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4 Nonlinear Analysis and Response Behavior INTRODUCTION As stated in Chapter 3, useful information about the expected earthquake response of concrete dams can be obtained from linear dynamic analysis based on assumed elastic behavior. However, a severe earthquake may cause significant damage to concrete dams, as was observed at Koyna Dam, and such damage usually is associated with important changes in structural stiffness. Consequently, if significant damage occurs, the actual performance of the dam can be predicted only by a nonlinear analysis that takes account of these stiffness changes. For this reason a rigorous evaluation of concrete dam seismic performance should consider the behavior of the materials and components in the system, which is characterized by nonlinear force-displacement relationships, in conjunction with nonlinear response analysis procedures. Additionally, realistic consideration of the nonlinear behavior of dams may indicate mechanisms that limit the earthquake response and provide an added margin of safety against failure. After definition of the seismic input, a comprehensive safety evaluation of a concrete dam requires (1) identification of the behavior of materials and components under dynamic loads, (2) mathematical models that can represent the nonlinear behavior, (3) efficient numerical procedures for computing the nonlinear earthquake response of the dam system, and (4) criteria to assess acceptable performance and evaluate failure modes. In this chapter the nonlinear behavior important in the earthquake response of concrete dams and the requirements for nonlinear analysis are identified. In addition, recent developments in modeling and analysis of such dams are summarized, and further research needed to improve the seismic safety evaluation is outlined. 61

62 CHARACTERISTICS OF NONLINEAR BEHAVIOR Concrete exhibits a complicated nonlinear relationship between stress and strain that is dependent on loading rate and history. The nonlinear behavior of mass concrete becomes significant as stresses approach the compressive strength or the comparatively small tensile strength. Because concrete dams are designed to resist their primary loadsgravity and hydrostatic through compressive stress fields, the tensile stresses induced by such static loads are minimal (or nonexistent). Moreover, when current design criteria for static loads are followed, the computed compressive stresses are found to be much less than the compressive strength of the concrete. However, strong earthquake ground motion can produce large dynamic stresses in dams, both compressive and tensile, and the combination of static and dynamic stresses may exceed the linear response range of the concrete, particularly with regard to the tensile stresses. Experiments demonstrate that concrete behavior is essentially linear under cyclic compressive loads up to approximately 50 to 60 percent of the compressive strength (4-1~. For example, assuming a compressive strength of 4,000 psi, nonlinear behavior is important for compressive stresses greater than about 2,000 to 2,400 psi. But linear elastic dynamic analyses of gravity and arch dams show that compressive stresses rarely exceed this range during typical earthquakes; consequently, the nonlinear behavior of concrete in compression, including hysteretic energy dissipation, can generally be neglected in the earthquake response analysis of dams. Tensile stresses in a dam produced by earthquake ground motion can be resisted only by the tensile strength of concrete, because reinforcement generally is not provided in the body of a dam. The tensile strength of concrete is an order of magnitude less that its compressive strength, and linear analyses demonstrate that the tensile limit may be exceeded at widespread different locations in a dam during an earthquake (3-8, 3-11~. For example, in a recent study (4-2) a gravity dam monolith subjected to a wide range of earthquake ground motions was analyzed assuming linear behavior. A moderate earthquake with a peak ground acceleration of 0.25 g induced tensile stresses greater than the tensile strength of typical concrete in both the top part of the dam and the heel, indicating the potential for cracking. As the stress in a dam approaches the tensile strength of the concrete, microcracks (which are always present in concrete) coalesce to form a crack surface. However, it is important to note that the tensile cracking caused by moderate earthquakes may not be deleterious to the performance of a dam, because the dynamic forces open and then recluse the cracks during a cycle of vibration. After the earthquake, static loads will generally return the stresses to compression, leaving the cracks in a closed condition and maintaining the dam's stability. Because tensile cracking in concrete is fundamentally a

63 fracture process that dissipates energy as the cracks propagate, the amplitude of dynamic response may be decreased by the cracking if the dam remains stable during the earthquake. During severe ground motion, however, it is possible for tensile cracks to propagate completely through a dam, potentially leading to dynamic instability and to uncontrolled release of water. Both linear analyses and model studies of arch dams have indicated related failure modes: extensive tensile cracking that results in the formation of a semicircular or rectangular notch near the crest, leading to dynamic instability of the notched portion. Although it is not certain that such a failure can actually develop in an arch dam due to an earthquake, or that crack propagation completely through a dam section necessarily results in instability, the major goal of a safety evaluation is to determine whether unstable response could develop due to credible levels of earthquake ground motion. Realistic analytical modeling of tensile cracking in mass concrete should recognize the variation in properties of the materials in a dam, the incremental construction procedure, and the effect of pore water on crack development. The practice adopted in some designs, as mentioned in Chapter 3, of varying the concrete strength over the dam height may affect the locations where tensile cracks develop. Also, contraction joints between the monoliths, as well as horizontal planes of weakness that may exist at lift joints, can be a major influence on crack location. The presence of pore water in saturated mass concrete affects the stress state and the initiation and propagation of cracks, particularly in the lower portions of a dam and at the interface with the foundation rock. Generally only the static pore water pressure is considered in an earthquake analysis, as a constituent of the combined (static plus dynamic) state of stress. The earthquake response is associated with dynamic total stresses that include both intergranular and pore pressure components, but typically it is assumed that the pore water does not migrate during the cyclic pressure changes because of their short duration. Consequently, it is usually assumed that the earthquake does not significantly influence the static pore pressure effects. It is recognized that some sort of hydraulic fracture mechanism might result at the wetted face of the dam due to dynamic pore pressures during the earthquake, but there is no evidence that such hydraulic fracturing has actually occurred. Thus, present practice assumes that pore pressures have no direct effect on dynamic cracking and that the cracking behavior may be characterized simply in terms of the dynamic total stresses. The preceding comments on concrete cracking pertain to both gravity and arch dams; however, the stress state in many gravity dams tends to be essentially uniaxial (cantilever stresses), whereas the stresses in arch dams are considered to be biaxial, involving components in both the cantilever and arch directions. On the other hand, the development of arch-direction tensile stresses is inhibited by vertical contraction joints that are provided

64 between monoliths, because the joints are unable to resist any significant net tension. The joints tend to open and close during a severe earthquake (3-17), and this nonlinear response mechanism has two consequences in the earthquake response of arch dams. First, the intermittent opening temporarily reduces the arch-direction resistance, causing transfer of load to both cantilever action and inertial resistance. The load picked up by cantilever bending may then lead to flexural overstress and failure of the monoliths. The second potential consequence of the joint opening-closing is compressive failure of the joint itself. Model tests have shown that loss of joint integrity is possible in arch dams (4-3), because as a joint opens the compressive stresses in the portion of the joint remaining in contact may increase dramatically, possibly crushing the concrete. On the other hand, the nonlinear behavior of the joints may limit the earthquake response of large (long-period) dams by further lengthening the vibration period, thereby reducing the dynamic amplification that is a function of the vibration properties of the dam and characteristics of the ground motion (3-17~. The behavior of the foundation rock supporting a dam is typically nonlinear, because the rock is often fractured and discontinuous. The nonlinear behavior of foundation rock will affect the static and dynamic response of a dam. For example, an elastic modulus based on small strain overestimates the stiffness of fractured foundation rock because of the rock's inability to resist large tensile stresses. During a severe earthquake the forces acting at the abutments of an arch dam also can increase significantly. In some cases it may be possible for the abutments to fail because of shear failure along planes of weakness in the foundation rock. Questions about foundation stability are a safety concern for gravity dams as well. Thorough consideration of the potential for foundation rock failure is often difficult, because information on subsurface rock conditions is limited, particularly in the case of older dams. However, foundation stability must always be studied intensively, because experience shows that actual failures of concrete dams generally are initiated in the foundation rock. The response predicted by a linear model of a concrete dam depends on both the spatial variation of earthquake ground motion (as described in Chapter 2) and the temporal variation. Analytically, the response of a linear dam system can be considered as the summation of responses to harmonic components of the ground motion specified for one or more support degrees of freedom. The nonlinear response of concrete dams may depend on other characteristics of the earthquake ground motion. For example, it may be affected by the duration of the ground motion and the amplitude of incremental ground velocity, particularly in the propagation of tensile cracks and the integrity of joints. Crack propagation under dynamic loads is affected by the number of loading cycles, which is related to the duration of ground motion. Similarly, the integrity of a joint may depend on the number of

65 times it opens and closes. A large increment of ground velocity, typical of ground motion near an earthquake source, will impart an impulse to a dam that can initiate and propagate tensile cracks or affect the dynamic stability after extensive cracking. For these reasons a realistic safety evaluation based on nonlinear analysis would require a detailed seismotectonic study of a dam site and the development of appropriate site-specific ground motion records. The nonlinear behavior mechanisms mentioned above (tensile cracking of concrete, loss of joint integrity, and foundation failure) result from dam vibration in response to earthquake ground motion. Overall sliding and overturning of a dam due to ground motion, the traditional criteria for static stability, are generally unrealistic failure modes during earthquakes. Indirect earthquake failures of dams due to fault displacement or overtopping also are important considerations, but they are not directly related to dynamic response. MATERIAL MODELS AND RESPONSE ANALYSIS As discussed in Chapter 3, linear dynamic analysis of a concrete dam system is valuable in understanding the general characteristics of earthquake response. However, if the linear analysis shows repeated tensile stresses significantly greater than the tensile strength, the concrete can be expected to crack, and the linear response results would no longer be valid. The ultimate earthquake behavior cannot be predicted from such linear models, because the strength of the materials is not represented, nor is the redistribution of forces due to tensile cracking, joint opening-closing, or foundation instability. A realistic nonlinear earthquake response evaluation of a concrete dam requires analysis of mathematical models that include tensile crack propagation in concrete, cyclic displacement behavior of joints and abutments, and deformability of the foundation rock. Because these phenomena induce nonlinear relationships between resisting forces and displacements, nonlinear methods of response analysis are required. These are considerably more complicated than linear analysis. A rigorous nonlinear response analysis should include the important darn-water and dam-foundation rock interaction effects mentioned in Chapter 3, together with nonlinear models of the dam concrete, joints, and foundation rock. The finite element method is presently recognized as the best approach for spatial discretization of the equations of motion for concrete dams because of its ability to represent arbitrary geometry and to incorporate arbitrary variations of material behavior. Finite elements can also be used to discretize the reservoir water and foundation rock, although alternatives such as boundary integral elements may be more efficient in modeling large reservoir and foundation regions. When concrete dam systems exhibit nonlinear response to severe earthquake

66 ground motion, the principle of superposition is no longer valid, and frequency- domain analysis may not be applied directly. Usually, the equations of motion are solved in the time domain using a time-stepping procedure. Because of the large amount of computation required to obtain the nonlinear dynamic response of a concrete dam system, it is mandatory to truncate the size of the water and foundation rock domains; however, the boundaries should be modeled in a manner that allows radiation of energy from the system. Methods for reducing the number of generalized coordinates while still representing the nonlinear behavior of a dam system may be explored (4-4~. The use of a substructure formulation also may provide an efficient approach for nonlinear earthquake analysis of dams. Reference 4-5 presents a numerical method in which a structure is divided into linear and nonlinear substructures. The method was used to analyze a concrete arch dam, including opening of vertical joints, which were modeled as local nonlinearities using gap elements. Other dynamic analysis procedures for concrete structures are described in reference 4-6. Finally, evaluation of the dynamic stability of portions of a dam after extensive cracking requires numerical procedures that properly represent the impact and sliding of the sections and that conserve momentum. An important decision in the analysis of the earthquake response of concrete dams is whether to use a two-dimensional or a three-dimensional model of the dam system. The complicated geometry of arch dams and associated valleys or canyons necessitates use of a three-dimensional model to represent their complex resistance mechanisms. Two-dimensional models are often employed for analysis of gravity dam monoliths with the resmctions discussed in Chapter 3, and nonlinear analysis of two-dimensional models is considerably less complicated than that for three-dimensional models. The complete nonlinear response analysis of concrete dams introduces considerations not normally required in a linear analysis. As mathematical models become more representative of true nonlinear material behavior, the numerical results may be sensitive to additional parameters of the model. Further research should be able to evolve mathematical models that can be expressed in terms of measured material properties with confidence, but the cracking of concrete is likely to be very sensitive to its failure-strain limit as well as to the static state of stress and strain that exists at the time the earthquake occurs. This initial state is primarily due to the gravity load and hydrostatic pressure, but it is also affected by shrinkage and temperature strains that accumulate during the incremental construction process and after completion up to the time of the earthquake. Diurnal and seasonal temperature variations control the temperature gradient through the dam, and this may have a major effect on the initial stress and strain distribution, particularly for arch dams, as does creep of the concrete over the service life of the dam. The importance of thermal and shrinkage strains (and possibly creep) is

67 evidenced by the cracks that developed in several monoliths in the Dworshak and Richard B.- Russell dams under static loading conditions (4-7~. An aerial photograph of the Richard B. Russell Dam is shown in Figure 4-1. The realistic analytical evaluation of earthquake response based on nonlinear behavior must recognize the distribution of stress and strain prior to an earthquake. This implies that for a given earthquake several analyses of a dam may be necessary, corresponding to the initial conditions at different times during the expected service life. The challenge of nonlinear dynamic analysis of concrete dams lies in developing mathematical models that represent the true behavior of concrete and joints and in incorporating these models in efficient numerical procedures. Although researchers and practitioners have investigated limited aspects of the nonlinear behavior of concrete dams, a great deal of innovative analytical research must yet be done to develop practical nonlinear response analysis procedures; in addition, the results of such analyses must be verified by careful experimentation before the procedures can be fully accepted for earthquake safety evaluation of concrete dams. CRITERIA FOR SAFETY EVALUATION Although the criteria for evaluating the seismic performance of concrete dams are discussed in detail in Chapter 6, it is pertinent to note here that performance criteria based on nonlinear response evaluations are especially important. Traditionally, a no-tension stress criterion has been used in the design of concrete dams. However, microcracking is always present in concrete, and the acceptance of moderate tensile cracking that does not impair the function of a dam is a realistic point of view for earthquake loads. The complete nonlinear earthquake analysis of new and existing dams is not likely to be undertaken in the near future, and linear analysis will remain the normal practice for some time to come. The U.S. Army Corps of Engineers (4-~) has proposed an evaluation procedure that uses several linear dynamic analyses to estimate the extent of cracking by varying the level of acceptable tensile stresses and modifying the viscous damping ratio to represent energy losses associated with cracking. If cracking is clearly indicated by large tensile stresses, a separate stability analysis must be performed based on the estimated elevation of the cracks. An evaluation (4-9) of the proposed criteria shows that, although they are conservative in predicting the presence of tensile cracking, incorrect elevations of the potential cracks are indicated. Further work is definitely required to improve the criteria for acceptable tensile cracking.

69 SUMMARY OF RECENT RESEARCH To understand the observed damage to Hsinfengkiang, Koyna, and Pacoima dams, experimental and analytical studies have been undertaken to determine the importance of nonlinear behavior in the earthquake response of concrete dams. Much of the work in concrete dams is related to or based on research in other applications of plain and reinforced concrete, for which excellent descriptions are available in references 4-10 and 4-11. Experimental Research Experimental investigations of the nonlinear behavior and failure mechanisms for concrete dams are discussed in Chapter 5 of this report, but the testing of the materials used in such dams is discussed briefly in the following paragraphs. Most mathematical models for tensile cracking are based on data from direct uniaxial, split-cylinder, and flexural tension tests. The measurement of tensile behavior is affected by the type of test and by the size, configuration, and curing conditions of the specimen; this fact clouds the selection of parameters for material models (4-12~. Most tests use plain concrete specimens of small aggregate that may not properly represent the effect of the large aggregate used in typical mass concrete. Tests of core samples properly taken from actual dams are more representative of the mix and in situ properties of the concrete. Because of the sensitivity of tensile tests to the loading and measurement system, optical interferometry techniques have been used to provide a continuous measurement of deformation in test specimens (4-13~. The results show the nonuniform distribution of strain longitudinally and transversely in the specimens and strain discontinuities in the microcracked and fractured regions. Analyses of the test results indicate that the energy dissipated in the fracture process is due to separation of the two sides of the crack, while the extension of the crack front produces relatively little energy dissipation. There have been few experimental investigations of the dynamic tensile behavior of concrete. Many mathematical models are based on early uniaxial tests (4-14) that showed increased tensile strength and stiffness with increasing monotonic strain rate. Recent biaxial tests of hollow concrete cylinders have been performed (4-15) using an impulsive loading system. The biaxial tensile strength in tension-compression loading also increased with decreasing rise time of the impulsive load, and the strain at failure was essentially independent of the rise time, which confirms the conclusions from earlier studies. These results suggest that a cracking criterion for concrete should be based on a maximum tensile strain instead of tensile strength. The dependence of the behavior on the loading rate, which is characteristic of a viscoelastic or viscoplastic material, should be accounted for in the mathematical

70 model. These trends are also indicated in a review of uniaxial tension tests of concrete specimens (4-14), where it is recommended that the tensile strength be increased by 50 percent to include the effect of strain rates typical in the earthquake response of dams. Models for Concrete Cracking Early work in evaluating the static behavior of reinforced concrete members, including tensile cracking, used the finite element method with predefined cracks modeled by separations between elements (4-16~. The modeling of tensile cracking by discrete gaps in the mesh was extended to allow automatic locations of cracks (4-17~. When the average stress between two elements exceeded the tensile strength of the concrete, the common nodal points were separated, altering the finite element mesh. This discrete-crack approach involves a number of computational difficulties because the finite element mesh is redefined at each loading stage, and it is difficult to determine in which direction a crack will propagate during the next load increment. Moreover, in this approach the crack propagation is dependent on the size, shape, orientation, and order of the elements used in the mesh. To overcome problems with the discrete crack approach, the tensile cracks can be considered "smeared" over an element (4-18~. In the smeared-crack approach the discontinuous displacement field caused by cracking is averaged over the element and represented by the continuous displacement functions used to derive the element. A crack is assumed to form in any element in which the principal stress reaches the tensile strength of the concrete in the direction perpendicular to this stress; then the isotropic model of the concrete is modified to an orthotropic one with zero stiffness in the tensile direction but with possible shear transfer across the crack. Such transfer of shear stresses across cracks is important in mass concrete because of the large aggregate size and the probability of aggregate interlocking, but the smeared crack approach can represent shear transfer only approximately, because there is no direct information on the width and distribution of the smeared cracks. The tensile strength used to determine crack initiation can be modified to account for a multiaxial stress state (4-19), and the smeared-crack model behavior can be easily incorporated into nonlinear finite element analysis procedures. This requires only modification of the tangent stiffness matrix for the current state of cracking in an element and release of stress perpendicular to newly formed cracks. The smeared-crack approach has been criticized because the numerical results are not objective with respect to the finite element mesh (4-20~. As the element size decreases, the zone of fracture decreases, and the force required to propagate the crack can decrease to a negligible value. To remedy this lack of objectivity, the theory of fracture mechanics has been

71 used to modify the smeared-crack approach in the so-called blunt-crack approach (4-20, 4-21, 4-22~. This approach recognizes that tensile cracking is a fracture process in which the concrete at the crack front exhibits strain softening (decreasing stress with increasing strain) as microcracks coalesce to extend the crack. The crack front has a characteristic width, usually related to the aggregate size in the concrete mix. The criterion for crack initiation uses the fracture energy for the material (energy released in formation of a crack with unit area), and objectivity is achieved by selecting the strain-softening modulus, and possibly the tensile strength, to give the fracture energy of the material. An important advantage of the blunt-crack model is that it includes the effect of aggregate size, which is important when attempting to compute the response of large mass concrete dams using material properties obtained from testing small specimens. In a finite element analysis the crack band-width can be assumed to be the element size (within certain limits), and the strain-softening modulus then depends on the fracture energy and tensile strength of the concrete (4-20~. The blunt-crack approach has been used in a similar form in reference 4-23, and it is related to the line (or fictitious) crack theory (4-24~. One problem with the smeared-crack approach is the difficulty in representing impact and sliding of sections in the dam after extensive crack propagation, because the discontinuous displacements across the crack are not well defined; in some cases a discrete-crack has been combined with a smeared-crack model to represent such behavior. Analytical Research Numerical computation of the nonlinear earthquake response of concrete dams has received more attention than physical testing of models and materials. An early investigation of the nonlinear response of gravity dam monoliths used a biaxial failure model for concrete (4-25~. The nonlinear compressive and tensile behavior of concrete was recognized by modifying an equivalent uniaxial stress-strain relationship (tension and compression) in accordance with the current state of stress. A smeared-crack approach was used to represent tensile cracking based on a strength criterion for crack initiation. Analysis of Koyna Dam, neglecting water and foundation rock interaction effects, showed that tensile cracks formed near the top of the dam close to the change in downstream slope, but the cracks did not propagate through the cross section. Including strain rate effects in the concrete model stiffened the dam, increasing the participation of the higher-vibration modes and producing more extensive cracking. An interesting finding was that the amount of tensile cracking was very sensitive to variations in the assumed concrete tensile strength; variations such as those observed in a set of typical tensile cracking experiments produced dramatically different amounts of cracking. The study also showed that tensile cracking predicted by this

72 material model did not dissipate a significant amount of energy, suggesting that nonlinear behavior may not reduce the earthquake forces developed in the dam. However, this conclusion may not be correct, because the fracture of concrete may not have been properly accounted for in an objective manner. Another series of studies (4-26, 4-27) involved analysis of a finite element model of a complete gravity dam-water-foundation rock system. The concrete was modeled as a rate- and history-dependent material using an elast~c- viscoplastic relationship with a Mohr-Coulomb failure surface for compression and tension. The material model used viscoplastic work to measure accumulated damage in the dam; the anisotropy due to tensile cracking was not considered. Analysis of Koyna Dam showed a substantial residual displacement with large energy dissipation during the strong-motion part of the input record. It appears, however, that the energy dissipation is from viscoplastic work as the concrete softens in tension, which would not occur if cracking were allowed; thus, the significance of these results may be questioned. Two recent studies illustrate the application of smeared-crack models for gravity dams. In the first application the crack-band model, described previously, was incorporated into a nonlinear dynamic analysis procedure for dam- water systems (4-28~. The analysis of a 400-ft-high gravity dam monolith with a full reservoir due to a horizontal ground motion with a maximum acceleration of 0.36 g showed extensive cracking of the concrete, as shown in Figure 4-2. The cracking is located near the stress concentration caused by the change in geometry of the downstream face and the dam-foundation interface. The distributed crack zone indicated in Figure 4-2 is in conflict with the experimental results, summarized in Chapter 5, which show crack propagation in a narrow zone. In another analytical study (4-29, 4-30) the crack-band model was used with several modifications: (1) a number of features designed to eliminate crack spreading, including a special formulation of the finite element to eliminate spurious stiffness, and (2) user control over the elements that are susceptible to cracking during the response analysis. The latter modification allows control over the direction of the crack, but it also indicates the lack of theoretical knowledge of crack propagation in concrete dams. Figure 4-3 shows the crack that forms in the same 400-ft-high gravity dam monolith with full reservoir when subjected to horizontal and vertical ground motions with maximum acceleration of 0.50 g and 0.30 g, respectively. The narrow crack zone extends from the stress concentration at the downstream face, turning down near and parallel to the upstream face. In addition, a crack formed near the base of the dam. Although the analysis showed that the dam remained stable, many numerical difficulties were encountered. As noted above, the alternate approach for modeling tensile cracking is to represent the formation, propagation, and closure of discrete cracks in the concrete. The discrete-crack approach has been used in the finite element

73 / / / / / / / / / / / / ~ _ _ /~x ~ ~ ~ ~ ~ ~ ~ ·. FIGURE 4-2 Cracking calculated in 400-ft-high monolith of Pine Flat Dam with full reservoir due to 0.36-g peak horizontal ground motion; solid lines show cracks open at one instant of iune, dashed lines show cracks that opened previously and then closed (4-28). earthquake analysis of gravity dam monoliths, neglecting interaction with the water and foundation rock (4-31~. Each crack was monitored, and the topology of the element mesh was redefined to represent the current state of cracking. Analysis of Koyna Dam showed that the top part of the dam would become unstable as a tensile crack propagated across the cross section due to an artificial earthquake with a peak acceleration of 0.50 g. As in the earlier research mentioned above, the extent of cracking was very sensitive to the assumed value for concrete tensile strength. The response was also

74 node 156 i-1- 'l / / I ~ . node 1 77~ //~7~ ~ 7 ~ ///n\l I- / / / / - -it 89 97 O' ~ , L: . . 1~ :: :::::: : ~ ~- 1~37~-~ ::: -I I.... ~1~:~ - ~ ,.: ~1~53~ ::: ~ ::~: :~::::~: ~~ I ... I: ::~ :: ::::~: f~:~1 6~:~ !:~::~: ~~:~ ~:~ i:::: . '~:~ i: ~ Aim: :::::: I ~:~ : Am: :~ ' ' If: 1~&~-~:~:~ ~ i:: ~- ,: 1: : : : r I :: ~ ::: i:::: : ~ : i: : , / ~ / / / 1 car If,,'. /, / / / . ,,,, / / I I .,////11 ///7~/ 7 //////1 /// / / / 1 1 4/////1 1 1 node 208 node 21 7 FIGURE 4-3 Cracking calculated in system of Figure 4-2 due to combined horizontal and vertical ground motions with peak values of 0.50 g and 0.30 g, respectively; cracked elements are shown in black (4-30). dependent on the finite element mesh size and orientation, although the effect of aggregate interlock across the cracks was relatively small. Another approach (4-6) used fracture mechanics techniques to overcome the sensitivity of the tensile stress field to the finite element discretization near a crack. Applying the technique to a monolith of Pine Flat Dam, including compressible water in the reservoir, showed that a tensile crack propagated from the upstream face at the change in slope, but stopped short of the downstream face as the compressive stresses arrested the crack growth. The upper portion of the dam appeared to remain stable; however, the cracking seemed extensive for the relatively small peak ground acceleration of 0.1 g. Other applications of fracture mechanics concepts to concrete dams are described in references 4-32 and 4-33. There have been two particularly noteworthy earthquake studies of arch dams that included the nonlinear behavior of construction joints. The first used an approximate representation of the discrete cracks formed by the joints (4-34~. The stiffness of the elements in the vicinity of the joints was modified to represent the current state of tensile stresses near the joints. This approach has also been used to represent opening of a horizontal joint

75 in a gravity dam (4-35~. Another study (4-36) used a simple discrete-crack model of the joint that required only one finite element through the thickness of the arch. The analytical results for a single arch rib showed the expected opening-closing behavior and large compressive stresses in the portion of the joint in contact. Later work (4-37) extended the crack model to represent vertical construction joint opening and horizontal joint opening at lift surfaces in arch dams. In one of the most complete nonlinear analyses to date, the earthquake response of Pacoima Dam (Figure 24) was computed, including dam-foundation interaction (with massless foundation rock) and dam-incompressible water interaction (4-38~. An example of the calculated response of Pacoima is shown in Figure 4-4. The earthquake ground motion for this study was the Pacoima Dam record scaled to a maximum horizontal acceleration of 0.50 g and vertical ground acceleration of 0.30 g. Figure 4-4 shows the joint stresses and displaced shape of the crest arch and crown cantilever at two instants of time during the response. The response results, as exemplified by this figure, showed that the vertical contraction joints in the upper part of the dam and the joint assumed at the dam-foundation interface opened under moderate ground motion. The loss of arch stiffness due to vertical joint opening also caused horizontal joint opening in the upper parts of the cantilevers. During severe ground motion some cantilever blocks lifted off their supports, and large compressive cantilever stresses developed, possibly invalidating the assumption of no joint slip and linear behavior of the concrete in compression. Nonlinear analysis of concrete dams allows for the inclusion of other phenomena that may affect the earthquake response. One such possible effect is cavitation of the impounded water, in which gaseous regions form if the absolute pressure in the water becomes less than the vapor pressure. The possibility of cavitation has been shown analytically and observed in model tests (4-3~. The formation and collapse of gaseous regions in the water would alter the hydrodynamic pressure acting on the upstream face of a dam and hence change the dynamic response. One analytical study of cavitation (4-39) for a gravity dam monolith, assuming incompressible water, showed that impact of the water resulting from collapse of the cavitation bubble can increase tensile stresses in the top part of the dam by 20 to 40 percent. In contrast, an evaluation of dam-water interaction including compressible water concluded that cavitation does not significantly affect the maximum stresses due to earthquake ground motion (4-40~. Recent Walk illUluuill~ cumpreSSIDIllly also confirms the latter conclusion, in which it was shown that cavitation has a very small effect on peak displacements and stresses in a dam (4-41~. However, cavitation can double the peak acceleration at the dam crest, which may affect appurtenances and facilities at the crest. To date, there has been no research into cavitation effects for arch dams, where dam-water interaction effects may be more important than . A 1_ ~ A ~ _ _ ~ . ... , . ^. .

76 ~ of .~4 ,4 ~ ~ o :~: -it. i-O'rl , alla ~ ~ ~ ~ . ~ ~ o _Z o- 5 A- E ~ ~ , ~ to .~::, : are 5 f sO ~ :°

77 in gravity dams; however, it has been noted that cavitation may have a beneficial effect by limiting the upstream hydrodynamic force acting to open the vertical joints (4-39~. RESEARCH NEEDS Recognizing the importance of evaluating the earthquake safety of concrete dams and the limited knowledge that exists concerning the nonlinear behavior of concrete dams, the following items should be addressed in a future comprehensive research effort: 1. Material Testing of Mass Concrete Additional data on the behavior of mass concrete under dynamic loads is urgently needed and will require an extensive physical testing program. The testing should emphasize the tensile cracking of the mass concrete under multiaxial stress states that are representative of the in situ stresses, including strain rate effects. The test program should recognize the special properties of the mass concrete used in dam construction, such as its large aggregate size. Concrete samples should be representative of the actual curing conditions in dams. To accomplish this, the testing should include laboratory samples as well as cores from actual dams, and some specimens should be taken from roller-compacted concrete dams. To perform these tests it may be necessary to develop or adapt equipment to provide biaxial cyclic loads on large specimens. 2. Development of Materials Models for Concrete Using the data obtained from testing of mass concrete specimens, it is important to develop realistic mathematical models for tensile cracking under dynamic loads. Identification of a set of parameters that are predictive of nonlinear dynamic behavior of concrete would be an important advance. The material model should allow multiaxial stress states, including criteria for tensile cracking and propagation of cracks, strain rate effects, and shear stress transfer by aggregate interlock. The models must make reasonable trade-offs between fitting the experimental data, satisfying conditions such as objectivity, and providing computational efficiency. Further study of the smeared-crack approach and use of fracture mechanics principles should be pursued. With advances in computational techniques and computer hardware, the discrete-crack approach for dynamic response may provide an alternative.

78 3. Modeling of Other Nonlinear Mechanisms Additional research is required to develop and apply models of construction joint behavior in the earthquake analysis of concrete dams. Such models must represent the redistribution of forces as the joints open and close and allow for degradation of the joint resulting from a large number of loading cycles. Models for concrete and rock abutments that include potential shear failure modes are also necessary. Although not directly related to dynamic response, analytical procedures must be used for determining the state of stress and strain in a concrete dam at any point in timedue to gravity and hydrostatic loads, shrinkage, temperature variations, temperature gradients, and creep for accurate assessment of nonlinear dynamic response of dams at the time of an earthquake. 4. Numerical Procedures for Computing Nonlinear Response Once realistic mathematical models for concrete, joints, and foundation rock are available, they can be incorporated into several classes of numerical procedures for nonlinear dynamic response analysis of concrete dam systems, including interaction with the impounded water and flexible foundation rock and the effects of reservoir-bottom shock-wave absorption. Research should identify methods for reducing the computational effort, while still representing the important nonlinear behavior of the system, so that nonlinear analysis can be used for routine design and evaluation of dams. Although the finite element method and time integration of the equations of motion appear to be the most useful techniques for nonlinear dynamic analysis of concrete dams, other discretization methods and procedures for solving nonlinear equations of motion should not be precluded. The use of algorithms that take advantage of the vector and parallel processors in supercomputers should be pursued. 5. Parametric and Detailed Response Studies With the development of efficient numerical procedures, response analyses of typical concrete dams can be used to improve understanding of the nonlinear behavior during earthquake ground motion. In particular the studies should identify the significance of tensile cracking and joint opening with respect to either precipitating failure or limiting dynamic response. It is important to determine the sensitivity of the response to parameters describing the nonlinear models. Postcracking stability in gravity and arch dams and postearthquake joint integrity in arch dams (as well as in some gravity dams) must be evaluated. In addition, the influence of the following factors should be determined: (1) ground motion characteristics, particularly amplitude,

79 frequency content, duration, and occurrence of velocity pulses, (2) water compressibility with respect to nonlinear response, and (3) modeling issues, such as degree of spatial discretization, extent of reservoir and foundation rock domains, and radiation conditions at the boundaries. 6. Dynamic Testing of Dam Models In parallel with the analytical studies, further dynamic testing of dam models is essential for verifying nonlinear behavior and calibrating mathematical models and numerical results. If at all possible, testing of gravity dam and arch dam models including water and foundation rock should be undertaken. New, larger earthquake simulators such as that recently installed at the research laboratory of the Ministry for Water Conservancy and Hydroelectric Power in Beijing, China, may be required for realistic testing of concrete dam systems. 7. Identification of Design Criteria Realistic design criteria for concrete gravity and arch dams should be developed using the results of analytical and experimental studies. Although nonlinear earthquake analysis may not become a standard practice in design offices, design criteria should recognize the tensile strength of concrete and postcracking stability of gravity and arch dams and identify acceptable limits of such behavior. Comparison of the predicted nonlinear response with the more easily calculated linear response would be useful to determine the limits of assessing nonlinear behavior based on the results of linear elastic analysis. Current design criteria should be evaluated in this regard. 8. Investigation of Earthquake-Resistant Design Measures Nonlinear response analysis capabilities will allow investigation of innovative measures for increasing the earthquake safety of concrete dams. Based on the results of nonlinear analyses, it may be possible to alter the geometry of dams and foundation abutments to improve resistance to earthquakes. Different jointing schemes, including use of joint materials that dissipate energy, should be investigated. It must be emphasized that these research needs are not only important in improving the seismic safety of concrete dams but would also represent significant advances in other applications of earthquake engineering. Similarly, research in the nonlinear dynamic behavior of other structural systems will have a positive effect on the study of concrete dams.