Stochastic Approach

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Sensitivity Analysis for Soil-StructureInteraction Phenomenon UsingStochastic ApproachM. Moghaddasi a , J. G. Chase b , M. Cubrinovski a , S. Pampanin a &A. Carr aa Department of Civil and Natural Resources Engineering , Universityof Canterbury , Christchurch , New Zealandb Department of Mechanical Engineering , University of Canterbury ,Christchurch , New ZealandAccepted author version posted online: 12 Apr 2012.Publishedonline: 14 Sep 2012.

To cite this article: M. Moghaddasi , J. G. Chase , M. Cubrinovski , S. Pampanin & A. Carr (2012)Sensitivity Analysis for Soil-Structure Interaction Phenomenon Using Stochastic Approach, Journal ofEarthquake Engineering, 16:7, 1055-1075, DOI: 10.1080/13632469.2012.677570

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Journal of Earthquake Engineering, 16:1055–1075, 2012Copyright © A. S. Elnashai & N. N. AmbraseysISSN: 1363-2469 print / 1559-808X onlineDOI: 10.1080/13632469.2012.677570

Sensitivity Analysis for Soil-Structure InteractionPhenomenon Using Stochastic Approach

M. MOGHADDASI1, J. G. CHASE2, M. CUBRINOVSKI1,S. PAMPANIN1, and A. CARR1

1Department of Civil and Natural Resources Engineering, University ofCanterbury, Christchurch, New Zealand2Department of Mechanical Engineering, University of Canterbury, Christchurch,New Zealand

This article analyses 1.36 million realistic soil-structure interaction (SSI) scenarios in a systematicfashion to define the correlation between soil, structural, and system parameters and interactioneffects on the structural response. In the analyses, a soil-shallow foundation-structure model thatsatisfies design building code requirements is utilized. It has been identified that soil shear wavevelocity, shear wave velocity degradation ratio, structure-to-soil stiffness ratio, and structural aspectratio combined with the system stiffness are the key parameters whose variation significantly affectsvariation in structural response. The critical range of variation of these parameters resulting in adetrimental SSI effects is also defined.

Keywords Soil-Structure Interaction; Sensitivity Analysis; Stochastic Approach; Influential Soil-Structure Parameters; Critical Range of Variation

1. Introduction

It has been clearly demonstrated in the accompanying article [Moghaddasi et al., 2012] thatthe structural response of a soil-structure system to seismic forces is strongly affected by theimpact of uncertainty in soil and structural parameters accompanied with the inherent ran-domness of the ground motion. For the single-degree-of-freedom (SDOF) model assumed,it has also been shown that soil-structure interaction (SSI) effects cannot be always safelyignored, given the likelihood of having amplification in the structural response due to foun-dation flexibility. Thus, at least for critical scenarios, SSI effects have to be taken intoaccount in the seismic design procedure.

However, significant complexities and variation in the structural response make theidentification of the critical scenarios a challenging task. A considerable step towardsidentification of these critical scenarios is to: (1) define the correlation between differ-ent parameters and the observed variation in response modification factors; and (2) tocomprehensively characterize and quantify the scenarios causing either reduction or ampli-fication in the structural response. Thus, those scenarios causing the greatest likelihood ofexceeding demand can be more precisely defined.

Received 11 July 2011; accepted 14 March 2012.Address correspondence to M. Moghaddasi, Department of Civil and Natural Resources Engineering,

University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand; E-mail: [email protected]

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1056 M. Moghaddasi et al.

In this context, Veletsos and Nair [1975] and Bielak [1975] showed that the differencebetween seismically induced linear response of a fixed-base and a flexible-base system isstrongly affected by structural aspect ratio, soil Poison’s ratio, soil hysteretic damping ratio,a dimensionless parameter expressing the relative stiffness of foundation and structure,and a dimensionless parameter representing soil-to-structure mass. Following this study, amore comprehensive investigation was carried out by Ciampoli and Pinto [1995]. They con-cluded that structural response of a nonlinear system does not show any systematic depen-dencies on the parameters regulating SSI phenomena, and it is statistically reduced due tofoundation flexibility. Later on, Stewart et al. [1999a,b] used a comprehensive database ofrecorded data, and concluded that ratio of structure-to-soil stiffness has the greatest influ-ence on the structural response of a soil-structure system to seismic forces. In addition,it has been established that structural aspect ratio, foundation embedment, and foundationflexibility are the other parameters with significant effect on inertial interaction. Finally,based on the framework of dimensional analysis, Zhang and Tang [2009] showed that SSIeffects are highly dependent on the structure-to-pulse frequency, foundation-to-structurestiffness ratio, and foundation damping ratio. They also presented certain limits for thesecontrolling parameters to distinguish whether or not SSI effects were significant.

Given the existing controversy in the previous findings and the assumed limitationsin the analysis performed, it is believed that the most rational way to identify critical SSIscenarios is to make use of a probabilistic approach. With this in mind, the results of acomprehensive probabilistic simulation explained in detail in the accompanying paper aresummarized in this article, and then used to: (1) define the correlation and dependencybetween structural response modification factors and model parameters; (2) identify thekey parameters having a significant effect on the structural response; (3) present the trendof variation of SSI effects due to change of these parameters; and finally (4) quantify thecritical range of variation of the parameters of consequence that causes detrimental SSIscenarios (i.e., scenarios with amplified structural response).

This probabilistic analysis is a critically important step towards understanding andreliably characterizing the complex SSI problem. It should also be noted that the presentedoutcomes are limited to a SDOF system, and that the scenarios presented do not considerthe extreme conditions, such as those imposed by liquefiable soils or near-fault effects onthe ground motion. However, the cases covered represent a significant majority of typicaldesign cases and scenarios.

2. Methodology and Monte Carlo Simulation

2.1. Outlines of the Procedure for the Probabilistic Study

A Monte Carlo technique was used to study sensitivity of inelastic seismic structuralresponse of an established rheological soil-shallow foundation-structure system to differ-ent model parameters. Four measures of structural response are considered: (1) structuraldistortion us that represents the horizontal displacement of the structure relative to the foun-dation; (2) structural drift dr that is the summation of foundation rocking and normalizedstructural distortion by the effective height; (3) total displacement utot that represents thedisplacement measured at the roof level including lateral displacement resulted from foun-dation response and structural distortion; and (4) structural acceleration as that representsthe total transmitted acceleration (or force) to the structure. A large number of nonlin-ear time-history simulations were run over models with randomly selected parametersusing a suite of recorded ground motions that were then scaled. Parameters of these sys-tems were systematically defined by a random process carefully ensuring realistic models,

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Sensitivity Analysis for Soil-Structure Interaction 1057

as well as covering a specific period range in the design spectrum. The period range ofTFB = 0.2, 0.3 . . . 1.8 s was considered to: (1) represent fixed-base (FB) structures withtotal height of 3–30 m and (2) satisfy the period-height relationship adopted in the NewZealand Standard [NZS1170.5, 2004] for seismic design of structures. For each period con-sidered TFB, 1,000 corresponding soil-foundation-structure (SFS) models were generatedby assembling the randomly defined parameters for the SFS system and using commonlyaccepted relationships between various model parameters. The number 1,000 was chosenwith the intention to: (1) give the best fit statistical distribution for the randomly selectedparameters and (2) increase the confidence level of the Monte-Carlo simulation comparedto the exact solution [Fishman, 1996]. All the nonlinear time-history simulations werecarried out using the finite element program “Ruaumoko 2D” [Carr, 2009].

2.2. Dynamic Soil-Foundation-Structure Model

The soil-foundation-structure (SFS) system investigated in this study is constituted froma rheological soil-foundation element and a structure (Fig. 1), following the substructuretechnique. The structure is modelled as a yielding SDOF system with a stiffness-degradingforce-deformation hysteresis rule as Takeda (bilinear envelope with strain hardening andstiffness degradation) assuming a 5% post-yield stiffness and unloading and reloadingparameters of γ = 0.3 and δ = 0.2, respectively. This SDOF representation is an approx-imate model of a multi-story building vibrating in its fundamental natural mode. Theconsidered SDOF structure is assumed to have the same period and viscous damping coef-ficient as that of the corresponding FB system and is characterized by its mass ms, lateralstiffness ks, coefficient of relative viscous damping cs, and effective height he.

The soil-foundation element was modelled by a lumped-parameter model representinga rigid circular footing resting on the soil surface and having a perfect attachment to the soil.Moreover, the foundation was assumed to have no mass or mass moment of inertia aboutthe horizontal axis. For evaluating the dynamic soil impedances incorporating soil nonlin-earity, the frequency-independent coefficients of a rheological Cone model [Wolf, 1994]were modified using the conventional equivalent linear method [Seed and Idriss, 1970].

FIGURE 1 Soil-shallow foundation-structure model for horizontal and rocking foundationmotions (color figure available online).

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To avoid more complication in time-domain analysis, soil material damping was assumedto be viscous instead of hysteretic. In the model presented, the horizontal degree of free-dom and the rocking degree of freedom are considered as representatives of translationaland rocking motions of foundation, respectively, and the effects of vertical and torsionalmotions are ignored.

2.3. Uncertainty in System Parameters and Randomness in Ground Motions

In seismic analysis, there are two recognized principal sources of uncertainty that need tobe addressed: (1) model parameters and (2) input ground motion. In this research, bothtypes of uncertainties are covered. A brief overview of the stochastic selection process ispresented here, but for more detailed information the interested reader is referred to theaccompanying article.

2.3.1. Selection of Uncertain Model Parameters. All soil parameters defining the soil-foundation element were considered as uncertain parameters. Initial soil shear wavevelocity (Vs)0, shear wave velocity degradation ratio (Vs)sec/(Vs)0, where (Vs)sec repre-sents the degraded shear wave velocity, soil mass density ρ and Poisson’s ratio υ weredefined as independent parameters. For each of them, a realistic range was defined, andthen 1000 uniformly distributed values were assigned to that range.

Randomly varying structural parameters include: structural effective height he, founda-tion radius r, and structural mass ms. Depending on these randomly generated parameters,the values for structural stiffness ks and structural damping cs were then calculated.To achieve realistic SFS models, the selection of these structural parameters was con-strained by commonly accepted relationships for the structure and/or the whole SFSsystem. These relationships can be found in Table 1 of Moghaddasi et al. [2012].

2.3.2. Selection of Ground Motions. Forty large-magnitude, small-distance groundmotions recorded on stiff/soft soil, i.e., soil type C with Vs = 180 – 360 m/s and soiltype D with Vs < 180 m/s to a depth of 30 m based on USGS soil geomatrix classifica-tion, were used as input ground motions in the Monte Carlo simulations. The time-historieswere obtained from the PEER and NGA databases available online at the link http://peer.berkeley.edu/nga.

The number 40 was chosen to reduce the variance in the response due to record-to-record variability and obtain an estimate of median response within a factor of ±0.1 witha 95% probability [Shome et al., 1998]. In addition, these recorded ground motions wereselected to satisfy the constrains of: (1) magnitude in the range of 6.5–7.5; (2) closestdistance to fault rupture in the range of 15–40 km; and (3) peak ground acceleration (PGA)greater than 0.1 g.

The selected records were then scaled to distribute PGAs within the range of 0.3–0.8 g,assuming that a nonlinear behavior of the structure will be caused by those intensity levels.Given the rigorous scaling criteria and recommendations in NZS 1170.5, all scaling factorswere chosen to be between 0.3 and 3.0.

3. Correlation Between SSI Effects and Model Parameters

To investigate the correlation between SSI induced modification in structural response andrandomly selected model parameters, the variation in the structural response modificationfactors, for all considered groups of models categorized based on TFB, was examined as afunction of:

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● soil parameters: (1) soil mass density ρ; (2) Poison’s ratio υ; (3) initial soil shearwave velocity (Vs)0; and (4) shear wave velocity degradation ratio (Vs)sec/(Vs)0

● structural parameters: (1) structural effective height he; (2) foundation radius r;(3) structural mass ms; and (4) structural aspect ratio he/r

● soil-structure system parameters: (1) ms/ρr2; (2) ms/ρπr2he; (3) σ = (Vs)secTFB/he;and (4) ϕ = he

(Vs)secTFB(he/r)0.25.

Note that the combined model parameters selected are based on the previous studies inliterature, such as Ciampoli and Pinto [1995], Stewart et al. [1999a], and Wolf [1994],which introduce the regulating parameters in SSI phenomenon. In this context, he/r rep-resents the structural aspect ratio. This parameter is not a complete parameter in termsof describing both soil and structural characteristics. However, it is a geometric param-eter of immediate engineering and design significance, as well as easily and robustlydetermined. Furthermore, ms/ρr2 and δ are measures of structure-to-soil mass ratio, andσ is a representative of structure-to-soil stiffness ratio. Finally, the combined effect ofσ and he/r is captured in φ, a parameter suggested by Ciampoli and Pinto [1995] tobe the best suited in measuring the limit condition where SSI effects are not worth toconsider.

In the following, the possibility of having either a linear or nonlinear correlation isanalyzed and discussed. For presentation purpose, structural response modification factorsare denoted as: (us)SSI/(us)FB, drSSI/drFB, (utot)SSI/(utot)FB, and (as)SSI/(as)FB, representingmodification in structural distortion, structural drift, total displacement, and structuralacceleration due to SSI effects, respectively.

3.1. Linear Correlations

The existing linear dependency of the structural response modification factors on theconsidered parameters is presented through Pearson correlation coefficient ρ. Pearson cor-relation coefficients are obtained by dividing the covariance of two variables by the productof their standard deviations and represent the level of correlation across a range [Dowdyand Wearden, 1983]. Assuming X is the calculated structural response modification fac-tor with mean and standard deviation values of μX and σ X , and Y is the model parameterwith mean and standard deviation values of μY and σ Y , the Pearson correlation coefficientbetween these two random variables is defined:

ρ(X, Y) = COV(X, Y)

σXσY= E[(X − μX)(Y − μY )]

σXσY(1)

where E represents the expected value and COV means covariance.The Pearson correlation coefficient is +1 in a perfectly increasing (positive) linear

relationship, and -1 in the case of a perfectly decreasing (negative) linear relationship.It approaches zero when there is less of a correlation between variables. In all other cases,it gives values between -1 and +1 indicating the degrees of linear dependence betweenthe variables. If the variables are independent, the Pearson correlation coefficient is zero.However, the converse is not always true.

The Pearson correlation coefficients representing the linear dependency between struc-tural response modification factors and the adopted soil, structural, and soil-structuresystem parameters are presented in Figs. 2, 3, and 4, respectively. Note that in the spectral

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–1

–0.5

0

0.5

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

ρ ν (Vs)0

(Vs)

sec/(V

s)0

ρ[(u

s) SSI/(

u s) FB,Y

]

TFB

(s)

TK

–1

–0.5

0

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1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

ρ ν (Vs)0

(Vs)

sec/(V

s)0

ρ[dr

SSI/d

r FB,Y

]

TFB

(s)

TK

(a) (b)

(c) (d)

–1

–0.5

0

0.5

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

ρ ν (Vs)0

(Vs)sec

/(Vs)0

ρ[(u

tot) SS

I/(u to

t) FB,Y

]

TFB

(s)

TK

–1

–0.5

0

0.5

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

ρ ν (Vs)0

(Vs)

sec/(V

s)0

ρ[(a

s) SSI/(

a s) FB,Y

]

TFB

(s)

TK

FIGURE 2 Pearson correlation coefficient spectra representing the correlation betweenstructural response modification factors and soil parameters.

–1

–0.5

0

0.5

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

he

r ms

he/r

[(u s) SS

I/(u s) FB

,Y]

TFB

(s)

TK

–1

–0.5

0

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

he

r ms

he/r

[dr SS

I/dr FB

,Y]

TFB

(s)

TK

(a) (b)

–1

–0.5

0

0.5

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

he

r ms

he/r

[(u to

t) SSI/(

u tot) FB

,Y]

TFB

(s)

TK

–1

–0.5

0

0.5

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

he

r ms

he/r

[(a s) SS

I/(a s) FB

,Y]

TFB

(s)

TK

(c) (d)

ρρρ

ρ

FIGURE 3 Pearson correlation coefficient spectra representing the correlation betweenstructural response modification factors and structural parameters.

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–1

–0.5

0

0.5

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

ms/ρr^3 δ ϕ σ

ρ[(u

s) SSI/(

u s) FB,Y

]

TFB

(s)

TK

–1

–0.5

0

0.5

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

ms/ρr^3 δ ϕ σ

ρ[dr

SSI/d

r FB,Y

]

TFB

(s)

TK

–1

–0.5

0

0.5

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

ms/ρr^3 δ ϕ σ

ρ[(u

tot) SS

I/(u to

t) FB,Y

]

TFB

(s)

TK

–1

–0.5

0

0.5

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

ms/ρr^3 δ ϕ σ

ρ[(a

s) SSI/(

a s) FB,Y

]

TFB

(s)

TK

FIGURE 4 Pearson correlation coefficient spectra representing the correlation betweenstructural response modification factors and soil-structure system parameters.

format presented, all considered scenarios are taken into account and categorized based onvalues of fundamental structural period TFB.

3.1.1. Correlation between SSI Effects and Soil Parameters. In terms of soil parameters, asshown in Fig. 2, only (Vs)0 and (Vs)sec/(Vs)0 have a more pronounced linear correlation with(us)SSI/(us)FB and (as)SSI/(as)FB. The existence of linear correlation for the other differentscenarios can almost always be ignored. In addition, the existing correlations are strongerfor (us)SSI/(us)FB compared to those for (as)SSI/(as)FB. However, note that the correlationsobserved are not a very strong linear correlation as the Pearson correlation coefficients arenot very close to ±1 (ρ ∼ 0.5). It should also be noted that, the existing linear correla-tions (or Pearson correlation coefficients) are almost unchanged for all periods considered,meaning, the dependency of (us)SSI/(us)FB and (as)SSI/(as)FB on (Vs)0 and (Vs)sec/(Vs)0 areindependent from TFB.

3.1.2. Correlation between SSI Effects and Structural Parameters. If the linear correlationbetween structural parameters and structural response modification factors are considered,no significant correlation exist as all Pearson correlation coefficients are small compared to±1. This result is illustrated in Fig.3. Therefore, no linear trend can be defined to correlatethe variation in structural parameters to the modification in structural response due to SSIeffects.

3.1.3. Correlation between SSI Effects and Soil-Structure System Parameters. When soil-structure system parameters are taken into account (Fig. 4), σ and ϕ are the only parameters

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that have a more pronounced linear relationship with (us)SSI/(us)FB and (as)SSI/(as)FB.In addition, the correlation between ϕ and (us)SSI/(us)FB or (as)SSI/(as)FB is more likelyto be linear compared to that for σ , as the corresponding Pearson correlation coeffi-cients are closer to ±1. It is also noted that the evaluated Pearson correlation coefficients,representing the dependency of structural response modification factors to σ and ϕ, areindependent from the periods considered. Parameters σ and ϕ also show a small linearcorrelation with drSSI/drFB and (utot)SSI/(utot)FB when stiff structures (TFB ≤ 0.6 s) areconsidered.

3.1.4. General Comments. Finally, considering the increasing (positive) or decreasing(negative) linear relationship between structural response modification factors and soil,structural, and soil-structure system parameters, it is concluded that (us)SSI/(us)FB and(as)SSI/(as)FB are increased when: (1) (Vs)0 increases; (2) smaller degradation in shearwave velocity occurs; (3) σ increases; or (4) ϕ decreases. These trends are discussed inmore detail in Sec. 4.

3.2. Nonlinear Correlations

To examine the possibility of having a nonlinear correlation between the response mod-ification factors and the soil, structural, and soil-structure system parameters, the graphsshowing data distribution are considered. In these graphs, the response modification fac-tors for each selected group of models with a specific period are represented based on thevariation of a certain parameter. By examining all possible scenarios, it has been seen thatthe measured structural response modification factors only have an obvious relationshipwith σ and ϕ. The graphs are noisy for all other cases. Therefore, parameters having no lin-ear correlation with the response modification factors (i.e., ρ, υ, he, r, ms, he/r, ms/ρπr2he

and ms/ρr3) also have no distinguishable nonlinear dependency. In other words, SSIinduced modification in structural response does not have a significant linear or nonlinearcorrelation with any of these parameters.

To avoid presenting unnecessary information, only the graphs showing the varia-tion of (us)SSI/(us)FB and (utot)SSI/(utot)FB with σ and ϕ are illustrated and discussedin the following. In these graphs, the Pearson correlation coefficients previously pre-sented are also shown to specify the existing relationship in terms of being a linear ornonlinear.

Figure 5 illustrates the relationship between (us)SSI/(us)FB and σ for different TFB val-ues. Clearly, there is a strong directionality in the data presented. In this context, as thePearson correlation coefficients are small, the existing relationship can be considered as anonlinear correlation. Equally, Fig. 6 shows the relationship between (us)SSI/(us)FB and ϕ.In this case, in addition to the strong directionality, it can be seen that the Pearson correla-tion coefficients are located in the range of ρ ∼= 0.5 − 0.7. Thus, the existing dependencybetween (us)SSI/(us)FB and ϕ is better to be assumed as a linear correlation rather than anonlinear type. These observations are in complete agreement with that has been presentedin Sec. 3.1.3.

If the relationship between (utot)SSI/(utot)FB and σ is examined (Fig. 7), it can be clearlyconcluded that the existing correlation between (utot)SSI/(utot)FB and σ is nonlinear, respect-ing the small values of Pearson correlation coefficients observed. In addition, when thecorrelation between (utot)SSI/(utot)FB and ϕ is considered (Fig. 8), only a nonlinear correla-tion can be distinguished when ϕ < 0.5. Above this value, the dependency vanishes veryquickly.

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FIGURE 5 Correlation and dependency between structural distortion modification factorsand σ = (Vs)secTFB/he (color figure available online).

FIGURE 6 Correlation and dependency between structural distortion modification factorsand ϕ = he

(Vs)secTFB(he/r)0.25 (color figure available online).

3.3. Linear and Nonlinear Correlations in Summary

In summary, on the basis of the results presented for both linear and nonlinear cor-relations, it can be concluded that structural response modification factors due to SSIeffects do not show a systematic dependency on the model parameters except for(Vs)0, (Vs)sec/(Vs)0, σ = (Vs)secTFB/he and ϕ = he

(Vs)secTFB(he/r)0.25.

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FIGURE 7 Correlation and dependency between total displacement modification factorsand σ = (Vs)secTFB/he (color figure available online).

FIGURE 8 Correlation and dependency between total displacement modification factorsand ϕ = he

(Vs)secTFB(he/r)0.25 (color figure available online).

4. Variation of SSI Effects with Model Parameters

To quantify the variation of structural response modification factors due to change in themodel parameters of consequence, a robust statistical presentation was adopted. In thisregard, all the examined scenarios are considered together, regardless of the initial group-ing based on TFB. This approach is acceptable since the observed correlation between

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the response modification factors and (Vs)0, (Vs)sec/(Vs)0, σ and ϕ are not changing, butare almost the same for all considered fundamental structural periods. Thus, the varia-tion of structural response modification factors due to change in the model parameters ofconsequence is independent of the TFB values.

To carry out this quantification, the existing dependency of the response modifica-tion factors to the parameters considered were presented for the 5th, 50th, 75th, and 95th

percentiles representing different levels of likelihood. The distance between the 5th and95th percentile boundary lines shows the possible variation in the response. The larger thedistance, the higher the variation. The line assigned to the 50th percentile values showsthe central trend of the response, and the boundary lines assigned to the 75th, and 95th

percentiles are seen as the response trend at the higher levels of probability.

4.1. Dependency on Initial Shear Wave Velocity

The dependency of structural response modification factors on (Vs)0 is shown in Fig. 9.Clearly, if smaller values of (Vs)0 or softer soil conditions are considered, larger variationin the response modification factors are expected. This variation in terms of dr and utot

could result in either a large reduction or amplification in the response. However, for us andas, this variation causes the possibility of a large reduction or only very small amplification.

This observation can be explained noting that a large foundation response is expectedat softer soil conditions. As a result, a large rigid body motion accompanied with a largedamping can be imposed to the SFS system that, in turn, is responsible for a significantamplification in dr and utot, and a significant reduction in us and as. In addition, the sce-narios with a decreased dr and utot might be correspondent to the cases where foundationimposed structural displacement is not as significant as structural distortion, which is alsodecreased due to SSI effects.

0

0.5

1

1.5

2

100 200 300 400

5th Prct.50th Prct.75th Prct.95th Prct.

(us) SS

I/(u s) FB

(Vs)0 (m/s)

(Vs)0 (m/s)

(Vs)0 (m/s)

(Vs)0 (m/s)

TK

0

0.5

1

1.5

2

100 200 300 400

drSS

I/dr FB

TK

(a) (b)

0

0.5

1

1.5

2

100 200 300 400

(uto

t) SSI/(

u tot) FB

TK

0

0.5

1

1.5

2

100 200 300 400

(as) SS

I/(a s) FB

TK

(c) (d)

FIGURE 9 The effects of soil shear wave velocity on structural response modificationfactors.

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As (Vs)0 increases, the variation in the response modification factors reduces signifi-cantly, such that the response modification factors approaches to 1.0. This trend is becausean increase in (Vs)0 corresponds to the stiffer foundation condition. Consequently, SFS sys-tems are forced to a more similar behavior to that of the corresponding fixed-base systems.

In Fig. 9, it is also shown that, at the median values, only (us)SSI/(us)FB is stronglyaffected by the variation of (Vs)0. In this context, when smaller values of (Vs)0 areconsidered, smaller values of (us)SSI/(us)FB or higher reduction in us are expected.However, if higher levels of probability are considered, different interpretations appear.For (us)SSI/(us)FB and (as)SSI/(as)FB, the 75th and 95th percentiles do not show a significantdependency on the change of (Vs)0. It thus indicates that the maximum expected modifi-cation in us and as is independent from (Vs)0. In other words, the maximum modificationin us and as may occur for any values of (Vs)0. However, different probabilities have to beconsidered.

When modification in dr and utot at the 75th and 95th percentiles is considered, anincrease in (Vs)0 tends to sharply decrease the response modification factors. This obser-vation indicates that foundation imposed structural response is significantly reduced due tothe increase of (Vs)0 and, thus, the likelihood of having a large amplification in dr and utot

decreases sharply.

4.2. Dependency on Shear Wave Velocity Degradation Ratio

The effects of (Vs)sec/(Vs)0 on the structural response modification factors are shown inFig. 10. The trends and conclusions are very similar to those indicated for (Vs)0, notingsmaller degradation ratio corresponds to the scenarios with smaller shear wave velocityor softer soil conditions. Clearly, wider variation in the structural response modificationfactors are expected when higher degradation in shear wave velocity occurs, i.e., whensmaller values of (Vs)sec/(Vs)0 are considered. This trend is justified as higher levels of

0

0.5

1

1.5

2

0.1 0.2 0.3 0.4 0.5 0.6 0.7


(us) SS

I/(us) FB

(Vs)sec

/(Vs)0

TK

0

0.5

1

1.5

2

0.1 0.2 0.3 0.4 0.5 0.6 0.7

drSS

I/dr FB

TK

(Vs)sec

/(Vs)0(a) (b)

0

0.5

1

1.5

2

0.1 0.2 0.3 0.4 0.5 0.6 0.7

(uto

t) SSI/(u

tot) FB

TK

(Vs)sec

/(Vs)0

0

0.5

1

1.5

2

0.1 0.2 0.3 0.4 0.5 0.6 0.7

(as) SS

I/(as) FB

TK

(Vs)sec

/(Vs)0(c) (d)

FIGURE 10 The effects of shear wave velocity degradation ratio on structural responsemodification factors.

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degradation will result in the more flexible foundations, as well as more added damping tothe system. Consequently, the response of the SFS system gets more dissimilar to that ofthe corresponding fixed-base condition.

In this context, similar to that has been presented for (Vs)0, a large reduction or ampli-fication in dr and utot may occur, while only a large reduction or a very small amplificationis expected for us and as. Obviously, the reduction is due to the large amount of damp-ing added to the system, and the amplification is due to the large foundation motionimposed. In addition, at the median values, only (us)SSI/(us)FB is sensitive to the variation of(Vs)sec/(Vs)0, while at the 75th and 95th percentiles, only drSSI/drFB and (utot)SSI/(utot)FB aresignificantly influenced by (Vs)sec/(Vs)0. It has also been noted that the maximum expectedmodification in us and as is independent from (Vs)sec/(Vs)0.

4.3. Dependency on Parameter σ

The dependency of structural response modification factors on σ is discussed next. Notethat, as defined in Sec.3, σ combines the effects of three parameters including: (1) degradedsoil shear wave velocity (Vs)sec; (2) fundamental structural period TFB; and (3) structuraleffective height he. Since (Vs)sec is directly related to (Vs)0 and (Vs)sec/(Vs)0, σ is thenhighly influenced by these two parameters. Therefore, similar trends to those have beenobserved for (Vs)0 and (Vs)sec/(Vs)0 might be expected.

The results are presented in Fig.11. Clearly, the variation in the response reduces verysharply with the increase of σ , such that the variation can be practically ignored whenσ > 20. In addition, since the response modification factors approach to 1.0, it can beconcluded that SSI does not have any reduction or amplification effects on the structuralresponse after σ > 20.

This observation is in complete agreement with that has been previously presentedfor (Vs)0 and (Vs)sec/(Vs)0. Specifically, because smaller σ can be correspondent to thescenarios with the softer soil condition or larger shear wave velocity degradation.

0

0.5

1

1.5

2

0 5 10 15σ σ

σ σ

20 25 30 35


(us) SS

I/(u s) F

B

TK

0

0.5

1

1.5

2

0 5 10 15 20 25 30 35

drS

SI/d

r FB

TK

(a) (b)

0

0.5

1

1.5

2

0 5 10 15 20 25 30 35

(uto

t) SSI

/(u to

t) FB

TK

0

0.5

1

1.5

2

0 5 10 15 20 25 30 35

(as) S

SI/(

a s) FB

TK

(c) (d)

FIGURE 11 The effects of σ = (Vs)secTFB/he. on structural response modification factors.

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Smaller σ may also be related to the scenarios with greater structural stiffness orgreater structural height, i.e. smaller TFB or greater he. In both cases, foundation motionis dominant and, thus, larger variation in SSI effects can be expected. After all, it should benoted that it is the combined effect of soil stiffness, structural period and structural heightthat could result in the trends observed due to the variation of σ .

For scenarios with σ < 20, if the median values are considered, only (us)SSI/(us)FB and(as)SSI/(as)FB are affected by the variation of σ . As a result of this dependency, smaller val-ues of σ cause higher reduction in the response modification factors. However, if higherlevels of probability are considered, (us)SSI/(us)FB and (as)SSI/(as)FB increase when σ

increases and approach to 1.0, while drSSI/drFB and (utot)SSI/(utot)FB reduce sharply to 1.0.

4.4. Dependency on Parameter ϕ

Finally, the dependency of structural response modification factors on ϕ is presented in thissection. As defined in Sec. 3, ϕ is a parameter combining the effects of: (1) degraded soilshear wave velocity (Vs)sec; (2) fundamental structural period TFB; (3) structural effectiveheight he; and (4) structural aspect ratio he/r. Similar to σ , φ is also highly subjective tothe effects of (Vs)0 and (Vs)sec/(Vs)0, but in the reverse order.

The results for the dependency of structural response modification factors on ϕ areillustrated in Fig. 12. Obviously, the degree of variation is less for smaller ϕ that, in turn,corresponds to the scenarios with greater (Vs)sec (stiffer soil conditions), greater TFB and he

(more flexible structures), and greater he/r (greater structural aspect ratio). In addition, theresponse modification factors approaches to 1.0 when smaller ϕ is considered.

When ϕ increases, the response modification factors for (us)SSI/(us)FB and(as)SSI/(as)FB decrease even for large percentile values. In this regard, at the 95th per-centiles, the likelihood of having amplification in us and as is negligible after ϕ >

0.5.

0

0.5

1

1.5

2

0 0.5ϕ ϕ

ϕ ϕ

1 1.5


(us) SS

I/(us) FB

TK

0

0.5

1

1.5

2

0 0.5 1 1.5

drSS

I/dr FB

TK

(a) (b)

0

0.5

1

1.5

2

0 0.5 1 1.5

(uto

t) SSI/(u

tot) FB

TK

0

0.5

1

1.5

2

0 0.5 1 1.5

(as) SS

I/(as) FB

TK

(c) (d)

FIGURE 12 The effects of ϕ = he(Vs)secTFB

(he/r)0.25 on structural response modificationfactors.

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In contrast, if the dependency of drSSI/drFB and (utot)SSI/(utot)FB on ϕ is considered,the variation in the response modification factors increases with ϕ. This increase resultsin a wide degree of variation, with the amount of amplification being greater than that ofreduction. Observing a large amplification in dr and utot, but only a large reduction in us

and as can be due to the significant foundation response occurring as ϕ increases. Largefoundation response, consequently, reduces the transmitted displacement (or force) to thestructure, while it increases the structural responses including rigid body motion due tofoundation flexibility.

It should be also noted that scenarios with ϕ > ∼0.5 – 0.7 are the unusual cases, i.e.tall and narrow stiff structures located on very soft soils, that are considered as outliersin the response modification spectrum shown in Moghaddasi et al. [2012]. This fact isalso distinguished in Figs. 7 and 8 as the number of data points presented is significantlyreduced when ϕ > ∼0.5 – 0.7. It basically means only a small number of models generatedsatisfy this condition. Therefore, the results presented for ϕ > ∼0.5 – 0.7 can be practicallyignored.

5. Risk of Detrimental SSI Effects Based on Model Parameters

To evaluate the risk of detrimental SSI effects, as compared to fixed-base assumptions,on structural response due to variation in the model parameters of consequence, two mainaspects are analyzed: (1) the probability of having amplification in the response of the SFSsystem as compared to the response of the corresponding FB structure, and (2) the levelof increase in the response due to SSI consideration. For this purpose, two amplificationlevels (A.L.) were taken into account: 1.1 and 1.2, and the probability of having scenarioswith the response modification factors greater than each level were calculated. For theconsidered amplification levels, the corresponding values of median percentage increase inthe response (Med[P.I.]) were also evaluated.

5.1. Amplification Risk due to Variation of Initial Shear Wave Velocity

The risk of detrimental SSI effects on structural response due to variation in (Vs)0 ispresented in Fig. 13. On the left side, the probabilities of amplification in the structuralresponse are presented, whereas on the right side, the corresponding values of medianpercentage increase are shown.

As expected from the results presented in Sec. 4.1, the probability of amplification inthe response is very small for us and is almost negligible for as, whereas the probability ofamplification in dr and utot cannot be simply neglected. It is also noted that the probabilityof amplification in the response is reduced as the higher level of amplification is considered.

The probability of amplification in the response decreases when (Vs)0 increases.It implicitly indicates that when structures on stiffer soil conditions are considered, theprobability of having detrimental SSI effects will be less critical. If (Vs)0 changes from100–350 m/s, the observed probability values at A.L.=1.1 vary from 10%–5% for us;40%–15% for dr; and 50%–20% for utot, and are almost 0% for as. Equally, these prob-abilities at A.L.=1.2 are in the range of: 5%–1% for us; 30%–5% for dr; and 35%–5%for utot.

If the values of median percentage increase in the response are taken into account,a reduction is also expected when σ = (Vs)secTFB/he increases. Therefore, the degree ofamplification in the structural response decreases similar to the probability of amplificationwhen stiffer soil conditions are considered. The corresponding values of median percent-age increase are in the range of: 20%–15% for us; 30%–15% for dr; 30%–15% for utot;

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0

20

40

60

80

100

50 100 150 200 250 300 350 400

A.L.=1.1A.L.=1.2

Pr[(

u s) SSI/(

u s) FB>A

.L.]

(%

)

(Vs)0 (m/s)

TK

0

20

40

60

80

100

50 100 150 200 250 300 350 400

Med

[P.I

.] (%

)

(Vs)0 (m/s)

TK (us)SSI

/(us)FB

>A.L.

(a) (b)

0

20

40

60

80

100

50 100 150 200 250 300 350 400

Pr[(

dr) SS

I/(dr

) FB>A

.L.]

(%

)

(Vs)0 (m/s)

TK

0

20

40

60

80

100

50 100 150 200 250 300 350 400M

ed[P

.I.]

(%)

(Vs)0 (m/s)

TK drSSI

/drFB

>A.L.

(c) (d)

0

20

40

60

80

100

50 100 150 200 250 300 350 400

Pr[(

u tot) SS

I/(u to

t) FB>A

.L.]

(%

)

(Vs)0 (m/s)

TK

0

20

40

60

80

100

50 100 150 200 250 300 350 400

Med

[P.I

.] (%

)

(Vs)0

(m/s)

TK (utot

)SSI

/(utot

)FB

>A.L.

(e) (f)

0

20

40

60

80

100

50 100 150 200 250 300 350 400

Pr[(

a s) SSI/(

a s) FB>A

.L.]

(%

)

(Vs)0 (m/s)

TK

0

20

40

60

80

100

50 100 150 200 250 300 350 400

Med

[P.I

.] (%

)

(Vs)0 (m/s)

TK (as)SSI

/(as)FB

>A.L.

(g) (h)

FIGURE 13 Risk of detrimental SSI effects based on variation in initial soil shear wavevelocity: (left) probability of amplification in the response; (right) level of increase in theresponse.

and 15%–10% for as when A.L.=1.1 is considered. Equally, for A.L.=1.2, the values ofmedian percentage increase are in the range of: 30%–25% for us; 40%–30% for dr; 45%–30% for utot; and 30%–20% for as. It should be noted that the values presented for as aremostly related to the outliers (uncommon soil-structure scenarios) as the probability ofamplification in as is negligible. This point is also valid for the results followed in case of(Vs)sec/(Vs)0, σ and ϕ.

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5.2. Amplification Risk due to Variation of Initial Shear Wave VelocityDegradation Ratio

The risk of detrimental SSI effects on structural response due to variation in (Vs)sec/(Vs)0 isshown in Fig. 14. Regardless of the values, the trends and conclusions are similar to thosefor (Vs)0. In this comparison, smaller degradation ratios have to be considered instead ofsmaller shear wave velocities.

The probabilities of amplification in the response and the corresponding values ofmedian percentage decrease when (Vs)sec/(Vs)0 increases or smaller soil shear wave veloc-ity degradation is considered. In this context, when (Vs)sec/(Vs)0 changes from 0.2–0.7,the probability of amplification at A.L.=1.1 vary from: 10%–5% for us; 35%–20% for dr;40%–20% for utot, and are almost 0% for as. Equally, these probabilities at A.L.=1.2 arein the range of: 2%–1% for us; 30%–15% for dr; and 30%–15% for utot.

The corresponding values of median percentage increase are in the range of: 20%–15%for us; 30%–15% for dr; 30%–20% for utot; 15%–10% for as at A.L.=1.1 and 30%–25%for us; 40%–30% for dr; 45%–30% for utot; and 25%–20% for as at A.L.=1.2.

5.3. Amplification Risk due to Variation of σ

The risk of detrimental SSI effects on structural response due to variation in σ is shownin Fig. 15. Clearly, an increase in σ yields smaller probability values such that after σ

> 20 any amplification more that 10%, and after σ > 15 any amplification more that 20%in the structural response may be practically ignored. In addition, an increase in σ is accom-panied with a decrease in the values of median percentage increase. These observationsonce more highlight that SSI effects are less critical for soil-structure scenarios includingstiffer soil condition, greater structural stiffness, and smaller structural height.

When σ changes from 2–20, the probability values at A.L.=1.1 vary from: 10%–0%for us; 35%–0% for dr; 45%–0% for utot, and are almost 0% for as. Equally, these probabil-ities at A.L.=1.2 are in the range of: 5%–0% for us; 35%–0% for dr; and 30%–15% for utot.The corresponding values of median percentage increase are in the range of: 20%–10% forus; 35%–10% for dr; 35%–10% for utot; about 10% for as at A.L.=1.1, and 35%–25% forus; 45%–25% for dr; 45%–20% for utot; and about 20% for as at A.L.=1.2.

5.4. Amplification Risk due to Variation of ϕ

Finally, Fig.16 illustrates the risk of detrimental SSI effects on the structural response due tovariation in ϕ. In this case, two different trends are seen for the probability of amplificationin us and as compared to the probability of amplification in dr and utot. The probabilityof amplification in us is very small and decreases as ϕ increases, while the probability ofamplification in as is almost negligible. In addition, an amplification in us can be practicallyignored when ϕ > 1.0. In contrast, the probability of amplification in dr and utot increasesas ϕ increases. The reason behind this increase is previously described in Sec. 4.4.

When ϕ changes from 0.1–1.5, the probability values at A.L.=1.1 vary from: 8%–0%for us; 20%–50% for dr; and 25%–60% for utot, and are almost 0% for as. Equally, theseprobabilities at A.L.=1.2 are in the range of: 2%–0% for us; 5%–45% for dr; and 8%–50%for utot.

In terms of the median percentage increase, an increasing trend exists for us, dr, andutot when ϕ increases, whereas the values of median percentage increase for as are almostconstant for all ϕ values. The values of median percentage increase are in the range of:15%–25% for us; 15%–65% for dr; 15%–80% for utot; and about 15% for as at A.L.=1.1,and 25%–35% for us; 25%–90% for dr; 25%–95% for utot; and about 15% for as atA.L.=1.2.

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0

20

40

60

80

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7

A.L.=1.1A.L.=1.2

Pr[(

u s) SSI/(u

s) FB>A

.L.]

(%)

TK

(Vs)sec

/(Vs)0

0

20

40

60

80

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Med

[P.I.

] (%

)

TK (us)SSI

/(us)FB

>A.L.

(Vs)sec

/(Vs)0(a) (b)

0

20

40

60

80

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Pr[d

r SSI/d

r FB>A

.L.]

(%) TK

(Vs)sec

/(Vs)0

0

20

40

60

80

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7M

ed[P

.I.] (

%)

TK drSSI

/drFB

>A.L.

(Vs)sec

/(Vs)0(c) (d)

0

20

40

60

80

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Pr[(

u tot) SS

I/(uto

t) FB>A

.L.]

(%)

TK

(Vs)sec

/(Vs)0

0

20

40

60

80

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Med

[P.I.

] (%

)

TK (utot

)SSI

/(utot

)FB

>A.L.

(Vs)sec

/(Vs)0(e) (f)

0

20

40

60

80

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Pr[(

a s) SSI/(a

s) FB>A

.L.]

(%)

TK

(Vs)sec

/(Vs)0

0

20

40

60

80

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Med

[P.I.

] (%

)

(Vs)sec

/(Vs)0

TK (as)

SSI/(a

s)FB

>A.L.

(g) (h)

FIGURE 14 Risk of detrimental SSI effects based on variation in shear wave velocitydegradation ratio: (left) probability of amplification in the response; (right) level of increasein the response.

6. Conclusions

A comprehensive statistical evaluation of SSI effects has been conducted using a simpli-fied soil-shallow foundation-structure model with equivalent soil domain and nonlinearstructural behavior following the Takeda type hysteretic model. The key findings can besummarized as follows:

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0

20

40

60

80

100

0 5 10 15σ

σ σ

σ σ

σ σ

σ20 25 30 35

A.L.=1.1A.L.=1.2

Pr[

(us) SS

I/(u s) F

B>

A.L

.] (

%)

TK

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Med

[P.I

.] (%

)

TK (us)SSI

/(us)FB

>A.L.

(a) (b)

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Pr[

(dr)

SSI/(

dr) F

B>

A.L

.] (

%)

TK

0

20

40

60

80

100

0 5 10 15 20 25 30 35M

ed[P

.I.]

(%)

TK drSSI

/drFB

>A.L.

(c) (d)

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Pr[

(uto

t) SSI/(

u tot) F

B>

A.L

.] (

%)

TK

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Med

[P.I

.] (%

)

TK (utot

)SSI

/(utot

)FB

>A.L.

(e) (f)

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Pr[

(as) SS

I/(a s) F

B>

A.L

.] (

%)

TK

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Med

[P.I

.] (%

)

TK (as)SSI

/(as)FB

>A.L.

(g) (h)

FIGURE 15 Risk of detrimental SSI effects based on variation in σ = (Vs)secTFB/he: (left)probability of amplification in the response; (right) level of increase in the response.

1. From all considered soil, structural, and soil-structure system parameters only(Vs)0, (Vs)sec/(Vs)0, σ = (Vs)secTFB/he, and ϕ = he

(Vs)secTFB(he/r)0.25 have a pro-

nounced correlation with structural response modification factors due to SSIeffects.

2. An increase in (Vs)0, (Vs)sec/(Vs)0 and σ cause a less variation in the resultedstructural response modification factors. In addition, as these parameters increase,the response modification factors approach 1.0, indicating the behavior of thesoil-structure systems are more similar to the behavior of the correspondingfixed-base system.

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0

20

40

60

80

100

0 0.5ϕ ϕ

ϕ ϕ

ϕ ϕ

ϕ ϕ

1 1.5

A.L.=1.1A.L.=1.2

Pr[(

u s) SSI/(u

s) FB>A

.L.]

(%)

TK

0

20

40

60

80

100

0 0.5 1 1.5

Med

[P.I.

] (%

)

TK (us)SSI

/(us)FB

>A.L.

(a) (b)

0

20

40

60

80

100

0 0.5 1 1.5

Pr[(

dr) SS

I/(dr)

FB>A

.L.]

(%)

TK

0

20

40

60

80

100

0 0.5 1 1.5M

ed[P

.I.] (

%)

TK drSSI

/drFB

>A.L.

(c) (d)

0

20

40

60

80

100

0 0.5 1 1.5

Pr[(

u tot) SS

I/(uto

t) FB>A

.L.]

(%)

TK

0

20

40

60

80

100

0 0.5 1 1.5

Med

[P.I.

] (%

)

TK (utot

)SSI

/(utot

)FB

>A.L.

(e) (f)

0

20

40

60

80

100

0 0.5 1 1.5

Pr[(

a s) SSI/(a

s) FB>A

.L.]

(%)

TK

0

20

40

60

80

100

0 0.5 1 1.5

Med

[P.I.

] (%

)

TK (as)

SSI/(a

s)FB

>A.L.

(g) (h)

FIGURE 16 Risk of detrimental SSI effects based on variation in ϕ = he(Vs)secTFB

(he/r)0.25:

(left) probability of amplification in the response; (right) level of increase in the response.

3. Considering the median values (the 50th percentiles), an increase in (Vs)0,(Vs)sec/(Vs)0 and σ results in an increase in the structural distortion modificationfactors, while modification in dr, utot, and as are only weakly sensitive to theparameters considered.

4. The likely maximum modification in us and as are independent from the variationof (Vs)0, (Vs)sec/(Vs)0, and σ . However, if dr or utot is considered, a sharp reduc-tion in the maximum modification factors is observed due to an increase in (Vs)0,(Vs)sec/(Vs)0, and σ .

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5. When ϕ increases, structural distortion modification factors and structural acceler-ation modification factors reduce very sharply at the median and likely maximumvalues. In contrast, the variation in structural drift modification factors and totaldisplacement modification factors gets more significant as ϕ increases. This vari-ation mostly results in amplification in structural drift and total displacement thanreduction.

6. In terms of quantification of the risk, the probability of amplification in the responseis very small for us and is almost negligible for as, whereas the probability ofamplification in dr and utot cannot be simply neglected.

7. An increase in (Vs)0, (Vs)sec/(Vs)0, and σ results in a reduction in the prob-ability of amplification in the structural response modification factors, as wellas a reduction in the values of median percentage increase. Specifically, whenσ > 20, detrimental SSI effects on the structural response can be practicallyignored.

8. An increase in ϕ reduces the probability of amplification in us such that the ampli-fication can be practically ignored when ϕ > 1.0. However, the probabilities ofamplification and the corresponding values of median percentage increase for dr orutot rise when ϕ increases.

References

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Stewart, J. P., Seed, R. B., and Fenves, G. L. [1999b] “Seismic soil-structure interaction in build-ings. II: empirical findings,” Journal of Geotechnical and Geoenvironmental Engineering 125(1),38–48.

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Stochastic Approach