Computational Methods in Earthquake Engineering

Preface

The book provides an insight on advanced methods and concepts for design andanalysis of structures against earthquake loading. It consists of 25 chapters coveringa wide range of timely issues in Earthquake Engineering. The goal of this Volume isto establish a common ground of understanding between the communities of EarthSciences and Computational Mechanics towards mitigating future seismic losses.Due to the great social and economic consequences of earthquakes, the topic is ofgreat scientific interest and is expected to be of valuable help to the large numberof scientists and practicing engineers currently working in the field. The chapters ofthis Volume are extended versions of selected papers presented at the COMPDYN2009 conference, held in the island of Rhodes, Greece, under the auspices of the Eu-ropean Community on Computational Methods in Applied Sciences (ECOMASS).

In the introductory chapter of Lignos et al. the topic of collapse assessment ofstructures is discussed. The chapter presents the analytical modeling of componentbehaviour and structure response from the early inelastic to lateral displacements atwhich a structure becomes dynamically unstable. A component model that capturesthe important deterioration modes, typically observed in steel members, is calibratedusing data from tests of scale-models of a moment-resisting connection. This con-nection is used for the two-scale model of a modern four-story steel moment frameand the assessment of its collapse capacity through analysis.

The work of Adam and Jager deals with the seismic induced global collapse ofmulti-story frame structures with non-deteriorating material properties, which arevulnerable to the P–� effect. The initial assessment of the structural vulnerabilityto P–� effects is based on pushover analyses. More information about the collapsecapacity is obtained with the Incremental Dynamic Analyses using a set of recordedground motions. In a simplified approach equivalent single-degree-of-freedom sys-tems and collapse spectra are utilized to predict the seismic collapse capacity of thestructures.

Sextos et al. focus on selection procedures for real records based on the Eurocode8 (EC8) provisions. Different input sets comprising seven pairs of records (hori-zontal components only) from Europe, Middle-East and the US were formed incompliance with EC8 guidelines. The chapter deals with the study of the RCbridges of the Egnatia highway system and also with a multi-storey RC buildingthat was damaged during the 2003 Lefkada (Greece) earthquake. More specifically,

v

vi Preface

the bridge was studied using alternative models and accounting for the dynamicinteraction of the deck-abutment-backfill-embankment system as well as of thesuperstructure-foundation-subsoil system. The building was studied in both the elas-tic and inelastic range taking into consideration material nonlinearity as well as thesurrounding soil. The results permit quantification of the intra-set scatter of the seis-mic response for both types of structures, thus highlighting the current limitationsof the EC8 guidelines. Specific recommendations are provided in order to eliminatethe dispersion observed in the elastic and the inelastic response though appropriatemodifications of the EC8 selection parameters.

Assimaki et al. study how the selection of the site response model affects theground motion predictions of seismological models, and how the synthetic motionsite response variability propagates to the structural performance estimation. Forthis purpose, the ground motion synthetics are computed for six earthquake sce-narios of a strike-slip fault rupture, and the ground surface response is estimatedfor 24 typical soil profiles in Southern California. Next, a series of bilinear single-degree-of-freedom oscillators is subjected to the ground motions computed usingthe alternative soil models and the consequent variability in the structural responseis evaluated. The results show high bias and uncertainty in the prediction of the in-elastic displacement ratio, when predicted using the linear site response model forperiods close to the fundamental period of the soil profile.

The chapter of Kappos et al. addresses the issue of pushover analysis of bridgessensitive to torsion, using as case-study a bridge whose fundamental mode is purelytorsional. Parametric analyses were performed involving consideration of founda-tion compliance, and various scenarios of accidental eccentricity that would triggerthe torsional mode. An alternative pushover curve in terms of abutment shear ver-sus deck maximum displacement (that occurs at the abutment) was found to be ameaningful measure of the overall inelastic response of the bridge. It is concludedthat for bridges with a fundamental torsional mode, the assessment of their seismicresponse relies on a number of justified important decisions that have to be maderegarding: the selection and the reliable application of the analysis method, theestimation of foundation and abutment stiffnesses, and the appropriate numericalsimulation of the pertinent failure mechanism of the elastomeric bearings.

Pardalopoulos and Pantazopoulou investigate the spatial characteristics of astructure’s deformed shape at maximum response in order to establish deformationdemands in the context of displacement-based seismic assessment or redesign ofexisting constructions. It is shown that the vibration shape may serve as a diagnos-tic tool of global structural inadequacies as it identifies the tendency for interstoreydrift localization and twisting due to mass or stiffness eccentricity. This chapterinvestigates the spatial displaced shape envelope and its relationship to the three-dimensional distribution of peak drift demand in reinforced concrete buildings withand without irregularities in plan and in height. A methodology for the seismicassessment of rotationally sensitive structures is established and tested through cor-relation with numerical results obtained from detailed time history simulations.

The chapter of Cotsovos and Kotsovos summarises the fundamental properties ofconcrete behaviour which underlie the formulation of an engineering finite element

Preface vii

model that is capable to realistically predict the behaviour of (plain or reinforced)concrete structural forms for a wide range of problems from static to impact loading,bypassing the problem of re-calibration. The already published evidence that sup-port the proposed formulation is complemented by four typical case-studies. Foreach case-study, the numerical predictions are computed against experimental datarevealing good agreement.

The chapter of Wijesundara et al. investigates the local seismic performances offully restrained gusset plate connections through detailed finite element models of asingle storey single-bay frame that is located at the ground floor of the four storeyframe. The chapter presents a design procedure, proposing an alternative clearancerule for the accommodation of brace rotation. Local performances of FE models arecompared in terms of strain concentrations at the beams, the columns and the gussetplates.

Vielma et al. propose a new seismic damage index and the corresponding damagethresholds. The seismic behavior of a set of regular reinforced concrete buildingsdesigned according to the EC-2/EC-8 prescriptions for a high seismic hazard levelare studied using the proposed damage index. Fragility curves and damage prob-ability matrices corresponding to the performance point are then calculated. Theobtained results show that the collapse damage state is not reached in the buildingsdesigned according the prescriptions of EC-2/EC-8 and that the damage does notexceed the irreparable damage limit-state for the buildings studied.

The application of discrete element models based on rigid block formulations tothe analysis of masonry walls under horizontal out-of-plane loading is discussed inthe chapter of Lemos et al. The problems raised by the representation of an irregularfabric as a simplified block pattern are addressed. Two procedures for creatingirregular block systems are presented. One using Voronoi polygons and anotherbased on a bed and cross joint structure with random deviations. A test problemprovides a comparison of various regular and random block patterns, showing theirinfluence on the failure loads.

Papaloizou and Komodromos discus the computational methods appropriatefor simulating the dynamic behaviour and the seismic response of ancient mon-uments, such as classical columns and colonnades. Understanding the behaviourand response of historic structures during strong earthquakes is useful for theassessment of conservation and rehabilitation proposals for such structures. Theirseismic behaviour involves complicated rocking and sliding phenomena that veryrarely appear in modern structures. The discrete element method (DEM) is utilizedto investigate the response of ancient multi-drum columns and colonnades duringharmonic and earthquake excitations by simulating the individual rock blocks asdistinct rigid bodies.

The study on the seismic behaviour of the walls of the Cella of Parthenon whensubjected to seismic loading is presented in the chapter of Psycharis et al.. Giventhat commonly used numerical codes for masonry structures or drum-columns areunable to handle the discontinuous behaviour of ancient monuments, the authorsadopt the discrete element method (DEM). The numerical models represent in detailthe actual construction of the monument and are subjected to the three components

viii Preface

of four seismic events recorded in Greece. Time domain analyses were performedin 3D, considering the non-linear behaviour at the joints. Conclusions are drawnbased on the maximum displacements induced to the structure during the groundexcitation and the residual deformation at the end of the seismic motion.

The chapter of Dolsek studies the effect of both aleatory and epistemic(modelling) uncertainties on reinforced concrete structures. The Incremental Dy-namic Analysis (IDA) method, which can be used to calculate the record-to-recordvariability, is extended with a set of structural models by utilizing the Latin Hy-percube Sampling (LHS) to account for the modelling uncertainties. The resultsshowed that the modelling uncertainties can reduce the spectral acceleration capac-ity and significantly increase its dispersion.

The chapter of Taflanidis discusses the problem of the efficient design of addi-tional dampers, to operate in tandem with the isolation system. One of the mainchallenges of such applications has been the explicit consideration of the nonlinearbehavior of the isolators or the dampers in the design process. Another challengehas been the efficient control of the dynamic response under near-field groundmotions. In this chapter, a framework that addresses both these challenges isdiscussed. The design objective is defined as the maximization of the structuralreliability. A simulation-based approach is implemented to evaluate the stochasticperformance and an efficient framework is proposed for performing the associateddesign optimization and for selecting values of the controllable damper parametersthat optimize the system reliability.

Mitsopoulou et al. study a robust control system for smart beams. First the struc-tural uncertainties of basic physical parameters are considered in the model of acomposite beam with piezoelectric sensors and actuators subjected to wind-typeloading. The control mechanism is introduced and designed to keep the beam inequilibrium in the event of external wind disturbances and in the presence of modeinaccuracies using the available measurement and control under limits.

Panagiotopoulos et al. examine through simple examples the performance andthe characteristics of a methodology previously proposed by the authors on avariationally-consistent way for the incorporation of time-dependent boundary con-ditions in problems of elastodynamics. More specifically, an integral formulationof the elastodynamic problem serves as basis for enforcing the corresponding con-straints, which are imposed via the consistent form of the penalty method, e.g. aform that complies with the norm and inner product of the functional space wherethe weak formulation is mathematically posed. It is shown that well-known andbroadly implemented modelling techniques in the finite element method such as“large mass” and “large spring” techniques arise as limiting cases of this penaltyformulation.

Sapountzakis and Dourakopoulos study the nonlinear dynamic analysis of beamsof arbitrary doubly symmetric cross section using the boundary element method.The beam is able to undergo moderate large displacements under general bound-ary conditions, taking into account the effects of shear deformation and rotaryinertia. The beam is subjected to the combined action of arbitrarily distributed or

Preface ix

concentrated transverse loading and bending moments in both directions as wellas to axial loading. To account for shear deformations, the concept of shear de-formation coefficients is used. Five boundary value problems are formulated andsolved using the Analog Equation Method. Application of the boundary elementtechnique yields a nonlinear coupled system of equations of motion. The evalua-tion of the shear deformation coefficients is accomplished from the aforementionedstress functions using only boundary integration.

The chapter of Papachristidis et al. presents the fiber method for the inelasticanalysis of frame structures when subjected to high shear. Initially the fiber approachis presented within its standard, purely bending, formulation and it is then expandedto the case of high shear deformations. The element formulation follows the as-sumptions of the Timoshenko beam theory, while two alternative formulations, acoupled and a decoupled are presented. The numerical examples confirm the accu-racy and the computational efficiency of the element formulation under monotonic,cyclic and dynamic/seismic loading.

A simplified procedure to estimate base sliding of concrete gravity dams inducedby an earthquake is proposed in the chapter of Basili and Nuti. A simple mechan-ical model is developed in order to take into account the sources that primarilyinfluence the seismic response of such structures. The dam is modelled as an elastic-linear single-degree-of-freedom-system. Different parameters are considered in theanalysis such as the dam height, foundation rock parameters, water level, seismic in-tensity. As a result, a simplified methodology is developed to evaluate base residualdisplacement, given the dam geometry, the response spectrum of the seismic input,and the soil characteristics. The procedure permits to assess the seismic safety ofthe dam with respect to base sliding, as well as the water level reduction that isnecessary to render the dam safe.

Papazafeiropoulos et al. provided a literature review and results from numericalsimulations on the dynamic interaction of concrete dams with retained water andunderlying soil. Initially, analytical closed-form solutions that have been widelyused for the calculation of dam distress are outlined. Subsequently, the numericalmethods based on the finite element method, which is unavoidably used for compli-cated geometries of the reservoir and/or the dam, are reviewed. Numerical resultsare presented illustrating the impact of various key parameters on the distress andthe response of concrete dams considering the dam-foundation interaction.

Motivated by the earthquake response of industrial pressure vessels, Karamanoset al. investigate the externally-induced sloshing in spherical liquid containers. Con-sidering modal analysis and an appropriate decomposition of the container-fluidmotion, the sloshing frequencies and the corresponding sloshing (or convective)masses are calculated, leading to a simple and efficient method for predicting thedynamic behavior of spherical liquid containers. It is also shown that consideringonly the first sloshing mass is adequate to represent the dynamic behavior of thespherical liquid container within a good level of accuracy.

Jha et al. introduce a bilevel model for developing an optimal MaintenanceRepair and Rehabilitation (MR&R) plan for large-scale highway infrastructure ele-ments, such as pavements and bridges, following a seismic event. The maintenance

x Preface

and upkeep of all infrastructure components is crucial for mobility, driver safetyand guidance, and the overall efficient functioning of a highway system. Typically,a field inspection of such elements is carried out at fixed time intervals to determinetheir condition, which is then used to develop optimal MR&R plan over a givenplanning horizon.

Frangopol and Akiyama present a seismic analysis methodology for corrodedreinforced concrete (RC) bridges. The proposed method is applied to lifetime seis-mic reliability analysis of corroded RC bridge piers, and the relationship betweensteel corrosion and seismic reliability is presented. It is shown that the analyticalresults are in good agreement with the experimental results regardless of the amountof steel corrosion. Moreover, after the occurrence of crack corrosion, the seismicreliability of the pier is significantly reduced.

Life cycle cost assessment of structural systems refers to an evaluation procedurewhere all costs arising from owing, operating, maintaining and ultimately disposingare considered. Life cycle cost assessment is considered as a significant assessmenttool in the field of the seismic behaviour of structures. Therefore, in the chapter byMitropoulou et al. two test cases are examined and useful conclusions are drawnregarding the behaviour factor q of EC8 and the incident angle that a ground motionis applied on a multi-storey RC building.

Bal et al. examine vulnerability assessment procedures that include code-baseddetailed analysis methods together with preliminary assessment techniques in orderto identify the safety levels of buildings. Their chapter examines the effect of fouressential structural parameters on the seismic behaviour of existing RC structures.Parametric studies are carried out on real buildings extracted from the Turkishbuilding stock, one of which was totally collapsed in 1999 Kocaeli earthquake.Comparisons are made in terms of shear strength, energy dissipation capabilityand ductility. The mean values of the drop in the performance are computed andfactors are suggested to be utilized in preliminary assessment techniques, such asthe recently proposed P25 method that is shortly summarized in the chapter.

The aforementioned collection of chapters provides an overview of the presentthinking and state-of-the-art developments on the computational techniques in theframework of structural dynamics and earthquake engineering. The book is targetedprimarily to researchers, postgraduate students and engineers working in the field.It is hoped that this collection of chapters in a single book will be a useful tool forboth researchers and practicing engineers.

The book editors would like to express their deep gratitude to all authors for thetime and effort they devoted to this volume. Furthermore, we would like to thankthe personnel of Springer Publishers for their kind cooperation and support for thepublication of this book.

Athens Manolis PapadrakakisJune 2010 Michalis Fragiadakis

Nikos D. Lagaros

http://www.springer.com/978-94-007-0052-9

Contents

Collapse Assessment of Steel Moment Resisting Frames UnderEarthquake Shaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Dimitrios G. Lignos, Helmut Krawinkler, and Andrew S. Whittaker

Seismic Induced Global Collapse of Non-deteriorating FrameStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Christoph Adam and Clemens Jager

On the Evaluation of EC8-Based Record Selection Proceduresfor the Dynamic Analysis of Buildings and Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Anastasios G. Sextos, Evangelos I. Katsanos, Androula Georgiou,Periklis Faraonis, and George D. Manolis

Site Effects in Ground Motion Synthetics for StructuralPerformance Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Dominic Assimaki, Wei Li, and Michalis Fragiadakis

Problems in Pushover Analysis of Bridges Sensitive to Torsion . . . . . . . . . . . . . . 99Andreas J. Kappos, Eleftheria D. Goutzika, Sotiria P. Stefanidou,and Anastasios G. Sextos

Spatial Displacement Patterns of R.C. Buildings Under SeismicLoads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123Stylianos J. Pardalopoulos and Stavroula J. Pantazopoulou

Constitutive Modelling of Concrete Behaviour: Need forReappraisal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147Demetrios M. Cotsovos and Michael D. Kotsovos

Numerical Simulation of Gusset Plate Connection with RhsShape Brace Under Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .177K.K. Wijesundara, D. Bolognini, and R. Nascimbene

xi

xii Contents

Seismic Response of RC Framed Buildings DesignedAccording to Eurocodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201Juan Carlos Vielma, Alex Barbat, and Sergio Oller

Assessment of the Seismic Capacity of Stone Masonry Wallswith Block Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .221Jose V. Lemos, A. Campos Costa, and E.M. Bretas

Seismic Behaviour of Ancient Multidrum Structures . . . . . . . . . . . . . . . . . . . . . . . . . .237Loizos Papaloizou and Petros Komodromos

Seismic Behaviour of the Walls of the ParthenonA Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265Ioannis N. Psycharis, Anastasios E. Drougas, and Maria-EleniDasiou

Estimation of Seismic Response Parameters ThroughExtended Incremental Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .285Matjaz Dolsek

Robust Stochastic Design of Viscous Dampers for BaseIsolation Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .305Alexandros A. Taflanidis

Uncertainty Modeling and Robust Control for SmartStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .331A. Moutsopoulou, G.E. Stavroulakis, and A. Pouliezos

Critical Assessment of Penalty-Type Methods for Imposition ofTime-Dependent Boundary Conditions in FEM Formulationsfor Elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .357Christos G. Panagiotopoulos, Elias A. Paraskevopoulos,and George D. Manolis

Nonlinear Dynamic Analysis of Timoshenko Beams . . . . . . . . . . . . . . . . . . . . . . . . . . .377E.J. Sapountzakis and J.A. Dourakopoulos

Inelastic Analysis of Frames Under Combined Bending,Shear and Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .401Aristidis Papachristidis, Michalis Fragiadakis,and Manolis Papadrakakis

Seismic Simulation and Base Sliding of Concrete Gravity Dams . . . . . . . . . . . . .427M. Basili and C. Nuti

Contents xiii

Dynamic Interaction of Concrete Dam-Reservoir-Foundation:Analytical and Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .455George Papazafeiropoulos, Yiannis Tsompanakis,and Prodromos N. Psarropoulos

Numerical Analysis of Externally-Induced Sloshingin Spherical Liquid Containers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .489Spyros A. Karamanos, Lazaros A. Patkas,and Dimitris Papaprokopiou

A Bilevel Optimization Model for Large Scale HighwayInfrastructure Maintenance Inspection and SchedulingFollowing a Seismic Event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .515Manoj K. Jha, Konstantinos Kepaptsoglou, Matthew Karlaftis,and Gautham Anand Kumar Karri

Lifetime Seismic Reliability Analysis of Corroded ReinforcedConcrete Bridge Piers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .527Dan M. Frangopol and Mitsuyoshi Akiyama

Advances in Life Cycle Cost Analysis of Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . .539Chara Ch. Mitropoulou, Nikos D. Lagaros,and Manolis Papadrakakis

Use of Analytical Tools for Calibration of Parameters in P25Preliminary Assessment Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .559Ihsan E. Bal, F. Gulten Gulay, and Semih S. Tezcan

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .583


Seismic Induced Global Collapseof Non-deteriorating Frame Structures

Christoph Adam and Clemens Jager

Abstract In a severe seismic event the destabilizing effect of gravity loads, i.e.the P-delta effect, may be the primary trigger for global collapse of quite flexiblestructures exhibiting large inelastic deformations. This article deals with seismicinduced global collapse of multi-story frame structures with non-deteriorating mate-rial properties, which are vulnerable to the P-delta effect. In particular, the excitationintensity for P-delta induced structural collapse, which is referred to as collapse ca-pacity, is evaluated. The initial assessment of the structural vulnerability to P-deltaeffects is based on pushover analyses. More detailed information about the collapsecapacity renders Incremental Dynamic Analyses involving a set of recorded groundmotions. In a simplified approach equivalent single-degree-of-freedom systems andcollapse capacity spectra are utilized to predict the seismic collapse capacity of reg-ular multi-story frame structures.

Keywords Collapse capacity spectra � Dynamic instability � P-delta

1 Introduction

In flexible structures gravity loads acting through lateral displacements amplifystructural deformations and stress resultants. This impact of gravity loads on thestructural response is usually referred to as P-delta effect. For a realistic buildingin its elastic range the P-delta effect is usually negligible. However, it may becomeof significance at large inelastic deformations when gravity loads lead to a negativeslope in the post-yield range of the lateral load-displacement relationship. In such

C. Adam (�)University of Innsbruck, Department of Civil Engineering Sciences, 6020 Innsbruck, Austriae-mail: [email protected]

C. JagerUniversity of Innsbruck, Department of Civil Engineering Sciences, 6020 Innsbruck, Austriae-mail: [email protected]

M. Papadrakakis et al. (eds.), Computational Methods in Earthquake Engineering,Computational Methods in Applied Sciences 21, DOI 10.1007/978-94-007-0053-6 2,c� Springer Science+Business Media B.V. 2011

21

[email protected]

[email protected]

22 C. Adam and C. Jager

Fig. 1 Normalized bilinearcyclic behavior of a SDOFsystem with and withoutP-delta effect

α − θ

α1

1

θ

with P-delta

no P-deltaf

μ

a situation large gravity loads combined with seismically induced large inelasticdeformations amplify the lateral displacements in a single direction. The seismiccollapse capacity of the structure is exhausted at a rapid rate, and the system isno longer able to sustain its own gravity loads. Additionally, material deteriorationaccelerates P-delta induced seismic collapse.

A profound insight into the P-delta effect on the inelastic seismic response ofstructures is given e.g. by Bernal [1], Gupta and Krawinkler [2], Aydinoglu [3],Ibarra and Krawinkler [4], and Lignos and Krawinkler [5]. Asimakopoulos et al. [6]and Villaverde [7] provide an overview on studies dealing with collapse by dynamicinstability in earthquake excited structures.

In an inelastic single-degree-of-freedom (SDOF) system the gravity load gen-erates a shearing of its hysteretic force-displacement relationship. Characteristicdisplacements (such as the yield displacement) of this relationship remain un-changed, whereas the characteristic forces (such as the strength) are reduced. Asa result, the slope of the curve is decreased in its elastic and post-elastic branchof deformation. The magnitude of this reduction can be expressed by means of theso-called stability coefficient [8]. As a showcase in Fig. 1 the P-delta effect on thehysteretic behavior of a SDOF system with non-deteriorating bilinear characteristicsis visualized. In this example the post-yield stiffness is negative because the stabilitycoefficient � is larger than the hardening ratio ˛.

Fundamental studies of the effect of gravity on inelastic SDOF systems subjectedto earthquakes have been presented in Bernal [8] and MacRae [9]. Kanvinde [10],and Vian and Bruneau [11] have conducted experimental studies on P-delta inducedcollapse of SDOF frame structures. Asimakopoulos et al. [6] propose a simple for-mula for a yield displacement amplification factor as a function of ductility and thestability coefficient. Miranda and Akkar [12] present an empirical equation to esti-mate the minimum lateral strength up to which P-delta induced collapse of SDOFsystems is prevented. In Adam et al. [13–15] so-called collapse capacity spectrahave been introduced for the assessment of the seismic collapse capacity of SDOFstructures.

In multi-story frame structures gravity loads may impair substantially the com-plete structure or only a subset of stories [2]. The local P-delta effect may inducecollapse of a local structural element, which does not necessarily affect the stabilityof the complete structure. An indicator of the severity of the local P-delta effect is

Seismic Induced Global Collapse of Non-deteriorating Frame Structures 23

the story stability coefficient, Gupta and Krawinkler [2]. Alternatively, Aydinoglu[3] proposes the use of the geometric story stiffness instead of the story stabilitycoefficient. However, a consistent relationship between the local P-delta effect andthe global P-delta effect, which characterizes the overall impact of gravity loads onthe structure, cannot be established due to dynamic interaction between adjacentstories in a multi-story frame structure [2].

In several papers, see e.g. Takizawa and Jennings [16], Bernal [17], Adam et al.[18], it is proposed to assess the global P-delta effect in frame structures by meansof equivalent single-degree-of-freedom (ESDOF) systems. If the story drifts remainrather uniformly distributed over the height, regardless of the extent of inelasticdeformation, a global assessment of the P-delta effect by means of ESDOF systemsis not difficult. Thereby, it is assumed that P-delta is primarily governed by thefundamental mode. As recently shown [19] this assumption holds true also for tallbuildings. However, if a partial mechanism develops, the global P-delta effect willbe greatly affected by the change of the deflected shape, and it will be amplified inthose stories in which the drift becomes large [1, 3]. In such a situation an adequateincorporation of P-delta effects in ESDOF systems is a challenging task.

In this paper a methodology is presented, which allows a fast quantification of theglobal P-delta effect in highly inelastic regular MDOF frame structures subjected toseismic excitation. Emphasis is given to the structural collapse capacity. Resultsand conclusions of this study are valid only for non-deteriorating cyclic behavior,i.e. strength and stiffness degradation is not considered.

2 Structural Vulnerability to Global P-Delta Effects

2.1 Assessment of the Vulnerability to Global P-Delta Effects

Initially, it must be assessed whether the considered structure is vulnerable to P-deltaeffects. Strong evidence delivers the results of a global pushover analysis [2]. Dur-ing this nonlinear static analysis gravity loads are applied, and subsequently thestructure is subjected to lateral forces. The magnitude of these forces with a prede-fined invariant load pattern is amplified incrementally in a displacement-controlledprocedure. As a result the global pushover curve of the structure is obtained, wherethe base shear is plotted against a characteristic deformation parameter. In generalthe lateral displacement of the roof is selected as characteristic parameter. It is as-sumed that the shape of the global pushover curve reflects the global or the localmechanism involved when the structure approaches dynamic instability.

In Fig. 2 the effect of gravity loads on the global pushover curve of a multi-storyframe structure is illustrated. Figure 2a shows the global pushover curve, wheregravity loads are either disregarded or of marginal importance. The pushover curveof Fig. 2b corresponds to a very flexible multi-story frame structure with a strongimpact of the P-delta effect leading to a reduction of the global lateral stiffness. In


xNxNy

V

V0y

xi

NxN

xi

xN

FN

V

V

iFi

FN

Fi

no P-delta effect

F

a

b

N

i

xNxNy

V

VPy

P-delta effect included

F

Fig. 2 Multi-story frame structure and corresponding global pushover curves. (a) Pushover anal-ysis disregarding the P-delta effect. (b) Pushover analysis considering the P-delta effect

very flexible structures gravity loads even may generate a negative post-yield tan-gent stiffness as shown in Fig. 2b [20]. If severe seismic excitation drives such astructure in its inelastic branch of deformation a state of dynamic instability maybe approached, and the global collapse capacity is attained at a rapid rate. Fromthese considerations follows that a gravity load induced negative post-yield tangentstiffness in the global pushover curve requires an advanced investigation of P-deltaeffects [2]. It is emphasized that collapse induced by static instability must be inves-tigated separately.


2.2 Example

Exemplarily, the structural vulnerability to P-delta effects of a generic single-bay15-story frame structure according to Fig. 3a is assessed. All stories are of uniformheight h, and they are composed of rigid beams, elastic flexible columns, and ro-tational springs at the ends of the beams. Nonlinear behavior at the componentlevel is modeled by non-degrading bilinear hysteretic behavior of the rotationalsprings (compare with Fig. 3b) to represent the global cyclic response under seis-mic excitation. The strength of the springs is tuned such that yielding is initiatedsimultaneously at all spring locations in a static pushover analysis (without gravityloads) under an inverted triangular design load pattern. To each joint of the framean identical point mass is assigned. The bending stiffness of the columns and thestiffness of the springs are tuned to render a straight line fundamental mode shape.Identical gravity loads are assigned to each story to simulate P-delta effects. Thisimplies that axial column forces due to gravity increase linearly from the top to thebottom of the frame. The frame structure has a fundamental period of vibration ofT1 D 3:0 s, which makes it rather flexible. The base shear coefficient, defined as

N = 15

a

b

i

rigidelastic

elasto-plastic

EJi

m

h PKi

1

Ki

αKiM

θ

m

P

xN

Fig. 3 (a) Generic 15-story frame structure. (b) Bilinear hysteretic loop of the rotational springs


Fig. 4 Global pushover curves of a 15-story frame structure based on a linear load pattern consid-ering and disregarding P-delta effects

ratio between yield base shear Vy and total weight W.� D Vy=W /, is � D 0:1.For additional dynamic studies structural damping is considered by means of massproportional Rayleigh damping of 5% of the first mode.

Figure 4 shows normalized base shear against normalized roof drift relations ofthis structure as a result of static pushover analyses utilizing an inverted triangularload pattern both considering and omitting gravity loads, respectively. Axial gravityloads are based on a ratio of life load plus dead load to dead load of 1.0, i.e. coef-ficient # D 1:0. Both global pushover curves exhibit a sharp transition from elasticto inelastic branch of deformation. This behavior can be attributed to specific tun-ing of the yield strength as specified above. The graphs of this figure demonstratethe expected softening effect of the gravity loads. Both elastic and inelastic globalstiffness decrease. For this particular structure the presence of gravity loads leads toa negative stiffness in the post-yield range of deformation. From this outcome it canbe concluded that this frame structure may become vulnerable to collapse inducedby global P-delta effects.

From the global pushover curve without P-delta a global hardening ratio ˛S of0.040 can be identified, which is larger than the individual hardening coefficients ˛

of the rotational springs of 0.03.As outlined by Medina and Krawinkler [20] there is no unique global stability

coefficient for those structures, which cannot be modeled a priori as SDOF systems.The global force-displacement behavior represented by the global pushover curveexhibits in its bilinear approximation an elastic stability coefficient and an inelas-tic stability coefficient, compare with Fig. 4. Recall that a stability coefficient is ameasure of the decrease of the structural stiffness caused by gravity loads.


0 1 2 3 40

5

10

15

xN/ xNy

stor

y

15-story frame

α = 0.03ϑ = 1.0linear load pattern

elastic

inelastic

Fig. 5 Deflected shapes of a 15-story frame structure from a pushover analysis

For the actual example problem the following elastic stability coefficient �e andinelastic stability coefficient �i can be determined: �e D 0:061; �i D 0:085. Thenegative slope of the normalized post-yield stiffness is expressed by the difference˛S � �i D �0:045.

In Fig. 5 corresponding displacement profiles of the frame structure in presenceof P-delta effects are depicted. As long as the structure is deformed elastically thedeflected shapes are relatively close to a straight line. However, once the structureyields there is a concentration of the maximum story drifts in the lower stories. Asthe roof displacement increases, the bottom story drift values increase at a rapid rate[20]. This concentration of the displacement in the bottom stories is characteristicfor regular frame structures vulnerable to the P-delta effect. Comparative calcula-tions have shown that the displacement profiles are close to a straight line even inthe inelastic range of deformation when gravity loads are disregarded.

3 Assessment of the Global Collapse Capacity

3.1 Incremental Dynamic Analysis

Incremental Dynamic Analysis (IDA) is an established tool in earthquake engineer-ing to gain insight into the non-linear behavior of seismic excited structures [21].Subsequently, the application of IDAs for predicting the global collapse capacity ofmulti-story frame structures, which are vulnerable to P-delta effects, is summarized.


For a given structure and a given acceleration time history of an earthquakerecord dynamic time history analyses are performed repeatedly, where in each sub-sequent run the intensity of the ground motion is incremented. As an outcome acharacteristic intensity measure is plotted against the corresponding maximum char-acteristic structural response quantity for each analysis. The procedure is stopped,when the response grows to infinity, i.e. structural failure occurs. The correspond-ing intensity measure of the ground motion is referred to as collapse capacity ofthe building for this specific ground motion record. There is no unique definition ofintensity of an earthquake record. Examples of the intensity measure are the peakground acceleration (PGA) and the 5% damped spectral acceleration at the struc-ture’s fundamental period Sa.T1/.

Since the result of an IDA study strongly depends on the selected record, IDAsare performed for an entire set of n ground motion records, and the outcomes areevaluated statistically. In particular, the median value of the individual collapse ca-pacities CCi ; i D 1; : : : ; n, is considered as the representative collapse capacity CCfor this structure and this set of ground motion records,

CC D med hCCi ; i D 1; : : : : ; ni (1)

3.2 Example

In the following the global collapse capacity of the generic 15-story frame structurepresented in Sect. 2.2 is determined. The collapse capacity is based on a set of 40ordinary ground motion records (records without near-fault characteristics), whichwere recorded in California earthquakes of moment magnitude between 6.5 and 7,and closest distance to the fault rupture between 13 and 40 km on NEHRP site classD (FEMA 368, 2000). This set of seismic records, denoted as LMSR-N, has strongmotion duration characteristics insensitive to magnitude and distance. A statisticalevaluation of this bin of records and its characterization is given in [14].

In Fig. 6 IDA curves are shown for each record with light gray lines. For thisexample the normalized spectral acceleration at the structure’s fundamental period,

Sa.T1/

g�(2)

is utilized as relative intensity measure. This parameter is plotted against the nor-malized lateral roof displacement xN ,

xN

Sd .T1/(3)

where Sd is the 5% damped spectral displacement at the fundamental period ofvibration.


20 1 3 4 5 6 7 80

2

4

6

8

10

12

14

16

xN/ Sd(T1)

median

CC15DOF

bilinear hysteretic loop

15-story frame

α = 0.03

ϑ = 1.0

LMSR-N set

Sa(

T1)

/g/γ

Fig. 6 IDA curves for 40 ground motion records. Median IDA curve. Median collapse capacityCC15DOF of a generic 15-story frame with a fundamental period of vibration of 3:0 s

Subsequently, an arbitrary IDA curve is picked from the entire set and its behav-ior discussed exemplary. When the relative seismic intensity is small the structureis deformed elastically. With increasing intensity the normalized roof displacementbecomes smaller because energy is dissipated through ductile structural deforma-tions. However, at a certain level of intensity the IDA curves bends at a rapid ratetowards collapse. When the IDA curve approaches a horizontal tangent, the col-lapse capacity of the structure for this particular accelerogram is exhausted. Theentire set of IDA curves shows that the IDA study is ground motion record specific.To obtain a meaningful prediction of the global collapse capacity the median IDAcurve is determined, which is shown in Fig. 6 by a fat black line. The median IDAcurve approaches a horizontal straight dashed line. This line indicates the relativemedian collapse capacity CC15DOF of this 15-degree-of-freedom (15DOF) structuresubjected to the LMSR-N bin of records:

CC15DOF D 10:5 (4)

Figure 7 shows time histories of normalized interstory drifts of the frame structurein a state of dynamic instability induced by a single seismic event. The correspond-ing ground motion record “LP89agw” is included in the LMSR-N bin. It can beseen that after time t D 15 s the ratcheting effect dominates the dynamic responseof the bottom stories, i.e. the deformation increases in a single direction. Becausethe displacements grow to infinity, collapse occurs at a rapid rate. The largest in-terstory drift develops in the first story. With rising story number the relative story


–0.6

–0.4

–0.2

0.0

0.2

0 10 20 30 40

time t [s]

inte

rsto

ry d

rift (

x i–x i

–1)/

h


15-story frame

α = 0.03

ϑ = 1.0

record 1: LP89agw

Sa(T) / g / γ = 10.0

story

15

1

10

5

Fig. 7 Global collapse of the 15-story frame structure induced by an individual ground motionrecord: time history of normalized interstory drifts

–0.20

–0.16

–0.12

–0.08

–0.04

0.00

0 5 10 15 20 25 30 35 40

time t [s]

stor

y di

spla

cem

ent x

i/H


15-story frame

α = 0.03

ϑ = 1.0

record: LP89agw

Sa(T1) / g / γ = 10.0

1

2

3

4

56

15

7 - 14

story

Fig. 8 Global collapse of the 15-story frame structure induced by an individual ground motionrecord: time history of normalized story displacements

displacements become smaller. In the upper stories a residual deformation remainsin opposite direction. This behavior can be attributed to higher mode effects.

The corresponding story displacements are depicted in Fig. 8. They are nor-malized by the total height H of the structure. With increasing story number the


interstory drifts accumulate to larger story displacements. However, the largest storydisplacements do not occur at the roof .i D 15/ thanks to higher mode effects asillustrated above.

4 Simplified Assessment of the Global Collapse Capacity

For large frame structures with many DOFs and a large set of ground motion recordsthe IDA procedure is computational expensive. Thus, it is desirable to provide sim-plified methods for prediction of the global collapse capacity of structures sensitiveto P-delta effects with sufficient accuracy.

Because in regular frame structures P-delta effects are mainly controlled bylateral displacements of the lower stories it is reasonable to assume that theseeffects can be captured by means of ESDOF systems even in tall buildings inwhich upper stories are subjected to significant higher mode effects [18]. Appli-cation of an ESDOF system requires that shape and structure of the correspondinglarge frame are regular. Thus, the following considerations are confined to regularplanar multi-story frame structures as shown in Fig. 9a, which furthermore exhibitnon-deteriorating inelastic material behavior under severe seismic excitation.

D(t)

h

P*

L

ka*, ζ

b

L*

a

xi

xNN

i

xg(t) xg(t)

x = φxN

Fig. 9 (a) Multi-story frame structure, and (b) corresponding equivalent single-degree-of-freedomsystem


4.1 Equivalent Single-Degree-of-Freedom System

The employed ESDOF system is based on a time-independent shape vector ¥,which describes the displacement vector x of the MDOF structure regardless ofits magnitude,

x D ¥ xN ; �N D 1 (5)

and on global pushover curves of corresponding pushover analyses applied to theoriginal structure disregarding and considering vertical loads, respectively. The lat-eral pushover load F is assumed to be affine to the displacement vector x,

F D ¥ FN (6)

Examples of such global pushover curves are shown in Figs. 2 and 4. Details of theproposed ESDOF system can be found in Fajfar [22] and Adam et al. [18].

According to [18] and [22] displacement D of the ESDOF system (Fig. 9b) isrelated to the roof displacement xN as follows,

D D m�

L� xN ; L� D ¥T M e; m� D ¥T M ¥ (7)

M is the mass matrix of the original frame structure, and e denotes the influencevector, which represents the displacement of the stories resulting from a static unitbase motion in direction of the ground motion Rxg .

The backbone curve of the ESDOF spring force f �S is derived from the base

shear V of the global pushover curve (without P-delta effect) according to [18, 22]

f �S D m�

L� V (8)

In contrast to a real SDOF system no unique stability coefficient does exist for anESDOF oscillator, since the backbone curve of the ESDOF system is based on theglobal pushover curve [1, 20]. As illustrated in Fig. 10 a bilinear approximation of

Fig. 10 Global pushovercurves with and withoutP-delta effect and theirbilinear approximations

1

1

xNxNy

V

V0y

VPy

αSKS

(αS−θi)KS

θeKS

(1−θe)KS

θiKS

no P-delta effect

with P-delta effect

11


Fig. 11 Backbone curveswith and without P-deltaeffect and auxiliary backbonecurve

DDy

f*

αSka*

αSk0*

ka*

k0*θaka*

θek0*

θik0*

f*0y

f*ay

(αS − θi)k0* =

(αS− θa)ka*

auxiliary backbone curve

no P-delta effect

with P-delta effect

the backbone curve renders an elastic stability coefficient �e and an inelastic stabil-ity coefficient �i . Analyses have shown that �i is always larger than �e; �i > .>/�e

[20]. Thus, loading of the ESDOF system by means of an equivalent gravity load,which is based on the elastic stability coefficient �e, leads to a “shear deformation”of the hysteretic loop of the ESDOF system, where the post-tangent stiffness is over-estimated. Consequently, the hazard of collapse would be underestimated. Ibarraand Krawinkler [4] propose to employ an auxiliary backbone curve, which featuresa uniform stability coefficient �a, compare with Fig. 11. In [4,18] the parameters ofthe auxiliary backbone curve are derived as:

�a D �i � �e˛S

�; k�

a D �

1 � ˛S

k�0 ; f �

ay D �

1 � ˛S

f �0y ; � D 1��e C�i �˛S (9)

Subsequently, an appropriate hysteretic loop is assigned to the auxiliary backbonecurve, which is sheared by �a when the ESDOF system is loaded by the equivalentgravity force P � [14]:

P � D �a k�a h (10)

This situation is illustrated in Fig. 12, where exemplarily a bilinear hysteretic curveis assigned to the auxiliary backbone curve. Now, the normalized equation of motionof the auxiliary ESDOF system can be expressed in full analogy to a real SDOFsystem as [14]

1

!�2a

R�� C 2�1

!�a

P�� C � Nf �S � �a�� D � Rxg

g �� (11)

with

�� D D

Dy

; Nf �S D f �

aS

f �ay

; !�a D

rk�

a

L� (12)

In Eqs. 11 and 12 �� is the non-dimensional horizontal displacement of massL� of the ESDOF, and Dy characterizes the yield displacement. Nf �

S denotes the


( )

DDy

ka*

αSka*

θaka*

f*

αs − θa ka*

D(t)

xg(t)

h

P*

L*

ka,ζ*

hysteretic loop with P-delta effect

auxiliary hysteretic loop

Fig. 12 Auxiliary equivalent single-degree-of-freedom system with bilinear hysteretic behavior

non-dimensional spring force, which is the ratio of the auxiliary spring force f �aS

and its yield strength f �ay � !�

a represents the circular natural frequency of the aux-iliary ESDOF system, and k�

a is the corresponding stiffness. The equivalent baseshear coefficient �� of the ESDOF system is calculated from the base shear coeffi-cient N� of the MDOF system according to [18]

�� D N��MDOF

; N� D Vy

Mg; �MDOF D L�2

m�M(13)

Vy is the base shear at the yield point, and M the (dynamic effective) total mass ofthe MDOF structure.

4.2 Collapse Capacity Spectra

Adam et al. [13–15] propose to utilize collapse capacity spectra for the assessmentof the collapse capacity of SDOF systems, which are vulnerable to the P-delta effect.In [15] it is shown that the effect of gravity loads on SDOF systems with bilinearhysteretic behavior is mainly characterized by means of the following structuralparameters:

� The elastic structural period of vibration T

� The slope of the post-tangential stiffness expressed by the difference � �˛ of thestability coefficient � and the strength hardening coefficient ˛

� The viscous damping coefficient � (usually taken as 5%)

In [15] design collapse capacity spectra are presented as a function of these pa-rameters. As an example in Figs. 13 and 14 collapse capacity spectra and thecorresponding design collapse capacity spectra, respectively, are shown for SDOFsystems with stable bilinear hysteretic behavior [15]. They are based on the LMSR-N set of 40 ground motions. Here, the collapse capacity CC is defined as the medianof the 40 individual collapse capacities CCi ; i D 1; : : : ; 40,


Fig. 13 Collapse capacity spectra of single-degree-of-freedom systems with bilinear hys-teretic loop

Fig. 14 Design collapse capacity spectra of single-degree-of-freedom systems with bilinear hys-teretic loop


0

5

10

15

0 1 2 3 4 5

0.02

0.80

0.20

0.10

0.08

0.06

0.40

0.04

LMSR-N set

period T [s]

CC

ζ = 0.05


0.044

T1

7.6

θ–α

Fig. 15 Application of design collapse capacity spectra to an equivalent single-degree-of-freedomsystem

CC D med hCCi ; i D 1; : : : : ; 40i (14)

which are for these spectra the 5% damped spectral acceleration at the period ofvibration T , where structural collapse occurs [15],

CCi D Sa.T /jig�

(15)

Application of design collapse capacity spectra is simple: an estimate of the elas-tic period of vibration T , stability coefficient � and hardening ratio ˛ of the actualSDOF structure need to be determined. Subsequently, from the chart the correspond-ing collapse capacity CC can be read as shown in Fig. 15.

4.3 Application of Design Collapse Capacity Spectrato Multi-Story Frame Structures

ESDOF systems allow the application of design collapse capacity spectra for as-sessing the collapse capacity of multi-story frame structures. Thereby, T and � � ˛

of a SDOF system are replaced by the fundamental period T1 of the actual MDOFsystem (without P-delta), and the difference of the auxiliary stability coefficient andhardening coefficient �a � ˛S . ˛S is the hardening coefficient taken from the globalpushover curve without P-delta effect. From the design collapse capacity spectrum


a prediction of the related collapse capacity CC is obtained. The actual collapsecapacity of the ESDOF system, i.e. the normalized median intensity of earthquakeexcitation at collapse, is subsequently determined from, compare with Eq. 13,

CCESDOF D CC

�MDOF(16)

This outcome represents an approximation of the collapse capacity CCMDOF of theactual MDOF building,

CCMDOF � CCESDOF (17)

4.4 Example

In an example problem the application of ESDOF systems and collapse spectra forthe prediction of the global collapse capacity of multi-story frame structures is illus-trated. For this purpose the generic 15-story frame structure of Sect. 2.2 is utilized.Recall that the fundamental period of this structure is T1 D 3:0 s, and the elas-tic stability coefficient, the inelastic stability coefficient and the hardening ratio,respectively, are: �e D 0:061; �i D 0:085; ˛S D 0:040. The auxiliary stabilitycoefficient according to Eq. 9 is �a D 0:084, and thus �a � ˛S D 0:044. Coefficient�MDOF , Eq. 13, is derived as: �MDOF D 0:774.

Application of design collapse capacity spectra as illustrated in Fig. 15 rendersthe collapse capacity CC D 7:6. Division by the coefficient �MDOF results in thecollapse capacity of the ESDOF system,

CCESDOF D 7:61

0:774D 9:83 (18)

Comparing this outcome with the result of the IDA procedure on the actual 15-storyframe structure according to Eq. 4, CC15DOF D 10:5, reveals that CCESDOF is forthis example a reasonable approximation of the collapse capacity.

In addition, Fig. 16 shows the collapse capacity of the 15-story frame for differentmagnitudes of gravity loads, i.e. the ratio ª of life plus dead load to dead load isvaried from 1.0 to 1.6. The latter value is considered only for curiosity. Median, 16%percentile and 84% percentile collapse capacity derived from IDAs are depicted byblack lines. These outcomes are set in contrast to the median collapse capacity froma simplified assessment based on ESDOF systems and collapse capacity spectrarepresented by a dashed line. It can be seen that in the entire range the simplifiedprediction of the collapse capacity underestimates the “exact” collapse capacity. Inother words, the simplified methodology renders for this example results on theconservative side. Note that the modification of the fundamental period T1 by P-delta is not taken into account.


Fig. 16 Collapse capacity of a 15-story frame structure for different magnitudes of gravity loads.Comparison with simplified assessment (dashed line)

Fig. 17 Collapse capacity of a 15-story frame structure for different hardening ratios of the bilin-ear springs. Comparison with simplified assessment (dashed line)

The same holds true when the hardening ratio of the bilinear springs is variedfrom 0.0 to 0.03, compare with Fig. 17. Application of ESDOF systems combinedwith collapse spectra renders median collapse capacities smaller than the actualones. As expected it can be observed that the collapse capacity rises with increasingpost-yield stiffness.


5 Conclusions

The vulnerability of seismic excited flexible inelastic multi-story frame structuresto dynamic instabilities has been evaluated. In particular a simplified methodologyfor assessment of the global collapse capacity has been proposed, which is basedon equivalent single-degree-of-freedom systems and collapse capacity spectra. Theresult of an example problem presented in this study suggests that the applicationof equivalent single-degree-of-freedom systems and collapse capacity spectra is ap-propriate to estimate the seismic P-delta effect in highly inelastic regular multi-storyframe structures provided that they exhibit non-deteriorating inelastic material be-havior under severe seismic excitation.

References

1. Bernal D (1998) Instability of buildings during seismic response. Eng Struct 20:496–5022. Gupta A, Krawinkler H (2000) Dynamic P-delta effect for flexible inelastic steel structures.

J Struct Eng 126:145–1543. Aydinoglu MN (2001) Inelastic seismic response analysis based on story pushover curves

including P-delta effects. Report No. 2001/1, KOERI, Istanbul, Department of EarthquakeEngineering, Bogazici University

4. Ibarra LF, Krawinkler H (2005) Global collapse of frame structures under seismic excitations.Report No. PEER 2005/06, Pacific Earthquake Engineering Research Center, University ofCalifornia, Berkeley, CA

5. Lignos DG, Krawinkler H (2009) Sidesway collapse of deteriorating structural systemsunder seismic excitations. Report No. TB 172, John A. Blume Earthquake Engineering Re-search Center, Department of Civil and Environmental Engineering, Stanford University,Stanford, CA

6. Asimakopoulos AV, Karabalis DL, Beskos DE (2007) Inclusion of the P- effect indisplacement-based seismic design of steel moment resisting frames. Earthquake Eng StructDyn 36:2171–2188

7. Villaverde R (2007) Methods to assess the seismic collapse capacity of building structures:state of the art. J Struct Eng 133:57–66

8. Bernal D (1987) Amplification factors for inelastic dynamic P- effects in earthquake analysis.Earthquake Eng Struct Dyn 15:635–651

9. MacRae GA (1994) P- effects on single-degree-of-freedom structures in earthquakes. Earth-quake Spectra 10:539–568

10. Kanvinde AM (2003) Methods to evaluate the dynamic stability of structures – shake tabletests and nonlinear dynamic analyses. In: EERI Paper Competition 2003 Winner. Proceedingsof the EERI Meeting, Portland

11. Vian D, Bruneau M (2003) Tests to structural collapse of single degree of freedom framessubjected to earthquake excitation. J Struct Eng 129:1676–1685

12. Miranda E, Akkar SD (2003) Dynamic instability of simple structural systems. J Struct Eng129:1722–1726

13. Adam C, Spiess J-P (2007) Simplified evaluation of the global capacity of stabilitysensitive frame structures subjected to earthquake excitation (in German). In: Proceed-ings of the D-A-CH meeting 2007 of the Austrian association of earthquake engineer-ing and structural dynamics, September 27–28, 2007, Vienna, CD-ROM paper, paper no.30, 10 pp


14. Adam C (2008) Global collapse capacity of earthquake excited multi-degree-of-freedom framestructures vulnerable to P-delta effects. In: Yang YB (ed) Proceedings of the Taiwan – Aus-tria joint workshop on computational mechanics of materials and structures, 15–17 November2008, National Taiwan University, Taipei, Taiwan, pp 10–13

15. Adam C, Jager C (submitted) Seismic collapse capacity of basic inelastic structures vulnerableto the P-delta effect

16. Takizawa H, Jennings PC (1980) Collapse of a model for ductile reinforced concrete framesunder extreme earthquake motions. Earthquake Eng Struct Dyn 8:117–144

17. Bernal D (1992) Instability of buildings subjected to earthquakes. J Struct Eng 118:2239–226018. Adam C, Ibarra LF, Krawinkler H (2004) Evaluation of P-delta effects in non-deteriorating

MDOF structures from equivalent SDOF systems. In: Proceedings of the 13th World Confer-ence on Earthquake Engineering, 1–6 August 2004, Vancouver BC, Canada. DVD-ROM paper,15 pp, Canadian Association for Earthquake Engineering

19. Adam C, Jager C (2010) Assessment of the dynamic stability of tall buildings subjected tosevere earthquake excitation. In: Proceedings of the International Conference for highrisetowers and tall buildings 2010, 14–16 April 2010, Technische Universitat Munchen, Munich,Germany. CD-ROM paper, 8 pp

20. Medina RA, Krawinkler H (2003) Seismic demands for nondeteriorating frame structures andtheir dependence on ground motions. In: Report No. 144, John A. Blume Earthquake Engineer-ing Research Center, Department of Civil and Environmental Engineering, Stanford University,Stanford, CA

21. Vamvatsikos D, Cornell CA (2002) Incremental dynamic analysis. Earthquake Eng Struct Dyn31:491–514

22. Fajfar P (2002) Structural analysis in earthquake engineering – a breakthrough of simpli-fied non-linear methods. In: Proceedings of the 12th European Conference on EarthquakeEngineering, CD-ROM paper, Paper Ref. 843, 20 pp, Elsevier


Documents

Computational Methods in Earthquake Engineering