MDOF Effects on the Accuracy of Nonlinear Static Seismic (Pushover) Procedures

T. Boen1 and T. N. Tjhin2

1Senior Advisor, World Seismic Safety Initiative (WSSI)

2Structural Engineer, Buckland & Taylor Ltd. ABSTRACT This paper discusses Multi-Degree-of-Freedom (MDOF) effects on the results of current nonlinear static (pushover) techniques for evaluation and design of structures. The apparent inadequacy of these techniques in estimating peak seismic response quantities influenced by the MDOF effects is illustrated. Improvements in the analysis procedures to adequately account for the MDOF effects are also presented. KEYWORDS earthquake engineering, seismic effects, nonlinear response, seismic analysis, dynamic analysis, seismic design INTRODUCTION Nonlinear Static Procedures (NSPs), such as the Capacity Spectrum Method (CSM) in the ATC-40 document (ATC 1996) and the Displacement Coefficient Method (DCM) as described in the FEMA-273/274 reports (ATC 1997a, 1997b) and standardized in the FEMA-356 report (ASCE 2000), have recently gained popularity for estimating the response of structures subjected to seismic ground motions. Originally presented for use in the performance-based seismic evaluation of existing buildings, these analysis methods have also been applied to design of new buildings, as recommended by Fajfar (1999), Chopra and Goel (1999), and Aschheim and Black (2000). The NSPs are generally believed to be superior than Linear Elastic Procedures (LSPs), such as the classic equivalent static lateral force procedures and modal superposition techniques, because

1

they explicitly consider inelasticity of yielding-expected structural components in resisting moderate and large earthquake intensities. Furthermore, the NSPs are more appealing than Nonlinear Dynamic Procedures (NDPs), which are considered to be the most sophisticated of all available seismic analysis methods, as they yield single-valued estimates of response quantities (e.g., lateral displacements, interstory drifts, member forces and moments, and plastic hinge rotations) for design or evaluation. The NSPs use “Equivalent” Single-Degree-of-Freedom (ESDOF) representations of structures to estimate roof peak displacement and the response quantities associated with this roof displacement. These ESDOF systems are generally based on the fundamental mode of response, thus systematically excluding the effects of higher modes (in the case of elastic response) or Multi-Degree-Of-Freedom (MDOF) effects (in the case of nonlinear response) on response quantities. The significance of higher modes was discussed in both the ATC-40 and FEMA-274 reports and in some recommendations (e.g., Reinhorn 1997). This paper discusses the effects of higher modes or MDOF effects on the results of NSPs. The accuracy of NSPs relative to the results obtained from the NDPs is presented, drawn from the recent ATC-55 MDOF studies (ATC 2005). Recent improvements in the analysis procedures to effectively account for higher modes are also presented. OVERVIEW OF NONLINEAR STATIC PROCEDURES General The NSPs conceptually consist of three steps, i.e., (1) capacity evaluation, (2) seismic demand estimation, and (3) performance evaluation. These steps are graphically summarized in Figure 1 with respect to the ATC-40 and FEMA-356 procedures. Capacity Evaluation In this step, a capacity (pushover) curve, consisting of a plot of base shear as a function of lateral displacement at a control node, is generated on the basis of a nonlinear static analysis of a mathematical model of the structure under consideration (Figure 1(a)). In the nonlinear static analysis, the structural model is subjected to monotonic increasing lateral forces or displacements until either a target displacement is exceeded or the building collapses. The structural model explicitly accounts for the effects of inelasticity of structural components, in addition to other modeling techniques typically employed in linear analysis procedures. The center of mass at roof is typically selected as the control node. Different vertical distributions of lateral forces are possible in developing a capacity curve. The applied force at any floor is proportional to the mass and displacement associated with a shape vector at the floor under consideration. Recognized shape vectors in the ATC-40 are (1) concentrated load, (2) code distribution, (3) first mode, (4) adaptive, and (5) multiple modes. In the FEMA-356, two separate nonlinear static analyses with two different shape vectors are to be conducted. The first shape vector is selected from the following set: (1) code distribution, (2) first mode, and (3) SRSS. The second load vector is either (1) uniform or (2) adaptive. For each

2

response quantity of interest, the larger value obtained from the two analyses is used in the performance evaluation. A short description of each shape vector is given in the later section. Once available, the pushover curve is converted to a capacity spectrum (or capacity diagram), i.e., the ESDOF representation of capacity curve. As shown in Figure 2, the ESDOF formulation of the ATC-40 is slightly different from that of the FEMA-356. This discrepancy is discussed later below.

Figure 1: Basic steps in the NSPs

Seismic Demand Estimation This step involves the estimation of a target displacement of the control node, i.e., the peak displacement at the roof, in conjunction with an elastic (2% or 5% damping) smoothed spectrum. Presented in ADRS (Acceleration Displacement Response Spectrum) format, the elastic spectrum represents hazard at the site. This step is where the ATC-40 procedure fundamentally differs from the FEMA-356 procedure. The CSM of the ATC-40 uses highly damped elastic spectrum, accounting for hysteretic energy dissipation at the maximum excursion, to estimate the target displacement, , of the performance point (Figure 1(b)). The equivalent damped elastic spectrum is derived from the elastic (2% or 5% damping) spectrum using the approach by Newmark and Hall (1982).

p∆

3

The Displacement Coefficient Method (DCM) is adopted in the FEMA-356 for estimating the target displacement, (Figure 1(b)). As the name implies, this approach determines p∆ p∆ by applying modification factors to the ESDOF elastic displacement, e∆ , i.e.,

*3210 ep CCCC ∆=∆ , (1)

where the description of all the modification factors are given in the Notation section. The ESDOF elastic displacement, , is determined on the basis of a period corresponding to the effective stiffness of the ESDOF system. The effective period is assumed to be the fundamental period of the structure.

*e∆

It is worth noting that the use of highly damped elastic spectra in the CSM of the ATC-40 was highly criticized as there is no robust physical justification for using equivalent viscous damping to represent hysteretic energy dissipation at peak displacement (Krawinkler 1995). Furthermore, there may not be clear relationship between the secant period associated with the performance point and dynamic response of the inelastic system. Consequently, improvements in the representation of this seismic demand have been proposed. Fajfar (1999), for example, suggests the use of inelastic spectra in lieu of the damped spectra. Where only smoothed spectrum is available, an established R-µ -T relationship (e.g., Nasser and Krawinkler 1995, Vidic et al. 1994) is applied to the spectrum. Black and Aschheim (2000) introduce Yield Point Spectra (YPS) as an alternative. Similarly, the coefficients , , and in the DCM of the FEMA-356 are subject to debate, and recent improvements have accordingly been recommended, e.g., in FEMA-440 (ATC 2005).

1C 2C 3C

Performance Evaluation Once the target displacement is estimated, the ensuing response quantities of interest are determined. These response quantities are then compared to the corresponding performance criteria set based on the performance objectives (Figure 1(c)). ESDOF Representations Figure 2 graphically summarizes the ESDOF representations per ATC-40 and FEMA-456. As shown in the figure, both methods use the same formulation to relate displacement of the ESDOF system, , and roof displacement of the MDOF system, *∆ ∆ , i.e.,

1

*

Γ∆

=∆ , (2)

where = modal participation factor with the top modal amplitude set to 1.0. The two methods, however, are slightly different in relating the MDOF base shear and the ESDOF force. The ATC-40 uses the relationship derived from modal superposition method as

1Γ

4

WVC

1

* 1α

= , (3)

where = modal mass effective coefficient. The FEMA-356 approximates the relationship as 1α

WVC 1

* Γ= , (4)

i.e., is assumed. If higher mode ESDOF representations are to be considered, 11 /1 α≡Γ iΓ is a poor approximation to , where i (> 1) is the mode number. iα/1

Figure 2: Load-deformation responses of MDOF and ESDOF systems

If an elastic mode shape vector and its mode shape properties are used as the shape vector of the lateral force distribution in generating the capacity curve and to calculate 1Γ and , the ATC-40 approach results in a match between the period of vibration of ESDOF system and the period associated with the mode shape. The same condition, however, does not apply to the FEMA-356 approach because of the approximation of Eq. (4).

1α

Lateral Load Patterns for Generating Pushover Curves As discussed in the previous section, various lateral load patterns are possible in developing capacity curves. The pushover methods differ in whether the shape vector remains proportional to an initial shape (which may be the first mode or another displacement pattern) or evolves as the onset of material nonlinearity causes softening of the structure, and in whether one or multiple modes are considered. Shape vectors in the lateral force distributions are described now; most of which are illustrated in Figure 3 for the structural model of Figure 5.

5

Figure 3: Examples of shape vectors for the 3-story steel moment-resistant frame of Figure 5

Single-Mode Shape Vectors Inverted Triangular. The inverted triangular pattern uses a shape vector that increases linearly with height. Uniform. This pattern uses a shape vector that is uniform with height, assuming assumes that the acceleration in the MDOF model is constant over its height. Sometimes termed “rectangular,” this shape is usually used to obtain a conservative base shear estimates. Code. The shape vector of the “code” load pattern follows the code formula. The shape varies from an inverted triangular shape for periods less than 0.5 sec to a parabolic shape for periods greater than 2.5 sec as a means to consider higher mode effects. The lateral force coefficient for the x-th floor, , is given by (e.g., FEMA-368 (BSSC 2001)) vxC

∑=

= n

i

kii

xxvx

hw

hwC

1

, (5)

where and are the weights of the i-th or the x-th floor, and are the height of the i-th or the x-th floor above the base, and is an exponent that varies linearly with period from 1.0 for

iw xw ih xhk

T < 0.5 sec to 2.5 for T > 2.5 sec. Since lateral forces are equal to the product of floor mass and the amplitude of a shape vector at each floor, the shape vectors are proportional to . kh First Mode. The shape vector of this load pattern follows the elastic first mode shape. SRSS. The SRSS (Square-Root-of-the-Sum-of-the-Squares) load pattern has a profile of lateral force distribution necessary to generate the SRSS story shear profile. The SRSS story shear profile is determined based on an SRSS combination of the modal story shears. Elastic spectral amplitudes and modal properties are used in determining the modal story shears even when nonlinear response is anticipated. A minimum of 90% of modal mass sufficient number of modes to represent at least 90% of the mass is generally included.

6

Adaptive. Varying shape vectors are utilized to recognize that softening of the capacity curve reflects a reduction in stiffness. Different adaptive shape vectors have been proposed, including, among others, Bracci et al. (1997), Gupta and Kunnath (2000), and Elnashai (2001). Multiple Mode Procedures Recognizing the importance of MDOF effects on the NSPs, researchers have recommended multiple mode procedures which consider response in several modes (e.g., Sasaki et al. 1998, Chopra and Goel 2001, Jan et al. 2004). Response in each mode may potentially be nonlinear, and potential interactions among the modes are not considered. Perhaps, the most recognized method in this category is the so-called Multimode Pushover Analysis (MPA) procedure by Chopra and Goel (2001). In this procedure, response quantities are estimated as SRSS combinations of the response quantities determined in each independent modal pushover analysis. Pushover analyses in each mode use lateral forces proportional to the mode shape amplitude and mass at each floor, and the mode shapes are assumed constant throughout the analysis. An R-µ -T or R- -T relationship may be used to estimate the peak displacement response in each mode, based on an ESDOF system, when nonlinear response is expected.

1C

0

0.1

0.2

0.3

0.4

0.5

-1 0 1 2 3

Base Shear/Weight

Roof Displacement/Height (%)

Mode 1

Mode 2Mode 3

4

SAP2000DRAIN-2X

Figure 4: Reversal of the third mode pushover curve for the steel frame of Figure 5

The implementation of this method can be hampered by the existence of reversal in the higher mode capacity curves. Typical higher mode capacity curves display softening behavior similar to that observed in first mode pushover analyses. In some cases, however, the higher mode force patterns cause the roof displacement to first increase in one direction and then reverse, resulting in oddly-shaped capacity curves (Figure 4). Changes in the rate of increase in the roof displacement or outright reversals lead to these peculiar higher mode pushover curves. This behavior can occur whether or not P-Delta effects are considered in the analysis. Recognizing that the roof displacement may not always be the best index as a basis for establishing the properties of ESDOF systems, Hernández-Montes et al. (2004) developed an alternative index, known as an energy-based displacement. Tjhin et al. (2005) recently published

7

results suggesting that first mode estimates of peak roof displacement are improved when the energy-based displacement is used in place of the roof displacement to establish the properties of the first mode ESDOF system. ACCURACY OF NONLINEAR STATIC PROCEDURES NSPs are more advanced than the LSPs as they consider redistribution of forces due to yielding and provide insight into the expected failure mechanism. The NSPs are also known to adequately provide peak floor and roof displacements. However, since they are based on ESDOF approximations, it is reasonable to expect that NSPs may not be adequate in predicting response quantities affected by MDOF effects, such as interstory drifts, story shears, and overturning moments. The effects of higher modes on NSP results can be demonstrated using modal formulations for elastic response as follows. The vector of modal peak displacements for the i-th mode, , is determined as

i∆

iidii TS φ∆ )(Γ= , (6)

where = spectral displacement associated with the period , and = the i-th mode shape vector. The corresponding modal lateral force vector, , is

)( id TS iT iφ

iF

iiaii TS MφF )(Γ= , (7) where = the mass matrix, and = spectral acceleration associated with the period . Because

M )( ia TS iT

iΓ and typically assume their largest values for the first mode, higher mode contributions to displacements typically are minor. By contrast, structures with fundamental periods in the constant velocity portion of the response spectrum will have higher mode spectral accelerations that may be several times the fundamental mode spectral acceleration, causing higher modes to contribute more to lateral forces than to displacements.

dS

Figure 5: The 3-story (regular and weak-story) steel frames of the ATC-55 project

8

Figure 6: The 9-story (regular and weak-story) steel frames of the ATC-55 project

Figure 7: The 8-story structural wall of the ATC-55 project

The inadequacy of the NSPs in predicting the interstory drift and force-type responses was confirmed in the recent ATC-55 MDOF study. In this study, the accuracy of the NSPs with

9

regard to MDOF effects was evaluated by comparing peak global response quantities obtained from the NSPs for four steel frames and one structural wall with those obtained from nonlinear dynamic analyses at different values of peak roof drift, representing elastic response and various degrees of nonlinear response. The steel frames used in the ATC-55 study (Figures 5 and 6) are two 3- and 9-story moment-resistant frames designed as part of the SAC joint venture (2000) as reported in FEMA-355C and a modification to these buildings to induce weak story mechanism into the first story. The selected structural wall (Figure 7) is one of the two longitudinal walls of the midrise building described in the ATC-40.

Figure 8: Peak floor displacement response

The NSPs evaluated include those with single mode load patterns and the MPA procedure by Chopra and Goel (2001). Because of the reversal found in some of the higher mode pushover curves, the MPA procedure was modified such that the higher mode capacity curves are of linear shape. The adaptive pushover was based on varying first mode shape adjusted at drift increments of 0.5% of the height of the building.

10

The global response quantities included in the evaluation are peak floor displacements, peak interstory drifts, peak story shears, and peak overturning moments. The interstory drifts were computed as the largest difference in the displacements of adjacent floors, and were normalized by the height of the story. Story shears were computed as the sum of the shears in the columns. Overturning moments at each floor were computed as the overturning moment for the floor above plus the product of the story shear and the story height for the story. For the dynamic analysis results, peak values over the height of a structure often occurred at different instants in time.

Figure 9: Peak interstory drift response

The ground motions for the nonlinear dynamic analyses consisted of 11 ordinary Site Class C motions and 4 near-fault motions. The ordinary ground motions were scaled such that the resulting peak roof drifts matched the predetermined roof drifts of the NSPs. For the near-fault motions, the structural models in pushover analysis were pushed until the drifts matched the peak drifts obtained from nonlinear dynamic analyses. Figures 8-11 show examples of peak response estimates obtained from the study. These figures also compare the corresponding dynamic response quantities with the deterministic estimates

11

made using the pushover methods. The minimum, maximum, mean, median, and mean plus and minus one standard deviation (SD) values of the dynamic response quantities are plotted for each floor or story with the deterministic results obtained using the pushover techniques for comparison.

Figure 10: Peak story shear response

As can be seen in Figure 8, NSPs generally provide good estimates of peak floor displacement response, even when the roof drifts are very high, i.e., 4% for the steel frames and 2% for the structural wall.

12

By contrast, the NSPs provide poor estimates of peak interstory drift response, as illustrated in Figure 9 for the elastic response (at 0.5% drift for the steel frames). Similar observations can be obtained for story shear response (Figure 10) and overturning moment response (Figure 11). The NSPs have the propensity to underestimate these response types. The inclusion of higher mode response as in the multiple mode procedure does not help capture the effects of higher modes accurately.

Figure 11: Peak overturning moment response

IMPROVEMENTS ON NONLINEAR STATIC PROCEDURES Numerous proposals to address the higher mode effects in the NSPs have been submitted since the publications of ATC-40, FEMA-273/274, and FEMA-356. Most of the proposals are based on (1) the use of adaptive load patterns (e.g., Bracci et al. 1997, Gupta and Kunnath 2000, and Elnashai 2001), (2) the use of multiple mode load patterns (e.g., Sasaki et al. 1998, Chopra and Goel 2001), and (3) the combination of the two (e.g., Aydinoğlu 2003). More recommendations for improvements are expected to be come, as a result of extension the NSPs for evaluating local responses, such as plastic hinge rotations (e.g., Jan et al 2004).

13

One promising improvement to be discussed herein is the so-called Scaled Nonlinear Dynamic Procedure (Scaled NDP) (Aschheim et al 2004), as described in the FEMA-440 document. The idea of this approach was conceived during performing the MDOF study of the ATC-55 project. As briefly discussed in the previous section, the study assumed that the NSPs could be counted on to estimate the roof displacement accurately. Ground motions considered to represent those that could occur at a particular site were scaled (iteratively) so that the peak roof displacement observed in each dynamic analysis was equal to a predetermined value. The dispersion in the response quantities obtained in this way was found to be relatively small. In many cases, it was observed that the results of a single time history response of a multi-degree-of-freedom model are better than those obtained from any pushover methods (See Figures 8-11). The approach retains the appeal of the NSPs in estimating the target displacement and as a simple tool to understand the nonlinear response and failure mechanism, while it simplifies the ground motion selection in the current NDPs. The basic steps are as follows. Step 1. The target displacement is estimated using a conventional NSP with lateral forces being applied in proportion to the first mode shape amplitude and mass at each floor level. The target displacement may be estimated using procedures recommended in the FEMA-440. Step 2. A set of n (≥ 3 recommended) ground motion records representing the characteristics of the hazard (e.g., magnitude, distance, site class) are selected. For each record, a nonlinear dynamic analysis is conducted, with the record scaled iteratively until the peak displacement of the control point is equal to the estimate determined in Step 1. Peak values of the response quantities of interest from the results of each analysis are extracted, and the sample mean, nx , of each peak quantity of interest is computed. Step 3. To address sampling error, arising from the limited number of observations of each quantity, and to estimate peak response quantities at the mean plus κ standard deviation level, multiply the sample mean, nx , by )COV1( κ+c , where COV = the coefficient of variation of the response quantity for a given level of roof drift, and is given by c

1COV28.11−

⎟⎠⎞

⎜⎝⎛ −=

nc , (8)

based on the assumption that the response quantities are normally distributed. The term κ is 0 where estimates of the true mean are sought. The coefficient 1.28 is the value of the standard normal distribution at the 90th percentile, representing a confidence level of 90%. Using Eq. (8), there is a 90% probability that nxc )COV1( κ+ exceeds the true mean plus standard deviations. κ

CONCLUSIONS From the discussion presented in this paper, one may conclude that NSPs (1) provide useful estimates of peak floor and roof displacements, (2) serve as a tool that can be used to evaluate

14

redistribution of forces as the structure under consideration goes into nonlinear range, and (3) provide insight into the expected mechanism. However, the NSPs have to be used with cautions as they are inadequate in predicting response quantities affected by higher modes, such as story shears and moments. Current procedures are not sufficient to capture these effects. As a result, other analytical approaches are needed for estimating response quantities that have significant contributions from higher modes. The scaled NDP discussed briefly herein offers an alternative to overcome this problem. It should be noted that the accuracy of NSPs is not only dependent on the MDOF effects. Other factors, such as seismic demand characterization, elastic displacement modification to include nonlinear response, representation of hysteretic characteristics, structural modeling uncertainties, representation of P-Delta effects in ESDOF systems, use of ESDOF systems for estimating displacement demands, influence seismic motion characteristics on ESDOF responses, affect the results of NSPs and have been subjects of much debate. These topics are beyond the scope of this paper; the readers are referred elsewhere, e.g., FEMA-440 document. NOTATION The following symbols are used in this paper: c = scale factor (parameter in the Scaled NDP);

0C = modification factor to relate ESDOF displacement and roof displacement (parameter in the DCM of FEMA-356);

1C = modification factor to relate expected maximum inelastic displacements to displacements calculated for linear elastic response (parameter in the DCM of FEMA-356);

2C = modification factor to represent the effect of hysteresis shape on the maximum displacement response (parameter in the DCM of FEMA-356);

3C = modification factor to represent increased displacement due to P-Delta effects (parameter in the DCM of FEMA-356);

*C = force coefficient of ESDOF system; viC = lateral force coefficient for the i-th floor (parameter in the DCM of FEMA-356);

COV = coefficient of variation (parameter in the Scaled NDP); iF = modal lateral force vector;

g = acceleration of gravity;

effh = *

11 mhm

n

iiii∑

=

φ = effective height of ESDOF system;

ih = height of the i-th floor of an n-story building, measured from the base; k = lateral stiffness of MDOF system; also coefficient in code formula of lateral force

distribution according to FEMA-368; *k = lateral stiffness of ESDOF system; im = lumped mass of the i-th floor of an n-story building;

15

*m = ∑ =seismic mass of ESDOF system; =

φn

iiim

11

M = mass matrix; also the number of ground motion records (parameter in the Scaled NDP);

n = number of stories; R = ratio of the elastic strength demand to the yield strength coefficient;

aS = pseudo spectral acceleration;

dS = spectral displacement; T = first-mode period of vibration;

iT = period of vibration for the i-th mode; V = lateral base shear of MDOF system subjected to monotonic increasing lateral forces;

yV = lateral base shear of MDOF system at yield; *yV = yield strength of the ESDOF model;

iw = weight of the i-th floor of an n-story building; W = seismic weight of MDOF system;

nx = mean of response quantity (parameter in the Scaled NDP);

1α = ( ) ⎟⎠

⎞⎜⎝

⎛ φ∑∑==

n

iii

n

ii mmm

1

21

1

2* = ∑=

Γn

iimm

1

*1 = first-mode effective mass coefficient;

∆ = lateral roof displacement subjected to monotonic increasing lateral forces; y∆ = lateral roof displacement at yield;

*∆ = lateral displacement of ESDOF system subjected to monotonic increasing lateral forces;

*y∆ = yield displacement of the ESDOF system;

e∆ = elastic roof displacement; *e∆ = elastic ESDOF displacement;

p∆ = target roof displacement;

y∆ = roof displacement at yield;

i∆ = vector of modal peak displacements for the i-th mode;

1iφ = first mode shape amplitude at the i-th floor of an n-story building;

iφ = the i-th mode shape vector;

1Γ = ∑∑==

φφn

iii

n

iii mm

1

21

11 = first-mode modal participation factor;

iΓ = modal participation factor for the i-th mode; κ = standard deviation level (parameter in the Scaled NDP); and µ = displacement ductility factor.

16

REFERENCES American Society of Civil Engineers (ASCE) (2000). Prestandard and Commentary for the Seismic Rehabilitation of Buildings. FEMA-356, Federal Emergency Management Agency, Washington, DC. Applied Technological Council (ATC) (1996). Seismic Evaluation and Retrofit of Concrete Buildings. ATC-40, Redwood City, CA. Applied Technological Council (ATC) (1997a). NEHRP Guidelines for the Seismic Rehabilitation of Buildings. FEMA-273, Vols. 1 and 2, Federal Emergency Management Agency, Washington, DC. Applied Technological Council (ATC) (1997b). Commentary on the Guidelines for the Seismic Rehabilitation of Buildings. FEMA-274, Federal Emergency Management Agency, Washington, DC. Applied Technological Council (ATC) (2005). Improvement of Nonlinear Static Seismic Analysis Procedures. FEMA-440, Federal Emergency Management Agency, Washington, DC. Aschheim, M., and Black, E. (2000). Yield Point Spectra for Seismic Design and Rehabilitation. Earthquake Spectra, Earthquake Engineering Research Institute, 16:2, 317-335. Aschheim, M. A., Tjhin, T. N., Comartin, C., Hamburger, R., and Inel, M. (2004). The Scaled Nonlinear Dynamic Procedure, Proceedings of the 2004 ASCE Structures Congress, 1-8. Aydinoğlu, N. M. (2003). An Incremental Response Spectrum Analysis Procedure Based on Inelastic Spectral Displacements for Multi-Mode Seismic Performance Evaluation,” Bulletin of Earthquake Engineering, 1, 3-36. Black, E. F., and Aschheim, M. A. (2000). Seismic Design and Evaluation of Multistory Buildings Using Yield Point Spectra, CD Release 00-04, Mid-America Earthquake Center, University of Illinois, Urbana, IL. Bracci J. M., Kunnath S. K., Reinhorn, A. M. (1997). Seismic Performance and Retrofit Evaluation for Reinforced Concrete Structures, ASCE Journal Structural Engineering, 123:1, 3-10. Building Seismic Safety Council (BSSC) (2001). NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures, Part 1 – Provisions. FEMA-368, Federal Emergency Management Agency, Washington, DC. Chopra, A. K., and Goel R. K. (1999). Capacity-Demand-Diagram Methods Based on Inelastic Design Spectrum, Earthquake Spectra, 15:4, 637-656.

17

Chopra, A. K., and Goel, R. K. (2001). A Modal Pushover Analysis Procedure to Estimate Seismic Demands for Buildings: Theory and Preliminary Evaluation, PEER-2001/03, Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA. Elnashai, A. S. (2001). Advanced Inelastic Static (Pushover) Analysis for Earthquake Applications, Structural Engineering and Mechanics, 12:1, 51-69. Fajfar, P. (1999). Capacity Spectrum Method Based on Inelastic Demand Spectra, Earthquake Engineering and Structural Dynamics, 28:9, 979-993. Gupta B., and Kunnath, S. K. (2000). Adaptive Spectra-Based Pushover Procedure for Seismic Evaluation of Structures, Earthquake Spectra, 16:2, 367-392. Hernández-Montes, E., Kwon, O.-S., and Aschheim, M. (2004). An Energy-Based Formulation for First- and Multiple-Mode Nonlinear Static “Pushover” Analyses. Journal of Earthquake Engineering, 8:1, 69–88. Jan, T. S., Liu, M. W., and Kao, Y. C. (2004). An Upper-Bound Pushover Analysis Procedure for Estimating the Seismic Demands of High-Rise Buildings. Engineering Structures, 26:1, 117-128. Krawinkler, H. (1995). New Trends in Seismic Design Methodology. Proceedings of the 10th European Conference on Earthquake Engineering, Balkema, Rotterdam, Vol. 2, 821-830. Nassar, A. A., and Krawinkler, H. (1991). Seismic Demands for SDOF and MDOF Systems. Report No. 95, John A. Blume Earthquake Engineering Center, Stanford University, Stanford, CA. Newmark, N. M., and Hall, W. J. (1982). Earthquake Spectra and Design, Earthquake Engineering Research Institute, Berkeley, CA. Reinhorn, A. (1997). Inelastic Technique in Seismic Evaluations. Seismic Design Methodologies for the Next Generation of Codes, P. Fajfar and H. Krawinkler, eds., Balkema, Rotterdam, 277-287. SAC Joint Venture (2000). State of the Art Report on Systems Performance of Steel Moment Frames Subject to Earthquake Ground Shaking, FEMA-355C, Federal Emergency Management Agency, Washington, DC. 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 Pushover Analysis.” Proceedings of the 6th US National Conference on Earthquake Engineering, Earthquake Engineering Research Institute, Seattle.

18

Tjhin, T. N., Aschheim, M., and Hernández-Montes, E. (2005). Estimates of Peak Roof Displacement Using “Equivalent” Single Degree of Freedom Systems. Journal of Structural Engineering, ASCE, 131:3, 517–522. Vidic, T., Fajfar, P., and Fischinger, M. (1994). Consistent Inelastic Design Spectra: Strength and Displacement. Earthquake Engineering and Structural Dynamics, 23, 502-521.

19