A Review of Existing Pushover Methods for 2-D RC Buildings

I.U.S.S.

Istituto Universitario di Studi Superiori

Università degli Studi di Pavia

EUROPEAN SCHOOL OF ADVANCED STUDIES IN REDUCTION OF SEISMIC RISK

ROSE SCHOOL

A REVIEW OF EXISTING PUSHOVER METHODS FOR 2-D

REINFORCED CONCRETE BUILDINGS

A Individual Study Submitted in Partial Fulfilment of the Requirements for the PhD Degree in

EARTHQUAKE ENGINEERING

By

MANUEL ALFREDO LÓPEZ MENJIVAR

Supervisor: Dr. RUI PINHO

September 2004

The dissertation entitled “A review of existing pushover methods for 2-D reinforced concrete buildings”, by Manuel Alfredo López Menjivar, has been approved in partial fulfilment of the requirements for the Doctor of Philosophy Degree in Earthquake Engineering.

Rui Pinho

Stelios Antoniou

Abstract

i

ABSTRACT

It is well known, within the Earthquake engineering community, that the most accurate method of seismic demand prediction and performance evaluation of structures is nonlinear time history analysis. However, this technique requires the selection and employment of an appropriate set of ground motions and having a computational tool able to handle the analysis of the data and to produce ready-to-use results within the time constrains of design offices; clearly, a simpler analysis tool is desirable. One method that has been gaining ground, as an alternative to time history analysis, is the nonlinear static pushover analysis. The purpose of the pushover analysis is to assess the structural performance by estimating the strength and deformation capacities using static, nonlinear analysis and comparing these capacities with the demands at the corresponding performance levels. Traditionally, conventional (i.e. non-adaptive) has been used and implemented in design codes. Although it provides crucial information on response parameters that cannot be obtained with conventional elastic methods (either static or dynamic), this method is not exempt from some limitations such as the inability to include higher mode order effects or progressive stiffness degradation. Therefore, the need for a fully adaptive procedure that overcomes the above deficiencies is readily noted. A revision of the “non-conventional” pushover procedures recently proposed was carried out, from which advantages and disadvantages of each approach were identified. It was noted that adaptive pushover methodology constitute a viable option to nonlinear dynamic analysis since it solves the inherent deficiencies that conventional pushover analysis possesses. Additionally, the possibility of using an alternative modal combination to the quadratic modal combination rules (i.e. SRSS and CQC) has been proposed and assessed. The technique is named Direct Vectorial Addition, DVA. The feasibility of this alternative modal combination is tested by using case models of actual buildings. To verify the accuracy of the different pushover schemes the results obtained by these nonlinear static methods are compared to those from the nonlinear time history analysis and the standard error is used to evaluate the accuracy of the comparisons.

Acknowledgements

ii

ACKNOWLEDGEMENTS

Index

iii

INDEX

ABSTRACT................................................................................................................................................................I

ACKNOWLEDGEMENTS ............................................................................................................................................II

INDEX...................................................................................................................................................................III

LIST OF TABLES ...................................................................................................................................................... V

LIST OF FIGURES.................................................................................................................................................... VI

1. INTRODUCTION ....................................................................................................................................................1 1.1 Scope ............................................................................................................................................................1 1.2 Objectives.....................................................................................................................................................2 1.3 Outline of this Dissertation...........................................................................................................................2

2. NONLINEAR STATIC ANALYSIS (PUSHOVER) – OVERVIEW....................................................................................4 2.1 Introduction ..................................................................................................................................................4 2.2 Generalities...................................................................................................................................................5

2.2.1 Description of the traditional pushover analysis................................................................................5 2.3 Development and assessment of the method ................................................................................................9

3. DESCRIPTION OF CASE STUDIES..........................................................................................................................13 3.1 Introduction ................................................................................................................................................13 3.2 Modelling ...................................................................................................................................................13

3.2.1 Analytical tool .................................................................................................................................13 3.2.2 Modelling of the members ...............................................................................................................13 3.2.3 Case study one: RM15.....................................................................................................................14 3.2.4 Case study two: The ICONS frame .................................................................................................18

3.3 Nonlinear dynamic analysis........................................................................................................................20

Index

iv

3.3.1 Case study one: RM15.....................................................................................................................20 3.3.2 Case study two: The ICONS frame .................................................................................................21

4. DESCRIPTION AND ASSESSMENT OF PUSHOVER METHODOLOGIES ......................................................................24 4.1 Introduction ................................................................................................................................................24 4.2 Measuring the accuracy of the pushover techniques ..................................................................................24 4.3 Non-adaptive non-modal pushover analysis...............................................................................................25 4.4 Non-adaptive modal pushover analysis ......................................................................................................26 4.5 Force-based adaptive pushover procedures ................................................................................................29 4.6 Displacement-based adaptive pushover procedures ...................................................................................34 4.7 Possible drawbacks due to the use of quadratic modal combination rules .................................................36 4.8 The ICONS frame.......................................................................................................................................39

4.8.1 Non-adaptive non-modal pushover analysis ....................................................................................39 4.8.2 Pushover analysis using DVA .........................................................................................................40

5. CONCLUSIONS....................................................................................................................................................44 5.1 Summary ....................................................................................................................................................44 5.2 Future Research..........................................................................................................................................45

6. REFERENCES ......................................................................................................................................................46

Index

v

LIST OF TABLES

Table 3.1 Definition of the considered structural system, after Antoniou and Pinho [2004a]..............................15

Table 3.2 Member cross-sectional dimensions for regular buildings, after Mwafy [2001] ..................................15

Table 3.3 Material properties used in the assessment, after Mwafy [2001]..........................................................16

Table 3.4 Calculated floor gravity loads and masses............................................................................................18

Table 3.5 Characteristics of the records employed by Antoniou and Pinho [2004a]............................................21

Table 4.1 Standard error of the pushover schemes in Figure 4.1..........................................................................25

Table 4.2 Standard error of the pushover schemes in Figure 4.4..........................................................................28

Table 4.3 Standard error of pushover schemes shown in Figure 4.7 ....................................................................34

Table 4.4 Standard error of pushover schemes shown in Figure 4.8 ....................................................................36

Table 4.5 Standard error of pushover schemes shown in Figure 4.7 and Figure 4.9 ............................................38

Table 4.6 Standard error of pushover schemes shown in Figure 4.8 and Figure 4.10 ..........................................39

Table 4.7 Standard error of the pushover schemes in Figure 4.11 and Figure 4.12..............................................40

Table 4.8 Standard errors of the profiles shown in Figure 4.13 to Figure 4.14 ....................................................41

Table 4.9 Standard errors of pushover schemes shown in Figure 4.15 and Figure 4.16.......................................42

Index

vi

LIST OF FIGURES

Figure 2.1 Pushover curves of the MDOF and equivalent SDOF systems, after Alvarez-Botero and

López-Menjivar [2004] .......................................................................................................................6

Figure 2.2 Pushover curves of the equivalent SDOF and bilinear systems, after Alvarez-Botero and López-Menjivar [2004]. ......................................................................................................................6

Figure 2.3 Displacement profiles of the ICONS simplified model under constant-force and constant-displacement distributions...................................................................................................................8

Figure 3.1 Fibre plasticity discretization in a reinforced concrete section. ........................................................14

Figure 3.2 Location of Gauss points along the member length. .........................................................................14

Figure 3.3 Plane and sectional elevation of the 12-storey regular frame building set, after Mwafy [2001]................................................................................................................................................15

Figure 3.4 Adopted modelling approach, after Mwafy [2001] ...........................................................................17

Figure 3.5 Elevation and plan views of the frame, after Carvalho et al. [1999] .................................................19

Figure 3.6 Reinforcement detail of the columns, after Carvalho et al. [1999]....................................................19

Figure 3.7 Scheme of vertical loads for nonlinear analysis, after Carvalho et al. [1999]...................................20

Figure 3.8 Elastic response spectra of the four records (5% equivalent viscous damping). ...............................21

Figure 3.9 Respose spectra of input motion: (a) acceleration; (b) displacement. ...............................................22

Figure 3.10 Artificial acceleration time histories for (a) 475 year (Acc-475) and (b) 975 year (Acc-975) return period......................................................................................................................................22

Figure 3.11 Analytical and experimental top frame displacement: (a ) Acc-475; (b) Acc-975............................23

Figure 3.12 Analytical and experimental drift profiles: (a ) Acc-475; (b) Acc-975 .............................................23

Figure 4.1 Comparison of drift profiles from triangular and uniform pushovers with the envelope of maximum inter-storey drifts from dynamic analysis for the RM15 frame........................................25

Figure 4.2 The multimodal pushover procedure [after Paret et.al., 1996]..........................................................26

Index

vii

Figure 4.3 Properties of the nth mode inelastic SDOF system derived from the corresponding pushover curve [after Chopra and Goel, 2001].................................................................................................27

Figure 4.4 Comparison of drift profiles from MPA with the envelope of maximum inter-storey drifts from dynamic analysis for RM15 frame ...........................................................................................28

Figure 4.5 Graphical representation of loading force vector calculation with incremental updating .................32

Figure 4.6 Graphical interpretation of incremental updating using tangent stiffness .........................................33

Figure 4.7 Comparison of drift profiles from FAP (SRSS) with the envelope of maximum inter-storey drifts from dynamic analysis for RM15 ............................................................................................33

Figure 4.8 Comparison of drift profiles from DAP (SRSS) with the envelope of maximum inter-storey drifts from dynamic analysis for RM15 ............................................................................................36

Figure 4.9 Comparison of drift profiles from FAP (DVA) with the envelope of maximum inter-storey drifts from dynamic analysis for RM15 ............................................................................................38

Figure 4.10 Comparison of drift profiles from DAP (DVA) with the envelope of maximum inter-storey drifts from dynamic analysis .............................................................................................................39

Figure 4.11 Comparison of drift profiles from triangular and uniform pushovers with the drift profile at the time step of maximum inter-storey drift from dynamic analysis using the a) Acc-475 record and b) Acc-975 record............................................................................................................40

Figure 4.12 Comparison of drift profiles from triangular and uniform pushovers with the envelope of maximum inter-storey drifts from dynamic analysis using the a) Acc-475 record and b) Acc-975 record..................................................................................................................................40

Figure 4.13 Comparison of drift profiles from FAP with the drift profile at the time step of maximum inter-storey drift from dynamic analysis using the a) Acc-475 record and b) Acc-975 record .........41

Figure 4.14 Comparison of drift profiles from FAP with the envelope of maximum inter-storey drifts from dynamic analysis using the a) Acc-475 record and b) Acc-975 record ....................................41

Figure 4.15 Comparison of drift profiles from DAP with the drift profile at the time step of maximum inter-storey drift from dynamic analysis using the a) Acc-475 record and b) Acc-975 record .........42

Figure 4.16 Comparison of drift profiles from DAP with the envelope of maximum inter-storey drifts from dynamic analysis using the a) Acc-475 record and b) Acc-975 record ....................................42

Chapter 1: Introduction

1

1. INTRODUCTION

1.1 SCOPE

The best way to assess the performance and to predict the demand on a structure subjected to earthquake action is nonlinear time history analysis. However, for this technique to be reliable some parameters need to be clearly defined. A set of characteristic ground motions that may affect the area [Bommer et al., 2003] and a mathematical tool able to handle all analyses often exceed the capabilities of a design office which works under tight time constraints. Thus, a simpler yet reliable method of structural analysis is desirable.

During the last decade, the nonlinear static pushover analysis has been gaining ground among the structural engineering society as an alternative mean of analysis. The purpose of the pushover analysis is to assess the structural performance by estimating the strength and deformation capacities using static, nonlinear analysis and comparing these capacities with the demands at the corresponding performance levels. The assessment is based on the estimation of important structural parameters, such as global and inter-storey drift, element deformations and internal forces. The analysis accounts for the geometrical nonlinearity and material inelasticity, as well as the redistribution of internal forces. Typically, the traditional procedure is to push the structure with a constant triangular or uniform distribution of forces. Although it provides crucial information on response parameters that cannot be obtained with conventional elastic methods (either static or dynamic), this method is not exempt from some limitations such as the inability to include higher mode order effects or progressive stiffness degradation. Therefore, the need for a fully adaptive procedure that overcomes the above deficiencies is readily noted.

An appealing adaptive pushover method is the one proposed by Antoniou and Pinho [2004a; 2004b]. In this adaptive pushover technique, the lateral load distribution is not kept constant but is continuously updated during the process, according to modal shapes and participation factors derived by eigenvalue analysis carried out at each analysis step. The method is multimodal and accounts for softening of the structure, its period elongation, and the modification of the inertial forces due to spectral amplification. Two variants of the method currently exist: Force-based Adaptive Pushover (FAP) and Displacement-based Adaptive Pushover (DAP). Moreover, of the two variants, the latter has already proved to provide results very similar to those obtained from the time history analysis when the response of seismically design buildings has been analysed [Antoniou and Pinho, 2004b].

Nevertheless, the adaptive pushover may not be exempt of limitations the main of which could be the way how different modes are combined. The modal combination rules used are SRSS and CQC that remove the contribution of negative quantities, since they are squared, rendering results that are always positive. It is known that during an earthquake the deform shape of a building is not always positive instead it can be composed of a wide range of displacements, with different signs, along the height of the structure. The possible flaw is evident when the response of 3D structures is studied. In this case, torsional modes are almost wiped away and their contribution to the overall force vector is completely modified because the signs of the modal displacements are removed by the present modal combination rules. This became obvious during the research performed by López-Menjivar [2003] on


2

3D reinforced concrete buildings. Thus, a new approach to combine the contributions of the modes is clearly needed.

However, it was observed in a previous research [Antoniou and Pinho, 2004b], that DAP might overcome the flaw mentioned in the precedent paragraph in the case of seismically design 2D building structures. Therefore, more study is needed before stating a definite conclusion.

1.2 OBJECTIVES

The nonlinear static pushover analysis is an alternative method, with respect to the nonlinear time history analysis, of demand prediction and performance evaluation that has been used in the last decade. Nevertheless, it has been made obvious that, despite its efficiency and applicability, it exhibits significant limitations [Krawinkler and Seneviratna, 1998].

Recently, many attempts have been proposed to improve the capability of the pushover analysis. Innovations such as considering high mode order effects, stiffness degradation and input motions have been developed. One of those current methods is an adaptive pushover analysis elaborated by Antoniou and Pinho [2004a, 2004b]. By making an extensive review of the current pushover methods, the main aims of this research are the following: The first one is to discuss and assess different pushover methodologies by employing a typical seismically designed concrete frame structure. The second purpose is to explore the possibility of using and alternative modal combination ruled, named DVA, to the usual SRSS and CQC rules to further improve the capabilities of the above mentioned adaptive approach.

1.3 OUTLINE OF THIS DISSERTATION

This research is organized into five chapters. It covers several aspects ranging from the motivation behind the selection of the research topic, the literature review, the development of a new modal combination scheme, to be used in an adaptive pushover analysis, to the testing of different pushover methods and the validation of this new scheme by applying it to two case studies.

Chapter 1 is devoted to the presentation of the research topic and the identification of the general scope and specific objectives.

The nonlinear static pushover analysis procedure, which has recently attracted considerable attention as a tool for seismic assessment, is described in detail in Chapter 2. Its merits and drawbacks are also discussed, along with essential issues, such as target displacements and the applied load pattern.

The study employs two reinforced concrete buildings. The first structure, RM15, belongs to a set of buildings already employed in a by Antoniou [2002] and includes frames that are seismically design. The second case study is a four-storey bare frame known to have a soft storey at the third floor, named ICONS frame hereafter. The latter is attributed to a drastic change in strength and stiffness at this level through a reduction in both the reinforcement content and the section dimensions in the columns between the second and third storeys, coinciding also with the location of lap-splicing. Structural characteristics of each building are illustrated in Chapter 3.

A review of recent research carried out worldwide on pushover analysis is presented in Chapter 4 with particular emphasis placed on attempts to extend its applicability and efficiency. One such successful attempt, the adaptive pushover analysis proposed by Antoniou and Pinho [2004a, 2004b] is described and scrutinized here. An attempt is made at identifying a limitation of the above adaptive pushover method following which a scheme, the Direct Vectorial Addition (DVA) is proposed and expounded as a solution. Along the development of the chapter, a case study is used to compare the response


3

obtained from different pushover analyses to obtain preliminary conclusions. The chapter closes with the application of DVA to another case study.

Chapter 5 is a summary of the most important aspects discussed in the dissertation, a thorough presentation of the main findings and conclusions, and the identification of the possible projects for future research are also outlined.

Chapter 2: Nonlinear static analysis (Pushover) −Overview

4

2. NONLINEAR STATIC ANALYSIS (PUSHOVER) – OVERVIEW

2.1 INTRODUCTION

The use of the nonlinear static analysis, named as pushover analysis hereafter, dates back to the 1970’s but only after gaining importance during the last 10-15 years had dedicated publications started to appear on the subject. Initially the majority of them concentrated on discussing the range of applicability of the method and its advantages and disadvantages, compared to elastic or non-linear dynamic procedures [e.g. Lawson et al., 1994; Krawinkler and Seneviratna, 1998].

Only very recently, there have been efforts to extend pushover analysis to take into account higher mode effects [Paret et al., 1996; Sasaki et al., 1998; Moghadam and Tso, 2002; Chopra and Goel, 2001, 2002]. Furthermore, there have also been some attempts to derive fully adaptive procedures [Bracci et al., 1997; Gupta and Kunnath, 2000; Requena and Ayala, 2000; Elnashai, 2000; Antoniou et al., 2002; Aydinoğlu, 2003] with update force distributions that take into account the strength and stiffness state of the building at each step.

The potential of the pushover analysis has been recognised in the last decade and it has found his way into seismic guidelines [ATC, 1997; SEAOC, 1995; CEN 1995]. It is expected to gain more popularity in the future and it is already included in some codes [PCM, 2003].

The purpose of the pushover analysis is to assess the structural performance by estimating the strength and deformation capacities using a static, non-linear analysis algorithm and comparing these capacities with the demands at the corresponding performance levels. The assessment is based on the estimation of important structural parameters, such as global and interstorey drift or element deformation and forces. The analysis accounts for the geometrical nonlinearities and material inelasticities, as well as the redistribution of internal forces, Hence it provides crucial information on response parameters that cannot be obtained with conventional elastic methods (either static or dynamic).

Response characteristics that can be obtained with the pushover analysis include [Krawinkler and Seneviratna, 1998]:

- Realistic force demands on potentially brittle elements, such as axial demands on columns, moment demands on beam-to-column connections or shear forces demands on short, shear dominated elements.

- Estimates of the deformation demands on elements that have to deform inelastically, in order to dissipate energy.

- Consequences of the strength deterioration of particular elements on the overall structural stability.


5

- Identification of the critical regions, where the inelastic deformations are expected to be high.

- Identification of strength irregularities in plan or elevation that cause changes in the dynamic characteristics in the inelastic range.

- Estimates of the interstorey drifts, accounting for strength and stiffness discontinuities. In this way, damage on non-structural elements can be controlled.

- Sequence of the member’s yielding and failure and the progress of the overall capacity curve of the structure.

- Verification of the adequacy of the load path, considering all the elements of the system, both structural and non-structural.

Clearly, these benefits come at the cost of additional analysis effort, associating with incorporating all-important elements and modelling their inelastic load-deformation characteristics.

2.2 GENERALITIES

2.2.1 Description of the traditional pushover analysis

The subsequent description abides by, in a rough manner, proposals contain in the EC-8 [CEN, 1995], Italian Building code [PCM, 2003] and NEHRP guidelines [ATC,1997]. The Nonlinear Static Procedure (commonly known as pushover analysis) is described as follows: “A model directly incorporating inelastic material response is displaced to a target displacement and resulting internal deformations and forces are determined”. A certain load pattern is selected and the intensity of the lateral load is monotonically increased. The sequence of cracking, plastic hinging and failure of the structural components throughout the procedure is observed, until either the target displacement is exceeded or the building collapses. The target displacement is intended to represent the maximum displacement likely to be experienced during the expected ground motion.

The static pushover analysis has no robust theoretical background. It is based on the assumption that the response of the multi-degree-of-freedom (MDOF) structure is directly related to the response of an equivalent single-degree-of-freedom (SDOF) system with appropriate hysteretic characteristics. This implies that the dynamic response of the MDOF system is determined by a single mode only and that the shape {Φ} of that mode is constant, throughout the time-history, regardless of the level of deformation.

Presuming that the vector {Φ} is known, the transformation factor, Γ can be computed thorough equation 2.1.

{ } { }{ } { }Φ⋅⋅Φ

⋅Φ=ΓM

1MT

T . (2.1)

where, M is the mass vector of the MDOF system.

The force-deformation characteristics of the equivalent SDOF system can be determined from the results of the nonlinear pushover analysis of the MDOF system by employing the equation 2.2

Γ= ndd * Γ= bFF * (2.2)


6

where, Fb and dn are the base shear force and control node displacement of the MDOF system. The procedure derives a base shear vs. top displacement curve for the equivalent SDOF system (Figure 2.1).

PUSHOVER CURVES OF THE MDOF AND EQUIVALENT SDOF SYSTEMS

0

200000

400000

600000

800000

1000000

1200000

0 50 100 150 200 250 300 350 400 450 500

DISPLACEMENT (mm)

BAS

E S

HE

AR (N

)

MDOF System Equivalent SDOF System Figure 2.1 Pushover curves of the MDOF and equivalent SDOF systems, after Alvarez-Botero and López-

Menjivar [2004]

From such plot an idealized elasto-perfectly plastic force-displacement relationship can be constructed, Figure 2.2. The yield force Fy

*, which represents the ultimate strength of the idealized system, is equal to the peak force of the pushover curve of the equivalent SDOF system. The initial stiffness of the idealized elastic-perfectly plastic system is determined in such a way that that the areas under the actual and the idealized system force-deformation curve are equal, see Figure 2.2. Based on this assumption, the yield displacement of the idealized SDOF system dy

* is given by:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−=

*

***

y

mmy

F

Ed2d (2.3)

where, Em* is the area under the equivalent SDOF system pushover curve.

EQUIVALENT SDOF SYSTEM AND BILINEAR SYSTEM

0

250000

500000

750000

1000000

0 100 200 300 400

Displacement [mm]

Bas

e Sh

ear [

N]

Equivalent SDOF System Figure 2.2 Pushover curves of the equivalent SDOF and bilinear systems, after Alvarez-Botero and

López-Menjivar [2004].

Although these considerations are apparently incorrect, sensitivity studies have shown that the modification factor Γ can be considered constant for small to moderate changes in {Φ} and that rather accurate predictions can be attained, if the structural response is dominated by the fundamental mode [Krawinkler and Seneviratna, 1998; Lawson et al., 1994; Fajfar and Gaspersic, 1996].


7

The maximum displacement of the equivalent SDOF system, dt*, subjected to the expected ground

motion can now be found by means of elastic spectra, inelastic spectra or time-history analysis. The expected deformation level of the MDOF structure, dt, can be estimated by the equation:

*tt dd Γ= (2.4)

Several critical parameters of the procedure are worthy of consideration, namely the target displacement, the shape of the load distribution, as well as its nature (forces or displacements).

2.2.1.1 Target displacement The target displacement of pushover analysis should approximate the maximum level of deformation that is expected during the design earthquake. It can be calculated by any procedure that accounts for the effects of non-linear response on displacement amplitude.

It was explained in the previous section that it is assumed that the target displacement for the MDOF structure can be estimated from the displacement demand of the equivalent SDOF system, through the use of the selected shape vector {Φ} (usually corresponding to that of the fundamental mode) and equation (2.4). Therefore, a method is sought to determine the target displacement of the SDOF system.

For increased sophistication, dynamic time-history analysis of the SDOF model can be used, assuming simple hysteretic rules. However, several other approaches exist. The N2 method [Fajfar and Gaspersic, 1996] utilises the strength reduction factor R, the period T of the SDOF model and R-µ-T relationships to calculate the ductility demand µ = dt

*/d*y and , consequently, the displacement

demand dt*. Similar procedures have been presented by Krawinkler and Seneviratna [1998]. The

Capacity Spectrum Method estimates the displacement demand by comparing graphically the lateral capacity of the system with highly damped spectra (or inelastic spectra as modified by Fajfar [1999]) in the Acceleration-Displacement Response Spectrum format.

2.2.1.2 Applied forces vs. applied displacements Considering earthquake loading as a set of imposed energy input, ground displacements and deformations of the structural members rather than a set of lateral forces seems a much more rational approach. After all, the fact that earthquake input has been modelled as forces rather than displacements can only be explained by historical reasons, related to the developments of contemporary engineering methods in countries of low seismic hazards, like England and Germany, where the most significant actions are the vertical gravity loads. Had the modern engineering made its initial step in earthquake-prone regions like New Zealand, California or Southern Europe, today’s code provisions would probably be based on deformations. Therefore, applying displacements rather than force patterns in the pushover procedures appears to be more appropriate and theoretically correct [Priestley, 1993].

However, displacement-based pushover analysis suffers from significant inherent deficiencies. Due to the constant nature of the applied patterns, it can be conceal important structural characteristics, such as strength irregularities. This is illustrated by means of an example in Figure 2.3, where the ICONS frame model, which will be described in section 3.2.4, has been pushed to the same target displacement with constant displacement and constant force patterns (triangular distributions). Although the interstorey drift at the soft-storey during an earthquake is expected to be larger than the other storeys, the displacement-based pushover yields equal drifts for all the storeys.


8

0

1

2

3

4

0 50 100 150 200 250

Displacement (mm)

Stor

ey

Constant force profile Constant displacement profile

Figure 2.3 Displacement profiles of the ICONS simplified model under constant-force and constant-

displacement distributions.

Hence, to apply displacements, rather than forces, requires adaptiveness meaning to update the displacement patterns, according to the structural properties of the analysed model, such as the stiffness of the mass distribution. The applied displacements at every step would be determined by modal analysis or any other method that explicitly accounts for the structural characteristics at the current level of inelasticity, in a way that approximates the expected dynamic deformations. On one side, such procedure would be theoretically more rigorous and match the new trends for displacement-based design and assessment, and, alternatively, it would expose the structural weakness that are concealed with fixed-displacement patterns and yield accurate results both at the local and the global level [Antoniou and Pinho, 2004b].

2.2.1.3 Lateral load patterns The lateral load patterns should approximate the inertial forces expected in the building during an earthquake. Although, clearly, the inertia force distributions will vary with the severity of the earthquake and with time, usually an invariant load pattern is used. This approximation is likely to yield adequate predictions of the element deformation demands for low to medium-rise framed structures, where the structure behaviour is dominates by a single mode. However, pushover analysis can be grossly inaccurate for structure of larger periods, where higher mode effects tend to be important. Moreover, Mwafy and Elnashai [2000] and Lawson et al. [1994] observed that pushover procedures are particularly poor in predicting the response of frame-wall structures, probably due to significant period shift and change of inertia force distribution upon yielding of the wall base.

Several investigations [Mwafy and Elnashai, 2000; Gupta and Kunnath, 2000] have found that, whereas in the elastic range force distributions of a triangular or trapezoidal shape provide a better fit to dynamic analysis results, at large deformations the dynamic envelopes are closer to the uniformly distributed force solutions. The above happens after the structure has sustained significant damage at a particular storey level, favouring a pattern akin to a SDOF system.

Since the constant distribution methods are incapable of capturing such variations in characteristics of the structural behaviour under earthquake loading, the use of at least two different patterns has been proposed. Various codes and guidelines [CEN, 1995; PCM, 2003; ATC, 1997] suggest that use of a “uniform” pattern, where the lateral forces are proportional to the local masses at each floor level, and a “modal” pattern, which is determined by a modal combination using a sufficient number of modes and an appropriate spectral shape. Alternatively, the “triangular”, in which the accelerations are proportional to the storey heights, rather than the “modal” pattern, may be utilized.


9

Different suggestions have been made in the past. These include the use of lateral loads proportional to the deflected shape of the structure [Fajfar and Fichinger, 1988] or proportional to the storey shear resistances at the previous step [Bracci et al., 1997], whereas Gupta and Kunnath [2000]. Similarly, Requena and Ayala [2000] suggested the derivation of the forces through modal combinations using the square root of the sum of squares (SRSS) method, taking into account a predefined number of modes of interest.

2.3 DEVELOPMENT AND ASSESSMENT OF THE METHOD

Inelastic static analysis in Earthquake engineering has been first employed by Gulkan and Sozen [1977] and Saiidi and Sozen [1981]. In their Q-model, Saiidi and Sozen suggested the use of the moment-curvature relationships of the individual members to derive the top-level displacement vs. shape moment curve of the MDOF (as opposed to the base shear that is normally used nowadays). The curve is idealized with a bilinear curve to derive the force displacement characteristics of the SDOF system. The deflected shape at yield is assumed to describe the characteristic vibration shape of the building {Φ}. A similar procedure is followed by the N2 method [Fajfar and Fischinger, 1998; Fajfar and Gaspersic, 1996] which after carrying out the pushover analysis relates the quantities Q* of the SDOF system with the quantities Q of the MDOF structure through the equation (2.5).

{ } { }{ } { }

Qmm

Q1M

MQQii

2ii

T

T⋅

Φ⋅Φ⋅

=⋅⋅⋅Φ

Φ⋅⋅Φ=Γ=∑∑* (2.5)

and it uses R-T-µ relationships to calculate the displacement demands.

A number of publications has recently reviewed the merits and deficiencies of the method. Lawson et al. [1994] discuss in some detail the range of applicability and the expected realism for various structural systems, and highlight the encountered difficulties. Four steel structures, which heights vary from two to 15 storeys, have been analysed using DRAIN-2DX under three different patterns of static loading (uniform, triangular and modal using the SRSS method to combine the modal shapes and their spectral amplifications). The results were compared against dynamic analysis using seven different earthquake records. The correlation between static and dynamic responses was good for the two and five storey buildings. However, the predictions of the static analysis were inadequate for the 10 and 15 storey buildings. The authors concluded that pushover analysis might be grossly inaccurate for tall buildings, where high mode effects are important. Surprisingly, the pushover analyses with the modal distribution yielded unacceptably poor results, which has been attributed to an exaggeration of higher-mode effects by the SRSS combination method. Further, unsuccessful attempts to correlate the hysteretic energy of the dynamic analysis with the area under the pushover curve showed that this area is a poor indicator of cumulative damage effects. The paper also discusses the issue of the target displacement, at which the building should be pushed. The procedure proposed by Qi and Moehle [1991] and Miranda [1991] is presented together with a simple method, whereby the target displacement of the MDOF system is calculated from its fundamental period and the spectral ordinate corresponding to it. Both methods provided good estimates of the target displacements. The authors close their paper with some recommendations on why, when and how to use pushover analysis. It is advisable to use the method, when there is doubt about the efficiency of simple elastic code-based procedures, for the seismic evaluation of existing structures and the design of retrofit schemes, but always with great care and good engineering judgment for the interpretation of results.

On the same basis, Faella [1996] compares the response of three, six and nine storey buildings subjected to artificial and real earthquakes with pushover analysis, and concludes that static analysis can, indeed, identify collapse mechanisms and critical regions, yielding reasonable estimates for the interstorey drifts. Although the effect of different load patterns is not investigated, confining the observations to the triangular distribution, the importance of accurate determination of the target displacement is stressed. It is also suggested that, since the sum of the maximum dynamic drifts is


10

larger than the maximum dynamic roof displacement, due to the random nature of earthquake loading, “it is advisable to compute the static interstorey drift for a maximum roof displacement higher than the dynamic one”. Finally, the author points out difficulties with static-dynamic comparison when the strong-motion input is rich in long period frequencies.

In the course of describing recent trends in seismic design methodology, Krawinkler [1995] presents pushover analysis as a simplified performance evaluation method that can also be used as a design tool. He discusses the theoretical limitations of the method, as well as the procedures for the estimation of the targer displacement, and emphasises the fact that pushover analysis “cannot disclose performance problems cused by changes in the inelastic dynamic characteristics, due to higher mode effects”. The latter is one of the most significant problems of conventional static procedures, as discussed in previous sections, but can be solved or mitigated with the application of adaptive patterns.

Furthermore, Krawinkler and Seneviratna [1998] summarise the basic concepts of the method stressing that its theoretical background is not rigorous, being based on the assumption that the MDOF response is related to the response of a SDOF oscillator. The conditions, under which pushover analysis can provide adequate information, are identified, and the important issues of the target displacement and the lateral load pattern are discussed. Like Lawson et al. [1994] the authors suggest that use or more than one invariant load pattern, and consider the application of adaptive pattern rather attractive. Moreover, they mention that the selection of the load pattern is more important than that of the target displacement and that to their belief “the load pattern issue is the weak point of the pushover analysis procedure”. Presenting an example of a successful pushover analysis they discuss the limitations of the method, cases in which the pushover predictions are inadequate or even misleading, and suggest areas of future development. Not surprisingly, “the higher mode effects, once a local mechanism has formed,” is considered the most important problem to be solved.

Ken and D’Amore [1999] assess the accuracy of the method, in comparison with inelastic time-history procedures using DRAIN-2DX to analyse an instrumented six storey steel building built in 1977. They conclude, as expected, that not all dynamic analyses of the same structure under a set of different earthquake records are predicted by pushover analysis. The roof displacement vs. base shear curve is considered too simplistic and inadequate, as it cannot describe the dynamic nature of the response during earthquakes, especially when a predefined and fixed transverse load vector is used. However, the superiority of the method compared to the code-based procedures is recognised. Moreover, as an improvement the authors suggest the use of pushover analysis supplemented by the use of 3-D nonlinear dynamic analysis.

Finally, Naeim and Lobo [1999] present the common mistakes committed during a pushover analysis. Amongst other issues, the authors discuss the importance of the loading shape function, the selected performance objectives, P-∆ effects and gravity loading, shear failure mechanisms and the post peak behaviour. In particular, for the latter, they note that “if an analysis package cannot model structural failure, then either the push has to be stopped at the onset of the first hinging, or extreme care must be exerted on the interpretation of the post-failure behaviour as reported by the program”, mentioning that most of the programs are not capable to model adequately the post peak response.

As already mentioned, there are good reasons for using pushover analysis rather than simplified elastic methods for estimating the deformation demands. Moreover, the simplicity of the method makes it a more attractive approach for everyday practice than nonlinear time-history analysis. Traditional pushover analysis can be extremely useful tool, if used with caution and acute engineering judgment, but, as discussed in the previous sections, it exhibits significant shortcomings and limitations, which are summarised below:


11

1. The theoretical background of the method is not robust and it is difficult to defend. As mentioned earlier, an important implicit assumption behind pushover analysis is that the response of a multi-degree-of-freedom structure is directly related to an equivalent single-degree-of-freedom system. Although in several cases the response is dominated by the fundamental mode, this can by not means be a generalised statement. Moreover, in dynamic time-history analysis the shape of the fundamental mode itself may vary significantly depending on the level of inelasticity and the location(s) of damage.

2. As a consequence of point 1, the deformation estimates obtained from a pushover analysis may be highly inaccurate for structures where higher mode effects are significant. The method, as currently prescribed in codes and seismic guidelines, explicitly ignores the contribution of the higher modes to the total response. In the cases where this contribution is significant, the pushover estimates may be totally misleading.

3. It is difficult to model three-dimensional and torsional effects. Pushover analysis is very well established and has been extensively used with 2-D models. However, little work has been carried out for problems that apply specifically to asymmetric 3-D systems, with stiffness or mass irregularities. Therefore, it is by no means clear how to derive the load distributions and how to calculate the target displacement for the different frames of the building. Moreover, there is no consensus regarding the application of the lateral force in one or both horizontal directions.

4. Traditional pushover analysis is still force-based, due to the inability of displacement-based pushover to capture important structural weakness, such as strength irregularities. However, as discussed in 2.2.1.2, it would be theoretical more rigorous to find a way to apply displacement distributions that are appropriately updated at different deformation levels since displacement are better related to structural damage.

5. The progressive stiffness degradation that occurs during the cyclic non-linear earthquake loading of the structure is not taken into account. This degradation leads to changes in the periods and the modal characteristics of the structure that affect the loading attracted during earthquake ground motion.

6. Being a static method, pushover analysis concentrates on the strain energy of the structure, neglecting other sources of energy dissipation, which are associated with the dynamic response, such as the kinetic and the viscous damping energy. Moreover, it neglects duration effects and cumulative energy dissipation demand.

7. Only horizontal earthquake load is considered. The vertical component of the earthquake loading, which can be in some cases of great importance is ignored, since no method has been proposed up to now on how to combine pushover analysis with actions that account for the vertical ground motion.

8. A separation between the supply and the demand is implicitly in the method. This is clearly incorrect, as the inelastic structural response is load-path dependent and the structural capacity is always associated to the earthquake demand.

Obviously, pushover analysis lacks many important features of dynamic non-linear analysis and it will never be a substitute as the most accurate tool for structural analysis and assessment. Nevertheless, several possible developments can considerably improve the efficiency of the method. Indeed, from the summary of pitfalls short listed above, points 1 to 5 can be overcome with the derivation of a fully adaptive procedure, which accounts for both higher mode contributions and the alterations of the local resistance and the modal characteristics, once a local mechanism has formed. In this way, the stiffness degradation, the period elongation and the higher mode effects can be


12

explicitly considered. Moreover, finding a way to incorporate, somehow, the expected ground motion in the analysis will provide site-specific results that apply to certain areas with particular seismic hazard characteristics. This can be achieved with the utilisation of spectra representative of these areas. Finally, the application of adaptive displacement rather than force patterns will provide a both conceptually appealing and accurate tool to replace “traditional” force-based approaches.

Chapter 3: Description of case studies

13

3. DESCRIPTION OF CASE STUDIES

3.1 INTRODUCTION

In order to assess different pushover methodologies two buildings have been selected, RM15 and the ICONS frame. The former has been already used in previous research [Antoniou, 2002] and the latter is a model of a full scaled structure constructed for pseudo dynamic testing [Carvalho et al., 1999]. Subsequently, a detailed description of the analytical tool, the modelling approach as well as the nonlinear dynamic analyses used is presented.

3.2 MODELLING

3.2.1 Analytical tool

The finite element analysis program SeismoStruct [SeismoSoft, 2004] is utilized to run all analysis. The structural package is able to predict the large displacement behaviour of space frames under static or dynamic loading, taking into account both geometric nonlinearities and material inelasticity. SeismoStruct accepts static loads (either forces or displacements) as well as dynamic (accelerations) actions and has the ability to perform eigenvalues, nonlinear static pushover (conventional and adaptive), nonlinear static time-history analysis, nonlinear dynamic analysis and incremental dynamic analysis.

3.2.2 Modelling of the members

Structural members have been discretised by using a beam-column model based on distributed plasticity-fibre element approach. The model takes into account geometrical nonlinearity and material inelasticity. Sources of geometrical nonlinearity considered are both local (beam-column effect) and global (large displacement/rotation effects). Since a constant generalized axial strain shape function is assumed in the adopted cubic formulation of the element, it results that its application is only fully valid to model the nonlinear response of relatively short members and hence a number of elements (usually three to four per structural member) is required to accurate model the structural frame members. Material inelasticity is explicitly represented through the employment of a fibre modelling approach which allows for the accurate estimation of structural damage distribution, the spread of material inelasticity across the section area and along the members length. In the fibre model the sectional stress-strain state or beam-column elements is obtained through the integration of the nonlinear uniaxial stress-strain response of the individual fibres in which the section has been subdivided. If a sufficient number of fibres is employed, the distribution of material nonlinearity across the section area is accurately modelled, even in the highly elastic range, see Figure 3.1


14

R C Section U nconfined C oncrete F ibres

Steel F ibresC onfined C oncrete F ibres

Figure 3.1 Fibre discretisation in a reinforced concrete section.

The spread of inelasticity along member length then comes as a product of the inelastic cubic formulation suggested by Izzudin [2001]. Two integration Gauss points per element are used for the numerical integration of the governing equations of the cubic formulation, Figure 3.2. If a sufficient number of elements is used the plastic hinge length of structural members subjected to high levels of material inelasticity can be accurately estimated.

Figure 3.2 Location of Gauss points along the member length.

It is worth mentioning that shear strains across the element cross section are not modelled; in addition, warping strains and warping effects are not considered in the current formulation, either. Additionally, the elastic torsional rigidity is used in the formulation of the nonlinear frame elements; this clearly involves some degree of approximation for the case of reinforced concrete sections.

No viscous damping was considered in any dynamic analysis, since energy dissipation through hysteresis is already implicitly included within the nonlinear fibre model formulation of the inelastic frame elements, and non-hysteretic type damping was assumed to be negligible within the scope of the present endeavour.

3.2.3 Case study one: RM15

The description of RM15 has mainly been taken from Antoniou and Pinho [2004a] and Mwafy [2001]. The RM15 model is part of a set of structures that three different structural configurations were employed: a 12 storey regular frame, an eight storey irregular frame and a dual wall-frame system. Additionally, different ductility classes and design ground acceleration were considered, resulting in a total of 12 models. The latter represent common reinforced concrete structures and are based on buildings designed and detailed at the University of Patras [Fardis, 1994], seemingly according to the 1995 version Eurocode 8 [CEN, 1995]. Subsequently, they were modelled by Mwafy [2001] under a framework of different project, and were then adapted by Rovithakis [2001]. The Rm15 belongs to the group of regular buildings and their general characteristics are defined in Table 3.1.


15

Table 3.1 Definition of the considered structural system, after Antoniou and Pinho [2004a]

Structural System

Number of Storeys

Structure Reference

Ductility Level

Design PGA (g)

Behavior Factor (q)

Tuncracked

(s) RH30 High 5.00 0.697 RM30 Medium

0.30 3.75 0.719

RM15 Medium 3.75 0.745 Regular Frame 12

RL15 Low 0.15

2.50 0.740

The geometric characteristics of the structural system is illustrated in Figure 3.3. The overall plan dimensions is 15m by 20m. The total height is 36 metres, with equal storey heights of 3m. The lateral force resisting system is a moment frame, wall-frame sub-set has both a central core extending over the full height and moment frames on the perimeter. The floor system is solid slab. The member cross-section sizes are given in Table 3.2.

Table 3.2 Member cross-sectional dimensions for regular buildings, after Mwafy [2001]

Columns Beams b× h

S R Internal External Corner

X-dir.

(1st storey)

X-dir.

(Long.) Z-dir.

Slab

RH30 .80×.80 .70×.70 .70×.70 .35×.65 .35×.65 .35×.65 0.14

RM30 .80×.80 .70×.70 .70×.70 .35×.60 .35×.60 .35×.60 0.14

RM15 .80×.80 .70×.70 .70×.70 .30×.60 .30×.60 .30×.60 0.14

RL15 .80×.80 .70×.70 .70×.70 .30×.60 .30×.60 .30×.60 0.14

3 x

5.0

= 15

.0 m

5 x 4.0 = 20.0 m

12 x

3.0

= 3

6.0

m

X

Z

Solid slabs

Figure 3.3 Plane and sectional elevation of the 12-storey regular frame building set, after Mwafy [2001]

Mean values of material strengths are utilised in the analyses rather than the characteristics values used in the design, an approach consistent with ‘assessment’. These values are presented in Table 3.3.


16

Table 3.3 Material properties used in the assessment, after Mwafy [2001]

Material parameter Values used in analysis

Mean compressive strength, fcm 33 N/mm2

Mean tensile strength, fct 2.6 N/mm2

Crushing strain, εc 0.0022

Con

cret

e gr

ade

C25

/30

Modulus of elasticity, Ec 30.5 kN/mm2

Yield strength, fy 585 N/mm2

Ultimate strength, fu 680 N/mm2

Ultimate strain, εsu 0.094

Stee

l S50

0

Young’s modulus, Es 200 kN/mm2

Reinforced concrete column-section and T-section are used for modelling of columns and beams, respectively. The contribution of the slab width to the beam has a significant effect on the stiffness and hence on the overall response of the buildings. Several values for the flange width are recommended by seismic codes. Due to the fact that the participation of the slab to the beam stiffness is less than the participation to the flexural strength, as a result of the moment reversal and the low contribution of the flange in tension [Paulay and Priestley, 1992]. It is suggested [ACI 318, 1995] to use an effective slab width equal to one half of that recommended for gravity load design. It is also recommended by EC8 to reduce the effective slab width for buildings subjected to seismic forces due to inelastic effects. The beam width plus 7% of the clear span of the beam on either side of the web is the effective flange width that is adopted in the model employed in the design of the buildings [Fardis, 1994] as well as in the current analysis. This provides values between the conservative flange width of EC8, which is intended for design purposes, and the full slab or the width recommended for gravity load design.

On the structure level 1064 elements are used to model the 12-storey regular frame. This includes the number of cubic elasto-plastic elements, the joint elements connecting the frames in the orthogonal direction, the cubic elastic elements and the shear spring elements, as subsequently discussed. On the member level, seven elements are used to model each beam. Three of them are elasto-plastic elements representing the beam between the faces of the columns. The lengths of these elements are determined in accordance with the distribution of transverse and longitudinal reinforcement specified in the design. On the basis of the arrangement of the transverse reinforcements, the confinement factors for the cross-section of these elements are evaluated using the mean values of material strengths. Two rigid elements are utilised to connect the beam ends with the framing columns (the length between the face and the centreline of the vertical elements, as shown in Figure 3.4(a).


17

(a) Beam column connection

External Frames

Internal Frames

Global X

Global ZGlobal Y

Zero length jointelements connecting

the frames inthe orthogonal

direction

Beam criticallengthsShear spring

with zerolength

Rigid arms

Column

L0.58 L0.21 L 0.21 L

Gauss sections

Column

d2

D1

d1B2B1

D2

b2b1

Concrete andsteel layers

Kobe

Loma Prieta

0 10 sec

0 15 sec

0 15 sec

(c) Decomposition of beam T-section into fibres

Steelfibres

Unconfined concrete fibre

Monitoringpoint

Confinedconcrete

fibre

Art-rec1

Regular frame building

Irregular frame building

(b) Cubic elasto-plastic element

a

b

c

Figure 3.4 Adopted modelling approach, after Mwafy [2001]

Under earthquake load, beam-column joints are subjected to high shear stresses that could lead to diagonal cracking and significant shear deformation. The total joint deformations can be considerable as a consequence of this cyclic load. Typically, 20% of the interstorey drift due to earthquake forces may be originate from joint deformations [Paulay and Priestley, 1992]. Towards this end, two shear spring elements are introduced in the present study to represent the shear stiffness of the beam-column connection. A simple linear elastic force-deformation relationship is utilised to calculate the shear stiffness of the joint. On the other hand, the structural mesh employs three cubic elasto-plastic elements for modelling of each vertical element, with the exception of the core. The method adapted to model the core is described below.

Gravity loads are applied as point loads at beam nodes. To account for inertia effects during dynamic analysis, masses are distributed in the same pattern adopted for the gravity loads and are represented by lumped 2D mass elements. The numerically-dissipative Hilber-Hughes-Taylor α-integration scheme is utilised to integrate the equations of motion.

According to the data used in the design [Fardis, 1994], the following static loads per unit area are considered to calculate the total gravity loads on the frames, - Slab self-weight, Wf = 2.0 kN/m2

- Live load, Q = 2.0 kN/m Using the appropriate coefficients, Ψ2 and φ, from the design code, the vertical loads are combined with seismic actions by applying the following rules: - At top floor: 1.0 G + 0.3 Q + EL - For other storeys: 1.0 G + 0.15 Q + EL where, G is the dead load and EL is the seismic load. Gravity loads are applied as point loads at nodes. To account for inertia effects during dynamic analysis, masses are calculated in a manner consistent


18

with the gravity loading combinations and represented by a lumped 2D mass element. Table 3.4 summarises the total gravity loads and mass values for different storeys of the building.

Table 3.4 Calculated floor gravity loads and masses

Gravity load (kN) Group Structure

reference 1st floor Other Top floor TotalTotal mass

RH30 3199.96 3199.96 2900.04 38099.60 3883.8

RM30 3147.96 3147.96 2847.92 37475.48 3820.1

RM15 3058.24 3058.24 2758.24 36398.88 3710.4 1

RL15 3058.24 3058.24 2758.24 36398.88 3710.4

3.2.4 Case study two: The ICONS frame

The second case study is a four-storey bare frame known to have a soft storey at the third floor. The latter is attributed to a drastic change in strength and stiffness at this level through a reduction in both the reinforcement content and the section dimensions in the columns between the second and third storeys, coinciding also with the location of lap-splicing. Such characteristics are common in buildings designed predominantly for gravity-loading and the failure of a storey mid-way up a building has been observed in past devastating earthquakes such as the Kobe earthquake of 1995 (e.g. EERI, 1997).

The four-storey, three-bay reinforced concrete bare frame, Figure 3.5, was design and built at the European Laboratory for Structural Assessment (ELSA) of the Joint Research Centre (JRC) at Ispra, Italy. The full-scale model was constructed for pseudo-dynamic testing, under the auspices of the EU-funded ECOEST/ICONS programme. The frame was designed essentially for gravity loads and a nominal lateral load of 8% of its weight. The reinforcement details attempted to mirror the construction practices used in southern European countries in the 1950’s and 1960’s.

The columns were non ductile, smooth reinforcing bars were used, capacity design principles were ignored and lap splicing occurred in critical regions, Figure 3.6. Detailed information about the frame, testing and set up of the experimental rig can be found at Carvalho et al. [1999] and Pinho and Elnashai [2000].


19

Figure 3.5 Elevation and plan views of the frame, after Carvalho et al. [1999]

Figure 3.6 Reinforcement detail of the columns, after Carvalho et al. [1999]

The materials considered at the design phase were a low strength concrete class C16/20 (CEN, 1991) and smooth reinforcement steel class Fe B22k (Italian standards). The latter refers to smooth bars with a yield stress of 235 MPa and ultimate strength of 365 MPa.

The vertical loads considered in the design consisted of the self-weight of the slab and transverse beams, finishes, infill walls and the quasi-permanent static load. Figure 3.7 shows the scheme of vertical loads applied to the structure.


20

Figure 3.7 Scheme of vertical loads for nonlinear analysis, after Carvalho et al. [1999]

In total, 112 inelastic frame elements, capable of representing progressive cracking and spread of inelasticity, are used to model the bare frame. All member sections are represented explicitly and, for the purpose of strain/stress evaluation, are subdivided into a number of fibres (200-300) varying according to the section size. The length of the elements varies according to their location, with smaller elements being used in the vicinity of beam-column connections where large levels of inelastic deformation are expected.

All connections are assumed rigid and fully-fixed boundary conditions are adopted at the base of the building. Effective slab widths of 1.0 m and 0.65 m were adopted for the long and short spans, respectively, following the formulae presented in Eurocode 2 [CEN, 1991].

Vertical loads and masses are applied at each beam node and at the beam-column joints mirroring the load and mass distribution presented previously in Figure 3.7.

Concrete is represented by a uniaxial constant confinement model (Mander et al., 1988) and is calibrated using the concrete characteristics values obtained during testing. As for the reinforcement bars, the Menegotto-Pinto [1973] steel model, with an isotropic hardening constitutive relationship (Filippou et al. 1983), was adopted.

3.3 NONLINEAR DYNAMIC ANALYSIS

3.3.1 Case study one: RM15

Four input time-histories, consisting of one-artificially generated accelerogram [Campos-Costa and Pinto, 1999] and three natural records (Loma Prieta earthquake, USA, 1989), were employed for the dynamic analysis of the study carried out by Antoniou [2002]. The selection of these four records aimed at guaranteeing that all models would be subjected to a wide-ranging type of earthquake action, in terms of frequency content, peak ground acceleration, duration and number of high amplitude cycles. Indeed, the original PGA of the records varies between 0.12g and 0.93g, the spectral shapes are markedly distinct and provide high amplifications at different periods, Figure 3.8, and the ratio between the significant duration (defined as the interval between the build up of 5% and 95% of the total Arias Intensity [Bommer and Martinez-Pereira, 1999]) and the total duration ranges from 22% to 72%. The characteristics of the records are summarised in Table 3.5, while their elastic response spectra, for an equivalent viscous damping of 5%, are shown in Figure 3.8.


21

Table 3.5 Characteristics of the records employed by Antoniou and Pinho [2004a]

Record PGA (g)

PRA (g)

5% AI Threshold

95% AI Threshold

Total Duration

ttotal

Significant Duration

teff

teff/ttotal

A975 0.30 1.28 2.32 s 12.75 s 15.0 s 10.43 s 69.5% Emeryville 0.25 0.90 11.23 s 20.16 s 40.0 s 8.93 s 22.3% Hollister 0.12 0.50 1.02 s 9.52 s 33.2 s 8.50 s 25.6%

Loma 0.93 4.25 1.44 s 8.68 s 10.0 s 7.24 s 72.4%

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

Period (s)

Acc

eler

atio

n (g

)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

Period (s)

Acc

eler

atio

n (g

)

(a) Record A975 (b) Record Emeryville

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4

Period (s)

Acc

eler

atio

n (g

)

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4

Period (s)

Acc

eler

atio

n (g

)

(c) Record Hollister (d) Record Loma

Figure 3.8 Elastic response spectra of the four records (5% equivalent viscous damping).

3.3.2 Case study two: The ICONS frame

3.3.2.1 Earthquake input A collection of artificial records was available for use in the pseudo-dynamic experimental programme. The records were derived following a probabilistic seismic hazard analysis carried out at JRC (Campos-Costa and Pinto, 1999). For the purpose of the experimental programme, the severity class “Moderate-High”, typical of Southern European countries, was chosen. A set of hazard-consistent time histories was artificially generated to fit the uniform risk spectra (URS) for return periods of 100, 475, 975 and 2000 years.

In Figure 3.9, the acceleration and displacement elastic response spectra for all accelerograms, computed for an equivalent viscous damping of 5%, are shown. These indicate peak acceleration response for periods of vibration of up to 0.5 seconds, with values ranging from 0.3g to 1.1g. Figure 3.10 shows the artificial Acc-475 (475 years return period) and Acc-975 (975 years return period) accelerograms; only these two records will be considered in the nonlinear dynamic and pushover


22

analyses carried out herein. As is common with artificial records, a wide range of frequencies is present in the accelerograms.

0 1 2 3 4

Period [s]

0.00

0.20

0.40

0.60

0.80

1.00

1.20A

ccel

erat

ion

[g]

2000 years 975 years 475 years 100 years

0 1 2 3 4

Period [s]

0

50

100

150

200

250

300

Dis

plac

emen

t [m

m]

2000 years 975 years 475 years 100 years

(a) (b)

Figure 3.9 Respose spectra of input motion: (a) acceleration; (b) displacement.

Time (s)

0 2 4 6 8 10 12 14

Acc

eler

atio

n (g

)

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

ACC-475

Time (s)

0 2 4 6 8 10 12 14

Acce

lera

tion

(g)

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

ACC-975

(a) (b)

Figure 3.10 Artificial acceleration time histories for (a) 475 year (Acc-475) and (b) 975 year (Acc-975) return period

3.3.2.2 Comparison of experimental and analytical results The bare frame specimen of Figure 3.5 was subjected to a pseudo-dynamic test using the Acc-475 input motion and subsequently to a second test carried out with the Acc-975 input motion. The results of the tests showed that the deformation demand concentrated in the 3rd storey for the Acc-475 test and that collapse of the 3rd storey was almost reached for the Acc-975 test (Pinho and Elnashai, 2000).

Nonlinear dynamic analyses of the model described in Section 3.2.4 have been carried out for the two records by placing them consecutively in order to reproduce the testing sequence and better predict the behaviour of the frame. The top frame displacements, obtained for the experimental test, for the Acc-475 record and the Acc-975 record are shown in Figure 3.11a and Figure 3.11b, respectively. In addition, the top displacement response obtained by numerical simulation of each test has been superimposed. It may be concluded that the analytical prediction agrees with the results obtained through testing.

Inter-storey drift is a crucial parameter in terms of structural response since it is closely related to the damage sustained by buildings during seismic action. The soft-storey in the test case frame can only be identified through observation of the inter-storey drift profile. Figure 3.12 shows the drift profiles at the peak displacement from both the analytical and experimental time-histories; it is clear that the analytical model is able to predict the soft-storey at the third floor. Further refinement of the analytical model could perhaps produce a closer match between the analytical and experimental drift profiles but this is not within the scope of the present exercise. It suffices that the nonlinear fibre


23

analysis can predict the soft-storey at the third floor and thus this will be the reference to which all other analytical analyses will be compared. Note that no post-test calibration has been carried out. The structure was modelled as it is. The differences between Analysis versus experiment are likely to be due to the fact the third storey developed a shear failure mechanism, not yet incorporated in the program used.

TOP LEVEL DISPLACEMENTS

-100

-50

0

50

100

0 2 4 6 8 10 12 14

TIME (S)

DIS

PLA

CE

MEN

TS (m

m)

EXPERIMENTAL ANALYTICAL

TOP LEVEL DISPLACEMENTS

-150

-100

-50

0

50

100

150

0 2 4 6 8 10 12 14

TIME (S)

DIS

PLA

CE

MEN

TS (m

m)

EXPERIMENTAL ANALYTICAL (a) (b)

Figure 3.11 Analytical and experimental top frame displacement: (a ) Acc-475; (b) Acc-975

0

1

2

3

4

0 0.2 0.4 0.6 0.8

Drift (%)

Stor

ey

Analytical

Experimental

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Drift (%)

Stor

ey

Analytical

Experimental

(a) (b)

Figure 3.12 Analytical and experimental drift profiles: (a ) Acc-475; (b) Acc-975

There are two drift profiles, which may be obtained from the dynamic analysis: the profiles corresponding to the time step when the maximum inter-storey drift at any level is reached and the envelope of maximum drift in all storeys. The former profile corresponds to a real state of the structure, structural equilibrium is thus conserved and so comparison of this profile with a pushover would seem more logical. The latter envelope is a combination of drifts occurring at different time steps and so is not a state within which the structure ever exists, however, it is more useful from a design or assessment point of view and so it would be desirable to obtain a pushover that could capture such a profile giving the maximum drift at all storeys.

Chapter 4: Description and assessment of pushover methodologies

24

4. DESCRIPTION AND ASSESSMENT OF PUSHOVER METHODOLOGIES

4.1 INTRODUCTION

In the following sections different pushover approaches, ranging from traditional to adaptive, will be presented. At the same time, the performance of selected pushover methodologies will be assessed. The evaluation will be performed by using the RM15 and the ICONS frames. Additionally, both case studies will be employed to examine the performance of the direct vectorial addition, DVA, when applied to the adaptive pushover method followed by Antoniou and Pinho.

4.2 MEASURING THE ACCURACY OF THE PUSHOVER TECHNIQUES

Note that, as showed by Antoniou and Pinho [2004a], it is the drift profile, rather than the so-called pushover curve, that should be employed in comparison analyses. Because structural damage is directly related to local deformations, rather than to forces, the interstorey drift has been selected as the parameter to compare the different analyses. The inter-storey drift profiles obtained from each pushover analysis are to be compared to the drift profiles from the nonlinear dynamic analysis and the standard error of the pushover results, with respect to the dynamic, calculated.

The standard error of the profile is found using the following formula:

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛∆

∆−∆=

n

1i iD

iPiDn1100Error(%) (4.1)

where, ∆iD is the inter-storey drift at a given level i from the dynamic analysis, ∆iP is the corresponding inter-storey drift from the pushover analysis and n is the number of storeys. The closer the standard error is to zero, the closer the pushover response is to the nonlinear dynamic response.

For the case of RM15, the inter-storey drift profiles obtained from each pushover analysis are to be compared to the drift profiles from the nonlinear dynamic analysis and the standard error of the pushover results, with respect to the dynamic, calculated. The envelope of maximum drift in all storeys will be the interstorey drift profile to be employed. This envelope is a combination of drifts occurring at different time steps and so is not a state within which the structure ever exists; however, it is more useful from a design or assessment point of view. The profile has been obtained at 2.5% total drift since at this percentage of global drift the structure is already deep within the inelastic range. The ground motion record employed to perform the dynamic analysis is Hollister. The selected record was deliberately chosen as that for which all the force-based variants, adaptive or fixed, failed to provide acceptable response predictions. Due to the effects of higher modes, highly amplified by the selected ground motion, the dynamic drifts tend to increase from the structure’s mid-height upwards, a feature that all force-based static techniques struggle to reproduce


25

Additionally, for the ICONS frame case, the results of selected pushover analyses presented herein will be compared with both of the profiles described in section 3.3.2. For the first profile mentioned previously, the drift at each storey will be found from the pushover analysis at a roof displacement level corresponding to the roof displacement at the time step when the maximum inter-storey drift in the dynamic analysis is reached. For the second type of drift profile, the drift at each storey will be obtained from the pushover analysis at a roof displacement corresponding to the maximum roof displacement obtained during the dynamic analysis.

4.3 NON-ADAPTIVE NON-MODAL PUSHOVER ANALYSIS

The current state of the practice e.g. FEMA 356 (ASCE, 2000) requires a pushover curve to be computed by applying monotonically increasing lateral loads to the structure with an invariant vertical profile. The shape of this profile might follow a triangular or uniform distribution. Such pushover analyses have been carried out on the RM15 frame and are compared with the drift profile described previously, Figure 4.1. Immediately, it is apparent that both the triangular and uniform pushover analyses fail to estimate dynamic drift profile. The drift at the lower storeys are largely overestimated while the upper ones are underestimated. The standard errors are presented in Table 4.1 wherein it may be noted that the triangular pushover analysis produces lower standard errors, however, they are still of an unacceptable magnitude.

2.50%

0

2

4

6

8

10

12

0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00%

INTERSTOREY DRIFT

STO

REY

DYNAMIC

TRIANGULARUNIFORM

Figure 4.1 Comparison of drift profiles from triangular and uniform pushovers with the envelope of

maximum inter-storey drifts from dynamic analysis for the RM15 frame

Table 4.1 Standard error of the pushover schemes in Figure 4.1

Standard error

Uniform 51.07

Triangular 91.60

Considering the problems and limitations of constant force distributions, the use of enhanced adaptive patterns, which are updated at every step to account for the progressive stiffness degradation that occurs during the actual earthquake loading, seems an attractive alternative. In this way, the changes in the modal characteristics, the period elongation and the different spectral amplifications can be considered.


26

4.4 NON-ADAPTIVE MODAL PUSHOVER ANALYSIS

In an attempt to consider higher mode effects, Paret et al. [1996] and Sasaki et al. [1998] suggested the simple, yet efficient, multi-mode pushover procedure (MMP) illustrated in Figure 4.2. This method comprises several pushover analyses under forcing vectors representing the various modes deemed to be excited in the dynamic response.

The individual pushover curves are converted to the Acceleration-Displacement Response Spectrum (ADRS) format and the Capacity Spectrum Method is utilised to compare the structural capacity with the earthquake demand. In this way, it becomes apparent which mode is more critical and where damage is likely to occur. The procedure is intuitive, and does indeed provide qualitative information and identify potential problems due to higher modes that conventional single mode pushover analysis fails to highlight. However, the effects of higher modes cannot be easily quantified, since the method does not provide estimation of the response.

Figure 4.2 The multimodal pushover procedure [after Paret et.al., 1996]

A refinement of the multi-mode pushover procedure is the PRC method (Pushover Results Combination), which was proposed by Moghadam [2002]. According to this method, the maximum seismic response is again estimated by combining the results of several pushover analyses, which are carried out using load patterns that match the modal shapes of a predefined number of modes. The final structural response is calculated form the pushover analyses results, according to the following combination rule:

∑=

⋅=N

1iii RR β (4.2)

where R is the final estimation of the response at a particular location, βi is the modal participation factor of mode i and Ri is the value of the corresponding response from the pushover analysis that utilises a load pattern with the shape of mode i. Usually, the first three or four modes are considered.

A similar procedure, also based on MMP, is the MPA (Modal Pushover Analysis) method suggested by Chopra and Goel [2002]. The procedure consists of the following steps:

1. Perform eigenvalue analyses to compute the natural frequencies and modes φn.

2. For a selected number of modes, derive the top displacement vs. base shear curve, employing force distributions sn

* according to sn*=m.φn, where m is the mass matrix of the structure.


27

3. Idealise the pushover curve as bilinear curve (Figure 4.3)

4. Convert the idealized curves to the Fsn/Ln – Dn relation; where,

*nn

sny

M

V

LF bny= , and

rnn

rnyny

uD

ϕΓ= .

Mn* = LnΓn is the effective modal mass

∑

∑

=

==ΓN

1k

2knk

N

1kknk

nm

m

ϕ

ϕ, and ∑

==

N

1kknkn mL ϕ

5. Compute the peak deformation of the nth inelastic SDOF system through response history or spectral analysis. The elastic period of vibration of the nth system is:

sny

nynn F

DL2T π=

6. Calculate the peak roof displacement associated to the nth mode.

7. From the pushover analysis extract values of the desired response rn (floor displacements, storey drifts, plastic hinge rotations, etc.) at the peak roof displacement.

8. Determine the total demand by combining the peak modal response using the SRSS rule. Use as many modes as required for sufficient accuracy. Typically, two or three modes are enough, according to the authors.

Figure 4.3 Properties of the nth mode inelastic SDOF system derived from the corresponding pushover

curve [after Chopra and Goel, 2001]

The researches further stated that when compared to inelastic dynamic analysis the MPA gave rather good estimates for global response parameters, such as interstorey drifts or floor displacements. The results were superior to the results of the pushover analysis with the fixed force distributions suggested by FEMA, which greatly underestimated drift demands and led to unacceptable large errors. However, it was found inadequate to compute local response quantities, such as plastic rotations, although this has been a characteristic problem of all the pushover procedures employed and not specifically of the suggested multi-modal variation.


28

It has been shown in the previous section the failure of traditional pushover analysis in predicting the dynamic drift profile of the RM15. Such outcome calls for an analysis technique which is an improvement to the non-adaptive non-modal pushover analysis, using a triangular or uniform distribution of forces. Note that the Modal Pushover Analysis (MPA) is actually a complete procedure, incorporating already estimation of the seismic demand. It is certainly rigorous and the relationship between ductility and damping has been thoroughly dealt with; thus, it is gaining attention from the seismic engineering community. Since it is a such well developed method seems to be a reasonable choice, among the non-adaptive modal pushover analyses, to be employed to explore the improvement that this analysis might have over conventional pushover methods. Thus, the drift profiles computed by MPA (with the first three modes considered) are compared to those from nonlinear dynamic analysis Figure 4.4 and Table 4.2 presents the standard error for the aforementioned analysis.

The drift profile obtained from MPA predicts very well the dynamic drift shape for the lower storeys; however, completely underestimates the drifts at the top storeys failing to capture the largest drifts that are developed in the frame. The standard error is smaller than those from the uniform and triangular pushovers; therefore, a relative improvement for this particular case study has obtained. However, it is stressed that only a single frame has been used herein and so general conclusions on the accuracy of any single method of pushover analysis should not be made. Note, also, that the current application follows the author’s interpretation of the MPA with the possibility of miscalculation not being excluded.

2.50%

0

2

4

6

8

10

12

0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00%

INTERSTOREY DRIFT

STO

REY

MPADYNAMICTRIANGULARUNIFORM

Figure 4.4 Comparison of drift profiles from MPA with the envelope of maximum inter-storey drifts from dynamic analysis for RM15 frame

Table 4.2 Standard error of the pushover schemes in Figure 4.4

Standard error

MPA 36.21

Triangular 51.07

Uniform 91.60


29

Overall, the above multi-modal procedures are a significant improvement of conventional pushover analysis. They are theoretically more robust and conceptually more attractive, since they explicitly consider the response of mode than one mode and the expected ground motion, and thus yield results that are closer to rigorous inelastic time history analysis. In particular MPA has been extensively developed and verified, and provides already a full approach for assessment of structures, since procedures to estimate demand are included. However, the need to carry out three or four different analyses, as well as calculations to attain satisfactory accuracy may be a significant drawback that undermines the main objective of pushover analysis, which is to be simple and efficient replacement of detailed dynamic analysis.

4.5 FORCE-BASED ADAPTIVE PUSHOVER PROCEDURES

When multiple modes are taken into account, it is customary, and rational, to combine the effects, rather than the applied forces. However, this is only valid when one consider elastic systems (e.g. elastic modal analysis and MPA), where the principle of superposition holds. For the case of nonlinear systems, the combination of the effects obtained from different pushover analyses, each of which corresponding to a given mode, leads to internal force which do not correspond to an equilibrium state. Nonetheless, Gupta and Kunnath [2000] proposed a methodology in which the applied load is constantly updated, depending on the instantaneous dynamic characteristics of the structure; in addition, a site-specific spectrum can be used to define the loading pattern. According to the method, eigenvalue analysis is carried out before each load increment, utilising the current structural stiffness state. The number of modes of interest that will be taken into account is predefined and the storey forces for each mode are estimated by:

)( jSWF aiijjij ϕΓ= (4.3)

where, i is the storey number, j is the mode number, N is the number of the modes considered in the analysis, Φij is the mass normalised mode shape value for the ith storey and the jth mode, Wi is the weight of the ith storey, Sa(j) is the spectral amplification of the jth mode and Γj is the modal participation factor for the jth mode computed by using

∑∑==

=ΓN

1k

2kjk

N

1kkjkj mm φφ

The modal base shears Vj are calculated and combined using SRSS to compute the building base shear V. The storey forces are then uniformly scaled using the ratio Sn between the predetermined base shear increment ∆V and the base shear calculated from the modal analysis V, as represented by equations 4.4 and 4.5.

jjnj VVVVSV ∆==' (4.4)

ijijnj FVVFSF ∆==' (4.5)

A static analysis is then carried out for each mode independently. The calculated action effects for each mode are combined with SRSS and added to corresponding values form the previous step. At the end of the step, the structural stiffness state is assessed, in order to be used in the eigenvalue analysis of the next step. The estimates of interstorey drifts and the sequence of the formation of local or global collapse mechanism presented in the paper were satisfactory. However, the utilisation of SRSS to combine the responses, derived for each mode, implies that equilibrium is not satisfied at the end of each step.


30

More conceptually sound is instead to carry out single nonlinear adaptive pushover analysis, whereby the modal combination is applied to the forces, rather than the responses. Along this line of thinking, Bracci et al. [1997] were the first to introduce such procedure that utilises fully adaptive patterns. The analysis starts by assuming an initial lateral load pattern {Fi}, usually triangular, whereas the additional loads {∆Fi} imposed in subsequent increments are calculated from the instantaneous base shear and storey resistances of the previous load step, according to the equation:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∆+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛−=∆ +

−

−+

kb

ki1k

b1kb

1ki

kb

kij1k

iV

FV

V

F

V

FVF (4.6)

where, i is the storey number, k is the increment number, Vb is the base shear and ∆Vb is the increment of the base shear.

The procedure was implemented in the dynamic analysis package IDARC [Valles et al., 1996] leading to the attainment of apparently promising results. Lefort [2000] implemented an extended version of this method, which employed an additional force scaling equation, equation 4.7, to account for higher mode contributions, obtaining, however, limited-accurate response predictions.

( )

( )old

ibN

1L

m

1j

2jjLL

m

1j

2jjii

i FV

W

W

F +∆⋅

Γ

Γ

=

∑ ∑

∑

= =⋅

=⋅

ϕ

ϕ (4.7)

where, i is the storey number, j is the mode number, Γj is the modal participation factor of mode j, Φij is the mass normalised mode shape value for the ith storey and the jth mode, m is the number of modes considering in the analysis, N is the number of storey in the building and ∆Vb is the increment of the base shear.

Further, Requena and Ayala [2000] discussed two variations of adaptive pushover (referred as approaches 2-A and 2-B) and compared them with a modal fixed pattern scheme (approach 1). According to approach 2-A the force applied to the different storey levels is derived by the combination of the contributions of the considered modes using SRSS and equals to:

( )∑

∑

∑

=

=

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⋅

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=N

1j

2

ijijN

1K

2kjk

N

1kkjk

i mSm

mF αφ

φ

φ (4.8)

where, økj is the modal shape for the kth storey and the jth mode and Sαj is the pseudo-spectral acceleration mode j. The alternative approach 2-B accepts the existence of an “equivalent” fundamental mode iφ , which is determined through a combination of vibration modes using the SRSS rule:

( )∑=

Γ=N

1j

2jiji φφ (4.9)

Γj is the participation factor of mode j defined as ∑∑==

=ΓN

1k

2kjk

N

1kkjkj mm φφ

The distribution of the lateral loads is defined according to:


31

∑=

=N

1kkk

iii

m

mF

φ

φ (4.10)

It is possible, when the structural stiffness changes, due to the formation of plastic hinges, to calculate a new distribution of the lateral loads that reflects the current state of inelasticity. The new load distribution may be used for the subsequent steps of the analysis, until the stiffness state of the structure changes again. These two alternative adaptive methods are appealing indeed. They are theoretically rigorous and they explicitly account for higher modes and spectral contributions. However, and unfortunately, the analytical results presented by the authors were only limited and the accuracy procedures could not be effectively assessed or judged.

A methodology conceptually identical to the 2-A method proposed by Requena and Ayala, was also suggested by Elnashai [2001]. The difference being that its implementation within a fibre analysis framework allowed for a continuous, rather than discrete, force distribution update to be carried out. The work of Elnashai, however, was confined to the application to simplified, not-necessarily realistic, “stick-models”. Antoniou and Pinho [2004a, 2004b] extended and verified the method to the case of actual reinforced concrete frames and implemented it in a finite element package that is freely available from the internet [SeismoSoft, 2004].

In the adaptive pushover approach followed by Antoniou and Pinho, the lateral load distribution is not kept constant but is continuously updated during the process, according to modal shapes and participation factors derived by eigenvalue analysis carried out at each analysis step. The method is multimodal and accounts for softening of the structure, its period elongation, and the modification of the inertial forces due to spectral amplification. Two variants of the method exist: Force-based adaptive pushover (FAP) and Displacement-based adaptive pushover (DAP).

The algorithm can be structured in four main stages; (i) definition of the nominal load vector and inertia mass, (ii) computation of the load factor, (iii) calculation of the normalised scaling vector and (iv) updating of the loading force vector. Whilst the first step is carried out only once, at the start of the analysis, its three remaining counterparts are repeated at every equilibrium stage of the nonlinear static analysis procedure, described as it follows.

The loading vector shape is automatically defined and updated by the solution algorithm at each analysis step, for which reason the nominal vector of forces, P0, must always feature a uniform (rectangular) distribution shape, in height, so as not to distort the load vector configuration determined in correspondence to the dynamic response characteristics of the structure at each analysis step. In addition, it is noteworthy that the adaptive pushover requires the inertia mass M of the structure to be modelled, so that the eigenvalue analysis, employed in updating the load vector shape, may be carried out.

The magnitude of the load vector P at any given analysis step is given by the product of its nominal counterpart P0,, defined above, and the load factor λ at that step (see Equation 2.15). The latter is automatically increased, by means of load control or response control strategy, until a predefined analysis target, or numerical failure, is reached.

P = λ.P0 (4.11)

The normalized modal scaling vector, F , used to determine the shape of the load vector (or load increment vector) at each step, is computed at the start of each load increment. In order for such scaling vector to reflect the actual stiffness state of the structure, as obtained at the end of the previous load increment, an eigenvalue analysis is carried out. To this end, the Lanczos algorithm [Hughes,


32

1987] is employed to determine the modal shape and participation factors of any given predefined number of modes, after which the total storey forces Fij can be determined as:

Fij = Γj Φij Mi (4.12)where, i is the storey number and j is the mode number, Γj is the modal participation factor for the jth mode, Φij is the mass normalized mode shape value for the ith storey and jth mode, and Mi is the mass of the ith storey.

As defined in the above equation, the lateral load pattern is spectrum independent and is only determined by the modal properties of the system. If a spectral shape is considered, Equation 4.12 becomes:

Fij = Γj Φij Mi Sa(j) (4.13)where, Sa(j) is the spectral amplification of the jth mode.

The lateral load profiles of each vibration mode are then combined by using either the Square Root of the Sum of the Squares (SRSS), if the modes can be assumed as fully uncoupled, the Complete Quadratic Combination (CQC) method, if cross-coupling of the modes and respective viscous damping is to be considered, or a weighted vectorial addition. Further, since only the relative values of storey forces (Fi) are of interest in the determination of the normalised modal scaling vector, F , which defines the shape, not the magnitude, of the load or load increment vector, the forces obtained from the modal combination rules, presented above, are normalized with respect to the total value

∑=

i

ii F

FF (4.14)

Once the normalised scaling vector and load factor have been determined, and knowing also the value of the initial nominal load vector, the loading vector Pt at a given analysis step t is obtained by adding to the load vector of the previous step, Pt-1 (existing balance loads), a newly derived load vector increment, computed as the product between the current load factor increment ∆λt, the current modal scaling vector F and the nominal vector P0:

Pi = Pt-1+∆ λt iF P0 (4.15)

The procedure, named as incremental updating by Antoniou and Pinho [2004a], is schematically represented in Figure 4.5.

∆Pt = ∆λt ×

normalised shape at step t

×

nominalload vector

=

new incrementof forces

Pt=Pt-1+∆Pt=

existing forces

+

new incrementof forces

=

new forces applied at step t

∑=1

Figure 4.5 Graphical representation of loading force vector calculation with incremental updating

In incremental updating, the tangent stiffness is used not to calculate the whole load P, but the incremental quantity ∆P to increase P of the previous analysis step. As in nonlinear time-history analysis, the analysis is run in incremental fashion, through piecewise linearization, as schematically shown in Figure 4.6. This approach can be viewed like “shifting” the origin of the system to the actual state of force and displacement, as a sort of pre-loaded state. Starting from that point, and from the


33

actual stiffness state of the system (tangent in “global coordinates”, but also secant in the “shifted coordinates”), the increment of load ∆P is calculated according to the spectrum scaling. The load on the structure is not dependent as whole on the actual stiffness state, but it is increased according to the actual stiffness state.

dk+∆dk

O 't

O 'k

dk dt+∆dtdt

Fk+∆Fk

Fk

Ft+∆FtFt K t

K k

O Figure 4.6 Graphical interpretation of incremental updating using tangent stiffness

It is noted how the computation of the response in terms of piece-wise linear increments renders fully valid the use of tangent stiffness to compute the vibration characteristics of the system, as well as spectral amplification.

Since it is a conceptually sound method, because the modal combination is applied to the loads rather than to responses, exhibits a displacement-based variant and is implemented in a freely available finite element package, the adaptive pushover adopted by Antoniou and Pinho seems to be a good selection to be applied to the example case. The force-based adaptive pushover analysis has been employed in the present section. The loads from all modes have been combined by using the SRSS rule. Analysis with spectral amplification has been undertaken by employing the elastic response spectra for the Hollister record, record used in the dynamic nonlinear analysis. The results of the analysis with FAP are presented in Figure 4.7 and the standard errors are shown in Table 4.3.

2.50%

0

2

4

6

8

10

12

0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00%

INTERSTOREY DRIFT

STO

REY

FAP-SRSSDYNAMICTRIANGULARUNIFORM

Figure 4.7 Comparison of drift profiles from FAP (SRSS) with the envelope of maximum inter-storey

drifts from dynamic analysis for RM15


34

Table 4.3 Standard error of pushover schemes shown in Figure 4.7

Standard error

FAP-SRSS 93.32

Triangular 51.07

Uniform 91.60

The results obtained through FAP appear to be similar to those from conventional pushover analysis, and the standard errors presented in Table 4.3 are comparable. The reason for the disappointing results could be the use of the SRSS rule, which always leads to a positive load profile that pushes the structure. A possible solution to this problem is explored in section 4.7.

4.6 DISPLACEMENT-BASED ADAPTIVE PUSHOVER PROCEDURES

In non-adaptive pushover, practical problems arise when one pushes the structure with a fixed pattern of displacements: the relative displacement between consecutive floor levels is fixed, thus structural characteristics such as strength irregularities and soft-storeys are concealed and the results are misleading. Thus, if displacement loading is to be employed in pushover runs, a method to realistically update the deformation patterns, as the stiffness of the structure changes, is required The employment of displacements rather than forces is conceptually more desirable and follows the present drive towards deformation-based design and assessment methods. Hence, an innovative algorithm has been proposed that will allow displacement-based pushover analysis to be carried out without the aforementioned shortcomings [Antoniou and Pinho, 2004b].

The implementation of the proposed displacement-based algorithm can be structured in four main stages; (i) definition of the nominal load vector and inertia mass, (ii) computation of the load factor, (iii) calculation of the normalised scaling vector and (iv) updating of the loading displacement vector. Whilst the first step is carried out only once, at the start of the analysis, its three remaining counterparts are repeated at every equilibrium stage of the nonlinear static analysis procedure, as described in the following subsections.

Steps (i) and (ii) above are identical to those described for the Force-based adaptive pushover algorithm (FAP), with the only difference consisting in the fact that the load vector now consist of displacements (U), rather than forces (P). In general terms, the former is obtained, at each step of analysis, through Equation 2.20, where λ represents the load factor, determined by means of load control algorithm and U0 is the nominal vector.

U = λ.U0 (4.16)It is noted that in DAP, applied loads and response deformations can be considered as effectively coincident, since both consist of displacement. Hence, the employment of the slightly more elaborate response control algorithm (see Antoniou and Pinho, 2004) is not required here, in view of the fact that load and response control types lead to the attainment of equal results. For this reason, the simpler load control algorithm (see Antoniou and Pinho, 2004) is used.

The normalized modal scaling vector, D , used to determine the shape of the load vector (or load increment vector) at each step, is computed at the start of each load increment. In order for such scaling vector to reflect the actual stiffness state of the structure, as obtained at the end of the previous load increment, an eigenvalue analysis is carried out. To this end, the Lanczos algorithm [Hughes,


35

1987] is employed to determine the modal shape and participation factors of any given predefined number of modes. Modal loads can be combined by using either the Square Root of the Sum of the Squares (SRSS) or the Complete Quadratic Combination (CQC) method. The procedure for the computation of the scaling displacement vector, D, through displacement- or drift-based approach is described next.

Displacement-based scaling refers to the case whereby storey displacement patterns Di are obtained directly from the eigenvalue vectors, as described in Equation 4.17, where i is the storey number and j is the mode number, Γj is the modal participation factor of the jth mode, Φij is the mass normalized mode shape value for the ith storey and the jth mode, and n stands for the total number of modes. The first approach is analogous to the force-based adaptive procedure, making use of the maximum storey displacements calculated directly by modal analysis to determine the scaling displacement vector:

( )∑∑==

Γ==n

1j

2ijj

n

1j

2iji DD φ (4.17)

The maximum displacement of a particular floor level, however, being essentially the relative displacement between that floor and the ground, provides insufficient insight into the actual level of damage incurred by buildings subject to earthquake loading. On the contrary, inter-storey drifts feature a much clearer and direct relationship to horizontal deformation demand on buildings. Hence, an alternative scaling scheme, whereby maximum inter-storey drift values obtained directly from modal analysis, rather than from the difference between not-necessarily simultaneous maximum floor displacement values, are used to compute the scaling displacement vector, has also been developed.

In such an interstorey drift-based scaling technique, the eigenvalue vectors are thus employed to determine the inter-storey drifts for each mode ∆ij, as shown in Equation 4.18, while the displacement pattern Di at the ith storey is obtained through the summation of the modal-combined inter-storey drifts of the storeys below that level, i.e. drifts ∆1 to ∆i:

∑=

∆=i

1kkiD with ( )[ ]∑∑

=−

=−Γ=∆=∆

n

1j

2j1ijij

n

1j

2iji ,, φφ (4.18)

It is nonetheless noted that, although the summation of modal inter-storey drifts constitutes an improvement over direct combination of displacements, it is still approximate, since it assumes that all drift maxima at different storeys occur at the same time, which is of course unrealistic. Work carried out by Antoniou and Pinho [2004b] indicates that interstorey drift scaling is better; thus, it is the scaling technique that have been adopted.

Since only the relative values of storey displacements (Di) are of interest in the determination of the normalised modal scaling vector, D , which defines the shape, not the magnitude, of the load or load increment vector, the displacements obtained by Equation 4.18 are normalised so that the maximum displacement remains proportional to the load factor, as required within a load control framework:

i

ii D

DDmax

= (4.19)

Once the normalised scaling vector, tD , and load factor Γt or load factor increment ∆Γt have been determined, and knowing also the value of the initial nominal load vector U0, the loading vector Ut at a given analysis step t can be updated using the alternative previously introduced, incremental updating, as give in Equation 4.20:

Ut = Ut-1+∆ λt tD U0 (4.20)


36

Because of the reasons exposed in the previous section, the Displacement-based variant of the adaptive pushover method adopted by Antoniou and Pinho is applied to the example case. As it was for FAP, the loads from all modes have been combined using the SRSS rule. Spectral amplification has been utilized in the present example. As it was stated above the interstorey drift scaling has been adopted. The results DAP are presented in Figure 4.8 and the standard errors are shown in Table 4.4.

2.50%

0

2

4

6

8

10

12

0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00%

INTERSTOREY DRIFT

STO

REY

DAP-SRSSDYNAMICTRIANGULARUNIFORM

Figure 4.8 Comparison of drift profiles from DAP (SRSS) with the envelope of maximum inter-storey


Table 4.4 Standard error of pushover schemes shown in Figure 4.8

Standard error

DAP-SRSS 25.58

Triangular 51.07

Uniform 91.60

It is clearly observed that the displacement-based adaptive pushover did manage to provide much improved approximation of the highly irregular dynamic deformation profile envelope. In effect, even though the attainment of fully accurate predictions was not observed, DAP was capable of predicting the trend of the dynamic drift envelopes, a feature beyond the capabilities of force-based pushovers. At the same time, the standard error of DAP is the smallest obtained so far from the studied pushover methods. The reason for this satisfactory outcome may lies in that although SRSS is used to combine the contributions of each mode, thus implying permanently positive storey drift profiles, DAP drift profiles do feature changes of their respective gradient (i.e. the trend with which drift values change from one storey to the next), introduced by the contributions the higher modes. When such gradient variations imply a reduction of the drift of a given storey with respect to its two adjacent floor levels, then the corresponding applied storey horizontal force must also be reduced, in some cases to the extend of sign inversion.

4.7 POSSIBLE DRAWBACKS DUE TO THE USE OF QUADRATIC MODAL COMBINATION RULES

Analysis methods, which employ mode shapes to compute actions or responses, invariable they use SRSS and CQC modal rules to combine the contributions from different structural modes. Modal Pushover Analysis [Chopra and Goel, 2001] and the methodology proposed by Antoniou and Pinho


37

[2004a, 2004b] are examples where the modal combination is performed by using any of mentioned rules.

Two shortcomings of the modal combination rules can be pointed out: the first one is that the result obtained does not fulfil equilibrium; the second limitation is that signs are lost during the combination process eliminating the contribution of negative quantities. The latter, it is believed, played an important role during the research performed by López-Menjivar on 3D reinforced concrete buildings. Such problem has been previously identified by Priestley [2003] and by Antoniou and Pinho [2004a]. Therefore, an alternative modal combination is needed. The proposal developed in this research is to add the participation of each mode directly, in an algebraic fashion, without removing any sign; thus, the contribution from each mode is fully taken with no modification. Such procedure will be named, hereafter, Direct Vectorial Addition (DVA).

The procedure can be expressed in the following way. The modal storey loads are computing by using the approach proposed by Antoniou and Pinho [2004a, 2004b]:

jaiijjij SM ,φΓ=Ψ (4.21)

where, i is the storey number, j is the mode number, Ψij is the modal storey loads which either could be forces or displacements, Γj is the modal participation factor of mode j, øij is the mass normalised mode shape value for the ith storey and the jth mode, Mi is the mass of the ith storey, Sa,j is the response spectrum ordinate, either acceleration or displacement, corresponding to the period of vibration of the jth mode.

The lateral load profiles of each vibration mode are then combined using the Direct Vectorial Addition, DVA, in the following fashion:

∑=

Ψ=Ψn

1jijji η (4.22)

where, Ψi is the storey loads which either could be forces or displacements, ηj is the participation factor of each mode which could be either positive or negative. This participation factor, ηj, can take a value ranging from zero, no mode participation, to 1, full mode participation.

The idea of using modal addition, to compute responses of structures, is not strange to Earthquake Engineering field. Priestley [2003] sketched some ideas about vectorial addition by using a finite number of modes to obtain response quantities of wall buildings.

More recently, Kunnath [2004] proposed to perform various pushover analyses. Each pushover would have an invariant load distribution vector obtained from the addition of load patterns which have been computed from the deform shape of each individual mode. The contribution of each modal load pattern can be scaled and shift the sign as well. It is showed that by this approach, envelopes that contain the dynamic response of structures can be constructed. However, the author does not express a definite way or rule to combine the modal load vectors and points out the need for more research to achieve a unique set of modal factor adequate to match the dynamic response in any case.

A manner to test the suitability of the proposed method, The Direct Vectorial Addition (DVA), is to try it on the example case comparing results with both nonlinear time history analysis and non-adaptive pushovers. This process will be exposed and described next.

The direct vectorial addition of the first three modes will be applied to combine the loads/response, be they forces or displacements, in all the following analyses. For the force-based pushover a weighting of 100% to the first three modes will be applied (positive in the first and third modes and negative in


38

the second mode), whilst for the displacement-based pushover 100% weighting to the first mode, -10% to the second mode and 10% to the third mode will be considered.

The force-based adaptive algorithm has been applied to the RM15 frame with a vectorial addition of the modal loads at each adaptive step. The inter-storey drift profile obtained from FAP, with a vectorial sum of the modes, is compared to the drift profiles from nonlinear dynamic analysis and traditional pushovers in Figure 4.9. A clear improvement is observed when DVA is used in FAP since the adaptive pushover drift profile is now closer to the dynamic drift envelope than its non-adaptive counterparts. The enhancement is also noticed by comparing the standard errors presented in Table 4.5.

2.50%

0

2

4

6

8

10

12

0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00%

INTERSTOREY DRIFT

STO

REY

FAP-DVADYNAMICTRIANGULARUNIFORM

Figure 4.9 Comparison of drift profiles from FAP (DVA) with the envelope of maximum inter-storey


Table 4.5 Standard error of pushover schemes shown in Figure 4.7 and Figure 4.9

Standard error

FAP-DVA 38.00

FAP-SRSS 93.32

Triangular 51.07

Uniform 91.60

The displacement-based adaptive pushover has again been applied to the case study frame using the inter-storey drift scaling technique but this time with a vectorial addition of the modal loads, as opposed to using the SRSS rule. As was the case of FAP, DAP has managed to improve the prediction of the dynamic interstorey drift when is DVA is used. The dynamic profile envelope is closely matched by the drifts obtained from DAP, Figure 4.10. The superior performance, in predicting the dynamic drifts, by employing DAP is further reinforced by the standard error calculations presented in Table 4.6.


39

2.50%

0

2

4

6

8

10

12

0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00%

INTERSTOREY DRIFT

STO

REY

DAP-DVADYNAMICTRIANGULARUNIFORM

Figure 4.10 Comparison of drift profiles from DAP (DVA) with the envelope of maximum inter-storey

drifts from dynamic analysis

Table 4.6 Standard error of pushover schemes shown in Figure 4.8 and Figure 4.10

Standard error

DAP-DVA 14.81

DAP-SRSS 25.58

Triangular 51.07

Uniform 91.60

To further test the applicability of the DVA in predicting the response obtained from nonlinear time history analysis the new methodology will be applied and assessed by using the ICONS frame. The idea to use the ICONS frame is to examine the performance of DVA, when applied to the adaptive pushover method followed by Antoniou and Pinho, for a single case study building known to have structural characteristics that may not be reproduced using conventional pushover approaches.

4.8 THE ICONS FRAME

4.8.1 Non-adaptive non-modal pushover analysis

The triangular and uniform loading profile pushovers been carried out on the case study frame and are compared with the two types of dynamic drift profile, described previously, as presented in Figure 4.11 and Figure 4.12. Immediately it is apparent that both the triangular and uniform pushover analyses fail to estimate the soft-storey in the third floor of the frame. The drifts at the lower storeys are also largely overestimated with both force distributions. The standard errors are presented in Table 4.7 wherein it may be noted that the triangular pushover analysis produces lower standard errors, however, they are still of an unacceptable magnitude.


40

0

1

2

3

4

0 0.2 0.4 0.6 0.8

Drift (%)

Stor

eyDynamic

Triangular

Uniform

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Drift (%)

Stor

ey

Dynamic

Triangular

Uniform

(a) (b)

Figure 4.11 Comparison of drift profiles from triangular and uniform pushovers with the drift profile at the time step of maximum inter-storey drift from dynamic analysis using the a) Acc-475 record and b)

Acc-975 record

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1

Drift (%)

Stor

ey

Dynamic

Triangular

Uniform

0

1

2

3

4

0 0.5 1 1.5 2

Drift (%)

Stor

eyDynamic

Triangular

Uniform

(a) (b)

Figure 4.12 Comparison of drift profiles from triangular and uniform pushovers with the envelope of maximum inter-storey drifts from dynamic analysis using the a) Acc-475 record and b) Acc-975 record

Table 4.7 Standard error of the pushover schemes in Figure 4.11 and Figure 4.12

Profile at time step of maximum inter-storey

drift

Envelope of max. inter-storey drifts

Acc-475 Acc-975 Acc-475 Acc-975

Uniform 50 % 52 % 36 % 47 %

Triangular 23 % 30 % 29 % 25 %

4.8.2 Pushover analysis using DVA

The direct vectorial addition of the first three modes will be applied to combine the loads/response, be they forces or displacements, in all the following analyses. For the force-based pushovers a weighting of 100% to the first three modes will be applied (positive in the first mode and negative in the higher modes), whilst for the displacement-based pushovers 90% weighting to the first mode, -10% to the second mode and -2.5% to the third mode will be considered.

The force-based adaptive pushover (FAP) has been applied to the case study frame with a vectorial addition of the modal loads at each adaptive step. The inter-storey drift profile obtained from FAP, with a vectorial sum of the modes, is compared to the drift profiles from nonlinear dynamic analysis in Figure 4.13 and Figure 4.14. It can be observed that the agreement between the profiles is very close both in both shape and magnitude. The force-based adaptive pushover, with direct modal combination, manages to capture the soft-storey mechanism in the third floor for both records, and


41

matches closely the drift levels in all other storeys. The superior behaviour predicted by FAP is further reinforced by the standard error calculations presented in Table 4.8. An improvement to the non-adaptive pushover results is observed through the consideration of the changing stiffness characteristics of the structure with adaptive pushover.

0

1

2

3

4

0 0.2 0.4 0.6 0.8

Drift (%)

Stor

ey

Dynamic

Adaptive

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Drift (%)

Stor

ey

Dynamic

Adaptive

(a) (b)

Figure 4.13 Comparison of drift profiles from FAP with the drift profile at the time step of maximum inter-storey drift from dynamic analysis using the a) Acc-475 record and b) Acc-975 record

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1

Drift (%)

Stor

ey

Dynamic

Adaptive

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Drift (%)

Stor

ey

Dynamic

Adaptive

(a) (b)

Figure 4.14 Comparison of drift profiles from FAP with the envelope of maximum inter-storey drifts from dynamic analysis using the a) Acc-475 record and b) Acc-975 record

Table 4.8 Standard errors of the profiles shown in Figure 4.13 to Figure 4.14

Profile at time step of maximum inter-storey drift


Acc- 475 Acc-975 Acc-475 Acc-975

Triangular 23 % 30 % 29 % 25 %

Uniform 50 % 52 % 36 % 47 %

FAP with DVA 11 % 10 % 14 % 6 %

Subsequently, the displacement-based adaptive pushover (DAP) has been applied to the case study frame using the inter-storey drift scaling (IDAP) technique with a vectorial addition of the modal loads as the modal combination rule.

Figure 4.15 show the inter-storey drift profiles from DAP as well as those from dynamic analysis at the time step of maximum inter-storey drift. The adaptive pushovers have again managed to predict the drift shape of the dynamic analysis for both records. The soft-storey is anticipated with DAP the drift ratios in all other floors are closely predicted. The comparison of the results obtained by DAP with the maximum envelope of inter-storey drift from dynamic analysis is presented in Figure 4.16.


42

The standard errors are presented in Table 4.9 for DAP. A significant improvement to the results is noted by the use of vectorial addition of the modes rather than the SRSS rule. By referring back to Table 4.8, it can be observed that slightly higher errors have been obtained with displacement-based adaptive pushover than force-based adaptive pushover but they are now much more comparable.

0

1

2

3

4

0 0.2 0.4 0.6 0.8

Drift (%)

Stor

ey

Dynamic

Adaptive

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Drift (%)

Stor

ey

Dynamic

Adaptive

(a) (b)

Figure 4.15 Comparison of drift profiles from DAP with the drift profile at the time step of maximum inter-storey drift from dynamic analysis using the a) Acc-475 record and b) Acc-975 record

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1

Drift (%)

Stor

ey

Dynamic

Adaptive

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Drift (%)

Stor

ey

Dynamic

Adaptive

(a) (b)

Figure 4.16 Comparison of drift profiles from DAP with the envelope of maximum inter-storey drifts from dynamic analysis using the a) Acc-475 record and b) Acc-975 record

Table 4.9 Standard errors of pushover schemes shown in Figure 4.15 and Figure 4.16

Profile at time step of maximum inter-storey drift


Acc- 475 Acc-975 Acc-475 Acc-975

Triangular 23 % 30 % 29 % 25 %

Uniform 50 % 52 % 36 % 47 %

DAP with DVA 20 % 8 % 18 % 8 %

It has been observed that all adaptive pushover variants with vectorial modal addition, as the modal combination employed, were able to predict the third floor soft-storey and produced much lower standard errors than the traditional pushover analyses. These conventional pushover methods failed to reproduce both the local and global dynamic response of the frame, falling short to predict the soft-storey at the third floor.

Finally, in order to conclude the present chapter, it must be highlighted that general conclusions on the accuracy of any method cannot be made from a couple of buildings and that this would require statistical studies to be carried out. Therefore, the next step of the research is to apply the newly proposed methodology to a group of buildings that display different characteristics such as height, number of storeys and bays subjected to several ground motions. The response obtained by using the


43

proposed method of modal combination will be compared to results obtained through a rigorous nonlinear time history analysis.

Chapter 5: Conclusions

44

5. CONCLUSIONS

5.1 SUMMARY

The main objectives of this research were (i) to discuss and assess different pushover methodologies and (ii) to explore the advantages that the use of an alternative modal combination rule could bring when applied to adaptive pushover methodology proposed by Antoniou and Pinho. The first aim was carried out by employing a typical seismically designed concrete frame structure in order to evaluate the results obtained by using different pushover approaches. The second objective has been attain by proposing a modal combination scheme, named Direct Vectorial Addition (DVA), which attempts to overcome the problem presented by the current quadratic modal rules, which is to change the sign of modal quantities involved turning them into positive values

The initial phase of this research reiterated that currently the most accurate procedure of demand prediction and performance evaluation of structures is the non-linear time history analysis. Nevertheless, the reliability of this method hinges on the unambiguous definition of certain indispensable parameters such as, a set of site-specific ground motions [Bommer et al., 2003], appropriate and accurate force-deformation relationships of various structural components and a robust mathematical tool capable of handling all analysis data and producing results within the time constrains of a design office. Consequently, a simpler yet reliable technique of structural analysis is desirable.

An alternative to the nonlinear time history analysis is the nonlinear static pushover analysis. The purpose of the pushover analysis is to assess the structural performance by estimating the strength and deformation capacities using static, nonlinear analysis and comparing these capacities with the demands at the corresponding performance levels. Although it provides crucial information on response parameters that cannot be obtained with conventional elastic methods (either static or dynamic), this method is not exempt from some limitations such as the inability to include higher mode order effects or progressive stiffness degradation. Therefore, the need for a procedure that overcomes the above deficiencies is readily noted.

A revision of the state-of-the-art pushover procedures was carried out, from which advantages and disadvantages of each approach were identified. At the same time, the accuracy of traditional pushover analyses, modal pushover analysis and adaptive pushover algorithm in predicting the seismic response characteristics of a non-seismically designed reinforced concrete frame model has been studied. Comparisons between drift profiles obtained through static (pushover) and dynamic nonlinear fibre analysis have been made and the standard error has been computed to test the accuracy of each method for a test case.

From the above comparative study, it was noted that the adaptive pushover methodology constitute a viable alternative to nonlinear dynamic analysis since solve the inherent deficiencies that conventional pushover analysis possesses. One methodology is particularly appealing which is the adaptive

Chapter 5: Conclusions

45

pushover approach proposed by Antoniou and Pinho [2004a, 2004b]. In this method, the lateral load distribution is not kept constant but is continuously updated during the process, according to modal shapes and participation factors derived by eigenvalue analysis carried out at each analysis step. The method is multimodal and accounts for softening of the structure, its period elongation, and the modification of the inertial forces due to spectral amplification. The above methodology has proved to give similar results to those obtained from nonlinear dynamic analysis, for 2D frame systems, especially when its displacement-based variant has been employed. Nevertheless, it was expressed that the methodology might have a drawback, due to the way the modal combination is carried out, to compute and update the load vector.

The adaptive pushover of Antoniou and Pinho uses the SRSS, or CQC, as modal combination rule. These rules transform the contribution of negative quantities turning them into positive values; therefore, this particular characteristic may mislead the response when flexible and tall buildings are analysed. The above fact affects mainly the force-based variant of the method for reasons already discussed (Section 4.5). Thus, it is logical to assess if an alternative method will lead to better results.

The Direct Vectorial Addition, DVA, was presented as an alternative to the usage of SRSS. The DVA directly combine the modes in algebraic fashion without changing the sign of the modal contributors. The proposed modal combination was tested by using the seismically designed reinforced concrete frame model, employed in the evaluation of different pushover schemes. In addition, the DVA was tested with a well-known case study, the ICONS frame where it was observed that the failure behaviour of such structure could be fully captured by using DVA.

5.2 FUTURE RESEARCH

It is highlighted that general conclusions on the accuracy of any of the methods presented in previous chapters cannot be made from two case studies and that this would require statistical studies to be carried out. Therefore, the next step of the research is to apply the newly proposed methodology to a group of buildings that display different characteristics such as height, number of storeys and bays subjected to several ground motions. The response obtained by using the proposed method of modal combination will be compared to results obtained through a rigorous nonlinear time history analysis. As it was done for the case of RM15 and ICONS frame, the similarity will actually be quantified by using the Standard error of each pushover respect to the Time history analysis. It is noted that each pushover result will be compared to Nonlinear Dynamic Analysis output obtained for each single accelerogram, as opposed to the statistical average of all dynamic cases. In doing it so, it is believed that a much more demanding and precise assessment of the static procedures is effectively carried out, since structural response peculiarities introduced by individual input motions are not smoothed out through results averaging.

Chapter 6: References

46

6. REFERENCES

ACI318 [1995]. “Building code requirements for structural concrete ACI318-95, and commentary ACI318R-95,” American Concrete Institure, Detroit, MI, USA.

Albanesi, T., Biondi, S. and Petrangeli, M. [2002]. “Pushover analysis: An energy based approach.” Proceedings of the 12th European Conference on Earthquake Engineering, Paper 605. Elsevier Science Ltd.

Alvarez-Botero J.C. and López-Menjivar M.A. [2004] “Assessment of the ‘Sede comunale Piazza Europa’ building, Comune di Vagli Sotto, Provincia di Luca, Regione Toscana (Technical Report),” EUCentre, Universitá degli Studi di Pavia. 49 pag.

American Concrete Institute (ACI) [1984]. “Earthquake effects on reinforced concrete structures,” Publication SP-84, US-Japan research, American Concrete Institute, Detroit, MI, USA.

Antoniou, S. [2001]. “Pushover analysis for seismic assessment of RC structures,” Transfer Report, Department of Civil Engineering, imperial College London, United Kingdom.

Antoniou, S. [2002]. “Advanced inelastic static analysis for seismic assessment of structures,” PhD Thesis, Imperial College London, University of London, United Kingdom.

Antoniou S., Rovithakis A. and Pinho R. [2002] “Development and verification of a fully adaptive pushover procedure,” Proceedings Twelfth European Conference on Earthquake Engineering, London, UK, Paper No. 822.

Antoniou, S. and Pinho, R. [2004a]. “Advantages and limitations of adaptive and non-adaptive force-based pushover procedures.” Journal of Earthquake Engineering, 8(4), pp. 497-522.

Antoniou, S. and Pinho, R [2004b]. “Development and verification of a displacement-based adaptive pushover procedure,” Journal of Earthquake Engineering, Vol. 8, No. 5, pp. 643-661.

ATC [1997]. NEHRP “Guidelines for the seismic rehabilitation of buildings.” FEMA Report No. 273, Federal Emergency Management Agency, Applied Technology Council, Washington D.C.

Aydinoğlu, M.N. [2003]. “An incremental response spectrum analysis procedure based on inelastic spectral displacements for multi-mode seismic performance evaluation,” Bulletin of Earthquake Engineering, No. 1, pp. 3-36.

Bommer, J.J. and Martinez-Pereira, A. [1999] “The effective duration of earthquake strong motion,” Journal of Earthquake Engineering, 3(2), pp 127-172.


47

Bommer, J.J., Acevedo, A.B. and Douglas, J. [2003]. “The selection and scaling of real earthquake accelerograms for use in seismic design and assessment,” Proceedings of the ACI International Conference on seismic bridge design and retrofit, American Concrete Institute. USA.

Bracci, J.M., Kunnath, S. K. and Reinhorn, A.M. [1997]. “Seismic performance and retrofit evaluation of reinforced concrete structures.” Journal of Structural Engineering, 123(1), pp. 3-10.

Campos-Costa, A. and Pinto A.V. [1999]. “European seismic hazards scenarios – An approach to the definition of input motion for testing and reliability assessment of Civil Engineering structures,” JRC Special publication No.X.99.XX, Joint Research Centre, Ispra, Italy.

Carvalho, E.C., Coelho, E. and Campos-Costa, A. [1999] “Preparation of the full-scale tests on reinforced concrete frames,” ICONS Report, Innovative Seismic Design Concepts for New and Existing Structures, European TMR Network, LNEC.

CEN [1991]. “European Prestandard ENV 1992-1-1: Eurocode 2 – Design of concrete structures, Part 1: General rules and rules for buildings”, Comite Europeen de Normalisation, Brussels, Belgium.

CEN [1995]. European Prestandard ENV1998-1-1: Eurocode 8 – Earthquake resistant design of structures, Part 1: General rules and rules for buildings. Comite Europeen de Normalisation, Brussels, Belgium.

Chopra, A.K. and Goel, R.K. [2001]. “A modal pushover analysis procedure to estimate seismic demands for buildings: Theory and preliminary evaluation.” PEER Report 2001/03, Pacific Earthquake Engineering Research Center, College of Engineering, University of California, Berkeley, January.

Chopra, A.K. and Goel, R.K. [2002]. “A modal pushover analysis procedure for estimating seismic demands for buildings.” Earthquake Engineering and Structural Dynamics, Vol. 31, pp. 561-582.

Earthquake Engineering Research Institute [1997] “The EERI Annotated Slide Collection”, EERI, Oakland.

Faella, G. and Kilar, V. [1998]. “Asymmetric multi-storey R/C frame structures: push-over versus nonlinear dynamic analysis.” Proceedings of the Eleventh European Conference on Earthquake Engineering, A.A., Ballkema, Rotterdam.

Fajfar, P. and Gasperic, P. [1996]. “The N2 method for the seismic damage analysis of RC buildings.” Earthquake Engineering and Structural Dynamics, Vol 25:31-46.

Fajfar, P. [1999]. “Seismic assessment and retrofit of RC structures.” Proceedings of the Eleventh European Conference on Earthquake Engineering – Invited Lectures, Paris, France, pp. 237-249.

Fardis, M.N. [1994] “Analysis and design of reinforced concrete buildings according to Eurocode 2 and 8. Configuration 3, 5 and 6,” Reports on Prenormative Research in Support of Eurocode 8.

Fardis, M.N. and Panagiotakos, T.B. [1997]. “Seismic design and response of bare and masonry-infilled reinforced concrete buildings. Part I: Bare structures,” Journal of Earthquake Engineering, Vol. 1, No. 1, pp 219-256.

Filippou F.C., Popov E.P. and Bertero V.V. [1983] "Modelling of R/C joints under cyclic excitations," Journal of Structural Engineering, Vol. 109, No. 11, pp. 2666-2684.

Gupta, B. and Kunnath, S.K. [2000]. “Adaptive spectra-based pushover procedure for seismic evaluation of structures.” Earthquake Spectra, 16(2), pp. 367-391.


48

Hughes, T.J.R. [1987]. The Finite Element Method, Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Inc.

Izzuddin, B.A. [2001] “Conceptual issues in geometrically nonlinear analysis of 3-D frame structures,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, pp. 1029-1053.

Kilar, V. and Fajfar, P. [1996]. “Simplified push-over analysis of building structure.” Proceedings of the Eleventh World Conference on Earthquake Engineering, Paper 1011, Pergamon, Elsevier Science Ltd, Acapulco, México.

Kilar, V. and Fajfar, P. [1997]. “Simple push-over analysis of asymmetric buildings.” Earthquake Engineering and Structural Dynamics, Vol. 26, pp.233-249.

Kilar, V. and Fajfar, P. [2001]. “On the applicability of pushover analysis to the seismic performance evaluation of asymmetric buildings” European Earthquake Engineering, XV(1), pp. 20-31, Jan-Apr.

Kilar, V. and Fajfar, P. [2001]. “On the applicability of pushover analysis to the seismic performance evaluation of asymmetric buildings,” European Earthquake Engineering, Vol.. XV, No. 1, pp. 20-31, Jan-Apr.

Krawinkler, H. and Seneviratna, G.D.P.K. [1998]. “Pros and cons of a pushover analysis of seismic performance evaluation.” Engineering Structures, 20(4-6), pp 452-464.

Kunnath, S.K. [2004]. “Identification of modal combination for nonlinear static analysis of building structures,” Computer-Aided Civil and Infrastructure Engineering, v. 19, pp 246-259.

Lawson, R.S., Vance, V. and Krawinkler, H. [1994]. “Nonlinear static push-over analysis - why, when, and how?.” Proceedings of the Fifth U.S. National Conference on Earthquake Engineering, Earthquake Engineering Research Institute, Oakland, California, I, Vol I:283-292.

Lefort, T. [2000]. “Advance pushover analysis of RC multi-storey buildings.” Engineering Seismology and Earthquake Engineering Section, Imperial College of Science, Tehnology and Medicine. Report No. 91/2, March 1991.

López-Menjivar, M.A. [2003] “3D Pushover of irregular reinforced concrete buildings,” Master’s Thesis. ROSE School, Universitá degli Studi di Pavia, pp. 72.

Mander J.B., Priestley M.J.N. and Park R. [1988] "Theoretical stress-strain model for confined concrete," Journal of Structural Engineering, Vol. 114, No. 8, pp. 1804-1826

Menegotto M. and Pinto P.E. [1973] "Method of analysis for cyclically loaded R.C. plane frames including changes in geometry and non-elastic behaviour of elements under combined normal force and bending," Symposium on the Resistance and Ultimate Deformability of Structures Acted on by Well Defined Repeated Loads, International Association for Bridge and Structural Engineering, Zurich, Switzerland, pp. 15-22.

Moghadam, A.S. and Tso, W.K. [1996]. “Damage assessment of eccentric multistory buildings using 3-D pushover analysis.” Proceedings of the Eleventh World Conference on Earthquake Engineering, Paper 997, Pergamon, Elsevier Science Ltd, Acapulco, México.

Moghadam, A.S. and Tso, W.K. [1998]. “Pushover analysis for asymmetrical multistory buildings.” Proceedings of the Sixth U.S. National Conference on Earthquake Engineering, Earthquake Engineering Research Institute, Oakland, California, 13 pag.


49

Moghadam, A.S. and Tso, W.K. [2000a]. “Pushover analysis for asymmetric and set-back multistory buildings.” Proceedings of the Twelfth World Conference on Earthquake Engineering, Paper 1093. New Zealand Society for Earthquake Engineering, Upper Hutt, New Zealand.

Moghadam, A.S. and Tso, W.K. [2000b]. “3-D pushover analysis for damage assessment of buildings.” Journal of Seismology and Earthquake Engineering, 2(3), pp.23-3.

Moghadam, A.S. and Tso, W.K. [2002]. “A pushover procedure for tall buildings.” Proceedings of the 12th European Conference on Earthquake Engineering, Paper 395. Elsevier Science Ltd.

Mwafy, A.M. and Elnashai, S.A. [2000]. “Static pushover versus dynamic-to-collapse analysis of RC buildings.” Engineering Seismology and Earthquake Engineering Section, Imperial College of Science, Technology and Medicine. Report No. 00/1, January.

Mwafy, A.M. [2001]. “Seismic performance of code-designed RC buildings,” PhD Thesis. Department of Civil and Environmental Engineering, Imperial College of Science, Technology and Medicine, London.

Paulay, T and Priestley, M.J.N. [1992]. “Seismic design of reinforced concrete and masonry buildings,” John Wiley and Sons, New York, 744 pag.

Paret, T.F., Sasaki, K.K., Elibeck, D.H. and Freeman, S.A. [1996]. “Approximate inelastic procedures to identify failure mechanism from higher mode effects” Proceedings of the Eleventh World Conference on Earthquake Engineering, Paper 966, Pergamon, Elsevier Science Ltd, Acapulco, México.

Penelis, G. and Kappos, A.J. [2002]. “3-D pushover analysis: The issue of torsion” Proceedings of the 12th European Conference on Earthquake Engineering, Paper 015. Elsevier Science Ltd.

Pinho, R. and Elnashai, A.S. [2000]. “Dynamic collapse testing of a full-scale four storey RC frame”, ISET Journal of Earthquake Technology, Paper No. 406; 37(4), pp. 143-164

Presidente del Consiglio dei Ministri [2003]. Norme Tecniche per il Progetto, la Valutazione e l'Adeguamento Sismico Degli Edifici, Ordinanza no. 3274.

Priestley, M.J.N. [1993]. “Myths and fallacies in earthquake engineering-conflicts between design and reality,” Bulletin of New Zealand National Society for Earthquake Engineering, Vol. 26, No. 3, pp. 329-341.

Pristley, M.J.N. [2003]. Myth and Fallacies in Earthquake Engineering, Revisited, IUSS Press, Pavia, Italy.

Requena, M. and Ayala, G. [2000]. “Evaluation of a simplified method for the determination of the non-linear seismic response of RC frames.” Proceedings of the Twelfth World Conference on Earthquake Engineering, Paper 2109. New Zealand Society for Earthquake Engineering, Upper Hutt, New Zealand.

Rovitahkis, A. [2001]. “Verification of adaptive pushover analysis procedures,” MSc. Dissertation, Department of Civil and Environmental Engineering, Imperial College London, United Kingdom.

Sasaki, K.K., Freeman, S.A. and Paret, T.F. [1998]. “Multi-mode pushover procedure (MMP) – a method to identify the effects of higher modes in a pushover analysis.” Proceedings of the Sixth U.S. National Conference on Earthquake Engineering, Earthquake Engineering Research Institute, Oakland, California, 12 pages.

SEAOC [1995]. “Performance based seismic engineering of buildings.” Vision 2000 Committee Structural Engineering Association of California, Sacramento, California.


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SeismoSoft [2004]. “SeismoStruct- A computer program for Static and Dynamic Non-linear Analysis of Framed Structures [online].” Available from URL: http://www.seismosoft.com

Valles, R.E., Reinhorn, A.M., Kunnath, S.K., Li, C. and Madan, A. [1996]. “IDARC2D Version 4.0: A computer program for the inelastic analysis of buildings.” Technical Report NCEER 96-0010, State University of New York at Buffalo, Buffalo, NY.

Documents

A Review of Existing Pushover Methods for 2-D RC Buildings