedded within the concrete mesh in order to create a perfect bond

between steel and concrete. Steel was modeled as an elasto-plastic material with hardening, while

the PRM (Pontiroli-Rouquand-Mazars) damage model that uses one scalar damage variable [157]

was used to represent concrete in the numerical simulation and was implemented in the FE code

using a Fortran subroutine. The kinematic contact algorithm was adapted to model the contact

between the slab and the block using hard contact for the normal interaction and a Coulomb-type

friction law for the tangential reaction. The explicit time integration method was used to solve the

problem and the Hillerborg regularization method [81] was integrated to reduce mesh dependence

eects.

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Figure 2.28: Damage distribution in and around the repaired zone [19]

Figure 2.29: Comparison of numerical and experimental results: (a) Calibration on friction param-

eter, (b) Vertical displacement at a point near the impact zone [19]

An analysis of two successive impacts at the same point and under the same conditions was

performed. After the rst impact, the damaged zone was repaired and the damage of elements

corresponding to this zone was reduced to zero. However, the potential damage on the remaining

part of the slab around the repaired part is taken into account by introducing three damaged zones

in the model. The damage distribution was supposed to have a cone shape that starts at the impact

point and goes through the slab thickness (Figure 2.28). The eect of gravity was included in the

model through a step of duration of 0.15 sec that allows a numerical gravity application before the

impact step.

A parametric study on the tangential friction at the slab and the impactor interface was carried

out since this parameter has a considerable inuence on the sliding of the impactor corner in contact

with the slab. An optimal value of 0.15 was found to be in agreement with experimental results

and permitted to correctly identify the experimental impact phase (Figure 2.29.a). Following the

calibration on the friction parameter, a comparison between experimental and numerical results of

the vertical displacement evolution at several points of the slab proves the accuracy of Berthet-

Rambaud et al. model in predicting the overall structure behavior. Figure 2.29.b depicts the

vertical displacement of a point close to the impact zone and shows that slab oscillations were very

well simulated. However, numerical amplitude values were found to be high after the rst oscillation,

which may be related to a general damping problem or to the complexity of boundary conditions.

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Figure 2.30: Details of nite element model of Mokhatar and Abdullah [137]

The aim of the study performed by Mokhatar and Abdullah [137] was to verify the accuracy of

numerical modeling technique in assessing the response of RC slabs and the steel reinforcement

failure mechanism when subjected to impact loading. A FE model was carried out using ABAQUS

in order to gain a better understanding of impacted slabs behavior and was validated with Slab 3

experimental results of Chen and May tests [34] described in section 2.3.1.2. The model was divided

into four main parts: the concrete region of slab, the steel reinforcement, the steel support and the

steel projectile (Figure 2.30). The concrete region was modeled using eight-node continuum solid

elements (C3D8R) while the steel support was considered as rigid and modeled with undeformable

discrete rigid elements (R3D4). These two parts were connected to each other using the tie contact

technique after assembling all elements. A proper bond action using the embedded technique was

created between the solid elements of the concrete region and the two-node beam elements of steel

reinforcement. The interaction between the steel impactor and the concrete region was dened as

surface-to-surface contact with the kinematic contact method for mechanical constraint formulation

and a friction coecient of 0.2. Material properties, as well as stress-strain and damage curves were

chosen based on several experimental works [34, 87, 169]. The increasing of concrete compressive

and tensile strengths due to impact load was considered with a rate of 1.5.

The FE model was validated with experimental results using impact force-time history and the

nal crack pattern. First, a mesh sensitivity was performed in order to nd the suciently rened

mesh which provides a reasonably accurate result (Figure 2.31.a). Then, the transient impact load

was estimated for three constitutive models of concrete, the Concrete Damage Plasticity (CDP)

considered having a brittle-cracking behavior, and the Drucker-Prager and the Cap-Plasticity models

characterized by their ductile behaviors. The results show that ductile models can evaluate the RC

slabs behavior under dynamic impact loading and provide a good agreement with experimental

results rather than brittle-cracking model (Figure 2.31.b). However, it can be seen from (Figure

2.32) that the nal crack pattern obtained using the CDP model shows a good correlation with

experimental results, but no eect of spallation could be observed which does not correspond to the

actual slab failure mode.

The numerical analysis performed by Trivedi and Singh [175] is among other numerical studies

that focus on identifying the global and local failure modes of RC slabs subjected to impact using

ABAQUS. Due to symmetry, Trivedi and Singh simulated one fourth of RC slabs of Zineddin and

Krauthammer experimental tests [189] using a 3D inelastic nite element model (Figure 2.33.a).

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Figure 2.31: Comparison of numerical and experimental transient impact force with dierent: (a)

Mesh densities, (b) Constitutive concrete models [137]

Figure 2.32: Final crack pattern of Slab3 bottom face: (a) Experimental result, (b) Numerical result

[137]

As mentioned in section 2.3.1.1, slabs were impacted by a cylindrical drop hammer which was not

introduced in Trivedi and Singh study, hence no contact modeling was presented. Alternatively,

a time dependent pressure loading applied at the slab center was considered to model the impact

loading and was distributed over a circular area with the same diameter as the hammer. The pressure

loading values were calculated by distributing the peak load over the impacted zone area, while their

amplitude curves in term of time were obtained using load-time histories available in Zineddin and

Krauthammer study (Figure 2.33.b). The amplitude values represent the ratio of the load value at

a certain time to the peak load.

Slabs were supposed to be fully constrained at the edges (no displacement is allowed in the three

directions), while displacement was restrained vertically on the x-plane of symmetry and horizontally

on the y-plane of symmetry. The concrete region of slabs was modeled as a 3D deformable part

and meshed with 3D continuum elements, whereas reinforcement was considered as a deformable

wire part and meshed with linear truss elements. No slip between concrete and steel was assumed,

hence reinforcement was embedded in the concrete part dened as the host component. Material

behaviors were incorporated in the FE model using the CDP model for concrete and a plasticity

model for steel.

Two approaches for post-cracking softening model were used to dene tension stiening for CDP

model, namely the limiting strain and fracture energy based approaches. Results showed that the

fracture energy cracking criterion gave mesh independent results which were in agreement with

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Figure 2.33: Details of Trivedi and Singh FE model: (a) One fourth of a slab with 2 meshes of

longitudinal and transversal steel bars, (b) Time-amplitude curve in case of a hammer dropped with

a height of 610 mm [175]

those obtained in experiments, but results using the strain criteria were inconsistent due to a mesh

sensitivity problem. Therefore, Trivedi and Singh limited their discussions to the fracture energy

approach and recommended its use as a robust method to study the behavior of RC structures

under impact. Besides the tension stiening, the eect of strain rate was investigated in the aim of

developing an adequate nonlinear dynamic FE model. Based on Tedesco et al. study [172], Trivedi

and Singh used a rate of 1.5 for concrete in compression and a rate in range of 1.7 to 1.75 for concrete

in tension. These rates are functions of the strain rate and allow taking into account the increase

in compressive and tensile strengths due to a dynamic loading. However, no signicant change was

observed in numerical results due to the use of strain rate, since the problem was associated with

low strain rates of the order of 10−4 to 10−2 s−1 .

Trivedi and Singh studied several criteria in order to predict the failure modes of slabs, such as

inection point, limiting strain based failure, bi-axial failure, shear failure, strain in steel and tensile

damage. Only some results using the rst two criteria are presented here:

Miyamoto et al. identication by inection point using the slab deformed shape [131]. As can

be seen in gure 2.34.a, for a RC slab with one mesh of longitudinal and transversal steel bars

impacted with a height of 610 mm, the formation of an inection point in the deected prole

indicates the transition from exure to punching shear failure which occurs at the slab center

under the impact region. A exural mode can be identied by a smooth deection curve with

parabolic shape (Figure 2.34.b).

Limiting strain based failure criterion by evaluating strain variations in XZ and YZ planes

at various elements of concrete selected at the top and the bottom of slabs, and localized at

and distant from the impact zone. Based on this criterion, a slab has a mixed mode failure

of exure and shear when strain in top elements localized at the impact zone exceeds the

compressive limiting strain and the top distant elements show tensile behavior. In addition,

bottom elements near the impact zone and distant elements have tensile and compressive

strain, respectively. When a slab fails in exure, elements at the impact zone show a crushing

failure at the top surface and cracking failure at the bottom surface (Figure 2.35).

37

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Figure 2.34: Identication of slabs failure mode by inection point criterion: (a) Transition from

exure to punching shear, (b) Flexural mode [175]

Figure 2.35: Identication of a slab exural failure mode by strain based failure criterion: (a) Top

elements, (b) Bottom elements [175]

As a strongly heterogeneous material, concrete shows a complex nonlinear mechanical behavior

characterized by softening response in tension and low conned compression. Softening is dened

as decreasing stress with increasing deformation beyond the concrete compressive strength, and

results in a reduction of the unloading stiness of concrete and irrecoverable deformations localized

at cracks. However, the connement of concrete with reinforcement results in a signicant increase

in the strength of compressed concrete and in a ductile hardening response described by increasing

stress with increasing deformations. In the current study, uniaxial compressive and tensile stress-

strain curves based on experimental observations are adopted and dened in terms of concrete

properties in compression and tension, respectively, as follows:

Where σc represents the concrete compressive stress values calculated, Ec is the Young's modulus

of concrete, fc is the maximum compressive strength of concrete, fcu is the compressive stress of

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concrete corresponding to the ultimate strain, εc represents the concrete compressive strain value

corresponding to the stress σc , εc1 is the compressive strain of concrete at the compressive strength

fc , and εcu is the ultimate strain of concrete in compression.

Where σt represents the concrete tensile stress values calculated, ft is the tensile strength of

concrete, εt represents the concrete compressive strain value corresponding to the stress σt , and εcr

is the tensile strain at concrete cracking at the tensile strength ft .

The properties of concrete in uniaxial compression are obtained from cylinder tests or cube tests. The

nonlinear stress-strain behavior under uniaxial compressive stress, shown in Figure 2.36, is divided

into three regions: linear elastic, non-linear plastic (hardening region) and post-peak stress (softening

region). The rst region, in which the concrete behaves almost linearly, is observed during 30-60%

of the maximum uniaxial compressive strength fc . At this level of strain, the specimen deformation

is recoverable and localized cracks are initiated but they do not propagate. This could be linked to

the balance of energy present in the concrete specimen which is in this case less than the energy

required to create new cracks.

In the plastic regime the response is typically characterized by stress hardening followed by

strain softening beyond the peak stress. Beyond the limit of elasticity, the deformation is no longer

recoverable and the stress-strain curve begins to deviate from a straight line. The concrete behaves

in a non-linear manner up to the peak stress and the crack system continues to grow slowly with the

increasing of the applied load. During this phase, cracks cause an increase in volume and a small

lateral expansion associated with Poisson's ratio eect is observed.

Immediately after the peak stress, the concrete undergoes strain softening and the lateral expan-

sion increases dramatically with the cracks propagation. The energy released by the propagation of

a crack is greater than the energy needed for propagation. The load carrying capacity of the test

specimen decreases and the stress-strain curve starts to decrease until a crushing failure occurs at

the ultimate strain. The shape of the stress-strain curve depends on the concrete strength which

continues to behave in a linear manner up to a higher stress level than that for normal strength

concrete. Also higher strength concrete must have a steeper descending curve and exhibits a brittle

failure mode due to the fact that the specic fracture energy of concrete in compression does not

increase much with the concrete strength.

When the concrete specimen is unloaded from any point of the strain softening branch of the

stress-strain curve, the nonlinear behavior can be easily noted from the residual deformation present

in the stress-strain curve. The concrete stiness is reduced because of the increase of the internal

damage which causes irrecoverable deformation in loading.

Compression stress-strain curve without detailed laboratory test results The way in

which the concrete behaves can be accurately determined on the basis of uniaxial compression test

results. However, problems arise when no such test results are available and when the only available

quantities for the analysis are the compressive strength and the modulus of elasticity of concrete.

Many uniaxial stress-strain relationships for concrete in compression were proposed in the literature,

such as the expressions developed by [3, 4, 31, 49, 99, 116, 154, 178, 180].

Desayi & Kirshnan formula [49] Among many formulas proposed in the literature to rep-

resent nonlinear stress-strain characteristics of concrete, Desayi and Krishnan suggested a simplied

equation that describes the ascending and descending parts of concrete compressive stress-strain

curve in a single expression as follows:

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Figure 2.36: Uniaxial compressive stress-strain curve for concrete [4]

Ec εc

σc = (2.15)

1 + ( εεc1c )2

Madrid parabola [31, 32] Considered as a good relation by CEB, the conventional Madrid

parabola is a parabolic equation that allows dening the uniaxial stress-strain curve of concrete in

compression as given below:

1 εc

σc = Ec εc [1 − ( )] (2.16)

2 εc1

However, Kmiecik and Kaminski [94] mentioned that this function is not accurate enough to

correctly describe the behavior of concrete. They also suggested, in case of parabolic relations, using

a lower value of initial modulus Ec so as to ensure that the parabola curve passes through the correct

value of fc .

Eurocode 2 curve (EN 1992-1-1) [4, 54] According to Eurocode 2 (EC2), concrete acts

elastically up to 0.4fc . Then, the compressive stress σc for hardening and softening parts is dened

as a function of uniaxial strain εc by using the concrete compressive strength fc and expressed by:

kη − η 2

σc = fc (2.17)

1 + (k − 2)η

with k = 1.1 Ec εc1 /fc and η = εc /εc1 .

EC2 provides a formulation of the strain at peak stress in term of the compressive strength (fc

in MPa), which can be used for normal and high strength concrete:

A constant value εcu = 0.0035 can be used for concrete with a characteristic value fck less than

55 MPa, however for fck ≥ 55 MPa the ultimate strain value should be replaced by:

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Table 2.3: Values of εc0 parameter for high strength concrete [54]

fck (MPa) 50 60 70 80 90

and the descending branch of stress-strain curve for high strength concrete (fck ≥ 50 MPa)

should be formulated by:

h i

σc = fc / 1 + {(η1 − 1) / (η2 − 1)}2 (2.20)

where η1 = εc /εc1 and η2 = (εc1 + εc0 ) /εc1 . εc0 is the parameter for high strength concrete and

its value can be taken from Table 2.3. Concrete compressive strength can be estimated from its

characteristic value by fc = fck + 8.

Pavlovic et al. curve [154] Pavlovic et al. found that the EC2 compressive stress-strain

curve is not accurate to estimate the crushing strength of concrete when high crushing strains are

expected, as in the case of slabs subjected to impact. Plasticity curve in EC2 considers concrete

compression behavior only up to the ultimate strain εcu1 limited to a maximum value of 0.0035,

which may lead to unrealistic overestimation of concrete strength. Consequently, Pavlovic et al.

suggested an extension of the compressive stress-strain curve beyond the EC2 ultimate strain and

proposed a new value. The extension was performed with a sinusoidal descending curve between

two points, the rst point corresponds to EC2 ultimate strain (εcu1 , fcu1 ) while the second is the end

of the sinusoidal part at Pavlovic et al. ultimate strain (εcu2 , fcu2 ). The sinusoidal part is dened

by the following equation:

1

σc = fc − + , εcu1 < εc < εcu2 (2.21)

β β. sin(αt2 π/2) α

where µ = (εc − εcu1 )/(εcu2 − εcu1 ) and β = fc /fcu1 . At the end of the sinusoidal descending

part at strain εcu2 , concrete strength was reduced to fcu2 by a factor α = fc /fcu2 . Pavlovic et al.

adopted a value of 20 to the reduction factor α and 0.03 to the ultimate strain εcu2 . Factors αt1 and

αt2 , which represent the governing tangents angles at the starting and end points of the sinusoidal

curve respectively, were chosen in order to obtain a stress-strain curve with a smooth overall shape,

hence αt1 = 0.5 and αt2 = 1.0.

Chinese code curve (GB50010-2002) [3] Chinese code for design of concrete structures

is considered as the only code among many design codes that allows dening uniaxial concrete

strength at high strains. The ascending and descending parts of stress-strain curve are described by

two dierent softening parameters related to the material strength and their values generally range

from 0.4 to 2.0. According to the design Chinese code, the uniaxial compressive stress-strain curve

of concrete can be estimated using the following equations:

(

= fc αa η + (3 − 2αa )η 2 + (αa − 2)η 3 , η ≤ 1

σc

(2.22)

= fc η/ αd (η − 1)2 + η , η > 1

σc

where η = εc /εc1 , while αa and αd represent the ascending and descending softening parameters

and they are expressed in term of the concrete strength (fc in MPa) as follows:

αa = 0.0475fc (2.23)

αd = 0.0485fc (2.24)

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Wang and Hsu curve [180] Wang and Hsu proposed two equations to describe the ascending

and descending parts of concrete stress-strain curve in compression. These equations are expressed in

terms of concrete compressive strength fc , concrete strain at maximum compressive stress εc1 , which

is considered having a value of 0.002; and a softened coecient ζ that is a function of reinforcement

ratios in x and y directions:

2

εc εc

σc = ζfc 2 ζεc1 − ζεc1 , if ζεεcc1 ≤ 1

2 (2.25)

εc /ζεc1 −1

σc = ζfc 1 − , if ζεεcc1 > 1

4/ζ−1

ζ represents the reduction in compressive stress resulting from locating reinforcing bars in the

compressed zone, hence the value ζ = 1.0 is considered in case of plain concrete behavior when

no reinforcement is taken into account. It should be noted that, in order to avoid any potential

numerical problem when using the following concrete stress-strain curve in a FE analysis, Wang and

Hsu suggested a minimal value of 0.2ζfc of the compressive stress in the descending part.

Majewski curve [116] Although Eurocode 2 assumes a limit of 0.4fc for the linear elastic

part of concrete stress-strain curve, Majewski found that a linear elasticity limit should increase

with the uniaxial concrete compressive strength. Elasticity limit elim is considered as a scale factor

and calculated as a percentage of stress to concrete strength according to this formula (fc in MPa):

Consequently, Majewski adopted a linear stress-strain relation for the initial stresses (σc ≤

elim fc ). Beyond this limit, hardening and softening parts were described by a nonlinear relation

that depends on several parameters, including the elasticity limit:

(

σc = Ec εc , if σc ≤ elim fc

(elim −2) 2 (elim −2) 2 e2 (2.27)

σc = fc 4(e ( εc )2 − fc 2(e

lim −1) εc1

( εc ) + fc 4(elim

lim −1) εc1

lim

−1) , if σc > elim fc

On the basis of experimental results, Majewski [116] proposed the following formulas to approx-

imate the values of εc1 and εcu :

(

εc1 = 0.0014[2 − exp(−0.024fc ) − exp(−0.140fc )]

(2.28)

εcu = 0.004 − 0.0011[1 − exp(−0.0215fc )]

Kratzig and Polling curve [99] Based on the recommendations of the Model code 1990 [2],

the concrete compressive stress-strain relation suggested by Kratzig and Polling was divided into

three part to describe the phases of elasticity, hardening and softening of concrete behavior. Thus,

three formulations were derived and expressed as follows:

σc = Ec εc , if σc ≤ fc /3

Eci fεc −(εc /εc1 )2

σc = 1+(Eci

c

εc1

−2) εεc

fc , if f c/3 < σc ≤ f c (2.29)

fc c1

2 −1

2+γc fc εc1

− γc εc + γ2εc εc1c

σ

c = 2fc , if σc > fc

The rst two equations describe the ascending part up to the maximum compressive strength

fc at εc1 by assuming an elasticity limit equal to one third of fc . In the hardening equation, the

modulus Eci was used to guarantee the stress-strain curve to pass through the point (εce , fc /3)

where εce denotes the elastic strain of concrete corresponding to the end of linear elastic phase. The

modulus Eci can be dened in terms of initial Young's modulus Ec and compressive strength fc :

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2

1 fc fc 3

Eci = − + Ec (2.30)

2Ec εc1 εc1 2

The third equation represents the descending part of concrete stress-strain curve and includes

the descent function γc that permits to take account for the descending part dependency on the

specimen geometry, which ensures mesh independent results in numerical analysis. γc controls the

area under the stressstrain curve, it depends on the constant material parameter Gcl that represents

the localized crushing energy and on the characteristic length lc derived from the respective element

mesh. Consequently, the descent function can be expressed by:

π2f ε

γc = h c c1 i (2.31)

2 Gcl

lc − 21 fc εc1 (1 − bc ) + bc Efcc

where bc is a constant factor assumed equal to the ratio of the plastic strain to the inelastic

pl

strain (εc = bc εin

c ) with 0 < bc ≤ 1. Values bc = 0.7 and Gcl = 15KN/m were found by [21, 118]

to t well with experimental data from literature. For solid elements, the characteristic length is

1/3

calculated by lc = Ve where Ve represents the element volume.

Wahalathantri curve [178] Wahalathantri et al. developed a complete stress-strain curve for

concrete under uniaxial compression by only using compressive strength, they also suggested slight

modications from the original version of Hsu and Hsu [85] in order to be used and comparable with

the damaged plasticity model in Abaqus. Modications were made only for concrete with maximum

compressive strength up to 62 MPa. In the ascending part, concrete acts elastically up to 0.5fc .

Beyond this limit, compressive stress values can be calculated in terms of the compressive strength

fc , the strain at peak stress εc1 and a parameter β which depends on the stress-strain curve shape:

!

β (εc /εc1 )

σc = fc (2.32)

β − 1 + (εc /εc1 )β

with

1

β= (2.33)

1 − [fc / (εc1 Ec )]

Wahalathantri et al. formula can be used only for stresses between 0.5fc at elastic strain εce

in the ascending part and 0.3fc at ultimate strain εcu in the descending part. Wahalathantri et al.

suggested to iteratively calculate εcu through the equation 2.32 for σc = 0.8fc . It should be noted

that in the above equations, σc and Ec are expressed in kip/in2 ( 1M P a = 0.145037743kip/in2 ).

2.4.2.1 Uniaxial tensile stress-strain curve

Indirect tensile tests as cylinder splitting and bending prisms are often preferred over direct tension

test and are used as an indirect way to determine concrete properties in tension (Figure 2.37). The

mechanical behaviors of concrete under the action of tensile and compressive loadings have many

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Figure 2.37: Uniaxial tensile stress-strain curve for concrete [156]

similarities. In general the limit of elasticity is observed to be about 60-80% of the uniaxial tensile

strength which corresponds to the onset of micro-cracking in the concrete material. Above this

level the response of concrete exhibits highly nonlinear behavior and the formation of microcracks

is represented macroscopically with a softening stress-strain response. The tensile stress drops

gradually with increasing deformations until a complete crack is formed. The reduction in strength

of the test specimen resulting from the opening of initial cracks in concrete leads to increase the

stress-strain curve nonlinearity. When the concrete specimen is unloaded from any point of the strain

softening branch of the stress-strain curve, the unloading response is weakened and the material

elastic stiness appears to be damaged.

Tensile stress-strain curve without detailed laboratory test results In the present section,

several uniaxial stress-strain relationships for concrete in tension proposed in the literature are

detailed, such as the expressions developed by [3, 83, 99, 178, 180].

Wang & Hsu curve [180] Wang and Hsu divided concrete behavior in tension into two

ascending and descending parts, the rst describes the elastic phase while the second indicates the

softening stress-strain response and is given in terms of cracking strength of concrete ft , strain at

concrete cracking εcr and the rate of weakening n:

σt = Ec εt , if εt ≤ εcr (2.36)

εcr n

σt = ft ( ) , if εt > εcr (2.37)

εt

Wang and Hsu adopted a value of n = 0.4. However, this parameter can be used to calibrate

the relation for a given simulation since tension stiening may considerably aect numerical results,

hence Kmiecik and Kaminski[94] proposed a range of values from 0.4 to 1.5. This tensile stress-

strain curve is characterized by a sharp change at the cracking strain, which may result in some

numerical diculties during a FE analysis. For similar cases, Wang and Hsu suggested to dene a

short plateau at the peak point.

Chinese code curve (GB50010-2002) [3, 187] According to the Chinese code for design of

concrete structures, the uniaxial tensile stress-strain curve of concrete includes ascending stiening

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and descending softening parts. When concrete is subjected to tension, it is assumed to exhibit

initially a linear elastic response up to the uniaxial tensile strength ft . After the tensile stress reaches

ft and as stresses increase, concrete response becomes nonlinear and the softening stress-strain curve

recommended by the Chinese code is given in term of the peak tension strain εcr corresponding to

ft :

(ε/εcr )

σt = ft (2.38)

αt ( εεcr − 1)1.7 + ε

εcr

αt is the coecient at the descent stage of the uniaxial tension stress-strain curve of concrete

and determined by:

αt = 0.312ft (2.39)

Wahalathantri curve [178] Wahalathantri et al. selected the experimentally validated tensile

stiening model of concrete proposed by Nayal and Rasheed [143] in order to dene the stress-strain

curve of concrete in tension. Nayal and Rasheed model consists of two linear descending parts and

has the ability to accurately simulate the response during primary and secondary cracking stages.

However, Wahalathantri et al. indicated that using Nayal and Rasheed curve for damaged plasticity

model under tension in Abaqus leads to runtime errors due to a sudden vertical stress drop at

the cracking tensile strain εcr . To avoid this problem, Wahalathantri et al. introduced a tensile

stress-strain curve with three linear descending parts characterized by a gradual reduction in stress:

The proposed modied part where stress is gradually reduced from maximum tensile stress ft

to 0.77ft between strain of values εcr and 1.25εcr ,

The primary cracking part where stress decreases steadily to 0.45ft between strain values

1.25εcr and 4εcr ,

Kratzig and Polling curve [99] Kratzig and Polling considered a linear and elastic response

of concrete in tension until the maximum strength ft is reached. They assumed that at strain εcr

a crack is initiated and occurs as tensile stress in concrete exceeds ft . Thereafter, a descending

exponential curve was adopted to represent softening behavior of concrete in tension after crack

initiation. Their tensile stress-strain relation after cracking can be expressed as:

In the above equation, γt is a shape parameter that controls the area under the stress-strain

curve and depends on the fracture energy Gf and the characteristic length lt :

in which ε0cr is the strain after which crack expands and is estimated by the following expression:

Gf

ε0cr = (2.42)

lt ft

The estimation of fracture energy and characteristic length values is detailed in the next section.

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Figure 2.38: Concrete stress-crack opening curve with: (a) Linear softening branch [7], (b) Bi-linear

softening branch [78], (c) Tri-linear softening branch [170]

Another approach could be also used to dene concrete behavior in tension, namely the energy

balance approach that showed a reasonable agreement with results from a tensile test according to

Hillerborg et al. study [81]. Hillerborg et al. assumed that the response of concrete under tension is

linear until the fracture surface is reached and a linear softening branch beyond cracking was adopted

(Figure 2.38.a). Consequently, concrete post-cracking behavior is governed by the equivalent fracture

energy criterion that is based on the amount of energy absorbed by the formation of a unit area of

crack surface. Cracks arise in planes perpendicular to the direction of maximum principle tensile

stress when the maximum tensile strength of concrete is reached, they present a highly irregular

shape by following the weakest path in the material. In fracture mechanics experiments, the fracture

energy is determined as the ratio of the total energy that is supplied to fracture a specimen (area

under the load-deformation relation) to the fractured cross-sectional area [83]. In Hillerborg et al.

frictitious crack model, the fracture energy Gf is assumed as a material property and presented

by the area under the stress-crack opening relation:

ˆ

Gf = σdw (2.43)

Nowadays, this model is presented in most nite element codes and used to ensure mesh inde-

pendence of results since the amount of absorbed energy is not very sensitive to the mesh size [81].

This enables to perform nite element analysis with a rather coarse mesh and treat complicated

problems with a limited computational cost.

Descending branch after cracking can be also presented by a bi-linear or tri-linear tension soften-

ing curve. Haidong [78] adopted in his analysis a bi-linear softening branch with a maximum crack

width of 4.6Gf /ft at which no stress is transferred by the opened crack (Figure 2.38.b). Tajima et

al. [170] used a tri-linear softening branch with fracture energy as depicted in Figure 2.38.c.

The simplied concrete equation for uniaxial tensile loading of Hordijk [83] is based on the

principle of the frictitious crack model proposed by Hillerborg et al. [81] and depends on the

specimen geometry. The corresponding tensile stress-strain relation consists of a loading linear part

up to the strength ft and a nonlinear descending part which is derived from the following stress-crack

opening relation:

(" # )

w 3 −c2 ww

w 3 −c

σt = ft 1 + c1 e c − (1 + c1 )e 2 (2.44)

wc wc

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The relation between crack opening and stress was found to be the most important input pa-

rameter for the nonlinear fracture mechanics analyses of concrete [156]. In the previous equation,

the crack opening is expressed as a product of inelastic strain εin

t and characteristic length lt :

−1

w = lt εin

t = lt εt − σt Ec (2.45)

Tajima et al. [170] indicated that the denition of the characteristic length of concrete in compression

and tension are similar, hence w is smeared over the average element length lt which can be calculated

q

1/3 3 3Ve

for 3D problems in term of the element volume by lt = Ve according to [118] or lt = 2 4π

according to [170]. Hillerborg et al. [81] dened the characteristic length in terms of concrete

Gf Ec

Young's modulus, fracture energy and tensile strength with lt = .

ft2

Hordijk used a test series on normal-weight concrete from the literature in order to determine

the unknown variables in the stress-crack opening relation. The best t was obtained for c1 = 3,

c2 = 6.93 and wc = 160µm where wc represents the critical crack opening.

Fracture energy without detailed laboratory test results Several relations were proposed

in the literature to estimate the fracture energy in case of lack of experimental data [2, 15, 83, 146].

Hordijk energy [83] The fracture energy of Hordjik can be calculated in terms of the tensile

strength ft and the critical crack opening wc by the integration of equation 2.44:

" ( 3 ) #

1 c1 1 1 3 6 6 1

+ c31 3

Gf = ft wc 1+6 − + 2+ 3+ 4 + 1 + c1 exp(−c2 ) (2.46)

c2 c2 c2 c2 c2 c2 c2 2

By replacing c1 and c2 by their values as mentioned previously, the fracture energy can be

estimated by:

Gf = 0.195wc ft (2.47)

CEB-FIP energy [2, 158] CEB-FIP model code proposed a relation between the fracture

energy and compressive strength of concrete. Fracture energy Gf is calculated for mode I in N/mm

in term of coecient αf that depends on the maximum aggregate size in concrete dmax :

0.7

fc

Gf = α f (2.48)

10

with fc representing the compressive strength of concrete in MPa and αf = (1.25dmax + 10) .10−3 ,

where dmax is introduced in mm and may vary between 2 and 32 mm. As can be seen, Gf has

generally a tendency to increase with fc and dmax . In the present study, a value of 16 mm is adopted

for dmax .

Oh-Oka et al. energy [146, 170] Oh-Oka et al. also determined the fracture energy of

concrete in N/mm in term of the compressive strength according to the following formula:

0.23fc + 136

Gf = (2.49)

1000

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Bazant and Oh energy [15, 134] Based on numerous experimental results, Bazant and Oh

proposed an equation of the fracture energy of concrete in terms of Young's modulus Ec , tensile

strength ft and the maximum size of coarse aggregate dmax :

Gf = (2.50)

Ec

Gf is calculated in kgf /cm, Ec and ft are given in kgf /cm2 (kgf /cm2 = 0.0980665 M P a) and

dmax is in cm.

Considered as a composite material, reinforced concrete consists of steel reinforcement embedded in

plain concrete. Steel is used in RC structures to carry tensile forces and to recover the low tensile

strength of concrete. Inclusion of steel reinforcement is very eective to control the development

of cracks by distributing cracks uniformly over the cracked regions, but it does not prevent the

cracking of concrete under tension. The behavior of reinforcing steel may control the response of

RC structures subjected to impact, thus the prediction of fundamental characteristics and behavior

of steel is necessary. In the current study, uniaxial steel stress-strain curves based on experimental

observations are adopted and dened in terms of steel properties as follows:

represents the steel stress values calculated,

is the deformation modulus of steel at the hardening phase, fy is the yield strength of steel, fu is

the ultimate stress of steel corresponding to the ultimate strain, εs represents the steel strain value

corresponding to the stress σs , εy is the yield strain of steel at the yield strength fy , εsh is the strain

of steel at which strain hardening initiates, and εu is the ultimate strain of steel.

Steel behavior can be determined from coupon tests of bars loaded monotonically in tension. These

tests allow obtaining a typical stress-strain curve for reinforcing steel in tension to failure (Figure

2.39). Steel response exhibits an initial linear elastic region, a yield plateau, a strain hardening

region and a post-ultimate stress region. The compression stress-strain curve of reinforcing steel is

assumed to be equal and opposite of its curve in tension.

The linear elastic region is up to the steel yield strength fy and steel stiness is governed by the

elastic modulus Es at low strains. Stress in this region is expressed as:

σ s = E s εs (2.52)

After the strain εy corresponding to the yield strength, steel response is referred as the yield

plateau or Lüder's plateau. At the beginning of this region, steel behavior shows a slight drop in

strength below the initial yield strength. Then, stress remains steady at this lower yield strength

and steel behaves nonlinearly and plastically. Steel stress-strain curve is generally idealized and

assumed to be horizontal in the yield plateau region with an average strength equal to the material

yield strength. Thus, the idealized stress-strain relationship in this region is:

σs = fy (2.53)

The yield plateau size depends on the steel tensile strength, e.g. the yield plateau is considerably

shorter for high-strength, high-carbon steels than for low-strength, low-carbon steels [179].

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Figure 2.39: Experimental tensile stress-strain curve for reinforcing steel [112]

The strain-hardening region starts at the idealized point of coordinates (εsh , fy ) and ends at the

ultimate point of coordinates (εu , fu ). At this phase, stress increases again with strain and exceeds

the yield strength by 30 to 60 % at the ultimate point, and stiness decreases with permanent plastic

deformation. The ratio of ultimate strength to yield strength depends on the steel specication and

grade [112]. At the ultimate point, the maximum tensile load is resisted and the tensile stress-strain

curve of steel has zero slope.

Finally, in the post-ultimate region, stress reduces with strain until steel reinforcement fracture

occurs and load capacity is lost. However, steel stress-strain curve is assumed to end at the ultimate

point since its shape at the post-ultimate region is related to the gauge location and the length over

which the data are collected [185].

2.5.2 Idealizations

This section discusses idealized stress-strain curves proposed in the literature to describe steel be-

havior. For practical application, typical stress-strain curve of steel can be idealized in two dierent

ways depending on the desired level of accuracy. For the rst idealization, steel is considered as a

linear elastic, perfectly plastic material with a bilinear stress-strain curve. The rst branch is linear

elastic up to yield strength fy and the second branch represents the yield plateau with a stress

equal to fy . In this idealization, the stress increase due to strain hardening is neglected. This ide-

alization was found to be sucient to analyze RC structures under service load conditions [36] and

very satisfactory in the case of low-strength, low-carbon steels characterized by a strain hardening

much greater than the yield strain [171]. The design equations of the ACI code are based on this

assumption [171]. However, this approximation underestimates steel stresses at high strains in cases

where the onset of yielding is directly followed by steel hardening [171]. According to Kwak and

Filippou [101], an elastic-perfectly plastic curve may lead to numerical convergence problems when

used in a FE analysis in cases where a RC structure is highly aected by reinforcing steel hardening.

On the contrary, this idealization was adopted by Kratzig and Polling [99] to derive an elasto-plastic

damage model for reinforced concrete and was found to be successfully used to model RC structures.

The second idealization assumes a linear elastic, linear plastic steel behavior with hardening.

The stress-strain curve consists of two linear branches, the elastic branch is up to yield strength fy

while stress varies in the plastic branch between fy and fu for strain varying between εy and εu ,

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respectively. This approximation is generally more used to estimate steel stresses at strains higher

than yield strain and considered as more accurate since it takes into account the strain hardening

eect. According to [171], this approximation would be more appropriate than the rst idealization to

model the steel behavior and allows assessing the corresponding ductility of a structure investigated

under high strains. Several researchers adopted the second idealization to represent steel behavior

[21, 28, 101]. Kwak and Fillippou [101] indicated that assuming a gradually increasing plastic

hardening branch immediately after yielding for the steel stress-strain curve does not aect the

accuracy of results. Britel and Mark [21] used a value of Es1 = 1111 M P a for the deformation

modulus of plastic-hardening branch, while Cao et al. [28] estimated Es1 in term of steel Young's

modulus Es as:

In order to predict the ultimate punching shear strength of slab-column connections, the second

idealization was also used by Theodorakopoulos and Swamy [173] by assuming a value of Es1 =

5 GP a 1.2 fy . However, the elastic

to express the strain hardening eect up to a stress equal to

part before yielding was divided into two linear branches, a branch up to 0.8 fy with a modulus of

elasticity Es = 200 GP a and a branch between stresses of 0.8fy and fy with a modulus equal to:

0 0.2fy

Es = (2.55)

0.002 + 0.2fy /E

The yield strain can be calculated using the following expression:

εy = 0.002 + fy /E (2.56)

Stress-strain curve of reinforcing steel bars embedded in concrete is slightly dierent from that

of a bare steel bar due to tension-stiening eect caused by the surrounding concrete bonded to

steel. During a tension-stiening phenomenon, the bond action between concrete and reinforcement

develops a number of cracks in the structure and results in the redistribution of tensile loads from

concrete to steel. The response of a RC structure with tension-stiening is more complicated and

stier than the response with a brittle failure [171]. The development of bond between concrete and

steel is represented by three stress transfer mechanisms, namely mechanical interaction, chemical

adhesion and surface friction [36]. The mechanical interaction is the main mechanism that maintains

the composite interaction between the two materials and for which load transfer can be idealized as a

continuous stress eld developed at the deformed surface of reinforcement and concrete [171]. Bond

stresses located along the surface between reinforcement and the surrounding concrete eectively

transfer tensile stresses between reinforcing bars and concrete. Steel and the surrounding concrete

are considered as full bonded and behave as a unit as long as the loading applied to the RC structure

does not exceed the bond stress capacity. Stresses at the interface of concrete and steel increase

with increasing the load applied, which results in a localized bond stress exceeding the bond capacity

and a localized damage that propagates gradually to the surrounding concrete. Consequently, steel

starts to yield at cracked sections and a signicant movement between the reinforcing steel and the

surrounding concrete may occur. The quality of bond between reinforcing steel and concrete has a

considerable inuence on cracks distribution and aects their width and spacing. For this reason,

the interface properties between concrete and steel are very important to analyze RC structures

and hence steel can be included in numerical analyses with an equivalent stress-strain relation that

considers bond eects by assuming a perfect bond between concrete and steel elements.

The main dierence between a bare steel bar behavior and that of a steel bar embedded in

concrete is related to the stress value at which steel yields. For an embedded steel bar, stresses

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Figure 2.40: Average tensile stress-strain curve for reinforcing steel embedded in concrete [16]

in reinforcing steel between cracks is less than the yield stress at cracks since tensile forces can

still partially be resisted by the concrete matrix located between cracks. Therefore, yielding of an

embedded bar traversing several cracks starts at an average value fn which is less than the yield

strength of a bare steel bar fy (Figure 2.40). In addition, stress-strain relationship of reinforcing

steel bars embedded in concrete can be obtained by averaging stresses and strains between cracks. A

bilinear average stress-strain relation was proposed by Belarbi and Hsu [16] based on experimental

results:

(

σs εs , if |εs | ≤ εn

= Es

(2.57)

σs = fy (0.91 − 2B) + 0.02 + 0.25B εεys , if |εs | > εn

As illustrated in Figure 2.40, the average behavior of steel embedded in concrete still maintains

an initial elastic phase up to a stress value fn at which yielding of steel bars starts. The second

branch is expressed in term of a parameter B that determines the dierence between fn and fy .

The parameter B is a function of the ratio ft /fy and the reinforcement steel ratio ρs limited to a

minimum value of 0.25%:

with ft is the tensile strength of concrete. In Figure 2.40, εn is the limiting boundary strain

dened as follows:

The average stress-strain curve proposed by Belarbi and Hsu that incorporates the eect of ten-

sion stiening was adopted by several researchers [117, 158, 171] to dene the behavior of reinforcing

steel embedded in concrete.

2.6 Conclusion

In civil engineering eld, RC structures are often subjected to some extreme dynamic loadings due to

accidental impacts of rigid bodies that may occur during their service life with a very low probability

of occurrence. In the available design codes of civil engineering, the design of RC structures subjected

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to impact is generally based on approximate static method. However, the transient dynamic analysis

has to be performed accounting for the main physical processes involved and accurate models are

needed to describe and predict the structural behavior of RC members subjected to impact loading.

This study focuses on RC slabs subjected to accidental dropped object impact during handling

operations within nuclear plant buildings. The objective of this chapter is to provide a general

overview of research programs which have been carried out in regards to analyzing structures under

general loading conditions, evaluating global response of RC slabs under drop-weight impact loading

conditions. A classication of impact response types of RC slabs according to the impactor mass and

velocity was provided. Then types of impact event and the transient behavior of RC slabs subjected

to impact are described and identied. In order to properly predict the behavior of RC slabs

under impact, impact dynamics, type of impact, failure modes and energy consideration should be

examined and identied. The impact response of RC slabs depends on several parameters, including

slab dimensions, material properties and impact conditions. In this chapter, uniaxial compressive

and tensile stress-strain curves of concrete and steel based on experimental observations in the

literature are adopted.

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Chapter 3

steps

3.1 Introduction

This chapter discusses the choice of deterministic software used in the present study to simulate

the problem of RC slabs which are subjected to accidental dropped object impact during handling

operations within nuclear plant buildings. First, the basic phases of nite element analysis are

described and the dierent steps which are necessary to create a nite element model in Abaqus

are presented. Furthermore, plasticity constitutive models used to represent concrete and steel in

numerical simulations are discussed by describing and identifying their fundamental parameters,

and contact algorithms used for modeling the interaction between two bodies are presented. Other

numerical features which are necessary for simulating impact analysis are also discussed.

As an alternative of experimental approach, nite element analysis (FEA) represents an advanced

engineering tool to design structural and civil engineering applications. Based on the nite element

method, FEA is a numerical technique for obtaining approximate solutions of partial dierential

equations as well as of integral equations. These equations may be generated from complex structures

with complicated features, for which it is dicult to nd an analytical solution. Thus, the simulation

procedure should be performed with an intensive attention in order to improve the quality of FEA

studies and obtain models that comply with the physical properties of the actual structure as far as

possible. Computational tools based on the nite element method are increasingly improved with the

development of computer technology, thus FEA is widely used as a powerful and helpful technique

to assess complex engineering problems. Once the numerical model is validated with experimental

results, FEA enables an accurate calculation of stress distributions and an evaluation of the inuence

of model parameter variations.

A nite element model is based on dividing the complex structure into several small elements

of nite dimensions that are connected to associated nodal points. Each element is assigned with a

specic geometric shape and appropriate material properties and has its own functions of dependent

variables. These functions can be interpolated using shape functions and describe the element

response in terms of the value of dependent variables at a set of nodal points. These functions

and the actual element geometry are used to determine the equilibrium equations that express the

displacement occurring at each node in term of the external forces acting on the element. The

unknown displacements are determined by solving a system of nite element equations that are

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generated, with the element stiness matrices, through the equilibrium equations and stress-strain

relationships.

Generally, FEA procedure involves the following three basic phases:

The pre-processing phase represents the most consuming in time among the three phases, it

is the phase where the numerical model of the physical problem should be dened and ready

to be submitted to the solution or analysis phase. It consists in dividing the model into a

number of discrete interconnected elements with common discrete nodes and applying certain

appropriate boundary conditions with reasonable assumptions. Suitable nite element types

should be selected correctly and a suciently rened mesh should be created, especially in

regions where high stresses are expected. The following process of creating the mesh, choosing

suitable elements connected with their respective nodes, and dening boundary conditions is

referred to as discretization of the physical problem.

The solution/Analysis phase represents the phase of dening the analysis type and submitting

the nite element model to nite element code solver. The analysis type can be either static or

dynamic depending on the structure response to applied loads. A numerical output database

will be generated by solving a series of linear or nonlinear equations. In this phase, numerical

output can be controlled in the nite element code solver and quantities such as displacements,

stresses, reactions, or other variables can be determined.

Post-processing phase represents the nal phase that provides a visualization environment of

results by using a post-processing software. Results can be displayed in contour plots or other

approaches which assist to get a better evaluation and interpretation of the simulation results.

However, at this stage, the task of interpreting results should be performed carefully in order

to verify the accuracy of FEA and assess whether the assumptions adopted during simulation

are satised and accurate. These assumptions related to the structure geometry, boundary and

loading conditions, material properties, and interaction between dierent parts of the model

will potentially aect the results, as well as mesh sizes that may have important eects on

predicted solutions.

The three basic phases of FEA procedure also can be used to describe the main framework of any

available FE software. Hence, almost all FE software are based on the same fundamental principle

of FE method and have the same general purpose that includes three components, namely, the

pre-processor, the processor and the post-processor. A pre-processor establishes the FE model and

the input data for the processor that solves the dierent equations generated during analysis. A

post-processor helps in reviewing and interpreting results, as well as checking the validity of the

solution.

Several commercial software packages are developed to assist in reducing the complexities of FEA

application. Due to advancement of computer technology, FE computer software have evolved to

become so powerful to conduct sophisticated analyses and assist in design process stages. However,

every commercial computer software is developed with its own capabilities, features and algorithms

that are dierent from others. Most FE code are computer-aided design (CAD) software developed

with a graphical user interface (GUI) that signicantly reduces the actual application of FEA and

provides a powerful tool for engineers to perform analyses with minimal eort and without knowing

the governing equations or the limitations of the analysis process. Recent computer-aided engineer-

ing (CAE) software oer a robust and reliable technology to automate critical intensive tasks and

integrate CAD. They permit to graphically generate complex 3D geometries and to automatically

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create rened mesh by merely indicating the desired mesh density. Most FE code are written using

programming languages such as Fortran, C and C++ that provide, with very less additional code,

the advantage of writing subroutines for purposes of adding new elements to the element library and

dening unique material models.

Choosing among the currently available commercial software is a very important task that is

based on requirement of applications and involves several criteria such as analysis type, ease of use,

eciency, software limits, computational cost, technical support and training. Furthermore, the

accuracy of numerical results is very dependent on the prociency and the fundamental theoretical

knowledge of the engineer in FEA. Most of the time, engineers will interact only with the pre- and

post- processor components of the software, which may lead to the danger of using its tools as a

black box in case of the engineer does not understand what happens in this essential part between

the two processors. Mistakes like accepting results with an inappropriate mesh, not considering

reasonable assumptions for an analysis and misrepresenting model properties can still be easily

made despite all the modern and sophisticated improvements in FE software. For example, the

automatic mesh generators that are provided in most commercial software oer a robust tool for

meshing complex geometries, but if misused, they would result in a loss of accuracy and a dramatic

increase in computational time that may even exceed available hardware and software resources.

Therefore and before starting an analysis, a FE model must be extracted from the real problem in

order to solve the problem using the most ecient modeling method with the less computational

time and a high degree of reliability. Likewise, the selection of the right FE software is absolutely

crucial, as well as understanding its limits and the computational resources.

The present study focuses on RC slabs subjected to low impact velocity, thus the FE software

chosen should include the ability of creating 3D geometries, applying impact conditions and dening

nonlinear material behaviors. In addition, it should be capable of simulating contact and interaction

between dierent parts of the structure, which is highly required for an impact problem. Given the

many available software, the well-known commercial FE software, Abaqus, is used to illustrate how

to simulate the geometric and material properties of the slab and impactor, as well as to dene

contact, boundary and initial conditions. Data input for FEA with Abaqus can be done through:

Abaqus Keyword edition by using an input le written with a text editor (.inp), or

Abaqus is a series of powerful engineering programs that are based on the FE method with a high

quality of pre- and postprocessing capabilities. The user-friendly nature of Abaqus oers the possibil-

ity of modeling and solving even the most complicated engineering problems. Simulations ranging

from relatively simple and linear to highly dicult and nonlinear can be performed easily with

Abaqus by only providing the engineering data such as the structure geometry, material behaviors,

loads, boundary and initial conditions. Abaqus provides complete numerical solutions and controls

their eciency by automatically choosing appropriate values for load increments and convergence

tolerances and continually adjusting them during the analysis in order to obtain accurate results.

Abaqus is divided into modules that permit to build a FE model by performing a successive

passage through modules. Each module denes a fundamental step in the modeling process. In the

present section, the dierent steps which are necessary to create a FE model in Abaqus and the data

that must be included are presented:

Part module: The structure geometry can be divided into multiple parts, each part represents

a main component of the model and is created out of a two-dimensional sketch.

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Property module: Mechanical behaviors and properties of materials can be specied by using

the Abaqus material library that contains an extensive list of linear and nonlinear material

models, or by creating new material models through several subroutines that are provided in

Abaqus and programmed in Fortran language. The behavior of most typical engineering ma-

terials can be simulated with Abaqus, which includes metals, concrete, composites, polymers,

and so on. Section properties of the physical structure are also dened in this module. Each

section is associated with a material and assigned to a correspondent part of the geometry.

Assembly module: Geometry of the model is dened by creating instances of each part which

is independent of each other. Instances are positioned relative to one another in a global

coordinate system in order to assemble the model.

Mesh module: Generating nite element meshes can be actually done either on the part or

the assembly module. Meshing involves choosing appropriate element types and mesh density.

Various types of geometrical elements that can model virtually any geometry are available in

Abaqus element library. All elements use numerical integration and must refer to a section

property denition.

Conguring analysis module: The analysis procedure type must be dened in order to solve

the numerical problem. Abaqus can be used to solve general static and dynamic response

problems and consists of two main analysis products, Abaqus/Standard and Abaqus/Explicit.

Abaqus/Standard allows carrying out static analyses where the structure has a long-term

response to the applied load and solves a coupled system of equations implicitly at each incre-

ment. Abaqus/Explicit is particularly used to simulate transient dynamic events and uses an

explicit formulation to perform a large number of small time increments without the need of

solving a system of equations at each increment. The whole analysis procedure may comprise

one or multiple steps. Thus, as many steps as needed must be created for the case load by

providing the required parameters and controlling the output data requested for each step.

Load module: Specifying boundary and initial conditions and applying loads to the structure

are step-dependent. Loads can be applied in Abaqus with many dierent ways which include

concentrated or distributed loads, pressures and loads per unit volume such as due to gravity

or acceleration. Zero-valued, including symmetry conditions, or non-zero boundary conditions

can be imposed to model regions where the displacements and/or rotations are known. Most

loads or boundary conditions can follow an amplitude curve varying with time. Non-zero initial

conditions can be also specied as predened elds, these include initial stresses, velocities or

temperatures.

Interaction and constraints module: Interaction and constraints between dierent parts or

surfaces of the model can be dened. Abaqus provides two algorithms for simulating contact,

general contact algorithm that allows dening contact between all model regions and contact

pairs algorithm that describes contact between two surfaces. Interaction properties are used

to enhance contact modeling, normal or tangential behavior can be dened with friction or

contact damping. Abaqus includes also several constraint types such as rigid bodies, coupling

constraints and surface-based ties. Constraints are used to couple a group of nodes motion to

the motion of other nodes.

Output control module: A large amount of output variables can be generated with an Abaqus

simulation, they depend on the analysis and element types used. Output are specied with

Field and History output managers that generate a default output request, but also enable to

control and manage the analysis requested output in order to only produce results that are

required to be interpreted and avoid excessive disk space.

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Job module: A job is associated with the model and submitted to the solver for analysis.

Submission of input les can be performed through Abaqus/CAE or manually from the com-

mand prompt. Abaqus oers the possibility of checking incorrect or missing data through a

data check analysis that can be used before running a simulation and helps in minimizing the

probability of errors in the model.

Visualization module: This module is only available in Abaqus/CAE which writes the analysis

results to an output database le with the extension (.odb). The evaluation of results are

performed interactively using the program Abaqus/viewer that allows viewing and displaying

results related to the model through a variety of options including color contour, X-Y and

deformed shape plots. Note that three other types of output are available in Abaqus and

results can be written to the Abaqus data le (.data) in tabular form, to the Abaqus restart

le (.res) that uses restart data to continue the analysis, and to Abaqus results le (.l) that

can be used with third-party software for subsequent postprocessing. Moreover, a Python

script can be used to read data from a (.odb) le and gather numerical results as tables, plots

or animations.

Selecting an appropriate approach to solve the equation governing the response of the nite element

analysis of a specic problem is very important since an incorrect solution and a highly computa-

tional analysis may result from choosing the wrong simulation method. Two methods are available in

Abaqus to solve a nite element problem, namely the implicit and explicit methods. Both methods

consist of a numerical direct time integration scheme to evaluate the unknown displacement at the

end of the time step. Implicit method is generally used for static, harmonic or modal analyses in

which the time dependency of the solution is not a signicant factor, while explicit method is used

for blast and impact problems involving high deformation and time dependent solution. However,

Abaqus allows using the implicit method to perform dynamic analysis of problems in which inertia ef-

fects are considered, and using the explicit method quasi-static problems that experience convergence

diculties in implicit analysis methods. Implicit direct integration is provided in Abaqus/Standard,

while explicit direct integration is provided in Abaqus/Explicit [5].

The direct-integration dynamic procedure provided in Abaqus/Standard oers a choice of implicit

operators using the implicit Hilber-Hughes-Taylor operator for integration of the equations of motion,

while Abaqus/Explicit uses the central-dierence operator. In an implicit dynamic analysis the

integration operator matrix must be inverted and a set of nonlinear equilibrium equations must

be solved at each time increment. In an explicit dynamic analysis displacements and velocities

are calculated in terms of quantities that are known at the beginning of an increment. Therefore,

the global mass and stiness matrices need not be formed and inverted, which means that each

increment is relatively inexpensive compared to the increments in an implicit integration scheme.

However, the size of the time increment in an explicit dynamic analysis is limited because the central-

dierence operator is only conditionally stable, whereas the implicit operator options available in

Abaqus/Standard are unconditionally stable. Thus, there is no such limit on the size of the time

increment that can be used for most analyses in Abaqus/Standard (accuracy governs the time

increment in Abaqus/Standard).

The explicit scheme is only stable if the size of the time step is smaller than the critical time

step size for the structure being simulated. This very small time step size requirement for stability

thereby makes explicit solutions recommendable for short transient situations. But, even though

the number of time steps in an explicit solution may be orders of magnitude greater than that of

an implicit solution, it may be more ecient than an implicit solution since no matrix inversion

is required. Very Large deformation problems such as impact analysis can result in millions of

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degrees of freedom eectively increasing the size of stiness matrix. The implicit scheme requires

considerable computation time, disk space, and memory because of the iterative procedure involving

the inversion of a large global stiness matrix (determined by the number of degres of freedom).

The computational cost for problems based on the implicit scheme is roughly proportional to the

square of the number of degrees of freedom, while the analysis cost for problems based on the

explicit scheme rises only linearly with problem size. Hence there is a need for an explicit method

which would prevent the inversion of stiness matrix. It is possible to solve complicated, very

general, three-dimensional contact problems with deformable bodies in Abaqus/Explicit. Problems

involving stress wave propagation can be more ecient computationally in Abaqus/Explicit than in

Abaqus/Standard.

3.6.1 Concrete constitutive model

Modeling concrete within nite element packages is a very complicated task due to the complexity

of its behavior. For an accurate simulation of RC structures, both elastic and plastic behavior of

concrete in compression and tension should be taken into account using a proper material model in

nite element analysis. Simulation of compressive behavior should include elastic behavior of con-

crete, as well as strain hardening and softening regimes. Simulation of concrete under tension should

include tension softening, tension stiening and interaction between concrete and steel elements.

There are dierent theories available in the literature to model the constitutive behavior of con-

crete, namely elasticity theory, viscoelasticity theory, viscoplasticity theory, classical plasticity the-

ory, endochronic theory, fracture mechanics, continuum damage mechanics and stochastic approach

[2, 96]. Among these, the most commonly used frameworks are based on plasticity, continuum

damage mechanics and combinations of plasticity and damage mechanics.

Plasticity theory was widely used to describe concrete behavior [72, 122]. Stress-based plasticity

models obviously have advantages over elastic theory since they allow modelling concrete behavior

when subjected to triaxial stress states, as well as representing hardening and softening regimes.

The main characteristic of these models is a plastic yield surface that corresponds at a certain stage

of hardening to the strength envelope of concrete and includes a nonassociative ow rule [72, 122].

The expression of the yield surface includes a hardening-softening function and inelastic strains are

considered in order to represent realistically the deformations in concrete [73, 96, 122]. Nevertheless,

such models do not incorporate a damage process to capture the variations of the elastic unloading

stiness upon mechanical loading accurately. As a result, they are not able to describe unilateral

eects and the reduction of material stiness due to microcracks [88, 96] (Figure 3.1.a).

Conversely, continuum damage mechanics models are based on the concept of a decrease of

the elastic stiness [73, 88]. The continuum damage theory was also widely used to model the

nonlinear behavior of concrete [33, 120]. For these models, phenomena such as strain softening,

stiness decrease and unilateral eects due to progressive microcrackings and microvoids are taken

into account [88, 96]. The mechanical eect of these phenomena is represented by a set of internal

state variables that describes the decrease of the elastic stiness at the macroscopic level [88, 120].

Nevertheless, such models do not consider the plastic strain (Figure 3.1.b), hence they are unable to

describe irreversible deformations nor predict dilatancy behavior under multiaxial loadings [73, 88].

On the other hand, several models were proposed in literature as a combination of isotropic

damage and plasticity in order to model both tensile and compressive behaviors of concrete [88, 104]

since neither pure elastic damage models nor pure elastoplastic constitutive models can satisfactorily

describe the concrete behavior. Both models fail in capturing the evolution of unloading stinesses

accurately [88, 152]. Thus, damage-plastic models are used to overcome this problem since both

irreversible deformations and microcrackings contribute to the nonlinear behavior of concrete (Figure

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Figure 3.1: Unloading response of: (a) elastic plastic, (b) elastic damage, (c) elastic plastic damage

models [88]

3.1.c). Damage plasticity models are formulated in the eective (undamaged) stress space combined

with a strain based damage model, which is numerically more stable and more appropriate to

represent tension and compression behaviors of concrete [73, 152]. Plastic yield function is no longer

expressed in term of the applied stress, but it is written in term of the eective stress that represents

the stress in undamaged material in between the microcracks [88].

Several plasticity models which have been implemented in ABAQUS were widely used to repre-

sent concrete in numerical simulations, namely the Drucker-Prager/Cap model, the brittle cracking

model and the concrete damage plasticity model. These models require multiple and complex pa-

rameters, which are generally obtained from material tests.

Plasticity models available in ABAQUS are decomposed into an elastic part and a plastic part.

They are usually formulated in terms of a yield surface, a ow rule and evolution laws that dene

the hardening. The yield function represents a surface in eective stress space, for which a failure

or damage state can be determined. For a plastic-damage model the yield function arrives at:

The states of stress corresponding to material failure are on this surface while the states of safe

behavior are inside.

The plastic ow is governed by a ow potential function G(σ̄) which is also dened in the eective

stress space according to nonassociative ow rule:

∂G(σ̄)

ε̇pl = λ̇ (3.2)

∂ σ̄

For an accurate simulation of the response of RC slabs under impact, a realistic representation of

concrete behavior under dynamic loads is necessary. The concrete model adopted for numerical

analysis should include the hardening behavior that occurs under compressive loading, the softening

behavior that results from the formation of microcracks, the damage that describes the decrease in

material stiness, stiness recovery related to crack opening and closing under cycling loading, and

rate dependence. Among dierent models available in Abaqus, CDP model represents a convenient

model to simulate concrete behavior due to its capabilities to represent irreversible deformations but

also stiness degradation [6]. A comparison to experimental results has proved the accuracy of this

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model to capture the complete behavior of concrete up to failure, and to be the most stable regime

for modeling concrete nonlinear behavior [135]. Therefore, the CDP model is used for concrete to

study the response of RC slabs under impact in the present study

CDP model is a continuum, plasticity-based, damage model for concrete that uses concepts

of isotropic damaged elasticity in combination with isotropic tensile and compressive plasticity to

represent the inelastic behavior of concrete. It is based on the models proposed by Lubliner et al.

[113] and Lee and Fenves [104], and designed for applications in which the concrete is subjected to

arbitrary loading conditions, including cyclic loading. The model assumes that the main two failure

mechanisms are tensile cracking and compressive crushing of the concrete material. Two hardening

variable related to concrete failure mechanisms under tension and compression, respectively, are

used in the CDP model in the aim of controlling the evolution of the yield surface. The model takes

into account the degradation of the elastic stiness induced by plastic straining both in tension and

compression by introducing two independent scalar damage variables for tension and compression,

respectively. The elastic behavior of concrete in CDP model is assumed to be isotropic and linear.

The plastic ow potential function and the yield surface in CDP model make use of two stress

invariants of the eective stress tensor, namely the hydrostatic pressure stress (or the eective

q

hydrostatic stress), p̄ = − 13 trace (σ̄) and the Mises equivalent eective stress, q̄ = 3

2 (S̄ : S̄)

where S̄ is the eective stress deviator (or deviatoric part of the eective stress tensor), dened as

S̄ = σ̄ + p̄I.

The CDP model assumes a nonassociated potential plastic ow that allows a realistic modeling

of the volumetric expansion under compression for concrete [152]. The ow potential G used for this

model is the Drucker-Prager hyperbolic function:

p

G= (ft tan ψ)2 + q̄ 2 − p̄ tan ψ (3.3)

where ft is the uniaxial tensile strength of concrete. The parameters ψ and determine the shape

of the ow potential surface: ψ is the dilation angle measured in the p − q plane at high conning

pressure, while is referred to as the eccentricity of the plastic ow potential surface. The dilation

angle represents the angle of inclination of the failure surface towards the hydrostatic axis in the

meridian plan [94]. The ow potential eccentricity aects the exponential deviation of G from the

linear Drucker-Prager ow potential [118] (Figure 3.2) and denes the rate at which the function

approaches the asymptote (the ow potential tends to a straight line as the eccentricity tends to

zero)[187].

The nonassociative ow rule, which is used in the CDP model requires a yield surface that

determines states of failure or damage. The model uses a yield condition based on the loading

function proposed by Lubliner et al. [113], with the modications proposed by Lee and Fenves [103]

to account for dierent evolutions of strength under tension and compression. The yield function

used in CDP model is expressed in the eective stress space and depends on two hardening variables

εpl

t and εpl

c :

1

F = (q̄ − 3αp̄ + β(εpl ) < σ̄max > −γ < −σ̄max >) − σ̄c (εpl

c )=0 (3.4)

1−α

with εpl

t and εpl

c are the plastic strain of concrete in tension and compression, respectively. In

this expression,σ̄max represents the maximum principal eective stress and <.> is the symbol of

1

the Macaulay bracket which is dened by < x >= 2 (|x| + x).

Moreover, α is a dimensionless material constant and depends on the ratio of concrete strengths

under biaxial and uniaxial compression as follows:

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Figure 3.2: Drucker-Prager hyperbolic function of CDP ow potential and its asymptote in the

meridian plane [94]

(fb /fc ) − 1

α= , 0 ≤ α ≤ 0.5 (3.5)

2(fb /fc ) − 1

A biaxial laboratory test is necessary to dene the value of α [100] (Figure 3.3.a). Typical

experimental values of α are within 0.08 and 0.12 [113], while Abaqus allows a range of values

between 0 and 0.5 [6]. fb represents the biaxial compressive strength of concrete and fc is its

uniaxial compressive strength.

β is a material function of the two hardening variables εpl

t and εpl

c and includes the ratio of the

biaxial to uniaxial compressive strength. it is expressed in terms of the eective tensile stress σ̄t and

eective compressive stress σ̄t as follows:

σ̄c (εpl

c )

β= (1 − α) − (1 + α) (3.6)

σ̄t (εpl

t )

The coecient γ represents a dimensionless material constant included in the CDP model only

for the stress states of triaxial compression when σ̄max < 0 [187] to better predict the concrete

behavior in compression under connement [152]. γ is calculated in term of the coecient Kc that

controls the failure surface in the deviatoric cross section:

3(1 − Kc )

γ= (3.7)

2Kc − 1

Kc is the ratio of the second invariant on the tensile meridian to that on the compressive meridian

at any given value of the pressure invariant p [6]. Physically, it represents the ratio of the distances

between the hydrostatic axis and respectively the compression meridian and the tension meridian

in the deviatoric cross section [94] (Figure 3.3.b). In Abaqus, Kc must satisfy the condition 0.5 ≤

Kc ≤ 1.0. For Kc = 1, the deviatoric cross section of the failure surface becomes a circle.

The constitutive equation under uniaxial compression for the CDP model takes the following form

[6, 113]:

c ) = E(εc − εc ) (3.8)

where dc is the damage variable in the compression zone, it can take value from zero, representing

the undamaged material, to one, representing the total loss of strength. Ec is the initial (undamaged)

elastic stiness of the material, while E = (1 − dc )Ec is the degraded elastic stiness in compression.

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Figure 3.3: Yield surface for the CDP model in: (a) plane stress, (b) the deviatoric plane corre-

sponding to dierent values of Kc [6]

σc

σ̄c = = Ec (εc − εpl

c ) (3.9)

(1 − dc )

where εpl

c is the equivalent plastic strain in compression.

Inelastic strains εin

c are used in the CDP model to describe the hardening rule. As illustrated in

Figure 3.4, the compressive inelastic strain is dened as the total strain minus the elastic strain εce

corresponding to the undamaged material,

εin

c = εc − εce (3.10)

where

σc

εce = (3.11)

E0

The stress-strain relation under uniaxial tension is similar to that in compression and takes the

following form [6, 113]:

t ) = E(εt − εt ) (3.12)

where dt is the damage variable in the tension zone and E = (1 − dt )Ec is the degraded elastic

stiness in tension.

The eective tensile stress is dened as:

σt

σ̄t = = Ec (εt − εpl

t ) (3.13)

(1 − dt )

where εpl

t is the equivalent plastic strain in tension. The values of dt range from 0 to 1 representing

the undamaged material and the total loss of strength, respectively.

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Figure 3.4: Response of concrete to uniaxial loading in compression for CDP model [6]

The strain-softening behavior for cracked concrete is dened by the phenomenon called tension

stiening which is required in the CDP model. The tension stiening eect depends on several

factors such as the density of reinforcement, the quality of the bond between the rebar and the

concrete, the relative size of the concrete aggregate compared to the rebar diameter, and the mesh

([7]).

The tension stiening can be specied in two ways:

t which is used in the CDP

model for numerical analyses. The strain after cracking is dened as the dierence between

the total tensile strain and the elastic strain εte for the undamaged material (Figure 3.5):

εck

t = εt − εte (3.14)

where

σt

εte = (3.15)

Ec

The specication of a postfailure stress-strain relation introduces mesh sensitivity in the results,

especially in cases with little or no reinforcement.

A fracture energy cracking criterion which is adequate to reduce the concern of the mesh

sensitivity. Hillerborg [81] denes the energy required to open a unit area of crack (Gf ) as

a material parameter, using brittle fracture concepts. CDP model assumes a linear loss of

strength after cracking (Figure 2.38.a). The cracking displacement at which complete loss

of strength takes place is, therefore, wc = 2Gf /ft . Descending branch after cracking can be

also presented by a bi-linear or tri-linear tension softening curve. In Abaqus, the fracture

energy cracking model is introduced by specifying the postfailure stress as a tabular function

of cracking displacement.

3.6.2.4 Damages

The variation of the elastic modulus of concrete can be due to two damage mechanisms of concrete

in tension and compression, respectively [27]. In uniaxial tension, plastic strains are very small

compared to those in compression, and concrete nonlinear response induces a lot of damage due to

mircrocraking. Positive strains controls the tension damage by producing microcrak opening and

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Figure 3.5: Response of concrete to uniaxial loading in tension for CDP model [6]

growth. In uniaxial compression, important plastic strains are observed with little damage. The

compression damage mechanism is the crushing in material which induces the volume fraction of

voids due to irreversible plastic strains. Thus, tensile damage and compressive damage of concrete

provide quite dierent responses, as indicated in some experimental observations [104].

To account for this dierence, the CDP model considers two independent damage variables,

namely the tensile damage dt and compressive damage dc . As previously mentioned, these damages

can take values ranging from zero, for the undamaged concrete, to one, for the fully damaged

concrete. Experiments to determine the damage variable in terms of the equivalent plastic strains

are very dicult to perform, thus many researchers studied the decrease of elastic stiness and

proposed expressions to tensile and compressive damages [27, 48, 73, 96, 104, 113, 118, 120]. Damage

evolution of concrete in tension and compression can be derived from uniaxial stress strain curves

and formulated as increasing functions of the inelastic strains. Pavlovic et al. [154] as well as

Jankowiak and Lodygowski [87] compared undamaged and damaged concrete responses in order to

determine the expressions of tensile and compressive damages, respectively, as follows:

σt

dt = 1 − (3.16)

ft

σc

dc = 1 − (3.17)

fc

However, Korotkov et al. [98] and Wahalathantri et al. [178] assumed the tensile damage variable

to be equal to the ratio of the cracking strain to the total tensile strain, and similarly dened the

compressive damage variable as the ratio of the crushing strain to the total compressive strain.

In Abaqus, the accuracy of damage evolution should be evaluated by ensuring that no negative

and/or decreasing plastic strain are produced with the increase of stress [7]. Tensile and compressive

plastic strains can be expressed in Abaqus for the CDP model in terms of the cracking and crushing

strains, respectively, as follows:

dt σt

εpl ck

t = εt − (3.18)

(1 − dt ) Ec

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dc σc

εpl in

c = εc − (3.19)

(1 − dc ) Ec

Negative and/or decreasing plastic strains are indicative of incorrect damage evolutions, which

leads to generate error message in Abaqus before the analysis is performed [7].

The uniaxial behavior of steel shows that steel can be modeled as an elasto-plastic material with

isotropic strain hardening characterized by an increase in stress with the increase of inelastic strain.

Thus, plasticity models are found to be the simplest and most computationally ecient models for

predicting steel behavior.

In Abaqus, the von Mises yield criterion, associative ow rule and isotropic hardening are typi-

cally used for modeling structural steel. The von Mises yield criterion can be written as:

where fy is the yield stress, R(p) is the isotropic hardening stress and q̄ is the von Mises equivalent

stress (see 3.6.2.1).

The evolution of the plastic strain is governed by the following plastic ow rule:

∂f

ε̇pij = λ̇ (3.21)

∂σij

with ε̇pij is the plastic strain rate and λ̇ is the plastic multiplier.

Abaqus provide a wide variety of element types available, it is important to select the correct

element for a particular application. Choosing an element for a particular analysis can be simplied

by considering specic element characteristics: rst- or second-order, full or reduced integration,

hexahedra/quadrilaterals or tetrahedra/triangles. By considering each of these aspects carefully,

the best element for a given analysis can be selected.

In this study, hexahedral solid elements with 8 nodes with reduced integration (C3D8R) are

used during the simulations of the problem involving contact-impact of reinforced concrete (Figure

3.6). These solid (or continuum) elements in Abaqus can be used for complex nonlinear analyses

involving contact, plasticity, and large deformations. These elements use linear interpolation in each

direction and are often called linear elements or rst order elements. These elements have only

three displacement degrees of freedom and are stress/displacement elements. Hourglassing can be

a problem with rst-order, reduced-integration elements (such as C3D8R) in stress/displacement

analyses. Since the elements have only one integration point, it is possible for them to distort in

such a way that the strains calculated at the integration point are all zero, which, in turn, leads

to uncontrolled distortion of the mesh. First-order, reduced-integration elements in Abaqus include

hourglass control, but they should be used with reasonably ne meshes. There are advantages

involved with the use of the hexahedral elements since they provide solution of equivalent accuracy

at less computational cost and allow a better convergence rate and a more uniform mesh compared

to tetrahedral elements. The computational eciency of reduced-integration element is very high

compared to the fully integrated element, especially in the problems involving contact-impacts.

Three dimensional truss elements having two degrees of freedom (T3D2) are used to represent the

reinforcement in FEM (Figure 3.6). Truss elements are used in two and three dimensions to model

slender structures that support loading only along the axis or the centerline of the element. No

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Figure 3.6: Finite elements used for the problem involving contact-impact of reinforced concrete [5]

moments or forces perpendicular to the centerline are supported. A 2-node straight truss element,

which uses linear interpolation for position and displacement, has a constant stress. The cross-

sectional area associated with the truss element as part of the section denition. When truss elements

are used in large-displacement analysis, the updated cross-sectional area is calculated by assuming

that the truss is made of an incompressible material, regardless of the actual material denition.

Truss elements have no initial stiness to resist loading perpendicular to their axis. The most

common applications of trusses at large strains involve yielding metal behavior in which cases the

material is eectively incompressible.

4-node R3D4 rigid elements are used to dene the surfaces of rigid bodies presented in the

FEM (Figure 3.6). R3D4 elements can be used in three-dimensional analysis to dene master

surfaces for contact applications. Rigid elements must always be part of a rigid body. By default in

Abaqus/Explicit, rigid elements do not contribute mass to the rigid body to which they are assigned.

To dene the mass distribution, you can specify the density of all rigid elements in a rigid body.

When a nonzero density and thickness are specied, mass and rotary inertia contributions to the

rigid body from rigid elements will be computed in an analogous manner to structural elements.

Abaqus/Explicit provides two algorithms for modeling contact and interaction problems: the gen-

eral contact algorithm and the contact pair algorithm. Both algorithms can be used with three-

dimensional surfaces. The general contact algorithm allows very simple denitions of contact with

very few restrictions on the types of surfaces involved. The contact pair algorithm has more re-

strictions on the types of surfaces involved and often requires more careful denition of contact.

The general contact and contact pairs algorithms in Abaqus/Explicit dier by more than the user

interface; in general they use completely separate implementations with many key dierences in the

designs of the numerical algorithms. Contact denitions are greatly simplied with the general con-

tact algorithm which allows generating contact forces to resist node-into-face. To dene a contact

pair, pairs of surfaces that will interact with each other must be indicated.

Abaqus/Explicit uses two dierent methods to enforce contact constraints with the contact pair

algorithm, namely the kinematic contact algorithm that uses a kinematic predictor/corrector contact

algorithm to strictly enforce contact constraints and the penalty contact algorithm that has a weaker

enforcement of contact constraints but allows for treatment of more general types of contact. There

are dierences in the way in which the kinematic and penalty contact algorithms enforce contact

constraints (Figure 3.7). Using Abaqus analysis guide [5], the dierences can be described as follows:

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Figure 3.7: Constraints with the contact pair algorithm: a) Kinematic contact, b) Penalty contact

[5]

ered in the analysis, Abaqus/explicit carries out a predictor phase and a corrector phase in

each time increment. In the predictor phase the kinematic state of the model is advanced by

ignoring any contact conditions. This can result in overclosure or penetration. In the corrector

phase of the time increment, an acceleration correction is applied to the slave and master nodes

to correct for this predicted penetration, while conserving momentum. This correction results

in a nal conguration for increment in which the slave node is exactly in compliance with the

master surface.

The penalty algorithm uses an explicit approach to enforce contact constraints. In contrast

to the kinematic algorithm, a corrector phase is not processed for an increment. Rather,

an interface spring is inserted automatically between the slave node and the master face

in increment to minimize the contact penetration. The force associated with the interface

spring is equal to the spring stiness multiplied by the penetration distance. Therefore, a

small residual penetration will exist since contact forces are not generated unless there is some

amount of penetration at the beginning of the increment.

In a mechanical contact simulation the interaction between contacting bodies is dened by as-

signing a contact property model to a contact interaction, including a constitutive model for the

contact pressure-overclosure relationship that governs the motion of the surfaces and a friction model

that denes the force resisting the relative tangential motion of the surfaces. The contact pressure-

overclosure relationship used in this study is referred to as the hard contact model (Figure 3.8.a).

Hard contact implies that the surfaces transmit no contact pressure unless the nodes of the slave

surface contact the master surface. No penetration is allowed at each constraint location and no

limit to the magnitude of contact pressure can be transmitted when the surfaces are in contact.

The classical isotropic Coulomb friction model is used to model the friction between the contacting

bodies in terms of the stresses at the interface of the bodies (Figure 3.8.b). The basic concept of

the Coulomb friction model is to relate the maximum allowable frictional (shear) stress across an

interface to the contact pressure between the contacting bodies. In the basic form of the Coulomb

friction model, two contacting surfaces can carry shear stresses up to a certain magnitude across

their interface before they start sliding relative to one another. The friction coecient needs to be

dened for the mechanical tangential interaction.

3.9 Constraints

3.9.1 Approaches to represent steel in RC numerical analysis

Typically, three distinct approaches are available in nite element analysis to represent steel rein-

forcement in a 3D nite element model of a RC structure, namely smeared approach, embedded

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Figure 3.8: Contact interaction properties: a) Hard contact model, b) Coulomb friction model [5]

approach and discrete approach. These approaches allow the interactions between the concrete ele-

ments and the reinforcement elements by specifying the constraints on the degrees of freedom of the

relevant nodes.

In the smeared or distributed approach (Figure 3.9.a), the distribution of reinforcing steel is

assumed to be uniform over the concrete elements in the appropriate direction. As a result, the

total stiness of a smeared RC nite element depends on the stiness of concrete and that assigned

to steel reinforcement. In addition, the material properties in a smeared nite element are estimated

according to composite theory based on individual properties of concrete and reinforcement. In this

approach, reinforcing bars are considered in the model using a ratio that depends on the size of

the concrete structure and the amount of steel used. Therefore, it is not necessary to discretely

simulate the bars as they are distributed in the nite model assuming a complete compatibility

between steel and concrete. Abaqus allows modeling this approach using the rebar option to dene

layers of uniaxial reinforcement in concrete elements as smeared layers with a constant thickness.

The thickness of layers is equal to the ratio of each reinforcing bar area to the spacing between

bars [5]. This approach is usually applied for large structural models when modeling details of the

reinforcement is not essential to evaluate the overall response of the structure [138, 169], or when

the structure is heavily reinforced with a huge number of reinforcing bars and an exact denition of

every single reinforcing bar is dicult [36].

The embedded approach allows an independent choice of concrete mesh, it is generally used to

create a bond and arbitrarily dene the reinforcing steel regardless of the mesh shape and size of

the concrete element (Figure 3.9.b). Each reinforcing bar is considered as an axial member and its

stiness is evaluated independently from the concrete elements. The displacements of the reinforce-

ment elements are compatible with those of the surrounding concrete elements. In this approach,

the identication of the intersection points of concrete elements with reinforcement bars leads to ad-

ditional nodes created in the model and located at the intersection points. This results in increasing

the number of degrees of freedom in the analysis, and hence the computational time becomes more

important. Abaqus allows the embedded element technique to model reinforcing bars as a group of

elements embedded in 'host' solid elements that represent the concrete elements in a RC structure.

Abaqus searches for the intersection nodes of the host and embedded elements and constrains their

translational degrees of freedom to the interpolated values of the corresponding degrees of freedom

of the host element [5]. This approach is generally found to be very advantageous in cases where

modeling the reinforcement of a structure is complex [138], and to be more ecient than the discrete

approach in term of the computational eort and improving the simulated interaction [169, 183].

The discrete approach is based on the use of separate elements to represent the reinforcing steel.

Reinforcement can be modeled using truss or beam elements connected to the concrete mesh nodes in

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Figure 3.9: Approaches to represent steel in RC numerical analysis: a) smeared approach, b) em-

bedded approach, c) discrete approach [138]

Figure 3.10: Tie constraint to tie two surfaces together in Abaqus [5]

the way that the mesh discretization boundary of concrete elements must overlap the direction and

location of the steel reinforcement elements (Figure 3.9.c). As a result, concrete and reinforcement

elements share common nodes and they are superimposed at some specied same regions. Perfect

bond is usually assumed between concrete and steel. However, in cases where slip of reinforcing

bars with respect to the surrounding concrete is of concern, ctitious spring elements of no physical

dimension can be used to model the bond-slip between reinforcing steel and concrete. These elements

are dened as linkage elements to connect concrete nodes with reinforcement nodes having the same

coordinates, and are determined by their mechanical properties and a specic bond-slip relationship

[36, 138].

A surface-based tie constraint in Abaqus is used to dene a fully constrained contact behavior by

providing a simple way to tie two surfaces together for the duration of a simulation. It uses a master-

slave formulation to prevent slave nodes from separating or sliding relative to the master surface. As

a result, an assignment of slave and master surfaces should be carried out to dene the contacting

surfaces. Typically, a rigid surface is always considered as master surface, and slave surface mesh

must be suciently rened to interact with the master surface. In case where two deformable

surfaces need to be tied, master surface should be that of the material with higher Young's modulus

or larger cross-sectional area [7]. This type of constraints enables to tie translational and rotational

motions between two surfaces, hence each of the nodes of the slave surface is constrained to have the

same motion as the node of the master surface to which it is closest. Abaqus eliminates the degrees

of freedom of the slave surfaces nodes that are constrained and allows the tie constraints only in

cases where the two surfaces to tie are close to one another (Figure 3.10) [5].

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3.10 Output

For the analysis of RC slabs subjected to impact, a large amount of output can be requested and

evaluated in Abaqus:

Energies: kinetic energies for whole model and slab (ALLKE), internal energy (ALLIE), en-

ergy dissipated by plastic deformation (ALLPD), recoverable strain energy (ALLSE), articial

strain energy (ALLAE) and energy dissipated by viscous eects (ALLVD). When concrete

damage is taken into account in simulations, the damage dissipation energy (ALLDMD) is

also evaluated.

Distributions of stresses (S11, S22, S33) on the upper and bottom surfaces of the RC slab at

dierent given times of the simulation.

Distributions of accelerations (A11, A22, A33) on the upper and bottom surfaces of the RC

slab at dierent given times of the simulation.

Distributions of velocities (V11, V22, V33) on the upper and bottom surfaces of the RC slab

at dierent given times of the simulation.

Distributions of displacements (U11, U22, U33) on the upper and bottom surfaces of the RC

slab at dierent given times of the simulation.

Distributions of stresses (S11) in the upper and lower reinforcement layers at dierent given

times of the simulation.

Graphical visualization of the cracking patterns in the slab obtained by assuming that cracking

initiates at points where the tensile equivalent plastic strain is greater than zero and the

maximum principal plastic strain is positive.

Displacements, velocities and accelerations for the impactor, as well as for dierent nodes of

the slab.

Evolution of contact load (CFN3) between the impactor and the slab in terms of time and

slab displacement at the point of impact.

Compressive damage variable (DAMAGEC), tensile damage variable (DAMAGET) and the

equivalent plastic strain (PEEQ) at dierent elements positioned at the center and the corner

of the impacted face of the slab.

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Chapter 4

methods

The concept of security is very old and inseparable from the state of knowledge of nature's laws

and the concept of lifetime. It is simultaneous to construction practices and related to uncertain

events as well as to the notion of codes and regulations. The early builders built for their personal

use and were therefore interested in what the structure fullls the functions that had been assigned.

By referring to the history of constructions, many structures are no longer in use because of a lack

of knowledge about nature's laws, in particular materials and loadings, but mostly because of the

time elapsed since their application. It was only in the 18th century that numerical approaches have

replaced by empirical approaches. Numerical approaches allowed reducing the concept of security

to a deterministic notion since they are based on the analysis of stresses: stress is obtained by a

"safety factor" that comprises a certain number of unknowns and limits them to a single factor.

In the fties, Freudenthal became the head of the safety probabilistic approach that has made

a signicant progress in the domain of structures [63, 64, 65, 66]. This approach is based on the

uncertain aspect of loadings on structures, material properties and mechanical models used for

design. Thus, any probabilistic approach requires the acquaintance of a risk that is not identied

and does not exist in deterministic approaches. The control of risk would rst result from expert

knowledge and algorithmic knowledge (rules and techniques). Both knowledge increase exponentially

with human development. The reliability theory and its tools were initiated in the modern era based

on probability theory. Recently, probabilistic approaches were introduced in the domain of civil

engineering to control risks related to design choices. This control can be included in design process

by combining a stochastic data model to a mechanical data model for a mechanical behavior [106]

(couplage mécano-abiliste).

Reliability is dened as the probability so that an equipment could be used without failure for

a given period of time, under specied operational conditions (Grand Larousse). Considered as

the ability of a structure to achieve a required function under certain given conditions and within

a given timeframe, reliability can be applied to any system that is expected to correctly carry

out its functions, e.g. a civil engineering structure. Reliability theory is based on methods using

probabilities to assess failure probability that is mathematically considered as the complement of

the reliability of the unit. A reliability approach of a mechanical system allows examining the

statistical dispersion of quantities, rejecting values out of tolerance and verifying their variability.

Many methods based on probability theory were developed, they permit to calculate the probability

of failure of structures on the one hand, and to study the inuence of variability of design parameters

on the structural behavior on the other hand. Uncertainties are modeled in terms of the mean, the

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variance and the probability density of input variables. Reliability approach was widely used in order

to ameliorate the design of RC structures. Arafah [12] studied the reliability of RC beam sections by

considering a limit state in exion and using Monte Carlo method. Araujo [13] presented a reliability

analysis of RC columns by combining a FEM with MC in order to calculate the mean and standard

deviation of the ultimate load applied to column. Val et al. [177] proposed a probabilistic method to

evaluate the reliability of RC frames with respect to an ultimate limit state. The method consisted in

coupling a nonlinear FEM with FORM by considering all the possible sources of uncertainty. Low et

al. [110] performed a reliability study on RC slabs subjected to explosion. Two deterministic models

were considered: a FEM and an analytical model of one degree of freedom. The comparison of these

two models was performed in a probabilistic framework by estimating the failure probability with

MC in the case of FEM, and with FORM/SORM in the other case. Two failure criteria related to

the slab displacement and the deformation in reinforcement were considered. The simplied model

was used in a parametric study to evaluate the eect of various parameters on the slab reliability.

In the current study, the platform OpenTURNS (version 13.2) is used under Linux environment

to perform reliability approach for several types of RC structures. The name OpenTURNS is for open

source Treatment of Uncertainty, Risk 'N Statistics. OpenTURNS was developed as a C++ library

and a Python TUI by EDF R&D, Airbus Group and Phimeca Engineering at the beginning of 2005,

with the collaboration of EMACS in 2014 [14]. OpenTURNS is an open source software dedicated to

uncertainty propagation by probabilistic methods, it includes a set of ecient mathematical methods

for reliability analysis and can be easily connected to any external computational code. These

features allow using OpenTURNS for dierent engineering and industrial problems. In addition, a

complete documentation is provided for OpenTURNS and can be found on its dedicated website

www.openturns.org.

In this chapter, the principles of structural reliability analysis, as well as the types of uncertainty

related to the structure and the deterministic model used, are presented. Next, the steps of a

reliability analysis are detailed and the methods used in OpenTURNS for uncertainty quantication,

uncertainty propagation and sensitivity analysis are described. Finally, statistical descriptions of

random variables intervening in RC structures are examined according to several studies in the

literature.

A civil engineering structure is a complex system composed of more or less simple elements behaving

in series or in parallel. This structure should provide a number of functions (mechanical strength,

durability, ...) with a certain level of reliability. In general, structural analysis and design are based

on deterministic methods and attempt to assure reliability by the application of safety factors.

However, due to many sources of uncertainties associated with loads, material properties, geometry

and modeling approach of structures, probabilistic concepts have to be used. Thus, structural

reliability analysis should be considered and functions are performed by not exceeding a certain

threshold level.

The aim of structural reliability analysis is to determine the probability that the structure does

not perform its functional requirements. Structural reliability theory is based on the probabilistic

modeling of the uncertainties and provides methods to estimate the probability of failure. The failure

of the structure is dened through the concept of limit state which represents a boundary between

satisfactory and unsatisfactory performance of the structure.

Sorensen [165] has provided the main steps in a reliability analysis:

2. Dening failure modes of the structure, i.e. exure, shear, torsion, fatigue, local damage.

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3. Decomposition of failure modes into series systems or parallel systems of single components,

in the case where failure modes consist of more than one component. A series system leads

to the failure of the entire system if one of the system components fails, while for a parallel

system no failure is induced as long as not all the system components fail.

by random variables and specifying the probability distribution types and statistical parame-

ters for these variables and the correlation between them.

5. Denition of limit state functions corresponding to each component in failure modes (ultimate

and serviceability limit states). Limit states are formulated in terms of the basic random

variables.

6. Estimating the probability of failure for each failure mode (simulation methods, approxima-

tions methods, response surface methods).

7. Performing sensitivity analyses in order to evaluate the sensitivity of the probability of failure

with respect to stochastic variables.

According to [153], the acceptable probabilities for structural failure are in order of 10−3 for service-

ability limit states and 10

−6 for ultimate limit states.

Sorensen [165] showed that the estimated probability of failure should be considered as a nominal

measure of reliability since the information used in reliability analyses are generally not complete.

Reliability analysis requires the knowledge of uncertainties related to the structure or system studied.

Nevertheless, before proceeding to reliability analysis and collecting information about uncertainties,

an engineer needs to know that engineering systems are exposed to several types of uncertainty and

each type requires a dierent process to be used in the evaluation of reliability. The uncertainties

in structures or systems can be classied into ve groups [165, 59], namely physical uncertainties,

measurement uncertainties, statistical uncertainties, model uncertainties and others.

Physical uncertainties are typically associated with loads, geometry of the structure and mechanical

properties of materials. A perfect knowledge of geometric dimensions is impossible since their

variability highly depends on errors in manufacturing and construction process. The uncertain aspect

of material properties is due to heterogeneity that describes the variability of material characteristics

from one point to another even in the same structural element. Loadings that can be classied as

permanent, unstable or accidental actions represent also an uncertain aspect in time since their

time history cannot be predicted in advance with certainty. Consequently, reliability analysis allows

representing these uncertainties by random variables and/or stochastic processes and elds of which

statistical parameters must be evaluated properly. In most cases, random variables dened by

probability distributions are sucient to model the uncertain quantities. A stochastic process can

be dened as a random function of time in a way that the value of the stochastic process is considered

as random variable for any given point in time. Stochastic elds are dened as a random function of

space and the value of the stochastic eld is a random variable for any given point in the structure.

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4.3.2 Measurement uncertainties

Measurement uncertainties are caused by inaccurate measurements of variable values and are at-

tributed to uncontrolled factors and imperfections in test conditions. Because of several uctuations

in the environment, test procedures, tools and so on, repeated measurements of the same physical

variable do not give the same value. In addition, resistance of actual physical structures includes

more uncertainties than structures tested under restrained laboratory conditions as they are sub-

jected to more uncertain environments that may be sometimes very severe. Thus, an engineer should

respond to this type of uncertainty by collecting a large number of observations, which may be lim-

ited to the availability of resources such as money and time, as well as to the scale of the structure

considered. The procedure of collecting observations can provide sucient information about the

measured quantity variability and may lead to a certain level of condence in the value used for

design.

Uncertainties in reliability analysis can also result from statistical parameters that are signicantly

dependent on sampling methods, variance estimation and choice of probability distribution type for

random variables. These parameters should be properly assessed and a condence interval should

be estimated and associated for each parameter. Statistical uncertainties are due to incomplete

statistical information owing to limited sample sizes of observed quantities. Statistical parameters

can be considered as random variables and are used to t an input data sample to a mathematical

function [163].

Model uncertainties are associated with structural idealizations of deterministic models used to

predict the physical behavior of structures and their response. Mechanical models are based on

several assumptions and can be performed using an empirical, analytical or numerical approach.

However, these assumptions considered in deterministic models can lead to inaccuracy results and

simple models do not take into account all possible factors that inuence the structural behavior.

Therefore, mechanical models introduce another type of uncertainty related to modeling error. They

are considered as an approximate prediction of the structural behavior since they always show some

dierence compared to actual response. The dierence between calculated and actual responses

can be used to develop a statistical description of the modeling error that can be introduced as an

additional random variable in reliability analysis.

Other sources of uncertainty that cannot be covered by reliability analysis can arise from the de-

nition of certain parameters, the performance of structures, human factors, i.e. the competence and

experience of builders and engineers.

The aim of deterministic models is to analyze structural behavior and predict the physical phenom-

ena observed by evaluating mechanical loads or stresses (S) and resistances (R). Nevertheless, a

validation of these models is necessary in order to reduce the dierence between the response of

the actual structure and that of the model. Input data of a calculation model are generally known

with only some signicant digits in spite of their uncertain aspect. By neglecting uncertainties the

calculated results may deviate signicantly from reality, hence they should be taken into account

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Figure 4.1: Limit state function [59]

during analysis. This can be accomplished by coupling the deterministic model with a probabilis-

tic model that allows modeling uncertainties and performing structural reliability analysis. Design

requirements become more complex by introducing uncertainties as the structure need to satisfy dif-

ferent criteria of failure, security and durability. The design should involve all relevant components

so that the structure does not reach its limit state. For example, a structure must be designed in

such a manner that its resistance is greater than the applied loads. In this case, the structure limit

state can be expressed with a single random variable as Z = R − S. Figure 4.1 illustrates a volume

integral of a two dimensional joint probability density function, where the two random variables are

the load and the resistance, and failure is dened as the event when the load exceeds the resistance.

Reliability approaches have the advantage of providing a realistic examination of uncertainties

and controlling their eect on structures response. However, their application shows some limita-

tions as they replace the reassuring notion of safety factor and conrm the acceptance of risks.

In addition, limitations also arise as a result of lack of probabilistic knowledge and reasoning in

engineering, lack of sucient data, the diculty of considering human errors and the novelty of de-

velopment of tools and software to analyze reliability problems. A reliability approach is based on a

statistical knowledge of basic random variables such as geometric properties, material characteristics

and actions. Each variable requires a coecient of variation (COV) and a probability distribution

dened by statistical parameters such as mean and standard deviation. Then by using probabilis-

tic methods, such as simulation or approximation methods, failure probability can be assessed by

studying all possible failure modes in order to make better decisions for an economical design. For

each mode, the equilibrium between loads and resistances is dened by a specic failure criterion

represented by a limit state function (g(X)) which is expressed in term of input random variables,

but also depends on the deterministic model and the threshold of the output variable considered. A

limit state function represents the state beyond which a structure can no longer perform the function

for which it was designed and separates safe domain Ds from failure domain Df , the latter corre-

sponds to the domain for which the limit state function in negative (g(X) ≤0 ) (Figure 4.1). Let's

rst consider the simplest case of Z = R − S, safe domain corresponds to values that are reasonable

and physically possible and acceptable, while failure domain corresponds to inadmissible values of

the variable Z.

In practice, probabilities are rarely known and need to be estimated, which represent the most

dicult part of reliability analysis. But before addressing this issue, uncertainties in structures

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Figure 4.2: PDF of Gauss, Lognormal, Uniform and Weibull distributions with same mean and

standard deviation [107]

or systems need to be quantied. This section describes how to model and quantify uncertainties

in OpenTURNS, as well as the essential parameters and characteristics to include in a reliability

analysis. Uncertainties are generally introduced in a probabilistic model as input variables assigned

with a random aspect. Thus, the mathematical modeling of random variables is an essential task

in any probabilistic formulation. A random variable and a particular realization of this variable are

denoted in this dissertation with an uppercase letter X and a lowercase letter x , respectively. Only

continuous random variables are considered in this study.

The probability density function (PDF), fX (x) , allows modeling the uncertainty of a random

variable. The cumulative distribution function (CDF), FX (x) , allow estimating directly the proba-

bility that a random variable is found to have a value less or equal to a specic value. For continuous

distributions, CDF represents the area under the PDF from minus innity to the value desired and

is given by:

ˆ x

P (X ≤ x) = FX (x) = fX (x)dx (4.1)

−∞

PDF is hence expressed as:

dFX (x)

fX (x) = (4.2)

dx

Figures 4.2 and 4.3, respectively, show the probability density and cumulative distribution func-

tions for four types of probability distributions of a random variable (Gaussian, Lognormal, Uniform

and Weibull).

To dene PDF, some parameters are needed to be identied, such as mean, variance, coecient

of variation, etc. The number of parameters depends upon the nature of uncertainty and the type

of distribution. Therefore, the mean µX of a continuous random variable X with PDF fX (x), also

known as the rst central moment, can be calculated as:

ˆ +∞

µX = x fX (x)dx (4.3)

−∞

The corresponding variance of X, also known as the second central moment and denoted as

Var (x ) , indicates how data are distributed about the mean and can be estimated as:

ˆ +∞

V ar(X) = (x − µX )2 fX (x)dx (4.4)

−∞

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Figure 4.3: CDF of Gauss, Lognormal, Uniform and Weibull distributions with same mean and

standard deviation [107]

The standard deviation, denoted as σX , is expressed in the same units as the mean value and

is calculated as the square root of the variance:

p

σX = V ar(X) (4.5)

as the ratio of the standard deviation to the mean. It is a nondimensional parameter that allows

indicating the degree of dispersion in the random variable. A high COV indicates a signicant

uncertainty and an important aspect of randomness in the variable. For a deterministic variable,

COV (X ) is equal to zero.

σX

COV (X) = (4.6)

µX

From a practical point of view and in order to correctly formulate a reliability problem, it is necessary

to model uncertainties jointly and to not limit the analysis to random variables modeled separately.

Therefore, a reliability analysis should involve the modeling of multiple random variables. This can

be accomplished by considering the joint probability density function fX (x) , which allows more

information to be provided. In case of multiple random variable, X represents the random vector

of input variables and x is a particular observation of this vector. Each variable in X is a random

variable that can be related to structure parameters such as geometry, material properties or loads.

fX (x) permits to calculate the probability that each of random variables exists in any particular

range of specic values of that variable and satises the following condition:

ˆ

fX (x1 , x2 , ..., xn )dx1 dx2 ...dxn = 1 (4.7)

Rn

In case of independent variables, the joint density is the product of the probability distributions

of variables. For two random variables, the joint probability density function can be described by a

three-dimensional plot as show in Figure 4.1.

Covariance and correlation analyses are generally used to study the dependence or independence

between two random variables in the aim of extracting as much information as possible for a reliability

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Figure 4.4: Correlation of two random variables

analysis. The covariance of two random variables X1 and X2 , COV (X1 , X2 ) , represents the second

moment about their respective means µ X1 and µ X2 , and indicates the degree of linear relationship

between X1 and X2 . COV (X1 , X2 ) can be calculated as:

´ +∞ ´ +∞

with µ X1 X2 = −∞ −∞ x1 x2 fX1 X2 (x1 , x2 )dx1 dx2 . Statistically independent variables have a

covariance equal to zero.

The correlation coecient, ρ X1 X2 , also represents the degree of linear dependence between two

random variables and its values range between -1 and +1. ρ X1 X2 can be expressed as a nondimen-

sional parameter by:

Cov(X1 , X2 )

ρ X1 X2 = (4.9)

σX1 σX2

Figure 4.4 depicts the physical properties of the correlation coecient. In Figure 4.4.a, the two

random variables are uncorrelated, the correlation coecient is expected to be zero and no linear

relationship exists between them. Figure 4.4.b indicates a non perfect linear relationship between

the two variables, with a correlation coecient value that is expected to range between 0 and 1. In

this case, the relationship between X1 and X2 is positive since X2 increases as X1 . However, in the

case of a nonlinear relationship between two random variables (Figure 4.4.c), ρ X1 X2 is equal to zero.

In the present study, the correlation between random variables is not taken into consideration

due to the lack of information provided on the relationship between the dierent parameters of the

structures considered.

The aim of a reliability analysis is to assess the probability of a variable of interest exceeding a

certain threshold. Variables of interest represent the output variables of the deterministic model

considered in the study. The propagation of uncertainties of input variables is performed through

the deterministic model towards output variables, hence these variables are aected by the random-

ness of the basic input variables and have also a random character. Therefore, a description of the

randomness of a variable of interest is essential to provide the necessary information on its uncer-

tainty. OpenTURNS allows evaluating the statistical characteristics of an output variable through

a dispersion analysis and a distribution analysis. In engineering applications concerning structural

reliability, a variable of interest can be related to forces (resistance of structure), deections, stresses,

strains, energies, etc. An output random variable is denoted in this dissertation as Y .

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4.4.2.1 Dispersion analysis

The purpose of a dispersion analysis is to characterize statistical parameters of one or more quantities

of interest. This involves determining the statistical moments (e.g. mean and standard deviation)

of the output variable, which depends on the accuracy desired and the method used. The quality

of this estimation depends on the output sample size, the amount of uncertainty present in the

corresponding variable and the condence of level required for the prediction. A condence interval

corresponds to a range of values in which the mean of a variable are located. Two dierent methods

can be used in OpenTURNS in order to estimate statistical moments of a variable of interest:

Monte Carlo method which is a numerical integration method using sampling. This method

consists in performing a sucient number of random simulations of the input vector X and

evaluating the value of Y for each simulation. Then, the mean and standard deviation of the

output sample can be estimated as:

N

1 X

µY = yi (4.10)

N

i=1

N

" #1/2

1 X

σY = (yi − µY )2 (4.11)

N

i=1

The condence interval that represents the mean estimation uncertainty permits to control the

dierence between the estimated and exact values. Its size decreases with the increase of the number

of simulations and the convergence of the estimators to exact values can be assured if the sample

size tends to innity. However, the only limitation of this method resides in cases where the number

of simulations is not suciently high, for example in case of a deterministic model with a signicant

computational eort. This can result in an important uncertainty for the estimations of µY and

σY , hence it is necessary to choose another propagation method, such as the quadratic combination

method, to estimate the central uncertainty of Y.

Quadratic combination method which enables to assess the central dispersion of an output vari-

able by using a Taylor decomposition of Y in the vicinity of the mean point (µX1 , µX2 , ..., µXn )

of random vector X. This method depends on the order of the Taylor decomposition. Open-

TURNS allows obtaining the estimations of µY and σY with a Taylor approximation at rst or

second order, which may lead to inaccurate results in highly nonlinear deterministic models.

However, quadratic combination method does not require any assumptions on the PDF type

of input variables, it only depends on their mean value and their dispersion.

The aim of a distribution analysis is to determine the PDF of output variables. The distribution of

a variable of interest can be determined using its values calculated by the physical model through a

sucient number of simulations. The determination of the distribution that ts an output sample

can be established graphically by plotting a histogram (Figure 4.5). A histogram can be developed

by arranging the values of Y in increasing order and subdividing them into several equal intervals

in each of which the number of observations is drawn. Then a distribution that ts the histogram

needs to be selected to model the behavior of the randomness. OpenTURNS proposes 17 parametric

distribution types to represent the uncertainty of a continuous variable, including normal, lognormal,

Gamma, Weibull, etc. The values of the distribution parameters are estimated in terms of the

output sample statistical moments. The probability density functions available in OpenTURNS

are listed in Appendix A with their formulas and their statistical parameters. Once the type of

distribution is selected and its parameters are evaluated, a statistical test of goodness-of-t must

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Figure 4.5: Representation of a sample of an output variable by a histogram [107]

be conducted to verify if the output sample and the probability distribution chosen are in good

agreement. OpenTURNS provides Chi-squared goodness-of-t test for discrete distributions and

Kolmogorov-Smirnov test for continuous distributions. Graphical statistical tests such as QQ-plot

can be also used to verify graphically if the assumed probability distribution is appropriate to

describe the output variable Y.

Kolmogorov-Smirnov test compares the observed CDF of the sample and that of an assumed

parametric distribution by estimating the maximum dierence between the two CDF as follows:

where FY (y) is the theoretical CDF of the assumed distribution and FYS (y) is the sample CDF.

Dn is a random variable and its CDF is related to a risk of error α as:

Dnα is the critical value of Kolmogorov-Smirnov test which is identied by default in OpenTURNS.

The assumed distribution is considered as accepted is the maximum dierence Dn is less or equal to

Dnα . OpenTURNS introduces the notion of p-value which is equal to the limit error probability

α

under which the assumed distribution is rejected (p-value= P (Dn ≤ Dn ) and compares it to the

p-value threshold α. Thus, the assumed distribution is accepted only if p-value is greater than the

value α (a value of 0.05 is used in this study).

QQ-plot is based on estimates of the quantiles. The principle is to estimate the α-quantiles

S

that correspond to the output sample qY (α) and the probability distribution chosen qY (α). Then,

S

the points (qY (α), qY (α)) are plotted and their positions according to the diagonal are veried.

Consequently, the assumed distribution is considered as acceptable if these points are close to the

diagonal.

In case where no parametric distribution t the output sample, OpenTURNS provides a non-

parametric statistical technique that allows representing the probability distribution of a random

variable graphically and without referring to any statistical model, namely the Kernel Smoothing

method. This method depends on the size of the sample.

The aim of structural reliability is to determine the probability that the structure does not perform

its functional requirements. This probability or failure probability Pf may be determined by the

integral (4.14) in term of the limit sate function:

ˆ

Pf = P [g(X) < 0] = fX (x)dx (4.14)

g(x)≤0

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In this equation fX (x) is the joint probability density function of the random variables X which

represent the uncertainties included in the reliability analysis and g(X) is the limit state function

which is related to a specic failure mode. The integration is performed over the failure domain,

for which g(X) is smaller or equal to zero. Integral 4.14 is considered as the fundamental equation

for reliability analysis, however it is not easy to calculate because of the complexity of the failure

domain Df and the joint probability density function fX (x), which may involve a large number

of variables. In addition, the estimation of failure probability becomes even more dicult when

complex deterministic models such as nite element models are considered. In order to assess failure

probability, rst, the limit state function should be dened for a special failure mode in term of

input random variables, then several probabilistic methods can be used to solve the integral 4.14:

Direct integration method allows estimating failure probability only in some particular cases

where joint density functions have simple shapes.

Numerical integration method may lead to signicant errors for very low probabilities and the

integration domain is not always bounded as well.

Probabilistic simulation methods, such as Monte Carlo (MC), Latin Hypercube Sampling

(LHS), Directional Simulation (DS) and Importance Sampling (IS), are based on a number of

samples of random variables corresponding to failure

Approximation methods are based on an approximation of the limit state function and an

identication of the most probable point that permits to estimate failure probability. The

limit state function is linearized in the First Order Reliability Method (FORM), while it is

approximated by a quadratic surface for the Second Order Reliability Method (SORM)

Response surface methods replace the initial deterministic model by an approximation response

surface expressed by a function whose values can be computed more easily.

The response and failure modes of an engineering structure depend on the type and the magnitude

of the applied load, as well as on the strength and stiness of the structure. Melchers [121] indicates

that limit states which describe the structural failure modes are generally divided in (Table 4.1):

1. Ultimate limit states which are mostly related to the loss of load-carrying capacity, e.g. exces-

sive plasticity, rupture due to fatigue, deterioration, etc.

2. Damage limit states which are related to loss of strength under repeated loads and can be

caused by excessive cracking or permanent inelastic deformation.

3. Serviceability limit states which are related to normal use of the structure and can be expressed

in terms of deections or vibrations.

The application of FORM and SORM rst requires the application of an isoprobabilistic transforma-

tion that aims to simplify the joint probability density function while failure probability retains its

estimated value. This transformation associates random variables in the physical space with stan-

dardized and independent random variables that dene the basic vectors of the standard normal

space.

The limit state function should also be expressed in the standard space and turns into H(U).

Then the integral of failure probability in equation 4.14 becomes:

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Table 4.1: Typical limit states for structures [121]

Ultimate (safety) Collapse of all Tipping or sliding, rupture, progressive

or part of structure collapse, plastic mechanism, instability,

corrosion, deterioration, re, fatigue

Damage Excessive or premature cracking, deformation

(often included in above) or permanent inelastic deformation

Serviceability Disruption of normal use Excessive deections, vibrations, local

damage, etc.

ˆ ˆ

Pf = fX (x)dx = fU (u)du (4.15)

g(x)≤0 H(u)≤0

where fU (u) is the density function of the distribution in the standard space, this distribution

is spherical and invariant by rotation. The Rosenblatt transformation [102] is the isoprobabilistic

transformation used by default by OpenTURNS when random variables are independent and can

only be used if the density function fX (x) of all random variables is well known.

The second step consists calculating the reliability index of Hasofer-Lind β [106] by solving the

following constrained optimization problem:

v

u n

uX

β = min(t u2i )H(ui )≤0 (4.16)

i=1

Geometrically, β is the distance of the design point from the origin of the standard space (Figure

4.6).

OpenTURNS uses the algorithm of Rackwitz-Fiessler [159] to solve this type of problem. This

algorithm is the most widely used in structural reliability as it can be simply formulated and gives

satisfactory results in practice. Resolving this problem allows assessing the value of β, as well as the

coordinates of the design point P∗ in the standard space. This point represents the most likely point

of failure, or in other words the point on the limit state surface nearest to the origin. Afterwards, the

limit state function is approximated by a Taylor expansion of rst order for FORM and of second

order for SORM around the design point.

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Figure 4.7: Approximation of limit state surface by FORM [75]

FORM and SORM require a reduced number of calls of the deterministic code and their com-

putational eort is independent of the magnitude of failure probability. They ensure a reasonable

accurate estimate of failure probability [50], while SORM improves the accuracy of result and gives a

very good estimate for large values of β [164]. FORM and SORM lose their accuracy when the limit

state surface is highly nonlinear in the vicinity of the design point (main curvatures at the failure

surface at the design point are large) [164], or when there may exist other important local optimum

points (the contribution of the global optimum point P∗ is not taken into account) [1]. In this case,

failure probability is only a function of the geometric properties of the limit state surface (the design

point position and the main curvatures). It is recommended not to use FORM and SORM in the

case of several failure criteria.

FORM FORM approximates the failure domain to the half-space dened by means of the hyper-

plane which is tangent to the limit state surface at the design point (Figure 4.7). This approximation

of rst order permits to calculate failure probability using equation (4.17), where Φ is the standard

normal cumulative distribution function (CDF).

Pf ≈ Φ(−β) (4.17)

SORM SORM consists in approximating the failure surface to a quadratic surface at the design

point, with the same main curvatures as the true failure surface at the design point (Figure 4.8). To

∂2f

calculate the main curvature κi , the Hessian matrix (Hij (f ) = ∂xi ∂xj ) should be determined since

the terms of second orders are considered. Failure probability can be calculated by the Breitung

formula (Equation 4.18) [26] which estimates the probability with an asymptotic analysis, in the

sense β→∞ with βκi xed.

n−1

Y

Pf ≈ Φ(−β) (1 + βκi )−1/2 (4.18)

i=1

This expression seems as the product of the approximation of FORM by a correction factor

expressed in terms of the principal curvatures at the most likely point of failure.

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Figure 4.8: Approximation of limit state surface by SORM [75]

Simulations methods are the most basic and simple methods to apply in order to assess failure

probability, they enable to evaluate that takes a limit state function for a representative sample of

input variables. A large number of samples are generated according to the probability density of

each random variable and the correlation coecients that link them. Simulation methods require

calculating the simulation coecient of variation in order to ensure their convergence. Results are

always expressed with a condence interval which allows examining if the number of simulations

or samples is sucient to obtain an accurate estimation of failure probability. Therefore, they

necessitate a signicant computational eort as they need a huge number of calls of the deterministic

model. Furthermore, they show a very slow convergence since the number of simulations should be

greater to the inverse of failure probability and the accuracy of results is proportional to samples

size. No particular assumptions are needed for the limit state function, hence simulation methods

remain valid in all cases even in cases where limit state function is very complicated and represents

high nonlinearities and curvatures.

Monte Carlo (MC) Despite its computational eort, Monte Carlo is the most eective and

widely known method to determine failure probability. This method represents a very powerful tool,

but also the most reliable technique for engineers for evaluating reliability of complicated structures

with only a basic knowledge of probability and statistics. It consists in estimating failure probability

in term of the number of simulation cases, among the total number of generated simulations, which

indicate failure (Figure 4.9). An estimate of failure probability by MC can be provided by the

expression:

N

1 X

Pf ≈ I(xi ) (4.19)

N

i=1

In this equation, N is the number of generated samples of the random vector X, I(xi ) describes

the indicator function so that I(xi ) = 1 if g(xi ) ≤ 0 and I(xi ) = 0 otherwise. The accuracy of the

estimate certainly depends on the number of simulations and the magnitude of failure probability.

For a small value of N the estimation of Pf may be subjected to a considerable uncertainty, thus

a large number of simulations may be required to reduce this uncertainty to an acceptable level. A

large number of samples is also required in case of low probabilities. Therefore the eciency of Monte

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Figure 4.9: Principle of Monte Carlo method, illustration in the standard space [46]

Carlo should be evaluated to study the error associated with the number N. OpenTURNS uses the

Central Limit Theorem in order to calculate the percentage error by means of a 95% condence

interval. Results provided by Monte Carlo indicate that the estimated value of Pf is very likely

(with a probability of at least 0.95) in the condence interval [Pf,inf , Pf,sup ].

The concept of Monte Carlo appears to be simple, however, its application in structural reliabil-

ity analysis where very low probabilities usually appear may be impossible for complex structures

that necessitate a huge computational eort during deterministic analyses. Therefore, and in order

to achieve eciency of simulation methods, the number of simulations needs to be greatly reduced

[79]. For this purpose, several variance-reduction techniques were developed to increase eciency

by reducing the variance or the error of the estimate of probability of exceeding a threshold. The

expected or mean value are not inuenced by these alternative techniques and the number of sim-

ulations required to assess failure probability decreases signicantly. The most commonly used

variance-reduction methods are importance sampling, directional simulation and Latin hypercube

sampling.

Importance sampling (IS) Importance sampling consists in generating samples that lead more

frequently to failure instead of extending them uniformly over all possible values of random variables.

The distribution of sampling points is concentrated in the region of most importance that mainly

contributes to failure probability (Figure 4.10). In order to obtain results in the desired region,

samples are simulated from a new probability density function hU (u) instead of the original joint

probability density function fX (x) and the probability of failure is given by:

N

1 X fX (ui )

Pf ≈ I(ui ) (4.20)

N hU (ui )

i=1

In this expression,

samples and centered at the design point, U is the random vector of samples based on the new

density function such that hU (u) > 0 in the failure domain [1].

Although the eciency of this method depends on the selection of the density hU (u) (the opti-

mum importance sampling density function gives the minimum variance, i.e. the variance of Pf is

equal to zero), importance sampling is generally recognized as the most ecient variance reduction

technique. It is very eective for estimating low failure probabilities and allows obtaining more

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Figure 4.10: Principle of importance sampling method, illustration in the standard space [46]

information on failure domain since samples are concentrated in the region of high density of failure

probability around the design point. Importance sampling produces accurate and satisfactory results

if the design point is correctly identied and reduces signicantly the number of simulations for an

appropriate choice of hU (u) (OpenTURNS uses a normal importance distribution, centered on the

standard design point). In addition, the number of samples aects the eciency of the sampling

method and sucient simulations in the failure region are needed. Indeed, the number of simulations

required to assess an accurate estimation of failure probability increases with the dimension of input

random vector. As the estimation of failure probability by importance sampling method depends

on the identication of the design point, this method shows the same disadvantages of FORM and

SORM related the nonlinearity of the limit state surface and the number of optimum points.

Directional simulation (DS) Directional simulation is also frequently used for the structural

reliability assessment. This method is more economical than Monte Carlo in term of computational

eort and requires no additional information. Directional simulation is based on the concept of

conditional probability and implicates the rotational symmetry of multi-normal probability density in

the standard space. The simulations are performed radially, hence random directions are generated in

the space instead of generating random simulation points and determining whether these simulations

are in failure or safe domain (Figure 4.11). Random directional vectors are generated according to a

uniform density distributed on the unit hypersphere and are expressed in polar coordinates. For each

random direction, the intersection of the direction and the limit state surface is determined iteratively

and the distance ρi from the origin to the limit state surface is searched. Failure probability is

evaluated in the conditional directional by a Chi-Square probability function with n degrees of

2

freedom (χn ) as follows:

N

1 X

1 − χ2n (ρi )2

Pf ≈ (4.21)

N

i=1

OpenTURNS requires the use of a numerical solver at every step of directional simulation method,

this solver necessitates a certain number of calls of the external deterministic code in addition to the

number of simulations required by directional simulation to assess failure probability. As a simulation

probabilistic method, directional simulation has the characteristic that failure probability converges

to an accurate value if the number of simulations increases. However, in the case of large number of

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Figure 4.11: Principle of directional simulation method, illustration in the standard space [46]

random variables, Monte Carlo requires less simulations than directional simulation to converge to

a solution of similar accuracy [74].

Latin hypercube sampling (LHS) Latin hypercube sampling is also another simulation based

technique to evaluate reliability. This method is based on a stratied sampling strategy, hence it

allows a better cover of the domain of input variables variation. The sampling procedure consists in

dividing the domain of each random variable into several intervals and assigning the same probability

to all intervals. The number of intervals depends on the sample size generated for each variable.

From each interval, a value is selected randomly with respect to the probability density in the

interval. A combination of random variable intervals permit to form the hypercube. In this method,

all the areas of the space are represented by input values, which results in a smoother sampling of

the probability distributions. The estimator of failure probability with Latin hypercube sampling

method is similar to that followed in Monte Carlo method and the governing is given in Eq.(4.19).

Latin hypercube sampling can also be combined with importance sampling method in the aim

of reducing the failure probability variance. In this case, simulations are generated uniformly in an

hypercube centered at the design point (Figure 4.12) and failure probability can be estimated by

the following equation:

N

V (p) X

Pf ≈ I(xi ) fX (ui ) (4.22)

N

i=1

OpenTURNS does not allow assessing failure probability by combining Latin hypercube sampling

with importance sampling, but also it requires input variables to be independent so that this method

could be applicable. Latin hypercube sampling methods can give comparable results to Monte Carlo,

but with a reduced number of calls of the deterministic model. It also improves the manner of

generating samples and enables a better exploration of the domain variations of input variables [1].

As all simulation probabilistic methods, a small number of simulations may not give accurate results,

while large sample size may not be convenient for deterministic models that are time consuming.

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Figure 4.12: Principle of Latin hypercube sampling method, illustration in the standard space [46]

Response surface methods are generally used in cases where the physical model requires a huge

computational eort in order to perform the reliability analysis. The principle of response surface

methods consists in replacing the deterministic model with an approximate polynomial model, called

response surface or meta-model. The approximate model is characterized and parametrized by a

certain number of coecients and its output values can be estimated more easily. Several techniques

are provided in OpenTURNS to choose the response surface type, namely the Taylor expansion,

the Least Square method and the polynomial chaos expansion. Once the type of response surface

is selected, its coecients are estimated through a nite number of simulations of the deterministic

model.

The Taylor expansion method allows replacing the initial model by an approximate model in

a restricted domain of the input variables. Its concept consists in performing a 1st or 2nd order

polynomial expansion of the deterministic model around a certain point. The approximation by

Taylor response surface is very useful to study central tendencies and determine the PDF of the

variable of interest. In this case, the expansion should be performed around the point of mean values

of input variables. However, if the aim is to assess the probability of exceedance of a threshold, the

Taylor expansion should be performed around the design point and its accuracy must be justied.

The Least Square method enables to replace the initial model limit state function by a linear

response surface of type:

g̃(X) = AX + B (4.23)

where A is the matrix of the coecients to be determined, and B is the constant term. The

coecients can be determined from a sample of output values obtained with the initial model and

calculated for dierent values of the input variables. The Least Square method allows determining

these coecients by minimizing the quadratic error between the output values of the initial model

and the approximate output values obtained by the response surface for the same input values.

The polynomial chaos expansion allows approximating the initial deterministic model by a re-

sponse surface considered as a projection of the physical model in a truncated Hilbertian orthonormal

basis:

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P

X −1

g̃(X) = αi ψi ◦ T (X) (4.24)

i=0

where T is an isoprobabilistic transformation which transforms the input random vector into

a reduced vector of independent components. In OpenTURNS, the Nataf transformation is used

if the input random vector has a Normal copula, while the Rosenblatt transformation is used for

all other cases [1]. ψi represents a multivariate polynomial basis in the Hilbertian space associated

with a particular polynomial family. OpenTURNS provides 4 polynomial family types, namely

Hermite, Legendre, Jacobi and Generalized Laguerre orthogonal polynomials. The polynomial chaos

is described by a degree P and a nite number of coecients αi that represent the unknown quantities

of the problem. These coecients are evaluated in OpenTURNS using least squares strategy through

a nite number of realizations of the input physical vector. The principle consists in selecting

the realizations according to a convenient experimental design and minimizing the square residual

between the initial model and meta-model responses.

In order to construct a basis of the meta-model with polynomial chaos expansion, three strategies

are implemented in OpenTURNS:

The Fixed strategy corresponds to a polynomial chaos basis with a specic number of terms

and depends on the degree of the polynomial chaos and the number of input random variables.

The Sequential strategy does not take into consideration the number of terms nor the degree of

the polynomial chaos. First, an initial basis is proposed and then updated iteratively depending

on a convergence criterion based on the dierence between the physical model response and

the polynomial chaos prediction.

The Cleaning strategy consists in generating an initial basis with a nite number of terms and

eliminating the non-ecient terms (i.e. the terms associated to coecients less than a given

value). Then, a new term is generated and the polynomial basis is updated according to the

new coecient. The procedure is repeated iteratively until only the most signicant terms are

considered.

Reliability results are sensitive to the used reliability analysis method [68, 70, 69]. Thus, probabilistic

method should be properly chosen for a particular problem. Two methodologies were proposed by

[68, 70, 69] in order to compare approximation and simulation probabilistic methods in term of their

eciency in assessing the same results of failure probability. The rst consisted in coupling the

probabilistic methods directly to the nite element model, while response surface method was used

to simplify the model. The approximated model was also coupled to the same probabilistic methods.

The inuence of random variables PDF was also investigated.

The physical problem examined in [68, 70, 69] is performed within the context of performance

based seismic engineering design. The nite element model is based on a nonlinear static pushover

analysis in [68, 70], while the problem is studied assuming a linear elastic behavior in [69]. Ghoul-

bzouri et al. [68] studied the reliability of two-story RC building structure taking the uncertainties

of three input variables into account. OpenSees was used to estimate failure probability using three

dierent types of probability distribution for random variables (lognormal, normal and gamma).

Goulbzouri et al. [70] developed their reliability analysis to the case of four-story RC building

structure. OpenSees was also used to assess failure probability and the same random variables were

considered and modeled using a lognormal distribution. Another reliability analysis four-story RC

building structure was also performed by Goulbzouri et al. [69] using FERUM and considering

15 random lognormal variables related to the structure geometry, material properties and loading

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conditions. The limit state function examined in the three references was related to the horizontal

displacement at the roof of the structure in order to fulll the requirements of the Moroccan seismic

code RPS 2000.

Goulbzouri et al. found that the direct coupling of probabilistic methods with the nite element

model does not predict the same results as the approximated response surface coupled to the same

probabilistic methods. This latter underestimates in general failure probability, but it allows per-

forming the reliability analysis with less computational eort compared to the direct coupling with

the nite element model. The computational eort of response surface method depends only on

the time needed to identify the approximated function. In addition, FORM does not give the same

results with respect to IS for both methodologies proposed. In general, FORM overestimates failure

probability when coupled with the nite element model, but shows good accuracy when coupled to

the approximated model. Goulbzouri et al. recommended the use of IS in the case of nite element

model, although it is hugely time consuming in comparison to FORM. Goulbzouri et al. indicated

also the importance of specifying appropriate probability density functions to model uncertainties

and that is not sucient to describe random variables only by mean values and COV. However, the

choice of PDF has a reduced eect when surface response based reliability analysis is performed. It

should be noted that these conclusions could not be generalized and should be used with attention

for other problems when assessing failure probability.

The objective of reliability sensitivity analysis is to identify input variables that mostly contribute

to the variability of the variable of interest and that of failure probability. Sensitivity analysis is

important to gain more information about the deterministic model and its probabilistic behavior,

especially for models with large number of input variables and those that represent a high nonlinear

response and several failure modes. Results from sensitivity analysis can be used to ameliorate

structural design in engineering, for examples the dimension of a design problem can be reduced

by determining the variables that have a negligible probabilistic eect, the probabilistic response

can be also improved by reducing the uncertainty in random variables. Several methods permit to

estimate the sensitivity of input random variables:

Quadratic combination method that is based on a Taylor expansion of the output variable

and evaluates the importance factors (IF) of input variables on the estimator of the output

variable mean:

∂g

IFxi = (µX ) (4.25)

∂xi

FORM permit to calculate direction cosines that describes the sensitivity of each standard

variable Ui on the reliability index around the design point.

∂β ∗

IFxi = (u ) (4.26)

∂ui

However, it is impossible to compare one factor to another since variables have dierent physical

units. Therefore, importance factors are multiplied by normalized factors in OpenTURNS, which

enables the comparison of these factors independently of the original units of the deterministic model

variables.

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4.6 Statistical descriptions of random variables

In order to study the reliability of RC structures, a statistical description of intervening random

variables must be provided. The basic information required is the probability distribution (PDF) of

each random variable and its statistical parameters, such as the mean and the coecient of variation

(COV). According to [91], the mean and COV of variables should be representative of values that

would be expected in actual structures in-situ. A sucient data are required in order to obtain

a reasonable estimate of the probability distribution. However, in other cases where the data do

not exist it is dicult to develop statistical description for variables. Hence, a literature review is

conducted in this section in order to examine statistical descriptions of random variables intervening

in RC structures according to several studies in the literature.

In practice, the strength of concrete in a structure may dier from its specied design strength

[123] and may not be uniform throughout the structure [91]. According to [91], the major sources

of variations in concrete strength are due to variations in material properties and proportions of

concrete mix, variations in mixing, transporting, placing and curing methods, variations in testing

procedures and the rate of loading, and nally the size eects.

Mirza et al. [123] have derived a relationship for the COV in-situ compressive strength as:

2

Vc = (Vccyl + 0.0084)1/2 (4.27)

where Vccyl is the COV of the cylinder tests. For average control Vccyl is about 0.15 and 0.12

for 20.68 MPA and 34.47 MPa concretes, respectively, and Vc can be taken as 0.18 and 0.15. The

concrete strength fc is assumed to follow a normal distribution and the COV of the concrete in-

situ compressive strength can be considered to vary between 0.15 and 0.18. These values were also

suggested by [57, 114, 115]

A reliability analysis was performed by Jonsson [91] on RC beams in bending and concrete

oshore structures which are exposed to extreme environmental load such as icebergs and waves.

The mean value of the compression strength considered in this study and presented in table 4.2 is

adjusted for the rate of loading.

In the paper of Val et al. [177], a probabilistic method for reliability evaluation of plane frame

structures with respect to ultimate limit states was proposed. The mean value of the compressive

strength was estimated by:

fcm = fc + 8 (4.28)

Val et al. assumed that the compressive strength has a normal distribution, thus the COV may

be obtained by:

8

COVfc = (4.29)

1.645fcm

Sun [168] has studied the eect of corrosion on the reliability of bridge girders. In this study and

according to [115], the concrete strength fc is assumed to follow a normal distribution with COV of

0.16.

Choi and Kwon [35] have used Monte Carlo simulation in order to assess the variability of

deections of RC beams and one-way slabs with known statistical data and probability. Their results

showed that the variability of deections can be high due to the random nature of the parameters

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related to the concrete and steel strength, as well as the structural dimensions. The parameters

were considered to be independent except for the modulus of elasticity and the tensile strength of

concrete which were considered correlated with the compressive strength of concrete. The values of

variables in this study were given in 'psi' for strengths and in 'in' for dimensions. The mean of the

concrete compressive strength was given by:

Braverman et al. [25] have performed a study to evaluate, in probabilistic terms, the eects of age-

related degradation on the structural performance of reinforced concrete members at nuclear power

plants. The paper focuses on degradation of reinforced concrete exural members (beams and slabs)

and shear walls due to the loss of steel reinforcing area and loss of concrete area (cracking/spalling).

As shown in Table 4.2, the mean value of the in-situ concrete strength and its COV considered in

this paper for RC slabs were based on [58, 115].

In order to combine the dynamic analysis of the structural slab subjected to blast loading with

its various aspects of uncertainties, a reliability study has been performed via the setting up of

displacement and strain performance functions [110]. This paper presents also results from a para-

metric investigation of the reliability of RC slabs under blast loading. The results showed that the

variation of Young's modulus of concrete has a largest inuence on the failure probability of the

slab, followed by the yield strength of steel bar and the crushing strength of concrete. The mean

values of random variables adopted in this study present the dynamically enhanced values with an

enhancement factor of 1.4 for the concrete compressive strength.

Low and Hao [111] have used two loosely coupled SDOF systems to model the exural and

direct shear responses of one-way RC slabs subjected to explosive loading. Incorporating the ef-

fects of random variables of the structural and blast loading properties, as well as the strain rate

eect caused by rapid load application, failure probabilities of the two failure modes were analyzed.

Failure probabilities of the two failure modes were evaluated by considering statistical variations of

material strengths, structural dimensions, and blast loading properties. Strain rate eect on mate-

rial strengths were also taken into account in the analysis. Therefore, the mean values of random

variables considered in this study present the dynamically enhanced values with an enhancement

factor of 2.3 for the concrete compressive strength.

The aim of the study of Firat and Yucemen [62] was to analyze the dierent sources of uncer-

tainties involved in the mechanical properties of materials and geometrical quantities of RC beams,

columns and shear walls in Turkey. They have showed that the in-situ compressive strength of

0

concrete (fck ) can be expressed in term of the compressive strength of the cylindrical specimens (fc )

(Eq. 4.31):

0

fck = Nfc .fc (4.31)

where Nfc is the overall bias in fc and is equal to 0.72. In this study, the COV of concrete

strength is found to be equal to 0.18.

Wisnieski et al. [182] have presented probabilistic models of ultimate shear and bending re-

sistance of RC and prestressed concrete bridge cross-sections. Statistical material and geometrical

parameters were dened according to [144, 145, 181]. All random variables were considered normally

distributed. Remaining parameters were considered as deterministic and their representative values

were taken from [4]. Wisnieski et al. have indicated that the value of COV of concrete compressive

strength for plant cast concrete (COV = 0.09) is smaller than the one for concrete cast in-situ (COV

= 0.12). In addition, they showed that the concrete compressive strength is correlated with the

concrete elasticity modulus and its tensile strength, with coecients of correlation C = 0.9 and C

= 0.7, respectively.

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Table 4.2: Summary on compressive strength of concrete according to various references

fc (M P a) fc (M P a)

Mirza et al. (1979) [123] 20.68 19.03 0.18 Normal -

27.58 23.37 0.18 Normal -

34.47 27.77 0.15 Normal -

Jonsson (1992) [91] - 27.77 0.18 Normal [123]

Val et al. (1997) [177] 30 38 0.13 Normal -

8

- fc + 8 1.645fcm - [30]

Braverman et al. (2001) [25] 27.58 24.49 0.16 Normal [58, 115]

Low and Hao (2001) [110] - 51.2 0.11 Normal [123]

Low and Hao (2002) [111] - 84.08 0.11 Normal [123]

Firat and Yucement (2008) [62] C14 to C30* 29.87 0.105 - -

Wisnieski et al. (2009) [182] 30 30 0.12 Normal [144, 145, 181]

fc (psi) fc (psi)

Choi and Kwon (2000) [35] - 0.675fck + 1100 0.176 Normal [123, 126, 139]

* TS Turkish standards: Requirements for design and construction of RC structures

According to [123], the COV of the in-situ tensile strength of concrete can be taken equal to 0.18

which is the value assumed for the compressive strength, and it was assumed that the tensile strength

follow a normal distribution.

Several equations exist in literature in order to calculate the tensile strength of concrete in term

of its compressive strength (Table 4.3).

According to [30], the concrete tensile strength may be estimated on the basis of the concrete

compressive strength using the following relationship:

fc 2/3 fc

0.95( ) ≤ ft ≤ 1.85(( )2/3 (4.32)

fco fco

where ft is the concrete tensile strength in MPa, fc is the concrete compressive strength in MPa

and fco is 10 MPa.

According to [177], the value of the tensile strength may be estimated from its relationship with

fc :

where αft was taken as an independent normal random variable and the COV of ft was obtained

from:

Choi and Kwon [35] have provided an equation of the tensile strength and the in-situ compressive

strength of concrete (the strengths are given in psi):

p

ft = 8.3 fck (4.35)

In addition, they have considered the tension stiening eect of concrete in their study in order

to include the eect of cracking in the analysis. The tension stiening eect of concrete can be

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Figure 4.13: Linear stepped model for tension stiening [35]

estimated by a tension stiening parameter, βt dened as βt = εtu /εti . This parameter was used

to characterize the post-peak of the tensile stress-strain relationship and was also considered as a

random variable in order to model the variability of post-tensile response of concrete, which was

represented as a linear reduction in the tensile stress as shown in Figure 4.13. This parameter was

also considered as a random variable in the study of [177] and it was assumed that it follows a normal

distribution with a mean equal to its nominal value and a COV equal to 0.15 since no available data

are available for this parameter.

A study on the cover failure in concrete structure following concrete deterioration has been

carried out by Choo et al. [37]. The purpose of this study was to analyze the relationship between

the range of the steel expansion and the crack creation and propagation. The tensile strength was

directly calculated from the compressive strength of concrete using the following equation:

ft = 0.18fc0.7 (4.36)

Pandher [153] has studied the reliability-based Partial safety factors for compressive strength of

concrete with partial replacement of cement by Fly ash having bers. Pandher has assumed that

the tensile strength of concrete is approximately one tenth of its compressive strength.

Although several equations are available in the literature to estimate the static modulus of elasticity

of concrete (Table 4.4).

In [11], the concrete elastic modulus was also estimated on the basis of the compression strength:

Ec = 33ρ1.5

c fc

0.5

(4.37)

where fc is the compressive strength (psi), Ec is the elastic modulus (psi), ρc is the weight density

of the concrete (lb/f t ).

3

Jonsson [91] has proposed to use the following approximate expression of the Young's modulus

in the cases where only the compressive strength of the concrete is known:

p

Ec = 5500 fc (4.38)

In the paper of Val et al. [177], the statistical variation of the modulus of elasticity of concrete

was expressed through the variability of the compressive strength of concrete and a coecient αE :

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Table 4.3: Summary on tensile strength of concrete according to various references

fc (M P a) ft (M P a)

Mirza et al. (1979) [123] 20.68 2.11 0.18 Normal -

27.58 2.34 0.18 Normal -

34.47 2.52 0.18 Normal -

Val et al. (1997) [177] - αft fc 2/3 Eq.(4.34) Normal [4]

30 αft = 0.3 COVαf = 0.15 Normal -

Braverman et al. (2001) [25] 27.58 2.46 0.18 Normal [58, 115]

Choo et al. (2008) [37] - 0.18fc0.7 - - -

Pandher (2008) [153] - fc /10 - - -

Wisnieski et al. (2009) [182] 30 (C = 0.7) 2.0 0.2 Normal [144, 145, 181]

50 (C = 0.7) 2.85 0.2 Normal

fc (psi) ft (psi)

√

Choi and Kwon (2000) [35] - 8.3 fck 0.218 Normal [123, 139]

stiening βt

Choi and Kwon (2000) [35] 3.0 0.11 Normal [139]

Val et al. (1997) [177] 0.6 0.15 Normal [30]

fc and αE were considered to be independent random variables and the COV of Ec was obtained

through the following relation:

p

Ec = 4.73 fc (4.41)

In order to understand the eects of their variability on the reliability of RC slabs subjected to

impact, the variability of the mechanical properties of reinforcing steel needs to be studied. According

to [126], the variations of yield strength and modulus of elasticity of steel may be caused by varying

rolling practices and quality control measures used by dierent manufacturers, as well as possible

variations in cross-sectional area, steel strength, and rate of loading.

Dierent researches have been carried out to evaluate the statistical parameters for yield strength of

reinforcement (Table 4.5). In [114], the probability distribution for the yield strength of steel bars

and stirrups was assumed to be lognormal. The study of Mirza and MacGregor [126] was based

on a sample that included 3947 bars taken from 13 sources. They have found that the probability

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Table 4.4: Summary on Young's modulus of concrete according to various references

fc (M P a) Ec (M P a)

Jonsson (1992) [91] 27.77 24530 0.1 Normal [123]

√

- 5500 fc - - [43]

Val et al. (1997) [177] 30 - 0.08-0.1 Normal [123]

- αE (0.1fc )1/3 Eq.(4.40) - [30]

30 αE = 2.15 × 104 COVαE = 0.05 Normal -

Braverman et al. (2001) [25] 27.58 26200 0.18 - [58, 115]

Low and Hao (2001) [110] 51.2 31200 0.1 Normal [123]

Low and Hao (2002) [111] 84.08 38500 0.1 Normal [123]

√

Choo et al. (2008) [37] - 4.73 fc (GP a) - - -

Wisnieski et al. (2009) [182] 30 33000 0.08 Normal [144, 145, 181]

50 37000 0.08 Normal

fc (psi) Ec (psi)

ACI (1992) [11] - 33ρ1.5

c √ fc0.5 - - -

Choi and Kwon (2000) [35] - 60400 fck 0.119 Normal [123, 139]

distribution of yield strength of steel could be represented by either a beta distribution or normal

distribution.

In [125], the stress-strain curve for reinforcing bars was assumed to be composed of three regions:

1. Region I, for which 0 < ε ≤ fy /Es , consists of a straight line from zero stress to the yield

stress

f = Es ε (4.42)

2. Region II, for which fy /Es < ε ≤ 0.01, consists of a straight horizontal line and a stress equal

to the yield stress

f = fy (4.43)

3. Region III, for which 0.01 < ε ≤ εu , consists of a parabola in term of the ultimate stress

1/2

ε − 0.01

f = fy + (fu − fy ) (4.44)

εu − 0.01

where fu , εu , Es and fy are the ultimate strength, ultimate strain, modulus of elasticity, and yield

strength of reinforcement.

The ultimate strain of reinforcing bars was assumed to be a deterministic variable and the

ultimate strength was assumed to be 1.55 times the yield strength of reinforcing bars. The probability

distribution of the yield strength reinforcing bars was assumed to follow a beta distribution with a

mean value of 460 MPa and a COV of 0.09.

The aim of the study of Bournonville et al. [23] was to evaluate the variability of the physical

and mechanical properties of reinforcing steel produced throughout the United States and Canada

and to develop expressions to represent the probability distribution functions for yield strength of

reinforcing bars. The representation of the statistical distribution of the yield strength was expressed

using a beta distribution which was found to provide a reasonably accurate description of the yield

strength distribution. The general form of the beta probability density function is:

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Table 4.5: Summary on yield and ultimate strengths of steel according to various references

fy (M P a) fy (M P a)

Mirza and MacGregor (1979) [125] 415 460 0.09 Beta [126]

27.58 23.37 0.18 Normal -

34.47 27.77 0.15 Normal -

Jonsson (1992) [91] - 445.34 0.093 LogNormal [129]

Braverman et al. (2001) [25] 415 455 0.1 LogNormal -

Bournonville et al. (2001) [23] - - - Beta -

Low and Hao (2001) [110] - 556 0.08 Normal [126]

Low and Hao (2002) [111] 415 - - Normal [126]

Nowak and Szeszen (2003) [145] - 595 0.08 Normal -

Firat and Yucement (2008) [62] - 0.9fy 0.09 - -

Wisnieski et al. (2009) [182] 500 500 0.05 Normal [144, 145, 181]

fy (M P a) fu (M P a)

Mirza and MacGregor (1979) [125] - 1.55fy - - [126]

Wisnieski et al. (2009) [182] 500 575 (C = 0.85) 0.05 Normal [144, 145, 181]

f − LB α U B − f β

P DF = Cx( ) x( ) (4.45)

D D

The variable f represents the yield strength value, and the variable D represents the dierence

between the values for the upper bound UB and the lower bound LB . The values of the parameters

presented in equation 4.45 were evaluated for dierent grades and sizes of reinforcement bars. For

A 615 Grade 75 reinforcement, the average of the probability distribution of yield strength was

estimated for all bar sizes, with:

C = 37335, α = 2.59, β = 1882.86, LB = 75000, U B = 4000000, D = 3925000.

Nowak and Szeszen [145] have focused on the yield strength distribution of steel reinforcing bars

through 416 samples of Grade 60 reinforcement. It was found that, regardless of the bar size, a

normal distribution provides a good representation for the yield strength of steel.

Wisnieski et al. [182] showed that the yield strength of steel is correlated with its ultimate

strength, with a coecient of correlation C = 0.85.

The modulus of elasticity of steel has been found to have a small dispersion and to be more or less

insensitive to the rate of loading or the bar size. But it was assumed to be a random variables in

some researches (Table 4.6). According to [124], the probabilistic distribution of the modulus of

elasticity for reinforcing bars can be considered as normal with a mean value equal to the specied

value and a coecient of variation of 0.033. In [125], the modulus of elasticity of reinforcing bars

was taken to be normally distributed with a mean value of 200000 MPa and a standard deviation

of 6600 MPa.

4.6.3 Dimensions

The variability of geometrical properties of RC slabs is generally a consequence of inaccuracies in

the construction process and the curing operation of concrete [62, 91]. Therefore, the dimensional

characteristics of RC members constructed at site are dierent from the values specied in the

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Table 4.6: Summary on elasticity modulus of steel according to various references

fy (M P a) Es (M P a)

Mirza and MacGregor (1979) [125] - 200000 0.033 Normal [126]

Mirza et et al. (1980) [124] - specied value 0.033 Normal -

Jonsson (1992) [91] 445.34 200000 0.033 Normal [129]

Ref. Nominal of Mean of COV PDF Main Ref.

fy (psi) Es (ksi)

Choi and Kwon (2000) [35] - 29200 0.024 Normal [126, 139]

design of the structure. Mirza and MacGregor [127] demonstrated that variations in dimensions

can signicantly aect the size and the strength of concrete members, and thus dimensions must be

considered as random variables. They also indicated that all suggestions for geometric properties

should be considered as preliminary since they are based on interpretation of available data. Most

researchers have indicated that the probability distributions for the geometric properties should be

taken as normal [124, 115] (Table 4.7). According to Hong et al. [82] and Sun [168], the COV

of dimensions decreases as the RC member size increases and for massive elements the eect of

uncertainty in dimensions is negligible. The paper of Udoeyo and Ugbem [176] focuses on a study

of dimensional variations of structural members measurements which were carried out on beams,

columns and slabs. Based on this study, normal distributions are recommended to represent the

probability distributions of all dimensions and slab thickness presents nominal values ranging from

150 to 180 mm and a coecient of variation between 0.0183 and 0.0689.

According to [91], the diameter of reinforcement can be considered one of the dimensional random

variables involved in evaluating reliability of RC structures and it was modeled with a lognormal

distribution with a mean equal to 1.0048 times the nominal diameter and a COV of 0.02. Mirza

and MacGregor [128] also considered a lognormal distribution for the diameter of bars with a mean

equal to 1.01 times the nominal diameter and a COV of 0.04.

Another dimensional variable which may be considered as a random variable is the cover of

concrete. In [25] , the mean of the cover was taken equal to 4.45 cm with a COV of 0.36.

4.7 Conclusion

The aim of structural reliability analysis is to determine the probability that the structure do not per-

form its functional requirements. A reliability approach is based on a statistical knowledge of basic

random variables such as geometric properties, material characteristics and actions. Failure proba-

bility can be assessed using probabilistic methods such as simulation and approximation methods, or

response surface methods allowing the approximation of the deterministic model with a polynomial

function. The rst order (FORM) and second order (SORM) approximation of limit state function

methods consist rst in transforming random variables of the physical space into standardized and

independent random variables that dene the basic vectors of the standard normal space. Next, the

most likely point of failure is identied which represents the point on the limit state surface nearest

to the origin in the standard space and also called the design point. Afterwards, the limit state

function is approximated by a Taylor expansion of rst order for FORM and of second order for

SORM around the design point. FORM approximates the failure domain to the half-space dened by

means of the tangent hyperplane to the limit state surface at the design point, and provides accurate

results in case of linear limit state functions in the standard space. SORM approximates the failure

surface to a quadratic surface at the design point and gives a very good estimate for large values

of β. Probabilistic simulation methods are commonly used to estimate failure probability. Random

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Table 4.7: Summary on dimensions of RC members according to various references

Mirza et al. (1980) [124] nominal - Normal -

MacGregor et al. (1983) [115] nominal - Normal -

Jonsson (1992) [91] beam width 0.013 Normal [129]

= 50 cm

beam width 0.005 Normal

= 13 cm

Udoeyo and Ugbem (1995) [176] 150 to 180 mm 0.0183 to 0.0689 - -

Low and Hao (2001) [110] 1x3x0.2 m 0.05 Normal [127]

Low and Hao (2002) [111] as designed 0.03 Normal [127]

Wisnieski et al. (2009) [182] Slab thickness 0.04 Normal [144, 145, 181]

= 25 cm

dA

Mirza and MacGregor (1982) [128] 1.01dA 0.04 LogNormal -

Jonsson (1992) [91] 1.0048dA 0.02 LogNormal -

35.34 mm 0.024 Normal [126]

c

Braverman et al. (2001) [25] 4.45 cm 0.36 - -

Jonsson (1992) [91] 4.85 0.087 Normal [129]

simulations are generated and the structure response is evaluated for each simulation. Monte Carlo

is the most simple simulation technique. Special methods, such as importance sampling, directional

simulation and Latin hypercube sampling are developed to optimize the strategies of sampling in the

aim of reducing the computational eort related to the number of calls of the external deterministic

model.

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Chapter 5

5.1 Introduction

A detailed step-by-step procedure for creating FE models of impacted slabs with Abaqus is described

in this chapter. Abaqus/CAE is used to visualize the FE model geometry and properties, as well

as to view results. However, Abaqus Python edition is used to create the model since a parameter-

based model is denitely required to perform reliability analyses that necessitate a written code

of the deterministic model to be coupled with probabilistic methods. The model Python script

includes variation of geometric and mesh sizes, material properties, number of steel bars and initial

conditions. Python script permit to easily take into consideration variations of the model geometric

parameters and to generate mesh in term of these variations which is not possible with an input le

(.inp).

A 3D nonlinear explicit nite element (FE) analysis was proven to be eective for transient and

dynamic impact analysis [138]. 3D models allow simulating the behavior of 3D structures more

realistically than 2D models, especially that nowadays, computational resources are not an issue

anymore. However, an engineer should keep in mind that complicated models are not necessarily

more trustworthy than simpler models since they may contain large numbers of errors. The aim

should not be to develop a model with an exact representation of the physical problem or the most

accurate model possible, but rather to develop the simplest model that provides an idealized repre-

sentation of the problem and though enables to represent the structural components and behaviors

with accuracy. The assumptions adopted during a FE model construction should be addressed cor-

rectly in order to verify and evaluate FE results quality. For this reason, valid answers should be

provided to this type of questions: Is the analysis static or dynamic? Do the material behaviors

include nonlinearities and plasticity? What are the main physical loadings applied? How to dene

boundary conditions? Which are the suitable FE elements to use? How to create a regular mesh?,

etc.

In general, the analysis of the response of a RC slab subjected to impact loading is complex

due to the many non-linearities involved. Dierent factors contribute to the nonlinear behavior of

the reinforced concrete such as the nonlinear stress-strain response, the damage due to crushing

and tensile failure, the eect of the loading rate, the interaction between the concrete and the

reinforcement. In order to accurately simulate the structure and obtain detailed information from

nite element approach, all these factors and contact algorithms must be correctly incorporated into

the nite element model. Section 5.2 is included as an illustrative example to show how to develop a

FEM of an impacted slab in Abaqus by performing a successive passage through its modules. This

section describes in detail the 3D FE model of Chen and May RC slabs [34] subjected to impact

using Abaqus v6.11. The analysis is performed with an explicit conguration that allows a better

representation of impact problems. In order to improve the modeling process and optimizing the

postprocessing of results, the model is parametrized through a code developed using the Python

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programming language. This code can easily be read using Abaqus/CAE and allows obtaining

a better consistency between the geometry and the quality of mesh. Python script enables to

systematically prioritize the accuracy of results in the regions of interest. The model developed is

then validated with the experimental results of Chen and May tests. The model adopted is used to

model RC slabs which are subjected to accidental dropped object impact during handling operations

within nuclear plant buildings. Then, a simplied analytical model is also used for these slabs. It

consists of two degrees of freedom mass-spring system which accounts for potential viscous damping.

A frequency decrease approach is used to describe the degradation of slab stiness.

5.2.1 Creating model

All input data should be specied in consistent units in Abaqus. In addition, selecting a coordinate

system to use during the FE analysis is of high importance in order to create the FE model and

interpret results. In this study, all variables used to simulate the FEM of impacted slabs are given

in the SI system of units and results are automatically displayed with SI units. Model assembly and

results are displayed according to the default global coordinate system in Abaqus. This coordinate

system consists in a right-handed Cartesian system with z-axis perpendicular to xy-plane. Positive

values of z-axis are oriented upward in the vertical direction. In this dissertation, the 1-axis, 2-axis

and 3-axis denote the x-axis, y-axis and z-axis, respectively. The origin (x1 , x2 , x3 ) of the global

coordinate system is located at the center of the lower surface of slabs.

Numerical simulations of RC structures require an accurate representation of the geometry of all

the structural components that constitute the feature of the structure. Every component may aect

the structural response and excessive simplications in geometry will obviously lead to inaccurate

results. Therefore, experience and engineering judgement are necessary to develop an appropriate

model for a specic type of problem. FE analysis allows considering complex geometries and a

physical representation of the FEM close to the actual structure. The complex mechanical behavior

of RC slabs under impact necessitates an adequate 3D simulation in terms of slab and impactor

characteristics.

In Abaqus, Part module is used to divide the structure geometry into multiple parts. Each part

denes a main component of the model and needed to be meshed separately. The Part module

permits to create deformable, discrete rigid, analytical rigid, or Eulerian parts. The simulation of

Chen and May slabs has 4 main components, namely the slab, the reinforcement, the impactor and

the supports (for more details about these components geometry, see section 2.3.1.2). Slab part

represents the slab concrete region surrounded by a steel frame and reinforcement part includes

longitudinal and transversal bars of one layer of reinforcement. Impactor part describes the striker

geometry used in experiments and support part represents the shape of cylinder components on

which the slab is supported.

The rst part created for Chen and May impacted slabs problem represents the slabs concrete region

surrounded by a steel frame. A three-dimensional, deformable solid part is created by sketching the

2D square geometry of dimension equal to the total width of slab and frame. The sketch is placed

on a square sheet of size equal to the approximate size specied to create the part. The sketcher

is dened by a certain number of units square depending on the approximate size. The sketch

created for slab is then extruded in the z-axis direction with a depth equal to the slab thickness.

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Figure 5.1: Part of slab

The geometries of slab concrete region and steel frame are continuous, hence they are merged in the

same part and not separated in two dierent parts in order to reduce the problem. The merging

process permits to facilitate the desired meshing technique and to avoid creating constraints to tie

the two parts together, which would results in reducing the computational cost of numerical analysis.

Therefore, the full geometry of slab is partitioned with many partitions to create the steel frame

around the concrete region, which allows the simplication of the meshing process leading to an

easier discretization and a higher quality of structured mesh.

The full geometry of slab represents a solid single cell. Partitions are performed by cutting the

cell with a plane that passes completely through the cell. The method used to dene the cutting

plane in the aim of creating the frame geometry is the PartitionCellByPlanePointNormal method.

In this method, partition is created by selecting a point on the cutting plane then selecting an edge

or datum axis that denes the normal to this plane. Figure 5.1 shows a three-dimensional view of

the part which is used to develop the typical geometry of Chen and May slabs and highlighted the

partition scheme adopted. The section in dark gray represents the steel frame, while the light gray

represents the concrete part of slabs. Details of the geometry and dimensions of Chen and May

slabs are illustrated in Figure 2.14. Slabs have two dierent geometries: 760 mm square slabs with

a thickness of 76 mm and 2320 mm square with a thickness of 150 mm. Steel frame that surround

the concrete region is considered with a width of 17.5 mm.

The geometry of reinforcement is considered identical to that of the experiments of Chen and May

(Figure 2.14). Reinforcement part includes longitudinal and transversal bars of one layer of rein-

forcement. It consists of 11 longitudinal and 11 transversal bars in the case of 760 mm square slabs,

and of 15 longitudinal and 15 transversal bars in the case of 2320 mm square slabs. The reinforce-

ment is modeled as a 3D deformable wire part using the wire option in Abaqus. The 2D sketch

is created by modeling a single transversal bar and a single longitudinal bar, then using the linear

pattern option available in Abaqus. The spacing of reinforcement bars is equal to 60 mm for 760

mm slabs and 142 for 2320 mm slabs.

Figure 5.2 shows a three-dimensional view of the modeling of longitudinal and transversal re-

inforcement in 760 mm square slabs. It should be noted that no shear reinforcement is present in

Chen and May slabs, otherwise an additional part should be considered to incorporate stirrups in

the FEM.

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Figure 5.2: Part of reinforcement

Impactors used in Chen and May experiments have a complex shape. They are composed of two

sections, the rst consists of the mass with which slabs are impacted while the second represents the

section of impactor that enters in interaction with RC slabs. Slabs were tested with two dierent

shapes of the interaction impactor section, a steel hemispherical section of 90 mm diameter was used

to test Slabs 1,2,4,5 and 6, while a steel cylindrical section of 100 mm diameter was used to test

Slab 3. Details of this section shape are presented in Figure 2.15.

In the FEM, impactors geometry is considered identical to that of the experiments of Chen and

May. A three-dimensional, deformable solid part is created by sketching the 2D impactor prole and

then revolving it by an angle of 360° about the y-axis of the impactor part coordinate system. The

mass and interaction sections of the impactor are considered in one part in order to avoid creating

constraints to tie the two sections together. However, the impactor is partitioned with a cutting

plane in order to distinguish both sections and facilitate the generation of mesh. The cutting plane

is dened with a point and a normal vector, and the partition scheme adopted for hemispherical

impactors is illustrated in Figure 5.3. The upper part represents the mass section is highlighted in

dark gray, while the bottom part represents the interaction section highlighted in light gray.

In Chen and May experiments, slabs were supported on 4 cylindrical supports (Figure 2.14.c).

Thus, in order to have a better representation of the actual structure and develop realistic structural

supports, these latter are developed in the FEM as rigid bodies having a cylindrical shape with a

diameter equal to the steel frame width and a thickness of 10 mm. The shape of rigid body does not

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Figure 5.4: Part of support

Table 5.1: Linear mechanical properties of concrete and steel materials for Chen and May slabs

Concrete 2500 35000 0.2

Steel 7850 200000 0.3

change during simulation. The following part models the geometry of a single cylindrical support as

a 3D deformable, discrete rigid part (Figure 5.4). A circle of 17.5 mm of diameter is created in the

sketcher, and then extruded in the z-axis with a depth equal to 10 mm. In Abaqus, a rigid body

must refer to a reference point that governs its motion.

Material properties greatly inuence the structural response of RC slabs and play an essential role

in dissipating impact energy. Thus, proper material models capable of representing both elastic

and plastic behaviors of materials are required for the nite element analysis. In Abaqus, Property

module is used to dene materials and sections, as well as to assign sections to each element of mesh

of the relevant part. Material properties can be modeled in Abaqus as isotropic, orthotropic, and

anisotropic. In the current study, materials are assumed to be homogeneous and isotropic having

the same properties in all directions. This results in only two independent quantities to identify

for each material, namely the Young's modulus and Poisson's ratio. In addition, density must be

dened for all materials used in any dynamic analysis where inertia eects are important.

Abaqus includes an extensively material library that can be used to model most engineering

materials, including metals, polymers, concrete, fabrics and composites [7]. It also allows creating

new material models through subroutines. For concrete, Abaqus provides three dierent constitutive

models, namely the smeared crack concrete model, the brittle cracking model and the concrete

damaged plasticity model. Among these crack models, the concrete damaged plasticity model is

adopted in the present study as it is the only model that can be used in both Abaqus/Standard and

Abaqus/Explicit, which make it very useful for the analysis of RC structures under both static and

dynamic loadings. Furthermore, CDP model is the only model that allows representing the complete

inelastic behavior of concrete in both tension and compression including damage characteristics. The

classical metal plasticity model available in Abaqus is used to dene the elastic-plastic behavior of

steel, which permit to express the steel stress as a tabular function of plastic strain.

Table 5.1 shows the linear mechanical properties of concrete and steel materials used in the FE

model for all Chen and May slabs, while Table 5.2 shows their nonlinear mechanical properties that

are used to dene the stress-strain curves for each slab materials.

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Table 5.2: Nonlinear mechanical properties of concrete and steel materials for Chen and May slabs

fc (MPa) ft (MPa) fy (MPa)

2 50 4.06 560

3 50 4.06 560

4 50 4.06 560

5 37 2.93 1035

6 45 2.93 1035

The fundamental constitutive parameters for CDP model consists of four parameters which identify

the shape of the ow potential surface and the yield surface in the three-dimensional space of stresses,

namely ψ , , Kc and the ratio fb /fc . A biaxial failure in plane state of stress and a triaxial test of

concrete are necessary to identify these four parameters, while a uniaxial compression and uniaxial

tension tests are needed to be carried out to describe the evolution of the stress-strain curves of

concrete (the hardening and the softening rule in tension and compression).

represents the angle of inclination of the failure surface towards the hydrostatic axis, measured in

the meridional plane (Figure 3.2). Jankowiak and Lodygowski [87] supposed that ψ ranges from

34° to 42°, while Kmiecik and Kaminski [94] indicated that a value of 40° is usually assumed in

simulations.

In the CDP model the plastic potential surface in the meridional plane assumes the form of a

hyperbola (Figure 3.2). The ow potential eccentricity is a small positive value which expresses the

rate of approach of the plastic potential hyperbola to its asymptote. In ABAQUS user's manual [7],

it is recommended to assume = 0.1, which implies that the material has almost the same dilation

angle over a wide range of conning pressure stress values. Increasing the value of provides

more curvature to the ow potential, implying that the dilation angle increases more rapidly as the

conning pressure decreases. For = 0, the surface in the meridional plane becomes a straight line.

Yield surface parameters Experimental results for biaxial loading on concrete reported by

Kupfer et al. [100] are the most reliable to determine the ratio of biaxial strength to compression

strength (fb /fc ) which represents the point in which the concrete undergoes failure under biaxial

compression. Kupfer et al. [100] found that this ratio ranges from 1.10 to 1.20, while Lubliner et al.

[113] reported a range of 1.10 to 1.16. Jankowiak and Lodygowski [87] indicated that fb /fc ratio is

sensitive to the change of parameters ψ and . ABAQUS user's manual [7] species a default value

of 1.16.

Physically, Kc is interpreted as a ratio of the distances between the hydrostatic axis and respec-

tively the compression meridian and the tension meridian in the deviatoric cross section (Figure

3.3). Typical values range for Kc is between 0.64 and 0.8 [113]. According to experimental results,

Majewski [116] indicates that the value of Kc increases slowly with the decrease of mean stress and

that it is equal to 0.6 for mean normal stress equal to zero. ABAQUS user's manual [7] recommends

to assume Kc = 2/3. This ratio must always higher than 0.5 and when the value of 1 is adopted,

the deviatoric cross section of the failure surface becomes a circle (as in the classic DruckerPrager

strength hypothesis).

Table 5.3 summarizes the parameters of yield surface and ow potential adopted in this study

according to several research previously mentioned.

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Table 5.3: Parameters of the damaged plasticity model

Value 38 0.1 1.12 0.0666 0.0

C1 Desayi and Kirshnan curve [49]

C2 Madrid parabola [31]

C3 Eurocode 2 curve [4]

C4 Pavlovic et al. curve [154]

C5 Chinese code curve [3]

C6 Wang and Hsu curve [180]

C7 Majewski curve [116]

C8 Kratzig and Polling curve [99]

C9 Wahalathantri curve [178]

Compression behavior The CDP model has the potential to develop complete stress-strain

curves of concrete for compression and tension separately based on experiment results. In this

study, no experimental results are available to perform an analysis of the stress-strain curve for

the concrete. Therefore, the expressions considered to describe the stress-strain curve are based on

several studies in literature (see section 2.4.1) with some modications in order to t the curve with

the behavior of CDP model.

The stress-strain relations proposed by [3, 31, 49, 154, 178, 180] represent parabolic stress-strain

relations for both the ascending and descending curve (see 2.4.1). They show that the nonlinear

elastic behavior of concrete occurs almost from the beginning of the compression process. The CDP

model requires that the elastic behavior of the material to be linear, therefore the nonlinear elastic

behavior in the initial stage of these curves was neglected and replaced in this study by a linear

branch. A linear elasticity limit can be assumed as 0.4fc according to Eurocode 2 [4]. Beyond this

limit, the strain-hardening and strain-softening regimes keep their parabolic shapes according to the

relations proposed by [3, 31, 49, 154, 178, 180]. The stress-strain relations proposed by [4, 99, 116]

can directly be used in the CDP model since their elastic behavior is linear.

The evolution of the scalar damage variable for compression is determined using equation 3.17.

Compressive stress and damage of concrete is included in Abaqus as tabular functions of the crushing

strain. Table 5.4 summarizes the compressive concrete behaviors considered for the CDP model in

this study. The rst column contains the symbol attributed to each compressive stress-strain curve.

Tensile behavior The choice of tension stiening parameters is important since too little tension

will introduce unstable behavior in the overall response of the model and it will be dicult to

obtain numerical solutions. In order to model the behavior of concrete in tension in the FEM,

several expressions for stress-strain curve, stress-crack opening curve and fracture energy are adopted

from the literature (see section 2.4.2.1). Expressions of tensile stress-strain curves developed by

[3, 99, 180, 178] can be directly used in the CDP model since their initial elastic behavior is linear.

Tensile stress is included in Abaqus as tabular functions of the cracking strain.

As previously discussed in section 3.6.2.3, the tension stiening can be specied by a fracture

energy cracking criterion by directly specifying the fracture energy Gf as a material property. Thus,

tensile behavior of concrete is also modeled with stress-crack opening curves. Three types of concrete

stress-crack opening curve are used with linear, bilinear and trilinear descending softening branches,

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Table 5.5: Tensile concrete behaviors for CDP model

Stress-strain curve

T1 Wang and Hsu curve (n=1) [180]

T2 Wang and Hsu curve (n=0.5) [180]

T3 Wang and Hsu curve (n=1.5) [180]

T4 Chinese code curve [3]

T5 Wahalathantri curve [178]

T6 Kratzig and Polling curve [99]

Stress-crack opening curve

D1 Hillerborg et al. curve [81]

D2 Haidong curve [78]

D3 Tajima curve [170]

Fracture energy

G1 Hordijk energy [83]

G2 CEB-FIP energy [2]

G3 Oh-Oka et al. energy [146]

G4 Bazant and Oh energy [15]

respectively (Figure 2.38). Finally, the tensile behavior of concrete is modeled by considering only the

fracture energy as material property. According to ABAQUS user's manual [7], typical values of Gf

range from 40 N/m for a typical construction concrete (with a compressive strength of approximately

20 MPa) to 120 N/m for a high-strength concrete (with a compressive strength of approximately 40

MPa). The relations proposed by [2, 15, 83, 146] are adopted in this study to estimate the fracture

energy.

The evolution of the scalar damage variable for tension is determined using equation 3.16. Dam-

age of concrete are included in Abaqus as tabular functions of the cracking strain. Table 5.5 sum-

marizes the tensile concrete behaviors considered for the CDP model in this study. The rst column

contains the symbol attributed to each tensile description of concrete.

Idealizations are usually used to model the reinforcing steel for a numerical analysis. In this study,

ve stress-strain curves are proposed. The rst assumes a linear elastic, linear plastic steel behavior

with hardening and is based on the idealization proposed [173]. The parameters of this curves

related to ultimate strength, as well as yield and ultimate strains are presented in section 2.5.2. For

other curves, the values of yield strain and ultimate strain of steel are considered from the literature

with εy = 0.25[119] and εu = 0.4 [171]. The idealization proposed by [171] models a linear elastic,

perfectly plastic material with a yield plateau of stress equal to the yield strength of steel. The

stress-strain curve proposed by [16] represents the average behavior of steel embedded in concrete

with a bilinear stress-strain curve taking hardening of steel into considerations. For more details,

see section 2.5.2.

In Abaqus, stress-strain curve of steel is included as tabular function of the plastic strain. Table

5.6 summarizes the stress-strain curves considered for the reinforcing steel in this study. The rst

column contains the symbol attributed to each stress-strain curve of steel.

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Table 5.6: Idealized stress-strain curves for reinforcing steel

S1 Theodorakopoulos and Swamy curve [173]

S2 Taqieddin curve [171]

S3 Kratzig and Polling curve [28]

S4 Cao et al. curve [99]

S5 Belarbi and Hsu curve [16]

Dening sections is an essential step to create nite element models in Abaqus. A denition of section

requires information about properties of a part or a region of a part, such as the cross-sectional area

and the material type to be assigned to this region. Each section is created independently and must

refer to a material name. However, a single section can be assigned to several regions or parts as

necessary and a single material can be referred to as many sections as necessary. Once sections are

created, section properties can be then assigned to the homogeneous set that contains the region

or part in question. As a result, the properties of the material referring to a particular section are

also assigned to the region or part considered, as well as to all relevant instances in the assembly.

Assigning sections to instances is automatically performed in Abaqus and elements related to those

instances have the same section properties specied. Property module in Abaqus allows several types

of sections, including homogeneous solid sections, shell sections, beam sections and truss sections.

A homogeneous solid section is the simplest section type that can be dened since the only

information required in this case are a material reference and the section thickness for 2D regions.

Truss sections are used to model slender 2D or 3D structures that provide axial strength but no

bending stiness. Thus, a truss section is specied and assigned to reinforcement part. It consists

of steel material and a cross-sectional area equal to one steel bar area. Slabs 2 and 3 are reinforced

with steel bars of 6 mm diameter, Slab 4 is reinforced with steel bars of 8 mm diameter, while Slabs

5 and 6 are reinforced with steel bars of 12 mm diameter.

3 other sections are created to model Chen and May slabs under impact. A homogeneous solid

section that refers to concrete material is assigned to the concrete part of slabs highlighted with

light gray in Figure 5.1. A second homogeneous solid section is also created and assigned to the steel

frame surrounding the concrete region of slabs (section in dark gray in Figure 5.1). This section

refers to steel material and is also used to describe properties of the bottom section of the impactor

part (section in light gray in Figure 5.3). Due to a lack of information, impactor and frame materials

are assumed to have the same mechanical properties and behavior as steel reinforcement. The third

homogeneous solid section created refers also to steel material and is assigned to the upper section

of the impactor part (section in dark gray in Figure 5.3), but the density used for steel material

of this section is magnied for each slab case in order to obtain the value of mass with which the

impactor hits the corresponding slab (Table 2.1).

In the case of rigid bodies, no section can be dened. They are simulated with point mass and

rotary inertia property features available in Abaqus. A mass must be assigned to a discrete rigid

part that refers to a reference point and whose motion is governed by the motion applied to this

point.

Assembly module is used in order to create the nal geometry of the FEM by dening instances of

parts which are independent of each other. Each part created in Abaqus is oriented in its own local

coordinate system independently of other parts of the model. When an instance of a part is created,

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Figure 5.5: Reinforcement in the model assembly

Abaqus positions the instance in a way that the origin of the part sketch corresponds to the origin of

the default coordinate system of the assembly. Thus, instances should be positioned relative to each

other in the global coordinate system in order to properly assemble the model. Multiple instances

can be created for a single part and the model contains only one assembly despite the fact that it

may involve several parts.

For Chen and May slabs problem, 8 instances are dened. An instance refers to the part of

slabs, the center of the lower surface of slabs corresponds to the origin of the global coordinate

system. As the slabs are reinforced with an upper and bottom layers of reinforcement, 2 instances

are created using the reinforcement part. The rst describes the bottom layer of reinforcement and

is translated with a distance equal to the concrete cover, while the second represents the upper layer

and is translated with a distance equal to e − 2c (e is the slab thickness and c is the concrete cover)

(Figure 5.5). Concrete cover is equal to 12 mm in the case of 760 mm square slabs, and 15 mm in the

case of 2320 mm square slabs (Figure 2.14). Thereafter, 4 instances are dened in the assembly in

order to create the cylindrical supports on which the slab part is supported. Each support instance

is translated and positioned at a corner of intersection of the steel frame with the concrete region

of slab part. Finally, a rotation of 90° around the x-axis then a translation of a distance equal to

e + 0.001 are performed in order to position the impactor instance in the assembly. The impactor

instance is positioned in the way that the y-axis of the local coordinate system of the impactor part

coincides with the z-axis of the global coordinate system of the assembly. This position corresponds

to Chen and May experiments and allows applying the appropriate initial conditions.

The nal model of Slab 2 is displayed in a three-dimensional view (Figure 5.6). The triad in the

lower-left corner of the gure indicates the orientation of the global coordinate system, the origin of

this system is located at the center of the lower surface of slabs.

Generating meshes in Abaqus is performed in the Mesh module that provides a variety of tools

and meshing techniques to simulate structures of dierent geometries. Meshing techniques are

used to control mesh characteristics and automatically generate a mesh that meets the need of a

particular structure analysis. Free, structured and swept techniques are available in Abaqus with the

possibility of using the medial axis or advancing front meshing algorithm. The automated process of

these techniques results in a mesh that exactly conforms to the original geometry of the structure.

Partition toolset can be used in the Mesh module to create partition for structures of complex

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Figure 5.6: Assembly of the model after creating instances

geometries in the aim of dividing the model into regions with simpler geometries. Seeding tools are

used to specify the mesh density for a part or a selected region of a part by creating seeds along its

edges. In Mesh module, the element type to be assigned to the mesh must be chosen by specifying

the element family, geometric order and shape. It should be noted that if an instance part is created

as independent, the mesh must be performed in the Assembly module. However, the mesh must be

operated in Part module for any dependent instance part.

In the current study of impacted RC slabs, A 3D solid element, the eight-node continuum element

(C3D8R) is used to develop the mesh of the concrete slab and the steel frame, as well as to create

the mesh of the impactor. The reinforcement is modeled with two nodes linear 3D truss elements

(T3D2), while the rigid parts of supports are meshed with four-node tetrahedral elements (R3D4)

using free meshing technique with the medial axis algorithm (Figure 5.7). Finite elements size of

reinforcement SizeElReinf is considered equal to 0.01 m, and cylindrical supports are modeled

with a seed equal to 0.0025 m.

5.2.6.1 Slab

4 dierent types of mesh are used to model the slab part (Mesh1, Mesh2, Mesh3, Mesh4) and

then compared in the aim of choosing the appropriate mesh that gives the most similar results to

those of experiments. For all mesh types, the steel frame is modeled with 3 elements in width.

Dierent partition schemes are considered in order to emphasize on the impact zone and allow a

better transformation of energy and force from the impactor to the slab.

Mesh1 Thus, the rst mesh consists of several partitions that permit to mesh the concrete region

of slab into three dierent densities of mesh (Figure 5.8). Partitions are performed in 3D by cutting

the solid slab cell with a plane that passes completely through the cell. The sweep meshing technique

with the medial axis algorithm are used to generate the mesh of the slab. The region at the center

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Figure 5.8: Partitions used to create Mesh1 and Mesh2

Figure 5.9: Mesh1 and Mesh2 of slab in xy-plane and in the direction of thickness

represents the zone where the impact loading is applied, it has a dimension equal to half of the slab

width and is modeled with ner mesh in order to accurately capture the contact force. Dimensions

in width and length of elements at this region are governed by the number of elements ne in the

thickness direction, and they are equal to e/2ne (e is the slab thickness). For Mesh1, elements along

the thickness of the slab are assumed to have the same size of e/ne (Figure 5.9). The corner regions

of slab have a coarse density mesh with elements of an aspect ratio of 1 and a size of e/ne. For Slab

2 and ne = 5, elements at the impact region have dimensions of 7.6x7.6x15.2 mm, while those at

corner regions have dimensions of 15.2x15.2x15.2 mm.

Mesh2 Mesh2 is the same as Mesh1 with only one dierence related to the size of elements in the

thickness direction. For Mesh2, elements along the thickness of the slab have dierent size, but a

bias command is used in order to have ner mesh at the upper surface of the slab. This surface is

where the impact loading is applied, hence it is preferred to be modeled with ne mesh allowing a

better representation of the impact zone in the FEM and more accurate results. Consequently, the

thickness of the slab is modeled with ne elements and a bias ratio of 2 (Figure 5.9).

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Figure 5.10: Partitions used to create Mesh3

Mesh3 For Mesh3, the slab is partitioned with 2 cutting planes in x- and y- direction, respectively

(Figure 5.10). Then, the edges of the concrete region of slab as well as those of the steel frame

are meshed using the bias command with a maximum element size of 20 mm at the slab corner

and a minimum element size of 1 mm at the center of the slab. The basic aim of this mesh type

is to concentrate the ne mesh density at the impact point since the region that is highly aected

during an impact analysis represents the contact zone of the impactor and the slab. For Mesh3,

the thickness is also meshed with ne elements and a bias ratio of 2 (Figure 5.11) and the mesh is

generated using the structured meshing.

Mesh4 The concrete region of slab and the steel frame are partitioned in 3D with 2 cutting planes

in x-direction and 2 cutting planes in y-direction in order to create an square that refers to the

impact zone. Next, this square is partitioned in both diagonal directions to be able to create ner

mesh at the impact point (Figure 5.12). The diagonals of the impact zone are meshed using the bias

command with a maximum element size SizeElImpactZoneM ax of 10 mm at the square corner and

a minimum element size SizeElImpactZoneM in of 1 mm at the center of the slab. The concrete

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Figure 5.12: Partitions used to create Mesh4

region around the impact square, as well as the steel frame, are meshed with elements having the

same size along all edges. The size of elements of slab edges SizeElSlab are taken equal to 10 mm,

which permits to have an uniform mesh in xy-plane for all the slab except the impact zone. A zoom

view of the impact square mesh with biased seeding along its diagonals is illustrated in Figure 5.13,

this mesh allows having ne mesh density only in the impact zone where the loading is applied. For

Mesh4, the thickness is also meshed with ne elements and a bias ratio of 2 (Figure 5.13) and the

mesh is generated using the sweep meshing technique with the medial axis algorithm.

The Python script written for Mesh4 allows changing the size of the impact square by only

specifying the desired dimension dimpact , then the mesh can be automatically generated according

to the seeds provided for slab and impact square edges by running the script in Abaqus/CAE (Figure

5.14). Another important advantage of Python script written for Mesh4 is that it allows studying

the eect of impact position by only specifying the coordinates of impact point in x-direction ximpact

and in y-direction yimpact . The mesh can be then automatically generated according to the impact

square by running the script in Abaqus/CAE, and the impactor will be always located above the

center of the impact square. (Figure 5.15). The script enables to perform the impact analysis of RC

slabs at any point on the slab.

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Figure 5.14: Mesh4 with dierent sizes of impact region

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Figure 5.16: The impactor mesh used with Mesh1, Mesh2 and Mesh3 of slab

5.2.6.2 Impactor

The mesh of impactor is generated using the sweep technique with the medial axis algorithm and a

constant seed of 10 mm considered along all the impactor edges (Figure 5.16). The following mesh

type is used for the impactor in cases where Mesh1, Mesh2 or Mesh3 are used to model the slab.

A second mesh type is considered for the impactor and used in the model with Mesh4 of the slab.

In this case and in order to allow a better contact between the slab and the impactor, more partitions

are needed. First, the impactor must be partitioned with a cutting plane passing through the region

of intersection of its cylindrical and hemispherical parts. Next, the hemispherical part is partitioned

to create a ner mesh at the center of the impactor surface that enters in contact with the slab.

The mesh of the hemispherical part is generated using the structured meshing technique, while the

cylindrical part is meshed using the sweep meshing technique with the medial axis algorithm. The

size of elements in the cylindrical part is considered the same as that in the previous mesh, while

the hemispherical part is meshed using the bias command with a maximum element size of 10 mm

and a minimum element size of 1 mm.

Abaqus can be used to solve dierent types of problems such as static, dynamic, quasi-static, seismic,

etc. Analyses are congured in Abaqus through one or multiple steps to describe the various loading

conditions of the problem. Steps are created using the Step module and two types of analysis steps

can be dened in Abaqus, namely general response analysis steps and linear perturbation steps.

General analysis steps can be used to analyze the overall structural response that can be linear or

nonlinear. Linear perturbation steps are used to calculate the linear perturbation response of the

problem such as extraction of natural frequency, transient modal analysis and response spectrum

analysis.

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In order to determine the dynamic response of RC slabs under low velocity impact of dropped

objects, only one dynamic/explicit step is created for the simulation as the problem consists of a

single event. In this step, initial conditions are applied to the impactor and the time period is

set to 0.03 s with a maximum time increment of 1e-6. The current analysis consists of two steps,

the impact step and the initial step generated automatically in Abaqus and in which the boundary

conditions are applied.

As previously mentioned in section 3.4, Abaqus provides several ways to introduce loads, boundary

conditions and initial conditions in nite element analysis. Specifying boundary and initial conditions

and applying load conditions to the structure is step-dependent, which means that the step or steps

at which these conditions become active should specied in the FEM.

To simulate the motion of the dropped object, the impactor is dropped at the center of Slab 2

and Slab 3 with a velocity of 6.5 m/s, while the impactor hits Slab 4 with an initial velocity of 8.0

m/s. Slab 5 and Slab 6 are subjected to an impact velocity of 8.7 m/s and 8.3 m/s, respectively.

Velocity values are considered as in experimental tests of Chen and May (Table 2.1). In the nite

element model of each slab, the impactor hits the slab with a constant velocity that is applied to

each of the nodes of the impactor in the direction perpendicular to the slabs and opposite to z-axis

direction. In this study, the eect of impact loading of dropped objects on the behavior of RC slabs

using a nite element analysis is examined in the presence of gravity load. Therefore, the analysis

of Chen and May slabs is carried out by applying a gravity load of 9.81 m/s2 on the whole model.

It should be noted that gravity load does not act on rigid bodies included in the FEM, namely the

four cylindrical supports on which the slab is supported. Gravity load is not prescribed as a function

of time nor dened according to an amplitude curve. In this case, Abaqus assumes that the loading

is applied instantaneously at the beginning of the step. Gravity load is created at the impact step

and acts in the negative z-axis direction.

The motion of a rigid body is governed by that of its reference point. Therefore, boundary

conditions used to constrain the cylindrical support instances are applied to their reference points

that are located at the center of the bottom circular surface of supports, respectively. The same

boundary conditions are applied to each of the cylindrical supports on which the slab is supported,

thus it is more convenient to group the reference points of these rigid bodies into a single set of

geometry. The ENCASTRE boundary condition is used to fully constrain the movement of the

relevant reference points and set their degrees of freedom to zero. Translational displacements

of the reference points in the x-, y- and z-directions, as well as their rotations about the x-, y-

and z-axes, remain restrained during the whole analysis. An additional boundary condition is

applied to the model in order to ensure the motion of the impactor in the vertical direction only.

Thus, a set of geometry that includes the upper circular surface of the impactor is created and the

displacement/rotation boundary condition is used to prescribe its displacement and rotation degrees

of freedom. A nonzero displacement in the z-axis direction is prescribed to the impactor upper

surface set, while the remaining degrees of freedom are constrained to zero. Boundary conditions of

supports and impactor are applied at the initial step.

In the Assembly module, the various parts of the model are positioned relative to each other in order

to create the nal geometry of the model but they are still unconnected. Therefore, interaction and

constraints between parts or surfaces need to be dened to connect instances properly to each other.

As previously mentioned in section 3.4, Abaqus provides two algorithm to simulate contact between

dierent regions in the model and several constraints types to couple the motion of a group of nodes

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to that of other nodes. In the current study, the embedded and tie constraints are used to represent

the reinforcing steel bars in the model and to tie the four cylindrical supports to the slab bottom

surface, respectively. Both algorithms for simulating contact are used and compared, namely general

contact algorithm and contact pairs algorithm.

The embedded approach is used to create bond between the two instances of steel reinforcement

and the slab instance and overcome the mesh dependency of the discrete approach. In this case, it

is not necessary to adapt the concrete mesh to overlap the common reinforcement nodes. Concrete

elements of slabs are grouped in a set of geometry and considered as the host elements. The set

of truss reinforcement elements are embedded in the set of three-solid elements of concrete that

constrain the translational degrees of freedom of the embedded reinforcement nodes. The embedded

constraint available in Abaqus couples the nodal degree of freedom automatically assuming a full

bond action between the reinforcement and concrete elements with no relative slip.

Four surface-to-surface tie constraints are used to create proper interaction between the discrete

rigid elements of cylindrical supports and the solid elements of the steel frame and the concrete region

of slab. Each tie constraint connects the upper surface of a cylindrical support to the corresponding

quarter of the slab bottom surface. The upper surfaces of supports are designated to be the master

surface as they represent the rigid surfaces, while the bottom surfaces of slab are dened as slave

surface. As cylindrical supports are positioned at the intersection points of the slab concrete region

and the steel frame, the slave surfaces of slab must include the bottom surfaces related to the slab

concrete region as well as those related to the steel frame. A tie constraint is mesh-independent,

the master surfaces are modeled with a coarser mesh. No relative motion between master and slave

surfaces is allowed throughout the whole analysis, thus shear interaction is avoided between the

elements of the selected surfaces.

To analyze the impact problem of RC slabs, the contact algorithms available in Abaqus are

used and compared in order to model the contact between the slab and the impactor. The rst

algorithm is the general contact algorithm which allows very simple denitions of contact with very

few restrictions on the types of surfaces. The second algorithm is the contact pair algorithm that

represents more restrictions on the types of surfaces involved. The contact pair algorithm necessitates

the denition of the contact surfaces. Thus the surface-to-surface contact is used to describe the

interaction between the upper surface of the slab and the impactor hemispherical surface. Both

algorithm need to dene the mechanical interaction property model in Abaqus/Explicit, while the

contact pair algorithm requires more denition such as the sliding formulation and the contact

constraint enforcement method.

Concerning the interaction properties, the hard contact pressure-overclosure relationship is used

to dene the mechanical normal interaction of the contact model. The hard contact implies that any

contact pressure can be transmitted between surfaces when they are in contact, while no contact

pressure is transmitted when surfaces separate. The frictional behavior between the slab and the

impactor is dened using the penalty friction formulation to dene the mechanical tangential inter-

action. These formulation allows some relative motion between the surfaces in contact. A uniform

coecient of friction of 0.45 is assumed for contact surfaces.

For contact pair algorithm, the nite sliding formulation which allows any arbitrary motion of

surfaces is used to account for the relative motion of the upper surface of the slab and the impactor

surface. The position of surfaces in contact changes during the analysis, then the element faces and

nodes that are in contact change. In order to enforce contact constraints, both the pure master-slave

kinematic and penalty contact methods are used and compared in the aim of nding the best method

to model the contact in the FEM. The impactor surface is considered as the master surface since the

surfaces attached to rigid bodies must always be designated as master surfaces in Abaqus/Explicit.

The slave surface in this case must be the upper surface of the slab since this latter represents the

deformable body and must be with the ner mesh in order to avoid signicant penetration for hard

contact.

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5.2.10 Output requests

In order to compare numerical results to those obtained from the experiments of Chen and May,

several Field and History output are dened in the Step module, but only the contact force versus

time is presented in this study. The aim is to compare the force-time histories of dierent slabs

with experimental results in order to nd the nite element features that allow a good agreement

with experiments. Therefore, the contact force on the master surface of the impactor in the global

z-direction (CFN3) is evaluated. It should be noted that a Butterworth lter of order 6 is used in

the model as Chen and May used a Butterworth lter with a cut-o frequency of 2000 Hz to lter

results and reduce noise. In Abaqus, ltering of contact force-time history can be applied only in the

case of general contact. For kinematic and penalty contact, results are presented with no ltering

as it is not allowed for contact pairs in Abaqus/Explicit.

The nal step in a nite element analysis in Abaqus is to create a job that permits to analyze the

model. Therefore, the Job module is used to create a job and submit it for analysis.

First, the FEM of Chen and May slabs is validated for Slab2 using the C6T1S1 simulation. (C refers

to the compression stress-strain curve, T refers to the tension stress-strain curve and S refers to the

steel stress-strain curve).

The problem of impact is a very complex problem that involves several nonlinearities. Thus, it is

important to create a mesh that allows an accurate transfer of impact energy to the slab. The

region of concern in this type of problem lies at the impact where the impactor hits the slab and

damage under compression caused by impact is concentrated in this zone. This region needs more

attention and more rened mesh is necessary as the main physical mechanisms of RC slabs under

impact occur at this region. 4 types of meshes are used in this study as indicated in section 5.2.6.

Mesh1 allows a more rened mesh at the impact zone compared to the remaining regions of the slab.

Mesh2 is the same as Mesh1 with the only dierence that elements in thickness direction do not have

the same size, more rened elements are used at the upper surface where the impact phenomenon

occurs. Mesh3 is the mesh allowing the most rened mesh at the impact point among the meshes

considered for Chen and May slabs, elements size varies linearly in the edges direction as well as in

the thickness direction. However, an excessive renement could lead to distortion problem due to

the aspect ratio of very small elements at the impact point. For this reason, Mesh4 is developed

in order to provide a rened mesh at the impact zone with an acceptable aspect ratio of elements.

Mesh4 allows a rened mesh only at the selected zone of impact, while elements at the slab part

surrounding the impact square have the same size.

For all mesh types, the three-dimensional model of Slab2 with 5 elements in the thickness consists

of:

3168 truss elements of type T3D2 with 2970 nodes to represent reinforcement,

460 solid elements of type C3D8R with 648 nodes to represent the impactor (except for Mesh4),

828 rigid elements of type R3D4 with 836 nodes to represent the cylindrical supports.

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Figure 5.18: Comparison of dierent mesh types proposed for Slab2

The impactor is modeled with 736 solid elements with 953 nodes when Mesh4 is used, as a dierent

mesh is considered for the impactor in this case. Although the mesh of impactor used with Mesh1,

2 and 3 presents a better aspect ratio for elements, the mesh used with Mesh4 is more convenient to

represent the concentrated forces acting on hemispherical impactor geometry. The concrete region

of Slab2 as well as the steel support consist of:

67280 solid elements of type C3D8R with 82134 nodes for Mesh1 and Mesh2,

69620 solid elements of type C3D8R with 84966 nodes for Mesh3,

33020 solid elements of type C3D8R with 40566 nodes for Mesh4.

As can be seen, Mesh4 allows reducing the number of elements and nodes, and hence the number of

degrees of freedom in the numerical analysis, in comparison to other proposed meshes. Consequently,

the computational eort is reduced. The comparison of experimental force-time curve of Slab2 with

those obtained with the dierent mesh types in this study shows the eciency and accuracy of

developing Mesh4 in predicting the behavior of the slab (Figure 5.18). The peak impact force is

accurately estimated and is obtained at the same time as experimental results, while the curve shape

in the after peak phase shows some dierence in comparison to experimental results. The accurate

value of peak force obtained can be attributed to a good estimation of contact stiness using general

contact algorithm. Impact duration does match with the tests. In numerical analysis, the slab and

the impactor remain in contact for about 12 ms, while the contact duration is estimated by 14 ms

in experiments.

The inuence of the contact algorithm used to model the contact phenomenon between the slab

and the impactor is also studied. Figure 5.19 shows the contact force-time curve obtained using

general contact algorithm and contact pair algorithm. The kinematic and penalty methods used to

enforce constraints in contact pair algorithm are also compared. As can be seen, both algorithms

provide good results which are in agreement with experimental results. The general contact slightly

presents more accurate results, especially for the initial linear phase of the curve and the peak value

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Figure 5.19: Comparison of contact algorithms available in Abaqus for Slab2 meshed with Mesh4

of contact force. The reason is that Abaqus allows using lters to lter the force-time output curve

only when the contact is dened as general algorithm. For contact pair algorithm, force-time curves

are presented with no ltering. The dierence between general contact and contact pair algorithms

is that the rst permits to specify automatically the contact surfaces in the nite element model,

which can be very useful for complex contact problems. The contact pair algorithm necessitates

specifying each of the surfaces that are in interaction.

A mesh sensitivity analysis is conducted in order to choose a convenient mesh that is suciently

rened to converge to an accurate solution with the minimum computational eort. Therefore,

several sensitivity analyses are performed:

The number of elements in the thickness direction is investigated, and it is found that 5

elements are sucient to obtain numerical results that are in good agreement with experiments.

Using 7 or 9 elements in the thickness direction does provide a more improvement in results

(Figure 5.20).

The maximum and minimum size of elements in the impact zone is studied. Elements are

generated in this zone using the bias seeding option. It can be seen that there is no need to

use very small elements at this region and that a maximum size of 0.01 m and a minimum size

of 0.001 are sucient to converge to a unique solution (Figure 5.20).

The size of elements surrounding the impact region is varied using a ner, intermediate and

coarser mesh. Numerical results obtained with elements of seed of 0.01m are in good agreement

with experimental results, while smaller elements do not give more accurate results (Figure

5.21).

The size of elements in reinforcement is studied. It can be seen that the size of these elements

has no eect on results (Figure 5.21).

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Figure 5.20: Sensitivity of mesh in term of the number of elements in the thickness direction and

the size of elements in the impact zone for Slab2 meshed with Mesh4

Figure 5.21: Sensitivity of mesh in term of the size of elements in the slab and the size of elements

in the reinforcement for Slab2 meshed with Mesh4

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5.3.4 Choice of materials behavior

In this study, the dynamic problem of impact is investigated in order to predict the structural

behavior of RC slabs. Although the impactor is modeled as a deformable body in the FEM of Chen

and May slabs, the response of impactor is not of concern. The reason for which the impactor is

modeled as deformable body is related to the aim of allowing the best quality of transmission of

impact energy between the impactor and the slab. The response of slab is more important in terms of

deformations and damages. The upper surface of the slab is the rst element that enters in collision

with the impactor, this surface is under compression due the compressive wave generated by the

impactor at the moment of impact. Then, the compressive wave progresses in the slab thickness and

reects from the bottom surface as a tensile wave. Thus, it is important to rst specify the behavior

of concrete in compression, as it is the rst physical mechanism involved during an impact event.

Thereafter, its behavior in tension can be studied.

The most dicult parameters to identify for the CDP model are those related to the behavior

of concrete in tension and compression. In order to accurately dene the concrete behavior in a

numerical analysis of an impact problem, several relationships that describe the compressive stresses

in concrete in term of compressive strain are considered. Those stress-strain curves are based on

several experimental results from the literature (see Chapter 2). Force-time curves obtained using

these curves in the model are depicted in Figures 5.22-5.24. It can be seen that compressive stress-

strain curves with a higher resistance to crushing allow a better estimation of the peak impact

force (C6 and C9), while compressive stress-strain curve with more softening allow a linear plateau

as experiments and a better estimation of the shape of the force-time curve after the rst peak.

Consequently, the curve C6 is adopted since it provides the best t with experimental results.

Subsequently, the behavior of concrete in tension is studied. CDP model permits to describe

the tensile behavior of concrete using a stress-strain curve, a stress-cracking displacement curve and

a fracture energy value. Several cases from the literature are considered and compared (Figures

5.25-5.28). It can be seen that for stress-strain curves T1 gives results which are in good agreement

with experiments. However, the trilinear stress-cracking displacement curve D3 seems to be the

most accurate way to describe concrete in tension in the CDP model. It provides a good estimation

of the peak impact force, it also predicts correctly the plateau in the force-time curve. Curves based

on the fracture energy cracking criterion seem to be more ecient as they permit a better assessing

of deformation and cracks opening. These curves are more adequate to reduce the concern of the

mesh sensitivity, however they results in more computational eort compared to tensile stress-strain

curves.

Figures 5.29-5.30 show that the steel behavior has no signicant eect on the prediction of

force-time curves.

C6T1S1 and C6D3S1 simulations with the relevant constitutive behaviors of materials are found

to give the most accurate in the case of Slab2. Thus, both simulations are used to validate the

model for the other slabs of Chen and May experiments (Figures 5.31-5.32). As can be seen, using

stress-cracking displacement curve to describe the tensile behavior of concrete in numerical analysis

provides more accurate results for all slabs then using tensile stress-strain curve.

In the case of Slab3, a at impactor is used in experiments as well as in the FEM. The peak impact

force is overestimated in this case indicating the diculty of general contact algorithm in accurately

modeling the interaction between the slab and the impactor when a at surface is considered for

the impactor. In the case of hemispherical impactor, the general contact algorithm allows a better

interaction between the two bodies in collision (Slab2, Slab4, Slab5). Slab4 is impacted with higher

velocity than in the case of Slab2 and Slab3, the FEM allows a good prediction of the peak value for

higher velocity values. For Slab3, 4 and 5 the times at which the impact force reaches its peak are in

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Figure 5.22: Comparison of concrete compressive stress-strain curves for Slab2

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Figure 5.23: Comparison of concrete compressive stress-strain curves for Slab2

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Figure 5.24: Comparison of concrete compressive stress-strain curves for Slab2

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Figure 5.25: Comparison of concrete tensile stress-strain curves for Slab2

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Figure 5.26: Comparison of concrete tensile stress-strain curves for Slab2

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Figure 5.27: Comparison of concrete tensile stress-cracking displacement curves for Slab2

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Figure 5.28: Comparison of fracture energy values for Slab2

good agreement with experiments. However, in the case of Slab6, the peak force is underestimated. It

should be mentioned that only for Slab6 the load was determined from the accelerations measured by

an accelerometer attached to the dropping mass in experiments, while a load cell is used to measure

the loads for other slabs. This indicates the importance of using the appropriate output variable in

the nite element analysis.

5.4.1 Finite element model (FEM)

The following section discusses the FEM of a RC slab which is subjected to accidental dropped

object impact during handling operations within nuclear plant buildings. From the previous section

where the FEM proposed for impact problems is validated with experiments from the literature, an

accurate FEM of impacted slabs should introduce dierent factors that involve several nonlinearities

in the response of impacted RC slabs. The factors that should be correctly incorporated into in the

model and that contribute to the nonlinear behavior of the reinforced concrete are the nonlinear

stress-strain response, the damage due to crushing and tensile failure, the interaction between the

concrete and the reinforcement, and the contact algorithm. In order to accurately simulate the

structure and obtain detailed information from nite element approach, all these factors and contact

algorithms must be correctly incorporated into the nite element model. However, since the FEM of

slab in nuclear plant is used for reliability analysis, the aim is to choose nite element features that

allow the minimum computational cost, but also provide an accurate representation of the impact

phenomenon. The slab is studied in a reliability framework for 2 cases:

Assuming a nonlinear behavior of steel and concrete materials. The choice of stress-strain

curves adopted is based on those used in the FEM of Chen and May slabs by considering the

relations that give a good agreement with experimental results with the minimum computa-

tional eort.

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Figure 5.29: Comparison of steel stress-strain curves for Slab2

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Figure 5.30: Comparison of steel stress-strain curves for Slab2

The slab is considered as simply supported on all the four sides and consists of a 4.85 m wide with

a length of 8.1 m and a thickness of 0.5 m. Each of the upper and lower slab steel layers consists of

20 longitudinal and 12 transversal reinforcement bars of 20 mm diameter. In order to evaluate the

transmission of impact energy to the slab during an impact phenomenon and consider the interaction

between the slab and the impactor, the impactor is incorporated into the model as a sphere rigid

body with 3600 kg mass and 30 cm diameter, dropped at the center of the slab with a velocity of 7.7

m/s. Table 5.7 shows the linear mechanical properties of concrete and steel materials used in the

FE model for the slab in nuclear plant, while Table 5.8 shows its nonlinear mechanical properties

that are used to dene the stress-strain curves for each slab materials.

Finite element features used for elastic case are detailed in Chapter 6. Only the FEM of nonlin-

ear case is detailed in this section. Mesh 4 is adopted from the study of Chen and May slabs as it is

the most suitable to use in the case of an impact problem. This mesh allows a better simulation of

the contact between the slab and the impactor and hence a better estimation of impact force. A 3D

solid element, the eight-node continuum element (C3D8R) is used to develop the mesh of the con-

crete slab. Abaqus/Explicit provides three alternative kinematic formulations for the C3D8R solid

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Figure 5.31: Comparison of C6T1S1 and C6D3S1 simulations for Slab3 and Slab4

Figure 5.32: Comparison of C6T1S1 and C6D3S1 simulations for Slab5 and Slab6

Table 5.7: Linear mechanical properties of concrete and steel materials for the slab in nuclear plant

Concrete 2500 35000 0.2

Steel 7850 200000 0.3

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Table 5.8: Nonlinear mechanical properties of concrete and steel materials for the slab in nuclear

plant

fc (MPa) ft (MPa) fy (MPa)

in nuclear plant 40 3.5 500

element, namely the average strain formulation, the orthogonal formulation and centroid formula-

tion. This latter is selected for the nonlinear case in the aim of reducing the nite element analysis

computational eort since it is the most economical among the three formulations. As illustrated

in Figure 5.33.a, elements along the slab thickness have dierent size with ner elements located at

the upper surface of the slab. The thickness is modeled with ve elements in the thickness direction

and a bias ratio of 2. The slab is modeled with two dierent densities of mesh. The impact zone

is modeled with ner mesh, it consists of a square of 1.0 m with a maximum element size of 10 cm

and a minimum element size of 1 cm. The mesh around the impact zone is considered uniform in

the xy-plane with a size of 10 cm. The reinforcement is modeled with truss elements (T3D2) of 10

cm of size and the embedded approach is used to create the bond between the steel reinforcement

and concrete. This approach allows independent choice of concrete mesh and arbitrarily denes the

reinforcing steel regardless of the mesh shape and size of the concrete element. The impactor is

meshed with four-node tetrahedral elements (R3D4) having a constant seed of 2 cm.

The contact pair algorithm is selected in order to model the contact and interaction problems

between the slab and the impactor since the model involves only two contact surfaces, namely the

upper surface of the slab and the impactor surface. In addition, it is more ecient in term of

computational cost when used with kinematic contact constraints. The nite sliding formulation

which allows any arbitrary motion of surfaces is used to account for the relative motion of the upper

surface of the slab and the impactor surface. The contact constraints are enforced with a pure

master-slave kinematic contact algorithm which does not allow the penetration of slave nodes into

the master surface (surfaces attached to rigid bodies must always be dened as master surface in

Abaqus/Explicit). Two interaction properties are presented in the model, a hard contact normal

interaction dened as hard contact and a penalty tangential interaction with a friction coecient of

0.45.

Stress-strain curves of concrete in tension and compression considered are those proposed by

Wang and Hsu [180], while steel is modeled according to Theodorakopoulos and Swamy curve [173].

The FEM of the slab consists of 32072 nodes and more than 27400 elements (21800 elements of type

C3D8R, 4448 elements of type T3D2, 1152 elements of type R3D4), and a simulation takes a total

time of 43 minutes for an impact step of 0.1 s. The Python script written for this enables to easily

change the impact position, as well as add stirrups (Figure 5.33.b), in the FEM (the main FEM of

this slab does not include shear reinforcement).

In order to simulate the impact of a dropped object on RC slab, two simplied analytical models

of the slab are used. The two models consist in a two degrees of freedom mass-spring system

which accounts for nonlinearity of RC by a stiness degradation approach. The second model also

accounts for potential viscous damping. The input parameters that involve in these models are slab

geometry, reinforcement ratio, concrete and steel densities, impactor mass and velocity. Outputs

of these models are the time-history curves of slab displacement at impact point, slab velocity at

impact point, impactor displacement and impactor velocity.

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Figure 5.33: Finite element model of slab: a) meshing, b) reinforcement

This model describes the interaction between the slab and the impactor via two springs (Figure

5.34). The parameters which represent the dynamic of the system are:

From the mechanical equilibrium of the two masses, two coupled ordinary dierential equations of

second-order are derived and solved in time by an approximated method, namely the Runge-Kutta

fourth order method:

(

d2 xi kc kc

dt2

+m xi − m xs = 0

i i (5.1)

d2 xs kc kc +ks

dt2

− ms xi + ms xs = 0

with the initial conditions

!

kc kc

−

xi mi mi

Assuming that X = and M =

kc (kc +ks ) , the system of equations for the

xs −m m s s

mass-spring model without damping can be written in a matrix form as:

d2 X (5.3)

dt2

+ MX = 0

In this model, viscous damping is taken into consideration in addition to the slab and impactor

stinesses and masses (Figure 5.35). It is assumed that the structural damping is to be of the

viscous type according to [20], i.e., the damping force is opposite but proportional to the velocity.

The impedance cc accounts for the viscosity of the contact, while cs accounts for the viscosity of

the slab.

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Figure 5.34: Mass-spring model without damping

d 2 xi

(

kc kc cc cc

dt2

+m xi − m xs + m ẋs − m ẋs = 0

i i i i (5.4)

d 2 xs

dt2

− ms xi + ms xs + − ms ẋi + (csm+cs c ) ẋs = 0

kc kc +ks cc

with the same initial conditions considered for the previous model and the equations are also

solved in time by the Runge-Kutta fourth order method.

! !

cc cc kc kc

−m −m

xi mi i mi

Assuming that X= , C= cc (cs +cc ) , and M= kc

i

(kc +ks ) , the system

xs −m ms −m ms

s s

of equations for the mass-spring model with damping can be written in a matrix form as:

d2 X

dt2

+ C dX

dt + M X = 0

(5.5)

In order to dene the equivalent two-degrees of freedom systems presented in the previous section,

it is necessary to evaluate the parameters of these models. The equivalent system is selected so that

the deection of the concentrated mass is equal to the deection of the slab at the impact point.

Eective mass of slab In order to obtain the slab mass of the equivalent system, the mass of

the real slab is multiplied by a transformation factor αM [20].

msequi = αM × ms (5.6)

According to [42, 56], the equivalent mass of the slab can be obtained by equating the kinetic

energy for the equivalent and real system:

¨ ¨ 2

1 duimp 2 1 du(x, y)

( αM ρdA) × ( ) = ρ dA (5.7)

2 dt 2 A dt

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˜ 2 (x, y)dA

A ρφ

˜

αM = (5.8)

ρdA

In the above equations, uimp (t) is the deection of the slab at the impact point and u(x, y, t)

u(x,y,t)

represents the deection at any point on the slab of x- and y-coordinates; φ(x, y, t) = uimp(t) is the

deected shape function; A is the slab area and ρ represents the mass per unit area of the slab.

According to [20], the mass transformation factor, for two-way slabs with simple and xed

supports, is evaluated on the basis of an assumed deected shape of the actual structure which

is taken to be the same as that resulting from the static application of the dynamic loads. The

approximation of the deected shape is based on the classical plate theory and is considered to

satisfy the boundary conditions of simply supported slabs:

∞ X

∞

X mπx nπy

u(x, y, t) = Wmn (t) sin sin (5.9)

a b

m=1 n=1

In the following, only the rst mode will be considered, that is, m = n = 1. Thus, the deected

shape will be:

πx πy

u(x, y, t) = W1 (t) sin sin (5.10)

a b

The maximum deection at the impact point of the slab (x = a/2, y = b/2) is therefore uimp =

W1 (t).

After substituting the uimp (t) and u(x, y, t) in Equation 5.7, the equivalent mass of the slab is

found to be equal the fourth of its real mass.

Slab stiness A frequency decrease approach is used to describe the RC's degradation which is

due to cracks in concrete and yielding in steel. This approach allows considering the nonlinearities

presented in RC. The slab stiness decrease is identied by quasi-static pushover tests in term of

the displacement of the slab at the center. The initial value of slab stiness is estimated to be equal

to 1.8e8 N/m for the main case of study with a thickness of 0.5 m. Figure 5.36 shows the variation

of the initial value of slab stiness in term of the slab thickness, as well as the stiness decrease in

term of displacement for several values of thickness.

Contact stiness In impact dynamics analysis, a detailed description of the contact between the

impactor and the structure during impact is not required and statically determined contact laws can

be used [9]. Thus, a contact law relating the contact force to the indentation is needed to dene

the contact between slab and impactor. The indentation is dened as the dierence between the

displacement of the impactor and the displacement of the slab at the impact point.

The local indentation α = xb − xd is represented by the spring (kc ) which can be linear or

nonlinear. Thus during the loading phase of the impact, the impact force can be obtained by using

a linearized contact law (q = 1) or an Hertzian contact law (q = 3/2).

Fc = kc αq (5.11)

For Hertzian contact between a plate and a hemispherical indentor, the contact stiness is given

by [10]:

4

kc = ER1/2 (5.12)

3

where R represents the impactor radius [149] and E is given by

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Figure 5.36: Slab stiness decrease in term of displacement for several values of thickness and the

variation of the stiness initial value in term of the thickness

1 1 − νs2 1 − νi2

= + (5.13)

E Es Ei

In the present study, we assumed that the impactor is rigid so E is in this case equal to:

1 1 − νs2

= (5.14)

E Es

By substituting the parameters by their values in the above equations, the contact stiness is

found to be equal to 7.29e9 N/m for a linearized contact law and 1.88e10 N/m3/2 for a Hertzian

contact law.

Impedances Biggs [20] indicated that the inclusion of damping in multidegree analysis involves

some rather troublesome problems since there is little theoretical means for determining the nature

of the damping. The damping was assumed to be viscous, thus the damping force applied to a mass

is directly proportional to the velocity of the structure at the impact point and may be expressed

by:

Fv = −cẋ (5.15)

Biggs also indicated that the magnitude of c is extremely dicult to determine. Thus, in order

to be sure that the damping coecients are reasonable, it is convenient to introduce the concept of

critical damping which represents the amount of damping that would completely eliminate vibration.

For a SDOF, the critical damping is given by:

√

ccr = 2 km (5.16)

while for multiple degree of freedom, the critical damping matrix is dened by [186]:

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Ccr = 2M−1/2 K M−1/2 = 0 (5.17)

According to [184], the contact impedance cc is only valid during contact and damping mechanism

is due to the energy transfer from the impactor to the structure. Denitions of the impedances of

impactor and plates were given by [149] for small-mass impact models for composite plates. The

impedance of the plate is expressed by:

p

cs = 8 ρD∗ (5.18)

where ρ is the mass per unit area of the slab and D∗ the equivalent bending rigidity which is

dened as:

D∗ ≈

p p

D11 D22 (A + 1)/2 where A = (D12 + 2D66 )/ D11 D22 (5.19)

By substituting the parameters by their values in the above equations, the slab impedance is

found to be equal to 5.52e6 N/(m/s).

The impactor impedance is dened for an Hertzian contact (q=3/2) by:

5 1/5

cc = kc2/5 ( mi )v0 (5.20)

4

and for a linear contact (q=1) by:

p

cc = kc mi (5.21)

By substituting the parameters by their values in the above equations, the contact impedance is

found to be equal to 1.26e7 N/(m/s) for a linearized contact law and 3.01e6 N/(m/s) for a Hertzian

contact law.

nuclear plant

5.5.1 Comparison of FEM and MSM results

The aim of using a mass-spring model is to develop a simplied model economical in computational

cost, but also allows an accurate and precise estimation of the structural response. Thus, both

MSM models proposed for the slab in nuclear plant assuming a nonlinear behavior of material are

compared with its FEM. The values of MSM parameters identied previously are used as preliminary

values to initially present the model, then they are calibrated in order to nd the best t with FEM

results. As a result, the values adopted for the MSM are as follows:

The initial value of slab stiness varies according to the case studied in terms of the input

variable values.

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Figure 5.37: Comparison of mass-spring models with and without damping

First, a comparison of the two mass-spring models used in this study is proposed (Figure 5.37).

The following comparison permits to show the importance of taking the potential viscous damping

in the analytical analysis of an impact problem. As can be seen, the mass-spring model without

damping overestimates the slab and impactor displacements, as well as their velocities. The reason

is that this type of models does not accurately model the dissipation of energy related to the plastic

deformation of materials, formation of cracks, friction and damping. Considering only the stiness

degradation approach to describe the reinforced concrete degradation due to cracks in concrete and

yielding in steel seems then insucient, hence viscous dampers with a constant impedance should

be used to provide a better dissipation of energy in the system. Thus, the mass-spring model with

damping gives more accurate results.

Accordingly, several cases are studied to compare the MSM accounting for viscous damping and

the FEM by varying several input variables involved in the MSM, including the impactor mass and

velocity for the same kinetic energy, the impactor velocity, the impactor mass, the slab thickness,

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the steel diameter and the concrete density. 38 cases are investigated and detailed in Annex B.

The cases studied enable to determine the limitations of the MSM proposed, as well as the range

of values of its variables for which the simplied model is veried. As a result, the two degrees of

freedom mass-spring model which accounts for potential viscous damping and for nonlinearity of RC

by a stiness degradation approach can be used for the following conditions:

Low-velocity high-mass impacts when the same impact energy is considered (v = 2.5 m/s, m =

34151 kg , v = 5.0 m/s, m = 8537 kg , v = 7.7 m/s, m = 3600 kg , v = 10.5 m/s, m = 1936 kg ).

For these cases, the maximum values of slab displacement, impactor displacement and slab

velocity, as well as the shape of the corresponding curves, are in good agreement with numerical

results. The simplied model shows less eciency in predicting the shape of the impactor

velocity in term of time for low-velocity high-mass impacts, although it allows a good estimation

of the impactor velocity maximum value. In the cases of high-velocity low-mass impacts, the

MSM allowed assessing the maximum values of slab and impactor displacements. However, the

shape of their curves in term of time shows a great dierence with respect to those obtained

with the FEM for the after contact phase. For these cases, the MSM estimates accurately the

maximum value and the curve shape of the impactor velocity. The maximum values of slab

velocity is not properly assessed with the MSM for high-velocity low-mass impacts, but the

shape of slab velocity-time curve is predicted with precision for the after contact phase.

Low-velocity impacts when the same mass is considered (v = 2.5 m/s, v = 5.0 m/s, v =

7.7 m/s, v = 10.5 m/s). In these cases, the maximum values and the curves shape of the slab

and impactor displacements show a good agreement with numerical results, respectively. The

MSM is very ecient in estimating the slab and impactor velocities (maximum values and

curves shape) for all cases studied by varying the initial velocity value, including high-velocity

impacts.

3600 kg , m = 5000 kg , m = 6000 kg . For these cases, the MSM results are in good agreement

with numerical results for all the output variables examined. The maximum values and the

curves shape are properly assessed. However, for low-mass impacts (m ≤ 1000 kg ) the MSM

loses its accuracy and cannot be used to estimate impacted RC slab response as it is governed

by more localized deection.

Thick slabs (e ≥ 0.4 m). For these cases, the MSM results are in good agreement with numer-

ical results for all the output variables examined. This may be related to the fact that thin

slabs are more subjected to penetration and perforation failure modes than thick slabs for the

same impact conditions, and the MSM does not allow predicting these types of failure modes.

All values of steel reinforcing bars considered. In these cases, the maximum values and the

curves shape of all output variables show a good agreement with numerical results.

All values of concrete density considered. In these cases, the maximum values and the curves

shape of all output variables show a good agreement with numerical results. It should be

noted that the MSM presents a higher precision and accuracy for high concrete density (ρc ≥

3500 kg/m3 ).

In the present section, only two cases are presented. Figure 5.38 represents the slab displacement at

the impact point, slab velocity at the impact point, impactor displacement and impactor velocity in

term of time for dierent values of masses and velocities with the same impact energy. The gure

shows that the response of the slab varies with the mass and the velocity of the impactor. As can be

seen, the MSM provides results which are in good agreement with numerical results, especially for

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Figure 5.38: Comparison of FEM and MSM for the same kinetic energy

the phase where the slab and the impactor are in contact. When the impactor is separated from the

slab, the MSM does not permit to accurately predict the numerical results. For large masses and

low velocities the time of contact between the slab and the impactor, as well as the displacement

at the impact point, is higher than the cases of small masses and higher velocities. For these latter

cases, when the impactor strikes the slab the damage initiates and the impactor rebound earlier.

These results, which agree with [149], demonstrate that the slab response to impact is inuenced by

impactor mass and impact duration and not only by impact velocity.

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Chapter 6

structures

6.1 Introduction

The determination of limit state functions is necessary to perform structural reliability analysis. But

it is typically impossible to dene an explicit expression of a limit state function, and to assess the

probability of failure with analytical integration, especially in cases of complex structures modeled

with the nite element method. To address this problem, numerous probabilistic methods can be

used to estimate failure probability as previously mentioned in Chapter 4. Selection of the most

suitable method for a particular type of problem is not apparent since the accuracy and computa-

tional eort of probabilistic methods depend on several factors, such as degree of nonlinearity of the

limit state function, type of random variable distributions, number of random variables and their

variance [53].

First, a simple application to a RC beam is performed in the aim of mastering the basics of

OpenTURNS and examining the probabilistic methods proposed in OpenTURNS to estimate failure

probability in structural reliability analysis in terms of their accuracy, precision and computational

eort. This application is used to address the issue of solving reliability problems in the domain of

civil engineering and to propose solutions based on the case studied.

Next, the problem of RC slabs subjected to low velocity impact is addressed. The aim is to

address the issue of computational cost of reliability analysis and to propose computational strategies

allowing the accurate assessment of the failure probability for minimum computational time. The

rst strategy consists in using deterministic analytical models involving low computational time. The

second one consists in choosing an appropriate probabilistic method where the failure probability is

assessed from a small number of simulations. Probabilistic methods, such as Monte Carlo, FORM,

SORM and importance sampling, are adopted from the beam application and used to calculate

the probability of failure. The choice of random variables and their distributions, as well as failure

criteria are discussed according to several studies in the literature. Firstly, the problem of impacted

slab is studied assuming a exural mode of failure and an elastic behavior for steel and concrete.

Several deterministic models are used and combined with a suitable probabilistic method. Studying

slabs under elastic behavior is important to examine the limitations of deterministic models and

to adopt the most convenient procedure to use in case of nonlinear behavior. Following this, the

problem of impacted slab is studied using deterministic models that take into account the nonlinear

material properties. In both case, elastic and nonlinear behaviors, a parametric study is performed

to identify the inuence of deterministic model parameters on the reliability of RC slabs under low

velocity impact.

Finally, a third application is also considered in the aim of presenting a procedure to be followed

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to study the response of very complicated structures in a reliability framework. For this purpose,

the prestressed concrete containment building of the Flamanville nuclear power plant is used. The

deterministic model of the containment building consists of a numerical model in which the full

process of aging is taken into account, including relaxation of the reinforcement, creep and shrinkage

of concrete. This application allows examining stress evolution with time in the containment building

during periodic surveillance testing carried out 20 years after its implementation. Polynomial chaos

expansion is used to simplify the physical model and study the height at which the containment is

under tension.

In order to illustrate the principle of each step of a structural reliability analysis presented in Chapter

4 and compare the available probabilistic methods in terms of their accuracy and computational

eort, the case of a RC beam modeled with CASTEM is studied. Combining CASTEM with

OpenTURNS is possible using the wrapper interface that allows linking any external code to the

OpenTURNS library and using it through the library functionalities.

The rst step in reliability analysis consists in providing a deterministic model (analytical or nu-

merical) that correctly represents the structure studied in order to predict the structural response.

The case of a RC cantilever beam is proposed from experiments in the framework of the LESSLOSS

project [108] (Figure 6.1). The beam under investigation is xed at only one end and subjected to a

concentrated load at the free end. A numerical model based on a multiber FE approach is proposed

to examine the response of the cantilever beam. Multiber nite elements are used to develop the

mesh of the beam (Figure 6.2), they permit to describe the behaviors of concrete and reinforcement

with a one-dimensional way and are assumed to follow the kinematics of the Timoshenko beam

theory which allows taking into account shear deformation eects in the cross section. Materials are

supposed to have an elastic behavior.

The variables which may be subjected to statistical variations in RC cantilever beams under an end

load condition can be classied into ve categories:

1. Dimensional variables related to the geometric properties of the beam: length and section

width

2. Dimensional variables related to the position of reinforcement and its size: reinforcement

section width and its eccentricity

3. Variables related to the concrete properties: Young's modulus and Poisson's ratio of concrete.

4. Variables related to the steel properties: modulus of elasticity and Poisson's ratio of steel.

For this application, the probabilistic characterization of random variables is not based on the

literature review detailed in Chapter 4. The aim of this example is to study the eect of type of

random variable distributions, number of random variables and their variance on the estimation of

failure probability. Thus, rst, all random variables are assumed to have a lognormal distribution

with a mean value equal to the nominal value and a COV of 0.1. Table 6.1 represents the random

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Figure 6.1: RC beam tested experimentally in the framework of the LESSLOSS project [108]

Figure 6.2: Discretization of multiber beam into elements, nodes and degrees of freedom [76]

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Figure 6.3: Geometrical properties of the multiber beam

Table 6.1: Random variables of the multiber beam and their descriptions

Geometric properties of the beam

Lb Beam length in z-direction 2.75 m 0.1 Lognormal

hB1 Width of beam section 0.25 m 0.1 Lognormal

Position and size of reinforcement

dbar Width of reinforcement section 0.02 m 0.1 Lognormal

ExcA Eccentricity of reinforcement 0.1 m 0.1 Lognormal

Concrete properties

Ec Young's modulus of concrete 25 GPa 0.1 Lognormal

νc Poisson's ratio of concrete 0.2 0.1 Lognormal

Reinforcement properties

Es Modulus of elasticity of steel 200 GPa 0.1 Lognormal

νs Poisson's ratio of steel 0.3 0.1 Lognormal

Loading variables

F orce1 Maximum force applied 1000 N 0.1 Lognormal

variables adopted in the present study in the case of multiber beam, their probability distribution,

their mean and COV.

Failure of structure is investigated according to a displacement failure criterion. Failure probability

is estimated for the following limit state function:

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Table 6.2: Importance factors of beam random variables

Quadratic 0.3529 0.3228 0.0128 0.0124 0.0203 0.0 0.0032 0.0 0.0396 0.2358

combination

FORM 0.3753 0.4015 0.0243 0.0236 0.0223 0.0 0.0061 0.0 0.047 0.0997

g = fmax − f (6.1)

where fmax is the threshold of the displacement of beam at the free end and f is the displacement

at the free end calculated by CASTEM. fmax is considered as an additional random variable to the

variables presented in Table 6.1 and also modeled with a lognormal distribution and a COV of 0.1.

6.2.4.1 Sensitivity analysis

A detailed sensitivity analysis is performed in this section in order to identify the random variables

that mostly contribute to the variability of the displacement of cantilever beams. 10 random variables

are considered, including the threshold of displacement. The aim is to select the most signicant

variables which control the beam response. Importance factors are calculated using Quadratic

combination method and FORM and detailed in Table 6.2.

In order to verify the accuracy of these estimations, several sensitivity studies are performed.

First, a study of mechanical sensitivity that enables to estimate the variation of the limit state

function when variable change is conducted. This study is purely deterministic and measures the

sensitivities with respect to the design variables [107]. In order to be able to make comparisons, a

non-dimensional quantity, namely the mechanical elasticity S̄i , is calculated as:

∂g(X) xr

S̄i = (x = xr ) (6.2)

∂xi g(xr )

where xr are particular representative values of dierent input variables, such as the mean.

Mechanical sensitivity allows distinguishing stress variables from resistance variables and is very

useful to select the random variables to take into consideration. The sign of S̄i values that are

calculated with CASTEM determines the type of variables. If S̄i is a positive value then the variable

is a resistance variable, while a negative value indicates a stress variable. The mechanical elasticity

multiplied by the COV of the variable permit to study the variability of the corresponding variable.

This can be achieved by comparing the product S̄i × COVi to the variable mean. The variability of

a variable is considered high in the case where |S̄i × COVi | is higher than its mean (mean of this

product is equal to 0.0746). Table 6.3 presents the stress and resistance variables for the cantilever

beam and shows if their variability aects highly the beam response.

Then a statistical sensitivity analysis is performed by calculating omission factors ζi that express

the relative error of the reliability index when a random variable is replaced by a deterministic value

[107]:

βXi

ζi = (6.3)

β

βXi is the reliability index calculated for the case where all variables are considered as random,

except Xi which is considered as deterministic. β is the reliability index in case where all variables

are random. An omission factor value close to unity indicates that the corresponding variable has

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Table 6.3: Stress and resistance variables of the cantilvever beam and their variability inuence

Lb 2.75 m -2.07 Stress 0.207 high

hB1 0.25 m 2.01 Resistance 0.201 high

dbar 0.02 m 0.0069 Resistance 0.0007 low

ExcA 0.1 m 0.293 Resistance 0.0293 low

Ec 25 GPa 0.499 Resistance 0.0499 low

νc 0.2 - -0.0006 Stress 0.0000 low

Es 200 GPa 0.195 Resistance 0.0195 low

νs 0.3 - -0.0001 Stress 0.0000 low

F orce1 1000 N -0.693 Stress 0.0693 low

fmax 0.0015 m 1.69 Resistance 0.169 high

Table 6.4: Omission factors of the cantilvever beam variables and their inuence on Pf

Lb 2.7785 1.309 high

hB1 2.8962 1.365 high

dbar 2.1564 1.016 medium

ExcA 2.1559 1.016 medium

Ec 2.1467 1.012 medium

νc 2.1221 1.000 low

Es 2.8735 1.354 high

νs 2.1221 1.000 low

F orce1 2.1811 1.028 medium

fmax 2.1220 1.000 low

no eect on the failure probability estimation. Table 6.4 shows omission factors of random variables

of the cantilever beam, as well as their inuence on the estimation of failure probability for a

displacement criterion.

Next, the variability of variables in terms of their mean and standard deviation are evaluated.

The reliability index is calculated for the case where all random variables, except for one variable,

have the same mean and standard deviation as mentioned in Table 6.1. The variation of this variable

is assessed for four cases by multiplying its mean or standard deviation by 2 or by 4. The variable

sensitivity is considered high if the calculated reliability index varies by more than 50% compared to

β [44] (β = 2.1221). Table 6.5 shows the variability of each random variable of the cantilever beam

according to their mean and standard deviation values.

As mentioned earlier, the mechanical and statistical sensitivity studies are performed in order

to verify the accuracy of results provided by Quadratic combination and FORM with respect to

importance factors. Table 6.6 summarizes the results of these studies and shows the inuence of each

variable on the estimation of failure probability for a displacement criterion in case of a cantilever

beam. As can be seen, the most inuential variables are related to the beam dimensions (hB1 and

Lb), which is in good agreement with results found by Quadratic combination and FORM (Table

6.2). In addition, these results show that mean and standard deviation values have an important

eect on the estimation of failure probability. Thus, they must be properly identied in structural

reliability analysis in order to assess accurate estimation.

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Table 6.5: Variability of the cantilever beam variables according to their mean and standard deviation

values

Random Variable Inuence of the mean value Inuence of the standard deviation value

µ1i = 2 × µi µ2i = 4 × µi Inuence σ1i = 2 × σi σ2i = 4 × σi Inuence

Lb 5.7891 9.9750 high 1.4167 1.1869 high

hB1 1.5688 4.0158 high 1.9994 1.1435 high

dbar 4.0489 6.2843 high 2.0344 1.0806 medium

ExcA 4.0182 6.3017 high 2.0358 1.7097 medium

Ec 3.1490 4.3862 medium 2.0475 1.7433 medium

νc 2.1204 2.1182 low 2.1221 2.1221 low

Es 2.8766 4.0282 low 2.0987 2.0092 low

νs 2.1219 2.1211 low 2.1221 2.1221 low

F orce1 0.4860 0.8769 high 1.9885 1.6975 medium

fmax 3.8401 5.4505 high 1.9531 1.3468 high

Variable type Inuence ζi inuence µi inuence σi inuence inuence

Lb Stress high high high high high

hB1 Resistance high high high high high

dbar Resistance low medium high medium medium

ExcA Resistance low medium high medium medium

Ec Resistance low medium medium medium medium

νc Stress low low low low low

Es Resistance low high low low low

νs Stress low low low low low

F orce1 Stress low medium high medium medium

fmax Resistance low low high high medium

After examining the most inuential input variables, the sensitivity analysis performed in the pre-

vious section allowed reducing the number of random variables to two (beam width and length).

The next step is to analyze the dispersion and the distribution of the output variable, which is in

this case the displacement at the free end of the beam. For this purpose, 100000 random simula-

tions of the input vector are generated using the numerical integration Monte Carlo method (see

section 4.4.2.1). Displacement values are determined for each simulation, they are then arranged in

increasing order and subdivided into several equal intervals. Their percentage frequency values are

calculated for each interval of displacement in order to plot the histogram that enables to determine

the distribution of the variable of interest (Figure 6.4). The displacement probability distribution

is rst represented graphically using the Kernel Smoothing method. As can be seen, the number

of simulations used is sucient to obtain a very smooth Kernel density estimate and displacement

values are distributed around a mean value of 0.614 mm with a standard deviation of 0.254 mm.

Note that the estimated density captures the peak that characterizes the mode, and represents a

nonparametric alternative to the tting of a parametric probability density function.

Another parametric way to determine the distribution that ts the output sample of the beam

displacement can be also used. It consists in selecting a parametric distribution to model the

randomness behavior of the variable of interest and verify if it is in good agreement with the output

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Table 6.7: Results of Kolmogorov-Smirnov test to verify beam displacement distribution

Beta r = 3.2035 0.05 4.48e−60 False

t = 17.467

a = 9.74208e−5

b = 0.00314393

Normal µ = 0.00065616 0.05 0.0 False

σ = 0.000274354

Lognormal µlog = −7.6105 0.05 8.96e−64 False

σlog = 0.509836

γ = 9.74208e−5

Logistic α = 0.00065616 0.05 0.0 False

β = 0.000151259

Gamma k = 4.14759 0.05 2.98e−12 False

λ = 7423.12

γ = 9.74208e−5

Gumbel α = 4674.8 0.05 1.05e−05 False

β = 0.000532686

sample. Among parametric distribution types proposed in OpenTURNS to represent the uncertainty

of a continuous variable, 6 types are assumed and veried using statistical tests of goodness-of-t.

The output sample distribution is compared with Beta, Normal, Lognormal, Logistic, Gamma, and

Gumbel distributions. The parameters of these distributions are estimated in terms of the output

sample statistical moments and veried using Kolmogorov-Smirnov test (see section 4.4.2.2). A p-

value threshold of 0.05 is used and compared to the p-value calculated. The assumed distribution

is considered as accepted, and thus the Kolmogorov-Smirnov test succeeds to nd a distribution

that ts the output sample, only if the p-value calculated is greater than the threshold. Table 6.7

shows the distributions selected to verify if they t the beam displacement sample, as well as their

statistical parameters estimated in terms of the mean and standard deviation of the displacement.

As shown, Kolmogorov-Smirnov test fails for all the distribution types selected, hence none of them is

appropriate to describe the beam displacement density. Figure 6.5 provides a graphical comparison

of the output sample distribution with the parametric distributions selected and shows that the

Gumbel distribution gives the best t. These results are also found using the QQ-plot with quantiles

estimated at 95%. The quantile points estimated with Gumbel distribution of parameters α = 4674.8

and β = 0.000532686 are the closest to the diagonal (Figure 6.6).

The aim of this section is to discuss and compare probabilistic methods proposed in OpenTURNS (see

section 4.4.3) in order to examine their accuracy, precision and computational eort. These methods

are used to evaluate failure probability of the beam displacement criterion under the inuence of

several parameters, such as the number of random variables, their coecient of variation and the type

of their probability distribution. Parametric cases considered are detailed in Tables 6.8-6.10 with

the corresponding values of failure probability estimated with dierent methods (FORM, SORM,

MC, IS, DS and LHS). All cases studied are divided into several subset as following:

3 subset of cases with the same random variables number: 2 random variables (hB1, Lb), 3 ran-

dom variables (hB1, Lb, fmax ), and 7 random variables (hB1, Lb, Ec , dbar, ExcA, Es , F orce1).

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Figure 6.4: Beam displacement distribution estimated with Kernel Smoothing method

Figure 6.5: Comparison of the beam displacement sample to several parametric distributions

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Figure 6.6: QQ-plot test to graphically compare the beam displacement sample to several parametric

distributions

Random variables are selected according to the sensitivity analysis results by considering the

most inuential parameters.

2 subset of cases where random variables taken into consideration are presented with iden-

tical distribution type (normal, lognormal), and 1 subset that contains all cases with mixed

distributions (i.e. cases where random variables are modeled with dierent distribution types).

5 subset of cases where random variables taken into consideration are presented with the same

COV (0.05, 0.1, 0.2, 0.3, 0.5), and 1 subset that contains all cases with mixed COV (i.e. cases

where random variables have dierent COV values).

Pf (method)

For each subset, the mean value and COV of

Pf (M C) ratio are calculated for all methods. MC is

assumed to be the method that gives the most accurate results, hence other methods are compared

to MC and examined in terms of their accuracy and precision. Accuracy is estimated in term of the

Pf (method)

mean value of

Pf (M C) ratio and refers to closeness to MC results. Precision is estimated in term

Pf (method)

of the COV of

Pf (M C) ratio and refers to the degree of variation of results. A particular method

is supposed to give the same results as MC in the case where the corresponding failure probabilities

ratio has a mean value of 1.0 and a COV of 0.

It should be noted that, although a sucient number of cases is necessary to make better con-

clusions on probabilistic methods eectiveness, the number of cases studied is limited to 27 and not

all possible cases are considered because of the computational eort needed by dierent probabilistic

methods to estimate failure probability for each case. However, this procedure can be useful to have

a general understanding of each method limitations and how to select an appropriate method for a

specic type of problem.

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Table 6.8: Failure probability estimated with dierent methods for all the cases studied of the cantilever beam (number of random variables=2)

Case Random variables Distribution COV FORM SORM MC IS DS LHS

1 hB1, Lb Lognormal 0.05 3.26E-06 3.03E-06 6.00E-06 2.86E-06 3.29E-06 2.00E-06

2 hB1, Lb Lognormal 0.1 0.01176 0.01098 0.01004 0.00967 0.01237 0.011486

3 hB1, Lb Lognormal 0.2 0.12427 0.11593 0.12273 0.12656 0.24328 0.11294

4 hB1, Lb Lognormal 0.3 0.21348 0.19846 0.18263 0.18459 0.21168 0.19014

5 hB1, Lb Lognormal 0.5 0.29791 0.27239 0.24675 0.25535 0.21762 0.21467

6 hB1 Lognormal 0.1 0.13678 0.13414 0.12865 0.13254 0.15701 0.13162

Lb Lognormal 0.3

7 hB1, Lb Normal 0.1 0.01147 0.01072 0.00974 0.01036 0.00982 0.01115

8 hB1 Normal 0.1 0.01308 0.01265 0.01199 0.01274 0.01249 0.01334

Lb Lognormal 0.1

9 hB1 Lognormal 0.1 0.00975 0.00885 0.00828 0.00809 0.01081 0.00852

Lb Normal 0.1

Table 6.9: Failure probability estimated with dierent methods for all the cases studied of the cantilever beam (number of random variables=3)

Case Random variables Distribution COV FORM SORM MC IS DS LHS

152

fmax Lognormal 0.05

11 hB1, Lb Lognormal 0.1 0.01470 0.01378 0.01291 0.01717 0.01316 0.01318

fmax Lognormal 0.1

12 hB1, Lb Lognormal 0.2 0.13781 0.12905 0.11029 0.10655 0.11422 0.13766

fmax Lognormal 0.2

13 hB1, Lb Lognormal 0.3 0.23292 0.21762 0.22649 0.21090 0.18678 0.18043

fmax Lognormal 0.3

14 hB1, Lb Lognormal 0.5 0.32725 0.30371 0.28458 0.27807 0.38822 0.34040

fmax Lognormal 0.5

15 hB1 Lognormal 0.1 0.13937 0.13684 0.11772 0.25863 0.14539 0.20151

Lb Lognormal 0.3

fmax Lognormal 0.1

16 hB1, Lb Normal 0.1 0.01421 0.01369 0.01446 0.01466 0.01322 0.01313

fmax Normal 0.1

17 hB1 Normal 0.1 0.01561 0.01554 0.01519 0.01637 0.01997 0.01704

Lb Lognormal 0.1

fmax Normal 0.1

18 hB1, Lb Normal 0.1 0.01449 0.01356 0.01298 0.01349 0.01311 0.01328

fmax Lognormal 0.1

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Table 6.10: Failure probability estimated with dierent methods for all the cases studied of the cantilever beam (number of random variables=7)

Case Random variables Distribution COV FORM SORM MC IS DS LHS

19 hB1, Ec Lognormal 0.05 3.76E-06 3.61E-06 6.00E-06 3.58E-06 3.94E-06 2.59E-06

dbar, ExcA, Es Lognormal 0.05

Lb Lognormal 0.05

F orce1 Lognormal 0.05

20 hB1, Ec Lognormal 0.1 0.01798 0.01557 0.01648 0.01899 0.01643 0.01632

dbar, ExcA, Es Lognormal 0.1

Lb Lognormal 0.1

F orce1 Lognormal 0.1

21 hB1, Ec Lognormal 0.2 0.14949 0.13680 0.11741 0.11175 0.13177 0.14121

dbar, ExcA, Es Lognormal 0.2

Lb Lognormal 0.2

F orce1 Lognormal 0.2

22 hB1, Ec Lognormal 0.3 0.24779 0.22652 0.18385 0.21789 0.19206 0.21644

dbar, ExcA, Es Lognormal 0.3

Lb Lognormal 0.3

F orce1 Lognormal 0.3

23 hB1, Ec Lognormal 0.5 0.34869 0.33052 0.26618 0.71120 0.24515 0.28015

153

Lb Lognormal 0.5

F orce1 Lognormal 0.5

24 hB1, Ec Lognormal 0.1 0.04633 0.04056 0.03791 0.03459 0.04606 0.03702

dbar, ExcA, Es Lognormal 0.3

Lb Lognormal 0.1

F orce1 Lognormal 0.2

25 hB1, Ec Normal 0.1 0.01798 0.01634 0.01640 0.01867 0.01755 0.01659

dbar, ExcA, Es Normal 0.1

Lb Normal 0.1

F orce1 Normal 0.1

26 hB1, Ec Lognormal 0.1 0.01716 0.01692 0.01648 0.01567 0.01796 0.01745

dbar, ExcA, Es Lognormal 0.1

Lb Normal 0.1

F orce1 Normal 0.1

27 hB1, Ec Normal 0.1 0.01836 0.01709 0.01651 0.01766 0.01727 0.01621

dbar, ExcA, Es Normal 0.1

Lb, F orce1 Lognormal 0.1

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Computational eort The computational eort required for a reliability analysis depends on

the computational time needed by the physical model to evaluate the response of the structure

considered, as well as on the number of calls of the deterministic model needed to assess failure

probability. In the current study, minimizing the computational eort of a reliability analysis is of

a great concern, especially for deterministic models, such as nite element models, that are very

time consuming to evaluate the output variable values. Therefore, the computational eort of each

probabilistic method is examined in term of the number of calls of the deterministic model in order

to analyze their eciency and help to initially select an appropriate method that provides accurate

results and reduces considerably the reliability analysis computational cost for a specic type of prob-

lem. The number of calls of the deterministic model is highly dependent on the magnitude of failure

probability in the case of simulation methods, and on the convergence criterion of AbdoRackwitz

algorithm to nd the design point in the case of approximation methods.

By comparing the number of calls of the deterministic model required by the various methods

for the beam problem, it is clear that approximation methods can reduce this number tremendously

and independently of the failure probability magnitude and ensures a reasonable accurate estimate

of failure probability for all cases studied in comparison with MC (Tables 6.8-6.10). However, a

degree of verication for a specic type of problem is necessary to ensure that FORM and SORM

provides convenient results, notably in case of highly nonlinear limit state functions. In the com-

putations performed by means of simulation probabilistic methods for cantilever beam, the COV

that represents the desired degree of precision on failure probability estimate is xed at 10% with a

condence level of 95% and the number of simulations required to assess failure probability is limited

to a maximum allowed value of 100000 simulations. It should be noted that the number of calls of

deterministic method for simulation methods depends on the convergence criterion in term of this

COV and a more signicant number of simulations is needed for less values of this COV. MC is

well-known to be the most eective and robust tool to estimate failure probability, but this method

becomes very time consuming when computing small failure probabilities due to the large number of

calls of the deterministic model required in such a case. As can be seen in Tables 6.8-6.10, for cases

1, 10 and 19, failure probability is of the order of 1E-6 and values provided by MC are not accurate

since the maximum allowed value of 100000 simulations is not sucient to reduce the uncertainty

on the estimate of failure probability to an accepted level. Thus, the eciency of MC is not veried

for these cases and a larger number of simulations is necessary to assess an accurate estimate. Al-

ternatively, simulation methods based on a variance-reduction principle are more ecient in term of

accuracy and provide similar results as FORM and SORM for cases with low failure probabilities. In

addition, the number of simulations is signicantly reduced by 97.5% for IS (number of simulations

< 2500) and 69% for DS (number of simulations < 31000). LHS is as much time consuming as

MC (number of simulations > 100000) but leads to an accurate estimate of probability exceeding a

displacement threshold.

For other cases studied, the number of calls of deterministic model is presented in term of failure

probability magnitude for dierent probabilistic methods (Figure 6.7). As can be seen, FORM

and SORM are the less time consuming methods and the corresponding number of calls does not

depend on the failure probability magnitude. For DS, the number of calls of the deterministic model

varies according to the random radial directions generated. For failure probabilities higher than

0.01 and in the cases of large number of random variables, MC requires less simulations than DS to

converge to results of similar accuracy. LHS gives comparable results to MC for all cases studied,

but no reduction in computational eort is observed, as expected, for any magnitude of failure

probability. Considering the duration of computations associated to IS, it is clear that, among

simulation probabilistic methods, IS is the less time consuming since it requires only a limited

number of evaluations of the deterministic model, and the number of simulations decreases with

the increase of failure probability. But, for low failure probabilities, IS is more time consuming in

comparison with approximation methods. IS loses its accuracy in the cases of random variables with

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Figure 6.7: Number of calls of beam FEM in term of failure probability magnitude for dierent

probabilistic methods

Eect of random variables number In order to compare the accuracy and precision of proba-

bilistic methods in assessing similar results to MC and examine the eect of the number of random

variables considered, all cases studied are divided into 3 subset of 2, 3 and 7 random variables. Cases

1, 10 and 19 are not considered for the following steps of the reliability analysis of the cantilever

beam since MC failed to provide an accurate estimate of failure probability due to an insucient

Pf (method)

number of simulations. As previously mentioned, the mean value and COV of the ratio

Pf (M C)

estimate the accuracy and the precision of probabilistic methods in comparison to MC, respectively.

A mean value close to 1.0 indicates that the relevant method provides similar results to MC and a

COV close to 0 indicates an estimation of failure probability with high precision.

As can be seen in Figure 6.8, simulation and approximation methods give good results for most

cases and failure probabilities are estimated with high accuracy in comparison to MC, the mean

of probabilities ratio for each subset of random variables number is identical. For approximation

methods, accuracy decreases as the number of random variables increases. This may be due to

the fact that, as the number of random variables increases, the limit state surface becomes more

complex with potentially other local optimum points and AbdoRackwitz algorithm may converge

to a local but not a global optimum point. For both methods (FORM and SORM) there is a

signicant dierence in precision for the subset of 2 and 7 random variables. For a higher number of

random variables, overall precision of SORM is observed to be low with a high degree of variation

Pf (SORM )

(COV = 0.175). For IS, precision decreases with the number of random variables. Low

Pf (M C)

precision as well as low accuracy results are particularly found for random variables with high and

mixed COV (cases 14, 15, 23 and 24). It should be noted that the case 23 with 7 random variables

of a COV of 0.5 is not considered for IS to represent the mean and COV of probabilities ratio.

Pf (IS)

Otherwise, the ratio

Pf (M C) is found to have a COV of 0.439. When LHS and DS are considered,

precision increases signicantly with the number of random variables and a the estimate of failure

probability is achieved with a COV less than the desired degree of variation of 0.1.

According to the sensitivity analysis performed to identify the random variables that mostly

contribute to the variability of the displacement of cantilever beams, the variation of failure proba-

bility in terms of the force applied at the free end of the beam and the number of random variables

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Figure 6.8: Eect of the number of random variables on the accuracy and precision of dierent

probabilistic methods in comparison to MC

The rst corresponds to the case where all variables are considered as random variables (10

variables).

The second corresponds to the case of the 7 variables that aect the most the estimate of

failure probability with respect to their omission factor (Table 6.4).

The third represents the case of the most signicant variables identied by Quadratic combi-

nation method (3 variables, Table 6.2).

The fourth represents the case with the 2 most inuential variables related to the beam di-

mensions (Table 6.6 ).

Statistical description of these variables is detailed in Table 6.1. As can be seen in Figure 6.9, a good

agreement is found between the estimated values of failure probabilities in the case where all random

variables are taken into account and the case where the number of random variables is reduced to

two after performing a sensitivity analysis. The variation of failure probability is presented in term

of the force applied at the free end of the beam, which indicates that the dierence between failure

probabilities estimated for each case is not signicant. To emphasize the importance of considering

all the most inuential parameters in a reliability analysis, the case of one random variables is

studied. In this case, only the beam width hB1 is represented as a random variable and its length

Lb is considered as a deterministic variable. Results with one random variable shows a substantial

dierence in comparison to cases where the 2 most inuential variables are modeled as random

variables. Figure 6.9 illustrates the importance of performing a sensitivity analysis since a bad

choice of random variables may lead to erroneous values of Pf . In addition, sensitivity analysis helps

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Figure 6.9: Variation of failure probability in terms of the force applied at the free end of the beam

and the number of random variables considered

reducing the number of random variables, which subsequently reduces the computational eort of a

reliability analysis.

Eect of the type of random variables distribution In order to compare the accuracy and

precision of probabilistic methods in assessing similar results to MC and examine the eect of the

type of random variables distribution, all cases studied are divided into 3 subset of normal, lognormal

and mixed distributions (mixed term refers to cases where random variables are not modeled with

the same probability distribution type). For this step, cases 1, 10 and 19 are also not considered as

MC provides inaccurate results. The mean values and COV of probabilities ratio are also compared

to MC results in order to estimate the accuracy and the precision of probabilistic methods with

respect to the type of distribution.

As can be seen in Figure 6.10, simulation and approximation methods give good results for most

cases and failure probabilities are estimated with high accuracy in comparison to MC, except for DS

that shows low accuracy in cases containing distributions other than normal. FORM and SORM

show lower accuracy and lower precision in the case of random variables modeled with Lognormal

distribution, this is due to the isoprobabilistic transformation required by approximation methods

to simplify the joint probability function. This transformation associates random variables in the

physical space with standardized and independent random variables in the standard space. Thus,

the transformation of random variables with normal probability distribution is more precise and

accurate. Considering simulation probabilistic methods, IS gives best results for accuracy as well

as for precision. Case 23 with 7 random variables of a COV of 0.5 is also not considered for IS to

represent the mean and COV of probabilities ratio. Precision of LHS and DS is signicantly aected

by the type of random

variablesdistribution with a high degree of variation in case of Lognormal

Pf (DS, LHS)

distribution (COV

Pf (M C) ' 0.25).

Another study of the eect of the type of random variable distributions is performed. For this

purpose, the main case of study is considered with two random variables, namely hB1 and Lb.

Both random variables are modeled with the same type of distribution by considering all parametric

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Figure 6.10: Eect of the distribution type of random variables on the accuracy and precision of

dierent probabilistic methods in comparison to MC

The values of distribution statistical parameters are estimated in terms of the statistical moments

for each random variable. First, the mean and COV values of the beam displacement are evaluated

in order to examine which distributions provide the same results for the variable of interest. Table

6.11 shows that the displacement mean value is not aected by the type of probability distribution,

but its variance is highly dependent on the distribution type and the displacement COV may reach

a value of 12.22 in the case where random variables are modeled with Logistic distribution. Among

the probability distributions considered, only 8 distributions allow the same statistical characteriza-

tion of the beam displacement that represents the output variable of the problem. Afterwards, the

inuence of these 8 probability distributions on the beam reliability is studied. Figure 6.11 indicates

that reliability depends signicantly on the probability distribution type used for random variables.

Results obtained in the cases of Beta, Gamma, Lognormal and Normal distributions show a good

agreement regardless of the probabilistic method used and the dierences between the estimated

values of failure probability are inconsiderable. The results obtained in the present section illustrate

the necessity of selecting the appropriate probability distribution type to model variable uncertain-

ties. Thus, it is not sucient to provide the mean and COV of random variables, but an appropriate

PDF must also be specied for each variable.

Eect of COV In order to compare the accuracy and precision of probabilistic methods in assess-

ing similar results to MC and examine the eect of the COV of random variables, all cases studied

are divided into 6 subset of 0.05, 0.1, 0.2, 0.3, 0.5 and mixed distributions (mixed term refers to

cases where random variables are modeled with dierent COV values). The mean values and COV

of probabilities ratio are also compared to MC results in order to estimate the accuracy and the

precision of probabilistic methods with respect to the COV of random variables.

Cases 1, 10 and 19 are considered in this step, they represents the cases of a COV equal to

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Table 6.11: Failure probability estimated with dierent methods for dierent types of probability

distribution (number of random variables=2)

Type Displacement Pf

µ (mm) σ (mm) COV FORM SORM MC IS DS LHS

Beta 0.614 0.254 0.42 0.01154 0.01060 0.01069 0.01133 0.01187 0.00968

ChiSquare 0.614 5.206 8.48 0.44222 0.44215 0.39102 0.35001 0.49413 0.34021

Exponential 0.614 0.254 0.42 0.02135 0.01763 0.01594 0.01842 0.01702 0.02245

Gamma 0.614 0.254 0.42 0.01157 0.01079 0.00941 0.00955 0.01061 0.01185

Gumbel 0.614 0.254 0.42 0.01533 0.01408 0.01192 0.01502 0.03935 0.01499

Histogram 0.614 0.415 0.72 0.04571 - 0.02605 0.03206 - 0.027282

Laplace 0.614 0.254 0.42 0.01054 0.01438 0.01551 0.01286 0.01585 0.01249

Logistic 0.614 12.79 20.84 0.47817 0.47311 0.07802 0.06679 0.04044 0.08630

Lognormal 0.614 0.254 0.42 0.01176 0.01098 0.01004 0.01001 0.01237 0.00892

NonCentral 0.614 7.92 12.89 0.46761 0.46361 0.12517 0.10236 - 0.10101

Student

Normal 0.614 0.254 0.42 0.01147 0.01071 0.00974 0.01216 0.00982 0.00838

Rayleigh 0.614 1.33 2.16 0.26306 0.22506 0.25424 0.29613 0.26058 0.27137

Student 0.614 12.22 19.91 0.46794 0.46302 0.11042 0.09995 0.06498 0.09391

Triangular 0.614 0.809 1.32 0.24727 0.23364 0.21294 0.19674 0.21188 0.21763

Truncated 0.614 0.254 0.42 0.011471 0.10691 0.00974 0.01215 0.00982 0.00838

normal

Uniform 0.614 1.14 1.86 0.35463 0.32210 0.26335 0.22915 0.24028 0.24516

Weibull 0.614 0.254 0.42 0.01621 0.01694 0.01567 0.01494 0.01888 0.01735

Figure 6.11: Eect of the type of random variables distribution on the estimation of failure proba-

bility for dierent probabilistic methods

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Figure 6.12: Eect of the COV of random variables on the accuracy and precision of dierent

probabilistic methods in comparison to MC

0.05 and low failure probability values. As can be seen in Figure 6.12, the dierence between MC

and other probabilistic methods considered is huge due to the inaccurate results obtained by MC.

For this subset failure probabilities are estimated with low accuracy for MC due to an insucient

number of simulations. However, IS and DS are more ecient in term of accuracy and provide

similar results as FORM and SORM for cases with low failure probabilities. For other subset, it can

be seen that the accuracy and precision of simulation and approximation methods degrades with

the increase of the random variables COV. The most inaccurate and imprecise results are obtained

with IS for COV ≥ 0.3 and DS for COV ≥ 0.2. IS for COV ≤ 0.2, FORM and SORM give best

results for accuracy and precision in term of the COV of random variables.

In order to have a better understanding on the eect of COV of random variables, the case of

7 random variables modeled with Lognormal distribution is considered assuming that all random

variables have the same COV value. The variation of failure probability is rst presented in term of

the mean of the force applied at the free end of the beam for dierent values of COV (Figure 6.13),

then presented in term of the COV for dierent values of the mean of the force (Figure 6.14). MC

is used to assess failure probability with a maximum allowed value of 1E6 simulations.

Figure 6.13 shows that the inection point of curves corresponds to the case where the limit state

function is equal to zero (G = fmax − f = 0). At this point, the value of the displacement threshold

is equal to the mean value of the calculated displacement and failure probability is equal to 0.5.

The corresponding force value is approximately of the order of 2800 N. Figure 6.14 shows failure

probability values as a function of the COV. The linear equation of Pf = 0.5 describes the limit

state function that corresponds to the point of intersection of the curves presented in the previous

gure. This gure permits to evaluate the range of low failure probabilities in term of the mean

value of the force applied at the free end of the beam. It can be seen that, for a force less than 200

N and a COV less than 0.4, failure probability represents values less than 0.005.

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Figure 6.13: Variation of failure probability in term of the mean of the force for dierent values of

COV

Figure 6.14: Variation of failure probability in term of the COV for dierent values of the mean of

the force

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Figure 6.15: Comparison of the beam displacement distributions estimated with the initial model

and Taylor expansion

6.2.4.4 Metamodeling

After examining the most inuential input variables, the number of random variables is reduced to

two in this section (beam width and length). The next step consists in evaluating the validity and

eciency of response surface methods provided in OpenTURNS, namely the Taylor expansion, the

Least Square method and the polynomial chaos expansion. The principal of such methods consists

in replacing the initial model with an approximate polynomial model in the aim of simplifying

reliability analysis of complex structures and reducing the computational eort to assess failure

probability of exceeding a threshold. In this section, 2 random variables are considered and modeled

with Lognormal probability distribution and a COV of 0.1. First, the distribution of the output

variable, which is in this case the displacement at the free end of the beam, obtained with the initial

model is analyzed and compared to that obtained with the available response surface types (Figures

6.15-6.17). For each type, 100000 random simulations of the input vector are generated with the

initial model of the beam and using the numerical integration Monte Carlo method. Displacement

values are determined for each simulation, then polynomial coecients of the approximate model are

estimated through the nite number of simulations of the deterministic model. The displacement

probability distribution is rst represented graphically using the Kernel Smoothing method. The

number of simulations used is sucient to obtain a smooth Kernel density estimate in the case of

the initial model and displacement values are distributed around a mean value of 0.614 mm with

a standard deviation of 0.254 mm. As can be seen, Least Square method and Taylor expansion

of 1st order fails in describing the beam displacement density, while the best t is obtained with

polynomial chaos expansion for cleaning and xed strategies.

Furthermore, approximate models obtained from several response surface types are combined

with dierent probabilistic methods in order to study the eciency of response surface methods in

assessing failure probability. For each type, failure probability is compared with that obtained in

the case of the beam FEM combined with the dierent probabilistic methods (Figure 6.18). Least

Square method, Taylor expansion and polynomial chaos expansion with sequential strategy are

found to signicantly underestimate failure probability, while results provided by polynomial chaos

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Figure 6.16: Comparison of the beam displacement distributions estimated with the initial model

and Least Square method

Figure 6.17: Comparison of the beam displacement distributions estimated with the initial model

and polynomial chaos

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Figure 6.18: Comparison of failure probability estimated with the initial model and several response

surface types

with xed or cleaning strategy show good agreement with those obtained in the case of the beam

FEM combined with a particular probabilistic method. It should be noted that the computational

eort needed to assess failure probability in the case of approximate models combined with any

probabilistic method is negligible (time of order of several seconds).

The problem of the cantilever beam is used as a simple example in the aim of mastering the basics of

OpenTURNS and examining the probabilistic methods proposed in OpenTURNS to estimate failure

probability in structural reliability analysis. The principle of each step of a reliability analysis is

illustrated and the available probabilistic methods are compared in terms of their accuracy, precision

and computational eort. The FEM of the RC beam is modeled with CASTEM and combined with

OpenTURNS using the wrapper interface.

This application allows drawing some preliminary conclusions which highly served the purposes

of other types of applications in order to initially select an appropriate probabilistic methods, given

computational eort constraints. Although Monte Carlo method is a robust method for several

structural reliability analysis, it is often impractical to use due to the large number of simulations

required to reach the desired level of accuracy. Comparing other probabilistic methods to MC, IS is

the method that gives best results in terms of accuracy and precision for dierent subset examined,

except for the subset of random variables with high COV (COV=0.5). For this subset, IS loses its

accuracy in estimating MC failure probabilities. DS is signicantly aected by the number of random

variables and is less precise for high random variables cases (7 random variables). DS is the method

that gives the highest values of the COV of probabilities ratio leading to less condence in failure

probabilities estimated with this method. LHS gives comparable results to MC, but no reduction

in computational eort is observed. For FORM and SORM, failure probabilities are estimated with

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high accuracy in comparison to MC, except in cases of high COV (COV>0.2). In term of reliability

analysis computational eort, FORM and IS are the less consuming methods in comparison to

other methods. However, these conclusions could not be generalized for every problem in reliability

analysis and some degree of verication should be conducted to ensure the eciency of methods

used.

In this section, a reliability analysis of RC slabs under low velocity impact is presented assuming a

exural mode of failure and an elastic behavior for steel and concrete. The behavior of RC slabs

in bending has two major phases before the failure, the rst is related to the increase of deection,

the second is related to steel yielding [60]. Probabilistic methods used in this section to study the

reliability of impacted slabs are the most known methods in the reliability domain, namely FORM,

SORM, MC and IS. Several criteria of failure are considered, they are related to the displacement of

slab at the impact point, the peak force of impact, stresses in reinforcement, and the impact energy

absorbed by the slab. To study the dynamic eect of impact applied to RC slabs, 3 deterministic

models are used and evaluated: a 3D FEM simulated with Abaqus/Explicit, a simplied model based

on the plate theory [160] and a two degrees of freedom mass-spring system [29]. The aim is to select

the most appropriate model to use in RC slabs reliability analysis, while accurately representing the

mechanical behavior of the slab studied and reducing the computational cost to the minimum. The

model chosen is then used in a parametric study in order to investigate the eect of several input

parameters of the deterministic model on the calculation of failure probability.

6.3.1.1 Finite element model

This section describes the 3D FE model of the RC slab subjected to impact within nuclear plant

buildings using ABAQUS/Explicit. Geometry of slab and impactor, reinforcement and material

properties, as well as initial and boundary conditions are present in section 5.4. A 3D solid element,

the eight-node continuum element (C3D8R) is used to develop the mesh of the concrete slab. As

illustrated in Figure 6.19.a, the mesh of the concrete slab contains ve elements in the thickness di-

rection with three dierent densities of mesh. The elements at the center of the slab have dimensions

of 5x5x10 cm, while the elements at the corners have dimensions of 5 cm in three directions. Thus,

the elements of regions between the center and corner regions are of 5x10x107 cm dimensions. The

reinforcement is modeled with truss elements (T3D2) (Figure 6.19.b) and the embedded approach

is used to create the bond between the steel reinforcement and concrete. This approach allows inde-

pendent choice of concrete mesh and arbitrarily denes the reinforcing steel regardless of the mesh

shape and size of the concrete element.

The impactor is meshed with four-node tetrahedral elements (R3D4). The denition of contact

properties and interaction types is very important for impact problems, but in this case of study this

denition is limited to a normal interaction and a hard contact with nite sliding. The contact pair

algorithm is used in order to model contact between the upper surface of the slab and the impactor

surface. The contact constraints are enforced with a pure master-slave kinematic contact algorithm

which does not allow the penetration of slave nodes into the master surface (surfaces attached to

rigid bodies must always be dened as master surface in Abaqus/Explicit).

At rst, nonlinearities due to concrete and steel behaviors are not considered, both materials

are assumed to have an elastic behavior. The only nonlinearity present in the FEM is due to the

contact algorithm between the impactor and the slab. The FEM of the slab consists of 53000 nodes

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Figure 6.19: Finite element model of slab: a) meshing, b) reinforcement

and more than 44900 elements, and a simulation takes a total time of 25 minutes for an impact step

of 0.1 s.

The Classical Laminated Plate Theory is used to develop an analytical solution for RC slabs. This

theory is based on the kinematic assumptions of Kirchho-Love for thin plates with small deections.

The approximate Navier solution is used to estimate the deection of the mid-surface of the slab

(w0 ). Thus, the deection of slab can be expressed as:

∞ X

X ∞

w0 (x, y, t) = Wmn (t) sin(αx) sin(βy) (6.4)

m=1 n=1

The impactor in this model is replaced by a concentrated force q(x, y, t) applied at the impact point

and expressed as following:

∞ X

X ∞

q(x, y, t) = Qmn (t) sin(αx) sin(βy) (6.5)

m=1 n=1

4Q0 (t)

where Qmn (t) = ab sin(αx0 ) sin(βy0 ) for impacted slab cases (x0 , y0 ) are the coordinates of

the impact point.

The aim is to determine the coecients Wmn (t). This can be achieved by integrating the following

constitutive relation over the slab surface:

∂ 4 w0 ∂ 4 w0 ∂ 4 w0

2

∂ ẅ0 ∂ 2 ẅ0

− D11 + 2 (D12 + 2D66 ) 2 2 + D22 +q(x, y, t) = I0 ẅ0 −I2 + (6.6)

∂x4 ∂x ∂y ∂y 4 ∂x2 ∂y 2

For this case of study, the coecients Wmn (t) can be estimated as:

ˆt

4 sin(αx0 ) sin(βy0 )

Wmn (t) = Q0 (τ ) sin(wmn (t − τ ))dτ (6.7)

abwmn I0

0

where a and b are the slab length and width, respectively. Values of Wmn (t) are determined

analytically assuming a sinusoidal evolution in time of impact loading [105].

This model is developed using Matlab, it consists of the same input variables as the FEM except

those related to the impactor. Impactor velocity, mass and radius are replaced in this model by an

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impact force which is estimated by means of the FEM. The impact time is taken equal to that of

the FEM impact step.

A simplied model of two degrees of freedom mass-spring system is developed using Matlab. This

model was proposed by CEB [29] to study the behavior of slabs subjected to hard impact. In this

case, the kinetic energy of the impactor is absorbed by the structure deformation, and the slab

local behavior as well as its overall deformation must be considered. The model consists of two

masses, that represent the impactor and slab masses, respectively, and of two springs that describe

the stiness of slab and the contact force, respectively. The dierential equations of equilibrium

of the two masses are previously detailed in Section 1.3.2.2. The rigidity of slab and the contact

force-time are determined by means of the FEM. The equivalent mass of slab in the MSM is obtained

by multiplying the mass of the real slab with a transformation factor. The slab mass, as well as

the transformation factor, highly depend on slab dimensions, material densities, bars diameter, and

number of longitudinal and transversal reinforcement bars. Impactor velocity, mass and radius are

the same as those used for the FEM. It should be noted that the transformation factor is the only

parameter calibrated for the MSM in order to nd the best t with the FEM.

Assuming an elastic behavior for steel and concrete, the variables which may be subjected to statis-

tical variations in RC slabs under impact can be classied into four categories:

1. Dimensional variables related to geometric properties of the slab: length, width and thickness

of the slab.

2. Variables related to the impactor and its kinetic energy: radius, mass and velocity of the

impactor.

It should be noted that the length and width of slab are not introduced into deterministic models to

model their relevant uncertainties, but they are considered to cover a wide range of slab geometries.

The probabilistic characterization of random variables is based on some literature reviews. Table

6.12 represents the random variables adopted for this application, their probability distribution, their

mean and COV. Mean values of random variables are taken equal to their nominal values, and their

COV is determined according to references cited in the fourth column. Probability distributions are

chosen according to the reliability based code calibration of JCSS (Joint Committee on Structural

Safety) [89]. JCSS code contains several rules and basics that are essential for the design of new

structures or evaluating the response evolution with time of already existing structures in a reliability

framework. The modulus of elasticity of steel is considered as deterministic variable due to its small

dispersion [126]. It should be noted that the following probabilistic characterization was adopted

before performing the literature review detailed in section 4.6.

Gay and Gambelin [67] described the problem that an engineer could expect during the analysis and

design of RC structures under certain loading conditions. This problem involves two fundamental

components:

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Table 6.12: Random variables of the RC slab and their statistical descriptions (elastic behavior)

Geometric properties of the slab

Length, a Normal 8.1 m 0.05 [127]

Width, b Normal 4.85 m 0.05 [127]

Thickness, e Normal 0.5 m 0.05 [127]

Impactor variables

Radius, r Normal 15 cm 0.05 [38]

Mass, m Lognormal 3600 kg 0.05 [38]

Velocity, v Lognormal 7.7 m/s 0.1 [38]

Concrete properties

Young's modulus, Ec Lognormal 35 GPa 0.1 [123]

Density, ρc Lognormal 2500 kg/m

3 0.1 lack of information

Reinforcement properties

Bars diameter, dA Lognormal 20 mm 0.1 lack of information

Stiness design for which the structure must not endure displacements higher than a particular

threshold dened in design specications (serviceability limit states in Eurocode).

Load-carrying capacity design for which the structure must continue to perform its functional

requirements under the applied loadings (ultimate limit states in Eurocode).

The rst aspect to consider for the application of impacted RC slabs assuming an elastic behavior

of materials is the displacement of slab at the impact point. Thus, the rst criterion is dened in

term of this displacement:

g1 = umax − u (6.8)

In this equation, umax is the displacement threshold and u represents the maximum displacement

of slab at the impact point. umax is considered according to Eurocode specications [142] equal to

a

250 , with a is the length of slab. The values of u are calculated according to the deterministic model

used and the values of input random variables generated in OpenTURNS. The RC slab is modeled

assuming an elastic behavior of materials, however this assumption does is not representative of the

actual response and the maximum displacement of slab is underestimated of approximately 60% in

this case [18]. Therefore, a coecient of 1.5 is introduced in the expression of the displacement failure

criterion in order to take into account plastic deformations related to actual materials behavior. The

displacement criterion is then expressed as:

[18] supposed that the failure of SDR protective slabs initiates when yielding of main reinforcement

occurs. Thus, a second failure criterion can be expressed in term of steel stresses as following:

g2 = fy − σs (6.10)

where fy is the yield strength of reinforcement and σs is the stress in reinforcement. For this

ultimate limit state function, failure probability can be assessed only through the combination of

slab FEM with OpenTURNS, stresses in reinforcement cannot be evaluated using the mass-spring

model nor that based on the plate theory. This criterion can be dened in such a way that stresses

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in any nite element of reinforcement should not reach the yield strength of steel, but exceeding

the threshold of stresses in one steel element does not necessarily lead to slab failure. Therefore the

variable of interest of steel stress is replaced by the variable related to the number of reinforcement

FE elements nA in which yielding occurs, and a threshold nAmax equal to the number of FE elements

in 3 longitudinal bars and 3 transversal bars is assigned.

g2 = nAmax − nA (6.11)

An energy criterion for RC slabs subjected to impact was dened by Miyamoto and King [130].

They indicated that this criterion would be the most ecient method of designing RC structures

under impact loads, especially for a exural failure mode. The criterion consists of that structural

failure is likely to occur if the slab is not capable of absorbing all of the energy transmitted during

impact collision. A part of the impactor kinetic energy

I

Ekin0 is transferred to slab as kinetic energy

S

and irrecoverable strain energy Eplas1 that results from the formation of cracks, friction and damping.

Therefore, a third ultimate limit state function corresponding to this criterion is considered assuming

that the slab should absorb 80% of the impact energy:

I S

g3 = 0.8Ekin0 − Eplas1 (6.12)

Another important criterion related to the impact force should also be examined. The evolution

in time of impact force represents one of the main problems of impacts on RC structures. It depends

on two colliding bodies geometry, their stinesses and the initial conditions of the impactor. Thus,

a fourth ultimate limit state function is considered:

g4 = Fmax − F (6.13)

where Fmax represents the ultimate load-carrying capacity of slab and from is determined a limit

load analysis, F is the peak of impact force calculated with the slab FEM.

6.3.4.1 Comparison of deterministic models

In order to be able to use simplied analytical model in reliability analysis, their eectiveness in

predicting the response of slab should be rst veried. Thus, mass-spring model and that based on

plate theory are compared to the nite element model. Table 6.13 shows the displacement of slab

at the impact point obtained with dierent models developed to evaluate the behavior of impacted

RC slabs assuming an elastic behavior of materials. It can be seen that the mass-spring model

allows a better estimation of displacement at the center of slab for dierent values of velocity with

a maximum error of only 1.5%. However, the analytical model based on plate theory is less precise

and estimate the displacement with a maximum error of 5%.

Next, deterministic models are compared in a reliability framework. Failure probability is as-

sessed using FORM method by combining the dierent deterministic models to OpenTURNS for

several values of velocity. Figure 6.20 shows that failure probability increases with the mean of

velocity for the displacement criterion. Results obtained with MSM are in good agreement with

failure probabilities estimated by coupling FORM with the FEM. In addition, the MSM is found

to be more eective in term of computational cost. Thus, it is selected to study the eect of input

variables on the estimate of failure probability in the case of impacted slabs with elastic behavior of

materials.

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Table 6.13: Comparison of displacement of slab at the impact point for dierent deterministic models

and dierent values of velocity (elastic behavior)

(m/s) u3 (mm) u3 (mm) Error (%) u3 (mm) Error (%)

7.7 14.85 14.11 4.95 14.62 1.54

10.5 18.29 17.97 1.74 18.24 0.27

13.2 21.51 21.23 1.32 21.57 0.27

15.5 24.26 24.09 0.67 24.23 0.14

19.7 29.24 28.88 1.24 29.19 0.17

24.4 34.32 33.36 1.94 34.13 0.55

Figure 6.20: Failure probability with dierent deterministic models and displacement criterion

For this application, two types of probabilistic methods are used and compared, namely approxi-

mation methods and simulation methods. As previously discussed in section 4.4.3, approximation

methods such as FORM and SORM are based on an approximation of the limit sate function, while

simulation methods such as MC and IS are based on a number of samples of random variables to

calculate failure probability. These methods are combined with the MSM and results are compared

to failure probabilities obtained with the FEM combined with FORM in order to choose the most

suitable probabilistic method to use in the parametric study. The MSM is used due to its reduced

computational time which allows combining it with simulation probabilistic methods, and because

it permits to accurately predict the physical response of slab as proven in the previous section.

Table 6.14 shows that FORM coupled with the MSM gives the closest values of failure probabilities

to those estimated with the numerical model. MC overestimates failure probabilities for velocities

equal to 10.5 and 13.2 m/s, this may be related to the fact that MC necessitates a huge number of

simulation for low probabilities range and that the transformation factor is considered as constant

parameter for all samples generated by MC and is not calibrated in terms of slabs variables, such as

slab dimensions, material densities and bars diameter.

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Table 6.14: Comparison of probabilistic methods combined with the MSM for the displacement

criterion

(m/s) FORM FORM SORM MC IS

7.7 0.00016 0.00021 0.00001 - 0.00043

10.5 0.03469 0.06715 0.06542 0.023611 0.06924

13.2 0.44949 0.48965 0.49107 0.75781 0.47835

15.5 0.86815 0.83744 0.99508 0.84211 0.98103

19.7 0.99906 0.99149 0.99999 0.96552 -

24.4 1.0 1.0 1.0 1.0 1.0

A comparison of dierent failure criteria dened in section 6.3.3 is important for a better under-

standing of the behavior of RC slabs subjected to impact with an elastic behavior of materials.

Figure 6.21 shows failure probability in term of the mean of impact velocity for the dierent failure

criteria considered for this application. It can be seen that the slab is more resistant to displacement

and steel yielding, while it is less resistant to the contact force for the same values of impact velocity

mean. In this case and for an optimal design, it is necessary to evaluate the maximum contact force

value that the slab can sustain, as well as the impactor velocity for which this maximum value of

force is reached. Once this velocity is determined, the performance of slab under displacement and

steel yielding criteria is subsequently veried.

In this gure, the performance of slab regarding its capacity of dissipating impact energy is also

examined. Unlike other criteria, failure probability decreases with the impact energy for this criterion

as a result of the energy dissipated by steel and slab deformations that increase with velocity. The

performance of slab in dissipating energy can be improved by considering the nonlinear behavior of

concrete and steel, which allows modeling concrete damage and the onset of plastic deformations.

It should be noted that this criterion does not indicate a failure mode of slab, it is used to study the

energy transmitted to RC slabs during an impact phenomenon.

The following parametric study is performed in the aim of investigating the eect of dierent input

variables on the estimation of failure probability for the application of impacted RC slabs. For this

purpose, the MSM is selected because, among the deterministic models proposed for this problem,

it is the most ecient in term of computational eort for a reliability analysis. In addition, the

MSM model is veried to accurately predict the displacement of slab at the impact point. Thus, the

parametric study is carried out by combining the MSM with FORM for the displacement criterion

and the eect of several input variables on the reliability of impacted RC is examined. Variables

such as slab dimensions, concrete density, steel bars diameter, impactor mass and rigidity of slab

are considered and failure probability is estimated for dierent values of the mean of the variable

studied.

Slab length Failure probabilities are calculated for several mean values of the slab length that

varies between 2.35 and 12 m. Figure 6.22 shows that, for a simply supported slab, reducing

the slab length leads to an increase in failure probability for the displacement criterion and that

this probability depends on the impactor velocity. Thus, an optimal design of RC slabs requires to

properly determine the structure dimensions and choose the appropriate values depending on impact

velocity in order to improve the performance of slab. It can be seen that a low variation in length

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Figure 6.21: Comparison of dierent failure criteria in term of impact velocity mean

Figure 6.22: Eect of the mean of slab length for the displacement criterion

leads to an important variation in failure probability, this is due to the fact that the displacement

threshold used to dene the displacement failure criteria is directly related to the slab length and

taken equal to L/250. This variation is less signicant in the case of slabs with higher length value

since the displacement threshold is higher.

Slab thickness Figure 6.23 shows failure probabilities of slabs having the same length of 8.1m and

dierent mean values of thickness. It can be seen that the slab thickness directly aects the reliability

of RC slabs, and that the nominal value of 0.5 initially chosen for slab thickness is sucient to ensure

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Figure 6.23: Eect of the mean of slab length for the displacement criterion

the reliability of slab only in the case of impact velocity less than 7.7 m/s (Pf = 10−3 − 10−4 ). For

higher velocities, a higher value of thickness should be considered. The results observed indicate

that for long and thin slabs, bending failure mode dominates. Thus, the displacement of slab at

the impact point is more signicant and failure probability of exceeding a constant displacement

threshold increases with a reduction in thickness.

Concrete density The eect of the mean of concrete density is small in comparison to that of

the slab length and thickness, especially in the cases where v=7.7 m/s and v=24.4 m/s (Figure

6.24). Therefore, this variable can be considered as deterministic variable in the case of low failure

probabilities.

Steel bars diameter In order to study the eect of steel bars diameter on the reliability of

impacted RC slabs, dierent diameters of reinforcement are considered while retaining the same

number of longitudinal and transversal bars. Figure 6.25 shows that the diameter of bars has an

inconsiderable inuence on failure probability for all the impact velocity values considered. However,

this inuence should be examined in the case where the plastic behavior of steel is introduced in the

analysis of RC slabs.

Impactor mass Figure 6.26 illustrates the variation of failure probability in term of the mean

value of the impactor mass. It can be seen that the mass has the same inuence as the impactor

velocity and failure probability increases with the mean value of mass. For v=7.7 m/s, failure

probability remain in the domain of low probabilities for a mass less than 6000 kg. However, for

velocities higher than 7.7 m/s, the eect of slab becomes more signicant and failure probability

varies greatly in term of the mean value of mass. Increasing the impactor mass and velocity lead

to an increase in impact energy, so a slab with the same dimensions and material properties lose its

performance to sustain the impact force applied.

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Figure 6.24: Eect of the mean of concrete density for the displacement criterion

Figure 6.25: Eect of the mean of steel bars diameter for the displacement criterion

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Figure 6.26: Eect of the mean of impactor mass for the displacement criterion

Figure 6.27: Eect of the mean of slab stiness for the displacement criterion

Slab rigidity One of the most important parameters in a mass-spring model is the rigidity of

the spring that represents the stiness of slab. Figure 6.27 shows that increasing the slab stiness

results in a decrease of failure probability and improve the slab performance for a displacement

criterion. This can be due to the fact that a more rigid slab is subjected to smaller displacement,

hence its failure probability for a displacement criterion is lower. RC slabs stiness can be increased

depending on the slab dimensions and material properties.

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6.3.5 Conclusion remarks

In this section, an application of reliability analysis to RC slabs under low velocity impact assuming

a exural mode of failure and an elastic behavior for steel and concrete is presented. Several criteria

of failure related to the displacement of slab at the impact point, the peak force of impact, stresses

in reinforcement, and the impact energy absorbed by the slab are considered. Probabilistic methods

such as FORM, SORM, MC and IS are used to assess failure probability. In order to study the

dynamic eect of impact applied to RC slabs, 3 deterministic models are used: a 3D FEM simulated

with Abaqus/Explicit, a simplied model based on the plate theory and a two degrees of freedom

mass-spring system. The aim is to select the most appropriate model to use in RC slabs reliability

analysis, while accurately representing the mechanical behavior of the slab studied and reducing the

computational cost to the minimum. The analytical MSM is seen to be very eective in predicting

the same values of slab displacement at the impact point as the numerical model, as well as in

estimating failure probabilities very similar to those obtained with the FEM for a displacement

criterion. The MSM is more convenient than numerical models to use in reliability analysis in term

of computational eort. Thus, it is used in a parametric study in order to investigate the eect of

several input parameters of the deterministic model on the estimation of failure probability of RC

slabs. The impact velocity is the most inuential variable, followed by the slab length, thickness and

stiness. Results obtained for this type of problem can be useful for the design of RC slabs subjected

to impact since they permit to identify the values of input variables for which failure probability is

not in the range of low probabilities.

Studying the response of impacted slabs assuming an elastic behavior of materials represents a

preliminary step in the procedure of evaluating the reliability of RC slabs under low velocity impact.

This application is initially considered to simplify the problem and to avoid complicating structural

reliability analysis of slabs as a rst step. However, it is clear that nonlinearities due to materials

behavior have a signicant eect on RC slabs response, in particular that cracking in concrete

under tension and plastic deformations in concrete at the impact zone and in reinforcement reduce

signicantly the peak of contact force obtained with an elastic model. The eect on nonlinearity

due to material properties is discussed in the next section.

behavior

6.4.1 Deterministic models

Deterministic FEM model used for the slab subjected to accidental dropped object impact within

nuclear plant building assuming a nonlinear behavior of materials is detailed in section 5.4, while

the simplied model used consists of a two degrees mass-spring model which is detailed in section

5.4.2.

In the case when the nonlinear behavior of materials is considered in the FEM, the variables which

may be subjected to statistical variations in RC slabs under impact can be classied into ve cate-

gories:

1. Dimensional variables related to the geometric properties of the slab: length, width and thick-

ness of the slab.

2. Dimensional variables related to the position of reinforcement and its size: concrete cover and

diameter of bars.

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3. Variables related to the concrete properties: Young's modulus, density, compressive strength

and tensile strength of concrete.

4. Variables related to the steel properties: modulus of elasticity and yield strength of steel.

5. Variables related to the impactor and its kinetic energy: radius, mass and velocity of the

impactor.

In this study, the probabilistic characterization of random variables is based on some literature

reviews mentioned in previous section 4.6 and in which the statistical descriptions of material prop-

erties and geometry variables for reinforced concrete structures are relatively well documented. For

variables related to the impactor, statistical descriptions that represent the variation of accidental

dropped objects impact during handling operations within nuclear plant buildings are not avail-

able in the literature. Table 6.15 represents the random variables adopted in the present study,

their probability distribution, their mean and COV, as well as the variables to be considered as

deterministic.

As previously mentioned, stress-strain curves of concrete in tension and compression considered

to study the reliability of RC slabs under impact assuming a nonlinear behavior of materials are

described as stress-strain relations. These relations vary in terms of the tensile strength, ft , and the

compressive strength, fc , respectively. Thus, the variation of these curves depends on the statistical

description of ft and fc .

Compressive strength of concrete Mirza et al. [123] estimated the mean value of the com-

pressive strength to be equal to 0.8fc assuming that fc follows a normal distribution and the COV

can be considered to vary between 0.15 and 0.18.

Tensile strength of concrete Tensile strength mean value is chosen according to Pandher [153]

assuming that it is approximately one tenth of the concrete compressive strength. The COV is taken

equal to 0.18 according to Mirza et al. [123] that studied the variation of concrete tensile strength

in-situ. Mirza et al. found that ft follows a normal distribution.

Young's modulus The elastic modulus of concrete is generally considered as a normally dis-

tributed variable with a COV of 0.1 [123] . Collins and Mitchell [43] approximated the Young's

√

modulus in MPa by Ec = 5500 fc .

As previously mentioned, stress-strain curve of steel considered to study the reliability of RC slabs

under impact assuming a nonlinear behavior of materials are described as stress-strain relations.

These relations vary in term of the yield strength, fy . Thus, the variation of this curve depends on

the statistical description of fy .

Yield strength of steel The distribution of steel yield strength is chosen according to Mirza

and MacGregor [125] assuming that it follows a normal distribution. The mean of yield strength is

generally considered equal to the nominal value with a COV of 0.08-0.1.

due to its small dispersion with respect to other variables.

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Table 6.15: Deterministic variables, random variables and their statistical descriptions

Geometric properties of the slab

Length, a 8.1 m Deterministic [82, 168]

Width, b 4.85 m Deterministic [82, 168]

Thickness, e 0.5 m 0.5 0.05 Normal [127]

Position and size of reinforcement

Concrete cover, c 3.5 cm Deterministic -

Bars diameter, dA 20 mm 20 0.024 Normal [126]

Concrete properties

Compressive strength, fc 40 MPa 0.8fc 0.15 Normal [123]

Density, ρc 2500 kg/m3 Deterministic -

Tensile strength, ft 3.5 MPa fc /10√ 0.2 Normal [153]

Young's modulus, Ec 35 GPa 5500 f c 0.08 Normal [91, 123]

Reinforcement properties

Yield strength, fy 500 MPa 500 0.05 Normal [126]

Modulus of elasticity, Es 200 GPa Deterministic [110]

Impactor variables

Radius, r 15 cm Deterministic lack of information

Mass, m 3600 kg 3600 0.1 Lognormal lack of information

Velocity, v 7.7 m/s 7.7 0.1 Lognormal lack of information

6.4.2.3 Dimensions

Only the thickness of slab is taken as random variable in the case of impacted slabs assuming a

nonlinear behavior of materials. It is considered as normally distributed with a mean value equal

to its nominal value and a COV of 0.05 [127]. The length and width are described as deterministic

variables since slab dimensions are important, thus the eect of their uncertainty is negligible [82].

Steel bars diameter is also considered as a dimensional random variable and is modeled as a normal

distribution with a mean equal to the nominal diameter and a COV of 0.024. The variation of

concrete cover is not taken into account in this study.

Several output variables may be obtained as result from a nite element model. These variables

allow dening limit state functions corresponding to the failure criteria studied. However, the only

variable of interest considered for impacted slabs assuming a nonlinear behavior of materials is the

displacement of the impact point on the front face of the RC slab, u. Thus, a serviceability limit

state function can be written as:

g = umax − u (6.14)

where umax is the displacement threshold and u represents the maximum displacement of the slab

at the impact point. Since no information available in the literature to choose the value of umax , a

deterministic parametric study is performed in order to estimate the displacement for which yielding

in tensile steel occurs. The displacement of steel yielding is evaluated for dierent impact energies,

and it is found that the stresses in reinforcement exceed the yield strength for a displacement of 50

mm as mean value.

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Table 6.16: Factorial experiment factors and mass-spring model parameters

Factors e dA v m

Low level 0.3 m 6 mm 2.5 m/s 2000 kg

High level 0.7 m 30 mm 13.2 m/s 6000 kg

ms ks cs kc cc

kg N/m N.s/m N/m N.s/m

0.015Mslab f (e, dA , v, m) 1.7e5 2e9 1e5

Table 6.17: Failure probability with FE and mass-spring models and displacement criterion

v (m/s) m (kg) FORM IS MC

Constant impact energy

7.7 3600 0.6824 0.6731 0.6557

10.5 1936 0.5475 0.5167 0.5311

13.2 1225 0.4150 0.7539 0.3895

Constant mass

5.0 3600 0.1861 0.2152 0.1853

10.5 3600 0.7073 0.7288 0.7226

Constant velocity

7.7 2000 0.2992 0.2976 0.2646

7.7 5000 0.7247 0.6883 0.6927

6.4.4.1 Comparison of strategies

In order to nd the best t with the FEM and estimate the eect of the slab and impactor properties

on the mass-spring model parameters, a 2k factorial experiment is designed in the aim of expressing

the slab stiness in terms of slab thickness, steel diameter, mass and velocity of the impactor. Table

6.16 represents the low and high level for the specied factors, but also the values of the mass-spring

model parameters which are found to give similar results to the FEM.

As discussed earlier, the two strategies proposed in this study to reduce the computational eort

of reliability analyses consist in using simulation probabilistic methods with the mass-spring model

and FORM with the nite element model. A comparison of these two strategies leads roughly to the

same value of failure probability, except in the case of an impact velocity of 13.2 m/s and a mass

of 1225 kg (Table 6.17. This may be related to the samples generated by IS and MC and which

do not correspond to the mass and velocity selected range for the design of experiments. However,

the second strategy is more time consuming since at least 36 simulations of the FEM are needed

in order to assess the failure probability with an average time of 45 minutes per simulation. As a

result, the rst strategy is more eective in term of computational cost but cannot be used if the

simplied model is not proven to be able to predict the behavior of the RC slab and give the same

results as the FEM.

Impactor velocity Figure 6.28 shows failure probabilities for dierent values of velocity mean for

the displacement criterion. As can be seen, failure probability increases with the impact velocity for

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Figure 6.28: Eect of the mean of impactor velocity for the displacement criterion (m=3600 kg)

the displacement criterion and impact duration increases with the impact energy. The displacement

at impact point can be related to the overall deection of slab, to the motion of shear cone during

impact, or to spalling that occurs at the upper surface of slab. RC slabs are observed to undergo

exural failure under low impact velocities [119] along with scabbing. The local response of scabbing

can dominate the slab behavior for impacts with low velocity as they are classied as short impact

time according to [149, 150]. Furthermore, shorter impact duration leads to less critical yielding

in steel [188], hence less displacement values are observed. However, under high impact velocities

shear type failure occurs, but the impactor could perforate the slab if its velocity is suciently

high. Therefore, higher displacements are observed at the impact point leading to an increase in

failure probability with impact velocity. According to [149] (see Figure 2.3), high-velocity impacts

can be associated with short impact times or long impact times depending on the slab stiness and

material properties. Consequently, for long impact times the displacement at impact point can be

involved due to an overall structural response that is aected by the boundary conditions and slab

size, and governed by several waves reection through the thickness reaching slab boundaries. For

short impact times, the local damage failure modes that govern the slab response are caused by

exural and shear waves with no signicant propagation through the thickness. As can be seen, it is

very dicult to predict the slab behavior in term of the mean of impactor velocity as the response of

RC slabs under impact is signicantly inuenced by several parameters, including slab and impactor

characteristics. Displacement at the impact point in case of perforation are more important than in

case of exural failure, thus it is important to design RC slabs to withstand smaller displacement

values under impact to prevent perforation and scabbing.

Impactor mass As previously discussed, slabs subjected to large-mass low-velocity impacts have

dierent behavior than slabs subjected to small-mass high-velocity impacts. This is can be seen

in Figure 6.29 that shows how the reliability of RC slabs subjected to impact is inuenced by the

mean value of the impactor mass. The mass has the same inuence as the impactor velocity and

failure probability increases with the mean value of mass. Values of maximum displacement and

impact duration are more signicant for higher masses and yielding in steel occurs leading to more

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Figure 6.29: Eect of the mean of impactor mass for the displacement criterion

damage in the slab for higher impact energy. Impacts of large mass and low initial velocity involve

an overall response of the RC slab as the dominating exural waves reach the boundaries during the

impact duration. This type of impact is classied as long impact times according to Olsson [149]

and local damage has small eect on slab response. Conversely, impacts of small mass and high

velocity cause deformation at a small zone surrounding the impact point leading to more localized

deection. Thus, the slab undergoes smaller displacements for smaller mass and failure probabilities

for the displacement criterion increases with the mean of the impactor mass. Accordingly, impacts

with v=5 m/s and m > 3600 kg can be classied as large-mass low velocity impacts using Abrate

classication [9]. However, this classication does not satisfy the criterion of Olsson [149] that

indicated that small-mass impacts correspond to impactor masses less than one fth of the mass of

the slab. The ratio of the mean of the impactor mass to the mean of the slab mass is always less

than 1/5 for all the mean values of the impactor mass considered. According to Olsson criterion all

cases studied must be classied as small-mass impacts. This is in contrast with Abrate classication,

hence it is necessary to take into account all parameters present in an impact problem in order to be

able to properly estimate the behavior of impacted RC slabs. It should be noted that small masses

are more critical for a given impact energy because they result in higher impact loads and less energy

absorbed by the slab.

Slab thickness Figure 6.30 shows how the reliability of RC slabs subjected to impact is inuenced

by the mean value of the slab thickness. Other values of random variables remain the same as dened

in Table 6.15, including the length and width of the slab and the reinforcement ratio. As illustrated

in this gure, increasing the thickness leads to lower failure probabilities for the dierent mean values

of impact velocity considered as the slab deection becomes less signicant for larger thicknesses.

Increasing the thickness results in increasing the slab stiness as well as the compression concrete

area, which leads to higher impact force and shorter impact times. In this case, a few compressive

wave reections through the thickness are observed and the damage is initiated at the early stages

of impact. According to Abrate's criterion [9] that classify impact events according to the ratio of

impact velocity versus the speed of propagation of compressive waves through the thickness direction,

the impact problem with higher thickness can be classied as high-velocity impact depending on the

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Figure 6.30: Eect of the mean of slab thickness for the displacement criterion

impact velocity. For v=5.0 m/s, the problem can be classied as low-velocity impact for thickness

value less than 0.5 m as the stress wave propagation through the thickness has no signicant role and

the slab would likely fail due to its overall bending behavior. For these thickness values and a velocity

equal to 5.0 m/s, the response of slab is inuenced by its dimensions and boundary conditions and

the impact duration is described as long impact times according to Olsson [149] (see Figure 2.3). The

reinforcement has more inuence on the slab resistance during exural failure mode, and the slab

may undergo higher displacements at the impact point. For v=5.0 m/s and thickness values higher

than 0.5 m, the impact can be classied as high-velocity impact according to Abrate's criterion [9]

and the impact duration can be described as short to very short impact times according to Olsson

[149] (see Figure 2.3). For v=7.7 m/s, the problem can be considered as high-velocity and short

impact time for thickness value higher than 0.6 m. For v=10.5 m/s, the problem can be considered

as high-velocity and short impact time for thickness value higher than 0.8 m. It should be noted that

increasing the slab thickness improves the slab performance to perforation and results in increasing

the ballistic limit dened as the lowest initial velocity of the impactor causing complete perforation

[9].

fect of material parameters on failure probability estimation and how they contribute to the strength

of RC structures. Only the mean value of the compressive strength is considered since those of the

tensile strength and the modulus of elasticity are estimated in term of the compressive strength.

Thus, an increase in the mean value of the compressive strength leads directly an increase in tensile

strength and modulus of elasticity mean values. As can be seen in Figure 6.31, failure probability

falls o with the mean value of the compressive strength. The compressive strength contributes sig-

nicantly to the slab rigidity, so that increasing the compressive strength allows lower displacement

at the impact point of the slab and yielding in steel is reduced. Nevertheless, increasing slab rigidity

leads to higher peak value for the impact force and punching shear failure mode could occur. A

high strength concrete improves the slab performance in term of perforation compared to normal

strength concrete since higher impactor velocities are needed to perforate the slab in this case. The

slab resistance to penetration can be enhanced by increasing the compressive strength of concrete.

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Figure 6.31: Eect of the mean of concrete compressive strength for the displacement criterion

Therefore, lower displacement values are observed for higher compressive strength mean values lead-

ing to lower failure probabilities. However, it can be seen that for v=5.0 m/s and v=7.7 m/s, using

concrete with compressive strength greater than 50 MPa does not seem to give a signicant im-

provement in slab reliability for the displacement criterion. Thus, using the appropriate concrete

properties is important for an optimal design of impacted RC slabs to resist local failure modes,

including spalling, scabbing, penetration and perforation. As in this reliability analysis increasing

the concrete compressive strength means an increase in its tensile strength, scabbing takes place in

a later phase of impact for the same impact conditions. This is due to the fact that scabbing occurs

when the tensile stresses generated by the tensile reected wave produced during impact become

equal or higher than the concrete tensile strength, and higher value of tensile strength delays the

scabbing local failure mode.

Yield strength of steel The inuence of yield strength of steel on failure probability is inves-

tigated for the displacement criterion proposed for this application. As can be seen in gure 6.32,

the yield strength of steel has no inuence for v=5.0 m/s and v=7.7 m/s because the displacement

criterion proposed in this study is chosen in order to not allow yielding in steel. The displacement

threshold is estimated for slabs with reinforcement of yield strength of 500 MPa. Thus, for higher

yield strength mean value, failure probability decreases for v=10.5 m/s since the problem is classied

in this case as long impact times and the overall response of slab dominates.

Reinforcement ratio In order to study the eect of reinforcement ratio on the impacted RC

slabs reliability, failure probability is calculated by considering several mean values of the steel bars

diameter (Figure 6.33). Other values of random variables remain the same as dened in Table 6.15,

including the slab geometry and impact conditions input variables. In addition, the eect of numbers

of transversal and longitudinal bars on failure probability for the displacement criterion is studied

(Figure 6.34). Those numbers are presented in the study as deterministic variables. Reinforcement

ratio depends on the steel bars diameter value as well as on the number of reinforcing bars in the

slab, thus increasing the steel diameter or the number of reinforcing bars leads to an increase in

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Figure 6.32: Eect of the mean of steel yield strength for the displacement criterion

reinforcement ratio. It can be seen that the number of transversal bars has no considerable eect

for v=5.0 m/s and v=7.7 m/s, but it inuences the magnitude of failure probability for v=10.5 m/s

up to 20 transversal reinforcing bars per layer. However, longitudinal reinforcing bars have more

inuence on the slab deection, hence a higher number of longitudinal bars reduces the displacement

at the impact point and results in lower failure probabilities. The slab stiness increases with the

reinforcement ratio and the exural failure mode represents the most critical parameter to design

longitudinal reinforcement, especially under high impact velocities. Thus, it can be concluded that

the 20 longitudinal bars initially chosen for each of the upper and bottom reinforcement layers are

not sucient to ensure its reliability for the displacement criterion nor to have an optimal design

under bending mode. This may be also the reason why steel bars diameter has no signicant eect

on the slab reliability, hence it will be interesting to examine its eect in case where the slab is

reinforced with more than 20 longitudinal bars per layer. Although reinforcement ameliorates the

ductility of RC slabs due to its plasticity, the steel ratio should be chosen with attention since a

strongly reinforced slab may fail under punching shear failure mode when subjected to high velocity

impacts. Consequently, it is important to design RC slabs under impact in a way which reduces the

reinforcement ratio, but also ensures that no failure or large deection would occur during impact.

Stirrups As previously indicated, the Python script written for the slab subjected to dropped

object within nuclear power plant enables to add stirrups to the FEM. Therefore, the eect of

reinforcing the slab with shear reinforcement on failure probability is also examined (Figure 6.35).

It is found that for low impact velocities, stirrups do not improve the performance of slab for

the displacement criterion proposed in the case where nonlinearities due to material behaviors are

considered in the FEM. This may be due to the thickness of slab that can be described as thick

slab. In the case of thick slabs subjected to low velocity impacts, cracks are rst induced at the

impact zone and then progress through the thickness without leading to a reversed shear cracks

pattern. However, higher velocities result in shear type failure mode for thick and stier slabs. This

can be illustrated in Figure 6.35 and, as a consequence, failure probability decreases for values of

mean velocity greater or equal to 10.5 m/s if shear reinforcement is considered in the FEM. It can

be also expected that stirrups have more inuence on the slab displacement in case of thin slabs

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Figure 6.33: Eect of the mean of steel diameter for the displacement criterion

Figure 6.34: Eect of numbers of transversal and longitudinal bars for the displacement criterion

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Figure 6.35: Eect of stirrups for the displacement criterion

under low velocity impact since they reduce shear cracks reversed from the slab bottom surface due

to bending stresses. If a RC slab is designed to resist punching shear failure mode under impact,

it is necessary to properly determine shear reinforcement which can improve the slab resistance by

varying the failure mode from punching shear mode into an overall exural failure mode

Boundary conditions To investigate the eect of varying boundary conditions of the slab on its

reliability under an impact loading, four boundary conditions are examined as follows:

The bottom four edges of the slab are considered as simply supported (pinned edges).

The four faces of the slab in the xz and yz planes are considered as simply supported (pinned

faces).

The bottom four edges of the slab are considered as clamped (xed edges).

The four faces of the slab in the xz and yz planes are considered as clamped (xed faces).

As can be seen in Figure 6.36, boundary conditions has a negligible eect on slab response for low

velocities (v=5.0 m/s and v=7.7 m/s). This is due to the fact that in the case of impactors dropped

at low velocity of a few m/s on large slabs, main exural and shear waves do not reach the boundary,

which can be associated with high-velocity impacts according to [148] and very short impact time

according to [149]. As a result, failure probability increases with the mean of velocity as the slab

undergoes higher displacements and tends to fail under exural failure mode. For v=10.5 m/s, the

displacement of slab at the impact point is more important for the pinned edges boundary condition

as the slab shows considerable rise in its rigidity for pinned faces and xed boundary conditions.

In this case, the problem is found to be dependent of boundary conditions and the time that main

waves need to reach structure boundaries is less than the impact duration. Thus, it can be classied

as low-velocity impact according to Abrate's criterion [9] and long impact time according to [149].

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Figure 6.36: Eect of boundary conditions for the displacement criterion

Impact position As previously indicated, the Python script written for the slab subjected to

dropped object within nuclear power plant enables to easily change the impact position in the FEM.

Therefore, failure probability corresponding to displacement criterion is estimated for ten dierent

impact point positions in the directions of the width, the length and the diagonal (Figure 6.37). The

results show that the most critical point for the response of slabs subjected to impact loading is at

the center of the slab. The more the impact point is close to supports, the more the slab is rigid

and consequently, the displacement is lower.

The present case of study proposes strategies in order to reduce the computational time of reliability

analyses. The rst strategy consists in using analytical models instead of nite element models. Due

to the low computational time of the simplied model proposed, probabilistic methods like Monte

Carlo and importance sampling which require too large number of simulations of the deterministic

model are coupled with this model. However, this strategy does not allow taking into account all

random variables that aect the slab response to impact loading and can be only used when the

simplied model is veried with the nite element model. Alternatively, the second strategy consists

in assessing failure probability from a small number of simulations using FORM with the nite

element model. Despite its computational cost, this strategy allows a better understanding of the

slab behavior when subjected to low velocity impact. The results of failure probability show high

values which are not acceptable for structural failure, this conrms that this type of loading must

be considered in the design of RC slabs. Therefore the slab dimensions, material parameters and

reinforcement ratio must be modied and evaluated in order to get better reliability of RC slabs.

The long term behavior of a prestressed concrete structure like nuclear power plants, is aected

by the delayed deformations that could initiate cracks, induce prestressing losses and result in a

redistribution of stresses in the structure. They could also be the cause that the structure does

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Figure 6.37: Eect of impact position for the displacement criterion

not perform its functional requirements. The evolution of creep and shrinkage deformations is the

main cause of the loss in prestress in cables due to the reduction in the prestressing force and,

consequently, the crack resistance of the interior wall of the nuclear power plant reduces. Thus,

studying the behavior of a nuclear power plant is important to evaluate the safety of the nuclear

facilities, i.e. maintaining the stability of the containment under dierent types of internal or external

aggressions during service or accidental loadings. A nuclear power plant consists of a RC internal

containment building and a prestressed external containment building (Figure 6.38). The external

containment building in which the nuclear reactor is enclosed from natural or accidental aggressions,

while the internal containment building protects the surrounding environment in case of a nuclear

accident. The internal containment building could be covered with a waterproof coating in order to

control the rate of leakage.

A prestressed containment building is designed to resist the increase of internal pressure in case

of accidental situations. It is tested at least once every 10 years in order to experimentally verify

the rate of leakage. These periodic surveillance tests are performed in the aim of examining the

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Figure 6.38: Containment building [161]

containment behavior in case of an accident and measuring the rate of leakage that increases when

the loss in prestress in cables becomes more signicant. During these tests and over the very long

term, in cases where delayed deformation are underestimated during the design of structure, tensile

stresses may appear in a common zone of the containment under accidental loadings. Thus, it is

necessary to examine and predict the delayed behavior of the containment building in order to control

the tension zone. For this purpose, numerical tools are very helpful since real scale experiments are

impossible for such type of structures.

A FEM is performed by IOSIS [24] to model the containment behavior of the Flamanville nuclear

power plant during a periodic surveillance testing carried out 20 years after its implementation using

the FE software ASTER. The model takes into account the full process of aging, including relaxation

of the reinforcement, creep and shrinkage of concrete. The aim is to estimate the evolution with

of orthoradial stress in term of the structure height and determine at which height tensile stresses

occur. The aim of this section is to conduct a reliability study based on IOSIS FEM by combining

ASTER used to describe the physical model with OpenTURNS. Polynomial chaos expansion is used

to simplify the physical model and to perform several analyses including dispersion, distribution and

sensitivity analyses.

The evolution of containment building is modeled for t ] tc ; tV D1 [, where tc is the date at which the

internal containment is prestressed and tV D1 is the date of the rst periodic surveillance testing (20

years after its implementation). Time increment is estimated in term of years, which makes a total

of 15 time increments between tc and tV D1 . The connement building of Flamanville nuclear power

plant consists of a prestressed cylindrical concrete vessel of 45 m of diameter and 65.05 m of height

(Figure 6.39). The vessel is capped with a dome and is supported on a at foundation slab. The

plate (gousset) linking the bottom of the internal containment to the foundation slab has height of

3.6 m and an extra thickness of 1.10 m.

In order to reduce the computational cost of the deterministic analysis, the geometry considered

in the model is simplied as illustrated in Figure 6.40. The model is limited to an angle of 22.22 gr

(symmetry with respect to vertical plans) and the containment building is modeled up to the height

h = +30.0 m which is approximately equal to half of its total height. Thus, the selected zone is

located between the levels of −0.6 m and +30.0 m, and the azimuth of 377.778 and 400.0 gr. In this

case, the length of sector at the neutral ber is equal to 8 m. The origin O is located at the center

of the containment, at z=0 corresponding to the level 0.0 m.

Two types of prestressed cables are used, namely the vertical cables of the vessel that are vertical

straight lines in the common zone and the horizontal cable of the vessel. Taking into account the

above simplication of geometry of the selected zone, ten vertical cables are modeled and located

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Figure 6.39: Geometry of the containment building of the Flamanville nuclear power plant

at the radius Rmoy = 22.95 m, while horizontal cables are located either at Rint = 22.75 m, Rext =

23.15 m or Rmoy = 22.95 m. They are modeled independently in the plate zone, and as groups of

2, 3 or 4 cables in the common zone. Overall, 34 horizontal cables are considered in the model, 15

independent cables, 8 groups of 2 cables, 1 group of 3 cables and 10 groups of 4 cables.

6.5.1.1 FE software

ASTER (Analyse des Structures et Thermomécanique pour des Etudes et des Recherches) is a nu-

merical software developed by EDF and R&D. It is based on the theory of continuum mechanics

and uses the nite element method to solve dierent types of problems, including mechanical, ther-

mal, acoustics, seismic, etc. Prestress can be simulated in ASTER using the DEFI_CABLE_BP

command that calculates the initial proles of tension along the prestressed cables of a concrete

structure. It uses specic data for prestress, such as the tension applied to tendon ends, the initial

force applied at the active anchors of cables and other parameters that characterize the materials

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and anchors. Furthermore, the basic creep of concrete can be modeled in ASTER using the model

of Granger based on GRANGER_FP_INDT relation. This relation does not take into account

the eect of temperature nor that of aging on the mechanical properties of concrete. The various

parameters of Granger model are:

τi are the coecients related to the 'delay' of the basic creep function

The curve of sorption-desorption that gives the humidity in term of the drying of concrete

Concrete is modeled as a visco-elastic material with a Young's modulus of 37000 MPa, a density

of 2500 kg/m3 and a Poisson's ratio of 0.2. The prestress is adopted in such type of structures

in order to avoid tension zones and thereby reduce cracking in concrete. Therefore, it is designed

so that concrete in the containment building can be fully compressed when fullling its functional

requirements. For this case of study, the prestress is achieved with vertical and horizontal cables

of 37T15and a cross-section of 5143 mm2 . Cross-sections of horizontal cables modeled as groups of

2, 3 or 4 cables are multiplied by 2, 3 or 4 respectively. The failure tensile stress is equal to 1814

MPa while the initial tensile stress at the anchorage is taken equal to 1429 MPa. Steel material of

cables is modeled as a linear elastic material with a Young's modulus of 190000 MPa and a Poisson's

ratio of 0.3. The prestress is evaluated taking into account the loss in tension along cables due to

rectilinear and curvilinear friction, as well as to the relaxation in prestressing steel (2.5 % at 1000

hours, 3.0% at 3000 hours). Losses due to the creep and the shrinking of the concrete, as well as that

resulting from the retreat of anchorage are also considered in the FEM. The relaxation of cables is

estimated to be equal to 2.5 % at 1000 hours and 3.0% at 3000 hours, while the retreat of anchorage

is supposed to be equal to zero for horizontal cables and 0.008 m for vertical cables. The model of

Granger is used to represent the basic creep of concrete in the FEM.

The selected zone of study necessitates choosing appropriate boundary conditions in order to take

into account the symmetry (Figure 6.41):

For the surface Surf1 located at the azimuth of 400 gr, displacement in x-direction is restrained

so that dx = 0.

For the surface Surf2 located at the azimuth of 377.78 gr, displacement in x-direction is re-

strained so that dx = −dy tan α.

For the upper surface Surf3 located at the level z = +30.0 m, rotations about x-,y- and z-axes

are zero and displacement dz in z-direction is constrained to an identical value for all the

surface nodes. The eect of the weight of the upper part of the containment as well as the

dome part is considered in the model by applying a compression stress upon this surface equal

to σzz = 1.51 MPa.

For the surface Surf4 located at the level z = −0.6 m, the selected zone is clamped at the

bottom. In addition, the foundation slab shrinkage is dened as a boundary condition of

displacement in the plate base.

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Figure 6.41: Mesh of the selected zone of study of the containment building

6.5.1.4 Mesh

The FE mesh of the internal containment includes the concrete part of the structure, as well as

the prestressed cables that signicantly contribute to rigidity and the nonlinear behavior of the

containment. The concrete part is modeled using solid FE with 8 nodes and linear integration,

while prestressed cables are modeled with two nodes linear truss elements. Passive reinforcement

are not taken into account in the FEM since their inuence on the structural rigidity is relatively

small. As previously mentioned, the dome and the foundation slab are replaced in the model by

equivalent boundary conditions.

The mesh is performed in the way that concrete nodes overlap with those of horizontal cables,

but vertical cables are meshed independently. Therefore, the discretization in space of horizontal

cables and the concrete part is the same and a node is located every 0.8 m. Vertical cables, as well as

the concrete part in the z-direction, are discretized in space with a node every 0.4 m. Consequently,

the FEM of the selected zone of the internal containment consists of 9200 nodes and 7900 elements,

including 6800 solid elements assigned to concrete and 1100 truss elements assigned to steel cables.

Figure 6.41 shows the mesh adopted for the selected zone study.

The internal containment structure shows three dierent thicknesses related to the slab foundation,

the plate and the common part of the containment. The dierence of thickness along the structure

necessitates the prediction of drying shrinkage and creep of concrete in terms of the geometry and

boundary conditions of each zone of dierent thickness. Delayed strains in concrete may aect the

sustainability of concrete structures for several reasons. Shrinkage strains εr may induce cracking in

concrete until causing steel reinforcement corrosion, while creep strains εrf may result in a signicant

reduction in the prestressing force.

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Concrete may be subjected to several types of delayed strains during its lifetime. In the case

where there is no water exchange between the concrete structure and its surrounding environment,

3 types of delayed strains are observed:

evaluated in term of water/cement ratio. Autogenous strains develops during hardening of

concrete in the early days of casting.

Early-age thermal shrinkage εth occurs due to excessive dierences within a concrete structure

or its surroundings.

Basic creep εf p is dened as the creep that occurs under loading conditions and increases with

the increase of stress and duration of loading.

In the case where water exchange occurs, 2 other types of delayed strains of concrete are observed:

Drying shrinkage strain εrd is a function of the migration of the water through the hardened

concrete and develops slowly.

Drying creep strain εf d represents the extra creep component at drying over the sum of shrink-

age and basic creep.

Consequently, the total strain that describes the time dependent behavior of concrete can be ex-

pressed as follows:

The variables which may be subjected to statistical variations in containment building can be clas-

sied into four categories:

1. Variables related to the concrete properties: Young's modulus and Poisson's ratio of concrete.

4. Variables related to the steel properties: multiplying factors for horizontal and vertical cables.

23 random input variables are considered, they correspond to the physical phenomena observed in

such type of structures, including relaxation of prestressing reinforcement, creep and shrinkage of

concrete. Random variables are considered as independent since no experimental data are available

to determine the correlation between them. All random variables are assumed to have a Lognormal

probability distribution with a mean value equal to the nominal value of the relevant variable and a

COV equal to 0.1. This choice is based on the fact that all input variables taken into account in the

probabilistic model should have positive random values. In addition, the COV is chosen the same

for all variables in order to study the inuence of each variable mean on the estimation of the height

corresponding to the tension zone without considering their eect of variation. It has be shown in the

reliability analysis of RC cantilever beam that the COV value highly aects the probabilistic response

and a variable with a higher COV has a signicant eect on the dispersion of output variables as

well as on the estimate of failure probability. Table 6.18 represents the random variables adopted

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Table 6.18: Random variables of the containment building and their descriptions

Concrete properties

Ec Young's modulus of concrete 37 GPa 0.1 Lognormal

νc Poisson's ratio of concrete 0.2 0.1 Lognormal

Reinforcement properties

αh Multiplying factor for horizontal cables 10615000 0.1 Lognormal

αv Multiplying factor for vertical cables 7210500 0.1 Lognormal

Shrinkage in concrete

retb Shrinkage coecient 1e-5 0.1 Lognormal

Creep in concrete

J1 -J5 Creep coecients 0.57048258e-12 0.1 Lognormal

J6 -J8 Creep coecients 33.8049253e-12 0.1 Lognormal

Figure 6.42: Horizontal and vertical cables considered in the reliability analysis of containment

building

in the present study in the case of containment building, their probability distribution, their mean

and COV.

The horizontal cables presented in the reliability analysis of containment by their multiplying

factors are shown in Figure 6.42(a). Figure illustrate the vertical cables considered, they are located

on both sides of the line over which the post-processing of stress is performed (2 cables on the left

side and 2 cables on the right side).

The aim of periodic surveillance testing is to measure the global leakage rate in the internal contain-

ment, which represents the percentage of decrease in dry air masses in the containment per day. A

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containment should ensure a leakage rate less than 1.5% of the total mass of dry air per day. During

periodic surveillance testing, tensile stresses may appear in some particular zone of the containment.

Furthermore, in very long term, beyond the expected period of time for which the containment is

designed, such tensile stresses may also occur in the common zone of the containment in cases where

delayed strains were underestimated during the design process.

The eect of dierential shrinkage is observed in the internal containment for height less than

5.8 m, which leads to tensile orthoradial stresses that reduce the compressive stress in concrete

generated from prestressing forces. For this application, the limit state function is dened using the

criterion of maximum height, that is failure occurs when the height of zone under tension is greater

than the threshold hmax (hmax = 5.8 m).

g = hmax − h (6.16)

Polynomial chaos method with cleaning strategy is used to study the reliability of the containment

building of Flamanville nuclear power plant. As previously demonstrated in the reliability study

of cantilever beam, the cleaning strategy is found to be the most ecient strategy, among those

available in OpenTURNS for polynomial chaos, to estimate failure probability and analyze the

dispersion of variables of interest. Two steps are necessary to perform the study:

The rst consists in coupling OpenTURNS and the initial deterministic code directly. The ini-

tial deterministic code is an ASTER FEM which is called by OpenTURNS for each assessment

of the limit state function denoted in this case gmodel .

The second is based on an approximation of the function gmodel using the polynomial chaos

expansion which enables to substitute the containment initial deterministic model with an

analytical expression. The approximated limit state function is denoted in this case gM etaM odel .

The aim is to analyze the eciency of using polynomial chaos expansion for problems with an

important number of random variables and a limit state which is not easy to evaluate. Thereafter,

x-axis corresponds to the number of calls to the initial deterministic model in the case a direct

combination between OpenTURNS and ASTER, while it indicates the number of sample points in

the design of experiments in the case of an approximate limit state function.

This case of study presents a deterministic FE model for the selected zone of the containment that

allows estimating the height at which the structure is under tension. Figure 6.43 shows the variation

of the average tensile stress induced in concrete in term of the height. The average stress is estimated

over the containment thickness for a given value of height. It can be seen that tensile stresses occur

in the zone of height 4.906 m, which corresponds to a case belonging to the safe domain according

to the failure criterion expressed in equation 6.16. This criterion is related to the height of zone

under tension with a height threshold of 5.8 m.

Figure 6.44 shows the variation of the mean of the tension zone height in term of the number of

simulations of the initial deterministic model that represents also the number of sample points used

to evaluate the approximate limit state function. The height mean is calculated with the initial

limit state function (gmodel ) using the numerical integration method Monte Carlo, and with the

approximate limit state function (gM etaM odel ) using Monte Carlo and the quadratic combination

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Figure 6.43: Variation of the average tensile stress in term of height

methods. It is assumed that convergence is achieved when the dierence between the height mean

values obtained with gmodel and gM etaM odel , respectively, is of the order of 10−2 . As can be seen,

1000 simulations are sucient to obtain the desired convergence and a good agreement between

calculated and approximated mean values of the tension zone height.

The COV of variation of the tension zone height is calculated as the ratio of the height standard

deviation to its mean value. The higher the COV is, the more the dispersion about the mean is

important. In other words, the estimation of mean value is more precise for lower values of COV.

Thus, the number of simulations used should be suciently high to reach the precision desired.

COV is expressed in percentage so the dispersion of variables of dierent units can be comparable.

However, this does not represent a problem in this case of study since the same variable is compared

using dierent methods to assess the output of models. Figure 6.44 shows that the most accurate

estimation of the mean of tension zone height is obtained using the quadratic combination method

with the meta-model as it allows reducing the COV to minimum. This is due to the fact the

quadratic combination method depends only on the limit state function, but does not depend on

the number of simulations or calls of the model as in the case of Monte Carlo numerical integration

method. This latter necessitates a sucient number of simulations in order to converge to the exact

value of mean.

In order to verify the eciency of polynomial chaos expansion in approximating accurately the

limit state function of the initial deterministic model, the sample of height obtained with the initial

model function gmodel and that obtained with the approximated function gM etaM odel for the same

realizations of the input physical vector are compared. The aim is to compare two statistical samples

having the same population and the same size of 1000.

A histogram is considered as a graphical tool with which the identication of the distribution of a

variable can be established. It can be used to verify if the meta-model provides a good estimation of

the height output sample obtained with the initial FEM. In order to facilitate the comparison, both

histograms obtained through the initial model and the approximate model, respectively, are plotted

on the same graph (Figure 6.45). x-axis corresponds to the height values in m, while y-axis indicates

the frequency values in %. Examining the histograms of the output samples allows some preliminary

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Figure 6.44: Mean and COV of height in term of the number of simulations

Figure 6.45: Histogram of tension zone height samples obtained with initial model and meta-model

for 1000 simulations

conclusions to be drawn. First, it can be concluded that both samples have approximately the same

mean and that height values are distributed around this mean of 4.906 m. The number of intervals

as well as their width depend on the sample size which is equal in this case to 1000. They are

calculated automatically in OpenTURNS in order to have the best display of the output variable

and to accurately evaluate the real distribution of any sample, whist it is not possible to adjust

the properties of a histogram in OpenTURNS. For this reason, frequencies of each output sample

are represented in dierent class intervals. However, the number of intervals as well as their width

remain the same for both samples since they have the same size.

Furthermore, cumulative distribution functions are plotted on the same graph for both samples

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Figure 6.46: CDF of tension zone height samples obtained with initial model and meta-model for

1000 simulations

(Figure 6.46). As can be seen, the CDF of height sample obtained with the approximate model is

below that obtained with the initial model for all values of height. Both distributions are in good

agreement and are similar between height values of 4.75 m and 5.25 m, i.e. around the mean value

of 4.906 m.

The probability distribution functions of output samples are estimated using Kernel Smoothing

method that enables to represent those distributions graphically. As can be seen in Figure 6.47,

the distribution function resulting from the meta-model is smoother than that resulting from the

initial model, especially around the mean value of the height, or in other words, in the zones of high

probability density. This problem can be solved by increasing the number of sample points in the

design of experiments or by using a higher order for the polynomial chaos expansion [155]. A higher

order of polynomial chaos permits a better representation of the third and fourth-order statistical

moments that measure the skewness and the kurtosis of a distribution, respectively.

The graphical statistical test QQ-plot is used with quantiles estimated at 95% to compare the

distributions of the height sample obtained with the initial model and that obtained with the ap-

proximated model. Figure 6.48 shows that quantile points are very close to the diagonal, which

means that the distributions of the two samples are almost identical. It should be noted that the

probability distribution obtained with the initial FEM model is not properly estimated since only

1000 random simulations are used to create the output sample. Thus, a higher number of random

simulations is needed for a better estimation.

The aim of this application is to study the likelihood of failure of the containment under tension

stresses at a certain height. Subsequently, only the limit state function approximated by the polyno-

mial chaos method is considered and failure probability is estimated by combining this function to

dierent probabilistic simulation and approximation methods. The eect of the number of sample

points in the design of experiments is examined in the aim of illustrating the importance of using

a sucient number of points to accurately estimate failure probability with a limit state function

approximated by polynomial chaos expansion. Figure 6.49 shows failure probability and reliability

index estimated with gM etaM odel that is coupled to dierent probabilistic methods for the failure

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Figure 6.47: PDF of tension zone height samples obtained with initial model and meta-model for

1000 simulations

Figure 6.48: Comparison of initial model and meta-model output samples using QQ-plot

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Figure 6.49: Failure probability and reliability index estimated with dierent probabilistic methods

depending on the number of points of the design of experiments

criterion related to the height of tension zone. Number of points in the design of experiments varies

from 100 to 1000 and 5 design of experiments are considered. As can be seen, results obtained using

simulation probabilistic methods are more precise than those obtained using approximation proba-

bilistic methods. The precision of methods increases with the number of points that allows verifying

if failure probabilities, as well as reliability indices, are accurately evaluated and if the convergence

to exact values is achieved. The convergence criterion of Mohammadkhani [136] is used in this study,

it consists in calculating the dierence of reliability indices of two successive iterations (2 dierent

number of sample points). Mohammadkhani indicated that the convergence is reached when the

dierence of reliability indices is less than 10−3 or 10−6 depending on the probabilistic model and

numerical model considered. According to this criterion, the convergence between 600 points and

1000 points is achieved. In the case of 1000 points, only IS provides the same failure probability

compared to Monte Carlo. Approximation methods do not give satisfactory results, which may

due to the presence of large curvatures at the failure surface approximated with polynomial chaos

expansion. As it is known, FORM and SORM do not allow a good estimate of failure probability

in the case of highly nonlinear limit state functions.

Figure 6.50 illustrates the convergence of failure probability assessed using dierent simulation

methods coupled the limit function approximated with 1000 points in the design of experiments.

Convergence of simulation methods is evaluated for a condence interval at level 0.95, which indicates

that the true value of failure probability is, with a probability close to 1, within the range of

condence interval. x-axis in Figure 6.50 corresponds to the number of calls to the meta-model by a

simulation method (number of simulations), while y-axis indicates the estimate of failure probability

assessed using this method. Simulation methods can be classied according to their condence

interval and the number of simulations needed to assess failure probability. It can be seen that MC

shows a high precision as it provides the tighter condence bounds for failure probability, but it

necessitate a signicant number of simulations (more than 60000 simulations). However, IS is more

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Figure 6.50: Convergence at level 0.95 of the estimate of failure probability for dierent probabilistic

simulation methods

ecient in this case as it provides a tight condence interval with less number of simulation (1200

simulations).

Figure 6.51 shows the importance factors associated to the dierent random variables considered in

the study of Flamanville internal containment. Importance factors are calculated using the quadratic

combination method and FORM, respectively. The limit state function estimated with 1000 points

using the polynomial chaos expansion is used for the sensitivity analysis. It can be seen that the

shrinkage coecient of concrete has the most important inuence among other random variables in

the containment probabilistic model for the considered failure criterion of tension zone height. On

the contrary, the multiplying factors of horizontal cables (15, 17, 19 and 21), as well as Young's

modulus and Poisson's ratio of concrete has less inuence on the estimator of height mean and

failure probability, respectively. Other random variables considered in the probabilistic model has

no inuence.

Figure 6.52 shows the eect of the number of points in the design of experiments on the shrinkage

coecient importance factor. It can be seen that, for higher number of points, the importance

factor calculated with the quadratic combination method and which indicates the eect of shrinkage

coecient on the estimator of height mean, trends to converge towards a steady value. However, the

importance factor calculated using FORM and which describes the sensitivity of shrinkage coecient

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Figure 6.51: Importance factors of containment random variables using quadratic combination

method and FORM

Figure 6.52: Importance factor of concrete shrinkage coecient in term of number of points in the

design of experiments

on the estimate of failure probability, trends to converge towards a steady value for a number of

points equal to 600.

6.5.4.6 Convergence?

In order to verify the eciency of the meta-model approximated with polynomial chaos expansion,

the output sample of height obtained with the approximated model is compared to that obtained with

the initial FEM. The points (hmodel , hM etaM odel ) are plotted on the same graph and their position

with respect to the diagonal is examined. A point on the diagonal indicates that hmodel = hM etaM odel .

A good approximation of the initial model can be deduced in the case where the points are close

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to the diagonal. In the other case, the number of points in the design of experiments should be

increased. Figure 6.53 shows that the number of points (hmodel , hM etaM odel ) that are close to the

diagonal increases with the number of points in the design of experiments. However, widely spaced

points from the diagonal indicate that the meta-model does not estimate accurately the height value

obtained with the FEM for the relevant values of input variables. Therefore, a higher number of

point in the design of experiments (>1000) is needed in order to reach the desired convergence.

The application of reliability analysis to a containment building permits to investigate the eective-

ness of the polynomial chaos method in estimating the initial physical model with an approximated

model. The comparison is performed on the estimation of statistical moments of the output variable

(dispersion analysis), as well as on the determination of its probability distribution (distribution

analysis). The reliability analysis results are obtained by combining the meta-model with various

approximation and simulation methods. Simulation methods are more accurate than approximation

methods and allow more precise failure probability values. Monte Carlo and importance simulation

provide the same value of failure probability and reliability index. However, the precision obtained

should be veried since no reference results are available for comparison.

In conclusion, the eciency of the polynomial chaos method is related to several factors, including

the limit state function to be estimated, the degree of the polynomial chaos, the number of points

in the design of experiments and the variation of random variables. Dierent methods are used to

estimate the reliability of the containment. The eciency of the response surface method used in

this study necessitate the convergence of results towards a steady value for all types of conditions.

These conditions can be related to the physical complexity and the precision of the mechanical

model, geometric properties of limit state function, number of random variables considered, ...

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Figure 6.53: Comparison of initial model and meta-model for dierent number of points in the design

of experiments

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Chapter 7

7.1 Conclusions

For an optimal and robust design that guarantees a real prediction of the behavior of structures,

uncertainties should be taken into account and be propagated in the structural deterministic analysis.

Probabilistic approaches were developed to correctly model the variation of input variables. The

aim of these approaches is to assess failure probability of structures according to one or several

criteria and to study the inuence of uncertain variables on the structure response. Structural

reliability analysis is based on the principle of combining a stochastic model with a deterministic

model. Stochastic models should include the probabilistic characteristics of random input variables

including a suitable probability distribution and appropriate values of their mean and coecient of

variation. Deterministic models allow predicting structural response and evaluating the variables

of interest, they can be based on analytical, empirical or numerical deterministic approaches. The

combination of mechanical and stochastic models can be very time consuming. Thus, the issue of

computational eort of reliability analysis is addressed and strategies are proposed to accurately

assess failure probability with the minimum computational cost.

This research discusses the use of reliability analysis for three dierent civil engineering applica-

tions of dierent degrees of complexity. The platform OpenTURNS is used to perform the reliability

analysis of the RC structures considered in the present study and propagate uncertainties in their

physical models. Thus, the reliability of three applications are examined: a RC multiber cantilever

beam subjected to a concentrated load at the free end, RC slabs which are subjected to acciden-

tal dropped object impact during handling operations within nuclear plant buildings, a prestressed

concrete containment building.

The physical problem of reinforced concrete (RC) slabs subjected to impact is classied as low

velocity impact event and considered as one of the most dicult nonlinear dynamic problems in

civil engineering because it involves several factors, including the nonlinear behavior of concrete, the

interaction eects between the concrete and reinforcement, and contact mechanics between the slab

and the impactor. The complexity of impact problems also lies in the dynamic response of RC slabs

and the time-dependent evolution of impact velocity. Several parameters may signicantly inuence

the impact response of RC slabs:

Slab dimensions and boundary conditions control the stiness of the slab.

Material properties have a signicant inuence on the slab transient response by aecting the

contact and overall slab stinesses.

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Impactor characteristics including impact velocity, shape, position, mass and rigidity inuence

the impact dynamics.

A detailed step-by-step procedure for creating FE models of impacted slabs with Abaqus is de-

scribed in this study. the FEM models correctely dierent factors that contribute to the nonlinear

behavior of the reinforced concrete such as the nonlinear stress-strain response, the damage due to

crushing and tensile failure, the eect of the loading rate, the interaction between the concrete and

the reinforcement. The analysis is performed with an explicit conguration that allows a better

representation of impact problems. The model developed is validated with the experimental results

of Chen and May tests and used to model RC slabs which are subjected to accidental dropped object

impact during handling operations within nuclear plant buildings. Numerical results obtained are

in good agreement for several slabs tested.

The reliability analysis of beam allows concluding that selection of the most suitable method

for a particular type of problem is not apparent since the accuracy and computational eort of

probabilistic methods depend on several factors, such as degree of nonlinearity of the limit state

function, type of random variable distributions, number of random variables and their variance. In

addition, it is found that FORM and IS are the less consuming methods in comparison to other

methods.

The reliability analysis of impacted RC slabs shows that analytical MSM can be very eective

in predicting the same values of slab displacement at the impact point as the numerical model,

as well as in estimating failure probabilities very similar to those obtained with the FEM for a

displacement criterion. Studying the response of impacted slabs assuming an elastic behavior of

materials represents a preliminary step in the procedure of evaluating the reliability of RC slabs

under low velocity impact. This application is initially considered to simplify the problem and to

avoid complicating structural reliability analysis of slabs as a rst step. However, it is clear that

nonlinearities due to materials behavior have a signicant eect on RC slabs response, in particular

that cracking in concrete under tension and plastic deformations in concrete at the impact zone and

in reinforcement reduce signicantly the peak of contact force obtained with an elastic model.

The application of reliability analysis to a containment building permits to investigate the eec-

tiveness of the polynomial chaos method in estimating the initial physical model with an approxi-

mated model. It is found that the eciency of the polynomial chaos method is related to several

factors, including the limit state function to be estimated, the degree of the polynomial chaos, the

number of points in the design of experiments and the variation of random variables.

7.2 Perspectives

For Chen and May experiments [34], only the transient impact force-time history obtained with

numerical model is compared to experimental results. However, Chen and May provide more results

such as accelerations, transient reinforcement strains, and damages on the upper and bottom surfaces

of slabs. Thus, it is better to study also the eciency of the FEM proposed for Chen and May slabs

in predicting other experimental results.

Other studies exists in the literature that studied experimentally the behavior of RC slabs under

impact. Zineddin and Krauthammer tests [189], as well as Hrynyk tests [84], are very important in

term of gaining a better understanding of the impact behavior of RC slabs. Therefore, validating the

FEM for dierent slab geometries and reinforcement ratio under dierent impact conditions seems

very important to study the limitation of the proposed FEM, especially that Hrynyk tests include

successive impacts on the same slab. They also tested the impact behavior of steel ber reinforced

concrete, which represents an important problem to investigate numerically.

The behaviors of concrete and steel are considered in the FEM similar to their static behavior.

However, an impact problem is a dynamic problem and concrete has dierent behavior under impact

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than under static load. Thus, using subroutines in Abaqus to implement a new concrete model that

takes into account its behavior under impact would provide more accurate prediction of the structural

response.

In the current study, the evolution in time and in space of uncertainties related to loading and

material properties, respectively, are not taken into account. These types of variation necessitate

modeling the random functions as stochastic eld if the evolution in space is studied, and as stochastic

process if the evolution in time is considered. The latest version of OpenTURNS allows more

probabilistic methods such as kriging and subset methods, it also allows the denition of randomness

as stochastic eld or stochastic process. Thus, it would be interesting to compare the new available

methods with those used in this study in order to examine their eciency and precision in assessing

failure probability. In addition, the description of the evolution in time of uncertainties related to

loading and that in space related to material properties permits a better presentation of the physical

problem in reliability analysis if data are available.

In this study, dierent failure criteria are examined, but independently. However a system

analysis is necessary since output variables considered in structural reliability analysis does not

reach their maximum values simultaneously. For example, the contact force reaches its maximum

value at the rst phases of contact while that of displacement occurs at the end of contact. Yielding

of steel and energy dissipation may occur at any time during the impact phenomenon. Thus, it is

necessary to study the reliability of slabs in the nonlinear case for several failure criteria, separately

or using a system analysis.

Several relationships are presented in this study to describe the behavior of concrete in tension

and compression based on research from the literature. It is found that these curves aect highly

the response of the slab, thus a reliability analysis can be performed in order to investigate the eect

of the tensile and compressive stress-strain curves of concrete on the probability of failure for one or

several failure criteria.

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