ICCBT 2008 - C - (06) – pp69-80

ICCBT2008

Nonlinear Static Pushover Analysis in Earthquake Engineering: State of Development

M. Seifi*, Universiti Putra Malaysia, MALAYSIA

J. Noorzaei, Universiti Putra Malaysia, MALAYSIA M. S. Jaafar, Universiti Putra Malaysia, MALAYSIA E. Yazdan Panah, Universiti Putra Malaysia, MALAYSIA

ABSTRACT Owing to the simplicity of nonlinear static pushover (NSP) analysis compared to nonlinear dynamic time-history analysis, currently proliferation in usage of NSP method is observed among society of civil engineers. Conceptually, NSP method relies on pushing the structure with incremental static lateral load by considering material inelasticity and geometric nonlinearity. In this study, initially the fundamental concept of NSP method was declared. Subsequently, the state of development of pushover analysis from pilot studies to the ones that found its way to FEMA-356, as the most prevalent practical code, to the state-of-art among novel proposed methods was presented and more than 10 types of innovative methods was investigated. The results criticized the pros and cons of each method. Eventually, the superior method in aspect of accuracy in accompany with simplicity of computational procedure was unveiled. Keywords: Nonlinear Static Pushover, Conventional Method, Adaptive Method, Earthquake Engineering *Correspondence Author: Mehrdad Seifi, Universiti Putra Malaysia, Malaysia. Tel: +60176103642. E-mail: mehrdadcivil@gmail.com

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1. INTRODUCTION During the past two decades major changes have been made in the field of earthquake engineering. Based on conventional elastic methods of seismic design a pseudo-capacity has been provided to resist the prescribed lateral force. It had been postulated that the structure is capable to resist the imposing load of earthquake by means of yielding into the inelastic range, energy absorption and ductile behavior. However, significant destructions and massive human and economic losses of engineered buildings during major seismic events such as Northridge, California (1994), Kobe, Japan, (1995) clarified the deficiency of such methods. Owing to this fact that these approaches do not provide insight of how the structure will actually perform under severe earthquake conditions. Consequently, distinguishing the actual performance of the structure through the performance-oriented procedures and guidelines termed as “Performance-Based Design Engineering”, (PBDE) has been frequently highlighted [9, 18]. This novel area has two major differences with conventional perspective on earthquake engineering. The first one is the direct connection of design to structural performance. The other one is to design the structure for multiple performance objectives. In simple words performance-based design is a designing strategy of how a structure fulfills stated performance targets when subjected to either particular or generalized earthquake ground motion. The performance targets may be a level of stress not to be exceeded, a load, a displacement, a limit state or a target damage state [8, 19]. The best way to assess the performance of structure subjected to earthquake action is nonlinear time history (NLTH) analysis. Nonlinear dynamic as the most rigorous analyzing method utilizes the combination of ground motion records with a detailed structural model. Therefore, it endows with relatively low uncertainty in results. In spite of that, the calculated responses are very sensitive to the characteristics of the individual ground motion used as seismic input; therefore, several analyses are required using different ground motion records. Besides, a mathematical tool able to handle all analysis often exceeds the capabilities of a design office which works under tight time constraints. Investigating for a reliable, less time-consuming method, results in “Nonlinear Static Pushover” (NSP) method [6]. Nonlinear static analysis called “pushover” hereinafter, per se is not a recent development and its genesis traced back to the 70’s decade. However, recent procession of performance-based design has brought the nonlinear static pushover analysis procedure to the forefront. In last decade, the majority of researches had focused on the range of applicability and merit and demerits of pushover methods. After revealing the deficiencies of the preliminary pushover methods, efforts have been made to improve it [26]. In the mid of 90’s, the potential of pushover analysis are certified and it found its way to seismic guidelines for instance SEAOC, 1995; FEMA 273/274, 1997, ATC-40, 1997[4, 13, 14, 32]. Nowadays, it gain more popularity and already included in some codes such as FEMA 356/357, 2000; ATC-55, 2005; FEMA-440, 2006. Preparation of a framework for predicting structural performance is the main goals of this method [3,15-17]. The pushover provides characteristic information that cannot be extracted via elastic static or dynamic analysis namely [20, 23]:

• Realistic force demands on potentially brittle elements.

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• Estimates of the deformation demands on inelastically deformed elements, in order to dissipate energy.

• Particular elements strength deterioration effects on the overall structural stability. • Identification of the crucial regions with considerable inelastic expected deformations. • Identification of plan or elevation strength irregularities. • Estimation of the interstorey drifts that account for strength or stiffness discontinuities. • Verification of the sufficiency of the load path, considering both structural and non-

structural elements of the system. • Sequence of the member’s yielding and hinge formation and the progress of the overall

capacity curve of the structure.

2. THEORETICAL BACKGROUND OF CONVENTIONAL PUSHOVER In the conventional pushover analysis it was assumed that the response of the multi-degree-of-freedom (MDOF) system could be represented by an equivalent-single degree of freedom (ESDOF) system [23]. This implies that the response is controlled by a single mode, and that the shape of this mode{ }Φ remains constant throughout the time history response, regardless to the level of deformation. Therefore, differential equation of an MDOF system by defining

{ } txX Φ= as relative displacement of MDOF system where tx roof displacement is can be written as: [ ] [ ] { } gtt xMFxCxM &&&&& }1{}{}{ −=+Φ+Φ (1) Where M and C are mass and damping matrices, F represents the storey force vector and

gx&& denotes the ground acceleration. If ESDOF system displacement x defined as:

{ } { }{ } { } tT

T

xMMx

1ˆ

ΦΦΦ

= (2)

By pre-multiplying the equation (1) by{ }TΦ , and substituting x from Eq.2, following equation are obtained for the response of ESDOF system:

gxMFxCxM &&&&& ˆˆˆˆˆˆ −=++ (3)

Where, M , C and F representing the properties of ESDOF system and are defined by:

{ } { }1ˆ MM TΦ= (4)

{ } { }{ } { }

{ } { }ΦΦΦ

ΦΦ=M

MCCT

T 1ˆ (5)

(6) { } FF TΦ=ˆ The force-deformation characteristics of the equivalent SDOF system can be determined by

the results of nonlinear pushover analysis of the MDOF system, which usually produces a base shear (V) vs. roof displacement diagram ( tx or tδ ) as shown in figure 1.

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Figure 1. Multilinear and bilinear static base shear vs. roof displacement response of MDOF structure [33]. The curve of base shear against roof displacement must be represented by a bilinear relationship in order to define “yield” strength ( yV ) effective “elastic” stiffness yye xVK = , and hardening (or softening) stiffness, es KK α= for the structure as shown in figure 1 [16]. 3. FUNDAMENTAL CONCEPTS OF PUSHOVER ANALYSIS Generally, in pushover method the structure is loaded with a predetermined or adaptive “lateral load pattern” and is pushed statically to “target displacement”. The lateral load might be considered as “force” or “displacement”. The loading is monotonic with the effects of the cyclic behavior and load reversals and with damping approximations. With increase in the magnitude of the loading, weak links and failure modes of the structure are found [19, 20]. These basic issues of pushover analysis are described here: Lateral load pattern: for a performance evaluation the load pattern selection is likely to be more critical than the accurate determination of the target displacement. It plays an important role due to the fact that it is supposed to deform the structure in a similar manner experienced in earthquake occurrence. Conventionally, as shown in figure 2 an inverted triangular or uniform shape is used consistent with the codified static lateral force distribution but use of adaptive load shape is on the increase. The importance of the loading shape increases when the response is not dominated by the single mode [23, 29].

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Target displacement: the target displacement estimates global displacement expected due to the design earthquake corresponding to the selected performance level. A convenient definition of target displacement is the roof displacement at the center of mass of the structure. It can be calculated by any procedure that accounts for the effects of non-linear response on displacement amplitude [19, 23, 26]. Applied forces vs. applied displacements: essentially and as a consequence of the historical developments of an understanding of structural dynamics current seismic design is based on force rather than displacement, Although nowadays design procedures have become more rigorous in their application, this basic force-based approach has not changed significantly since its inception in the early 1900s [21]. On the other hand, structural displacements are a main cause of damage in structures subjected to earthquakes. Simple techniques for estimating structural displacements enable development of a design approach based explicitly on expected displacements, therefore, displacement-based approach is assumed as an effective to design and control of structures [19]. 4. STATE OF DEVELOPMENT OF PUSHOVER ANALYSIS The pushover analysis in earthquake engineering has been first applied by, Gulkan and Sozen (1974) where “equivalent” SDOF or “substitute” structure is derived to represent the MDOF system. In this method it was assumed that the single mode is dominant and the shape of this mode through the analysis is constant. Definitely, both of the assumptions are incorrect [36]. Shibata and Sozen (1976) proposed a method, where the designing forces of earthquake-resistant reinforced concrete structures has been determined by a modified linear model and recognizing the effect of energy dissipation in the nonlinear range [34]. Thereupon, a procedure at the level of linear spectral response analysis, with explicit options about the levels of inelastic response in different elements of a multistory reinforced concrete structure was provided. Afterwards, simplified inelastic analysis procedures for MDOF systems put forward by Saiidi and Sozen (1981), Fajfar and Fischinger (1988) and Qi and Moehle (1991) [27].

Figure 2. Conventional lateral load distribution [33]

(a) Triangular (b) Uniform

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Krawinkler (1995) introduced capacity spectrum method by application of pushover analysis. Also, the method is introduced as instrumental tool in design of structures. Along the lines of investigations, Bracci et al. (1995) during experimental and analytical study dealt with the behavior of gravity load designed reinforced concrete frame subjected to simulated seismic motion and introduced a preliminary adaptive method [7]. According to Faella and Killar (1996) that implement the pushover analysis, confined to triangular loading, for 3, 6 and 9 storey buildings and compared the results versus dynamic analysis. They concluded that pushover analysis is rational in aspect of inter-storey drift and collapse mechanism. [7, 27, 30]. Krawinkler and Seneveritna (1998) entitled pushover analysis as popular tool for seismic performance evaluation of existing and new structures. They presented the precision of this approach and more importantly identified the cases in which the pushover predictions are misleading. In their study, the lateral load patterns and also the target displacement issues have been investigated and it has been accepted as true that performance evaluation is more related to load pattern selection than the accurate determination of the target displacement. The writers believe that the use of invariant patterns may lead to misleading predictions, they suggested on use of adaptive patterns. Eventually, they announced, “further improvements of the pushover procedure need to focus on the load pattern issue” [23]. Sasaki et al. (1998) offered the “multi-mode pushover” that tries to incorporate higher modes by considering multiple pushover curves derived from different modal force patterns. The “Adaptive Pushover Method” developed by Gupta and Kunnath (2000) uses a varying load pattern that moves the structure back and forth by combining modes at different stiffness states of the structure [25, 10]. Fajfar (2000) developed N2 method as a variant of capacity spectrum method used by ATC-40. This method combines the pushover analysis of MDOF model with the response spectrum of equivalent SDOF system. In N2 method selection of an appropriate lateral load distribution depends on the mass matrix (M) and displacement mode shape (Φ ) could be assumed as an advantage of the method in comparison to the conventional invariant load [12]. Subsequently, Moghadam and Tso (2000) offered “MT method” to account for torsional effects on irregular buildings. The target displacements are obtained by an elastic spectrum analysis and since the top displacement of different resistant elements are different, many target displacement should be computed. To obtain the capacity curve, triangular lateral load distribution along the height of the building was assumed (figure2. a). For testing the credibility of the approach, uniform moments resisting frame, set-back moment resisting frame and uniform wall-frame as 3 different building configurations were used. Moreover, an ensemble of 10 artificial ground motion records, with response shape similar to Newmark-Hall design spectrum, was developed to run the nonlinear time history (NLTH) analysis. The same approach was repeated by different load distribution from the triangular one. Based on the writer’s statement this method works well for uniform moment resisting frame, although the results for the other two types of structures are not well-correlated with NLTH analysis results. Furthermore, a fixed load pattern were adopted neglecting the changes in the mode shapes due to plastic hinges formation [28].

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“Modal pushover analysis” (MPA) that could be assumed as the first serious effort to overcome the deficiencies of conventional pushover was recommended by Chopra and Goel (2001). MPA is essentially the extension of single mode pushover analysis to multi-mode response, and use of the theory of response spectrum analysis to combine the modal contributions [10]. Accordingly, adaptive pushover analysis (APA) that certainly could be addressed as a revolution among modern pushover methods proposed by Elnashai (2001). The APA method accounting for spread of inelasticity, geometric nonlinearity, full multi-modal, spectral amplification and period elongation, within a framework of fiber modeling of materials. These developments lead to static analysis outcome become closer than ever to inelastic time-history analysis. It is concluded that there is great scope for improvements of this simple and powerful technique that would increase reliability in its employment as the primary tool for seismic analysis in practice [11]. “Method of Modal Combination” (MMC) was suggested by Kalkan and Kunnath (2004). In this method, spatial invariant lateral load distribution is determined from:

(7) ( )∑=

ΦΓ=n

innannnj TSmF

1,ξα

where: nα = Modification factor that can be assumed positive or negative

nΓ = modal participation factor of the n th mode as;

[ ] [ ]{ }( )nT

n Mtm /Φ=Γ in which [ ] [ ][ ]ΦΦ= mM Tn

m = mass matrix nΦ = mode shape corresponding to the n th mode

aS = spectral acceleration. During their study 3 regular 2D frames involving 6 and 13 storey buildings instrumented during Northridge earthquake (1994) and also not-instrumented 4-storey building by using the Northridge (1994) and Loma Perieta (1989) earthquake record were evaluated. In order to model the structures precisely, “fiber element modeling” was adopted for the whole of structural members. The results of this method in aspects of inter-storey drift and roof drift ratio were compared to the code-based (FEMA-356; 2000) method, MPA method (Chopra and Goel; 2001) add NLTH analyze as reference. Accordingly, it was shown that at least for the low and mid-rise structures the outcomes are approximately precise and more logical in comparison to the other mentioned methods. However, improvement in aspect of invariant load distribution should be made [22]. In 2004, Antoniou and Pinho extended and modified the adaptive pushover analysis (APA) which was previously proposed by Elnashai (2001). In “Force-based Adaptive Pushover” (FAP) analysis recommended by Antoniou and Pinho (2004) eigenvalue analysis carried out at each analysis step and accordingly, lateral load distribution continuously updated based on initial stiffness of structure. This fully multi-modal method accounts for plastic hinge constitution and softening of structure and enhanced in aspects of modification of inertia forces due to spectral amplification and period elongation [1]. Apart from force-distribution, they introduced an innovative method addressed as “Displacement-based Adaptive Pushover (DAP) in 2004, this approach overcame the intrinsic

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deficiency of conventional fixed pattern displacement pushover which was considered of no use. This new method were fully in line with recent seismic design/assessment trends where the direct use of displacements, as opposed to forces, is preferred as a recognition of the conspicuous evidence that seismic structural damage is in fact induced by displacements. Furthermore, the additional modeling and computational effort, with respect to conventional pushover procedures, is not protracted [26, 27]. Generally, the adaptive pushover approaches include 4 major steps namely as: i) Definition of inertia mass and the nominal load vector ii) Computation of the load factor iii) Generation of the normalized scaling vector iv) Updating of the loading force vector. Barros and Almeida (2005) delivered a new multi-modal pushover method. In this method the load pattern (LP) was on the base of relative participation of each mode of vibration in the elastic response of structure subjected to ground motion.

(8) ...221

11 +ΦΓ+ΦΓ=ΦΓ= ∑

=

n

iiiLP

where nΓ is the modal participation factor of the n th mode and nΦ is the corresponding mode shape. The method was tested on 3D models including a one-storey, one bay symmetric structure, one-storey, one-bay stiffness asymmetric structure, and three-storey, two-bay mass asymmetric structure and the results compared with the conventional FEMA-273 pushover method and NLTH analysis. The prototypes are very small and due to very limited results presented by the authors, it is difficult to have clear-cut idea about the soundness of method though, the outcome signified the importance of higher mode effects. Also the authors suggested; “it will be necessary to perform some experimental studies to validate these computational results” [5]. Aschehim et al. (2006) compared the results of the “Modified Modal Pushover Analysis” (MMPA) and “Energy-based MPA” method as two new generation of MPA method that was primarily developed by Chopra and Goel (2001). In the modified version of MPA the elastic contribution of the higher modes that have been estimated by mean spectrum of the scaled ground motion for the drift level of interest are combined by SRSS rule with the contributions from the first mode. In the next method, energy-based pushover, corresponding capacity curve with each modal pushover is found out based on the incremental work in the analysis. These two methods were compared for a set of five buildings subjected to scaled ground motions. The investigators brought to a close that the results of the both methods are not consistently trustworthy [2]. A new modal combination method addressed as “Factored Modal Combination” (FMC) was developed by Park et al. (2007). In FMC method the directions as well as the absolute value of each modal contribution were taken into consideration. In this method for defining the peak response 0r of a design parameter superposition of modal contribution are operated.

(9) ∑ +++== ...303020201010000 rRrRrRrRr nn where, 10 0 ≤≤ nR is modal combination factor and 0nr is peak response of the n th natural mode. In this method multiple storey load profiles were determined by combining the spectrum modal responses multiplied by modal combination factors. By taking a quick look it is identified that FMC method follows the basic concept of MMC method proposed by Kunnath and Kalkan (2004), nevertheless, introducing 10 0 ≤≤ nR make the FMC method capable to produce various earthquake loads [31].

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In this article FMC method applied for 2D models with and without vertical irregularity and the results were compared to conventional pushover method by employing the inverted triangular load pattern and incremental dynamic response at approximately 0.5%, 1.0% and 2.0% drift of the total height of structures based on the suggestions of ATC-2005. Eventually, the authors concluded: “As expected, the inverted triangular analysis (ITA) did not accurately predict the results of the nonlinear time-history analysis. Generally, the ITA underestimated both the story shears and the interstory drifts. Moreover, due to the high complexity of this type of structures the NSPs should be applied conservatively. Particularly, the earthquake load profiles of high-rise buildings and buildings with irregular plans and sections are complicated by the effect of higher dynamic modes [31]. 5. CRITICAL DISCUSSION Through the proceeding of pushover analysis several challenges were addressed before the procedure can be upon agreed and also it seems that still there are some issues that should be investigated. Generally, next to the initial genesis of pushover analysis the epoch of that could be categorized into two phases:

1. 1992 to 1999 that allocates to the Criticism of preliminary methods 2. 2000 to present that could be addressed as the period of novel ideas in order to

overcome the deficiencies of pushover analysis.

5.1 Criticism Period (1992-1999) In this period there are some publications that reviewed the pros and cons of the preliminary proposed pushover analysis. Lawson et al. (1994) applied the analysis with different recommended types of constant load distribution for 4 steel structures with heights vary from 2 to 15 stories. The authors concluded that when higher mode effects are important, the prediction of conventional pushover is grossly inaccurate [27]. A controversial point raised from the review made by Krawinkler that “it must be emphasized that the pushover analysis cannot disclose performance problems caused by changes in the inelastic dynamic characteristics due to higher mode effects” where it could be assumed as the major problem of conventional pushover methods [23]. Some of the studies certified the constant load pattern as Achilles' heel of the conventional pushover analysis procedure. According to Kim and D’Amore investigation in 1998, it was proved that, traditional pushover based on single mode shape are inadequate to determine the seismic behavior of some types of structures [23, 35]. Naeim (1999) explained the common pitfalls of pushover analysis. In this critical review some of the important issues for a meaningful pushover analysis such as, importance of loading or displacement shape function, role of gravity load, effect of P-Δ load, shear failure mechanism, rebar development and lap lengths and puzzlement between pushover and earthquake loading have been come into picture [29].

5.2 Novel Ideas Period (2000-present) During the current decades, some methods with varying degree of rigor and success has been proposed to overcome the significant shortcomings of conventional pushover. Generally the major novel approaches could be categorized into three divisions including:

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a. N2 and MT methods b. MPA family approaches c. Adaptive pushover analysis methods

The “N2 method” proposed by Fajfar in 2000, like other approximate methods, subject to limitation and for that time being, its application was restricted to planar analysis of structures [12]. Killar and Fajfar (2002) investigated the possibility of extension of N2 method for irregular 3-D structures during a comparative study among N2, MT and NLTH analysis. The presented results clarified that the N2 method misjudges the displacement calculation in center of mass, flexible edge and stiff edge of irregular structures. Besides it was revealed that MT method is not being able to characterize the inelastic behavior of irregular structures where noticeable redistribution is expected during nonlinear range [27]. A critical study by Kalkan and Kunnath, 2004 highlights the drawbacks of the MPA method. By scrutinizing the procedure of this method, it is clear that MPA is not a static method but adoption of NLTH analysis for pushover analysis, since it requires numerous runs of SDOF NLTH analysis for adopted ground motion to recognize target displacement. Regarding the rigorous MPA procedure it seems that one of the main goals of pushover as respects decreasing the time of analysis has been lost. In addition, running the pushover analyses independently in each mode and denying the other modes contribution in the formation of plastic hinges results in distinct inaccuracy in estimating of plastic hinges rotation, a significant parameter for comparing acceptance criteria in performance-based evaluation [25]. It is worth mentioning that while the accuracy deficit of MPA method has been modified by the other methods like MMPA and energy-based pushover, the general use of these methods for design and evaluation attributable to their computationally demanding procedure is cumbersome [33]. In 2005 the limits of applicability of conventional and initial adaptive pushover analysis proposed by Elnashai, 2001 were investigated by Papanikolaou et al. for a number of R/C structures [30]. Afterwards, The reliability of the modified methods including FAP and DAP were tested through the study on RC structures, subjected to 4 various acceleration records and the results were satisfactory [27]. Moreover, during the study by the authors in university Putra Malaysia, 2007, the reliability of all types of the modified APA methods were scrutinized. The results certified that “interstorey drift-based scaling” approach as a division of DAP method has significant accuracy comparing to the NLTH results [33]. 6. CONCLUSIONS Based on the review on the state of development of pushover analysis following conclusions can be drawn:

i. Pushover analysis is a solution for complicated problems of estimating the capacity and deformation problems for certain types of structures.

ii. A majority of studies have been performed on 2D R/C frame structures. Thus, the applicability of pushover for 3D structures, steel structures and high-rise frames is not examined throughly and more investigation on these aspects is required.

iii. The conventional code-based method has many deficiencies according to neglecting higher mode contribution, stiffness degradation and the period elongation, but this method is the most well-known method in society of practical engineers.

iv. In recent decade, several methods such as APA, N2, MPA, MT, MMC and etc. have been proposed to overcome the deficiencies of the conventional method.

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v. Among several modified method “Adaptive Pushover Analysis” (APA) proposed by Antoniou and Pinho in 2004, seems to be more rational, since the others are complicated theoretically. Also they have protracted procedures.

vi. Based on the study of the authors, 2007 the “interstorey drift-based scaling adaptive pushover method” could be nominated as the most precise type of pushover analysis.

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