The dynamic analysis of a concrete gravity dam is a reasonably complex problem. The response of a dam subjected to dynamic loading is a combined effect of the interaction among dam, reservoir and foundation systems and reservoir bottom absorption. Dam–foundation and dam–reservoir interactions are two important aspects in the dynamic analysis of dams. The first one can be considered by simplified or rigorous methods. More realistic results may be obtained by using rigorous methods, which require finite element or boundary element modeling of foundation domain and result in increasing computation time and computer storage requirement.

The present study deals with seismic analysis of Koyna dam considering the effect of dam-foundation interaction and dam-reservoir interaction with reservoir bottom absorption effect. EAGD software has been used to numerically evaluate the response of a two-dimensional dam-water-foundation system to earthquake ground motion. Through a comprehensive investigation it was shown that the effects of dam-water-foundation interaction and reservoir bottom absorption has a profound influence on the response of concrete gravity dams to horizontal and vertical ground motion. The water impounded in the reservoir is idealized as a fluid domain of constant depth and infinite length in the upstream direction, and the dissipation of hydrodynamic pressure waves in the reservoir bottom materials is modeled approximately by a boundary condition that partially absorbs incident hydrodynamic pressure waves. A parametric study has been carried out for dam-water interaction effect on dam response with different reservoir depths and Dam-foundation interaction effect on the results of dam response with different damping values. Results from non-linear analysis of the dam show high tensile stresses in the heel as well as in the change of slope at downstream of the dam. The damping characteristics contribute a significant influence on the dynamic response of dam foundation system. Wave reflection coefficients play an important role in seismic analysis, the maximum displacement increases because of added damping and time period increases because of refraction of hydrodynamic pressure waves.

**Key words:**

Seismic analysis of gravity dam, Dam-reservoir-foundation interaction, Reservoir bottom absorption effect, Reservoir depth, Effect of damping, Finite element analysis.

**Introduction:**

Concerns about the seismic safety of dams have been growing during the recent years, partly because the population at risk in locations downstream of major dams continues to expand and also because it is increasingly evident that the seismic design concepts in use at the time most existing dams were built were inadequate. The distress of a concrete dam is affected by several parameters such as the compressibility of the impounded water, the various dynamic interactions which can be incorporated in the general term dam-reservoir-foundation interaction, the possible existence of a sedimentary material at the bottom of the reservoir and the selection of an appropriate upstream boundary condition to represent the infinite extent of the reservoir in the upstream direction. The impact of these factors has been investigated in the present study of numerical analysis for the evaluation of the seismic distress and response of concrete dam and their interaction with the retained water and the foundation.

EAGD-84 software is used evaluate the response of a two-dimensional dam-water-foundation system to earthquake ground motion. The method is the general response history analysis (RHA) procedure presented by Fenves and Chopra for determining the earthquake response of concrete gravity dams including the effects of dam-water-foundation interaction and reservoir bottom absorption is outlined. The analysis procedure is based on the substructure method, wherein the dam, water and foundation region are modeled as three different substructures of the complete system.

The present study aims at the seismic responses of realistic gravity dam, i.e., Koyna dam due to earthquakes including dam-foundation interaction, reservoir-dam interaction and reservoir bottom absorption effect. In particular, the earthquake response of idealized concrete gravity dam is computed using both horizontal and vertical components of the recorded Taft (California) ground motion, with a maximum ground acceleration of 0.18g in the horizontal component. Since this is the only record used in the analysis, the response results obtained are not meant to be general, and they mainly depend on the characteristics of this particular excitation. The Taft ground motion, however, is a typical moderate earthquake, particularly in the short-period range of its spectrum, which is the main range of interest in the analysis of concrete gravity dams. Both dam-reservoir and dam-foundation rock interactions were considered in the analysis.

**Dam-foundation modeling**

The dam monolith is idealized as an assemblage of planar, four node non conforming finite elements. The finite element is obtained by dividing the dam cross section into quadrilateral elements connected at nodal points. Elements in the shape of parallelograms with an aspect ratio near the unity give the most accurate results. The elastic properties of the materials in the dam can be defined independently for each finite element. Energy dissipation in the dam and concrete is represented by constant hysteretic damping factor ?_s. A viscous damping ratio ?, the same for all the natural vibration modes of the dam on rigid foundation rock with empty reservoir, corresponds to a constant hysteretic damping factor of ?_s= 2?. Forced vibration field tests on dams indicate that the viscous damping ratio is in the range of 1 to 3 percentage, fairly independent of the vibration mode number. A constant hysteretic damping factor of ?_s= 0.1, which corresponds to a 5 percent viscous damping ratio in all vibration modes of the dam, is reasonable value for the much larger motion and higher stresses expected in a dam during strong earthquake ground motion.

**Dam-reservoir modeling**

The water impounded in the reservoir is idealized as a fluid domain of constant depth and infinite length in the upstream direction. The elevation of the free- surface is the only parameter specified for the impounded water. The computer program uses the following properties for the impounded water: the velocity of pressure waves C=1438.656 m/sec and unit weight of water 9.81 kN/m^3. The reservoir bottom is assumed to be horizontal.

**Reservoir bottom absorption:**

The absorptiveness of the reservoir bottom materials is characterized by the wave reflection coefficient a, which is defined as the ratio of the amplitude of the reflected hydrodynamic pressure wave to the amplitude of vertically propagating pressure wave incident on the reservoir bottom. A wave reflection coefficient of unity indicates that pressure waves are reflected from reservoir bottom without attenuation; a wave reflection coefficient of zero indicates that vertically propagating pressure waves are fully absorbed into the reservoir bottom materials without reflection. The materials at the bottom of the reservoir determine the value of the wave reflection coefficient a according to the following equation [Fenves and Chopra,1987]a = (1-k)/(1+k) where k= C/?_r C_r, C is the velocity of pressure waves and ? is the density of water , C_r= v(E_r )/v(?_r ), and E_ris the young’s modulus of elasticity and ?_r is the density of the reservoir bottom materials.

The dam is analyzed considering the horizontal and vertical components of the ground motion recorded at Taft Lincoln School Tunnel during the Kern County, California earthquake of July 21st 1952; both components are shown in Figure 1.

**Details of the present study:**

The cross section and finite element discretization of the problem are presented in Figures 2 and 3 respectively. Numerical data of the dam is shown in Table 1.

**Results And Discussion**

The results of the computer analyses consist of the time histories of horizontal and vertical displacements at selected nodal points. The maximum horizontal displacement at the crest of the dam and maximum principal stresses at four critical locations like heel, toe, crest and change of slope at downstream side of the dam monolith are summarized for the dam supported on flexible foundation rock. The excitation for the dam-reservoir-foundation system is defined by both horizontal and vertical components of free- field ground motion in the plane of monolith of the dam transverse to the dam axis. Results obtained from the analysis are compared with the permissible tensile(+) and compressive stresses(-) of the Koyna dam.

**Analysis for different reservoir depth ratios:**

Seismic analysis has been carried out to understand the effect of reservoir depth on stress distribution of Koyna dam considering different values of reservoir depth ratios H/H_s(H is reservoir depth and H_s is maximum height of the reservoir) of 0.55, 0.60, 0.80, 0.95. The horizontal displacement response of crest along with its time period with variation in reservoir depth is shown in Table 2. From Table 2, it is observed that the maximum crest displacement and time period at its occurrence has significant change when H/H_s is greater than 0.80. The maximum crest displacement increases with increase in reservoir depth.

Fig. 4 shows that maximum horizontal crest response histories of dam with variation in reservoir depth. From Fig. 4 it is observed that the horizontal displacement response increase after H/H_s = 0.80 because increase in water depth increases the hydrodynamic pressure which increases the crest displacement and decreases the time period of dam.

The maximum and minimum principal stresses at specified locations of Koyna dam for variation in reservoir depth due to combined ground motion is shown in Table 3. From Table 2 it is observed that the maximum tensile and compressive stresses in the monolith can be seen at change of slope at the downstream side of dam for H/H_s = 0.95. The maximum tensile stress(5.68 MPa) exceeds the permissible tensile strength of used concrete by approximately 2 times. The maximum compressive stress is 6.567 MPa which is less than the permissible compressive strength of concrete. So it is safe in compression.

The maximum and minimum principal stress variation along the base of the dam is shown in the Figs. 5 and 6. From these figures it is observed that the heel is subjected to maximum tensile stress when H/H_s=0.95 and a maximum compressive stress when H/H_s=0.60. The maximum principal stresses vary from tension at heel to compression at toe. The minimum principal stresses de-crease from heel to toe. This is because of increase in water depth increases the hydrodynamic pressure which increases tensile stresses at heel. Fig. 7 shows the first three mode shapes of Koyna dam for varying reservoir depth due combined ground motion.

**Variation with hysteretic damping factors**

Effect of hysteretic damping factor is studied for seismic analysis of Koyna dam. Damping factors are taken for study are as follows 0.10 (5%), 0.12 (6%), 0.16 (8%), 0.2 (10%). Generally 0.1 is taken as damping ratio since 5% of viscous damping ratio is considered for foundation and concrete. The displacement response of horizontal crest of dam along with its time period with variation in damping factors is shown in Table 4. From Table 4 it is observed that maximum crest displacement decreases with increasing in damping factors and there is no significant change in time period.

Fig. 8 shows the maximum horizontal crest displacement response histories of Koyna dam for variation in damping factors due to combined ground motion. From these figures, It is observed that maximum crest displacement decreases with increase in damping factors and there is no significant change in time period.

The maximum and minimum principal stresses at specified locations of Koyna dam for variation in damping factors due to combined ground motion is shown in Table 5. From Table 5 it is observed that the maximum tensile and compressive stresses develop at change of slope on downstream side of the dam for N_f= 0.10. The maximum tensile stress(5.43 MPa) exceeds the permissible tensile strength of concrete by approximately 2 times. The maximum compressive stress observed is 6.49 MPa which is less than the permissible compressive strength of concrete.

The maximum and minimum principal stress variation along the base of the dam is as shown in the Figs. 9 and 10. From these figures it is observed that the tensile and compressive stresses decrease from heel to toe. Because of increasing in damping restricts the movement of dam which reduces tension at heel. Fig.11 shows the first three mode shapes of Koyna dam for 5% damping factor due to combined ground motion.

**Reservoir bottom absorption**

Effect of reservoir bottom absorption is studied by varying a (wave reflection coefficient) from 0 to 1. Where a = 0 represent complete absorption of earthquake and wave pressure by reservoir bottom sediments and a = 1 means complete reflection of earthquake and wave pressure by reservoir bottom sediments. Other values of a like 0.50, 0.70 are also studied for this effect. The displacement response of horizontal crest of dam along with its time period with variation in wave reflection coefficient values is shown in Table 6. From Table 6, it is observed that when a increases from 0 to 1 leads to decrease in reservoir bottom absorption capacity effect which leads to increase in maximum crest displacement and time period.

Fig. 12 shows the maximum horizontal crest displacement response histories of Koyna dam for variation in wave reflection coefficients. From these figures, it is observed that as reservoir bottom becomes more reflective (i.e. as a values increases) the maximum displacement increases because of added damping due to reservoir bottom absorption and its time period increases because of increasing refraction of hydrodynamic pressure waves into the reservoir bottom material and propagation of pressure waves upstream through the impounded water.

The maximum and minimum principal stresses at specified locations of Koyna dam for variation in wave reflection coefficients due to both horizontal and vertical components of Taft ground motion is shown in Table 7. From Table 7 it is observed that the maximum tensile and compressive stresses develop at a = 1 at downstream side where change of slope occurs. The maximum tensile stress (5.43 MPa) exceeds the permissible tensile strength of concrete by approximately 2 times. The maximum compressive stress observed is 6.491 MPa which is less than the permissible compressive strength of concrete.

The maximum and minimum principal stress variation along the base of the dam is shown in the Figs. 13 and 14. From these figures it is observed that the tensile stresses reduce from heel to toe. Maximum principal stresses increase from a = 0 to 1. The minimum principal stresses increase from heel to toe. This is due to reservoir bottom absorption which reduces the added hydrodynamic effect.

**Conclusions:**

Based on these response results, it is observed that the earthquake response of dams is increased by dam-water interaction and decreased by reservoir bottom absorption with the magnitude of these effects depending on the flexibility of the foundation rock and on the component of ground motion.

Effect of reservoir depth is studied and it is observed that maximum tensile stress occurs at downstream side of the dam where there is change of slope which is approximately 2 times the tensile strength. But it is safe in compression

The damping characteristics contribute a significant influence on the dynamic response of dam foundation system. From the comparison of displacement response of dam and time period for varying damping factors it is observed that, there is no significant change in the time periods but response is higher for decreasing damping factors which shows that as damping ratio decreases, response of dam is increased. Maximum tensile stress exceeds by approximately 2 times the permissible tensile strength at change of slope on downstream side. But it is safe in compression.

Wave reflection coefficients play an important role in seismic analysis the maximum displacement increases because of added damping and time period increases because of refraction of hydrodynamic pressure waves. Maximum tensile stress is more than tensile strength by approximately 2 times. But it is safe in compression. So the Koyna dam is safe in compression for the all the cases studied.

Results from non-linear analysis of the dam show high tensile stresses in the heel as well as in the change of slope at downstream of the dam. Thus the tensile stresses can cause cracking at these zones.

**References:**

- Anil K. Chopra (1978) “Earthquake resistant design of concrete gravity dams,” Journal of the Structural Division, Vol. 104, pp. 953-971.
- Anil K. Chopra & Gupta (1982) “Hydrodynamic and Foundation Interaction Effect in Frequency Response Functions for Concrete Gravity Dams,” Earthquake Engg. , Structure Div., Vol 10, pp. 89-105.
- Anil K. Chopra, Gregory Fenves (1987) “Simplified Earthquake Analysis of Concrete Gravity Dams,” Journal of Structural Engineering, Vol.113, ISSN 0733-9445, August.
- Brijesh singh and Pankaj Agarwal (2009) “Seismic Response Of High Concrete Gravity Dam Including Dam-Reservoir- Foundation Interaction Effect,” Journal of South Asia Disaster Studies, Vol. 2, December.
- Fenves, G. and A. K. Chopra (1984) “Earthquake analysis of concrete gravity dams including reservoir bottom absorption and dam-water-foundation rock interaction,” Earthquake Engineering and Structural Dynamics, Vol. 12, No. 5, pp.663-680.
- Fenves, G. and A. K. Chopra (1984) “EAGD-84: A computer program for earthquake response analysis of concrete gravity dams,” Earthquake Engineering Research Center, University of California, Berkeley, Report No. UCB/EERC-84/11.
- Fenves, G. and A. K. Chopra (1983) “Effects of reservoir bottom absorption on earthquake response of concrete gravity dams,” Earthquake Engineering and tructural Dynamics, Vol. 11, No. 6, pp. 809-829.
- Fenves, G. and A. K. Chopra (1987) “Simplified earthquake analysis of concrete gravity dams: Separate hydrodynamic and foundation interaction effects,” Journal of Engineering Mechanics, Vol. 111, No. 6,pp.783-806.
- MATLAB, version (2012), The Math Works Inc., software.
- Rajib Sarkar and L. Stempniewski (2007) “Influence of reservoir and foundation on the nonlinear dynamic response of concrete gravity dams”, Journal of Earthquake Technology, Vol. 44, pp377-389, June.

Dr. T. V. Praveen

Professor, Dept. of Civil Engineering

Andhra University, College of Engineering (A)

Visakhapatnam

Birhane G. Hagos

Research Scholar, Dept. of Civil Engineering

Andhra University, College of Engineering (A)

Visakhapatnam

Dr. I. Siva Parvathi

Assistant Professor, Dept. of Civil Engineering

Andhra University, College of Engineering (A)

Visakhapatnam

D. Priyanka

Assistant Professor, Dept. of Civil Engineering

MVJ College of Engineering

Bangalore