random vibration analysis for the seismic response of ... · the necessity for random-vibration or...

GAZETAS, G., DEACHAUDHURY, A. & GASPARM, D. A. (1981). Ghotechnique 31, No. 2, 261-277

Random vibration analysis for the seismic response of earth dams

G. GAZETAS,* A. DEBCHAUDHURY* and D. A. GASPARINI*

The randomness of earthquake ground motions and the sensitivity of earth dams to details of the excitation make random vibration methods of analysis attractive and economical tools with which one can directly predict statistics of the response to potential earthquakes. A new random vibration formulation is introduced in this Paper and employed to study characteristics of the dynamic behaviour of earth dams modelled as inhomogeneous shear beams and excited by strong motions consisting of vertical shear waves. Unrealistic simplifying assumptions of classical random vibration theories are avoided with this method which properly accounts for the frequency content and time evolution in intensity of the excitation. Results are presented in the form of variation with time, and distribution with depth from the crest, of statistics of displacements, accelerations, shear strains and seismic coefficients on potential sliding masses. Key factors that influence the dynamic behaviour are identified and their effect is demonstrated through a number of parametric plots. A set of dimensionless graphs is finally presented whereby one can readily obtain estimates of peak accelerations, seismic coeficients, displacements and strains to be experienced by a dam during an earthquake whose maximum acceleration and predominant frequency have been evaluated. These plots can be useful engineering tools in preliminary design calculations.

Le caracttre altatoire des mouvements provoques par des tremblements de terre ainsi que la sensibilite des barrages en terre aux caracteristiques de l’excitation font des methodes d’analyse bastes sur des vibrations aleatoires des outils tconomiques et attrayants permetant de prevoir directement les statistiques de la response aux tremblements de terre potentiels. Cet article prisente une nouvelle formule pour le calcul des vibrations aleatoires qui sert a etudier les caracteristiques du comportement dynamique de barrages en terre mod&es sous fonne de poutres de cisaillement non homogtnes et excites par d’importants mouvements composes d’ondes de cisaillement vertical. Cette methode permet d’tviter les hypotheses peu realistes et simplificatrices que I’on rencontre dans les theories classiques sur les vibrations aleatoires car elle tient bien compte des frequences et de l’evolution, en fonction du temps, de I’intensitt de l’excitation. Les resultats sont present& sous la forme de variations en fonction du temps et de la distribution en fonction de la profondeur a partir de la Crete des statistiques de dtplacements, d’accelerations, de deformations de cisaillement et de coefficients sismiques concernant des masses glissantes potentielles. Les facteurs cles qui influencent le comportement

Discussion on this Paper closes 1 September, 1981. For further details see inside back cover. *Case Western Reserve University, Cleveland, Ohio.

dynamique sont identifies et leurs effets sont illustres a l’aide dun certain nombre de traces parametriques. Enfin, I’article prisente une serie de graphiques adimensionnels permettant d’obtenir facilement des previsions relatives aux accelerations maximales, aux coefficients sismiques, aux deplacements et deformations que subira un barrage lors dun tremblement de terre dont I’acceleration maximale et la frequencc predominante ont Ctt evaluees. Ces tracts peuvent itre trts utiles a l’ingenieur pour les calculs preliminaires de projets.

NOTATION

square root of mean squared (r.m.s.) absolute acceleration maximum r.m.s. acceleration within a dam (i.e. crest acceleration). average shear wave velocity of a dam r.m.s. relative displacement maximum r.m.s. displacement within a dam (i.e. crest displacement) shear modulus maximum shear modulus, at the base of the dam height of the dam r.m.s. seismic coefficient power spectral density (PSD) function of frequency 0 fundamental natural period of the dam r.m.s. horizontal shear strain maximum r.m.s. horizontal shear strain within a dam critical damping ratio of the dam in the ith mode ground critical damping ratio ith natural period of the dam predominant ground frequency

INTRODUCTION

The last decade has witnessed significant advances of random vibration theory as it is related to problems of structural dynamics (Clough & Penzien, 1975; Vanmarcke, 1976; Clarkson, 1977). Yet, as Christian (1979) notes in a recent state-of- the-art paper, few attempts have been made to introduce random vibration methodology into the broad field ofsoildynamics. These attempts include: the works of Donovan (1971) and Faccioli (1972) who studied the liquefaction potential of a site using the Palmgren-Miner fatigue criterion with an

261

262 G. GAZETAS, A. DEBCHAUDHURY AND D. A. GASPARINI

assumed probability distribution of the peak seismic shear stresses (Rayleigh distribution); the work of Faccioli (1976) who developed an equivalent linear random vibration formulation to study the one-dimensional amplification of soil deposits with Ramberg-Osgood-Masing constitutive relations; and the soil-structure interaction study of Romo-Organista, Lysmer & Seed (1977) who used a finite element code to generate deterministic transfer functions and then employed classical random vibration theory to derive root-mean-squared (r.m.s.) and peak values of the response.

It appears that the only attempt so far to use probabilistic techniques in assessing the seismic stability of earth dams has been reported by Singh 8~ Khatua (1978). They developed a random vibration methodology that utilizes stochastic linearization and performs an iterative, step-wise linear analysis in connection with a finite element discretization of the dam. Thus statistics of the earthquake-induced shear stresses are obtained, and estimates are made of the expected cumulative liquefaction damage using the aforementioned fatigue criterion and laboratory data in a semi- probabilistic fashion.

Several possible explanations can be offered for the apparent reluctance of the geotechnical profession to apply random vibration methodology in soil dynamics problems. Besides the fact that, in general, engineers do not have the same familiarity with the concepts of random vibration as they do with deterministic dynamics, one must not forget that the earthquake excitation at a particular site is currently described deterministically, e.g. through a design response spectrum or a set of recorded accelerograms. This makes the methods of deterministic analysis (response spectrum or time- integration) even more attractive. More decisive, however, are the following two alleged limitations of the classical random vibration theory. The first is that only simple dynamic models can be studied with the theory; thus, in many cases, important aspects of the problem are at best only crudely reflected in the analysis (e.g. inelastic response). In contrast deterministic methods can handle the most sophisticated dynamic models. The second is that the majority of available random vibration solutions are based on the assumption of ‘stationarity’ (i.e. the vibrations are endless) at least for the earthquake excitation. However, earthquakes are of transient nature and the ground motion initially builds up and eventually decays with time. The response of flexible structures (like tall earth dams) may be particularly sensitive to such an unrealistic assumption of stationarity, especially for motions of not very long duration (e.g. motions not far from the epicentre).

Here and elsewhere (Gazetas, DebChaudhury & Gasparini, 1981), a study of the seismic response and safety of earth dams is presented using a novel random vibration methodology that directly obtains statistics (averages, standard deviations, probability distributions) of pertinent response quantities due to stochastically described earthquake excitations. The method avoids to a large degree the aforementioned drawbacks of some of the presently used random vibration theories, in that it models the dam essentially as accurately as current deterministic methods do and it treats the input earthquake motion as a non-stationary process characterized by a finite build-up time, a period of more or less uniform intensity and a period of decay.

The necessity for random-vibration or probabilistic type analyses is first illustrated in the Paper through two case-histories followed by a brief description of the dynamic model and the method of analysis. More emphasis is placed on the presentation of a series of studies whereby the importance of key parameters characteristic of the earth-dam or ground-excitation modelling is demonstrated. It is believed that these studies can provide the designer with a significant insight into the dynamic behaviour of embankment dams and an appreciation of the factors that control their safety. Moreover, dimensionless plots are offered (Figs 11-13) that can be readily used in probabilistic or even in preliminary deterministic analyses to evaluate r.m.s. or peak accelerations, displacements, shear strains and seismic coefficients on potential sliding masses. The Authors believe that these plots will be useful engineering tools in the hands of the profession.

WHY RANDOM VIBRATION ANALYSIS OF EARTH DAMS?

In evaluating the dynamic performance of earth dams the engineer is faced with uncertainties in all steps of the analysis. As a result, a wide range of stresses, displacements and other response quantities can be determined depending on which analytical procedure is used, what properties are assigned to the soil and what accelerograms are chosen to describe the input motion. Even when one single analytical model with fixed soil properties is used, one can get a significant variability in the response due to a number of exciting accelerograms, although the latter may have a common peak acceleration, common predominant frequency and common duration. Consequently, in order to predict the seismic safety of an embankment dam reliably it is necessary to perform a series of analyses using several input motions to obtain a possible range of responses and estimate the maximum response. Random

RANDOM VIBRATION ANALYSIS OF EARTH DAM SEISMIC RESPONSE 263

Table 1. Characteristics of five Japanese eartbquakes

Earthquake Epicentral distance: km

230 0.75 390 300 1.75 5.40 150 0.83 3.75 240 6QO 19.50 240 2.60 7.00

Maximum Maximum ground crest

acceleration: acceleration: gal gal

(4

100 r : . .

O’C Frequency: cycles/s

(b)

Fig. 1. Variability in the elastic response of the Sannokai Dam in Japan to five earthquakes (from Okamoto et al., 1969)

vibration analysis can provide this maximum response directly and thus offers an economic alternative to multiple deterministic analyses.

To demonstrate the dependence of response quantities, such as accelerations and permanent deformations, on the unpredictable details of the excitation, two case studies are briefly presented, involving the recorded motions at the Sannokai Dam and the calculated permanent deformations of a rigid block sliding on a vibrating plane.

The vibrations of the Sannokai Dam, an earthfill dam in Japan which is 37 m high by 140 m long, were recorded during five distant earthquakes whose characteristics are shown in Table 1 (from Okamoto et al., 1969). Figure l(b) plots the ratio (FR) of the Fourier Amplitude Spectra calculated from accelerograms perpendicular to the dam axis, recorded at the crest (A) and the abutment(B). For each particular frequency, FR expresses the

amplification of a corresponding sinusoidal motion when it is ‘filtered’ through the dam; it depends mainly on how close this frequency is to the natural frequencies of the dam, on the effective internal damping of the soil and the details of the excitation. As can be seen from Table 1, the five earthquakes occurred at similar long distances from the dam and produced very small and not very different peak accelerations. Therefore, the dam is expected to have deformed in the linear elastic range and only minor differences in soil damping and dam frequencies should be anticipated. Yet, the scatter of the recorded FR spectrum is huge (Fig. l(b)) and is primarily attributed to differences in the details of the five ground motions.

Figure 2, adapted from Franklin & Chang (1977), displays the residual relative displacements of a rigid block sliding on a vibrating plane ground- surface as a function of the ‘yield’ acceleration (i.e. the maximum acceleration that can be transmitted before the frictional capacity of the interface is exceeded). The ground motion consists of nine accelerograms recorded during the San Fernando 1971 earthquake on soil sites located at epicentral distances ranging from 28.6 to 40 km, as shown in Fig. 2. The records are normalized so that they all have common peak acceleration (0.5g) and common peak velocity (30 in/s). Thus the nine records have practically identical peak parameters, predominant frequency and significant duration and, with the present state of knowledge, they can be considered as equivalent.

Nevertheless, the variability in the resulting permanent displacements is enormous and is solely due to the different (but unpredictable)details of the scaled motions, since the characteristics of the dynamic system (i.e. the mass of the block and the yield acceleration) are uniquely defined. The above method is the one used in assessing the permanent deformation of potential sliding masses in earth dams, with the time-histories of the average acceleration used as ground excitation (Newmark, 1965; Sarma, 1975; Makdisi & Seed, 1978).

Thus it is obvious that analyses based on a single design accelerogram can lead to anything from very conservative to unsafe earth dams. On the other


700

600

c 500 . ‘” 5 E 8 m 400 a .E D

c” 0 u, 300 E ,”

E

$ 200

100

0

(

I I /

1 0,025 005 010

70-

0

60- q

Symbol ;alteck record

DO56 FO81 FO88 DO57 DO58 E072 DO59 DO65 E083

[IMy1

0 15 0.25 0.50 I

, E c

I :picentral hstance:

km

28.6 32.9 34.1 37.1 39.5 39.8 40 ,o 40.0 40.0

fv&x~mum rwstance coefflclent

Maximum seIsmlC COeffIClent

Fig. 2. Variability in permanent sliding displacements of a rigid block subjected to nine records of the San Fernando 1971 earthquake which were scaled to have common peak velocity and common peak acceleration (adapted from Franklin & Cbang, 1977)

hand, deterministic time-history analyses even with a limited number ofground excitations (e.g. three or four) is an expensive operation and can give confidence only in the average (expected) response-not in the probability of rare events which are of greatest engineering interest. Instead random vibration analysis offers a viable economic alternative and deserves special attention.

ELASTIC SYSTEM AND METHOD OF ANALYSIS Dynamic model of dam

It is desirable to use a simple but reliable dynamic model for the lateral vibrations of earth dams during earthquakes. To this end the so-called shear-slice model (Ambraseys, 1960a) has been preferred to a more accurate two-dimensional plane-strain finite element formulation, although the developed random vibration method can be readily applied with both models. The chosen model treats the dam as a one-dimensional shear

beam with variable cross-section (triangular or truncated-wedge) and yields horizontal accelerations and shear stresses due to uniformly distributed motions at the base. The two key assumptions of the model are that only horizontal shear deformations take place and the resulting shear stresses are uniform on any horizontal plane. Although the latter assumption violates the physical requirement of zero normal and shear stresses on the two slopes of the dam, and the former neglects the tensile and compressive stresses that develop due to wave-reflections at the faces of the dam, the shear-slice model predicts with satisfactory accuracy horizontal accelerations, displacements and shear stresses.

To account for the variation of soil modulus G with effective confining pressure c,,’ a new version of the shear-slice model has been adopted in our analyses, hereafter referred to as inhomogeneous shear model (Gazetas, 1981a, b). This model


Table 2. Natural frequencies, mode shapes and participation factors of the inhomogeneous and the homogeneous shear beam theories

Quantity Inhomogeneous beam: Homogeneous beam:

G = G, i2’3, p = constant G = constant, p = constant

Natural frequency I w.=L4;

Modal displacement shape I

U, = &sin[nn(l-il”)] I u, = Jo@. 3

Modal shear-strain shape

y. = - & &(sin [nx(l - Zzis)]

+ nni2’3 cos [wc( 1 -V)]

Modal seismic coefficient shape I

k n = 3 Jc1+41.2) I k, = $3(1+41,2)

Participation factor I I-” = 2

nn I I

Z. = z/H; c = ,/(c/p), (? = average modulus = 6/7G,; b. = nth root of Jo(b) = 0; J, and J, = Bessel functions of the first kind, 0 and 1 brder.

considers the average G across a horizontal plane as an increasing function of the distance from the crest

G(z) = G,(z/H)“~ (1)

in which G, = the average modulus at the base (i.e. at z = If). This relationship has been substantiated by considerable direct and indirect field evidence, i.e. measurements of s-wave velocities and observations of the response of dams during earthquake or man-induced vibrations (Abdel-Ghaffar & Scott, 1979; Okamoto et al., 1969; Gazetas, 1981a, b).

Closed-form solutions have been obtained for the natural frequencies, modal shapes of displacements, strains and seismic coefficients as well as participation factors of an embankment dam whose modulus varies according to equation (1) and has a constant material density. Table 2 portrays the corresponding formulas and compares them with results of the homogeneous shear beam theory (Ambraseys, 1960a).

Ground excitation The simplest possible stochastic model for a

seismic ground motion is the so-called stationary white noise (SWN) whose intensity and frequency content does not vary with time and which contains all frequency components equally (constant power spectral density (PSD)). Because of its simplicity it has enjoyed significant popularity in spite of the fact that it only crudely approximates reality. Two

significant improvements over the SWN have been adopted herein.

Firstly, the frequency content is described through a Tajimi-Kanai PSD function’

1+ 4ifz(w/~,)z _~ s(w) = [ 1 - (w/w#]2 + 4~,*(w/w,)* so (2)

where cr and wf are parameters to be chosen in order to fit the local data and So is a measure of ground intensity. Equation (2) is plotted in Fig. 3(a). Because of the similarity of equation (2) with the formula that gives the acceleration of a 1-dof oscillator having natural frequency u.+, viscous damping ratio & and excited by a sinusoidal motion So sinwt, wf and ir may be interpreted as the predominant ground frequency and the ground damping, respectively. By analysing 140 accelerograms Lai (1979) found of to vary from about 5.7 rad/s to 51.7 rad/s while & ranged from 0.1 to 09.

Secondly, the intensity of motion may vary with time in any specified way; thus slow build-up and decay ofseismic intensity can be treated realistically. Figure 3(b) shows two intensity envelopes that were used in the analyses reported herein. One is intended to represent a typical earthquake, while the other (boxcar) simulates a suddenly applied stationary excitation. The latter is commonly assumed in most classical random vibration methods of analysis.

I The concept of PSD function is clarified in Appendix 1.


Kanai-Tajiml psd function

WhItenoIse psd function

wiw, (4

Ttme: s

ib)

Fig. 3. Stochastic description of ground excitation: (a) power spectral density function; (b) intensity envelopes

Method of analysis

The random vibration formulation described by Gasparini (1979) and Gasparini & DebChaudhury (1980) was used directly to obtain evolutionary statistics of the response of earth dams subjected to random excitation of the form described above. It is fundamentally the time domain stochastic approach originally introduced by Wang & Uhlenbeck (1945) and further developed in connection with optima1 control theory (Bryson & Ho, 1969); only a bare outline of the theory is presented here.

The method is analogous to a conventional modal time-history analysis. A dam is decoupled into modes and modal state vectors are defined consisting of modal displacement and velocity. Then a set of differential equations are formed governing the time-evolution of the modal mean vector and the modal covariance matrix (i.e. the modal state vector statistics). Exact analytical solutions of these equations for non-stationary excitation have been derived by Gasparini (1979) and Gasparini & DebChaudhury (1980). Therefore modal responses to any non-stationary excitation can be obtained analytically.

Once evolutionary modal state vector statistics are computed, any response can be obtained by modal superposition. For earthquake excitation, the response quantities fluctuate around zero mean values; evolutionary variances of responses are the only non-trivial statistics. For any zero-mean process d the following relationship is true.

Var(d) = E[d*] = dT (3)

where the symbol E[ ] means expected (i.e. mean)

value and d is defined as the standard deviation or r.m.s. value of the process d. The r.m.s. value is the most significant statistical measure of a response quantity and knowledge of its variation with time is the key to estimating the probability distribution of maximum response (Crandall & Mark, 1963; Vanmarcke, 1976).

The developed formulation is exact in the sense that no simplification is made in order either to form or to solve the system of governing differential equations. Thus the method is as accurate as the currently used deterministic dynamic procedures that are based on modal superposition (e.g. Ambraseys & Sarma, 1967; Makdisi & Seed, 1978).

Problem parameters

A series of analyses have been performed and are reported in order to study the evolution with time and the distribution in space (within the dam) of r.m.s. values of pertinent response quantities such as displacements, accelerations, shear stresses and strains, and seismic coefficients on potential sliding masses. The following parameters related to the ground excitation or the dynamic model of the dam are identified and their influence on the response is studied in detail.

(a) the ‘predominant ground frequency’ wr which defines the location of the peak in the PSD function; an extreme value is wr = co that transforms the KanaiLTajimi into a white- noise spectrum

(b) the ground damping & which controls the sharpness of the peak in the PSD function

(c) the intensity envelope which determines the type of non-stationarity under consideration

(d) the fundamental natural period of vibration of the dam: T= 2.58H/C

(e) the type of variation of the soil modulus G within the dam (inhomogeneous or homogeneous models)

(f) the internal damping ii corresponding to each mode i of the dam

RESPONSE CHARACTERISTICS

The scope of this section is threefold: to demonstrate the analytical capability of the developed non-stationary random vibration theory; to present a comprehensive study of the previously identified parameters that affect the response; and to develop dimensionless plots that can be used in preliminary engineering analyses quickly to obtain estimates of accelerations, displacement and stresses in the dam. To this end a study is made of the response characteristic of a typical 90m tall earth dam subjected to strong stochastic ground excitations, all of which have a common total duration of 20s and a common intensity of PSD function, S, = 200cmZ/s3. The predominant


Y I I

0 10 20 Time: s

(a) lb)

11

d: cm 0 2 4 6

0.2

0.4

’ 0.6

0.8

1

1

a: g

0 0.1 0.2 0 ,3

z 0.4 Z

0.6

0.8

0.4 -?

0.6

7 !

0.8

Fig. 4. R.m.s. values of displacements, accelerations, shear strains and seismic coefficient of a 90 m-tall earth dam subjected to strong stochastic excitations having different frequency contents

268 G. GAZETAS. A. DEBCHAUDHURY AND D. A. GASPARINI

frequency or, the ground damping & and the intensity envelope are variables whose significance is investigated. A recent analysis of 140 seismic records (Lai, 1979) suggests that representative (i.e. most probable) values for wr and [r are 67~ rad/s and 40x, respectively. Excitations whose PSD function corresponds to the above values ‘of wr and & are thereby considered as standard motions agsainst which the effects of other wr and <r values are compared and evaluated.

To get an idea of how strong are the above ground motions (i.e. with S, = 200cm2/s3, or = 67r, ir = 0.40), one should roughly estimate the standard deviation of their acceleration and velocity. From Crandall & Mark (1963)

CT 1 a [ qfq1+41f2)

f 1 112

=O.l12g (4a)

and

Gl [ 1

l/Z

uN 4w,i, Y 1: 4.6 cm/s (4b)

As a comparison, for example, the motion recorded on the Caltech DO56 accelerograph during the San Fernando 1971 earthquake (M = 6.6) at an epicentral distance of 29 km, had S, N 185.87 cm2/s3, 0,-6.6rr, [,=0.41, u==O.l12g, 0,2:5.3cm/s. Clearly, this record could be considered as a sample of the stochastic excitation used in our analyses. The peak acceleration and velocity of the Caltech record was approximately 0.3Og and 23cm/s, respectively. It is therefore a strong motion record, similar to the El Centro 1940 NS record whose corresponding maxima were 0.33 g and 34 cm/s.

Consistent with the intensity of excitation, a shear wave velocity of 180m/s and a first mode critical damping ratio of 15% were selected for the dam. Somewhat larger damping ratios were assigned to the higher modes in accordance with a large body of experimental evidence (Okamoto et al., 1969; Ambraseys & Sarma, 1967; Abdel-Ghaffar & Scott, 1979). In summary, therefore, the results reported herein apply to typical tall earth dams subjected to strong earthquake ground motions.

Time-evolution of r.m.s. values

Figure 4(a) shows a typical output of the random vibration theory: the evolution with time of the r.m.s. values of the top displacement, top acceleration, maximum shear strain and seismic coefficient on a very shallow potential sliding mass. Clearly such plots give a more complete picture of the response than a single steady-state r.m.s. value acting at an equivalent stationary strong-motion duration that the classical random vibration theory provides (Vanmarcke, 1976; Clough & Penzien, 1975). Three curves are shown in each plot, corresponding to three different values of the pre-

Y

Fig. 5 Significance of the relative magnitudes of predominant ground frequency and fundamental frequency of the dam

dominant ground frequency wf = rr, 6~ and co, and a single intensity envelope (also shown at the bottom of the first plot for easy reference). The following conclusions can be drawn.

Firstly, predominant ground frequency wf has-a very small effect on the top relative displacement d, and maximum shear strain $,,, whereas its influence is appreciable on the top absolute acceleration a^, and (consequently) the near-the-top seismic coefficient k. Qualitatively to explain these differences one should observe thatthe fundamental frequency of the dam, wi = (77r/9) C/H N 1,56n, lies between n and 67~. Thus, as can be judged from Fig. 5, only the fundamental mode of the dam receives considerable amount of seismic energy during the wf = rc motions. Instead the three higher modes appear to get most of the input power during the standard, wf = 67c excitation. When it is remembered that higher modes are more significant in determining maximum accelerations rather than displacements (e.g. for a given maximum displacement S, maximum accelerations increase with frequency, S, = w2 S,, i.e. they become larger for higher modes), the differences in Fig. 4(a) are obvious. The wf = co (i.e. the white-noise) PSD function leads to r.m.s. responses that are similar to those of the standard function. This is rather accidental, however, since in this case the two PSD functions are nearly equal at the fundamental frequency of the dam, the standard PSD is larger at the 2nd, 3rd and 4th natural frequencies, while the white noise is larger at the 5th and 6th frequencies (only six modes were used in all the analyses).

Secondly, the magnitude ofthe induced deformations is large, as expected from such strong motions. For example, the r.m.s. value of the maximum shear strain f,,, is of the order of O.l’A. In order to use the published experimental data regarding strain- dependence of soil modulus and damping, an equivalent sinusoidal shear strain amplitude ye must be obtained. To this end the shear-strain


energy of the equivalent uniform motion is made equal to the average expected value of strain energy of the stochastic response. In mathematical terms

1 z fy,’ = - s

EC?(t)1 dt 7 0

where T = 20 s is the duration of motion. Numerical evaluation of the above integral, with E[y2(t)] = [$t)12 corresponding to the standard excitation (Fig. 4(a)), is straightforward and leads to y, N 0.09%. Using the experimental damping-strain curve suggested by, for example, Fig. 8 of Makdisi & Seed (1978), one can read for the above value of strain an equivalent damping ratio: <, N 1.5%, i.e. as high as the first modal damping in the Authors’ analyses. Similarly, the modulus reduction against y curve from Makdisi & Seed (1978) yields for the equivalent modulus G,-G,/3, where G, is the small strain (< 10m4 %) modulus. Since an s-wave velocity of 180m/s was assumed here, one can postulate that an initial velocity of approximately

180 J!ir 3 12 m/s would be necessary to make the parameters used in the analyses strain-compatible. Indeed, 312 m/s is a very realistic velocity for a 90 m tall modern earth dam. The 83.3m high Santa Felicia Dam in California, for example, was found to have an average s-wave velocity ofabout 300 m/s (Abdel-Ghaffar & Scott, 1979). The results reported here are based on analyses which used moduli and damping ratios consistent with the high level of shear strains (N 10m3) that developed under strong stochastic excitation.

Thirdly, because of the high damping in the system (c, = 15x, & = 18%) stationarity is almost reached during or shortly after the constant intensity part of the excitation (i.e. the horizontal branch ofthe intensity envelope). This is reflected in the nearly time-independent responses approximately 4-6 s after the beginning of the motion.

Spatial distribution of r.m.s. values

Figure 4(b) portrays the variation with depth from the crest of the largest r.m.s. values of the relative displacement, absolute acceleration, shear strain and seismic coefficient on potential sliding masses. Again three curves are shown in each plot, corresponding to wf values of 71,6x and co and the same intensity envelope. The following can be seen.

Firstly, predominant ground frequency wf has a strong influence on the exact distribution within the dam of accelerations, strains and seismic coefficients while its effect on the distribution of relative displacements is negligible. There is sharp amplification near the top of the dam of r.m.s. accelerations and seismic coefficients, during the standard and the white-noise (0, = 6n and co) excitations. The top-to-bottom amplification ratio (AR) of the r.m.s. accelerations

AR = 6,/ci, (6)

is about 4, with both excitations. On the other hand, the or = n spectrum leads to a very slow attenuation of accelerations with depth and AR is a little over 2. Recognizing the relative importance of the higher modes versus the fundamental mode is the key to explaining the above differences. The higher modes have a whip-lash effect on the dam and whenever they receive considerable seismic energy (as, for example, with the wI = 67~ and wf = 03 motions) sharp amplification of accelerations occurs in the upper third of the dam.

The need properly to assess the frequency content of the design excitation (i.e. of of the PSD function) is vividly demonstrated by the distribution of seismic coefficients with depth. The designer might not even suspect the possibility of shallow, near-crest sliding failures or intolerable permanent deformations ifhe bases his analyses on earthquake motions having predominant frequencies less than, or about equal to, the fundamental frequency of the dam.

Secondly, the r.m.s. values of the maximum shear strain f,,, were previously found to be relatively insensitive to wr (Fig. 4(a)). But the depth from the crest at which f,,, occurs is different for the three PSD functions, as seen in Fig. 4(b); it is about 0.20H for wf = 671,0.35H for wf = 03 and 030H for wI = n. The variation of $ with depth becomes increasingly uniform as its maximum moves away from the crest. Thus if one neglects the very small values of y just under the crest, the error committed by replacing the actual distribution ofstrains with a constant strain throughout the dam is smallest with the wf = 71 excitation and largest with the wf = 67~ excitation. But even in the latter case the error is quite small, less than 20%. This observation has a practical significance; it implies that shear modulus and damping ratio change approximately uniformly with depth during a strong earthquake. Hence the initial variation of modulus with depth (equation (l)), which is the basis of the dynamic (inhomogeneous) model, remains valid even when large non-linear deformations take place. One can therefore readily apply the presented theory to perform strain-compatible equivalent linear analyses using experimental G/G, and c against y curves (such as those used in Makdisi & Seed, 1978) without a need to resort into expensive finite element formulations.

Effect of intensity envelope and material damping

The effects of the two envelopes shown in Fig. 3(b) were studied in parallel with the effect of modal damping ratios ii of the dam. Figure 6 summarizes the results in the form of evolutionary r.m.s. values of crest displacements and accelerations experi-

270 Ci. GAZETAS, A. DEBCHALIDHURY AND D. A. GASPARINI

Time: s Time: s

Fig. 6. Effects of soil damping and intensity envelope on evolutionary r.m.s. responses

Fig. 7. Effect of ground damping on the variation of r.m.s. acceleration with depth

enced by the previously described 90m high dam having four different values of fundamental modal damping, 0.015, 0.05, 0.15 and 0.25, and subjected to the standard stochastic excitation. Thus the complete range of possible damping ratios and possible time-variations in the intensity of potential ground motions has been investigated.

Except when damping ratios are unusually small (less than 573, stationary responses are nearly reached with both intensity envelopes. However, with the standard envelope near-stationary values develop only for a very short time interval, after the constant intensity part of the excitation. With the boxcar envelope stationarity is achieved shortly after the initial build-up period and persists for the rest of the motion. This difference is significant in relation to questions regarding peak values, permanent deformations and fatigue-type damage that

are considered elsewhere (Gazetas et al., 1981). Accelerations reach stationarity faster than dis-

placements do, especially at low damping ratios. For example, at [I = 0.015, even with the boxcar intensity envelope, top displacement increases monotonically during the 20 s of excitation, never achieving stationarity. In contrast, top acceleration remains practically constant after 6 s of shaking.

Finally, it may be observed that, for the standard intensity envelope, the higher the damping is, the steeper the attenuation with time of the last portion of the evolutionary r.m.s. values becomes.

Effect of ground damping

Since if controls the sharpness of the peak in the input PSD function one expects its influence to be dependent on the location of the fundamental frequency w, of the dam relative to the predominant ground frequency wr. Thus, with the wf = 6a excitation, as & decreases the PSD function becomes sharper and the top response increases. The opposite is true with the wf = II excitation. In general, however, the discrepancies resulting from different if values are small and for all practical purposes should be neglected. Figure 7 compares the spatial distribution of r.m.s. accelerations experienced by the standard dam when subjected to wI = 67t stochastic motions with three different values of if, 0.20, 0.40 and 060. The differences are too small to deserve any further discussion here.

Effect of soil inhomogeneity

Figures 8 and 9 demonstrate how important it is to account for the proper variation of soil modulus within the dam, as was done in this work by using equation (1) to describe the increase of G with mean effective stress. The two figures compare the time- evolution and spatial distribution of the r.m.s. responses of two dams modelled either as inhomo-


I = Inhomogeneous dam G = G,i2j3

H = Homogeneous dam G = constant

z

2 cm

0 2 4 6

0.2

0’4

0.6

0.1

F%

c 2 0.05

n k

0 0.1 0.2 0.: II II

0.21 y/ I

0.8

10

Time: s

(a) (b)

0.8

1

y^:% 0 005 0 .l

Fig. 8. Comparison of r.m.s. responses of homogeneous and inbomogenews dam (H = 90 m; c’ = 180 m/s)


I = Inhomogeneous dam G = G, 72/s

Ii = Homogeneous dam G = constant B-

E 2-

‘G

0.6 -

i: cm

0 1 2 3 4

H

; 0.5 r 1

;: g

“l-+7-

z” 05

1 !

0 0.05 0.1

H T Time: s

(a)

Fig. 9. Compnrisoa of r.m.~ responses of homogeneous and inbomogeneous dam (If = 30m; c = 18Om/s)


H: m H: m L 1 I I t I I I 0 1 2 0 1 2

T,: s T, : s

Fig. 10. Crest acceleration and crest displacement spectra for four different stochastic excitations

geneous or as homogeneous shear beams with the same average soil properties. The two dams have the same average s-wave velocity (C = 180 m/s) and the same modal damping ratios (ii = 0.15, cZ = 0.18, etc.) but they differ in height. The first is a 90m tall dam and thus has a fundamental frequency w1 N 1.56 rad/s (i.e. it is the standard dam examined so far); the second, being only 30 m high, has a frequency three times larger (i.e. o 1 N 4.70 rad/s).

The formulas that give the natural frequencies, modal displacements, strains, seismic coefficients and participation factors of an inhomogeneous and a homogeneous dam are presented in Table 2. Gazetas (1980a, b) gives a more detailed discussion on differences and similarities in natural frequencies and shapes of the two models. Figures 8 and 9 compare the responses of the two models to the standard stochastic excitation. The message is clear, irrespective of dam stiffness.

Firstly, the homogeneous model seriously under- predicts near-crest displacements, accelerations and seismic coefficients (sometimes by more than 50%). However, it predicts similar or slightly higher responses at the lower half of the dam.

Secondly, the shear strain in the homogeneous dam increases almost linearly with distance from

the crest up to a depth of about 3/4 of the dam height, remaining practically constant thereafter. In contrast, in the inhomogeneous dam, the shear strain attains its maximum at about l/4 of the height from the crest, slowly decreasing thereafter. Thus one can safely approximate the actual pattern of the shear strain in this dam with a uniform distribution (i.e. constant strain throughout the dam), whereas such a simplification with the homogeneous model would seem rather arbitrary. This concept is further considered in Gazetas et al. (1981).

On the basis of the above comparisons and the field evidence presented by Gazetas (1981a) the Authors believe that continuing use of the homogeneous model in practice is unjustified, especially in view of the simpler formulas of the inhomogeneous theory (sinusoids against Bessel functions).

Effect offundamental period of the dam

The resilience of a dam can be conveniently represented by its fundamental period of oscillation T1 which is a function of the average s-wave velocity C and of the height H of the structure

Tl = 2.58H/C

Higher modes will have periods T” at fixed ratios of

G. GAZETAS, A. DEBCHAUDHURY AND D. A. GASPARINI

4-

Fig. 11. Normaliied crest accelerations and displacements versus the dimensionless period T, wp (soil damping 15%)

. * kli,

0 o-5 1

; 05

Fig. 12. Variation of seismic coefficient with height of potential sliding mass for various values of T, w(

T,, T. = TJn (see Table 2), so that for a given value of Ti the full spectrum of the higher natural periods of the dam is known.

Figures l&13 summarize the results of a series of studies aimed at revealing the dependence of accelerations, displacements, strains and seismic coefficients upon the fundamental period of the dam, for a wide range of stochastic excitations.

In Fig. 10 are displayed r.m.s. displacement and acceleration spectra for four excitation PSD functions (wr = n, 2rr, 67~ and co; ir = 0.40; standard intensity envelope) and one set of modal damping ratios (ii = 0.15, [I = 0.18, [3 = 0.23, c4 = 0.30, is = 0.35 and & = 040). It can be seen that relative

01 I 1 107 2on

Tl Wf

Fig. 13. Normalized peak r.m.s. shear strain versus the dimensionless period TI wf

displacements d^ increase monotonically with Ti and the shape of the 2, against T1 curves is insensitive to the frequency content of the input motion (i.e. to wr). On the other hand, the shape of the r.m.s. absolute top acceleration spectrum, &, against T,, changes with the predominant ground frequency c+. In the range of major engineering interest, high dams will experience smaller top accelerations than low dams during seismic motions with or >4rc, whereas almost the opposite seems to be true during motions with very small wr (< 27r).

In order to account directly for the dependence of the top acceleration spectrum on wr, the dimensionless amplification ratio AR, defined in equation (6), is plotted in Fig. 11 as a function of the dimensionless frequency parameter Ti wr. The results fall within a relatively narrow band and consequently a single amplification spectrum can be approximately constructed by drawing the average curve among the data points. The very existence of such a spectrum is of appreciable practical significance. It can be used not only to obtain r.m.s. values in a probabilistic analysis but also directly to estimate the peak crest acceleration in a simplified deterministic analysis if the design peak ground acceleration abmax - and the predominant ground frequency wC are known. Although AR is a ratio of r.m.s. values rather than peaks, it can be argued that approximately

for a particular dam (see Vanmarcke, 1977). Thus, Fig. 11 can be used even for deterministic seismic response analyses. If no information is available on the basis of which estimates of of can be made, use


of a typical value wr 1: 612 is recommended. AR is not sensitive to small deviations of wr (except, perhaps, in very low dams) and the error would be insignificant compared with possible errors, e.g. in determining the soil properties or the peak ground acceleration. Ambraseys & Sarma (1967) have presented similar design spectra by averaging deterministically obtained spectra for a number of strong earthquakes; they modelled the dam as a homogeneous shear beam, however.

Along the same lines, and with the same degree of approximation, the distribution with depth of the r.m.s. seismic coefficient normalized by the r.m.s. top acceleration is a unique function of the dimensionless frequency parameter Tl wp, as illustrated in Fig. 12. Again this figure can be used even for deterministic safety analyses (at least in preliminary design computations) by approximating

k k A.!?%_ (8) a mmax 43

Thus by combining Fig. 11 and Fig. 12 the peak seismic coefficient k,,, operating on any potential sliding mass within the dam can be estimated and then used in connection with the statically determined yield acceleration k, of the same sliding mass to read maximum permanent sliding displacements 6 from nublished design curves of the form

Mammax f,z) against kJk,,, (Sarma, 1975; Makdisi & Seed. 1978: Gazetas. 1980).

Conversely; in Gazktas et al. (1981), a more rigorous, random vibration methodology is presented that leads to direct evaluation of probability distributions of seismic coefficients and permanent displacements.

The crest displacement spectra of Fig. 10 are also replotted in Fig. 11 in the form of variation of the displacement ratio (DR) (in ml/‘)

^

DR =&JO’ (9)

as a function of the dimensionless frequency parameter Tl cop The displacement ratio was determined by trial and error so that the results would fall within a narrow band. Indeed the data points define a unique line with practically no scatter. Therefore, using this line (Fig. 1 l), predictions can be made of the r.m.s. crest displacement of the dam relative to the ground, with very good accuracy. Relative displacements increase almost linearly with distance from the base of the dam (e.g. Fig. 4(b)). Thus it is possible with Fig. 11 to get rough estimates of the expected differential displacements between, for example, soil outlet works such as spillways, sluiceways, buried pipes and the body of the dam.

Finally, the r.m.s. shear strain spectra have been

reduced to a single plot (depicted in Fig. 13) which relates the shear-strain ratio (SR) (in m- ‘)

(10)

with the dimensionless frequency parameter Tl cop Again SR was determined so that the results would fall within a narrow band. The maximum deviation from the mean in this plot is only about 5%. The r.m.s. value of the maximum shear strain $,,, may therefore be evaluated readily and with good accuracy from the mean curve presented in Fig. 13, after the r.m.s. crest acceleration 8, has been determined on the basis of Fig. 11.

CONCLUDING REMARKS

An exact analytical random vibration formulation has been introduced to obtain statistics of the dynamic response of earth dams to strong earthquakes. The dams are modelled as one-dimensional, viscously damped, inhomogeneous shear beams and the ground excitation, consisting exclu- sively of vertical shear waves, is described through a Kanai-Tajimi power spectral density function and an arbitrary variation in intensity with time. Results are presented in the form of time variation and spatial distribution of r.m.s. (i.e. standard deviations) of absolute accelerations, relative displacements, shear strains and seismic coefficients on potential sliding masses. Key factors that influence the dynamic behaviour are identified and their effect is illustrated in a series of parametric studies.

It appears that the dimensionless frequency parameter Tl wf, which determines the size of the predominant ground frequency relative to the natural frequencies of the dam, is the single most important factor that controls the response. Three ratios, AR, DR and SR (equations (7), (9) and (lo)), which for a given dam and a known r.m.s. ground acceleration yield r.m.s. values of crest acceleration, crest displacement and maximum shear strain, respectively, are with good accuracy unique functions of Tl cop Similarly, the distribution of seismic coefficients with the depth from the crest of the potential sliding mass is also a function of Tl of only. The importance of this observation makes the graphs in Figs 1 l-l 3 useful engineering tools that can be used even with deterministic preliminary- type analyses.

Inhomogeneity that results from the dependence of soil modulus on effective confining pressure has proven to be a very important factor and should be considered in any dynamic analysis of earth dams. This can be conveniently done by using the close- form solutions (Table 2) developed by Gazetas (1981a, b) for a dam with shear modulus increasing as a 2/3-power of the distance from the crest.

276 G. GAZETAS. A. DEBCHAUDHURY AND D. A. GASPARINI

Limitations of procedure and needed research

The results presented in this Paper are based on an idealized model of the dam (one-dimensional inhomogeneous shear beam). Several phenomena that the model ignores may, in certain cases, have an effect on the dynamic behaviour of the dam.

Rocking and vertical vibrations take place even during horizontal ground excitation (vertical s- waves), as a result of flexural-type deformations and of wave reflections on the slopes of the dam. Clearly a two-dimensional (e.g. plane-strain) model is needed to study the effect of such vibrations on the behaviour of the dam. Nevertheless, horizontal accelerations and seismic coefficients are, in most practical cases, only slightly influenced by the two phenomena.

Moreover, ground excitation does not merely consist of vertical s-waves. Inclined (travelling) and surface waves participate in the motion giving rise to a difference in phase between the motions of various points under the foundation. As a result, ground excitation has in effect rocking and tor- sional, in addition to translational, components. Because of the large dimensions of the base of a dam (as compared with those of other structures) the effect of the two rotational components may be significant. Chopra et al. (1969) have presented a study of the effect of rocking components but more work is needed before definitive conclusions can be drawn.

The canyon geometry is another crucial parameter of the problem. Dams in relatively narrow canyons undergo three-dimensional deformations which has so far received only approximate treat- ment. For example, Ambraseys (1960) and Gazetas (1980) have studied the shear vibrations of a homogeneous or inhomogeneous dam built in a rectangular canyon. Computer limitations and expense of the sophisticated (finite element) methods that can rigorously account for a truly three-dimensional geometry are the primary cause of the lack of systematic study of this phenomenon.

Dam-foundation interaction may significantly alter dynamic deformations and stresses in earth dams founded on alluvial deposits. The shear beam model can be modified to estimate crudely such interaction effects (e.g. Ambraseys, 1960b), but a finite element formulation is necessary for a more rigorous analysis (Idriss, Mathur & Seed, 1974).

Finally, the non-linear hysteretic nature of soil is commonly modelled in an empirical way, i.e. through iterative linear analyses using strain- compatible moduli and damping ratios. In this context, the random vibration formulation developed by the Authors can be readily extended to treatsoilnon-linearity.Theapproachisillustrated in Gazetas et al. (1981).

The cost of the analyses reported here is at least

an order of magnitude less than corresponding deterministic analyses. For example, one complete run takes about 20 sin the VAX- 1 l/780 computer of the University Deterministic time-history analysis with a single accelerogram takes about 60 s and at least three such analyses must be performed for a rough estimate of r.m.s. values.

APPENDIX 1 POWER SPECTRAL DENSITY FUNCTION

The power spectral density denoted by S(o), in which o is the frequency in rad/s, describes how the energy in a random vibration is distributed among the frequencies of vibration. If one considers a particular ground accelerogram a,(t) as being the random outcome from a large number of seismic waves hitting a site

a,(t) = C Aisin(oit+4) i= I

the total power of a,(t) is z;= 1 A,*/2. The power of a random process characterized by a PSD function S(o) is (2 S(w)do. Thus the ordinate S(o;) ofthe PSD function at & can’be related to the amplitude of the corresponding sinusoid

S(wi) = lim A,‘/2Ao A”-0

For more information on the concept and use of PSD function the reader is referred to Crandall& Mark (1963) Clough & Penzien (1975) and Vanmarcke (1976).

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