Direct finite element method for nonlinear earthquake analysis of concrete dams including dam water foundation rock interaction

Transcription

1 Arnkjell Løkke Doctoral thesis Doctoral theses at NTNU, 218:252 Doctoral theses at NTNU, 218:252 NTNU Norwegian University o Science and Technology Thesis or the Degree o Philosophiae Doctor Faculty o Engineering Department o Structural Engineering ISBN (printed ver.) ISBN (electronic ver.) ISSN Arnkjell Løkke Direct inite element method or nonlinear earthquake analysis o concrete dams including dam water oundation rock interaction

2 Arnkjell Løkke Direct inite element method or nonlinear earthquake analysis o concrete dams including dam water oundation rock interaction Thesis or the Degree o Philosophiae Doctor Trondheim, September 218 Norwegian University o Science and Technology Faculty o Engineering Department o Structural Engineering

3 NTNU Norwegian University o Science and Technology Thesis or the Degree o Philosophiae Doctor Faculty o Engineering Department o Structural Engineering Arnkjell Løkke ISBN (printed ver.) ISBN (electronic ver.) ISSN Doctoral theses at NTNU, 218:252 Printed by NTNU Graisk senter

4 ABSTRACT Evaluating the seismic perormance o concrete dams requires nonlinear dynamic analysis o two- or three-dimensional dam water oundation rock systems that include all the actors known to be signiicant in the earthquake response o dams. Such analyses are greatly complicated by interaction between the structure, the impounded reservoir and the deormable oundation rock that supports it, and the act that the luid and oundation domains extend to large distances. Presented in this thesis is the development o a direct inite element (FE) method or nonlinear earthquake analysis o two- and three-dimensional dam water oundation rock systems. The analysis procedure applies standard viscous-damper absorbing boundaries to model the semi-unbounded luid and oundation domains and speciies at these boundaries eective earthquake orces determined rom a ground motion deined at a control point on the ground surace. Part I develops the direct FE method or 2D dam water oundation rock systems. The underlying analytical ramework o treating dam water oundation rock interaction as a scattering problem, wherein the dam perturbs an assumed "ree-ield" state o the system, is presented, and by applying these concepts to a bounded FE model with viscous-damper boundaries to truncate the semi-unbounded domains, the analysis procedure is derived. Stepby-step procedures or computing eective earthquake orces rom analysis o two 1D reeield systems are presented, and the procedure is validated by computing requency response unctions and transient response o an idealized dam water oundation rock system and comparing against independent benchmark results. This direct FE method is generalized to 3D systems in Part II o this thesis. While the undamental concepts o treating interaction as a scattering problem are similar or 2D and 3D systems, the derivation and implementation o the method or 3D systems is much more involved. Eective earthquake orces must now be computed by analyzing a set o 1D and 2D systems derived rom the boundaries o the ree-ield systems, which requires extensive bookkeeping and data transer or large 3D models. To reduce these requirements and acilitate implementation o the direct FE method or 3D systems, convenient simpliications o the procedure are proposed and their eectiveness demonstrated. Part III o thesis proposes to use the direct FE method or conducting the large number o nonlinear response history analyses (RHAs) required or Perormance Based Earthquake Engineering (PBEE) o concrete dams, and discusses practical modeling considerations or two o the most inluential aspects o these analyses: nonlinear mechanisms and energy dissipation (damping). The indings have broad implications or modeling o energy dissipation and calibration o damping values or concrete dam analyses. At the end o Part III, the direct FE method is implemented with a commercial FE program and used to compute i

5 the nonlinear response o an actual arch dam. These nonlinear results, although limited in their scope, demonstrate the capabilities and eectiveness o the direct FE method to compute the types o nonlinear engineering response quantities required or PBEE o concrete dams. Keywords: Concrete dams; nonlinear earthquake analysis; dam water oundation rock interaction; absorbing boundaries; response history analysis; three-dimensional analysis ii

6 ACKNOWLEDGMENTS First and oremost I would like to express my deepest gratitude to my main academic advisor Proessor Anil K. Chopra at UC Berkeley or sharing his extensive insight into the theory o dynamics o structures and earthquake analysis o concrete gravity dams; or his steady guidance, encouragement and challenging o my ideas; and or being a mentor and role model. His contributions during our countless meetings and discussions over the last three years are grateully acknowledged. I would also like to thank my advisors at NTNU: Proessor Svein Remseth or making it possible or me to write this thesis or NTNU in collaboration with the UC Berkeley and to Adjunct Proessor Amir Kaynia or his guidance and support, and or always inding the time to answer my many questions. I am also grateul to several other individuals who have all contributed to this research in dierent orms: Dr. Ushnish Basu or his advice and contributions to the development o the direct FE method and its numerical implementation. Dr. Frank McKenna or his help with implementing 2D and 3D luid soil structure interacting models in OPENSEES. Dr. Neal Simon Kwong or providing a critical outside view o my research, and or making lunch time discussions at campus such an enjoyable experience. Last but not least, I am orever grateul to my loving wie Sarah. Your encouragement, patience and dedication to seeing this through, even when it meant moving hal way around the world, means everything to me. Without your support, this PhD thesis would have never happened. iii

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8 TABLE OF CONTENTS INTRODUCTION... 1 Motivation... 1 Objectives and scope... 2 Organization o this thesis... 2 PART I DIRECT FE METHOD FOR NONLINEAR EARTHQUAKE ANALYSIS OF TWO-DIMENSIONAL DAM WATER FOUNDATION ROCK SYSTEMS Introduction... 6 System and ground motion Semi-unbounded dam water oundation rock system Modeling o unbounded domains Governing equations Dam and oundation domain Fluid domain Dam water oundation rock system Dam oundation rock system Dam oundation rock interaction as a scattering problem Viscous-damper absorbing boundaries Equations o motion Free-ield earthquake motion Computing eective earthquake orces Bottom boundary Side boundaries Relation to the Domain Reduction Method Numerical validation Dam on rigid oundation rock Dam oundation rock system Ignoring eective earthquake orces at side boundaries Can oundation mass be ignored? Dam water system Dam water interaction as a scattering problem Viscous-damper absorbing boundary Equations o motion Computing eective earthquake orces at Γ r Numerical validation v

9 Hydrodynamic orces on rigid dam Dam water system Ignoring eective earthquake orces on luid boundary Γ r Dam water oundation rock system Dam water oundation rock interaction as a scattering problem Equations o motion Approximating water oundation rock interaction Dam water oundation rock system Summary o procedure Numerical validation Frequency response unctions or dam response Response to transient motion Inluence o water oundation rock interaction Conclusions PART II DIRECT FE METHOD FOR NONLINEAR EARTHQUAKE ANALYSIS OF THREE-DIMENSIONAL DAM WATER FOUNDATION ROCK SYSTEM Introduction... 5 System and ground motion Semi-unbounded dam water oundation rock system Earthquake excitation Equations o motion Governing equations Interaction as a scattering problem Approximating water oundation rock interaction Final equations o motion Computing eective earthquake orces Forces at bottom boundary Forces at side boundaries Computing orces at side boundaries: uniorm canyon Computing orces at side boundaries: arbitrary canyon geometry Forces at upstream luid boundary Numerical validation o the direct FE method Reproducing ree-ield motion in oundation rock Free-ield motion at lat box surace (the lat box test) Free-ield motion at canyon surace Dynamic response o Morrow Point Dam System analyzed vi

10 EACD3D-8 model or substructure method Frequency response unctions or dam response Response to transient motion Frequency response unctions or spatially uniorm motion Dam on rigid oundation rock Dam oundation rock system Dam water system Dam water oundation rock system Simpliications o the direct FE method Using 1D analysis to compute eective earthquake orces at side oundation boundaries Ignoring eective earthquake orces at side oundation boundaries Avoiding deconvolution o the surace control motion Ignoring eective earthquake orces on upstream boundary o luid domain Summary o procedure Conclusions PART III MODELING AND PRACTICAL IMPLEMENTATION OF THE DIRECT FE METHOD FOR PERFORMANCE BASED EARTHQUAKE ENGINEERING OF CONCRETE DAMS Introduction Perormance based earthquake engineering o dams The Paciic Earthquake Engineering Research Center PBEE ramework Seismic hazard analysis Structural analysis Damage analysis Loss analysis Modeling o uncertainty Modeling o concrete dams by the direct FE method Modeling o nonlinear mechanisms Cracking o concrete Opening and closing o vertical contraction joints Sliding and separation at lit joints and concrete-rock interaces Discontinuities in the oundation rock Modeling o energy dissipating mechanisms Material damping Radiation damping Energy dissipation at the reservoir boundaries Calibration o damping values vii

11 4 5 Nonlinear earthquake analysis o Morrow Point Dam System and ground motion FE model o dam water oundation rock system Nonlinear modeling parameters Static and dynamic loads Implementation o the direct FE method with ABAQUS Results rom nonlinear dynamic analysis Conclusions CONCLUSIONS AND FUTURE WORK Summary and conclusions Future work REFERENCES NOTATION APPENDIX A SELECTION OF DOMAIN SIZE FOR TWO-DIMENSIONAL DAM WATER FOUNDATION ROCK SYSTEMS APPENDIX B THE DOMAIN REDUCTION METHOD FOR SEISMIC INPUT IN SOIL STRUCTURE-INTERACTION ANALYSES APPENDIX C COMPUTING FREQUENCY RESPONSE FUNCTIONS IN THE TIME DOMAIN APPENDIX D APPLYING UNIFORM GROUND MOTION IN THE DIRECT FE METHOD APPENDIX E COMPUTING BOUNDARY TRACTIONS FROM 1D STRESS-STRAIN RELATIONS viii

12 INTRODUCTION Motivation Concrete dams are critical structures that provide important services in the orm o power generation, lood control, irrigation and recreation. There are also catastrophic consequences or lie and property in the case o a ailure o these structures. However, most existing dams in seismic regions were designed by methods that are now considered inaccurate and obsolete. The damage sustained by the ew concrete dams that have been subjected to intense ground motions, e.g., Koyna Dam in India, Hsinengkiang Dam in China, Seidrud Dam in Iran, and Pacoima Dam in the United States, together with the growing concern o the seismic saety o critical acilities, has led to considerable interest in reevaluating existing dams using modern analysis and experimental procedures. In recent years, interest has also increased on how to apply the principles o Perormance Based Earthquake Engineering (PBEE) to concrete dam evaluations. Earthquake analysis o concrete dams is greatly complicated by interaction between the structure, the impounded reservoir and the deormable oundation rock that supports it, and the act that the luid and oundation domains extend to large distances. To overcome the diiculties in modeling dam water oundation rock interaction and semi-unbounded domains in the inite element method (FEM), the dam engineering proession has oten employed an expedient solution: the oundation is modeled in limited extent and assumed to have no mass, hydrodynamic eects are approximated by an added mass o water moving with the dam, and the design ground motion at the surace is applied directly to the bottom ixed boundary o the oundation domain. This modeling approach has become popular in actual dam engineering projects because it is easy to implement in commercial FE codes; however, research has demonstrated that such oversimpliied analyses can overestimate stresses by as much as a actor o 2 to 3, thus leading to overly conservative design o new dams and the incorrect conclusion that an existing dam is unsae and needs retroitting. Clearly, this situation is not satisactory. Evaluating the seismic saety o concrete dams requires accurate and robust analysis procedures that recognize dam water oundation rock interaction and the semi-unbounded sizes o the water and oundation domains, and consider nonlinearities such as opening and closing o vertical contraction joints and cracking o concrete during intense earthquake motion. However, recognizing that most dam engineers may or various reasons be predisposed to use a particular FE program, it is also important 1

13 that such analysis procedures are general enough to be used with any FE program without requiring undue eorts rom the user. Objectives and scope The overall objective o this research is to develop a procedure or nonlinear response history analysis (RHA) o semi-unbounded dam water oundation rock systems that is accurate, robust, and general enough to be used with any commercial FE program without modiication o the source code. In particular, the objectives are to: Extend the "standard" inite element model o dam water oundation rock systems to include absorbing boundaries at the upstream end o the luid domain and bottom and side boundaries o the oundation domain to model their semi-unbounded geometries. Develop a practical procedure or determining the seismic input to the analysis procedure starting rom a ree-ield control motion speciied on level ground. Validate the accuracy o the analysis procedure by comparing against analytical and semi-analytical unbounded domain models using the substructure method o analysis or 2D and 3D dam water oundation rock systems. Develop guidelines or implementation o the procedure in commercial FE sotware, as well as recommendations or practical modeling choices such as the sizes o the oundation and luid domains to be included in the FE model, modeling o nonlinear mechanisms, and modeling o energy dissipating mechanisms (damping). Demonstrate the useulness o the analysis procedure or conducting nonlinear RHA o concrete dams in commonly used commercial FE programs. Advocate use o more accurate analysis procedures or earthquake analysis o concrete dams in the dam engineering community. Organization o this thesis This thesis is written in three parts, each corresponding to a paper that has been published in international peer-reviewed journals (Parts I and II) or has been submitted or such publication (Part III). The content in this thesis is more extensive than the journal papers however, because it also contains several parts (derivations, validation results, etc.) that were let out o the papers due to space limitations. Part I presents the analytical ramework underlying the direct FE method, develops the analysis procedure or earthquake analysis o two-dimensional dam oundation rock, dam water, and ultimately dam water oundation rock systems, and derives the equations o motion or these systems. Then, procedures or computing eective earthquake orces starting rom a ree-ield ground motion deined at a control point on the oundation surace are 2

14 developed. Several examples are presented to validate the accuracy o the direct FE method applied to a wide range o 2D systems. Part II generalizes the direct FE method to three-dimensional systems, which is undamentally more challenging to analyze because o their complicated 3D geometry. The equations o motions or 3D systems with absorbing boundaries are derived, procedures or computing eective earthquake orces rom a set o 1D and 2D analyses are presented, and several numerical examples are presented to validate the accuracy o the direct FE method applied to 3D systems. To acilitate implementation o the procedure or 3D systems, convenient simpliications are proposed and their eectiveness demonstrated. Part III proposes to use the direct FE method or conducting the large number o nonlinear RHAs required or PBEE o concrete dams. A brie introduction to PBEE in the context o concrete dams is presented, the most signiicant nonlinear mechanisms that can develop in concrete dams are discussed, and the various types o energy dissipation (damping) in the dam water oundation rock system are reviewed. Recommendations or how to model these eatures in the direct FE method are presented. Finally, the capabilities o the direct FE method are demonstrated by computing the nonlinear earthquake response o Morrow Point Dam using a commercial FE code. This thesis also includes ive appendices: Appendix A presents guidelines or selecting an appropriate size or the oundation domain to be included in 2D dam water oundation rock models. Appendix B outlines the use o the Domain Reduction Method or seismic input to soil structure interaction analyses and demonstrates that when based on the same assumptions the DRM and direct FE method will give identical results. Appendices C and D provides details on the computational procedures used to calculate requency response unctions in the time domain and to apply uniorm ground motion in the direct FE method, respectively. Lastly, Appendix E presents equations or computing eective earthquake orces rom one-dimensional stress-strain relations. In addition, two contributions rom this research are not documented in this thesis. The irst is the development o several sets o scripts and computer code to implement the direct FE method or nonlinear RHA and to acilitate the benchmark analyses using the substructure method or 2D and 3D systems implemented in the Fortran77 programs EAGD84 and EACD3D-8. These scripts will be made publicly available online through the NISEE e- library websites. The second is the objective o advocating the use o more accurate analysis procedures in the dam engineering community. This has been achieved through presentations at international conerences and workshops on concrete dams and earthquake engineering. 3

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16 PART I Direct inite element method or nonlinear earthquake analysis o two-dimensional dam water oundation rock systems

17 PART I: INTRODUCTION 1 Introduction Evaluating the seismic perormance o concrete gravity dams requires dynamic analysis o two- or three-dimensional dam water oundation rock systems that include all signiicant actors in the earthquake response o concrete dams [1]: dam oundation rock interaction including inertia eects o the rock [2,3]; dam water interaction, including water compressibility and energy absorption at the reservoir bottom [4 6]; radiation damping due to the semi-unbounded sizes o the reservoir and oundation domains [3,7]; and nonlinear behavior o the dam and oundation rock [8 16]; Analysis procedures based on the substructure method have been available since 1984 or 2D requency-domain analysis o dam water oundation rock systems [17]. This method models the semi-unbounded domains rigorously and speciies the ground motion directly at the dam oundation rock interace; however, it is restricted to homogeneous material properties and simple geometry o the reservoir and oundation domains. More importantly, the substructure method is restricted to linear behavior o the entire system. Thus, nonlinear eects such as cracking o concrete and separation and sliding at joints and interaces cannot be modeled. The direct method o analysis on the other hand, models the entire system directly in the time-domain using inite elements (FEs). Such analyses are oten conducted using commercial sotware that ignores one or several o the above actors to acilitate nonlinear dynamic analysis. For many years, the dam engineering proession used a FE model that included a limited extent o oundation rock, assumed to have no mass, and approximated hydrodynamic eects by an added mass o water moving with the dam. The design ground motion typically deined at a control point on the ree surace was applied at the bottom ixed boundary o the oundation domain without modiication. These approximations are attractive because they simpliy the analysis greatly, however, such a model solves a problem that is very dierent rom the real problem on two counts: (1) the assumptions o massless rock and incompressible water implied by the added mass water model are unrealistic, as research has demonstrated [1]; and (2) applying ground motion speciied at a control point on the ree surace to the bottom boundary o the FE model contradicts recorded evidence that motions at depth generally dier signiicantly rom surace motions. In recent years, some engineers have shited away rom this approach. To eliminate these unrealistic assumptions, the FE model o the dam must be extended to comprise a oundation domain that includes mass, stiness, and material damping appropriate or rock, and a luid domain that includes water compressibility and reservoir bottom absorption. The semi-unbounded oundation and luid domains must be reduced to bounded sizes with appropriate radiation conditions at the domain boundaries to allow propagation o outgoing waves. Development o such absorbing boundaries is a vast ield 6

18 PART I: INTRODUCTION with rich literature [18 34]. The earthquake motion cannot be speciied directly at the model truncations as this would render any absorbing boundary ineective. Instead, eective earthquake orces are computed rom the earthquake motion and applied either directly at the absorbing boundaries [35 37] or via a layer o elements interior o the boundaries [38 4]. Utilizing these concepts, a direct inite element procedure or nonlinear analysis o dam water oundation rock systems was developed by Basu [41]. Here, the high perorming Perectly Matched Layer (PML) [25] was used as the absorbing boundary, and the Eective Seismic Input (ESI) method [38], also known as the Domain Reduction Method (DRM) [4] was used to apply the eective earthquake orces. Although the procedure rigorously incorporates all the above actors signiicant in the earthquake response o dams, the PML boundary and DRM procedure are currently not available in most commercial FE codes; the only exception is LS-DYNA [42]. Thus, this procedure is not accessible to researchers and practicing engineers who, or various reasons, preer other FE codes. These limitations can be overcome by modeling the absorbing boundaries by viscous dampers [18] and speciying the eective earthquake orces directly at these boundaries. Both o these eatures are available in almost every commercial FE code and are thereore chosen herein. A variation o such a procedure initiated by the US Bureau o Reclamation, wherein eective earthquake orces on the side boundaries are ignored, is oten used in the dam engineering proession [43 45]. The ollowing chapters develops the ormulation or a direct inite element method or nonlinear earthquake analysis o semi-unbounded dam water oundation rock systems that incorporates all signiicant actors or the earthquake response o dams, while ensuring broad applicability by using the well-known viscous damper as the absorbing boundaries. Derivation o the analysis method is ounded on the idea o treating interaction as a scattering problem [38,46], and ollows a similar outline as the procedure developed by Basu [41] using PML-boundaries and DRM or seismic input. In Chapter 2, the system and ground motion is deined, and the governing equations or each o three subdomains are presented. In Chapters 3 and 4, the direct FE method is developed or dam oundation rock and dam water systems, respectively, by utilizing the concept o treating interaction as a scattering problem and ormulating the radiation condition or viscous-damper boundaries in a convenient way. These two procedures are integrated to ormulate the analysis procedure or the combined dam water oundation rock system in Chapter 5. At the end o each o the Chapters 3 5, the analysis method is validated by computing the response o idealized dam oundation rock (Chapter 3), dam water (Chapter 4), and dam water oundation rock (Chapter 5) systems and comparing against results obtained using the substructure method. In Chapter 5, the importance o including water oundation rock interaction is also discussed. A shortened version o this part o the thesis has been published in the journal Earthquake Engineering and Structural Dynamics: Løkke, A., and Chopra, A.K. (217). Direct inite element method or nonlinear analysis o semi unbounded dam water oundation rock systems. Earthquake Engineering & Structural Dynamics, 46.8,

19 PART I: SYSTEM AND GROUND MOTION 2 System and ground motion 2.1 Semi-unbounded dam water oundation rock system The idealized, two-dimensional dam water oundation rock system (Figure 2.1) has three parts: (1) the gravity dam with nonlinear properties; (2) the oundation rock, consisting o a bounded region adjacent to the dam that can be nonlinear and inhomogeneous, and a semiunbounded region that is restricted to be linear; and (3) the luid domain, consisting o a bounded region o arbitrary geometry adjacent to the dam that may be nonlinear, and a uniorm channel, unbounded in the upstream direction, that is restricted to be linear. Thus, nonlinear eects such as concrete cracking, sliding and separation at construction joints, lit joints, and concrete-rock interaces, and cavitation in the luid may be considered in the analysis. The earthquake excitation is deined at a control point at the surace o the oundation rock by two components o ree-ield ground acceleration (Figure 2.1): the horizontal x y component ag () t transverse to the dam axis, and the vertical component ag () t. The surace o the oundation rock is assumed to be at the same elevation in the ar upstream and downstream directions; this geometric restriction is introduced to deine a convenient reeield state o the oundation rock in Section 3.1. Prismatic channel with linear luid Irregular part o luid domain Sediments Nonlinear dam Control point x a () t g y a () t g Nonlinear part o oundation rock Unbounded, linear oundation rock Figure 2.1: Semi-unbounded dam water oundation rock system: (1) the dam itsel; (2) the oundation rock, consisting o a bounded, nonlinear region and a semi-unbounded, linear region; and (3) the luid domain, consisting o an irregular, nonlinear, region, and a semi-unbounded prismatic channel with linear luid. Figure adapted rom Re. [41]. 8

20 PART I: SYSTEM AND GROUND MOTION 2.2 Modeling o semi-unbounded domains The dam water oundation rock system in Figure 2.1 is modeled by a FE discretization o a bounded system with viscous-damper boundaries to represent the semi-unbounded oundation and luid domains. The earthquake motion cannot be speciied directly at these model truncations as this would render any absorbing boundary ineective. Instead, eective earthquake orces are computed rom the earthquake excitation and applied at the absorbing boundaries. The size o the oundation and luid domains included in the FE model is determined by the ability o the absorbing boundaries to absorb outgoing (scattered) waves rom the dam. I an advanced boundary such as the PML [25] is used, a small domain is suicient to model the luid and oundation domains (Figure 2.2a). In contrast, the simple viscous-damper boundary applied in this ormulation requires much larger domains (Figure 2.2b). Advanced absorbing boundary, e.g. PML (a) Simple absorbing boundary, e.g. viscous dampers (b) Figure 2.2: Dam water oundation rock system with truncated oundation and luid domains: (a) small domain sizes with advanced absorbing boundary; (b) large domain sizes with simple absorbing boundary. 2.3 Governing equations Dam and oundation domain The equations o motion governing the vector o total displacements r t in the FE model o the dam with a truncated oundation domain and absorbing boundary Γ (Figure 2.3) are t t t t t t st + + ( ) = h + b + + (2.1) mr cr r R R R R where m and c are the mass and damping matrices, respectively; r ( t ) is the vector o internal orces which may be nonlinear in the dam and adjacent part o the oundation rock; R t and h Rt are the vectors o hydrodynamic orces acting at the dam water interace Γ and b h st water oundation interace Γ b, respectively; R is the vector o static orces, including sel- 9

21 PART I: SYSTEM AND GROUND MOTION t weight, hydrostatic pressures, and static oundation reactions at Γ ; and R are the orces associated with the absorbing boundary Γ, which include the eect o the excitation caused by seismic waves propagating rom a distant earthquake source to the dam site, and the radiation condition at the boundary. Expressions or the orces R t, h Rt and t b R will be derived later. Water b h Dam and nonlinear oundation rock Linear and homogeneous oundation rock Absorbing boundary, Figure 2.3: Schematic FE model o the dam and oundation rock, with absorbing boundary Γ to truncate the semi-unbounded oundation domain Fluid domain The water is modeled as a linear inviscid, irrotational and compressible luid with hydrodynamic pressures p governed by the acoustic wave equation: 1 C 2 p= p 2 (2.2) where C is the speed o pressure waves in water. Hydrodynamic pressures are caused by acceleration o the boundaries in contact with the reservoir: the upstream dam ace Γ h and the reservoir bottom Γ b (Figure 2.4). These pressures are related to the total accelerations r t at the luid solid interace by the boundary conditions: p n = ρn r h h h, at Γ h t p n + b qp = ρnb rb, at Γ b (2.3a) (2.3b) where n h and n b are the outward normal vectors to the luid at Γ h and Γ b, respectively; and ρ is the density o water. The second term on the let hand side o Equation (2.3b) is associated with the absorption o hydrodynamic pressure waves in sediments deposited at the 1

22 PART I: SYSTEM AND GROUND MOTION reservoir bottom. This wave absorption is modeled in an approximate way by a boundary condition that allows partial absorption o incident hydrodynamic waves, where the damping coeicient q is given by 1 α qc = (2.4) 1 + α where α is the reservoir bottom relection coeicient [6,17]. This simpliied model is chosen herein to allow or a meaningul comparison with the substructure method [17] in the numerical validations presented at the end o Chapters 3 5. The eects o reservoir bottom sediments can alternatively be included using more sophisticated methods, or example by directly modeling the thickness and extent o sediments discretized by inite elements with a viscoelastic [47,48] or poroelastic [49,5] material model. At the ree water surace the boundary condition is simply p = ; eects o surace waves are not included as these have little inluence on the dynamic response o concrete dams [51]. Lastly, an appropriate radiation condition must hold at the absorbing boundary Γ r. p = Absorbing boundary, r Sediments t n ṙ b b b h t n ṙ h h Dam Foundation rock Figure 2.4: Schematic FE model o the luid domain highlighting the various boundary conditions at the reservoir boundaries. Discretizing the luid domain using inite elements and deining p t as the vector o total hydrodynamic pressures where the superscript t has been added or consistency with the notation or the total displacements r t the standard discretization process results in ρ h b r (2.5) sp t T T + bp t + hp t = Q + Q r t + H t where s, b and h are the corresponding "mass", "damping" and "stiness" matrices o the t luid [52], and H r is the vector o orces associated with the absorbing boundary Γ r. In contrast to the "standard" ormulation [52], the damping matrix b does here not include the t eects o the radiation condition because these are represented by the orces H r. Also included in this term are the orces exerted on the absorbing boundary due to excitation o the 11

23 PART I: SYSTEM AND GROUND MOTION part o the luid domain upstream o Γ r that has been eliminated; an expression or these orces will be derived in Chapter 4. The matrix Q h relates hydrodynamic pressures in the luid to accelerations in the dam at the dam water interace Γ h according to the boundary condition o Equation (2.3a): Q = N n N Γ (2.6) T d h h h h Γh where N h and N h are the shape unctions o the dam and luid nodes, respectively, on the interace Γ h. The matrix Q b is constructed the same way, but integrated over Γ b Dam water oundation rock system t t The hydrodynamic orces R h and R b in Equation (2.1) that act on the dam and oundation rock, respectively, can be expressed in terms o the hydrodynamic pressures p t as [52] R t = Q p t t = h h h R Q p t (2.7) b b b Substituting this equation into Equation (2.1) and combining with Equation (2.5) gives the equations o motion or the dam water oundation rock system with truncated oundation and luid domains: t t m r c r + T T ρ( + ) t t Qh Qb sp bp t t t st r ( ) ( Qh + Qb) r R + R + + = t t h p Hr (2.8) where the coupling matrices Q h and Q b have non-zero entries only on the interaces Γ h and Γ b, respectively. Expressions or the unknown orces R t and Ht r associated with the absorbing boundaries Γ and Γ r will be derived in the subsequent chapters. 12

24 PART I: DAM FOUNDATION ROCK SYSTEM 3 Dam oundation rock system 3.1 Dam oundation rock interaction as a scattering problem Dam oundation rock interaction may be treated as a scattering problem in which the dam perturbs the ree-ield motion in the oundation rock. Procedures based on this idea have been developed or analysis o soil structure interaction systems [38 4], and o dam water oundation rock systems using PML absorbing boundaries and the DRM or seismic input [41]. In this chapter, these ideas will be utilized to ormulate an analysis procedure or the dam oundation rock subsystem with absorbing boundaries modeled by viscous dampers. Consider the linear oundation rock in its ree-ield state, i.e., beore the dam was constructed or excavation had started (Figure 3.1a). This domain is separated into two + subdomains: Ω denotes the region interior o the uture absorbing boundary Γ, and Ω is the semi-unbounded exterior region. The vector o ree-ield displacements at nodes in both subdomains is denoted by r (Figure 3.1a); a procedure to determine this motion will be presented in Section 3.4. Future absorbing : r boundary, : r t Seismic waves (a) : r Seismic : r t r waves (b) Figure 3.1: Illustration o dam oundation rock interaction as a scattering problem: (a) oundation rock in its + ree-ield state with displacement ield deined by r in Ω Ω ; (b) dam oundation rock system with displacement ield deined by the total motion r t t + in Ω and the scattered motion r r in Ω The dam oundation rock system is also separated into two subdomains (Figure 3.1b): + Ω denotes the dam and oundation region interior o the absorbing boundary Γ, and Ω is the semi-unbounded exterior region, the latter is identical to the exterior region in the reeield system. Following the approach irst proposed by Herrera and Bielak [46], the displacement ield in the dam oundation rock system is deined by the variables: r t, in the interior region Ω (3.1a) 13

25 PART I: DAM FOUNDATION ROCK SYSTEM t r r, in the exterior region + Ω (3.1b) where r t t is the vector o total displacements governed by Equation (2.1), and r r + represents the scattered motion in the exterior region Ω, i.e., the perturbation o the reeield motion caused by the presence o the dam. This substitution o variables in Ω will + subsequently allow ormulation o the governing equations or the absorbing boundary in a way that the orces R t in Equation (2.1) can be determined rom the ree-ield motion r. 3.2 Viscous-damper absorbing boundaries A set o continuously distributed viscous dampers enorces the one-dimensional radiation condition [18]. Assuming that incident waves impinge perpendicular to the boundary, this radiation condition is σ + ρ Vu = τ + ρ Vw = s p (3.2a) (3.2b) where σ () t and τ () t are the normal and tangential tractions; ut () and wt () are the normal and tangential displacements (Figure 3.2); ρ is the density o the oundation medium; and V p and V s its pressure-wave velocity and shear-wave velocity. The viscous damper is a perect absorber o body waves that arrive normal to the boundary, but only a partial absorber or body wave impinging at an arbitrary angle and or surace waves. However, the accuracy is generally acceptable provided the boundary is placed at suicient distance rom the wave source [2]. The viscous-damper boundary simulates the semi-unbounded oundation region where the displacements were deined by the scattered motion (Equation 3.1b), i.e., = t u u u and = t + w w w. Because the oundation rock in Ω is assumed to be linear, it ollows that the boundary tractions associated with the scattered motion are σ = σ t σ and τ = τ t τ. Substituting or the scattered motion and the corresponding tractions in Equation (3.2) and rearranging terms one obtains: + Ω t t σ = σ ρ vp u u t t τ = τ ρ vs w w (3.3a) (3.3b) Thus, the total tractions on the absorbing boundary consist o two parts: the ree-ield tractions, and the product o a damper coeicient and the scattered motion. Temporarily or convenience o notation u and w is used instead o 14 t r or displacements

26 PART I: DAM FOUNDATION ROCK SYSTEM In a discretized model the distributed dampers can be lumped at the boundary nodes, resulting in discrete viscous dampers with coeicients (Figure 3.2): c = Aρ V, normal to the boundary p p c = Aρ V, tangential to the boundary s s (3.4a) (3.4b) where A is the tributary area (tributary length in a 2D model) or the boundary node. In inite element notation, Equation (3.3) can be written or the boundary Γ as R = R c r r t t (3.5) where R is the vector o nodal orces consistent with the ree-ield tractions and c is the t matrix o damper coeicients c p and c s. The vectors R, R, and matrix c contain nonzero entries only or nodes on Γ., w, u A c p c s Foundation rock Absorbing boundary, Figure 3.2: Deinition o damper coeicients c p and c s or lumped viscous damper on Γ. 3.3 Equations o motion t t Substituting Equation (3.5) in Equation (2.1), noting that Rh Rb in the absence o water in the reservoir, and rearranging terms, the inal equations o motion or the dam oundation rock subsystem with truncated oundation domain are obtained: mr t st + c + c r t + ( r t ) = R + P (3.6) where the eective earthquake orces acting on the boundary Γ are P = R + c r (3.7) Observe by comparing Equations (3.6) and (2.1) that the unknown orces R t associated with the absorbing boundary Γ have now been expressed in terms o the viscous damper 15

27 PART I: DAM FOUNDATION ROCK SYSTEM t orces cr and the eective earthquake orces P. The latter consists o two parts: (1) R, the orces consistent with the ree-ield tractions at Γ, and (2) the damper orces cr determined rom the spatially varying ree-ield motion at Γ. Working with the scattered displacements in Ω + has thus enabled derivation o Equation (3.7) or the eective earthquake orces in terms o the ree-ield displacements and tractions. 3.4 Free-ield earthquake motion The ree-ield motion r required to compute the eective earthquake orces P can be determined by various methods. The standard procedure is to deine the ground motion at the control point (Figure 2.1) to be consistent with a design spectrum. This target spectrum may be the Uniorm Hazard Spectrum (UHS) determined by probabilistic seismic hazard analysis (PSHA) [53], or a Conditional Mean Spectrum (CMS) [54]. Recorded ground motions are selected, scaled and modiied to "match" in some sense the target spectrum; alternatively synthetic motions may be developed or an earthquake scenario. These methods are well developed or a single component o ground motion; work on extending these methods to two- or three components acting simultaneously is in progress. To determine the required ree-ield motion at the boundary Γ rom the ground motion at the control point it is necessary to introduce assumptions on the type o seismic waves and their incidence angle. The simplest assumption, oten used or site response analyses and soil structure interaction analyses, is vertically propagating SH-waves and P- waves [55,56]. This is clearly a major simpliication o the actual seismic wave ield, that generally consists o a superposition o vertically and horizontally propagating SH-, SV- and P-waves, and horizontally propagating surace waves. This assumption is oten justiied on the basis that most sites are located relatively ar away rom the earthquake source, and that the gradual sotening o rock and soil towards the earth's surace leads to diraction o seismic waves towards vertical incidence [57]. It is not obvious that this assumption is appropriate or concrete dams sited on competent bedrock, but at the present time it seems to be the only pragmatic choice. Under the assumption o vertically propagating waves and homogeneous or layered rock, the ree-ield motion r at Γ can be obtained by deconvolution o the ground motion k ag () t at the control point using standard requency-domain procedures [55]; sotware such as SHAKE [58] or DEEPSOIL [59] can be utilized or this purpose. In principle, the deconvolution analysis can provide directly the motion at every nodal point on Γ ; however, such an implementation may become cumbersome i output rom the deconvolution analysis is required at a large number o elevations. An alternative method that overcomes this problem is presented in the next section. 16

28 PART I: DAM FOUNDATION ROCK SYSTEM 3.5 Computing eective earthquake orces Bottom boundary It was assumed in Section 3.4 that the earthquake motion is caused by vertically incident seismic waves propagating up rom an underlying elastic medium. Because the ree-ield oundation-rock system (Figure 3.1a) is assumed to be linear and homogenous or horizontally layered, the boundary tractions at the bottom o the truncated oundation domain can be expressed as the sum o tractions due to the incident and relected seismic waves: σ = σi + σ (3.8) R where σ I and σ R are the normal tractions due to the incident (upward propagating) and relected (downward propagating) seismic waves, respectively. At the boundary, the radiation condition must be satisied or both the incident and relected waves: σ ρ Vu = I p I σ + ρ Vu = (3.9) R p R where u I and u R are the displacements at the boundary in the normal direction corresponding to the incident and relected seismic waves. The ree-ield velocity u at the boundary is the sum o the incident and relected waves, i.e., u = u + I u R. Substituting or u R in Equation (3.9), and inserting the result in Equation (3.8), a new expression or the ree-ield boundary tractions σ is obtained: σ = ρ Vp 2u I u (3.1a) It ollows that a similar expression can be derived or the tangential tractions τ : τ = ρ Vs 2w I w (3.1b) Such expressions were irst derived by Joyner and Chen [6]. Expressing Equation (3.1) in inite element notation to obtain R = c 2r I r, substituting the results into Equation (3.7) and cancelling terms, the inal expression or the eective earthquake orces at the bottom o the oundation domain is obtained: P = 2c r I (3.11) where r I is the motion at Γ due to the incident (upward propagating) seismic waves. This equation has the advantage that it requires only the motion r I o the incident wave, thus avoiding computation o the ree-ield tractions required i directly using Equation 17

29 PART I: DAM FOUNDATION ROCK SYSTEM (3.7). Furthermore, the incident motion r I is easily computed as 1/2 the outcrop motion at the bottom boundary, which is extracted directly rom the deconvolution analysis. The procedure to compute P rom Equation (3.11) is summarized in Box 3.1. Box 3.1: Computing P at bottom boundary o oundation rock. 1. Determine the outcrop motion at the bottom oundation-rock boundary by 1D k deconvolution o each component o the surace control motion ag () t, k = x, y. 2. Compute the incident motion r I as 1/2 the outcrop motion at the bottom boundary determined in Step 1 and obtain r I by taking the time derivative o r I. 3. Calculate the eective earthquake orces P at the bottom boundary rom Eq. (3.11) using r rom Step 2. I Side boundaries The ree-ield motion r (and its time derivatives) required to compute the eective earthquake orces P at the side boundaries can be obtained directly rom the deconvolution analysis; ree-ield tractions can then be computed rom 1D stress-strain relations and these stresses converted to orces. Alternatively, both quantities can be computed by an auxiliary analysis o the oundation rock in its ree-ield state (Figure 3.1a). Analysis o this system reduces to a single column o oundation-rock elements with a viscous damper at its base that is subjected to the orces o Equation (3.11) and analyzed to determine r and R at each nodal point along the height. The procedure is summarized in Box 3.2 and illustrated in Figure 3.3a. Although straightorward, both o these approaches requires the orce histories P at all nodal points on the side boundaries to be stored or later use in setting up Equation (3.7). Clearly, such "book-keeping" may become cumbersome to implement or large models, especially or a 3D system [37]. These diiculties can be avoided by introducing ree-ield boundary elements in the orm o 1D oundation-rock columns at the side boundaries that are solved in parallel with the main FE model (Figure 3.3b) [61]. However, such elements are currently not available in most commercial FE or inite dierence codes, the only exceptions are FLAC [62] and PLAXIS [63]. The relected motion must equal the incident motion at every rock outcrop (stress-ree boundary), hence is the incident motion exactly equal to 1/2 the outcrop motion. 18

30 PART I: DAM FOUNDATION ROCK SYSTEM Tied DOFs ( r r ) let right Apply orces P rom Eq. (3.7) to side boundaries Record and R ṙ P Apply orces rom Eq. (3.11) Apply orces P rom Eq. (3.11) to bottom boundary (a) Viscous damper boundary Free-ield boundary elements Transer o orces Apply orces P rom Eq. (3.11) to bottom boundary (b) Viscous damper boundary Figure 3.3: Two methods or application o eective earthquake orces to side boundaries o oundation domain: (a) auxiliary analysis o 1D column to compute r and R ollowed by direct application o P ; (b) use o ree-ield boundary elements. 19

31 PART I: DAM FOUNDATION ROCK SYSTEM Box 3.2: Computing P at side boundaries o oundation rock. 1. Determine the outcrop motion at the bottom oundation-rock boundary by 1D k deconvolution o each component o the surace control motion ag () t, k = x, y. 2. Calculate the eective earthquake orces P at the bottom boundary rom Equation (3.11), with the motion r I due to the incident (upward propagating) seismic wave computed as 1/2 the outcrop motion extracted rom the deconvolution analysis. 3. Develop a FE model or the ree-ield oundation-rock system: a single column o elements that has the same mesh density as the main FE model at the side boundaries, with viscous dampers applied at the base in the x- and y-directions (Figure 3.3a). 4. Compute the ree-ield velocities r and orces R at each node over the height by analyzing the oundation-rock column subjected to orces given by Eq. (3.11) at its base. 5. Calculate the eective earthquake orces P at the side boundaries rom Eq. (3.7) using r and R rom Step Relation to the Domain Reduction Method The Domain Reduction Method (DRM) [4] is a two-step methodology or modeling earthquake response where large contrasts exist between the physical scales o the background model and a smaller localized eature. The method overcomes the issues o scale dierence by subdividing the original problem into two simpler ones where a local eature perturbs the ree-ield motion in a larger background domain. This same idea was utilized in deriving the equations o motion or the dam oundation rock system, Equations (3.6) and (3.7), earlier in this chapter. Although initially developed or large scale geological simulations, the DRM has also been successully applied to speciy the seismic input in soil structure interaction problems in a layer o FEs interior o the absorbing boundary [41,64]. This has the advantage that it completely de-couples the boundary condition rom the method o seismic input, unlike the direct FE method developed above where the eective earthquake orces were derived assuming viscous-damper boundaries. Thus, DRM can be used with any advanced boundary condition and the domain sizes reduced accordingly (Figure 2.2). When developing a general procedure or earthquake analysis o concrete dams however, this beneit is outweighed by two disadvantages o using DRM: (1) implementation o DRM requires modiication o the FE source code, eectively limiting the procedure to users o LS-DYNA, which is the only commercial FE program requently used by dam engineers where DRM is available; and (2) speciying seismic input or a dam water 2