Earthquake Analysis of Arch Dams: Factors to Be Considered

Transcription

1 Earthquake Analysis of Arch Dams: Factors to Be Considered Anil K. Chopra, M.ASCE 1 Downloaded from ascelibrary.org by on 5/8/18. Copyright ASCE. For personal use only; all rights reserved. Abstract: The factors that significantly influence the three-dimensional analysis of arch dams are identified: the semiunbounded size of the reservoir and foundation-rock domains, dam-water interaction, wave absorption at the reservoir boundary, water compressibility, dam foundation rock interaction, and spatial variations in ground motion at the dam-rock interface. Through a series of example analyses of actual dams, it is demonstrated that (1) by neglecting water compressibility, the stresses may be significantly underestimated for some dams or overestimated for others; () by neglecting foundation-rock mass and damping, the stresses may be overestimated by a factor of to 3; and (3) spatial variations in ground motion, typically ignored in dam engineering practice, can have profound influence on the earthquakeinduced stresses in the dam. This influence obviously depends on the degree to which ground motion varies spatially along the damrock interface. For the same dam, this influence would differ from one earthquake to the next, depending on the location and depth of the causative fault relative to the dam site. How to select damping values for the dam concrete and foundation rock is also addressed. DOI: 1.161/(ASCE)ST X American Society of Civil Engineers. CE Database subject headings: Arch dams; Dam foundations; Ground motion; Earthquakes. Author keywords: Arch dams; Dam-water foundation interaction; Spatially varying ground motion. Introduction Most existing dams in seismic regions were designed using methods that are now considered simplistic and inaccurate. The damage sustained by the few dams that have been subjected to intense ground motions, e.g., Koyna Dam in India, Hsinfengkiang Dam in the People s Republic of China, Sefidrud Dam in Iran, and Pacoima Dam in the United States, together with the growing concern of the seismic safety of critical facilities, has led to considerable interest in reevaluating existing dams using modern analysis and experimental procedures. Over the past years, the seismic safety of many dams has been evaluated, and some have been upgraded to improve their seismic resistance [United States Bureau of Reclamation (USBR) 1998a, b, 1999, ]. Evaluating the seismic performance of arch dams requires dynamic analysis of three-dimensional dam-water foundation-rock systems that recognize the semiunbounded size of the impounded water and foundation-rock domains, and consider nonlinearities associated with opening or slipping of vertical contraction joints and cracking of concrete during intense earthquake motion. The objective of this paper is to discuss analysis procedures that are appropriate and the factors to be included in estimating seismic demands on concrete arch dams. 1 Horace, Dorothy, and Katherine Johnson Chair in Engineering, Dept. of Civil and Environmental Engineering, Univ. of California, Berkeley, CA chopra@ce.berkeley.edu Note. This manuscript was submitted on June 3, 1; approved on April, 11; published online on January 17, 1. Discussion period open until July 1, 1; separate discussions must be submitted for individual papers. This paper is part of the Journal of Structural Engineering, Vol. 138, No., February 1, 1. ASCE, ISSN /1/5 14/$5.. Linear Analysis Procedures Based on the substructure method, a linear dynamic analysis procedure was developed to determine the earthquake response of arch dams and implemented in the computer program EACD-3D-96 (Tan and Chopra 1995b, 1996b). This procedure enables threedimensional analysis of a concrete arch dam-water foundationrock system, including (1) dam foundation rock interaction; () dam-water interaction considering compressibility of water and partial absorption of hydrodynamic pressure waves by sediments invariably deposited at the reservoir boundary bottom and sides; and (3) the semiunbounded extent of the impounded water and foundation-rock domains. The significance of these factors in dam response has been investigated previously (Tan and Chopra 1995a, 1996a). The preceding analysis procedure and the computer program EACD-3D-96 have recently been extended to determine dam response to free-field ground motion that varies spatially along the dam-rock interface (Wang and Chopra 1), resulting in the computer program EACD-3D8 (Wang and Chopra 8). Ground motion typically recorded at a few locations at the dam-rock interface around the canyon is interpolated to define the excitation at all nodes in the finite-element idealization of the dam that are located at the interface (Alves 4; Chopra and Wang 8). This extended analysis procedure includes dam-water-foundation interaction effects to the same degree of rigor as in Tan and Chopra (1995b). In contrast, earlier studies of dam response to spatially varying excitation have ignored foundation inertia and damping have and assumed water to be incompressible (Mojtahedi and Fenves ; Alves 4). Finite-element analyses of dams conducted in professional practice are often based on commercial software, ignoring these factors: hydrodynamic effects are represented by an added water mass moving with the dam, implying that water compressibility and reservoir boundary absorption are neglected; and foundation rock is usually assumed to be massless, implying that foundation inertia and JOURNAL OF STRUCTURAL ENGINEERING ASCE / FEBRUARY 1 / 5 J. Struct. Eng., 1, 138(): 514

2 damping material and radiation are ignored. These approximations are attractive because they simplify the analysis greatly, but they render inaccurate results, as demonstrated subsequently. Downloaded from ascelibrary.org by on 5/8/18. Copyright ASCE. For personal use only; all rights reserved. Dam-Water Foundation Interaction Effects Starting in 1996, the USBR embarked upon a major program to evaluate the seismic safety of dams. Among the several dams investigated was the famous Hoover Dam, a 1-m (75-ft)-high curved gravity dam. Seismological and geological investigations concluded that the Mead Slope fault, at a distance of approximately 1.61 km (1 mi), was capable of generating an earthquake strong enough to cause a peak acceleration of approximately :8 g at the site. Assuming the foundation rock to be massless and neglecting water compressibility assumptions necessitated by the limitations of the computer software available at the time led to very large tensile stresses. At 6 m ( ft) below the dam crest, the induced stresses on the downstream and upstream faces greatly exceeded the tensile strength of the concrete, indicating that the dam would crack through the thickness (USBR 1998b). These results did not seem credible to USBR engineers, who believed that this dam, with its cross section similar to gravity dams, should perform much better because it is curved in plan and confined in a narrow canyon. To resolve this apparent inconsistency between results of computer analyses and expectations based on engineering experience and judgment, a comprehensive investigation into the significance of dam-water foundation-rock interaction effects was undertaken. The discussion of dam-water foundation-rock interaction effects that follows is based on earthquake analyses of four arch dams: Deadwood Dam, a 5-m-high single curvature dam; Monticello Dam, a 93-m-high double curvature dam; Morrow Point Dam, a 14-m-high double curvature dam; and Hoover Dam, a 1-m-high curved arch gravity dam. Ground motions consistent with the seismic hazard for each of the four dam sites were selected. Defining the selected ground motion as free-field motion, assumed to be spatially uniform around the dam foundation rock interface (or canyon surface), all analyses were implemented by the EACD- 3D-96 computer software (USBR 1998a, 1998b, 1999, ). Implications of Neglecting Foundation-Rock Inertia and Damping The earthquake-induced stresses in each of the four dams computed under two conditions are first compared, considering the following: (1) dam foundation-rock interaction; and () foundation-rock flexibility only (Figs. 1 4). For these two conditions, the largest arch stress on the upstream or downstream face of the dam is 3.3 MPa (476 psi) versus 5.8 MPa (844 psi) for Deadwood Dam; 5. MPa (73 psi) versus 9.7 MPa (141 psi) for Monticello Dam; 4.6 MPa (665 psi) versus 9. MPa (1336 psi) for Morrow Point Dam; and 5. MPa (758 psi) versus 15. MPa (4 psi) for Hoover Dam. These results demonstrate that if only foundation-rock flexibility is considered, the stresses are overestimated by a factor of approximately for the first three dams and by a factor of approximately 3 for Hoover Dam. Dam foundation rock interaction lowers the fundamental resonant frequency of the dam and increases the overall damping in the system. The lowering of the frequency is almost entirely attributable to foundation flexibility with negligible influence of foundation mass and damping (Tan and Chopra 1995a). However, the computed response is larger if only foundation flexibility is considered, because the increase in overall damping caused by foundation damping material and radiation is ignored. This Fig. 1. Peak values of tensile arch stresses in Deadwood Dam, considering (a) dam-foundation interaction; (b) foundation-rock flexibility only (image courtesy of Larry K. Nuss) Fig.. Peak values of tensile arch stresses in Monticello Dam, considering: (a) dam-foundation interaction; (b) foundation-rock flexibility only (image courtesy of Larry K. Nuss) energy loss cannot be modeled by increasing the damping value for the dam in a dam-foundation model that considers foundation flexibility only, because there is no rational basis (short of full damfoundation interaction analysis) to determine the quantitative increase in damping. Because such overestimation of stresses may lead to overly conservative designs of new dams and to the erroneous conclusion 6 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / FEBRUARY 1 J. Struct. Eng., 1, 138(): 514

3 Downloaded from ascelibrary.org by on 5/8/18. Copyright ASCE. For personal use only; all rights reserved. Fig. 3. Peak values of tensile arch stresses in Morrow Point Dam, considering (a) dam-foundation interaction; (b) foundation-rock flexibility only (image courtesy of Larry K. Nuss) that an existing dam is unsafe and thus requires upgrading, it is imperative that dam foundation-rock interaction effects be included in earthquake analysis of concrete dams. Implications of Neglecting Water Compressibility Next, the earthquake-induced stresses in two of the four dams are compared. The stresses are computed under two conditions: with water compressibility considered and with water compressibility neglected. In the latter case, reservoir boundary absorption effects are implicitly neglected. By neglecting water compressibility and reservoir boundary absorption, the stresses may be significantly underestimated, as in the case of Monticello Dam (Fig. 5), or considerably overestimated, as in the case of Morrow Point Dam (Fig. 6); note that these discrepancies vary with the location on the dam surface. Thus, water compressibility and reservoir boundary absorption should be included in the analysis of arch dams. It is not possible to identify the system parameters for which dam response will be overestimated (or underestimated) by neglecting water compressibility. This is because water compressibility modifies the resonant frequency, increases the overall damping (because of the radiation of energy upstream or through the reservoir bottom), and significantly influences the shape of the frequency response curve (Tan and Chopra 1995a). Thus, the influence of water compressibility on the earthquake response of a dam would depend on the frequency characteristics of the ground motion as well as the vibration periods of the dam, increasing the response in some cases but reducing it for others, as observed previously. Fig. 4. Peak values of tensile arch stresses in Hoover Dam, considering (a) dam-foundation interaction; (b) foundation-rock flexibility only (image courtesy of Larry K. Nuss) Fig. 5. Peak values of tensile arch stresses in Monticello Dam computed under two conditions: (a) water compressibility considered; (b) water compressibility neglected (image courtesy of Larry K. Nuss) JOURNAL OF STRUCTURAL ENGINEERING ASCE / FEBRUARY 1 / 7 J. Struct. Eng., 1, 138(): 514

4 Downloaded from ascelibrary.org by on 5/8/18. Copyright ASCE. For personal use only; all rights reserved. Fig. 6. Peak values of tensile arch stresses in Morrow Point Dam computed under two conditions: (a) water compressibility considered; (b) water compressibility neglected (image courtesy of Larry K. Nuss) Influence of Spatial Variations in Ground Motions Structural response to spatially varying ground motions may be split into two parts: quasi-static and dynamic (Chopra 7). The quasi-static component is the response caused by static application of the time-varying displacements prescribed at the nodal points (in the finite-element model) located at the dam-rock interface. How significantly the dam response is affected by spatial variations in the excitation is closely tied to the importance of the quasi-static component of the response. To illustrate this concept, the response of Pacoima Dam to spatially varying ground motions recorded at the dam-rock interface during the earthquakes of January 13, 1, and January 17, 1994, was determined by the EACD-3D8 computer program; gravity effects were not included. Starting with the ground motions recorded at three locations on the dam-rock interface, shown in Fig. 7, the motions at all nodes on the interface were determined by interpolating or extrapolating the records to define the spatially varying input motions in the EACD-3D8 computer program. January 13, 1, Earthquake The quasi-static component is a significant but not dominant part of the displacement response of Pacoima Dam to the January 13, 1, earthquake records. Fig. 8, which identifies the quasi-static component in the history of displacements at the center of the crest, shows that the quasi-static component is not a major part of the displacement in the radial direction (the direction of largest response), whereas it is dominant in the displacements in the tangential and vertical directions. At the crest center, the ratio of the peak values of the quasi-static component and the total displacement is 78% for the tangential response and 9% for the vertical response, but only 45% for the radial response. Therefore, the spatial variations in ground motions are expected to significantly influence, but not dominate, the stresses in the dam, Fig. 7. Accelerograph (Channels 1 17) locations at Pacoima Dam [reprinted from Mojtahedi and Fenves (), with permission from California Geological Survey] 8 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / FEBRUARY 1 J. Struct. Eng., 1, 138(): 514

5 Downloaded from ascelibrary.org by on 5/8/18. Copyright ASCE. For personal use only; all rights reserved. 4 (c) Vertical (a) Radial (b) Tangential Time, sec as shown in Fig. 9, which presents the peak values of the tensile stresses in the cantilever direction on the downstream face of the dam; similar figures for arch and cantilever stresses on both faces of the dam are available in Chopra and Wang (8). Presented are stresses caused by four different excitations. The first three are spatially uniform excitations defined by ground motions recorded at the base of the dam (Channels 9 11), the right abutment (Channels Quasi-static Fig. 8. Quasi-static and total displacement histories at crest center of Pacoima Dam caused by spatially varying ground motion during the January 13, 1, earthquake: (a) radial component; (b) tangential component; (c) vertical component [reprinted from Chopra and Wang (1), with permission from Wiley] (a) Total 1 14), and the left abutment (Channels 15 17); see Fig. 7. The fourth excitation is defined as the recorded (and interpolated or extrapolated) spatially varying ground motions. The stresses caused by spatially varying excitation may be smaller or larger than those caused by spatially uniform ground motion, depending on the intensity of the latter. They are larger when compared to the stresses caused by the base motion the (b) (c) (d) Fig. 9. Peak values of tensile cantilever stress (MPa) on the downstream face of Pacoima Dam caused by the January 13, 1, earthquake: (a) spatially uniform excitation defined by Channels 9 11; (b) spatially uniform excitation defined by Channels 1 14; (c) spatially uniform excitation defined by Channels 15 17; (d) spatially varying excitation [reprinted from Chopra and Wang (1), with permission from Wiley] JOURNAL OF STRUCTURAL ENGINEERING ASCE / FEBRUARY 1 / 9 J. Struct. Eng., 1, 138(): 514

6 Downloaded from ascelibrary.org by on 5/8/18. Copyright ASCE. For personal use only; all rights reserved. least intense of the three spatially uniform excitations but are generally smaller than the stresses caused by the abutment motions. Comparing the four parts of Fig. 9 also reveals that spatial variations in ground motion significantly influence, but not dominate, the stresses in Pacoima Dam caused by the January 13, 1, earthquake. January 17, 1994, Northridge Earthquake The quasi-static component is dominant in the displacement response of Pacoima Dam to the January 17, 1994, Northridge earthquake records; the spatial variations in ground motion therefore have profound influence on the computed stresses in the dam. Fig. 1 identifies the quasi-static component in the history of displacements at the center of the dam crest. The ratio of the peak values of the quasi-static component and the total displacement at the crest center is 87, 93, and 97% for the radial, tangential, and vertical directions, respectively. Consequently, the spatial variations in ground motion are expected to profoundly influence the stresses in the dam. Fig. 11, which presents the peak value of the tensile stresses in the arch direction on the downstream face of the dam, confirms this expectation; similar figures for arch and cantilever stresses on both faces of the dam are available in Chopra and Wang (8). Presented are the stresses caused by four different excitations. The first three are spatially uniform excitations defined by ground motion at the base of the dam (Channels 9 11), the right abutment (Channels 1 14), and the left abutment (Channels 15 17); see Fig. 7. The fourth excitation is defined as the recorded (and interpolated or extrapolated) spatially varying ground motion. The distribution pattern of stresses caused by the three spatially uniform excitations is similar, although the magnitude of stresses caused by ground motions recorded at the base of the dam is much smaller than that caused by motions at the left or right abutment; the stresses caused by the two abutment excitations are similar in magnitude. Comparing the stresses caused by spatially varying and spatially uniform excitations shows that spatial variations in ground motion had a profound influence on the magnitude and the distribution of arch stresses (Fig. 11). In particular, note the stress concentration near the top of the left abutment [Fig. 11(d)]. Spatial variations in ground motion cause much larger cantilever stresses on both faces (compared with all three spatially Displacement, cm uniform excitations) in portions of the dam adjacent to the dam foundation rock contact (Chopra and Wang 8). Comparison of Computed and Recorded Response Mauvoisin Dam Earthquake records from M3.6 to M4.9 earthquakes centered 1 km away from three well-instrumented arch dams in Switzerland provided a rare set of data, although for very low intensity ground motion, for evaluating whether it is reasonable to assume the foundation rock as massless, as is common in engineering practice. Finite-element models of the dam- foundation rock system, assuming foundation rock to be massless, were developed and their properties calibrated against ambient tests and forced vibration tests; this calibration led to damping ratios of 3% for the dams (Proulx et al. 4). The response of the Mauvoisin and Puntdal-Gall dams caused by ground motion recorded in the free field away from the dam base, and of the Emosson Dam caused by base motion, were computed. Although the models were calibrated against data from ambient or forced vibration tests, the computed motions at the dam crest were much larger than the earthquake records. To achieve reasonable agreement between computed and recorded motions, the damping ratio had to be increased to 15% at Emosson and to 8% at Mauvoisin and Punt-dal-Gall. Because these damping values are unrealistically large, it was concluded that the assumption of massless foundation rock in the preceding analysis that implied neglecting foundation material and radiation damping was unrealistic. Using the calibrated damping ratio of 3% for the dam and reasonable damping values for the foundation rock, subsequent analysis included dam foundation rock interaction effects (considering foundation inertia, material damping, and radiation damping). These analyses, implemented on the EACD-3D-96 computer program, led to computed responses that were closer to the measured responses, but significant discrepancies remained. Recently, the response of the Mauvoisin Dam to the recorded spatially varying ground motion was computed by the EACD- 3D8 program and compared with recorded motions. A finer finite-element model was developed, and the damping properties of the dam and foundation rock were selected to calibrate with 8 4 Radial -8 4 Tangential -8 4 Vertical Quasi-static Total Time, sec Fig. 1. Quasi-static and total displacement histories at crest center of Pacoima Dam caused by spatially varying ground motion during the Northridge earthquake: (a) radial component; (b) tangential component; (c) vertical component [reprinted from Chopra and Wang (1), with permission from Wiley] 1 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / FEBRUARY 1 J. Struct. Eng., 1, 138(): 514

7 (a) (b) Downloaded from ascelibrary.org by on 5/8/18. Copyright ASCE. For personal use only; all rights reserved. the overall damping of 3% for the dam-water-foundation-rock system measured from forced vibration tests. The computed displacement responses of the dam are reasonably similar to the recorded displacements, but the agreement is far from perfect. The peak values of the computed displacements in the stream direction the direction of the largest response are very close to the recorded value (Fig. 1); at a node near the crest center it is 9% of the recorded value, and at a node near the crest right quarter point it (c) Fig. 11. Peak values of tensile arch stress (MPa) on the downstream face of Pacoima Dam caused by the Northridge earthquake: (a) spatially uniform excitation defined by Channels 9 11; (b) spatially uniform excitation defined by Channels 1 14; (c) spatially uniform excitation defined by Channels 15 17; (d) spatially varying excitation [reprinted from Chopra and Wang (1), with permission from Wiley] Displacement, mm SM3: stream SM3: cross-stream SM3: vertical (d) is 11% of the recorded value. However, the computed displacement history does not agree as well; although it is very close to that recorded over some time segments, the two differ significantly during other time segments (Fig. 1). Recognizing that the recorded ground motions provide an incomplete description of the earthquake excitation and that no attempts were made to adjust the published data for parameter values for the mass and stiffness properties of the concrete and rock Computed Recorded Time, sec Fig. 1. Comparison of recorded and computed displacements (stream, cross-stream, and vertical components) at crest center of Mauvoisin Dam [reprinted from Chopra and Wang (1), with permission from Wiley] JOURNAL OF STRUCTURAL ENGINEERING ASCE / FEBRUARY 1 / 11 J. Struct. Eng., 1, 138(): 514

8 Downloaded from ascelibrary.org by on 5/8/18. Copyright ASCE. For personal use only; all rights reserved. (Proulx et al. 4), the agreement between computed and recorded motions is good considering the complexity of the system analyzed. Pacoima Dam The response of the Pacoima Dam to the spatially varying ground motions recorded during the 1 earthquake is determined by the EACD-3D8 computer program. The computed displacements compare well with the recorded displacements (Fig. 13); the agreement is much better than that obtained earlier in the case of Mauvoisin Dam. The time variation of the computed displacements is close to the recorded response over its entire duration; however, the peak displacement at Channels 1 and is overestimated. Recognizing that the recorded ground motions provide only an incomplete description of the earthquake excitation and that no attempts were made to adjust the published data (Alves 4; Alves and Hall 6) for the mass and stiffness parameters of concrete and rock, the agreement between computed and recorded displacements is satisfactory. A similar comparison of responses computed by linear analysis against motions recorded during the Northridge earthquake is not Displacement, mm appropriate because the ground shaking was intense enough to cause significantly nonlinear behavior in the dam, as manifested by the opening of vertical contraction joints and the cracking of concrete. However, a linear analysis by the EACD-3D8 computer program predicted large arch stresses in the thrust block between the dam and the left abutment and in the portion of the dam adjacent to the thrust block [see Fig. 11(d) and additional figures in Chopra and Wang (8)], suggesting that cracking would occur in these areas, which is what actually happened during the earthquake; see Fig. 14. Selection of Damping Values In the case of the Mauvoisin Dam, damping values for the dam alone and foundation rock separately were selected for the EACD-3D8 model to achieve damping in the overall dam-water foundation-rock system that is close to the measured value (Darbre et al. ) of approximately %. For this purpose, frequency response functions for the response at the crest center caused by spatially uniform excitation at the interface in the stream and cross-stream directions were 4 Channel 1: Radial component at crest right third point Computed Recorded Channel : Radial component at crest center Channel 3: Vertical component at crest center Channel 4: Tangential component at crest center Channel 5: Radial component at crest left quarter point Channel 6: Radial component at 8% height right third point Channel 7: Radial component at 8% height center Channel 8: Radial component at 8% height left quarter point Time, sec Fig. 13. Comparisons of recorded and computed displacements at Channels 1 8 of Pacoima Dam caused by the January 13, 1, earthquake [reprinted from Chopra and Wang (1), with permission from Wiley] 1 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / FEBRUARY 1 J. Struct. Eng., 1, 138(): 514

9 Downloaded from ascelibrary.org by on 5/8/18. Copyright ASCE. For personal use only; all rights reserved. Fig. 14. A joint opened and cracks occurred in and adjacent to the thrust block of Pacoima Dam during the Northridge earthquake (image by Anil K. Chopra) determined for several combinations of damping values for the dam and foundation rock. In one such combination, a viscous-damping ratio of 1% (in all vibration modes) of the dam alone and 3% for the foundation rock was assumed. Determined from the resonance curve (by the half-power bandwidth method) the viscous-damping ratio was.% in the first symmetric mode and 1.5% in the first antisymmetric mode (Chopra and Wang 8), indicating that the chosen damping values for concrete and rock provide an overall damping of approximately % for the dam-water foundation-rock system, consistent with experimental data. In the case of the Pacoima Dam, damping values for the dam alone and foundation rock were selected for the EACD-3D8 model to achieve damping in the overall dam-water foundationrock system consistent with the measured values of 6.% in the first symmetric vibration mode and 6.6% in the first antisymmetric vibration mode (Alves 4). For this purpose, the frequency response functions caused by spatially uniform excitation were computed for several combinations of damping in the dam and foundation rock. In one such combination, a viscous damping ratio of % (in all vibration modes) of the dam alone and 4% for the foundation rock was assumed. Determined from the resonance curve (by the half-power bandwidth method) the viscous damping ratio was 7.% in the first symmetric mode and 6.7% in the first antisymmetric mode (Chopra and Wang 8), values that are close to the measured values. The preceding analysis of damping values for the two substructures dam concrete and foundation rock have important implications for engineering practice. For example, the damping values of 1 and 3% for the two substructures, respectively, in the case of the Mauvoisin Dam, and and 4%, respectively, for the Pacoima Dam, combined to provide damping in the overall dam-water foundation system that is consistent with measured damping for these dams. Therefore, the current practice of invariably specifying a viscous damping ratio of 5% for the concrete dam alone and a similar value for the foundation rock separately should be abandoned, because it is likely to lead to excessive damping in the overall damwater foundation-rock system unless the foundation rock is much softer than concrete and thus underestimate the response of the dam. Nonlinear Analysis Although the importance of dam-water-foundation rock interaction effects (including reservoir boundary absorption) and spatial variations in ground motion demonstrated in this paper was based on results of linear analysis of arch dams, they should obviously also be included in nonlinear analyses. Such analyses are necessary to consider the opening and closing of contraction joints between cantilevers and the cracking of concrete caused by overstressing in tension. Such nonlinear analyses may be accomplished by analyzing a finite-element model of the dam-water foundation-rock system, recognizing that a very large finite-element model is necessary to simulate the semiunbounded size of the foundation rock and impounded water domains if the absorbing boundary selected to simulate semiunbounded domains is simple, e.g., viscous dampers. For example, a finite-element model developed by the USBR to analyze Morrow Point Dam included over, brick elements to model the dam, over 1, brick elements to model a large foundation domain, and over, elements to model the fluid domain; an integration time step of 1 6 to 1 5 s was selected for the explicit time-stepping procedure in LSDYNA software (Scott et al. 8). Obviously, computer solution of such a large nonlinear finiteelement system using a very short time step is extremely time consuming, both in data preparation and computer implementation. Work is now in progress to incorporate the perfectly matched layer (PML) model, a novel absorbing boundary, into LSDYNA (Basu and Chopra 4). PML modeling of the foundation and fluid domains would greatly reduce the size of the computer model by minimizing the number of finite elements in these two domains, permitting a detailed, fine mesh to model the dam, the region of primary interest. This capability is expected to save computational effort and time by at least an order of magnitude. Conclusions Through a series of example analyses of actual dams, it is demonstrated that (1) by neglecting water compressibility, stresses may be significantly underestimated for some dams and overestimated for others; () by ignoring foundation-rock mass and damping, stresses may be overestimated by a factor of 3; and (3) spatial variations in ground motion, typically ignored in dam engineering practice, can have profound influence on earthquake-induced stresses in the dam. This influence obviously depends on the degree to which ground motion varies spatially along the dam-rock interface. This influence would differ from one earthquake to the next for the same dam, depending on the location and depth of the causative fault relative to the dam site. Therefore, earthquake analysis of arch dams should include the following factors: the semiunbounded size of the reservoir and foundation-rock domains, dam-water interaction, hydrodynamic wave absorption at the reservoir boundary, water compressibility, dam foundation rock interaction, and spatial variations in ground motion at the dam-rock interface. However, finite-element analyses of dams conducted in professional practice often ignore these factors. These simplistic analyses lead to unacceptably inaccurate estimates of earthquake-induced stresses. Stresses are grossly overestimated by assuming foundation rock to be massless, leading to overly conservative designs of new dams and to the erroneous conclusion that an existing dam is unsafe, thus requiring upgrading. Analyses ignoring spatial variations in ground motion fail to identify the zones of the dam that are likely to be critically stressed and damaged. Not only should these factors be included in engineering practice, but also the damping values assigned to the dam alone and foundation rock separately must be consistent with damping in the JOURNAL OF STRUCTURAL ENGINEERING ASCE / FEBRUARY 1 / 13 J. Struct. Eng., 1, 138(): 514

10 Downloaded from ascelibrary.org by on 5/8/18. Copyright ASCE. For personal use only; all rights reserved. overall dam-water foundation-rock system measured from forced vibration tests or system identification methods applied to recorded motions during earthquakes. In particular, the current practice of invariably specifying 5% viscous damping for the concrete dam and a similar value for the foundation rock separately should be abandoned because in many cases it will lead to excessive damping in the overall dam-water foundation rock system and hence underestimation of response. Acknowledgments I am grateful to Jean Proulx and Georges Darbre for providing the data and recorded motions from Mauvoisin Dam, to S. W. Alves for advising on interpolating spatially varying ground motions at the dam-rock interface from recorded motions at a few locations, and to Larry K. Nuss for providing Figs References Alves, S. W. (4). Nonlinear analysis of Pacoima Dam with spatiallyuniform ground motion. Rep. No. EERL 4-11, California Institute of Technology, Pasadena, CA. Alves, S. W., and Hall, J. F. (6). System identification of a concrete arch dam and the calibration of its finite element model. Earthquake Eng. Struct. Dyn., 35(11), Basu, U., and Chopra, A. K. (4). Perfectly matched layers for transient elastodynamics of unbounded domains. Int. J. Num. Methods Eng., 59(8), Chopra, A. K. (7). Dynamics of structures: Theory and applications to earthquake engineering, 3rd Ed., Prentice Hall, Englewood Cliffs, NJ. Chopra, A. K., and Wang, J.-T. (8). Analysis and response of concrete arch dams including dam-water-foundation rock interaction to spatiallyvarying ground motions. UCB/EERC8-3, Earthquake Engineering Research Center (EERC), Univ. of California, Berkeley, CA. Chopra, A. K., and Wang, J. T. (1). Earthquake response of arch dams to spatially varying ground motion. Earthquake Eng. Struct. Dyn., 39(8), Darbre, G. R., De Smet, C. A. M., and Kraemer, C. (). Natural frequencies measured from ambient vibration response of the arch dam of Mauvoisin. Earthquake Eng. Struct. Dyn., 9(5), Mojtahedi, S., and Fenves, G. (). Effect of contraction joint opening on Pacoima Dam in the 1994 Northridge earthquake. Data Utilization Rep. CSMIP/-5 (OSMS -7), California Strong Motion Instrumentation Program (CSMIP), Univ. of California, Berkeley, CA. Proulx, J., et al. (4). Analytical and experimental investigation of damping on arch dams based on recorded earthquakes. Paper No. 68. Proc., 13th World Conf. on Earthquake Engineering, International Association for Earthquake Engineering (IAEE), Tokyo. Scott, G., et al. (8). Current methodology for nonlinear analysis of arch dams using explicit finite element techniques. Proc., 13th World Conf. on Earthquake Engineering, International Association for Earthquake Engineering (IAEE), Tokyo. Tan, H., and Chopra, A. K. (1995a). Dam-foundation rock interaction effects in frequency-response function of arch dams. Earthquake Eng. Struct. Dyn., 4(11), Tan, H., and Chopra, A. K. (1995b). Earthquake analysis of arch dams including dam-water foundation rock interaction. Earthquake Eng. Struct. Dyn., 4(11), Tan, H., and Chopra, A. K. (1996a). Dam-foundation rock interaction effects in earthquake response of arch dams. J. Struct. Eng., 1(5), Tan, H., and Chopra, A. K. (1996b). EACD-3D-96: A computer program for three-dimensional earthquake analysis of concrete dams. Rep. No. UCB/SEMM-96/6, Univ. of California, Berkeley, CA. United States Bureau of Reclamation (USBR). (1998a). Linear elastic analysis addressing structural issues for the modification decision analysis for Deadwood Dam. Technical Memorandum DEA- D811-MDA-TM-98-1, USBR, Denver, CO. United States Bureau of Reclamation (USBR). (1998b). Linear elastic dynamic structural analysis including mass in the foundation for Hoover Dam. Technical Memorandum HVD-MDA-D811-97, USBR, Denver, CO. United States Bureau of Reclamation (USBR). (1999). Linear elastic static and dynamic structural analysis of Monticello Dam, USBR, Denver, CO. United States Bureau of Reclamation (USBR). (). Static and dynamic linear elastic structural analysis (EACD3D96), Morrow Point Dam, USBR, Denver, CO. Wang, J. T., and Chopra, A. K. (8). EACD-3D8: A computer program for three-dimensional earthquake analysis of concrete dams considering spatial-varying ground motion. UCB/EERC8-4, Earthquake Engineering Research Center (EERC), Univ. of California, Berkeley, CA. Wang, J. T., and Chopra, A. K. (1). Linear analysis of concrete arch dams including dam-water-foundation rock interaction considering spatially varying ground motions. Earthquake Eng. Struct. Dyn., 39(7), / JOURNAL OF STRUCTURAL ENGINEERING ASCE / FEBRUARY 1 J. Struct. Eng., 1, 138(): 514