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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.

[email protected]

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


Joshi S.G et al.,



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).


Joshi S.G et al.,



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.


Joshi S.G et al.,



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.


Joshi S.G et al.,



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.


Joshi S.G et al.,



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


Joshi S.G et al.,



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).


Joshi S.G et al.,



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


Joshi S.G et al.,



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.


Joshi S.G et al.,



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.


Joshi S.G et al.,



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.


Joshi S.G et al.,



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.


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R & D Conf. on Water and Energy for 21st Century, Central Board of Irrigation &

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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

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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

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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

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