a method for coupled arch dam-foundation-reservoir seismic behaviour analysis

* Correspondence to: R. J. Ca( mara, Departamento de Barragens, LaboratoH rio Nacional de Engenharia Civil, Av. DoBrasil 101, 1799 Lisboa Codex, Portugal.

Received 4 June 1998Revised 29 June 1999

Copyright ( 2000 John Wiley & Sons, Ltd. Accepted 14 September 1999

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2000; 29: 441}460

A method for coupled arch dam-foundation-reservoir seismicbehaviour analysis

R. J. Ca( mara*

LaboratoH rio Nacional de Engenharia Civil, Portugal

SUMMARY

This paper presents a method for coupled arch dam}foundation}reservoir seismic behaviour analysis. Thedam is discretized by "nite elements (FE) and the foundation and reservoir are discretized by boundaryelements (BE). The opening of contraction joints and the spatial variability of the seismic action is taken intoaccount. The study of Pacoima dam by this method is also presented. The computed results show that nocracks were to be expected due to the vibrations induced during the Feb. 9, 1971 San Fernando earthquake.Copyright ( 2000 John Wiley & Sons, Ltd.

KEY WORDS: seismic response; arch dams; non-linear; contraction joints; interaction; hybrid method

1. INTRODUCTION

The linear elastic behaviour of arch dams during low-intensity earthquakes is a reasonableapproximation. The in"nite dimension of the foundation originates radiation damping. In thepresent work, the dam's foundation is analysed by a boundary element method in the frequencydomain. The canyon is assumed 3D, with an uniform cross-section. In the same way, the reservoiris assumed in"nite, with an uniform cross-section. It is also analysed by a boundary elementmethod, in the frequency domain. The dam is assumed to be a thin shell and is discretized bytriangular elements. So, both the dam and reservoir only need the dam's middle surfacediscretization, and the foundation is de"ned by the insertion. The earthquake is characterised bya power density spectrum of accelerations on a half-space point. The spatial variability is takeninto account by a boundary element method analysis of a SH wave model with an arbitraryspatial incidence on the canyon.

The behaviour of arch dams during high-intensity earthquakes may be idealized as non-linear.In the present model, the non-linear behaviour is assumed to be due to the no-tension contractionjoints. The earthquake is assumed being de"ned by an acceleration time history. A stress transfer

technique is used, where the unbalanced forces in the time domain are transformed to thefrequency domain, allowing a frequency analysis of the system, going afterwards to the timedomain and then performing a new evaluation of the unbalanced forces. The scheme is repeateduntil convergence is attained. The spatial variability of the earthquake is also considered in thefrequency domain by the adopted wave model.

In order to take into account the opening of contraction joints, Dowling and Hall [1] useda step by step time scheme, but assumed uniform earthquakes. The in#uence of the spatialvariability of the earthquakes was also taken into account by Mojtahedi and Fenves [2], butassumed rigid water and massless foundation. The present method takes into account all of theseaspects.

2. FOUNDATION ANALYSIS BY THE BOUNDARY ELEMENT METHOD

The canyon cross-section was assumed uniform. Therefore, according to Zhang and Chopra [3],it is possible to use the transform G

ijtowards the co-ordinate x

1(along the canyon axis), of the

displacement's fundamental solution G*ij

(component i), due to a point load (along xj) applied on

the full space,

Gij(x

0, x

04, k)"P

=

~=

G*ij(x, x

04) exp(!jkx

1) dx

1(1)

for a receiver at x"(x1, x

2, x

3) and a source at x

04"(0, x

02, x

03), where k is the wave number

and x04"(x

2, x

3).

The transform nF of the stresses nF* due to G*, at a face of unitary normal n,

nFij(x

0, x

04, k)"P

=

~=

nF*ij(x, x

04) exp(!jkx

1) dx

1(2)

may be directly computed from Gij(x

0, x

04, k) for the elastic constants E and l.

Let be M;M (x0, k)N and MtN (x

0, k)N the Fourier transforms along x

1of the displacements and

stresses at the boundary surface M; (x)N e Mt(x)N. Then

M;(x)N"P=

~=

M;M (x0, k)N exp(!jkx

1) dk (3)

and

Mt(x)N"P=

~=

MtN (x0, k)N exp(!jkx

1) dk (4)

The transformed representation theorem is obtained replacing Equations (3) and (4) in the 3Drepresentation principle, changing the order of integration and taking into account Equations (1)

442 R. J. CA) MARA

Copyright ( 2000 John Wiley & Sons, Ltd Earthquake Engng. Struct. Dyn. 2000; 29:441}460

Figure 1. Foundation model.

and (2),

[C(x0s

)] M;M (x04

, k)N#P!1#W!1

)

[ nF (x0, x

04, k)]T (;M (x

0, k)N d!

"P!1#W!1

)

[G(x0, x

04, k)]TMtN (x

0, k)N d! (5)

where !1#e!1

)are, respectively, Figure 1 (according to Ref. [3]), the boundary lines of the canyon

and half-space cross-section and [C(x04

)] is the matrix that corrects the values of displacements ata boundary point x

04, when the line has di!erent right and left derivatives. For equal derivatives

the matrix is diagonal with diagonal values equal to 0.5.Based upon the representation theorem, the following discretized equations are obtained:

[A(k)] M;M N"[B(k)] M¹N

COUPLED ARCH DAM-FOUNDATION-RESERVOIR ANALYSIS 443


where M;M N are the transformed nodal displacements at !1#X!1

), M¹N are the transformed nodal

stresses at the interface between the dam and foundation !iand

[A (k)]"A[C (x

1)]

)

)

)

[C(xMn

)] B#A[AI (x

1, k)]

)

)

)

[AI (xMn

, k)] B[B(k)]"A

[BI (x1, k)]

)

)

)

[BI (xMn

, k)] B[AI (x

04, k)]"

1

2

M+j/1

lj P

1

~1

[ nF (q, xN04

, k)]Ti[N (q)] dq

[BI (x04

, k)]"1

2

M2+

j/M1

lj P

1

~1

[G(q, xN04

, k)]Tj[HM (q, k)]

jdq

Mn

is the number of nodal points and M the number of elements at !1#X!1

). M

12M

2are the

elements at !1#. l

jis the length of the element j. The relationship between the transformed "elds of

displacements M;M (x0, k)N

jand stresses MtN (x

0, k)N

jat element j, with the transformed nodal

displacements M;M Njand nodal stresses M¹N

j(nodal points of surface elements at !

i, intercepted by

the line x"x0; For the thin shell elements quadrilaterals with two sides parallel to x

1axis are

used) are assumed:

MtN (x0, k)N

j"[HM (x

0, k)]

jM¹N

j

M;M (x0, k)N

j"[N]M;M N

j

solving for transformed displacements:

M;M N"[A(k)]~1 [B(k)]M¹N

Numerical evaluation of

M;(x)N"P=

~=

M;M (x0, k)Ne~+kx dk, x3!

)X!

#

444 R. J. CA) MARA


is

M;(x)N"+q

wqM;M (x

0, k

q)N e~+kqx

where wq

are weighting coe$cients. The displacement vector at nodal point j of !i, with

co-ordinates xj"(x

j, y

j, z

j) is expressed by

MrNjN"+

q

wqM;M (x

0j, k

q)Ne~+kqxj

where

M;M (x0j

, kq)N"M;M

jN

In the present case the interpolation between two points at !1#

is not necessary, because thepoints at !

iare directed along the x

1-axis, Figure 1. Taking into account that the nodal

displacements are combinations of the nodal stresses M¹ N, we have

M;M (x0j

, kq)N"[E(x

0j, k

q)] M¹N

Therefore, the displacements in the frequency domain, MrNjN, at nodal point j are

MrNjN"[ fI

j] M¹N

with

[ fIj]"+

q

wq[E (x

0j, k

q)] e~+kqxj

assembling for the Minodal points at !

i

MrN N"[ f ]M¹N

being [ f ] a square matrix of rank 3Mi

assuming the stresses obtained by linear interpolation

Mt(x)N"[h(x)] M¹N

related to nodal forces MRM N by

MRM N"[/] M¹N



with

[/]"P!i

[h(x)]T [h (x)] d!

then

MRM (w)N"[/][ f (w)]~1 MrN (w)N

and the 3Mi]3M

iimpedance matrix is obtained:

[SK (w)]"[/][ f (w)]~1

or by symmetrization we obtain the impedance matrix for 3D displacements (&g' refers to ground,and &b' to nodes at structure}rock interface):

[S'""

(w)]3D"12

([SK (w)]#[SK (w)]T)

The displacements at each pair of points (one upstream;U(L)m

, one downstream;D(L)m

, ¸ beingthe distance between them) are related to the medium displacements ;S

mand to rotations hS

m(S 2 from shell) by

;U(L)m

";Sm#(!1)m

¸

2hSn

;D(L)m

";Sm!(!1)m

¸

2hSn

(m"2, 3; n"3, 2)

;U(L)1

";D(L)1

";S1

where the downstream}upstream direction is directed along the x1-axis, and the downwards

direction along the x3-axis. The discretization of !

i, Figure 1, is done by symmetrical points to the

shell insertion, and directed along the x1-axis. In matrix notation

MrN N"[A]MrNS

so that

[S'""

(u)]"[A]T[S'""

(u)]3D[A]

According to Ref. [3] the solution of the scattering wave problems is based on Equation (5)replacing MtN (x

0)N by !Mtn

&&(x

0)N, which represents the "ctitious stresses along the cross-section of

the canyon, due to the &free-"eld' (indice &! ' ) wave stresses at the half-space. In this expression M;M Nare the scattering displacements.

446 R. J. CA) MARA


3. RESERVOIR ANALYSIS BY THE BOUNDARY ELEMENT METHOD

The reservoir cross-section was assumed uniform. The hydrodynamic pressure of reservoir'sfundamental solution satis"es rigid bottom boundary conditions and zero hydrodynamic pres-sure at free surface. The fundamental solution P*, due to a source d (n) (where d is the distributionof Dirac) in the reservoir, may be computed from the modes of vibration t

n(x

1, x

2) (with natural

frequencies "n; C

0is the sound wave propagation velocity) of a cross-section where x

3is the

reservoir axis. According to Ca( mara [4]:

P* (x, n)"+n

tn(x

1, x

2)t

n(m

1, m

2)

2sn

exp(!D m3!x

3Ds

n) (6)

where

s2n"

"2n!u2

C20

with the source at n and the receiver at x. u is the frequency of the harmonic vibrations. Thevibration modes problem are expressed by

2+

m/1

L2tn(x

1, x

2)

Lx2m

"!

"2n

C20

tn(x

1, x

2) (7)

satisfying the boundary conditions

tn(x

1, x

2)"0 at the free surface

and

2+

m/1

Ltn(x

1, x

2)

Lxm

jm(x

1, x

2)"0 at the bottom

jm(x

1, x

2) is the m component of the unitary normal to the boundary line of the cross-section at

(x1, x

2). The orthogonality conditions are

PA

tm(x

1, x

2)t

n(x

1, x

2) dx

1dx

2"d

mn

Let P* and P be, respectively, the fundamental solution and the solution of the pressure waveequation, for harmonic vibrations, satisfying at the free surface P"0 and where the normalpressure gradient is "xed at the structure}#uid interface ). Taking into account the conventions



Figure 2. Reservoir's conventions.

shown in Figure 2, we obtain

PV~Vm

A3+

m/1

L2P*

Lx2m

#k2P*B P dv#PVmA

3+

m/1

L2P*

Lx2m

#k2P*B P dv

"!P)P*

LP

Ljds#P)

LP*

LjP ds

The summation in <!<m is equal to zero because

3+

m/1

L2P*

Lx2m

#k2P*"0 if x3Om

3

The summation in <m , when the limit of <m , is equal to zero becomes

PVmA

3+

m/1

L2P*

Lx2m

#k2P*B P dv

"!+nPA1

P (x1, x

2, m

3) t

n(x

1, x

2)t

n(m

1, m

2) dx

1dx

2

On the other side,

+nPA1

P (x1, x

2, m

3)t

n(x

1, x

2)t

n(m

1, m

2) dx

1dx

2

448 R. J. CA) MARA


!+nPA1

P (x1, x

2, x

3(x

1, x

2))

tn(x

1, x

2)t

n(m

1, m

2)

2e~(m3~x3(x1,x2))sn dx

1dx

2

#+nPA2

P (x1, x

2, x

3(x

1, x

2))

tn(x

1, x

2)t

n(m

1, m

2)

2e~(x3(x1,x2)~m3)sn dx

1dx

2

"+nPA1`A2

[g (x1, x

2)]t

n(x

1, x

2)t

n(m

1, m

2) dx

1dx

2"g(m

1, m

2)

where

g(x1, x

2)"P(x

1, x

2, m

3)!1

2P(x

1, x

2, x

3(x

1, x

2)) e~(m3~x3(x1,x2))sn if x3A

1

g(x1, x

2)"

P(x1, x

2, x

3(x

1, x

2))

2e~(x3(x1,x2)~m3)sn if x3A

2

On the contact line between A1

and A2, x

3(x

1, x

2)"m

3, so g(x

1, x

2)"1

2P (x

1, x

2, m

3). If the

point is on the surface g (m1, m

2)"1

2P (m

1, m

2, m

3).

So taking into account

+nPA1

P (x1, x

2, m

3)t

n(x

1, x

2)t

n(m

1, m

2) dx

1dx

2

!+nPA1

P (x1, x

2, x

3(x

1, x

2))

tn(x

1, x

2) t

n(m

1, m

2)

2e~(m3~x3 (x1,x2))sn dx

1dx

2

#+nPA2

P(x1, x

2, x

3(x

1, x

2))

tn(x

1, x

2)t

n(m

1, m

2)

2e~(x3(x1,x2)~m3)sn dx

1dx

2

"

1

2P(m

1, m

2, m

3)

we obtain the representation theorem,

1

2P (m

1, m

2, m

3)"!+

nP)

2+

m/1

P (x1, x

2, x

3)Lt

n(x

1, x

2)

Lxm

tn(m

1, m

2)

2sn

e~D m3~x3Dsn j

mds

#+nP)

LP(x1, x

2, x

3)

Ljtn(x

1, x

2)t

n(m

1, m

2)

2sn

e~Dm3~x3Dsn ds

This result is by no means trivial because the summation in m is from 1 to 2 and not 3. In orderto discretize the motion equations, the vibration modes of the reservoir cross-section arenumerically computed from a "nite element discretization of the cross-section. Triangularelements with three nodes are used, the pressures being obtained by linear interpolation with area



shape functions ¸i. The vibration modes t@

nisatisfy the orthogonality conditions

t@niKM

ijt@

mj"d

mn"2

n

t@niMM

ijt@mj"d

mn(i, j"1, 2,2) (m, n"1, 2,2)

Here KM and MM are respectively, the "nite elements &sti!ness' and &mass' matrices of thediscretized 2-D cross-section compatible with the boundary discretization of the shell}#uidinterface. Taking into account the normalization used for the fundamental solution we obtain

tni"

t@ni

C0

tm(x

1, x

2)"

3+i/1

¸itmi

The discretized global motion equations, obtained from the representation theorem areexpressed by

1

2P

i"G

ij ALP

LjBj

!H@ijPj

where

Gij"+

kP)

k

+l

+n

¸j¸l

tnltni

2sn

e~Dmi3~x3Dsn ds

H@ij"+

kP)

k

+l

+n

2+

m/1

¸j

blm/

tnltni

2sn

e~Dmi3~x3Dsnj

mds

(k"1,2, n3 of elements), (n"1,2, n3 of vibration modes), (i, j"1,2, n3 of nodal points), )kis

the area of element k and (mi1, mi

2, mi

3) are the co-ordinates of node i. b

lm//"L¸

l/Lx

m. The

summation in )kfollows the scheme: index 1 takes only the nodal points of )

k; the contribution of

each vibration mode n is added being the value of the summations assembled at the (i, j) positionof the matrixes H@

ijand G

ij. In matrix notation

[G] GLP

LjH"[H]MPN (8)

Taking into account that

GLP

LjH"owM;G jN"[A]M;G N

4)%--

where ow

is the mass per unit volume of water, M;G jN are the accelerations at nodal points of theshell, directed along the unit vectors normal to the shell}#uid interface and pointing the exterior

450 R. J. CA) MARA


of the reservoir, and M;G N are the accelerations of nodal points of the shell, each with threecomponents. The pressures MPN correspond to forces MFN

4)%--applied on the shell

[B]MPN"MFN4)%--

Solving (8) for MPN

MPN"[H]~1[G] GLP

LjH"[H]~1[G][A]M;G N

so that

MFN4)%--

"[S(u)]M;N4)%--

where the impedance matrix is given by

[S (u)]"!u2[B][H(u)]~1[G(u)][A]

4. COUPLED ARCH DAM}FOUNDATION}RESERVOIR ANALYSIS BYTHE HYBRID FREQUENCY}TIME METHOD

In order to solve the motion equations of the coupled arch dam}foundation}reservoir, takinginto account no tension contraction joints, the hybrid frequency}time method is used, accordingto Wolf [5]. This method is based on the following scheme:

(1) the FFT (Fast Fourier Transform) of the static forces (Dead Weight and HydrostaticPressure; MF

4(t)N"H(t)MF

4N) and of the &free-"eld' displacements, ;'

"(t) (&g' denotes ground, the

degree of freedom of the interface dam}foundation are referred by &b' while the others are referredby &s'. H (t) is the Heaviside function), are computed:

MF4(t)NPMF

4(u)N

M;'"(t)NPM;'

"(u)N

(2) the impedance matrix [S0] is evaluated, where [¹]"[K]!u2[M] ([K] is the sti!ness

matrix, [M] the mass matrix of the structure and u the frequency of harmonic vibrations),

[S0(u)]"C

[¹44(u)]

[¹"4

(u)]

[¹4"

(u)]

[¹""

(u)]#[S'""

(u)]Dat the discrete frequencies u

k, until the Nyquist frequency. The discrete frequencies are

uk"

2nk

¹

, k"0,2, N/2



with ¹"N*¹ (¹ is the duration of the periodic time history of accelerations. It is assumed to bethe summation of N time intervals of duration *¹). The impedance matrix of the reservoir,relating the forces applied by the reservoir on the structure with normal accelerations of theupstream face, is then added to the structure impedance matrix.

(3) the vector of &real forces', MRN, applied on the structure is evaluated at the N discretefrequencies,

MR (u)N"GMF

4(u)N

[S'""

(u)]M;'"(u)NH

The discrete Fourier pair transform is

Xk"

1

N

N~1+r/0

xrexp(!j2nkr/N) (k"0,2 , N!1)

and

xr"

N~1+k/0

Xkexp( j2nkr/N) (r"0,2, N!1)

so

Xm"X*

N~m(m"0,2, N!1)

only being necessary the knowledge of Xm

until N/2. Here (* ) indicates the complex conjugate. Inorder for the time series x

rto be real, it is necessary that Imag(X

0)"0 and that Imag (X

N@2)"0.

(4) the inverse of the impedance matrix, [S0], is evaluated at the N/2 discrete frequencies,

[S0(u)]P[S

0(u)]~1

Then the following iterations ( j ) are done:

(1) the time domain pseudo-forces MP0N are computed,

MPj0(t)N"!P

S

[B]TMp0N dS

( j"0; MPj0(t)N"M0N)

being the integrals evaluated over the joint surfaces and Mp0N being the stresses at the open points

of the joints (detected by the existence of a normal tension), and [B] the operator relating nodaldisplacements with strains (in this case the di!erence between the displacements of two joint facesat the given point).

452 R. J. CA) MARA


(2) the unbalanced forces MQN are applied,

MQj (t)N"!MPj0(t)N

(3) after being transformed to the frequency domain,

MQj (t)NPMQj (u)N

(4) After solving the system of equations for displacements at the N/2 discrete frequencies

M;j (u)N"[S0(u)]~1(MR(u)N#MQj(u)N)

(5) the displacements are transformed from frequency domain to time domain, and then returnto step (1) in order to evaluate the pseudo-forces, until the scheme converges (the criterion consistsin limiting the relative error of unbalanced forces in time domain),

M;j (u)NPM;j(t)N

The degrees of freedom at the joints of the coupled system are the mean displacements androtations and the di!erence of displacements and rotations between the nodes at the two jointfaces. They relate the mean displacements, and so the mean normal accelerations to the shellsurface to the hydrodynamic pressures of the reservoir.

A segmenting procedure was adopted, with transition zones at the end of each time segment.The decay "lter used in time domain for unbalanced forces was according to Wolf [5]

f (t)"2At!t

0t1!t

0B3!3 A

t!t0

t1!t

0B2#1

t0

and t1

are, respectively, the beginning and the end of the transition zone. When convergencewas reached in a time step, no change of unbalanced forces up to this step takes place in nextiterations. In spite of this procedure, instability was noticed due to high-frequency peak ofaccelerations during joint closure. So, in order to avoid the causes of instability, a low-pass "lterwas used in the frequency domain. The criteria of convergence used in the computer schemeassumes that the relative error of the norm of unbalanced forces and moments are both less than5]10~4.

It was assumed that the stresses at the joint face are

pi"K

i

2+j/1

3+n/1

(Nijn

*;jn#N@

ijn*h

jn) if the normal stress p

3is negative

pi"0 if the normal stress p

3is positive (tension)

(i"1, 2, 3), K3"E/e, K

1"K

2"G/e



Figure 3. Structure and actions model.

*; and *h are, respectively, di!erence of displacements and rotations between pairs of nodalpoints, each on its side of the joint surface. &e' is the joint thickness. N

ijnand N@

ijnare shape

functions compatible with thin shell elements based on Hermite polynomials. The joint sti!nesswas computed by numerical integration (4]6 Gauss points per joint element).

5. EXAMPLE

The seismic behaviour of Pacoima arch dam was analysed by the present method. The dam'sresponse was studied under 5 s of the S75W component of Feb. 9, 1971 San Fernando earthquake[6], recorded near the left abutment of the dam. The dam was analysed in the four followinghypotheses:

For hypothesis I, the assumed static actions are the Dead Weight and Water at crest level andfor hypothesis II, dead weight and water at 1

3of dam's height. Both cases assumed the dam with

contraction joints and take as input a time history of accelerations. Considering as static actionsdead weight and water at 1

3of dam's height, hypothesis III assumes the input as a time history of

accelerations and hypothesis IV assumes the input stochastic as a density power spectrum ofaccelerations. Both III and IV assumed the dam without contraction joints.

All four hypotheses assumed an input of horizontal accelerations at a point on the half-spaceand distributed along the canyon by a SH wave model with vertically incidence, being thedisplacements in downstream}upstream direction.

The concrete and rock mass were assumed [2] with an elastic behaviour, characterized bya Young's Modulus of 24 GPa and a Poisson's ratio of 0.2. The unit mass was assumedhomogeneous with 2400 Kg/m3. The hysteretic damping of concrete was assumed 2]0.1 and thehysteretic damping of rock mass was assumed with half the value for concrete. No damping wasused in the joints. The normal and the tangential sti!ness of contraction joints were assumed 240and 100 GPa/m, respectively.

The adopted structure and action models are both presented in Figures 3 and 4. The dam is notsymmetric due to the left abutment. The peak tensile stresses for di!erent hypotheses are

454 R. J. CA) MARA


Figure 4. Density power spectrum of accelerations.

presented in Figures 5 and 6. The peak accelerations at dam's body and the joints motion forhypothesis II are presented in Figure 7 (the analysed point is located at crest central cantilever).

For the non-linear analysis a time discretization of 256 time steps (of 0.02 s) was assumed,which was divided into eight segments (of 32 time steps) with transition zones after each segment(of 15 time steps). With this time discretization and for the adopted mesh the computation takesabout 40 h of CPU on an IBM Risc 6000 model 39H computer. The equilibrium was obtained ateach time step belonging to each of the adopted eight segments by means of an iterative proceduretaking into account the unbalanced forces from each time step till the end of the transition zone.The total number of iterations divided by the number of time steps is about 100 (obviously, ineach segment, the number of iterations is greater for the initial time steps than for the last). Thehard disk storage was approximately 300 MB. So a larger mesh was not adopted, neithera greater time interval for accelerations history, which would allow assuming zero accelerationsduring the "nal part of the history. The adopted method considers actions and responses periodic,so that zero "nal part of accelerations time history would allow the structure to stop by dampinge!ect, before a new period begins. However, even with these approximations, we expect to capturethe main aspects of the dam's response on the safe side and obtain a comparison for elasticbehaviour with the stochastic stationary hypotheses.

On the other hand, the high damping used covers the fact that in the adopted model there is nodissipation of energy at the joints. Therefore the ampli"cation ratio, about 2.5, of accelerationsmeasured at the dam during the Northridge earthquake was matched. The adopted joint modelassumes no slip (when closed) as a consequence of the in#uence of the shear boxes. When thecontraction joints open, the slip allowed depend on the geometry of the shear boxes and so, inorder to facilitate the computations, it was assumed no limit on the slip, depending on thereliability of the model on the results obtained. In order to take into account a more realistic jointbehaviour, one should adopt the sub-structure scheme, as done by Fenves [7], where the jointsand their neighbourhood are discretized in three dimensions. However, the CPU time would beprohibitive for the computer used.



Figure 5. Peak tensile stresses for hypotheses I and II.

As the unbalanced time forces have high-frequency components that originate divergence ofthe scheme, we must cut o! those forces above a certain frequency limit (as a "rst trial we set thislimit at 12 Hz).

From the computed results we conclude that

(i) The hoop compressive peak stresses are higher when no tension contraction joints are takeninto account in dam's behaviour.

(ii) For water level at 13of the dam's height (as occurred during the San Fernando earthquake) no

cracking is expected due to the seismic vibrations.

456 R. J. CA) MARA


Figure 6. Peak tensile stresses for hypotheses III and IV.

(iii) The average of stresses peak values occurring during consecutive 5 s time intervals (as shownby the stochastic computations for hypothesis IV) it is not much di!erent from the 5 s sampleused for hypothesis III. In order to be the same would be necessary to compute the responseof the dam to several earthquakes with the same power density and to average the peakvalues. So the response of the dam to the accelerations time history sample used isrepresentative of earthquakes with similar spectral properties.

(iv) The computed peak accelerations at dam's body including the ratio from the abutments tothe top and bottom of central cantilever are about the ratio observed [2] during the 1994Northridge earthquake.



Figure 7. Peak accelerations at dam's body and motions of the joints for hypothesis II.

(v) The computed joint motion is reasonable, as we may conclude from the performance of theprototype during the 1994 Northridge earthquake [8]. We must note the strong impact dueto the closure of the contraction joints, yet with peak compressive stresses lower than theexpected concrete uniaxial compressive dynamic strength. The joint deformations whenclosed are assumed elastic and are compatible with the expected joint thickness (about 2 cmthick). The nodal joint compressions (closure times joint rigidity) depend on the correctevaluation of the di!erence of displacements at local joint axis (must be completely compat-ible with used joint sti!ness hypothesis).

458 R. J. CA) MARA


A global equilibrium (referred to nodal forces and moments) was assumed between shellelements (straight cross-sections with linear equivalent stress distributions due to assumed elasticbehaviour, without any ruptures) and the joint elements (bilinear stress distribution at crosssection characterised by a diagram with zero tensile stresses). Based on simple equilibriumconsiderations (and on the performance of used "nite elements), the compressive hoop stressescomputed at joints are more reliable then the hoop stresses computed at shell elements.The computed low values of joint openings may depend both on the number of joints assumed atthe dam model and on the coarse mesh adopted and Kircho! hypothesis of shell elements. In fact,the imposed restrictions on the possible deformed shapes may originate lower radial displace-ments. The computed high values of compressive hoop stresses may depend on the zero tensilestrength hypothesis assumed at contraction joints.

6. CONCLUSION

The adopted method takes into account the in#uence of the reservoir and foundation, idealised asin"nite media. The radiation e!ect of elastic waves (foundation) and of pressure waves (reservoir)are better modelled than with traditional methods ("nite elements with asymptotic radiationboundary conditions) although this aspect is only formal. It also takes into account the spatialvariability of the earthquake and the in#uence of no-tension contraction joints on dam'sresponse. The use of "nite element method with time discretization (solving hyperbolic equations)is usually conditioned by the time and space steps necessary to accurately represent thepropagation of stress and displacement pulses. The method presented has the accuracy of static"nite elements due to the use of harmonic solutions (solving elliptic equations).

The chosen example of Pacoima arch dam under a severe earthquake shows, as expected, thatthe maximum peak hoop tensions are lower than the ones computed for the hypothesis ofneglecting the in#uence of contraction joints. The maximum peak cantilever tensions obtainedwhen the water is about 1

3the dam height would not originate horizontal cracks near the central

top zone, as observed during the San Fernando earthquake. According to Sharma [8, 9], thecracking that occurred at the left abutment may be explained by permanent foundation displace-ments acting as static settlements. The computed compressive stresses at mid-section of the jointelements were lower than the concrete uniaxial dynamic strength.

ACKNOWLEDGEMENTS

The author thanks to the Director of National Laboratory of Civil Engineering for allowing the researchwork presented, and to the Foundation for Science and Technology for partial "nancial support of theproject. The author also thanks to the reviewers, whose points of view contributed to a substantialimprovement of the paper.

REFERENCES

1. Dowling MJ, Hall JF. Non-linear seismic analysis of arch dams. Journal of Engineering Mechanics, ASCE 1989; 115:4.2. Mojtahedi S, Fenves GL. Response of a concrete arch dam in the 1994 Northridge, California earthquake. 11th=orld

Conference on Earthquake Engineering, Acapulco, 1996.



3. Zhang L, Chopra AK. Computation of spatially varying ground motion and foundation rock impedance matrices forseismic analysis of arch dams. Report ;CB/EERC-91/06, May 1991.

4. Ca( mara RC, Coupled arch dam-foundation-reservoir seismic behaviour. Safety evaluation for rupture scenarios, Ph.D.¹hesis, F.E.U.P., Oporto, 1992. (in Portuguese).

5. Wolf JP, Soil-Structure-Interaction Analysis in ¹ime Domain, Prentice-Hall: Englewood Cli!s, NJ: 1988.6. California Institute of Technology, Strong-motion earthquake accelerograms. Digitised and plotted data. Report

EER¸ 72-51, 1973.7. Fenves GL, Mojtahedi S, Reimer RB. ADAP-88. A computer program for non-linear analysis of concrete arch dams.

Report ;CB/EERC-89/12, Nov. 1989.8. Sharma RP, Jackson HE, Kumar S. Response of Pacoima dam to the January 17, 1994 Northridge Earthquake, 60th

Annual USCOLD Lecture Series, Seismic Design and Performance of Dams, Los Angeles, 1996.9. Sharma RP, Jackson HE, Kumar S. E!ects of the January 17, 1994 Northridge earthquake on Pacoima arch dam and

interim remedial repairs, 19th ICOLD Meeting, Florence, 1997.

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a method for coupled arch dam-foundation-reservoir seismic behaviour analysis

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