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~2 Analysis of Linear Response PRELIMINARY COMMENTS The ability to evaluate the effects of earthquake ground motion on concrete dams is essential to assessing the safety of existing dams, to determining the adequacy of modifications planned to improve old dams, and to evaluating designs for proposed new dams. However, the prediction of the performance of concrete dams during earthquakes is one of the most challenging and complex problems found in the field of structural dynamics because of the following factors: 1. Dams and the retained reservoirs are of complicated shapes, as dictated by the topography of the sites. 2. The response of a dam is influenced to a significant degree by the interaction of the motions of the dam with the impounded water and the foundation rock; thus, the deformability of the foundation and the earthquake- induced response of the reservoir water must be considered. 3. A dam's response may be affected by variations in the intensity and frequency characteristics of the earthquake motions over the width and height of the canyon; however, this factor cannot be fully considered at present because of the lack of instrumental data to define the spatial variation of ground motion, as discussed in Chapter 2. When evaluating the earthquake behavior of concrete dams, it is reasonable in most cases to assume that the response to low- or moderate-intensity earthquake motions is linear. That is, it is expected that the resulting deformations of the dam will be directly proportional to the amplitude of the applied ground shaking. Such an assumption of response linearity greatly simplifies 36

37 both the formulation of the mathematical model used to represent the dam, reservoir water, foundation rock system and also the procedures used to calculate the response. The results of linear analysis serve to demonstrate the general character of the dynamic response, and the amplitudes of the calculated strains and displacements indicate whether the assumption of linearity is valid. In the case of a major earthquake it is probable that the calculated strains would exceed the elastic capacity of the dam's concrete, indicating that damage would occur; in this case a much more complicated nonlinear analysis would be required to determine the expected extent of damage. However, a linear analysis still can be very valuable in helping to understand the nature of the dynamic performance and in deciding whether a nonlinear analysis will be necessary. In many cases a reasonable estimate of the expected degree of damage can be made from the linear analysis, even though the results suggest that a slight to moderate degree of cracking or other form of nonlinearity is to be expected. In this chapter both the earliest, rather primitive, techniques of estimating earthquake performance are described and also the refined, modern, linear computer analysis procedures that are presently recommended for seismic safes', evaluations of concrete dams. STATIC ANALYSIS Traditional Analysis and Design Because most dams in the United States were built prior to the development of modern computer analysis procedures, earthquake effects were accounted for in the designs by using methods that are now considered oversimplified. In particular, the dynamic behavior of the dam, reservoir water, foundation rock system was not recognized in defining the earthquake forces used in traditional design methods (3-1, 3-2~. Thus, the forces associated with the inertia of the dam were expressed simply as the product of a seismic coefficient- taken to be constant over the surface of the dam, with typical values of 0.05 to O.l~and the weight of the dam per unit surface area expressed as a function of location. Seismic water pressure in addition to hydrostatic pressure was specified in terms of the seismic coefficient and an additional pressure coefficient; the latter was evaluated based on the assumptions that the dam was rigid and had a plane vertical upstream face and that water compressibility effects were negligible (3-3, 3-4~. Generally, interaction between the dam and the foundation rock was not considered in evaluating the aforementioned earthquake forces, but in the seismic stress analysis of arch dams the flexibility of the foundation rock sometimes was recognized through the use of Vogt coefficients (3-1~. Stresses in gravity dams with ungrouted construction joints were usually determined by treating the concrete

38 blocks as vertical cantilever beams; for the analysis of arch dams, and sometimes for gravity dams in narrow canyons or with keyed joints, the trial load analysis procedure was usually used. Traditional design criteria (3-1, 3-2) require that the compressive stresses not exceed either one-fourth of the specified compressive strength or 1,000 psi. Tensile stresses were usually not permitted in gravity dams or, if they were, were limited to such a small value that cracking was not considered possible; in arch dams the tensile stresses were required to remain below 150 psi. In the design of gravity dams it was generally believed that stress levels were not a controlling factor, so the designer was concerned mostly with satisfying criteria for overturning and sliding stability. Earthquake Performance of Koyna Dam As mentioned in Chapter 1, Koyna Dam in India is one of two concrete dams that have suffered significant earthquake damage (3-5, 3-6~. A photograph of this dam is shown in Figure 3-1, and it is useful to discuss its earthquake performance in some detail in this report because it was designed by the traditional static analysis procedure using a seismic coefficient of 0.05. Even though a "no-tension" criterion was satisfied in the design procedures considering seismic as well as all other forces, the earthquake of 11 December 1967 caused important horizontal cracks on the upstream and/or downstream faces of a number of nonoverflow monoliths near the elevation at which there is an abrupt change in slope of the downstream face. Although the dam survived the earthquake without any sudden release of water, the cracking appeared serious, and it was decided to strengthen the dam by constructing buttresses on the downstream face of the nonoverflow monoliths; the overflow monoliths were not damaged. To understand why the damage occurred, the dynamic response of the tallest nonoverflow monolith was calculated, assuming linear behavior. The results indicated large tensile stresses on both faces, with the greatest values near the elevation of the downstream-face change of slope. These calculated stresses (shown in Figure 3-2), which exceeded 600 psi on the upstream face and 900 psi on the downstream face, were about two to three times the estimated 350-psi tensile strength of the concrete at that elevation. Hence, significant cracking consistent with what was observed could have been expected during an earthquake of this intensity. The maximum compressive stress in the monolith (not shown in Figure 3-2) was about 1,100 psi, well within the compressive capacity of the concrete. A similar analysis of the nonoverflow monoliths indicated that little or no cracking should have occurred there, which also is consistent with the observed behavior.

39 FIGURE 3-1 Koyna Dam, India, was damaged by a magnitude 6.5 earthquake in December 1967 (3-5, 3-6). Limitations of Traditional Procedures It is apparent from the preceding discussion that the dynamic stresses that develop in gravity dams due to earthquake ground motion bear little resemblance to the results given by standard static design procedures. In the case of Koyna Dam the earthquake forces based on a seismic coefficient of 0.05, uniform over the height, were expected to cause no tensile stresses; however, the earthquake caused significant cracking in the dam. The discrepancy is the result of not recognizing the dynamic amplification effects that occur in the dam's response to earthquake motions. The typical design seismic coefficients, 0.05 to 0.10, used in designing concrete dams are much smaller in the typical period range for such dams than are the ordinates of the pseudoacceleration response spectra for intense earthquake motions, as shown in Figure 3-3. It is of interest to note that the seismic coefficients used for dams are similar to the base shear coefficients specified for buildings. However, the Uniform Building Code provisions (3-7) are based on the premise that the structures should be able to:

40 200 300 400 500 600 600 1 1 00 psi 43OoO 600 iN\ N: 11~ 100 Envelope Values of Maximum Principal Stresses for Koyna Dam due to Koyna Earthquake 2900 290 _ 3500 350 L 4100 410 Compressive Tensile Concrete Strength, PSI FIGURE 3-2 Maximum stresses in the tallest Koyna Dam monolith calculated by input of the Koyna earthquake record, compared with the estimated strength of the concrete (3-6). 1. resist minor levels of earthquake ground motion without damage; 2. resist moderate levels of earthquake ground motion without structural damage, but possibly some nonstructural damage; and 3. resist major levels of earthquake ground motion . . . without collapse, but possibly with some structural as well as nonstructural damage. While these may be appropriate design objectives for buildings, major dams should be designed more conservatively, and this intended conservatism is reflected in the no-tension or at most small-tension limitation used in traditional methods for designing dams. What the traditional methods fail to recognize, however, is that in order for dams to satisfy these criteria during earthquakes, consideration must be given to their dynamic behavior. For linearly elastic structures the dynamic aspect of the response is indicated by the response spectra and by the dynamic displacement patterns, which are conveniently expressed in terms of free-vibration-mode shapes.

41 0.6 of LL 0.5 LL LL O 0.4 LD C: O 03 On LLJ On ~ 0.1 LL o Average Aceleration Response Spectrum . g ~Peak Ground Acceleration = 0.339, Damping = 5% l ~ I \ / \ Seismic Base Shear Coefficient \ for Buildings: 1/1 5~/ T Seismic Coefficient 0.2 for Concrete Gravity Dams _ ~ . , , , ~ I ~ ~ ~ I , , I I I 0 1 2 3 VIBRATION PERIOD (in seconds) FIGURE 3-3 Companson of seismic coefficients used in traditional design with the response spectrum of a strong earthquake (3-29). In linear analyses the effective modal earthquake forces may be expressed as the product of a seismic coefficient (which depends on the earthquake pseudoacceleration response spectrum and the vibration period of the mode, and varies according to the shape of the mode) and the unit weight of concrete. The seismic coefficient associated with forces in the fundamental mode of a gravity dam varies with height, somewhat as shown in Figure 3- 4. For the first two modes of a symmetric arch dam (fundamental symmetric and antisymmetric modes), the coefficient may vary over the dam face, as shown in Figure 3-5. These figures are in sharp contrast with the uniform distribution of seismic coefficient that has been assumed traditionally and that has led to erroneous distribution of lateral forces and hence of stresses in the dam. One of the results of assuming a heightwise-uniform seismic coefficient is Hat calculated stresses in gravity dams are found to be greatest at the base of the dam. This has led to the concept of decreasing the concrete strength with increases of elevation, as has been done for some dams (e.g., Koyna Dam in India and Dworshak Dam in the United States). However,

42 ..... - · e e e e e e e e e · e · · e · e e e e · e e e ~ ·.-.- · · ~ e - · ~ · e e ~ e · ~ · e - · e e ~ ~ · e · ~ - ·--~e · ~ · e - · · · e ~ e · e e e e · · e - ·~ · ~ ~ e ~ ·.... · e e e e · - ~ · ~ ~ e - · ~ ~ e · e · e · - . . . · . . . · - . . . ..... Traditional Design Procedure _ .... ~ _ · ~ e · · · e · · e · · A · · e e ~ ., · e e e e' · · . · . . .` · -..' · · e. · · ~ e · / :1 .! ·1 _- Fundamental Vibration Mode FIGURE 3-4 Vanaiion of seismic coefficient along height of gravity dams: traditional constant value versus dynamic vanai~on (3-29). the results of dynamic analysis of Koyna Dam (Figure 3-2), as well as the location of the earthquake-induced cracks, demonstrate that the largest stresses actually occur at the two faces in the upper part of gravity dams. Therefore, those are the regions of gravity dams where the highest-strength concrete should be provided if the designer chooses to vary the concrete strength within the dam. Another undesirable consequence of specifying a heightwise constant seismic coefficient is that the detrimental effects of concrete added near the dam crest are not apparent, as has been shown by analytical study of Pine Flat Dam (3-31~. This typical gravity dam, shown in Figure 3-6, was built by the U.S. Army Corps of Engineers in California. As can be seen in the photograph, it was widened at the crest to provide for a roadway. The resulting added crest concrete may appear to have a beneficial effect in reducing the stresses predicted by traditional static analysis and also may serve useful functions in providing freeboard above the maximum water level, in resisting the impact of floating objects, and in affording the roadway. However, because of the sharp increase of seismic coefficient in the crest region (Figure 3-4), the crest mass may cause a dramatic increase in the dynamic stresses approximately doubling them in the earthquake response of Pine Flat Dam, as shown in Figure 3-7. An interesting consequence of this type of unfavorable response mechanism was seen in monolith 18 of Koyna Dam, which suffered the worst damage during the earthquake; it is believed that this exaggerated damage resulted from an elevator tower that extended 50 ft above the top of the block and therefore was subjected to greatly increased inertial forces.

43 ,,,,, ,~J~J'i~,~,,.~,,,,,,, ,,,,,,, ,,,,, ~ \ 1 ~ L1~1 ~ Ad ~ In., ,,, ,,,,, /,,.,,,.;,---. \ it' l l as' ~ r _ 7 G - 3 ~,-~. /,,,,,~,,~/,,, ,.... . ~ 7 / FUNDAMENTAL SYMMETRIC VIBRATION MODE 7 / 7 \ / 7 / FUNDAMENTAL ANTISYMMETRIC VIBRATION MODE FIGURE 3-5 Variation of seismic coefficient over face of arch dams (3-12).

44

45 Structural Section 600 400 200 100 i 100 3Q~ ~100 psi ACTUAL SECTION p1~ ~0 Additional Crest Mass 3001 200 W00 ~ \\~200 ^00 STRUCTURAL SECTION \ 100 FIGURE 3-7 Effects of nonstructural crest mass on maximum principal stresses calculated in the tallest Pine Flat Dam monolith due to Koyna earthquake input (3-31). Traditional design earthquake loading for concrete dams includes seismic water pressures in addition to hydrostatic pressure, as specified by various formulas (3-2, 3-9~. These formulas differ somewhat in detail and in numerical values but not in the underlying assumptions, because they are all based on the classical results of Westergaard (3-3) and Zangar (3-4~. In a typical formula the seismic water pressure is given as Pe = CSwH, where C is a coefficient that varies nonlinearly from zero at the water surface to about 0.7 at the reservoir bottom, S is the specified seismic coefficient, w is the unit weight of water, and H is the total depth of water. For a seismic coefficient of 0.10 the additional pressure at the base of the dam is 7 percent of the hydrostatic pressure; values at higher elevations also are small. These additional pressures have little influence on the computed stresses and hence on the geometry of the gravity section that satisfies the standard design criteria. On the other hand, when the true dynamic behavior of the dam is considered' including dam-water interaction and water compressibility, numerous analyses have demonstrated that hydrodynamic effects are significant in the

46 earthquake response of concrete dams and can lead to important stress increases, especially in arch dams (3-S, 3-10, 3-11~. It is apparent, therefore, that hydrodynamic effects are not properly modeled in the traditional design procedures. Finally, it is evident that the static overturning and sliding stability criteria that have been used in traditional gravity dam design procedures have little meaning in the context of the oscillatory responses produced in dams by earthquake motions. DYNAMIC ANALYSIS Arch Dam Analysis Forerunner Recognizing the limitations of the static seismic coefficient method, design and research engineers became interested in dynamic analysis procedures to reliably predict the earthquake response of dams. In one of the earliest (1963) dynamic analyses applied to arch dams (3-12), the uniform seismic coefficient used in static methods was replaced by a spatially varying coefficient computed for the first two vibration modes of a symmetric dam (first symmetric and first antisymmetric mode), based on the modal periods and shapes and on an earthquake response spectrum. For ground motion in the upstream- downstream direction, the hydrodynamic effects were based on Westergaard's formula for a rigid gravity dam; for cross-stream motions a formula based on the work of Zienkiewicz and Nath (3-13) was applied. The stress analysis for each mode was done by the trial load method, and the modal stresses were combined appropriately. Finite Element Modeling The procedures for earthquake analysis of dams began to change rapidly in the late 1960s with the adoption of finite element modeling procedures, with advances in methods of dynamic analysis, and with the increasing availability of large-capacity, high-speed computers. One of the earliest applications of this new technology to analysis of the earthquake response of an arch dam was reported in 1969 (3-14~. In this investigation the dam was modeled as an assemblage of three-dimensional finite elements on a rigid base, and the impounded water was modeled as a mesh of incompressible liquid elements. The dynamic response was calculated by the mode-supe~position method. Subsequently, the Bureau of Reclamation funded the development of a computer program based on similar concepts intended specifically for the static and dynamic analysis of the Bureau's arch dams (2-5~. This program, called ADAP (Arch Dam Analysis Program), modeled the dam body by

47 solid elements designed to represent the arch dam geometry efficiently. In addition, the program provided for modeling the flexibility (but not the mass) of the foundation rock, either by means of Vogt coefficients, which treat the foundation as an elastic half space, or by a finite element mesh representing a specified block of foundation rock. When the project was planned, it also was intended to model the interaction with water in the reservoir by means of a mesh of incompressible liquid elements, as mentioned above. However, this final phase of the research was not funded, and in practice the impounded water was modeled by Westergaard-type added masses modified to take account of the doubly curved upstream face (3-15~. In the mid-1970s a dynamic finite element analysis procedure, including a different Westergaard-type added mass, was described in Section 4-32 of a 1976 Bureau of Reclamation publication on gravity dams (3-16) and in Section 4-56 of a 1977 Bureau of Reclamation publication on arch dams (3- 12~. In the latter publication the same added hydrodynamic mass was used in response analysis for both the upstream-downstream and the cross-stream components of ground motion; the obvious inconsistency of this assumption was recognized in Section 4-55 of the publication, but no better alternative was proposed. Foundation Model Deficiency The two principal limitations of the finite element modeling of a dam- foundation system, mentioned in Chapter 2 of this report, were revealed by extensive work done during the past 20 years in the seismic analysis of nuclear power plants. First, the boundary hypothesized at some depth to define the foundation rock region included in the analysis is typically assumed to be rigid. For concrete dam sites, where similar competent rock usually extends to great depths and there is no obvious "rigid" boundary such as may be assumed at a soil-rock interface, the assumption of a rigid boundary may result in serious distortion of the foundation interaction effects; this distortion results mainly because energy loss (damping) associated with radiation of vibration waves beyond the assumed foundation block is not properly represented. These effects may significantly reduce the seismic stresses in the dam (3-~. Second, the earthquake input is usually applied to the dam-foundation model as prescribed motions of the rigid foundation block boundary, as discussed in Chapter 2. Clearly, the earthquake motions at depth in the foundation rock will not be the same as the free-field motions recorded at the ground surface; hence, the dynamic analysis procedure should be formulated to use some different specification of the seismic input, as described in Chapter 2. Ideally, the earthquake input should be specified as spatially varying motion at the dam-foundation rock interface, but this has usually

48 not been possible for lack of appropriate instrumental records from past earthquakes. Limitations of Reservoir Water Interaction Models As mentioned above, the traditional static analysis procedures represent the effect of the impounded water during earthquake excitation by means of added masses calculated from Westergaard's classical formula. However, Westergaard's added mass result applies rigorously only to the case of a rigid dam with planar vertical face, and the analysis underlying the result implicitly neglects the effects of water compressibility. Although the concept has long been used in practical design, these limitations often have not been understood; extensive studies during the past two decades have been devoted to obtaining better understanding of the dam-water interaction problem. From the first it was evident that the Westergaard rigid dam assumption is not consistent with the concept of dynamic earthquake response analysis, so efforts were made to formulate procedures for dynamic analysis considering dam-water interaction arising from dam flexibility. Results of these studies have contributed greatly to the understanding of dam-water interaction (3- 10~. The first attempt to deal with water interaction for dams having curved faces made use of a liquid finite element model of the reservoir, as mentioned above (3-14~. Because this numerical approach eliminated the geometric limitation inherent in the Westergaard formulation, it was adopted subsequently in the ADAP computer program (3-15~. Using this extended program, it was shown that the geometry of the dam face and of the reservoir boundaries can have a significant effect on the hydrodynamic forces applied to the dam (3-15~. In this work the water was assumed to be incompressible, and because this assumption greatly simplifies the analysis of dam-water interaction, it continues to be used in some research (3-26) and in many practical earthquake response analyses of concrete dams (3-18~. On the other hand, earlier work on gravity dams indicated that the compressibility of impounded water might play an important part in the interaction mechanism (3-10, 3-19~; further research along these lines has confirmed the significant contribution of water compressibility in the earthquake response of most concrete dams. The key parameter that determines the significance of water compressibility in the earthquake response of dams is the ratio of the fundamental natural frequency of the impounded water to the fundamental natural frequency of the dam alone. If this ratio is large enough (greater than 2 for gravity dams), the impounded water affects the dam response essentially as an incompressible fluid. However, this frequency ratio usually is less than 2 for dams with realistic values for the elastic modulus of concrete; hence, water compressibility is expected to be important in the earthquake response of both arch dams (as

49 snown in Figure 3-8) and gravity dams (figure not included). Water compressibility has less influence on the earthquake response of dams with lower concrete moduli, to the point of becoming negligible where the modulus is taken to be unrealistically low, as shown in Figure 3-9. However, even for such unrealistically flexible concrete, the compressibility can have a significant effect on the response to vertical ground motion (Figure 3-9~. These data and additional results (3-20) indicate that water compressibility is expected to be significant in the response of most concrete dams. Thus, the typical added mass representation of hydrodynamic effects that is based on assumed incompressible water (as, for example, in the ADAP computer program) may lead to inaccurate results whether the added mass is calculated from some modification of Westergaard's classical result for a straight dam or is based on a three-dimensional finite element analysis of the liquid domain. On the other hand, the influence of water compressibility will vary from one dam to another, and in some cases the effect may not be large. Comparison of the top two stress contour plots of Figure 3-10 shows that water compressibility had little effect on the calculated seismic response of Monticello Dam (3- 26) when the reservoir bottom was assumed to be rigid. In that case the compressibility was the only important difference between the analyses done by ADAP and those done by a similar program named EACD-3D (2-3~. This Monticello Dam example and also the results shown in Figure 3-11 for Morrow Point Dam demonstrate another limitation of incompressible water formulations. That is, they cannot recognize the potential absorption of hydrodynamic pressure waves into the sediments and rock at the reservoir boundaries. These results show that when reservoir boundary wave absorption effects are considered, significant reductions of the dam stresses may be indicated. It is apparent from these figures that these wave absorption effects can be significant in the response of arch dams and are particularly important in the response to the vertical component of the earthquake motions. PRESENT KNOWLEDGE AND CAPABILITIES Conclusions About Interaction Effects The extensive research that has been devoted during the past 20 years to evaluating hydrodynamic and foundation interaction effects has led to greatly increased understanding of these phenomena. It is now generally recognized that: 1. The earthquake response of concrete dams is increased significantly by interaction with the impounded water, with the hydrodynamic contribution being especially large in response to the vertical earthquake component. 2. Hydrodynamic effects usually are more significant in the earthquake

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51 z z o CL o LL cD ~n cn o a: z z o o u] ~: o LL cn lL 1io .~ l lo o o o ~ o o o o - ~ 1 Lo c ~-s s ~ f>- o o o Lr) u) o G 111 ~ o LU 1 m CD cn o C) - . >o a: LD cn J _ o o ~7 o o 11 ~ o o LLJ CD cn L~ o . o > U] cn LL (sa40u! u!) 1S380 w~a 1v 1N3wao~lasla 1vlavs cn o C' a (n . _ LL] E~ o - oD ~a . o o~ o o- ~ cn o ~ o a' co ~ - - - o ~ o o o o cO - ~ - ~ - l - ~ o ~ ;- D ' - cPq t~ ~ o o c~ - a5 3 o 11 U3 . ~ - o C) ~ C) _ C) C~ o - ~5 . . o o~ ~ ~: ~ Ct Pt _` _ ,= C) ~ ~4 C~ 43

~ Incompressible water \~ ~\~\3350~//~//50 ( \ J >) )~'(~w or/: ~ / Compressible water, = 1 ~\sO~-s\1°0~35O~/~/~/~/? TV ~50 Compressible water, = 0.9 two \/ J an ~ - Compressible water, = 0.5 FIGURE 3-10 Calculated effects of water compressibility and reservoir boundary absorption on upstream face maximum envelope stress contours for Monticello Dam subjected to Morgan Hill earthquake record (3-26).

53 UPSTREAM COMPONENT Full Reservoir Rigid Reservoir Boundary, a= 1 Cantilever Stress Arch Stress \ 500 \ 400 \ 300 100 1 50~ Full Reservoir Rigid Reservoir Boundary, a= 1 Cantilever Stress Arch Stress 200~\ Full Reservoir Rigid Reservoir Boundary, a= 1 Cantilever Stress Arch Stress ~ ~ /~ 20'0 ~O)J /0~ 300 / ~ WAHOO/ ~ / i7400 ~ Em/ ~ VERTICAL COMPONENT iOO \ ~/oo Ado / / /uu [/ 400 :~ ~200 Full Reservoir Rigid Reservoir Boundary, a = 1 Full Reservoir Rigid Reservoir Boundary,a= 1 Cantilever Stress Arch Stress ~800 A\ 5 100 / . ~ CROSS-STREAM COMPONENT Full Reservoir Rigid Reservoir Boundary,a= 1 Cantilever Stress Arch Stress Cantilever Stress Arch Stress 100 ~25 \ \50\ \ 7511 ~JI ~0 / FIGURE 3-11 Calculated effects of reservoir boundary absorption on upstream face maximum envelope stress contours for Morrow Point Dam due to Taft earthquake record (3-11).

54 response of a slender arch dam than for a massive gravity dam. 3. The assumption of water incompressibility that is commonly used in practical analysis may lead to errors on either the conservative side or the unconservative side for upstream-downstream earthquake motion, but it is more likely to be unconservative in predicting response to the vertical and cross-stream components of motion (3-10, 3-20, 3-21~. 4. An important deficiency of the incompressible water approximation is that the hydrodynamic-wave absorption effects of the underlying rock or reservoir boundary sediments cannot be taken into account, and it has been shown that neglecting boundary wave absorption may lead to unrealistically large estimates of seismic response. 5. Neglecting the dynamic interaction of gravity dams with deformations of the foundation rock also will generally lead to overestimation of the seismic response. Because arch dams resist the reservoir water pressures and the thermal and earthquake forces, at least in part, by transmitting them by arch action to the canyon walls, dam-foundation rock interaction also is likely to be significant in the earthquake response of arch dams, possibly more so than in the case of gravity dams. The preceding observations lead to the conclusion that dam-reservoir water interaction, including water compressibility and pressure wave absorption at the reservoir boundaries, and dam-foundation rock interaction all should be considered in the earthquake response analysis of concrete dams. Dynamic Analysis Procedures and Computer Programs Two-Dimensional Analysis Analysis procedures and computer programs that take account of all of these factors have been developed for evaluating the earthquake performance of concrete dams idealized as two-dimensional systems (2-4, 3-22), and these procedures and programs should be used in those cases where two- dimensional analyses are appropriate because of their computational efficiency. In gravity dams with plane veridical joints the monoliths tend to vibrate independently, as evidenced by the spelled concrete and water leakage at the joints of Koyna Dam during the 1967 earthquake (3-5, 3-6~; hence, a two-dimensional plane-stress model of the individual monoliths is usually appropriate for predicting the response of such dams to moderate or intense earthquake motions. In some cases of gravity dams built in a broad valley, especially roller-compacted concrete dams built without vertical joints, a plane-strain idealization may be adopted in place of the plane-stress model. On the other hand, if the gravity dam has effectively keyed contraction joints or is located in a narrow canyon, the assumption of independent response of the blocks may not be appropriate.

ss One computer program that has been developed to perform the earthquake analysis of two-dimensional dam-water-foundation rock systems, called EAGD- 84, is based on a substructure formulation of the problem (2-4, 3-22~. In this program a finite element idealization is used to model arbitrary geometry and variations of material properties of the dam; consequently, both overflow and nonoverflow sections and also appurtenant structures can be modeled satisfactorily. The impounded water is treated as a continuum in order to efficiently represent its large extent as well as the radiation of hydrodynamic pressure waves upstream. The effects of alluvium and silts that accumulate at the bottom of the reservoir and of the underlying rock are modeled approximately by a boundary that may partially absorb the incident hydrodynamic pressure waves; more rigorous methods for treating this effect without such a simplifying approximation have recently become available (3-23~. The foundation rock supporting the dam is idealized as a viscoelastic half-plane continuum that, as mentioned earlier, accounts for the energy radiation effects of the dam-foundai~on rock interaction. With this computer program a complete interaction analysis can be performed of the dynamic response of a gravity dam to the upstream and vertical components of free-field earthquake motion, with both components acting simultaneously at the dam-foundation rock interface. Three-Dimensional Analysis As mentioned earlier, computer programs employing finite element idealizations for the earthquake analysis of arch dam-water-foundation systems have been in use for as long as two decades. ADAP (2-5), the first widely available program developed specifically for dynamic analysis of arch dams, also uses a mesh of finite elements to model the foundation rock. However, as mentioned earlier, these elements are assumed to be massless, so they model the foundation flexibility but do not account for wave propagation in the rock and the consequent radiation damping effect. Also mentioned earlier is the fact that the liquid finite elements used in recent versions (3- 15) of this program are assumed to be incompressible. Thus, the water compressibility effects and hydrodynamic wave absorption effects of the reservoir boundary, which as stated earlier can be significant in the seismic response, are not considered in ADAP. These limitations are overcome in a computer program that is based on the substructure method and was developed recently for the three-dimensional analysis of concrete dams. The program, named EACD-3D (2-3, 3-24), accounts for dam-water interaction, including the effects of compressibility and reservoir boundary pressure wave absorption, using procedures analogous to those employed in the two-dimensional program EAGD-84. So far, however,

56 it has not been possible to take full account of the dam-foundation rock interaction. All three substructuresdam, reservoir water, and supporting rock are idealized as finite element systems to represent the complicated dam geometry and site topography, but special techniques were introduced to efficiently recognize the great upstream extent of the reservoir. The massless finite element model of the foundation rock is similar to that used in ADAP, and thus it also is deficient in representing radiation energy loss. On the other hand, recent research in Japan (3-27) has been directed toward modeling of arch dam-foundation rock interaction, but it has not yet advanced sufficiently to be of use in practical arch dam earthquake response analyses. Both EACD-3D and ADAP can be used to perform a complete dynamic analysis of a concrete arch dam subjected to the simultaneous action of upstream, vertical, and cross-stream components of the free-field motion specified at the interface between dam and foundation rock. In the model with massless foundation rock the free-field surface motion at the dam-rock interface is the same as the motion at the rigid boundary of the foundation block. In principle, spatial variation of the input earthquake motions could be specified either for the free-field input used in EACD-3D or for the foundation block boundary input used in ADAP, and it is evident that such spatial variation does occur across dam sites, as discussed in Chapter 2. However, reliable descriptions of the earthquake motions to be expected at such locations are not available at present, and "multiple-support" excitation is seldom used for arch dam analysis, even though it is technically feasible (3-25). Selection of numerical values for the parameters necessary to describe a dam-water-foundation rock system for analysis by the aforementioned computer programs should be based on appropriate experimental tests. Clearly, the properties of the reservoir water present no problem, and the properties of the concrete comprising the dam can be defined adequately by standard procedures. Evaluation of the elastic modulus and damping of the foundation rock is not so simple, but, as discussed in Chapter 5, numerous field measurement studies have demonstrated that vibration properties calculated using typical finite element system models agree well with measured values. However, techniques are not presently available to determine the reservoir wave reflection coefficient, which is seen in Figures 3-10 and 3-11 to have a major influence on the seismic stresses. Until such measured data are available, it is suggested that values for this coefficient be calculated based on the properties of the water and the underlying rock, as described in references 3-22 and 3-24. This approach neglects the unknown cushioning effect of reservoir boundary sediments and thereby will probably overestimate the earthquake response in most cases.

57 The procedures applied in the analysis of arch dams can be used to evaluate the earthquake response of other types of dams that also must be modeled as three-dimensional systems; buttress dams (including both flat slab and multiple arch dams) are a typical example. In the early decades of this century, buttress dams were often used in preference to gravity dams because they require about 60 percent less concrete. But as stated in a Bureau of Reclamation publication (3-28, p. 10), "the increased formwork and reinforcing steel required usually offset the savings in concrete.... [Hence, this] type of construction usually is not competitive with other types of dams when labor costs are high." However, even though new dams of this type are not being built in the United States, it still is necessary to carry out seismic safety evaluations of the existing structures. Littlerock Dam in California, shown in Figure 3-12, is a typical multiple arch water supply and flood control dam for which a seismic safety evaluation has been performed. Simplified Dynamic Analysis Procedures While the response history analysis procedures described above are appropriate for final-stage analyses of the seismic safety of existing dams and proposed new dams, simplified analysis procedures would be preferable for the preliminary evaluation or design stages. In response to this need a simplified procedure was developed in 1978 for the analysis of gravity dams in which the maximum response due to the fundamental mode of vibration was represented by equivalent lateral forces computed directly from the earthquake design spectrum (3-29~. Recently, this simplified two-dimensional analysis of the fundamental mode response has been extended to include the effects of dam-foundation rock interaction and wave absorption at the reservoir bottom (3-30), in addition to the effects of the compressible water-dam interaction considered in the earlier procedure. Also now included in the simplified procedure is a "static correction" method to approximate the response contributions of the higher-vibration modes. The simplified procedure is sufficiently accurate for preliminary design and safety evaluation of gravity dams. While many of the basic concepts underlying the procedure may be applicable to arch dams, the extension of such a method to treat three-dimensional systems is likely to be very difficult for several reasons: (1) the geometry of arch dams varies considerably from one site to another, thereby reducing the possibility of defining "standard" values for vibration properties and parameters, and (2) their response is generally not dominated by a single mode of vibration.

58 it:: ::k .i. i .. FIGURE 3-12 Littlerock Multiple Arch Dam, Califomia, completed in 1924 for use in irrigation, has a maximum height of 168 ft and length of 800 ft.

s9 RESEARCH NEEDS Although considerable progress has been made in the past 20 years, much additional research needs to be done to improve the reliability of methods for the seismic analysis, design, and safety evaluation of concrete dams. To meet this objective the following tasks should be pursued: 1. Instrumentation Major dams in seismic areas of the United States should be instrumented to measure their responses during future earthquakes. The instrumentation should be designed to provide adequate information on the characteristics and spatial variation of the ground motion at the site, on the response of the dam, and on the hydrodynamic pressures exerted on the dam. Because of the urgent need for such data, dams in highly seismic regions of other countries also should be considered for instrumentation. This effort should be coordinated with the plans for the seismic arrays recently installed in Taiwan, India, and the People's Republic of China under cooperative agreements with the United States. 2. Field Forced-Vibration Tests Forced-vibration tests should be conducted on selected dams using more than one water level where feasible and the resulting hydrodynamic pressures and motions of the structures and their foundations should be recorded and analyzed. 3. Evaluation of Analytical Methods for Response Analysis Existing analytical methods for computing the response of all types of concrete dams to earthquakes should be evaluated by comparing calculated results with the responses recorded during forced-vibration tests and, more importantly, during actual earthquakes when significant ground motions are recorded at appropriate dam sites. If necessary, the methods should be refined and the computer programs needed for their implementation prepared in a form convenient for application in engineering practice. 4. Improvement of Arch Dam Analysis The methods used for input of the earthquake motions in present methods of earthquake analysis of arch dams urgently need improvement. Similarly, improvements are needed in procedures used to account for the interaction between arch dams and their supporting foundation rock.

60 5. Simplified Analysis Procedures Simplified analysis procedures should be developed that are suitable for the preliminary phase of design and safety evaluation of arch dams. 6. Evaluation of Dynamic Sliding and Rocking Response of Gravity Dams Analysis procedures should be developed to determine the dynamic sliding and rocking response of gravity dam monoliths. Utilizing these procedures, rational stability criteria should be derived, replacing the traditional sliding and overturning criteria that do not recognize the oscillatory response of dams during earthquakes.