Post on 25-Feb-2018
INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING
Volume 6, No 1, 2015
© Copyright by the authors - Licensee IPA- Under Creative Commons license 3.0
Research article ISSN 0976 – 4399
Received on May, 2015 Published on August 2015 11
Analyzing the effect of foundation inhomogeneity on the seismic response
of gravity dams Joshi S.G1, Gupta I.D2, Murnal P.B3
1- Associate Professor, Civil Eng. Dept., KLEIT, Hubli, Karnataka, India.
2- Honorary Fellow, Indian Institute of Technology, Roorkee, Uttarakhand, India.
3- Professor, Appld. Mech. Dept., Govt. College of Eng., Aurangabad, Maharashtra, India.
sharadgjoshi@rediffmail.com
doi:10.6088/ijcser.6002
ABSTRACT
Very comprehensive investigations are carried out in this paper to show that the soil-structure
interaction effects on the seismic response of gravity dams can be modeled accurately simply
by approximating a horizontally layered foundation with an equivalent homogeneous
foundation. For this purpose, a dam cross section along with a finite size of foundation block
is analyzed by standard finite element method (FEM) for a large number of hypothetical
cases of foundation inhomogeneity defined by two and three layers with different
combinations of thicknesses and impedance contrasts using FEM software ABAQUS. The
displacements and the maximum principal stresses induced under excitation by typical
horizontal and vertical acceleration time histories of a real earthquake are compared with the
corresponding results obtained for an equivalent homogeneous foundation in each case. Very
good matching between the two results has established that the heterogeneity of foundation
block does not much govern the soil structure interaction effects on the dam response, and the
use of an equivalent homogeneous foundation is sufficient in practical engineering
applications. The conclusions are not expected to be dependent on the characteristics of the
input excitation.
Keyword: Gravity dam, soil-structure interaction, acceleration time history, impedance ratio,
foundation inhomogeneity.
1. Introduction
Soil-structure interaction (SSI) is known to influence the response of gravity dams
considerably with decreasing modulus of elasticity of the foundation rock (Ef) with respect to
the modulus for the dam material (Ed). The ratio Ef/Ed of the two moduli represents the
impedance contrast between the foundation rock and the dam. Lin et al. (1988, 2004) have
reported reduction in the stresses in the dam body with decrease in the impedance contrast
owing to lengthening of the fundamental period and increase in the damping ratio of the
dam-foundation system. Lokke and Chopra (2013) have given simple relationships for the
period lengthening ratio and the addition to the damping for different values of Ef/Ed. Some
other studies (e.g.; Sarkar et al., 2007; Heirany and Ghaemian, 2012) have shown that
displacement at the top increases with decrease in the ratio of modulus of foundation rock
and concrete.
The widely used standard FEM approach analyses the dam and a sufficiently large portion of
the foundation together to account for the soil-structure interaction effects. The reason given
in support of using this approach is that it considers the effect of the inhomogeneities in the
foundation. However, to model the SSI effects accurately, in this approach, the free-field
Analyzing the effect of foundation inhomogeneity on the seismic response of gravity dams
Joshi S.G et al.,
International Journal of Civil and Structural Engineering 12
Volume 6 Issue 1 2015
ground motion is required to be deconvoluted and applied at the base of the foundation block
with its mass taken into account (Joshi et al., 2014). As the deconvolution can be performed
for a horizontally layered foundation block only, the foundation has to be idealized in terms
of horizontal layers to use the standard FEM approach. On the other hand, for homogeneous
foundation, the sub-structure approach (Fenves and Chopra, 1984; Gupta, 2010) can be used
to get more accurate and physically realistic estimate of the soil structure interaction effects
in a very convenient way by replacing the effect of the foundation by a frequency-dependent
complex stiffness matrix connected with the base nodes (Chopra et al., 1976; Dasgupta and
Chopra, 1979). This paper has therefore carried out investigations to show that a horizontally
layered foundation can be idealized by an equivalent homogeneous foundation, so that the
sub-structure approach could be applied.
In the present investigations, displacements and maximum principal stresses in a dam cross
section are computed using FEM approach for several different cases of horizontally layered
foundation and comparing the results with those obtained for the corresponding cases of
equivalent homogeneous foundation. The results in both the cases are computed by applying
the free-field ground motion as it is at the base of the foundation block. This is done for
reasons of simplicity, as the purpose is only to examine the matching between the results for
the inhomogeneous and the equivalent homogeneous foundations and not to quantify the SSI
effects accurately for a particular case. This also helps to avoid the unrealistic differences
between the two results due to the approximations in the process of deconvolution of ground
motion. Very good qualitative and quantitative matching between the results on the dam
responses for inhomogeneous and equivalent homogeneous foundations has established that a
layered foundation can well be replaced by an equivalent homogeneous foundation. The
equivalent homogeneous foundation can be used to analyze the response of dam by
substructure approach, which is able to model both the soil structure interaction and
hydrodynamic effects more accurately and in a more realistic way.
2. Formulation of the problem
To compute the numerical results in the present study, an atypical dam cross section, with
geometry and dimensions similar to that of the tallest non-overflow section of Koyna dam,
India has been considered along with a foundation block of size 5B×2H, where B is the base
width and H the height of the dam (ICOLD, 2001). This section has a height of 103 m and
base width of 70.2 m with unusually large freeboard and top width, which is atypical of a
gravity dam. The lateral size of the foundation block considered is equal to two times the
base width on either side of the dam base and it extends below the dam base to a depth of two
times the dam height. Thus, the foundation block has a size of 206 m × 351 m. Figure 1
shows a 2D FEM idealization for the dam and the foundation block used to compute the
response by standard FEM method.
The foundation block is assumed to be fixed rigidly at the base and only horizontal
displacements are permitted on the vertical sides. The large extent of the foundation
considered (5B ×2H) is expected to minimize the wave reflections from the base into the dam.
Commercial FEM solver program ABAQUS (Version 6.14; 2014) is used to analyze the dam
response. The dam section is discretised into 66 elements and the foundation block into 714,
approximately equal sized first order plane stress quadrilateral iso-parametric elements
(CPS4R) with two degrees of freedom at each node and analyzed under 2D elastic condition.
As explained before, the horizontal and vertical components of the free-field ground motion
are applied at the base of the foundation block, without performing any deconvolution, to
Analyzing the effect of foundation inhomogeneity on the seismic response of gravity dams
Joshi S.G et al.,
International Journal of Civil and Structural Engineering 13
Volume 6 Issue 1 2015
compute all the present results. To compute the numerical results, density of the dam concrete
(γd) has been taken as 2650 kg/m3, which is the average value for the entire section as per the
completion report of the Koyna project. The dynamic values of Young’s modulus of elasticity
(Ed) and Poisson’s ratio (νd) for the dam concrete are taken as 44,950 MPa and 0.2,
respectively; which are the experimental values based on the testing of cores extracted from
different locations in the dam body (CWPRS, 2002; Gaikwad et al., 2003). The viscous
damping ratio (ξ) of 5% has been assumed for the present analysis.
Two different scenarios comprising two and three horizontal layers are considered to
synthesize a large number of cases of foundation heterogeneity. The various cases of
inhomogeneous foundation are generated by varying the thickness and the elastic modulus of
the individual layers, keeping the total thickness of the foundation block fixed at 206 m. The
density (γf), Poisson’s ratio (νf), and damping ratio (ξ) for the foundation are also kept
constant equal to 2800 kg/m3, 0.2, and 5%, respectively. The modulus of elasticity (Ef) for
various foundation layers is defined as factor of the modulus of elasticity (Ed) of the dam
concrete, which can be considered to represent the impedance contrast. With )/( dfi EE as
the impedance contrast and hi as the thickness of the ith foundation layer, the impedance
contrast for the equivalent homogeneous foundation is defined by the following expression:
∑∑=
i
i
i
dfiieqdf hEEhEE )/()/(
In the standard FEM analysis, the input acceleration time histories of ground motion are
applied at the base of the foundation block. As the input excitation gets amplified due to the
inertial force of the foundation, the free field design ground motion is required to be
deconvoluted to get accurate estimation of the response with SSI effects accounted. As
explained before, the objective of the present study is served by applying the free-field
ground accelerations directly at the base of the foundation without deconvolution. To
compute the numerical results, the longitudinal horizontal and the vertical components of 10th
December 1967 Koyna earthquake with Magnitude 6.5 and focal depth 12.0 km, recorded at
foundation gallery of the dam at epicentral distance of 12.6 km have been used. These
components are characterized by corrected peak ground acceleration (PGA) values of 0.49g
and 0.24g, respectively. The time history of the ground accelerations of both these
components of input ground motion are shown in Figure 2.
Figure 1: 2-D FEM idealization for the tallest non-overflow section of Koyna dam along
with the foundation block (Abaqus model).
Analyzing the effect of foundation inhomogeneity on the seismic response of gravity dams
Joshi S.G et al.,
International Journal of Civil and Structural Engineering 14
Volume 6 Issue 1 2015
Figure 2: Horizontal and vertical components of Koyna dam accelerogram used as input
excitation for computation of illustrative results.
3. Results and discussion for two layer scenario
The two layer scenario is illustrated schematically in Figure 3. The various cases of
inhomogeneous foundation in this scenario are generated by varying the thickness of the top
layer to 10, 20, 30 and 70 m, and considering three different combinations of the impedance
contrasts Ef1/Ed and Ef2/Ed of the two layers for each case of thickness. The values of Ef1/Ed
are taken as 0.5, 1.0 and 2.0 with the corresponding Ef2/Ed as 1.0, 2.0 and 5.0, respectively.
This gives a total of 12 different cases of foundation heterogeneity as listed in Table-1. For
each of the 12 cases in Table-1, the impedance contrast (Ef/Ed)eq for the equivalent
homogeneous foundation block of thickness 206 m is also given.
Figure 3: Heterogeneous foundation with two horizontal layers.
Analyzing the effect of foundation inhomogeneity on the seismic response of gravity dams
Joshi S.G et al.,
International Journal of Civil and Structural Engineering 15
Volume 6 Issue 1 2015
Table 1: Foundation heterogeneity modeled by two horizontal layers with different
combinations of thicknesses and moduli of elasticity.
Case
No
First Layer Second Layer Equivalent
(Ef/Ed)eq h1 (m) Ef1/Ed h2 (m) Ef2/Ed
1 10 0.50 196 1.00 0.9757
2 10 1.00 196 2.00 1.9514
3 10 2.00 196 5.00 4.8544
4 20 0.50 186 1.00 0.9514
5 20 1.00 186 2.00 1.9029
6 20 2.00 186 5.00 4.5146
7 30 0.50 176 1.00 0.9272
8 30 1.00 176 2.00 1.8544
9 30 2.00 176 5.00 4.5631
10 70 0.50 136 1.00 0.8301
11 70 1.00 136 2.00 1.6642
12 70 2.00 136 5.00 3.9806
The displacements and maximum principal stresses throughout the dam section are
computed for all the 12 cases listed in Table-l for both the layered and the equivalent
homogeneous foundation. Figure 4 shows for all the 12 cases the maximum absolute
displacements on the upstream face of the dam for the two layered heterogeneous foundation
compared with those for the corresponding equivalent homogeneous foundation. The
displacements for the equivalent homogeneous foundation are, in general, seen to have
similar trend and very good numerical agreement with the results for the heterogeneous
foundation in all the widely varying cases.
From the results in Figure 4, it is seen that the maximum displacement amplitudes as well as
the difference between the exact and approximate displacement responses decrease with
increasing modulus of elasticity of the upper layer for a fixed thickness of upper layer. On the
other hand, for fixed modulus of the upper layer, the maximum displacement amplitudes as
well as the difference between the two results is seen to increase with increase in the
thickness of the upper layer. These observations indicate that the decrease in the impedance
contrast and increase in the thickness of the upper layer result in higher displacement
response due to increased SSI effects and larger differences between the exact and
approximate results. These differences can perhaps be reduced by assigning higher weight to
the upper layer in defining the equivalent impedance contrast. But, the differences are so
small for all the cases that the approximation of equivalent homogeneous foundation can be
considered very good for practical purposes. The use of equivalent homogeneous foundation
is also able to provide accurate estimates of the stresses in the dam as shown the next.
Analyzing the effect of foundation inhomogeneity on the seismic response of gravity dams
Joshi S.G et al.,
International Journal of Civil and Structural Engineering 16
Volume 6 Issue 1 2015
Figure 4: Comparison of the exact and approximate results of maximum horizontal
displacements on upstream face of the dam for different cases of heterogeneities of two
layered foundation.
Analyzing the effect of foundation inhomogeneity on the seismic response of gravity dams
Joshi S.G et al.,
International Journal of Civil and Structural Engineering 17
Volume 6 Issue 1 2015
Figure 5: Comparison of Maximum Principal stresses on the upstream face of the dam for
various cases of 2-layered heterogeneous foundation with the corresponding cases of
equivalent foundation.
The comparison of the absolute maximum values of the maximum principal stresss on the
upstream face of the dam obtained for the layered foundation with those for the
Analyzing the effect of foundation inhomogeneity on the seismic response of gravity dams
Joshi S.G et al.,
International Journal of Civil and Structural Engineering 18
Volume 6 Issue 1 2015
corresponding equivalent homogeneous foundation are shown in Fig. 5 for all the 12 cases
listed in Table-1. Similar to that for the displacement response, the equivalent homogeneous
foundation is seen to provide the maximum principal stresses in very good agreement with
those based on the layered foundation. However, unlike the displacement response, the stress
amplitudes are not seen to be affected that significantly by increasing impedance contrast
and thickness of the top layer. The maximum principal stress on the upstream face of the
dam is seen to increase monotonically from almost zero at the top to some large value at the
base of the dam with a very strong hump in-between, near the neck of the dam at elevation
of about 70 m.
Very high concentration of stresses on both upstream and downstream sides near the
neck of the dam is generally obtained in gravity dams due to sudden change in the
downstream slope (Burman, et al., 2010). For the case of Koyna dam, this was in fact the
location where extensive cracking occurred during the 1967 Koyna earthquake (Chopra and
Chakrabarti, 1973). The Koyna dam section is of atypical shape with unusually high
concentration of mass above the neck due to large freeboard and top width. To have more
detailed investigation of the ability of the equivalent homogeneous foundation to predict
accurately the stresses in the dam section, Fig. 6 compares the stress contours based on exact
two layered fondation (left side plots) with those based on the approximate equivalent
homogeneous foundation (right side plots) for several typical cases listed in Table-1. These
are the instantaneous stress contours at the moment when the stress value has the largest
absolute value on the downstream point of change of slope.
Figure 6: Comparison of the distributions of major principal stress in the dam section for
several cases of the two layered heterogeneous foundation (left hand side plots) with those for
the corresponding equivalent homogeneous foundation (right hand side plots).
Analyzing the effect of foundation inhomogeneity on the seismic response of gravity dams
Joshi S.G et al.,
International Journal of Civil and Structural Engineering 19
Volume 6 Issue 1 2015
From the results in Figure 6, the distributions of major principal stress obtained using layered
and the equivalent homogeneous foundation are seen to be strikingly similar and in very good
agreement for widely differing cases of inhomogeneous foundation. The distribution of
stresses in both the cases is seen to be physically very realistic with a zone of high stress near
the neck. This further confirms the applicability of the equivalent homogeneous foundation in
response analysis of gravity dams. As explained before, such high values of stress were also
indicated around this height on the upstream face (Figure 5).
4. Results and discussion for three layer scenario
To further strengthen the conclusions made from the results for two layer scenario, a three
layered scenario is considered the next for the foundation in homogeneity as shown
schematically in Figure 7. In this scenario, the thicknesses of the layers are kept fixed as
h1=20 m, h2=50 m and h3=136 m, and six different cases of heterogeneous foundation are
synthesized by selecting different combinations of the impedance contrasts Ef2/Ed and Ef3/Ed
for the lower two layers with the impedance contrast of top layer Ef1/Ed kept fixed as 0.5. The
results for the two-layered cases were seen to be controlled more significantly by the
impedance contrast of the top layer, and thus the differences between the exact and
approximate results were also attributed to the impedance contrast of the equivalent
homogeneous foundation. To investigate this in further details, the impedance contrast of the
top layer is fixed at a low value and varied only for the lower two layers in the three layer
scenario. The details of the six cases of three layer scenario along with the impedance
contrast for the equivalent homogeneous foundation are given in Table-2.
Figure 7: Heterogeneous foundation with three horizontal layers.
Table 2: Foundation heterogeneity modeled by three horizontal layers with fixed thicknesses
and different combinations of moduli of elasticity. Case
No.
h1=20m h2=50m h3=136m Equivalent
Ef/Ed Ef1/Ed Ef2/Ed Ef3/Ed
1 0.50 1.00 2.00 1.6117
2 0.50 1.00 3.00 2.2718
3 0.50 1.00 5.00 3.5922
4 0.50 2.00 3.00 2.5146
5 0.50 2.00 5.00 3.8349
6 0.50 3.00 5.00 4.0776
Analyzing the effect of foundation inhomogeneity on the seismic response of gravity dams
Joshi S.G et al.,
International Journal of Civil and Structural Engineering 20
Volume 6 Issue 1 2015
Figure 8: Comparison of the exact and approximate results of maximum horizontal
displacements on upstream face of the dam for different cases of heterogeneities of three
layered foundation.
Similar to Figure 4 for the two-layered scenario, the absolute maximum displacements on the
upstream face of the dam for the six cases of three-layered scenario are shown in Figure 8.
The matching between the exact and approximate displacement response for the three layered
scenario is seen to be equally good. The results for absolute maximum values of maximum
principal stresses on the upstream face of the dam for the six cases of three layer scenario are
also presented in Figure 9. The matching between the results for the layered and equivalent
homogeneous foundation can be considered similar to that for the two-layer scenario in
Figure 5. These observations establish that the use of equivalent homogeneous foundation is
applicable in general.
Analyzing the effect of foundation inhomogeneity on the seismic response of gravity dams
Joshi S.G et al.,
International Journal of Civil and Structural Engineering 21
Volume 6 Issue 1 2015
Figure 9: Comparison of Maximum Principal stresses on the upstream face of the dam for
various cases of 3-layered heterogeneous foundation with the corresponding cases of
equivalent foundation.
Similar to Figure 6 for the cases of two layer scenario, Figure 10 shows the contours of stress
distributions within the body of the dam at the instant when the stress value at the point of
change of slope on the downstream face has the highest value for several cases listed in Table-
2. The pattern of stress distribution and their amplitudes are found to be similar and in good
agreement in all the cases. The slight deviations in the matching between the two results are
towards somewhat higher values of stress or larger areas for higher stresses in case of the
equivalent homogeneous foundation. This represents slightly higher conservatism, justifying
the use of equivalent homogeneous foundation approach for practical engineering applications.
However, as mentioned before, the differences between the two results for all the cases of two
as well as three layered scenario cannot be considered significant to be of any practical
importance.
Analyzing the effect of foundation inhomogeneity on the seismic response of gravity dams
Joshi S.G et al.,
International Journal of Civil and Structural Engineering 22
Volume 6 Issue 1 2015
Figure 10: Comparison of the distributions of major principal stress in the dam section for
several cases of the three layered heterogeneous foundation (left hand side plots) with those
for the corresponding equivalent homogeneous foundation (right hand side plots).
5. Conclusion
The inhomogeneities in the foundation of gravity dams are generally believed to have
significant effect on the dam response, particularly on the levels and distribution of stress
within the dam body. The standard FEM approach is therefore used to analyze the response
of gravity dams with a finite portion of the foundation block with inhomogeneities modeled
together with the dam. However, to get accurate and realistic estimate of the soil structure
interaction effects in this approach, it is necessary to deconvolute the free-field ground
acceleration and apply it at the base of the foundation block with its mass taken into account.
As it is difficult to perform the deconvolution for a foundation with general inhomogeneities,
horizontally layered foundation is commonly assumed in practical applications. To obviate
the need of deconvolution, the present paper has shown that a layered foundation can be well
approximated by an equivalent homogeneous foundation, for which the dam response can
also be estimated more conveniently using the sub-structure approach.
Analyzing the effect of foundation inhomogeneity on the seismic response of gravity dams
Joshi S.G et al.,
International Journal of Civil and Structural Engineering 23
Volume 6 Issue 1 2015
A large number of cases of inhomogeneous foundations are synthesized by varying the
thicknesses and impedance contrasts for two and three layered foundation. An equivalent
homogeneous foundation with the same dimensions and a single value of the impedance
contrast, obtained by thickness weighted average, is also considered for each case. The
displacement and maximum principal stresses in the dam body computed for the cases of
heterogeneous and the equivalent homogeneous foundation are found to be in very good
agreement, indicating that the actual heterogeneity of foundation does not affect the dam
response to any significant extent.
The foundation heterogeneities may perhaps be important to get the stress distribution within
the foundation itself, which may be needed to assess the requirement for its strengthening, if
any. However, this purpose may well be served more conveniently by a simple static analysis
only. The present study has shown that the equivalent homogeneous foundation is a good
approximation for the purpose of dynamic response analysis of the dam. With the
homogeneous foundation it is possible to use the substructure approach of response analysis,
which is able to model the soil-structure interaction effects in a more accurate and realistic
way and obviates the need for deconvolution of the free-field ground acceleration (Joshi et al.,
2014). The sub-structure approach is also able to model more accurately the hydrodynamic
forces with the reservoir bottom absorption effects taken in to account (Fenves and Chopra,
1984; Gupta, 2010; Gupta and Pattanur, 2013).
6. References
1. Abaqus (2014), Abaqus Theory Manual, Version 6.14, Dassault Systèmes Simulia
Corp., Providence, RI, USA.
2. Burman, A., D. Maity and S. Sardeep (2010), Iterative analysis of concrete gravity
dam-nonlinear foundation interaction, Journal of Engineering. Science and
Technology., 2(4), pp 85-99.
3. Chopra A.K. and P. Chakrabarti (1973), The Koyna earthquake and damage to
Koyna dam, Bulletin of Seismological. Society of. America., 63(2), pp 381-397.
4. Chopra, A.K., P. Chakrabarti and G. Dasgupta (1976), Dynamic stiffness matrices
for viscoelastic half-plane foundation, Journal. of Engineering. Mechanics. Division.,
ASCE., 102(EM3), pp 497-514.
5. CWPRS (2002), A Report on the Estimation of Dynamic Modulus of Elasticity for
the Materials of Koyna dam, Central Water and Power Research Station, Pune, India.
6. Dasgupta, G. and A.K. Chopra (1979), Dynamic stiffness matrices for half planes,
Journal. of Engineering. Mechanics. Division., ASCE, 105(5), pp 729-745.
7. Fenves, G. and A.K. Chopra (1984), EAGD-84: A Computer Program for
Earthquake Analysis of Concrete Gravity Dams, Report No. UCB/EERC-84/11,
Univ. of California, Berkeley, California, USA.
8. Gaikwad, V.V., V.M. Kulkarni and S.G. Joshi (2003), Determination of dynamic
modulus of elasticity for Koyna dam concrete – a unique experiment, Procs. 4th Intl.
Analyzing the effect of foundation inhomogeneity on the seismic response of gravity dams
Joshi S.G et al.,
International Journal of Civil and Structural Engineering 24
Volume 6 Issue 1 2015
R & D Conf. on Water and Energy for 21st Century, Central Board of Irrigation &
Power, Aurangabad, India, pp 440-452.
9. Gupta, I.D. (2010), Investigation of Dynamic Response Characteristics of Gravity
Dams by Sub-Structure Approach, ISH Journal of Hydraulic Engineering, 16(2), pp
1-12.
10. Gupta, I.D. and L.R. Pattanur (2013), Investigation of reservoir bottom absorption
effects on stochastic seismic response of gravity dams, ISH Journal. of Hydraulic
Engineering., 18(3), pp 1-9.
11. Heirany, Z. and M. Ghaemian (2012), The effect of foundation’s modulus of
elasticity on concrete gravity dams behavior, Indian Journal of. Science. and
Technology., 5(5), pp 2738-2740.
12. ICOLD (2001), Design Features of Dams to Resist Ground Motion, Bulletin 120,
International Commission on Large Dams, Paris.
13. Joshi, S.G., I.D. Gupta, L.R. Pattanur and P.B. Murnal (2014), Investigating the
effect of depth and impedance of foundation rock in seismic analysis of gravity dams,
International Journal of Geotechnical and Earthquake Engineering, 5(2), pp 1-18.
14. Lin, G., R. Zhang and F. Wang (1988), Structure-foundation interaction effects on
seismic load reduction of concrete gravity dams, Proceedings of . Ninth World
Conference on Earthquake Engineering., August 2-9, 1988, Tokyo-Kyoto, Japan,
Vol. VIII, pp 371-376.
15. Lin, G., Z. Hu, S. Xiao and J. Li (2004), Some problems on the seismic design of
large concrete dams, 13th World Conference on Earthquake Engineering., Vancouver,
B.C., Canada, August 1-6, 2004, Paper No. 1085.
16. Lokke, A., and A.K. Chopra (2013), Response Spectrum Analysis of Concrete
Gravity Dams Including Dam-Water-Foundation Interaction, PEER Report 2013/17,
University of California, Berkeley.
17. Sarkar, R.., D.K. Paul and L. Stempniewski (2007), Influence of reservoir and
foundation on the nonlinear dynamic response of concrete gravity dams, ISET
Journal of Earthquake Technology., 44(2), pp 377-389.