simplified analysis of conc gravity dams inc foundation flexibility (1989) - report (82)
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SIMPLIFIED ANALYSIS OF CONCRETE GRAVITY
DAMS .,CLUDING FOUNDATION FLEXIEILITY
James M. Nau
Department of Civil EngineeringNorth Carolina State UniversityRaleigh. North Carolina 2769b
0 T0 3 1c9
September 1989
Final Report
i - :::.DEPARTMENT OF THE ARMY
US Army Corps of Engineers
Washington, DC 20314-1000
, ,Contract No. DACW39--88-K-OO,33
CTjWork Unit 315,88
. StructUres Laboratory
89T0 3 01
IJSmy ng epW tebry1989 mntSato. HI AI OII TO~i/-/lil il,:09Hall F-rinal Reprtr Msip 38-19
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I e Ia
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1 TITLE Include Security Classification)
Oip li: ied Analysis of Concrete Gravity Dams Including Foundation Flexibility
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'Fina rtiDort FROM TO September 1989 83
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7 COSATI CODES 18 SUBJECT TERMS Continue on reverse if necessary and identifyby 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 bje tiv of this study was to develop ;I modcl for the flexihle foundation rock
lt-, lth a dam and to inll rporatt. 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 ,.
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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. .. 2
2.2 Earthquake Induced Loads and Stress Calculations .. ....... 4
3. Development of the Foundation Flexibility Matrix .. .........
3.1 Theory of Flamant. ...................... 7
3.2 Procedure............................9
4. Parametric Study .......................... 12
4.1 Objective ........................... 124.2 Selection of Dams and Response Parameters. .......... 12
4.3 Results of Parametric Study. ................ 14
5. Summary and Conclusions. ...................... 17
6. References ............................ 19
7. Appendix ............................. 56
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iii
List of Tables
Table Page
1. The Coefficient Fmn for the Half-Plane Problem . . . 20
2. Flexibility Coefficients for Column 12 of the
Foundation 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 Stresseson the
UpstreamFace . . . 25
7. Maximum Principal Stresses on the Downstream Face . . 26
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iv
List of Figures
Figure Page
1. Dam-Water-Foundation Rock System ... .......... . 27
2. Standard Mode Shape and Fundamental Period
for the Dam on a Rigid Foundation and Empty
Reservoir ......... ...................... . 28
3. Standard Values for R1 , the Ratio of Fundamental
Vibration periods of the Dam with and without water . 29
4. Standard Values for Rf, the Period Lengthening
Ratio 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 Depth
of Water for H/Hs=I and Various Values of
R2 = %rs/ .r....... ...................... .32
7. Finite Element Mesh Generated by SDFDAM for
Dam S130 ......... ...................... 33
8. Flamant Isotropic, Elastic Half-Plane:
(a) Vertical and Horizontal Relative
Displacements Due to a Uniformly Loaded Strip
(b) Location of Nodes and Load for theFlamant Equation ...... .................. . 34
9. Location and Direction of Coefficients for
Column 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 of
the 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 the
Computation of Response of Dam S130 (Case 4) . . . . 40
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Figure p ge
15. Effect of Foundation Modulus, Ef (Case 4) ..... 41
16. Effect of Foundation Modulus, Ef (Case 8) ..... 42
17. Effect of Foundation Modulus, Ef (Case 12) ..... 43
18. Effect of Foundation Modulus, Ef (Case 16) ..... 44
19. Effect of Foundation Modulus, Ef (Case 20) ..... 45
20. Effect of Foundation Modulus, Ef (Case 24) ..... 46
21. Effect :,f cundation Modulus, Ef (Case 25) ..... 47
22. Effect of Foundation Modulus, Ef (Case 26) ..... 48
23. Effect of Foundation Modulus, Ef (Case 27) ..... 49
24. Effect of Foundation Modulus, Ef (Case 28) ..... 50
25. Effect of Foundation Modulus, Ef (Case 29) ..... 51
26. Effect of Foundation Modulus, Ef (Case 30) ..... 52
27. Effect of Foundation Modulus, Ef (Case 31) ..... 53
28. Effect of Foundation Modulus, Ef (Case 32) ..... 54
29. Comparison of Fundamental Mode Response of
SDFDAM and EAGD-84 (Case 4) .... ............. 55
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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
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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.
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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 AVY)
pl yAs) = impulsive water pressure,
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y(v) = shape of the fundamental mode of
vibration of the dam on a rigid foundation
with empty reservoir,
Ws = fundamental resonant frequency of the dam on a
rigid foundation, including hydrodynamic
effects.
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-foundation
rock interaction (Fig. 4).
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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 foundation
with 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,
gp WwH
ts
I n cos((2n - 1) ir/2] (5)
nl(2n - 1) /i- 1 2 i.)2
X- (2(
(2n - 1)2 s
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where
i = y/H,
Or = C/2H = the fundamental resonant frequency
for impulsive pressure in water,
os = 2t/RITs = the fundamental resonant frequency
of 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
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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.
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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) 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)
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where e is the base for the natural logarithm. Now, the equation for wmn can
be rewritten as
(m-n+0.5)a
Wrn - 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,
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nUm = - F , (13)
where
Hn = magnitude of the horizontal load, and
FMN= Fmn.
For vertical displacements,
1-v
WMN= ± Hn (14)2Ef
except where x = 0, wMN = 0.
For plane strain Ef and Vf are replaced by Ef/(l-vf 2 ) 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
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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.
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f V~f)F12,12 nf
At x/a = 40,
A 2 (m - n) - 1 - 2 4 n -0) - 1 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 -7
fl 2 , 1 2 6.2 1 (10.764) - 4.066 x 10 in., and12,12 (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 - 0.252 -7S2,12 6 (10.764 - 6.602) = 1.572 x 10 in.,n(7.9xlO)
1 - Vf - 2V f2 1 - .25 - 2(.25)2
fl, 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.
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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 ofdams 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
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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.
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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/E 5 = . The stresses of perhaps greatest concern
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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
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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.
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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
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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.
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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 Earthquake
Analysis of Concrete Gravity Dams, Report No. UCB/EERC-84/II, Earthquake
Engineering Research Center, University of California,
Berkeley, California, Aug., 1984.
5. Fenves, G., and Chopra, A. K., Simplified Analysis for Earthquake
Resistant 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 Report
70-20, Department of Civil Engineering, University of California,
Berkeley, 1970.
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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. 31
17 -9.052
18 -9.167
19 -9.275
20 -9.378
Table 1. The Coefficient Fmn for the Half-Plane Problem
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Flexibility Coefficient
Location (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 the
Foundation Rock Flexibility Matrix
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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 0.0
Sd 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/ft 3 ) 144.0 144.0 144.0 144.0
UPSTREAM DOWNSTREAM
BASE
Table 3. Properties of Dams Used for the Parametric Study
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Case Ef/E s Ef Earthquake Dam
1 1/2 1.5 EQ1 S130
2 1 3.0 EQ1 S130
3 2 6.0 EQ1 S130
4 c c EQ1 S130
5 1/2 1.5 EQ2 S130
6 i 3.0 EQ2 S130
7 2 6.0 EQ2 S130
8 0 0 EQ2 S130
9 1/2 1.5 EQ1 S200
10 1 3.0 EQ1 S200
11 2 6.0 EQ1 S200
12 c 0 EQI S200
13 1/2 1.5 EQ2 S200
14 1 3.0 EQ2 S200
15 2 6.0 EQ2 S200
16 0 0 EQ2 S200
17 1/2 1.5 EQ1 S300
18 1 3.0 EQ1 S300
19 2 6.0 EQ1 S300
20 0 0 EQI S300
21 1/2 1.5 EQ2 S300
22 1 3.0 EQ2 S300
23 2 6.0 EQ2 S30024 c c EQ2 S300
25 1/2 2.5 EQI D638
26 1 5.0 EQ1 D638
27 2 10.0 EQ1 D638
28 0 0 EQI D638
29 1/2 2.5 EQ2 D638
30 1 5.0 EQ2 D638
31 2 10.0 EQ2 D638
32 0 0 EQ2 D638
Note: Ef in million psi
Table 4. Cases Considered in this Study
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Natural Period of Viscous Damping Spectral Accele-
Case Vibration (sec) Factor ration Value (ci)
1 0.163 0.139 1.360
20.145 0.094 1.8103 0.134 0.067 2.030
4 0.122 0.043 1.720
5 0.163 0.139 0.500
6 0.145 0.094 0.517
7 0.134 0.067 0.670
8 0.122 0.043 0.684
9 0.261 0.138 1.360
10 0.232 0.093 1.740
11 0.215 0.066 2.230
12 0.195 0.041 2.090
13 0.261 0.138 0.52514 0.232 0.093 0.637
15 0.215 0.066 0.639
16 0.195 0.041 0.703
17 0.402 0.138 1.520
18 0.357 0.092 1.890
19 0.331 0.065 1.710
20 0.301 0.040 2.150
21 0.402 0.138 0.438
22 0.357 0.092 0.515
23 0.331 0.065 0.60924 0.301 0.040 0.753
25 0.682 0.137 0.687
26 0.607 0.091 0.696
27 0.562 0.064 0.898
28 0.511 0.039 1.710
29 0.682 0.137 0.507
30 0.607 0.091 0.717
31 0.562 0.064 0.853
32 0.511 0.039 0.904
Table 5. Natural Periods, Viscous Damping Factors, and Spectral
Acceleration Values
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Maximum Principal Stress on the
Upstream Face (psi)
SDFDAM BLOCK EAGD-84 Ratio Ratio
Case (i) (2) (3) (M / 3) (2)/(3)
1 376 401 251 1.50 1.60
2 514 519 266 1.93 1.95
3 581 578 352 1.65 1.64
4 486 495 478 1.02 1.04
5 114 131 65 1.75 2.02
6 119 135 111 1.07 1.22
7 166 176 135 1.23 1.30
8 170 180 193 0.88 0.93
9 541 567 443 1.22 1.28
10 704 706 487 1.45 1.4511 915 917 671 1.36 1.37
12 855 860 657 1.30 1.31
13 182 190 120 1.52 1.58
14 231 233 141 1.64 1.65
15 231 234 147 1.57 1.59
16 259 259 211 1.23 1.23
17 772 772 444 1.74 1.74
18 972 932 674 1.44 1.38
19 875 889 760 1.15 1.17
20 1112 111.8 742 1.50 1.51
21 187 203 165 1.13 1.23
22 229 237 178 1.29 1.3323 280 280 215 1.30 1.30
24 358 347 214 1.67 1.62
25 584 838 638 0.92 1.31
26 592 845 872 0.68 0.97
27 782 992 1100 0.71 0.90
28 1548 1676 1866 0.83 0.91
29 414 441 431 0.96 1.02
30 612 632 573 1.07 1.10
31 740 758 776 0.95 0.9832 788 805 795 0.99 1.01
Table 6. Maximum Principal Stresses on the Upstream Face
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Maximum Principal Stress on the
Downstream Face (psi)
SDFDAM BLOCK EAGD-84 RATIO RATIO
Case (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 1.72
8 252 295 270 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
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FWATER
ABSORPTIVE RESERVOIR RIGID8BOTTOM MATERIALS D--M BASE
/9
Figure 1. Dam-Water Foundation Rock System
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I0
00.9
as
Ea 0.7
~0.6-
05-Vibrat..in Period T lA. a
Q4 - Hsa height of dam in ft.
o E elastic modulus ofconcrete in psi
*Q3
.2 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 Period
for the Dam on a Rigid Foundation and Empty
Reservoir. After Chopra (1978)
. .. = = eee e ee I II III
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1.7
E - 7xl O6 psi
1.6o 0
0
SE:5x10
6
E E 1.4o o E z4xIO
0 0 E:3x 106
"0 vo 1.3- Ez2xlO6
0 0
02 E<I x I06
0 000
- >
it
1.0
0.4 0.5 0.6 0.7 0.8 0.9 1.0
Total Depth of Water,H
Height of Dam,Hs
Figure 3. Standard Values for R. the Ratio of Fundamental
Vibration Periods of ihe Dam with and without
Water. After Chopra (1978).
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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).
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0.35
0.30 -
7f •0.50
0.25 -
4A.0 0.25
4 0.20 0.1
eC
z 0.15
.10
S015-
00
0
0.05
0 I 2 3 4 5
ES /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. After
Fenves and Chopra (1986).
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1.0
0 0.8-
0
5.60.99
S- 0.4
a.. 0.97>w
0 .
0.7.
0 0.2 0.4 0.6 0.8
gp wH
Figure 6. Standard Plots for Variation of p over Depth or Water
for H/H = 1 and Various Values ol R2 = s w . After
Chopra T1978).
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331
Figure 7. Finite Element Mesh Generated by SDFDAM for DAM S130
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(m-n)a
a/2 I a/2
I a) m
Equationt
n m
(a)
Equationt
• ~ ~~ ~ ~ ~ ~ ~ Bs ofIlI
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C -
-4
-4
U -
44
0-4
04-
0
-44
a) 4J-f cc
C4-4
co044
443
CD 0.
44
cs;4-4
0
CC.)
4
C4
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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
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HORIZONTAL RESPONSE SPECTRA. 5 DAMPING
3-
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
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EAGD-84( EQ 1)G o o U P TR EA M 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 -. 1 4 DT-.02
EAGD-84( EQ 1)DOWNSTrREAM FACE S 130 -RIGID)
700-
600-
3 00
2 00-
100-
0-1 11 1 9 1
0 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 Response
of Damn S130 (Case 4)
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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-
.. 24 0
~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 Response
of Dam S200 (Case 12)
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EAGD-84( EQ 1)GooUPTREAM FACE (S130 -RIGID)
a.
w 400-
300-
200
100-
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 Sl30 -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 of
Response of Damn S130 (Case 4)
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RIGID ( EQ 1)UPSTREAM FACE ( 5130 )
700 -
600-
(L
500-
UJ
wUn(nwLkr 400-
I--
a. 300-a
Z
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 130 )
700
I. 60 0
14J
u 500
.j 400
Z 300
0.
> 20 0
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)
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RIGID ( EQ 2 )UPSTREAM FACE S130 )
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 200
EnU7 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)
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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 FI 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 (Case 12)
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RIGID ( EQ2 )UPSTREAM FACE ( S200 )
50 0
400
a-
w 300
I- ,,
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
Figure 18. Effect of Foundation Modulus, Ef (Case 16)
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RIGID EQ 1 )UPSTREAM FACE S 300)
2.2
2-
1 8
c.. 1.6 -
w ,+
n 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)
2.1
2
1.9-
CL 1.7-
a 1.5-w 1.4-
r ' 1.2-
Lo0.9-
Z'-~ 0.8w 0.7-IL
S 0.6
0.5-
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
Figure 19. Effect of Foundation Modulus, Ef (Case 20)
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RIGID ( EQ 2 )700 UPSTREAM FACE S300 )8O0 -
700-
a. 600-
CA,u 500-iJJ
j 400
z 400CL
a-0
200
100-I
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
Figure 20. Effect of Foundation Modulus, Ef (Case 24)
. . . . " •• l I l I If
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Ef/Es 1/2 (EQ 1)UPSTREAM FACE D638
1.6
1.5-
1 .4-
a- 1.2 -
Eli .w
M0.9
0.8
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 ( EQiDOWNSTREAM 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, Ef(Case 25)
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Ef/Es= 1 EQi 1UPSTREAM FACE CD638)
1.4-
(. 1.2 -
(4 1.1
nl
0.8-
0 L00.7 -
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
Figure 22. Effect of Foundation Modulus, Ef (Case 26)
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Ef/Es =2 (EQi )UPSTREAM FACE ( D638 )
2.1
2
1.9
1.8'.7
13. 1.6
1.5-
L&J 1.4 -
(n 1.3
W- v~ 1.25 1.1
0 .9_'- 0.8
r 0.70. = 0.6
0.5
0.40.3
0.2
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
0.8
n 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
Figure 23. Effect of Foundation Modulus, Ef (Case 27)
. . . . ' '= a I I I I If
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RIGID EQ 1 )UPSTREAM FACE D 638)
4-
.3.5-
.L -
1i-
CA
22
0
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
CA
LA 2.5 -
-j 2
0 .5-
0
0 200 400 600
ELEVATION. FT.a SDFDAM + EAGD-84 BLOCK
Figure 24. Effect of Foundation Modulus, E (Case 28)
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Ef/Es - 1/2 (EQ 2)UPSTREAM FACE ( D638 )
0.9
a. 0.8
w 0.7-&n
L ' 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 CD638 )
80 0
700
a.uai 600-
w
"w 500-
05
-J 400-
i-L) 300z
a- 200-
100-
0 200 400 600
ELEVATION. FT.0 SOFDAM . EAGD-84 o BLOCK
Figure 25. Effect of Foundation Modulus, Ef (Case 29)
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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
-- 0.60
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
0.30.2-
0.1
0 200 400 600
ELEVATION. FT.a SDFDAM + EACD-84 * BLOCK
Figure 26. Effect of Foundation Modulus, Ef (Case 30)
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Ef/Es =2 EQ2 )UPSTREAM FACE (D638)
1 9
1.8-
1.7-
1.6-
0 1.5 -
ui 1.4 -w n 1.3 -
W7J 1.2-
IL O 0.9
G 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-
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
Figure 27. Effect of Foundation Modulus, E (Case 31)
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RIGID EQ 2 )2.1UPSTREAM FACE (0638)
2
1.9
1.8-
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 1.4
a
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, Ef(Case 32)
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RIGID -1 MODE EQ 1)
700- UPSTREAM FACE (S1 30)
Soo
-; 500-
w
400-
I- 300-
IL
200-
100-
0o 20 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 608O 0 I 100 120
ELEVATION. PT.
a SDFDAM + EAGD-84
Figure 29. Comparison of Fundamental Mode Response of SDFDAM
and EAGD-84 (Case 4)
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7. APPENDIX
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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 Sl30)
600
400-
-
V 200-
100-
0
0 20 40 60 80 100 120
ELEVATION. FT.cSOFOAM 4 -AGD-84 *BLOCK
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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. 400-0
zcr 300-
S200-
100-
0 20 40 60 80 100 120
ELEVATION, FT .SSDFDAM + EAGD-84 BLOCK
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Elf/Es =2 EQ 1Bo -UPSTREAM FACE 5 130)
700-
5~00
400 -
w
200
1J 0-
0.
0 20 40 60 s0 100 120
ELEVATION, FT .r~SOFOAM + EAGO-84 BLOCK
Elf/Es ==2 ( EQiDOWNSTREAM 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
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Ef/Es=1 /2 EQ 2)UPSTREAM FACE ( S130 )
190 -
180 -
170-
160-
150-
E- 140-
vi 1-30
Ln 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 )
22 0
200180
O~160
W 140
120
-J 100
C.a 80-
w 60-
x 40-
20'
0
-20 , , , I
0 2 40 60 80 100 120
ELEVATION. FT.E3 SOFOAM + EACD-84 . BLOCK
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Ef/Es=1 EQ2 )UPSTREAM FACE S130)
200-
190-
1SO0
170-
S150-
a 140-w 130-
120-
r 110-
100-
z 70-w 60-
. 50
40
301
20-
10-
0 20 40 60 80 100 120
ELEVATION. FT.13 SDFDAM + EAGD-84 BLOCK
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
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Ef/Es -- 2 (EQ 2)UPSTREAM FACE ( S130 )60-
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 130
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
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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 S200)
900
B00
S700-a-
w500-
a. 400-
S300a-
200-
100-
0-
0 20 40 60 s0 100 120 140 160 180 200
ELEVATION. FT.a SDFDAM + EAGO-84 BLOCK
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Ef /Es= 1 (EQ 1 )UPSTREAM FACE S 200 )
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 DFDAM EAGD-84 o BLOCK
Ef /Es= 1 ( EQ 1)DOWNSTREAM FACE S 200 )
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
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Ef/Es =2 EQ 1UPSTREAM FACE CS200)
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1.5
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66
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400-
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67
Ef/Es= 1 EQ2 )UPSTREAM FACE CS200)
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400-
350
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68
Ef/Es=2 EQ2 )
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450-
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250-
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69
Ef/Es 1 /2 EQ 1)UPSTREAM FACE CS300)
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0.3-
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1.4
1.3
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70
Ef /Es= 1 EQi 12.1UPSTREAM FACE S 300)
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1.8-1.7-
1.4-
1.3-
1.2-
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0.6-
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0.4-
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1.2-
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0.3-
0.2-
0.1
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71
Ef/Es =2 EQ 1UPSTREAM FACE S 300)
1.7-
1.6-
1.5-
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1.3
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0.4-
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1.6
1.4-
S 1.3-
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0.4-
0.3-
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72
Ef/Es 1 /2 EQ 2)500UPSTREAM FACE (S300)
500-
400
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w
S200-
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340
320-
300-
Vi 260-a-240
Lw 220-En
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100-1180-0
40-20
0 40 80 120 160 200 240 280
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73
Ef/Es= 1 EQ2 )UPSTREAM FACE CS300)
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500-
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Ef/Es = 1 ( EQ2 )DOWNSTREAM FACE S 00)
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150-
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Ef/Es= 2 EQ2 )UPSTREAM FACE CS300)
700-
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7-200-
100-
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Ef, /Es=-2 ( EQ2 )DOWNSTREAM FACE S300)
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