Non-linear seismic ground response analysis …scientiairanica.sharif.edu/article_4257_35e948b97f49c2125755bb9f6a... · Non-linear seismic ground response analysis considering two-dimensional

Scientia Iranica A (2018) 25(3), 1083{1093

Sharif University of TechnologyScientia Iranica

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

Non-linear seismic ground response analysis consideringtwo-dimensional topographic irregularities

N. Soltani� and M.H. Bagheripour

Department of Civil Engineering, Faculty of Engineering, Shahid Bahonar University of Kerman, Kerman, Iran.

Received 5 June 2016; received in revised form 17 September 2016; accepted 10 December 2016

KEYWORDSTopographicirregularity;FEM;Viscous boundary;Non-linear method.

Abstract. In the occurrence of an earthquake, local site conditions such as soilcharacteristics, dimension of topographic irregularities, seismic bedrock depth, etc. as wellas characteristics of incident wave have important e�ects on seismic ground response. In thisstudy, Finite Element Method (FEM) coupled with viscous boundaries is used to evaluatethe e�ect of empty two-dimensional valleys on ampli�cation or attenuation of seismic waves.Parametric studies are carried out and the e�ects of dimension of the topography, frequencyof the incident wave, and bedrock depth on the seismic ground response are considered usingnon-linear method in a time domain analysis. Results are shown by means of horizontaland vertical ampli�cation ratios in valley span and its surrounding area. It is concludedthat displacement variation on ground surface due to topographical e�ects is a considerablefactor to select a site location or design structures in the valley mount and its surroundingarea.© 2018 Sharif University of Technology. All rights reserved.

1. Introduction

Evaluation of surface ground motion due to seismicexcitation at bedrock is one of the most importantissues in geotechnical earthquake engineering. Incommon seismic events, body waves travel from thesource mostly across a bedrock and �nally end insoil layers. This is while most of the changes inthe characteristics of ground motions occur in thesoil layers [1]. These changes are generally studiedwith focusing and scattering of the phenomenon ofseismic waves. Due to the drastic variation of thenature of seismic waves passing through soil layers, itis very important to incorporate realistic and preciseseismic excitation models into the analysis of structural

*. Corresponding author.E-mail addresses: [email protected] (N. Soltani);[email protected] (M.H. Bagheripour)

doi: 10.24200/sci.2017.4257

seismic response. Hence, vulnerability of structurescan be a function of an important factor known asthe site seismic response. Ground response analysescan properly satisfy the needs for realistic and preciseseismic excitation in analysis of structures or soil-structure interaction.

Site e�ects are generally divided into two cate-gories: e�ects of local deposits and those of topography.In this regard, the e�ect of topographic irregularities onground motions is of great importance. Recent studieshave indicated that topographic irregularities (e.g.,mountain ridges or valley notches) cause signi�cantchanges to strong ground motions during earthquakes.Investigations into many earthquakes in the past havealso indicated the e�ect of surface topographic changeson ground response. Evidences of topographic e�ectsare available in Alaska 1964 [2], Canal Beagle Chile1985 [3,4], Northridge 1994 [5,6], Athens 1999 [7],Umbria-Marche 1997 [8], and many others. Thesigni�cance of local topographic irregularities is shown

1084 N. Soltani and M.H. Bagheripour/Scientia Iranica, Transactions A: Civil Engineering 25 (2018) 1083{1093

by the fact that earthquake causes vast damages tosome certain regions and only slight damages to others.

Generally, numerical studies of the e�ect of topo-graphic irregularities on the ground response are mul-tidimensional analyses, which use various approachessuch as Finite Element Method (FEM) (e.g., [9,10]);Boundary Element Method (BEM) (e.g., [11,12]);Spectral Element Method (SEM) (e.g., [13]); FiniteDi�erence Method (FDM) (e.g., [14]); and hybridmethods such as coupled Finite and In�nite Ele-ment method (FE-IFE) (e.g., [15,16]), and coupledFinite and Boundary Element Method (FEM-BEM)(e.g., [17,18]). Detailed explanation of multidimen-sional analyses and comparison of results may be foundin some references (e.g., [9,19,20]).

Extensive researches have been conducted to op-timize ground seismic response analysis (e.g., [21-29]).For example, Lermo and Ch�avez-Garc��a [30] referredto the limitations of the classic method of spectralanalysis, especially limitations imposed on �eld oper-ation and the process of recording data. They alsoproposed a new method to study site e�ects. Followingwell-known Nakamura method, they used the ratio ofthe spectral amplitude of the horizontal to the verticalcomponent of minor earthquakes. They also comparedtheir results with those of the classic spectral approach.LeBrun et al. [31] carried out an experimental studyon the topographic e�ects of a large hill with a heightof 700 m, width of 3 km, and length of up to 6 kmon seismic analysis. In the course of investigations, 7seismographs were installed on a hill and a total of 58earthquake records were obtained. They studied theground motion with di�erent methods: the ClassicalSpectral Ratios (CSR) and the horizontal to verticalspectral ratios were calculated both based on noise, socalled Nakamura's method (HVNR), and then based onearthquake data, so called Receiver Function technique(RF). The comparison between these two methodsshowed that the H/V method was able to suggest thefundamental frequencies of a hill. Fu [32] investigatede�ect of scattering surface waves by comparison ofdi�erent theories. He also examined these waves bystudy of the propagation of SH waves through two-dimensional models. He compared accuracy of di�erenttheories according to incident wavelength and dimen-sion of model. Bouckovalas and Papadimitriou [14]analyzed the e�ect of topography on seismic waves us-ing the Finite Di�erence Method (FDM). Their studywas based on a site with uniform slope with a visco-elastic soil medium. The study was also conducted onthe e�ects of di�erent parameters on seismic groundmotion, which was based on the vertical propagationof SV waves. Kamalian et al. [17] studied the e�ects oftopography on a medium with heterogeneous materials.They used two-dimensional modeling method basedon FEM-BEM. They also modeled distant boundaries

using con�ning elements. They showed that their pro-posed method needed smaller time step than the BEMscheme. They also discussed the e�ective dimensionof irregularities on seismic ground motion. Gatmiriet al. [18] investigated the e�ect of alluvial valleyson the ampli�cation or attenuation of seismic waves.They also used a coupled model of FEM-BEM in whichnearby �eld was modeled using FEM while the far �eldwas modeled using BEM. They veri�ed the accuracyof their method by numerical study and discussed thatarti�cial waves developed at the truncation points ofthe model would vanish easier if an optimized methodwas used. Asgari and Bagheripour [33] performed anon-linear one-dimensional analysis of ground responseusing Hybrid Frequency-Time Domain (HFTD) ap-proach. They used the advantages of both the timedomain and the frequency domain methods to optimizethe solution procedure. They showed the accuracy oftheir method by di�erent illustrative examples. DiFiore [10] considered the e�ect of incident wave fre-quency and gradient of slopes as a form of topographicirregularities on the ampli�cation or attenuation ofseismic waves using FEM. Investigations showed thatampli�cation of seismic wave increased with increasein the gradient of slope. Also, analysis of 0.5 to 32 Hzincident wave frequencies revealed that the largestampli�cation of seismic waves was seen in frequenciesbetween 4 to 12 Hz. Bazrafshan Moghaddam andBagheripour [34] proposed a new method for non-linearanalysis of ground response using HFTD approach.This method was based on a non-recursive process andused a matrix-notation. Comparison of the results ofthe proposed method and those obtained by SHAKEand NERA software using records collected on thesites of several actual earthquakes showed accuracy ande�ciency of the proposed method. Tripe et al. [35]conducted a study, using time domain approach, intothe contribution of slopes of homogeneous and linear-elastic soils to the ampli�cation or attenuation ofseismic waves using FEM.

One of the important problems in ground responseanalysis is the limitation or inability of numericalmethods to simulate in�nite boundaries and developmathematical models for the passage of seismic wavesas well as reduction in re ected waves from the bound-aries into the models developed for soil layer. Such ade�ciency greatly in uences the results.

It should be noted that one-dimensional groundresponse analysis methods are suitable for horizontalor gently sloping grounds and, perhaps, for soil pro-�les having a set of parallel layers. However, otherproblems such as slopes; non-linear ground surfaces;topographic irregularities; and presence of heavy andsti� structures, buried structures, and tunnels requiretwo- or even three-dimensional analysis [1].

In the present study, non-linear time domain

N. Soltani and M.H. Bagheripour/Scientia Iranica, Transactions A: Civil Engineering 25 (2018) 1083{1093 1085

analysis is carried to investigate the e�ect of two-dimensional valleys on the ampli�cation or attenuationof seismic waves using PLAXIS, which is one of themost powerful and applicable pieces of software to sim-ulate in�nite or semi-in�nite soil and rock medium andis based on FEM. Using this software, one may applyproper boundaries to the FE models to simulate thein�nite extent of a soil medium and prevent re ectingof arti�cial waves to the model.

Results are presented in non-dimensional am-pli�cation diagrams for both horizontal and verticaldirections. In this investigation, a realistic numeri-cal simulation of the medium is conducted to avoidre ection of seismic waves at boundaries using specialboundary conditions known as \Viscous Boundaries".In fact, these boundaries are coupled with the rest ofmodel's area simulated by FEM.

Application of such boundaries proposed in thisresearch develops a powerful FE tools, namely, FE-IFE, which diminishes di�culties caused by couplingof in�nite elements with �nite elements. Such anapproach also helps developing a more rational methodand, hence, achieving more acceptable results [9].

Amongst di�erent types of seismic sites, emptyvalleys have been given less attention than other topo-graphic irregularities. However, these types of valleysare important in theoretical studies and engineeringapplications since they may have large numbers ofinhabitants due to various life resources. In addition,some of the important structures like dams, bridges,and brie y many infrastructures have been built inthese types of valleys. In the following, the theoreticalapproach adopted in this study is discussed.

2. Proposed theoretical approach

2.1. Equation of motion in multidimensionalanalysis

The basic equation of motion for a system a�ected bya dynamic load in a �nite element approach can bederived as:

[M ] f�ug+ [C] f _ug+ [K] fug = ffg ; (1)

where [M ], [C], and [K] are mass, damping, andsti�ness matrices, respectively. Also, f�ug, f _ug, fug,and ffg are acceleration, velocity, displacement, andforce vectors, respectively, which vary with time. Inthe following, like in Yoshida's manner [20], matrix andvector notations are not used for simplicity.

Generally, in the engineering practice and espe-cially in the case of ground response analysis, relativedisplacement is considered; thus [20]:

u = ur + Iub: (2)

In Eq. (2), ur is the relative displacement vector relatedto the displacements of the base, ub is the displacement

vector for the base, and I is a vector whose componentsare unity when the directions of the degree of freedomand loading are the same. These components arereduced to zero when those directions are di�erent.Substituting Eq. (2) into Eq. (1), one can be inferredthat:

M (�ur+I�ub)+C ( _ur+I _ub) +K (ur + Iub) = 0: (3)

Since ub is a rigid displacement in the domain, we have:

Kub = 0: (4)

Considering movement of domain in the air, one canobtain:

C _ub = 0: (5)

Further, the following relation can be inferred usingEqs. (3) to (5):

M �ur + C _ur +Kur = �MI�ub: (6)

The subscript r is further omitted for simplicity inthe following, assuming that u indicates relative dis-placement. Mass matrix can be formed using twodi�erent formulations: consistent mass matrix, inwhich the same interpolation function is applied in itsdevelopment, and lumped mass matrix, in which moresimpli�ed interpolation function is used. However, thelatter is a diagonal matrix and is easier to apply invarious problems [20]. Detailed discussion on massmatrix formulations can be found in various references(e.g., [36]). In this study, lumped mass matrix formu-lation is adopted.

2.2. Evaluation of damping matrixIn �nite element formulation, damping matrix is oftenformulated as a Rayleigh damping using mass andsti�ness matrices:

C = �M + �K: (7)

In the above equation, � and � are coe�cients, whichare obtained in the following. Such a formulation fordamping ensures that when the �rst part of the aboveequation is dominant, more low-frequency vibrationsare damped; conversely, if the second part is dominant,more high-frequency vibrations are damped [37].

Substitution of Eq. (7) into Eq. (6) leads to [20]:

�u+ (�+ �!20) _u+ !2

0 _u = 0: (8)

Also, according to Eqs. (6) to (8), for Single-Degree-Of-Freedom (SDOF) system, it can be inferred that:

� = �/2!0 + �!0/2: (9)


For multi-degree-of-freedom system, by separationfrom the group of the SDOF system, one may reachthe following relation:

�i = �/2!i + �!i/2; (10)

where subscript i shows the ith mode parameters.The important range of deformations in the en-

gineering practices occurs in the low-frequency modes.Since there are only two parameters in the Rayleighdamping, only two conditions can be satis�ed at maxi-mum. When the damping ratios in the �rst and secondmodes are determined, two parameters are obtainedas [20]:

� = 2!1!2�1!2 � �2!1

!22 � !2

1; (11a)

� = 2��2!2 � �1!1

!22 � !2

1

�: (11b)

The sti�ness part of Rayleigh damping relation ismore important than the mass part; therefore, onecan evaluate � from the damping at the predominantperiod considering � = 0 [20].

2.3. Numerical integration schemeGenerally, in seismic ground response problems, meth-ods are categorized into linear, equivalent linear, andnon-linear with di�erent soil behavior assumptions.Analyses by these methods can be carried out eitherin time or frequency domain. The equivalent linearprocedure is among the frequency domain methodsand most of the non-linear methods use time domainapproach.

In the time domain analysis, equation of motionis solved by integrating intervals in a small amountof time, which is called step by step time integrationmethod. In this method, time intervals should be smallin order to accurately follow loading variation as wellas to fully account for changes in materials properties.

Time integration schemes can be categorized intoexplicit and implicit approaches. Despite some limita-tions, explicit scheme is relatively simple to formulate.Implicit scheme is rather complicated; however, theprocess of calculation is more trustworthy and accu-rate. Among implicit schemes, Newmark method isoften adopted.

In this study, in order to solve the equation ofmotion in time domain, displacement and velocity intime, t+�t, can be derived using Newmark method asfollows [37]:

ut+�t=ut+ _ut�t+ ((1=2�a)�ut + a�ut+�t)�t2; (12a)

_ut+�t = _ut + ((1� b)�ut + b�ut+�t)�t; (12b)

ut+�t = ut + �u: (12c)

In the above equations, �t is time step and coe�cientsa and b determine the accuracy of the numerical timeintegration. The applicable range for a and b to ensurea stable solution is as follows [37]:

b � 0:5; (13a)

a � 1/4(0:5 + b)2: (13b)

Interested readers can �nd various recommended valuesfor a and b in di�erent references. However, commonvalues for them in [20,37] as a = 0:3025 and b = 0:60are adopted here.

Eq. (12) can also be rewritten as [37]:

�ut+�t = �0�u� �2 _ut � �3�ut; (14a)

_ut+�t = _ut + �6�ut + �7�ut+�t; (14b)

or:

�ut+�t = �0�u� �2 _ut � �3�ut; (15a)

_ut+�t = �1�u� �4 _ut � �5�ut; (15b)

in which the coe�cients �0 to �7 are introduced in thetime step and in a and b integration coe�cients.

Using the implicit scheme, Eq. (1) at t + �t isgenerally written as follows [37]:

M �ut+�t + C _ut+�t +Kut+�t = f t+�t: (16)

Using Eqs. (14) to (16) and further mathematicaloperations, then by simpli�cation, one can obtain:

(�0M + �1C +K)�u = F t+�text +M(�2 _ut + �3�ut)

+ C(�4 _ut + �5�ut)� F tint: (17)

According to Eq. (12), �u can be obtained and addedto the earlier displacement (ut).

2.4. Boundary conditions and elementcharacteristics

Among the methods employed for the analysis ofground response, FEM is one of the most powerfulones because of capability of simulating complicatedgeological and geotechnical conditions, which are thematter of concern in this paper. Nevertheless, accuratesimulation of boundary conditions and radial attenua-tion of wave energy is of particular importance to FEdynamic analysis. Application of boundaries with anyconstraint may lead to so called \trap box" e�ect forseismic waves in the model and, hence, to �ctitiousresults.

In static deformation analysis, the vertical bound-aries of the mesh are often �ctitious boundaries and,


thus, they do not a�ect the deformation behavior ofthe environment to be modeled [37]. However, fordynamic analysis, boundaries have to be placed farenough, farther than to be the boundaries in staticanalysis. Further, introduction of boundaries at fardistance virtually means large mesh required in FEmodel with excessive elements, which also entails extracalculation time and core memory. As the size ofthe divisions decreases, the in uence of boundaryconditions becomes a prime concern.

Since seismic ground response analysis domain ismainly of rectangular shape, it can be assumed thatmost of the waves traveling in horizontal directionare surface waves and their wavelengths are largerthan body waves; thus, one can use wide and thinelements [20]. In addition, adoption of large elementsfor an FE model �lters high-frequency componentswhose short wavelength cannot be simulated withnodal points with long intervals.

Considering the mechanism governing the prop-agation of seismic waves, it has been found that sizeof elements should be less than 1/5 to 1/6 of thewavelength corresponding to the highest frequencycontent of the input motion [20].

2.4.1. Viscous boundariesWhen viscous boundaries are applied, equivalentdampers are used instead of commonly used boundaryconstraints. Such equivalent dampers absorb stressesinduced on the boundary. It further means that suchdampers act when stress waves are travelling outwardsfrom the domain.

Components of absorbed normal and shearstresses, when an equivalent damper is introduced inx direction, are de�ned as follows:

�n = � c1�vp _ux; (18a)

� = � c2�vs _uy: (18b)

In the above equations, � is the density of materi-als, vp and vs are respectively the velocity of thecompressional and shear waves, and c1 and c2 arerelaxation coe�cients that are applied to the modelto enhance the performance of the viscous boundaries.The interesting point is that if incident compressionalwaves reach the model's vertical boundaries, c1 andc2 coe�cients are reduced to unity (c1 = c2 = 1).However, in presence of shear waves, the dampinge�ect of viscous boundaries would not be adequateif coe�cients c1 and c2 are neglected. In fact, thee�ect of these boundaries is increased directly withincrease in c2 value. Recent studies have shown thatapplication of c1 = 1 and c2 = 0:25 would optimize theabsorbing e�ect of these boundaries [37]. Fundamentalformulation of these viscous boundaries is based on theprocedure described in [38].

Figure 1. Schematic image of the model and 15-nodeelement used in this research.

3. Problem de�nition

To investigate the e�ect of topographic irregularitiesand site e�ect parameters on seismic ground response,a valley environment is adopted as shown in Figure 1.To evaluate the ampli�cation ratio due to interactionbetween topography and soil layer, a two-dimensionalFEM approach is considered for parametric study. Soillayer is assumed overlain rigid bedrock and the adoptedseismic excitation is an in-plane vertically propagatingSV wave induced on bedrock. The obtained responseis investigated at the ground level at various points forthe e�ect of soil layer and topographic irregularities onampli�cation or attenuation of seismic waves.

15-node elements are used to model the soilmedium because they provide more accurate resultssince they bene�t from a better interpolation scheme(Figure 1). These elements have 2 degrees of freedomde�ned at every node and have 12 Gaussian points.

The accuracy of the proposed method is shownby comparison of the results obtained using the cur-rent method with those of FE-IFE method, includingnon-dimensional diagrams for horizontal and verticaldisplacement amplitudes, through the valley span andits surrounding area [9]. The satisfactory agreementbetween the results of two methods prove an acceptableperformance of viscous boundaries in simulation ofsemi-in�nite and in�nite environments. Hence, the ex-isting model is capable of simulating similar conditions.

3.1. Frequency content of seismic excitationTwo-dimensional site e�ects are important where thedimension of topography is approximately equal to thewavelength of the seismic wave [39]. In earthquakeengineering, it has been shown that the frequencycontent of a strong earthquake almost ranges from0.1 to 20 Hz. Further, the velocity of seismic wavesnear the ground surface lies between 0.1 to 3 km/s;topographies having dimensions larger than tens ofmeters to several kilometers usually behave as in two-dimensional site response models (e.g., [1,10,40]).

In order to facilitate study of the e�ect of the


input harmonic wave's frequency content, a non-dimensional parameter known as the dimensionlessfrequency, ao, is de�ned based on the following relation:

ao =!h�Vs

; (19)

where, ! is the angular frequency of the incident waveinduced on the bedrock, h is the maximum valleydepth, and Vs is the velocity of shear wave travellingthrough the soil medium.

To simplify evaluation of the impact of di�erentparameters on seismic ground response analysis, resultsare shown in terms of ampli�cation factor in horizontaland vertical directions as follows:

HA = ux=uo; (20a)

VA = uy=uo; (20b)

where, HA and VA are, respectively, the horizontaland vertical ampli�cation. In the above equations, uxand uy denote displacement amplitude along x and ydirections, respectively; also, uo is the amplitude of theincident wave. The incident wave amplitude is selectedso that the maximum input acceleration (in depth)is equal to 0.35 g. Diagrams of seismic ampli�cationhelp in understanding the pattern of response alongthe valley span and its surrounding area.

4. Parametric study and discussion

As seen in Figure 2, the valley considered in this studyis symmetric, whose maximum depth is h, while Hrefers to depth of the bedrock. L and l are, respectively,halves of the top and bottom spans of valley and areadopted in this study by �xed values of 100 m and 50 m.

Poisson's ratio is assumed to be constant andequal to 0.33, while soil's modulus of elasticity isadopted as 2:4� 107 kN/m2. The unit weight of soil isconsidered to be 23.54 kN/m3. The dimensions of thevalley in real situation are basically within the studiedrange [15].

In this study, the e�ects of valley ratio (h=H),frequency content of incident wave, and the depth ofthe bedrock on seismic ground response are considered.Generally, in this paper, because of the geometric sym-metry and the vertically planar SV seismic excitation,the results for half of the valley are shown.

Figure 2. Con�guration of the adopted valley.

4.1. E�ect of valley ratioIn order to evaluate the e�ect of valley ratio, constantvalues of h=H = 0, 1/4, 2/4, and 3/4 are adopted. Inthis part, the depth of the bedrock is 200 meters.

Figure 3(a) shows that in the case of free �eld,ampli�cation of the horizontal component nearly dou-bles, which re ects the extent of outcrop motion tothe corresponding bedrock. In other shape ratios,as the distance from the valley center increases, theconditions of free �eld are also observed. Therefore,ground surface motions are always ampli�ed comparedwith other points at depths.

As the parameter h=H increases, ampli�cationof the horizontal component of displacement in thecenter of the valley is increased. However, in all cases,except for the free �eld condition, which has a uniformampli�cation, the locus of maximum HA occurs in thevalley center. Increasing h=H causes HA to reach adouble quantity in the farther distances with respectto the center. It is important to note that the areaa�ected by topographical problem varies depending ontopographic dimensions. Therefore, model dimensions

Figure 3. Seismic ampli�cation comparison for di�erentvalley shape ratios (ao = 1).


are determined based on topographic dimensions. Itfurther implies that by increase in the area of the valley,free �eld condition is attained in a farther distance.

According to Figure 3(b), although the incidentwave is considered to be an SV wave propagating invertical direction, the vertical component of the outputwave at the ground surface does not vanish even in thecase of free �eld condition. However, in the case oftopographic irregularities, VA is larger than the samevariable in free �eld condition. This phenomenon canbe attributed to the interference of incident waves andtheir consecutive re ections in soil medium. At thevalley center, due to the assumed symmetry of thevalley and the upward propagation of incident wave,VA is virtually reduced to zero.

Increasing h=H causes the maximum VA to in-crease correspondingly. As seen in Figure 3(b), themaximum occurs in h=H = 3=4 and it continues toa relatively large distance from the edge of the valley.However, the locus of the maximum shifts from thebody to top of the valley and it is sustained over fartherdistances as the depth of the valley increases. As wellas HA, increasing h=H causes the surrounding area toundergo more ampli�cation.

It is found that increase in the depth of thevalley not only increases the maximum values of HAand VA, but also sustains the ampli�cation of thetopography over farther distances. In other words,the e�ect of topographic irregularity on seismic groundmotion becomes more considerable. Di�erent behaviorswith di�erent depths of the valley can be attributedto 2 major factors: �rst, increase in valley's bodyslope, demonstrated by increase in depth while top andbottom spans of valley are �xed, causes increase in am-pli�cation. The reason behind such a phenomenon hasbeen explained earlier by the current investigators [9].Second, increasing the valley depth and, consequently,an increase in its area cause more ampli�cation bysurface waves and their interference with incident andre ected waves.

4.2. E�ect of frequency content of input waveAs mentioned earlier, frequency content of input waveis here referred to as ao parameter. In this study,constant values of ao = 0:25, 0.5, and 1 are adopted fornon-dimensional frequencies of seismic excitation andthe shape ratio of the valley is considered as h=H = 1=2(where h = 100 m and H = 200 m). The wavelengthsof the incident waves are considered 1, 2, and 4 timesthe valley depth.

As can be inferred from Figure 4(a), HA increasesin almost all parts of valley mount and its surroundingarea as frequency content decreases. However, infrequency ratios of ao = 0:25 and 0.5, the locus ofmaximum takes place at the top of the valley, butin ao = 1, it happens at its bottom. In other

words, frequency content of the incident wave is oneof the e�ective parameters to determine the locus ofmaximum HA. Decreasing the frequency content causesmore oscillation in HA and this component reaches avalue double the input wave in the farther distancesfrom the center.

According to Figure 4(b), maximum VA occursat top of the valley in di�erent frequencies. Decreasingthe frequency causes VA at farther distances from thevalley center to reach a minimal value.

It should be noted that in addition to the surfacetopographical and mechanical properties of soil layer,di�erent frequency contents of the incident wave due tochange in ao are also e�ective in determination of modeldimensions to attain free �eld condition. However, thisfrequency content also a�ects the quantities and lociof maximum and minimum HAs and VAs. It furtherimplies that for sites having the same topographi-cal but di�erent seismological conditions, numericallysimulated model may not have the same dimensions,since di�erent frequency contents of seismic input wave

Figure 4. Seismic ampli�cation comparison for di�erentwave characteristics (h=H = 1=2).


necessitate di�erent extensions of the model to reachfree �eld condition.

4.3. E�ect of bedrock depthIn order to study the e�ect of bedrock depth, dimensionof topography is kept constant while various ratios forH=h, namely, 1.5, 2, 2.5, 3, and 3.5, are adopted. Inthis part, the height of valley is considered constantand equal to 100 m.

According to Figure 5(a), it can be inferred thatan increase in the bedrock depth causes HA to decreasein the bottom of the valley; however, it does not varyin the top of the valley. This phenomenon can beattributed to sharp angle at the bottom of topography,which may cause the waves to be trapped and re ectedin multiple manners. This e�ect vanishes by increas-ing the depth of the bedrock. It is also importantto note that the e�ect of topographic dimension onthe ground seismic response is not considered as anabsolute impact. In fact, it is rational to evaluatetopographical dimension in relation to the bedrock

Figure 5. Seismic ampli�cation comparison for di�erentbedrock depths (ao = 1).

depth, since the ratio of the topographical dimensionto the bedrock depth is important. Generally, in a par-ticular topographical environment, increase in bedrockdepth signi�cantly reduces the e�ect of topographicalconditions on the ground response.

Investigation into Figures 5(a) reveals that whenthe bedrock depth becomes more than double the valleyheight, the locus of maximum HA moves to the top ofthe valley; however, in other cases, it would be at thebottom of the valley.

It is noted that variation in soil layer thicknessresults in alternation of natural frequency of the soillayer, which, in turn, induces a signi�cant impact onthe HA component of the response, particularly at thebottom of the valley. The e�ect of this parameter isvery evident on the VA as will be discussed in thefollowing.

Evaluation of Figure 5(b) shows that in allbedrock depths considered here, the locus of maximumVA occurs at the top of the valley. However, atH=h = 1:5; 2:5, and 3.5, its value is larger than otherratios considered.

Since the natural frequency of the soil layerdepends mostly on the shear wave velocity and thethickness of the soil layer, it is seen that in H=h =1:5; 2:5, and 3.5, the natural frequency of the soilmass occurs almost close to the frequency of harmonicincident load and, as a result, resonance occurs. Thisphenomenon causes the maximum amplitudes to reachthe highest value in these 3 states. It is noteworthythat because the wave energy dissipation is consideredas the damping ratio in the model, the ampli�cationsreach a local maximum, but never attain an in�nitevalue.

In order to investigate the sensitivity of soil massvariation (due to change in bedrock depth), variousvalues for the H=h ratio in the range of 1:5 < H=h <2 are considered for the simulated model. For thispurpose, the H=h values of 1.5, 1.7, 1.8, 1.9, and 2 areadopted and compared upon the results reached. Ascan be inferred from Figure 6, increasing the bedrockdepth causes HA and VA to decrease from the valuecorresponding to the �rst natural frequency.

According to the results, variations of displace-ment are seen inside the valley as well as in thesurrounding environment. Displacement variationsare caused by surface waves and their interferencewith incident and re ected waves. Consequently, suchvariations can have a signi�cant e�ect on the seismicresponse of structures, which may be constructed in thevalley and its surrounding area . Therefore, seismicdesign in such environments requires complementaryinvestigations.

It is deduced from the discussion given in preced-ing sections that topographical e�ect on seismic groundresponse may directly a�ect the selection of appropriate


Figure 6. Seismic ampli�cation comparison forsensitivities of bedrock depths (ao = 1).

site locations for important projects such as dams,pipelines, etc. It further implies that comprehensivestudies, especially with regards to topographical im-pact on seismic response, are required to optimize thesafety factor and, hence, reduce the seismic risk andthe cost of projects. Evaluation of the results alsoreveals that, for a structure built inside and aroundsteeper valley banks, stronger HA and VA shouldbe considered, irrespective of the fact that incidentearthquake wave only has the horizontal accelerationcomponent.

5. Conclusion

In this study, the seismic response of a site wasconsidered regarding topographical irregularities. Hor-izontal and vertical ampli�cation factors due to two-dimensional topographic e�ect were evaluated using arealistic model based on FEM coupled with viscousboundaries. The solution was based on fully non-linearsoil behavior in time domain. Using the numerical

approach allowed a parametric study of e�ects of valleyratio, frequency of incident wave, and bedrock depth onHA and VA.

Basic conclusions drawn from the numerical sim-ulation are as follow:

� The loci of maximum and minimum HA and VAdepend not only on the geometric characteristics ofthe site, but also on the frequency content of the in-cident wave. It is shown that di�erent sites with thesame geometric characteristics may require di�erentseismic ground responses and special considerationsaccording to seismic zone of the site;

� It is also concluded that the severity of the e�ectsof the incident wave frequency on seismic responseare controlled by the geometric characteristics of thesite;

� Dimension of the mesh domain in numerical studiesin seismic ground response analysis depends on bothgeometric characteristics and seismic zone of theinterested site;

� To obtain an optimized model size and to reach free�eld condition, dimension of the valley should notbe regarded as a unique parameter. Rather, thesedimensions should be normalized with respect tobedrock depth;

� It is concluded that displacement variation onground surface due to topographical e�ects is animportant parameter to select the site location ordesign of important structures, especially those withlinear behavior.

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Biographies

Navid Soltani received his BS degree in Civil En-gineering from Yazd University in 2007 and his MSdegree in Geotechnical Engineering from Shahid Ba-honar University of Kerman in 2010. In 2012, hebecame a PhD student at Shahid Bahonar Universityof Kerman. He has published 12 journal and confer-ence papers as well as several research projects withdi�erent governmental and private organizations. Hisresearch interests include geotechnical earthquake en-gineering, especially seismic ground response analysis;soil-structure interaction; and numerical methods.

Mohammad Hossein Bagheripour received hisBS degree in Civil Engineering from Shahid BahonarUniversity of Kerman in 1988. He worked as consultingengineer in civil projects in south-eastern Iran beforemoving to Australia to continue his postgraduate stud-ies. In 1993, he received his MEng Degree from theUniversity of Sydney and was also awarded a PhDdegree in Geotechnical Engineering in 1997 for hiscontinuing research work on jointed rock mechanics.Immediately after graduation, he returned to Kerman,Iran, where he is currently Professor in the CivilEngineering Department of Shahid Bahonar Universityof Kerman. His research interests include soil androck mechanics in general, and earthquake geotechnicalengineering in particular. He has published numerouspapers in various international and national journalsand presented many others at conferences.

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