Contents lists available at ScienceDirect

Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

An improved replacement oscillator approach for soil-structure interactionanalysis considering soft soils

Yang Lua,⁎, Iman Hajirasoulihab, Alec M. Marshallc

a College of Architecture and Environment, Sichuan University, PR ChinabDepartment of Civil & Structural Engineering, The University of Sheffield, UKc Department of Civil Engineering, University of Nottingham, UK

A R T I C L E I N F O

Keywords:Soil-structure interactionSoft soilSeismic designReplacement oscillatorNonlinear analysis

A B S T R A C T

This paper aims to improve the effectiveness of the replacement oscillator approach for soil-structure interaction(SSI) analysis of flexible-base structures on soft soil deposits. The replacement oscillator approach transforms aflexible-base single-degree-of-freedom (SDOF) structure into an equivalent fixed-base SDOF (EFSDOF) oscillatorso that response spectra for fixed-base structures can be used directly for SSI systems. A sway-rocking SSI modelis used as a baseline for assessment of the performance of EFSDOF oscillators. Both elastic and constant-ductilityresponse spectra are studied under 20 horizontal ground motion records on soft soil profiles. The effects offrequency content of the ground motions and initial damping of the SSI systems are investigated. It is concludedthat absolute acceleration spectra, instead of pseudo-acceleration spectra, should be used for EFSDOF oscillatorsin force-based design of SSI systems. It is also shown that using an EFSDOF oscillator is not appropriate forpredicting the constant-ductility spectra when the initial damping ratio of the SSI system exceeds 10%. Based onthe results of this study, a correction factor is suggested to improve the accuracy of the replacement oscillatorapproach for soft soil conditions.

1. Introduction

The preliminary design of typical building structures in currentseismic design codes and provisions is mainly based on elastic spectrumanalysis, where the inelastic strength and displacement demands areestimated by using modification factors, such as the constant-ductilitystrength reduction factor Rμ (i.e. reduction in strength demand due tononlinear hysteretic behaviour) and inelastic displacement ratio Cμ[1–3]. The spectral shapes of elastic response spectra and modificationfactors in most seismic design codes and provisions (e.g. [3,4]) arederived by averaging the results of response-history analyses performedon single-degree-of-freedom (SDOF) oscillators using a number ofearthquake ground motions [5–7]. In engineering practice, the fre-quency content of a ground acceleration motion at a soft soil site isoften characterized by a predominant period [8] as an influentialparameter for estimating the seismic response of buildings.

It is well known that spectral accelerations for soft soil sites attaintheir maximum values at specific periods TP, which correspond to theresonance between the vibration of buildings and the amplification ofseismic waves travelling upwards through various soil deposits [9].However, most current seismic codes adopt design acceleration spectra

that are smoothed by the averaging of a number of spectra whose peakordinates may occur at significantly different values of TP. As a con-sequence, averaging these dissimilar spectra leads to a flatter spectrumfor soft soil profiles than for rock and stiff soil sites, while disregardingthe frequency content of the ground motions [7].

Xu and Xie [10] developed the concept of a Bi-Normalized ResponseSpectrum (BNRS) by normalizing the spectral acceleration Sa and theperiod of the structure T by the Peak Ground Acceleration (PGA) andthe spectral predominant period TP of each ground excitation, respec-tively. Based on analyses performed using 206 free-field records of theChi-Chi earthquake (1999), they found that the BNRS curves werepractically independent of site class or epicentre distance, and thusrepresented a good substitute for the code-specified design spectra thatare based on simple averaging of spectral values. In a follow-up study,Ziotopoulou and Gazetas [7] demonstrated that BNRS can preserve theresonance between soil deposits and excitations, thereby reflectingmore realistically the effects of the frequency content of the groundmotion.

Comprehensive studies have been carried out in the past threedecades to calculate values of constant-ductility strength reductionfactor Rμ and inelastic displacement ratio Cμ for fixed-base structures

https://doi.org/10.1016/j.engstruct.2018.04.005Received 18 September 2017; Received in revised form 1 March 2018; Accepted 3 April 2018

⁎ Corresponding author.E-mail address: yang.lu@scu.edu.cn (Y. Lu).

Engineering Structures 167 (2018) 26–38

0141-0296/ © 2018 Elsevier Ltd. All rights reserved.

T

[11,12]. It has been shown that Rμ and Cμ usually reach their maximumand minimum values, respectively, at the predominant period of theground motion Tg, which is defined as the maximum ordinate in therelative velocity spectrum calculated for an elastic SDOF system havinga 5% damping ratio. It has also been observed that, in the vicinity of Tg,maximum inelastic displacements are sometimes smaller than theelastic displacement demands. It should be noted that the predominantperiod is mainly a characteristic of soft soils.

The studies discussed above all assumed that the structures wererigidly supported, adopted a viscous damping ratio between 2 and 5%,and disregarded the effects of soil stiffness and damping within the soildomain, also known as soil-structure interaction (SSI) effects. However,it is well known that SSI can significantly affect the seismic response ofsuperstructures, especially those on soft soil profiles [13,14]. Khosh-noudian et al. [15] and Khoshnoudian and Ahmadi [16,17] investigatedthe effects of SSI on the seismic performance of nonlinear SDOF andmulti-degree of freedom (MDOF) systems and proposed empiricalequations to predict the inelastic displacement ratios. However, theresults of their studies were mainly based on pulse-like near-fieldearthquakes, and therefore, may not be directly applicable for othertypes of earthquake ground motions.

For design purposes, an SSI system is usually replaced by anequivalent fixed-base SDOF (EFSDOF) oscillator (also called replace-ment oscillator) having an elongated period of Tssi, an effective initialdamping ratio of ξssi and an effective ductility ratio of μssi. Inelastic andlinear EFSDOF oscillators were adopted by Mekki et al. [18] and Mo-ghaddasi et al. [19], respectively, by using inelastic spectra andequivalent linearization to facilitate a design procedure for nonlinearflexible-base structures. Similarly, Seylabi et al. [20] developed a linearEFSDOF oscillator based on equivalent linearization. Since previousstudies have shown that using inelastic response spectra can providemore accurate design solutions for nonlinear systems compared toequivalent linearization (e.g. [21,22]), the current study is focused oninelastic EFSDOF oscillators.

The effectiveness of the EFSDOF oscillator approach for seismicdesign of structures located on soft soil sites is evaluated in this paper. Asway and rocking SSI model, which provides sufficient accuracy formodelling the dynamic soil-structure interaction in engineering prac-tice (e.g. [13,14]), is used as a reference to assess the accuracy of theresults obtained using the EFSDOF oscillators. The effects of both SSIand frequency content of seismic excitations on elastic and inelasticresponse spectra are investigated using the adopted SSI models and theEFSDOF oscillators for 20 far-field earthquake ground motions recordedon soft soil sites. The results are then used to improve the EFSDOFoscillator for predicting constant-ductility spectra of flexible-basestructures on soft soil profiles. The current study, for the first time,proposes improvements to the replacement oscillator approach andexplicitly includes the effect of frequency content of ground motions onsoft soils in SSI analysis. The paper provides a description of theadopted SSI model and key design parameters, as well as the EFSDOFoscillator. Limitations of the EFSDOF oscillator approach for highlydamped SSI systems are identified and some modifications are sug-gested to improve predictions. The strengths and potential applicationsof the improved EFSDOF approach to SSI procedures in performance-based design are also addressed.

2. Soil-structure interaction model

For the SSI model adopted in this study, the superstructure isidealized as an equivalent SDOF oscillator having a mass ms, massmoment of inertia Js, effective height hs, and lateral stiffness ks. In re-sponse to seismic loading, the oscillator is assumed to exhibit elastic-perfectly plastic behaviour as an energy dissipation mechanism, inaddition to having a viscous damping ratio of ξs in its elastic state. Thisnonlinear hysteretic model can simulate the seismic behaviour of non-deteriorating structural systems such as buckling-restrained braced

frames and moment resisting steel frames. The superstructure re-presents either a single-storey or a multi-storey building correspondingto its fundamental mode of vibration.

The dynamic behaviour of the shallow foundation is simulated usinga discrete-element model, which is based on the idealization of ahomogeneous soil under a rigid circular base mat as a semi-infinitetruncated cone [23]. The accuracy of this model has been validatedagainst more rigorous solutions [24,25]. Fig. 1 shows the SSI modelused in this study, which consists of a superstructure and a foundationwith sway and rocking components defined by Wolf [25] as follows:

=−

=kρv r

νc ρv πr

82

,hs

h s

22

(1)

=−

=kρv r

νc

ρv πr83(1 )

,4θ

sθ

p2 3 4

(2)

⎜ ⎟= ⎛⎝

− ⎞⎠

= − ⎛⎝

⎞⎠

M ν πρr M ν π ρrvv

0.3 13

, 9128

(1 )θ φp

s

5 2 52

(3)

where kh, kθ and ch, cθ correspond to the zero-frequency foundationstiffness and high-frequency dashpot coefficient for the sway androcking motions, respectively. The circular foundation beneath the su-perstructure is assumed to be rigid, with a radius r, mass mf and cen-troidal mass moment of inertia Jf. For simplicity, the superstructure isassumed to be axisymmetric with its mass uniformly distributed over acircular area of radius r. Therefore, the moment of inertia J of either thesuperstructure or the foundation is equal to mr2/4, m being the corre-sponding mass of the foundation mf or the superstructure ms. Thehomogenous soil half-space is characterized by its mass density ρ,Poisson’s ratio ν, as well as the shear and dilatational wave velocities vsand vp. An additional rocking degree of freedom φ, with its own massmoment of inertia Mφ is introduced so that the convolution integralembedded in the foundation moment-rotation relation can be satisfiedin the time domain. The matrix form of the equations of motion of theSSI model shown in Fig. 1, subjected to a ground acceleration time-history, is given in Appendix A. The authors implemented the nonlineardynamic analyses in MATLAB [26]; results were obtained in the timedomain using Newmark’s time-stepping method. In order to solve thenonlinear equations, the modified Newton-Raphson’s iterative schemewas utilized. The performance of the linear SSI model was verifiedagainst results obtained using the foundation impedance functions [27];for inelastic structures the model was verified using the central differ-ence numerical integration method [28].

Note that soil incompressibility leads to a high value of vp (i.e. vp→

Fig. 1. Soil-structure interaction model.

Y. Lu et al. Engineering Structures 167 (2018) 26–38

27

∞ as ν→ 0.5), which consequently results in an unrealistic over-estimation of the rocking damping at high frequencies (see Eq. (2)). Toaddress this issue, an added mass moment of inertia Mθ was assigned tothe foundation rocking degree of freedom, while vp was replaced with2vs for 1/3 < ν≤ 0.5. The soil material damping ξg was evaluated atthe equivalent frequency of the SSI system ωssi and modelled by aug-menting each of the springs and dashpots with an additional dashpotand mass, respectively.

3. Modelling parameters

In this study, the following dimensionless parameters were used tocharacterize the important features of SSI systems:

(1) Structure-to-soil stiffness ratio a0:

=a ω hvs s

s0 (4)

where ωs = 2π/Ts is the circular frequency of the superstructure in itsfixed-base condition, with Ts being the corresponding natural period.

(2) Slenderness ratio of a building s:

=s hr

s(5)

(3) Structure-to-soil mass ratio m:

=m mρh r

s

s2 (6)

(4) Ductility demand μ:

=μ uu

m

y (7)

where um is the maximum earthquake-induced displacement and uy theyield displacement (see Fig. 2). For an SSI system, either a globalductility μssi or a structural ductility μs can be defined. The formercorresponds to the SSI system with maximum and yield displacementsspecified at the top mass of the superstructure relative to the ground;the latter corresponds to the structural distortion that excludes foun-dation rigid-body sway and rocking motions.

It has been shown that a0 and s are key parameters that control theseverity of SSI effects [13]. In engineering practice, a0 generally variesfrom 0, for buildings that are rigidly supported, to 3 for buildings builton very soft soil profiles [28]. In the current study, all superstructureswere assumed to have a slenderness ratio s less than or equal to 4, whilethe structure-to-soil mass ratio m was set to 0.5 and the foundation

mass was assumed to be ten percent of the structural mass (i.e. mf/ms= 0.1). The Poisson’s ratio ν was taken as 0.5 (for very soft soil inundrained conditions) and both elastic structural damping and soilhysteretic damping ratios were set to 5% (i.e. ξs = ξg=0.05) unlessstated otherwise. The stated parameter values are representative ofthose for common building structures (e.g. [13,28]).

Considering an SDOF oscillator with a simple elasto-plastic force-deformation relation depicted in Fig. 2, the constant-ductility strengthreduction factor Rμ and inelastic displacement ratio Cμ are defined as:

= =R VV

C uu

,μe

yμ

m

e (8)

where Ve and ue are maximum base shear and displacement of an os-cillator under seismic loading in its elastic condition, and um is themaximum displacement of a yielding oscillator with a reduced baseshear strength Vy under the same loading condition. Given the defini-tion of ductility demand given by Eq. (7), the inelastic displacementratio can be calculated by:

=Cμ

Rμμ (9)

Eq. (8) illustrates that Rμ and Cμ link the strength and displacementdemands of an inelastic system to its elastic counterpart, allowing theseismic demands of an inelastic SDOF oscillator to be determined di-rectly from an elastic design acceleration Sa or displacement Sd spec-trum. This will be explained in more detail in the following sections.

4. EFSDOF oscillator

The elastic response of a dynamic system under a specific groundmotion is mainly dependent on its natural period and damping ratio.Replacement of an SSI system by an equivalent fixed-base SDOF(EFSDOF) oscillator with an equivalent period Tssi and an effectivedamping ratio ξssi has been adopted by many researchers as a con-venient way to simplify SSI analyses (e.g. [29]). This approach is basedon the selection of Tssi and ξssi for the EFSDOF oscillator so that itsresonant pseudo-acceleration and the corresponding frequency areequal to those of the actual SSI system. Since EFSDOF oscillators ingeneral can provide accurate estimations of the deformation demandsof SSI systems, several studies have been devoted to the derivation ofTssi and ξssi for flexibly-supported structures [30,31]. Notable examplesinclude the Veletsos and Nair [31] simplified approximations, whichhave been coded in some of the current design provisions (e.g. [4]).More recently, Maravas et al. [32] developed exact solutions of ωssi andξssi given by:

=+

ω χξ1 4ssi

ssi2

(10)

= ⎡

⎣⎢ +

++

++

⎤

⎦⎥ξ χ

ξω ξ

ξω ξ

ξω ξ(1 4 ) (1 4 ) (1 4 )ssi

h

h h

θ

θ θ

s

s s2 2 2 2 2 2

(11)

where χ is defined by:

= ⎡

⎣⎢ +

++

++

⎤

⎦⎥

−

χω ξ ω ξ ω ξ

1(1 4 )

1(1 4 )

1(1 4 )h h θ θ s s

2 2 2 2 2 2

1

(12)

and the frequencies ωh, ωθ and damping ratios ξh, ξθ (including bothradiation damping and soil material damping) are calculated accordingto:

=−

=−

= =ωα k m ω

mω

α k J ωm h

ξ ωrv

βα

ξ ωrv

βα

, ,2

,2h

h h f

sθ

θ θ f

s sh

s

h

hθ

s

θ

θ

2 2

2 (13)

in which ω is the circular frequency of vibration. Closed-form expres-sions for αh, αθ, βh, βθ, defined as frequency-dependent coefficientsassociated with dynamic spring stiffness and dashpot damping, were

ve

vy

uy umue

v

u

Fig. 2. Elastic-perfectly plastic lateral force-displacement relation.

Y. Lu et al. Engineering Structures 167 (2018) 26–38

28

proposed by Veletsos and Verbic [27]. These frequency-dependentsprings and dashpots that characterize the dynamic foundation force-displacement relationships are termed “foundation impedance func-tions”. Fig. 3 presents a comparison between cone and impedancemodels as well as the Maravas et al. [32] solutions by expressing theperiod lengthening ratio Tssi/Ts and the effective damping ratio ξssi asfunctions of the structure-to-soil stiffness ratio a0 and the slendernessratio s. The procedures for determining Tssi/Ts and ξssi using the coneand impedance models are explained in Appendix A.

Due to the frequency dependence of ωh, ωθ, ξh and ξθ, the responseparameters shown in Fig. 3 were obtained iteratively until the fre-quency of vibration ω equalled ωssi, within an acceptable tolerance of0.1%. It should be mentioned that the Maravas et al. [32] method in-herently assumes that the structural damping is frequency independent.Therefore, if viscous damping is used, as was done in this study, thedamping ratios ξs in Eqs. (10)–(12) should be multiplied by ωs/ω.

In general, Fig. 3 shows good agreement between the three sets ofresults, which validates the use of the cone model as the baseline forevaluating the EFSDOF oscillator results. It is observed that slenderbuildings always have a greater period lengthening and a lower effec-tive damping when compared with short squatty structures. Softer soilprofiles (i.e. higher a0 values) also lead to greater period lengtheningand higher effective damping ratios for less slender structures (i.e.s= 1, 2). For SSI systems with slender superstructures (i.e. s= 3, 4), anincrease in the structure-to-soil stiffness ratio a0 can significantly in-crease the period lengthening, while it has a negligible influence on theeffective damping ratio. Note that these observations are based on theparameters considered in this study, which represent common buildingstructures located on soft soil profiles.

Fig. 4 schematically illustrates how the EFSDOF oscillator can beused to design flexible-base structures. For elastic systems, an SSIsystem shown in Fig. 4(a) can be replaced by a fixed-base oscillatorwith Tssi and ξssi shown in Fig. 4(b). As a result, the base shear anddisplacement demands of the flexible-base system can be obtained froma response spectrum derived for fixed-base structures with an effectivedamping ratio ξssi and an elongated period Tssi (or a reduced initialstiffness kssi).

If the superstructure exhibits nonlinear deformation, the maximumseismic lateral force imposed on the SSI system will be equal to the baseshear strength Vy of the superstructure. To measure the level of inelasticdeformation, either the global ductility μssi = ussi,m/ussi,y or the struc-tural ductility μs = us,m/us,y can be used. Based on the assumption thatthe energy dissipated by yielding of the SSI system (Fig. 4(c)) is equal tothat of the EFSDOF oscillator (Fig. 4(d)), the following relation betweenthe global and structural ductility ratios, with reference to Fig. 4(e), can

be obtained [33]:

⎜ ⎟= ⎛⎝

⎞⎠

− +μ TT

μ( 1) 1ssis

ssis

2

(14)

It should be mentioned that the energy dissipation due to elasticdamping was not accounted for in the derivation of Eq. (14). TheEFSDOF oscillator used in this study enables both global and structuralductility demands to be determined simultaneously. Therefore, dis-placement demands relating to either an SSI system (including therigid-body motions of the foundation) or the structural deformation canbe estimated using Eq. (14).

5. Response parameters

In this study, the linear and nonlinear dynamic response of around10,000 fixed-base and flexible-base SDOF structures (around 200,000response-history analyses) with a wide range of fundamental periods,target ductility demands and damping ratios were obtained under atotal of 20 ground motions, as listed in Table 1. The selected groundmotions were all recorded on soft soil profiles with relatively highsurface wave magnitudes (Ms > 6.1). These records were carefullychosen by FEMA [4] and are provided in Appendix C of that report.While the maximum response of an SSI system is strongly dependent onthe characteristics of the ground motion, it will be shown in this paperthat using appropriate normalizing parameters can significantly reducethe sensitivity of the results to the design ground motions. The suite ofrecords in Table 1 has been used to study the inelastic displacementratio for SSI systems where foundations were either bonded to the soil[14] or allowed to separate (uplift) [34]. However, the effect of thespectral predominant period was disregarded in those studies.

The current study investigates the accuracy of the EFSDOF oscillatorby comparing results with those of the corresponding SSI model illu-strated in Fig. 1. Note that for squatty buildings (e.g. s= 1), the ef-fective damping ratio ξssi can increase up to 25% (see Fig. 3), whereas itis usually around 5% for typical fixed-base structures. It is required byseismic provisions [1] that the effective damping ratio of a linear SSIsystem is higher than 5% but does not exceed 20%. Therefore, in thecurrent study, the damping ratios ξssi of the selected SSI systems, whichwere achieved using various combinations of a0 and s, were restrictedto the range of 5–20%. In the following sections, the response obtainedusing the SSI models and their EFSDOF oscillators are illustrated usingelastic acceleration spectra, constant-ductility strength reduction factor,and inelastic displacement ratio spectra.

a0

0 0.5 1 1.5 2 2.5 31

1.2

1.4

1.6

1.8

2

2.2T s

si/T

s s=4

s=1

=0.5 g=0.05

ImpedanceMaravasCone

(a)

a0

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

s=1

s=4

=0.5 g=0.05(b)

ssi

Fig. 3. Comparison of period lengthening ratio and effective damping of SSI systems (ν=0.5 and ξg= 0.05).

Y. Lu et al. Engineering Structures 167 (2018) 26–38

29

6. Elastic acceleration response spectrum

The average acceleration response spectra of the 20 selected groundmotions (Table 1) were calculated for the EFSDOF oscillators and theircorresponding SSI models considering different effective damping ra-tios, as shown in Fig. 5. To account for the frequency content of theground motions, the results are also presented using Bi-NormalizedResponse Spectrum (BNRS) curves where the predominant period TP

was measured for each acceleration record at its maximum spectralordinate value. It was found that the period TP was almost unaffectedby the initial damping level in the range of interest (i.e. ξ=5–20%); avalue of TP corresponding to 5% damping was therefore used for nor-malizing spectra with higher damping ratios.

In Fig. 5(a) and (b), the solid lines represent the SSI models, whereasthe dashed lines are the results obtained using the EFSDOF oscillators,both of which were obtained by averaging the peak absolute accel-eration of the structure (including ground accelerations) under the 20ground acceleration records. The dotted lines correspond to the average

pseudo-acceleration spectra of the EFSDOF oscillators. Fig. 5 shows thatfor SSI systems with low initial damping ratios of ξssi≤ 10%, usingeither absolute or pseudo-acceleration spectra of the EFSDOF oscillatorscan provide an accurate prediction of the peak absolute accelerations ofthe structural mass in the SSI models. However, the spectral accelera-tions of SSI models having higher initial effective damping ratios (i.e.ξssi = 15% and 20%) are generally higher than those of the EFSDOFoscillators, especially when spectral pseudo-accelerations are com-pared. The difference between absolute and pseudo-accelerationspectra is negligible in typical fixed-base building structures due totheir low structural damping ξs [35]. Therefore, the pseudo-accelera-tion spectra adopted by seismic codes can provide accurate seismicdesign of fixed-base buildings. In addition, damping in soil serves todissipate external energy to a structure, which is usually designed onthe basis of a pseudo-acceleration spectrum. However, using the spec-tral pseudo-acceleration of EFSDOF oscillators with high effectivedamping ξssi may result in a severely underestimated design base shearfor the actual flexible-base structures (explained in detailed in

Fig. 4. Equivalent fixed-base SDOF (EFSDOF) oscillator approach to design flexible-base structures.

Table 1Ground motions recoded on very soft soil profiles.

Date Event Magnitude (Ms) Station Component (degree) PGA (cm/s2)

10/17/89 Loma Prieta 7.1 Foster City (APEEL 1; Redwood Shores) 90, 360 278, 26310/17/89 Loma Prieta 7.1 Larkspur Ferry Terminal 270, 360 135, 9510/17/89 Loma Prieta 7.1 Redwood City (APEEL Array Stn. 2) 43, 133 270, 22210/17/89 Loma Prieta 7.1 Treasure Island (Naval Base Fire Station) 0, 90 112, 9810/17/89 Loma Prieta 7.1 Emeryville, 6363 Christie Ave. 260, 350 255, 21010/17/89 Loma Prieta 7.1 San Francisco, International Airport 0, 90 232, 32310/17/89 Loma Prieta 7.1 Oakland, Outer Harbor Wharf 35, 305 281, 26610/17/89 Loma Prieta 7.1 Oakland, Title & Trust Bldg. 180, 270 191, 23910/15/79 Imperial Valley 6.8 El Centro Array 3, Pine Union School 140, 230 261, 21704/24/84 Morgan Hill 6.1 Foster City (APEEL 1; Redwood Shores) 40, 310 45, 67

Y. Lu et al. Engineering Structures 167 (2018) 26–38

30

Appendix A). Therefore, for the force-based seismic design of SSI sys-tems, the absolute acceleration spectra should be used in EFSDOF os-cillators. This implies that for SSI analyses, damping reduction factorscompatible with absolute acceleration spectra should be adopted [36].

Fig. 5 also shows that the conventional acceleration responsespectra exhibit two subsequent peaks, whereas the BNRS curves reach a

distinct peak value at Tssi/TP≈ 1. As discussed earlier, a BNRS accountsfor the frequency content of the ground motions in the averaging pro-cess. The peak spectral ordinates of the BNRS for initial effectivedamping ratios of ξssi = 0.05, 0.1, 0.16 and 0.21 are, respectively, 1.22,1.17, 1.13 and 1.11 times higher than those of the conventional spectra.By using more ground motion records, the spectral shape in Fig. 5(a)

Tssi

(a)Sa

/PG

A

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

a0=3 s=4 [ ssi=0.05]

a0=2 s=1 [ ssi=0.15]a0=3 s=2 [ ssi=0.1]

a0=2.5 s=1 [ ssi=0.2]

SSIEFSDOF absoluteEFSDOF pseudo

0 0.5 1 1.5 2 2.5 30.5

1

1.5

2

2.5

3

3.5

4

Tssi/TP

(b)

Sa/P

GA

Xu and Xie (2004)Ziotopoulou and Gazetas (2010)

a0=3 s=4a0=3 s=2

a0=2 s=1a0=2.5 s=1

SSIEFSDOF absoluteEFSDOF pseudo

Fig. 5. Elastic acceleration spectra for flexible-base structures: (a) conventional format and (b) bi-normalized format.

R

(a)

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

=4

=2

=5

=3

Tssi (sec)

R

(b)

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

=4

=2

=5

=3

Tssi /Tg

C

Tssi (sec)

=5

=2

(c)

0 0.5 1 1.5 2 2.5 30

1

2

3

4

EFSDOF =5%

a0=3 s=4 a0=1 s=2

C

Tssi /Tg

=5

=2

(d)

0 0.5 1 1.5 2 2.5 30

1

2

3

4

a0=3 s=4 a0=1 s=2 EFSDOF =5%

Fig. 6. Conventional (a, c) and normalized (b, d) Rμ and Cμ spectra for SSI models and EFSDOF oscillators (5% damping ratio).

Y. Lu et al. Engineering Structures 167 (2018) 26–38

31

would become more similar to those adopted by seismic codes, where aflat segment is expected due to averaging and smoothing. In that case,the difference between the peak values for the conventional and bi-normalized spectra would be even more significant. In Fig. 5(b), thecurves associated with ξssi = 0.05 coincide with the shaded area thatenvelops the 5% damped BNRS obtained by Xu and Xie [10] and Zio-topoulou and Gazetas [7], demonstrating the consistency of the BNRS.

7. Constant-ductility strength reduction factor and inelasticdisplacement ratio

According to the definitions of the modification factors used for SSIsystems (shown in Fig. 4), Rμ and Cμ were calculated based on thedisplacements of the structural mass relative to the ground, which in-cluded the foundation rigid-body motions. The spectral predominantperiod for a specified ground motion Tg is defined as the period atwhich the maximum ordinate of the relative velocity spectrum (for adamping ratio of ξssi) occurs.

Fig. 6 compares the Rμ and Cμ spectra derived using the SSI modelsand EFSDOF oscillators. The a0 and s values of the SSI systems werechosen so that the effective damping ratio ξssi was approximately equalto 5%, which was then assigned to the EFSDOF oscillators. The resultsin Fig. 6 are the averaged Rμ and Cμ spectra obtained for all 20 groundmotions and are presented in both conventional and normalized for-mats. Similar to previous studies (e.g. [5,37]), the peaks and valleys aremore noticeable when using the normalized format (Fig. 6(b) and (d)).For instance, the normalized response spectrum curves indicate that, ata period ratio Tssi/Tg≈ 1, the peak displacement of an inelastic systemis on average smaller than its elastic counterpart (i.e. Cμ < 1), while

the constant-ductility strength reduction factor Rμ is always maximum.This important behaviour is not obvious from the conventional re-sponse spectra shown in Fig. 6(a) and (c).

Fig. 6 illustrates that the use of the EFSDOF oscillator is, in general,able to provide a reasonable estimate of Rμ and Cμ for SSI systems.However, for slender structures (e.g. a0= 3, s= 4) where periodlengthening becomes higher, the oscillator approach slightly under-estimates Rμ, which consequently leads to an overestimation of Cμ,especially when global ductility demands become higher. Since theEFSDOF oscillators work perfectly well for predicting the elastic re-sponse of the SSI system with a0= 3 and s= 4 (see Fig. 5(b)), theunderestimation of Rμ could be a result of a higher strength predictedby the EFSDOF oscillators than that required by the SSI models to sa-tisfy a target ductility demand. As will be discussed in the followingsections, due to a large period lengthening effect, a global ductility ratioμssi = 4 for an SSI system with a0=3 and s=4 corresponds to anunexpectedly high structural ductility ratio μs > 10, which is not usedin common practice. Therefore, the results for higher global ductilitydemands are not seen to be important for practical design purposes.Note also that it may not be practical for a common flexible-baseslender building to have a short elastic fundamental period (e.g. a0= 3,s= 4, Tssi < 0.5 in Fig. 6). These systems were mainly used to showthat the damping ratio values due to the combination of a0 and s (ratherthan their individual values) result in constant-ductility spectral shapes.

For a higher effective damping ratio ξssi = 10%, the performance ofthe EFSDOF oscillators is still excellent, as shown in Fig. 7. However, ingeneral, values of Rμ calculated by the oscillator approach are slightlyhigher than those from the SSI models. Fig. 7 also includes results forSSI systems with a larger soil material damping ξg= 10%; Rμ and Cμ

R(a)

0 0.5 1 1.5 2 2.5 30

2

4

6

8

=4

=2

=5

=3

Tssi (sec)

R

(b)

0 0.5 1 1.5 2 2.5 30

2

4

6

8

=4

=2

=5

=3

Tssi /Tg

C

Tssi (sec)

(c)

0 0.5 1 1.5 2 2.5 30

1

2

3

4=5

=2

EFSDOF =10%

a0=2 s=2 g=10%a0=3 s=2

C

Tssi /Tg

(d)

0 0.5 1 1.5 2 2.5 30

1

2

3

4

=5

=2

EFSDOF =10%

a0=2 s=2 g=10%a0=3 s=2

Fig. 7. Conventional (a, c) and normalized (b, d) Rμ and Cμ spectra for SSI models and EFSDOF oscillators (10% damping ratio).

Y. Lu et al. Engineering Structures 167 (2018) 26–38

32

predictions by the EFSODF oscillators for these cases are very good.Therefore, it can be concluded that an EFSDOF oscillator is a viablesubstitute for a lightly-to-moderately damped SSI system.

Fig. 8 presents results for a much higher initial damping ratioξssi = 20%, which is the upper limit of the overall damping of an SSIsystem suggested in seismic provisions [1]. It is shown that the EFSDOFoscillators, on average, over-predict the constant-ductility strength re-duction factor Rμ, and underestimate the inelastic displacement ratio Cμof the corresponding SSI systems. For the normalized Rμ spectra shownin Fig. 8(b), this over-prediction, which is up to 26%, is more pro-nounced when the Tssi/Tg ratio is smaller than 1.5.

It can be concluded from the above observations that the EFSDOFoscillators, over a wide range of normalized period, over- and under-estimate, respectively, Rμ and Cμ values for SSI systems with a highinitial damping ratio. Therefore, a correction factor can be introducedto improve predictions of the EFSDOF oscillators for highly damped SSIsystems. Note that for common building structures having a slendernessratio s greater than 2, the effective damping ratio is always lower than10%, regardless of a0 values (see Fig. 3), which means that the EFSDOFoscillator approach can be directly applied to these structures withoutany modification.

To improve the prediction of the seismic response of SSI systems, acorrection factor α is defined in this study as the ratio of Rμ predicted byan EFSDOF oscillator to that of the SSI model. According to Eq. (9), αcan also be used to modify the inelastic displacement ratio Cμ predictedby an EFSDOF oscillator:

= =α T T ξ μR T T ξ μ

R T T ξ μC T T ξ μ

C T T ξ μ( / , , )

( / , , )( / , , )

( / , , )( / , , )g

μ EFSDOF g

μ ssi g

μ ssi g

μ EFSDOF g

,

,

,

, (15)

The constant-ductility strength reduction factor ratios Rμ,EFSDOF/Rμ,ssi were calculated for each of the SSI systems which had initial ef-fective damping ratios varying from 11 to 20% at a 1% interval.Fig. 9(a) is an example of the results for SSI systems with a globalductility ratio μssi = 5. As expected, the correction factor becomesgreater for higher initial effective damping levels, and the averageddata exhibits, approximately, an ascending, a constant, and a des-cending trend, respectively, in spectral regions Tssi/Tg < 0.4,0.4≤ Tssi/Tg < 0.9, and Tssi/Tg≥ 0.9. Mean Rμ,EFSDOF/Rμ,ssi ratios forductility values from 2 to 5 are compared in Fig. 9(b), which showsthat, in general, greater correction factor values should be applied tomore ductile systems. Fig. 9(b) also illustrates the mean α spectra de-rived using both ratios of Rμ,EFSDOF/Rμ,ssi and Cμ,ssi/Cμ,EFSDOF, which arefairly similar and may be approximated using the following simplifiedpiecewise expression:

=

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

+ ⩽

< ⩽

− + < ⩽

>

=

−

−+

( )α μ

1 0.4

c 0.4 0.9

1.5 1 0.9 1.5

1 1.5

c

TT

TT

TT

TT

TT

TT

ξ

c 10.4

c 10.6

(0.12ln 0.3)

g g

g

g g

g (16)

Fig. 9(c) and (d) illustrate the proposed correction factor α fordifferent ductility levels and initial effective damping ratios calculatedusing Eq. (16). It is shown that higher modification factors are requiredfor SSI systems with higher ductility demands and initial effectivedamping ratios. Comparing Fig. 10 with data in Fig. 8 demonstrates

R(a)

0 0.5 1 1.5 2 2.5 30

2

4

6

8

=4

=2

=5

=3

Tssi (sec)

R

(b)

0 0.5 1 1.5 2 2.5 30

2

4

6

8

=4

=2

=5

=3

Tssi /Tg

C

Tssi (sec)

(c)

=5

0 0.5 1 1.5 2 2.5 30

1

2

3

4

EFSDOF =20%

a0=2.5 s=1 a0=3 s=1.2

=2

C

Tssi /Tg

(d)

=5

0 0.5 1 1.5 2 2.5 30

1

2

3

4

EFSDOF =20%

a0=2.5 s=1a0=3 s=1.2

=2

Fig. 8. Conventional (a, c) and normalized (b, d) Rμ and Cμ spectra for SSI models and EFSDOF oscillators (20% damping ratio).

Y. Lu et al. Engineering Structures 167 (2018) 26–38

33

that Rμ and Cμ spectra derived using modified EFSDOF oscillators are inmuch better agreement with those of the SSI models. Note that Eq. (16)is applicable to SSI systems having an initial damping ratio rangingfrom 11% to 20% and a global ductility ratio less than 5.

8. Structural and global ductility ratios

Although the global ductility μssi relates the displacement demandof an inelastic SSI system to its yielding displacement, the structuralductility μs is sometimes more important since it directly reflects theexpected damage in a structure. By using the global ductility μssi, thestructural ductility ratio μs can be calculated according to Eq. (14). Inorder to evaluate the effectiveness of this equation, the actual structuralductility ratios μs obtained by response-history analysis using the SSImodel (points) are compared with those calculated using Eq. (14)(lines) in Fig. 11. The presented results are the averaged values for the20 records (Table 1) considering four global ductility values μssi = 2, 3,4, and 5; the shaded areas illustrate the practical range of the designstructural ductility demands μs.

In general, Fig. 11 shows good agreement between Eq. (14) and theresults of response-history analyses, especially for lightly-damped SSIsystems with equivalent natural periods close to those of their fixed-base systems (e.g. Fig. 11(a)). For highly nonlinear structures, on theother hand, using Eq. (14) leads to an overestimation of μs. This isparticularly obvious for systems with a higher period lengthening ef-fect, as shown in Fig. 11(b), (c) and (e). However, it may not be im-portant for common buildings that are usually designed for a structural

ductility ratio of less than 8. Note that for a given global ductility ratio,the period lengthening effect is greater for structures with a higherstructural ductility ratio (see Eq. (14)). The results illustrated in Fig. 11generally demonstrate very good agreement between structural ducti-lity ratios μs obtained from the SSI model response-history analysis andthose calculated by Eq. (14). This is especially evident within theshaded areas that represent practical design scenarios.

9. Discussion

In the present study, elastic and constant-ductility response spectrafor soil-structure interaction systems were derived through response-history analyses performed using a selection of ordinary ground mo-tions recorded on very soft soil sites. The structure was modelled by anSDOF oscillator having an elastic-perfectly plastic hysteretic behaviour.The elasto-dynamic response of the soil-foundation system was simu-lated using the cone model. The results of this study highlighted theimportance of spectral predominant periods for soft soil conditions andwere used to improve the efficiency of the EFSDOF oscillator approach.Compared to existing SSI procedures based on EFSDOF oscillators, theimproved EFSDOF oscillator has the following advantages: (1) themodel explicitly includes the effect of frequency content of groundmotions on the seismic response of structures on soft soils through theuse of spectral predominate periods, and (2) the model provides im-proved estimation of constant-ductility strength reduction factor andinelastic displacement ratio of SSI systems with high initial effectivedamping ratios. The improved EFSDOF can be easily implemented in

(a)

0 0.5 1 1.5 2 2.5 30.9

1

1.1

1.2

1.3

ssi=11%

ssi=20%

=5Mean

Tssi /Tg

(b)

0 0.5 1 1.5 2 2.5 30.9

1

1.1

1.2

1.3 Mean for =2Mean for =3Mean for =4Mean for =5

Mean of all R dataMean of all C data

proposed

Tssi /Tg

0 0.5 1 1.5 2 2.5 30.9

1

1.1

1.2

=5=4=3=2

(c)

Tssi /Tg

=15%

0 0.5 1 1.5 2 2.5 30.9

1

1.1

1.2

=5=4=3=2

(d)

Tssi /Tg

=20%

Fig. 9. (a)–(b) Correction factor α obtained from response-history analyses, and (c)–(d) proposed analytical values of α as a function of period of vibration, effectiveductility ratio and effective damping ratio of an SSI system.

Y. Lu et al. Engineering Structures 167 (2018) 26–38

34

either force-based (using Rμ) or displacement-based (using Cμ) designfor SSI systems. The effects of near-fault directivity, the structuralhysteretic model, and higher modes were not considered in this studyand require further evaluation.

10. Conclusions

Around 200,000 response-history analyses were carried out usingfixed-base and soil-structure interaction models to study the elastic andinelastic response spectra of buildings on soft soil profiles. Based onresults for 20 ground motions recorded on very soft soil deposits, it wasshown that normalizing the equivalent period of an SSI system Tssi bythe corresponding predominant period resulted in more rational spectrafor seismic design purposes. In the elastic response spectra, Tssi is nor-malized by the spectral predominant period TP corresponding to thepeak ordinate of a 5% damped elastic acceleration spectrum, while fornonlinear structures Tssi should be normalized by the predominantperiod of the ground motion Tg at which the relative velocity spectrumreaches its maximum value.

It was shown that an actual SSI system could be replaced by anequivalent fixed-base oscillator having a natural period of Tssi, a viscous

damping ratio of ξssi, and a ductility ratio of μssi. It was concluded thatthe absolute acceleration spectra, instead of the pseudo-accelerationspectra, should be used for EFSDOF oscillators in force-based design ofSSI systems. The EFSDOF oscillator approach provided an excellentestimate of acceleration and inelastic spectra for lightly-to-moderatelydamped SSI systems. However, it was shown that the EFSDOF oscilla-tors, in general, overestimate the constant-ductility strength reductionfactor Rμ of SSI systems with high initial damping ratio (e.g. squattystructures founded on very soft soil profiles), which consequently leadsto an underestimation of inelastic displacement ratio Cμ. Based on theresults of this study, a correction factor was proposed to improve theefficiency of the EFSDOF oscillators to predict the Rμ and Cμ spectra ofSSI systems having initial effective damping ratios greater than 10%.

Finally, it was demonstrated that for any ground motion, thestructural ductility demand of a nonlinear flexible-base structure can becalculated, with good accuracy, from the global ductility demand of thewhole SSI system. The improved EFSDOF oscillator can thus be easilyimplemented in the performance-based design of structures on soft soilwith a target ductility ratio which is defined either for an SSI system orfor the structure alone.

R

(a)

0 0.5 1 1.5 2 2.5 30

2

4

6

8

=4

=2

=5

=3

Modified EFSDOF =15%

a0=3 s=1.48 a0=3 s=1.8 g=10%

Tssi /Tg

R

(b)

0 0.5 1 1.5 2 2.5 30

2

4

6

8

=4

=2

=5

=3

a0=3 s=1.2 a0=2.5 s=1

Modified EFSDOF =20%

Tssi /Tg

0 0.5 1 1.5 2 2.5 30

1

2

3

4

C

Tssi /Tg

(c)

Modified EFSDOF =15%

a0=3 s=1.48 a0=3 s=1.8 g=10%

=2

=5

0 0.5 1 1.5 2 2.5 30

1

2

3

4

C

Tssi /Tg

(d)

a0=3 s=1.2 a0=2.5 s=1

Modified EFSDOF =20%

=2

=5

Fig. 10. Improved performance of the modified EFSDOF oscillators.

Y. Lu et al. Engineering Structures 167 (2018) 26–38

35

Appendix A

The equation of motion of an SSI system subjected to a ground acceleration time-history u (t)g can be expressed in the following matrix form:

+ + = −M u t C u t K u t M R u t[ ]{ ¨ ( )} [ ]{ ( )} [ ]{ ( )} [ ]{ } ¨ ( )g (A1)

For nonlinear structures, the term “[M]{u(t)}” is replaced with restoring forces “{F(t)}”. In the frequency domain where the system is subjected toa harmonic motion having a frequency ω and an amplitude Ug, Eq. (A1) can be written as:

Tssi /Tg

(a)

s

0 1 2 30

2

4

6

ssi=5

ssi=2

a0 s ssi=5%

Tssi /Tg

s

0 1 2 30

5

10

15

ssi=5

ssi=2

(b)

a0 s ssi=5%

Tssi /Tg

s

(c) 0 1 2 30

5

10

15

ssi=5

ssi=2a0 s ssi=10%

Tssi /Tg

s

0 1 2 30

3

6

9

ssi=5

ssi=2

(d)

a0 s g=10% ssi=10%

s

(e)Tssi /Tg

ssi=5

ssi=2

0 1 2 30

5

10

15

a0 s ssi=15%

s

Tssi /Tg

0 1 2 30

3

6

9

12

ssi=5

ssi=2

(f)

a0 s ssi=20%

Actual Predicted Practical range of s

Fig. 11. Structural ductility ratios μs: response-history analysis using SSI model (points) versus results using Eq. (14) (lines).

Y. Lu et al. Engineering Structures 167 (2018) 26–38

36

− + + = −ω M iω C K U ω M R U ω( [ ] [ ] [ ]){ ( )} [ ]{ } ¨ ( )g2 (A2)

where i is the imaginary unit satisfying i2=−1.For the cone model shown in Fig. 1, the mass, damping coefficient, and stiffness matrices are given by:

=

⎡

⎣

⎢⎢⎢⎢⎢

+

+ + + −

− +

⎤

⎦

⎥⎥⎥⎥⎥

M

mm c ξ ω

J J M c ξ ω c ξ ω

c ξ ω M c ξ ω

[ ]

0 0 0/ 0 0

/ /

Sym. / /

s

f h g

s f θ θ g θ g

θ g ϕ θ g

0

0 0

0 0(A3)

=

⎡

⎣

⎢⎢⎢⎢

− −+ +

+ + −

⎤

⎦

⎥⎥⎥⎥

C

c c c hc c k ξ ω c h

c c h k ξ ω cc

[ ]

02 / 0

2 /Sym.

s s s s

h s h g s s

θ s s θ g θ

θ

0

20

(A4)

=

⎡

⎣

⎢⎢⎢⎢

− −+

+

⎤

⎦

⎥⎥⎥⎥

K

k k k hk k k h

k h k[ ]

000

Sym. 0

s s s s

h s s s

s s θ2

(A5)

where the frequency ω0 equals ω and ωssi in the frequency and time domains, respectively. The displacements and influence coefficients are definedas:

= =u u u θ φ R{ } [ , , , ] , { } [1,1,0,0]ssi hT T (A6)

where uh is the foundation swaying displacement, and the displacement of the structural mass relative to the ground ussi = uh+ θhs+ us, withreference to Fig. A1.

For the impedance model depicted in Fig. A1(a):

=⎡

⎣

⎢⎢ +

⎤

⎦

⎥⎥

Mm

mJ J

[ ]0 0

0Sym.

s

f

s f (A7)

=⎡

⎣

⎢⎢

− −+

+

⎤

⎦

⎥⎥

Cc c c h

c β rk v c hc h β rk v

[ ] /Sym. /

s s s s

s h h s s s

s s θ θ s2

(A8)

=⎡

⎣

⎢⎢

− −+

+

⎤

⎦

⎥⎥

Kk k k h

k α k k hk h α k

[ ]Sym.

s s s s

s h h s s

s s θ θ2

(A9)

where closed-form expressions for αh, αθ, βh and βθ were proposed by Veletsos and Verbic [27] as frequency-dependent dynamic modifiers to thefoundation swaying and rocking stiffness. These dynamic modifiers are also functions of the soil Poisson’s ratio ν and hysteretic soil damping ratio ξg.

= =u u u θ R{ } [ , , ] , { } [1,1,0]ssi hT T (A10)

Unlike the pseudo-acceleration spectra for design of fixed-base buildings with external energy dissipation systems, the absolute accelerationspectra for EFSDOF oscillators should be used for flexible-base structures. This point can be addressed by comparing the equations of motion for thestructural mass in both the SSI system and the EFSDOF oscillator shown in Fig. A1:

+ + + =u u ω ξ u ω u( ¨ ¨ ) 2 0ssi g s s s s s2 (A11)

+ + + =u u ω ξ u ω u( ¨ ¨ ) 2 0sdof g n sdof n sdof2 (A12)

where the elastic dynamic properties of the EFSDOF oscillator are characterized by its circular frequency of vibration =ω k /mn sdof s and a viscous

uh ushsug

hs

ms, Js

mf, Jf

ks

kh k

ussi

ug

usdof

(a) (b)

ksdof

ms

Fig. A1. (a) SSI impedance model; and (b) equivalent fixed-base SDOF (EFSDOF) oscillator.

Y. Lu et al. Engineering Structures 167 (2018) 26–38

37

damping ratio of ξ. Provided that usdof of the EFSDOF oscillator is an accurate estimation of ussi of the SSI system, the absolute acceleration of theEFSDOF oscillator +(u u )sdof g equals that of the SSI system +(u u )ssi g . Due to a low structural damping ratio of ξs = 5%, the base shear demand ofthe flexible-base structure can be calculated using either the pseudo-acceleration ω us

2s,max or the absolute acceleration +max(u u )ssi g which equals

the spectral absolute acceleration of the EFSDOF oscillator +max(u u )sdof g . If, however, the spectral pseudo-acceleration of the EFSDOF oscillatorω un

2sdof,max is used, the design base shear may be underestimated, due to high damping effects ξ= ξssi≫ 5%.For the SDOF oscillator illustrated in Fig. A1(b), Eqs. (A1) and (A2) reduce to the corresponding single equation of motion. The resonant response

of this SDOF oscillator subjected to the harmonic motions U (ω)g satisfies the following expressions [43]:

=−

=−

ω UU ξ ξ

ω ωξ¨

12 1

,1 2

n res

gn

res2

2 2 (A13)

where Ures is the resonant amplitude of the displacement usdof.Similarly, it may be assumed that Eq. (A13) also holds for the SSI systems shown in Figs. 1 and A1(a), whereby ωssi, ξssi, and ussi correspond,

respectively, to ωn, ξ, and usdof of the SDOF oscillator. With this assumption, the equivalent natural frequency ωssi and the effective damping ratio ξssican be solved by Eq. (A13) at the resonant response of the interacting system.

References

[1] ASCE/SEI 7-16. Minimum design loads and associated criteria for buildings andother structures. Reston, Virginia: American Society of Civil Engineers; 2017.

[2] ASCE/SEI 41-17. Seismic evaluation and retrofit of existing buildings. Reston,Virginia: American Society of Civil Engineers; 2017.

[3] CEN. Eurocode 8: Design of structures for earthquake resistance. Part 5:Foundations, retaining structures and geotechnical aspects. EN 1998-5:2004.Brussels, Belgium: Comité Européen de Normalisation; 2004.

[4] FEMA 440. Improvement of nonlinear static seismic analysis procedures.Washington DC: Applied Technology Council; 2005.

[5] Ruiz-García J, Miranda E. Inelastic displacement ratios for evaluation of structuresbuilt on soft soil sites. Earthq Eng Struct Dyn 2006;35:679–94.

[6] Vidic T, Fajfar P, Fischinger M. Consistent inelastic design spectra: strength anddisplacement. Earthq Eng Struct Dyn 1994;23:507–21.

[7] Ziotopoulou A, Gazetas G. Are current design spectra sufficient for soil-structuresystems on soft soils? Adv. Performance-Based Earthq. Eng., Springer; 2010, p.79–87.

[8] Rathje EM, Abrahamson NA, Bray JD. Simplified frequency content estimates ofearthquake ground motions. J Geotech Geoenviron Eng 1998;124:150–9.

[9] Mylonakis G, Gazetas G. Seismic soil-structure interaction: beneficial or detri-mental? J Earthq Eng 2000;4:277–301.

[10] Xu L, Xie L. Bi-normalized response spectral characteristics of the 1999 Chi-Chiearthquake. Earthq Eng Eng Vib 2004;3:147–55.

[11] Miranda E, Bertero VV. Evaluation of strength reduction factors for earthquake-resistant design. Earthq Spectra 1994;10:357–79.

[12] Miranda E, Ruiz-Garca J. Evaluation of approximate methods to estimate maximuminelastic displacement demands. Earthq Eng Struct Dyn 2002;31:539–60.

[13] Ghannad MA, Jahankhah H. Site-dependent strength reduction factors for soil-structure systems. Soil Dyn Earthq Eng 2007;27:99–110.

[14] Khoshnoudian F, Ahmadi E, Nik FA. Inelastic displacement ratios for soil-structuresystems. Eng Struct 2013;57:453–64.

[15] Khoshnoudian F, Ahmadi E, Sohrabi S. Response of nonlinear soil-MDOF structuresystems subjected to distinct frequency-content components of near-fault groundmotions. Earthq Eng Struct Dyn 2013;13:1809–33.

[16] Khoshnoudian F, Ahmadi E. Effects of pulse period of near-field ground motions onthe seismic demands of soil-MDOF structure systems using mathematical pulsemodels. Earthq Eng Struct Dyn 2013;42:1565–82.

[17] Khoshnoudian F, Ahmadi E. Effects of inertial soil-structure interaction on inelasticdisplacement ratios of SDOF oscillators subjected to pulse-like ground motions. BullEarthq Eng 2014;13:1809–33.

[18] Mekki M, Elachachi SM, Breysse D, Nedjar D, Zoutat M. Soil-structure interactioneffects on RC structures within a performance-based earthquake engineering fra-mework. Eur J Environ Civ Eng 2014;18:945–62.

[19] Moghaddasi M, MacRae GA, Chase JG, Cubrinovski M, Pampanin S. Seismic designof yielding structures on flexible foundations. Earthq Eng Struct Dyn 2015.

[20] Seylabi EE, Jahankhah H, Ali Ghannad M. Equivalent linearization of non-linearsoil-structure systems. Earthq Eng Struct Dyn 2012;41:1775–92.

[21] Chopra AK, Goel RK. Direct displacement-based design: use of inelastic vs. elasticdesign spectra. Earthq Spectra 2001;17:47–64.

[22] Fajfar P. Capacity spectrum method based on inelastic demand spectra. Earthq EngStruct Dyn 1999;28:979–93.

[23] Ehlers G. The effect of soil flexibility on vibrating systems. Bet Und Eisen1942;41:197–203.

[24] Meek JW, Wolf JP. Cone models for homogeneous soil I. J Geotech Eng1992;118:667–85.

[25] Wolf JP, Deeks AJ. Foundation vibration analysis: a strength of materials approach;2004.

[26] The Mathworks Inc. MATLAB – MathWorks. www.mathworks.com/products/matlab; 2016.

[27] Veletsos AS, Verbič B. Vibration of viscoelastic foundations. Earthq Eng Struct Dyn1973;2:87–102.

[28] Lu Y, Hajirasouliha I, Marshall AM. Performance-based seismic design of flexible-base multi-storey buildings considering soil-structure interaction. Eng Struct2016;108:90–103.

[29] Avilés Ja, Pérez-Rocha LE. Influence of foundation flexibility on Rμ and Cμ factors. JStruct Eng 2005;131:221–30.

[30] Avilés J, Suárez M. Effective periods and dampings of building-foundation systemsincluding seismic wave effects. Eng Struct 2002;24:553–62.

[31] Veletsos AS, Nair VV. Seismic interaction of structures on hysteretic foundations. JStruct Div 1975;101:109–29.

[32] Maravas A, Mylonakis G, Karabalis DL. Simplified discrete systems for dynamicanalysis of structures on footings and piles. Soil Dyn Earthq Eng 2014;61:29–39.

[33] Avilés J, Pérez-Rocha LE. Soil-structure interaction in yielding systems. Earthq EngStruct Dyn 2003;32:1749–71.

[34] Ghannad MA, Jafarieh AH. Inelastic displacement ratios for soil–structure systemsallowed to uplift. Earthq Eng Struct Dyn 2014;43:1401–21.

[35] Chopra AK. Dynamics of Structures. Pearson Education; 2012.[36] Lin YY, Chang KC. Study on damping reduction factor for buildings under earth-

quake ground motions. J Struct Eng 2003;129:206–14.[37] Miranda E, Ruiz-Garcia J. Influence of stiffness degradation on strength demands of

structures built on soft soil sites. Eng Struct 2002;24:1271–81.

Y. Lu et al. Engineering Structures 167 (2018) 26–38

38