Joint Research Centre - I-21020 Ispra (VA) Italy - Tp 480Telephone: (+39-332) 78-9989 exchange 78-9111. Telefax: 78-9049Telex: 380042 EUR I - 380058 EUR IEUROPEAN COMMISSIONDIRECTORATE GENERAL XII - JRCSCIENCE, RESEARCH AND DEVELOPMENT - JOINT RESEARCH CENTRESAFETY TECHNOLOGY INSTITUTEAPPLIED MECHANICS UNITOctober 97SEISMIC BEHAVIOUR OF REINFORCED CONCRETE BRIDGES. MODELLING, NUMERICAL ANALYSIS AND EXPERIMENTAL ASSESSMENTByJoo Paulo Sousa Costa de Miranda GuedesACKNOWLEDGEMENTS iACKNOWLEDGEMENTSThis work was developed under the Human Capital and Mobility programme of theEuropean Commission in the European Laboratory for Structural Assessment of theJoint Research Centre at Ispra, Italy; the software and all experimental data necessary toperform the work were provided by this laboratory.The author is therefore grateful to the European Commission for the financial support.The participation of the laboratory staff is also acknowledged. In particular, I am espe-cially grateful to Artur Pinto for all support and friendship during the almost four yearsthat we worked together at the Joint Research Centre. Likewise, I am also grateful toPierre Pegon for his precious help.I thank the head of the Applied Mechanics Unit, Professor Jean Donea, and all grantholders and visitors in Italy with whom I had the chance to have very helpful discus-sions, in particular: Antonio Arde, Didier Combescure and Alfredo Campos Costa, andAlice Bernard for the tremendous work of reading and correcting the final text. Thanksfor your friendship too.To Professor Raimundo Delgado at the Faculdade de Engenharia da Universidade doPorto I thank for the supervision of this work.I am also grateful to all friends in Ispra for the wonderful time I spent in Italy. I alsothank the friends in Portugal that, in spite of the distance, always stood by me. Specialwords go to Angela Pereira, Eduarda Mesquita (thanks for the cover...) and Alfredo DeLos Reyes: thank you for all you have done for me.Finally, I thank my parents, my brothers and my grandmother for their unconditionalsupport. I dedicate this work to them.iiABSTRACT iiiABSTRACTThis work was developed within the Prenormative Research in support of the EuroCode8 (PREC8) programme of the European Commission. EuroCode 8 (EC8) is the provi-sional European standards for the design of civil engineering structures in seismic proneareas. The programme included a series of Pseudo-dynamic tests on a set of ReinforcedConcrete (R/C) bridges designed according to EC8. The Pseudo-dynamic method isbased on a hybrid formulation that combines the numerical integration of the equationsof motion of a structure and the experimental measurement of the corresponding restor-ing forces. Application to bridges can be made using substructuring techniques where thedeck is simulated numerically and only the piers are physically tested in the laboratory.This was the first Pseudo-dynamic testing campaign performed in the world using thesubstructuring technique on large scale structures.Numerical tests using a fibre model were performed to predict the bridges response.These preliminary results allowed us not only to verify the ability of the model to predictthe behaviour of these structures under cyclic loading, but also to establish the maximumforces provided by the control system in the laboratory. The need to improve the fibremodel for the non-linear behaviour in shear derived from the comparison between theexperimental response and the numerical results. A strut-and-tie formulation coupledwith the classic fibre model was developed. This formulation is based on the analogybetween an R/C structure damaged with diagonal cracking and a truss made of concretediagonals and steel ties.Subsequently, the numerical analyses were repeated using the new model and the resultswere compared with the experimental response. These new results were analysed anddiscussed in detail. In order to understand the irregularity issue, a new set of bridges withincreasing degree of irregularity was designed and their numerical response was evalu-ated for seismic actions of growing intensity.iv ABSTRACTRESUMOEste trabalho foi desenvolvido no mbito do programa de Investigao Prenormativa daComisso Europeia de Suporte ao EuroCdigo N. 8 (EC8). Este cdigo pretende ser ostandard Europeu para o dimensionamento de estruturas de Engenharia Civil sujeitas aaces ssmicas. O programa incluiu uma campanha de ensaios Pseudo-dinmicos reali-zados em pontes de beto armado projectadas de acordo com o referido cdigo. Omtodo Pseudo-dinmico um mtodo hbrido que combina a integrao numrica dasequaes de movimento com a medio experimental das foras de restituio da estru-tura. Nos ensaios realizados o tabuleiro foi sub-estruturado e apenas os pilares foram tes-tados no laboratrio. Note-se que este foi o primeiro teste Pseudo-dinmico realizadonuma estrutura de grandes dimenses utilizando a tcnica de sub-estruturao.Testes preliminares realizados com o modelo de fibras permitiram antecipar o valormximo das forcas de restituio esperadas no laboratrio. Estes resultados permitiramainda verificar a capacidade do modelo para prever o comportamento de pilares de pontesujeitos a aces cclicas. A necessidade de desenvolver um modelo que representasseconvenientemente o comportamento no linear s foras de corte resultou da com-parao entre a resposta experimental e os resultados das previses numricas. Assim,acopulado ao modelo de fibras para as foras de flexo foi desenvolvido um modelo dotipo biela-tirante para as foras de corte. Esta formulao baseada na analogia entreuma estrutura de beto armado com fissurao diagonal e uma trelia constituda por ele-mentos diagonais de beto ligados por elementos longitudinais e transversais de ao.A simulao dos ensaios no laboratrio foi repetida utilizando o modelo desenvolvido eos resultados foram comparados com a resposta experimental. As diferenas ainda assimencontradas so discutidas e analisadas em detalhe. Para terminar, a resposta numricade uma nova srie de pontes com diferentes graus de irregularidade, foi projectada deacordo com o cdigo EC8 e analisada para aces ssmicas de intensidade crescente.ABSTRACT vRESUMLe travail ici prsent a t dvelopp dans le cadre de la Recherche Prnormative,comme support du EuroCode8 (EC8), des Commissions Europennes. EC8 est le stand-ard provisoire Europen pour le design de structures de gnie civile, dans les zones derisque sismique. Ce programme inclut des tests exprimentaux pour un certain nombrede structures de ponts, testes sous conditions Pseudo-dynamiques. La mthode Pseudo-dynamique est une mthode hybride qui combine l'intgration numrique des quationsdu mouvement d'une structure, et les mesures exprimentales des forces de restitutioncorrespondantes. De plus, le tablier a t substructur et seules les piles ont t testesphysiquement dans le laboratoire. En effet, ces expriences ont t la premire srie detests Pseudo-Dynamiques effectus avec une technique de substructuration.Des tests numriques realiss avec le modle de fibres ont t executs avant les expri-ences afin de prdire la rponse des structures. Ces rsultats prliminaires permettent nonseulement de vrifier la capacit du modle pour la prediction du comportement de pontsen bton arm soumis l'action de charges cycliques, mais aussi d'identifier les forcesmaximales admises par le systme de contrle du laboratoire. Le besoin d'amliorer unmodle pour l'tude du comportement au cisaillement non linaire survient de la com-paraison entre la rponse exprimentale et les rsultats des prdictions numriques. Ainsiune formulation strut-and-tie accouple au modle de fibres classique a t dvelop-pe. Cette formulation est base sur l'analogie d'une structure en bton arm endom-mage par des fissures diagonales, avec une poutre en treillis constitue de diagonales enbton et d'attaches en acier.La srie numrique a t rpte en employant le nouveau modle et les rsultats ont tcompars avec les rponses exprimentales. Les diffrences encore trouves sont ana-lyses dans ce travail. Finalement, une nouvelle srie de ponts est projecte et leurrponse des intensits sismiques croissantes est value.vi ABSTRACTTABLE OF CONTENTS viiTABLE OF CONTENTSSEISMIC BEHAVIOUR OF REINFORCED CONCRETE BRIDGES.MODELLING, NUMERICAL ANALYSIS AND EXPERIMENTAL ASSESSMENTACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiRESUMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivRESUM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vTABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiLIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiLIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxi 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 GENERAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 MOTIVATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 THE ELSA LABORATORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 DEVELOPED WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 OUTLINE OF THE THESIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 THE PSEUDO-DYNAMIC TEST METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 TESTING METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 NUMERICAL ALGORITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 -Newmark algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Central differences scheme (explicit) - - - - - - - - - - - - - - - - - - - - - 17-method (implicit) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 17Shing method (implicit) - - - - - - - - - - - - - - - - - - - - - - - - - - - - 18Modified Newmark method (explicit) - - - - - - - - - - - - - - - - - - - - - 20Operator-Splitting scheme (implicit non-iterative) - - - - - - - - - - - - - - - 202.4 IMPLEMENTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1 Characteristics of the control system . . . . . . . . . . . . . . . . . . . . . . . . 242.4.2 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 THE ELSA FACILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26viii TABLE OF CONTENTS2.5.1 Characteristics of the laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5.2 Architecture of the control system at the ELSA laboratory . . . . . . . . . . . . . 272.6 SUBSTRUCTURING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.7 ASYNCHRONOUS INPUT MOTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.8 ENGINEERING APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.9 FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 AN EXPERIMENTAL CAMPAIGN ON R/C BRIDGES . . . . . . . . . . . . . . . . . . . 413.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 THE PREC8 PROGRAMME - BRIDGES WORKING GROUP . . . . . . . . . . . . . . 433.3 THE TESTING CAMPAIGN AT THE ELSA LABORATORY . . . . . . . . . . . . . . . 443.3.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.2 The bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.3 Design of the bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3.4 Scaling of the structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3.5 Construction of the piers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4 THE TEST SET-UP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4.1 Test specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Piers - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 50Deck - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 513.4.2 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Rotations - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 53Horizontal displacements - - - - - - - - - - - - - - - - - - - - - - - - - - - 533.5 QUASI-STATIC TEST OF A SQUAT PIER . . . . . . . . . . . . . . . . . . . . . . . . . 543.5.1 Test objectives and testing procedure . . . . . . . . . . . . . . . . . . . . . . . . 543.5.2 Test specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.5.3 Processing of the test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Force-displacement diagrams and damage - - - - - - - - - - - - - - - - - - 55Flexural and shear deformations - - - - - - - - - - - - - - - - - - - - - - - 56Energy dissipation - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 56Ductility - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 57Other measurements - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 573.5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.6 THE PSEUDO-DYNAMIC TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.6.1 Input motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.6.2 The tests sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.6.3 Inertia and damping forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.7 TESTING RESULTS FROM THE FIRST SET OF BRIDGES . . . . . . . . . . . . . . . 633.7.1 Global response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.7.2 Piers force-displacement diagrams and damage . . . . . . . . . . . . . . . . . . 653.7.3 Ductility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.7.4 Earthquake performance of the bridges . . . . . . . . . . . . . . . . . . . . . . . 67Irregular bridges (B213) - - - - - - - - - - - - - - - - - - - - - - - - - - - 67Regular bridge (B232) - - - - - - - - - - - - - - - - - - - - - - - - - - - - 683.8 ULTIMATE CAPACITY OF THE PIERS . . . . . . . . . . . . . . . . . . . . . . . . . . 683.8.1 Cyclic testing of the bridge piers . . . . . . . . . . . . . . . . . . . . . . . . . . 693.8.2 Flexural and shear deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.8.3 The piers performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.9 TESTING RESULTS FROM THE SECOND AND THIRD SET OF BRIDGES . . . . . . 703.9.1 Bridge B213A isolated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Test set-up - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 71Test results - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 723.9.2 Bridge with asynchronous input motion . . . . . . . . . . . . . . . . . . . . . . 74TABLE OF CONTENTS ixTest set-up - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 75Test results - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 753.10 SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.11 FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4 THE FIBRE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.2 FIBRE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.2.1 Compatibility equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.2.2 Equilibrium equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.2.3 Tangent stiffness matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.3 CONCRETE CONSTITUTIVE LAWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.3.1 Monotonic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Compression stresses - - - - - - - - - - - - - - - - - - - - - - - - - - - - 129Tensile stresses - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1324.3.2 Cyclic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Preliminary model - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 135Improved model - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1374.4 STEEL CONSTITUTIVE LAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1424.4.1 Monotonic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1424.4.2 Cyclic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Basic model - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 144Inelastic buckling - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1464.5 NUMERICAL APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1484.5.1 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1494.5.2 Monotonic loading tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Reinforced concrete rectangular cross sections - - - - - - - - - - - - - - - 149Influence of the concrete tensile strength - - - - - - - - - - - - - - - - - - - 1514.5.3 Cyclic tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Uniaxial loading of a cantilever T beam - - - - - - - - - - - - - - - - - - - 153Biaxial loading of a R/C square section - - - - - - - - - - - - - - - - - - - 1554.6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5 NON-LINEAR SHEAR MODEL FOR R/C PIERS . . . . . . . . . . . . . . . . . . . . . . 1615.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615.2 NON-LINEAR SHEAR MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625.2.1 Shear mechanisms in R/C structural elements . . . . . . . . . . . . . . . . . . . 1635.2.2 Numerical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Chang model - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 166Priestley model - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 168A Strut-and-Tie model for 2D elements - - - - - - - - - - - - - - - - - - - 169Fibre based models - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1715.3 STRUT-AND-TIE MODEL IN FIBRE MODELLING . . . . . . . . . . . . . . . . . . . 1725.3.1 Non-linear shear modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172Equations of compatibility of displacements - - - - - - - - - - - - - - - - - 174Equations of equilibrium of forces - - - - - - - - - - - - - - - - - - - - - - 176Damage of the struts - - - - - - - - - - - - - - - - - - - - - - - - - - - - 178Constitutive laws - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 179Tensile strut - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 180Shear cracking angle - - - - - - - - - - - - - - - - - - - - - - - - - - - - 181Cross-sectional area of the struts - - - - - - - - - - - - - - - - - - - - - - 1865.3.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187x TABLE OF CONTENTS5.4 NUMERICAL IMPLEMENTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1885.4.1 Parameters of the model in CASTEM 2000 . . . . . . . . . . . . . . . . . . . . . 1895.4.2 Strain at the transverse steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189Equations of compatibility for incremental displacements - - - - - - - - - - - 190Equilibrium of internal forces - Iterative algorithm - - - - - - - - - - - - - - 1915.4.3 Shear tangent stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1935.4.4 Modelling and implementation remarks . . . . . . . . . . . . . . . . . . . . . . . 195Yielding of transverse steel - - - - - - - - - - - - - - - - - - - - - - - - - 195Torsional moments - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 196The snap-back phenomenon - - - - - - - - - - - - - - - - - - - - - - - - - 1975.5 MODEL VALIDATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2015.5.1 The experimental tests - some remarks . . . . . . . . . . . . . . . . . . . . . . . 2015.5.2 Numerical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203Model parameters - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 204Fibre discretization - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 205Struts inclination angle - - - - - - - - - - - - - - - - - - - - - - - - - - - 2065.6 RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2085.6.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Low strength pier - P1 - - - - - - - - - - - - - - - - - - - - - - - - - - - 209Medium strength pier - P3 - - - - - - - - - - - - - - - - - - - - - - - - - 211High strength pier - P5 - - - - - - - - - - - - - - - - - - - - - - - - - - - 2125.7 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2135.8 FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 6 NUMERICAL SIMULATION OF THE EXPERIMENTAL CAMPAIGN . . . . . . . . . 2456.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2456.2 TESTING PROCEDURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2476.2.1 The Pseudo-dynamic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2476.2.2 The test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2486.3 GEOMETRIC CHARACTERISTICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2486.3.1 Geometric characteristics of the bridges - Regularity parameter . . . . . . . . . . 2496.3.2 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2506.4 INPUT DATA AND NUMERICAL DAMPING . . . . . . . . . . . . . . . . . . . . . . . 2516.5 PRELIMINARY STUDIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2526.5.1 Material properties and piers finite element mesh . . . . . . . . . . . . . . . . . 2526.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2536.5.3 Flexibility of the foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2536.5.4 Shear contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2546.6 NUMERICAL PREDICTIONS OF THE PSEUDO-DYNAMIC TESTS . . . . . . . . . . 2566.6.1 Numerical simulations and interaction with the testing programme . . . . . . . . 256Bridge B213A - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 256Bridge B213B - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 257Bridges B213C and B232 - - - - - - - - - - - - - - - - - - - - - - - - - - 2586.6.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2586.6.3 Numerical results and comparison with the experimental response . . . . . . . . 2596.6.4 Conclusions from the predictive analysis . . . . . . . . . . . . . . . . . . . . . . 2616.7 THE POST-EXPERIMENT NUMERICAL ANALYSES . . . . . . . . . . . . . . . . . . 2636.7.1 Material properties and piers finite element mesh . . . . . . . . . . . . . . . . . 2636.7.2 The non-linear shear results - comparison with the experimental response . . . . . 2646.8 A NEW NUMERICAL CAMPAIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2666.8.1 Comparison between the two integration algorithms . . . . . . . . . . . . . . . . 2676.8.2 The influence of the behaviour model . . . . . . . . . . . . . . . . . . . . . . . . 2686.8.3 New material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268TABLE OF CONTENTS xi6.8.4 Analysis of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2706.9 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2706.10 FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 7 NUMERICAL ANALYSIS OF A NEW SET OF BRIDGES . . . . . . . . . . . . . . . . . 3037.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3037.2 DESIGN OF THE PREC8 BRIDGES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3047.3 DESIGN OF THE NEW SET OF BRIDGES . . . . . . . . . . . . . . . . . . . . . . . . . 3107.3.1 The regularity parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3107.3.2 The response spectrum analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3137.4 THE RESPONSE OF THE NEW BRIDGES . . . . . . . . . . . . . . . . . . . . . . . . . 3167.5 ANALYSIS OF THE RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3187.6 FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 8 CONCLUSIONS AND FUTURE RESEARCH . . . . . . . . . . . . . . . . . . . . . . . . . 3398.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3398.2 THE EXPERIMENTAL CAMPAIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3398.3 THE NUMERICAL TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3438.4 FUTURE RESEARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 9 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347ANNEXES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 A PHOTOGRAPHIC DOCUMENTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 B CASTEM 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367B.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367B.2 ADVANTAGES OF CASTEM 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367B.3 OPERATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368B.4 OBJECTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368B.5 THE GIBIANE LANGUAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369B.6 PROCEDURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369B.7 AN EXAMPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370B.8 FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 C NON LINEAR DYNAMIC ALGORITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381C.1 NEWMARK FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381C.2 -NEWMARK FORMULATION USING THE OPERATOR-SPLITTING METHOD . . . . . 384 D EXPERIMENTAL BENDING DISPLACEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 E TIMOSHENKO BEAM FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391E.1 COMPATIBILITY OF DISPLACEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391E.2 CONSTITUTIVE EQUATIONS - AN ELASTIC ISOTROPIC MATERIAL. . . . . . . . . . . . . 392E.3 EQUILIBRIUM EQUATIONS - VIRTUAL WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394E.4 FINITE ELEMENT FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395E.5 TANGENT STIFFNESS MATRIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400E.6 MASS MATRIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400xii TABLE OF CONTENTSLIST OF FIGURES xiiiLIST OF FIGURESSEISMIC BEHAVIOUR OF REINFORCED CONCRETE BRIDGES.MODELLING, NUMERICAL ANALYSIS AND EXPERIMENTAL ASSESSMENT 1 INTRODUCTION 2 THE PSEUDO-DYNAMIC TEST METHODFigure 2.1 -Pseudo-dynamic testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.2 -Digital control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 2.3 -Reaction-Wall facility at the ELSA Laboratory . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 2.4 -Pseudo-dynamic testing with substructuring. An isolated two floors building (see Figure 2.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 2.5 -Pseudo-dynamic test coupled with the substructuring technique . . . . . . . . . . . . . . . 39Figure 2.6 -Pseudo-dynamic testing with substructuring - A four piers bridge case. . . . . . . . . . . . 40 3 AN EXPERIMENTAL CAMPAIGN ON R/C BRIDGESFigure 3.1 -Full-scale scheme of the bridges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 3.2 -Test set-up. Pseudo-dynamic test with substructuring . . . . . . . . . . . . . . . . . . . . . 82Figure 3.3 -Geometrical characteristics of the piers (M - Medium, S - Short and T - Tall). Representation of the pier slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 3.4 -Reinforcement layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 3.5 -Stress versus strain diagrams of a concrete sample . . . . . . . . . . . . . . . . . . . . . . 85Figure 3.6 -Axial tensile stress versus strain diagrams for the steel bars . . . . . . . . . . . . . . . . . 86Figure 3.7 -Preliminary test on a squat bridge pier - Imposed top displacement time history. . . . . . . 86Figure 3.8 -Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 3.9 -Cyclic test on the 2.8m high pier. Force versus displacement diagrams: a) total curve, b) linear regime and first cracking, c) within the cracking zone, d) up to ductility 1.5, e) up to ductility demand 3.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 3.10 -Displacement and deformation profiles: a) total displacement, b) shear and bending displacements, c) flexural curvature, b) shear strain . . . . . . . . . . . . . . . . . . 89Figure 3.11 -Relative displacements due to: a) bending moments, b) shear forces . . . . . . . . . . . . 90Figure 3.12 -Flexural moment versus curvature and shear force versus shear strain diagrams . . . . . . 91Figure 3.13 -Relative energy dissipation: a) at the different slices, b) accumulated from the base up to the slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Figure 3.14 -Diagrams measuring: a) out-of-plane deformations, b) the vertical force and the axial displacement at the pier axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure 3.15 -Design accelerogram. Comparison with the EC8 response spectrum (5% damping) . . . . 93xiv LIST OF FIGURESFigure 3.16 -Non-linear elastic curves (broken line) adopted for the behaviour of the medium and the tall piers in the HLE tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Figure 3.17 -Piers: Test specimens and testing sequence . . . . . . . . . . . . . . . . . . . . . . . . . 95Figure 3.18 -Schematic representation of the performed pseudo-dynamic tests (see Figure 3.17) . . . . 96Figure 3.19 -Displacement time histories for bridge B213A. . . . . . . . . . . . . . . . . . . . . . . . 97Figure 3.20 -Displacement time histories for bridge B213B. . . . . . . . . . . . . . . . . . . . . . . . 98Figure 3.21 -Displacement time histories for bridge B213C. . . . . . . . . . . . . . . . . . . . . . . . 99Figure 3.22 -Displacement time histories for bridge B232 (regular) . . . . . . . . . . . . . . . . . . 100Figure 3.23 -Frequencies and mode shapes of the bridges . . . . . . . . . . . . . . . . . . . . . . . . 101Figure 3.24 -Envelop displacement profiles of the deck for the regular and the irregular bridges . . . 102Figure 3.25 -Power spectrum of the piers top displacements . . . . . . . . . . . . . . . . . . . . . . 103Figure 3.26 -Force-displacement diagrams for the bridge B213A. . . . . . . . . . . . . . . . . . . . 104Figure 3.27 -Force-displacement diagrams for the bridge B213B . . . . . . . . . . . . . . . . . . . . 105Figure 3.28 -Force-displacement diagrams for the bridge B213C . . . . . . . . . . . . . . . . . . . . 106Figure 3.29 -Force-displacement diagrams for the bridge B232 (regular) . . . . . . . . . . . . . . . . 107Figure 3.30 -Force-displacement diagrams for the short pier of the irregular bridges (B213A, B213B and B213C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Figure 3.31 -Flexural and shear deformations of the short piers for the HLE (10 peak values) . . . . . 109Figure 3.32 -Vulnerability functions - Demands for the two earthquake levels: a) pier drift ratio b) ductility c) energy dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Figure 3.33 -Force displacement diagrams for the piers tested cyclically . . . . . . . . . . . . . . . . 111Figure 3.34 -Moment-curvature diagrams - Short pier of the bridges 213A and 213B . . . . . . . . . 112Figure 3.35 -Moment-curvature diagrams - bridge 213C . . . . . . . . . . . . . . . . . . . . . . . . 113Figure 3.36 -Moment-curvature diagrams - bridge 213C . . . . . . . . . . . . . . . . . . . . . . . . 114Figure 3.37 -Moment-curvature diagrams - bridge 232 . . . . . . . . . . . . . . . . . . . . . . . . . 115Figure 3.38 -Numerical frequencies and mode shapes of the bridge models for the three design solutions: B213A, B213A-5Dev and B213A-1Dev. . . . . . . . . . . . . . . . . . 116Figure 3.39 -Stiffness tests on the bridge piers (and abutments) before the first Pseudo-dynamic test on each of the three design solutions . . . . . . . . . . . . . . . . . . . . . . . 117Figure 3.40 -Piers top diplacement time histories for the three design solutions and for the 1.2 DE . . 118Figure 3.41 -Force versus piers top displacement diagrams for the two isolated bridges and for the 1.2 DE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Figure 3.42 -Maximum displacements of the R/C piers top section and deck . . . . . . . . . . . . . . 120Figure 3.43 -Piers top diplacement time histories for the reference bridge B213A with asynchronous input motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Figure 3.44 -Force versus piers top displacement diagrams for the asynchronous input motion . . . . 122Figure 3.45 -Maximum displacement drifts and dissipated energy: synchronous versus asynchronous 123 4 THE FIBRE MODELFigure 4.1 -Fibre model. Deformation of the transverse section. . . . . . . . . . . . . . . . . . . . . 128Figure 4.2 -Confinement effect in the concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Figure 4.3 -Concrete response curves for monotonic loading . . . . . . . . . . . . . . . . . . . . . . 134Figure 4.4 -Concrete response under compression cyclic loading . . . . . . . . . . . . . . . . . . . . 135Figure 4.5 -Numerical model for the concrete behaviour under cyclic loading . . . . . . . . . . . . . 136Figure 4.6 -Numerical model for the concrete under tensile stresses (detail from Figure 4.5) . . . . . 137Figure 4.7 -Axial stress-strain constitutive law for concrete . . . . . . . . . . . . . . . . . . . . . . . 139Figure 4.8 -Steel bars under monotone increasing deformation . . . . . . . . . . . . . . . . . . . . . 143LIST OF FIGURES xvFigure 4.9 -Numerical model for the steel under cyclic loading . . . . . . . . . . . . . . . . . . . . . 146Figure 4.10 -Steel cyclic behaviour model for inelastic buckling . . . . . . . . . . . . . . . . . . . . 147Figure 4.11 -numerical results using the fibre model in CASTEM 2000 (Park and Paulay test) . . . . 150Figure 4.12 -Influence of the tensile parameters in the section global response . . . . . . . . . . . . . 152Figure 4.13 -Transverse section of the T beam - S1V3 test . . . . . . . . . . . . . . . . . . . . . . . 154Figure 4.14 -Force displacement curve on the top of the cantilever T beam . . . . . . . . . . . . . . 156Figure 4.15 -Column transverse section tested by Bousias. Distribution of fibres in the cross-section . 157Figure 4.16 -Loading path: axial force and horizontal displacements . . . . . . . . . . . . . . . . . . 158Figure 4.17 -Free top force versus displacement response curves in OZ direction . . . . . . . . . . . 158Figure 4.18 -Free top force versus displacement response curves in OY direction . . . . . . . . . . . 159 5 NON-LINEAR SHEAR MODEL FOR R/C PIERSFigure 5.1 -Diagonal cracking: dowel and interlock effect . . . . . . . . . . . . . . . . . . . . . . . 164Figure 5.2 -Chang model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Figure 5.3 -Contribution of axial load to shear strength (Priestly model [73]) . . . . . . . . . . . . . 169Figure 5.4 -Average stress-strain response of concrete. Modified model due to Vecchio and Collins . 171Figure 5.5 -Cracking pattern - internal acting forces. . . . . . . . . . . . . . . . . . . . . . . . . . . 173Figure 5.6 -Truss analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174Figure 5.7 -Compatibility of displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Figure 5.8 -Internal forces in the section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Figure 5.9 -Influence of the tensile behaviour law in the shear stress-strain response curve . . . . . . 181Figure 5.10 -Analogy with a membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Figure 5.11 -Cracking equilibrium model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Figure 5.12 -Cross-sectional area of the struts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Figure 5.13 -Shear fibre model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190Figure 5.14 -Influence of yielding of the stirrups in the shear stress-strain response curve . . . . . . . 197Figure 5.15 -Torsional moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Figure 5.16 -Numerical simulation of torsional moment resisting mechanisms . . . . . . . . . . . . . 199Figure 5.17 -The snap-back phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200Figure 5.18 -Profile of the irregular bride tested at the ELSA laboratory (scale 1:1) . . . . . . . . . . 201Figure 5.19 -Parameters of the confined and unconfined concrete . . . . . . . . . . . . . . . . . . . 203Figure 5.20 -Parameters of the longitudinal and transverse steel . . . . . . . . . . . . . . . . . . . . 204Figure 5.21 -Distribution of the fibres in the cross-section (bending + shear). . . . . . . . . . . . . . 205Figure 5.22 -Cracking angle computed through the cracking equilibrium model . . . . . . . . . . . . 207Figure 5.23 -History of displacements imposed at the top of the piers . . . . . . . . . . . . . . . . . 215Figure 5.24 -Numerical versus experimental peak displacement profiles for the four piers and the non-linear shear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216Figure 5.25 -Numerical versus experimental force versus total, flexural and shear displacement response curves for pier P5D for different transverse steel ratios . . . . . . . . . . 217Figure 5.26 -Numerical force versus total, flexural and shear displacement response curves for pier P5D for different meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218Figure 5.27 -Numerical versus experimental force versus total displacement response curves for pier P1D at two different height levels and using the non-linear shear model . . . . 219Figure 5.28 -Numerical versus experimental force-flexural and shear displacement response curves at 1.7m from the bottom, for pier P1D, using the non-linear shear model . . 220Figure 5.29 -Numerical versus experimental force versus total displacement response curves for pier P1D at two different height levels and using the linear elastic shear model . . . 221xvi LIST OF FIGURESFigure 5.30 -Numerical versus experimental force-flexural and shear displacement response curves at 1.7m from the bottom, for pier P1D, using the non-linear shear model . . 222Figure 5.31 -Numerical versus experimental force versus total displacement response curves for pier P3D at two different height levels and using the linear elastic shear model . . . 223Figure 5.32 -Numerical versus experimental force-flexural and shear displacement response curves at 1.7m from the bottom, for pier P1D, using the non-linear shear model . . 224Figure 5.33 -Numerical versus experimental force versus total displacement response curves for pier P3D at two different height levels and using the linear elastic shear model . . . 225Figure 5.34 -Numerical versus experimental force-flexural and shear displacement response curves at 1.7m from the bottom, for pier P3D, using the linear elastic shear model . 226Figure 5.35 -Numerical versus experimental force versus total displacement response curves for pier P3S at two different height levels and using the non-linear shear model . . . . 227Figure 5.36 -Numerical versus experimental force-flexural and shear displacement response curves at 1.7m from the bottom, for pier P3S, using the non-linear shear model . . 228Figure 5.37 -Numerical versus experimental force versus total displacement response curves for pier P3S at two different height levels and using the linear elastic shear model . . . 229Figure 5.38 -Numerical versus experimental force-flexural and shear displacement response curves at 1.7m from the bottom, for pier P3S, using the linear elastic shear model . 230Figure 5.39 -Numerical versus experimental force versus total displacement response curves for pier P5D at two different height levels and using the non-linear shear model . . . . 231Figure 5.40 -Numerical versus experimental force-flexural and shear displacement response curves at 1.7m from the bottom, for pier P5D, using the non-linear shear model . . 232Figure 5.41 -Numerical versus experimental force versus total displacement response curves for pier P5D at two different height levels and using the linear elastic shear model . . . 233Figure 5.42 -Numerical versus experimental force-flexural and shear displacement response curves at 1.7m from the bottom, for pier P5D, using the linear elastic shear model . 234Figure 5.43 -Numerical versus experimental force versus total, flexural and shear displacement response curves for pier P5D with different transverse critical cracking angles . . . 235Figure 5.44 -Numerical versus experimental force versus total displacement response curves for pier P1D at two different height levels and using the non-linear shear model . . . . 236Figure 5.45 -Numerical versus experimental force-flexural and shear displacement response curves at 1.7m from the bottom, for pier P1D, using the non-linear shear model . . 237Figure 5.46 -Numerical versus experimental force versus total displacement response curves for pier P3D at two different height levels and using the linear elastic shear model . . . 238Figure 5.47 -Numerical versus experimental force-flexural and shear displacement response curves at 1.7m from the bottom, for pier P1D, using the non-linear shear model . . 239Figure 5.48 -Numerical versus experimental force versus total displacement response curves for pier P3S at two different height levels and using the non-linear shear model . . . . 240Figure 5.49 -Numerical versus experimental force-flexural and shear displacement response curves at 1.7m from the bottom, for pier P3S, using the non-linear shear model . . 241Figure 5.50 -Numerical versus experimental force versus total displacement response curves for pier P5D at two different height levels and using the non-linear shear model . . . . 242Figure 5.51 -Numerical versus experimental force-flexural and shear displacement response curves at 1.7m from the bottom, for pier P5D, using the non-linear shear model . . 243Figure 5.52 -Numerical versus experimental peak displacement profiles for the four piers and the non-linear shear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244LIST OF FIGURES xvii 6 NUMERICAL SIMULATION OF THE EXPERIMENTAL CAMPAIGNFigure 6.1 -Numerical analysis assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273Figure 6.2 -Confined and unconfined concrete constitutive laws and model parameters for the piers . 273Figure 6.3 -Numerical model for the steel under cyclic loading . . . . . . . . . . . . . . . . . . . . . 273Figure 6.4 -Distribution of the fibres in the piers and deck cross-section . . . . . . . . . . . . . . . . 274Figure 6.5 -3D representation of the pier and block foundation with cubic finite elements . . . . . . . 274Figure 6.6 -Top force-displacement curve with and without flexible foundation . . . . . . . . . . . . 275Figure 6.7 -Numerical simulation of the flexible foundation . . . . . . . . . . . . . . . . . . . . . . 275Figure 6.8 -Experimental and numerical results for the squat pier (displacements at 1.70m high) . . . 276Figure 6.9 -Analytical and experimental time histories of the top piers displacement for bridge B213A (preliminary results) . . . . . . . . . . . . . . . . . . . . . . . . . . 277Figure 6.10 -Analytical and experimental time histories of the top piers displacement for bridge B213B (preliminary results) . . . . . . . . . . . . . . . . . . . . . . . . . . 278Figure 6.11 -Analytical and experimental time histories of the top piers displacement for bridge B213C (preliminary results) . . . . . . . . . . . . . . . . . . . . . . . . . . 279Figure 6.12 -Analytical and experimental time histories of the top piers displacement for bridge B232 (preliminary results) . . . . . . . . . . . . . . . . . . . . . . . . . . . 280Figure 6.13 -Time histories of the force at the top of the piers for bridge B213A. . . . . . . . . . . . 281Figure 6.14 -Time histories of the force at the top of the piers for bridge B213B . . . . . . . . . . . . 282Figure 6.15 -Time histories of the force at the top of the piers for bridge B213C . . . . . . . . . . . . 283Figure 6.16 -Time histories of the force at the top of the piers for bridge B232 (regular) . . . . . . . . 284Figure 6.17 -Force-displacement diagrams for bridge B213A (preliminary results) . . . . . . . . . . 285Figure 6.18 -Force-displacement diagrams for bridge B213B (preliminary results) . . . . . . . . . . 286Figure 6.19 -Force-displacement diagrams for bridge B213C (preliminary results) . . . . . . . . . . 287Figure 6.20 -Force-displacement diagrams for bridge B232 (preliminary results) . . . . . . . . . . . 288Figure 6.21 -Vertical forces in the 1.2xDE Pseudo-dynamic test for the three piers of bridge B213C . 289Figure 6.22 -Confined and unconfined concrete characteristics . . . . . . . . . . . . . . . . . . . . . 290Figure 6.23 -Longitudinal and transverse steel characteristics. . . . . . . . . . . . . . . . . . . . . . 290Figure 6.24 -Analytical and experimental time histories of the top piers displacement for bridge B213A using the non-linear shear model . . . . . . . . . . . . . . . . . . . 291Figure 6.25 -Analytical and experimental time histories of the top piers displacement for bridge B213B using the non-linear shear model . . . . . . . . . . . . . . . . . . . 292Figure 6.26 -Analytical and experimental time histories of the top piers displacement for bridge B213C using the non-linear shear model . . . . . . . . . . . . . . . . . . . 293Figure 6.27 -Analytical and experimental time histories of the top piers displacement for bridge B232 (regular). New materials and the improved model for the concrete . . 294Figure 6.28 -Force-displacement diagrams for bridge B213A. . . . . . . . . . . . . . . . . . . . . . 295Figure 6.29 -Force-displacement diagrams for bridge B213B. . . . . . . . . . . . . . . . . . . . . . 296Figure 6.30 -Force-displacement diagrams for bridge B213C. . . . . . . . . . . . . . . . . . . . . . 297Figure 6.31 -Force-displacement diagrams for bridge B232 (regular) . . . . . . . . . . . . . . . . . . 298Figure 6.32 -Force-displacement diagrams for bridge B213A. . . . . . . . . . . . . . . . . . . . . . 299Figure 6.33 -Numerical simulation of the a-Newmark method with the Operator Splitting (OS) scheme - data exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300Figure 6.34 -Analytical and experimental time histories of the top piers displacement for bridge B213A using the non-linear shear model . . . . . . . . . . . . . . . . . . . 301Figure 6.35 -Analytical and experimental time histories of the top piers displacement for bridge B213B using the non-linear shear model . . . . . . . . . . . . . . . . . . . 302xviii LIST OF FIGURES 7 NUMERICAL ANALYSIS OF A NEW SET OF BRIDGESFigure 7.1 -Design flexural moment versus curvature diagrams for the piers transverse sections . . . 309Figure 7.2 -Bridges profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311Figure 7.3 -Piers transverse section layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315Figure 7.4 -Properties of the confined and unconfined concrete at the piers . . . . . . . . . . . . . . 317Figure 7.5 -Numerical model for the steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317Figure 7.6 -Force-displacement diagrams for bridge B010 . . . . . . . . . . . . . . . . . . . . . . . 325Figure 7.7 -Force-displacement diagrams for bridge B213 . . . . . . . . . . . . . . . . . . . . . . . 326Figure 7.8 -Force-displacement diagrams for bridge B2(1.5)3 . . . . . . . . . . . . . . . . . . . . . 327Figure 7.9 -Force-displacement diagrams for bridge B223 . . . . . . . . . . . . . . . . . . . . . . . 328Figure 7.10 -Force-displacement diagrams for bridge B2(2.5)3 . . . . . . . . . . . . . . . . . . . . . 329Figure 7.11 -Force-displacement diagrams for bridge B233. . . . . . . . . . . . . . . . . . . . . . . 330Figure 7.12 -Force-displacement diagrams for bridge B23(2.5) . . . . . . . . . . . . . . . . . . . . . 331Figure 7.13 -Force-displacement diagrams for bridge B232. . . . . . . . . . . . . . . . . . . . . . . 332Figure 7.14 -Ratio between the energy dissipated by each pier and total dissipated energy. . . . . . . 333Figure 7.15 -Ductility demands of the piers for growing seismic actions (0.5% critical damping) . . . 334Figure 7.16 -Maximum ductility demands for the eight different bridge profiles and different earthquake intensity factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335Figure 7.17 -Total dissipated energy for the 1.0 and 2.0 earthquake intensities . . . . . . . . . . . . . 335Figure 7.18 -Ductility demands of the piers for growing seismic actions (5.0% critical damping) . . . 336Figure 7.19 -Maximum ductility demands of each bridge versus the regularity parameter . . . . . . . 337 8 CONCLUSIONS AND FUTURE RESEARCH ANNEXES A PHOTOGRAPHIC DOCUMENTATIONFigure A.1 -Reinforcing steel of the short pier (pier A1) . . . . . . . . . . . . . . . . . . . . . . . . 359Figure A.2 -Reinforcing steel of the piers plinths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360Figure A.3 -Positioning of the pier on the plinth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360Figure A.4 -Casting of the plinth with the pier suspended by the crane . . . . . . . . . . . . . . . . . 361Figure A.5 -Casting of the concrete plinth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361Figure A.6 -General view of the test set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362Figure A.7 -Top view of the test set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363Figure A.8 -Cracking patterns at the central piers of the irregular bridges - Lateral view. . . . . . . . 364Figure A.9 -Cracking patterns at the central piers of the irregular bridges - Front view . . . . . . . . 365Figure A.10 -Bridge B232 piers after the final cyclic test up to failure . . . . . . . . . . . . . . . . . 366 B CASTEM 2000Figure B.1 -The three-dimensional structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377Figure B.2 -The first vibration mode (top view). Superimposition on the structure . . . . . . . . . . . 377Figure B.3 -The first vibration mode (another view). Superimposition on the structure . . . . . . . . 378Figure B.4 -The second vibration mode (top view). Superimposition on the structure . . . . . . . . . 378Figure B.5 -The second vibration mode (another view). Superimposition on the structure . . . . . . . 379Figure B.6 -The deformed structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379LIST OF FIGURES xix C NON LINEAR DYNAMIC ALGORITHMS D EXPERIMENTAL BENDING DISPLACEMENTSFigure D.1 -Splitting of shear and bending displacements. . . . . . . . . . . . . . . . . . . . . . . . 387 E TIMOSHENKO BEAM FORMULATIONFigure E.1 -Beam element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393xx LIST OF FIGURESLIST OF TABLES xxiLIST OF TABLESSEISMIC BEHAVIOUR OF REINFORCED CONCRETE BRIDGES.MODELLING, NUMERICAL ANALYSIS AND EXPERIMENTAL ASSESSMENT 1 INTRODUCTION 2 THE PSEUDO-DYNAMIC TEST METHOD 3 AN EXPERIMENTAL CAMPAIGN ON R/C BRIDGESTable 3.1: Bridges characteristics (1:1 scale) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Table 3.2: Steel mechanical properties (average values from 3 tests) . . . . . . . . . . . . . . . . . . . 51Table 3.3: Concrete mechanical properties (average values from samples of the pier A1) . . . . . . . . 51Table 3.4: Concrete cubic compression strength (average values) . . . . . . . . . . . . . . . . . . . . 51Table 3.5: Deck cross-section geometrical and mechanical characteristics (1:2.5 scaled model) . . . . 52Table 3.6: Curvature ductility in the slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Table 3.7: Cauchy similitude relationships between Prototype (P) and Model (M) . . . . . . . . . . . 60Table 3.8: Scaling factors and maximum nominal input acceleration for the Pseudo-dynamic tests . . . 60Table 3.9: Global response for design earthquake and high level earthquake tests . . . . . . . . . . . . 64Table 3.10:Physical damage for the two earthquake-level tests . . . . . . . . . . . . . . . . . . . . . . 65Table 3.11: Ductility demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Table 3.12:Ultimate curvature ductility in the slices - (1:2.5) scale non-isolated bridges . . . . . . . . . 69Table 3.13:Maximum displacements [m] for the DE and HLE (1:2.5 scale bridges) . . . . . . . . . . . 73Table 3.14:Maximum displacements [m] for the asynchronous DE and HLE (1:2.5 scale bridges) . . . 76Table 3.15:Numerical frequencies of the ten first modes of vibration . . . . . . . . . . . . . . . . . . 101 4 THE FIBRE MODELTable 4.1: Steel reinforcement ratios [%] (see Figure 4.11) . . . . . . . . . . . . . . . . . . . . . . . 150 5 NON-LINEAR SHEAR MODEL FOR R/C PIERSTable 5.1: Concrete cubic compression strength (average values) . . . . . . . . . . . . . . . . . . . 202Table 5.2: Longitudinal and transverse reinforcement in the piers cross-section . . . . . . . . . . . . 203Table 5.3: Transverse steel ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205Table 5.4: Critical cracking angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206Table 5.5: Cracking angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207xxii LIST OF TABLES 6 NUMERICAL SIMULATION OF THE EXPERIMENTAL CAMPAIGNTable 6.1: Mode shapes and regularity parameters (see equation (6.1)) . . . . . . . . . . . . . . . . 250Table 6.2: Earthquake demands (preliminary values - numerical) . . . . . . . . . . . . . . . . . . . 260Table 6.3: DE demands (post-experimental values - linear and non-linear shear model) . . . . . . . . 265Table 6.4: HLE demands (post-experimental values - linear and non-linear shear model) . . . . . . . 266Table 6.5: OS Scheme numerical results versus experimental response . . . . . . . . . . . . . . . . 270 7 NUMERICAL ANALYSIS OF A NEW SET OF BRIDGESTable 7.1: Modal analysis of the (1:2.5) scale bridges . . . . . . . . . . . . . . . . . . . . . . . . . 307Table 7.2: Design base shear forces, base flexural moments and top displacements of the piers. . . . 308Table 7.3: Mode shapes and regularity parameters (see equation (6.1)) . . . . . . . . . . . . . . . . 312Table 7.4: Regularity parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312Table 7.5: Response spectrum analysis for the (1:2.5) scale bridges . . . . . . . . . . . . . . . . . . 314Table 7.6: Longitudinal steel ratios [%] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315Table 7.7: Response spectrum analysis for the (1:2.5) scale bridge B010 . . . . . . . . . . . . . . . 316Table 7.8: Earthquake demands (numerical values for the average strength of the materials) . . . . . 319Table 7.9: Earthquake demands (numerical values for the design strength of the materials) . . . . . . 320Table 7.10:Earthquake demands (numerical values for the 2.0DE and the 3.0DE accelerograms) . . . 321Table 7.11: Earthquake demands (numerical values for the 4.0DE accelerograms) . . . . . . . . . . . 322 8 CONCLUSIONS AND FUTURE RESEARCHINTRODUCTION 1 1 INTRODUCTION1.1 GENERALEarthquakes damage civil engineering structures every year and bridges are no excep-tion. Historically, bridges have proven to be vulnerable to earthquakes, sustaining dam-age to substructures and foundations. In some cases, due to unseating of the spans orlocal failure of the restrains, bridges are totally destroyed as superstructures collapsefrom their supporting elements [7]. The vertical component of the acceleration also con-tributes for this outcome. To prevent the falling out of bridge girders, restrainer tie platesare in general used as studied by Obata et al [57]. Shear failure or insufficient ductility inthe piers due to poor confinement of the core concrete and poor detailing of the trans-verse and longitudinal steel, are also observed in several bridge piers emphasizing theimportance of the shear strength in the design.As a matter of fact, these seismic events represent an important in situ source of informa-tion that should not be neglected and, furthermore, should always be present in theimprovement of design codes and guidelines. In fact, significant advances in seismicdesign and strengthening of bridges have occurred after large earthquakes. The 1971 SanFernando earthquake caused substantial damage and exposed a large number of deficien-cies in the design specifications of that time. During the Loma Prieta earthquake of Octo-ber 1989, the dramatic collapse of the Cypress Street Viaduct in Okland and the damageof many elevated freeway bridge structures in the San Francisco bay area, highlightedweaknesses in bent joints, lack of ductility in beams and columns and poor resistance tolongitudinal and transverse loads. Inspections (observed damage) after the earthquakerevealed shear cracking and spalling of concrete especially in outrigger knee joints on2 INTRODUCTIONbends of several reinforced concrete viaducts [83]. In some cases, the shear force con-centration in shorter piers caused by the irregular profile of some bridges was not prop-erly taken into account in the design.More recent events like the Northridge earthquake of January 1994 [2], [90], and theKobe earthquake of January 1995 [28], [68], stressed once again those deficiencies. Inparticular, shear failure was still observed in several bridge piers emphasizing the impor-tance of the design and the detailing of transverse steel to provide proper shear strengthcapacity. Changes in the design code and guidelines were and still are undertaken tocompensate for these shortcomings.Notice that, in agreement with the design codes, structures in seismic areas may experi-ence inelastic behaviour during a large earthquake. The most recent approaches arebased on the capacity of the structures to dissipate energy through the hysteretic behav-iour of the piers; a certain degree of damage is thus acceptable. The amount of inelasticbehaviour allowed in the design is measured through a behaviour factor q that reducesthe design forces computed through a linear elastic analysis. This factor depends on theimportance of the bridge, i.e. on the requirements for post-earthquake functionality.Thus, different levels of acceptable ductility and damage are usually provided in thedesign codes.A Pre-normative Research in support of EuroCode N. 8 (PREC8), the provisional Euro-pean standards for the design of civil engineering structures in seismic prone areas, waslaunched by the European Commission (EC) to cover the topics of the EuroCode N. 8(EC8) that needed to be clarified. Several working groups, each one dealing with differ-ent type structures, were established under this programme, namely: Reinforced concreteframes and walls, Infill frames, Bridges and Foundations and retaining walls. This activ-ity was performed jointly with eighteen research organisations in the European Uniongrouped together in the PREC8 on Human Capital and Mobility (HCM) network. Itincluded a large experimental campaign on different structural elements such as walls,frames and bridges. Moreover, a series of numerical analyses were also performed inorder to extrapolate the experimental results to other input actions and structures.INTRODUCTION 31.2 MOTIVATIONThe subject of the present work, the seismic behaviour of Reinforced Concrete (R/C)bridges, was the general topic under discussion by the Bridges Working Group (BWG)established within the PREC8 programme. Among other points, the programme forbridges included: the classification of structural regularity, the evaluation of behaviourfactors, the improvement of methods of analysis and of capacity design procedures andthe consideration of base isolation and asynchronous input motion. Notice that, althougha general design code exists for civil engineering structures, bridges exhibit some charac-teristics that make them quite different from ordinary buildings, thus demanding specialguidelines. Firstly, the mass of the bridge deck is an order of magnitude larger than themass of a typical floor system. Secondly, bridge structural systems are not as redundantas typical building structures. Thus, in June 1994 a Pre-Standard for the seismic designof bridges, EC8 Part 2: Earthquake Resistant Design of Bridges, was approved by theEuropean Committee for Standardization (CEN).The work follows closely this pre-normative research programme, responding to some ofthe needs of the bridges working group, namely: the analysis of the experimentalresponse of a set of bridges, with and without isolating devices, tested under synchro-nous and asynchronous dynamic loading, the evaluation of behaviour factors and struc-tural regularity and the development of numerical models for bending and shear forces tosimulate bridge type structures, in particular the piers. The irregularity issue is discussedbased on an expression developed by other authors that associate the combination of thedeck and piers mode shapes with a single parameter.1.3 THE ELSA LABORATORYThe European Laboratory for Structural Assessment (ELSA) in the Joint Research Cen-tre (JRC) of the EC at Ispra in Italy, was the main supporter of the work. The excellentconditions of this laboratory, providing the hardware and software and all the necessaryexperimental equipment, allowed experimental tests on reinforced concrete structures tobe carried out, and more suitable analytical models to represent the structures tested inthe laboratory to be improved. Notice that to check and calibrate a numerical model it isnot only necessary to have access to experimental data but also to have a good knowl-4 INTRODUCTIONedge of the characteristics of the specimens and testing conditions in the laboratory thatpermitted this data to be obtained. In the case of this work, the author had the opportunityto follow all the experimental tests performed on bridges in the ELSA laboratory, partic-ipating actively in the post-treatment of the results.In this laboratory, the experiments are referred to as Pseudo-dynamic tests. The methodassociated to these tests is a hybrid method that combines the numerical integration ofthe equations of motion of a structure and the experimental measurement of the corre-sponding restoring forces. Since the inertial forces are simulated numerically, a reducedhydraulic power is required and the test is performed with a time scale enlarged withrespect to real time. The equations of motion are solved on-line using a step-by-stepnumerical integration algorithm using the physical forces from the specimen and theinertial and the damping forces from the analytical model. Thus, to perform such a test itis necessary to possess a rigid structure to sustain the reaction forces applied to the struc-tures. The European Laboratory for Structural Assessment has the most important reac-tion-wall facility in Europe.Notice that a campaign of experimental tests on structures representing different config-urations and design solutions is the ideal means to verify the adequacy of design strate-gies. However, the costs associated with an experimental campaign with such largenumber of test specimens make such programmes very expensive. Moreover, theprogresses in non-linear modelling in the recent years confine the testing activity to spe-cial cases, verifying extreme design solutions or the behaviour of critical regions, andusing the results as the basis for model calibration. It must be pointed out that numericalresults are also used for predicting the response of the structures in the laboratory, thushelping to prepare the experimental campaign.Therefore, the numerical analysis is complementary to the experimental analysis; thenumerical models that are used for simulating structures not tested in the laboratory arecalibrated with the experimental results. In the ELSA laboratory, the numerical analysisare performed with the computer code CASTEM 2000, an object oriented finite elementcode developed by a group of researchers of the Commissariat l'Energie Atomique(CEA) at Saclay in France. Thanks to the generality of its data and the different levels ofINTRODUCTION 5modelling available, different analyses can be done within the same computational envi-ronment. Moreover, such computer code, being completely opened to the user, allowsnew improvements to be implemented.1.4 DEVELOPED WORKThanks to the conditions of the laboratory described in the previous paragraphs, the workcarried out in this thesis includes the experimental and the numerical analyses of bridges.The experimental campaign consisted of Pseudo-dynamic tests on a series of bridgestructures using a substructuring technique: the piers were tested in the laboratory andthe deck was simulated numerically on-line. The first part of the experimental campaignconcerned the test of four bridges: one irregular bridge and one regular bridge designedaccording to the EC8, and two others representing two alternative design solutions to theirregular bridge.The testing campaign in the ELSA laboratory was completed afterwards with three sup-plementary tests, namely: two bridges with isolating/dissipating devices and one bridgewith asynchronous input motion. To prepare the experimental campaign, a quasi-staticcyclic test was also performed on the short pier of the EC8 irregular bridge. The aim ofthis test was to evaluate the adequacy of the testing devices and instrumentation in thelaboratory and to obtain the experimental response of a squat pier damaged by a control-led displacement history up to failure. The experimental results allowed not only thestudy of the behaviour of the set of bridges tested in the laboratory, but also the calibra-tion of the analytical model adopted in the numerical analysis that preceded each experi-ment.To simulate the bridge piers, a fibre model was proposed. This model is in between thelocal and the global formulations. Although the algorithm computes the global deforma-tions at the level of the Gauss points of the structural elements, the response is given bythe integral of the local forces calculated at different points representing the differentmaterials in the transverse section. Thus, the fibre model can be regarded as a step fur-ther in the refinement of standard beam models. In fact, it uses the same algorithms tocompute the deformation of the longitudinal axis of the finite elements: three rotationsand three displacements at each node. The difference between this model and the stand-6 INTRODUCTIONard beam model is in the procedure that it follows to calculate the resisting forces;instead of considering a global constitutive law at the level of the transverse sections, thefibre model goes deeper in the cross-section and computes the deformation and the stressof a set of points forming a mesh in the transverse section. The model was implementedin the object oriented computer code CASTEM 2000.A series of numerical tests using the fibre model were performed before the experimentsto predict the response of the structures. These preliminary results allowed us to verifythe ability of the model to predict the behaviour of R/C bridges under dynamic loading.Furthermore, the maximum forces admitted by the control system in the laboratory wereestablished taking into account these preliminary numerical results: an alarm systemstops the Pseudo-dynamic procedure whenever the imposed forces go above that limit.After the experimental campaign, the fibre model was improved, in particular to takeinto account the non-linear behaviour of R/C squat piers under high shear forces. Theexperimental tests performed in the ELSA laboratory on piers under these conditionsshowed a quite different behaviour compared to piers under predominant bendingmoments would be expected. Therefore, a strut-and-tie formulation coupled with theclassic fibre model for bending forces was developed to represent the response of suchelements. This formulation is based on the analogy of a R/C structure damaged withdiagonal cracking with a truss made of concrete diagonals and steel ties. The experimen-tal data were exhaustively used for checking and calibrating this model. Subsequently, the numerical campaign was repeated using the fibre model with the inclu-sion of the non-linear behaviour in shear and the results were compared with the experi-mental response. Topics like the flexibility of the foundation block and the importance ofthe shear displacements to the global response of the squat piers, are discussed. The dif-ferences still found between the numerical and the experimental results are analysed.Finally, the fibre model was used again for studying the behaviour of a set of bridges rep-resenting different configurations in between the two bridge profiles tested at the ELSAlaboratory: the so-called regular and irregular profiles, in order to extract conclusionsabout the regularity issue. The all set of bridges was designed for a behaviour factor of2.5 and the responses were analysed for the accelerogram used in the experimental cam-INTRODUCTION 7paign multiplied by growing intensity parameters.1.5 OUTLINE OF THE THESISApart from the introduction, the work is divided into seven chapters. The Pseudo-dynamic technique, its potentialities, the advantages and disadvantages in relation toother testing methods, are described in chapter 2. The implementation procedures andthe control system in the ELSA laboratory at Ispra are also detailed in this chapter. Thepost-treatment of the experimental data and the analysis of the results is presented inchapter 3. Notice that this was the first Pseudo-dynamic testing campaign performed inthe world using a substructuring technique on large-scale structures.The description of the fibre model implemented at the ELSA laboratory is done in chap-ter 4. A series of numerical tests used for checking the model and highlighting its poten-tialities are also presented. The cracking of concrete and the simulation of bi-axialbending actions are also subjects under discussion in this chapter. The new model withthe strut-and-tie formulation for shear forces, coupled with the fibre model for bendingforces, is described in chapter 5. At the end of this chapter, the numerical model isapplied to a series of piers tested in the ELSA laboratory and the responses are comparedto the experimental results.In chapter 6, the results for the bridges when both the predictive model and the non-lin-ear shear model are adopted, are analysed and compared with the experimental response.This chapter is divided into three parts: the pre-experimental tests, the post-experimentaltests and, at the end, a supplementary set of numerical tests performed in order to analysemore in detail the differences still found between the post-experimental and the experi-mental results.The numerical results of a new set of R/C bridges designed according to the EC8 andrepresenting different configurations in between the so-called regular and irregular pro-files tested in the ELSA laboratory, are analysed in chapter 7. The final conclusions andfuture research are drawn in chapter 8.8 INTRODUCTIONTHE PSEUDO-DYNAMIC TEST METHOD 9 2 THE PSEUDO-DYNAMIC TEST METHOD2.1 INTRODUCTIONThe Pseudo-dynamic test method is a hybrid method which combines the numerical inte-gration of the equations of motion of a structure and the experimental measurement ofthe corresponding restoring forces. Since the inertial forces are simulated numerically, areduced hydraulic power is required [26] and the test is performed with a time scaleenlarged with respect to the real time. This essentially means that the same equipment asconventional quasi-static tests is used; hydraulic actuators impose a history of prescribeddisplacements on the specimen and load-cells on the actuators measure the force neces-sary to apply the corresponding displacements. The equations of motion are solved on-line through a step-by-step numerical integration algorithm, using the physical forcesfrom the specimen and the inertial and damping forces from the analytical model.In such a test, the structure is condensed on a reduced number of degrees-of-freedom(d.o.f.), those controlled by the actuators in the laboratory during the experiment. Thisstatic condensation performed indirectly in the test, only preserves exactly the static(elastic) behaviour of a structure, while it may alter considerably its dynamic behaviourdue to the spurious mass and damping redistribution it induces in the structure. The useof reduced matrices not only imposes that the dynamic behaviour should be mainly con-trolled by the lowest modes of vibration, but also that the lowest frequencies of thewhole system should be as close as possible to the frequencies of the reduced system.Moreover, those modes of vibration should be well represented by the reduced d.o.f.The concepts of the Pseudo-dynamic technique appear during the late 60s contempora-10 THE PSEUDO-DYNAMIC TEST METHODneously to the installation of servo-controlled hydraulic actuators in structural testinglaboratories. The Japanese were the first researchers to study and apply this technique tostructural engineering [51]. In fact, the largest facility in the world belongs to the Build-ing Research Institute (BRI) of the Ministry of Construction of Japan. The European Laboratory for Structural Assessment (ELSA) at Ispra has the mostimportant reaction-wall facility in Europe and has been and will be used in the future forresearch activities on the design of civil engineering structures in seismic areas [25]. TheELSA facility has been available since 1993. The experimental campaign that suppliedmost of the data used in this work was actually performed in this laboratory using thePseudo-dynamic technique.The technique, the implementation procedures and the control system as they areinstalled in the ELSA laboratory at Ispr