Modi ed seismic design lateral force distribution for the

Scientia Iranica A (2020) 27(3), 1050{1065

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringhttp://scientiairanica.sharif.edu

Modi�ed seismic design lateral force distribution forthe Performance-Based Plastic Design (PBPD) of steelmoment structures considering soil exibility

A. Gholamrezatabara, B. Ganjavib, G. Ghodrati Amiria;�, and M.A. Shayanfara

a. School of Civil Engineering, Iran University of Science & Technology, Narmak, Tehran, P.O. Box 16765-163, Iran.b. Department of Civil Engineering, University of Mazandaran, Babolsar, Iran.

Received 25 May 2017; received in revised form 21 November 2017; accepted 18 August 2018

KEYWORDSSeismic design lateralforce;Performance-basedplastic design;Steel momentstructures;Soil-structureinteraction;Optimum shearproportion factor.

Abstract. It is well known that structures designed by conventional seismic designcodes experience large inelastic deformations during strong ground motions. A realisticestimation of force distribution based on inelastic response is one of the important stepsin a comprehensive seismic design methodology in order to represent expected structuralresponses more accurately. This paper presents an extensive parametric study to investigatethe structural damage distribution along the height of the Steel Moment-Resisting Frames(SMRFs) designed based on the state-of-art constant-ductility Performance-Based PlasticDesign (PBPD) approach considering soil exibility e�ects when subjected to 20 strongground motions. To this end, the e�ects of fundamental period, target ductility demand,and base exibility level are investigated and discussed. Based on the numerical results ofthis study, simpli�ed equations are proposed for practical purposes to re�ne and modifythe lateral force distribution pattern already suggested by researchers based on the studyof inelastic behavior developed for �xed- and exible-base structures by using relativedistribution of maximum story shears of the selected structures subjected to variousearthquake ground motions. It is demonstrated that the proposed equations can adequatelyestimate the optimum values of the shear proportioning factor in both �xed-base and soil-structure systems.© 2020 Sharif University of Technology. All rights reserved.

1. Introduction

The design lateral forces and design story shearsfrom the equivalent lateral force recommended by thecurrent code-speci�ed seismic design procedure (e.g.,IBC 2009 [1], ASCE/SEI 7{10 [2], NEHRP 2009 [3],UBC [4]) are primarily based on elastic analysis. In

*. Corresponding author.E-mail addresses: [email protected] (A.Gholamrezatabar); [email protected] (B. Ganjavi);[email protected] (G. Ghodrati Amiri);[email protected] (M.A. Shayanfar)

doi: 10.24200/sci.2019.21616

this method, the structural elements are designedbased on equivalent static forces, and the shape ofthe fundamental mode of the structure is dominant todetermine the height-wise distribution of these seismicdesign static forces. Establishing such code-compliantlateral load distributions patterns may not providean accurate representation of the story shear strengthdemands and explicitly lead to seismic performanceassessment criteria [5{12]. Chopra [13] conducted thenonlinear dynamic analysis of several shear-buildingmodels subjected to the El-Centro Earthquake of 1940to evaluate the ductility demands corresponding toeach story. The models were designed in accordancewith the seismic force patterns speci�ed by Uniform

A. Gholamrezatabar et al./Scientia Iranica, Transactions A: Civil Engineering 27 (2020) 1050{1065 1051

Building Code [4]. It was concluded that this distri-bution pattern did not lead to equal ductility demandin all stories. Moghaddam [14] conducted the sameanalysis of a number of shear buildings with thespeci�ed yield strength distributed by UBC-97 [4] sug-gestion pattern. They showed that the code-compliantlateral load distribution did not lead to a uniformheight-wise distribution of ductility. Moghaddam andMohammadi [15] proposed a design lateral load patternfor the seismic design of shear-building structures toachieve uniform deformation distribution. In anotherinvestigation, they developed a new concept to op-timize the distribution pattern for the performance-based seismic design approach [11]. However, theirstudy was based on the results of shear-building struc-tures that might not be applicable to more realisticbuilding structures such as moment-resisting framesthat are basically designed based on the \strong-column weak-beam" design philosophy. Several otherstudies focused on moment-resisting frame that aimedto develop new lateral load patterns to control theamount of the global structural damage and to achieveprede�ned performance objectives and, �nally, providehigher performance levels exposed to seismic groundmotions. Leelataviwat et al. [8] proposed improvedload distribution using the concept of energy balanceapplied to moment-resisting frames with an intendedyield mechanism. Lee and Goel [16] primarily discussedthe discrepancy between the earthquake-induced shearforces and the forces determined by lateral load dis-tribution patterns. They applied the same concept topropose load pattern in accordance with the UniformBuilding Code [4], which was a function of the massand fundamental period of the structure. Goel etal. [17] applied the method successfully to a varietyof common steel framing systems and reinforced con-crete Moment Frames (MFs). Through the resultsof extensive inelastic static and dynamic analyses,they showed that the frames could develop desiredstrong column-sway mechanisms and that the storydrifts and ductility demands were well within thetarget values, thus meeting the desired performanceobjectives. Park and Medina [18] proposed a seismicdesign methodology for moment-resisting frames basedon uniform structural damage distributed along theheight. They concluded that, based on the proposedapproach, designs were expected to provide increasedprotection against global collapse and loss of life duringa strong earthquake event. Chao et al. [19] primarilyreviewed the lateral force distributions used in thecurrent seismic codes by conducting the nonlineardynamic analysis of several frame structures. Theydemonstrated that code lateral force distributions didnot represent the maximum force distributions thatmight be induced during the nonlinear response ofmotions and might make inaccurate predictions of

deformation and force demands. Their comprehensivestudies lead to the development of a new seismicdesign lateral load distribution based on the inelasticbehavior of a structure and, also, a new methodologycalled Performance-Based Plastic Design (PBPD) forthe seismic design of a wide range of frame systems in-cluding moment-resisting frames, eccentrically-bracedframes, special truss-moment frames, and reinforcedconcrete frames. In these investigations, performancelimit states are pointed out by the predictable globalyield mechanism and the pre-designated target driftlimit. The design base shear at each performancelevel is derived from an energy-based method, wherethe energy required to push the structure up to thetarget drift is calculated as a fraction of elastic inputenergy that is obtained from the selected elastic designspectra [16,17,19]. However, they did not incorporatethe target ductility demand in the design processdirectly.

All of the above-mentioned research studies arebased on the �xed-base structures without consid-ering the e�ect of soil exibility, i.e., Soil-StructureInteraction (SSI). Several studies have been performedto investigate the e�ect of SSI on the seismic re-sponses of structures [20{24]. The results of thesestudies demonstrated that structures supported bysoil-foundation might be a�ected by SSI signi�cantroles due to wave propagation in the soil medium.Based on the concept developed for �xed-base shearstructures, Ganjavi and Hao [25] proposed new opti-mum design lateral loading patterns for the seismicdesign of elastic soil-structure systems through theintensive dynamic analysis of multistory shear-buildingmodels subjected to a group of 21 arti�cial earth-quakes adjusted to the soft soil design. Ganjavi etal. [26] also parametrically investigated the adequacyof code-speci�ed lateral loading patterns for the seismicdesign of elastic and inelastic soil-structure systemsbased on the analysis of shear buildings consideringSSI e�ects. Due to the challenges of code-speci�edlateral load distributions and the need to improve theseismic performance of exible-base buildings on softsoils, they [26] proposed an optimum seismic designmethodology for nonlinear shear buildings located onsoft soils based on the concept of uniform damagedistribution. However, their study was also based onthe results of shear building structures that might notbe applicable to more realistic building structures suchas SMRFs.

This study evaluates lateral force distributions bythe nonlinear dynamic analysis of constant-ductilitySMRF structures designed according to the PBPDprocedure located on alluvium soil considering SSIe�ects. The aim of this study is to parametricallyinvestigate height-wise structural damage (ductilitydemand) distribution designed based on the conven-

1052 A. Gholamrezatabar et al./Scientia Iranica, Transactions A: Civil Engineering 27 (2020) 1050{1065

tional PBPD approach for constant-ductility �xed-and exible-base SMRF structures. Moreover, theadequacy of load patterns already proposed in PBPD isinvestigated through the height-wise distribution of theinter-story ductility demand ratio subjected to variousstrong ground motions.

2. Analytical model based on PBPD approach

The PBPD method is based on two key performancelimit sates including pre-selected target drift and yieldmechanisms [8]. These two design parameters controlthe degree and distribution of structural damagesdirectly. In this approach, the determination of designbase shear, lateral force distribution, and plastic de-sign corresponding to the speci�ed performance levelare the three main components of design. For aspeci�ed hazard, the design base shear is calculatedby equating the work needed to push the structuremonotonically up to the target drift to the energyrequired by an equivalent Elastic-Plastic Single-Degree-Of-Freedom (EP-SDOF) system to achieve the samestate. Moreover, the height-wise distribution of lateraldesign forces is developed based on the concept ofthe relative story shear distributions and is consistentwith the results of the inelastic dynamic response [19].Finally, the proposed plastic design procedure is per-formed to detail frame members in order to achieve theintended yield mechanism.

2.1. Design base shearAs explained earlier, the design base shear as a keyelement in the PBPD method is calculated by equatingthe work needed to push the structure monotonicallyup to the target drift to that required by an equivalentEP-SDOF system to achieve the same state. For ide-alized Elastic-perfect Plastic (EP) behavior and usingthe value of pseudo-velocity or substituting pseudo-acceleration, the work energy can be calculated throughEq. (1) [27]:

Ee+Ep= E= (1=2MS2v)=

12 M

�T2�Sa:g

�2

; (1)

where Ee and Ep are the elastic and plastic componentsof the energy needed to push the structure up to thetarget drift, respectively. Sv is the design pseudo-spectral velocity; Sa is the pseudo-spectral accelera-tion; T and M are the natural period and total mass ofthe system, respectively. is the energy modi�cationfactor, which is related to the structural ductility factor(�s) and the ductility reduction factor (R�), and canbe obtained by the following equation:

=2�s � 1R2�

: (2)

Based on the spectra proposed by Newmark and Hall

Figure 1. Energy modi�cation factor, , versus theperiod.

(1982) [27], the energy modi�cation factor ( ) can beobtained through Eq. (2), as shown in Figure 1 [16].

The work-energy equation can be rewritten in thefollowing form:

12

�Wg

��T2�� VyWg�2

+ Vy

NXi=1

�ihi

!�p

=12 �Wg

��T2�Sag

�2

: (3)

By simplifying Eq. (3), the ratio of (Vy=W ) can bewritten through Eq. (4):

VyW

=��+

p�2 + 4 S2

a

2; (4)

where � is a dimensionless parameter given by Eq. (5):

� =�h� � �P 8�2

T 2g

�; (5)

where �p represents the plastic rotation at the targetdrift ratio, and h� stands for

PNi=1 �ihi, where �i is

the proportioning factor of the equivalent lateral forceat level i.

2.2. Lateral force distributionA new design lateral force distribution for the plasticdesign was obtained based on the results of inelasticdynamic responses and maximum story shears alongthe height of structural systems, de�ned as Eq. (6) [19]:

Vi =

Pnj=i wjhjPnj=1 wjhj

!0:75T�0:2

Vy; (6)

where w and h are the seismic weight and height abovethe base, respectively. T is the fundamental period andVy represents the design base shear. The equation ismore consistent with the results of inelastic analysisthan with the code-speci�ed seismic load pattern. Itcan be shown that the ratio Vi=Vn, designated as


shear distribution factor, �i, can be obtained throughEq. (7) [19]:

ViVn

= �i =

Pnj=i wjhjwnhn

!0:75T�0:2

; (7)

where Vi and Vn are the story shear at level i and toplevel, respectively, and �i is the shear proportioningfactor at level i. Hence, the lateral force at level i, Fi,can be expressed as:

Fi = (�i � �i+1)Vn: (8)

2.3. Plastic design procedureThe provided design approach is capable of achievingthe satisfactory performance of structures under asevere earthquake by means of a pre-de�ned controlledmechanism. The procedure develops a strong-column-weak beam mechanism and a stable hysteretic responsewithin an acceptable margin of target drift [8]. Byapplying the principle of virtual work for the beammechanism (Figure 2), the required beam strength ateach level can be obtained through Eq. (9):

nXi=1

2�iMpbr+2Mpc=nXi=1

Fihi=nXi=1

(�i��i+1)hiFn;(9)

where Mpbr and Mpc are the plastic moment of beamsand the required plastic moment of columns in the �rststory, respectively (Figure 2). Leelataviwat et al. [8]proposed the plastic moment of the �rst-story columnsto avoid the pre-de�ned mechanism as in Eq. (10):

Mpc =1:1V h1

4; (10)

where V is the total base shear, h1 is the height ofthe �rst story, and coe�cient 1.1 is the overstrengthfactor to account for possible overloading due to strainhardening. However, as the main goal of this approach,an attempt has been made to prevent the formationof plastic hinges in the columns except at the columnbases of the structure. Hence, the column shouldbe designed for the exural moment greater than the

sum of the exural strength of the beams at the samejoint. To ensure that the strong column-weak beammechanism is achieved, columns should be designedfor the sum of the nominal plastic moment of beamsmultiplied by the over-strength factor (�). Moreover,to include the beams yielding overstrength, the appliedforce at each level, Fi, must be updated as follows(Eq. (11)):

Fiu = (�i � �i+1)Fnu; (11)

where Fnu is the updated force at the roof level andcan be determined by the equilibrium equation for onecolumn as Eq. (12):

nXi=1

(�i � �i+1)hiFnu = Mpc +nXi=1

�iMpbi; (12)

where Mpc is the plastic moment at the base of theframe (Eq. (10)), and � and Mpbi are the overstrengthfactor and the nominal plastic moment of beam atlevel i, respectively. After updating the lateral forces,design moments of the column can be determinedby developing the column as a cantilever based onEq. (13):

Mc(h) =nXi=1

�i�iMpbi �nXi=1

�iFiu(hi � h); (13)

where Mc(h) is the moment in the column at height habove the ground, and �i is equal to 1 for h � hi andzero for else. The axial force in the column at heighth above the ground, Pc(h), can be obtained throughEq. (14):

Pc(h) =nXi=1

2�iMpbi

L+ Pcg(h); (14)

where L is the span length of the beams, and Pcg(h) isthe gravity axial force acting at height h. By applyingthe explained approach, the values of Mc(h) and Pc(h)of the column element can be obtained according tothe plastic analysis procedure and, then, it can bedesigned as a beam-column element by appropriate

Figure 2. One-bay frame with (a) selected mechanism and (b) frame with soft-story mechanism.


Table 1. Design parameters for 4-, 8-, 12-, 16-story steel Performance-Based Plastic Design (PBPD) frames used tocalibrate �i.

Design parameters 10% in 50 yearshazard

Story height and beam span forsingle-bay moment-resisting frames

Number of Stories 4, 8, 12, 16Sa 0.36 gYield drift ratio �y 1%Target drift ratio �u 2%, 4%, 6%Inelastic drift ratio �p = �u � �y 1%, 3%, 5%� = �u=�y 2, 4, 6R� 2, 4, 6 0.75, 0.438, 0.306

�i =�ViVn

�bb = 0:75T�0:2

Figure 3. Selected 4-, 8-, 12-, 16-story moment frames.

design provisions. Finally, it should be mentionedthat, as a well-known numerical modeling strategy, thesuperstructure frame elements followed the lumped-plasticity modeling approach to computing their non-linear response regarding the Rayleigh damping andrigid diaphragm assumption. For computer modeling,beams and columns were modeled as non-degradingquasi-elastoplastic (i.e., at a strain-hardening ratio of2%). A moment-curvature relationship that considers

the axial load- exural bending interaction was con-sidered to model the hysteretic behavior of the steelcolumns. However, slab contribution to the beam'sbending capacity was neglected in this study. Thedesign parameters and elevation views of designed MFsbased on PBPD approaches are listed in Table 1. Thesizes of beams and columns were selected using AISC-LRFD speci�cations [28] assuming A572 GR.50 steelfor all members, as shown in Figure 3.


3. Base exibility model

In this study, a cone model was proposed to simulatethe dynamic behavior of an elastic homogeneous soilhalf-space, as shown in Figure 4 [29]. The modelis developed using one-dimensional wave propagationtheory and can represent a circular rigid foundationwith mass mf and mass moment of inertia If restingon homogeneous half-space soil. The cone model iswidely used for modeling both surface and embed-ded foundations and, in lieu of the rigorous elasto-dynamical approach, it can provide su�cient accu-racy for engineering design purposes [30]. The soil-foundation system is modeled by an equivalent lineardiscrete model based on the cone model approach withfrequency-dependent coe�cients [29]. The foundationis considered as a circular rigid disk (the exibilityof the foundation is not taken into account). Thecomponents of motions for the following half-spacewere modeled through two transitional and rotationalDegree Of Freedoms (DOFs). The coe�cient of swayand rocking springs and dashpots representing theassociated motions are summarized as in Eqs. (15)and (16) [29]:

kh =8�V 2

s r2� � ; ch = �VsAf ; (15)

k' =8�V 2

s r3

3(1� �); c' = �VpIf ;

M' =9��r5

128(1� �)

�VpVs

�2

; (16)

where kh, k', ch, and c' are the sway sti�ness,

sway viscous damping, rocking sti�ness, and rockingviscous damping, respectively. �, �, Vp, and Vsstand for the density, Poisson's ratio, and dilatationaland shear wave velocities of soil, respectively, and ris the radius of the equivalent circular foundation.Moreover, for the vertical and rocking motions in thecase of nearly incompressible soil (1=3 < � < 1=2),an additional tramped mass moment of inertia �M'equal to �M' = 0:3�(� � 1=3)�r5 is added to If ,which is connected to the foundation and moves asa rigid body in the phase with the foundation forthe rocking degree of freedom. An internal rotationalDOF ', with a mass moment of inertia m', wasde�ned to incorporate frequency dependency of soildynamic sti�ness. It is worth noting that, based onthe current seismic provisions such as NEHRP 2003 [3]and FEMA 440 [31], the soil strain level related tothe degraded shear wave velocity to approximate thesoil nonlinearity e�ects on soil-foundation elements isconsidered [32].

It is shown that, for a speci�c earthquake, theseismic response of a soil-structure system dependson the dynamic characteristics of the structure andsoil beneath it. The SSI e�ective parameters thatare known as non-dimensional key parameters and canbest describe the seismic response of the superstructurein a complex soil-structure system are de�ned asfollows [33].

A dimensionless frequency as an index for thestructure-to-soil sti�ness ratio is de�ned as a0 =!�x �H=vs, where !�x denotes the natural frequencyof the �xed-base structure. �H is the e�ective heightof the structure corresponding to the fundamentalmode properties of Multi-Degree Of Freedom (MDOF)

Figure 4. Typical multi-story Steel Moment-Resisting Frame (SMRF) building models: (a) Fixed-base model and (b) exible-base model.


building and can be obtained through Eq. (17):

�H =nXj=1

"mj'j1

jXi=1

hi

!#,nXj=1

mj'j1 ; (17)

where j is the number of stories, mj is the mass of thejth story, hi is the height from the base level to levelj, and 'j1 is the amplitude at the jth story of the �rstmode.

It is shown that a0 has the most signi�cant e�ectson the seismic response of the soil-structure system [34].Based on studies conducted to categorize the intensityof the soil-structure interaction e�ects due to the base exibility, a0 takes the value between zero for �xed-basestructures and 2 for the very exible-base models [35].

� Aspect ratio of the building �H=r is de�ned as thesecond interacting parameter with various e�ects onSSI practices [35];

� Ductility demand of structure is de�ned as � =um=uy, where um and uy are the maximum inter-story displacement and the yield inter-story dis-placement, respectively [36];

� Structure-to-soil mass ratio index is de�ned as �m =mtot=�r2H, where mtot and H are the total weightand total height of the structure, respectively;

� The ratio of the foundation-to-structure mass ismf=mtot, where mf is the mass of the rigid foun-dation;

� Material damping ratio of the soil (�0).

The �rst two interacting parameters are the key pa-rameters that de�ne the main SSI e�ect [31]. The thirdone controls the level of nonlinearity in the structure.Other parameters of less importance are assigned tosome typical values for conventional building struc-tures [35]. Hence, this study considers the foundation-to-structure mass ratio as 0.5 and the foundation massratio as %10 of the total mass of the superstructure.Poisson's ratio is set to 0.4 for the alluvium soil. Inaddition, the damping ratio of the soil material is setto 5%.

4. Nonlinear dynamic analysis procedure andselected ground motions

A series of 4-, 8-, 12-, and 16-story Steel Moment-Resisting Frames (SMRFs) are considered to inves-tigate the e�ect of various design load patterns onthe height-wise distribution of story shear forces. Allmodels are incrementally subjected to a group of strongground motions, and a step-by-step solution scheme isapplied during time history analysis. The steel frameswere analyzed based on the nonlinear dynamic analysisvia computer software OpenSees [37]. Rayleigh-type

damping was considered for the analysis, in which 5%of critical damping was assigned to the �rst two modesof vibration of the frames. Further, nonlinear static(pushover) analyses for each frame were performed toobtain relevant dynamic characteristics such as baseshear and roof drift at the �rst plastic hinge andyielding. It should be noted that pushover analysis wasconducted by the OpenSees software [37], consideringa slow ramp loading function and a triangular-invertedloading distribution as prescribed in the Mexican seis-mic design standards, similar to that of the ASCE-7-10 load pattern (2010) and utilized by Ruiz-Garc��aand Gonz�alez for steel MF structures [38]. Since theobjective of this study is to achieve a pre-speci�edconstant-ductility demand, it is necessary to evalu-ate the nonlinear maximum story drift demand forvarious building frames. Hence, it was decided toscale up the acceleration spectral ordinates such thatthe maximum inter-story ductility demand among allstories reached the speci�ed target value [38]. For theinelastic dynamic analysis, an ensemble of 20 earth-quake ground motions with di�erent characteristics wasutilized to incorporate the e�ect of ground motioncharacteristics recorded on alluvium (NEHRP site classD [39]). All the selected ground motions are obtainedfrom earthquakes at a magnitude greater than 6.5having the closest distance to fault rupture more than10 km without pulse-type characteristics. The mainparameters of the selected ground motions are givenin Table 2. For each record, the horizontal componentwith a larger peak ground velocity is de�ned as a strongcomponent. A ow chart is provided to show iterativeanalysis procedures for the constant-ductility PBPDapproach, as illustrated in Figure 5 [40].

5. Relative story shear distribution

The relative story shear distribution is de�ned as theratio of maximum earthquake-induced story shear forceat level i to that at top level n (i.e., � = Vi=Vn). Themean responses are obtained by averaging the resultsof the structures to each record. It is notable that storyshear Vx in any story is the sum of the lateral forcesabove that story; thus, the story shear distributionand lateral force distribution have a direct relationshipde�ned as Eq. (18):

Vx =nXi=x

Fi: (18)

In addition, the proposed equation (Eq. (18)) tends tosomewhat overestimate the optimal b value obtainedfrom the analyses. Eq. (19) is recommended and usedfor both cases of �xed- and exible-base SMRFs asfollows:�i = (Vi=Vn)b: (19)


Table 2. Characteristics of strong ground motions used in this study.

Event Record ID Station name Mag. Distance(km)

Ag(g)

Vg(cm/s)

Dg(cm)

1 RSN68.eq LA - Hollywood Stor FF 6.61 22.8 0.2 21.7 15.92 RSN162.eq Calexico Fire Station 6.53 10.5 0.3 22.5 9.93 RSN169.eq Delta 6.53 22.0 0.3 33 20.24 RSN174.eq El Centro Array #11 6.53 12.6 0.4 44.6 21.35 RSN721.eq El Centro Imp. Co. Cent 6.54 18.2 0.4 48.1 19.36 RSN728.eq Westmorland Fire Station 6.54 13.0 0.2 32.3 22.37 RSN752.eq Capitola 6.93 15.2 0.5 38 7.18 RSN776.eq Hollister - South & Pine 6.93 27.9 0.4 63 32.39 RSN777.eq Hollister City Hall 6.93 27.6 0.2 45.5 28.510 RSN778.eq Hollister Di�erential Array 6.93 24.8 0.3 44.2 19.711 RSN783.eq Oakland - Outer Harbor Wharf 6.93 74.2 0.29 41.8 9.612 RSN953.eq Beverly Hills - 14145 Mulhol 6.69 17.2 0.5 66.7 12.213 RSN960.eq Canyon Country - W Lost Cany 6.69 124.0 0.4 44.4 11.314 RSN1003.eq LA - Saturn St 6.69 27.0 0.4 41.6 5.015 RSN1077.eq Santa Monica City Hall 6.69 26.5 0.9 41.6 15.216 RSN1107.eq Kakogawa 6.9 22.5 0.3 26.9 8.817 RSN1116.eq Shin-Osaka 6.9 19.2 0.2 31.3 8.418 RSN1158.eq Duzce 7.51 15.4 0.3 58.9 44.119 RSN1203.eq CHY036 7.62 16.0 0.2 44.8 34.020 RSN3749.eq Fortuna Fire Station 7.01 20.4 0.3 38.1 16.7

Figure 5. Flowchart showing a general procedure forPerformance-Based Plastic Design (PBPB) approach tothe constant-ductility nonlinear dynamic analysis of �xed-and exible-base buildings.

Figure 6 shows relative story shear distributionsobtained from nonlinear dynamic analyses of 4-, 8-,and 16-story �xed-base SMRFs that are designed bythe conventional PBPD subjected to 20 individual

earthquake ground motions. The results are providedat two levels of low and high inelastic behavior. Inaddition, the average results of all the selected groundmotions along with shear force pattern having di�erentshear proportional values are provided and compared.As seen earlier, the distribution power b is a functionof the fundamental period of vibration and the level ofinelastic behavior, such that it decreases and increasesby increasing the period and ductility demand value,respectively. The dependency of the fundamentalperiod is also reported and consistent with theprevious studies [16,19]. However, nothing has beenyet reported on the e�ect of the target ductilitydemand on the b value. The results of this �gure showthat the optimum b value could vary from 0.67 to 1.1based on the values of T�x and �t. The results arediscussed in the upcoming section in greater detail.

6. E�ect of shear proportional factor

Frames with 4-, 8-, 12-, and 16-story levels were used to�nd an optimal distribution of the shear proportioningfactor through nonlinear dynamic analysis. Each framewas designed using four possible functions for the shearproportioning factor. These functions were assumed bythe values of b taken as 1, 0.75, 0.5, and 0.25. The mem-ber sizes of the frame structures are designed by select-ing 2% target drift and using the design parameters,as presented in the previous section. It was assumedthat the distribution of beam strengths followed thedistribution of shear proportioning factors. The resultsshowed the relative distribution of maximum story


Figure 6. Relative distribution of story shear Vi=Vn for �xed-base structures under strong ground motions.

shears of the selected frames and the upper bound dis-tribution of shear proportioning factors in the nonlineardynamic analysis results for various ductility demands.

For a 4-story frame structure with a shorterperiod (T = 0:72 s) and regardless of the values of theshear proportioning factor, the relative distributionsof maximum story shears are close together. The

relative story shear distribution using various valuesof b ranging from 0.7 to 0.9 represents an upper boundof the nonlinear dynamic analysis results at low to highductility ratios. As shown for the short-period framestructure, the required value of b increases considerablyas the level of inelasticity increases. To achieve morerealistic distributions of the shear proportioning factor


according to the period of the structures under strongmotions, results of 8-, 12-, 16-story building structuresare provided. The relative distributions of maximumearthquake-induced story shears for each case of 8- to16-story frames are shown in Figure 3. As shown ear-lier, the distribution of maximum earthquake-inducedstory shears of frames is related to the period and canbe approximated using optimal b values of the shearproportioning factor at various levels of inelasticity.As illustrated earlier, the optimal b value decreasesas the period increases, and it increases as the targetductility demand increases. Based on the mean resultsof the distribution of maximum story shears of framesobtained from 20 strong earthquakes, the optimal bvalue for each frame can be determined by the trendline at the upper bound of story shear distributionfor various ductility demands. For an 8-story (T =1:44 s) building, the optimal b value is obtained rangingfrom 0.65 to 0.7 for low to high ductility demands bycalculating the least square �t values. Further analysesby Chao and Goel (2005 and 2006a) [41,42] showedthat the relative story shear distribution using b = 0:75represented an upper bound of the nonlinear dynamicanalysis results. The same analysis was conducted inthe case of the 12- and (T = 2:1 s) 16-story (T =2:5 s) building frames with a longer period, and thecorresponding values of b were then obtained at variousductility ratios. Consequently, the optimal b value forthese cases and the recommended trend line for selectedframes are presented in Figure 7.

7. E�ect of ductility

The e�ects of ductility demand on the relative storyshear demand distribution are investigated using threeSMRF systems with 4, 8, 12, and 16 stories asrepresentative of low- to high-rise buildings. Results ofthe relative distribution of story shear at various lowto high levels of inelasticity (� = 2; 4; 6) are plotted inFigure 8. As demonstrated, generally, by increasing thelevel of inelasticity, the relative story shear increasesfor all models from shorter to long period cases. In theselected frames with various stories, a small di�erenceof story shear can be seen between moderate and highlevels of inelasticity (� = 4; 6) when compared with thecorresponding value of low ductility demand. Further,it is revealed that this trend is followed for low- andhigh-rise SMRF buildings. The b value correspondingto lower and higher ductility demands are obtainedbased on the relative distributions of maximum storyshears.

8. E�ect of base exibility

In order to examine the e�ect of soil exibility on thenonlinear response of SMRFs designed based on the

PBPD approach, Figure 9 is plotted. To this end,the individual, mean, mean plus standard deviation(mean + �), and di�erent shear proportion factors areshown at low (a0 = 1) and high (a0 = 2) levels ofbase exibility. As seen, by increasing the a0 value, theamount of the required parameter b decreases; however,in the case of predominant SSI e�ect (a0 = 2), therelative story shear demands of top stories increase.In addition, as can be seen, the mean envelope curvesof the maximum responses can adequately predict therequired relative shear force demands of all the selectedrecords that are obtained by changing the b values.In fact, for each soil-structure system, the optimumb values were computed based on nonlinear dynamicanalyses. It is also observed that the envelope curvescorresponding to the optimum values are very close tothe curves of +�. The results indicate that, similarto the �xed-base systems, the shear proportion factorb is dependent on both fundamental period and levelof inelastic demands, where the latter e�ect has notbeen taken into account for the conventional PBPDapproach. It can be observed that by increasing thestructural period and ductility demand, the b valuesdecrease and increase, respectively. Based on theoptimum values of b, a practical expression is proposedin the next section.

9. Dispersion of results and proposed practicalequation

The results presented in the previous sections areobtained from the mean responses of an ensemble of20 strong ground motions listed in Table 2. It is clearthat the application of the mean value provides anaverage, where its e�ciency depends on the dispersionof computed results. Based on observations, it isbelieved that the structural response of a structuralsystem with certain dynamic characteristics is stronglyrelated to the selected ground motion record. Inaddition, the dispersion of results shows the impact ofrecord-to-record variability on the nonlinear responsesof systems. Hence, as an e�ective tool to evaluatethe dispersion of obtained results, the Coe�cient OfVariation (COV) is utilized, which is de�ned as theratio of the standard deviation to the mean value.As illustrated in Figure 10, the COV distribution isprovided for the 8- and 16-story SMRF structures withvarious ductility demands. The derived spectra presentthe dispersion along the height of selected models. Itis observed that increasing the fundamental period ofstructures is accompanied by an increase in COV ofthe maximum story shear distribution in both �xed-and exible-base cases. The COV values of the resultsexhibit low dependency on the ductility demand forstructures with shorter periods. This is probablybecause the structural response is governed primarily


Figure 7. Comparison of relative shear distributions by various shear proportioning factors and �xed-base system.

by the fundamental vibration mode in a short periodrange; hence, the ductility demands along the buildingheight also follow the fundamental mode pattern.

The format for this design lateral force distribu-tion based on the inelastic state of a structure was

originally proposed by Lee and Goel [16] by using theshear distribution factor derived from the relative dis-tribution of maximum story shears of a large number ofSMRFs subjected to four selected earthquake records.They applied the least square �t method and proposed


Figure 8. Relative distribution of story shear Vi=Vn for designed Moment Frames (MFs) with various ductilities.

Figure 9. Relative distribution of story shear Vi=Vn for designed Moment Frames (MFs) for exible-base SteelMoment-Resisting Frames (SMRFs).

practical Eq. (19), in which b is de�ned as Eq. (20):

b = �T�0:2: (20)

T is the fundamental period. The value of parameter� was originally proposed as 0.5 by Lee and Goel

(2001) [16], which was later revised to 0.75 basedon more extensive nonlinear dynamic analyses of Ec-centrically Braced Frames (EBFs) and Special TrussMoment Frames (STMFs) by Chao and Goel [41,42].However, the level of inelastic behavior (i.e., ductil-ity demand) was not considered in their proposed


Figure 10. The Coe�cient Of Variation (COV) of the relative distribution of story shear Vi=Vn for various ductilitydemands.

equation. In this study, extensive nonlinear dynamicanalyses were carried out on SMRF systems designedbased on the PBPD approach and at di�erent levels ofinelasticity (i.e., constant ductility demand) to modifythe suggested lateral force distribution for variousductility demands. Based on the proposed equationfor various ductility demands, a more realistic designlateral force distribution was obtained that accounts forthe inelastic behavior of structures when subjected tostrong ground motions. The suggested optimum b valuefor the lateral force distribution that can be appliedto most of the conventional frame types is de�ned asEq. (21):

b = aT�c; (21)

where the parameters a and c are computed fromnonlinear regression analyses of numerical data for both�xed- and exible-base structures having two levels oflow and high inelastic behaviors, as shown in Figure 11.As can be seen, the proposed equations can adequatelyestimate the optimum values of b for both �xed-and(SSI) systems. In addition, the results clearly showthat the optimum b value increases with increasing theductility demand. Conversely, the b value decreaseswith increasing the a0 parameter, while, similar tothe �xed-base systems, it descends with increasing thefundamental period.

10. Conclusions

This study carried out extensive nonlinear dynamic

analyses on �xed- and exible-base Steel Moment-Resisting Frame (SMRF) systems designed based on anew constant-ductility Performance-Based Plastic De-sign (PBPD) approach at di�erent levels of inelasticityunder a group of 20 strong earthquake ground motionsto modify the suggested lateral force distribution forvarious ductility demands. The e�ect of inertial soil-structure systems was parametrically investigated onthe height-wise distribution of relative shear forcedemands. In addition, the adequacy of various lateralload patterns in ductility distribution of PBPD framebuildings was investigated. Based on the results, itwas revealed that the height-wise distribution of storyductility demands tended to be more uniform as thefundamental period of the structure increased. How-ever, by increasing the ductility ratios, the distributionof the ductility demand became more non-uniformalong the height of the structure.

Based on the proposed equation for various duc-tility demands, a more realistic design lateral forcedistribution was proposed, accounting for the inelasticbehavior of structures when subjected to strong groundmotions. The suggested lateral force distributioncan be applied to most of the conventional steelMF systems. The results showed that the suggestedmodi�cation for lateral force distribution for the typesof steel-framed structures investigated in this study wasmore rational and made a better prediction of inelasticseismic demands at various levels of inelasticity. Itwas demonstrated that the proposed equations couldadequately estimate the optimum values of shear pro-


Figure 11. Optimum b values versus structure period for �xed and exible Steel Moment-Resisting Frames (SMRFs),average of 20 earthquakes.

portioning factor b for both �xed and Soil-StructureInteraction (SSI) systems. In addition, the resultsclearly showed that optimum b value increased withincreasing the ductility demand. Conversely, the bvalue decreased with increasing the a0 parameter,while, similar to the �xed-base systems, it descendedwith increasing the fundamental period.

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Biographies

Abolfazl Gholamrezatabar is a PhD Student atIran University of Science & Technology, Tehran, Iran.


His current research interests include structural seismicperformance and seismic retro�tting.

Behnoud Ganjavi received his PhD degree in Earth-quake and Structural Engineering from the Universityof Western Australia in January 2013. Now, he worksas an Associate Professor of Earthquake Engineering atUniversity of Mazandaran. His current research inter-ests include earthquake engineering with an emphasison dynamic soil-structure interaction and optimumperformance-based seismic design.

Gholamreza Ghodrati Amiri received his PhDdegree in Earthquake Engineering from McGill Univer-

sity, Montreal, Canada in 1997. Now, he works as a FullProfessor and the Dean of School of Civil Engineering,Iran University of Science & Technology, Tehran, Iran.His current research interests include seismic perfor-mance, evaluation and retro�t of existing structures.

Mohsen Ali Shayanfar received his PhD degreein Earthquake Engineering from McGill University,Montreal, Canada, in 1997. Now, he works as anAssociate Professor of School of Civil Engineering,Iran University of Science & Technology, Tehran, Iran.His current research interests include the reliabilityof structures and the retro�tting and repairing ofstructures, especially buildings and bridges.

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