DYNAMIC ANALYSIS OF ARCH DAMS INCLUDING HYDRODYNAMIC EFFECTS

By John F. Hall1 and Anil K. Chopra,2 M. ASCE

ABSTRACT: A procedure is developed to analyze, under the assumption of lin­ear behavior, the earthquake response of arch dams including hydrodynamic effects. The dam and fluid domain are treated as substructures and modeled with 6nite elements. The only geometric restriction is that an infinite fluid do­main must maintain a constant cross section beyond some point in the up­stream direction. For such an infinite, uniform region, a finite element discre­tization over the cross section is combined with a continuum representation in the upstream direction. The fluid domain model approximately accounts for the effects of foundation flexibility on hydrodynamic pressures. Computed re­sponses of an arch dam to harmonic ground motion are presented.

INTRODUCTION

The substructure method has been employed in the earthquake anal­ysis of dam-fluid systems. Effects of water in the reservoir are included in the equations of motion of the dam through hydrodynamic forces which act on the upstream dam face. These hydrodynamic terms are computed from solutions to the wave equation over the fluid domain substructure subjected to appropriate boundary conditions. The method permits inclusion of water compressibility which has been shown to in­fluence the earthquake response of concrete gravity dams (1).

Explicit mathematical expressions for the hydrodynamic terms can be employed for simple reservoir geometries if the analysis is performed in the frequency domain. Due to water compressibility, the hydrody­namic terms are frequency dependent . This procedure was effective in the two-dimensional analysis of concrete gravity dams wi th an infinite reservoir of constant depth (1). The method has also been applied to arch dams where the upstream face is a segment of a circular cylinder contained within vertical banks of the reservoir which enclose a central angle of 90° and extend radially to infinity (8).

If the reservoir geometry is irregular, the hydrodynamic terms must be obtained by numerical discretization techniques, and several analyses for arch dams have been carried out in the time domain (6,9). Such so­lutions are expensive if the fluid domain mesh is long, and the mesh

Research Fellow in Civ. Engrg., California Inst, of Tech., Pasadena, Calif., formerly graduate student at the University of California, Berkeley, Calif.

2Prof. of Civ. Engrg., Univ. of California, Berkeley, Calif. Note.—Discussion open until July 1, 1983. To extend the closing date one

month, a written request must be filed with the ASCE Manager of Technical and Professional Publications. The manuscript for this paper was submitted for re­view and possible publication on August 19, 1981. This paper is part of the Jour­nal of Engineering Mechanics, Vol. 109, No. 1, February, 1983. ©ASCE, ISSN 0733-9399/83/0001-0149/$01.00. Proc. No. 17696.

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DYNAMIC ANALYSIS or ARCH DAMS INCLUDINGHYDRODYNAMIC Errscrs

By Iohn F. Hallland Anil K. Chopra,’ M. ASCE

Aasrnacr: A procedure is developed to analyze, under the assumption of lin-ear behavior, the earthquake response of arch dams including hydrodynamiceffects. The dam and fluid domain are treated as substructures and modeledwith nite elements. The only geometric restriction is that an infinite uid do-main must maintain a constant cross section beyond some point in the up-stream direction. For such an infinite, uniform region, a finite element discre-tization over the cross section is combined with a continuum representation inthe upstream direction. The uid domain model approximately accounts forthe effects of foundation exibility on hydrodynamic pressures. Computed re-sponses of an arch dam to harmonic ground motion are presented.

|NTFi0DUCTlONThe substructure method has been employed in the earthquake anal-

ysis 0f'dam- uid systems. Effects of water in the reservoir are includedin the equations of motion of the dam through hydrodynamic forceswhich act on the upstream dam face. These hydrodynamic terms arecomputed from solutions to the wave equation over the uid domainsubstructure subjected to appropriate boundary conditions. The methodpermits inclusion of water compressibility which has been shown to in-fluence the earthquake response of concrete gravity dams (1).

Explicit mathematical expressions for the hydrodynamic terms can beemployed for simple reservoir geometries if the analysis is performedin the frequency domain. Due to water compressibility, the hydrody-namic terms are frequency dependent. This procedure was effective inthe two-dimensional analysis of concrete gravity dams with an infinitereservoir of constant depth (1). The method has also been applied toarch dams where the upstream face is a segment of a circular cylindercontained within vertical banks of the reservoir which enclose a centralangle of 90° and extend radially to infinity (8).

If the reservoir geometry is irregular, the hydrodynamic terms mustbe obtained by numerical discretization techniques, and several analysesfor arch dams have been carried out in the time domain (6,9). Such so-lutions are expensive if the uid domain mesh is long, and the mesh

‘Research Fellow in Civ. Engrg., California Inst. of Tech., Pasadena, Calif.,formerly graduate student at the University of California, Berkeley, Calif.

ZProf. of Civ. Engrg., Univ. of California, Berkeley, Calif.Note.—Discussion open until Iuly 1, 1983. To extend the closing date one

month, a written request must be filed with the ASCE Manager of Technical andProfessional Publications. The manuscript for this paper was submitted for re-view and possible publication on August 19, 1981. This paper is part of the ]our-nal of Engineering Mechanics, Vol. 109, No. 1, February, 1983. ©ASCE, ISSN0733-9399/83/0001-0149/$01.00. Proc. No. 17696.

149

must be extremely long if pressure wave reflections from the upstream boundary are to be avoided. "Quiet" boundaries which satisfactorily transmit the pressure waves and which can be placed close to the dam do not seem possible in the time domain. However, a satisfactory dis­cretization technique for two-dimensional finite or infinite fluid domains has recently been developed for use in frequency domain analyses of embankment and concrete gravity dams. The fluid domain consists of a finite region of arbitrary geometry which may be connected to an in­finite portion of constant depth extending upstream (5).

The purpose of this paper is to extend the method of Ref. 5 to arch dams with realistic fluid domains of complicated geometry. The three-dimensional fluid domain consists of a finite region of arbitrary geom­etry which may be connected to an infinite region of uniform cross sec­tion. Finite element procedures are developed for evaluation of the hy-drodynamic terms. Using these techniques, results are presented for the response of a selected arch dam to harmonic ground motion.

SYSTEMS AND GROUND MOTION

In the three-dimensional analysis procedure presented in this paper, the dam can be of arbitrary shape. The reservoir may extend only a short distance upstream (Fig. 1(a)) or to a large enough distance so that it can be considered infinite for purposes of analysis (Fig. 1(b)). In the latter case, the reservoir cross section is assumed to be uniform beyond some point in the upstream direction. Behaviors within the elastic dam and compressible water are assumed to be linear.

The earthquake ground motion is defined by the upstream-down-stream, x, cross-stream, z, and vertical, y, components of acceleration. Interaction between the dam and the foundation rock is not considered, and the specified ground motion along the foundation boundary of the dam is assumed uniform. Interaction between the fluid and foundation medium (rock or silt), i.e., the influence of foundation flexibility on hy-drodynamic pressures, is approximately considered through a damping boundary condition (6) applied along the bottom and sides of the res­ervoir. The ground motion along the reservoir bottom and sides is de­scribed by free-field accelerations assumed uniform. The actual accel­eration of these boundaries depends on the interaction.

ANALYSIS OF DAM RESPONSE

Governing Equations of the Dam.—The arch dam is discretized with finite elements as shown in Fig. 2; shell elements are most appropriate (2). Undamped, free-vibration mode shapes <(>; of the dam are chosen as generalized coordinates to describe the motion of the dam with or without water. The vector v((t) of nodal displacements relative to the ground due to the component of ground motion in the € direction (( = x, y, or z) is approximately expressed as a linear combination of the first / modes of vibration of the dam:

v'(o = 2>;y/<o a) 150

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must be extremely long if pressure wave re ections from the upstreamboundary are to be avoided. “Quiet” boundaries which satisfactorilytransmit the pressure waves and which can be placed close to the damdo not seem possible in the time domain. However, a satisfactory dis-cretization technique for two-dimensional finite or infinite uid domainshas recently been developed for use in frequency domain analyses ofembankment and concrete gravity dams. The uid domain consists ofa finite region of arbitrary geometry which may be connected to an in-finite portion of constant depth extending upstream (5).

The purpose of this paper is to extend the method of Ref. 5 to archdams with realistic uid domains of complicated geometry. The three-dimensional uid domain consists of a finite region of arbitrary geom-etry which may be connected to an infinite region of uniform cross sec-tion. Finite element procedures are developed for evaluation of the hy-drodynamic terms. Using these techniques, results are presented for theresponse of a selected arch dam to harmonic ground motion.

Svsrems AND Gnouuo MOTIONIn the three-dimensional analysis procedure presented in this paper,

the dam can be of arbitrary shape. The reservoir may extend only a shortdistance upstream (Fig. 1(a)) or to a large enough distance so that it canbe considered infinite for purposes of analysis (Fig. 1(b)). In the lattercase, the reservoir cross section is assumed to be uniform beyond somepoint in the upstream direction. Behaviors within the elastic dam andcompressible water are assumed to be linear.

The earthquake ground motion is defined by the upstream-down-stream, x, cross-stream, z, and vertical, y, components of acceleration.Interaction between the dam and the foundation rock is not considered,and the specified ground motion along the foundation boundary of thedam is assumed uniform. Interaction between the uid and foundationmedium (rock or silt), i.e., the in uence of foundation exibility on hy-drodynamic pressures, is approximately considered through a dampingboundary condition (6) applied along the bottom and sides of the res-ervoir. The ground motion along the reservoir bottom and sides is de-scribed by free-field accelerations assumed uniform. The actual accel-eration of these boundaries depends on the interaction.

ANALYSIS OF DAM RESPONSE

Governing Equations of the Dam.—The arch darn is discretized withfinite elements as shown in Fig. 2; shell elements are most appropriate(2). Undamped, free-vibration mode shapes ¢]- of the dam are chosenas generalized coordinates to describe themotion of the dam with or.without water. The vector ve(t) of nodal displacements relative to theground due to the component of ground motion in the 6 direction (3= x, y, or z) is approximately expressed as a linear combination of thefirst] modes of vibration of the dam:

Iv‘(t) = 2 .1», Yf(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (1)

/‘=1

150

(a) FINITE FLUID DOMAIN

(b) INFINITE FLUID DOMAIN

FIG. 1.—Three-Dimensional Arch Dam-Fluid Systems

The mode shapes <|>; and corresponding natural frequencies <o; of the dam are computed from the eigenproblem

k t y = u>jm<|>; (2)

in which m and k = symmetric mass and stiffness matrices for the finite element system. Only degrees of freedom (DOF) of the dam for nodes not on the dam-foundation boundary are included in Eq. 2.

The steady state responses to harmonic ground acceleration aeg(t) =

emt can be expressed in terms of complex frequency response functions. Thus, a response quantity, say r(t), is given by

r{t) = f{m)eM (3)

in which f(w) = the complex frequency response function. The J complex frequency response functions Y-(u>) for the generalized

modal displacements Y/(f) are found by solving a set of / equations (Eq. 4) which is an extension of that presented earlier (5) for two-dimensional dam-fluid systems:

S(o>) Y€(o>) = L'(co); t = x,)),z (4)

in which Y'(co) = the vector of Y/(w); j =1,2, ...,] and

S;,(a))=-(o2{«t»{}TQ{(o))

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WATER

(O) FINITE FLUID DOMAIN

4 /°°

(I5) INFINITE FLUID D(:MAlN I

FIG. 1.—Three-Dimensional Arch Dam-Fluld Systems

The mode shapes 4), and corresponding natural frequencies of of thedam are computed from the eigenproblem

1<¢,=afm¢,- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(2)in which m and k = symmetric mass and stiffness matrices for the finiteelement system. Only degrees of freedom (DOF) of the dam for nodesnot on the dam-foundation boundary are included in Eq. 2.‘The steady state responses to harmonic ground acceleration a§(t) =

e""' can be expressed in terms of complex frequency response functions.Thus, a response quantity, say r(t), is given by

r(t) = f(u>) e"“’T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (3)in which i'(m) = the complex frequency response function.

The] complex frequency response functions Y (to) for the generalizedmodal displacements Yf(t) are found by solving a set of I equations (Eq.4) which is an extension of that presented earlier (5) for two-dimensionaldam-fluid systems:

S(<n) Y"(w) = L‘(@); e = x, y, Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (4)in which Y”(m) = the vector of Yf(w); j = 1, 2, ..., ] and

S,~.<<»> = -<»2{¢;1~T eta») '151

S^co) = -co2 Mt + m C, + Kj - co2 {0T Q{(co)

L/(a,) = . - ^ E < - { ^ } r Q i « ( < o )

E ' = [ m | m s ] e ' '. (5)

In Eq. 5, *)>{ lists the x, y, and z components of the dam's ; th mode shape for all nodes along the dam-fluid interface a-b-c-d-a (Fig. 2). M ; /

Cj, and Xy are the ;'th modal mass, damping and stiffness defined by

Mj^tfmty Cj = 2$ajMj K; = co2M,. (6)

in which (•• = the ;'th modal damping ratio. Ee = a vector of inertial forces on the dam arising from a unit acceleration of the dam as a rigid body in the € direction with the reservoir empty. m„ is a mass matrix coupling DOF on the dam-foundation boundary, with the DOF not on this boundary; mg is nonzero for consistent mass matrices only. The ith term of e€ equals the length of the component of a unit vector along € in the direction of the ith translational DOF. The vectors e1, ey, and ez, for ground motions in the x, y, and z directions contain ones in positions corresponding to x, y, and z translational DOF, respectively, with zeros elsewhere.

Hydrodynamic Force Vectors.—Hydrodynamic terms appear on both sides of Eq. 4, as added loads Q% (co) on the right, and added masses Q|(co) on the left, the latter also coupling the modal equations. The added load terms are associated with hydrodynamic pressures on the dam face due to ground accelerations while the dam is rigid. Added mass terms arise from hydrodynamic pressures due to motions of the dam relative to its base. The hydrodynamic terms depend on the exci­tation frequency, a consequence of the fluid compressibility. For an in­compressible fluid, the hydrodynamic terms become independent of frequency.

The hydrodynamic terms Q$(co) and Q (̂co) in Eqs. 4 and 5 are vectors listing the x, y, and z components of hydrodynamic forces on the dam at the dam-fluid interface with terms ordered to correspond to those of 4»|. The force vectors are computed from hydrodynamic pressures on the dam-fluid interface by the method of virtual work.

FiG. 2.—Finite Element Dam Model, Infinite Fluid Domain

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Sf]-(0)) = —<n2 M; + ito Ci + — <02

Lira) = ~—¢,TE‘ — {W Qt‘ <<»>E‘ = [m|mg] ee . . . . . . . . . . ..'. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (5)

In Eq. 5, ¢{ lists the x, y, and z components of the dam’s j th modeshape for all nodes along the dam- uid interface a—b—c—d-a (Fig. 2). M,-,Cf, and K; are the j th modal mass, damping and stiffness defined by

M,=¢,Tm¢, c,=2g,<»,M, 1<,=w§M, . . . . . . . . . . . . . . . . . . . . . . . ..(6)in which §- = the jth modal damping ratio. E‘? = a vector of inertialforces on the dam arising from a unit acceleration of the dam as a rigidbody in the if direction with the reservoir empty. m is a mass matrixcoupling DOF on the dam-foundation boundary, with the DOF not onthis boundary; mg is nonzero for consistent mass matrices only. The i thterm of ee equals the length of the component of a unit vector along t’in the direction of the ith translational DOF. The vectors ex, e”, and e’,for ground motions in the x, y, and z directions contain ones in positionscorresponding to x, y, and z translational DOF, respectively, with zeroselsewhere.

Hydrodynamic Force Vectors._—Hydrodynamic terms appear on boths_ides of Eq. 4, as added loads Q{f ((1)) on the right, and added massesQI(w) on the left, the latter also coupling the modal equations. Theadded load terms are associated with hydrodynamic pressures on thedam face due to ground accelerations while the dam is rigid. Addedmass terms arise from hydrodynamic pressures due to motions of thedam relative to its base. The hydrodynamic terms depend on the exci-tation frequency, a consequence of the uid compressibility. For an in-compressible uid, the hydrodynamic terms become independent offrequency. _ _ I ‘

The hydrodynamic terms Qff (co) and Q§(u>) in Eqs. 4 and 5 are vectorslisting the x, y, and z components of hydrodynamic forces on the damat the dam— uid interface with terms ordered to correspond to those of11>,’-. The force vectors are computed from hydrodynamic pressures onthe dam- uid interface by the method of virtual work.

WI ‘=FIG. 2.—Finlte Element Dam Model, lnflnite Fluid Domain

V

152

The hydrodynamic pressure distribution p(x,y,z,ut), in excess of the hydrostatic pressure, is governed by the three-dimensional Helmholtz equation, which is valid for small displacements, irrotational motion, and negligible viscous effects:

d2p d2p d2p to2

-f5 + - 4 + - 4 + - rp = 0 (7) dx2 dy2 dz2 C2V w

in which C = velocity of compression waves in water. Along the dam-fluid interface, the pressures should satisfy

dp w •—(s,r,co) = —a„(s,r) (8a) dn g

in which s, r = boundary coordinates as shown in Fig. 2; w = unit weight of water; g = acceleration of gravity; an (s, r) = normal component of acceleration of the upstream dam face; and n = the inward normal direction to a boundary. Along the reservoir bottom and sides

dp w -Ms',r',a)) = —an(s',r') + i(oqp(s',r',u>) (8b) dn g

in which s',r' = boundary coordinates as shown in Fig. 2; an(s',r') = normal component of free field acceleration of the reservoir bottom and sides; and q = a damping coefficient. Neglecting waves at the free sur­face of the water (y = H)

p(x,H,z,u) = 0 (9)

In addition to the boundary conditions of Eqs. 8 and 9, the pressures should satisfy the radiation condition for fluid domains extending to in­finity in the upstream direction.

The last term in Eq. 8b accounts approximately for interaction between the fluid and the foundation medium (rock or silt) by providing only a partial reflection of a hydrodynamic pressure wave which strikes the foundation boundary. For a plane wave normally incident to a plane boundary, the reflection coefficient ar, which is the ratio of reflected to incident pressure wave amplitude, can be related to q by

1-qC T 1+qC

By specifying ar, q can be obtained from Eq. 10 as

1 1 - a , C l + a r

(10)

(11)

Note that ar = 1, leading to q = 0, is the rigid foundation case. For the one-dimensional wave propagation mentioned above, q can be related to the foundation medium properties by

W (12) wrC,

in which wr and Cr = unit weight and compression wave velocity of the foundation medium.

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The hydrodynamic pressure distribution ;5(x,y,z,m), in excess of thehydrostatic pressure, is governed by the three-dimensional Helmholtzequation, which is valid for small displacements, irrotational motion,and negligible viscous effects:62;? 62? 82p (D2

———+-—+—--+—'=U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7axz 6]/2 azz Czp ( )

in which C = velocity of compression waves in water. Along the dam-uid interface, the pressures should satisfy

g—g(s, r, co) = —% a,,(s,r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8a)

in which s, r = boundary coordinates as shown in Fig. 2; w = unitweight of water; g = acceleration of gravity; an (s, r) = normal componentof acceleration of the upstream dam face; and n = the inward normaldirection to a boundary. Along the reservoir bottom and sides

§E(s’,r’,w) = ~-Z£a,,(s’,r’) + iwq;5(s’,r’,w) . . . . . . . . . . . . . . . . . . . . .. (8b)8n gin which s’,r' = boundary coordinates as shown in Fig. 2; a,,(s’,r’) =normal component of free field acceleration of the reservoir bottom andsides; and q = a damping coefficient. Neglecting waves at the free sur-face of the water (y = H)

(x,H,z,w)=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (9)

In addition to the boundary conditions of Eqs. 8 and 9, the pressuresshould satisfy the radiation condition for uid domains extending to in-finity in the upstream direction.

The last term in Eq. 8b accounts approximately for interaction betweenthe uid and the foundation medium (rock or silt) by providing only apartial re ection of a hydrodynamic pressure wave which strikes thefoundation boundary. For a plane wave normally incident to a planeboundary, the re ection coefficient ot,, which is the ratio of re ected toincident pressure wave amplitude, can be related to q by

1 —- qC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..10

a' 1+qC ()

By specifying cx,, q can be obtained from Eq. 10 as1 1 — or ‘

= —- ———-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11q C 1 + oz, ( )

Note that ex, = 1, leading to q = 0, is the rigid foundation case. For theone—dimensional wave propagation mentioned above, q can be relatedto the foundation medium properties by

w=——— . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12q w,C, ( )in which w, and C, = unit weight and compression wave velocity of thefoundation medium.

153

The hydrodynamic force vector Q{f (co) is obtained from the pressures p%(s,r,u>) along the dam-fluid interface arising from accelerations of the foundation in the direction t while the dam is rigid. This acceleration distribution is given by an(s'r) ~ e((s,r); s,r = s,r or s',r' (13)

in which e*(s,r) = a function defined over accelerating boundaries which gives the length of the component of a unit vector along € in the direc­tion of the inward normal n. Qy(co) is obtained from p|(s,r, co) on the dam-fluid interface which arises from accelerations of the dam in its ;'th vibration mode. Thus

an(s,r) = $(s , r ) (14a)

an(s',r') = 0 (14b)

in which <j>̂ (s, r) = a continuous function representation of the compo­nent of the )th mode shape normal to the dam-fluid interface.

Response Analysis.—The complex frequency response functions YJ{oi), j = 1, 2, . . . , / are obtained by solving Eq. 4 for a range of values of the excitation frequency w. Responses Yj(t) to an arbitrary ground acceleration ae

g(t) can be obtained by Fourier synthesis of the response to individual harmonic components using the Fast Fourier Transform algorithm (FFT). Nodal displacements of the dam are then found using the transformation of Eq. 1, and stresses can be computed using stress-displacement transformation matrices for each finite element.

The terms My, C;, Kjr ty, and E€ of Eq. 5 are computed from a finite element model of the dam; whereas, the hydrodynamic force vectors Q^(w) and Qy(co), are determined from an appropriate model of the fluid domain. Procedures for computing the hydrodynamic force vectors for three types of fluid domains modeled as finite element systems are presented in the next four sections.

FINITE FLUID DOMAINS OF IRREGULAR GEOMETRY

Solution of the boundary value problem (BVP) of Eq. 7, subject to the acceleration boundary condition Eq. 8 and the free surface condition Eq. 9 for finite fluid domains of irregular geometry (Fig. 3(a)), can be ob­tained numerically by the finite element method. In this approach, the fluid domain is divided into three-dimensional finite elements as shown in Fig. 3(b). The interelement hydrodynamic pressure is defined in terms of discrete values p,(co) at the nodal points using shape functions. The nodal pressures are the unknowns in the BVP, one DOF for each node below the fluid free surface (where the nodal pressures are zero) assem­bled into the vector p(o>). A finite element discretization of the BVP of Eqs. 7, 8, and 9 leads to the matrix equation (4):

( 0)2 \ W H + iaq B - —2 G I p(w) = - D (15)

in which H, B, and G are symmetric matrices analogous to stiffness, damping, and mass matrices arising in dynamics of solid continua; and

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The hydrodynamic force vector Q{f(u>) is obtained from the pressuresPg‘ (s, r, co) along the dam— uid interface arising from accelerations of thefoundation in the direction € while the dam is rigid. This accelerationdistribution is given by -

a,,(s,r) = ee(s,r); s,r = s,r or s’,r' . . . . . . . . . . . . . . . . . . . . . . . . . .. (13)in which cf (s, r) = a function defined over accelerating boundaries whichgives the length of the component of a unit vector along 6 in the direc-tion of the inward normal n. QI(m) is obtained from ;3§(s, r, (1)) on thedam- uid interface which arises from accelerations of the dam in its jthvibration mode. Thus

an (s, r) = ¢§(s, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1411)

a,,(s’,r’) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (1412)in which ch; (s, r) = a continuous function representation of the compo-nent of the thmode shape normal to the dam- uid interface.

_ Response Analysis.—The complex frequency response functionsYI(co), j = 1, 2, . . ., I are obtained by solving Eq. 4 for a range of valuesof the excitation frequency w. Responses Yf(t) to an arbitrary groundacceleration a§(t) can be obtained by Fourier synthesis of the responseto individual harmonic components using the Fast Fourier Transformalgorithm (FFT). Nodal displacements of the dam are then found usingthe transformation of Eq. 1, and stresses can be computed using stress-displacement transformation matrices for each finite element.

The terms M]-, Ci, Ki, (1),, and E 8 of Eq. 5 are computed from a niteelement model of the dam; whereas, the hydrodynamic force vectorsQ{f(w) and QI(w), are determined from an appropriate model of theuid domain. Procedures for computing the hydrodynamic force vectors

for three types of uid domains modeled as finite element systems arepresented in the next four sections. I

Fmm: F|.u|o Doumms o|= lnneeuran Geomernv I

Solution of the boundary value problem (BVP) of Eq. 7, subject to theacceleration boundary condition Eq. 8 and the free surface condition Eq.9 for finite uid domains of irregular geometry (Fig. 3(a)), can be ob-tained numerically by the finite element method. In this approach, theuid domain is divided into three-dimensional finite elements as shown

in Fig. 3(b). The interelement hydrodynamic pressure is defined in termsof discrete values p,»(w) at the nodal points using shape functions. Thenodal pressures are the unknowns in the BVP, one DOF for each nodebelow the uid free surface (where the nodal pressures are zero) assem-bled into the vector p(co). A finite element discretization of the BVP ofEqs. 7, 8, and 9 leads to the matrix equation (4):

2

H+it.>B—3c-; -(t.>)=9D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(15)q C2 P g

in which H, B, and G are symmetric matrices analogous to stiffness,damping, and mass matrices arising in dynamics of solid continua; and

154

P(«.Vt*i

(a) FLUID DOMAIM

(b) FINITE ELEMENT DISCRETIZATION

FIG. 3.—Finite Fluid Domain of Irregular Geometry

D = vector of nodal accelerations computed from the normal accelera­tions an(s,r) along the dam-fluid interface a-b-c-d-a and an(s',r') along the reservoir bottom and sides b-f-i-j-g-c-b, a-e-i-f-b-a, and d-h-j-g-c-d. The nonzero portion of B is a submatrix corresponding to nodes along the reservoir bottom and sides where the boundary condition of Eq. 8b is applied.

The unknown pressures p(co) can be determined by solving Eq. 15. For the case a = 0, p(a)) can also be determined using an eigenvector expansion. The eigenproblem associated with Eq. 15 for a = 0 is

H£ = 72G£ (16)

which upon solution yields real valued eigenvalues ym and eigenvectors £,„. The eigenvectors are orthogonal to H and G and are normalized so that

= 7*

?m G £,„ = 1

and they then satisfy ££ H £,„

Expanded in terms of the first M eigenvectors, p (w) becomes

w

p((o) = - z r ' z T D g

(17a)

(17b)

• (18)

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0Fllwn.w) ___...l“" I

" 0,, s ,ré-:5.

(GI FLUID DOMAIM

FINITE isremzur enoneit

"“'of:‘§irisitttii~

-

Ib) FINITE ELEMENT DISCRETIZATION

FIG. 3.—FInIte Fluid Domain of Irregular Geometry

D = vector of nodal accelerations computed from the normal accelera-tions an (s, r) along the dam- uid interface a-b-c-d-a and 11,, (s’,r’) along thereservoir bottom and sides b-f-i—j—g-c-b, a-e-i-f-I1-a, and d-h-j-g-c—d. Thenonzero portion of B is a submatrix corresponding to nodes along thereservoir bottom and sides where the boundary condition of Eq. 8b isapplied.

The unknown pressures p(w) can be determined by solving Eq. 15.For the case q == 0, p(w) can also be determined using an eigenvectorexpansion. The eigenproblem associated with Eq. 15 for q = 0 is

H§=y2G§ ................................................ ..(16)which upon solution yields real valued eigenvalues 'y,,, and eigenvectors§,,,. The eigenvectors are orthogonal to H and G and are normalized sothat

§,{,G§,,,=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (1711)and they then satisfy {L I-I L," = yf, . . . . .,. . . . . . . . . . . . . . . . . . . . . . . (17b)Expanded in terms of the first M eigenvectors, p(w) becomes

ZUp(@)=§z1""1zTD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (18)155

in which Z = [£,, £2, . . . , £M] and T = an M x M diagonal matrix with mth diagonal term = 7^ - (<o2/C2).

The variation of p (w) with excitation frequency 00 can be deduced for q = 0 by examining Eq. 18.. The amplitude of the mth eigenvector is real valued and resonates to infinity at excitation frequencies equal to the eigenfrequencies (ob

m = 7^. Thus, p(<o) is a real valued function of fre­quency unbounded at the <ab

m. This response behavior of a finite fluid domain is characteristic of any undamped finite solid. For q > 0, the behavior of p (w) is similar to that of a finite solid with nonproportional damping; i.e., bounded at all frequencies and complex valued for 00 > 0. The source of damping is the outward radiation of energy through the flexible foundation (foundation radiation damping).

INFINITE FLUID DOMAIN OF UNIFORM CROSS-SECTION

Boundary Value Problems.—The fluid domain of Fig. 4 extends to infinity along the x-axis with uniform y-z cross-section. The BVP of Eqs. 7, 8, and 9 is solved below for an acceleration ax(y,z) of the dam-fluid interface a-b-c-d-a and for an acceleration an(s',r') = a'n(r') of the reservoir bottom and sides, unvarying in the upstream direction. The coordinate s' is parallel to the x-axis, and r' follows the boundary around the y-z cross section. Solutions are carried out separately for these two accel­eration conditions which are shown in Figs. 4(«) and (c).

The governing Eq. 7 with the boundary conditions

dp W — (0,y,z,w)= —ax(y,z) OX S

(19a)

dp — (x,r',m) = icon »(x,r',co) dn

(19b)

INFINITE CHANNEL

NODAL LINE

a^y.z)

(a) ACCELERATION OF DAM-FLUID INTERFACE

(c) ACCELERATION OF RESERVOIR BOTTOM AND SIDES

(d) FINITE ELEMENT DISCRETIZATION OF FIG. c

FIG. 4.—Infinite Fluid Domain of Uniform Cross-Section

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in which Z = [§1, Q2, ..., QM] and F = an M X M diagonal matrix withmth diagonal term = 'yf,, — (oz/C2).

The variation of p(w) with excitation frequency w can be deduced forq = 0 by examining Eq. 18, The amplitude of the mth eigenvector is realvalued and resonates to infinity at excitation frequencies equal to theeigenfrequencies mi’, = ~)$,. Thus, p(w) is a real valued function of fre-quency unbounded at the o>',’,,. This response behavior of a finite uiddomain is characteristic of any undamped finite solid. For q > 0, thebehavior of p(u>) is similar to that of a finite solid with nonproportionaldamping; i.e., bounded at all frequencies and complex valued for to >O. The source of damping is the outward radiation of energy throughthe ‘ exible foundation (foundation radiation damping).

INFINITE F|.u|n DOMAIN o|= UNIFORM Cnoss-SEc'r|oNBoundary Value Problems.—The uid domain of Fig. 4 extends to

infinity along the x-axis with uniform y-z cross-section. The BVP of Eqs.7, 8, and 9 is solved below for an acceleration a,,(y,z) of the dam- uidinterface a—b-c-d—a and for an acceleration an (s',r’) = af,(r') of the reservoirbottom and sides, unvarying in the upstream direction. The coordinates’ is parallel to the x-axis, and r’ follows the boundary around the y—zcross section. Solutions are carried out separately for these two accel-eration conditions which are shown in Figs. 4(a) and (c).

The governing Eq. 7 with the boundary conditions

erIx (oly/Z, 03) = _%a;;(y/Z) - - - - - - ' ~ ' ' ' ' ' ' ' ' ' " ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

6-£(x,r',w) = ioq;5(x,r',w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (1911)

nou _iI1.v.1.wl mrmnz CHANNEL

uomu. rm:4-<1.<y.z> _

(b) CHANNELDISCRETIZATIONor FIG. <1

Ah /

c 1

(uIACCELI-LRATION orDAM-FLUID dINTERFACE

NODE

i>(l.v.1.<-1) / ‘0" 5 J I: FINITE Q‘

(c)ACCELERATION on r E‘-E"E"T \or RESERVOIR b /BOTTOM - Ia) FINITE =‘ND 9°55 ° ‘ ELEMENT DISCRETIZATION

or FIG. c

W‘I\IFIG. 4.—InIinite Fluid Domaln of Uniform Cross-Section

156

p(x,H,z,a>) = 0 (19c)

defines the first BVP. Eq. 7 with boundary conditions

dp f (0,y,z,a>) = 0 (20a) aX

dp W . -L(x,r',i>>) = a'n(r') + iwqp(x,r'ra>) (20b) an g

p(x,H,z,m) = 0 (20c)

defines the second BVP. First BVP.—The uniform cross section of the fluid domain allows the

x distribution of pressure to be separated from the y-z distribution. Thus

p(x,y,z,b>) = px(x,u>) fyz(y>z>«>) (21)

in which px(x,a>) must satisfy

-^-K2px = 0 . (22a)

and pyz(y,z,co) must satisfy

d Pyz , ° Pyz , , 2 .-. _ n OOM

^ + # + "^-° {22b)

in which K = a separation constant; and

X2 = K2 + ^ . ( 23 )

Boundary conditions include Eq. 19a and the separated conditions

^ ( r ' x o ) = i<»qpyz{r',<») • • • (24a) dn "

pyz(H,z,a) = 0 (24b)

The two-dimensional Eq. 22b with boundary conditions of Eq. 24 de­fines an eigenvalue problem. A finite element discretization of this ei­genvalue problem using a two-dimensional mesh (Fig. 4(d)) leads to the matrix equation (4):

[FT + mq W] «|i = X2G; I|I (25)

in which the matrices H', B', and G' are symmetric. The nonzero portion of B' is a submatrbc corresponding to nodes along the boundary a-b-c-d in Fig. 4(d). Only DOF for nodes below the free surface are included in Eq. 25.

The eigenvalues X„ and eigenvectors «(»„ determined from Eq. 25 are complex valued and dependent on the excitation frequency w unless q = 0; in which case they are real valued and frequency independent. The »|i„ are orthogonal and are normalized so that

«l£G'>„ = 1 ' , (26a)

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ri(x,H,z,o>) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (190)

defines the first BVP. Eq. 7 with boundary conditions6-g (0,y,z,o) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (2011)

g%(x,r’,u>) = —% af,(r') + iwq (x,r’,w). . . . . . . . . . . . . . . . . . . . . . . . .. (2011)

f1(X,H,Z,m) = O . . . . . ..; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (200)defines the second BVP. ‘

First BVP.—The uniform cross section of the uid domain allows thex distribution of pressure to be separated from the y-z distribution- Thus;7(x,y,z,w) = ;3x(x, w) ;7yz(y, z, (1)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)

in which )3x(x,a>) must satisfy

4273. _F'“K2Px=O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(22.1)

and pyz(y,Z,(l)) must satisfy2 82

98-5%! +»i§ + Alp), = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (22b)

in which K = a separation constant; and

A2 2 + (D2 I 2 I (23)= K — . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .C2

Boundary conditions include Eq. 19a and the separated conditions

a?pf(r',w)=imq yz(r',u>) . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(24a)

;3yz(H,z,o) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (2411)The two-dimensional Eq. 22b with boundary conditions of Eq. 24 de-

fines an eigenvalue problem. A finite element discretization of this ei-genvalue problem using a two-dimensional mesh (Fig. 4(d)) leads to thematrix equation (4): 2

[Hi+iwqB‘]\I1=A2G‘\l1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (25)in which the matrices I-I", B‘, and Gi are symmetric. The nonzero portionof Bi is a submatrix corresponding to nodes along the boundary a—b-c—din Fig. 4(d). Only DOF for nodes below the free surface are included inE . 25.

qThe eigenvalues An and eigenvectors 111,, determined from Eq. 25 arecomplex valued and dependent on the excitation frequency to unless q= 0; in which case they are real valued and frequency independent. The41,, are orthogonal and are normalized so that

of 010),, = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (2611)157

and they then satisfy

i|£(H'" + iwq B;) «|i„ = X2„ (26b)

The separated function in the y-z plane, pyz(y,z,u>) from Eq. 22b, is ex­pressed in discrete form as

pyz(co) = Ik -n„(co); n = 1, 2 (27)

Discretization in the x direction is inappropriate because the fluid do­main extends to infinity in that direction. Therefore, continuum solu­tions to Eq. 22a are employed. The K in Eq. 23 can take on only the values given by

^ - £ i = V* + «", • • (28)

Since the infinite fluid domain is excited at x = 0, px(x,u>) must decay with increasing x or travel from x = 0 to x = °°. Thus, it is of the form

px(x,u>) = e-«"x; n = 1, 2, (29)

in which the root with both |x„ and v„ positive is taken in computing K„ from Eq. 28. Including the first N terms in pyz and px leads to an ap­proximate expression for p(x,a>):

N

p(x,<*) = 2 +»e~™ %(<*) = * e(x) q(eo) (30) «=i

in which e(x) = an N x N diagonal matrix with nth diagonal term = e~K"x. If g = 0, then X„ is real; and K„ is real or imaginary, depending on whether w is less than or greater than \„C; i.e., K„ = |x„ or K„ = iv„.

The aforementioned formulation can be interpreted as a discretization of the fluid domain into channels of infinite length (Fig. 4(b)). The ith term of the vector p (x, w) in Eq. 30 represents the variation of pressure with x along the ith nodal line. The f|„(co) are determined to satisfy the discrete form of the boundary condition Eq. 19a (4):

dp w Gf- (0,w) = D* (31)

dx g

in which G' is the same matrix as in Eq. 25; and D* = a vector of nodal accelerations corresponding to the acceleration ax{y,z) of the dam-fluid interface. The f|„(co) are found and substituted back into Eq. 30 to yield

w p(x,co) = - 9 e(x) K -1 * r D* (32)

g in which K = an N x N diagonal matrix with nth diagonal term = K„. At x = 0, Eq. 32 reduces to

IV

p(0,w) = - * K 1 WrW (33) 8

For q = 0, \„ , and ijj„ are real valued. Then, from Eq. 32, the am­plitude of «Ji„ decays exponentially with increasing x at e'v"x (when w

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and they then satisfy

\I1,f(H‘ + iwq B’) ll!" = Af, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (2612)The separated function in the y-z plane, y,(y,z,w) from Eq. 22b, is ex-pressed in discrete form as

py,(t.>) = 41,, »q,,(@); 11 = 1, 2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (27)Discretization in the x direction is inappropriate because the uid do-

main extends to infinity in that direction. Therefore, continuum solu-tions to Eq. 22a are employed. The K in Eq. 23 can take on only thevalues given by I .

I 2K,,= )tf,—%=|J.n+iv,, .. . . . . . . . . . . . . . . . . . . . . . . . . .; . . . . . . . ..(28)

Since the infinite uid domain is excited at x = 0, ,,(x,u>) must decaywith increasing x or travel from x = 0 to x = w. Thus, it is of the form

;3,,(x,(n) = e"‘""; n = 1, 2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (29)

in which the root with both )1" and 11,, positive is taken in computing i<,,from Eq. 28. Including the first N terms in Pyz and x leads to an ap-proximate expression for p(x,<1>):

N

aw») = §_j ~11. -1. <<»> = 1' em no) ----------------------- -- <30),\|-Iin which e x) = an N >< N diagonal matrix with nth diagonal term =

e_“"". If q = 0, then A” is real; and |<,, is real or imaginary, depending onwhether w is less than or greater than )\,,C; i.e., K" = p.,, or |<,, = iv".

The aforementioned formulation can be interpreted as a discretizationof the uid domain into channels of infinite length (Fig. 4(b)).' The ithterm of the vector p(x,a>) in Eq. 30 represents the variation of pressurewith x along the ith nodal line. The f1,,(w) are determined to satisfy thediscrete form of the boundary condition Eq. 19a (4):

dp wG‘— , = —— " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .dx (0 w) g D (31)

in which G‘ is the same matrix as in Eq. 25; and D‘ = a vector of nodalaccelerations corresponding to the acceleration a,,(y,z) of the dam-fluidinterface. The ,,(m) are found and substituted back into Eq. 30 to yield

wp(x,u>)=§\Ife(x)K'1\I'TD" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (32)

in which K = an N ><' N diagonal matrix with nth diagonal term = Kn-At x = 0, Eq. 32 reduces to

p(0,<») =§\I»I<'1\rTo* . . . . .. ............................ (as)For q = 0, An, and III, are real valued. Then, from Eq. 32, the am-

plitude of mp, decays exponentially with increasing x at e"""" (when to

158

< X„C) or is a nondecaying harmonic e~'VaX (when to > X„C). At exci­tation frequencies equal to the eigenfrequencies u>'n = X„C, K„ = 0 and the amplitude of ty„ is infinite. The part of the amplitude that approaches infinity is real below to'„ and imaginary above. Thus, p(x,to) is real for o) < (*i, complex for to > (o\, and unbounded at frequencies equal to a>'n. The harmonic, nondecaying distribution with x in the amplitude of an eigenvector <Apn for to > m'n represents a radiation of energy in the infinite, upstream direction of the fluid domain. This fluid radiation damping is nonzero for to > toj and is responsible for the imaginary com­ponent of p(x,w). It does not, however, prevent the infinite resonances at frequencies oi'n above a>i because of the orthogonality of the eigenvec­tors; i.e., a resonating eigenvector i|i„ is unaffected by the fluid radiation damping associated with the lower eigenvectors.

For q > 0, the complex eigenvector i|i„ has an x distribution of e~^"xe~'v"x, an exponentially decaying harmonic. Thus, all energy is eventually radiated by the foundation. Since this foundation radiation damping occurs for all frequencies greater than zero, p(x,to) is bounded at all frequencies and complex valued for to > 0.

Second BVP.—The BVP of Eq. 7 with the boundary conditions of Eq. 20 is two-dimensional in the y- and 2-coordinates. Omitting the x vari­ations from these equations results in the two-dimensional Helmholtz equation for p(y,z,to)

d2p . d2p oy2

^2+j?+r2p = 0 (34)

dp W . and the boundary conditions — (r', to) = — a '„ (r') + i wo p (r', to) (35a)

dn g

p(H,z,to) = 0 (35b)

Solution of the above BVP can be obtained by the finite element method using a two-dimensional mesh (Fig. 4d)). The finite element discretization is the matrix equation (4)

H' + iaq W - —2 G' w

p(to) = - D ' (36)

in which H', B', and G! are the same symmetric matrices as in Eq. 25; p(to) = vector of unknown nodal pressures; and D' = vector of nodal accelerations computed from a'„(r') along the boundary a-b-c-d-a. Only DOF for nodes below the free surface are included in Eq. 36.

Alternatively to solving Eq. 36, p(to) can be determined using an ei­genvector expansion employing the complex valued and frequency de­pendent eigenvalues \„ and eigenvectors I|J„ resulting from the associ­ated eigenproblem Eq. 25 and which are available if the first BVP is being solved concurrently. In this alternative, p (to) is approximately expressed in terms of the first N eigenvectors as

p(to) = - * A"1 >PT D' (37) i

in which <P = [il^, i|i2, . . . , i}>N] and A = an N x N diagonal matrix with

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< AHC) or is a nondecaying harmonic e“””‘ (when to > AHC). At exci-tation frequencies equal to the eigenfrequencies wf, = )\,,C, i<,, = 0 andthe amplitude of mp" is infinite. The part of the amplitude that approachesin nity is real below mi, and_ imaginary above. Thus, p(x,w) is real formp < oil, complex for to > co'1, and unbounded at frequencies equal to101,. The harmonic, nondecaying distribution with x in the amplitude ofan eigenvector I11, for w > wt, represents a radiation of energy in theinfinite, upstream direction of the uid domain. This uid radiationdamping is nonzero for to > w'1 and is responsible for the imaginary com-ponent of p(x,w_). It doesnot, however, preventthe infinite resonancesat frequencies co; above oi’, because of the orthogonality of the eigenvec-tors; i.e., a resonating eigenvector III, is unaffected by the uid radiationdamping associated with the lower eigenvectors.

For q > 0, the complex eigenvector ll!" has an x distribution ofe""""e"”"", an exponentially decaying harmonic. Thus, all energy iseventually radiated by the foundation. Since this foundation radiationdamping occurs for all frequencies greater than zero, p(x, to) is boundedat all frequencies and complex valued for to > 0.

Second BVP.—The BVP of Eq. 7 with the boundary conditions of Eq.20 is two-dimensional in the y— and z-coordinates. Omitting the x vari-ations from these equations results in the two-dimensional Helmholtzequation for (y,z,w)62? p 62,5 (02 _ay2+aZ2+C2p 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (34)

8 .and the boundary conditions £ (r’, to) = —% a§,(r') + i wq p(r',w) (3511)

f1(f‘I,Z,u))= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (35b)Solution of the above BVP can be obtained by the finite element

method using a two-dimensional mesh (Fig. 4d)). The finite elementdiscretization is the matrix equation (4)

i - i “)2 i - w IH+1<.iqB-as p(co)=-§D . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(s6)

in which H‘, Bi, and G‘ are the same symmetric matrices as in Eq. 25;p(w) = vector of unknown nodal pressures; and D‘ = vector of nodalaccelerations computed from aj,(r’) along the boundary a-b-c-d-a. OnlyDOF for nodes below the free surface are included in Eq. 36. I

Alternatively to solving Eq. 36, p(w) can be determined using an ei-genvector expansion employing the complex valued and frequency de-pendent eigenvalues An and eigenvectors llln resulting from the associ-ated eigenproblem Eq. 25 and which are available if the first BVP is beingsolved concurrently. In this alternative, p(m) is approximately expressedin terms of the first N eigenvectors as

p(t.>) = g qr A“ IIIT 0* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (37)in which ‘I! = [$1, \|;2, ..., lI.IN] and A = an ‘N X N diagonal matrix with

159

nth diagonal term = X2, - co2/C2. Note that A = the square of K from Eq. 32.

The frequency variation of p(w) is similar to that of the finite fluid domain examined earlier. The eigenfrequencies coj, = X„C are the same as in the first BVP.

INFINITE FLUID DOMAINS OF IRREGULAR GEOMETRY

A solution scheme for the BVP of Eqs. 7, 8, and 9 is presented for the fluid domain of Fig. 5(a), where a finite region of irregular shape is con­nected to a region extending to infinity in the x direction with uniform y-z cross section. The plane of connection is the y-z cross section e-f-g-h-e. Normal accelerations of the dam-fluid interface a-b-c-d-a and reser­voir bottom and sides are an(s,r) and an(s',r'), respectively. Beyond e-f-g-h-e, an(s',r') = an{r'), unvarying in the x direction. In this region, s' is parallel to the x-axis, and r' follows the boundary around the y-z cross section.

The fluid domain is discretized, as shown in Fig. 5(b) and (c). The finite region is divided into three-dimensional finite elements as done earlier for the finite fluid domain. Within the infinite region, the channel discretization of the previous section is employed, matching the adjacent

(b) FINITE ELEMENT DISCRETIZATION OF IRREGULAR REGION

FiG. 5.—Infinite Fluid Domain of Irregular Geometry

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nth diagonal term == A3, — oz/C2. Note that A = the square of K fromEq. 32.

The frequency variation of p(w) is similar to that of the finite uiddomain examined earlier.'The eigenfrequencies mi, = )\,,C are the sameas in the first BVP. I *

INFINITE FLUID DOMAINS OF IRREGULAR GEOMETRY

A solution scheme for the BVP of Eqs. 7, 8, and 9 is presented for theuid domain of Fig. 5(a), where a finite region of irregular shape is con-

nected to a region extending to infinity in the x direction with uniformy-z cross section. The plane of connection is the y-z cross section e-f-g-h-e. Normal accelerations of the dam- uid interface a-b—c-d—a and reser-voir bottom and sides are an (s, r) and a,,(s’,r’), respectively. Beyond e-f-g—h—e, an (s ’,r’) = a,,(r’), unvarying in the x direction. In this region, s’is parallel to the x-axis, and r’ follows the boundary around the y-z crosssection. ,,

The uid domain is discretized, as shown in Fig. 5(b) and (c). Thefinite region is divided into three-dimensional finite elements as doneearlier for the finite uid domain. Within the infinite region, the channeldiscretization of the previous section is employed, matching the adjacent

ccGn(S,I’) on(SI1rI) : GA('1)

D -“<4 un(s',r')Io) IDEALIZED FLUID c I

DOMAIN

iat -ii sr-is,d~‘-"°3§a:§15*‘ ... _‘(‘riteIttip"\‘ith3 \*=a./

AI ‘I5f§§§§i§?i§§P§?§§§%*b"§§5§§ -

(c) CHANNEL OISCRETIZATIONOF INFINITE REGION

III) FINITE ELEMENT DISCRETIZATIDNOF IRREGULAR REGION

FIG. 5.--Infinite Fluid Domain of Irregular Geometry

160

three-dimensional mesh along e-f-g-h-e. The separated regions of Figs. 5(b) and (c), analogous to free bodies of solid continua, require that the normal accelerations along the plane of separation be preserved. The finite element matrix Eq. 15 is written for the finite region of Fig. 5(b) including only DOF for nodes below the free surface and is partitioned as follows:

[Hu _H21

(0 2

C2

H12

H22.

G n

_G 2 i

+ JCiM/

Gj2

G 2 2 .

B„ -B21

B12

B 2 2 .

\ rpi(«)i z(»)

(38)

in which nodes along e-f-g-h-e are identified by subscript 2 and remain­ing nodes by subscript 1. In Eq. 38, D! and D2 are acceleration vectors of group 1 and group 2 nodes computed from the accelerations of the exterior boundaries of the finite region; i.e., the dam-fluid interface a-b-c-d-a and the reservoir bottom and sides b-f-g-c-b, a-e-f-b-a, and d-h-g-c-d. D J M is an acceleration vector of group 2 nodes associated with the unknown x direction acceleration of the plane e-f-g-h-e. Eq. 38 without D%(<x>) is just a partitioned form of Eq. 15 written for a zero acceleration condition normal to e-f-g-h-e. However^ as part of the infinite fluid do­main of Fig. 5(a), e-f-g-h-e is an interface between two subregions and undergoes, as yet unknown, accelerations which contribute to the vector D2(w) in Eq. 38.

Consideration of the infinite region of Fig. 5(c) leads to an expression for D^w). The vector p2(w) of nodal pressures along e-f-g-h-e arises from two sources: the unknown acceleration D^w) and the acceleration a'n(r') along the floor and sides. Pressures at e-f-g-h-e due to D^w) and a'„(r') are given by Eqs. 33 and 37, respectively. Using superposition, p2 (a)) can be expressed as

10 p 2 ( a ) ) = - ^ K - 1 { K ~ 1 * r D , ' + * rD5(w)} (39)

If p2(w) is expressed by an eigenvector expansion using the first N eigenvectors i|»„

Then, from Eq. 39

(40)

w %(o)) = - K " 1 {K-1 ^ T D ' + iTD^co)} (41)

Multiplication of Eq. 41 by K yields the expression for D^co):

W TV

--*TDJ(o)) = K%(co) - - K - ' ^ D ' 8 8

(42)

Substitution of Eqs. 40 and 42 into Eq. 38 with a premultiplication of the second submatrix equation by *PT yields

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three-dimensional mesh along e-f-g-Ii-e. The separated regions of Figs.5(b) and (c), analogous to free bodies of solid continua, require that thenormal accelerations along the plane of separation be preserved. Thefinite element matrix Eq. 15 is written for the finite region of Fig. 5(b)including only DOF for nodes below the free surface and is partitionedas follows:

([1111 H12] + [B11 B12]

H2lIH22 B21IB22_

at c; o p (w) _ Q D ' A-E; - 8 {————D2_ 15%)} . . . . . . . . . . . . . . . . . . .. (as). . I

in which nodes along e-f-g—h-e are identified by subscript 2 and remain-ing nodes by subscript 1. In Eq. 38, D1 and D2 are acceleration vectorsof group 1 and group 2 nodes computed from the accelerations of theexterior boundaries of the finite region; i.e., the dam— uid interface a-b-c-d-a and the reservoir bottom and sides b—f-g-c-b, a—e-f—b—a, and d—h-g-c-d.D§(w) is an acceleration vector of group 2 nodes associated with theunknown x direction acceleration of the plane e—f-g—h-e. Eq. 38 withoutD-§(w) is just a partitioned form of Eq. 15 written for a zero accelerationcondition normal to e-f-g—h—e. However, as part of the infinite uid do-main of Fig. 5(a), e—f-g-I1-e is an interface between two subregions andundergoes, as yet unknown, accelerations which contribute to the vectorD§(w) in Eq. 38. I I

Consideration of the infinite region of Fig. 5(c) leads to an expressionfor D’§(w). The vector pl (co) of nodal pressures along e—f-g-h—e arises fromtwo sources: the unknown acceleration D§(w) and the accelerationa1}(r’) along the floor and sides. Pressures at e-f-g-h-e due to D§(w) anda§,(r’) are given by Eqs. 33 and 37, respectively. Using superposition,p2(w) can be expressed as . _

p,(<.>) =§\IIK‘1{K”1\IfTD"+ iirTD§(<.>)} . . . . . . . . . . . . . . . . . . . .. (39)

I If 132(0)) is expressed by an eigenvector, expansion using the first Neigenvectors III" I I I

p2(w) = \IIf|2(w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (40)

Then, from Eq. 39

112(0)) = -';i<"1{I<"‘ ii oi + iIrTo;(a)}- . . .1 . . . . . . . . . . . . . . . . . . . . . .. (41)

Multiplication of Eq. 41 by K yields the expression for D’§(co):

%’~IrTo;(w) =- K 2(o) —%K‘1\I'TDi . . . . . . . . . . . . . . . . . . . . . . . . . .. (42)

Substitution of Eqs. 40 and 42 into Eq. 38 with a premultiplication of thesecond submatrix equation by \I'T yields

161

qrT

<o2 " H u + mq B u - — G n

<o2

H21 + ibiq B21 - — G21 > r

a>2 " H12 + iwq B12 - — Gj2

">2 " H ^ + imcj B ^ ~ G^

9

9 + K

g l ^ D j + K ^ ^ D '

The pressure vector p^w) can be obtained by solving Eq. 43. For g = 0, several features of the frequency variation of p2(«) are ev­

ident. When the frequency is below the first eigenfrequency <a\ of the infinite region, no imaginary terms are present in Eq. 43, so Pi(co) is real valued. Above <a\, p^w) is complex valued due to fluid radiation damp­ing. Also, when a'n(r') is nonzero, Pi(w) becomes unbounded at each (x>'„ because of the infinite value attained by the nth diagonal term of the matrix K_1 on the right side of Eq. 43. However, none of the frequencies (o'n are eigenfrequencies of the complete fluid domain. These eigenfre-quencies must satisfy the eigenvalue problem associated with Eq. 43— this equation with q = 0 and a zero right side—and be real valued. Such frequencies, if they occur, will be less than Wj because, above wl7 the eigenproblem is complex valued (complex matrix K). At an excitation frequency equal to an eigenfrequency, p^co) is unbounded if q = 0. For q > 0, Pi(w) is bounded at all frequencies and complex valued for w > 0.

COMPUTATION OF HYDRODYNAMIC FORCE VECTORS

For the fluid domains of Figs. 3-5, computation of the hydrodynamic force vectors Q^(w); € = x, y, z; and Q{(w) of Eq. 5 proceeds as follows:

1. The boundary acceleration functions of Eqs. 13 and 14 are con­verted into acceleration vectors for use in Eqs. 15 or 18 (finite fluid do­main), Eqs. 33 and 37 (infinite fluid domain of uniform cross section), or Eq. 43 (infinite fluid domain of irregular geometry). For the finite fluid domain, these vectors are denoted by D(

0 and D ;, and their computation is described in Ref. 4. Use is made of the boundary portion of the finite element mesh along the dam-fluid interface and reservoir bottom and sides in Fig. 3(b). Vectors {D1}*, I = x, and {D*};- of Eq. 33 and vectors {Djf, {B2Y0, ( = x, y, z, and {Djy, {D2};. of Eq. 43 are computed similarly. In the latter case, {D2}; = {0}. {D!}f, i = y,z in Eqs. 37 and 43 is computed with ai

n(r') = ee(r'), and {D'}*, € = x, and {D!};. of Eq. 43 are also zero Vectors.

2. Using the acceleration vectors of step 1, hydrodynamic pressure vectors for a fluid domain are obtained by solving Eqs. 15 or 18, Eqs. 33 and 37, or Eq. 43. Pressures along the dam-fluid interface are assem­bled into p %(<»), i = x,y, z, and p^(co).

3. As also described in Ref. 4, the hydrodynamic force Vectors are computed from the pressures along the dam-fluid interface obtained in step 2.

EXAMPLE ANALYSIS

To demonstrate the results that can be obtained by the analysis pro-

(43)

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p (02 I (D2H11 + HM] Bu — '6; G11 H12 + iuiq B12 — -6; G12 ‘II {p1(w)}

T . ‘"2 » T . '~°2 'II2((°)\I! H21+z<nqB2,—EG21 ‘If H2Z+zuiqB22-—E;G22 \F+K

w D1=— ——---—-—-—+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43

g I\l!T D, + K-1 \IITD'I ' ( )

The pressure vector p1(<») can be obtained by solving Eq. 43.For q = 0, several features of the frequency variation of p2(w)vare ev-

ident. When the frequency is below the first eigenfrequency oil of thein nite region, no imaginary terms are present in Eq. 43, so p1(w) is realvalued. Above <»‘1,_p1(w) is complex valued due to uid radiation damp-ing. Also, when a§,(r’) is nonzero, p1(w) becomes unbounded at eachml, because of the infinite value attained by the nth diagonal term of thematrix K '1 on the right side of Eq. 43. However, none of the frequenciesmi, are eigenfrequencies of the complete uid domain. These eigenfre-quencies must satisfy the eigenvalue problem associated with Eq. 43-this equation with q = 0 and a zero right side—and be real valued. Suchfrequencies, if they occur, will be less than ml because, above L01, theeigenproblem is complex valued (complex matrix K). At an excitationfrequency equal to an eigenfrequency, p1(w) is unbounded if q = 0. Forq > 0, 131(0)) is bounded at all frequencies and complex valued for (1) > 0.

COMPUTATION OF HYDRODYNAMIC Fonce Vecroas E .For the uid domains of Figs. 3—5, pomputation of the hydrodynamic

force vectors Q{f(w); 6 = x, y, z ; and Q]f»(m) of Eq. 5 proceeds as follows:

1. The boundary acceleration functions of Eqs. 13 and 14 are con-verted into acceleration vectors for use in Eqs. 15 or 18 (finite uid do-main), Eqs. 33 and 37 (infinite uid domain of uniform cross section),or Eq. 43 (infinite uid domain of irregular geometry). For the finite uiddomain, these vectors are denoted by Di and Dj, and their computationis described in Ref. 4. Use is made of the boundary portion of the finiteelement mesh along the dam- uid interface and reservoir bottom andsides in Fig. 3(b). Vectors {D "}§, 6 = x, and {D"},- of Eq. 33 and vectors{D1},‘§, {D2}§, 6 = x, y, z, and {D1},-, {D2}, of Eq. 43 are computed similarly.In the latter case, {D2}, = {O}. _{D'}§, 6 = y, z in Eqs. 37 and 43 is computedwith i1'n(r') = ee(r’), and {D‘}§, 6 = x, and {D‘}]- of Eq. 43 are also zerovectors.

2. Using the acceleration vectors of step 1, hydrodynamic pressurevectors for a uid domain are obtained by solving Eqs. 15 or 18, Eqs.33 and 37, or Eq. 43. Pressures along the dam- uid interface are assem-bled into p{f(w), 6 = x; y, z, and p§(w). I

3. As also described in Ref. 4, the hydrodynamic force vectors arecomputed from the pressures along the dam— uid interface obtained instep 2.

EXAMPLE ANALYSIS pTo demonstrate the results that can be obtained by the analysis pro-

162

TABLE 1.—Dimensions of Morrow Point Dam

y, in feel (1) 465 372 279 186 93 0

T„, in feet (2)

0.0 28.9 46.3 52.9 49.0 34.4

Td, in feet (3)

12. 6.4 0.8

-2.6 2.7

17.2

Ru, in feet (4)

375.0 352.8 324.9 296.5 266.7 234.8

Rd,in feet (5)

363.0 316.1 258.0 210.8 171.3 136.6

9, in degrees

(6)

56.20 47.85 39.50 33.00 26.50 13.25

cedures developed in this paper, the response of Morrow Point Dam to harmonic ground motion is presented. The dam-fluid system is assumed symmetric with dimensions averaged from the two halves (Fig. 6, Table 1). The fluid domain is of the infinite, irregular type; the reservoir cross section is assumed uniform upstream of the y-z plane e-f-g-h-e. The water depth H at the upstream face of the dam equals the dam height.

Because the dam-fluid system is symmetric, dam and hydrodynamic pressure responses to upstream-downstream, x, and vertical, y, ground motions are also symmetric, and those due to cross-stream, z, ground

FIG. 6.—Morrow Point Dam, Infinite Fluid Domain

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I ‘TABLE 1.--Dimensions of Morrow Point DamW y, in

feet(1)

feet(2) (3)

R,,, infeet(4)

R), Infeet(5)

6, indegrees

(6)46537227918693

0

0.028.946.352.949.034.4

Tu, in T,,, infeet

12.6.40.8

-2.62.7

17.2

375.0352.8324.9296.5266.7234.8

363.0316.1258.0210.8171.3136.6

56.2047.8539.5033.0026.5013.25

cedures developed in this paper, the response of Morrow Point Dam toharmonic ground motion is presented. The dam- uid system is assumedsymmetric with dimensions averaged from the two halves (Fig. 6, Table1). The fluid domain is of the infinite, irregular type; the reservoir crosssection is assumed uniform upstream of the 3/-z plane e-f—g—h—e. TheWater depth H at the upstream face of the dam equals the dam height.

Because the dam— uid system is symmetric, dam and hydrodynamicpressure responses to upstream-downstream, x, and vertical, y, groundmotions are also symmetric, and those due to cross-stream, z, ground

are y=465' .o a v *1-B 5 h

y=372' KT4 _

y=279'

y=|B6'

F93. . I

‘ CROSS-SECTIONAT 9=0° Y CANYONF0, b ’ I C CROSS-SECTIONS

l<—>I * 92. la

___E _____..________..__._________,__-..-

l' I

UNIFORM'*" RESERVOIR

UPSTREAMDAM FACE

7,.

““._!I!/I‘Y//rel

ZQIOO $50e I Ii

75‘ ~\

e215'

CANVONCONTOUR5

/ SUPERIMPOSED PLANS ATes‘ ELEVATION INTERVALS

FIG. 6.-Morrow Point Dam, Infinite Fluid Domain

163

(a) MORROW POINT DAM (b) FLUID DOMAIN

FIG. 7.—Finite Element Meshes

motion are antisymmetric. Only the symmetric eigenvectors of the dam and fluid domain need be employed for the analysis of response to x and y ground motions, and only the antisymmetric eigenvectors are re­quired for the analysis of response to z ground motion. Furthermore, only half the dam and fluid domain need be considered in the analysis if appropriate boundary conditions along the plane of symmetry are em­ployed. For x and y ground motions, the components of dam displace­ment and fluid acceleration normal to the plane of symmetry are zero. The x and y components of dam displacement and the hydrodynamic pressures at the plane of symmetry are zero for z ground motion.

The finite element mesh of the dam employs shell elements (7) with quadratic shape functions as shown in Fig. 7(a). Properties of the mass concrete of the dam are elastic modulus Ed = 3 x 106 psi (2.07 x 1010

Pa), unit weight wd = 150 pcf (2,401.5 kg/m3), and Poisson's ratio v = 0.17. The damping ratio for all modes of vibration of the dam £; = 5%. The fluid domain mesh employs quadratic, three-dimensional finite ele­ments (Fig. 7(b)), and the plane e-f-g-h-e is placed as close to the dam as possible to minimize the number of DOF in the mesh. At the dam-fluid interface, the finite element mesh for the dam coincides with the fluid domain mesh. The water has unit weight w = 62.4 pcf (999 kg/ m3) and compression wave velocity C = 4,720 fps (1,438.7 m/s). The reflection coefficient ar is chosen as 0.90.

Results presented in Fig. 8 are responses of the dam to harmonic ground accelerations = e'°" in the upstream-downstream, vertical, and cross-stream directions. The absolute value (or modulus) of r€(w) the radial component of acceleration at the dam crest relative to the ground acceleration, is plotted against the normalized excitation frequency pa­rameter co/coj (x and y ground motion) or <o/a>5 (z ground motion), where <0j and w5 are the fundamental frequencies of the dam in symmetric and antisymmetric vibrations. When presented in this form, the plotted re­sults apply to similarly shaped systems of any height. The response of the dam is presented for four conditions: dam without water; dam with

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O , INFINITE cm:M

\V-"75:’..w-/

_4'*-lid"?

IiiIIIII...I2.‘I .

\"I"J.‘\4.%giIAi

A ‘

1

.___‘

N‘Zr

(a) iioniwi POINT DAM FLUID wt-IAIN

FIG. 7.—Flnlte Element Meshes

motion are antisymmetric. Only the symmetric eigenvectors of the damand fluid domain need be employed for the analysis of response to xand y ground motions, and only the antisymmetric eigenvectors are re-quired for the analysis of response to z ground motion. Furthermore,only half the dam and uid domain need be considered in the analysisif appropriate boundary conditions along the plane of symmetry are em-ployed. For x and y ground motions, the components of dam displace-ment and uid acceleration normal to the plane of symmetry are zero.The x and y components of dam displacement and the hydrodynamicpressures at the plane of symmetry are zero for z ground motion.

The finite element mesh of the dam employs shell elements (7) -withquadratic shape functions as shown in Fig. 7(a). Propgrties of the mags)concrete of the dam are elastic modulus E = 3 >< 10 psi (2.07 >< 10Pa), unit weight wd = 150 pcf (2,401.5 kg/fina), and Poisson's ratio v =0.17. The damping ratio for all modes of vibration of the dam §]- = 5%.The uid domain mesh employs quadratic, three-dimensional finite ele-ments (Fig. 7(b)), and the plane e-f-g-h-e is placed as close to the damas possible to minimize the number of DOF in the mesh. At the dam-uid interface, the finite element mesh for the dam coincides with the

‘d d ' . ' ' = .Hi‘) ..‘Z1““.‘i§§.l§§§li.,§}l§.$2a§Zi§3iy“E‘l“§%‘Bt F15. (f§3Z.‘§°f...(/93.9 -if!re ection coefficient oi, is chosen as 0.90.

Results presented in Fig. 8 are responses of the dam to harmonicground accelerations = e’“" in the upstream-downstream, vertical, andcross-stream directions. The absolute value (or modulus) of i7"(w) theradial component of acceleration at the dam crest relative to the groundacceleration, is plotted against the normalized excitation frequency pa-rameter to/to‘ (x and y ground motion) or oi/co“ (z ground motion), whereoi and oi‘; are the fundamental frequencies oflthe dam in symmetric andantisymmetric vibrations. When presented in this form, the plotted re-sults apply to similarly shaped systems of any height. The response ofthe dam is presented for four conditions: dam without water; dam with

164

i WATER

: N O N E

- INCOMPRESSIBLE - COMPRESSIBLE - COMPRESSIBLE

NEGLECTED NEGLECTED INCLUDED

1 W~ "•- 30-

^ tt}/ttls.

UPSTREAM-DOWNSTREAM GROUND MOTION

VERTICAL GROUND MOTION

8 = 13.25°

CROSS-STREAM GROUND MOTION

FIG. 8.—Radial Accelerations of the Dam Crest Due to Harmonic Ground Accelerations

full reservoir, neglecting both water compressibility and fluid-founda­tion interaction; and dam with full reservoir considering water com­pressibility and including fluid-foundation interaction in one case but not in the other. It is apparent that dam-fluid interaction, water com­pressibility, and fluid-foundation interaction have a significant effect on the response of this arch dam to harmonic ground motion. These effects are examined further in Ref. 4, wherein a comprehensive set of response results is presented.

The curves of Fig. 8 were obtained with the computer program EADFS (3) using the CDC 7600 computer. With the reservoir empty, the solution of the eigenproblem of the dam (Eq. 2) accounts for most of the com­putational effort (Table 2). Including the fluid domain, but neglecting water compressibility and fluid-foundation interaction, requires little ad­ditional effort. With compressible water, nearly all the computational effort is spent in the solutions at each excitation frequency of Eq. 43 (requiring only a small amount of complex valued arithmetic if fluid-foundation interaction is neglected, i.e., q = 0) and the eigenproblem Eq. 25 (only if fluid-foundation interaction is included, i.e., q > 0). Also

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1o.- _ -rou DATION9 -°° WM Wm“ Iiti’é%.i¢ri’<‘>~eo— -~»—— none —

——--- INODMPRESSIBLE NEGLECTED50, -— OOMPRESSIBLE NEGLECTED

' - ———- COMPRESSIBLE INCLUDED3 40- 1 I;-_|,, . - .'=;w- * " "

w- -'-I I I‘ .'-..‘_ ’:" :")\ I"/._*\io- .‘ Y," “ '~».___; _, w I‘-‘\__.

‘. _ . » ~- s°o H I. 2. 3. 4. w/w‘

UFSTREAM-DOWNSTREAM GROUND MOTION .

1o-6.0. Uso.-so.-

5= 4o.- '3>- .»"_: 1°‘ :1

20.— I I-Y \‘\ I

I ' H 7.-' J 7-IT, I:..

°,o I III. 2. 3. 4. °’/“IVERTICAL GROUND MOTION

7°‘ 9-13.25" pso.- I

50."

3 40.»J’ 1in. 30‘-

20.» \|O.— I ‘~.:;._

LY _A'l-;-< "'--.t_-- <'_". G

0 0 I 2. 3. 4. W/w‘I CROSS- STREAM GROUND MOTION

FIG. 8.—FiadiaI Accelerations of the Dam Crest Due to Harmonic GroundAccelerations

full reservoir, neglecting both water compressibility and uid-founda-tion interaction; and dam with full reservoir considering water com-pressibility and including uid-foundation interaction in one case butnot in the other. It is apparent that dam- uid interaction, water com-pressibility, and uid-foundation interaction have a significant effect onthe response of this arch dam to harmonic ground motion. These effectsare examined further in Ref. 4, wherein a comprehensive set of responseresults is presented.

The curves of Fig. 8 were obtained with the computer program EADFS(3) using the CDC 7600 computer. With the reservoir empty, the solutionof the eigenproblem of the dam (Eq. 2) accounts for most of the com-putational effort (Table 2). Including the uid domain, but neglectingwater compressibility and uid-foundation interaction, requires little ad-ditional effort. With compressible water, nearly all the computationaleffort is spent in the solutions at each excitation frequency of Eq. 43(requiring only a small amount of complex valued arithmetic if uid-foundation interaction is neglected, i.e., q =‘0) and the eigenproblemEq. 25 (only if uid-foundation interaction is included, i.e., q > 0). Also

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TABLE 2.—Solution Times (Computation Time) on the CDC 7600 for Computing Complex Frequency Response Functions

Case (D

Symmetric

(( = x, y)

Antisymmetric

(e = z)

Dam

DOF

(2)

220

210

Number of modes

(3)

12

12

Fluid Domain

DOF (4)

105 (group 1) 52 (group 2)

90 (group 1) 42 (group 2)

Number of «|i„

(5)

31

27

Water (6)

none

compress­ible

none

compress­ible

Fluid-foundation interaction

(7)

— neglected

included

• — neglected

included

Total compu­tation time

seconds (8)

14

102

772

13

71

496

included in Table 2 are the Central Processor times assuming 400 fre­quencies are needed to define the frequency response curves with q = 0 and 200 frequencies for the smoother curves with q > 0. Total solution times significantly increase when water compressibility is included and again when fluid-foundation interaction is included.

CONCLUSIONS

The substructure method in the frequency domain has been extended for response analysis of realistic arch dam-fluid systems subjected to upstream-downstream, cross-stream, and vertical components of ground motion. Responses to arbitrary ground motion can be obtained by Four­ier synthesis applied to the complex frequency response functions de­termined by the analysis procedures presented in this paper.

The arch dam and impounded water are treated as two substructures of the total system, and the dam response is represented as a linear combination of the first few natural modes of vibration of the dam alone. The arch dam is modeled as a system of finite elements, and the hy-drodynamic terms in the equations of motion for the dam are deter­mined from analysis of finite elements models of the fluid domain. This finite element procedure applies to either finite fluid domains or infinite domains consisting of an irregular region of finite size connected to a region of uniform cross section extending to infinity in the upstream direction. For such an infinite uniform region, a finite element discre­tization within the cross section combined with a continuum represen­tation in the infinite direction provides for the proper transmission of pressure waves. Water compressibility and fluid-foundation interaction are considered in the fluid domain model.

Utilizing the finite element procedures presented in this paper for analysis of fluid domains and transforming the displacements of the dam to modal coordinates, the substructure method provides an effec­tive approach to analysis of arch dams. The solution times to consider water compressibility and fluid-foundation interaction are significant but not exorbitant.

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TABLE 2.—S0Complex Frequency Response Functions

lution Times (Computation Time) on the CDC 7600 for Computing

Case(1)

NumberDOF of modes(2) (3)

DOF(4)

Numberof ‘Ill!

(5)Water

(6)

foundationinteraction

(7)

Totalcompu-Dam Fluid Domain F|uid_ ta on

timeseconds

(8)Symmetric

(6 = x, y)220 12 105 (group 1)

52 (group 2)31 none

compible

neglectedincluded

14

l'€SS- 102

772210 12 90 (group 1) 27 I'lO I16 13Antisymmetric

(6 = z) 42 (group 2) compreSS_ible

neglected 71included 496

included in Table 2 are the Central Processor times assuming 400 fre-quencies are needed to define the frequency response curves with q =0 and 200 frequencies for the smoother curves with q > O. Total solutiontimes significantly increase when water compressibility is included andagain when uid-foundation interaction is included.

CONCLUSIONS

The substructure method in the frequency domain has been extendedfor response analysis of realistic arch dam- uid systems subjected toupstream-downstream, cross-stream, and vertical components of groundmotion. Responses to arbitrary ground motion can be obtained by Four-ier synthesis applied to the complex frequency response functions de-termined by the analysis procedures presented in this paper.

The arch dam and impounded water are treated as two substructuresof the total system, and the dam response is represented as a linearcombination of the first few natural modes of vibration of the dam alone.The arch dam is modeled as a system of finite elements, and the‘hy-drodynamic terms in the equations of motion for the dam are deter-mined from analysis of finite elements models of the uid domain. Thisnite element procedure applies to either finite uid domains or infinite

domains consisting of an irregular region of finite size connected to aregion of uniform cross section extending to infinity in the upstreamdirection. For such an infinite uniform region, a finite element discre-tization within the cross section combined with a continuum represen-tation in the infinite direction provides for the proper transmission ofpressure waves. Water compressibility and uid-foundation interactionare considered in the uid domain model.

Utilizing the finite element procedures presented in this paper foranalysis of uid domains and transforming the displacements of thedam to modal coordinates, the substructure method provides an effec-tive approach to analysis of arch dams. The solution times to considerwater compressibility and uid-foundation interaction are significant butnot exorbitant.

‘I66

ACKNOWLEDGMENT

This research investigation was supported by Grants ATA74-20554 and ENV76-80073 from the National Science Foundation to the Univer­sity of California, Berkeley. The writers are grateful for this support .

APPENDIX.—REFERENCES

1. Chopra, A. K., and Chakrabarti, P., "Earthquake Analysis of Concrete Gravity Dams Including Dam-Water Foundation Rock Interaction," Earthquake Engi­neering and Structural Dynamics, Vol. 9, No. 4, 1981, pp. 363-383.

2. Clough, R. W., Raphael, J. M., and Majtahedi, S., "ADAP—A Computer Pro­gram for Static and Dynamic Analysis of Arch Dams," Report No. EERC 73-14, University of California, Berkeley, Calif.

3. Hall, J. F., "A Computer Program for Earthquake Analysis of Dam-Fluid Sys­tems," University of California, Berkeley (to be published).

4. Hall, J. F., and Chopra, A. K., "Dynamic Response of Embankment, Con­crete-Gravity and Arch Dams Including Hydrodynamic Interaction," Report No. UCB/EERC-80/39, University of California, Berkeley, Calif., Oct., 1980, 220 pp.

5. Hall, J. F., and Chopra, A. K., "Two-Dimensional Dynamic Analysis of Em­bankment and Concrete Gravity Dams Including Hydrodynamic Effects," Earthquake Engineering and Structural Dynamics, Vol. 10, No. 2, 1982.

6. Hatano, T. and Nakagawa, T., "Seismic Analysis of Arch Dams—Coupled Vibrations of Dam Body and Reservoir Water," Technical Report, Central Re­search Institute of Electric Power Industry, Tokyo, Japan, Nov., 1972.

7. Pawsey, S. F., "The Analysis of Moderately Thick to Thin Shells by the Finite Element Method," Report No. US SESM 70-12, Structural Engineering Labo­ratory, University of California, Berkeley, Calif., Aug., 1970.

8. Porter, C. S., and Chopra, A. K., "Dynamic Analysis of Simple Arch Dams Including Hydrodynamic Interaction," Earthquake Engineering and Structural Dynamics, Vol. 9, No. 6, 1981, pp. 573-597.

9. Priscu, R., et al., "New Aspects in the Earthquake Analysis of Arch Dams," Criteria and Assumptions for Numerical Analysis of Dams, D. N. Naylor, et al., ed.; Swansea, United Kingdom, Sept., 1975, pp. 710-722.

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AcmoweeoemsurThis research investigation was supported by Grants ATA74-20554

and ENV76-80073 from the National Science Foundation to the Univer-sity of California, Berkeley. The writers are grateful for this support.

APPENDIX.-—REFERENCES1. Chopra, A. K., and Chakrabarti, P., "Earthquake Analysis of Concrete Gravity

Dams Including Dam-Water Foundation Rock Interaction,” Earthquake Engi-neering and Structural Dynamics, Vol. 9, No. 4, 1981, ppl 363-383.

2. Clough, R. W., Raphael, I. M., and Majtahedi, S., ”ADAP—A Computer Pro-gram for Static and Dynamic Analysis of Arch Dams," Report N0. EERC 73-14, University of California, Berkeley, Calif.

3. Hall, I. F., "A Computer Program for Earthquake Analysis of Dam-Fluid Sys-tems,” University of California, Berkeley (to be published).

4. Hall, I. F., and Chopra, A; K., “Dynamic Response of Embankment, Con-crete-Gravity and Arch Dams Including Hydrodynamic Interaction,” ReportN0. LICB/EERC-80/39, University of California, Berkeley, Calif., Oct., 1980,220 pp.

5. Hall, I. F., and Chopra, A. K., “Two-Dimensional Dynamic Analysis of Em-bankment and Concrete Gravity Dams Including Hydrodynamic Effects,”Earthquake Engineering and Structural Dynamics, Vol. 10, No. 2, 1982.

6. Hatano, T. and Nakagawa, T., "Seismic Analysis of Arch Dams-—CoupledVibrations of Dam Body and Reservoir Water,” Technical Report, Central Re-search Institute of Electric Power Industry, Tokyo, Iapan, Nov., 1972.

7. Pawsey, S. F., ”The Analysis of Moderately Thick to Thin Shells by the FiniteElement Method,” Report No. US SESM 70-12, Structural Engineering Labo-ratory, University of California, Berkeley, Calif., Aug., 1970.

8. Porter, C. 5., and Chopra, A. K., ”Dynamic Analysis of Simple Arch DamsIncluding Hydrodynamic Interaction,” Earthquake Engineering and StructuralDynamics, Vol. 9, No. 6, 1981, pp. 573-597. I . .

9. Priscu, R., et al., "New Aspects in the Earthquake Analysis of Arch Dams,"Criteria and Assumptions for Numerical Analysis of Dams, D. N. Naylor, et al.,ed.; Swansea, United Kingdom, Sept., 1975, pp. 710-722.

167