II-Modified Capacity Spectrum Method

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A Modified Capacity Spectrum Methodwith Direct Calculation of SeismicIntensity of Points on Capacity CurveWu Jing a b , Liang Renjie a , Wang Chunlin a b & Zhou Zhen a ba The Key Laboratory on Concrete and Prestressed ConcreteStructures of Ministry of Education of China , Southeast University ,Nanjing, Chinab International Institute for Urban Systems Engineering , SoutheastUniversity , Nanjing, ChinaPublished online: 29 Apr 2011.

To cite this article: Wu Jing , Liang Renjie , Wang Chunlin & Zhou Zhen (2011) A Modified CapacitySpectrum Method with Direct Calculation of Seismic Intensity of Points on Capacity Curve, Journal ofEarthquake Engineering, 15:4, 664-683

To link to this article: http://dx.doi.org/10.1080/13632469.2010.505274

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Journal of Earthquake Engineering, 15:664–683, 2011Copyright © A. S. Elnashai & N. N. AmbraseysISSN: 1363-2469 print / 1559-808X onlineDOI: 10.1080/13632469.2010.505274

A Modified Capacity Spectrum Method with DirectCalculation of Seismic Intensity of Points

on Capacity Curve

WU JING1,2, LIANG RENJIE1, WANG CHUNLIN1,2,and ZHOU ZHEN1,2

1The Key Laboratory on Concrete and Prestressed Concrete Structures of Ministryof Education of China, Southeast University, Nanjing, China2International Institute for Urban Systems Engineering, Southeast University,Nanjing, China

A direct methodology for solving the seismic intensity of each point on the capacity curve is proposed.By utilizing the procedure, a continuous curve between the structural response and the seismic inten-sity, the structural response function, can be easily generated. Unlike previous procedures that searchfor the performance point of a determined seismic intensity, the proposed methodology easily drawsthe full curve without iterations. The procedure is applicable to both a smooth design spectrum andan actual response spectrum. Examples indicate the methodology is accurate and fast, and convenientto be combined with existing procedures, such as Modal Pushover Analysis.

Keywords Nonlinear Static Procedure; Next-Generation Performance-Based Seismic Design;Seismic Intensity; Structural Response Function; Performance Point; Capacity Spectrum Method

1. Introduction

In order to help engineers and designers work better with stakeholders to identify theprobable seismic loss of new and existing buildings as well as to select an appropri-ate performance level during the design stage, FEMA (Federal Emergency ManagementAgency) contracted with ATC (Applied Technology Council) to initiate a research programto develop Next-generation Performance-based Seismic Design procedures and guidelines[ATC, 2009; FEMA, 2006]. The main concept of this program is to divide the whole per-formance evaluation process into four independent but closely related functions: the hazardfunction, response function, damage function, and loss function. The response function,which establishes the relationship between the structural response and increasing seismicintensity using a reasonable methodology, plays an important role in this evaluation.

In general, there are two methods to investigate the structural response during seismicexcitation: nonlinear response history analysis (NL-RHA) and a nonlinear static proce-dure (NSP). Incremental dynamic analysis (IDA) [Vamvasikos, 2002] is a typical nonlineardynamic analysis method. The procedure involves exciting a structure with one (or more)ground motions, each of which is scaled to multiple levels of intensity. Thus, the structuralresponses at different levels of intensity that produce one (or more) curves of input-output

Received 12 January 2010; accepted 20 June 2010.Address correspondence to Wu Jing, The Key Laboratory on Concrete and Prestressed Concrete Structures

of Ministry of Education of China, Southeast University, Nanjing 210096, China; E-mail: [email protected]

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Direct Calculation of Seismic Intensity 665

measurements are obtained, where the input uses parameters that describe the groundmotion intensity measure (IM), and the output represents the structural damage measure(DM). The IDA curve reflects the structural response rule at different levels of seismicintensity, but each point in the curve requires a nonlinear response history analysis of thestructure. In such an analysis, convergence problems are often encountered when struc-ture demonstrates strongly nonlinear behavior. Although it has been adopted by FEMA350[FEMA, 2000a] and other standards as the latest method to evaluate the global collapsecapacity of steel frames, this method is difficult to extend because of the calculationconditions and the computing cost.

Relatively, the nonlinear static procedure, which is also called static pushover analysis(SPO), is easier to practice and can take into account the major nonlinear characteristics ofstructure. When combined with the capacity spectrum evaluation approach, it yields a betterunderstanding of the structural characteristics at a determined seismic intensity by search-ing for the “performance point” and extracting the response index. Based on these features,NSP is widely applied in the practice of Performance-Based Seismic Design (PBSD) andhas also been adopted by some standards and codes.

Based on the results of the nonlinear static procedure, this article derives a directmethod for calculating the seismic intensity of each point on the capacity curve by estab-lishing the relationship between the dynamic factor of the pseudo- (spectral-) accelerationand the ductility factor of the elasto-plastic system to determine the structural responsefunction curve.

2. Nonlinear Static Procedure and Seismic Intensity

The nonlinear static procedure involves subjecting a structure with vertical loads to mono-tonically increasing equivalent lateral forces that represent the inertia forces and adjustingthe stiffness of the calculation model as the structure enters the inelastic state until a pre-determined target displacement is reached. The equivalent lateral forces are defined by ashape vector:

{F} = λ [M] {φ}, (1)

where {F} is the equivalent lateral forces vector, λ is the loading scale factor, [M] is themass matrix of structure, and {ø} is the selected shape vector.

Based on the NSP results, a pushover curve of structure is plotted with the total baseshear V as ordinate against the lateral roof displacement �r as abscissa. In general, apushover curve describes the nonlinear behavior of a structure as it gradually changesfrom elastic to inelastic, but it cannot determine the structural response at certain seismicintensities. To achieve this, the “performance point” corresponding to the certain seismicintensity must be determined from the pushover curve. In the FEMA356 method [FEMA,2000b], the target displacement under a certain intensity is determined by multiplying theelastic response by a number of modification factors based on the statistical analysis, andthen the performance point is determined in the pushover curve. However, ATC-40 [ATC,1996] suggested combining the capacity curve converted from the pushover curve with thedemand curve to reach the performance point iteratively. According to Akkar’s research[Akkar, 2007], the former tends to overestimate the deformation. While, in the ATC-40capacity spectrum method (CSM), the inelastic hysteretic characteristic of the structure isregarded as an additional damping of the equivalent linear structure system; thus, the elasticresponse spectrum, which is typically assumed to have a 5% damping ratio, is modified bythe equivalent damping. The base shear and roof displacement are converted to the spectral

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acceleration and spectral displacement (Sa-Sd format) as follows, which is called capacitycurve:

Sa,i = Vi

αmM(2)

Sd,i = �r,i

γφr, (3)

where Sa,i and Sd,i are the spectral (or pseudo — pseudo will be used in the following text)acceleration and spectral displacement, respectively, at point i; am is the mass participationfactor of the shape vector; M is the total mass of the structure; γ is the modal participa-tion factor of the shape vector; and ør is the amplitude of the shape vector at the rooflevel.

It is evident that the shape vector is an important factor that influences the calcula-tion precision of the CSM. The traditional pushover analysis is based on the assumptionthat the response is controlled by the fundamental mode and that the mode shape remainsunchanged after the structure yields. A satisfactory prediction can be obtained for low- ormedium-rise structures in which higher modes contribute less and plastic development hasa diminished influence on mode changes [Krawinkler, 1998]. Chopra and Goel proposedmodal pushover analysis (MPA) [Chopra, 2002], which has greatly influenced subsequentresearches. In this method, pushover analyses are carried out separately for each significantmode, and the contributions from individual modes to the engineering demand parameters(EDPs) are combined using an appropriate combination rule (e.g., SRSS or CQC). Theupper-bound pushover analysis, proposed by Jan et al. [2004], is based on a single shapevector, which is a combination of the first mode shape and a scaled second mode shape.Kunnath [2004] used a modal-combination-based lateral force pattern to envelop the dis-placement demand. On the other hand, researchers have proposed several adaptive loadingpatterns to consider the change in the dynamic characteristics after the structure enters theinelastic range. Eberhard [1993] used the mode shape based on the secant stiffness of eachstage. Gupta proposed an adaptive pushover procedure in which the conventional responsespectrum analysis is applied at each pushover step [Gupta, 2000]. Mori [2006] utilized thedeflected shape that corresponded to the maximum drift of the pushover analysis to approx-imate the post-elastic first mode shape, and Kim [2008] suggested that the loading patternbe proportional to the total seismic masses at the floor and roof levels to estimate the peaklateral displacement demand. The adaptive modal combination (AMC) procedure [Kalkan,2006] and the improved MPA procedure [Mao, 2008] improved the MPA procedure inwhich the mode shapes remain invariant.

FEMA356 suggested that at least two lateral force distribution patterns should beapplied [FEMA, 2000b]. These two force patterns are selected from two groups separately.The first group includes an inverted triangle, the fundamental modal shape to determine theforce pattern, and the shear distribution derived from the modal combination. They mainlyapproximate the distribution pattern in the elastic range. The second group includes a lat-eral force distribution that is proportional to mass and an adaptive loading pattern. Thisgroup mainly considers the mode shift characteristic after yielding.

On the other hand, the demand curve of the ground motion is defined by the over-damping elastic response spectrum. The pseudo acceleration-spectral displacement formatis used as follows and is plotted with the capacity curve in one graph:

Sd,i = T2

4π2Sa,i. (4)

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Direct Calculation of Seismic Intensity 667

Note that, in general, the intersection of the capacity curve and the demand curve byarbitrary damping is physically meaningless. This is because the damping represented bythe demand curve may not represent the actual damping of the intersection. Only whenan appropriate point is selected on the capacity curve, and the demand curve intersectedin the determined point with the capacity curve behaves the same equivalent damping asthe point, that the particular point can be regarded as a “performance point”. ATC-40 sug-gests an iterative calculating method to search for the performance point for a given seismicintensity, as shown in Fig. 1. The estimated performance point (EPP), (Sd,es,Sa,es), can bedetermined by “equal displacement approximation”. That is, the inelastic spectral displace-ment is taken as that of the elastic displacement. Then, the hysteretic damping representedby the EPP is calculated to develop a reduced demand spectrum, which intersects withthe capacity curve at the trial performance point (TPP), (Sd,tr,Sa,tr). If the distance betweenEPP and TPP exceeds the acceptable tolerance, iteration is required to obtain a convergencesolution.

The iteration procedure presented in ATC-40 provides an illustration of how the capac-ity matches the seismic demand. However, existing researches pointed out that there isno direct physical relationship between the hysteretic energy dissipation of the maxi-mum excursion and equivalent damping, particularly for highly inelastic systems, and thatthe period associated with the secant stiffness may have little to do with the dynamicresponse of the inelastic system [Krawinkler, 1995; Fajfar, 1999]. Fragiacomo investi-gated the errors caused by a demand curve based on the single-degree-of-freedom (SDOF)system [Fragiacomo, 2006]. The research indicated that the smaller hysteretic loop has alower seismic response for the short period system and demonstrated that the concept ofdemand curve defined by an equivalent linear system and additional damping is inconsistentwith the actual situation. Some other researches included similar conclusions [Gencturk,

Sa,tr

Sd,esSd,tr

Sa,es

Sd

Sa

reduction

EPP (estimated performance point)

capacity curve

spectrum w

ith 5%

damping

TPP (trial performance point)

iteration

spectrum with 5% + hysteretic

damping of EPP

FIGURE 1 Illustration of the iterative procedure suggested by ATC-40 to search for theperformance point.

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2008; Akkar, 2007]. Therefore, Fajfar [1999] and Chopra [1999] suggested using theinelastic response spectrum to determine the demand curve; Chen et al. [2007] used thedemand curve generated by an inelastic response spectrum to design steel bridge piers andarch bridges with hysteretic dampers. Based on this consideration, the inelastic responsespectrum is used in the procedure proposed in this article.

The inelastic pseudo-acceleration response spectrum is defined as the maximumpseudo-acceleration response of a series of SDOF systems with the same ductility factorbut different elastic periods excited with a selected ground motion. This response spectrumis referred to as an actual response spectrum, which depends on the configuration and mag-nitude of the ground motion wave, inherent damping, the ductility factor and the post-yieldstiffness of the structure. This actual spectrum can be quickly generated by many commer-cial programs, such as Bispec [Hachem] and SeismoSignal [SeismoSoft]. Another type ofspectrum, which is referred to as a design spectrum, is based on the statistical analysis andsmooth simplification of the inelastic response spectrums of a group of ground motionswith similar soil conditions and is commonly regarded to represent the ground motion thatmay occur in the future on the site. In the PBSD process, the former is used to evaluate theresponse of a structure with a determined ground motion, and the latter is usually used fordesign purposes.

Independent of the selected shape function and response spectrum, an appropriatemethod should be adopted to obtain the response of a given seismic intensity of the equiv-alent SDOF system that is converted from the structure: the “performance point” on thecapacity curve. The performance point should have the same ductility factor as the demandcurve, which intersects the capacity curve; on the other hand, the demand curve is deducedfrom the elastic response spectrum with the specific ductility factor. It should be mentionedthat the ductility factor is undetermined before the performance point is solved out. Thus,an iteration process similar to the ATC-40 method is necessary to determine the position ofthe performance point accurately. Another method is to prepare a family of demand curveswith different ductility factors and then investigate every intersection of the demand curvewith the capacity curve to choose a minimum-error point. Kalkan [2006] suggested that theinterval of the ductility factors in this family should be about 0.25.

3. Seismic Intensity of Each Point on the Capacity Curve

In the frame of the Next-generation Performance-based Seismic Design, the structuralresponse function establishes the continuous relationship between the seismic intensityand the structural response. The spectral displacement under a certain shape function isthe most fundamental response index and is also the basic index in existing methods. Afterthe performance points of the interested modes at a given seismic intensity are determined,the EDPs of the structure can be obtained by extracting them from the analysis results andcombined according to a combination rule. The structural response function is plotted byconnecting the structural response at different intensities. As mentioned above, the tradi-tional methods require iteration or choosing one curve from several demand curves withdifferent ductility factors to determine the performance point, which results in a tediousand complex procedure. If we start from a point on the capacity curve to calculate the cor-responding seismic intensity, the calculation procedure is direct and convenient . Based onthe inelastic spectrum reduced from the smooth elastic spectrum with simplified formula-tions, Dolsek expanded the N2 procedure [Fajfar, 1988] to generate the structural responsefunction of a structure [Dolsek, 2005, 2007]. However, the methodology proposed in thisarticle deals with an actual spectrum respect to a determined ground motion as well as asmooth spectrum.

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Direct Calculation of Seismic Intensity 669

For a point on the capacity curve (Sd,i,Sa,i), the ductility factor μi is defined as follows:

μi = Sd,i

Sdy, (5)

where Sdy is the spectral displacement of the global yield point of the structure. To deter-mine the global yield point, the capacity curve is usually simplified as a bilinear curve thathas the same area with respect to the axis of spectral displacement, which is referred toas “the equal energy rule”. The post-yield stiffness can be obtained from the simplifiedbilinear curve simultaneously.

Now, given the ductility factor μi, which seismic intensity will produce the spectraldisplacement Sd,i? Commonly, the magnitude of the seismic intensity can be measured bythe peak of the ground motion acceleration Apg,i. Obviously, Apg,i is the only unknownquantity in the calculation.

For an elastic SDOF system with an initial period T and damping ratio ξ , the pseudo-acceleration is defined as the following:

Sae = ω

∣∣∣∣∫ t

0

··xg (τ ) e−ξω(t−τ ) sin ω (t − τ) dτ

∣∣∣∣max

, (6)

where ω is the angular frequency of the structure and ω = 2πT ; and xg (τ ) is the accel-

eration record of the ground motion. In addition xg (τ ) = Apg xg1(τ ) , where xg1 (τ ) is thenormalized ground acceleration record for which the peak acceleration is 1, and Apg is thepeak acceleration of the ground motion; thus,

Sae = Apgβ, (7)

where β is the dynamic factor of pseudo-acceleration for the elastic system, which is relatedto the period T , the damping ratio ξ and the configuration of the ground motion:

β = ω

∣∣∣∣∫ t

0

··xg1 (τ ) e−ξω(t−τ) sin ω (t − τ) dτ

∣∣∣∣max

. (8)

Correspondingly, for a similar elasto-plastic structural system with ductility factorμ and post-yield stiffness ratio α, the inelastic pseudo-acceleration can be expressed asfollows:

Sa = Sae

R (T , μ, α)= Apg

β (T , ξ)

R (T , μ, α), (9)

where R is the reduction factor of the inelastic pseudo-acceleration with respect to theelastic pseudo-acceleration, which is related to the ductility factor μ, the period T andthe post-yield stiffness ratio α. Note that the period T in Eq. (9) is the elastic period ofthe structure. Define βp as the dynamic factor of pseudo-acceleration for the elasto-plasticsystem, that is

βp = β (T , ξ)

R (T , μ, α)(10)

which yields

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Sa = Apgβp (T , μ, α, ξ). (11)

In Eq. (11), βp is related to the configuration of the ground motion, the period, theductility factor, the post-yield stiffness and the damping ratio. However, it is independentof the peak value of ground acceleration. Equation (11) implies the rule that the inelas-tic pseudo-acceleration response is proportional to the seismic intensity for one groundmotion. Therefore, the seismic intensity corresponding to the point (Sd,i,Sa,i) on the capacitycurve can be derived according to the following formula:

Apg,i = Sa,i

βp,i. (12)

According to Eq. (10), βp is essentially the inelastic pseudo-acceleration of the struc-ture at the scaled ground motion with unit peak acceleration and can be easily generatedfrom commercial programs. The damping ratio ξ , period T , and post-yield stiffness ratio α

are only related to structure characteristics, and the ductility factor μ and the configurationof the ground motion are known. The Apg–Sd curve can then be plotted by repeating thisprocess for each point on the capacity curve.

4. Dynamic Factor of Pseudo-Acceleration for Elasto-Plastic System

In the process stated above, the dynamic factor of pseudo-acceleration for an elasto-plasticSDOF system with different ductility factors should be prepared; this can be accomplishedin two ways.

The first is applicable to an actual response spectrum that corresponds to a speci-fied ground motion. This is essentially to calculate the pseudo-acceleration response ofthe equivalent SDOF system subjected to the scaled motion at unit peak acceleration withdifferent ductility factors. Using the initial period T , post-yield stiffness ratio α, dampingratio ξ of the SDOF system with an equivalent bilinear curve and the configuration of theground motion, the βp–μ curve can be obtained with the following steps.

1. Scale the amplitude of the specified ground motion to obtain xg1 (τ ), which is withunit peak acceleration.

2. Construct a SDOF system with unit mass and then determine the stiffness to matchits initial period to T . The damping ratio of the system is ξ , and the post-yieldstiffness ratio is α. Here, the use of unit mass results in that the maximum baseshear V of the system is just the pseudo-acceleration response Sa. And because theground motion is with unit peak acceleration, this value is also the dynamic factorof the pseudo-acceleration βp.

3. Choose a pair of yield displacement and yield force (Sdy,i,Say,i) and then subject thesystem to xg1 (τ ) for nonlinear response history analysis to calculate the maximumdisplacement response Sd,i and maximum base shear response Sa,i (which is alsoβp,i). According to Eq. (5), the ductility factor μi can be determined using Sd,i andSdy,i. The resulting pair of (βp,i,μi) is one point on the βp-μ curve.

4. Repeat step 3 with different yield forces and then connect the resulting points todraw the βp–μ curve.

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Direct Calculation of Seismic Intensity 671

βp

1.5

1.0

0.5

01.0 1.5 2.0 2.5 3.0

a

c

1

2

3b

μ

FIGURE 2 Relationship between βp and μ for a SDOF system with a period of 1 s that issubjected to El-Centro motion. The system has a damping of 5% and a post-yield stiffnessratio of 0.

In general, βp decreases as the ductility factor μ increases. However, it is interestingthat their relationship is not always monotonic. This result is primarily caused by the ran-domness of the frequency components of the ground motion. Figure 2 shows a segment ofthe curve between the two parameters for a SDOF system with a period of 1 s when excitedby the 1940 El-Centro ground motion. The system has a damping of 5% and a post-yieldstiffness ratio of 0. The motion is the north-south component of the motion recorded in El-Centro during the Imperial Valley, California earthquake of May 18, 1940 [Chopra, 2001]and is used in the examples in this paper as a typical ground motion. It can be observed thatthere are three values of βp,i, when μi is 1.5, which are marked by 1, 2, and 3. In particular,for point 2 in the figure, the increase of βp,i, which means a decrease in the seismic inten-sity (according to Eq. (12)), leads to an increase of the ductility factor and the structuraldisplacement response. This indicates that the solution is unstable. For design purposes,the conservative larger value of βp,i, (point 1) should be used to obtain a relatively minorseismic intensity. Thus, the curve can be generated by connecting points a and c. Therefore,there is a significant step in the βp–μ curve where the step distance depends on the intervalof μ. The step is shown as the dashed line in the figure. Note that the points interpolatedin the step region do not agree with the actual conditions but are relatively conservative.The use of an exact βp–μ curve can improve the calculation precision, while the simplifiedstepped curve will give approximate one to one relationship between seismic capacity andspectral displacement. In the numerical examples in this article, the former, which is shownas the solid line in the figure, is used. However, the latter can be easily generated usingcommercial programs, such as Bispec mentioned above.

Another way to generate the βp–μ curve is applicable to the design spectrum. Underthis condition, the dynamic factor of pseudo-acceleration for an elastic SDOF system, β, isdefined by the elastic design spectrum as follows:

β = Sae

Apg(13)

and the reduction factor R, which is the ratio of the inelastic pseudo-acceleration relativeto the elastic pseudo-acceleration, is a function of the ductility factor μ, initial period Tand post-yield stiffness ratio α. Many proposals suggested a reduction factor R [Newmark,

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1982; Vidic, 1994]. Then the βp–μ relationship may be calculated directly for the SDOFsystem with the specified initial period and post-yield stiffness ratio according to Eq. (10).For example, given the elastic design spectrum generated by the ground motions with apeak acceleration of 1 g and 84.1% guarantee (i.e., mean plus one standard deviation spec-trum) [Chopra, 2001], the pseudo-acceleration is defined as follows, with an initial periodthat varies from 0.66–to 4.12 s:

Sae = 1.8

T; (g) (14)

then, in this period range,

β = 1.8

T. (15)

Moreover, the reduction factor function proposed by Newmark and Hall [1982] is usedwith a post-yield stiffness ratio α of 0; thusz,

R ={ 1 T < Ta√

2μ − 1 Tb < T < Tc′

μ T > Tc

}, (16)

where Ta, Tb, and Tc are the period parameters related to the site soil characteristics; Tc′

is also related to the ductility factor μ [Chopra, 1999]. For rigid site soil, Tc = 0.66 s.Based on the above assumptions, the relationship between the dynamic factor of pseudo-acceleration βp and the ductility factor μ is defined as follows for a SDOF system with aninitial period of 1 s (Fig. 3a):

βp = 1.8

μ. (17)

The βp–μ relationship is period dependent. For example, Fig. 3b gives the smoothcurve respect to a structural system with period of 0.5 s. A suite of 20 ground motions arealso used to generate the βp–μ curves, and in company with the curve of mean plus onestandard deviation from the 20 curves, they are superimposed in Fig. 3. From the figurewe can see that in most occasions the smooth design spectrum is relative conservativecompared with the actual situations.

5. Calculating Process

By combining the methodology proposed by this article and the existing nonlinear staticprocedure, we can establish the structural response function quickly and visually. Thedetailed steps are as follows.

1. A pushover analysis is performed by subjecting a structure with vertical loads tomonotonically increasing lateral forces based on a selected shape vector. Individualpushover analysis is performed separately according to the structural modes if theinfluence of higher modes is taken into account; moreover, for adaptive loadingpatterns, the loading shape vector changes according to the stiffness of the structure.

2. For each mode shape, record the base shear and top displacement during pushoveranalysis to plot the V-�r curve as well as those EDPs concerned.

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Direct Calculation of Seismic Intensity 673

βp

Design spec.

Single acc.

1.0 1.5 2.0 2.5 3.0 µ

µ

Mean +

a. a system with initial period of 1s

3.0

2.0

1.0

3.0

2.0

1.0

1.0 1.5 2.0 2.5 3.0

βp

Design spec.

Single acc.

Mean +

b. a system with initial period of 0.5s

FIGURE 3 Relationship between βp and μ for a SDOF system with period of 1 s and 0.5 sfor 20 ground motion records, the median plus one standard deviation, and a smooth designspectrum.

3. Convert the V–�r curve to the Sa–Sd format using Eqs. (2) and (3) and plot it inthe first quadrant. For adaptive loading patterns, different shape vectors as well asthe corresponding mass participation factors, the modal participation factors andthe amplitudes of the roof are used in different controlling ranges while the multi-degree-of-freedom (MDOF) system is converted to a SDOF system. The EDP-Sd

curve is determined simultaneously.4. According to the capacity curve Sa–Sd, determine the Sa–μ curve and plot it in

the second quadrant (Fig. 4), where the ductility factorμ is taken as 1 before thestructure yields and is calculated according to Eq. (5) after the structure yields.

5. Determine the βp–μ curve according to the initial period, damping ratio and post-yield stiffness ratio, and plot them in the same coordinate system of the secondquadrant.

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674 W. Jing et al.

Sd

Apg,i=Sai

Sai

Sa,i

Sa,y

Sdy Sd,i

μi

μi

βp,i

βp,i

=Sd,i

Sdy

Sa(βp)

Sa–μ

βp,i(T,α,ξ)–μ

demand curve

Sa (A

pg,i ,μi ,α,ξ)–S

d

T= 2πSdy

Saysimplified capacity curve

actual capacity curve

μ μ = 1

FIGURE 4 Illustration of the calculation of the peak acceleration that corresponds to apoint on the capacity curve.

6. For each point on the capacity curve, calculate the seismic intensity Apg,i withequation (12) given Sa,i and βp,i in μi.

7. Repeat step 6 and plot the basic structural response curve Apg–Sd.8. For each point, using the Sd as an index, extract the seismic intensity from the Apg–

Sd curve and the structural response from the EDP-Sd curve. Then, the Apg-EDPcurve for this mode may be plotted.

9. For pushover analysis procedures involving several modes, repeat steps 2–8 foreach mode and determine the structural response curve for each one.

10. Combine the structural response of each mode by combination rules, such as SRSSor CQC, to obtain the total response with different seismic intensities.

6. Numerical Examples

Two examples are carried out in this article to describe and demonstrate the methodologystated above. The first example utilizes a SDOF system to describe the calculation pro-cess of the seismic intensity and spectral displacement response curve according to thecapacity curve and the inelastic response spectrum; furthermore, the results from the non-linear response history analysis and existing example are used as “exact” solutions forcomparison. The second example is a multi-story frame structure that is used to establishthe structural storey drift function curve by combining the methodology proposed in thispaper with MPA proposed by Chopra. This example demonstrates how the methodologycan be combined with the available NSPs, which account for higher modes and the modeshift due to structural plasticity.

6.1. Example 1

This is a SDOF system, with a mass of 3×105 kg, initial elastic period of 1 s, a yielddisplacement of 0.1116 m, a post-yield stiffness ratio of 0 (i.e., ideal elasto-plastic system)and a damping ratio of 5%, as shown in the insert of Fig. 5. In Fig. 5, the Sa–Sd curveis plotted in the first quadrant. The system is subjected to the El-Centro ground motionmentioned above, and the displacement response curve with different seismic intensities isrequired.

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Direct Calculation of Seismic Intensity 675

Sd(m)

S a(m

/s2 )

NL-RHA results, El-Centro(NS)

Apg

(m/s

2 )

Sa−SdSa−μ

βp1−μβp2−μ

βp,i

Apg2−μ

Apg1−μ

Apg2 −S

dApg1 −S

d

1g

(0.4464,4.406)[Chopra, 1999]

Sa,i

Apg,i

design spectrum, 1g

point i, El-Centro(NS)

subscript :

T = 1sξ = 5%Sdy=0.1116mα=0

m=3×105 kg

1 : actual response spectrum

2 : design spectrum

(βp)

μ 5 4 3 2 1

1

5

10

15

2

3

4

0.1 0.2 0.3 0.4 0.5 0.6

FIGURE 5 Procedure used to calculate the basic structural response curve.

First, the mentioned ground motion is scaled to a unit peak acceleration of 1 m/s2 toestablish the curve between the dynamic factor of pseudo-acceleration and the ductility fac-tor according to the structural initial period, damping ratio and post-yield stiffness, which isshown as the βp–μ curve in the second quadrant in Fig. 5. Simultaneously, the Sa–Sd curveis converted to a Sa–μ curve and is also plotted in the second quadrant. For each ductilityfactor μi, capture the values of Sa,i and βp,i from the curves separately. In addition, the peakground acceleration Apg,i that corresponds to point i may be calculated by Eq. (12). In orderto express the results, the graph is expanded to four quadrants, as shown in Fig. 5, where thedownward axis represents the seismic intensity (peak ground acceleration Apg is used), andthe Apg–Sd and Apg–μ curves can be plotted in the fourth and third quadrants, respectively.The above curves are represented by subscript 1 in Fig. 5. To demonstrate the accuracy ofthis method, several waves with different peak accelerations for the same ground motionrecord are used to conduct NL-RHA procedures, and the results are marked in the fourthquadrant in Fig. 5 with solid triangles.

In addition, the Apg–Sd curve calculated by the smooth βp–μ curve mentioned aboveis simultaneously plotted in Fig. 5 and is represented by subscript 2.

In Fig. 5, the first quadrant is the standard capacity curve; the second quadrant is usedto calculate the seismic intensity of each point; and the third and fourth quadrants areused to express the relationship of the ductility factor μ and spectral displacement Sd withincreasing seismic intensity, respectively.

From the Apg–Sd curve in the fourth quadrant, the structural response behavior with dif-ferent seismic intensities can be investigated visually. Note that the methodology proposedby this article is completely explicit and direct, without iterations or trial calculations. Atthe same time, the structural response function, which is fundamental in Next-generationPerformance-based Seismic Design, can also be generated directly or conveniently derived

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676 W. Jing et al.

from the plotted Apg–Sd curve. Moreover, the performance point with a given seismic inten-sity can be obtained by interpolating the Sa – Sd curve or other curves according to the givenApg quickly and conveniently. In Fig. 5, the resulting points defined by Apg which is equalto 1 g from the smooth design spectrum are marked with circles, which shows that they areidentical as [Chopra, 1999] (this example is equivalent to system 5 in the reference).

In Fig. 5, the Apg–Sd curve calculated with the design spectrum in the fourth quadrantlooks like a straight line with a constant slope. In fact, this phenomenon agrees with theactual situation. The ductility factor μ is 1 before the system yields; thus,

Apg = Sa

βp= 4π2Sd

1.8T2. (18)

After the system yields, we obtain

Apg = Sa

βp= Sayμ

1.8= SaySd

1.8Sdy. (19)

Because Say

Sdy= 4π2

T2 , it can be concluded that the Apg–Sd curves before and after yieldinghave the same slope. This result indicates that the seismic intensity is directly proportionalto the spectral displacement response in this example. Thus, the inelastic displacementresponse of this structure is equal to the response of the equivalent elastic structuresubjected to the same ground motion, which is consistent with the well-known “equaldisplacement rule” for long period structures.

6.2. Example 2

A simplified SAC-9 frame, with the analysis model and parameters shown in Fig. 6, is usedas an example. The frame is similar to that used by Chopra [2002] except for the following:(1) the splices of columns are at the storey level; and (2) P-� effect is not considered. Thestructure is subjected to the El-Centro ground motion of 1940.

The natural vibration periods, amplitudes in each story, mass participation factors, andmodal participation factors of the first three modes are shown in Table 1. These modeshapes are used in pushover analyses as separate loading shape functions. The story drifts,roof displacements, and base shear forces are obtained from the results, and then the baseshear-top displacement curves are converted to the Sa–Sd curves and plotted in the firstquadrant in Fig. 7, with each plot corresponding to a specific mode. The story drifts areplotted as well (the story drift of the first floor, which is denoted as d1, is adopted in thisexample). The capacity curves are simplified to bilinear lines and the Sa – μ curves areplotted. The key parameters of the simplified capacity curves are also shown in Table 2.

According to the natural periods, damping ratios (5% in this example) and post-yieldstiffness ratios of these modes, the curves between the dynamic factor of pseudo-acceleration and the ductility factor are established and plotted in the second quadrantseparately, as well as the Sa – μ curves. Utilizing Eq. (12), the Apg – μ and Apg–Sd curvescan be generated and plotted in the third and fourth quadrants using the same steps as inExample 1.

Note that the curves in the first and fourth quadrants in Fig. 7 share the same Sd axis.This feature indicates that when the curves are cut by a vertical line, the intersection points

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Direct Calculation of Seismic Intensity 677

B-1

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

W14

×50

0W

14×

455

W14

×37

0W

14×

283

W14

×25

7

Seismic Mass Beams5.35× 10 kg5

W24× 68

4.945× 10 kg5W27× 84

4.945× 10 kg5W30× 99

4.945× 10 kg5W36× 135

4.945× 10 kg5W36× 135

4.945× 10 kg5W36× 135

4.945× 10 kg5W36× 135

4.945× 10 kg5W36× 160

5.05× 10 kg5W36× 160

4.825× 10 kg5W36× 160

9.15m 9.15m 9.15m 9.15m 9.15m

3.65

m5.

49m

3.96

m3.

96m

3.96

m3.

96m

3.96

m3.

96m

3.96

m3.

96m

Columns: 345MPa

W14

×50

0W

14×

455

W14

×37

0W

14×

283

W14

×25

7

W14

×50

0W

14×

455

W14

×37

0W

14×

283

W14

×25

7

W14

×50

0W

14×

455

W14

×37

0W

14×

283

W14

×25

7

W14

×50

0W

14×

455

W14

×37

0W

14×

283

W14

×25

7

W14

×50

0W

14×

455

W14

×37

0W

14×

283

W14

×25

7

Connections:indicates a moment resisting connection.indicates a simple(hinged) connection. Beams: 248MPa

Materials:

FIGURE 6 Nine-story building.

represent the values of different parameters with the same seismic intensity. Thus, the rela-tionship between the seismic intensity and the structural response (story drift of the firstfloor in this example) of the first three modes can be easily drawn, as shown in Fig. 8.

Combine the responses of the first three modes by SRSS as follows:

ro ≈(

N∑n=1

r2no

)1/2

, 20

where ro is the total structural response, rno is the response of the n th mode, and N is thenumber of modes considered. The total response is also plotted in Fig. 8.

Figures 9 and 10 summarize the relationships between the story drift of each floor andthe seismic intensity under the first mode and SRSS combined results, where di representsthe story drift of the ith floor. These results can be used to formulate the storey drift config-uration of any seismic intensity, as shown in Fig. 11 for the 2.5 × El-Centro intensity. Thedesign spectrums, which use the elastic spectrum [Chopra, 2001] and reduction expression

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TAB

LE

1T

hepa

ram

eter

sof

the

first

thre

em

odes

Peri

odA

mpl

itude

sof

shap

eve

ctor

atflo

orle

vel

Part

icip

atio

nfa

ctor

s

Mod

enu

mbe

rT

(s)

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

αm

γ

12.

3568

0.51

60.

861

1.19

61.

556

1.88

52.

213

2.52

52.

825

3.05

0.44

750.

8302

20.

8314

−1.2

02−1

.855

−2.2

41−2

.318

−1.9

97−1

.209

0.04

41.

651

3.05

0.17

440.

1122

30.

4359

2.44

53.

171

2.58

50.

748

−1.4

79−3

.217

−2.9

74−0

.393

3.05

0.07

70.

0358

678

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Direct Calculation of Seismic Intensity 679

30

20

10

10

20

30

3.03.5 2.5 2.0 1.5 1.0 0.1 0.2 0.3

(a) 1st mode

d 1(m

)

(b) 2nd mode

Sd (m)

Apg

(m/s

2 )Sa−

SdSa−μ

βp−μ

A pg−μ A

pg −Sd

d1,i

Apg,i

d1−Sd

2.5x El Centro

(c) 3rd mode

Sd (m)

d 1(m

)

Apg

(m/s

2 )

Sa−SdSa−μ

1.6

0.12

0.08

0.04

1.2

0.8

0.4

2

4

6

8

3.0 2.5 2.0 1.5 1.0 0.5 0.1 0.2 0.3 0.4 0.5 0.6

βp−μ

A pg-μ

Apg −S

d

d1,i

Apg,i

d 1−S d

2.5x El Centro

S a(m

/s2 )

(βp)

μ

μ

10

8

6

4

2

5

10

15

3.03.5 2.5 2.0 1.5 1.0 0.1 0.2 0.3 0.4

2.5x El Centro

Sd (m)

d 1(m

)

Apg

(m/s

2 )

Sa−SdSa−μ

βp−μ

A pg−μ A

pg −Sd

d1,i

Apg,i

d1−Sd

μ

S a(m

/s2 )

(βp)

S a(m

/s2 )

(βp)

FIGURE 7 Procedure used to calculate the structural response curves of the 1st, 2nd, and3rd modes, respectively.

TABLE 2 The key parameters of the simplified bilinear capacity curves of the firstthree modes

Yield point

Mode number Say(m/s2) Sdy(m) Post-elastic stiffness ratio α

1 1.4252 0.2005 0.04262 7.3757 0.1291 0.07453 21.0022 0.1011 0.1296

recommended by Vidic et al. [1994], are utilized to generate the corresponding smoothspectrum results, and in company with the result calculated by NL-RHA, they are alsoplotted in this figure. The SRSS combination results respect to the actual ground motiondiffers slightly with the NL-RHA result. This may be caused by the simplification of bilin-ear model of the capacity curve and the MPA procedure. However, it is relative accurateagainst that from design spectrum, which is conservative in most situation.

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680 W. Jing et al.

1st m

ode

SRSS

3rd

mod

e

2nd

mod

e

Apg

(m/s

2 )10

8

6

4

2

0.03 0.06 0.09 0.12 d1(m)

FIGURE 8 Relationship between the seismic intensity and the story drift of the first floor.

12 345678910

8

6

4

2

0.03 0.06 0.09 0.12

Apg

(m/s

2 )

d1(m)

FIGURE 9 Relationship between the story drift of each floor and seismic intensity formode 1.

The continuous curves shown in Fig. 10 can be easily applied in the Next-generationPerformance-based Seismic Design procedure as one series of structural responsefunctions.

7. Conclusions

This article proposes a methodology to formulate the continuous structural response func-tion by establishing the inelastic pseudo-acceleration spectrum of unit peak accelerationseismic wave respect to ductility factor. The results lead to the following conclusions.

1. During seismic capacity evaluation based on NSP, an appropriate method shouldbe adopted to generate the structural response function. The structural response

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Direct Calculation of Seismic Intensity 681

Apg

(m/s

2 )

di(m)

42135678910

8

6

4

2

0.03 0.06 0.09 0.12

FIGURE 10 Relationship between the storey drift of each floor and seismic intensity bySRSS.

Floo

r le

vel

di(m)

1st mode by El-Centro(NS)SRSS by El-Centro (NS)NL-RHA

1st mode by the

9th

8th

7th

6th

5th

4th

3rd

2nd

1st

0.02 0.04 0.06 0.08 0.10 0.12

design spectrumSRSS by the design spectrum

FIGURE 11 Distribution of storey drift from 1st mode, SRSS mode combination, andNL-RHA result for 2.5 × El-Centro ground motion, and from the design spectrum byChopra [2001].

function also plays an important role in the Next-generation Performance-basedSeismic Design.

2. Unlike the traditional procedures that search for the “performance point” at a deter-mined intensity, the proposed procedure starts from a point on the capacity curve.This method leads to an explicit calculation with only one unknown parameter andthus avoids iterations or trial calculations.

3. According to the methodology proposed in this article, the structural responsefunction respect to different intensities can be drawn easily, which is beneficialfor determining the changing trends and structural response rule; in addition, it

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682 W. Jing et al.

can be used directly in the Next-generation Performance-based Seismic Designframework.

4. The performance point at a determined seismic intensity can be traced from theresponse curve quickly and conveniently.

5. The proposed methodology is applicable to both a smooth design spectrum and anactual response spectrum that corresponds to a determined ground motion.

6. The numerical examples indicate that the proposed procedure is accurate in theSDOF domain and can be combined easily with the existing NSP calculating pro-cedures to consider the higher modes and mode shape alteration as a result ofstructural inelastic development.

Acknowledgments

The authors would like to acknowledge financial supports from the National BasicResearch Program of China -973 Program (No.2007CB714200), the National NaturalScience Foundation (50878055), the National Key Technologies R&D Program of China(2008BAJ12B04, 2008BAJ12B05), and Natural Science Foundation of Jiangsu Province(BK2008313).

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