Evaluation of a multimode pushover procedure for asymmetric in plan buildings under biaxial seismic excitation

Abstract

In this paper a recently developed multimode pushover procedure for the approximate estimation of structural performance of asymmetric in plan buildings under biaxial seismic excitation is evaluated. Its main idea is that the seismic response of an asymmetric multi-degree-of-freedom system with $N$ degrees of freedom under biaxial excitation can be related to the responses of $N$ ‘modal’ equivalent single-degree-of-freedom (E-SDOF) systems under uniaxial excitation. The steps of the proposed methodology are quite similar to those of the well-known modal pushover analysis. However, the establishment of the (E-SDOF) systems is based on a new concept, in order to take into account multidirectional seismic effects. The proposed methodology does not require independent analysis in the two orthogonal directions and therefore the application of simplified superposition rules for the combination of seismic component effects is avoided. After a brief outline of the theoretical background and the application process, an extensive evaluation study is presented, which shows that, in general, the proposed methodology provides a reasonable estimation for the vast majority of the calculated response parameters.

1 Introduction

In recent years an increasing interest for pre-earthquake assessment and rehabilitation of existing buildings has been observed. For this purpose a linear or nonlinear analysis procedure has to be implemented in order to calculate some critical response parameters. Given that the structures designed according to older seismic codes (or even without taking into account any seismic code provisions) are expected to sustain extensive inelastic deformations under strong earthquake excitations, nonlinear dynamic analysis (NDA) is undoubtedly the most rational procedure. However, the application of NDA in current practice is complicated and involves many shortcomings related to its computational cost, to the selection and scaling of the accelerograms to be used, etc. Thus, many researchers tried to establish simplified nonlinear analysis methods for the approximate estimation of the inelastic performance of buildings under seismic excitations. As a result of these efforts, static pushover analysis (SPA) was established. Initially, SPA has been developed in some more or less similar variants called ‘conventional’ procedures. All of these variants are based on the assumption that the inelastic response of a structure can be related to the response of an equivalent single-degree-of-freedom (E-SDOF) system. SPA was shortly adopted by several seismic codes and prestandards (American Society of Civil Engineers 2008; Applied Technology Council 1996; European Committee for Standardization 2004) under the name ‘nonlinear static procedure’ (NSP) and became a very popular and useful tool for the earthquake resistant design of new, as well as the seismic rehabilitation of existing buildings.

However, as it has already been stressed by many researchers (e.g., Krawinkler and Seneviratna 1998; Goel and Chopra 2004), this procedure involves many shortcomings and can provide reasonable results only for low- and medium-rise planar systems. This is mainly due to the fact that the determination of the structural response is based on the assumption that its dynamic behavior depends only on a single elastic vibration mode. In addition, this elastic mode is supposed to remain constant despite the successive formation of plastic hinges during the seismic excitation. Also, the choice of the roof displacement as the target displacement instead of any other displacement is arbitrary and it is doubtful whether the capacity curve is the most meaningful index of the nonlinear response of a structure, especially for irregular and spatial systems. Therefore, various ‘advanced’ pushover procedures have been proposed to overcome some of these shortcomings, e.g., modal pushover analysis (MPA) (Chopra and Goel 2001), energy-based pushover analysis (Hernadez-Montes et al. 2004), etc.

Nevertheless, the aforementioned ‘advanced’ pushover procedures—in their initial version—can be rigorously applied only to very simple structures which can be modeled by planar models, since they do not take into account multidirectional seismic effects. It is well known that the very common in current practice plan-asymmetric buildings have to be designed or assessed for concurrent action of at least two horizontal components of the seismic excitation. In literature only few investigations concerning this issue can be found (e.g., Fujii 2007; Lin and Tsai 2008; Fajfar et al. 2005; Reyes and Chopra 2011a, b; Magliulo et al. 2012). For example, on the basis of several assumptions, Fujii (2007) determines two orthogonal principal directions of an equivalent single storey model of the multi-storey building under consideration and applies proper lateral loads simultaneously along them. The inelastic behaviour of the building is correlated to the response of the equivalent single storey model. Lin and Tsai (2008) use pushover analysis to establish three-degree-of-freedom modal sticks, each one corresponding to a vibration mode of a multi-storey asymmetric building under biaxial excitation. The response of the building is then determined by modal superposition of modal sticks’ responses, calculated by means of uncoupled modal response history analysis (Chopra and Goel 2001). On the other hand, some researchers (e.g., Fajfar et al. 2005; Reyes and Chopra 2011a, b; Magliulo et al. 2012) apply pushover analyses independently in two horizontal directions and use one of the widely used directional combination rules [e.g., American Society of Civil Engineers 2008, Section 3.2.7; European Committee for Standardization 2004, Section 4.3.3.5.1(6)] to take into account the multidirectional seismic effects. However, these rules are based on the superposition principle, while it is well known that this approach lacks a theoretical basis in the domain of inelastic response.

Recently, a new multimode pushover procedure for the approximate estimation of the seismic response of asymmetric in plan buildings under biaxial seismic excitation has been developed (Manoukas et al. 2012). Its main idea is that the seismic response of an asymmetric multi-degree-of-freedom (MDOF) system with $N$ degrees of freedom under biaxial excitation can be related to the responses of $N$ ‘modal’ E-SDOF systems under uniaxial excitation. The whole procedure is quite similar to the well-known MPA (Chopra and Goel 2001) as extended for asymmetric buildings (Reyes and Chopra 2011a, b; Chopra and Goel 2004). However, the establishment of the E-SDOF systems is based on an essentially different concept. In particular the properties of the E-SDOF systems are determined by proper equations which take into account bidirectional seismic effects. The proposed methodology does not require independent analysis in each direction of excitation, hence directional combination is avoided. Also, as it will be explained in more detail at the end of Sect. 3, it leads to smaller number of extreme values of the response parameters, so a reduction of the computational cost is achieved.

The preliminary evaluation of the proposed procedure, comprising applications to single-storey buildings consisting of quite simple structural system, indicated that, in general, provides conservative results and relatively small mean errors with regard to the NDA (Manoukas et al. 2012). The objective of this paper is the further evaluation of the procedure for multi-storey asymmetric in plan buildings with realistic structural system, in order to check its accuracy and to identify possible limitations or/and shortcomings.

Firstly, the theoretical background and the assumptions of the proposed methodology are briefly outlined. Secondly, the sequence of steps to be followed for its implementation is systematically presented. The accuracy of the proposed methodology is evaluated by an extensive parametric study, which comprises implementation of the procedure to ten multi-storey asymmetric in plan buildings with varying values of normalized eccentricity. The whole investigation shows that, in general, the proposed methodology provides a reasonable estimation of the vast majority of the response parameters. Finally, the paper closes with comments on results and conclusions.

2 Theoretical background

Concerning the linear range of behaviour, it has been demonstrated that the proposed methodology can accurately determine the modal response of MDOF systems under two proportional horizontal seismic components (Manoukas et al. 2012). However, in the nonlinear range some fundamental assumptions have to be made:

• The seismic response of a MDOF system can be expressed as superposition of the responses of appropriate SDOF systems just like in the linear range.

• Each SDOF system corresponds to a vibration ‘mode’ $i$ with ‘modal’ vector ${\varvec{\varphi }}_{i}$ (the quotation marks indicate that the application of the superposition principle is not strictly valid).

• The displacements ${{\varvec{u}}}_{i}$ and the inelastic resisting forces ${{\varvec{F}}}_{{{\varvec{s}}i}}$ are supposed to be proportional to ${\varvec{\varphi }}_{i}$ and ${{\varvec{M}}}{\varvec{\varphi }}_{i}$, respectively (where M is the mass matrix).

• The ‘modal’ vectors ${\varvec{\varphi }}_{i}$ are supposed to be constant, despite the successive development of plastic hinges.

• It is supposed that Rayleigh damping is present.

Of course, such assumptions violate the very logic of nonlinearity, as the superposition principle does not hold for nonlinear systems. However, keeping always in mind that our main intention is the development of an approximate simplified procedure, the recourse to these assumptions is inevitable. They must be thought as a fundamental postulate, which constitutes the basis on which many simplified pushover procedures are built (Manoukas et al. 2011).

The only additional assumption introduced is that the two horizontal seismic components $\ddot{u}_g (t)_X\, \hbox {and}\,\ddot{u}_g (t)_Y$ are proportional to each other, i.e.:

$$\begin{aligned} \ddot{u}_g \left( t \right) _Y =\kappa \ddot{u}_g \left( t \right) _X =\kappa \ddot{u}_g \left( t \right) \end{aligned}$$

(1)

where $\kappa$ is a constant factor. Of course, this is not true for recorded earthquake ground motions. However, this approximation is in accordance with the very common assumption adopted by seismic codes which specify that—within the framework of NSP as well as the linear analysis methods—the two horizontal seismic components are represented by the same design spectrum, while directional combination may be conducted using the percentage combination rule (e.g., American Society of Civil Engineers 2008, Section 3.2.7.1) which implies a constant factor (0.3) similar to $\kappa$. Obviously, the evaluation of this assumption, as well as the definition of specific values of $\kappa$ is beyond the objective of the present study.

Given the aforementioned assumptions, the nonlinear response of an $L$-story MDOF system with $N$ degrees of freedom (in the usual case of rigid diaphragms $N=3L$) to a biaxial earthquake ground motion ($\ddot{u}_g (t)_X\, \hbox {and}\, \ddot{u}_g (t)_Y =\kappa \ddot{u}_g (t)_X =\kappa \ddot{u}_g (t)$ along $X$ and $Y$ axes, respectively) is described by the following equation (for the sake of simplicity ($t$) is left out in all following expressions) (Manoukas et al. 2012):

$$\begin{aligned} {{\varvec{M}}}{\dot{{\varvec{u}}}} +{{\varvec{C}}}{\dot{{\varvec{u}}}}+{{\varvec{F}}}_{{\varvec{s}}} =-{{\varvec{M}}}({\varvec{\delta }}_{,X} +\kappa {\varvec{\delta }}_{,Y}){\ddot{u}}_{g} \Rightarrow {{\varvec{M}}} {\ddot{{\varvec{u}}}} +{{\varvec{C}}}{\dot{{\varvec{u}}}}+{{\varvec{F}}}_{{\varvec{s}}} =-{{\varvec{M}}}{\varvec{\delta }}_{,XY} \ddot{u}_{g} \end{aligned}$$

(2)

where ${{\varvec{u}}}, {\dot{{\varvec{u}}}}, {\ddot{{\varvec{u}}}}$ are the displacement, velocity and acceleration vectors of order $N$, M is the $N\times N$ diagonal mass matrix, ${{\varvec{C}}}$ is the $N\times N$ symmetric damping matrix, ${\varvec{F}}_{{\varvec{s}}}$ the resisting forces vector and ${\varvec{\delta }}_{,X}, {\varvec{\delta }}_{,Y}$ are the influence vectors that describe the influence of support displacements on the structural displacements for independent uniaxial horizontal seismic excitations along $X$ and $Y$ axes, respectively. Vector u is written as follows:

$$\begin{aligned} {{\varvec{u}}}=[{{\varvec{u}}}_X, {{\varvec{u}}}_Y, {\varvec{\theta }}_z ]^{T} \end{aligned}$$

(3)

where ${{\varvec{u}}}_{X}, {{\varvec{u}}}_{Y}, {\varvec{\theta }}_z$ are the vectors of order $L$ of displacements along $X$ axis, along $Y$ axis and rotations around $Z$ (vertical) axis, respectively. The influence vectors ${\varvec{\delta }}_{,X}$ and ${\varvec{\delta }}_{,Y}$ are:

$$\begin{aligned} {\varvec{\delta }}_{,X}&= \left[ {{\varvec{I}}}, {{\varvec{0}}}, {{\varvec{0}}} \right] ^{T} \end{aligned}$$

(4)

$$\begin{aligned} {\varvec{\delta }}_{,Y}&= [{{\varvec{0}}}, {{\varvec{I}}}, {{\varvec{0}}}]^{T} \end{aligned}$$

(5)

where I, 0 are vectors of order $L$ with each element equal to unity and zero, respectively. Due to the aforementioned assumptions, vectors u and ${{\varvec{F}}}_{{\varvec{s}}}$ can be expressed as the sum of the ‘modal’ contributions (Anastassiadis 2004; Chopra 2007):

$$\begin{aligned} {{\varvec{u}}}&= \sum _{i=1}^N {{\varvec{u}}}_i =\sum _{i=1}^N {\varvec{\varphi }}_iq_i, \end{aligned}$$

(6)

$$\begin{aligned} {{\varvec{F}}}_{{\varvec{s}}}&= \sum _{i=1}^N {{\varvec{F}}}_{{\varvec{si}}} =\sum _{i=1}^N {\alpha _i} {{\varvec{M}}}{\varvec{\varphi }}_i \end{aligned}$$

(7)

where $\alpha _{i}$ is a hysteretic function that depends on the ‘modal’ co-ordinate $q_{i}$ and the history of excitation (Anastassiadis 2004). By substituting Eqs. 6 and 7 into 2 and applying well-known principles of structural dynamics, $N$ uncoupled equations can be derived, each one corresponding to an E-SDOF system (Manoukas et al. 2012):

$$\begin{aligned} {M_{XYi}^{*}} \ddot{D}_i+ 2{M_{XYi}^{*}} \omega _{i} \zeta _{i} \dot{D}_i+V_{XYi} =-{M_{XYi}^{*}} \ddot{u}_g \end{aligned}$$

(8)

where $D_i =q_i /\nu _{XYi}, \dot{D}_i, \ddot{D}_i$ the displacement, velocity and acceleration of the ith $\left( {i=1\ldots N} \right)$ E-SDOF system, $\omega _{i}$ and $\zeta _{i}$ are the natural frequency and damping ratio of the elastic vibration mode $i$ and:

$$\begin{aligned}&\displaystyle V_{XYi} = V_{Xi} +\kappa V_{Yi} \end{aligned}$$

(9)

$$\begin{aligned}&\displaystyle {M_{XYi}^{*}} = {M_{Xi}^{*}} +\kappa (\nu _{Xi} L_{Yi} +\nu _{Yi} L_{Xi} )+\kappa ^{2} {M_{Yi}^{*}} \end{aligned}$$

(10)

$$\begin{aligned}&\displaystyle \nu _{XYi} = \nu _{Xi} +\kappa \nu _{Yi} \end{aligned}$$

(11)

where $V_{Xi}, V_{Yi}$ are the ‘modal’ base shears parallel to $X$ and $Y$ axes respectively, $M_{Xi}^{*}, M_{Yi}^{*}$ and $\nu _{Xi}, \nu _{Yi}$ are the effective modal masses and the modal participation factors of the elastic vibration mode $i$ due to independent uniaxial excitations along $X$ and $Y$ axes respectively, while $L_{Xi} ={\varvec{\delta }}_{,X}^{T}{{\varvec{M}}}{\varvec{\varphi }}_{i} \,\hbox {and}\, L_{Yi} ={\varvec{\delta }}_{,Y}^{T}{{\varvec{M}}}{\varvec{\varphi }}_{i}$.

Equation (8) shows that, due to the aforementioned assumptions, the nonlinear response of a MDOF system with $N$ degrees of freedom subjected to a biaxial seismic excitation $\ddot{u}_{gX}$ and $\ddot{u}_{gY} =\kappa \ddot{u}_{gX} =\kappa \ddot{u}_g$ along $X$ and $Y$ axes, respectively, can be expressed as the sum of the responses of $N$ SDOF systems under uniaxial excitation $\ddot{u}_g ,$ each one corresponding to a vibration ‘mode’ having mass equal to $M_{XYi}^{*}$, displacement equal to $D_{i}$ and inelastic resisting force equal to $V_{XYi}$,  i.e. the sum of ‘modal’ base shear parallel to $X$ axis plus ‘modal’ base shear parallel to $Y$ axis multiplied by $\kappa$ (see Eq. 9) (Manoukas et al. 2012).

3 The sequence of steps for the implementation of the proposed methodology

The application process of the proposed methodology resembles the one of MPA. However, the definition of the E-SDOF systems is essentially different, in order to take into account multidirectional seismic effects. In Table 1 the properties of the $i$th ‘modal’ E-SDOF system are tabulated, along with the properties that it would have in case of uniaxial excitation (parallel to $X$ axis). The displacement of the E-SDOF system for both cases is correlated to the roof displacement instead of any other displacement. As it has been pointed out in introduction, this choice is arbitrary and obviously not obligatory. However, it has been adopted by the vast majority of the researchers as well as by all modern seismic codes. The further investigation of its validity is beyond the objective of this paper.

Table 1 Properties of the $i$th E-SDOF system

The proposed methodology should be implemented for all possible combinations of the seismic components. In particular, the following four combinations should be examined:

$$\begin{aligned}&\ddot{u}_{gX} +\kappa \ddot{u}_{gY} \end{aligned}$$

(12)

$$\begin{aligned}&\ddot{u}_{gX} -\kappa \ddot{u}_{gY} \end{aligned}$$

(13)

$$\begin{aligned}&\ddot{u}_{gY} +\kappa \ddot{u}_{gX} \end{aligned}$$

(14)

$$\begin{aligned}&\ddot{u}_{gY} -\kappa \ddot{u}_{gX} \end{aligned}$$

(15)

The equations derived by the process presented in the previous paragraphs have to be modified proportionately for each combination. It can be easily proved—by simple implementation of the process— that the consideration of the four combinations with opposite sign (e.g., $-\ddot{u}_{gX} -\kappa \ddot{u}_{gY}$ instead of $\ddot{u}_{gX} +\kappa \ddot{u}_{gY})$ leads to identical properties for the E-SDOF systems, so they can be skipped.

The steps needed for the implementation of the proposed methodology are as follows (Manoukas et al. 2012):

• Step 1: Create the structural model.

• Step 2: Calculate $\nu _{XY1}$ (Eq. 11) and $M_{XY1}^{*}$ (Eq. 10) of the fundamental elastic vibration mode 1 for the first combination of seismic components $(\ddot{u}_{g\mathrm{X}} +\kappa \ddot{u}_{gY})$.

• Step 3: Apply to the structural model a set of lateral incremental forces (and moments) proportional to the vector M ${\varvec{\varphi }}_{1}$ of the fundamental elastic vibration mode 1 and determine the (resisting force)-(displacement) curve $V_{XY1}- u_{N1}$ of the MDOF system. $u_{N1}$ can be chosen to correspond to any degree of freedom, but usually the roof displacement parallel to $X$ or $Y$ axis is used.

• Step 4: Divide the abscissas of the $V_{XY1}- u_{N1}$ diagram by the quantity $\nu _{XY1} \varphi _{N1} = u_{N1}/D_{1}$ and determine the (resisting force)-(displacement) curve $V_{XY1}- D_{1}$ of the E-SDOF system.

• Step 5: Idealize $V_{XY1}- D_{1}$ to a bilinear curve using one of the well known graphic procedures (e.g., American Society of Civil Engineers 2008, Section 3.3.3.2.5) and calculate the period $T_{1}$ and the yield strength reduction factor $R_{1}$ of the E-SDOF system corresponding to mode 1, from the following equation:

  $$\begin{aligned} T_1 =2\pi \sqrt{\frac{m_1 Dy_1}{Vy_1}} \rightarrow S_a \left( {T_1 } \right) \rightarrow R_1 =\frac{m_1 Sa(T_1 )}{Vy_1} \end{aligned}$$

  (16)

  where $m_{1} = M_{XY1}^{*}, D_{y1}, V_{y1}$ are the mass, the yield displacement and the yield strength of the system, respectively, and $S_{a}(T_{1})$ is the spectral acceleration.

• Step 6: Calculate the target displacement of mode 1 using one of the well known procedures of displacement modification (e.g., American Society of Civil Engineers 2008, Section 3.3.3.3.2; Federal Emergency Management Agency-Applied Technology Council (ATC) 2004, Section 10.4). If the procedure is applied for research purposes using recorded earthquake ground motions, it is recommended to estimate the inelastic displacement of the E-SDOF system by means of NDA, instead of using the relevant coefficients (e.g., $C_1$ in American Society of Civil Engineers 2008 and Federal Emergency Management Agency-Applied Technology Council (ATC) 2004). This is due to the fact that the coefficient values given by codes are based on statistical processing of data with excessive deviation and, therefore, great inaccuracies may result (Manoukas et al. 2006).

• Step 7: Calculate the ‘modal’ values of the other response quantities of interest (drifts, plastic rotations, etc.) of mode 1 by conducting pushover analysis up to the already calculated target displacement.

• Step 8: Repeat steps 3–7 applying the incremental forces (and moments) in the opposite direction.

• Step 9: Repeat steps 2–8 for an adequate number of modes.

• Step 10: Calculate the extreme values of response parameters by utilizing one of the well established formulas of modal superposition (SRSS or CQC).

• Step 11: Repeat steps 2–10 for all possible combinations of the two horizontal components of the seismic excitation (Eqs. 12–15).

In general, if $n$ is the number of modes taken into account, $2n$ pushover analyses have to be implemented (step 3). Also, $2n$ target displacements (steps 4–6), $2n$ “modal” values (step 7) and $2^{n}$ extreme values (step 10) of response parameters have to be calculated for each combination examined ($2^{n+2}$ extreme values for all combinations). It is worth noticing that when a multimode pushover procedure (e.g., MPA) is applied independently in two orthogonal directions and the multidirectional seismic effects are taken into account by using the SRSS formula or ASCE 41-06 provision (Section 3.2.7), the number of the resulting extreme values of the response parameters is $2^{2n}$ or $2^{2n+1}$, respectively. Also, the proposed methodology does not require independent analysis in each direction of excitation. It is obvious that with increasing $n$ ($n \ge 2$), a significant reduction of the computational cost is achieved.

In case of taking into account the accidental eccentricity imposed by seismic codes, four different models should be analyzed and, as a consequence, the number of the aforementioned extreme values is quadrupled. However, it has been demonstrated that the influence of accidental design eccentricity to the inelastic response of buildings to strong earthquakes is not important (Stathopoulos and Anagnostopoulos 2010), so its omission is justified.

4 Evaluation study

4.1 Structural models

The implementation of the proposed methodology to single-storey buildings consisting of quite simple structural system produced satisfactory results (Manoukas et al. 2012). In the present study, the procedure is further evaluated for multi-storey asymmetric in plan buildings with realistic structural systems, in order to check its accuracy and to identify possible limitations or/and shortcomings.

In particular, an extensive parametric study is carried out comprising applications to five 3-storey and five 6-storey asymmetric in plan (but regular in elevation) reinforced concrete buildings with different values of normalized structural eccentricity. Each building is characterized by a string symbol comprising the letter ‘B’ (from the word ‘building’) and two numbers separated by a ‘-’. The first number (3 or 6) indicates the number of the building’s storeys, while the second indicates the value of the normalized eccentricity $e/r=e_X /r=e_Y /r$ (where $e_{X}, e_{Y}$ are the distances between center of mass CM and center of rigidity CR, and $r$ is the radius of gyration) ranging between 0.10 and 0.50. For example, the symbol B6-2 corresponds to a building having six storeys and normalized eccentricity equal to 0.2. The predefined values of normalized eccentricities are achieved by proper selection of the CM position.

The analysed buildings are shown in Fig. 1. Their structural system consists of moment frames in normal grid with bay width 5 m and storey height 3 m. The concrete is of class C16/20 $({\hbox {f}_{\mathrm{ck}} =16\,\hbox {MPa}})$ and the reinforcement steel bars S500 ($\hbox {f}_{\mathrm{yk}} =500\,\hbox {MPa})$ according to the Greek standards. The cross-sections’ dimensions and the reinforcement remain constant along the height of the building (Fig. 2). Each storey has a mass equal to 150 t and a mass moment of inertia equal to 4,062.5 $\hbox {tm}^{2}$.

Fig. 1

Plan and 3D structural models of the analyzed buildings

Fig. 2

Cross-sections of columns and beams

All analyses are performed using the program SAP 2000 v10.0.7. The modeling of the inelastic behavior is based on the following assumptions:

• Shear failure is precluded.

• The inelastic deformations are concentrated at the critical sections, i.e. at the ends of the frame elements (plastic hinges).

• Plastic hinges are modeled by bilinear elastic-perfectly plastic moments-rotations diagrams with practically unlimited available plastic rotations and yield moments calculated automatically by the program.

• The moment-axial force interaction is taken into account by appropriate interaction surface incorporated in SAP 2000.

4.2 Earthquake excitations

It is obvious that using of an as large as possible number of seismic excitations is desirable. It is not easy to define a specific minimum number over which the results can be considered as reliable. Given that according to seismic codes, when NDA is implemented a number of seven accelerograms is sufficient, many researchers use seven ground motions in order to evaluate their procedures (e.g., Fajfar et al. 2005). The whole investigation conducted here comprises a number of 12 accelerograms, which is considered adequate to obtain preliminary conclusions for the accuracy of the proposed methodology. These accelerograms correspond to strong earthquake motions recorded in Greece and are tabulated in Table 2. The excitations with relatively low ground accelerations (3, 4, 9 and 11) are scaled using an amplification factor equal to 1.5. Thus, the analyzed buildings sustain excessive nonlinear deformations for all excitations. It is considered that each ground motion acts simultaneously along the two horizontal axes of the buildings with the same intensity.

Table 2 List of seismic excitations

4.3 Analysis process

For each building two sets of pushover analyses are performed:

• One based on the proposed methodology (PM). Given that each ground motion acts simultaneously along the two horizontal axes with the same intensity, i.e. $\kappa = 1$ and $\ddot{u}_{gX} =\ddot{u}_{gY} ,$ the possible combinations of the seismic components are only two: $\ddot{u}_{gX} +\ddot{u}_{gY} \left( {\hbox {PM}+} \right)$ and $\ddot{u}_{gX} -\ddot{u}_{gY}$ (PM$-$).

• A second similar to MPA (Reyes and Chopra 2011a, b; Chopra and Goel 2004) (conventional procedure—CP), which comprises pushover analyses of the examined buildings for independent uniaxial excitations along $X$ and $Y$ axes and directional combination of the response quantities using the percentage combination rule. The assumptions and steps of the second procedure are nearly identical to those of the proposed method, except that step 11 is obviously skipped and in steps 2–4 $\nu _{Xi}, M_{Xi}^*, V_{Xi}\, \hbox {or}\, \nu _{Yi}, M_{Yi}^*, V_{Yi}$ are used in place of $\nu _{XYi}, M_{XYi}^{*}, V_{XYi}$.

In both sets of pushover analyses four translational vibration modes are taken into account (1st, 2nd, 4th and 5th). 1st and 4th modes dominate the response for excitation along $Y$ axis, while 2nd and 5th modes dominate the response for excitation along $X$ axis. As it has been demonstrated, using two pairs of translational modes is adequate for the determination of the response even of very tall buildings, while using the 3rd (torsional) and/or higher modes does not significantly improve the results (Reyes and Chopra 2011a). The ‘modal’ superposition is conducted by applying the CQC formula. In Table 3 the modal periods as well as the modal participating mass ratios of the modes taken into account are shown. The maximum ‘modal’ response of each E-SDOF system is calculated by means of NDA for each excitation. Then, the target roof displacement is estimated by multiplication of the resulting response by the quantity $\nu _{XYi} \varphi _{Ni}$ (PM) and $\nu _{Xi} \varphi _{Ni}$ or $\nu _{Yi} \varphi _{Ni}$ (CP). For each building, the floor displacements and storey drifts at the center of mass (CM), at the flexible side (C4) and at the stiff side (C9) of the plan are determined.

Table 3 Modal periods (T) and modal participating mass ratios ($\hbox {pmr}_\mathrm{x}, \hbox {pmr}_\mathrm{y}$)

The response values obtained by the two variants of pushover analysis are compared to the results of NDA, which is considered as the reference solution. For the latter analysis, each accelerogram is considered acting simultaneously along the two horizontal axes in all possible combinations $\left( {\ddot{u}_{gX} +\ddot{u}_{gY} ,\ddot{u}_{gX} -\ddot{u}_{gY} ,-\ddot{u}_{gX} +\ddot{u}_{gY}\,\hbox {and}-\ddot{u}_{gX}-\ddot{u}_{gY}} \right)$. For each response parameter $R_{j,s}$ estimated by the two applied variants of NSP for an excitation $j$, the error with regard to the NDA results $E_{j}$ is determined by the following relation:

$$\begin{aligned} E_j (\%)=100\frac{R_{j,s} -R_{j,d} }{R_{j,d} } \end{aligned}$$

(17)

where $R_{j,d}$ is the value of the response parameter obtained by NDA. Furthermore, the mean error ME for the 12 excitations used in this study and the corresponding standard deviation SD are determined using Eqs. (18) and (19) respectively:

$$\begin{aligned} {\textit{ME}} (\%)&= \frac{1}{12}\sum _{j=1}^{12} {E_j } =100\frac{1}{12}\sum _{j=1}^{12} {\left( {\frac{R_{j,s} -R_{j,d} }{R_{j,d} }} \right) } \end{aligned}$$

(18)

$$\begin{aligned} {\textit{SD}} (\%)&= \sqrt{\frac{1}{11}\sum _{j=1}^{12} {\left( {E_j -{\textit{ME}}} \right) ^{2}} } \end{aligned}$$

(19)

4.4 Results

In Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14 the mean errors for the 12 excitations (referring to the maximum values obtained by NDA) of floor displacements and storey drifts at the selected points (CM—center of mass, C4—flexible side, C9—stiff side) along $X$ and $Y$ axis are shown. Notice that the positive sign (+) means that response parameters obtained by PM or CP are greater than those obtained by NDA. Conversely, the negative sign ($-$) means that the response parameters are underestimated. It is apparent that the two combinations of PM (PM+ and PM$-$) provide an upper bound and a lower bound value for each response parameter. The exact value (NDA) in most cases (188 of 270 calculated floor displacements and 187 of 270 calculated storey drifts) lies in this range (Fig. 15a). For the vast majority of response parameters (246 of 270 calculated floor displacements and 217 of 270 calculated storey drifts) PM leads to conservative results (Fig. 15b). This means that the number of modes taken into account is adequate. Non-conservative values are observed mainly for the stiff side of the analyzed buildings. The mean errors of the more conservative combination (PM+ or PM$-$) for floor displacements range between $-$32 and 66 %, while for storey drifts between $-$51 and 63 %. In general, greater values of mean errors are observed for the response quantities of the 6-storey buildings, except the upper storeys’ drifts. This is more evident at the center of mass and at flexible side of the buildings. In comparison with CP, the absolute values of mean errors resulting from PM are smaller for 253 of 270 calculated floor displacements and for 236 of 270 calculated storey drifts (Fig. 15c). The mean errors of CP range between $-$23 and 138 % for displacements and between $-$32 and 139 % for drifts. The predominance of PM is more obvious for the response quantities at the mass center and flexible side of the analyzed buildings. Concerning the influence of the normalized eccentricity, the maximum mean errors for the response quantities at the center of mass and at flexible side of the buildings generally occur for $e/r=0.2$ and tend to be reduced with increasing eccentricity. For the response quantities at stiff side, no specific trend is observed. Finally, concerning the standard deviation of the results, the values range between 9 and 47 % for PM and 12 and 68 % for CP.

Fig. 3

Mean errors (%) of displacements at the center of mass (CM) of 3-storey buildings

Fig. 4

Mean errors (%) of displacements at the flexible side (C4) of 3-storey buildings

Fig. 5

Mean errors (%) of displacements at the stiff side (C9) of 3-storey buildings

Fig. 6

Mean errors (%) of storey drifts at the center of mass (CM) of 3-storey buildings

Fig. 7

Mean errors (%) of storey drifts at the flexible side (C4) of 3-storey buildings

Fig. 8

Mean errors (%) of storey drifts at the stiff side (C9) of 3-storey buildings

Fig. 9

Mean errors (%) of displacements at the center of mass (CM) of 6-storey buildings

Fig. 10

Mean errors (%) of displacements at the flexible side (C4) of 6-storey buildings

Fig. 11

Mean errors (%) of displacements at the stiff side (C9) of 6-storey buildings

Fig. 12

Mean errors (%) of storey drifts at the center of mass (CM) of 6-storey buildings

Fig. 13

Mean errors (%) of storey drifts at the flexible side (C4) of 6-storey buildings

Fig. 14

Mean errors (%) of storey drifts at the stiff side (C9) of 6-storey buildings

Fig. 15

Effectiveness of the proposed method (PM)

5 Conclusions

A recently developed multimode pushover procedure for the approximate estimation of the seismic response of asymmetric buildings under biaxial excitation is evaluated in this paper through an extensive parametric study. The main idea of the procedure is that the seismic response of an asymmetric building under biaxial excitation can be related to the responses of a series of “modal” E-SDOF systems under uniaxial excitation. The whole procedure is quite similar to the well-known MPA. However, the establishment of the E-SDOF systems is based on an essentially different concept. From the presentation and the evaluation of the proposed method the following conclusions are derived:

• The proposed methodology does not require independent analysis in each direction of excitation, so application of simplified directional combination rules is avoided.

• Two pairs of modes, one dominating the response for seismic excitation along $X$ axis and the other along $Y$ axis (2nd–5th and 1st–4th respectively in the buildings analyzed in the present study), are, in general, adequate to obtain a reasonable estimation of response values.

• Concerning the response parameters of the center of mass and the flexible side of the analyzed buildings, the proposed methodology provides for each response parameter an upper limit and a lower limit which in the vast majority of cases envelope the corresponding value obtained by NDA. Furthermore, the mean errors with regard to the NDA results are significantly smaller than those resulting from a multimode pushover procedure comprising independent analysis along two horizontal axes and directional combination of the results (conventional procedure). The maximum values of mean errors occur for $e/r=0.2$ and tend to be reduced with increasing eccentricity.

• Concerning the stiff side of the analyzed buildings (which dispose low torsional stiffness), the proposed methodology, although it gives generally better results than the conventional procedure, in many cases fails to provide a reasonable estimation of the response parameters. Similar trends have been observed during the application of various pushover procedures in the past and some modifications have been proposed. Relevant investigations are now in progress in order to develop an improved version of the proposed methodology which will be able to estimate the response parameters at the stiff side of asymmetric buildings with acceptable accuracy. This will be the objective of a forthcoming paper.

Finally, it is worth noticing that despite the fact that no restrictions are set to the development of the proposed methodology, generalization of the above conclusions for all types of asymmetric buildings requires further investigations, comprising application to a large variety of spatial structures and using an adequately high number of earthquake ground motions.