2-D soil-structure interaction in time domain by the SBFEM and two non-linear soil models

2-D soil-structure interaction in time domain by the SBFEM

and two non-linear soil models

Hossein Rahnema a,n,Sassan Mohasseb b,1,Behtash JavidShari?c

a Shiraz University of Technology,Shiraz,Iran

b University of Tehran,Tehran,Iran

c Fars Regional Electrical Company,Shiraz,Iran

a r t i c l e i n f o

Article history:

Received7May2015

Received in revised form

10January2016

Accepted15January2016

Keywords:

Soil-structure interaction

Sub-structuring method

Direct modeling

UCSD model

UCD model

a b s t r a c t

Effects of soil-structure interaction(SSI)have proven to be of more importance than to be ignored.Quite

a few methods of modeling and analysis exist to anticipate the real behavior of the structure when placed

on?exible soil rather than on rigid ground surface.Yet,how to model the soil needs to be inspected

carefully since probable deformations of soil may be at times far from predicted.In this study a newly

formed approach is inspected to suggest possible solutions to shortcomings of conventional ones,

compare two non-linear soil models,and implement strengths of newer methods.The soil-structure

system is modeled and analyzed once directly with the UCSD soil model and then compared with non-

linear sub-structuring method with the UCD model.Analyzes are performed in the time domain for both

cases.The soil is supposed to be comprised of sands with various density values.The Loma-Prieta

earthquake record(Loma-Prieta,1989)is used to carry out time domain analyzes and capture structural

responses.The interactional forces exerted to the near-?eld soil,which account for the interaction

between these two media as well as the radiation damping of the in?nite half-space,have replaced the

earthquake motion and the far-?eld has accordingly been truncated out.The non-linear near-?eld soil-

structure system has then been dynamically analyzed.Force outputs reveal a decrease when elastoplastic

SSI is considered;while displacement amplitudes are found to be greater for cases not involving SSI,or

involving elastic SSI.Changing the applied constitutive model for the soil as well as sand density from

loose to dense manifests changes in responses.As the soil gets denser,the SSI behavior gets closer to that

of the elastic case.Contrary to the sub-structuring method which usually,and conventionally,assumes

linear elastic behavior for the soil-structure system,direct modeling may predict non-linear responses of

the system and effects of the structure0s being placed upon an inelastic environment.

&2016Elsevier Ltd.All rights reserved.

1.Introduction

Soil-structure interaction(SSI)is an interdisciplinary?eld

involving soil and structural dynamics,earthquake engineering,

geomechanics,numerical methods,and many other(inter)related

?elds.The effect of soil on structural responses under static and

dynamic loads is of special interest when it comes to structures

such as nuclear power plants and offshore industries.Solution

techniques to the problem of SSI have advanced rapidly with the

evolution of powerful computers that are now able to handle large

amounts of computations with relatively fast speed.As more

complicated calculations get possible to be done rapidly,more

advanced solutions schemes appear to lead to more logical solu-

tions to the SSI problem.

The evolution of researchers'insight to SSI analysis gave rise to

the interest of solving the problem in time-domain.[34]is a good

reference on practical and exact formulation of time-domain SSI

considering conventional theories of wave propagation and seis-

mic motion effects on the soil.It soon got clear that the best way to

model the soil to behave closest to reality is to do it in time

domain.

soil does not usually react elastically and this is obvious to

researchers,but modeling the soil as an elastic-plastic material has

not been easy until relatively recently.The problem of dynamic SSI

has been tried to be solved via various modeling techniques.

After noticing the effects of loading on the underlying soil,

interaction effects within and near the structure were started to be

addressed.Unexpectedly heavy damages were recorded during

some earthquakes which were not easy to be accounted for with

then-at-hand information.In1999Athens Earthquake for

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Soil Dynamics and Earthquake Engineering

https://www.360docs.net/doc/b37954914.html,/10.1016/j.soildyn.2016.01.008

0267-7261/&2016Elsevier Ltd.All rights

reserved.

n Corresponding author.

E-mail addresses:rahnema@sutech.ac.ir(H.Rahnema),

smteam@gmx.ch(S.Mohasseb),b.javidshari?@sutech.ac.ir(B.JavidShari?).

1SMTeam technical director,Zurich,Switzerland.

Soil Dynamics and Earthquake Engineering88(2016)152–175

example,the town of Adàmes on the eastern cliff of the Ki?issos river canyon was subject to such damages.Assimaki et al.[40] examined the importance of soil stratigraphy and soil-structure interaction on the characteristics of ground surface motion.They carried out elastic two-dimensional wave propagation analyzes followed by non-linear time domain simulations including soil-structure interaction effects aggravating the motion.One common way of accounting for non-linearity of the media is to see it in the dissipation of energy as of modeling the damping of the system in the proper form.[31],for example,suggests a couple of veri?ed damping formulations to model non-linear response of the site under small and large strains.According to Kausel[37],it is the interaction of seismic waves with the structure and its effects on the soil nearby which ultimately matter.Traditional approaches were gradually replaced by conventional ones in a couple of dec-ades after renowned researchers contributed to the problem.The superposition method was made use of to inspect the problem of SSI,and computational methods began to grow along with laboratory testings.

The importance of inspecting SSI and examining its effect lies primarily in accurate estimation of static and especially dynamic properties of the structure.Structural simulations to solve numerous problems of interest may account for a decent imple-mentation of the soil system to yield results as reliable as possible. As a good example,Karapetrou et al.[66]assessed seismic vul-nerability of high-rise RC buildings considering SSI effects in the simulation in which they used?nite elements to model the underlying soil.They intended to record the effects of SSI on a problem which had been traditionally investigated assuming the structure to be?xed-base.They modeled SSI applying direct method considering linear elastic as well as non-linear soil beha-vior.The soil was modeled hyperbolically and the in?nity of the media was accounted for by Lysmer0s dashpots.

Many of the SSI models employed up through the early19700s were relatively"simple"in the sense that they were restricted to systems in which the foundation rested directly onto the surface of a homogeneous half-space,and the seismic motion in the free-?eld was invariant in horizontal planes,e.g.the motions resulted from waves propagating vertically in a laterally homogeneous soil. For such models,the intuitively obvious strategy of prescribing the free-?eld motion directly underneath the soil"springs"supporting the structure in a formulation in the frequency domain was both suf?cient and rigorous.However,when discrete methods of ana-lyzes,such as?nite elements,started being applied to SSI pro-blems,and especially when embedded structures began to be considered,substantial discrepancies were observed between the results of the numerical analyzes and the classical analytical method,which demanded an explanation as to why the differ-ences.This motivated the development of the so-called three-step solution,which provided the means to accomplish fully consistent comparisons between the results obtained by purely numerical models with?nite elements and those by the lumped parameter method based on foundation impedances or"springs"together with seismic motions prescribed underneath these springs.In this regard,SSI was assumed to be comprised of a summation of the effects of kinematic interaction,foundation stiffness and inertial interaction.In a series of analytical works,Lysmer and Richart[41] commenced solving the problem of SSI with numerical methods and suggested techniques to model the damping existing in the soil when the seismic wave reaches and affects the near-?eld of the soil-structure system.Applying this and such techniques could be,and still is,ef?cient enough to shift the inspections from the "sub-structuring"method to the"direct"[43].Hatzigeorgious and Beskos[39]performed3-D analyzes of tunnels considering the effects of SSI taking up such methods.The?nite element method was used to determine the seismic response of the soil-structure system in the time domain.The structure was a3-D tunnel the concrete material of which behaved inelastically,as did the sur-rounding rock.They used viscous absorbing boundaries in con-junction with the discretization of the rock medium.One major outcome of this study was that the SSI effect when the structure is surrounded by soft rock is negligible,while for a given seismic action,the damage of the liner of the tunnel increases with decreasing rock strength[39].Another problem that could vastly involve the inelastic behavior of the soil and its effects on the foundation of the structure,and hence on the structure,was the study of interaction of foundation-structure systems with seismi-cally precarious slopes,as was carried out by Kourkoulis et al.[63]. In their research,Kourkoulis et al.implemented a strain softening constitutive model in the numerical analyzes via FEM to account for the non-linear behavior of the soil comprising the slope crest near the structure.They showed that a frame structure founded on a properly designed raft could survive the combined effects of slope failure and ground shaking,even if the latter is the result of a strong base excitation ampli?ed by the soil layer.This work obviously involved non-linear SSI effect consideration which was professionally performed.

Another major shortcoming of the sub-structuring method was that it would primarily lead the calculations to be performed in the frequency domain which would deprive the equations to account for possible nonlinearities.Yet,the effect of the far-?eld on near-?eld responses is best imposed on the system through schemes involving advanced sub-structuring.To do this,SSI needs to be addressed once conventionally,i.e.via sub-structuring method,and once rigorously.The two procedures will then be combined to yield the novel approach.Hybrid approaches have been under development since the introduction of numerical methods of modeling unbounded(i.e.in?nite)media such as the In?nite Element Method(IEM),the Boundary Element Method (BEM)and the semi-analytical Scaled Boundary Finite Element Method(SBFEM),formerly known as the Consistent In?nitesimal Finite Element Cell Method(CIFECM).Pavlatos and Beskos[38] used a hybrid BEM/FEM scheme in the time domain to analyze elastoplastic structures under plane strain or plane stress condi-tions.Several isotropic hardening plasticity models were applied in the FEM part of the solution to account for the non-linear behavior of the near-?eld soil.Newmark0s step-by-step implicit time integration was followed and the scheme was proved through several examples to be effective and advantageous.The solution procedure suggested by Pavlatos and Beskos is in fact the basis of the solution technique of the present study.Newer tech-niques of numerical solution have replaced conventional ones and the soil has been modeled with newer constitutive models.FEM and BEM were implemented to analyze seismic ground motions of the Hyogo-Ken Nanbu earthquake happened in1995[60,65].The ?nite element-modeled soil,which was mostly soft alluvium and had ampli?ed the motion in Kobe,caused vast damages there[65]. Yerli et al.[61]parallelized the FEM and IEM to analyze SSI in two dimensions so that distributed computer systems may run huge models.In?nite elements were used to represent the soil extending to in?nity resulting in the need of a large?nite element mesh to de?ne the near-?eld for reliable predictions[61].Halabian and El Naggar[55]studied the effect of non-linear SSI on seismic response of slender structures using a hybrid FEM-SBFEM method. They inspected the placement of very massive structures on relatively soft soils.The non-linearity of the soil adjacent to the structure was observed to affect the response of the structure to dynamic loading.The SBFEM was employed to model the non-linear(i.e.near-?eld)zone of the soil;supporting the structure as a series of bounded media.The material properties of the bounded media were selected compatible with the average effective strains over the whole bounded medium using the SBFEM for the

H.Rahnema et al./Soil Dynamics and Earthquake Engineering88(2016)152–175153

unbounded media.The structure was represented by the FEM.The proposed method was used to model the dynamic response of a one-mass structure and a tower supported on a homogeneous stratum and excited by an earthquake.Halabian and El Naggar [55]found that the secondary soil non-linearity might increase or decrease the base forces of tall slender structures depending on the type of structure,frequency content of the input motion and the dynamic properties of the near-?eld soil.Bode et al.[56]used a mathematical approach to inspect SSI in the time domain.To calculate the transient response of general,three-dimensional structures resting on a homogeneous elastic half-space subjected to external loads and seismic motions,they proposed a formula-tion including the time domain formulation of the soil behavior and the coupling of the corresponding soil algorithm to the used ?nite element code.Their approach was based on the half-space Green 0s function for displacements elicited by Heaviside time-dependent surface point loads,which is in a sense similar to the unit impulse response used in the time domain SBFEM.

How to model the in ?nity of the soil medium is a major con-troversy in rigorous methods of modeling of SSI.Nakamura [58]proposed a solution method with a transformed energy trans-mitting boundary to analyze SSI non-linearly in the time domain.He transformed the frequency-dependent impedance into the time domain,which is again a rather similar approach to trans-forming the frequency domain dynamic stiffness into the time domain unit impulse response performed in the SBFEM.He then performed linear and non-linear time history earthquake response analyzes using the modeled boundary [58].Kobayashi et al.[53]too had carried out non-linear time-history analyses considering material and geometrical non-linearity.In fact,[58]was an update to [53]in that the boundary was modeled so as to match with the non-linear SSI problem.To couple the boundary with the near-?eld,Bransch and Lehmann [1]suggested a non-linear HHT-αmethod with elastic-plastic SSI in a coupled SBFEM/FEM approach.The method they presented was proposed to be used as a tool for accurate calculation of building ground deformations and the stability of the sub-soil subject to dynamic loading.Soil dynamics and wave propagation problems with the SBFEM is progressively being developed at the time being.Recently for example,Chen et al.[64]generalized the method for time domain analysis of wave propagation in 3-D unbounded domains by the SBFEM,which is one of the most appropriate ones to capture the effects of unboundedness of the domain.The same idea of displacement unit impulse response matrix representing the interaction force –displacement relationship on the near-?eld/far-?eld interface was made use of.In order to reduce the computational effort asso-ciated with the three dimensionality of the problem,the fully coupled unbounded domain was subdivided into multiple sub-domains and the displacement unit impulse response matrices of all subdomains were calculated separately.This approximation

introduced an error into the calculations which was reduced by placing the near-?eld/far-?eld interface far enough from the domain of interest.The developed SBFEM was proved to be effectively working via numerical examples and parametric stu-dies.As was mentioned,the SBFEM is being progressed for newer,more advanced problems and will help researchers disentangle from more complexities and complications of conventionally-based solutions in the future.1.1.Basic SSI equation

In the sub-structuring method and whenever this method is applied,since usually the at-hand recorded motions are those of the free-?eld,the ground motion,i.e.motion incorporating the excavation and its changes on the dynamic stiffness of the system,is calculated using Eq.(1).u g b eωTèé

??S g bb eωT à1?S f bb eωT f u f b eωTg e1Twhere u denotes displacement amplitudes when superscripted by

g for the ground (i.e.excavated)system while S stands for dynamic stiffness of the media either for the ground system (g )or the free-?eld (f ).b stands for the basemat while bb addresses the measured subscripted parameter at the base when a displacement equal to unity is imposed to the same level;and ωdenotes frequency.Fig.1illustrates the equation above in sub-structuring method of SSI,where the free-?eld is decomposed into the excavated part and the ground,the latter then bearing the structure through their common nodes [12].

The concept above is applied to extract the motion of the excavated ground (i.e.near-?eld)which will later be used to analyze the SSI problem.

1.2.The scaled-boundary ?nite element method

The structure including a ?nite region of adjacent soil is mod-eled by ?nite elements while the remaining unbounded soil is represented by boundary conditions on its interface with the structure,which leads to a model with a ?nite number of degrees of freedom [61].To do that,the semi-analytical Scaled Boundary Finite Element Method (SBFEM)[18]which is a combination of Finite Element and Boundary Element methods has been imple-mented to calculate the acceleration unit impulse response of the far-?eld.The latter is required to replace the far-?eld ground with forces induced by it when the earthquake motion is being applied to the system.The acceleration unit impulse response may be calculated through other approaches too such as the Boundary Element Method;however,although modeling may be a bit more complicated using SBFEM,resulting matrices will easier to work with since they are banded rather than sparse.Also,a fundamental solution is necessary when using BEM,which is not the case for

f

g

u f b S f bb

u g b S g bb

S e bb

e

Fig.1.Decomposition of free-?eld into ground and excavation.

H.Rahnema et al./Soil Dynamics and Earthquake Engineering 88(2016)152–175

154

SBFEM.The program SIMILAR developed by Wolf and Song [19]was made use of to carry out SBFEM calculations.The system is decomposed into three regions,namely the structure,the near-?eld soil and the far-?eld soil as shown in Fig.2.

Truncating the far-?eld out of the soil-structure system results in its replacement by forces produced on the hypothetical inter-face nodes which shake the near-?eld soil-structure system [18].Similar solution techniques have been exerted to solve SSI pro-blems.Kutanis and Elmas [9]inspected this issue making use of the same procedure [9]while framed structures had already been examined in elastoplastic SSI by J.Noorzaei et al.[13].Both studies yielded good correspondence with the originally linear sub-structure method,except for the discrepancies recorded for the non-linearity assumption of the media in the latter research which was logically accounted for.[20]introduces a practical yet approximate method for modeling the foundation underlying soil as an elastic media.This method is an improved method of sim-pli ?ed wave propagation theories application in the solution of SSI problems.A similar approach to [20]can be found in [24]which ban be taken for foundations of almost any shape incorporating six modes of vibration of the footing on saturated as well as unsatu-rated soils [24]can be regarded as an update to [35].Although these two works were published some time sooner than [20],they,especially [24],are in rather close resemblance in their outcomes with Cone Model.Genes and Ko?ak [4]published their pioneering work on SSI using coupled FEM and SBFEM and then generalized their work to layered media in 2005[6].Although very precise and reasonable results were fetched,manipulations were necessary in the solution scheme as a result of high computational expenses.Dynamic large-scale SSI systems with SBFEM in layered media were inspected in a parallelized coupled procedure by Genes [5].For all carried out works,the soil-structure system was in need of an input motion.An extensive review of linear and non-linear methods of magni ?cation and deconvolution of seismic motions to the depth and surface of a soil column are presented in details covering the works done so far in [8].Based on what is presented there,the time domain input motion is converted to frequency domain using a Fourier Transform,and then a frequency domain magni ?cation factor called a transfer function is calculated based on the soil material,thickness and damping.Then,introducing this 0multiplier 0,the time domain motion is calculated.Interaction forces are determined by convolution integrals convolving accel-eration unit impulse responses in discretized time and effective input motions [19].

The acceleration unit impulse response matrix is convolved with the adjusted accelerogram to yield the interaction forces,as shown in Eq.(2):

R et Tèé?Z t

M 1et àτT€u eτTèé??

d τe2T

where M 1is the acceleration unit impulse response which is achieved by transforming the dynamic stiffness from frequency domain to time domain [18].R (t )is the vector of interaction forces each line of which corresponding to a time step of earthquake pulses.t is the total time of the earthquake motion and τ

represents each time step,while €u

eτTdenotes the accelerogram components at each time step.Since the far-?eld is regarded as linear,Eq.(1)can be easily applied to the acceleration time history corresponding to this region.It should be noted that the accel-erogram baseline needs to be corrected so that the velocity (as well as displacement for far-?eld)reaches zero at the end of the earthquake event.Also,pulses smaller than 0.05g,(i.e.noises)should be ?ltered out of the record.

The dynamic stiffness matrix is de ?ned as in Eqs.(3)and (4)for bounded and unbounded media respectively [17,18].S ωj àá

?K àω2j m ti ωj C e3TS 1ωeT?i ωC 1tK 1tA l =i ω

e4T

where S and S 1is the dynamic stiffness matrix of the bounded

and unbounded domain respectively,(either free-?eld or ground),C 1and K 1are respectively radiation damping and static stiffness of the unbounded domain while K and C stand for static stiffness and damping matrices respectively,ωj is the frequency step for the analysis in the frequency domain,and A l is a constant.

The scaled boundary ?nite-element equation in unit impulse response follows from the inverse Fourier transformation of the corresponding relationship in dynamic stiffness.To be able to perform this transformation,[S 1(ω)]is decomposed into the sin-gular part,i.e.the value for ω-1,and the remaining regular part [S r 1(ω)],in which [S r 1(ω)]is in fact A l =i ω.The inverse Fourier transform yields:

S 1t eT?C 1_δet TtK 1δet TtS r

1t eTe5T

where δis the Dirac Delta function.Eq.(4)holds for the accel-eration unit impulse at each individual time step,i.e.dynamic

stiffness in time domain:

u j t eT?u ωj àá

e i ωj t e6TTo perform the transformation from frequency domain to time domain and vice versa,Fourier and inverse Fourier Transforma-tions are used (Eq.(7)and Eq.(8)):

P eωT?Z t1

à1

P et Te ài ωt dt e7T

P t eT?1=2π

Z

t1à1

P eωTe i ωt d ω

e8T

in which P may represent e.g.a time-varying force.On the other hand,substituting Eq.(5)in the interaction force –displacement relationship yields:

R t eT?C 1_u et TtK 1u et TtZ t

S 1r et àτTu eτTd τ

with the ?rst two terms on the right-hand side representing the

instantaneous response and the third term the lingering response.Alternatively,the response matrix to a unit impulse of accelera-tions can be calculated in the frequency domain:M 1ωeT?

1ei ωT

2

S 1eωT

e9T

After determining the interaction forces which are to be divi-ded between interface nodes,they are inserted in total equations of motion in interaction sense as in Eq.(10):M ss M sb M bs M bb "#€u t s t eT€u t b t eT()tK ss K sb

K bs K bb "#u t s t eTu t b t eT()?0f g àR t eTèé()

e10Twhere M ss and K ss are domain mass matrices and static stiffness

matrices respectively.Adding damping to the environment and thus to the formulation,along with embedding the SBFEM

Fictitious boundary

Finite region

Near-field

Fig.2.Decomposition of the soil-structure system into three sub-regions.

H.Rahnema et al./Soil Dynamics and Earthquake Engineering 88(2016)152–175155

formulation inside the total equation of motion will result in Eq.(11):

M ss M sb

M bs M bb "#€u t s t eT€u t

b t eT()t0? 0? 0? C 1? "

#_u t s t eT_u t b t eT

()tK ss K sb

K bs K bb "#u t s t eTu t b t eT

()t0f g R t 0S 1

r t àτeT??u t b τeTèéd τ()

?0f g

C 1? _u g b t eTèétK 1? u g b t eTèétR t 0S 1r t àτeT??u t b τeTèéd τ()

e11TIn the equation above [S 1r ]represents the dynamic stiffness

matrix of the unbounded media in time domain,[K ij ]is the static stiffness matrix of the elements of the bounded near-?eld,unbounded far-?eld and elements on the near-and far-?eld interface.τis each time step at which earthquake pulses are recorded.

Basics of SBFEM made use of in this study were introduced.It should be noted that SBFEM has been further developed in the last 10years.Two-phase fully saturated soils in the two dimensional space have been modeled using SBFEM taking into account Biot's coupled consolidation [7].The method has been improved to solve non-homogeneous anisotropic heat conduction problems [69].3D crack interaction may be analyzed now with SBFEM [70].Modeling wave propagation in semi-in ?nite domain is possible to be mod-eled ef ?ciently with SBFEM,as can be found in [71]in which complex inclined soil ?eld are explained how to be modeled in order to account for SSI.This semi-analytical method is also now being used to solve probabilistic fracture mechanics problems stochastically [72].Singularity at cracks and notches in arbitrarily laminated composites [73]and modeling of composite beams [74]have been recently done with SBFEM.Anisotropic soil is and apparently will be in more and more advanced schemes numeri-cally modeled under wave propagation using this method [75].Displacement unit impulse response using SBFEM is being applied (e.g.[74],[75])to address such problems.

A thorough review of boundary element methods can be found in [26]which represents common approaches of numerical methods for modeling the far-?eld in typical SSI problems.Basics and formulations of the Boundary Element Method (BEM),SBFEM,etc.are explained in this book which can be regarded as a useful collection of effective approaches of the problem of interest.Also,[28]is a good reference that discusses wave propagation analysis in in ?nite domains through SBFEM.1.3.UCSD 2soil

Quite a few models to anticipate soil and site response to cyclic loads have been lately proposed.A bounding surface hypo-plasticity model for sand incorporated into two-dimensional ?nite difference analysis was suggested by Wang et al.[46]to perform non-linear effective-stress analysis of soil structures.The solution scheme seems quite ef ?cient except for when the soil interacts with a structure in its vicinity,for which the scheme was not inspected.Lopez-Caballero and Farahmand-Razavi [29]looked into the problem of seismic SSI numerically when the system was under the effects of soil liquefaction.Soil nonlinearity as a result of the named phenomenon was imposed to the foundation of the structure,capturing bene ?cial as well as unfavorable effects of non-linear SSI on structural responses and settlements.The input motion of the earthquake included only the horizontal excitation and the super-structure was modeled as a single story-single bay reinforced concrete frame.The analyzes were carried out with

two-dimensional ?nite elements [49].Dupros et al.[47]used the Finite Element Method to inspect seismic wave propagation in three-dimensional non-linear inelastic geological media.The fea-sibility of large-scale modeling based on an implicit numerical scheme and a non-linear constitutive model is demonstrated in their work.To run their model calculations,as a result of time consumed to account for nonlinearities,parallel programing was made use of [50],which shows complications of numerical mod-eling of inelastic systems,especially when three-dimensional problems are of interest.Vasileios et al.[48]proposed a one-dimensional constitutive model developed for the non-linear ground response analysis of layered soil deposits involving small number of parameters needed.The model is very ef ?cient in reproducing almost any type of experimentally observed hys-teretic soil behavior.To account for inelasticity and liquefaction which is its most critical sense in soils,Zerfa and Loret [57]modeled earth structures in a coupled elastic-plastic scheme to account for their responses to dynamic loads.Prévost 0s multi-surface constitutive model was used to predict effective stresses and wave re ?ections towards the structure were avoided using a viscous boundary.Loose and dense sands were assumed under two earth dams all subject to an acceleration time history.Lique-faction was observed to propagate in the system and its repro-duction indicated the robustness of the constitutive model and ?nite element simulations [59].A study of cyclic mobility in saturated to dense cohesionless soils resulting in an ef ?cient plasticity model in presented in [3].Elgamal et https://www.360docs.net/doc/b37954914.html,ed newly developing ?ow and hardening rules to extend this study of theirs and introduce their famous UCSD model into OpenSees.A rather comprehensive account of other constitutive models common till 1996can be found in [32],while common and advanced-for-the-time techniques of accounting for the problem of SSI in years around those of [32]are presented in [36].It may be good to note that,although the knowledge obviously existed then,in works presented in [36]hardly could such constitutive models in their grown forms be found to account for the problem of SSI,which may be associated with the advancement of computer techniques and facilities to incorporate the ?nite element method with con-stitutive models to solve a problem of totally non-linear medium.In this study the UCSD soil model has been implemented for modeling the free-?eld pressure dependent sandy soil with ?nite elements.Desirable precision,pressure dependency and consistency with the equilibrium differential equation of the media are of char-acteristics of this model rendering it appropriate for this problem.Since SSI analyzes may be time consuming at times,and also as a result of indirect need of soil behavior in the modeling,a model which fetches the system with relatively fast calculation procedures is better to be implemented.Fig.3illustrates shear stress –strain behavior of this multi-yield surface constitutive model.

where G r ,B r and γmax are respectively low-strain elastic refer-ence shear modulus,reference bulk modulus and the maximum

shear strain at which the maximum shear strength is reached.p 0

r is the reference mean effective con ?ning pressure.Reference para-meters are de ?ned in such models to account for changes of mechanical properties with reference to the highest elastic deformation after which non-elastic strains begin to appear.All parameters indexed r are measured and utilized at same strains in

each step.σ10,σ20and σ30

are effective principal https://www.360docs.net/doc/b37954914.html,ing Eq.(12),octahedral shear strains may be achieved:

γ?2=3ξxx àξyy

2

tξyy àξzz 2tξxx àξzz àá2t6ξ2xy t6ξ2yz t6ξ2

xz

!1=2e12T

ξij are strain components parallel to j and normal to the i axis.

The three coordinate axes x ,y and z are considered based on which strains are assumed to be considered.τf is the maximum

2

University of California San Diego.

H.Rahnema et al./Soil Dynamics and Earthquake Engineering 88(2016)152–175

156

octahedral shear strength which is related to the effective con-?ning pressure by the internal friction angle of the soil as shown in Eq.(13):

τf ?2???

2p sin φ3àsin φp 0

e13T

Eq.13represents how the shear strength of the soil is mobi-lized in practice,where φis the mobilized internal friction angle of the soil.The octahedral stress is calculated by:

τ?1=3?eσxx àσyy T2teσyy àσzz T2teσxx àσzz T2t6σ2xy t6σ2yz t6σ2xz

1=2

e14T

σij are stress components parallel to j and normal to the i axis.A

hyperbolic relation holds for the backbone curve of non-linear stress –strain response in constant con ?ning stress condition:

τ?

G γ1tγ=γr p 0r =p 0

àád

e15T

d being a curv

e ?tting constant which is also used to de ?ne var-iations o

f G and B as a function of the current value of effective con ?nin

g pressure:

G ?G r p 0=p 0r àád

e16:a TB ?B r p 0=p 0r

àád e16:b T

To keep the constitutive model calibrated,the equations of γr (subscript r stands for 0reference 0in all denotations)must satisfy Eq.(17)in p 0r

τf r ?2???

2p sin φ3àsin φp 0r ?G r γMAX

1tγMAX =γr

e17T

Finally,Eq.(18)de ?nes the internal friction angle of the sandy soil:

sin φ?3???

3p eσm =p 0r T6t????3p eσm =p 0r T

e18Tσm being the product of the last modulus and strain pair in the

modulus reduction curve [21].Table 1represents values used in calculations carried out by this model.

About the permeability of the soil under the seismic load of earthquake,it is enough for the scope of this paper to give a clue that an equivalent permeability coef ?cient 3.41times the static permeability coef ?cient of the sand has been used in the calcula-tions.It should be noted that the mere subject of determination of soil permeability changes while under earthquake load is and has been the theme of a wide range of studies.More details on this issue along with the logic behind our adopted equivalent coef ?-cient can be found in [33].1.4.UCD 3soil

The other constitutive relation applied to inspect responses associated with linear and non-linear sub-structure method was the UCD material,which has been originally suggested for the sake of SSI sub-structuring solution scheme.Replacing the soil with springs having special characteristics spread under and around the foundation,the footing may be given lateral and/or vertical degrees of freedom of movement from which rocking can be inferred.The original aim has been to compare the rigorous method with sub-structuring,as well as exerting precision to the latter via considering possible non-linearity.

In this scheme,the p –y non-linear behavior includes the elastic p –y e portion as well as the plastic p –y p one.Radiation damping for the far-?eld is de ?ned as linearly behaving dampers.The plastic portion of the response is de ?ned by:

p ?p ult àep ult àp 0Tcy 50cy 50p p 0 "#n

e19Twhere p ult is the material ultimate strength in the direction of

loading,p 0is the value of p in the beginning of the plastic loading cycle,c is the tangential modulus control constant in the beginning of plastic yielding and n is the sensitivity factor to control the sharpness of the p –y p curve.y 50signi ?es displacement at which 50%of p ult is mobilized in monotonic loading in the y -direction (https://www.360docs.net/doc/b37954914.html,teral),z 0p the displacement in the z -direction (i.e.vertical)at the start of the current plastic loading cycle,and z p is the total plastic response in the z -direction.To use this model in the present

σ1'

σ2

'

σ3'

p 0'p '

32

σ2'

32

σ3'

γmax

G r

Fig.3.(a)USCD soil model in principal effective-stress space and (b)in deviatoric plane;(c)octahedral shear stress vs.shear strain [21].

Table 1

Suggested values for soil parameters in the UCSD model [21].

Loose sand (Dr ?15–35%)

Medium-dense sand (Dr ?65–85%)Dense sand (Dr ?85–100%)

Density (ton/m 3)

1.7

2.0

2.1

Reference shear modulus at p r '?80(kPa)

5.5?104 1.0?105 1.3?105Reference bulk modulus at p r '?80(kPa)

1.5?105 3.0?105 3.9?105Friction angle (degrees)293740Peak shear strain at p r '?80(kPa)

0.10.10.1Reference pressure (p r ')808080Pressure dependence coef ?cient

0.50.50.5Phase transformation angle (degrees)292727Porosity (e)

0.85

0.55

0.45

3

University of California Davis.

H.Rahnema et al./Soil Dynamics and Earthquake Engineering 88(2016)152–175157

solution scheme,conventional fundamentals of shallow founda-tions bearing capacity have been used as in Eq.(20)[15]:q ult ?cN c F cs F cd F ci tγD f N q F qs F qd F qi t0:5γBN γF γs F γd F γi

e20T

for whose bearing capacity factors,namely for shape,depth and inclination of the foundation,are calculated via Meyerhof 0s method [11].In the above formulation;q ult is the ultimate bearing capacity per unit area of the footing;c is the cohesion of the soil underlying the foundation (in case cohesive);B is dimension of the foundation and N c ,N q and N r are bearing capacity factors.F cs ,F cd and F ci are supposed shape,F qs ,F qd and F qi depth and F rs ,F rd and F ri ,inclination factors in each section of the equation.The lateral bearing capacity is calculated as the total resisting force on the embedded side of the footing,as in Eq.(21):p ult ?0:5γK p D 2f

e21T

in which p ult is the soil passive pressure per unit area and K p is the passive lateral pressure coef ?cient based on Coulomb [27].The footing 0s resistance against sliding will be calculated by:t ult ?W g tan δtA b c

e22T

where t ult is the frictional strength per unit area of the foundation and W g is the weight of the structure on the considered footing.δstands for the friction angle between soil and foundation usually assumed as one-third to two-third of the internal friction angle of the soil.A b is the footing area in contact with the underlying soil and hence the last term of the equation is the contribution of the coherence between the footing and the soil to the frictional strength,in case the soil is cohesive.Fig.4depicts the behavior pattern assumed for the footing.

Making use of this material to account for non-linear SSI,Ray-chowdhury [44]inspected the seismic response of low-rise steel moment-resisting frame buildings.The mobilization of the ultimate capacity and the associated energy dissipation which is especially the case in intense earthquake events cause the non-linear origin of SSI and was captured effectively.Raychowdhury showed that this phe-nomenon may reduce the force and ductility demands of the structure,provided that potential consequences such as excessive settlement are talked carefully.Mechanisms of seismically induced settlement of buildings with shallow foundations on lique ?able soils were studied by Dashti et al.[23]and Bray and Dashti [45]and can be served as a complementary work.Post-liquefaction reconsolidation settlement may be greatly detrimental to the structural system.Dashti et al.per-formed a series of centrifuge experiments involving buildings situated atop a layered soil deposit to identify the mechanisms involved in liquefaction-induced building settlement.They resulted that besides shaking intensity,soil 0s relative density and thickness and the building 0s weight and width,the earthquake strong shaking is of utmost importance.They ?nally stated that the development of high excess pore pressures,localized drainage in response to the high transient hydraulic gradients,and earthquake-induced ratcheting of the buildings into the softened soil are important effects that should be captured in design procedures estimating liquefaction-induced build-ing settlement [47].

Quite a number of other constitutive laws and solution schemes have been proposed to account for non-linear soil/site response due to earthquake excitation (refer to [51,52,54,55]for examples).Each of these researches,although precious in its time and useful for so many problems,had its own limitations in modeling soil non-linear response in direct dynamic interaction with the structure.

Irregular bounded soil

Fig.5.Near-?eld soil along with the RC frame placed upon:(a)Considered soil-structure system;(b)Model created in OpenSees.

Displacement

Force

k in

Fig.4.Behavior of footing on UCD soil,(a)horizontal and vertical soil-replacing springs;(b)force –displacement behavior of soil-replacing springs [42].

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2.System description

2.1.Finite element model

A6-story reinforced concrete frame whose foundation upper surface is2m below the surface of the ground is modeled on the near-?eld sandy soil to model the problem rigorously,while for the sub-structure scheme one non-linear Winkler foundation is considered as the footing under each column under which springs with force-varying non-linear constants are assumed.Fig.5 depicts a scheme of the problem for the direct solution method and Fig.6illustrates one for the sub-structure method.How to bind the interface nodes(i.e.farthest nodes of the near-?eld onto the nodes of the boundary)has been explained in summary in[30].

In Fig.5(a)the hatched boundaries indicate high-compaction regions of the soil either as a result of the overlying structure or the weight of overlying soil layers.It is obvious that the deeper the soil,the higher the density.As Fig.11later depicts,9-node ele-ments are implemented for soil?nite elements which results in a last node placed in the center of the quadrilateral element.Prop-erties of such elements and nodal applications will shortly be explained.The soil behavior is,as was already formulated,so-called irregular,i.e.it does not manifest only linear elastic defor-mations.Noticeable plastic deformations caused by the interaction of the soil and structure which affects structural responses take place only in the near-?eld soil bounded to the soil-structure

Footing No.1

Footing No.2Footing No.3

Footing No.4

Fig.6.(a)Structure placed on soil-replacing springs for sub-structure modeling,and(b)Winkler foundation model[14].

50.0 cm

50.0 cm

8Φ32

Φ10@15cm

Fig.7.Beam and column elements cross section.

Table2

Mechanical properties of concrete for non-linear structural behavior.

Mechanical properties Characteristic

strength(kPa)

Strain in

maximum

strength

Crushing

strength

(kPa)

Strain

before

crushing

Tension

strength

(kPa)

Core

concrete

24?1030.0024 5.6?1030.0150

Cover concrete 21?1030.0025?1030.0050

Table3

Mechanical properties of steel for non-linear structural behavior.

Mechanical

properties

Yield stress

(kPa)

Initial modulus of

elasticity(kPa)

Strain hardening

ratio

Reinforcing steel

values

420?1032?1080.01

H.Rahnema et al./Soil Dynamics and Earthquake Engineering88(2016)152–175159

system.This is why the near-?eld soil is discretised to be analyzed through the Finite Element Method together with the structure.The far-?eld,which is truncated out of the model,is replaced by the Scaled Boundary Finite Element Method analysis outcomes.Fig.5(b)illustrates the soil elements by the nodes created in OpenSees.To create the whole soil-structure model in OpenSees,commands and procedures explained in [10]were used.As is observed,a ?ner mesh is applied for soil elements closer to the structure.The depth with recognizable plastic behavior,i.e.the near-?eld depth,has been supposed to be equal to 60m in this study.

Stiffness properties of the springs replacing the soil under the structure (Fig.6)depend on Terzaghi 0s theory of ultimate bearing capacity exerted through the UCD material.

The cross section of beam and column elements of the struc-ture,which may behave non-linearly when exposed to (seismic)loads and hence modeled with ?bers along each element,is shown in Fig.7.It should be noted that in order for comparison with results of [2],the beams and columns are modeled as in Fig.7to be of equal axial and ?exural stiffness to those in [2].These stiffness coef ?cients denoted below Fig.7correspond to the case in which the structure manifests linear elastic behavior.An introductory solution to the same problem focusing only on some simple aspects has been presented in [22].

It is important to know that contrary to conventional methods which assume stiffness reduction factors resulting from cracks in reinforced concrete sections,no such factors will be needed when ?ber sections are used since stiffness reduction is automatically accounted for in each step of loading and hence the stiffness will be updated at the beginning of each next step.Table 2presents mechanical properties of the concrete used to model the RC frame.

Mechanical properties considered to model steel rebar of structural element sections,such that elements section properties correspond with amounts in Fig.7,are listed in Table 3.

The near-?eld soil domain discretised into ?nite elements is supposed to consist of medium-density sand (Depth o 23m,except for the soil right under the foundation)and dense sand (Depth 423m and a depth of 4m under the foundation).Fig.5and Table 4show the discretization and mechanical properties of the soil,respectively.

Table 5represents the geometry of the RC frame whose structural responses are of favor for the results of this study.

As mentioned earlier,soil elements are once modeled based on the UCSD material too.Each element of soil consists of nine nodes,each three placed on one side at equal distances the ninth of which resting in the center of the quadrilateral element.Fig.8presents a typical quadrilateral element and its node numbering order.

The nodes on the corners of these 9-node quadrilateral ele-ments record the ?uid pressure as well as displacements,while the other ?ve nodes record only displacements.One of the com-mon features of methods such as BEM an SBFEM is that they can be coupled with ?nite elements to create a complete model of the media of the problem.Each of these methods has its own tech-niques of coupling which should be accounted for when it is applied to the problem.As for SBFEM,a few coupling procedures are available (e.g.[1]and [25]).A thorough study of the Finite Element Method (FEM)and analysis procedures with plane ele-ments,etc.is presented in the very valuable book by Reddy [16].The UCSD soil model which is used o discretize the near-?eld is used to assume the near-?eld as non-linear and the far-?eld as almost linear.This is while the UCD soil model,with which no discretization of the soil into ?nite elements is assumed,is used to discretize the whole soil-structure system into two sub-structures.In this regard,the underlying soil,which consists of the whole soil medium (including the near-?eld as well as the far-?eld as one continuous medium)behaves nonlinearly,which means the total soil is almost linear under small deformations and non-linear under large deformations.As a result,no 0far-?eld 0exists by de ?-nition for this soil model.The SBFEM is readily usable for linear systems,and how to tune it for non-linear media (e.g.via Homo-topy (HAM),etc.)is far beyond the scope of this study.

Non-linear soil has been restricted to the near-?eld in this study.Attention should be given to the fact that it is true that considering the far-?eld soil to behave linearly and elastically may cause some possible inaccuracies in the responses,yet since SBFEM is basically prescribed for linear-elastic media and is only the subject of running researches at the present and those very recently performed (e.g.[68])to be generalized to non-linear problems,it is inevitable to assume such behavior for the far-?eld.As a result,the mentioned inaccuracies need to be mini-mized to yield acceptable results considering this assumption.

Table 4

Mechanical properties of near-?eld non-linear sandy soil.Soil properties Mass den-sity (ton/m3)Reference shear modulus (kPa)

Reference bulk modulus (kPa)

Friction angle

Phase transforma-tion angle

Peak shear strain Reference

pressure (kPa)

Pressure depen-dence coef ?cient (d)Porosity (e)

D o 23m 2.0 1.0?105 3.0?10537270.1800.50.7D 423m

2.1

1.3?105 3.9?105

40270.1

800.5

0.45

Table 5

Geometric properties of the RC frame.Number of stories Typical story

height (m)Basement story height (m)Number of intervals Length of intervals

(m)Foundation rigidity measure Load per unit length of beam elements (kNm à1)6

3

2

3

4

Rigid

50

Fig.8.Typical quadrilateral soil element [21].

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160

Bearing in mind that most of the interactional effects caused by the structure to the soil are from inertial forces and these forces are added to kinematic forces coming from near-and far-?elds into the system,(interaction induced)displacements and stresses during the earthquake event shall be much more considerable in the near-?eld compared to those in the far-?eld and it is then substantially more vital to assume the near-?eld to have the potential of non-linear,inelastic behavior than the far-?eld.Sup-posing the non-linear near-?eld to be large enough,the linearity assumption of the far-?eld introduces little error in the ?nal results of the analyzes.The reason is that one major cause of the non-linear behavior of the soil is the overload (i.e.the super-structure)the effect of which diminishes with depth as a result of the in ?nity of the soil half-space.Radiation damping through in ?nite soil is another effective factor that lessens the stresses caused by dynamic loads.On the other hand,since in this study only one structure is investigated (note that the problem of structure-soil-structure involving inter-structural effects through soil may also be addressed),non-linear behavior of the soil in a suf ?cient distance from the structure will be responsive.In addi-tion,a short reasoning can be found in [67]implying that the most probably non-linearly behaving depth of soil under a heavy structure is the depth containing the amount of soil with weight equal to the total weight of the structure,right under its footings

downwards.

Fig.10.Loma-Prieta effective input motions for sandy ?

eld.

Fig.9.Horizontal and vertical free-?eld acceleration time histories of the Loma-Prieta 1989earthquake,scaled to g [45].

H.Rahnema et al./Soil Dynamics and Earthquake Engineering 88(2016)152–175161

2.2.Motions

Horizontal and vertical free-?eld motions of Loma-Prieta earthquake (1989)records [45]shown in Fig.9are considered for the analyzes.

3.Effective input motions

Using the Fourier Transform,the Fourier amplitude spectrum has been obtained and applying the dynamic stiffness multiplier gained from SBFEM 4,the Fourier amplitude spectrum of the effective input motion to the interface yields.At last,through an inverse Fourier Transform,the resulting motion follows.Results are presented primarily for acceleration time histories.Fig.10represents the procedure for horizontal and vertical motions.

The multiplier,which is constant for both horizontal and ver-tical components of the motion,is apparently more in ?uential on

the strong motion component of the frequency content of the motion.It reduces higher frequencies much more than lower fre-quencies which are of greater amplitudes.So,although not totally ?ltered out,the high frequency component of the motion is sub-ject to more reduction altering from free-?eld to ground.

4.Results and discussion 4.1.Interaction forces

After determining the effective input motions to the interface of the media,the path will be open to carry out the calculation of interaction forces using convolution integrals (Eq.(2)).The most important factor needed to determine these forces is the accel-eration unit impulse responses.The virtual impulse time in which the peak of the impulse falls is supposed 1s.The progression with discretized time is presented in Fig.11for incompressible and compressible conditions of the soil that will later correspond to undrained and drained near-?eld conditions,respectively [52]

.

Fig.11.Acceleration unit impulse responses of the ground obtained from SBFEM for (a)incompressible sandy ?eld;and (b)compressible sandy ?

eld.

Fig.12.Interaction forces for incompressible and compressible sandy soil,(a)Horizontal motion;and (b)vertical motion,

4

The Scaled Boundary Finite Element Method.

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162

Fig.11reveals the difference between response origins of sites with and without drainage contrivances.The generalized structure refers to the structure plus the near-?eld soil,both of which may act non-linearly when exposed to cyclic or monotonic loads of any kind.It is observed that to a unit impulse,the incompressible site reveals its maximum reaction instantaneously unlike for the incompressible condition.Fig.12illustrates the interaction forces to the generalized structure-unbounded soil interface for the two mentioned materials,namely,compressible and incompressible sands respectively corresponding to drained and undrained conditions.

It can be conferred from the ?gures that the greater interaction force relates to the incompressible condition,which corresponds to liquefaction assumptions and experiments.In fact,liquefaction

may usually occur when the soil faces hardships draining the pore liquid (i.e.undrained condition)resulting in rapid increase of pore water pressure,which is the incompressible case of this study.On the other hand,the forces are smoother for the compressible case and the frequency content is much milder.The force time-histories show a very short time delay with the free-?eld motion which is sensible from a practical point of view.4.2.Structural responses

4.2.1.Displacements

Responses of the structure with ?xed-base as well as when subject to elastic and elastoplastic seismic soil-structure interac-tion have been recorded and are of interest.For the sake of com-parison,the cases of conventional elastic-plastic soil behavior have also been included in the results.Fig.13depicts displacement amplitudes in the height of the frame for the state in which no

SSI

Fig.13.Frame displacement amplitude,no SSI

assumed.

Fig.14.Frame displacement amplitudes,different sand states under the

foundation.

Fig.15.Frame displacement amplitudes when loose sand exists under the

foundation.

Fig.16.Frame displacement amplitudes when medium-dense sand exists under the foundation.

H.Rahnema et al./Soil Dynamics and Earthquake Engineering 88(2016)152–175163

is assumed.Fig.14,on the other hand,illustrates a comparison between structural response amplitudes for different states of sandy soil interacting with the structure.

Apparently,based on conventional methods no difference is observed whether the structure be placed in either sand or clay,as the curves cover one another for the whole height of the frame.As is observed from the graphs for the SSI including state,the denser the soil gets the less the displacements will be.Drainage of the soil results in bigger displacements for heights less than 9m and smaller displacements for the higher compared to the undrained case.When the soil is undrained,the movement of the stories above the ground will not necessarily be more than those below the surface including the basemat.No discrepancies are observed in displacement amplitudes from one state to another of the soil for conventional behavior assumptions,while,for the UCSD material which implements drainage of the soil,these dis-crepancies are observed.It is worth mentioning that for the ?xed-base structure,structural response amplitudes are not affected any differently by the lateral pressure from either sand or clay,and corresponding curves manifest excellent matching (Fig.14).The graphs show that with elements gaining the potential of non-linear behavior,response amplitudes will tend to get bigger

and

Fig.17.Frame displacement amplitudes when dense sand exists under the

foundation.

Fig.18.Stories displacement time-histories for inelastic structure,(a)drainage allowed;and (b)drainage not allowed.

H.Rahnema et al./Soil Dynamics and Earthquake Engineering 88(2016)152–175

164

this is apparently accounted for by non-resilient cracks appearing in the concrete after loading,making the behavior inelastic as a result of geometric as well as material non-linearity caused by changes in members moments of inertia along with behavior shifts in burden-attracting steel rebar.The problem of SSI is also of interest for different states of foundation-underlying sand inspected with one single constitutive law,namely UCD.The sand state is divided into three major categories:Loose,medium and dense.Fig.15illustrates a brief comparison between response amplitudes for different states

of

Fig.19.Stories displacement time-histories for (a)inelastic structure with no SSI;and (b)elastic structure in sandy

soil.

Fig.20.Stories displacement time-histories for (a)elastic structure on elastic foundation whose stiffness is derived from loose sand characteristics;(b)elastic structure on loose sand;and (c)inelastic structure on loose sand.

H.Rahnema et al./Soil Dynamics and Earthquake Engineering 88(2016)152–175165

Fig.21.Stories displacement time-histories for (a)elastic structure on elastic foundation whose stiffness is derived from medium-dense sand characteristics;(b)elastic structure on medium-dense sand;(c)inelastic structure on medium-dense

sand.

Fig.22.Stories displacement time-histories for (a)elastic structure on elastic foundation whose stiffness is derived from dense sand characteristics;(b)elastic structure on dense sand;and (c)inelastic structure on dense sand.

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166

soil-structure system when the sandy soil is assumed to be loose. Based on the results shown here,cracks propagating transversely through elements cross sections when the concrete frame is not supposed to react merely elastically leads naturally to larger dis-placement amplitudes as a result of gradual stiffness decrease through the progression of the seismic motion.

Amongst the curves in Fig.15,the ones representing responses of linear structure on elastic base,linear structure on non-linear foundation and non-linear structure on non-linear foundation are most informative.It can be inferred from these curves that for the linear elastic frame,which is mostly the case subject to medium intensity earthquakes when designing of commonplace structures has been performed based on current codes,on a non-linear inelastic foundation which accounts for the highly probable non-linearity of soils,structural response amplitudes may vary between those of totally elastic and totally inelastic systems. Compared to the linear?xed-base structure,on the other hand,a shift of about2cm is observed for all stories other than the basemat which seems to tend to remain constant in the height of the structure.On the basis of this comparison,the non-linear ?xed-base frame,which does not seem to be logical though a common assumption,underestimates responses for heights less than4m while shows overestimation for heights more than this.It should not be neglected that although the green curve is like an average-push of the?ve different possible structural responses, conservatively the closest curve to reality is that of the totally non-linear system and hence all other simulations more or less yield underestimation when compared to this curve.

For the medium-dense sand,on the other hand,discrepancies will be less and non-linearity of the foundation is less prominent. Fig.16depicts the displacement amplitudes for the?ve possible behaviors.

Apparently,as the height increases,not only do displacements get larger but also the gap between amplitudes of linear frames and non-linear ones grows bigger and for the totally non-linear systems gets more critical.Fig.17represents the same issue for the case of dense sand.It is observed that the denser the soil,the less SSI perceived,and SSI is least comprehensible for the densest state of sand existing under and around the basemat.

It is worthwhile mentioning that although the difference of response amplitudes for the linear?xed-base system and the totally non-linear one is something between4and8cm in the top story of the frame for all sand densities and compaction states,this amount may be controlling as it may govern the target displacement of the structure when performance-based design of the building is of interest.

4.2.2.Stories displacement time-histories

Fig.18depicts selected stories displacement time-histories for inelastic structures on drained and undrained sands along with responses of a point with coordinates corresponding to those of point B(see Fig.5)on the free-?eld,i.e.when no structure is placed on the assumed site.

It is important to note that overall displacement patterns are sig-ni?cantly different for the two drainage conditions of sand.For the drained sand a milder frequency content than for the undrained is observed in the response time history of the free-?eld as well as the coupled site.This con?rms the deduction that the drainage of the site soil leads to the?ltering of some frequencies of the motion.In addition,lateral displacements seem to be accumulative for the?rst story of the frame placed on undrained sand,while the same issue is right for vertical displacements of drained coupled site and free-?eld which manifests potential swelling in the underlying soil of the soil-structure system.The overall pattern of responses of the B point is not very different for the free-?eld compared to the soil-structure system, although not exactly similar,when the soil is provided with enough drainage;whereas for the undrained case responses vary signi?cantly after the establishment of the structure on the ground.For the sake of comparison,Fig.19illustrates responses of?xed-base inelastically behaving and embedded elastic frames,respectively.

Frequency content as well as maximum responses of above shown time-histories should be noticed.When members of the frame have the possibility of inelastic behavior,peak displacements will tend to be greater than when all responses are taking place in the elastic range.Non-linear inelastic behavior,on the other hand,leads to the ?ltering of some high frequency responses.

Fig.20(a)illustrates soil-structure interactional responses for different elasticity states.In this?gure,elastic properties of loose UCD sand are adopted to model the underlying soil as elastically behaving.In Fig.20(b),the structure is still elastic while the soil under the foundation has all characteristics of loose sand modeled by the UCD material.Fig.20(c)shows the system when all struc-tural elements too have the potential of inelastic behavior.

With the underlying soil being in the loose state,displace-ments,especially vertical ones in the bottom story,should be monitored in both elastic and inelastic cases.When the soil is assumed to be elastic,the overall?gure of response time-histories will be almost similar for all stories.The ratio of vertical to hor-izontal motions grows from top to bottom of the frames of

all Fig.23.Time history of base shears for different conditions of sand.

H.Rahnema et al./Soil Dynamics and Earthquake Engineering88(2016)152–175167

types.When the soil is assumed to behave inelastically,vertical displacements of the bottom story are almost ten times those on elastic foundation.As inelasticity increases in the subsystems,horizontal and vertical responses grow.Medium-dense sand effects on structural responses are depicted in Fig.21.

As the sand gets denser,displacements decrease.For the medium-dense sand with properties shown in Table 1instead of loose sand under the footings,peak displacements of the bottom story will decrease at least to half of those with the previous state for the case when the soil is assumed elastic.The ratio of vertical to horizontal displacements is still ascending from top to bottom of the frame.When the structure gets inelastic,maximum responses increase by about 20percent.

Fig.22shows structural response time-histories for when dense sand exists under the footings.

Again,with transmission of the underlying soil density to dense,displacements get smaller and the inelastic behavior gets closer to the elastic.In all cases,with possible inelasticity of the frame,displacements increase as a result of stiffness reduction during the earthquake motion.

4.2.3.Support reactions and base shears

The state and properties of the underlying soil not only affects the displacements of the super-structure,but will also result in different force distribution patterns from the ground to the structure.Vertical reactions should be inspected individually for each footing while horizontal reactions the summation of which is the base shear should be inspected after summing up the four reactions of the four footings.Fig.23depicts variations of base shear time history with changes in soil conditions.

The ?gure above states that the base shear time history trend for the undrained soil is different from that of other cases.Based on this ?gure,although for the UCSD soil the maximum base shear is less than that of other cases,it lasts longer and its mean varies slowly after it meets the maximum until the motion ends.The sustenance of this long lasting relatively high base shear may even cause severer damages to the structure especially for longer lasting quakes,although the peak is not as high.As is clear from the ?g-ure,the denser the soil gets the more the base shears will be and this manifests the favorable effects of SSI on structural forces,letting aside of possible changes in destruction patterns.Figs.24and 25show support reaction time-histories for each of the footings for drained and undrained sands modeled with UCSD.For both cases,the structure is assumed to potentially behave inelastically.Differences between these two series of support reaction time-histories state the difference of responses of a struc-ture on two differently behaving types of sand,namely drained and undrained.Note that all other properties of the soil-structure sys-tems including the geometry and mechanical properties of the media (both soil and structure)have been exactly the same and the only variation has been in drainage possibility of the sand.For one thing,the ?ltering of higher frequencies of the motion effects is noticeable when the soil is drained since as is observed the curves are all much smoother in the ?rst ?gure;for another,the alteration of the overall shape of time-varying reactions is of interest.It should be noted that for the earthquake record used in this study,PGAs of the horizontal and vertical motions were close and the frequency content of the vertical motion was much higher than that of the horizontal.The great vertical PGA along with the weight of the structure resulted in huge vertical support reaction time-histories,especially in the strong portion of the motion.

Discrepancies of reactions for each soil state are mostly noticeable in horizontal time-histories.Vertical shape and amount of reactions are rather similar with close peaks for all four

footings

Fig.24.Support reaction time-histories of inelastic structure on drained medium-dense UCSD sand:(a)Footing No.1;(b)Footing No.2;(c)Footing No.3;and (d)Footing No.4.

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Fig.25.Support reaction time-histories of inelastic structure on undrained medium-dense UCSD sand:(a)Footing No.1;(b)Footing No.2;(c)Footing No.3;and (d)Footing No.

4.

Fig.26.Support reaction time-histories of elastic structure conventionally embedded in sandy soil:(a)Footing No.1;(b)Footing No.2;(c)Footing No.3;(d)Footing No.4.

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Fig.27.Support reaction time-histories of inelastic structure with no SSI:(a)Footing No.1;(b)Footing No.2;(c)Footing No.3;and (d)Footing No.

4.

Fig.28.Support reaction time-histories of elastic structure on loose UCD sand:(a)Footing No.1;(b)Footing No.2;(c)Footing No.3;(d)Footing No.4.

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Fig.29.Support reaction time-histories of inelastic structure on loose UCD sand:(a)Footing No.1;(b)Footing No.2;(c)Footing No.3;(d)Footing No.

4.

Fig.30.Support reaction time-histories of elastic structure on medium UCD sand:(a)Footing No.1;(b)Footing No.2;(c)Footing No.3;and (d)Footing No.4.

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