INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, VOL. 6,219-241 (1982)

SHEAR VIBRATION OF VERTICALLY INHOMOGENEOUS EARTH DAMS

GEORGE GAZETAS* Department of Civil Engineering, Rensselaer Polytechnic Institute, Troy, N. Y. 12181, U.S.A.

SUMMARY

Closed-form solution to the problem of free vibrations of vertically inhomogeneous earth dams, modelled as truncated-wedge-shaped shear beams, has been obtained by implementing an inverse procedure in which the determination of the function describing the inhomogeneity constitutes part of the problem. The resulting cube-root variation of the shear-wave velocity with distance from the crest compares very favourably with measurements in two Japanese dams. The results of the method are presented in the form of natural periods, modal shapes and average seismic coefficients for a number of truncation ratios. Compared with an 'equivalent' homogeneous dam, the inhomogeneous experiences sharper amplification of modal displacements and greater average seismic coefficients near the crest and has natural periods which are closer to each other. This behaviour is in better agreement with the observed response of a 37 m-high dam during five earthquake motions.

INTRODUCTION

It is well understood that safety of an earth dam during earthquakes is controlled by the characteristics of its dynamic response. The latter depends on the intensity, frequency content and details of the exciting motion, as well as the geometric configuration and the material properties of the dam.

For not very strong earthquake motions, soil behaves nearly elastically and a viscoelastic one-dimensional model that treats the dam as a symmetrical homogeneous wedge-shaped shear beam has enjoyed significant popularity among engineers and researchers over the last years. The model, originally introduced in 1936' and further developed about twenty years

is only an idealization of reality and in fact violates the physical requirement of zero shear stresses on the two slopes (up and down-stream) of the dam. In addition the shear model cannot account for the compressive and tensile stresses that develop due to the multiple wave-reflections at the faces of the dam. These stresses have an influence on the overall dynamic response, as was shown by several

In spite of these limitations the shear-beam model has successfully predicted certain dynamic characteristics of actual earth dams observed during forced vibrations with large shaking machine^'"^ and during small and/or distant ear thq~akes.~ '" ' '~ Furthermore, comparisons with the results of two-dimensional (finite-element) analyses indicate that the error in estimating acceleration and inertia-force time histories with the one-dimensional model is negligible (usually less than about 10 per cent). More significant differences do, naturally, arise when stresses and deformations are compared, but this has minor consequences since it is primarily inertia forces and accelerations that are used to evaluate the seismic performance of dams

* Professor.

0363-9061/82/020219-23$02.30 @ 1982 by John Wiley & Sons, Ltd.

Received 11 November 1980 Revised 24 September 1980

220 G . GAZETAS

On the other hand, the using the ‘limiting equilibrium’ or the ‘displacement’ methods. one-dimensional method leads to closed-form solutions which are particularly convenient in performing parametric studies at a fraction of the cost of the more sophisticated finite-element methods.

The above justify the continued and the needed additional research and improvement of the viscoelastic shear beam model.

Shear-wave velocity (c) and equivalent critical damping ratio (@) are the two soil properties that should be reliably estimated in order to determine the response of the dam with the one-dimensional theory. As of today only homogeneous dams can be treated with this model, which means that an average value of c or @ has to be estimated when the dam consists of different zones. However, a non-uniform variation of shear-wave velocity with depth seems to be more realistic, since the effective normal stresses increase with vertical distance from the crest. Although the actual variation with depth (function of the soil type, method of construction and geometry of the dam) should be assessed for each particular dam by means of laboratory and/or field measurements, several researcher^'^.'^ have expressed their prefer- ence for a relationship of the form:

4.13-16

c ( x ) = ccr + a ( x / H ) P (1)

where ccr is the crest S-wave velocity, a and p positive constants and x the depth from the crest of the dam. The mathematical complexity of the problem, however, when such a variation-law is introduced has discouraged further development in this direction. Thus, today methods of earthquake resistant design of earth dams are based on the assumption of vertically homogeneous dams, even when numerical techniques are used (see Reference 14).

The main objective of this paper is to present a rigorous, closed-form solution to the problem of dynamic shear response of linearly-hysteretic two-dimensional truncated-wedge-shaped dams whose S-wave velocity varies in a manner similar to that described by equation (l), with p = (1/5)/(1/3). Both free and forced vibrations are studied. The results are presented in the form of natural frequencies, modal shapes and simultaneous seismic coefficients on potential sliding masses, for a wide range of truncation ratios and p =1/3. The importance of inhomogeneity is revealed by comparison with the corresponding results of the homogeneous theory’ in the light of some field evidence.

It is emphasized that, in consistency with the shear-beam assumption, by shear-wave velocity c ( x ) or shear modulus G(x) in this paper, we denote the average value of c or G over a horizontal plane at a distance x from the crest. Theoretical considerations (summarized in Appendix I of the paper) lead to the conclusion that, in an elastic two-dimensional wedge under plane-strain conditions, this average c ( x ) increases indeed in proportion with the cube-root of x. Moreover, it is seen that the variation of the shear-wave velocity at points along the (central) x axis of the wedge is not very different from the cube-root distribution that applies to the average velocities c ( x ) . Thus, in practice, in order to determine representative values of shear-wave velocities of existing earth dams, as well as to study the variation of such velocities with depth, field wave velocity measurements can be carried out either across the full width or along the crest axis of a dam. Such comprehensive geophysical measurements have been recently reported by Abdel-Ghaffar and Scott? by recording the arrival times of S-waves at several seismometer stations located on the slopes and the crest of the Santa Felicia Dam in California, shear-wave velocities were estimated at various depths. The waves were triggered by sledge hammer impacts at several locations on the up-stream slope and the crest of the dam. Analysis of their data revealed that an increase of shear-wave velocity with depth as sharp as the 1/3 power of x implies, seems most appropriate. More recently, Gazetas and

VIBRATION OF EARTH DAMS 22 1

Abdel-Ghaff arZ0 analysed comprehensive field measurements of twelve earth and rock-fill dams from Japan, U.S. and Yugoslavia; they found a remarkably good agreement between the recorded variations of velocity with depth and the herein advocated distribution according to the cube root of x. Additional evidence (direct and indirect) presented in the last sections of the paper further substantiates the reality of such a distribution. It is, therefore, believed that the presented theory can be a useful engineering tool in seismic design of earth dams and embankments.

It should be noted in passing that Rashid in 1961t (as reported by Seed16) analysed the response of a (complete) wedge whose shear-wave velocity increased as the 1/6 power of the depth. Such a variation is closer to the uniform distribution than the one proposed herein (see plot in Figure 2) and, as can be seen in Appendix I, does not adequately reflect the dependence of soil properties on the effective normal octahedral stress. Rashid's solution is compared with the one presented here later in the paper.

FREE VIBRATIONS

Instead of attempting to obtain a solution to the completely defined problem, an inverse procedure is adopted in which the function describing the variation of velocity with depth is not prespecified, but its determination constitutes part of the problem. A system of differential equations that transforms the governing equation of motion to the classical one-dimensional wave equation is first constructed. The solution of this system is combined with the known solution of the wave equation to obtain the solution to the inhomogeneous problem, while simultaneously the functional form of the shear-wave velocity is determined. Such a method of solution was first applied by Schreyer" to study one-dimensional wave propagation into an inhomogeneous half-space.

Let x and z be the coordinates axes for the truncated and symmetrical wedge shown in Figure 1. The shear model is essentially based on the following two simplifying assumptions: (a) only shear deformations take place; and (b) the resulting shear stresses on any horizontal

Figure 1. Dam geometry and forces acting on a differential slice

plane are uniformly distributed. Thus, when elastic free vibrations of the dam are considered, the dynamic equilibrium of a thin slice at depth n yields:'

t Unfortunately the original work, that was conducted for the requirements of a graduate a)urse at Berkeley, is not available in any form (personal communication with Seed, August 25, 1979). The solution was expressed in terms of the Bessel function lo.2 and led to mode shapes not very different from those of the classical homogeneous theory.

222 G. GAZETAS

in which u =horizontal displacement at depth x, t = time, and c’ = G/p is the yet unspecified S-wave velocity at depth x; G and p are the shear modulus and material density, respectively, at the same depth.

To account for the internal dissipation of energy in the material, linear hysteretic damping can be reproduced in equation (2) by assuming a complex value of the wave velocity of the form c* = c J(1+2i/3), where /3 is the critical damping ratio. However, since the effect of damping on the natural frequencies and modal shapes is usually very small, it is common practice to neglect it when analysing the free vibrations of earth dams (e.g. Seed and Martin’). Instead, damping ratio is used, for example, when performing modal superposition, to ‘read’ spectral values from the appropriate design spectra. This practice is followed herein, and, thus, c in equation (2) is considered as a real number. The analysis for complex c* is completely analogous, and presents no additional difficulty.

A general solution is sought for equation (2) that will comply with the boundary conditions of the free vibration problem. The latter can be expressed by the following equations:

u(H, t) = 0 (34 a

G(x ) - u(x, t ) ] = 0 ax x = H

corresponding respectively to zero displacements at the dam-rock interface and zero shear stresses at the uppermost free surface (crest).

Consider a transformation of the dependent variable, u, and the independent variable, x, of the form:

u(x, t ) = * (X)O(S t ) (44

with

and

x’= f(x)

+(x)>O, f(x)>O forx>O

f (0) = 0 The motivation of this transformation is the desire to obtain a governing differential equation for u(2, t) identical in form to the wave equation

a’u a 2 V at2 - co - ax“ _-

in which co = a reference ‘celerity’. If this task were realized, the general harmonic solution for u(2, t) could be written as:

u(2, t) = [A exp (iox’/co) + B exp ( - io2/co)] exp (iwr) (6) in which o = the frequency of vibration; i = J- 1; and A and B are integration constants that are later determined by satisfying the boundary conditions (equation (3)).

Now, substitution of equation (4a) and equation (4b) into the governing equation (2) yields:

+ 2 - - + c - + - - -u ( ax ac ax ax’ a2* x a*>c ax rl (7)

VIBRATION OF EARTH DAMS 223

In order to reduce equation (7) to the form of equation (5), f, 4 and c must satisfy the following equations:

2 2($) = c o 2

2(2c!? d f + 2 a" df$+c$7+-- a2f c af 4) = o (I ax ax ax ax ax x ax

ac a4 a'@ c a4 ax ax ax x ax

'(2 - -+c,+- -) = 0

To obtain non-trivial solutions equations (8a), (8b), (8c) for the wave velocity, c, and the transformation functions, 4 and f , it is sufficient to rewrite them in the form:

af ax C-=CrJ

a ax - ( c2xlb2 $) = 0

-(c2xZ)=o a ax

The system of the non-linear differential equations (gal, (9b) and (9c) can be directly solved in closed form. Compliance with the requirements of solution after a series of integrations:

(1 + e ~ ~ ) ~ / ~ c = CrJ

X

4 = (1 + e ~ ~ ) - ~ / ~

equation ( 4 ~ ) - leads to the following

in which 8 is an integration constant and co serves as a reference celerity.

and co are expressed as: Equations (10) can be written in a non-dimensional form if the integration parameters 8

8 = pH-' ( 1 W

co = C , . , , H ~ - ~ / ~ (1 1b)

where the parameter cm can be interpreted as the maximum value of c, i.e. at x = H. By combining equations (10) and (11) with the wave propagation solution (equation (6)) and calling y the x / H ratio (Figure 1) one obtains the general solution for the amplitude of vibrations:

@ ( y ) = ( l + p y ~ ) - l / 3 { ~ exp [i- - p OH -1/3 ( [ ~ + p y ~ ] ~ / ~ - ~ ) ] 2 Cm

224 G. GAZETAS

with

u ( Y, t ) = @( Y 1 exp ( i 4 (124

provided that the variation of the shear-wave velocity with depth is given by:

(13) c -213 -1 -= (1 +py2)2 i3 Cm

with y s y s l (13a)

where A = h/H and y = x / H are the truncation and depth ratios (Figure 1) and p is a parameter defining the exact shape of the velocity distribution with depth (Figure 2).

O?

.2

.4

.6

.8

I .

c/c, 0.5 1.0

--

-

-

-

-

I GROUND

y=- X H

........ p . - - u = 10'

- -- IJ = 104

U = 5 X I ~ '

Figure 2. Variation of S-wave velocity with distance from origin of coordinate axes

Alternatively, equation (12) and equation (13) can be interpreted as follows: the harmonic free vibrations of an inhomogeneous earth dam whose S-wave velocity varies with depth according to equation (13), are described by equation (12).

Figure 2 portrays in a dimensionless form the velocity, c, as function of depth, x, for several values of p for a dam with a truncation ratio, A, 0.05. The normalizing velocity, c,, can be interpreted as the maximum value of c, at the base of the dam. It is clear from this figure that the resulting variation of c is realistic and relatively insensitive to p. Indeed, for large values of p (say p 3 10') equation (13) can be approximated by:

C

Cm

---. 113 Y

VIBRATION OF EARTH DAMS 225

that is, S-wave speed increases as the cube-root of the depth-a reasonable variation for many types of soil and methods of construction, according to field evidence that is presented in the last sections of the paper. Equation (14) is exactly true if p is infinitely large.

On the other hand, for very small values of p (p < 500) equation (13) approximately yields:

provided that some truncation ( A aJ(3/p)) exists. This restriction is necessary in order to have a meaningful solution from a geotechnical point of view. Because otherwise, for small values of p, equation (13) describes a wave velocity decreasing (instead of increasing) with depth, from an infinite value at y = 0 to a minimum value at y = J(3 /p) . Clearly the solution for such a medium would be only of academic interest. Notice, however, that earth dams are always flattened at the crest and the above restriction ( A a d(3/p)) , in practice, is usually met, even for small values of p.

In the following sections p is taken as 00, since the resulting cube-root variation of velocity seems to give the best agreement with field observations. The analysis for the more general case of an arbitrary p is completely analogous, although the resulting formulae are slightly more complicated.

NATURAL FREQUENCIES AND MODAL SHAPES

To determine the constants of integration A and B in equation (12), the boundary conditions expressed by equations (3) are enforced. Noticing that as p + 00

and substituting equation (12) into equations (3) yields the following equations in A and B:

Aexp( ia )+Bexp(- ia )=0 (17a)

A(A-’/’-ia) exp (~uA’/~)+B(A-~/’+~Q) exp (-id’’’) = O (17b)

in which:

Equations (17) state an algebraic eigenvalue problem. For a meaningful solution the determinant of the coefficients in equation (15) must vanish. This condition leads after some straightforward algebraic operations to the following ‘period’ relation:

c o s Z ( a - d ) = l (194

in which

d = uA ’ I 3 - arctan (ah *I3)

Thereby it follows that the eigenvalues (a,) are the roots of

P(a,) = a,(l - A 2 / ’ ) + arctan (a,h’/’) - ntr = 0 n = 1,2,3, . . . (20)

Equation (20) is numerically solved using the Raphson-Newton method. Its roots for the first six modes of vibration (n = 1-6) and a wide range of truncation ratios ( A = 0-1.00) are given

226 G. GAZETAS

in Table I. Knowing a,, the natural periods are determined using equation (16):

37r H T , = - - an Cm

Table I. Roots (a,) of the period relation

Natural mode (n) Truncation

ratio 1 2 3 4 5 6

0.00 0.030 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0,550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.000

a 3.1504 3,1653 3.2294 3.3266 3.4546 3.6137 3.8069 4.0392 4.3182 4.6549 5.0654 5.5730 6,2130 7.0414 8.1515 9.7113

12.0574 15.9749 23-8199 47.3725

00

2 a 6.3460 6.4387 6.7721 7.2026 7.7121 8.3013 8.9801 9.7655

10-6821 11.7643 13.0612 14.6437 16.6190 19.1550 22.5325 27.2557 34.3349 46.1261 69.6974

140.3905 00

3 v 9,6041 9.8312

10.539 1 11,3641 12.2906 13-3309 14.5 079 15.8537 17,4115 19,2407 21.4238 24,0803 27.3888 31.6301 37.2719 45.1559 56,9651 76.6267

115.921 1 233,7513

00

4 a 12.9183 13,3069 14.4117 15.6308 16.9692 18.4546 20.1240 22.0249 24.2195 26,7914 29.857 1 33.5841 38.2230 44.1669 52.0708 63.1132 79,6504

107- 1805 162.1963 327-1621

00

5a 16.2748 16.8340 18.3348 19.9425 21.6879 23.6144 25.7730 28.2264 31.0554 34.3683 38.3151 43.1 113 49.0793 56.7248 66.8899 81.0898

102.3543 137.7522 208.4888 420.4896

00

6 a 19-6613 20.3926 22.2846 24.2767 26.4260 28.7914 31.4375 34.4420 37-9045 41.9574 46.7845 52-6493 59.9458 69.2923 8 1 *7 181 99.0752

125.0667 168-3321 254.7892 514.0247

00

Substitution of the relations (18) and (20) in equation (12) leads, after some operations, to mode shapes, a,( y), given by

(22) 1

an (Y) = 2/3 sin [an(l - Y ~ ’ ~ ) I Y

with

A S y S l

The first four mode shapes computed from equation (22) and normalized to a unit crest amplitude are portrayed in Figure 3 for a broad range of values of the truncation ratio (0.05 S A d 0.80).

IMPORTANCE OF INHOMOGENEITY

One way to study the effects of inhomogeneity on the dynamic response of earth dams is to compute the error involved in replacing the inhomogeneous by a homogeneous earth dam having the average shear-wave velocity, E, of the former. For example, the natural periods of

VIBRATION OF EARTH DAMS 227

1 .o Q!

0.0 0.5 o.or I I I I I I I I I 1

1 .o @*

-1 .o 0.0

Y Y

Y Y

-1 .o 0.0 1 .o

Figure 3. The first four normal mode shapes of the inhomogeneous earth dam

the homogeneous dam, F,, are given by:

- 2 ~ r H a, E

T,,=--

in which d,, n = 1,2, . . . , are the roots of the period relation for a homogeneous dam, Jo(a,,) Yl(hd,,) - J I ( ~ , , ) Yo(Aan) = 0, and are given by Ambraseys2 as functions of the trunca- tion ratio, A ; Jo, J1, Yo, Yl are the Bessel functions of the first (J) and second (Y) kind, zero and first order, respectively. The average shear-wave velocity can be obtained by using equation (14) for the variation of velocity with depth:

6 l - A 7 ’ 3 1

I? = 6’ cby dy/lA by dy = - 7 1 - A ’ Cm

228 G. GAZETAS

It is instructive to study the ratio of the natural periods of the two dams, T,/Tn, versus A. This is done in Figure 4 which portrays this ratio for the four first modes (obtained by using equations (21), (23) and (24)). It is clearly seen that the fundamental period (n = 1) is only slightly overestimated when an 'equivalent' homogeneous dam is considered. The error is barely 2 per cent for very small values of A and decreases as A increases, converging to zero (i.e. Tl = TI) as A tends to one. It can, therefore, be stated that the two models predict essentially identical fundamental periods.

0.9 I 0.0 0 .5 1

TRUNCATION RATIO, a Figure 4. Comparison of natural periods of an inhomogeneous and a homogeneous earth dam

Larger discrepancies are observed with the higher periods of the dam. The homogeneous model underpredicts these periods by 10 to 20 per cent in the range of small truncation ratios (say, A a0-1). This is so mainly because the natural periods of the inhomogeneous dam are closer to each other than the periods of the homogeneous dam, especially for low values of A. For example, for A = 0.05 a homogeneous dam has:

T1/Tzi-5*62/2*41=2.33

and

f1/F3 = 9-08/2.41= 3.77

(calculated from Table 1 of Reference 2) while the corresponding numbers for an inhomogeneous dam with the same A are (see Table I):

Tl/T2=6*4387/3.1653 -2.034

T1/7'3=9.8312/3.1653=3.106

This discrepancy may lead to different estimates for the response and safety of a dam, depending on the frequency characteristics of the 'design' ground motion.

Again, as A increases towards 1 the differences of the two theories die away-hardly a surprising result, in view of the fact that the average wave speed, F, tends to become equal to cm as A approaches 1; in other words, the velocity becomes practically uniform (see

VIBRATION OF EARTH DAMS 229

Figure 2). It is worth noting that for earth dams A typically varies between 0.02 to 0.10 but higher values (A = 0.2-0-5) are typical for other types of embankments (highway, flood levees, etc.).

When modal shapes are compared, it appears that the homogeneous shear-beam model does not predict as sharp a ‘deamplification’ with depth as the one resulting from the inhomogeneous model (equation (22)). The latter is caused mainly by the y-2’3 factor in equation (22) and becomes more important for low truncation ratios. This is illustrated in Figure 5 which compares the three first mode shapes predicted by the two models, for two values of A, 0.05 and 0.5. For A = 0.05 the inhomogeneous dam seems to deform almost like a ‘flexural’ beam in spite of the fact that only shear deformations take place. The response of the homogeneous dam, however, is closer to a uniform shear-beam response, especially in the fundamental mode (n = 1). As A increases to 0.5 the inhomogeneous dam experiences a less sharp decay of displacements with depth and thus its mode shapes are very similar with

Y

t

HOHCGENEOUS DAN - - - -

0.5 Y

1.0

Figure 5. Comparison of the three first normal mode shapes of an inhomogeneous and a homogeneous earth dam for 2 values of A (0.05 and 0.5)

230 G. GAZETAS

the shapes of the ‘equivalent’ homogeneous dam and even with the ones of a uniform homogeneous shear beam having velocity E.

The above differences in modal shapes can influence the safety of dams during earthquake- induced vibrations. If shear-wave velocity does in reality vary with depth according to ‘the cube-root’ rule (equation (14))’ shallow sliding failures or excessive displacement near the crest of the dam should be expected to be the predominant modes of failure. This seems to have actually been the case with some Japanese earth dams, as reported by Okamoto,” and by Seed and Martin.5 To demonstrate this danger better, the ‘average seismic coefficients’ operating during earthquakes on potential sliding masses of the inhomogeneous dam are compared with those of the ‘equivalent’ homogeneous dam.

AVERAGE SEISMIC COEFFICIENT DUE TO BASE EXCITATION

The seismic stability of earth and rock-fill dams that are composed of non-liquefiable materials is currently assessed in terms of the expected permanent displacements of potential sliding masses, assumed to occur along the direction of the failure surface^.'^ The ‘average seismic coefficient’ is related to the driving inertia force, F(t) , acting on such a sliding mass (Figure 6(a)):

in which W =total weight of the sliding soil. F( t ) can be considered as the sum of the inertia forces, hFi(t), acting on horizontal elementary slices i. By using the principle of modal superposition:

where A Wi =the weight of the elementary slice i ; Q n ( y i ) = the n modal displacement of the dam at yi (equation (22)); r, =the participation factor of the n mode in the overall motion of the dam (Appendix 11); and San(t) = the acceleration time history of a single-degree-of- freedom oscillator having the natural period, T., and the damping ratio, &, of the n vibrational mode and subjected to the ground motion. For a failure surface forming a triangular wedge with horizontal base at depth x = yH (Figure 6(a)), the ‘average seismic coefficient’ is a function

Equations (25) and (26) then yield a single formula that is presented only of y and r. in Appendix 11.

Figure 6(b) compares the variation with the depth of the sliding mass of the peak ‘average seismic coefficient’, k,,, developed when an inhomogeneous or a homogeneous dam is excited by the N-S component of the El Centro 1940 accelerogram. Results are portrayed only for two fundamental periods of the dam, 0-75 sec and 1-50 sec. They are representative of the results obtained for periods ranging between 0-20 sec and 2.0 sec. Fifteen modes were retained in equation (26) (n = 15), while the damping ratio was taken as 20 per cent of critical. The latter value is consistent with the 0.33g maximum ground acceleration of the El Centro record, as indicated by the results of strain-compatible equivalent linear finite-element analyses by Makdisi and Seed (Reference 14: Table I and Figure 2).

The conclusions from the comparison of the two models are summarized as follows: (1) the inhomogeneous model predicts approximately 30 to 70 per cent higher peak crest accelerations than the homogeneous model, depending on the fundamental period of the dam relative to the spectral content of the excitation; (2) km, decreases very sharply with the depth of the sliding mass in inhomogeneous dams, while showing a much smoother decrease in homogeneous

5.13.14

Y

0

0.5

1.0

0

0.5

Y

1.0

INHO

MOGE

NEOU

S DA

M

HOMO

GENE

OUS

DAM

---

.75

sec

Figu

re 6

. V

aria

tion

of p

eak

aver

age s

eism

ic co

effic

ient

with

dep

th of

slid

ing

mas

s due

to t

he E

l Cen

tro

1940

reco

rded

acc

eler

atio

n (A

= 0

, B =

20°/o

)

a U

P x h) w c

232 G. GAZETAS

dams; and (3) the differences of the two models persist only to a depth from the crest equal to about 1/3 of the dam height; below this level the two theories would predict almost the same likelihood of ‘failure’ of a particular dam.

On the basis of a large number of data obtained from analyses of homogeneous dams subjected to several ground motions, Makdisi and Seed14 proposed a standard design curve for the variation with depth of the peak average seismic coefficient, k,,,, normalized with the peak crest acceleration, Figure 7 compares this design curve as well as the range of the corresponding data with the average curve obtained by analysing inhomogeneous dams (periods ranging from 0.20sec to 2.0sec) subjected (1) to the N-S El Centro record and (2) to a motion whose response spectrum is that proposed by Seed et ~ 1 . ’ ~ for stiff soil conditions, with 0.08g peak ground acceleration. Although more analyses are needed for a quantitatively accurate conclusion, it is evident from Figure 7 that the ‘inhomogeneous’ and the

Y

Figure 7. Recommended average curves for the ratio of peak average seismic coefficient to peak crest acceleration

‘homogeneous’ design curves are drastically different. It should be remembered that the curves of Figure 7, in order to give k,,, must be multiplied with the peak crest acceleration, which is about 50 per cent higher in inhomogeneous dams.

EFFECT OF NON-LINEAR SOIL BEHAVIOUR

One might question the validity of the above conclusions for strong ground motions, given the non-linear hysteretic nature of soil deformation; not only the ‘effective’ equivalent damping ratio, but also the ‘effective’ shear modulus depend on the amplitude of shear strain, y. Non-uniform strain along the height of the dam reduces the soil modulus (or the wave-speed) by different amounts depending on the location, y. Thereby, at a particular moment, the assumed cube-root variation of velocity with depth will no longer be true, and the presented method invalid.

Fortunately, however, this is true only to a minor extent with inhomogeneous dams. As demonstrated in Figure 8, the maximum value of y experienced by the aforementioned 10 inhomogeneous dams due to the El Centro record and to the stiff-soil spectrum of Seed et ~ 1 . ‘ ~ is on the average nearly uniform. Largest and smallest strains differ by 40 per cent at

VIBRATION OF EARTH DAMS 233

most. This implies that shear modulus or velocity would approximately change uniformly with depth during a strong earthquake. Hence, to the same degree of approximation, the variation of velocity according to the cube-root of depth, would always be true. Strain-compatible equivalent linear analyses could therefore be performed using the presented theory and published GIGo and @ uersus y curves (e.g. Figure 8 of Reference 14). It can safely be argued that such analyses would not lead to any significant changes of the average curves in Figures 7 and 8, although individual responses might experience a small increase or decrease depending on the fundamental period of the dam relative to the frequency content of the ground excitation.

Y 1 Y”

0 1 2

y 0 . 5

\ \

inhomogeneous dams

homogeneous dams

1.0 L I Figure 8. Variation with depth of peak shear strain normalized to the peak strain at the base

SOME EVIDENCE ON THE VARIATION OF S-WAVE VELOCITY Although systematic laboratory and/or field measurements are necessary in order accurately to determine the wave speed of shear modulus along the height of a dam or embankment, some indication of their variation can be obtained by observing the time lags of wave arrivals at various points inside the dam where seismometers have been installed. Minami” reports such wave measurements on two dams in Japan, Makio and Togo Dams.

A cross-section of the Makio Dam showing the locations of the various seismometers appears in Figure 9. Makio is a rock-fill dam with a clayey central core and a height of 85 m. Its design was based on a pseudostatic limit equilibrium analysis with a seismic coefficient of 0.15 and a minimum factor of safety of 1.35. By comparing the time lags of the wave fronts during several earthquakes the following (average) shear-wave velocities were estimated: Near the top of the dam, c = 350 m/sec; and, near the ground, cm = 1000 m/sec.

From the geometry of the dam and the location of the seismometers:

H =L 85. +2.3 = 87.3 m and x =2 .+2 .3 = 4.3 m

(the origin of the coordinate axes is at the point of intersection of the two slopes of the dam). Consequently, the expected crest-to-bottom velocity ratio, assuming the cube-root-variation law, is

c/c, = (x /H)”~ = (4*3/87*3)’/3 = (0*05)’/3=0*37

which compares very favourably with the ‘measured’ ratio, 0.35.

234 G. GAZETAS

0 LOCATION OF SEISMOMTERS

Figure 9. Cross-section of the Makio Rockfill Dam”

Tag0 Dam is an earthfill dam with an inclined clay core and has a height of 31 m. It is founded on alluvial soil deposit. The estimated velocities near the top and near the bottom are 125 m/sec and 250 m/sec, respectively. From the geometry of the dam and the location of the ~eismometers’~

H = 3 3 m and x-2.5m

The cube-root-variation law, therefore, yields:

C / C ~ = ( 2 ~ 5 / 3 3 ) ” ~ = (0*076)”3 = 0.423

a value which indicates a slightly steeper increase of c with depth than that actually observed (c/c, 3 125/250 = 0.5).

The above evidence further supports the hypothesis that a cube-root-variation of the S-wave velocity with distance from the crest is more realistic for earth dams and embankments than a uniform shear velocity assumption.

A CASE STUDY

In geotechnical engineering practice no new analytical procedure can be accepted unless it is ‘experimentally’ substantiated. However, owing to the nature of the earthquake phenomenon (place and time of occurrence cannot be predetermined), there is very little recorded field performance by which the adequacy of the various analytical methods used in soil dynamics might be judged. For this reason a documented case study presented by Okamoto and his coworkers””2 is of significance and has been utilized to test the present theory.

The vibrations of the Sannokai Dam, an earth dam located in the northeastern part of the Japan Island, were recorded during five distant earthquakes that occurred in 1964. The dam is 37 m high and has a crest which is 140 m long by 12 m wide (Figure 10). It consists of four zones whose index properties are shown in Table 11, and it is founded on green tuff. Numerous accelerometers were installed at the crest, the down-stream slope and the abutments, and their location is indicated in Figure 10. The characteristics of the recorded motions (epicentral distance and maximum ground and dam accelerations), shown in Table 111, convince that the amplitudes of deformations that occurred in the dam were very small, corresponding to the linear elastic range of the soil.

The horizontal acceleration Fourier spectra of the five recorded motions exhibited three well-defined peaks at the frequencies of 2.8, 4.2 and 5*4cps, approximately. These were naturally interpreted as the first three natural frequencies of the dam. The average normalized

VIBRATION OF EARTH DAMS 235

LOCATION OF SEISMOMETERS

l=L

L- P 193.82m 4 7

Figure 10. The Sannokai Dam, in Japan”

Table 11. Soil properties of Sannokai

Section of the dam Index

properties A . B C D

Liquid limit 50.5 50.0 41.0 60.0 Plasticity index 12.0 7.5 3.5 29-1 Optimum moisture % 20.0 35.0 25.0 15-0 Water content % 20-0-30.0 35.0-50.0 25.0-35.0 15.0-25.0 ~~ ~ ~~ ~ ~~ ~

Table 111. Characteristics of the recorded earthquakes’””

Earthquake Date of Epicentral Maximum ground Maximum crest number occurrence distance (km) acceleration (gal) acceleration (gal)

1 Jan, 9,1964 230 2 Jan. 10,1964 300 3 Feb. 7,1964 150 4 May 7,1964 240 5 May 7,1964 240

0.75 1.75 0-83 6.0 2.6

3.9 5.4 3.75 19.5 7.0

236 G. GAZETAS

deflection curves at the first two natural frequencies (i.e. the mode shapes) are portrayed in Figure 11 (open circles). Also shown in this figure for comparison are the mode shapes predicted by the shear-beam theory for a homogeneous earth dam. For more details the reader is referred to Okamoto.'2

0

10.

E x 20. -

30,

0.

10.

E x 20. -

30.

I

D

0 AVERAGE OF RECORDED MOTIONS

INHOEfOCENEOUS SHEAR BEAM NODEI. ( H * = 3 7 m)

INHOIIOGENEOUS SHEAR BEAM MODEL (H*=32 m)

HOHOCENCOUS SHEAR B E M I MODEL ( H * = 3 7 m) - - - - - - - Figure 11. Predicted us. observed mode shapes of the Sannokai Dam

No similarity appears to exist between observed and predicted mode shapes. In fact the sharp deamplification with depth of the recorded first resonant shape led Okamoto and his coworkers initially to interpret it as the second mode of natural vibrations." Later, however, when even stronger recorded motions (during the Niigata, 1964, and the Off-Tokachi, 1968, earthquakes) exhibited similar mode shapes, they rejected their initial interpretation. Instead, they attributed the failure of the used homogeneous model to explain what actually happened in the field mainly to 'the nonuniform rigidity of the embankment.. . because of the different [amount of] consolidation within the dam body. . .'I2

Indeed when the dam is considered to be inhomogeneous with shear velocity varying with depth according to equation (14) and the present theory is used, the field performance can be better explained. Thus, from the geometry of the dam (Figure 10):

h = 2/193*82 = 0.01

and from equation (20):

a1 = 3.142 a2 = 6.284 and a3 = 9.426

VIBRATION OF EARTH DAMS 237

Substituting al and a2 in equation (23) yields the first two mode shapes:

and

which are also plotted (normalized to a unit crest displacement) in Figure 11. It is clear from Figure l l (a) that the agreement of equation (27a) with the observed first

resonant shape is satisfactory. This fact suggests that vertical inhomogeneity of the type proposed in this paper is indeed at least partly responsible for the acute decrease with distance from the crest of the recorded accelerations during the first resonance. Furthermore, it can be argued that owing to the geometry of the canyon (Figure 10) the ‘effective’ height of the dam for its lateral vibrations is actually smaller than its maximum height (37 m) considered in the analysis. Accounting for this phenomenon would lead to prediction of a sharper deamplification of the motion by the present theory. For example, an ‘effective’ height of 32 instead of 37m leads to much better agreement of the computed and recorded pattern of displacements, as shown in Figure ll(a).

In addition, notice that the seismometers were placed on the down-stream slope and not inside the dam (Figure 10). However, although the one-dimensional model assumes uniform distribution of horizontal displacements on any horizontal plane, a two-dimensional analysis of the vibrations of the dam cross-section would have predicted a sharper decrease with distance from the crest of the horizontal accelerations of the sloping surface than those of the central vertical axis of the dam. This can be seen, for example, in Figure 1 of Reference 6 in which the analysis was made by means of finite elements for a vertically homogeneous dam. It is, therefore, concluded that a milder deamplification with depth and, consequently, a better agreement with the presented theory would have been observed had the seismometers been placed near the centre of the dam.

Also plotted in Figure l l (a) is the mode shape of the solution by Rashid that was discussed at the begmning of the paper (1/6-power variation of the average velocity with depth). It is evident that the predicted deamplification of mode displacements with depth by this theory is not sharp enough to explain the observed behaviour, although the method constitutes an improvement over the classical homogeneous theory.

The second resonance occurred at a frequency, 4.2 cps, 1.5 higher than the first resonant frequency (243 cps). The shear-beam theory, though, predicts a frequency 2 times higher than the fundamental frequency for an inhomogeneous dam and 2.335 times for a homogeneous dam. This failure of the one-dimensional theories is hardly surprising since it is mainly vertical and not horizontal deformations that actually take place during the second r e s ~ n a n c e . ~ ’ ~ * ~ * ~ ~ It is indeed reported by Okamoto et al.” that at the frequency of 4.2 cps ‘the horizontal deformation is accompanied by a vertical component’. Naturally, therefore, the computed second mode shapes do not match the recorded second resonance shape (Figure ll(b)).

Notice, nevertheless, that the observed third resonant frequency, 5.4 cps, is 1.929 times higher than the observed first resonant frequency (2.8 cps). This suggests that the second natural mode of purely horizontal vibrations (1-Dim model) corresponds to and should be compared with the third rather than second observed resonance (2-Dim deformations). The mode shapes shown in the above mentioned Figure 1 of Reference 6 seem to support such an assumption. Thus again the inhomogeneous theory (predicting ratio of frequencies, 2) seems

’ to be in better accord with the observed behaviour (ratio 1.929) than the classical homogeneous

238 G. GAZETAS

theory (ratio 2.335). Unfortunately the third observed resonant shape is not reported in References 11 or 12, and thus the comparison cannot be completed.

The author realizes that ‘better’ prediction of the recorded field performance in this single case does not necessarily prove the adequacy of the method, especially for other types of embankment materials and of methods of construction. It does, nevertheless, provide justification for using the method in studying the dynamic behaviour and in evaluating the safety of earth dams during earthquakes.

CONCLUSIONS

‘Exact’, closed-form solutions for the one-dimensional vibrations of vertically inhomogeneous truncated-wedge-shaped earth dams have been obtained by implementing an inverse procedure in which the determination of the function describing the inhomogeneity constitutes part of the problem. The resulting shear-wave velocity varies approximately as the cube-root of the depth from the crest, provided that some truncation is present. Such a variation is likely to be more representative of actual field conditions than a uniform wave-speed throughout the dam, as is clearly indicated by the presented Japanese field measurements in Makio and Togo Dams, and by recent measurements with the Santa Felicia, California, Dam.

The results are presented in the form of natural periods and modal shapes for a number of truncation ratios. Comparisons with the periods and shapes of an ‘equivalent’ homogeneous earth dam, having the geometry and the average material properties of the inhomogeneous dam, reveal different behaviour of the two structures. Particularly interesting is the sharp increase of modal displacements near the crest of the inhomogeneous model, a trend that is also reflected in the variation with depth of the peak ‘average seismic coefficients’ acting on potential sliding masses.

The presented theory is finally evaluated by comparing its predictions with the recorded response of an earth dam in Japan. The agreement between observed and calculated first natural mode shape convinces for the usefulness of the theory. Inhomogeneity along the height of the dam seems to be an important factor in determining the response and in evaluating the safety of an earth dam. Other factors, however, such as the canyon geometry, the vertical deformation of the dam, as well as dam-foundation interaction, may also have an influence on the response and their proper consideration in the design (along with the inhomogeneity) seems necessary.

ACKNOWLEDGEMENT

The author wishes to acknowledge the help of Ahmed Abdel-Ghaffar who read an early draft of the paper and offered valuable comments.

APPENDIX I: ANALYTICAL EVIDENCE ON S-WAVE VELOCITY VARIATION IN AN ELASTIC WEDGE OF EARTH MATERIAL

It is well established that, in most soils (sands and clays), shear modulus is proportional to the square root of the effective normal octahedral stress, d,. As a result, in uniform and horizontal soil deposits, for example, the S-wave velocity is proportional to x1l4 (since c - GI/* and ab - x ) , where x is the depth from the surface. In earth dams, whose geometry can be approximated as a triangular wedge, it appears that crh and, consequently, c - x . Indeed, Figure 12(b) confirms this distribution for a uniform elastic wedge having 3Oo-slopes

1/3

VIBRATION OF EARTH DAMS 239

and a Poisson’s ratio Y = 0.3. To determine the spatial distribution of the normal octahedral stress, (TO = (cX +ay + mZ)/3, the results of elastic finite-element analysis published by Poulos and Davis” in the form of lines of equal stresses ( T ~ and uY were utilized. Owing to plane-strain conditions: mz = v(vX + ( T ~ ) and, thus, was estimated throughout the wedge. Figure 12(a) portrays the (normalized) variation of c across six horizontal lines, where c was computed as the fourth-root of (TO, according to the above arguments. Notice the relatively small differences in c from point to point across each horizontal line; with the exception of a small region near the slopes where c + 0, one can safely accept a more-or-less uniform variation of S-wave velocity at each level x from the crest. (This argument further supports the validity of the shear-beam model.) Upon averaging the velocities across each horizontal line, the points

C - cm

0.87

0 .94

1.00

AVERAGE c 4VERACE Cm

Figure 12. (a) Distribution of shear wave velocity along six horizontal cross-sections in a two-dimensional wedge under plane-strain conditions. (b) Variation with depth from the crest of the average shear-wave velocity over

horizontal planes

shown in Figure 12(b) are obtained. It is readily seen that the agreement with the cube-root-of- depth curve is excellent in this case, of an admittedly idealized earth-dam cross-section. The reader can also easily verify that along the vertical axis x, c increases in roughly the same manner (i.e. in proportion to x 1’3).

APPENDIX 11: AVERAGE SEISMIC COEFFICIENT

For a failure surface forming a triangular wedge with horizontal base at depth x = yH, the average seismic coefficient is given by:

240

in which:

G. GAZETAS

2 1 a, 1 -A2’ -sin [2a,(l -A2/3)]/2a, r, =- 3

where u ( f ) is ground acceleration time history. The other quantities involved have been defined in the text and the notation.

APPENDIX 111: NOTATION

The following symbols are used in this paper:

a,, d, = dimensionless frequency corresponding to the n th mode of vibration of the inhomogeneous dam (equations (18) and (20)) and the equivalent homogeneous dam (equation (24)), respectively.

p = critical damping ratio f, = participation factor of n mode

c = shear-wave velocity co = reference ‘speed’ (equation ( 5 ) )

E = average shear-wave velocity f = transformation function of x (equation (4b))

k = average seismic coefficient

cm, cm = shear-wave velocities at the base and crest of the dam

Jo, J1 = Bessel functions of the first kind, zero and first order

T, = n th natural period of the inhomogeneous dam ?;, = n th natural period of the equivalent homogeneous dam u =horizontal displacement x = depth (see Figure 1) y = x/H (see Figure 1)

y = parameter defining the variation of wave-speed with depth (equations (1 l), (13))

(I = transformation amplitude function of x (equation (4)) o = circular frequency

Yo, Yl = Bessel functions of the second kind, first and second order

a,, = n th modal shape

REFERENCES

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3. M. Hatanaka, ‘Fundamental consideration on the earthquake resistant properties of the earth dam’, Bull. 11

4. J . Krishna, ‘Earthquake resistant design of earth dams’, Earrhq. Eng. Sym., Roorkee Univ. (1962). 5. H. B. Seed and G. R. Martin, ‘The seismic coefficient in earth dam design’, J. Soil Mech. Found. Eng. Din, Proc.

ance’, Bull. Seism. SOC. Am., 50.45-56 (1960).

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VIBRATION OF EARTH DAMS 24 1

6. A. K. Chopra eral., ‘Earthquake analysis of earth dams’, Pmc. 4rh World Conf. Earrhq. Eng., Santiago, Chile (1960). 7. R. W. Clough and A. K. Chopra. ‘Earthquake stress analysis in earth dams’, J. Engng Mech. Div., Roc. Am.

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Eng. Res. Lab. Rep. EERL 78-02, Calif. Inst. of Techn. (1978). 10. W. 0. Keighley, ‘Vibrational characteristics of an earth dam’, Bull. Seism. Soc. Am., 56.6 (1966). 11. S. Okamoto er al.. ‘On the dynamical behavior of an earth dam during earthquakes’, Proc. 4th World Conf.

12. S. Okamoto, Introduction ro EarthquakeEngineering, Wiley, 1973. 13. N. N. Ambraseys and S. K. Sarma, ‘The response of earth dams to strong earthquakes’, Giorechnique. 17,

14. F. I. Makdisi and H. B. Seed, ‘Simplified procedure for estimating dam and embankment earthquake induced

15. W. M. Newmark, ‘Effects of earthquakes on dams and embankments’, GJorechnique, 15, 139-160 (1965). 16. H. B. Seed, ‘Stability of earth and rockfill dams during earthquakes’, in Embankmenf Dam Engineering, Eds.

Hinchfeld and Poulos. Wiley, 1973. 17. H. L. Schreyer, ‘One-dimensional elastic waves in inhomogeneous media’, J. Engng Mech. Div.. Proc. Am. Soc.

Civ. Eng., 103, No. 5 (1977). 18. H. B. Seed, C. Ugas and J. Lysmer, ‘Site-dependent spectra for earthquake resistant design’, Bull. Seism. Sac.

Am., 66,221-232 (1976). 19. I. Minami, ‘On vibration characteristics of fill dams in earthquakes’, Proc. 4rh World Conf. Earth. Eng., A-5,

Santiago, Chile (1969). 20. G. Gazetas and A. M. Abdel-Ghaffar, ‘Earth dam characteristics from full-scale vibrations’, Pmc. X Inr. Conf.

Soil Mech. Found. Engrg., Srockholm (1981) [accepted contribution of the U.S. National Committee]. 21. H. G. Poulos and E. H. Davis, Elastic Solutions for Soil and Rock Mechanics, Wiley, New York. 1974.

Earthq. Eng., Santiago, Chile (1969).

181-213 (1967).

deformations’, J. Geofech. EngngDiv., Proc. Am. SOC. Civ. Eng., 104, No. 7 (1978).