SIMPLIFIED ANALYSIS OF CONCRETE GRAVITY DAMS …

SIMPLIFIED ANALYSIS OF CONCRETE GRAVITYDAMS ".,CLUDING FOUNDATION FLEXIEILITY

James M. Nau

Department of Civil EngineeringNorth Carolina State UniversityRaleigh. North Carolina 2769b

0 GT0 3 1c9

September 1989

Final Report

i - :::.DEPARTMENT OF THE ARMY

US Army Corps of EngineersWashington, DC 20314-1000

, ,Contract No. DACW39--88-K-OO,33CTjWork Unit 315,88

. StructUres Laboratory

89T 10 3 01

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Oip pli: ied Analysis of Concrete Gravity Dams Including Foundation Flexibility

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7 COSATI CODES 18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number)-E) GROUP SUB.GPOI)P * Concrete gravity dams Simplified procedure, -

Foundation flexibility

Seismic Analysis

19 ABSTRACT (Continue on reverse if necessary and identify by block number) '

-1:e rbje tiv of this study was to develop ;I modcl for the flexihle foundation rocklt-, lth a dam and to inll orporatt. this model into the finite element procedure and a two-

itnsional model of the monolith, SDFDAM. To account for foundation flexibility, the

tpeorv of Flamant, assuming the foundation to be an isotropic, elastic half-plane, was used.IC Tildings of this study indicate that it is important to include the effects of

")ini-,tion flexibility in the seismic analysis of concrete gravity dams on the supposition

th rc liabl, foundatian properties can be obtained from field or laboratory measurements.

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Preface

This report describes a simplified analysis procedure for seismic analysis

of concrete gravity dams that includes flexible foundations. The research was

accomplished with funds provided to the Structures Laboratory (SL), US Army

Engineer Waterways Experiment Station (WES), by the Engineering and Construction

Directorate, Headquarters, US Army Corps of Engineers (HQUSACE), under

Structural Engineering Work Unit 31588. The Technical Monitor was Mr. Lucian

G. Guthrie, HQUSACE.

The research was accomplished for the Structural Mechanics Division (SMD),

SL, WES. Dr. J. M. Nau, North Carolina State University, conducted this

research under Contract No. DACW39-88-K-0033 and is the author of this report.

Dc. R. L. Hall, SMD, managed and coordinated the study under the general

supervision of Messrs. Bryant Mather, Chief, SL, James T. Ballard, Assistant

Chief, SL, and under the direct supervision of Dr. Jimmy P. Balsara, Chief,

SMD.

Commander and Director of WES during preparation of this report was

COL Larry B. Fulton, EN. Technical Director was Dr. Robert W. Whalin.

Di-

Ad_tD iat r ::: I',.

Contents

Page

Preface. ......................... . .. .. .. .. ....

List of Tables. ........................ .. .. . ..

List of Figures. ............................. iv

Conversion Factors, Non-SI to SI (metric) Units of Measurement .. ..... vi

Chapter

1. Introduction ...........................

2. Simplified Analysis of Fundamental Mode Response .. ........ 2

2.1 Equivalent Single Degree-of-Freedom System. .. ........ 22.2 Earthquake Induced Loads and Stress Calculations .. ....... 4

3. Development of the Foundation Flexibility Matrix .. .........

3.1 Theory of Flamant. ...................... 73.2 Procedure............................9

4. Parametric Study .......................... 12

4.1 Objective ........................... 124.2 Selection of Dams and Response Parameters. .......... 124.3 Results of Parametric Study. ................ 14

5. Summary and Conclusions. ...................... 17

6. References ............................ 19

7. Appendix ............................. 56

iii

List of Tables

Table Page

1. The Coefficient Fmn for the Half-Plane Problem . . . 20

2. Flexibility Coefficients for Column 12 of theFoundation Rock Flexibility Matrix .. ......... 21

3. Properties of Dams Used for the Parametric Study . . 22

4. Cases Considered in this Study ... ........... 23

5. Natural Periods, Viscous Damping Factors, and

Spectral Acceleration Values .... ............. 24

6. Maximum Principal Stresses on the Upstream Face . . . 25

7. Maximum Principal Stresses on the Downstream Face . . 26

iv

List of Figures

Figure Page

1. Dam-Water-Foundation Rock System ... .......... . 27

2. Standard Mode Shape and Fundamental Periodfor the Dam on a Rigid Foundation and EmptyReservoir ......... ...................... . 28

3. Standard Values for R1 , the Ratio of FundamentalVibration periods of the Dam with and without water . 29

4. Standard Values for Rf, the Period LengtheningRatio Due to Dam-Foundation Rock Interaction . . . . 30

5. Standard Values for 4f, the Added DampingDue to Dam-Foundation Rock Interaction . ....... . 31

6. Standard Plots for Variation of P, over Depthof Water for H/Hs=I and Various Values ofR2 = %rs/ .r ........ ...................... .32

7. Finite Element Mesh Generated by SDFDAM forDam S130 ......... ...................... 33

8. Flamant Isotropic, Elastic Half-Plane:(a) Vertical and Horizontal RelativeDisplacements Due to a Uniformly Loaded Strip(b) Location of Nodes and Load for theFlamant Equation ...... .................. . 34

9. Location and Direction of Coefficients forColumn 12 of Flexibility Matrix ... .......... 35

10. Horizontal Earthquake Time Histories .. ........ 36

11. Response Spectra for 5-Percent Viscous Damping . . . 37

12. Effect of Time Increment on the Computation ofthe Response of Dam S130 (Case 4) ... .......... . 38

13. Effect of Time Increment on the Computation ofthe Response of Dam S200 (Case 12) .. ......... . 39

14. Effect of Number of Modes Used in theComputation of Response of Dam S130 (Case 4) . . . . 40

V

Figure p age

15. Effect of Foundation Modulus, Ef (Case 4) ..... 41






21. Effect :,f Fcundation Modulus, Ef (Case 25) ..... 47








29. Comparison of Fundamental Mode Response of

SDFDAM and EAGD-84 (Case 4) .... ............. 55

vi

Conversion Factors, Non-SI to SI (metric)

Units of Measurement

Non-SI units of measurement used in this report can be converted to SI (metric)

units as follows:

Multiply By To Obtain

feet 0.3048 metres

inches 25.4 millimetres

pounds (mass) per cubic foot 16.01846 kilograms per cubic metre

pounds (force) per square inch 0.006894757 megapascals

1. INTRODUCTION

In the simplified method for seismic analysis of concrete gravity dams,

the hydrodynamic forces arising from the fundamental mode of vibration are

calculated and applied to the upstream face of the dam as an equivalent static

load (Chopra, 1978). The stresses throughout the dam are computed using the

finite element procedure and a two dimensional model of the monolith (Cole and

Cheek, 1986). In this analysis, the effects of foundation flexibility are

ignored. The objective of this study is to develop a model for the flexible

foundation rock beneath a dam and to incorporate this model into the Cole and

Cheek procedure, hereinafter referred to as SDFDAM. To assess the influence

of the added foundation flexibility, a parametric study is conducted.

Solutions obtained from SDFDAM are compared with those from the well-

established computer program EAGD-84 (Fenves and Chopra, 1984). The program

EAGD-84 provides a time history solution of the dam and includes all the

significant modes of vibration. Four dams are used in this study which range

in height from 130 feet to 638 feet. The maximum principal tensile stresses

on the upstream and downstream faces are compared to judge the suitability of

the simplified equivalent lateral force method, including dam-foundation rock

interaction.

* A table of factors for converting non-SI units of measurement to SI (metric)

units is presented on page vi.

2

SIMPLIFIED ANALYSIS OF FUNDAMENTAL MODE RESPONSE

2.1 Equivalent Single Degree-of-Freedom System

The fundamental mode of vibration has the greatest effect on the response

of short-period structures subjected to earthquake excitation. Since concrete

gravity dams fall into this category, using only the fundamental mode should

provide a good approximation in the calculation of the forces produced by an

earthquake. To simplify the approach further, only the response to horizontal

ground motion is considered. This response has been shown to be more

significant than the response from vertical ground motion. Figure 1 shows the

dam-reservoir-foundation system. The dam is supported on flexible foundation

rock and impounds a reservoir with a horizontal bottom. The reservoir is

assumed to be of infinite extent in the upstream direction, and the absorptive

effects of accumulated reservoir-bottom sediments are ignored.

In the simplified analysis procedure, an equivalent single degree-of-

freedom system is defined. This system has the same properties as the dam

with an empty reservoir, but is modified by an added mass for the hydrodynamic

effects. The mass per unit height of the equivalent system can be expressed

as,

ms(y) = ms(y) + ma(y) (1)

where

y = height above the base,

m s (y) = mass of the dam without water,

PI (y,0)

m (y) = = added mass of the water,a AV (Y)

pl(yAs) = impulsive water pressure,

3

y(v) = shape of the fundamental mode ofvibration of the dam on a rigid foundationwith empty reservoir,

Ws = fundamental resonant frequency of the dam on arigid foundation, including hydrodynamiceffects.

The analysis is simplified further by introducing a standard mode shape

and fundamental period. This approximation is possible since the cross-

sectional properties of concrete gravity dams do not vary significantly. The

standard mode shape is shown in Figure 2. As shown in this figure, the

equation for the fundamental period of the dam on a rigid foundation with an

empty reservoir is

Ts=1.4 HS- (2)

where

Hs = the height of the dam, in ft, and

Es = modulus of elasticity of the dam concrete,in psi.

The fundamental natural vibration period of the dam is lengthened by the

presence of water in the reservoir and by the flexible foundation rock. The

pericd of the fundamental mode is thus given by

Ts = RIRfTs, (3)

where R, = period lengthening ratio due to dam-water

interaction (Fig. 3), and

Rf = period lengthening ratio due to dam-foundationrock interaction (Fig. 4).

4

The damping for the equivalent SDOF system is given by

3 - + 4f, (4)s R 1 (Rf) s

where 4s = damping ratio of the dam on a rigid foundationwith empty reservoir, and

4f = damping ratio due to dam-foundation rock

interaction (Fig. 5).

Typically, ks is taken as 0.05. Values for 4f are given in Fig. 5. In this

figure, Tjf is the hysteretic damping factor for the foundation rock, taken as

0.1 in this study.

2.2 Earthquake InducuA Loads and Stress Calculations

The loads resulting from the horizontal ground motion can be approximated

by a set of equivalent horizontal static forces applied to the dam. These

earthquake induced loads consist of two parts: the hydrodynamic forces and

the inertial mass of the dam. The hydrodynamic effects are represented by an

added mass of water moving with the dam. This added mass depends on several

factors including compressibility of the water and the fundamental mode shape

and frequency of the dam. The impulsive pressure resulting from the 3tored

water during an earthquake can be computed in dimensionless form from,

gp1 W 2wH

ts

I ln cos((2n - 1) ir/2] (5)

nl(2n - 1) /i- 1 2 i.)2X- (2(

(2n - 1)2 s

5

where

i = y/H,

Or = C/2H = the fundamental resonant frequencyfor impulsive pressure in water,

os = 2t/RITs = the fundamental resonant frequencyof the dam and reservoir,

Iln = W(y)cos[(2n - l)ny/2]dy,

y = height above base,

H = height of the water,

C = the velocity of sound in water = 4720 fps,

w = unit weight of water = 62.4 pcf, and

g =acceleration of gravity.

From equation (5), normalized values of P, are computed as a function of

y for various values of the ratio 0s/0r = R2 , and the results are shown in

Figure 6. These plots were made for a full reservoir, i.e., H - Hs . To

obtain the impulsive pressure, Pl, when the reservoir is not full, the value

from Figure 6 is multiplied by the ratio of the height of the water to the

height of the dam squared (H/Hs)2. Thus, the impulsive pressure is

pl(y) = (value from Figure 6)wH(H/HS) 2. (6)

The la-teral earthquake forces over the height of the dam, including the

hydrodynamic effe.- are computed from

f(y)= - [ws(y)I(y) + gpl (Y's)]' (7)

m y

6

where

r = modal earthquake-excitation factor

m = generalized mass

Sa(Tsks) = ordinate of the earthquake spectrum in g,

Ws (y) = weight per unit height of the dam, lb/ft, and

W(y) = the fundamental mode shape of the dam.

Calculations for several cross sections and water levels show the value

of r/m* is about 4.0. This value is used throughout the study.

Because the cross-section of the dam is not symmetric, the earthquake

forces must be applied separately in both the upstream and downstream

directions. The principal tensile stresses are of most concern, since

concrete is much weaker in tension. To calculate the principal tensile

stresses on the upstream face, the earthquake-induced forces are applied to

the dam in the downstream direction. Likewise, to calculate the principal

tensile stresses on the downstream face, the earthquake-induced forces are

applied to the dam in the upstream direction. For each loading case, the

earthquake-induced forces are combined with the hydrostatic and gravity loads.

The computer program SDFDAM (Cole and Cheek, 1986) incorporates Chopra's

original simplified procedure for the analysis a concrete gravity dams.

SDFDAM uses a two-dimensional finite element method for the stress analysis of

the dam monolith. For this analysis, subroutines from the standard finite-

element package SAP (Wilson, 1970) are used. An automatic mesh generating

package is incorporated into SDFDAM. The only input parameters required for

generating the mesh are the dimensions that describe the cross section. An

example of the mesh generated by SDFDAM is shown in Figure 7.

7

3. DEVELOPMENT OF THE FOUNDATION FLEXIBILITY MATRIX

3.1 Theory of Flamant

To more accurately predict the behavior of a concrete gravity dam during

an earthquake, the flexibility of the foundation rock should be taken into

consideration. The classic theory of Flamant (Christian and Desai, 1977) is

used. This theory assumes an isotropic, elastic half-plane.

For a state of plane stress, the equation for the relative vertical

displacements, w.n, of points m and n, shown in Figure 8(a), resulting from a

vertically loaded strip is

(m-n+0.5)a

Wmn= 2Vn/(7CEfa) J ln(d/x)dx , (8)

where

a = width of loaded strip,

Vn = magnitude of the vertical load,

Ef - modulus of elasticity of foundation,

vf = Poisson's ratio of foundation,

m = distance from point n to point of desired

displacement, and

n = point beneath the center of loading.

To determine the constant d, a reference point is chosen for the deformed

surface. If the deflection Wnn is assumed to be zero,

d = a/(2e) (9)

8

where e is the base for the natural logarithm. Now, the equation for wmn can

be rewritten as

(m-n+0.5)aWrn - 2Vn/(nEfa) ln[a/(2ex)]dx (10)

.(m-n--0.5) a

which, after integrating, becomes

2Vnw = F (ii)

mn iEf mn,

where

Fin = A ln(A) - B ln(B),

A = 2(m - n) - 1, and

B = 2(m - n) + 1.

The coefficient Fmn depends upon m and n. This dependence may be

expressed in terms of x/a, the dimensionless distance from the loaded area.

Table 1 shows the variation of Fmn with x/a.

The horizontal displacement, umn, resulting from the vertical load is

constant and is given by

u - + f Vn (12)

except in the case of a point at the center of the loaded area (x = 0), where

the horizontal displacement is zero.

Similar considerations can be used to determine equations which will give

the vertical and horizontal displacements resulting from a horizontal (shear)

load over a length a, (Hn/a). For horizontal displacements,

9

nUm = - F , (13)

where

Hn = magnitude of the horizontal load, and

FMN= Fmn.

For vertical displacements,

1-vWMN= ± Hn (14)2Ef

except where x = 0, wMN = 0.

For plane strain Ef and Vf are replaced by Ef/(l-vf2 ) and Vf/(l-Vf).

3.2 Procedure

The foundation flexibility must be formulated in matrix form for

implementation into the computer program SDFDAM. SDFDAM automatically

generates a finite element mesh that has 11 equally spaced nodes at the base

of the dam. Because planar quadrilateral elements with two degrees of freedom

per node are used to model the dam, a total of 22 degrees of freedom result at

the dam-foundation interface. To create the flexibility matrix for the

foundation, a unit load is applied at each DOF at a time. As the unit load

is applied to a DOF, the displacements are calculated for every DOF. These

displacements are the flexibility coefficients and form a column in the

flexibility matrix. This procedure is repeated until the unit load has been

applied to each DOF and all the corresponding flexibility coefficients are

calculated. Since the nodes at the foundation are equally spaced, the

distance between each node can be set equal to the constant a. The load width

10

also is set equal to this distance a. Therefore, when either the vertical

load (Vn) or the horizontal load (Hn) is applied at a node, it will be equally

spaced a distance a/2 from each adjacent node, as shown in Figure 8(b). With

this arrangement, the ratio x/a used in the determination of F.n in the

Flamant equation will increase in increments of 1 from node to node away from

the load. From Table 1, it is observed that the F., coefficient increases at

a decreasing rate as the distance x/a increases. Directly beneath the center

of the load, Fmn is equal to 0, since all other displacements are relative to

it. To approximate the displacement at the load, a coefficient (Fmn) must be

computed at a ratio of x/a that is "far" from the load. In this study, an x/a

ratio equal to 40 was chosen to determine the displacement directly beneath

the load. Since the width of the dam at its base is equal to 10a, the

foundation at four dam widths away from the applied unit load is assumed not

to be influenced by the dam. To calculate the surface displacements for the

other nodes, the Fmn coefficients at these nodes are subtracted from the Fmn

coefficient for x/a equal to 40.

The only other variables required in the Flamant equations are the

horizontal (Hn) or vertical (Vn) load, the modulus of elasticity of the

foundation (Ef), and Poisson's ratio of the foundation (Vf). For purposes of

creating the flexibility matrix, Vn and Hn are always equal to unity.

Obviously, symmetry can be used in calculating the flexibility coefficients.

To illustrate how the flexibility matrix is developed, column 12 will be

derived. Figure 9 shows the locations and positive directions of the 22

degrees-of-freedom. In addition, Figure 9 shows the location aqd direction of

the unit load and flexibility coefficients, and the deformed shape after the

unit load is applied. Flexibility coefficients f1 ,12 , f2 ,12 , and f1 2 ,12 are

selected to be calculated here. Plane strain is assumed.

11

f V ~f 2)F12,12 nf F

At x/a = 40,

A 2 2(m - n) - 1 - 2 (4 n -0) - 1 7 79,

B 2(m - n) + 1 2(40 - 0) + 1 =81, and

Fin = A ln(A) - B ln(B) = -10.764

For Vf = 0.25 and E, = 7.9 x 106 psi, the flexibility coefficients become

(1I- (0.25) 2 3 -7fl 2 , 1 2 6 (.2 1 (10.764) - 4.066 x 10 in., and

12,12 n(7.9x 10 )

f f [F (x/a= 40) - F (x/a = 5)].2,12 IEF nm

From Table 1, for x/a 5, Fm = -6.602 and

=(1 - 0.25 2 -7S2,12 - 6 (10.764 - 6.602) = 1.572 x 10 in.,n(7.9xlO)

1 - Vf - 2V f2 1 - .25 - 2(.25)2fl,1 = (=91)

2 Ef (2)(7.9xi0)

fl,12= 3.956 x 10- 8 in

The results for the remaining coefficients for column 12 of the flexibility

matrix are shown in Table 2. Once all the flexibility coefficients are

calculated, the matrix is inverted and appropriately combined with the

stiffness matrix of the dam.

12

4. PARAMETRIC STUDY

4.1 Objective

The objective of the parametric study is to verify that the modified

version of SDFDAM produces acceptable results for the preliminary analysis of

concrete gravity dams. The principal stresses computed from the simplified

analysis of SDFDAM (Cole and Cheek, 1986) are compared with the time history

results of EAGD-84 (Fenves and Chopra, 1984). In addition, the stresses

computed from the block or layered model and elementary beam theory (Fenves

and Chopra, 1986) are evaluated and compared.

4.2 Selection of Dams and Response Parameters

Four dams are used in this study. These dams were previously used in a

study by the Corps of Engineers for verifying their program SDFDAM (Cole and

Cheek, 1986). Table 3 lists the dimensions and properties of each dam. The

"standard" dams designated as S130, S200, and S300 are dimensioned to be

typical of dams between 130 and 300 feet in height. These standard cross

sections represent over 90 percent of the dams built by the Corps in the

United States. The dam designated as D638 is the existing Dworshak dam

located in Clearwater, Idaho. This dam was chosen since its great height

presents an extreme case for checking the validity of the approximate

procedure used by SDFDAM. Since the major modification to SDFDAM was the

addition of foundation rock flexibility, the one parameter which will be

varied is the ratio Ef/E s . Four Ef/E s ratios are used for this study: 1/2,

1, 2, and - (rigid). The dam modulus, Es, remains constant for all dams, so

only the foundation modulus, Ef, is varied to obtain the desired ratio.

Two earthquakes, one of moderate and one of high intensity, were selected

for the study. The San Fernando earthquake recorded at the Pacoima Dam on

13

February 9, 1971 is selected as the high intensity earthquake and will be

referred to as EQ 1. This earthquake has a maximum acceleration of 1.17 g.

The Imperial Valley earthquake recorded at El Centro, California on May 18,

1940 represents the moderately intense earthquake, and will be referred to as

EQ 2. It has a maximum acceleration of 0.348 g. The horizontal accelerogram

for each earthquake is shown in Figure 10, and the response spectra for 5

percent viscous damping are shown in Figure 11. Results from SDFDAM (Cole and

Cheek, 1986), BLOCK (Fenves and Chopra, 1986), and EAGD-84 (Fenves and Chopra,

1984) are generated for all four dams, for both earthquakes, and for the four

Ef/E s ratios. Thus a total of 32 cases arise. Table 4 identifies each of

these cases. The period and damping for each case, calculated using Eqs. 3

and 4, are shown in Table 5. The spectral acceleration values are also shown

in Table 5.

Before EAGD-84 is run for each of the 32 cases, the parameters which

control the response computations must be carefully selected to ensure that

the computed dynamic response is accurate. Dam S130 was of particular concern

becanse of its low fundamental period, as shown in Table 5. To insure that

the proper time step is selected for this dam, time intervals (DT) of .005,

.01, and .02 seconds are used in EAGD-84 for the rigid foundation (Ef/Es =o ),

case 4. For these calculations, ten modes are combined. The maximum

principal stresses for the upstream and downstream faces of the dam are

plotted in Figure 12. The stresses for DT of .02 seconds are only slightly

greater than those for .005 and .01 seconds. The same investigation was

conducted for dam S200 with a rigid foundation (case 12) and a DT of .01 and

.02 seconds. Again there is good agreement in the results for these time

steps, which are shown in Figure 13. On the basis of these findings, a time

step equal to .02 seconds is used for all 32 cases.

14

Another response parameter of concern is the selection of the number of

modes. General guidelines are to use five modes if the foundation rock is

rigid, and ten modes if the foundation rock is flexible. To insure that the

correct selection is made for this parameter, the number of modes should be

increased until there is little change in the stresses on the upstream and

downstream faces of the dam. Again, the rigid foundation case for dam S130

(case 4) is used to investigate the effects of the number of modes. For these

analyses, 1, 2, 5, and 10 modes are included. For these calculations a time

step of .02 seconds is used. The results are shown in Figure 14. As shown in

this figure, the case with 1 mode generally overestimates the dynamic

response; the inclusion of higher modes reduces the response. Although the

recommendation to include five modes is appropriate for the rigid foundation,

ten modes are considered, for convenience, for all foundation conditions in

the parametric study.

4.3 Results of Parametric Study

The maximum principal stresses from EAGD-84, SDFDAM, and BLOCK for the

upstream and downstream faces are plotted and compared for all 32 cases.

Plots for the rigid base cases (4, 8, 12, 16, 20, and 24) for the three

standard dams, and all cases (25 through 32) for dam D638 are presented in

this section. These plots are shown in Figures 15 through 28. The figures

containing the plots of the other cases are shown in the Appendix. These

selections were made because all of the results for the three standard dams

show the same general trend. However, the results for dam D638 show a

departure from this pattern. For the three standard dams, the tensile

stresses from SDFDAM are greater than those from EAGD-84 for all ratios of

Ef/E s . The closest agreement in stresses from the two procedures is observed

for the cases in which Ef/E5 = . The stresses of perhaps greatest concern

15

are located near the top one-fourth of the dam, where the slope of the

downstream face changes abruptly. Tables 6 and 7 list the stresses at this

location on the upstream and downstream faces. For comparison, these tables

also give the ratios of stresses from SDFDAM and BLOCK to those of EAGD-84.

For the three standard dams, the ratio of the SDFDAM stress to the EAGD-84

stress, (1)/(3), ranges from a high of 2.11 in case 2 to a low of 0.88 in case

8. In other words, the approximate fundamental mode analysis may provide an

overestimate of the maximum principal stress by as much as 111 percent. Only

in case 8 is the stress from SDFDAM on the unconservative side; however, this

underestimate is insignificant. On the other hand, for the 638 ft Dworshak

Dam, D638, there are several cases that show SDFDAM to produce unconservative

results. These cases (cases 25, 26, 27, and 28) are for the higher intensity

earthquake, EQI. The maximum underestimate is about 30 percent on the

upstream face. Similar conclusions may be reached when the stresses computed

from the oversimplified block model are compared with those from EAGD-84. The

maximum overestimate is about 150 percent; in no case does the underestimate

exceed 10 percent. It is worthy to note that the stresses from BLOCK compare

favorably with those of SDFDAM on the upstream face, but exceed the SDFDAM

results on the downstream face. This result may be due to the limitation of

the elementary beam theory in predicting principal stresses near inclined

surfaces.

Because of the apparent conservative of SDFDAM for the majority of dams

considered in this study, one final investigation was conducted for dam S130

on a rigid foundation. Referring to Figure 14, which contains EAGD-84

solutions for various numbers of modes, the results show larger stresses over

about the top half of the dam when only 1 mode is considered. This

observation explains at least part of the overestimate in the equivalent

16

lateral force method, since all SDFDAM results in Figures 15-28 include one

mode only. To see how the results compare for one mode, Figure 29 is

presented. In this figure, the results from EAGD-84 (from Figure 14 for one

mode) are compared with the stresses from SDFDAM from Figure 15. These

results compare favorably, again indicating the general conservatism

introduced into the simplified method when only one mode is considered.

17

5. SUMMARY AND CONCLUSIONS

The US Army Corps of Engineers developed the computer code SDFDAM (Cole

and Cheek, 1986) for the analysis of concrete gravity dams subjected to

earthquakes. The approximate procedure reported by Chopra for the

determination of the earthquake-induced loads is incorporated into SDFDAM.

This procedure considers the response in the fundamental mode of vibration.

In the current version of SDFDAM, the stresses in the dam are computed under

the assumption that the foundation is rigid. Subsequent studies hAve shown

that the effects of dam-foundation rock interaction may be significant and

should be included in the analysis. The purpose of this study was to develop

and implement a procedure into SDFDAM to account for foundation flexibility.

The theory of Flamant, in which the foundation is assumed to be an isotropic

elastic half-plane, was used.

A parametric study was conducted to assess the validity of the modified

version of SDFDAM. The computer program EAGD-84 (Fenves and Chopra, 1984) was

used as the standard for this investigation. Four dam cross sections, ranging

in height from 130 to 638 ft, four foundation moduli, and two earthquake

ground motions were used. The maximum principal stresses on the ilpstream and

downstream faces of each dam were plotted and compared. In general, the

stresses for the 130 ft, 200 ft, and 300 ft dams obtained from SDFDAM are

greater than those from EAGD-84. Thus, for these dams, the simplified

procedure including foundation interaction effects provides conservative

estimates of the earthquake-induced stresses, regardless of the foundation

modulus and the intensity of the earthquake motion. It should be noted,

however, that the approximate stresses obtained from SDFDAM may exceed the

exact values of EAGD-84 by as much as 100 percent. This overestimate may be

18

attributeci in part to the inclusion of cnly one mode in the simplified

procedure.

For the 638 ft dam subjected to the high intensity earthquake, the

results of SDFDAM are not on the conservative side for all foundation

conditions. The stresses obtained from the approximate procedure of SDFDAM

are as much as 30 percent lower than those of EAGD-84. When subjected to the

earthquake motion of lesser intensity, however, the simplified procedure is

conservative for all foundation moduli. While it is difficult to draw general

conclusions from these limited results, it is evident that for the extreme

case of a high dam subjected to an intense earthquake, the simplified analysis

procedure may be inadequate.

Finally, the results of this study reveal, as expected, that the stresses

in a dam subjected to earthquake loading are a function of the foundation

modulus. Because foundation compliance alters the fundamental natural period

of the dam, the response is increased or decreased, depending upon the

frequency content of the ground motion. These findings indicate that it is

important to include the effects of foundation flexibility in the seismic

analysis of concrete gravity dams. Of course, this conclusion assumes that

reliable foundation properties can be obtained from field or laboratory

measurements.

19

6. REFERENCES

1. Chopra, A. K., "Earthquake Resistant Design of Concrete Gravity Dams,"Journal of the Structural Division, ASCE, Vol. 104, No. ST6, June, 1978,pp. 953-971.

2. Christian, J. T., and Desai, C. S. Numerical Methods In Geotechnical

Engineering, McGraw-Hill, Inc., New York, 1977.

3. Cole, R. A., and Cheek, J. B., "Seismic Analysis of Gravity Dams,"Technical Report SL-86-44, U.S. Army Engineer Waterways Experiment

Station, Vicksburg, Mississippi, Dec., 1986.

4. Fenves, G., and Chopra, A. K., "EAGD-84, A Computer Program for EarthquakeAnalysis of Concrete Gravity Dams," Report No. UCB/EERC-84/II, EarthquakeEngineering Research Center, University of California,Berkeley, California, Aug., 1984.

5. Fenves, G., and Chopra, A. K., "Simplified Analysis for EarthquakeResistant Design of Concrete Gravity Dams," Report No. UCB/EERC-85/10,Earthquake Engineering Research Center, University of California,

Berkeley, California, June, 1986.

6. Wilson, E. L., "SAP - A General Structural Analysis Program," SESM Report70-20, Department of Civil Engineering, University of California,Berkeley, 1970.

20

x/a Fin

0 0

1 -3.296

2 -4.751

3 -5.574

4 -6.154

5 -6.602

6 -6.967

7 -7.276

8 -7.544

9 -7.780

10 -7.991

11 -8.181

12 -8.356

13 -8.516

14 -8.664

15 -8.802

16 -8. 931

17 -9.052

18 -9.167

19 -9.275

20 -9.378

Table 1. The Coefficient Fmn for the Half-Plane Problem

21

Flexibility CoefficientLocation (fRxC) Coefficient (in.)

1 , 12 3.956 x 10 - 8

2 , 12 1.572 x 10- 7

3 , 12 3.956 x 10-8

4 , 12 1.741 x 10 - 7

5 , 12 3.956 x 10- 7

6 , 12 1.960 x 10 - 7

7 , 12 3.956 x 10-8

8 , 12 2.271 x 10 - 7

9 , 12 3.956 x 10 - 8

10 , 12 2.821 x 10- 7

11 ,12 0

12 , 12 4.066 x 10 - 7

13 , 12 -3.956 x 10- 8

14 , 12 2.821 x 10 - 7

15 , 12 -3.956 x 10- 8

16 , 12 2.271 x 10- 7

17 , 12 -3.956 x 10-8

18 , 12 1.960 x 10- 7

19 , 12 -3.956 x 10- 7

20 , 12 1.741 x 10- 7

21 , 12 -3.956 x 10- 8

22 , 12 1.572 x 10- 7

Table 2. Flexibility Coefficients for Column 12 of theFoundation Rock Flexibility Matrix

22

Dams

Dimensions, ft. S130 S200 S300 D638

HEIGHT 130. 200. 300. 638.W 14. 17. 21. 27.BASE 130.5 146.83 230.83 495.HWATER 115. 185. 285. 570.BH1 95. 158. 250. ---BH2 110. 175. 270. 585.Su 0.120 0.083 0.083 0.0Sd 0.710 0.667 0.700 0.800

Material Properties

Modulus of Elasticity(million psi) 3.0 3.0 3.0 5.0

Poisson's Ratio 0.2 0.2 0.2 0.2Unit Weight (lb/ft3 ) 144.0 144.0 144.0 144.0

UPSTREAM DOWNSTREAM

BASE

Table 3. Properties of Dams Used for the Parametric Study

23

Case Ef/E s Ef Earthquake Dam

1 1/2 1.5 EQ1 S130

2 1 3.0 EQ1 S1303 2 6.0 EQ1 S1304 c c EQ1 S130

5 1/2 1.5 EQ2 S130

6 i 3.0 EQ2 S1307 2 6.0 EQ2 S130

8 0 0 EQ2 S130

9 1/2 1.5 EQ1 S20010 1 3.0 EQ1 S20011 2 6.0 EQ1 S20012 c 0 EQI S200

13 1/2 1.5 EQ2 S200

14 1 3.0 EQ2 S20015 2 6.0 EQ2 S200

16 0 0 EQ2 S200

17 1/2 1.5 EQ1 S30018 1 3.0 EQ1 S300

19 2 6.0 EQ1 S30020 0 0 EQI S300

21 1/2 1.5 EQ2 S30022 1 3.0 EQ2 S30023 2 6.0 EQ2 S30024 c c EQ2 S300

25 1/2 2.5 EQI D63826 1 5.0 EQ1 D638

27 2 10.0 EQ1 D63828 0 0 EQI D638

29 1/2 2.5 EQ2 D63830 1 5.0 EQ2 D63831 2 10.0 EQ2 D63832 0 0 EQ2 D638

Note: Ef in million psi

Table 4. Cases Considered in this Study

24

Natural Period of Viscous Damping Spectral Accele-Case Vibration (sec) Factor ration Value (ci)

1 0.163 0.139 1.3602 0.145 0.094 1.8103 0.134 0.067 2.0304 0.122 0.043 1.720

5 0.163 0.139 0.5006 0.145 0.094 0.5177 0.134 0.067 0.6708 0.122 0.043 0.684

9 0.261 0.138 1.36010 0.232 0.093 1.74011 0.215 0.066 2.23012 0.195 0.041 2.090

13 0.261 0.138 0.52514 0.232 0.093 0.63715 0.215 0.066 0.63916 0.195 0.041 0.703

17 0.402 0.138 1.52018 0.357 0.092 1.89019 0.331 0.065 1.71020 0.301 0.040 2.150

21 0.402 0.138 0.43822 0.357 0.092 0.51523 0.331 0.065 0.60924 0.301 0.040 0.753

25 0.682 0.137 0.68726 0.607 0.091 0.69627 0.562 0.064 0.89828 0.511 0.039 1.710

29 0.682 0.137 0.50730 0.607 0.091 0.71731 0.562 0.064 0.85332 0.511 0.039 0.904

Table 5. Natural Periods, Viscous Damping Factors, and SpectralAcceleration Values

25

Maximum Principal Stress on theUpstream Face (psi)

SDFDAM BLOCK EAGD-84 Ratio RatioCase (i) (2) (3) (M /(3) (2)/(3)

1 376 401 251 1.50 1.602 514 519 266 1.93 1.953 581 578 352 1.65 1.644 486 495 478 1.02 1.04

5 114 131 65 1.75 2.026 119 135 111 1.07 1.227 166 176 135 1.23 1.308 170 180 193 0.88 0.93

9 541 567 443 1.22 1.2810 704 706 487 1.45 1.4511 915 917 671 1.36 1.3712 855 860 657 1.30 1.31

13 182 190 120 1.52 1.5814 231 233 141 1.64 1.6515 231 234 147 1.57 1.5916 259 259 211 1.23 1.23

17 772 772 444 1.74 1.7418 972 932 674 1.44 1.3819 875 889 760 1.15 1.1720 1112 111.8 742 1.50 1.51

21 187 203 165 1.13 1.2322 229 237 178 1.29 1.3323 280 280 215 1.30 1.3024 358 347 214 1.67 1.62

25 584 838 638 0.92 1.3126 592 845 872 0.68 0.9727 782 992 1100 0.71 0.9028 1548 1676 1866 0.83 0.91

29 414 441 431 0.96 1.0230 612 632 573 1.07 1.1031 740 758 776 0.95 0.9832 788 805 795 0.99 1.01

Table 6. Maximum Principal Stresses on the Upstream Face

26

Maximum Principal Stress on theDownstream Face (psi)

SDFDAM BLOCK EAGD-84 RATIO RATIOCase (i) (2) (3) ()/03) (2)/(3)

1 520 622 309 1.68 2.012 697 809 330 2.11 2.453 784 901 443 1.77 2.034 662 771 626 1.06 1.23

5 180 217 103 1.75 2.116 186 224 137 1.36 1.647 247 289 168 1.47 1.728 252 295 270 0.93 1.09

9 700 860 383 1.83 2.2510 903 1080 620 1.46 1.7411 1164 1366 850 1.37 1.6112 1098 1283 848 1.29 1.51

13 237 311 177 1.34 1.7614 314 376 205 1.53 1.8315 315 378 207 1.52 1.8316 327 415 302 1.08 1.37

17 995 1192 592 1.68 2.0118 1245 1460 746 1.67 1.9619 1123 1329 888 1.26 1.5020 1421 1650 906 1.57 1.82

21 263 325 239 1.10 1.3622 315 385 263 1.20 1.4623 378 450 288 1.31 1.5624 476 556 289 1.65 1.92

25 892 1172 893 1.00 1.3126 904 1180 1204 0.75 0.9827 1177 1378 1521 0.77 0.9128 2275 2286 2541 0.90 0.90

29 649 682 611 1.06 1.1230 933 933 805 1.16 1.1631 1116 1099 1012 1.10 1.0932 1185 1161 1155 1.03 1.01

Table 7. Maximum Principal Stresses on the Downstream Face

27

FWATER

ABSORPTIVE RESERVOIR RIGID8BOT TOM MATERIALS D--M BASE

/9

Figure 1. Dam-Water Foundation Rock System

28

I0

00.9as

Ea 0.7

~0.6-

05-Vibrat..in Period T lAN. a

Q4 - Hsa height of dam in ft.o E elastic modulus of

concrete in psi*Q3

.2o Q2

* 0.1

0 I I I I I I I I I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Mode Shape .,

Figure 2. Standard Mode Shape and Fundamental Periodfor the Dam on a Rigid Foundation and EmptyReservoir. After Chopra (1978)

. .. = = e eee e ee I II III

29

1.7

E - 7xl 1O6 psi

1.6o 0

0

SE:5x106

E E 1.4-o o E z4xIO 6

0 0 E:3x 106"0 vo 1.3- Ez2xlO 60 0

"02 E<I x I06

0 000

- >

it

1.0

0.4 0.5 0.6 0.7 0.8 0.9 1.0Total Depth of Water,H

Height of Dam,Hs

Figure 3. Standard Values for R., the Ratio of FundamentalVibration Periods of ihe Dam with and withoutWater. After Chopra (1978).

30

1.7

1.6 -

, 1.5 ,-

(p 1.4zzwI-

D 1.3 -zw-J

0

F 1.2w0-

1.1

1.00 I 2 3 4 5

Es/Ef

t I I I 1,I

o1 1/2 1/3 1/4 1/5

MODULI RATIO, Ef/E s

Figure 4. Standard Values for R , the Period Lengthening

Ratio Due to Dam-Foungation Rock Interaction.

After Fenves and Chopra (1986).

31

0.35

0.30 -

7f •0.50

0.25 -4A.0 0.25

4 0.20 0.1eC,

z 0.15

.10

S015-

000

0.05

0 I 2 3 4 5ES /Ef

I I I I i J

I 1/2 1/3 1/4 I/5

MODULI RATIO, Ef/E s

Figure 5. Standard Values for , the Added Damping Due

to Dam-Foundation Rock Interaction. AfterFenves and Chopra (1986).

32

1.0

0 0.8-

0

5 0.60.99

S- 0.4

a.. 0.97> w

0 .0.7 0.

0 0.2 0.4 0.6 0.8gp /wH

Figure 6. Standard Plots for Variation of p over Depth or Waterfor H/H = 1 and Various Values ol R2 = s w r. AfterChopra T1978).

331

Figure 7. Finite Element Mesh Generated by SDFDAM for DAM S130

34

(m-n)a

a/2 I a/2

I (a) m

Equationt

n m

(a)

Equationt

• ~ ~ ~ ~ ~ ~ ~ ~ Bs of IIlI

35

C"-

"-4

-4U"-

"44

0-4

04-

0

-44

a) 4J-f cc

C4-4

co044

443

CD 0

.

44

cs;4-4

0

CC.)

4

C4

36

EARTHQUAKE EQ 11.2

I1

0.8 IC3 0.6-

0.4-

0.2-

0 o

z -0.2-

S-0.4-

-0.6-

-0.8-

0 2 4 6 6 10 12 1'4 16s 18 20

T1ME. SEC.

EARTHQUAKE EQ 21.2-

1

0.6-

0 .

P 0.4-

0.2-

0Z -0.200

0 2 1 6 8 10 1 2 1'4 16S 18S 20

7IME. SEC.

Figure 10. Horizontal Earthquake Time Histories

37

HORIZONTAL RESPONSE SPECTRA. 5% DAMPING3-

2.8-

2.6

~ 2.4

2 2.2-0g 2-

-J 1.8.wIJ(" 1.6

1.4omn 1.2--

0.8- + + + +-0.6- .. 4.+++

4 +.- + -+ + + + +

0.4 ' ---- I

0 0.2 0.4 0.6 0.8

PERIOD. SEC.- PACOIMA DAM. S16E + ELCENTRO. SOOE

Figure 11. Response Spectra for 5-Percent Viscous Damping

38

EAGD-84( EQ 1)GooUPTREAM FACE ( 130 -RIGID)

500-

CL

400-

4 300-

zw 200-

100-

0-0 20 40 s0o 8 100 120

ELEVATION. Fr.

0 DTin.003 + DT'-. 01 4 DT-.02

EAGD-84( EQ 1)DOWNSTrREAM FACE S 130 -RIGID)

700-

600-

3 00-

2 00-

100-

0-1 11 1 9 10 20 40 so s0 100 120

ELEVATION. F.0DT-.003 + DT-.01 4DT-.02

Figure 12. Effect of Time Increment on the Computation of the Responseof Damn S130 (Case 4)

39

EAGD-84 ( EQ 2 )400 UPSTREAM FACE S200 - RIGID

350

I. 300-

LA 250-

200-

C-0150

ECL

10-

so-

0 20 40 60 S 100 120 140 160 180 200

ELEVATION. Fr.0 DT-.01 . DT-.02

EAGD-84 ( EQ 2 )320 DOWNSREAM FACE ( S200 - RIGID)

300-

280

260-

.. 240

~3220-S200-

180ISO-140

Z 120

100

60

40

20

0~0 20 40 0 s0 100 120 140 160 180 200

ELEVATION. FT.a DT-.01 + DT-.02

Figure 13. Effect of Time Increment on the Computation of the Responseof Dam S200 (Case 12)

40

EAGD-84( EQ 1)GooUPTREAM FACE (S130 -RIGID)

a.

w 400-

300-

200

1 00-

0.

0 20 40 60 so 100 120

ELEVATION. FT.a 1 MOD0E + 2 MODELS 5 MODES A 10 MODES

EAGD-84 ( EQ 1)Boo. DOWNSTREAM FACE Sl 330 -RIGID)

700-

in 500-

400-

Z300-(L

S200-

100-

0 20 40608 100 120

ELEVATION. FT.MODE + 2 MODES 0 5 MODES A 10 MODES

Figure 14. Effect of the Number of Modes Used in the Computation ofResponse of Damn S130 (Case 4)

41

RIGID ( EQ 1)UPSTREAM FACE ( 5130 )

700 -

600-

(L

500-

UJ

wUn(nwLkr 400-

I--

a. 300-aZ

a.200-

100-

I 0

0 20 40 60 80 100 120

ELEVATION. FT.n SDFDAM + EAGD-84 BLOCK

RIGID ( EQ 1 )DOWNSTREAM FACE S S130 )

700

I. 600

14J

u 500

.j 400

Z 3000.

> 200

100-

0 f I I I I I I I

0 20 40 60 80 100 120

ELEVATION. FT.13 SDFDAM + EAGD-84 o BLOCK

Figure 15. Effect of Foundation Modulus, Ff (Case 4)

42

RIGID ( EQ 2 )UPSTREAM FACE S 5130 )

260-

240

220-

ij 200-0.

V; 180-V) 160-cn,1 140-(n

120 -

c3 100-z& O-

60

40-

20~

0-

-20 -,I I I I I

0 20 40 60 80 100 120

ELEVATION, FT.n SDFDAM + EAGD-84 c BLOCK

RIGID (EQ 2)DOWNSTREAM FACE ( 3130 )

300,-

280-

260-

240-

Ct 220

a 200EnU7 180wI 160-(n-j 140 -

0. 120-

IL 80-

60-

40-

20-

0-20 , , ,

0 20 40 60 S0 100 120

ELEVATION, FT.n SDFDAM + EAGD-84 * BLOCK

Figure 16. Effect of Foundation Modulus, Ef (Case 8)

43

RIGID (EQ 1 )UPSTREAM FACE (S200)

1.4-

1.3-

1.2-

W 1En

w- 0.9-

S0.8

0.7-

S0.6-

0.5

S 0.4-

0.2-

0.1

0 1 F I I I I I

0 20 40 60 s0 100 120 140 160 180 200

ELEVATION, FT.0 SDFDAM + EAGD-84 BLOCK

RIGID (EQi 1DOWNSTREAM FACE CS200)

1.3-

1 .2

CL

En 0.9

0.7-

0.6-

iE 0.5-

. 0.4-

0.3-

0.2-

0.1

0

0 20 40 60 830 100 120 140 160 180 200

ELEVATION. F7.a SDFDAM +i E-AGD-84 BLOCK

Figure 17. Effect of Foundation Modulus, E f (Case 12)

44

RIGID ( EQ2 )UPSTREAM FACE ( S200 )

500

400

a-

w 300

I-C,,

200-

a.z010

0

-100-

0 20 40 60 80 100 120 140 160 180 200

ELEVATION, FT.U SDFDAM + EAGD-84 < BLOCK

RIGID ( EQ 2 )DOWNSTREAM FACE C 5200 )

400-

Cn 350-I.

.i. 3002

w 250-

o-L.) 150-K

(L 100-

50-

-50 f

0 20 40 60 80 100 120 140 160 180 200

ELEVATION. FT.0 SDFDAM - EAGD-84 o BLOCK


45

RIGID (EQ 1 )UPSTREAM FACE S 300)

2.2

2-

1 .8

c.. 1.6 -

w ,+nE 1.4--

1.2

0.4-

0.2-

0 40 so 120 160 200 240 280

ELEVATION. FT.0 SOFDAM + EAGD-84 BLOCK

RIGID (EQ 1 )DOWNSTREAM FACE CS300)

2.12

1.9-

CL 1.7-

a 1.5-w 1.4-

r"' 1.2-

Lo0.9-

Z'-~ 0.8w 0.7-IL

S 0.60.5-0.4-0.3-0.2-0.1

0

O 40 850 120 160 200 240 280

ELEVATION, FT.13 SDFOAM + EAGD-84 e BLOCK


46

RIGID ( EQ 2 )700 UPSTREAM FACE S 5300 )8O0 -

700-

a. 600-

CA,u 500-iJJ

j 400

z 400CL

a-0

200

100-I 0

0 40 80 120 160 200 240 280

ELEVATION. FT.

E3 SDFDAM + EAGD-84 * BLOCK

RIGID (EQ 2)DOWNSTREAM FACE ( S300)

600[

a-- 500

LLJcnEnw

400U-

.300

200-

100-

0 40 80 120 160 200 240 2BO

ELEVATION. FT.SOFDAM + EAGD-84 * BLOCK


. . . . " • • • l I l I If

47

Ef/Es 1 /2 (EQ 1)UPSTREAM FACE D 638

1.6

1.5-

1 .4-

a- 1.2 -

Eli .w

M0.90.8

IL 0 0.8-

0.6-0. 0.5 -

< 0.4-2 0.3-

0.2-

0.1

0 200 400 600

ELEVATION, FT.0 SDFDAM + EAGD-84 " BLOCK

Ef/Es= 1/2 ( EQi 1DOWNSTREAM FACE C0638)

1.4

1.3-

rn 1.2-0. 1.1

_- 0.7 -

0.60C 0.5 -

0

:0 0.4 -03

0.2-

0.1

0 200 400 600

ELEVATION, FT.0 SDFDAM + EAGD-84 BLOCK

Figure 21. Effect of Foundation Modulus, E f(Case 25)

48

Ef/Es= 1 (EQi 1UPSTREAM FACE CD638)

1.4-

(. 1.2 -

(4 1.1

(nl

0.8-

0.L0 0.7 -

cs 0.6-

IL 0.5-

S 0.4-

0.3-

0.2-

0.1

0 200 400 60

ELEVATION. FT.a SDFDM + EAGD-84 * BLOCK

Ef/Es= 1 ( EQi)DOWNSTREAM FACE CD638)

1.5-

1.4

1.3-

CL 1.2 -

vi 1.1

w.

P220 0.9-

0.8-

0007-

0.6-

CL 0.5-

0.4-

0.2-

0.1

0 200 400 500

ELEVATION. FT.13 SDFDAM + E-AGD-84 * BLOCK


49

Ef/Es =2 (EQi )UPSTREAM FACE ( D638 )

2.1

21.9

1.8'.7

13. 1.6

1.5-L&J 1.4 -(n 1.3W- v~ 1.2

5 1.1

0 .9_'- 0.8

r 0.70. = 0.6"

0.5

0.40.30.2

0.1

0 200 400 600

ELEVATION. FT.0 SDFDAM + EAGD-84 > BLOCK

Ef/Es -= 2 ( EQ I )DOWNSTREAM FACE ( D638 )

2

1.9

1 .8-1.7

C 1.6

0O- 1.5

ci 1.4u1 1.3

1.3

. . 1.2N8 1.1

O 0.9

0.8n 0.7

0.6< 0.5

0.4 -

0.3

0.2

0.1

0 200 400 600

ELEVATION. FT.0 SDFDAM + EAGD-84 BLOCK


. . . . ' ' = = a I I I I If

so

RIGID (EQ 1 )UPSTREAM FACE D 638)

4-

.3.5-

.L 3 -(1i-

CA

2 20

0.5-

0 200 400 600

ELEVATION, FT.SDFDAM + EAGD-84 BLOCK

RIGID (EQ 1 )DOWNSTREAM FACE D 6.38)

4

.3.5-

0- .3

CALA 2.5 -

-j 2

0 .5-

0

0 200 400 600

ELEVATION. FT.a SDFDAM + EAGD-84 B LOCK


51

Ef/Es - 1/2 (EQ 2)UPSTREAM FACE ( D638 )

0.9

a. 0.8

w 0.7-&nL ' 0.6-

Cnj 0.5-

90jE 0.4-S0.3-

0.

0.2-

0.1 -

0 -

-0.1 -f

0 200 400 600

ELEVATION, F7.a SDFDAM + EAGD-84 * BLOCK

Ef/Es= 1 /2 (EQ2)DOWNSTREAM FACE C D638 )

800

700a.uai 600-

w"w 500-05

-J 400-

i-L) 300-z

a- 200-

100-

0 200 400 600

ELEVATION. FT.0 SOFDAM . EAGD-84 o BLOCK


52

Ef/Es 1 (EQ 2)UPSTREAM FACE (D638)

1.7

1.6

1.5

1.4

C 1.3

a 1.2u') 1.1

0. .9-- 0.6

0

0.7zR 0.6-

0.5-

S 0.4-

0.3-

0.2-

0.10 -

0 200 400 600

ELEVATION. FT.a SOFOAM + EAGD-84 o BLOCK

Ef/Es = 1 ( EQ 2)DOWNSTREAM FACE D638 )

1.3

1.2

'7j 1.1CL

w0.9

0.8

0.6

rt 0.5-S0.4-

0.3

0.2-

0.1

0 200 400 600

ELEVATION. FT.a SDFDAM + EACD-84 * BLOCK


53

Ef/Es =2 (EQ2)UPSTREAM FACE (D638)

1 .91.8-1.7-1.6-

0 1.5 -

ui 1.4 -w&n 1.3 -W7J 1.2-

IL O 0.9G 0.8-

Fr 0.7a- 0.6

S 0.5-0.4-0.3-0.2-0.1

0 200 400 S00

ELEVATION, Fr.0 SDFDAM + F-AGD-84 BLOCK

Ef/Es =2 (EQ 2)1.7DOWNSTREAM FACE D 638)

1.7-

1.4-

M 1.3-

vi 1.2-U, 1.1

Ena 0.9

dQ30.8-Uj;0.7-

FF 0.6-(L 0.5-

0.4-

0.3

0.2-

0.1

0-0 200 400 600

ELEVATION. Fr.a SDFDAM + EAGD-84 BLOCK


54

RIGID (EQ 2 )2.1UPSTREAM FACE (0638)

2

1.91.8-

1.5-

Cn

1.2

Ll 1.1

660.9-Z" 0.8-w 0.7-a.

0.6-. ~ 0.5 -m 0.4-

0.3-0.2-0.1

0 200 400 600

ELEVATION. FT.D SDFOAM + F-AGD-84 BLOCK

RIGID ( EQ2)DOWNSTREAM FACE D 638)

1.8-

w 1.6 -

~I 1.4a

Z- 1.7-

x~ 0.9-

~O0.8-

S 0.5 -0.42-

0.1 -

0 200 400 600

ELEVATION. FT.D SDFDAM + F-AGD-84 BLOCK

Figure 28. Effect of Foundation Modulus, E f(Case 32)

55

RIGID -1 MODE (EQ 1)700- UPSTREAM FACE (S1 30)

Soo

-; 500-

w

400-

I- 300-

IL

200-

100-

0o 2 0 40 60 80 100 120

ELEVATION. Fr.U SDFOAM + EAGD-84

RIGID -i MODE (EQ 1)BooDOWNSTREAM FACE (S130)

700-

ii[. 600-Ui

500-

S400-

Z300-

S200-

100-

00 20 0 60 I8O 0 I 100 120

ELEVATION. PT.a SDFDAM + EAGD-84

Figure 29. Comparison of Fundamental Mode Response of SDFDAMand EAGD-84 (Case 4)

56

7. APPENDIX

57

Ef/Es1 /2 (EQi)60 -UPSTREAM FACE (5130)

500-

Ld 400-cn

30-

a-

0-)

0 20 40 s0 80 100 120

ELEVATION. FT~.SSDFDAM +~ EAGD-84 BLOCK

EfEs=1 /2 (EQi)DOWNSTREAM FACE Sl S30)

600

400-

-

V 200-

100-

00 20 40 60 80 100 120

ELEVATION. FT.cSOFOAM 4 -AGD-84 *BLOCK

58

EfEs1 (EQi)500 -UPSTREAM FACE CS130)

700-

0~600-

W~ 500-

S400-

C-)9 300-It

200-

100-

0 20 40 s0 s0 100 120o

ELEVATION. Fr.0SDFDAM + EAGD-84 '~BLOCK

Ef/Es - 1 ( EQ 1)DOWNSTREAM FACE 5 130)

900

800-

a. 700-

w600-

wF- 500-

a. 400-0z

cr 300-

S200-

100-

0 20 40 60 80 100 120

ELEVATION, FT.SSDFDAM + EAGD-84 BLOCK

59

Elf/Es =2 (EQ 1Bo -UPSTREAM FACE 5 130)

700-

5~ 00

4' 00-

w

200

1J 00-

0.

0 20 40 60 s0 100 120

ELEVATION, FT.r~SOFOAM + EAGO-84 BLOCK

Elf/Es ==2 ( EQi 1DOWNSTREAM FACE (l 130)

0.9

ui 0.7w(A~

(n

S0.5

Z-0.4-

a- 0.3-

0.2-

0.1

0 20 40 60 830 100 120

ELEVATION. FT.13 SOFOAM + EAGO-64 BLOCK

60

Ef/Es=1 /2 EQ 2)UPSTREAM FACE ( S130 )

190 -

180 -

170-

160-

150-

E- 140-

vi 1-30Ln 120 -(Aw 110-

I- 100-tn_j 90-

) 70-zi- 60-(n 50-

< 40-

: 30-

20-

10-0

- 10 I I I 1 I I

0 20 40 60 80 100 120

ELEVATION, FT.1 SDFDAM + EAGD-84 < BLOCK

Ef/Es =1 /2 ( EQ 2)DOWNSTREAM FACE 30 )

220

200

180

O~160

W 140

120

-J 100

C.a 80-

w 60-

x 40-

20'

0

-20 , , , I

0 20 40 60 80 100 120

ELEVATION. FT.E3 SOFOAM + EACD-84 . BLOCK

61

Ef/Es=1 (EQ2)UPSTREAM FACE S 130)

200-190-1SO0170-

S150-a 140-w 130-

120-r 110-

100-

z 70-w 60-

. 50

4030120-10-

0 20 40 60 80 100 120


Ef/Es =1 ( EQ2)DOWNSTREAM FACE C5130)

240-

220-

200-

180-

u1 160-

(A 140-

W 120-

< 100-

a- 6

S 40

20-

0 20 40 60 80 100 120

ELEVATION. FT.SSDFDAM + EAGD-84 'r BLOCK

62

Ef/Es -- 2 (EQ 2)UPSTREAM FACE ( S130 )260-

240

220

Uj 20002

(n0 130w

140

120

0 ;

d 100-

60

40

2 0-

20

0 20 40 60 80 100 120

ELEVATION. FT.n SDFDAM 4 EAGD-84 o BLOCK

Ef/Es 2 ( EQ 2)DOWNSTREAM FACE S S130

280-

260

240-

n~ 220-

fl 200-wwfn 180 -

I 160-

.j 140-

0120-

100-(L 80-

40-

20-

0

0 20 40 60 80 100 120

ELEVATION. FT.n SDFDAM - EAGO-84 < BLOCK

63

Ef/Es= 1 /2 (EQi 1UPSTREAM FACE (5200)

1.1-

0.8L&J

W 0.7-

0.6-

0.5-

0.4-

S 0.3-

0.2-

0.1

0 20 4.0 60 80 10 120 140 160 180 200

ELEVATION, FT.SSDFOAM + FAGD-84 BLOCK

Ef/Es 1 /2 ( EQ 1)DOWNSTREAM FACE S 200)

900

B00

S700-a-

w500-

a. 400-

S300-a-

200-

100-

0-0 20 40 60 s0 100 120 140 160 180 200

ELEVATION. FT.a SDFDAM + EAGO-84 BLOCK

64

Ef/Es= 1 (EQ 1 )UPSTREAM FACE S 5200 )

1.3

1.1 -

. 0.9-

. 0.9-

zt: 0.5-

(L 0.4-

0.3-

0.2-

0.1 -

0-0 20 40 60 B0 100 120 140 160 180 200

ELEVATION, FT.S SDFDAM EAGD-84 o BLOCK

Ef/Es= 1 ( EQ 1)DOWNSTREAM FACE S S200 )

1.2

1.1

Cn 0.9

0.8

S0.8tno

"3 0.5-..

0.3-

0.2

0.1

0-0 20 40 60 80 100 120 140 160 180 200

ELEVATION. FT.n SOFDAM + EAGD-84 € BLOCK

65

Ef/Es =2 (EQ 1UPSTREAM FACE CS200)

1.6

1.5

1.4-

1.3-

0.. 1.2-

(n 1a:u 0.8-

Z 0.85

(L 0.5-

0.4-

0.3-

0.2-

0.1

0-0 2 40 60 so 100 120 140 160 180 200

ELEVATION. Fr.o SDFOAM + EAGD-84 * BLOCK

Ef/Es 2 ( EQi 1DOWNSTREAM FACE S 200)

1.5

1.4

1.3-

1.2-a. 1.1

W-0 0.9-

0.5

0.70.60

0. .5

0.4-

0.3-

0.2-

0.1

0~0 20 40 60 80 100 120 140 160 180 200

ELEVATION, FT.0 SDFDAM + EACD-84 * BLOCK

66

Ef/Es = 1/2 (EQ 2)UPSTREAM FACE S 3200 )400-

350

~.300a.

cliw 250(nEnw1 200-U)

(L 150-

zw 100a.

50

--

0 20 40 60 80 100 120 140 160 180 200

ELEVATION. FT.n SDFDAM + EAGD-B4 , BLOCK

Ef/Es = 1/2 ( EQ 2 )DOWNSTREAM FACE S S200 )

320

300

280

260

(. 240

, 220w(n 200w 180

( 160

n 140CL[ 120

100-n Bo-I 80

60-40

200.

-20

0 20 40 60 80 100 120 140 160 180 200

ELEVATION. FT.a SDFDAM + EAGD-84 o BLOCK

67

Ef/Es= 1 (EQ2)UPSTREAM FACE CS200)

500-

450-

400-

350

cncn 300-

nt 250

ad 200-

z 150-0-

~<100-

50-

0 20 40 60 80 100 120 140 160 18 0 200

ELEVATION. FT.0SDFDAM + F-AGD-84 BLOCK

Ef/Es - 1 ( EIQ2)DOWNSTREAM FACE S 200)

350-

~i300-

w 250-U,w

200-t,

a~150-

100-

a--

£0

0 20 40 60 80 100 120 140 1601020

ELEVATION. FT.oSDFDAM + EAGD-84 BLOCK

68

Ef/Es=2 (EQ2)500- UPSTREAM FACE S 200)

450-

400-

(L350

LLIU,V)30

250-

200-

50-

0 20 40 o 1o 10 ;10 140 160 160o 200

ELEVATION. FT.b SDFDAM + EAGD-84 BLOCK

Ef/Es =2 (EQ2)400DOWNSTREAM FACE S 200)

350-

S300-

LJ250-U)

200

150-

zm 100-

50-

0-

0 20 40 60 80 1 00 120 140 160O 180 200

ELEVATION, Fr.o SDFDAM + E-AGD-84 BLOCK

69

Ef/Es 1 /2 (EQ 1)UPSTREAM FACE CS300)

1.7

1.6-

1.5-

1.4-

S 1.3-

S 1.2-

wo

a 0.9-a)

_ 0.8-

1- 0.7

Fr 0.6-a- 0.5-

< 0.4

0.3-

0.2-

0.1

0 40 80 120 160 200 240 280

ELEVATION. FT.0 SDFOAM +EACD-84 BLOCK

Ef/Es=1/2 (EQi)DOWNSTREAM FACE S 300)

1.4

1.3

Fn 1.2

(L

w-n 0.9-

0.8-

0.7-

5,0.6-zIr 0.5-

S0.4-0.3-

0.2-

0.1

0 40 80 120 160 200 240 280

ELEVATION. FT.aSDFDAM +EAGD-94 *BLOCK

70

Ef/Es= 1 (EQi 12.1UPSTREAM FACE S 300)

2-1.9-1.8-1.7-

1.4-1.3-1.2-

a 1.1

R 0.7-0.6-0.5-0.4-0.3-0.2-

0 40 so 120 160 200 240 280

ELEVATION. FT.aSDFDAM4 + I- - 4 *BLOCK

Ef/Es - 1 ( EQi 1DOWNSTREAM FACE S 300)

1.8 -1.7-1.6-

j ~ 1.5-

a. 1.3

1.2-

~'0.9-00.8-

0.7-a. 0.6-

0.5-0.4-0.3-0.2-0.1

0 40 s0 120 160 200 240 280

ELEVATION. FT.0 SDFOAM *EAGD-84 oBLOCK

71

Ef/Es =2 (EQ 1UPSTREAM FACE S 300)

1.7-1.6-

1.5-(Fl 1.4-

1.3W 1.27En

t2 1.1j50

g0 0.8-

~- 0.7-w 0.6-

S 0.5-

0.4-

0.3-0.2-

0.10-

0 40 80 120 160 200 240 280

ELEVATION, FT.r~SOFDAM + EAGD-84 BLOCK

Ef/Es =2 (EQi 1DOWNSTREAM FACE CS300)

1.6

1 .4-

S 1.3-S 1.2-

U1 1.1

(a0.9-

0.87

1" 0.6-a. 0.5-

0.4-

0.3-

0.2-

0.1

0 40 80 120 160 200 240 280

ELEVATION. FT.a SOFOAM + EAGO-84 . BLOCK

72

Ef/Es 1 1/2 (EQ 2)500UPSTREAM FACE (S300)

500-

400

w 300-U)w

S200-

a-

zw 100-

0.

0 40 80 120 180 200 240 280

ELEVATION. FT.a SOFOAM + F-AGO-84 0BLOCK

Ef/Es= 1/2 (EQ2)DOWNSTREAM FACE (S300)

340

320-

300-

Vi 260-a-240

Lw 220-En

w 200-

ISO

100-1180-60

40-

20

0 40 80 120 160 200 240 280

ELEVATION. FT.a SOPOAM + E-AGD-84 *BLOCK

73


600 - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

500-

a 400-w

w300-

0.200-

0~

- 100

0 40 80 120 160 200 240 280


Ef/Es = 1 ( EQ2)DOWNSTREAM FACE S 300)

400

350-

Fn 300-a.

w 250-

200-

150-

2r 100-

50-

50

0 40 80 120 160 200 240 280

ELEVATION. FT.0l SOFOAM + F-AGD-84 .BLOCK

74


700-

600-

0i-500-

I40400

7-200-

100-

0 40 so 120 160 200 240 280

ELEVATION. FT.tl SOFOAM + EAGD-84 * BLOCK

Ef,"/Es=-2 ( EQ2)DOWNSTREAM FACE S 300)

400-

w 300-

S200-

(L0.zw 100-0..

0

-100-0 40 s0 120 160 200 240 280O

ELEVA-ION, Fr.a SDFDAM + F-AGD-84 * BLOCK

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