Journal of Earthquake Engineering (2011)
SENSITIVITY ANALYSIS FOR SOIL-STRUCTURE INTERACTION
PHENOMENON USING STOCHASTIC APPROACH
M. Moghaddasi†*, M. Cubrinovski1, S. Pampanin1, J.G. Chase2, A. Carr1
1
Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, New Zealand
2
Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand
ABSTRACT
A significant contribution to the seismic design procedure including soil-structure interaction
(SSI) is a complete understanding of the scenarios causing detrimental SSI effects. In that
regard, this paper analyses several realistic SSI scenarios in a systematic fashion to define the
correlation between soil, structural, and system parameters and SSI effects on the structural
response. In the analyses, a soil-shallow foundation-structure model that satisfies various
requirements suggested by design building codes is utilized. The soil stratum is assumed to
be equivalent linear and the superstructure is considered to behave nonlinearly. Parameters of
the model are defined randomly via a rigorous Monte Carlo simulation while constraining the
process to generate realistic models. These randomly generated models are then subjected to
a suite of recorded earthquake ground motions with different characteristics. Specifically,
1.36 million nonlinear time-history simulations are run covering realistic variations and
combinations of soil, structure and earthquake ground motions. From the results, key
parameters whose variation significantly affects structural response due to SSI are identified.
The critical range of variation of these parameters resulting in a detrimental SSI effects is
also depicted in a comprehensive statistical presentation. The comprehensive representation
of the critical parameters provides a well-defined basis for incorporation of SSI in a seismic
design procedure.
†
Address correspondence to Masoud Moghaddasi, Department of Civil and Natural Resources Engineering,
University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand
Email: masoud.moghaddasi@gmail.com
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1 INTRODUCTION
It has been shown in the accompanying paper (Moghaddasi, Cubrinovski et al. 2011) that
structural response of a combined soil-structure system is strongly affected by the impact of
uncertainty in soil and structural parameters accompanied with the inherent randomness of
the input ground motion. Also indicated is that soil-structure interaction (SSI) effects can
only be safely ignored with 70% confidence, respecting the existence of 30% risk of having
amplification in the structural response due to foundation flexibility. Considering the existing
risk of amplification in the structural response gives the impression that for critical scenarios,
SSI effects has to be taken into account in the design procedure. However, significant
variation in the structural response makes the identification of the critical scenarios a
challenging task. A considerable step towards identification of those critical scenarios is to:
(i) define the correlation between different parameters and the observed variation in demand
modification factors; and (ii) to comprehensively characterize and quantify the scenarios
causing either amplification or reduction in the structural response.
In that regard, Veletsos and Nair (1975) and Bielak (1975) have shown that the
difference between seismically induced response of a fixed-base and flexible-base system is
strongly affected by structural aspect ratio, soil Poison’s ratio, soil hysteretic damping ratio, a
dimensionless parameter expressing the relative stiffness of foundation medium and the
structure, and a dimensionless parameter representing soil-to-structure mass. A more
comprehensive investigation of the effects of these parameters on the seismic structural
nonlinear response has been carried out later by Ciampoli and Pinto (1995). They concluded
that structural nonlinear demand does not show any systematic dependencies on the
parameters regulating SSI phenomena and it is statistically lower in the case of flexible
structures. Following those studies and based on the available strong motion data of a
comprehensive database, it has been concluded that ratio of structure-to-soil stiffens has the
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greatest influence on the seismic structural demand of a soil-structure system (Stewart,
Fenves et al. 1999; Stewart, Seed et al. 1999). Also, it has been founded that structural aspect
ratio, foundation embedment and flexibility are the other parameters with significant effect
on inertial interaction. Finally, based on the framework of dimensional analysis, Zhang and
Tang (Zhang and Tang 2009) have shown that SSI effects are highly dependent on the
structure-to-pulse frequency, the foundation-to-structure stiffness ratio, foundation damping
ratio. They also presented certain limits for these controlling parameters to distinguish
whether or not SSI effects will be significant.
Respecting all the previous studies, it is believed that the most rational way to identify
the critical SSI scenarios is to make use of a probabilistic approach. In that regard, the results
of a comprehensive probabilistic simulation explained with details in the accompanied paper
and also summarised herein are used to: (i) define the correlation and dependency between
structural seismic demand modification factors and model parameters; (ii) identify the key
model parameters having a significant effect on the structural seismic demand; (iii) present an
affection trend of the effective model parameters on the structural seismic demand; and
finally (iv) quantify the critical range of variation of the effective model parameters causing
SSI scenarios with the amplification effects on the structural response. This performed
probabilistic analysis is a critically important step towards understanding and reliably
characterizing complex problem of soil-structure interaction.
It also is important to note that the presented outcomes are limited to a SDOF system,
and it does not consider the extreme conditions such as those imposed by very soft
(liquefiable) soils or near-fault effects on the ground motion.
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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 structural seismic
demands of an established rheological soil-shallow foundation-structure (SFS) system to
different system parameters. Two measures of structural seismic demand are considered: (i)
structural distortion demand (𝑢) representing the horizontal displacement of the
superstructure relative to the foundation; and (ii) structural total displacement (𝑢𝑠𝑡𝑟 )
representing the summation of structural distortion and structural lateral displacement due to
foundation horizontal motion and rocking. A large number of nonlinear time-history
simulations were run over models with randomly selected parameters using a suite of scaled
recorded ground motions. Parameters of these systems were systematically defined by a
random process carefully ensuring to satisfy the requirements of realistic models and also
cover a common period range in the design spectrum. The period range of 0.2, 0.3 … 1.8 s
was considered to: (i) represent the fixed-base (FB) superstructures with total height of 3 −
30 m and (ii) satisfy the period-height relationship adopted in New Zealand Standard
(NZS1170.5 2004). For each considered period (TFB), 1000 SFS models were generated by
assembling the randomly defined parameters for the SFS system and using commonly
accepted relationships between various model parameters. The number 1000 was chosen with
the intention to: (i) give the best fit statistical distribution for the randomly selected
parameters and (ii) increase the confidence level of the Monte-Carlo simulation compared to
the exact expected solution (Fishman 1996). The procedure adopted in defining the
parameters is discussed with more details in (Moghaddasi, Cubrinovski et al. 2011). All the
nonlinear time-history simulations were carried out using a FEM code “Ruaumoko 2D” (Carr
2009).
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2.2 Dynamic Soil-Foundation-Structure Model
The interacting soil-structure system investigated in this study is constituted from a
rheological soil-foundation element and a superstructure (Figure 1), following the
substructure technique. The structure is modelled as a yielding single-degree-of-freedom
(SDOF) system by the force-deflection behaviour of Takeda type (elastoplastic with strain
hardening and stiffness degradation) with 5% post-yield stiffness and parameters of 𝛾 = 0.3
and 𝛿 = 0.2 (𝛾 and 𝛿 are defined in Figure 1). This SDOF representation is an approximate
model of a multi-story building vibrating in its fundamental natural mode. The considered
SDOF structure is assumed to have the same period and viscous damping coefficient as those
of the corresponding FB system and is characterized by height (ℎ𝑒𝑓𝑓 ), mass (𝑚𝑠𝑡𝑟 ), lateral
stiffness (𝑘𝑠𝑡𝑟 ), and damping (𝑐𝑠𝑡𝑟 ).
The soil-foundation element was modelled by a lumped-parameter model representing
a rigid circular footing resting on the soil surface and having a perfect bond to the soil.
Moreover, the foundation was assumed to have no mass and mass moment of inertia about
the horizontal axis. For evaluating the soil dynamic impedances incorporating soil
nonlinearity, the frequency-independent coefficients of a rheological Cone model (Wolf
1994) was modified using the conventional equivalent linear method (Seed and Idriss 1970).
To avoid more complication in time-domain analysis, soil material damping was assumed to
be viscous type instead of hysteresis. In the presented model, the horizontal degree of
freedom and the rocking degree of freedom are considered as representatives of translational
and rotational motions of foundation respectively, and the effect of vertical and tortional
motions are ignored. Details about the adopted soil-foundation element can be found in
(Moghaddasi, Cubrinovski et al. 2011).
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2.3 Uncertainty in System Parameters and Randomness in Ground Motions
In seismic analysis, there are two recognized principal sources of uncertainty which need to
be addressed: (i) model parameters and (ii) input ground motion, typically categorized as
epistemic and aleatory uncertainties respectively. In this research, both types of uncertainties
are covered. A brief overview of the stochastic selection process is presented here, but for
more detailed information the interested reader is referred to (Moghaddasi, Cubrinovski et al.
2011).
2.3.1 Selection of uncertain system parameters
All soil parameters defining the soil-foundation element were considered as uncertain
parameters. Initial soil shear wave velocity ((𝑉𝑠 )0), shear wave velocity degradation ratio
((𝑉𝑠 )⁄(𝑉𝑠 )0 ), where (𝑉𝑠 ) represents the degraded shear wave velocity, soil mass density (𝜌),
and Poisson’s ratio (𝜐), were defined as independent parameters; for each of them, a realistic
range was defined first, and then 1000 uniformly distributed values were assigned to that
range.
Randomly varying structural parameters include: structural effective height (ℎ𝑒𝑓𝑓 ),
foundation radius (𝑟), and structural mass (𝑚𝑠𝑡𝑟 ). Depending on these randomly generated
parameters, the values for the structural lateral stiffness (𝑘𝑠𝑡𝑟 ), and structural damping (𝑐𝑠𝑡𝑟 )
were then calculated. To achieve realistic SFS models, the selection of the introduced
structural parameters was constrained by commonly accepted relationship either for the
superstructure or for the whole SFS system.
2.3.2 Selection of ground motions
Forty different large magnitude-small distance ground motions recorded on stiff/soft soil
(type 𝐶 and 𝐷 based on USGS classification) were used as input motions in the simulations.
This number was chosen to reduce the variance in the response due to record-to-record
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variability and obtain an estimate of median response within a factor of ±0.1 with 95%
confidence (Shome, Cornell et al. 1998). The records were selected in such a way to satisfy
the constrains of: (i) the magnitude in the range of 6.5 − 7.5, (ii) the closest distance to fault
rupture in the range of 15 − 40 km, and (iii) the peak ground acceleration (PGA) greater than
0.1𝑔. The selected records were then scaled to have reasonably distributed PGAs within the
range of 0.3 − 0.8𝑔, assuming that a nonlinear behaviour of the superstructure will be caused
by those levels of intensity. Respecting rigorous scaling criteria and recommendations in
NZS 1170.5, all scaling factors were chosen to be less than 3.0.
3 RESULTS AND DISCUSSIONS
3.1 Correlation between Model Parameters and SSI Effects
To investigate the possible correlation and dependency between structural seismic demand
modification factors (i.e., 𝑢𝑆𝐹𝑆 ⁄𝑢𝐹𝐵 , and (𝑢𝑠𝑡𝑟 )𝑆𝐹𝑆 /(𝑢𝑠𝑡𝑟 )𝐹𝐵 ) and individual random
parameters, the variation in the demand modification factors were examined as a function of:
 soil parameters: 𝜌, 𝜐, (𝑉𝑠 )0, and 𝑉𝑠 /(𝑉𝑠 )0
 structural parameters: ℎ𝑒𝑓𝑓 , 𝑟, 𝑚𝑠𝑡𝑟 , and ℎ𝑒𝑓𝑓 ⁄𝑟
 soil-structure system parameters: 𝑚𝑠𝑡𝑟 ⁄𝜌 𝑟 3 , 𝛿 = 𝑚𝑠𝑡𝑟 ⁄𝜌 𝜋𝑟 2 ℎ𝑒𝑓𝑓 , 𝜑 =
ℎ𝑒𝑓𝑓 ⁄𝑉𝑠 𝑇𝐹𝐵 (ℎ𝑒𝑓𝑓 ⁄𝑟)
0.25
, and 𝜎 = 𝑉𝑠 𝑇𝐹𝐵 ⁄ℎ𝑒𝑓𝑓
for all considered groups of models categorized based on 𝑇𝐹𝐵 . In that attempt, the possibility
of having either linear or nonlinear correlation is investigated. Form the considered
parameters, ℎ𝑒𝑓𝑓 ⁄𝑟 represents structural aspect ratio. Clearly, this parameter is not a
complete parameter in terms of describing both soil and structural characteristics. However, it
is a geometric parameter of immediate engineering significance. 𝑚𝑠𝑡𝑟 ⁄𝜌 𝑟 3 and 𝛿 are
measures of structure-to-soil mass ratio, and 𝜎 is a representative of structure-to-soil stiffness
ratio. Finally, the combined effect of 𝜎 and ℎ𝑒𝑓𝑓 ⁄𝑟 is presented in parameter 𝜑.
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3.1.1 Linear correlations
The existing linear dependency of the demand modification factors to the considered
parameters is presented through Pearson correlation coefficient. Pearson correlation (𝜌)
coefficient is obtained by dividing the covariance of two variables by product of their
standard deviations. Assuming 𝑋 is the calculated structural seismic demand with mean and
standard deviation values of 𝜇𝑋 and 𝜎𝑋 , and 𝑌 is the model parameter with mean and standard
deviation values of 𝜇𝑌 and 𝜎𝑌 , the Pearson correlation coefficient between these two random
variables is defined as:
𝜌(𝑋, 𝑌) =
𝐶𝑂𝑉(𝑋,𝑌)
𝜎𝑋 𝜎𝑌
=
𝐸[(𝑋−𝜇𝑋 )(𝑌−𝜇𝑌 )]
𝜎𝑋 𝜎𝑌
(1)
where E is the expected value operator and COV means covariance. The Pearson correlation
is +1 in a perfect increasing (positive) linear relationship, is −1 in the case of a perfect
decreasing (negative) linear relationship, and approaches zero when less of a relation between
variables is expected. It gives some values in between −1 and +1 in all the other cases,
indicating the degrees of linear dependence between the variables. However, it is important to
note that if the variables are independent, Pearson correlation coefficient is zero, but the
converse is not always true (Dowdy and Wearden 1983).
To present the linear correlation between structural seismic demand modification
factors and the considered model parameters considering all possible scenarios, Figure 2
shows the correlation coefficients for all 𝑇𝐹𝐵 values. As illustrated in Figure 2(a)-(c), from all
considered soil, structural, and system parameters only (𝑉𝑠 )0 , 𝑉𝑠 /(𝑉𝑠 )0 , 𝜑 and 𝜎 have more
pronounced linear correlation with structural distortion demand modification factors (i.e.,
𝜌 > 0.5 𝑜𝑟 𝜌 < −0.5). Also, note that the existing linear correlation is slightly higher for
structures with longer periods; however, the difference is not that significant. Considering the
type of correlation (positive or negative) gives the impression that structural distortion
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demand modification factor is reduced when: (i) initial shear wave velocity decreases; (ii)
larger degradation in shear wave velocity occurs; (iii) expected value of 𝜑 increases; and (iv)
expected values of 𝜎 decreases.
When structural total displacement is considered as seismic demand [Figure 2(d)-(f)],
except for stiff structures (𝑇𝐹𝐵 ≤ 0.6 s), no significant linear correlation can be distinguished
for almost all model parameters. When stiff structures are considered, 𝜑 and 𝜎 can show a
small linear correlation.
3.1.2 Nonlinear correlations
To examine the possibility of having nonlinear correlation and dependency between the
demand modification factors and soil, structural, and system parameters, noisiness graphs
showing data directionality are considered. In these graphs, the demand modification factors
for each selected groups of models are presented based on the variation of a certain model
parameter. Examining all possible scenarios, it has been implied that the measured structural
seismic demand modification factors only have an obvious nonlinear relationship and
dependency with the parameters of 𝜑 and 𝜎; and the graphs are quite noisy for all the other
cases. It implicitly concludes that parameters showing almost no linear correlation with the
demand modification factors also have no distinguishable nonlinear dependency.
Avoiding the presentation of any unnecessary information, the noisiness graphs
showing the variation of 𝑢𝑆𝐹𝑆 ⁄𝑢𝐹𝐵 and (𝑢𝑠𝑡𝑟 )𝑆𝐹𝑆 /(𝑢𝑠𝑡𝑟 )𝐹𝐵 based on the variation of 𝜑 and 𝜎
are only illustrated and discussed in the following. Figure 3 shows the relationship between
𝑢𝑆𝐹𝑆 ⁄𝑢𝐹𝐵 and 𝜑 for different 𝑇𝐹𝐵 values; also shown in this figure is the previously presented
Pearson correlation coefficient. Clearly, there is a strong directionality in the presented data.
However, considering the Pearson correlation coefficients located in the range of 0.5 − 0.7
gives the coarse impression that the existing dependency might be assumed as a linear
relationship. This strong directionality is also recognised in the case of relation between
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structural distortion demand modification factor and 𝜎 (Figure 4). But in this case, the
Pearson correlation coefficients do not indicate any strong linear relationship.
If structural total displacement demand modification factors are considered, a nonlinear
correlation between (𝑢𝑠𝑡𝑟 )𝑆𝐹𝑆 /(𝑢𝑠𝑡𝑟 )𝐹𝐵 and 𝜑 can only be distinguished if 𝜑 ≤ 0.5 and after
that point, the dependency will vanish so quickly (Figure 5). However, if the correlation
between (𝑢𝑠𝑡𝑟 )𝑆𝐹𝑆 /(𝑢𝑠𝑡𝑟 )𝐹𝐵 and 𝜎 is examined, a strong nonlinear relationship exists, even
though, no linear correlation has been found (Figure 6).
In summary, on the basis of the presented results, it can be concluded that structural
inelastic demand does not show a systematic dependency on the soil-structure model
parameters except for (𝑉𝑠 )0 , 𝑉𝑠 /(𝑉𝑠 )0 , 𝜑 and 𝜎.
3.2 Variation of Structural Seismic Demand due to change in the Effective Model Parameters
To quantify the variation of structural seismic demand due to change in the recognised
effective model parameters, a robust statistical presentation was adopted. In that regard, the
variation in 𝑢𝑆𝐹𝑆 ⁄𝑢𝐹𝐵 , (𝑢𝑠𝑡𝑟 )𝑆𝐹𝑆 /(𝑢𝑠𝑡𝑟 )𝐹𝐵 were examined as a function of (𝑉𝑠 )0 , 𝑉𝑠 ⁄(𝑉𝑠 )0 , 𝜑,
and 𝜎 considering all the examined scenarios together, regardless of the initial grouping
based on 𝑇𝐹𝐵 . This approach is acceptable since the observed correlation between the demand
modification factors and the selected parameters are almost similar through all the period
values (i.e., the effects of model parameter variation on the structural seismic demand
modification factors are independent of the 𝑇𝐹𝐵 values). In order to carry out this
quantification, the existing dependency of the demand modification factors to the considered
parameters were presented through 5𝑡ℎ , 50𝑡ℎ , 75𝑡ℎ , and 95𝑡ℎ percentile lines representing
different levels of likelihood. The distance between 5𝑡ℎ and 95𝑡ℎ percentile boundary lines
shows the possible variation in the response. The larger this distance is, the higher variation is
expected. The line assigned to the 50𝑡ℎ percentile values shows the trend of the response in
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median terms, and the boundary lines assigned to 75𝑡ℎ , and 95𝑡ℎ percentile values show the
acceptable and high confidence levels.
Figure 7 illustrates the sensitivity of 𝑢𝑆𝐹𝑆 ⁄𝑢𝐹𝐵 to the controlling parameters. Clearly,
when 50% is considered as the expected confidence level, smaller values of (𝑉𝑠 )0 , 𝑉𝑠 /(𝑉𝑠 )0,
and 𝜎 or larger values of 𝜑 can result in smaller structural distortion demand modification
factors (or higher reduction in the structural distortion demand). As the values of (𝑉𝑠 )0 , 𝑉𝑠 /
(𝑉𝑠 )0, and 𝜎 increase or the values of 𝜑 decrease, the demand modification factor approaches
to unity. These observed trends are justified for the case of (𝑉𝑠 )0, 𝜑 and 𝜎 (note that 𝜑 and 𝜎
are highly influenced by (𝑉𝑠 )0 ), as an increase in the shear wave velocity can result in stiffer
foundation condition. Having the foundation stiffer makes the SFS system to show more
similar response to the corresponding fixed-base condition and consequently make 𝑢𝑆𝐹𝑆 ⁄𝑢𝐹𝐵
to approach to unity. If soil shear wave velocity degradation is considered, larger values of
𝑉𝑠 /(𝑉𝑠 )0 correspond to less degradation and as a result less damping is added into the soilstructure system. Thus, larger values of 𝑉𝑠 /(𝑉𝑠 )0 will cause the SFS system to behave in a
more similar way as the corresponding fixed-base condition.
It is also shown in Figure 7 that if smaller values of the effective parameters are
considered, larger variations in the demand modification factors is expected and as the values
of the parameters increase, the variation in the ratio of 𝑢𝑆𝐹𝑆 ⁄𝑢𝐹𝐵 reduces significantly.
Therefore, more attention in a design procedure has to be considered if that range of
parameters is encountered. Also noted in this figure is that the sensitivity of the demand
modification factors to the change of (𝑉𝑠 )0, 𝑉𝑠 /(𝑉𝑠 )0 and 𝜎 will be significantly reduced
considering 75𝑡ℎ and 95𝑡ℎ percentile values. It implicitly indicates that the maximum
expected modification in structural distortion demand is independent of the shear wave
velocity and shear wave velocity degradation ratio for acceptable and high confidence levels.
It also means that, the highest modification in the structural distortion demand may occur for
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any value of (𝑉𝑠 )0, 𝑉𝑠 /(𝑉𝑠 )0 and 𝜎 , even though different probabilities has to be considered.
This conclusion is not valid for the case of 𝜑 noting when the value of 𝜑 increases, structural
distortion demand modification factor decreases even for high levels of confidence levels.
Sensitivity of (𝑢𝑠𝑡𝑟 )𝑆𝐹𝑆 /(𝑢𝑠𝑡𝑟 )𝐹𝐵 to the considered controlling parameters is shown in
Figure 8. For the 50𝑡ℎ percentile values, the ratio of (𝑢𝑠𝑡𝑟 )𝑆𝐹𝑆 /(𝑢𝑠𝑡𝑟 )𝐹𝐵 is only weakly
sensitive to the considered parameters; meaning, in median terms, structural total
displacement demand is not influenced by the change of any model parameters. Furthermore,
median values are slightly higher than unity implying that SSI effects on structural total
displacement is always increasing and it has to be taken into account in a design procedure.
When higher levels of confidence are considered, different interpretations appear. For an
increase in the value of (𝑉𝑠 )0 , 𝑉𝑠 /(𝑉𝑠 )0, and 𝜎, the ratio of (𝑢𝑠𝑡𝑟 )𝑆𝐹𝑆 /(𝑢𝑠𝑡𝑟 )𝐹𝐵 tends to
decrease sharply while approaching to unity. It indirectly implies that the variation in the
structural total displacement demand modification factor decreases along with the reduction
in the possibility of having a large modification in the seismic demand. In contrast, when 𝜑
increases the observed variation in the ratio of (𝑢𝑠𝑡𝑟 )𝑆𝐹𝑆 /(𝑢𝑠𝑡𝑟 )𝐹𝐵 increases and consequently
the expected demand modification factor increase.
3.3 Risk Evaluation for SSI Based on Variation in Model Parameters
To evaluate the risk of detrimental SSI effects on structural seismic demand based on the
variation in the effective model parameters, two main aspects are analysed: (i) the probability
of having amplification in the demand of the SFS system as compared to the demand of the
corresponding FB structure; and (ii) the level of increase in the demand due to SSI
consideration. For this purpose, three amplification levels (A.L.) were taken into account as:
1.0, 1.1, and 1.2, and the probability of having scenarios with the demand modification factor
higher than each level were calculated. For each considered A.L., the corresponding median
values of the percentage increase in the response (Med[P.I.]) were also evaluated.
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The probability of having amplification in the structural distortion demand
(Pr[𝑢𝑆𝐹𝑆 ⁄𝑢𝐹𝐵 > 𝐴. 𝐿.]) is shown in Figure 9 for the variation in (𝑉𝑠 )0 , 𝑉𝑠 /(𝑉𝑠 )0 , 𝜑, and 𝜎. The
corresponding median percentage increases are presented in Figure 10. When amplification
level is considered to be 1.0, the probability of 𝑢𝑆𝐹𝑆 ⁄𝑢𝐹𝐵 > 1.0 increases as a result of an
increase in the value of (𝑉𝑠 )0 , 𝑉𝑠 ⁄(𝑉𝑠 )0, and 𝜎. Note that this increase is more significant in
the case of 𝜎. In contrast to the above mentioned, an increase in the value of 𝜑 will
significantly reduce the probability of amplification in the structural distortion demand. From
the observed trends, it can be concluded that the probability of having SSI with detrimental
effects is higher when structures are located on stiffer soils, and also when smaller shear
wave velocity degradation is expected. The probability of increase in the structural distortion
demand due to SSI ranges between 15% − 50% for (𝑉𝑠 )0 , 10% − 50% for 𝑉𝑠 ⁄(𝑉𝑠 )0, 10% −
90% for 𝜎 and 1% − 50% for 𝜑. It is important to note that even though the probability of
having detrimental SSI effects increases due to an increase in values of (𝑉𝑠 )0 , 𝑉𝑠 ⁄(𝑉𝑠 )0 , and 𝜎,
the corresponding values for median percentage increase decrease. It means that a higher
amplification in structural response due to SSI is expected when structures are located on
softer soils and obviously when less degradation is expected. Values of the median
percentage increase are in the range of 1% − 10% for (𝑉𝑠 )0 , 1% − 5% for 𝑉𝑠 ⁄(𝑉𝑠 )0, 5 −
20% for 𝜑, and 1-10% for 𝜎. Respecting these values, gives the impression that, in median
terms, SSI effects on structural distortion demand can be limited to 20%.
Provided that higher amplification levels are considered (i.e., A.L.=1.1 or A.L.=1.2),
the probability of amplification in the demand will be considerably reduced for all
parameters. Specifically, when 𝜎 > 20 and 𝜑 > 1.0 any detrimental effect of SSI on
structural distortion demand may be practically ignored. Furthermore, by taking the median
values of percentage increase into account, it can be concluded that the amplification in the
13 | P a g e
Sensitivity Analysis for Soil-Structure Systems
structural distortion demand caused by SSI effects is limited to 30% for all the effective
model parameters.
When structural total displacement is considered as the structural seismic demand and
A.L. is assumed to be unity, an increase in the value of all considered parameters results in an
increase in the probability of amplification (Figure 11). Thus, it can be concluded that
detrimental SSI effects on structural total displacement are more probable to occur when
stiffer soils are considered. The observed probability values are as 65% − 75% for (𝑉𝑠 )0 ,
60% − 75% for 𝑉𝑠 ⁄(𝑉𝑠 )0, 55% − 75% for 𝜑, and 60% − 90% for 𝜎. In terms of the
corresponding median values of percentage increase, a reduction is expected when the values
of (𝑉𝑠 )0 , 𝑉𝑠 ⁄(𝑉𝑠 )0, and 𝜎 increase. In contrast, an increase in the value of 𝜑 will result in
higher median values of percentage increase. This conclusion is sensible since the rigid body
motion components originating from the soil deformation are greater when softer soils are
considered. The observed median values of percentage increase are as 5% − 35% for (𝑉𝑠 )0,
5% − 20% for 𝑉𝑠 ⁄(𝑉𝑠 )0, 5% − 75% for 𝜑, and 1-30% for 𝜎. The observed high probabilities
of amplification in the structural total displacement accompanied with the high values of
median percentage increase strongly emphasize that the effect of SSI on the structural total
displacement cannot be simply ignored in a design procedure.
If higher levels of amplification are considered (i.e., A.L.=1.1 or A.L.=1.2), an increase
in the values of (𝑉𝑠 )0 , 𝑉𝑠 ⁄(𝑉𝑠 )0 , or 𝜎 equals to having smaller probability values, while an
increase in the value of 𝜑 still result in an increase in the probability values. For the case of
(𝑉𝑠 )0 , 𝑉𝑠 ⁄(𝑉𝑠 )0 , or 𝜎, the amplification in the system drift demand due to SSI effects is limited
to 50% while 100% amplification in demand might be expected for 𝜑.
14 | P a g e
Sensitivity Analysis for Soil-Structure Systems
4 CONCLUSIONS
1. From all considered soil, structural, and system parameters only (𝑉𝑠 )0 , 𝑉𝑠 /(𝑉𝑠 )0 , 𝜑 and
𝜎 have a pronounced linear correlation with structural distortion demand. However, no
significant linear correlation is noticeable between these parameters and system drift
demand.
2. In median terms (50% is considered as the expected confidence level), any increase in
the values of (𝑉𝑠 )0 , 𝑉𝑠 /(𝑉𝑠 )0, or 𝜎 results in an increase in the structural distortion demand
modification factor (uSFS /uFB ), whereas, an increase in the value of 𝜑 reduces the ratio
of 𝑢𝑆𝐹𝑆 ⁄𝑢𝐹𝐵 significantly. However, when higher confidence levels are taken account, the
sensitivity of the demand modification factors to the change of the parameters will be
reduced.
3. In median terms (50% is considered as the expected confidence level), structural total
displacement demand modification factor (drSFS /drFB ) is only weakly sensitive to the
considered parameters. But, when higher levels of confidence are considered, different
interpretations appear. For an increase in the value of (Vs )0 , Vs /(Vs )0, and σ, the ratio of
drSFS /drFB tends to decrease very sharply, in contrast, when the value of φ increases,
drSFS /drFB also increases.
4. When amplification level of unity is considered, an increase in the value of
(𝑉𝑠 )0 , 𝑉𝑠 ⁄(𝑉𝑠 )0 , or 𝜎 may result in an increase in the probability of amplification in the
structural distortion demand, and this increase is much more significant for the case of 𝜎.
However, an increase in the value of 𝜑 will significantly reduce that probability. If higher
amplification levels are considered, the probability of amplification in the demand will be
considerably reduced for all parameters. Specifically, when 𝜎 > 20 and 𝜑 > 1.0 any
detrimental effect of SSI on the structural distortion may be practically ignored.
15 | P a g e
Sensitivity Analysis for Soil-Structure Systems
5. When structural total displacement is considered as the structural seismic demand and
amplification level is assumed to be unity, an increase in the value of each considered
parameter results in an increase in the probability of amplification; while for higher levels
of amplification, an increase in the values of (𝑉𝑠 )0 , 𝑉𝑠 ⁄(𝑉𝑠 )0 , or 𝜎 equals to having smaller
probability values, and an increase in the value of 𝜎 still result in an increase in the
probability values.
16 | P a g e
Journal of Earthquake Engineering (2011)
Paper Title: Sensitivity Analysis for Soil-Structure Interaction Phenomenon using Stochastic Approach
Authors:
Moghaddasi, Cubrinovski, Pampanin, Chase, Carr
Figure 1. Soil-shallow foundation-structure model for horizontal and rocking foundation
motions.
17 | P a g e
Journal of Earthquake Engineering (2011)
Paper Title: Sensitivity Analysis for Soil-Structure Interaction Phenomenon using Stochastic Approach
Authors:
Moghaddasi, Cubrinovski, Pampanin, Chase, Carr
1
1
TKDS
/(u ) ,Y)
TKDS
str FB
0.5
str SFS
0
-0.5
-1
0.2 0.4 0.6 0.8
(a)
1
T
((u )
(u
SFS
FB
/u ,Y)
0.5

(Vs)0

V /(V )
s
-0.5
s 0
-1
0.2 0.4 0.6 0.8
1.2 1.4 1.6 1.8
(d)
FB
m
0
-0.5
r
h /r
1.2 1.4 1.6 1.8
1
T
1.2 1.4 1.6 1.8
0.5
str FB
eff
str SFS
FB
/u ,Y)
SFS
(u
-1
0.2 0.4 0.6 0.8
(b)
1
T
0
-0.5
-1
0.2 0.4 0.6 0.8
1.2 1.4 1.6 1.8
(e)
FB
FB
1
1
TKDS
0.5
((u )
str SFS
str FB
FB



/(u ) ,Y)
str
0
-0.5
-1
0.2 0.4 0.6 0.8
TKDS
3
m /r
0.5
/u ,Y)
1
T
TKDS
str
/(u ) ,Y)
eff
((u )
h
0.5
SFS
1.2 1.4 1.6 1.8
1
TKDS
(u
1
T
FB
1
(c)
0
1
T
FB
0
-0.5
-1
0.2 0.4 0.6 0.8
1.2 1.4 1.6 1.8
(f)
FB
Figure 2. Pearson correlation coefficients: (a-c) correlation between considered parameters
and structural distortion demand modification factors; (d-f) correlation between considered
parameters and structural total displacement demand modification factors.
18 | P a g e
uSFS ⁄uFB
Journal of Earthquake Engineering (2011)
Paper Title: Sensitivity Analysis for Soil-Structure Interaction Phenomenon using Stochastic Approach
Authors:
Moghaddasi, Cubrinovski, Pampanin, Chase, Carr
𝜑
Figure 3. Correlation and dependence between structural distortion demand and 𝜑 =
0.25
ℎ𝑒𝑓𝑓 ⁄𝑉𝑠 𝑇𝐹𝐵 (ℎ𝑒𝑓𝑓 ⁄𝑟) .
19 | P a g e
uSFS⁄uFB
Journal of Earthquake Engineering (2011)
Paper Title: Sensitivity Analysis for Soil-Structure Interaction Phenomenon using Stochastic Approach
Authors:
Moghaddasi, Cubrinovski, Pampanin, Chase, Carr
𝜎
Figure 4. Correlation and dependence between structural distortion demand and
𝜎 = 𝑉𝑠 𝑇𝐹𝐵 ⁄ℎ𝑒𝑓𝑓 .
20 | P a g e
(u𝑠𝑡𝑟 )SFS ⁄(u𝑠𝑡𝑟 )FB
Journal of Earthquake Engineering (2011)
Paper Title: Sensitivity Analysis for Soil-Structure Interaction Phenomenon using Stochastic Approach
Authors:
Moghaddasi, Cubrinovski, Pampanin, Chase, Carr
𝜑
Figure 5. Correlation and dependence between structural total displacement demand and 𝜑 =
0.25
ℎ𝑒𝑓𝑓 ⁄𝑉𝑠 𝑇𝐹𝐵 (ℎ𝑒𝑓𝑓 ⁄𝑟) .
21 | P a g e
(u𝑠𝑡𝑟 )SFS ⁄(u𝑠𝑡𝑟 )FB
Journal of Earthquake Engineering (2011)
Paper Title: Sensitivity Analysis for Soil-Structure Interaction Phenomenon using Stochastic Approach
Authors:
Moghaddasi, Cubrinovski, Pampanin, Chase, Carr
𝜎
Figure 6. Correlation and dependence between structural total displacement demand and
𝜎 = 𝑉𝑠 𝑇𝐹𝐵 ⁄ℎ𝑒𝑓𝑓 .
22 | P a g e
Journal of Earthquake Engineering (2011)
Paper Title: Sensitivity Analysis for Soil-Structure Interaction Phenomenon using Stochastic Approach
Authors:
Moghaddasi, Cubrinovski, Pampanin, Chase, Carr
1.5
1.5
TKDS
TKDS
1
u
SFS
/u
FB
uSFS/uFB
1
0.5
0
5th Prct.
50th Prct.
75th Prct.
95th Prct.
100
(a)
200
(Vs)0 (s)
300
0.5
0
0.1
400
1.5
0.4 0.5
Vs/(Vs)0
0.6
0.7
TKDS
/u
SFS
u
SFS
/u
FB
1
FB
1
u
0.3
1.5
TKDS
0.5
0
0
(c)
0.2
(b)
0.5
1

1.5
2
0.5
0
0
2.5
(d)
5
10
15

20
25
30
35
Figure 7. Sensitivity of structural distortion demand to: (a) initial shear wave velocity; (b) soil
0.25
shear wave velocity degradation ratio; (c) 𝜑 = ℎ𝑒𝑓𝑓 ⁄𝑉𝑠 𝑇𝐹𝐵 (ℎ𝑒𝑓𝑓 ⁄𝑟) ; (d) 𝜎 = 𝑉𝑠 𝑇𝐹𝐵 ⁄ℎ𝑒𝑓𝑓 .
23 | P a g e
Journal of Earthquake Engineering (2011)
Paper Title: Sensitivity Analysis for Soil-Structure Interaction Phenomenon using Stochastic Approach
Authors:
Moghaddasi, Cubrinovski, Pampanin, Chase, Carr
2.5
2.5
TKDS
TKDS
2
1.5
1.5
uSFS/uFB
(u )
str SFS
/(u )
str FB
2
1
5th Prct.
50th Prct.
75th Prct.
95th Prct.
0.5
0
100
(a)
200
(Vs)0 (s)
300
1
0.5
0
0.1
400
TKDS
0.6
0.7
TKDS
str FB
/(u )
str FB
1.5
str SFS
1
(u )
/(u )
0.4 0.5
Vs/(Vs)0
2
1.5
str SFS
(u )
0.3
2.5
2
0.5
(c)
0.2
(b)
2.5
0
5th Prct.
50th Prct.
75th Prct.
95th Prct.
1
0.5
0
0.5
1

1.5
2
0
2.5
(d)
0
5
10
15

20
25
30
35
Figure 8. Sensitivity of system drift demand to:(a) initial shear wave velocity; (b) soil shear
0.25
wave velocity degradation ratio; (c) 𝜑 = ℎ𝑒𝑓𝑓 ⁄𝑉𝑠 𝑇𝐹𝐵 (ℎ𝑒𝑓𝑓 ⁄𝑟) ; (d) 𝜎 = 𝑉𝑠 𝑇𝐹𝐵 ⁄ℎ𝑒𝑓𝑓 .
24 | P a g e
Journal of Earthquake Engineering (2011)
Paper Title: Sensitivity Analysis for Soil-Structure Interaction Phenomenon using Stochastic Approach
Authors:
Moghaddasi, Cubrinovski, Pampanin, Chase, Carr
100
100
40
20
80
80
FB
60
0
SFS
SFS
FB
60
Pr[u
TKDS
/u >A.L.] (%)
A.L.=1.0
A.L.=1.1
A.L.=1.2
Pr[u
/u >A.L.] (%)
TKDS
75
150
(a)
225
300
(V ) (m/sec)
40
20
0
0.1
375
s 0
100
0.7
40
/u >A.L.] (%)
TKDS
20
Pr[u
FB
60
SFS
FB
/u >A.L.] (%)
SFS
Pr[u
0.6
s 0
TKDS
(c)
0.3
0.4
0.5
Vs/(V ) (m/sec)
100
80
0
0.2
(a)
0
0.5
1

1.5
2
80
60
40
20
0
2.5
(d)
0
5
10
15

20
25
30
35
Figure 9. Probability of amplification in structural distortion demand based on variation in:
(a) initial shear wave velocity; (b) soil shear wave velocity degradation ratio; (c) 𝜑 =
0.25
ℎ𝑒𝑓𝑓 ⁄𝑉𝑠 𝑇𝐹𝐵 (ℎ𝑒𝑓𝑓 ⁄𝑟) ; (d) 𝜎 = 𝑉𝑠 𝑇𝐹𝐵 ⁄ℎ𝑒𝑓𝑓 .
25 | P a g e
Journal of Earthquake Engineering (2011)
Paper Title: Sensitivity Analysis for Soil-Structure Interaction Phenomenon using Stochastic Approach
Authors:
Moghaddasi, Cubrinovski, Pampanin, Chase, Carr
100
100
SFS
/u >A.L.
u
FB
60
40
75
150
225
300
(Vs)0 (m/sec)
40
0
0.1
375
0.3
0.4
0.5
V /(V )
s
0.6
0.7
s 0
100
u
SFS
/u >A.L.
u
FB
80
Med [P.I.] (%)
80
Med [P.I.] (%)
0.2
(b)
100
60
40
20
(c)
FB
20
(a)
0
0
/u >A.L.
60
20
0
SFS
80
Med [P.I.] (%)
80
Med [P.I.] (%)
u
A.L.=1.0
A.L.=1.1
A.L.=1.2
SFS
/u >A.L.
FB
60
40
20
0.5
1

1.5
2
0
0
2.5
(d)
5
10
15

20
25
30
35
Figure 10. Median of percentage increase in structural distortion demand based on variation
in: (a) initial shear wave velocity; (b) soil shear wave velocity degradation ratio; (c) 𝜑 =
0.25
ℎ𝑒𝑓𝑓 ⁄𝑉𝑠 𝑇𝐹𝐵 (ℎ𝑒𝑓𝑓 ⁄𝑟) ; (d) 𝜎 = 𝑉𝑠 𝑇𝐹𝐵 ⁄ℎ𝑒𝑓𝑓 .
26 | P a g e
Journal of Earthquake Engineering (2011)
Paper Title: Sensitivity Analysis for Soil-Structure Interaction Phenomenon using Stochastic Approach
Authors:
Moghaddasi, Cubrinovski, Pampanin, Chase, Carr
100
TKDS
A.L.=1.0
A.L.=1.1
A.L.=1.2
str FB
60
str SFS
40
20
0
75
150
(a)
225
300
(V ) (m/sec)
20
0.2
0.3
0.4
0.5
V /(V )
s
0.6
0.7
s 0
/(u ) >A.L.] (%)
100
TKDS
80
str FB
60
str SFS
40
Pr[(u )
str FB
/(u ) >A.L.] (%)
str SFS
Pr[(u )
40
(b)
100
(c)
60
0
0.1
375
s 0
20
0
TKDS
80
Pr[(u )
Pr[(u )
str SFS
str FB
80
/(u ) >A.L.] (%)
/(u ) >A.L.] (%)
100
0
0.5
1

1.5
2
TKDS
80
60
40
20
0
2.5
(d)
0
5
10
15

20
25
30
35
Figure 11. Probability of amplification in structural total displacement demand based on
variation in: (a) initial shear wave velocity; (b) soil shear wave velocity degradation ratio; (c)
0.25
𝜑 = ℎ𝑒𝑓𝑓 ⁄𝑉𝑠 𝑇𝐹𝐵 (ℎ𝑒𝑓𝑓 ⁄𝑟) ; (d) 𝜎 = 𝑉𝑠 𝑇𝐹𝐵 ⁄ℎ𝑒𝑓𝑓 .
27 | P a g e
Journal of Earthquake Engineering (2011)
Paper Title: Sensitivity Analysis for Soil-Structure Interaction Phenomenon using Stochastic Approach
Authors:
Moghaddasi, Cubrinovski, Pampanin, Chase, Carr
100
100
A.L.=1.0
A.L.=1.1
A.L.=1.2
str SFS
/(u ) >A.L.
(u )
str FB
60
40
75
150
225
300
(Vs)0 (m/sec)
40
0
0.1
375
0.3
0.4
0.5
V /(V )
s
0.6
0.7
s 0
100
(u )
str SFS
/(u ) >A.L.
(u )
str FB
str SFS
80
Med [P.I.] (%)
80
Med [P.I.] (%)
0.2
(b)
100
60
40
20
(c)
str FB
20
(a)
0
0
/(u ) >A.L.
60
20
0
str SFS
80
Med [P.I.] (%)
80
Med [P.I.] (%)
(u )
/(u ) >A.L.
str FB
60
40
20
0.5
1

1.5
2
0
0
2.5
(d)
5
10
15

20
25
30
35
Figure 12. Median of percentage increase in structural total displacement demand based on
variation in: (a) initial shear wave velocity; (b) soil shear wave velocity degradation ratio; (c)
0.25
𝜑 = ℎ𝑒𝑓𝑓 ⁄𝑉𝑠 𝑇𝐹𝐵 (ℎ𝑒𝑓𝑓 ⁄𝑟) ; (d) 𝜎 = 𝑉𝑠 𝑇𝐹𝐵 ⁄ℎ𝑒𝑓𝑓 .
28 | P a g e
Sensitivity Analysis for Soil-Structure Systems
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