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Effects of pulse period of near-field ground motions on the seismicdemands of soil–MDOF structure systems using mathematical

pulse models

Faramarz Khoshnoudian1 and Ehsan Ahmadi2,*,†

1Associate Professor, Faculty of Civil Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran,Iran

2Faculty of Civil Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

SUMMARY

In this paper, the effects of pulse period associated with near-field ground motions on the seismic demands ofsoil–MDOF structure systems are investigated by using mathematical pulse models. Three non-dimensionalparameters are employed as the crucial parameters, which govern the responses of soil–structure systems:(1) non-dimensional frequency as the structure-to-soil stiffness ratio; (2) aspect ratio of the superstructure;and (3) structural target ductility ratio. The soil beneath the superstructure is simulated on the basis of the Conemodel concept. The superstructure is modeled as a nonlinear shear building. Interstory drift ratio is selected asthe main engineering demand parameter for soil–structure systems. It is demonstrated that the contribution ofhigher modes to the response of soil–structure system depends on the pulse-to-interacting system period ratioinstead of pulse-to-fixed-base structure period ratio. Furthermore, results of the MDOF superstructures demon-strate that increasing structural target ductility ratio results in the first-mode domination for both fixed-basestructure and soil–structure system. Additionally, increasing non-dimensional frequency and aspect ratio ofthe superstructure respectively decrease and increase the structural responses. Moreover, comparison of theequivalent soil–SDOF structure system and the soil–MDOF structure system elucidates that higher-modeeffects are more significant, when soil–structure interaction is taken into account. In general, the effects offling step and forward directivity pulses on activating higher modes of the superstructure are more sever insoil–structure systems, and in addition, the influences of forward directivity pulses are more considerable thanfling step ones. Copyright © 2013 John Wiley & Sons, Ltd.

Received 1 June 2012; Revised 19 January 2013; Accepted 24 January 2013

KEY WORDS: near-field ground motions; mathematical pulse models; soil–structure interaction

1. INTRODUCTION

Recently, many investigations have been carried out to distinguish the consequences of near-field groundmotions on the inelastic demands of structures. Many researchers tried to estimate the extreme impacts ofnear-field ground motions on the seismic behaviors of structures and quantify their effects on designcriteria [1–13]. But, these studies have been performed on fixed-base structures. In other words, theeffect of soil flexibility has been neglected in previous investigations. It is prominent that soil–structureinteraction (SSI) effect has critical impacts on the dynamic behaviors of structures [14–16].

The substitution of actual near-field ground motions with simple pulse models has been studiedextensively. Many researchers suggested synthetic and mathematical pulse models to determineprimary influences of actual near-field ground motions and introduce an organized method for

*Correspondence to: Ehsan Ahmadi, Faculty of Civil Engineering, Amirkabir University of Technology (TehranPolytechnic), Tehran, Iran.†E-mail: [email protected]

Copyright © 2013 John Wiley & Sons, Ltd.

EARTHQUAKE ENGINEERING & STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2013; 42:1565–1582Published online 28 February 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.2287

including near-field effects. Such simplified simulations assist structural engineers to enhance theirknowledge of near-field ground motions. In 1998, Krawinkler and Alavi employed mathematicalpulse models with triangular shape to simulate real near-field ground motions [17]. Agrawal and Heapproximated near-field ground motion pulses through employing decaying sinusoidal curves [18].Makris made attempts to classify pulse components of near-field ground motions into three groups,and each group was modeled by a separate function [19]. Sine functions were utilized by Sasaniand Bertero to resemble velocity spectra of actual near-field ground motions with fling and forwarddirectivity effects [20]. Menun and Fu proposed a model whose parameters were derived from anonlinear regression analysis for a particular near-field record [21]. Bozorgnia and Mahininvestigated the reliability of using synthetic rectangular pulses and demonstrated that such pulsescan adequately capture nonlinear demands of structures [22]. In addition, some other models weredescribed to represent the pulse component of near-field records [23, 24]. Baker extracted forwarddirectivity pulses from actual near-field ground motions through wavelet analysis. To categorizenear-field ground motions, the ratio of the extracted pulse intensity to that of the original recordwas employed [25]. Ninety-one near-field ground motions were discerned as pulse-like motionsamong an ensemble of 3500 studied strong ground motions. Kalkan and Kunnath investigatedthe consequences of renowned features of near-field ground motions on the seismic responses ofstructures [11]. Simple sine pulses, previously used by Sasani and Bertero, were also adopted tosimulate forward directivity and fling step effects. In the case of higher-mode effects, the resultsindicated an adequate consistency between the results of synthetic pulses and real near-field groundmotions. Sehhati et al. revealed that the structural response of multi-story structures under near-field ground motions is governed by forward directivity pulses [12]. Also, it was noted thatsimplified pulses are capable to represent the effects of such pulses on structural responses. Xu andAgrawal demonstrated that the influences of mathematical pulse models are similar to the extractedpulse components and the pulse portions of actual near-field records are the main cause of themaximum elastic and inelastic demands of structures [13].

Forward directivity and fling step are the two salient characteristics of near-field ground motions. Inforward directivity effect, near-field ground motions are considerably influenced by orientation of rupturepropagation, whereas fling step refers to static displacement of ground surface. Such effects, owing tonear-field ground motions, cause structures to sense the maximum energy arising from fault rupture asstrong pulse-like motions. In the case of forward directivity, rupture generates forward with respect to thesite, and the orientation of rupture and the site coincide. As well, this effect may emerge as pulses withlong period and large amplitude [26]. Forward directivity will occur as fault rupture has the same velocityas shear-wave velocity of the site. In normal orientation of strike–slip faults, the displacement related tothis shear-wave velocity is the most significant. Not only, displacements but also other structuralresponses are affected by forward directivity effect. A modification factor of elastic spectral accelerationprediction law to account for pulse-like effects was proposed by Baker [27]. It was confirmed that thespectra associated with pulse components are larger than those determined at periods corresponding to thevelocity-pulse period. In this case, a modification factor for elastic spectral acceleration prediction wassuggested to take into consideration pulse-like effects. Influence on displacements is probably moreimportant for inelastic spectral response as addressed by Iervolino et al. [28]. They investigated theinelastic displacement ratios of near-source pulse-like ground motions and proposed a formula forcomputing inelastic displacement ratio using nonlinear regression analysis. Their studies revealed thatforward directivity effects impose large inelastic displacement ratios to the structures with small structure-to-pulse period ratios. On the other side, fling step effect is generally specified by a large-velocity pulseand a uniform gap in the displacement time history. It was noted that fling step usually occurs in theparallel orientation of strike–slip faults and was not interacted by forward directivity significantly [29].

Pulse periods due to forward directivity effects are a function of earthquake magnitude and affectdifferent range of structural periods. This note was first stated by Somerville and mathematicallystudied by Mavroeidis and Papageorgiou [30, 23]. In addition, near-field ground motions influencethe characteristics of their response spectra [31]. Such ground motions also have notable impacts ondamping modification factors [32]. Tang and Zhang proposed a new technique for extracting bothvelocity and acceleration pulses resulting from forward directivity effects of near-field groundmotions [33]. The extracted velocity pulses can exhibit correlation with the seismological

1566 F. KHOSHNOUDIAN AND E. AHMADI

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2013; 42:1565–1582DOI: 10.1002/eqe

parameters. They were employed in regression analysis, and empirical scaling laws were derived thatconnect the directivity pulse parameters to earthquake magnitude and rupture distance. A detailedoverview of pulse period prediction models can be found in Tang and Zhang’s study [33]. Near-fieldground motion containing strong pulses might enforce structures to attenuate considerable quantities ofthe input energy by small number of excursions with large displacements. Studies clarified that high-velocity pulses are able to impose sever nonlinear demands to multi-story structures [5]. More recently,Alavi and Krawinkler have revealed that intermediate period structures are probable to undergo largenonlinear demands too [34]. On the other hand, the soil beneath the structure significantly influencesthe dynamic properties of the structure. It is mentioned that the soil–structure system is certainly moreflexible than the typical fixed-base structure. Consequently, SSI effects elongate the fundamental periodof the structure as a result of the system’s stiffness reduction. Moreover, the capability of flexible soilto dissipate the energy of the superstructure is significant because of undeniable contribution of soilradiation damping, which can drastically influence the responses of the structure [35].

As discussed earlier, both near-field ground motion and SSI have notable consequences on theresponse of structures. In this regard, it would be necessary to study the effects of pulse periodassociated with near-field ground motions on the responses of multi-story structures including SSI.In the present paper, attempts are made to investigate the effects of pulse period (forward directivityand fling step pulses) on the seismic responses of nonlinear soil–MDOF structure systems. Near-field ground motions are simulated by using sinusoidal functions, which are capable of dictatingforward directivity and fling step effects to the soil–structure system with acceptable precision. Onthe other hand, SSI effects alter the dynamic properties of the structure significantly; hence, themain objective of this paper is to examine the effects of pulse-to-interacting system period ratio onthe seismic demands of soil–structure system. Moreover, higher-mode effects are investigated byvariation of key parameters of soil–structure system such as non-dimensional frequency, aspectratio, and structural target ductility ratio. The analysis is carried out in the time domain usingacceleration time histories of the mathematical pulse models. In addition, to gain better insight intothe higher-mode effects, the equivalent SDOF structures are analyzed, and the results of equivalentsoil–SDOF structure system is compared with those obtained from soil–MDOF structure system.

2. SOIL–MDOF STRUCTURE MODEL

As illustrated in Figure 1, the studied soil–structure system consists of a multi-story building and afoundation located on the soil medium. The superstructure is considered as a nonlinear n-story shearbuilding with equivalent circular plan. In this figure, mi, Ii, ri, and Hi symbols stand respectively forthe mass, the mass moment of inertia, the radius of the equivalent circular plan, and the height ofthe mass in the ith story from the foundation surface. The geometric features of all stories areassumed identical. The story height, hi, and the effective load (dead as well as live load) arepresumed 3.3m and 10 kN/m2 as for typical buildings. The foundation is regarded as a circular rigidbody, and the flexibility of the foundation is ignored. The mass and mass moment of inertia of thefoundation are described by m0 and I0, respectively.

To determine the foundation mass, two limitations are applied to the soil–structure system. First,the foundation uplift is prevented because of the design earthquake load that is in accordance withASCE7-10 [36]. Second, the practical ratio of the foundation mass, m0, to the overall mass of thesuperstructure, M, is taken into consideration for conventional buildings. In this case, 0.2⩽m0/M⩽ 0.5is selected for the studied soil–structure systems.

To impose the soil effects into the structure, a lumped-mass parameter model is employed. The soilis considered as a homogenous elastic half-space medium and is substituted with a simplified three-DOF system based on the Cone model concept. The Cone model was recommended in [37, 38] todetermine the dynamic stiffness and damping coefficients of the soil. In comparison with more time-consuming and strictly precise numerical methods, the Cone model requires only simple numericalcalculations with reasonable accuracy for engineering and practical purposes [39]. In the Conemodel, the soil is represented by a truncated semi-infinite cone. Coefficients representing stiffnessand damping ratio of the soil are obtained on the basis of the strength-of-materials approaches.

EFFECTS OF PULSE PERIODS ON DEMANDS OF SOIL–STRUCTURE SYSTEM 1567


The horizontal (sway), s, and the rocking (rotational), ’, DOFs are presented as the rigid body motionsof the foundation, respectively. us and ’Hn indicate the horizontal displacement components caused bythe sway and rocking motions at the roof story. un reflects the deformation that is associated with thestrain in the superstructure. To consider the frequency dependency of the rotational spring and dashpotcoefficients, the internal rotational DOF, θ, is ascribed to a polar mass moment of inertia, mθ, andconnected to the foundation node using a rotational dashpot. In the case of incompressible and nearlyincompressible soil (0.33< υ< 0.50), two features are imposed into the model: (1) the axial-wavevelocity, Va, is limited to two times the shear-wave velocity, 2Vs, and (2) a trapped mass moment ofinertia, ΔM’, which moves as a rigid body in the same phase with the foundation for the rocking DOF,is attributed to the foundation node. υ stands for Poisson’s ratio of the soil, which depends on the valueof shear-wave velocity. ΔM’ is added to I0 for the soil with Poisson’s ratio greater than 0.3 [39]. Thecoefficients of springs and dashpots for the sway and rocking motions are calculated using the formulaspresented in the following:

ks ¼ 8rVs2r= 2� υð Þ ; Cs ¼ prVsr

2 (1)

k’ ¼ 8rVs2r3= 3 1� υð Þð Þ ; C’ ¼ prVar

4=4 (2)

mθ ¼ 9p2=128� �

rr5 1� υð Þ Va=Vsð Þ2 ; ΔM’ ¼ 0:3p Va υ� 0:33ð Þrr5 (3)

r stands for mass density of the soil, which relies on the shear-wave velocity and is postulated 2.35 kg/m3

for shear-wave velocity higher than 750m/s and 1.95 kg/m3 for shear-wave velocity lower than 750m/s. ris the radius of the foundation.

Nonlinear hysteretic damping is adopted using frictional elements to consider material damping of thesoil. Meek and Wolf demonstrated that nonlinear hysteretic damping independent of frequency is moresuitable and can be introduced by frictional elements, which permit causal analysis in the time domain[40]. In this research, frictional elements are employed for analyzing the soil–structure system. Materialdamping of the soil is assumed 5%. In addition, modeling each component (beams and columns) andassigning plastic hinges to these components are disregarded. Strength drop in plastic hinges (drop

Figure 1. Nonlinear n-story shear building on flexible soil medium.



from hardening point to the softening one) can result in computational difficulty and possible non-convergence errors when employed as modeling input. In this regard, for the sake of simplicity andparametric study, the overall force-relative displacement of each story is presumed. This deformation-controlled force-relative displacement behavior is represented by bilinear hysteretic model, and thecorresponding strain hardening ratio is set equal to 0.05. Therefore, the idealized force-displacementcurve of each story reflects the hardening behavior of the structure and ignores softening.

In this research, to investigate higher-mode effects, 5-story, 15-story, and 25-story buildings areassumed as low-rise, intermediate-rise, and high-rise buildings, respectively. Periods of the fixed-basemulti-story structures are computed by using formulas given in ASCE07-10. For these aforementionedbuildings, the fundamental fixed-base periods are calculated as 0.7, 1.5, and 2.3 s, respectively. Also,stiffness is distributed over the height of the superstructure corresponding to the lateral load patternpresented in ASCE07-10. The nonlinearity level in the superstructure is represented by structural targetductility assuming 2, 4, and 8. Viscous damping ratio of the superstructure is determined in accordancewith Rayleigh’s damping concept and the damping ratio of each mode is assumed 5%.

3. EQUIVALENT SOIL–SDOF STRUCTURE SYSTEM

In this research, to estimate higher-mode effects for the studied systems, the equivalent soil–SDOFstructure system is defined in accordance with the recommendations of Federal EmergencyManagement of America (FEMA) 440 [41]. Figure 2 illustrates the equivalent soil–SDOF structuresystem. The equivalent soil–SDOF structure system is described by the first-mode properties of thesoil–MDOF structure system.

The equivalent mass is designated by me and is equal to the overall mass of the MDOFsuperstructure. ke stands for the equivalent stiffness and is calculated by using following equation:

ke ¼ meoe2 (4)

oe is the circular frequency of the first mode corresponding to the soil–MDOF structure system. He

represents the equivalent height and is assumed 0.7Hn, where Hn is the total height of the superstructure.

4. MATHEMATICAL PULSE MODELS USED IN THIS STUDY

In this investigation, the well-known effects of near-field ground motions on the inelastic demands ofnonlinear soil–MDOF structure systems are studied. To achieve this goal, the input excitation for the

Figure 2. Equivalent soil–SDOF structure system.



assumed systems should be selected accurately to draw reasonable conclusions. It is preferred to employsimple sinusoidal pulses instead of the actual near-field records. Such mathematical pulse models wereused by Sasani and Bertero [20] for the first time and later on by Kalkan and Kunnath [11]. It isirrational to expect that synthetic pulses can thoroughly exhibit all characteristics of actual records,especially for complicated frequency-content ground motions (such ground motions possess multi-peakvelocity spectra). However, Sasani and Bertero, and Alavi and Krawinkler have revealed that simplepulses can be used to capture the remarkable response properties of structures under near-field groundmotions within limitations [11, 34]. Alavi and Krawinkler investigated the effects of near-fault groundmotions on frame structures by using both actual records and idealized mathematical pulses, and it wasnoted that the most important features can be adequately captured by these pulses [34]. Sine functionswere employed by Kalkan and Kunnath, and they revealed that these pulse models can reflect the effectsof higher modes on structures with acceptable precision [11].

Idealized pulses, used in this paper, are described by sinusoidal functions as represented in Figure 3.Figure 3(a) illustrates a fling step type of motion where the ground motion exhibits a static offset at the

Figure 3. Idealized sinusoidal pulses: (a) fling step pulse, (b) forward directivity pulse [20, 11].



end of the displacement time history, whereas Figure 3(b) indicates a forward directivity type ofmotion. As shown in Figure 3, the duration of the forward directivity pulse is assumed 1.5 times theduration of the fling pulse. It is noted that Kalkan and Kunnath adopted such assumption in theiranalysis too [11]. However, it has been proved that the duration of the forward directivity pulse is inthe range of 1.5 to 2.5 times the duration of the fling pulse. The value of 1.5 used by the authorscoincides with the lower limit of this range. Parametric study is a persuasive motive to use suchmathematical pulse models instead of actual ground motions. To fulfill the objective of performing adeep sensitivity analysis, pulse amplitude and pulse period are selected as the fundamental inputparameters of the idealized pulse models. To track the effects of the pulse period, the pulse recordsare scaled in a way that the acceleration spectrum of each pulse coincide with the 5% dampeddesign spectrum with minimum error in the period range of 0.5 to 3.0 s (constant velocity range) byadjusting the pulse amplitude. This scaling technique has been employed previously by Alavi andKrawinkler, and Kalkan and Kunnath, and it was confirmed that such scaling method is necessary tomake more reliable comparison for the results obtained from different ground motions [34, 11]. Thedesign spectrum is constructed for soil type E and the maximum considered earthquake.

The acceleration, velocity, and displacement elastic response spectra of the pulsemodels with 5% dampingratio are depicted in Figure 4. These spectra are normalized by their maximum time-history values. It is notedthat the velocity and displacement spectra for the forward directivity pulse are more detrimental than the flingstep pulse. Even though the pulse period of the motion corresponds nearly to the maximum value of thevelocity response spectrum herein, it is not true in all cases of actual near-fault ground motions [11]. In thecase of ground motions with complex frequency contents, velocity spectrum contains multiple peaksinstead of a distinctive peak. Baker suggested that the proposed pulse period measured by wavelet analysismay be more closely correlated with the earthquake magnitude than the pulse period corresponding tothe peak of velocity spectrum [25]. So, the pulse periods obtained from velocity spectra can becompletely different from those extracted from wavelet analysis for multiple peak velocity spectra.

In this research, pulse-to-fixed-base structure period ratio (Tp/Tfix) is varied from 0.5 to 1.5 byincrements of 0.1. This range is in complete agreement with the range in which near-field effects canbe replaced by idealized pulses; moreover, at this range, the salient properties of structures can becaptured by such pulses with reasonable approximation [34]. The range of Tp/Tfix was selected between0.375 and 3 by Alavi and Krawinkler, and it was noted that the prominent response features of near-fault ground motions can be represented by simple idealized pulses in this range [34]. Also, Ghahariet al. proposed a threshold period of approximately 0.38 times the main pulse period above whichthe pulse type motion controls the response of structures [42]. In the present study, Tfix/Tp is variedfrom 2/3 to 2 (within the aforementioned range), and high-frequency ground motions can be ignored.

5. SOIL–MDOF STRUCTURE SYSTEMS USED IN THIS STUDY

It is well known that the response of soil–structure system depends on geometric and dynamicproperties of the structure and the beneath soil. These effects can be incorporated into the studiedmodel by the following non-dimensional parameters [16, 35]:

Figure 4. Elastic response spectra with 5% damping for fling step and forward directivity pulse [11].



a0 ¼ ofixHn=Vs ; S ¼ Hn=r (4)

Non-dimensional frequency parameter, a0, is introduced as an index for the structure-to-soilstiffness ratio. ofix is the circular frequency of the fixed-base structure. This index can have valuesof up to 3 for structures located on very soft soils; while this value reduced to small quantities,almost zero stands for fixed-base structures. In this study, this parameter is assumed 0, 1, 2, and 3 to

Figure 5. Distribution of IDRs for the fixed-base structures with structural ductility ratios of 2 and 8: (a) flingstep and (b) forward directivity pulses.



cover different levels of soil flexibility. Aspect ratio of the superstructure, S, represents the slendernessof the structure. In this paper, values of 1, 2, 3, and 4 are assigned to this parameter to cover a widerange of aspect ratios. These two mentioned parameters have a wider range of variations and aretypically considered as the key parameters of the soil–structure system [16, 43].

The n + 3-DOF soil–structure model used in this research has the capability to be analyzed in thetime domain. Herein, the model has been analyzed by direct step-by-step integration adopting BetaNewmark method. To achieve this goal, MATLAB code is developed to use for analyzing the soil–structure systems [44]. Twenty two acceleration time histories are used as the representatives ofnear-field ground motions based on the mathematical pulse models to impose the effects of flingstep and forward directivity pulses to the studied systems. An in-depth parametric study isconducted by employing the three-key non-dimensional parameters of structural target ductilityratio, non-dimensional frequency, and aspect ratio for the MDOF and equivalent SDOF structures.For each structure, first the yield base shear of the structure is calculated by iteration to reach thespecified structural target ductility in the soil–structure system within 1% of accuracy under theselected acceleration time history. Consequently, demands of all stories are calculated for the fixed-base structures as well as the soil–structure systems.

6. SEISMIC RESPONSE EVALUATION OF SOIL–MDOF STRUCTURE SYSTEM

Almost 3000 nonlinear time history analyses are performed by assuming nonlinear shear buildings withvarious structural target ductility ratios, non-dimensional frequencies, and aspect ratios. Interstory driftratio (IDR), defined as the relative displacement between two consecutive story levels normalized bythe story height, is used as the primary engineering demand parameter. The IDRs are obtained fromnonlinear time history analyses of the buildings subjected to the pulses with different pulse-to-fixed-base structure period ratios (Tp/Tfix). Because of the paper size limit, results are not presented in detail,and it is limited to the either graphical or tabular presentation to clarify the general conclusions.

Figure 5 illustrates the distribution of IDRs over the height for fixed-base case (a0 = 0) in which there isno interaction between the structure and the soil. Because of the multitude number of the graphs, forconvenience, the results for pulse-to-fixed-base structure period ratios (Tp/Tfix) of 0.5, 0.7, 1, 1.3, and 1.5are illustrated only. It is noted that maximum IDRs occur at the roof story with Tp/Tfix< 1, assumingstructural ductility ratio of 2 for both fling step and forward directivity pulses. The participation of highermodes in the structural responses is demonstrated. In the case of Tp/Tfix< 1, pulse periods are so close to

Table I. Maximum IDRs over all the stories for the fixed-base structures.

Pulse type Number of stories Structuralductility ratio

% maximuminterstory drift

Tp/Tfix

Fling step 5 2 3.5 0.54 3.9 0.58 4.8 0.5

15 2 4.0 14 4.3 18 4.5 1

25 2 4.3 14 4.6 18 4.8 1

Forward directivity 5 2 4.8 0.54 4.92 18 5.0 1

15 2 3.2 14 4.6 18 6.0 1

25 2 3.6 14 5.5 18 7.0 1



higher-mode periods of the structure and activate the higher modes of the structure. Another worthmentioning point is that activation of higher modes also happens for the 5-story building, which isshorter than the 15-story and 25-story buildings. Kalkan and Kunnath concluded that only forwarddirectivity pulses can trigger higher modes and fling step pulses tend to excite the first mode of thestructure [11]. However, Figure 5 illustrates that fling step pulses are capable of activating higher modestoo. The increase of Tp/Tfix results in the reduction of IDRs associated with the roof story. In this case,lower modes of the structure participate more significantly in the seismic responses of the structure incomparison with higher modes. Furthermore, at Tp/Tfix ratios close to 1, the maximum IDRs occur in thefirst story, which is illustrative of the fact that the response of the structure is in conformity with its firstmode. At Tp/Tfix ratios higher than 1, pulse period departs from all periods of the structural modes. Insuch condition, because of the proximity of the first-mode period of the structure to pulse period incomparison with the other modes, the first mode is dominant and the distribution of IDRs turns intouniform shape in upper stories because of the non-participation of higher modes in the seismic responseof the structure. Another important point is the effect of structural ductility ratio on the activation ofhigher modes. In low structural ductility ratios, periods obtained from the linear modal analysis (Tfix) areclose to the nonlinear periods of the structure,* and the contribution of higher modes to the structuralresponse becomes meaningful; but, as the structural ductility ratio increases, nonlinearity effects amplify,and the nonlinear periods of the structure diverge more from the linear modal periods. Therefore, pulseperiods are not close to the nonlinear periods of the structure to trigger their corresponding modes, andthe seismic response of the structure is in accordance with its first mode because of the significant modeparticipation factor. Consequently, the maximum IDR occurs at the first story. Table I displays themaximum value of the IDR over all the stories for the fixed-base structures. As it is evident from thistable, most of the maximum IDRs occur at Tp/Tfix ratio equal to 1. In general, forward directivity effectsare greater than the fling step ones in the critical story (the story which corresponds to the maximumIDR). This can be well justified by the fact that forward directivity has double momentum pulse incomparison with fling step pulse with one momentum pulse.

To discuss in detail about the effects of Tp/Tfix ratio on IDRs of the structures, the coefficients ofvariation (COVs) associated with IDRs are depicted in Figure 6. It is noted that increase in number of

*It is noted that nonlinear period of the structure refers to the period of the structure in nonlinear phase and can beobtained from performing modal analysis of the structure whose stiffness has been modified because of the nonlinearityeffects. This period changes as the system yields and the effective stiffness of the structure reduces and the structureexperiences large relative lateral displacement.

Figure 6. COVs of IDRs for the fixed-base structures.



stories results in more response discrepancy in lower stories compared with upper stories. The reason liesin the structural modes’ increment as a result of increasing story numbers. Therefore, the effects of Tp/Tfix ratio become more significant, and the seismic response of the structure depends more on thisratio. Also, the sensitivity of the structure to this ratio would be slight with enhancing structuralductility ratio, especially for the 15-story and 25-story buildings. In general, COVs of forwarddirectivity pulse is more meaningful than fling step pulse; on the other hand, structural responsesare more sensitive to forward directivity pulse than fling step one.

It is noted that the two key parameters related to the SSI effects, that is, non-dimensional frequencyand aspect ratio, are crucial in soil–structure systems. Figures 7–9 illustrate IDRs corresponding to the5-story, 15-story, and 25-story buildings with non-dimensional frequency of 3, aspect ratio of 2 and 4,and structural ductility ratio of 2 and 8. Non-dimensional frequency reflects interaction severitybetween the soil and the structure and raising this parameter leads to amplification of radiationdamping associated with soil–structure system and decrease its stiffness. Because of the growth ofradiation damping, the maximum IDRs of soil–structure system reduce. This parameter also promptsthe period of soil–structure system (Tssi) to elongate due to the reduction in the system’s stiffness. Asan example, because of the SSI effects, the period changes from 0.7, 1.5, and 2.3 s to 1.192, 2.429, and3.683 s, respectively for the 5-story, 15-story, and 25-story soil–structure systems with non-dimensionalfrequency of 3 and aspect ratio of 4. The maximum values of the IDR over all the stories are tabulatedin Table II for the soil–structure systems assuming non-dimensional frequency of 3.

Figure 7. Distribution of IDRs for the 5-story soil–structure systems with structural ductility ratios of 2 and8 and aspect ratio of 2 and 4.



By comparing Tables I and II, corresponding to the fixed-base structures and the soil–structuresystems respectively, it is noted that the maximum values of IDRs of the soil–structure systemsgenerally occur at Tp/Tfix equal to 1.5, whereas for the fixed-base structures, these values happen atTp/Tfix equal to 1. Therefore, the pulse periods, in which the soil–structure systems possess themaximum values of IDRs, are 1.5 times the periods of the fixed-base structures and are equivalentto 1.05, 2.25, and 3.45 s for the 5-story, 15-story, and 25-story buildings, respectively. So, theobtained Tp/Tssi ratios correspond to 0.88, 0.93, and 0.94, which are close to 1. It indicates that theseismic response of soil–structure system depends on Tp/Tssi instead of Tp/Tfix, and demandsassociated with soil–structure systems are primarily controlled by Tp/Tssi ratio. Tp/Tssi ratios aredifferent from 1, because the responses are computed at most for Tp/Tfix ratio of 1.5 anddetermination of the values higher than 1.5 would lead to the values of Tp/Tssi closer to 1.

In general, aspect ratio plays an important role in soil–structure systems. An increase in thisparameter will decrease the stiffness of the system like non-dimensional frequency, and therefore,the system’s period elongates. Slenderizing the superstructure significantly mitigates radiationdamping for soil–structure system, which well explains the escalation in IDRs of the system.

As shown in Figures 7–9, the seismic responses for the squatty structures are different from thoseobtained for the slender ones. For the squatty structures (S = 1, 2), the effects of Tp/Tfix ratios are

Figure 8. Distribution of IDRs for the15-story soil–structure systems with structural ductility ratios of 2 and8 and aspect ratio of 2 and 4.



inconsiderable, and the graphs pertaining to different ratios of Tp/Tfix are tight together, whereas for theslender structures (S=4), these effects are vice-versa. It is concluded that the radiation damping of soil–structure system usually increases as a result of an increase in soil flexibility, and the increment rate ismuch greater as the structure significantly becomes squatty [43]. Therefore, for squatty structures, theswift growth of the system’s damping ratio causes the input energy of pulse to dissipate with higherrate, and the variations of IDRs become inconsequential. In addition, the impacts of structural ductilityratio on the activation of higher modes for the soil–structure systems and the fixed-base structuresare the same. It is noted that slender structures (S=4) with fundamental period shorter than 1 s arenot common in practice.

Coefficients of variation of IDRs for the soil–structure systems along the height of the structure aredepicted in Figure 10. One important point is that COVs of IDRs for the squatty structures (S = 1, 2) aresmaller than those for the slender structures. As discussed earlier, it emanates from the growth rate ofdamping ratio, which is significant in squatty structures. Therefore, the discrepancy of IDRs becomesinsignificant for such cases. So, slenderizing the structure enlarges the discrepancy of the results, andthis conclusion is in accordance with the previous results. This observation is more meaningful forforward directivity pulses than fling step ones.

By comparing the COVs of IDRs of the fixed-base structures (Figure 6) with those obtained fromthe soil–structure systems (Figure 10), this point will be revealed that the COVs of the soil–structuresystems, especially the slender structures, have higher values than those of the fixed-base structures.

Figure 9. Distribution of IDRs for the 25-story soil–structure systems with structural ductility ratios of 2 and8 and aspect ratio of 2 and 4.



It means that the seismic demands of soil–structure system are more susceptible to pulse periods incomparison with fixed-base structure. The reason lies in the reduction of soil–structure system’sstiffness due to the SSI effects. Despite the reduction in IDRs of soil–structure system due to the highvalues of radiation damping ratio, the reduction of the system’s stiffness due to the soil flexibilitycauses the soil–structure system to be more responsive to pulse period. This sensitivity becomes morenotable in forward directivity pulses.

7. SEISMICRESPONSE EVALUATIONOF EQUIVALENT SOIL–SDOF STRUCTURE SYSTEMS

To gain a better insight into the capability of pulse-like motions in activating higher modes in soil–structure systems and comparing them with the same effect in fixed-base structures, the equivalentsoil–SDOF structure systems are idealized based on FEMA 440. To study higher-mode effects inboth the soil–structure system and fixed-base structure, Cm parameter is defined as the maximumIDR of the soil–MDOF structure system to the corresponding value of the equivalent soil–SDOFstructure system. Figures 11–13 illustrate the values of this parameter for the 5-story, 15-story, and25-story buildings, respectively. The variations trends of the three figures are in completecompatibility with the earlier discussions in Section 6. Generally, the increase of Tp/Tfix ratio

Table II. Maximum IDRs over all the stories for the soil–structure systems with non-dimensional frequency of 3.

Pulse type Aspect ratio Numberof stories

Structuralductility ratio

% peak valueof interstory drift

Tp/Tfix

Fling step S= 2 5 2 2.7 0.54 3.3 0.58 4.5 0.5

15 2 2.0 1.54 3.7 1.58 3.8 1.5

25 2 2.5 1.54 3.9 1.58 4.3 1.5

Forward directivity 5 2 2.6 0.54 3.0 1.58 4.2 1.5

15 2 2.0 1.54 3.5 1.58 5.0 1.5

25 2 2.5 1.54 4.0 1.58 5.7 1.5

Fling step S= 4 5 2 3.5 0.54 3.4 0.58 3.9 0.5

15 2 2.0 1.54 3.3 1.58 3.4 1.5

25 2 2.5 1.54 3.6 1.58 4.1 1.5

Forward directivity 5 2 2.8 0.54 3.0 1.58 4.1 1.5

15 2 2.0 1.54 3.4 1.58 4.7 1.5

25 2 2.5 1.54 3.8 1.58 5.4 1.5



Figure 11. Variations of Cm parameter for the 5-story building.

Figure 10. COVs of IDRs for the soil–structure systems.




reduces higher-mode effects, and consequently, Cm value decreases. According to the obtained results(Figure 11) for structural ductility ratio of 2, the higher modes are activated considerably comparedwith those for structural ductility ratio of 8. SSI effects cause the higher modes to triggerremarkably, and Cm value reaches about 6 in some cases. It is noted that this parameter, Cm, is moresignificant where Tp/Tfix ratios are smaller than 0.75. In addition, the ability of forward directivitypulse to activate higher modes is greater than the fling step, and the SSI effects amplify thisphenomenon. Moreover, this ability is more remarkable for the slender structures. It can be wellexplained by the rapid growth rate of the radiation damping ratio in squatty structures as well asamplification of mode participation factors regarding to slender structures. For example, in the caseof the 15-story building and non-dimensional frequency of 3, first-mode participation factor is about1.2 and 1.4 for respective aspect ratio of 2 and 4. It is well known that the structural ductilitydemand with considering SSI effects is generally smaller than the structural target ductility of thecorresponding fixed-base structure [45]. It implies that, particularly for small structural targetductility ratios, the required value of ductility in the superstructure including SSI effects may beequal to one, and the behavior of the structure would be close to linear. Reduction of structuralductility ratio emphasizes on considerable higher modes participation.

With the previous discussions, at high values of structural ductility ratio, the seismic demands arecompatible with the first mode, and only this mode is activated by pulse. In soil–structure systems,the sensitivity to pulse is more significant because of soil flexibility and reduction in the stiffness ofthe system. Figures 12 and 13 present the variations of Cm parameter for the 15-story and 25-storybuildings, respectively. The trend of results is analogous to the 5-story building but more severe sothat the value of Cm parameter reaches 8 in some cases. The DOFs increase as the number of storiesincrease, and consequently, contribution of higher modes becomes more considerable.

8. CONCLUSIONS

In this paper, the seismic demands of nonlinear soil–structure systems are investigated using mathematicalpulse models that dictate the well-known effects of near-field ground motions including fling step andforward directivity. The soil beneath the superstructure is simulated on the basis of the Cone modelconcept, and the MDOF superstructure is modeled as a multi-story nonlinear shear building. Structuraltarget ductility ratio, non-dimensional frequency, and aspect ratio are adopted as key parameters ofthe soil–structure system. The effects of pulse period are considered in the form of IDRs of soil–structure systems.

The results demonstrate that an increase in the structural ductility ratio results in migrating maximumIDRs from higher stories to lower ones and pulse period is able to activate lower modes. It is noted thatIDRs are more sensitive to pulse period for soil–structure systems because of the reduction of ductilityratio of soil–structure systems compared with fixed-base structures.

The non-dimensional frequency representing interaction severity leads to IDRs reduction due tothe soil radiation damping increase. Variations of Cm parameter obviously confirm that the role of




higher-mode effects in the responses is more considerable for soil–structure systems in comparisonwith fixed-base structures. In soil–structure system, the further aspect ratio prompts the IDRs toincrease as a result of the decrease in radiation damping ratio. For squatty structures, the sensitivityof IDRs to pulse periods is insignificant because of the rapid increase of damping ratio incomparison with slender structures. Variations of Cm parameter evidently demonstrate thatincreasing aspect ratio causes higher modes to contribute more to the responses of soil–structuresystems.

The variations of maximum IDRs from Tp/Tfix equal to 1 at fixed-base structure (a0 = 0) to Tp/Tfix equalto 1.5 at soil–structure system (a0 = 3) show that the seismic responses associated with soil–structuresystems depend on Tp/Tssi instead of Tp/Tfix. The effects of fling step and forward directivity pulses onactivating higher modes are commonly more significant for soil–structure systems, and in addition, theeffects of forward directivity is more considerable than fling step ones.

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