A multi-mode displacement-based pushover (MDP) procedure for seismic assessment of buildings

https://doi.org/10.1016/j.soildyn.2021.106950Get rights and content

Highlights

  • An enhanced multi-modal displacement-based pushover (named: MDP) procedure based on structural dynamics theory is developed.

  • The proposed procedure is straightforward and has the ability to account for the higher modes effect into the response.

  • It can take into account the sign reversal in the modal load vectors due to the contribution of higher modes.

  • The method requires less computational effort and time cost compared to adaptive procedures.

  • Application of the proposed procedure on steel moment frames verifies its sufficiency in accurately estimating seismic demands.

Abstract

A new enhanced multi-modal pushover procedure based on the structural dynamic theory with a straightforward and easy-to-use algorithm is developed in this study for estimating the structural demands. The proposed method, which is called the multi-mode displacement-based pushover (MDP) procedure, consists of multiple single-run non-adaptive pushover analyses conducted using a range of enhanced displacement-based lateral load patterns proposed herein. These lateral load distributions are constructed by numerically adding and subtracting the modal displacement vectors modified using modification factors that are a function of the effective modal participating mass ratio (αn). The final seismic demands in the proposed procedure are computed by enveloping the results obtained by several single-run pushover analyses. The MDP procedure has the ability to account for the higher mode effects into responses, sign reversal in the modal load vectors due to the contribution of higher modes as well as the non-concurrency of the maximum modal responses. Less computational effort and time cost, compared to the adaptive methodologies, is another advantage of the proposed procedure. The accuracy and efficiency of the MDP procedure in estimating the structural demands are validated by applying it on three steel special moment frames (SMFs) with different heights. The results demonstrate that the proposed procedure is successful in predicting the critical seismic demands with acceptable accuracy.

Introduction

The most appealing characteristic of nonlinear static procedures (NSPs) to practicing engineers is their simplicity and low computational effort in comparison to Nonlinear Time History (NTH) analysis and Incremental Dynamic Analysis (IDA). Nevertheless, NTH and IDA are widely recognized as the most robust analytical procedures to evaluate the seismic demands of structures responding in the nonlinear range [[1], [2], [3]]. Two inherent shortcomings that conventional NSPs suffer from are (1) the incapability to take the higher mode effects into consideration that restricts their application to low-rise buildings wherein the higher mode effects are not crucial on seismic response of the structure [2], [4], [5] and (2) the invariant loading pattern disregarding the instantaneous changes in dynamic properties of the structure during the pushover analysis including the progressive stiffness degradation of components and member yielding [1,2,[6], [7], [8], [9]]. These drawbacks spurred the proposal of enhanced multi-mode pushover methods. Among them, some approaches (esp. adaptive procedures) have a relatively complicated algorithm and cannot be implemented in every engineering software. If so, the application of a more precise and reliable method like NTH and IDA is more rational [10].

The modal pushover analysis (MPA) based on structural dynamics theory [2], its modified (MMPA) version [11], and its extended versions for unsymmetric-plan building [12] and for three-dimensional analysis of symmetric- or unsymmetric-plan buildings [13], the upper-bound (UB) pushover procedure [14], the mass proportional pushover analysis (MPP) [15], the consecutive modal pushover (CMP) procedure [3] and its modified version (MCMP) [16], the generalized pushover analysis (GPA) [17], the extended N2 method (EN2) [10], the envelope-based pushover analysis (EPA) [18], the cyclic pushover analysis (CPA) [19], the single-run multi-mode pushover (SMP) procedure [20], the modal shear-based pushover (MSP) method [21], dynamic load pattern (DLP) pushover procedure [22], the improved upper-bound method (IUB) [23], the multi-run multi-mode pushover procedure [24], and normalized multi-mode nonlinear static procedure (NMP) [25] are some of the non-adaptive (i.e., with invariant lateral load pattern) multi-mode force-based NSPs proposed in recent years to remedy the inherent drawbacks of the conventional pushover procedures. Additionally, several force-based adaptive pushover methods [[6], [7], [8], [9],[26], [27], [28], [29], [30], [31], [32]] have also been developed in recent years, wherein the applied load pattern is updated at each step of the analysis according to the modal properties of the structure to account for the changes in the dynamic characteristics of the structure due to the progressive stiffness degradation of components and members yielding. More recently, some advanced pushover procedures were also developed for use in more complicated structures. For instance, Zarrin et al. proposed the multi-mode N2 (MN2) pushover procedure [33] and the updated consecutive modal pushover (UCMP) procedure [34] for jacket-type offshore platforms.

It was demonstrated in recent years that a deformation-based approach is an effective tool for the design and assessment of structures [35,36]. Indeed, the theory of performance-based earthquake engineering (PBEE) methodology, which is an appropriate and sophisticated tool in earthquake engineering and widely recommended by today's code provisions [[37], [38], [39]], are basically stemmed from the displacement-based seismic design (DBSD) approach [40,41]. This reflects that the application of displacement loading, in lieu of force actions, in pushover methods would be theoretically a rigorous option for nonlinear static analysis of structures and features the highest potential to properly predict the inertia forces induced by dynamic analysis [42,43].

A displacement-based adaptive pushover (DAP) method developed by Antoniou and Pinho [42] is one of the theoretically robust deformation-based pushover procedures that, in contrast to its force-based counterpart (FAP), gives rise to better prediction of seismic demands of structures [[42], [43], [44]]. In the DAP procedure, the modal interstory drift profiles for each mode weighted by the spectral amplification at the instantaneous period of that mode are combined through a quadratic modal combination rule (e.g., SRSS, CQC). The applied displacement pattern at each story level is then computed through the summation of the modal-combined interstory drifts of the stories below that level. To overcome the limitation of the quadratic modal combination rules in taking into account the effects of sign inversion in the modal load vectors due to the contribution of higher modes, Abbasnia et al. [45] introduced a multi-run adaptive pushover procedure, so-called APAM, based on a new modal combination rule named the effective modal mass combination rule. It was shown that this method can improve the results of the DAP procedure. Later on, another displacement-based pushover procedure was proposed by Abbasnia et al. [46] whereby multiple response displacement spectra corresponding to the instantaneous ductility ratio of the structure at each loading step are used in order to account for the actual energy dissipation properties of the structure. In another investigation, Shafiee and Tehranizadeh [47] improved the DAP procedure and proposed a new dual-run displacement-based adaptive pushover method called Improved DAP (IDAP).

As stated, the aforementioned displacement-based pushover procedures take an adaptive algorithm in their solution. Despite the foregoing advantage of the adaptive pushover approach over the non-adaptive one, it is a laborious task and cannot guarantee an accurate estimation of seismic demands. Another point to note is that the adaptive pushover solution is a complex analysis that requires a lot of computational effort and time cost, and cannot be implemented in some structural analysis tools used in practice. In this regard, some researchers have developed enhanced non-adaptive displacement-based pushover procedures wherein the structure is pushed using an invariant loading pattern. One of these methods is the drift pushover analysis (DPA) proposed by Sahraei and Behnamfar [48]. In this method, the loading protocol is a predetermined drift-based profile established using a generalized interstory drift spectrum that considers the contributions of higher modes concurrently. Afterward, this method was extended by Behnamfar et al. [49] which is conceptually on the basis of the spectral analysis method. The accuracy of the DPA and its extended version (EDPA) in estimating story drifts is not examined. In another research, Amini and Poursha [50] introduced a non-adaptive displacement-based pushover (NADP) procedure wherein the seismic demands are estimated by enveloping the results derived from two or three single-run displacement-based pushover analyses. Although this method provides appropriate results for higher story levels, the outcome for lower levels is not accurate. This is attributed to the employment of a single-run displacement-based pushover analysis using a load pattern according to the first mode shape.

The above considered, the present study is intended to focus on the development and validation of a multi-mode displacement-based pushover procedure with non-adaptive lateral force distribution schemes. The proposed procedure takes the advantage of the factored modal combination method used by several investigators [45,[51], [52], [53]], and suggests a new method to take into account the higher mode effects in establishing the lateral load pattern. That is, a weighted vectorial addition and/or subtraction combination technique is employed in determining the lateral load distribution. Hence, unlike the quadratic modal combination rules (e.g., SRSS, CQC), the effect of sign inversion due to the contribution of higher modes is reflected in the applied loading vector proposed in this research. Moreover, the effect of the frequency content of a particular response spectrum on the load pattern is considered through the modification of the modal story displacements by the spectral displacement of ground motions. Briefly, the proposed method is a simple procedure that consists of some pushover analyses executed using a series of lateral load combinations constructed by algebraically adding and subtracting the modified modal load vectors. The final seismic demands are obtained by enveloping the peak responses resulting from several pushover analyses. The non-adaptive nature of the suggested method preserves the conceptual simplicity and computational attractiveness of the NSPs. In order to verify the accuracy of the proposed procedure, it is applied to three code-conforming steel moment frames (SMFs) with different heights. A new simple method based on the response spectrum analysis (RSA) concept is also proposed to determine the target roof displacement in the pushover analysis. The efficiency of the proposed method in estimating the seismic demands of buildings is examined by comparing its results with those resulting from nonlinear response history analysis (NL-RHA), treated as the most accurate solution. Moreover, a displacement-based pushover analysis using a load pattern according to the first mode shape (DMode-1) of the structure and the DAP procedure [42] are also performed for the sake of comparison. The validation of the proposed procedure is done in two earthquake levels, i.e. the design basis earthquake (DBE) and the maximum considered earthquake (MCE) levels according to ASCE 7–16 [54].

Section snippets

Description of the proposed pushover procedure

The proposed pushover procedure is elaborated in the following section. This section consists of three subsections including 1) the conceptual basis of modal response analysis, 2) displacement-based lateral load distribution proposed, and 3) details of the proposed method.

Verification of the proposed procedure

To what follows, the effectiveness and accuracy of the MDP procedure in predicting seismic demands of structures is appraised by comparing the seismic demands resulting from the proposed procedure with the relevant results corresponding to the nonlinear response history analysis (NL-RHA) as a benchmark solution. To this end, three generic steel special moment frame (SMF) structures with different heights are selected to examine the accuracy of the proposed methodology. Furthermore, a

Results and discussion

It is obvious that the effectiveness of each pushover procedure is influenced by the applied load pattern. As pointed out before, the MDP procedure is comprised of several single-run pushover analyses conducted by applying a range of proposed lateral load patterns. In order to specify the importance of each load pattern in the final seismic demand estimations of structures, the contribution of each load pattern in the maximum interstory drift ratios of the structures is firstly investigated in

Conclusions

This investigation aims to develop a simple multi-mode displacement-based pushover (MDP) procedure for estimating the seismic demands of structures. The displacement-based lateral load patterns utilized in the proposed method are generated by algebraically adding and subtracting the modified modal displacement vectors. To validate the proposed procedure, it is implemented on three SMF buildings with 6-, 12-, and 18-stories. The accuracy of the proposed procedure is examined by comparing its

Author statement

Aydin Daei: Conceptualization; Data curation; Formal analysis; Methodology; Software; Writing – original draft; Writing – review & editing, Mohamad Zarrin: Conceptualization; Formal analysis; Methodology; Software; Supervision; Writing – original draft; Writing – review & editing

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    • FOSM-based updated consecutive modal pushover procedure (FUCMP) for seismic uncertainty quantification of jacket offshore platforms

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      Final structural responses are determined by enveloping the results obtained by the multiple pushover analyses. Several other force-based (Jan et al., 2004; Sucuoglu and Selim Gunay 2011; Vafaee and Saffari 2017; Zarrin et al., 2021) and displacement-based (Sahraei and Behnamfar 2014; Behnamfar et al., 2016; Amini and Poursha, 2016; Jalilkhani et al., 2020; Daei and Zarrin, 2021) enhanced multi-mode pushover procedures were also proposed in recent years to consider the effects of higher modes on the seismic responses of structures. Another group of enhanced pushover procedures, known as adaptive pushover procedures, has also been proposed wherein the instantaneous changes in the dynamic properties of the structure due to the progressive yielding of structural members are also taken into account in the applied load pattern in addition to the effects of the higher modes.

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