Struct Multidisc Optim (2012) 45:861–874DOI 10.1007/s00158-011-0740-x

INDUSTRIAL APPLICATION

Modified-modal-pushover-based seismic optimum designfor steel structures considering life-cycle cost

Gang Li · Yi Jiang · Dixiong Yang

Received: 6 July 2011 / Revised: 29 September 2011 / Accepted: 28 October 2011 / Published online: 26 November 2011c© Springer-Verlag 2011

Abstract A modified-modal-pushover-based optimizationtechnique is presented to design steel moment resistingframe buildings for minimizing the life-cycle cost basedon the framework of performance based earthquake engi-neering. Modified modal pushover analysis (MMPA) pro-cedure capturing the higher mode effect well is utilized toanalyze the inelastic seismic demands of the structures sub-jected to the considered design earthquakes in terms of theChinese seismic code for buildings, especially for themedium- to high-rise buildings. Furthermore, the life-cyclecost is formulated as the summation of the initial mate-rial cost and the future expected damage loss, which canbe stated as a function of seismic performance levels andtheir corresponding failure probability by means of a sta-tistical model. Meanwhile, the damage loss is explicitlyand continuously expressed by the defined interstory driftindex using the fuzzy-decision theory. Moreover, the pow-erful adaptive simulated annealing algorithm is applied tosolve the discrete optimization problem due to the discrete-ness of standard steel sections. Finally, a 9-story planar steelframe is provided to illustrate the effectiveness of the pro-posed optimization design technique, which achieves notonly more cost-effective design but greatly improves therobustness of the optimum design as well.

G. Li (B) · Y. Jiang · D. YangState Key Laboratory of Structural Analysis of Industrial Equipment,Dalian University of Technology, Dalian, 116024, Chinae-mail: ligang@dlut.edu.cn

Y. Jiange-mail: jiangyi@mail.dlut.edu.cn

D. Yange-mail: yangdx@dlut.edu.cn

Keywords Structural optimization · Steel structures ·Earthquake · Modified modal pushover analysis ·Life-cycle cost · Interstory drift

1 Introduction

After the 1994 Northridge and 1995 Kobe Earthquakes,which lead to an awareness that the level of structuraland nonstructural damage that could occur in the code-compliant buildings may conflict with the public notionsof acceptable performance, performance based earthquakeengineering is recognized as the emerging trend in seis-mic engineering for designing the new structures, andretrofitting the existing ones (SEAOC 1995; ATC-40 1996;FEMA-273 1997; FEMA-356 2000; FEMA-445 2006;ATC-58 2009). However, one major shortcoming in the tra-ditional prescriptive design approach based on the linearelastic techniques is that it cannot well address the struc-tural inelastic deformations causing the future expected fail-ure loss.

In contrast to that, a good choice for performance basedseismic optimization design is to adopt a feasible and effec-tive procedure for estimating the seismic demands of thestructures even within the inelastic range. At present, non-linear response history analysis (RHA) remains extremelycomputational demanding in spite of the increasing com-puting power, thus the structural engineers are more likelyto apply nonlinear static methods before finally transition-ing to nonlinear history analysis. Besides of it, an ensembleof ground motions compatible with the seismic design spec-trum for the specific site must be considered (Chopra 2009,Section 6.9, 19.5), and such expensive analyses must berepeated for several excitations due to the variability ofthe plausible ground motions. Consequently, the nonlinear

862 G. Li et al.

static method is a viable alternative for structural engineersto determine the section types of steel structures or the mem-ber dimensions and reinforced ratios of reinforced concretebuildings with the inelastic deformations.

Pushover analysis procedure is becoming the currentstandard practice to evaluate the seismic demands, and isaccepted by more and more guideline documents and relatedcodes (FEMA-273 1997; FEMA-356 2000; FEMA-4402005; ATC-58 2009; National Standard 2010) due to itssimplicity and efficiency. For the past decade, researchers inearthquake engineering have made lots of efforts to developthe associated theory and application for pushover analysismethods. As far as the lateral load distribution is con-cerned, there are two basic types for pushover analysismethods in general: invariant and adaptive lateral load dis-tribution. Invariant lateral load distributions (Chopra andGoel 2002; Chopra et al. 2004; Hernandez-Montes et al.2004; Jiang et al. 2010) are made to simulate the iner-tia forces under the seismic hazards, which cannot accountfor the redistribution of inertia forces owing to the struc-tural yielding and the associated changes of the dynamicproperties of structures. Based on this invariant load dis-tribution, Chopra and Goel (2002) and Chopra et al. (2004)proposed the modal pushover analysis (MPA) and modifiedMPA (MMPA) procedures to deal with higher mode effectseffectively and efficiently. To overcome the limitations ofthe invariant load distribution, the adaptive force distri-butions (Gupta and Kunnath 2000; Antoniou and Pinho2004a, b; Kalkan and Kunnath 2006) that attempt to fol-low the time-variant distributions of inertia forces have beenproposed and investigated in detail. For instance, Guptaand Kunnath (2000) presented the adaptive spectra-basedpushover procedure, in which the adaptive lateral forcedistributions are scaled with the elastic spectral accelera-tion for site-specific ground motions. Antoniou and Pinho(2004a, b) investigated the advantages and limitations of theadaptive and non-adaptive force- and displacement-basedpushover procedures. Kalkan and Kunnath (2006) proposedthe adaptive modal combination procedure, which inte-grates the inherent merits of the capacity spectrum methodand the modal pushover procedure adopting the energy-based pushover curve. Although the significant higher modeeffects, theoretical foundation, and computational aspectneed to be investigated and improved further, the abovementioned pushover analysis procedures have the advan-tages of high computational efficiency, acceptable accuracyand easy operation. In addition, it is noted that more andmore literatures (Ganzerli et al. 2000; Liu et al. 2003, 2005;Zou and Chan 2005; Fragiadaki et al. 2006; Zou et al. 2007)on building structural optimization have made use of thepushover analysis methods. Nevertheless, there is still noworks on the inelastic structural optimization consideringthe higher modes effects, which are very significant for the

medium- to high-rise buildings. In order to bridge the gap,this investigation is to optimize the structure under earth-quakes considering the life-cycle cost using these advancedpushover methods.

Recently, many investigations on structural optimaldesign have considered the life-cycle cost consisting of twomain parts: the initial cost and future expected damage losscaused by earthquakes. The future expected failure conse-quences (i.e., damage, downtime, and death) may be sucha complex problem involving not only construction cost butalso the costs associated with social, economical, political,cultural, and ethical and many other respects. Frangopolet al. (1997) have presented a reliability-based approachfor the life-cycle cost design of the deteriorating rein-forced concrete T-girders from a highway bridge. Minimumexpected life-cycle cost design criteria proposed by Wenand Kang (2001a, b) was formulated and investigated toreduce seismic risks to socio-economically acceptable levelwith respect to the uncertainties of design loads and resis-tances. The design methodology, which considers the costsof construction, maintenance, and failure consequences,including the deaths and injuries, as well as discountingcost over time, was developed for a nine-story office build-ing subjected to earthquakes and winds (Wen and Kang2001a, b). Later applications for minimizing the life-cyclecost in structural optimum design were studied by someresearchers, such as Sarma and Adeli (2002), Liu et al.(2003, 2004, 2005), Fragiadaki et al. (2006), Zou et al.(2007), Li et al. (2009), Okasha and Frangopol (2009),and Mitropoulou et al. (2011). For example, Li and Cheng(2003) and Zou et al. (2007) formulate the life-cycle cost,including two parts: the initial material cost expressed interms of design variables and the expected future failure lossstated as a function of seismic performance levels and theirassociated failure probability by a statistical model. Fur-thermore, Mitropoulou et al. (2011) formulate the total costas the summation of the initial cost of a new or retrofittedstructure and the present value of the limit state cost, whichconsists of the damage repair cost, the loss of contentscost due to structural damage (interstory drift, floor accel-eration etc.), the loss of rental cost and the income loss,and the cost of injuries and human fatality. These litera-tures indicate the initial cost often inherently contradicts itscorresponding expected damage loss in the design lifetimeperiod. Huge economical losses caused by the structuraland nonstructural damage demonstrate that seeking a pre-ferred balanced point between the initial material cost andthe damage loss through the life-cycle cost analysis of struc-tures is quite necessary and important for making rationaldesign decisions.

Moreover, the codes for seismic design represent a con-sensus of the earthquake engineering profession on seismicdesign philosophy, in which buildings should be able to

Modified-modal-pushover-based seismic optimum design for steel structures considering life-cycle cost 863

resist minor but frequently occurring earthquakes withoutdamage, maintain function with reparable damage whensubject to moderate earthquakes, and behave no collapsewhen subject to rare and major earthquakes (NationalStandard 2010). Nowadays, it is widely recognized thatthe lateral displacement or interstory drift is a proper mea-sure index for quantifying the structural and nonstructuraldamage of multi-story building under various levels ofearthquakes. Although the interstory drifts can reflect theearthquake-resistant capacity for the multi-story building,interstory drift performance oriented structural seismic opti-mum design, particularly inelastic drift performance ori-ented, is still a challenging and difficult task in buildingseismic design (Park and Kwon 2003; Chan and Zou 2004;Zou et al. 2007, 2010).

This paper presents a general structural optimizationtechnique for performance based design of seismicallyexcited steel building structures. Optimization problems areformulated to address the life-cycle cost minimization usingthe MMPA procedure considering the higher mode effectseffectively, which the adaptive simulated annealing algo-rithm is applied to find the optimal solutions effectively.Furthermore, due to the discreteness of the damage statesdefined, the fuzzy-decision theory is employed to estimatethe damage loss of steel frame continuously with respect tothe specified probabilities of occurrence for the earthquakes.

2 Seismic optimum design procedure

2.1 Modified modal pushover procedure

This section describes the modal pushover analysis (MPA)and the modified MPA (Chopra and Goel 2002; Chopra et al.2004) procedures, and presents their relationship with theseismic optimum design procedure proposed in the paper.

The governing differential equation of motion for a non-linear multi-degree of freedom (MDOF) structures underthe horizontal earthquake ground motion, ug(t), can beformulated as

mu + cu + fs(u, u) = −mιug(t) (1)

where m and c mean the mass and classical damping matri-ces of the structure separately, ü, ü, and u stand for theacceleration , velocity and displacement response vector rel-ative to the ground, respectively; and the resisting vector,fs(u, u), indicating the force-displacement relationship andhistory of structure, depends implicitly on the displacementand velocity response vector. ι is the influence vector, whenthe seismic masses are lumped at the floor, the influencevector equals the unit vector, namely ι = 1.

According to the underlying assumptions of the MPA,the lateral force vector on the right hand of the (1) can beexpanded as the summation of the classical modal lateralforce distributions, sn as

mι =N∑

n=1

sn =N∑

n=1

�nmφn (2)

and the vector, φn , and the coefficient �n are respectivelythe nth mode shape and modal participation factor de-rived by

�n = Ln

Mn, Ln = φT

n mι, Mn = φTn mφn (3)

Although the classical modal analysis is not valid for theinelastic system, but the vibration properties of the inelasticsystem undergoing small vibrations approximates the natu-ral vibration periods and modes of the corresponding linearsystem. Hence, the displacement of the MDOF system canbe expanded in terms of the natural vibration modes of therespective linear system,

u(t) =N∑

n=1

φnqn(t) (4)

Substituting (4) into (1), pre-multiplying by φTn and ac-

cording to the mass- and damping- orthogonality of modes,the approximate modal equation for the MDOF system canbe written as

qn(t) + 2ζωnqn(t) + Fsn

Mn= −�nug(t) (5)

where

Fsn = Fsn(q, q) = φTn fs(u, u) (6)

where Fsn is the general resisting force relying on all modalcoordinates qn(t), which implies coupling of modal coor-dinates due to yielding of the structure. The coordinatesqn(t) of the above modal equations are mutably coupledunlike that of the linear system, and these equations givethe same results as the nonlinear MDOF governing (1) forthe responses in principle. ωn , and ζn denote the circularfrequency and damping ratios. However, it shows no advan-tages over (1). Therefore, when the co-ordinate qn(t) isexpanded as

qn(t) = �n Dn(t) (7)

where Dn(t) is the modal displacement co-ordinate forthe given ground motion, ug(t). Substituting (7), and (3)

864 G. Li et al.

into (5), another simple form of the modal equation for theMDOF system can be obtained

Dn(t) + 2ζωn Dn(t) + Fsn

Ln= −ug(t) (8)

And Chopra and Goel (2002) gave the approximate relation-ship between the modal pushover curve (Vbn − urn) and theabove modal equation properly, expressed by

Fsn

Ln= Vbn

Ln�n= Vbn

M∗n, Dn = urn

�nφn(9)

where Vbn and urn separately denote the nth modal baseshear and roof displacement, and M∗

n represents the ntheffective modal mass. Further, the roof displacement canbe achieved through the conversion of the (9), as follows:

urn = �nφn Dn (10)

Here the coupling effects between modal co-ordinatesarising from the yielding of the system are neglected. Whatis more, the Fsn/Ln − Dn relation is approximate by usingthe pushover curve (usually bilinear form) to solve themodal (8). Through the simple conversion of the pushovercurve, Dn can be determined as the peak value of Dn(t)obtained by inelastic RHA of the single degree of free-dom (SDOF) system for a given ground motion ug(t).Undoubtedly, Dn can also be conveniently derived from theinelastic design spectrum, compatible with the defined seis-mic hazard ug(t) for the specific site. Accordingly, this wayto obtain the value Dn can be used not only for design-ing the new structures efficiently, but for evaluating theexisting ones.

According to the inelastic design spectral displacement,Dn can be acquired through the elastic spectral displace-ment Dno multiplied by the inelastic deformation ratio C(Ruiz-Garcia and Miranda 2003) expressed as

Dn = C Dno (11)

where the elastic spectral displacement Dno depends onthe natural period Tn and damping ratio ζn . Meantime,the inelastic deformation ratio C relies upon the ductil-ity factor μ, the strength reduction factor Ry , the naturalperiod Tn and the characteristic period Tc associated tothe given site. Moreover, several empirical equations forthis ratio, defined as the ratio of peak deformations ofinelastic and corresponding linear SDF systems, were estab-lished by Ruiz-Garcia and Miranda (2003) and Chopra andChintanapakdee (2004) respectively, or directly through theinelastic design spectrum such as Newmark-Hall (Chopra

2009, Section 7.11), Krawinkler and Nassar (1992) andVidic et al. (1994) inelastic spectrum. In practice, there aresome different forms of empirical equations of the inelas-tic deformations ratio. However, one common conclusionis that when the period ratio, Tn/Tc, is larger than unit,the inelastic deformation ratio C approximates unit (slightlydifferent).

Furthermore, as for the higher modes, on the higher-mode contributions to the seismic demands behaving aslinearly elastic based on the assumption of modified MPA,the target pushover roof displacement can be estimated fromthe elastic design spectrum directly. Thus, an eigenvalueanalyses is used to determine the vibration properties of thestructures (i.e., natural periods Tn , the modes φn , and themodal participation factors �n). Meanwhile, the target dis-placement of the modal pushover analyses can be acquiredfrom the design spectrum easily. Then, the pushover analy-sis using modal load distributions (considering as much asenough) are performed directly, and the seismic responsesof each final modal pushover step (such as the interstorydrift, floor displacement, and so on) are extracted and com-bined through the selected combinations rules, e.g. thesquare-root-of-sum-of-squares (SRSS). In fact, for all con-sidered modes the modal target displacements is determinedfrom the elastic design spectrum directly. Fortunately, thispractice is validated by the recent practical modal pushoveranalysis (PMPA) (Reyes and Chopra 2011), which esti-mates seismic demands directly from the elastic designspectrum and is almost as accurate as the classic responsespectrum analysis procedure.

2.2 Formulation of optimum problems

For a building structure whose topology is predefined, theobjective function is to minimize the life-cycle cost f , con-sidering two parts in this paper: the initial material cost f1,and the expected damage loss f2 (viewed as the presentvalue). The objective function is presented as

Minimize : f = f1 + f2 (12)

First of all, the objective function of the initial materialcost f1 can be express explicitly for facilitating the numer-ical solution of the optimization problem. Moreover, theexpected damage loss stated as a function of seismic perfor-mance levels and their associated failure probability basedon a statistical model needs to be explicitly expressed, interms of the initial material cost f1 and the damage lossfunction Lr with respect to r th performance level using thefuzzy-decision theory (Zadel 2000; Li and Cheng 2003; Zouet al. 2007).

Modified-modal-pushover-based seismic optimum design for steel structures considering life-cycle cost 865

The initial material cost f1 of a steel frame structure ispresented as the summation of the columns’ and beams’cost,

f1 =∑

i

(Aci Lciρswc + Abi Lbiρswb) (13)

where wci , and wbi mean cost per mass for columns andbeams, respectively, ρs denotes density of steel, Aci andAbi stand for cross-section area of column and beam formember i , Lci and Lbi indicate length of the column andbeam for member i . When the section types are determined,the initial material cost of steel building generally dependson the weight of the steel members.

On the other hand, the expected future loss due to thestructural damage or failure consists of direct loss and indi-rect loss. Specifically, the direct loss is the cost of repair orreplacement of structural members, nonstructural compo-nents, and equipments, and so on. Whereas, the estimationof the indirect loss, which is caused by the structural mal-function, injuries and deaths, psychological and politicalinfluence, etc, is much complex. Obviously, the differentseismic risks lead to the respective different expected dam-age loss. In this investigation, the total expected loss isdefined as the summation of the product of the occur-rence probability of earthquake and corresponding expecteddamage loss (Li and Cheng 2003; Zou et al. 2007),

f2 =∑

r

Pr Lr (14)

where r denotes the seismic design level and Pr the occur-rence probability of the considered earthquake at the r thdesign level, which can be determined from specified coderequirements; Lr is the future expected structural damageor failure loss including direct and indirect losses under ther th design seismic event from a statistical model.

In addition, the buildings are categorized into fourclasses, A, B, C, and D, according to the Chinese Stan-dard for Classification of Seismic Protection of Buildings(National Standard 2008). Class A represents the most

important structure and Class D the structure with the lowestimportance. Class C represents the normal building. More-over, the states of seismic damage based on the Chinesecode practice include five states, i.e. negligible, slight, mod-erate, severe and complete collapse, which correspond toA1, A2, A3, A4, and A5, respectively (showed in Table 1).Hence, according to the classification of the damage states,the interstory index τ of building is defined as

τ = u

hR (15)

where u means the relative displacement, h is story height,and R is taken as 500 for frame structures and 1000 forshear-wall and frame-wall structures (Li and Cheng 2003;Zou et al. 2007; National Standard 2008, 2010). The inter-story index is defined to measure the state of damage.Finally, the future expected damage loss Lr is simplifiedand formulated (Li and Cheng 2003; Zou et al. 2007) as

Lr (A) =⎡

⎣5∑

j=1

L j(

A j)⎤

⎦

r

f1 (16)

where the subscript r and j correspond to the r th seismicdesign level and the five states of damage, respectively. AndL j = total structural damage loss (including direct and indi-rect losses) corresponding to the damage state A j , and canbe calculated from Table 1.

It should be noted that the set of damage states is discrete.Thus, the fuzzy-decision theory is introduced to assess thestructural loss effectively, which makes the structural losscontinuous over the range of interstory index. The member-ship function μ j (τ ) is used to describe the state of damagein terms of interstory drift index τ , defined as

μ (τ) = {μ1, μ2, μ3, μ4, μ5} (17)

5∑

j=1

μ j (τ ) = 1, μ j (τ ) ∈ [0, 1] (18)

Table 1 Relationships betweenstates of damage, interstory driftindex, and damage loss

State of damage Negligible Slight Moderate Severe Complete

A1 A2 A3 A4 A5

Interstory drift index τ <1.0 1.0–2.0 2.0–4.0 4.0–10.0 >10.0

Ratio of direct loss to material cost 0.02 0.10 0.30 0.70 1.00

Ratio of indirect loss to material cost

Class A 0.0 0.0 1.0–10.0 10.0–50.0 50.0–200.0

Class B 0.0 0.0 0.5–1.0 3.0–6.0 8.0–20.0

Class C 0.0 0.0 0.5 2.0 6.0

Class D 0.0 0.0 0.2 1.0 2.0

866 G. Li et al.

0 1 2 3 4 5 6 7 8 9 10 11 120.0

0.2

0.4

0.6

0.8

1.0

1.2

μ5

μ5

μ4

μ4

μ3μ3

μ2μ2μ1

A5

A4

A3

A2

a5

a4

a3

a2

Mem

bers

hip

func

tion

valu

e

Interstory drift index, τa

1

A1

μ1

Fig. 1 Membership functions

Figure 1 shows the membership functions correspondingto their states of damage, where the μ j (τ ) is a triangularfuzzy function. The middle point of each damage state A j

is viewed as the transformation point of each membershipfunction (i.e., the symbols a1, a2, a3, a4,a5 in Fig. 1),

a1 = 0.5; a2 = 1.5; a3 = 3.0; a4 = 7.0; a5 = 10.0 (19)

Further, the linear membership function describes mathe-matically between the membership function value and thevariable τ as follows:when τ < a1

μ1 (τ ) = 1, μ j (τ ) = 0 ( j = 2, 3, 4, 5) (20a)

when a j ≤ τ < a j+1

μ j (τ ) = (τ − a j+1

)/(a j − a j+1

)

μ j+1 (τ ) = (τ − a j

)/(a j+1 − a j

)

μk (τ ) = 0 ( j =1, 2, 3, 4; k =1, 2, 3, 4, 5; k �= j, j +1)

(20b)

when τ ≥ a5

μ5 (τ ) = 1, μ j (τ ) = 0 ( j = 1, 2, 3, 4) (20c)

Accordingly, using the above membership function μ j (τ )

for the damage state A j , the discrete structural damage

loss L j is transformed into a continuous function directlyexpressed by the interstory drift index τ as

Lr (τi ) =⎡

⎣5∑

j=1

μ j (τi ) L j(

A j)⎤

⎦

r

f1

(r = 1, 2, 3; i = 1, 2, · · · , Ni ) (21)

where the damage loss Lr (τi ) denotes the i th interstory driftat the r th design seismic level. Taking (20a), (20b), (20c)into (21), it can be further simplified into a multi-segmentanalytical linear function as:

Lr (τi ) = (α1τi + α2) f1 (r = 1, 2, 3; i = 1, 2, · · · , Ni )

(22)

Based on Table 1, the coefficients α1, α2 rely on not onlystate of damage but also building classes. Take a buildingbelonging to Class C (a normal building) for instance, thecoefficients value of expected failure loss can be calculated,displayed in Table 2. Consequently, after formulating theexplicit expression of the damage loss corresponding to thei th interstory drift at the r th seismic design criteria, the totaldamage loss f2 of building structure in (17) can be explicitlypresented in terms of the interstory drift index τ as

f2 = f1

3∑

r=1

⎧⎨

⎩pr ×⎡

⎣Ni∑

i

(α1τi + α2)

⎤

⎦

⎫⎬

⎭ (23)

Then, the performance based constraints of optimizationdesign model of building are addressed. Based on theChinese Seismic Code (2008), the predicted deformationsunder the minor and severe earthquake must not exceedthe allowable limits, respectively. The limits of the inelas-tic interstory drift ratio (IDR) under severe earthquake arepresented as

ui

hi= ui − ui−1

hi≤ θi (24)

where ui represents the relative displacement of the i thstory; ui and ui−1 denote story displacements of two adja-cent i th and (i-1)th floor levels; hi means the i th storyheight; and θi indicates the specified interstory drift ratio

Table 2 Coefficients ofexpected failure loss forClass C building

Coefficient τ< 0.5 0.5 ≤ τ < 1.5 1.5 ≤ τ < 3.0 3.0 ≤ τ < 7.0 7.0 ≤ τ < 10.0 τ ≥ 10.0

α1 0.000 0.080 0.4667 0.475 1.433 0.000

α2 0.020 −0.020 −0.600 −0.625 −7.333 7.000

Modified-modal-pushover-based seismic optimum design for steel structures considering life-cycle cost 867

limit. In order to simplify the inelastic target roof displace-ment for the fundamental mode, the first natural periodshould comply with the following requirement, T1 ≥ Tc.Fortunately, for most of the medium and tall building, thefundamental period T1 is greater than the characteristicperiod Tc of usual ground motions at the specified site witha distance far from the earthquake fault.

Lastly, the step-by-step procedure of MMPA-based struc-tural optimization for minimizing the life-cycle cost issummarized in Fig. 2. The modal vibration properties ofthe building structures can be determined from an eigen-value analysis (i.e., natural periods, Tn , the modes, φn , themodal participation factors �n , and the invariant height-wise lateral load distribution sn). Then the target roofdisplacements for the nth mode can be computed by (10).Thus displacement controlled pushover analysis are per-formed subject to the nth modal lateral load distribution sn

until the predetermined target roof displacement of build-ing is reached. Accordingly, we can extract easily the lastpushover step responses, such as the displacements, andinterstory drift ratios. After implementing the MMPA, if the

Structural modeling

Eigenvalue analysis

Calculate , , , ( 1,2, , )n n n n nT n Nφ φΓ = =s m

Determine th modal target displacement by Eq.(10)n

1 1if 1, ; if 2 , onnon D CD n N D D= = ≤ ≤ =

Perform displacement-based pushover analysis

subject to modal lateral load distribution ns

Extract the responses of the last modal pushover step

e.g. displacement , interstory drift ratio , etc.ininu IDR

= +1n n

Calculate the total responses by SRSS

e.g. displacement , interstory drift ratio , etc.iiu IDR

Does design meet optimum criteria?

Modify design based on optimization algorithm

No

StopYes

Fig. 2 Flowchart of MMPA-based structural optimization

convergence criteria are met, optimum design completes; ifnot, modify the design based on the adopted optimizationalgorithms. Herein, a kind of evolution based optimizationalgorithm, i.e. adaptive simulated annealing algorithm, isapplied to deal with the discrete variable optimal design ofsteel building, as described in detail in the following section.

2.3 Adaptive simulated annealing algorithm

Simulated annealing (SA) proposed by Kirpatrick et al.(1983) originates from the theory of statistical mechanics,with an analogy to the physical process of annealing a metal.In essence, the SA algorithm repeats the structural analy-sis through random change of the current design to producethe candidate design. Furthermore, the improved candi-date design is unconditionally accepted in the annealingprocess according to the Metropolis criterion (Metropoliset al. 1953). The SA algorithm often can produce a goodlocal though not necessarily global optimal solution withina reasonable computing time. Therefore, Adaptive simulateannealing (ASA) (Ingber 1989, 1993) algorithm is devel-oped to find the global optimum, which is well suited forsolving the highly nonlinear problems with little compu-tational effort, especially for discontinuous design spaces.What is more, for most of the practical problems a quickimprovement of the design is enough for the engineeringapplications.

Actually, the ASA algorithm adopts the very fast re-annealing schedule (Ingber 1989) relying on the numbersof the accepted and generated designs, which permits toadaptively change the sensitivities in the multi-dimensionaldesign space. Meanwhile, ASA approach remains similar toa bouncing ball that can bounce over mountains from valleyto valley adaptively. Then, the cost value of each accepteddesign is compared to that of the best design found so far. Ifthe two values differ by less than the convergence epsilonfor N consecutive times (where N is number of designsfor convergence check), the adaptive optimization is termi-nated. Accordingly, the general step-by-step procedure issummarized as follows:

(1) In a D-dimensional design space, a group of the initialdesign S0 is randomly produced with correspondingobjective function f0, and initial temperature T0.

(2) Generate a random change according to the previousdesign, and obtain a new possible design Si .

(3) Calculate the objective function fi and the acceptedprobability p. If fi+1 is less than fi , then the currentdesign is accepted unconditionally (i.e. p = 1); other-wise, if fi+1 is larger than fi , then the current designis accepted with a probability of p = exp(−( fi+1 −fi )/Ti ).

868 G. Li et al.

(4) Perform an annealing schedule, and in ASA algorithmlower the annealing temperature Ti+1 = mTi (0 <

m < 1), here, m = exp(−ck1/D), where c and krepresent two control parameters.

(5) If the criteria of the re-annealing schedule are met,conduct the re-annealing schedule adaptively; if not,it continues.

(6) If the convergence criteria are met, the optimizationprocedure completes; if not, go to step (2).

3 Numerical example

3.1 Description of structural systems

A 9-story planar steel frame is taken as an example toillustrate the effectiveness of the proposed optimal designtechniques. Figure 3 gives the structural schematic geom-etry and member grouping of the frame, which has fivetypes of column and beam respectively. Column splicesare located on the first, third, fifth, and seventh levels at1.83 m (6 ft) above the center line of the beam to columnjoint. Further, the ground floor seismic mass is assumed tobe 4.82 × 105 kg, the first floor 5.04 × 105 kg, the rooffloor 5.35 × 105 kg, and others 4.95 × 105 kg. The mod-ulus of the elasticity equals 200 GPa (29000 ksi), and theyield strength of columns and beams are 345 MPa (50 ksi)and 248 MPa (36 ksi), respectively. Moreover, the steelframe is assumed to have rigid connections and pinnedsupports at the under-ground floor. Gravity loads are con-sidered, including concentrated loads applied to the topnodes of columns (133.5 kN ground floor, 151.3 kN firstand roof floors, 142.4 kN other floors) and a uniformly

3.65m

5.49m

8@ 3.96m

5@ 9.15m

1C

1.83m

2C

3C

4C

5C

1B

1B

1B

2B

2B

2B

3B

3B

4B

5B

Fig. 3 Geometry and member grouping of the 9-story steel frame

yδ0cδ rδ

ek

s s ek kα=

c c ek kα=yFcF

F

δ

r yF Fλ=

Capping Point

Fig. 4 Backbone curve of the plastic hinge

distributed load of 17.5 kN/m applied to all beams. TheP-Delta effects due to gravity loads are also considered inthe pushover analysis. Additionally, the concentrated plasticrotational spring element, which follows a bilinear hys-teretic response based on the modified Ibarra Krawinklerdeterioration model (Ibarra and Krawinkler 2005; Lignosand Krawinkler 2010; OpenSees 2010), is distributed atthe ends of the columns and beams. The backbone of thehysteretic load-deformation curve is defined by three typesparameters: (1) the elastic (initial), strain-hardening andpost-capping stiffness, Ke, Ks = αs Ke, and Kc = αc Ke;(2) the yield, cap, residual deformation, δy , δc and δr ; (3)the yield, cap, and residual strength, Fy , Fc, and Fr = λFy

(Fig. 4). Notably, the nonlinear behavior of the plastic rota-tional spring neglects the strength, stiffness deterioration ofstructural components.

Two levels of earthquake loads, minor earthquake with apeak acceleration of 0.20 g (63%, probability of exceedancein 50 years) and severe earthquake with a peak accelerationof 1.14 g (2–3%, probability of exceedance in 50 years)are considered, shown in Fig. 5 according to the accel-eration response spectrum with 2% damping ratio of the

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.45

0.5

Seis

mic

coe

ffic

ient

, (

g)

Period, T (s)

Severe earthquake Minor earthquake

0.1

α

Fig. 5 Chinese seismic design response spectra

Modified-modal-pushover-based seismic optimum design for steel structures considering life-cycle cost 869

Table 3 Sections forcolumn and beam Location Standard section database (design space)

Column (1) W14x730, (2) W14x665, (3) W14x605, (4) W14x550, (5) W14x500,

(6) W14x455, (7) W14x426, (8) W14x398, (9) W14x370, (10) W14x342,

(11) W14x311, (12) W14x283, (13) W14x257, (14) W14x233, (15) W14x211,

(16) W14x193, (17) W14x176, (18) W14x176, (17) W14x145, (18) W14x132.

Beam (1) W36x232, (2) W36x210, (3) W36x194, (4) W36x182, (5) W36x170,

(6) W36x160, (7) W36x150, (8) W36x135, (9) W33x130, (10) W33x118,

(11) W30x116, (12) W30x108, (13) W30x99, (14) W30x90, (15) W27x84,

(16) W24x84, (17) W24x76, (18) W24x68, (17) W24x62, (18) W24x55.

Chinese seismic design code (National Standard 2010). Andthe elastic limit of IDR under minor earthquake is checkedafter implementing the optimization. The occurrence prob-ability pr for the severe earthquake is assumed to be 4.5%according to the design return period. Meantime, the frameexample is assumed to be a Class C building accordingto the Chinese Standard for Classification of Seismic Pro-tection of Buildings (National Standard 2008). Elastic andinelastic allowable interstory drift ratio limits respectivelyequal 1/250 and 1/50 (National Standard 2010). The unitcost of steel column and beam are assumed to be $723per ton and $695 per ton accordingly. As a result, basedon the values of the direct loss and indirect loss given inTable 1, the continuous damage loss function Lr can bederived using (23) and Table 2.

Moreover, the design convergence check is implementedin the ASA algorithm, in which the number of the objec-tive value of each accepted design compared with that ofthe best deign found so far is up to 15 times, and the toler-ance difference between the current objective value of theaccepted design and that of the best design found so faris less than the convergence epsilon set as 10−8. The re-annealing schedule should be conducted adaptively whenthe accepted design is up to 100 times or the generateddesign is up to 1000 times. In addition, the cross sections ofsteel members are W-shape available from the manuals ofthe American Institute of Steel Construction. More specif-ically, all column and beam design variables are selectedfrom W-sections members respectively sorting by cross-section areas and moment inertia) listed in Table 3. And

another group of constraints is introduced appropriate to theengineering problems, which guarantees that the stiffness ofthe lower floor is not less than that of the above floor and isalso quite important for the seismic concept design of build-ings including the regularity of building configuration, andstorey stiffness and strength (National Standard 2010), asexpressed by

Cm − Cm+1 ≤ 0, Bm − Bm+1 ≤ 0 m = 1, 2, 3, 4 (25)

where Cm , Bm (m = 1, 2, 3, 4, 5) are the design variables ofcolumns and beams.

3.2 Optimal results and discussions

Due to the randomness in the process of the optimization,the structural optimal design is implemented for three timesand the optimum results denoted by Opt 1, Opt 2, and Opt 3(ordered), including the design variables and objective val-ues are demonstrated in Table 4. Through the comparison ofthe optimum cost values the maximum difference is approx-imate to 1%, although it cannot be guaranteed the optimalresults are the global optimum. It is observed that there isthe small discrepancy for the former three modal periodsand lateral stiffness of optimally designed frame. What ismore, when the material cost is relatively less, the damageloss will be greater accordingly. Take Opt 2 for instance,the material cost is 135776 US$ then the estimation of thedamage loss is up to 162882 US$. The optimum height-wiseIDRs in Fig. 6 can be used to interpret this point. For Opt

Table 4 Optimal designvariable and objective valueconsidering three modes

No Column Beam f1 f2 f (obj.)

(C1, C2, C3, C4, C5) (B1, B2, B3, B4, B5) (US $) (US $) (US $)

Opt 1 (8, 11, 13, 15, 18) (2, 6, 11, 11, 12) 148470 149288 297758

Opt 2 (8, 12, 12, 12, 16) (7, 8, 15, 15, 16) 135776 162882 298458

Opt 3 (6, 11, 15, 18, 20) (4, 4, 10, 10, 12) 145948 154813 300761

870 G. Li et al.

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

1

2

3

4

5

6

7

8

9Fl

oor

Interstory Drift Ratio (%)

Opt 1 Opt 2 Opt 3

Fig. 6 Comparison of IDRs for Opt 1, 2, and 3

2, the IDRs of floor 2 and 7 are respectively up to 1.58%and 1.61%, greater than the maximum IDR correspondingto Opt 1 and Opt 3 (Table 5). Accordingly, the material cost(135776 US$) for Opt 2 is smaller than that (148470 US$and 145948 US$) for Opt 1 and Opt 3. Whereas, the dam-age loss (162882 US$) for Opt 2 is larger than that (149288US$ and 154813 US$) for Opt 1 and Opt 3. In addition, dur-ing the implementation of the MMPA, the first three modalresponse contributions are included, and Fig. 7 shows thefirst modal response contribution in terms of IDR of Opt 1,and MMPA fairly captures the distinct effect of high modes(see IDRs of floor 7, 8, and 9).

In the process of the optimization, when constraintsviolation happens it will be penalized through theequation “Penalty = base + multiplier � (constraintviolationexponent)” (Engineous Software, Inc. 2004), wherebase = 10, multiplier = 1.0, exponent = 2.0. In order tomake the penalty effective, the objective value is normalizeddivided by 1 million US$. Figure 8 displays the optimiza-tion histories of the feasible and best found so far objectivevalues for Opt 1, 2, and 3. It is seen that, lots of feasiblesolutions are found, which may help the decision makers

Table 5 The first three structural periods and maximum IDR for threeoptimal cases

No Period (s) Maximum IDR

1st 2nd 3rd Value (%) Floor

Opt 1 2.151 0.827 0.515 1.54 2

Opt 2 2.317 0.896 0.550 1.58 2

Opt 3 2.176 0.846 0.527 1.61 7

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

1

2

3

4

5

6

7

8

9

Floo

r

Interstory Drift Ratio of Opt 1 (%)

First three modal combined IDR Fundmental modal IDR

Fig. 7 Comparison of the first three modal combined and the funda-mental modal IDR of Opt 1

to make the final decision. Here, a feasible design for Opt3 (run counter = 984) is chosen, and the feasible designis (C1, C2, C3, C4, C5) = (6, 11, 12, 13, 16) and (B1,B2, B3, B4, B5) = (7, 8, 14, 15, 15). The first, secondand third periods of frame structure are 2.258, 0.871, and0.542 s, respectively. The objective value (life-cycle cost) fis 308150 US$, including the material cost f1 140475 US$and the damage loss f2 167675 US$.

In order to scrutinize the remarkable contribution of thehigher mode effect, a counterpart optimization problemonly using the conventional pushover method is formulated(objective function is also the life-cycle cost), and the opti-mum design is displayed in the first row in Table 6 and

0 200 400 600 800 1000 12000.25

0.30

0.35

0.40

0.45

0.50

0.55

Feasible (Opt 1) Better (Opt 1) Feasible (Opt 2) Better (Opt 2) Feasible (Opt 3) Better (Opt 3)

Lif

e-cy

cle

cost

, f (

mill

ion

US$

)

Runcounter

Fig. 8 Feasible and better designs during the optimization histories forOpt 1, 2, and 3

Modified-modal-pushover-based seismic optimum design for steel structures considering life-cycle cost 871

Table 6 Comparison of optimaldesign and objective valuefor 3 cases

Min: (obj.) Number of Column Beam f1 f2 f (total)

considered (C1, C2, C3, C4, C5) (B1, B2, B3, B4, B5) (US$) (US$) (US$)

modes

f 1 (9, 13, 15, 17, 19) (4, 7, 13, 14, 14) 130051 128515 258566

f1 3 (12, 13, 18, 19, 20) (7, 9, 14, 20, 20) 107546 222539 330085

f 3 (8, 11, 13, 15, 18) (2, 6, 11, 11, 12) 148470 149288 297758

7. It is observed that, the first, second and third peri-ods of building are 2.328, 0.901, and 0.559 s separately.Meanwhile, the life-cycle cost is 258566 US$, includingthe material cost 130051 US$ and the damage loss 128515US$. Compared with the optimum design (Opt 1, shown inthe last row in Table 6) of the original problem, the life-cycle cost is decreased by 13.2%, and the material costis decreased by 12.4% and the damage loss reduced by13.9% (Fig. 9). Figure 10 illustrates the IDRs of the opti-mum design only considering the contribution of mode 1(“Conventional Pushover-based”) compared with those ofthe original optimum design (Opt 1), in which “MMPA-based (mode 1)” means the response of mode 1 usingMMPA considering 3 modes, and “Conventional Pushover-based” means that pushover analysis using the fundamentalmodal load pattern. Through Fig. 10, it demonstrates thatneglecting the higher mode effects underestimates the IDRsof floor 7, 8, and 9, which makes the material cost and thedamage loss are underestimated significantly.

Besides, the damage loss for Opt 1 consists of the directloss 39810 US$ and the indirect loss 109478 US$, whilethat of Conventional Pushover-based optimization consistsof the direct loss 32749 US$ (underestimated by 17.7%)and the indirect loss 95766 US$ (underestimated by 12.5%)displayed in Fig. 11. The index of robustness IR = DirectLoss / (Direct Loss + Indirect Loss) for the building isutilized to qualify the robustness for the system proposedby Baker et al. (2008) and JCSS (2008). A robust systemis considered to be the one which the indirect loss doesnot contribute significantly to the total expected risk. If

Table 7 The first three periods and maximum IDR for thethree compared cases

Min: Number of Period (s) IDR

(obj.) considered 1st 2nd 3rd Maximum Floor

modes (%)

f 1 2.328 0.901 0.559 1.45 1

f1 3 2.650 1.063 0.666 1.96 4

f 3 2.151 0.827 0.515 1.54 2

the index of robustness IR is larger then the robustness ofthe building behaves better under the specific design haz-ard. And the index of robustness IR of the optimum designis decreased from 0.267 to 0.255, which means that theconventional pushover-based structural optimization lowerthe robustness of the optimum system effectively withoutconsidering the higher mode effects. It is shown that theconsiderable effect of higher modes of moderate and tallbuildings should be carefully incorporated into the seismicanalysis and optimal design.

Moreover, another optimization problem, which only thematerial cost is set as the objective function, is formulatedto indicate the significance of the damage loss. The opti-mum design is exhibited in the second row in Tables 6 and 7,and the first, second and third periods are 2.650, 1.063, and0.666 s respectively. For the found optimum design, the IDRof the fourth floor approximates the limit value of 2% andthe IDR constraint of the floor behaves active. Besides, thelife-cycle cost is up to 330085 US$, including the materialcost 107546 US$ and the damage loss 222539 US$. Com-pared with the optimum design (Opt 1), the life-cycle costis increased by 10.9%, where the material cost is decreasedby 27.6%, and nevertheless the damage loss increased by49.1% (Fig. 9). Further, Fig. 12 demonstrates the IDRs of

1 2 30

50000

100000

150000

200000

250000

300000

350000MMPA-based;obj: Life-cycle Cost

MMPA-based;Obj: Material Cost

Cos

t, f

(US$

)

Case

Damage loss f2

Material cost f1

Conventional Pushover-based;Obj: Life-cycle Cost

Fig. 9 Comparison of the material cost and damage loss for 3 cases

872 G. Li et al.

0.0 0.4 0.8 1.2 1.6 2.00

1

2

3

4

5

6

7

8

9Fl

oor

Interstory Drift Ratio (%)

Obj: Life-cycle Cost MMPA-based MMPA-based (mode 1) Conventional Pushover-based

Fig. 10 Comparison of the IDRs of the optimum design using conven-tional pushover-based method and that of Opt 1

the optimum structural design with the only material cost asthe optimization objective, and that of the original optimumdesign, which the responses of the respective fundamentalmode also displayed.

Finally, the damage loss for this case consists of thedirect loss 37814 US$ and the indirect loss 184725 US$(Fig. 11). Though the direct loss is reduced by 5.0%, theindirect loss is increased by 68.7% distinctively comparedwith Opt 1. The Index of robustness for the material cost ori-ented optimum design is merely 0.170, while life-cycle costoriented is up to 0.267 (Opt 1). It appears that the life-cyclecost oriented structural optimization not only achieves morecost-effective design than the material cost oriented solely,but improves the robustness of the building as well.

1 2 30

50000

100000

150000

200000

250000

300000

MMPA-based;obj: Life-cycle Cost

MMPA-based;Obj: Material Cost

Los

s, f

(U

S$)

Opt

Indirect Loss Direct Loss

Conventional Pushover-based;Obj: Life-cycle Cost

Fig. 11 Comparison of the direct and indirect loss for 3 cases

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.20

1

2

3

4

5

6

7

8

9

Floo

r

Interstory Drift Ratio (%)

MMPA-based Obj: Life-cycle Cost Obj: Life-cycle Cost (mode 1) Obj: Material Cost Obj: Material Cost (mode 1)

Fig. 12 Comparison of the IDRs of material cost oriented optimumdesign and that of Opt 1

4 Conclusions

Performance based optimization techniques with theMMPA method to estimate the seismic demands of the steelstructure under considered design earthquakes in terms ofthe Chinese seismic code design standards is proposed inthis paper, which provides an effective way for not onlyoptimizing the structural life-cycle cost as well as satisfyingthe drift performance design criteria.

The powerful adaptive simulated annealing algorithm isapplied to effectively achieve the optimal points (designs).Here, an appropriate statistical model for evaluating thedamage loss of the structures is utilized based on the fuzzy-decision theory. Furthermore, the higher mode effect ofmoderate and tall building can be captured fairly throughthe MMPA procedure. Simultaneously, the contributionsof higher modes are estimated and investigated in detail,which present significant effect on the life-cycle cost. It isillustrated that the contributions of the higher mode effectsare underestimated by 13.2% of the life-cycle cost for theillustrative example using the conventional pushover anal-ysis methods without considering the higher mode effects.Moreover, through the illustrated frame example consider-ing the life-cycle cost, there exist to be a preferred balancedesign, which represents a compromise between the mate-rial cost and the damage loss. Meanwhile, the life-cycle costoriented structural optimization can not only obtain morecost-effective design than only material cost oriented, butalso improve the robustness of the optimum design greatly.

Acknowledgments The authors are grateful to Prof. C.M. Chan ofDepartment of Civil Engineering, Hong Kong University of Scienceand Technology for his insightful suggestions. And we also would liketo acknowledge Dr. D.G. Lignos of Department of Civil Engineer-ing and Applied Mechanics, McGill University for his sincere helps

Modified-modal-pushover-based seismic optimum design for steel structures considering life-cycle cost 873

of the structural data files and contributions on the modified IbarraKrawinkler deterioration model. The supports of the National Nat-ural Science Foundation of China (Grant nos. 90815023 and51021140006), and the Fundamental Research Funds for the CentralUniversities (Grant no. DUT11LK06) are much appreciated. Also,we greatly appreciate the anonymous reviewers for their insightfulsuggestions and comments on the early version of this paper.

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