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This is to certify that the

dissertation entitled

Response of an Earth Dam to Spatially Varying
Earthquake Ground Motion

presented by

Mu—Tsang Chen

has been accepted towards fulfillment
of the requirements for

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Wye/«Al

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Date 3/1/95

 

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_.__.._ -—————

 

 

 

RESPONSE OF AN EARTH DAM TO SPATIALLY VARYING

EARTHQUAKE GROUND MOTION

By

Mu-Tsang Chen

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

1995

ABSTRACT

RESPONSE OF AN EARTH DAM TO SPATIALLY VARYING
EARTHQUAKE GROUND MOTION

By

Mu—Tsang Chen

The stochastic responses of the Santa Felicia earth dam to spatially varying earth-
quake ground motion (SVEGM) are analyzed. A space-time earthquake ground motion
model that accounts for both coherency decay and seismic wave propagation is used to
specify the support motions, and the results are compared with those for various simpliﬁed
excitations. In addition, a preliminary reliability analysis using the Mohr-Coulomb strength
criterion is performed. The effects of propagation speed and direction for SV/P and SH
waves and different coherency models on the responses of the dam are also investigated.
Finally, techniques to simplify the excitation for use with simpliﬁed mechanical models of
the dam are presented.

The Santa Felicia dam is represented as a three-dimensional inhomogeneous ﬁnite
element model using the I-DEAS VI.i computer program. The variation of shear modulus
with depth from the crest due to conﬁning pressure is taken into account. Finite element-
based random vibration analysis is performed and the statistical moments of the displace-
ment, strain and stress responses are computed. Statistical moments of the maximum shear
stress, which is non-linearly related to the Cartesian stresses, are computed using Monte

Carol simulation as well as an approximate ﬁrst order second moment method.

The results of the study indicate that the effect of SVEGM is signiﬁcant, especially
on the stress response of stiff material near the base of the dam. The assumption of fully
coherent support motions is found to slightly over-estimate the displacement and strain
responses but signiﬁcantly under-estimate the stress response near the base. The wave
passage effect is not as signiﬁcant as the coherency decay when the ground displacements
are not highly coherent. The sensitivity of the stress response to different coherency functions
at the base can be dramatic. For simpliﬁed analysis, a 2-D shear beam model should be
capable of yielding good results at locations distant from the base with a suitably modiﬁed
seismic excitation. For computing the stress response in stiff material near the base, a 3-D
ﬁnite element model is required, but costly dynamic analysis can be avoided if the ground

displacements are not highly coherent; a pseudo-static analysis sufﬁces in this case.

Q?) My famify

ACKNOWLEDGEMENTS

I would like to express my sincere appreciation to Professor Ronald Harichandran for
his insight, encouragement and assistance throughout this research. It is certain that this

work would not have been possible without him.

I would also like to thank my guidance committee members Professor Frank Hatﬁeld,
Professor Thomas Wolff and Professor Chi Lo for their generous support and constructive

suggestions during the period of this study.

Loving thanks is offered to my parents and my wife for their sacriﬁce and patience.

Finally, I gratefully acknowledge the ﬁnancial support provided by the National

Science Foundation under Grant No. BCS 912l292.

TABLE OF CONTENTS

 

 

LIST OF TABLES _ xii

LIST OF FIGURES xv
CHAPTER

1. INTRODUCTION ..... ....................... . ................................................. . ....... .1

2. LITERATURE REVIEW ............................................................................... 6

General ............................................................................................................... 6

2.2 Conventional Approaches ................................................................................... 7

2.2.1 Seismic design criteria .............................................................................. 7

2.2.1.1 Pseudo-static method ..................................................................... 7

2.2.1.2 Sliding displacement method ......................................................... 9

2.2.1.3 Liquefaction criterion .................................................................. l 1

2.2.2 Shear beam models ................................................................................. 12

2.2.2.1 One-dimensional shear beam theory ............................................ 12

2.2.2.2 Shear beam with variable rigidity ................................................ 15

2.2.2.3 Forced vibration analysis ............................................................. 18

2.2.2.4 Two-dimensional shear beam theory ........................................... 20

2.2.2.5 Relevant research ........................................................................ 23

2.2.3 Plane strain ﬁnite element model ............................................................ 25

2.2.4 Three-dimensional model ....................................................................... 28

2.2.5 Model testing .......................................................................................... 30

2.2.5.1 Elastic vibration test .................................................................... 30

2.2.5.2 Dynamic failure test ..................................................................... 31

2.2.6 Field measurement ................................................................................. 32

2.3 Recent Studies .................................................................................................. 34

vi

3. DETERMINISTIC RESPONSE OF THE SANTA FELICIA

EARTH DAM ................... . ........... ........................ ...36
3.1 Description of Santa Felicia Dam ..................................................................... 36
3.2 Model Assumptions .......................................................................................... 39

3.2.1 Geometry ................................................................................................ 39
3.2.2 Soil properties ......................................................................................... 39
3.2.3 Hydrodynamic effect ............................................................................. 39
3.2.4 Dam-foundation interaction ................................................................... 40
3.2.5 Radiation damping ................................................................................. 40
3.3 Modeling and Analysis using I-DEAS VI.i ..................................................... 40
3.3. 1 Introduction ............................................................................................ 40
3.3.2 Finite element model .............................................................................. 4]
3.3.3 Free vibration analysis ............................................................................ 41
3.3.4 Response to ground motion .................................................................... 43
3.4 Types of Failures in Earth Dams ....................................................................... 49
3.4.] Sliding failure ......................................................................................... 50
3.4.2 Liquefaction failure ................................................................................ 50
3.4.3 Longitudinal cracks ................................................................................ 52
3.4.4 Transverse cracks ................................................................................... 52
3.4.5 Embankment and foundation piping ....................................................... 53

4. STOCHASTIC RESPONSE ANALYSIS USING A F INITE ELEMENT

MODEL — THEORY ............ . ............. .. ....... . ................ .....54
4.1 Characteristics of Earthquakes .......................................................................... 54
4.2 Engineering Interpretation of Stochastic Responses ......................................... 54
4.3 Ground Motion Model ...................................................................................... 57
4.4 Finite Element Formulation of the Equations of Motion ................................... 58

4.4.1 Finite element discretization ................................................................... 58
4.4.2 Modal analysis ........................................................................................ 60
4.5 Random Vibration Analysis .............................................................................. 62

vii

4.6

4.7

4.8

4.5.1 Introduction ............................................................................................ 62

4.5.2 Variance of dynamic displacements ....................................................... 63
4.5.3 Variances of pseudo-static displacements .............................................. 65
4.5.4 Covariance of pseudo-static and dynamic displacements ....................... 66
4.5.5 Calculation of covariance matrix of nodal stresses ................................. 67
4.5.5.1 Dynamic response ....................................................................... 68
4.5.5.2 Pseudo—static response ................................................................. 68
4.5.5.3 Covariance of pseudo-static and dynamic responses ................... 69
Viscous Damping vs. Hysteretic Damping ....................................................... 69
4.6.1 Shear beam with viscous damping .......................................................... 70
4.6.1.1 Displacement response ................................................................ 71
4.6.1.2 Shear force ................................................................................... 71
4.6.1.3 Bending moment ......................................................................... 72

4.6.2 Shear beam with hysteretic damping ...................................................... 72
4.6.3 Discussion .............................................................................................. 74
Statistical Moments of Maximum Shear Stress and Strain ................................ 75
4.7.1 Introduction ............................................................................................ 75
4.7.2 Calculation of maximum shear stress for special cases ........................... 76
4.7.2.1 Uniaxial loading 12x ...................................................................... 76
4.7.2.2 Pure shear loading Ix). .................................................................. 79
4.7.2.3 2-D beam bending problem with stresses I, and Ix. ..................... 79

4.7.3 FOSM method for 3-D problems ..................................... l ....................... 8 5
4.7.3.1 Using Itil as basic variables .......................................................... 85
4.7.3.2 Using 1,2 as basic variables .......................................................... 88
Simulation ........................................................................................................ 91

5. STOCHASTIC RESPONSE OF THE SANTA FELICIA DAM

— CASE STUDY ..... ................ .....94
5.1 Ground Motion Model ...................................................................................... 94
5.2 Computation Steps ............................................................................................ 98
5.3 Numerical Results ........................................................................................... 100

5.3.1 Responses to general excitation ............................................................ lOO

viii

5.3.1.1 Displacement ............................................................................. 100

5.3.1.2 Stress ......................................................................................... 103

5.3.1.3 Strain ......................................................................................... 105

5.3.2 Modal contributions ............................................................................. 105
5.3.3 Responses to simplified excitations ...................................................... l 12
5.3.3.1 Displacement ............................................................................. l 12

5.3.3.2 Maximum shear strain ............................................................... l 14

5.3.3.3 Maximum shear stress ............................................................... 1 14

5.3.4 Correction factors for the FOSM method ............................................. 120
5.3.5 Response details over the dam .............................................................. 123
5.3.6 Contribution of response components to total response ........................ 129
5.3.6.1 Displacement ............................................................................. 129

5.3.6.2 Strain ......................................................................................... 131

5.3.6.3 Stress ......................................................................................... 131

5.4 Reliability against Local Failure under Gravity and Earthquake Loads .......... 137
5.4.1 Introduction ......................................................................................... 137
5.4.2 Gravity load .......................................................................................... 140
5.4.3 Results ................................................................................................. 141

5.5 Summary ........................................................................................................ 148

6. RESPONSE TO DIFFERENT GROUND MOTIONS

— PARAMETRIC STUDIES ...... ..... ..................... .150
6.1 Effect of Propagation Speed and Direction for SV/P Waves ........................... 150
6.2 Effect of Propagation Speed for SH Waves ..................................................... 152
6.3 Effect of Different Coherency Models ............................................................ 156

6.3.1 Coherency models ............................................................................... 159
6.3.1.1 Abrahamson’s model ................................................................ 159
6.3.1.2 Novak’s model .......................................................................... 161
6.3.1.3 Luco and Wong’s model ........................................................... 164
6.3.1.4 Hao’s model .............................................................................. 164

6.3.2 Responses to different coherency models ............................................ 167
6.3.2.1 Displacement and strain ............................................................. 167

6.3.2.2 Maximum shear stress ............................................................... 171

6.3.3 Discussion ............................................................................................ 178

7. SIMPLIFIED MODELING AND ANALYSIS TECHNIQUES .................. ....184
7.1 Introduction .................................................................................................... 184
7.2 One-dimensional Excitation Model ................................................................ 184
7.2.1 Fifteen strip subdivision of the base ...................................................... 186
7.2.1.1 Introduction ............................................................................... 186

7.2.1.2 Displacement and maximum shear strain responses .................. 190

7.2.1.3 Maximum shear stress response ................................................. 190

7.2.2 Effect of different number of strips on the boundary ............................. 194
7.2.2.1 Introduction ............................................................................... 194

7.2.2.2 Responses .................................................................................. 194

7.3 Conclusion and Discussion ............................................................................ 207
8. CONCLUSIONS AND RECOMMENDATIONS .................................... .....208
8.1 Summary and Conclusions ............................................................................. 208
8. 1. 1 Responses ............................................................................................. 209
8.1.2 Modal contributions ............................................................................. 210
8.1.3 Contribution of response components .................................................. 210
8.1.4 Reliability analysis ............................................................................... 211
8.1.5 Effect of wave propagation ................................................................... 212
8.1.6 Effect of coherency function ................................................................. 212

8.1.7 FOSM method for estimating the statistics of

the maximum shear stress ................................................................... 213

8.1.8 Simpliﬁed ground motion excitation model ......................................... 214

8.2 Recommendations for future research ............................................................ 215
8.2.1 Topographic effects .............................................................................. 215
8.2.2 2-D shear beam model and SVEGM ..................................................... 216
8.2.3 Nonlinear analysis ................................................................................ 216
8.2.4 Dynamic pore water pressure .............................................................. 216

APPENDIX A

 

 

Outline of the Use of I-DEAS Computer Software - - - - 217
A.1 Introduction .................................................................................................... 217
A2 Finite Element Modeling Procedures .............................................................. 217
A3 Steps for Free Vibration Analysis ................................................................... 219
A4 Deterministic Dynamic Analysis .................................................................... 220
A5 Generating [M] and [A] Matrices .................................................................... 221
A6 Techniques to Display Response Contours ..................................................... 224
A7 Gravity-induced Stresses ................................................................................ 226
APPENDIX B
Comparison between Seismograph Records and Computed Responses
for the Northridge Earthquake of January 17, 1994 -- -- - 227
BIBLIOGRAPHY ......................................................................................... 235

xi

TABLE 2.1.
TABLE 2.2.
TABLE 3.1.
TABLE 3.2.
TABLE 3.3.
TABLE 3.4.
TABLE 3.5.
TABLE 3.6.

TABLE 3.7.

TABLE 4.].

TABLE 4.2.

TABLE 4.3.

TABLE 4.4.

TABLE 5.1.
TABLE 5.2.
TABLE 5.3.

LIST OF TABLES

Response of l-D shear beam model .......................................................... 15
Velocity of transverse wave (m/sec) ......................................................... 17
Low strain shear moduli for Santa Felicia earth dam ................................ 38
Natural frequencies (Hz) compared with simpliﬁed model ...................... 38
Comparison of elapsed time for solving ten vibration modes ................... 42
Comparison of natural frequencies (Hz) using different mesh sizes ........ 42
Types of mode for the ﬁrst 20 modes ....................................................... 43
Natural Frequencies ................................................................................... 44

Maximum acceleration (ft/secz) response to different number

of modes .................................................................................................... 49

Comparison of responses using hysteretic damping and varying

viscous damping ........................................................................................ 74

Comparison of responses using hysteretic damping and constant

viscous damping ........................................................................................ 74

Comparison of stochastic moments of tmax using different approaches...79

Simulation results and correction factors (CF) for the mean

and standard deviation of tmax estimated by FOSM methods

using different basic variables ................................................................... 84
Double-ﬁlter autospectrum parameters ..................................................... 94
Coherency function parameters ................................................................. 95
Approximate elapsed time for calculation of integration .......................... 99

xii

TABLE 5.4.
TABLE 5.5.
TABLE 5.6.

TABLE 5.7.

TABLE 5.8.

TABLE 5.9.

TABLE 5.10.

TABLE 5.11.

TABLE 5.12.
TABLE 5.13.

TABLE 5.14.

TABLE 5.15.

TABLE 5.16.

TABLE 5.17.

TABLE 5.18.

TABLE 5.19.

Maximum u+3o nodal stress response for each component ................... 103
Maximum u+3o nodal strain response for each component ................... 105

Percent modal contribution to the dynamic x-displacement

response at the middle of the dam crest .................................................. 108

Percent modal contribution to the dynamic x-displacement

response at quarter length along the crest ............................................... 109

Percent modal contribution to the dynamic txy response - Node 275 ..... 1 10
Percent modal contribution to the dynamic ‘yxy response - Node 441 ..... 1 11

Comparison of maximum u+3o response for displacement,

maximum shear stress and maximum shear strain for four

different excitation models ...................................................................... l 14
Comparison of critical u+3o estimates of tnm (kPa) using

simulation and FOSM approaches with 1,2 as the basic variables .......... 120
Statistics of the correction factors CF,Ll and CFO ..................................... 122
Critical u+3o values of Tmax (kPa) in different material zones ............... 125
Coordinates of the nodes (ft) at which the 15 largest

displacement responses occur ................................................................. 129
Percentage contributions of the response components to the total

variance of the x-displacement ................................................................ 130
Nodal coordinates (ft) for the largest 15 variances of 715' ....................... 132
Percentage contribution of response components to the

total variance of the shear strain 7,). ........................................................ 132
Nodal coordinates (ft) for the largest 15 variances of Ix). ....................... 133

Percentage contribution of response components to the

total variance of the shear stress In. . ....................................................... 133

xiii

TABLE 5.20. Percentage contribution of response components to the total

variance of the shear stress Ix), at the critical node in the core ................ 134
TABLE 5.21. Physical properties of the core, shell and gravel ..................................... 140
TABLE 6.1. Maximum 11-1-30 responses to different apparent wave velocities .......... 152

TABLE 6.2. Maximum u+3o responses due to different wave types and speeds ....... 156

TABLE 6.3. Parameters in coherency function ( x10_4) ........................................... 167
TABLE 6.4. Maximum responses (u+3o) corresponding to different
coherency models .................................................................................... 171

TABLE 6.5. Maximum u+3o Tmax response (kPa) to different coherency models ..... 175

TABLE 7.1. Strip dimensions and reference point coordinates (ft) ............................ 188
TABLE 7.2. Estimated average separation (ft) between each pair of strips ................ 191

TABLE 7.3. Maximum 30' displacement responses using simpliﬁed

and general excitation models ................................................................. 198

TABLE 7.4. Critical u+3o maximum shear strain responses using

simpliﬁed and general excitation models ................................................ 198
TABLE 7.5. Critical 11-1-36 1mm responses (kPa) using simplified and
general excitation models ........................................................................ 205

TABLE B.l. Critical computed and recorded responses of the Santa

Felicia dam at the crest to the Northridge earthquake ............................ 234

xiv

Figure 2.1.
Figure 2.2.
Figure 2.3.
Figure 2.4.
Figure 2.5.
Figure 2.6.
Figure 2.7.
Figure 3.1.
Figure 3.2.
Figure 3.3.
Figure 3.4.
Figure 3.5.
Figure 3.6.
Figure 3.7.

Figure 3.8.
Figure 3.9.
Figure 3.10.
Figure 3.11.
Figure 3.12.

LIST OF FIGURES

Stability analysis for slope of an earth dam ................................................. 8
One-dimensional shear beam model .......................................................... l3
One-dimensional shear beam vibration modes .......................................... 16
Shear wedge in a rectangular canyon ......................................................... 20
2-D model vibration mode for m=1 and n=1 ............................................. 22
2—D model vibration mode for m=l and n=2 ............................................. 22
2-D model vibration mode for m=2 and n=2 ............................................. 22
Structural Details of Santa Felicia Earth Dam ........................................... 36
Santa Felicia earth dam model side view ................................................... 37
Illustration for different material zones ..................................................... 38
Finite Element Mesh .................................................................................. 41
Firth four vibration modes ......................................................................... 45
Vibration modes 5, 6, 7 and 8. ................................................................... 46

Accelerogram similar to the El Centro, California, earthquake

of May 18, 1940 ......................................................................................... 47
Velocity excitation ..................................................................................... 48
Displacement excitation ............................................................................. 48
Sliding failure of earth dam subjected to earthquake motions ................... 50
Liquefaction at the upstream side .............................................................. 51

Cracking caused by differential settlement between

embankment sections of dumped rock and rolled earth ............................. 52

XV

Figure 3.13.
Figure 4.1.
Figure 4.2.

Figure 4.3.
Figure 4.4.
Figure 4.5.
Figure 4.6.

Figure 4.7.
Figure 5.1.
Figure 5.2.
Figure 5.3.
Figure 5.4.

Figure 5.5.

Figure 5.6.

Figure 5.7.

Figure 5.8.

Figure 5.9.
Figure 5.10.

Typical transverse differential settlement cracks ....................................... 53
First upcrossing of level 11-1-30 in time history of response ....................... 56

Illustration of probability distribution of maximum

shear stress for l-D case ............................................................................ 78

Display of function F (Ix, In) = t ,/ 4 + In, ....................................... 80
Illustration of stress distributions of beam bending problem ..................... 83
Estimate of mean of Tmax for different number of samples ....................... 92

Estimate of standard deviation of ’CWM for different number of samples ..92

Probability density distribution of maximum shear stress ......................... 93
Fitted autospectra for ground motion accelerogram .................................. 94
Coherency vs. frequency ............................................................................ 96
Coherency vs. separation ........................................................................... 96

Accelerograms at the downstream side, middle of the base

and upstream side ....................................................................................... 97

30' x-displacement contours (ft) (a) plan view, (b) isometric view,
(c) cutting plane at z = 0, (d) cutting plane at x = 0 ................................... 101
3G x-displacement contours (ft) for general excitation case:

(a) total response (b) x component (c) y component (d) 2 component... 102

u+30‘ stress contours (lb/ftz) at the base for general excitation

case: (a) Ix (b) II). (c) TX: ((1) “Ema, ............................................................ 104
u+3o strain contours at the top free surface for general

excitation case: (a) yx (b) 7,). (c) sz (d) ymax ............................................ 106
30' x-displacement response (ft) using different excitation models ......... l 13
”+36 contours of 7m, at the top free surface for different

excitation models ..................................................................................... l 15

xvi

Figure 5.11.

Figure 5.12.

Figure 5.13.

Figure 5.14.

Figure 5.15.

Figure 5.16.

Figure 5.17.

Figure 5.18.

Figure 5.19.

Figure 5.20.

u+3o contours of ”cm, (lb/ftz) at the base for different excitation

models using the simulation approach ..................................................... 116

u+3o contours of 1'merb/ftz) at the base for different excitation

models using the FOSM approach with Itil as the basic variables ........... 118

u+3o contours of tmax (lb/ftz) at the base for different excitation

models using the FOSM approach with 1,2 as the basic variables ........... 119

u+3o contours of ymax at the top free surface using the

FOSM approach with ITiI ((a) and (b)) and ti2 ((c) and (d))

as the basic variables; (a) and (c) are for general excitation

model. (b) and (d) are for identical excitation model ................................ 121
u+3o 1mm contours (lb/ftz) at the base: (a) general excitation/

simulation, (b) general excitation/corrected FOSM, (c) identical
excitation/simulation, ((1) identical excitation /corrected FOSM ............. 124
“+36 contours of tmax (lb/ftz) at the top free surface due to (a)

general, (b)de1ayed, (c) identical and (d) coherency only excitations ...... 126

u+3o contours of 1mm (lb/ftz) on the cutting plane at z = 0

due to (a) general, (b)de1ayed, (c) identical and (d) coherency

only excitations ........................................................................................ 127
u+3o contours of ymax at the base due to (a) general,

(b)de1ayed, (c) identical and (d) coherency only excitations .................. 128

u+3o ”Cm contours (lb/ft2) at the base. (a) and (c) show the

total response, (b) and ((1) show the static response component .............. 135

u+3o 1m contours (lb/ftz) at heights of 70 ft. (a) and (c) are

total responses, (b) and (d) are static response components .................... 136

xvii

Figure 5.21.

Figure 5.22.

Figure 5.23.

Figure 5.24.

Figure 5.25.

Figure 6.1.

Figure 6.2.

Figure 6.3.

Figure 6.4.

Figure 6.5.

Figure 6.6.

u+3o rm, contours (lb/ftz) at heights of 125 ft ((a) and(b))
and 200 ft ((c) and (d)). (a) and (c) show total responses,

(b) and ((1) show static response components .......................................... 138

Safety State under gravity load (a) base view (b) top view

(c) XY cutting plane at z = 0 (d) YZ cutting plane at x = 0. .................... 142

Reliability index contours for identical excitation case:
(a) plan view, (b) bottom view, (c) XY cutting plane at z = 0

and ((1) Y2 cutting plane at x = 0 .............................................................. 143

Reliability index contours for general excitation case:
(a) plan view, (b) bottom view, (c) XY cutting plane at z = 0

and (d) YZ cutting plane at x = 0 ............................................................. 145

Accelerogram at the base of the Santa Felicia dam in the

upstream-downstream .............................................................................. 147
(a) SV/P waves at lower speed, (b) SV/P waves

at shallow incident angle .......................................................................... 150
Comparison of contours of 30' x-displacement response (ft)

to seismic waves at different apparent wave velocity .............................. 151
Comparison of u+3o Tmax (lb/ftz) response at the base to

different apparent wave velocity excitations ........................................... 153

SH waves propagating in the longitudinal direction of

Santa Felicia dam ..................................................................................... 154

Comparison of 30' x-displacement response (ft) between SV/P

wave and SH wave at an apparent wave velocity of 4.27 km/s ............... 155

“+30 contour of mm, (lb/ftz) for SV/P waves and SH waves with

an apparent velocity of 4.27 km/s ............................................................ 157

xviii

Figure 6.7.

Figure 6.8.
Figure 6.9.

Figure 6.10.
Figure 6.11.
Figure 6.12.
Figure 6.13.
Figure 6.14.

Figure 6.15.

Figure 6.16.

Figure 6.17.

Figure 6.18.
Figure 6.19.
Figure 6.20.
Figure 6.21.
Figure 6.22.

Responses to SH waves with an apparent velocity of 2 km/s

(a) top view x-displacement (ft) (b) x-displacement (ft) on YZ

cutting plane at x=0 (c) 1mm (lb/ftz) along the base (d) T

max
(lb/ftz) on YZ cutting plane at x=0 ........................................................... 158
Coherency vs. frequency for Abrahamson’s model ................................. 160
Coherency vs. separation for Abrahamson’s model ................................ 160
Match of Novak’s and Harichandran’s models (Case I) .......................... 162
Coherency vs. frequency for Novak’s model (Case I) ............................. 162
Match of Novak’s and Harichandran’s model (Case 11) .......................... 163
Coherency vs. frequency for Novak’s model (Case 11) ........................... 163

Luco and Wong’s low coherency decay model

plotted against frequency ......................................................................... 165

Luco and Wong’s low coherency decay model

plotted against separation ......................................................................... 165

Luco and Wong’s high coherency decay model

plotted against frequency ......................................................................... 166

Luco and Wong’s high coherency decay model

plotted against separation ......................................................................... 166
Hao’s coherency model for SMART-l Event 45 .................................... 168
Hao’s coherency model for SMART-1 Event 45 (at f = 5 Hz) ................ 168
Hao’s coherency model for SMART-1 Event 30 .................................... 169
Hao’s coherency model for SMART—1 Event 30 (at f = 5 Hz) ................ 169

36 x-displacement response (ft) using (a) Harichandran’s
(b) Abrahamson’s (c) Hao’s (Event 30) and (d) Novak’s

low coherency decay models ................................................................... 170

xix

Figure 6.23.

Figure 6.24.

Figure 6.25.

Figure 6.26.

Figure 6.27.

Figure 6.28.

Figure 6.29.
Figure 6.30.
Figure 6.31.
Figure 6.32.
Figure 6.33.

Figure 7.1.
Figure 7.2.

Figure 7.3.
Figure 7.4.
Figure 7.5.

Figure 7.6.

n+36 7mm response using (a) Harichandran’s (b) Abrahamson’s

(c) Hao’s (Event 30) and (d) Novak’s (Case 1) models ........................... 172
11+3G 1mm response (lb/ftz) using (a) Harichandran’s (b)
Abrahamson’s (c) Novak’s Case 11 and (d) Novak’s Case I models ....... 173

u+36 Static 17mm (lb/ftz) response using (a) Harichandran’s (b)

Abrahamson’s (c) Novak’s Case II and (d) Novak’s Case I models ....... 174

11+3G 1mm response (lb/ftz) using (a) Luco and Wong’s low

coherency decay (b) Luco and Wong’s high coherency decay ................ 176

n+36 Static tmax response (lb/ftz) using (a) Luco and Wong’s low
coherency decay (b) Luco and Wong’s high coherency decay ................ 177

Plot of static response integrand for Harichandran’s

ground motion model ............................................................................... 178
Plot of the static response integrand for Novak’s Case I model .............. 180
Plot of the static response integrand for Novak’s Case 11 model ............. 180
Plot of the static response integrand for Hao’s model—Event 30 ............. 181
Plot of the static response integrand for Hao’s model—Event 45 ............. 181

Plot of the static response integrand for Abrahamson’s

horizontal ground motion ......................................................................... 182
Illustration of truncated wedges in 2-D shear beam model ..................... 184

Illustration for calculation of average separation between

two rectangular areas ............................................................................... 185
YZ Cutting Plane at x=O .......................................................................... 187
Illustration of lS-strip division ................................................................ 187
Convergence of average separation between S1 and S1 1 ........................ 189

Plan and elevation views of the 36 x-displacement responses (ft)

using the 1-D ((a) and (c)) and general ((b) and (d)) excitation models. 192

XX

Figure 7.7.

Figure 7.8.

Figure 7.9.

Figure 7.10.

Figure 7.11.

Figure 7.12.

Figure 7.13.

Figure 7.14.

Figure 7.15.

Plan and elevation views of theu+ 36 ymax responses using
the 1-D ((a) and (c)) and general ((b) and (d)) excitation models ........... 193
Bottom and elevation views of the 11+ 36 tmax responses (lb/ftz)

using the 1-D ((a) and (c)) and general ((b) and (d)) excitation models. 195

114-36 'tmax responses (lb/ftz) inside the dam: (a) and (b) are XZ
cutting planes at mid-height; (c) and (d) are YZ cutting planes at x = 0 ..... 196

Plan and elevation views of the 36 x-displacement responses (ft)
using the l-D excitation model with 9 strips ((a) and (c))
and 21 strips ((b) and (d)) at the base ........................................................ 197

Plan and elevation views of the 11+30‘ 'ymax responses

using the l-D excitation model with 9 strips ((a) and (c))
and 21 strips ((b) and (d)) at the base ........................................................ 199

Bottom and elevation views of the 1.1+ 36 tmax responses (lb/ftz)

using the 1-D excitation model with 9 strips ((a) and (c))

and 21 strips ((b) and (d)) on the base ....................................................... 20]

Comparison of the 11+ 36 1' responses (lb/ftz) in the core

mar
and shell for the (a) general (b) 9-strip (c) 15-strip

(d) 21-strip excitation cases ..................................................................... 202

1t+36 12mm response (lb/ftz) using l-D excitation model:

(a) and (b) are for cutting planes at mid-height for the
9- and 21- strip case; (c) and (d) are for YZ cutting planes

at x = 0 for the 9- and 21- strip cases ......................................................... 203

The 1.1+30' tmax responses (lb/ftz) due to identical excitation:

(a) horizontal cutting plane at mid-height, (b) YZ cutting
plane at x = 0, (c) horizontal cutting plane at a height of 75 ft,

and (d) horizontal cutting plane at a height of 120 ft ................................ 204

xxi

Figure 7.16.

Figure A.1.

Figure A.2.

Figure B].
Figure B.2.

Figure 8.3.

Figure B.4.

Figure B.5.

Figure B.6.

Figure B.7.

Figure B.8.

Figure B.9.

1.1+30' Tmax contours (lb/ftz) on horizontal cutting planes at

heights of 75 ft ((a) and (b)) and 120 ft ((0) and (d)).
(a) and (c) are for 1-D excitation (15 strips),

(b) and (d) are for general excitations ...................................................... 206

Display of half of the Santa Felicia dam model

consisting of twenty—two objects ............................................................. 218
Partial dataset of an universal file ............................................................ 225
Locations of the recorders on the Santa Felicia dam ............................... 228

Ground accelerations of Northridge earthquake in the
upstream-downstream direction ............................................................... 229
Ground velocities of Northridge earthquake in the

upstream-downstream direction ............................................................... 229
Ground displacements of Northridge earthquake in the
upstream-downstream direction ............................................................... 230
Comparison of upstream-downstream displacement

records at the crest with computed response due to

upstream-downstream excitation ............................................................. 230
Comparison between the computed upstream-downstream

displacement responses at the crest of the dam having

damping ratios of 0.25 and 0.06 ............................................................... 231
Comparison between computed displacement response of

Santa Felicia dam with a damping ratio of 0.45 and

seismograph records ................................................................................. 232

Comparison between computed velocity response of Santa

Felicia dam with a damping ratio of 0.45 and seismograph records ....... 232

Comparison between computed displacement response of Santa

Felicia dam with a damping ratio of 0.45 and seismograph records ....... 233

xxii

1. INTRODUCTION

Earth dams have been used in various parts of the world for many centuries. However,
the behavior of earth dams under earthquake excitation has been investigated only during
the past six decades. Because the failure of earth dams can result in large property damage
and loss of life, design, analysis and construction efforts to build safe and economic
structures are very important. Over the years considerable research has been directed
towards the development of analytical and numerical techniques for evaluating the
response of earth dams subjected to earthquake ground motions. While the developments
have enabled the estimation of deformations, strains, accelerations and stresses generated
in earth dams in response to earthquake excitation, the limitations of the models used and
the assumptions made in the analysis should be carefully considered when evaluating the

predicted performance or type of damage.

In many earth-dam studies, the analysis techniques used are quite simplistic. For
example, one of the most commonly used analytical model, known as the “shear beam”
model, characterizes the dam as a uniform one-dimensional elastic beam, with the dynamic
response restricted to horizontal shear deformation in the upstream-downstream direction.
Although some of the more restrictive assumptions of the shear beam model have now been
relaxed, it still remains somewhat unsatisfactory. More realistic and sophisticated models
based on the ﬁnite element method have found increasing use since the mid-19605. For the
most part these have been restricted to two-dimensional plane strain model which assumes
that the dam is inﬁnitely long. With advances in computer performance and storage, three-
dimensional models can now be used in order to model the dam more realistically and to

obtain more accurate predictions of the responses to earthquake excitation.

Finite element models are successful in accounting for geometry irregularity and
material inhomogeneity. However, in most studies to date, the ground motion acting at the

base of a dam has been assumed to be identical at all location, or at best due to a simple

2

propagating wave. Recent research has shown (Harichandran and Vanmarcke 1986, Loh
1985, Abrahamson et al. 1990, Schneider et al. 1990) that significant variation of
earthquake ground motion exists over the base dimensions of large structures such as dams.
This spatially varying earthquake ground motion (SVEGM) is characterized by wave
propagation as well as by coherency loss. Studies on certain classes of structures have
shown that the coherency loss aspect of SVEGM is very signiﬁcant (Harichandran and

Wang 1990, Novak and Suen 1987, Sweidan 1990, Zerva 1988).

The Santa Felicia earth dam located in southern California has been selected for
assessing the effect of SVEGM since it has been studied extensively (Abdel-Ghaffar and
Scott 1979a, Abdel—Ghaffar and Scott 1979b, Abdel-Ghaffar and Scott 1981a, Abdel-
Ghaffar and Scott 1981b, Gazetas 1981, Abdel-Ghaffar and Koh 1982, Prevost et al. 1985,
Abdel-Ghaffar and Elgamal 1987, Elgamal and Abdel-Ghaffar 1987). The dam is analyzed
using a three-dimensional finite element model using Integrated Design Engineering
Analysis Software (I-DEAS) developed by SDRC (1987). Some structural properties such
as natural frequencies, mode shapes, modal masses, and stiffness and mass matrices are

obtained using I-DEAS.

In contrast to traditional dynamic analysis, which is based on deterministic concepts,
the theory of random vibration accounts explicitly for uncertainties that almost always exist
in the dynamic loads or inputs that act on the system, especially seismic ground motions.
Further, SVEGM is usually characterized as a random field and hence the responses of
structures subjected to this are most effectively computed using random vibration analysis.
Therefore, the responses of the Santa Felicia dam to SVEGM are computed in this study

using random vibration and probabilistic analysis.

Chapter 2 contains a review of conventional methods used to model earth dams and
techniques for earth-dam analysis and design. In general, there are two different

approaches, theoretical methods and numerical methods. The former is developed on the

basis of the shear beam theory by which the dynamic behavior of earth dams can be solved
analytically. The latter is based upon the ﬁnite element method which is a widely accepted
numerical procedure for solving the partial differential equations of motion governing the

system.

Traditionally, earthquake-resistant design of earth dams is performed with slope
stability analysis using the “limiting equilibrium” method. The method is described at the
beginning of the chapter. Following this, the applications of the shear beam model and the
finite element model are briefly reviewed. The limitations and disadvantages of each
method are also discussed. In addition, some sophisticated deterministic analyses treating
soil as viscoelastic material are reviewed. Finally, recent studies on the applications of

probabilistic methods to the seismic analysis of earth dams are outlined.

In Chapter 3, the deterministic response of the Santa Felicia dam to a simulated
earthquake excitation is computed. The constituent material properties and site conditions
of the dam are specified here. Some assumptions in the model such as geometry, soil
pr0perties, hydrodynamic effect, dam-foundation interaction and radiation damping effect

are also elucidated.

A three—dimensional finite element model is used throughout this study. The
modeling and analysis processes using I-DEAS are discussed briefly and further details are
given in the Appendix. The number of modes required to obtain accurate responses is
determined by assessing convergence of deterministic dynamics analysis results. The types
of vibration modes are shown and classiﬁed. In addition, the types of damage sustained by
earth dams due to earthquake motions, such as sliding, liquefaction, cracking and piping

failure, are also described.

Chapter 4 presents the background of the theories employed in this study. The
characteristics of earthquake ground motions and engineering interpretation of stochastic

responses are described in the beginning. A ground motion model which accounts for wave

travel and coherency loss is used (Harichandran and Vanmarcke 1986). In the second part,
finite element discretization and modal analysis are introduced. The theory of random
vibration is presented. An expression for the displacement response is obtained using

stationary random vibration analysis.

In addition to the displacement response, stress and strain are also of interest in the
analysis. The covariances of Cartesian stress and strain responses to SVEGM are
formulated using random vibration analysis. The maximum shear stress and strain indicate
where local yielding of dam material will occur. Since principal stresses (strains) are
related non-linearly to the Cartesian stresses (strains), the first-order second-moment
method (FOSM) is employed to approximately estimate the mean and variance of the
maximum shear stress from those of the Cartesian stresses. The accuracy of the FOSM
approach results is assessed by comparing them with results obtained using Monte Carlo
simulation. Another important factor which significantly affects the dynamic
characteristics of earth dams is damping. The effect of two types of damping, hysteretic

damping and viscous damping, are assessed using a one-dimensional shear beam model.

In Chapter 5, the parameters of the ground motion model used are given. The
computational steps of the stochastic response analysis are summarized. The computed
responses due to simplified excitations, such as identical and delayed excitations, as well
as due to general excitation are presented in this chapter. The results for the maximum shear
stress responses using the FOSM method and simulation are compared. Correction factors
are proposed for the mean and standard deviation of the maximum shear stress estimated

by the FOSM approximation.

Details of the responses such as the significance of the various modal responses and
the contributions of the three response components (the variance of the pseudo-static

response, the variance of the dynamic response, and the covariance between the static and

5

dynamic response) to the total response are examined. Finally, reliability against local

yielding under gravity and earthquake loads is assessed in this chapter.

Chapter 6 considers a series of parametric studies. The effect of seismic waves
travelling with a lower velocity or at a shallow incident angle is evaluated. The effect of
having SH waves rather than P/SV waves is assessed. Sensitivity of the responses to the
different coherency models proposed by Abrahamson (1993), Novak (1987), Luco and
Wong (1986) and H80 (1989) are studied in addition to the model proposed by Harichandran
and Vanmarcke (1986). The discrepancies among these models are presented and discussed

and comparisons between the effects of different coherency models are made.

In Chapter 7, a technique of simplifying the excitation in order to enable the use of a
2-D shear beam model is presented. The boundary of the dam where the ground motion
excitation is applied is divided into several strips. Each strip is considered to have a single
degree of freedom and hence fully-correlated excitation is assumed within each strip.
However, ground motion incoherence is still allowed between each pair of strips. The
displacement, maximum shear strain and maximum shear stress responses using this
simpliﬁed excitation model are compared with those using the general model. In addition,
the effect of using different numbers of strips on the boundary are evaluated and the

adequacy of this simplified model is discussed.

In the final chapter, Chapter 8, the conclusions of the research and recommendations

for future work in this area are presented.

2. Literature Review

2.1 General

In the past six decades or so, engineers have attempted to better understand the
dynamic behavior of dam structures under earthquake conditions. Theoretical analysis as a
shear beam, numerical analysis as a two-dimensional (2-D) or three-dimensional (3-D)
body using the finite element method (FEM), and full-scale or model tests are the principal

methods of analyzing the vibrations of earth dams.

The one-dimensional (1-D) shear beam model considers the dam as a plate obtained
by slicing it with two cross sections perpendicular to the dam axis and analyzing the in-
plane response of the plate as a wedge-shaped shear beam. The effect of both banks of the
valley is neglected, but it is said to be small when the length of the dam exceeds four times
the height (Hatanaka 1955). However, this limitation has been improved by using a 2-D
shear beam theory which accounts for the variation of deformation along the longitudinal
direction of the dam. The problem is therefore handled in a more realistic manner, but a
precise analysis in this case is rather difficult, especially for modeling a dam with

inhomogeneous material.

The FEM was initially used to analyze the plate model described above as a 2-D
plane—strain model (Clough and Chopra 1966). A dam with an infinite length is assumed in
the analysis. The popularity of the finite-element method stemmed mainly from three
factors: (a) its capability for handling any number of zones with different materials; (b) its
capability of rationally reproducing the 2-D dynamic stress and displacement field during
ground motion; and (c) its capability to account for the effects of excitation by horizontally
travelling waves. Nevertheless, the assumption of plane-strain conditions is inappropriate
for dams built in narrow valleys. In such cases, the stiffening effect due to the short length
may not be neglected. Recent advances in computer hardwares have facilitated the use of

3-D models (Martinez and Bielak 1980).

7

In experimental approaches, a scaled model of the dam is typically placed on a
shaking table and vibrated. The test includes an elastic vibration test for investigating the
vibration characteristics of the dam and a dynamic failure test for studying the conditions
of failure due to vibration. The most important consideration in a model test is for a
similarity to exist between the real structure and the model, but with a complex structure,
such as an earth darn, it is difficult to obtain similarities for all characteristics. Therefore,
the model is usually made so that the property being tested is similar to that of the actual
earth dam. Instead of model tests, full-scale forced-vibration tests (Petrovski et a1. 1974,
Abdel-Ghaffar and Scott 1981a) were performed to overcome the drawback in modeling.
Furthermore, ﬁeld measurement and observation (Seed et a]. 1978) of earth dam responses

during earthquakes can yield direct information for purposes of analysis and design.

2.2 Conventional Approaches
2.2.1 Seismic design criteria
2.2.1.1 Pseudo-static method

For a long time the standard method of evaluating the safety of earth dams against
sliding during earthquakes has been the so called “pseudo-static” method. This method is
based on the assumption that dams are absolutely rigid bodies ﬁxed on their foundations,
and experience a uniform horizontal base acceleration equal to the underlying ground
acceleration. The horizontal ground acceleration was obtained from seismic codes in terms
of a single peak value, giving a horizontal inertia force on a potential sliding mass in the
upstream-downstream direction. The force is determined as the product of a seismic
coefficient and the weight of the potential sliding mass. The stability of slopes can be
analyzed on a trial failure surface, considering the equilibrium of the weight, the inertia

force and the resisting force of the sliding block.

According to this method, a trial failure surface ABC shown in Fig. 2.1 is an arc of a

circle with its center at O and a radius of R. Considering the unit length of the embankment

8

perpendicular to the cross section shown, the forces acting on the trial failure surface are as

follows:
1. Weight of the wedge, W.

2. Inertia force on the wedge, khW, which accounts for the effect of an earthquake on

the trial wedge. The factor kh is the average coefficient of horizontal acceleration.

3. Resisting force per unit area, F R, which is the shear strength of the soil acting along

the trial failure surface, ABC.

The factor of safety with respect to strength, F S, is determined by

: Resisting moment about 0 _ FRiABC)R
5 Overtuming moment about 0 WLl + kh WLZ'

 

 

(2.1)

This procedure is repeated with several trial failure surfaces to determine the minimum
value of F s. The slope is considered to be stable if the minimum value of F5 is greater than
or equal to 1. The magnitude of kh used for the design of many dams in the past ranged from

0.05 to 0.15 in the United States (Das 1993).

 

 

 

 

 

 

Figure 2.1 Stability analysis for slope of an earth dam

9

Nevertheless, it is now clearly understood that earth dams behave as deformable
rather than rigid bodies, and their response to seismic base excitation is influenced by the
properties of the constituent materials, the geometry, and the nature of the base motion.
Early convincing evidence of such behavior has come from forced-vibration tests
(Keightley 1966), and from observations of response during earthquakes (Okamoto et a1.
1969), although it had been theoretically anticipated much earlier (Mononobe 1936). Other
serious drawbacks of the pseudo-static procedure have been thorough discussed by Seed

et a1. (1973) as follows:

1. The design method considers classical slope instability as the only potential mode
of failure. In fact, over the years, several types of seismic damage have been
observed in earth dams and embankments over the world, such as liquefaction ﬂow
failure, longitudinal cracks, differential crest settlement, loss of free board,

transverse cracks, and piping failure.

2. The horizontal inertia forces do not remain at a constant level for a significant time
duration or act along a single direction, but rather fluctuate rapidly in both
magnitude and direction. Thus, even if the factor of safety dropped momentarily
below unity the slope would not necessarily experience a gross instability but might
merely undergo some permanent deformation. This idea of performance controlled
by magnitude of deformations rather than by limiting value of pseudo-static factors
of safety has been exploited and evolved into a full-fledged standard practical
procedure of assessing the seismic safety of earth dams consisting of nonliqueﬁable

soils, which will be introduced in the next section.
2.2.1.2 Sliding displacement method
The sliding displacement method is primarily based on the original concept proposed
by Newmark (1965). It is assumed that sliding of an earth mass will begin during an

earthquake when the moment due to extraneous forces exceeds the resisting moment. The

earth mass which has begun to slide will be subject to gravitational force, seismic force and

10

resistance, but the sliding will progress. However, resistance will continue to act while the
seismic force reduces and reverses, so that the sliding of the earth mass is decelerated and
then stopped. When earthquake acceleration is again increased the sliding starts once more
and the same behavior is repeated. With each repetition, the total quantity of sliding is
increased, and when the earthquake is over there will be a certain degree of displacement
remaining. If the amount of sliding does not endanger the dam, the dam will be stable after

the earthquake.

To clarify this phenomenon, consider a slope, as shown in Fig. 2.1, subjected to an
earthquake, the stability of the slope will depend on the shear strength of the soil and the
average coefficient of horizontal acceleration. The factor of safety of the soil mass located
above the most critical surface ABC will become equal to 1 when kh is equal to ky. This
value of ky is defined as the coefficient of yield acceleration. If the soil wedge ABC is
subjected to given earthquake accelerograph, the velocity of the sliding wedge can be
determined by integration of the area under the acceleration versus time plot, in which the
acceleration exceeds ky. Similarly, the displacement can be derived from the velocity. With
time, the displacement of the wedge gradually increases. In most cases of embankment
stability, it can be shown (Seed, 1979) that when the crest acceleration does not exceed
0.75g, deformation of such embankments will usually be acceptably small if the dam has

F5 = 1.15 as determined by the pseudo-static analysis.

The sliding displacement method has gained acceptance more recently in the
determination of the displacement of slopes (Sarma 1975, Makdisi and Seed 1978). And it
was proved to be useful in cases where the yield resistance of the soil can be reliably
determined and does not signiﬁcantly decrease with time during an earthquake. Compacted
cohesive clays, dry sands and very dense saturated sand may belong in this category of soils

since no signiﬁcant pore water pressure builds up during cyclic loading.

11

2.2.1.3 Liquefaction criterion

When dams are constructed of loose or medium dense saturated sands, severe damage

and failures due to build-up of pore water pressure may occur. To evaluate the seismic

stability of such embankments, Seed (1979) developed an analysis procedure which

involves the following essential steps:

1.

2.

Estimate the initial static stresses in the darn by means of finite element analysis.

Determine the dynamic soil properties such as shear modulus, Poisson ratio and

damping characteristics as functions of the strain level.

Compute the stresses induced in the dam by the design ground excitation, using
plane-strain ﬁnite element analysis and the dynamic soil characteristics determined

in step 2.

Subject representative samples of the embankment materials to the combined
effects of the initial static stresses and the superimposed dynamic stresses, and
determine their effects in terms of the generation of pore water pressures and the

ensuing reduction in strength and development of strains.

Perform slope stability analyses and semi-empirically convert the strain potentials
to a set of "compatible" deformations. The stability and performance of the dam are
judged from the results of these stability analyses and/or the size and distribution

within the dam of the "compatible" deformations.

This procedure has been employed to explain several cases of liquefaction failures,

most notably those of the Sheffield and Lower San Fernando Dams. However, some

potential limitations of the liquefaction evaluation procedure have been criticized in recent

years. Regarding the definition of liquefaction as that stage in a cyclic laboratory test when

a peak pore pressure ratio reach 100%, it has been argued that development of a 100% pore

pressure ratio does not necessarily lead to zero shear strength. Second, the associated

undrained cyclic triaxial stress-controlled testing of anisotropically consolidated specimens

12

is used for evaluating the liquefaction potential. A particular limitation of the cyclic triaxial
test stems from the fact that the orientation of the major principal stress may rotate
intermittently by 900 during each cycle. In reality, it is more likely that the principal stress
directions rotate randomly during shaking, and only momentarily could the planes of

maximum static and dynamic shear stress coincide.

From a practical point of view, these methods are acceptable as preliminary design
criteria. However, for rigorous seismic design of earth dams, more comprehensive
investigations are required. Shear beam and finite element models have been developed to
estimate earthquake induced deformations and stresses in dams, and to be able to predict

various types of damage and failure.

2.2.2 Shear beam model

The shear beam model for earth dams was first introduced by Mononobe (1936). This
model was rigorously exploited in the 1960’s and 1970’s to conduct parametric studies, to
interpret the results of full-scale tests, and to obtain seismic coefficients for use in design.

The following major assumptions are made in the shear beam theory:

1. The dam is a beam with a variable wedge-shaped cross section. Only horizontal
lateral displacements and simple shearing deformations take place, and are

uniformly distributed along horizontal planes across the dam.

2. The dam consists of homogeneous material which can be described by a constant

shear modulus, or a modulus that varies along the height of the dam.

3. The dam is built in a rectangular canyon and is subjected to identical lateral base

motions.
2.2.2.1 One-dimensional shear beam theory

Based on the assumptions of the shear beam theory, closed-form solutions to the

vibration of the dam can be derived as summarized below. Fig. 2.2 shows a typical l-D

13

shear beam model. Considering the equilibrium of the inertia force and shearing force

acting on the shaded element in Fig. 2.2,

2
a u _ any _ 8(b1')

 

dy

where
u = displacement along the x axis;
p = density of material;
Q = shearing force; and

T = shearing stress.

At any y location,

awn -212 91-12 a:
By —ayr+bay—yt+bay

Thus, equation (2.2) becomes

 

 

 

 

B

Figure 2.2 One-dimensional shear beam model

V

(2.2)

(2.3)

(2.4)

14

Assuming the relation between shear stress and shear deformation to be

t = — (2.5)

= 0 (2.6)

This is the differential equation governing the vibration of the dam.

Let the natural frequency and the mode of vibration be 0) and (My), respectively. Then

assuming u to be u = (Demo! and substituting in equation (2.4),

7- 2
19.1519 29’.

+ (I) = 0 2.7

which is a typical Bessel’s equation.

Let Vs = x/G/p denote the velocity of shear waves. The general solution of
equation (2.7) with arbitrary constants C I and C 2 is:
my (Dy
(I>(y) = ClJo(—V—s) + C2Y0(Vs) (2.8)

where J" and Y n are Bessel functions of the first and second kind of order n, respectively.

Using the boundary conditions:

 

1:0 aty=0 :> E) =0 2 C2=0;
By )=0
u=0aty=H => <I>(H)=O => 10(wVH)=0.

S

. . . H
Therefore, the characteristic equation becomes 9:7— = Z1. where Zr are the roots of
5
Jo (Z) = 0.

From this the undamped natural frequencies of vibration are

(1). = ——sZ. (2.9)

15

and the vibration modes are

(2")
(Di(_v) = JO 7 (2.10)
The shearing stress will be
01'] Ziy ion
1"- = -—Z—J1('ﬁ)€ (2.11)

Modal analysis can now be used to compute the response of the dam to an earthquake.
In modal analysis, the earthquake excitation is distributed to the various normal modes, and
the sum of the modal responses is the total response. The proportion of excitation

distributed to each normal mode, i.e. participation factor, is given by

 

 

2 . (it) .
= KPHy‘IMyM} _ J-ZYJO H d.» ___2__ (212)
" 2 , 2 #1 2in 'ZMZ.) '

Assuming H = 275 ft and V5 = 1,000 ft/sec, the values of frequency, participation
factor, and vibrational shape function for the first four modes are listed in Table 2.1. Their

mode shapes are shown in Fig. 2.3.
2.2.2.2 Shear beam with variable rigidity

In the above theory the shear modulus is assumed to be constant over the entire dam,

but in most cases it is greater in the interior of the dam than near the surface because of the

TABLE 2.] Response of l-D shear beam model

 

 

 

 

 

 

 

i z, (0.- (Hz) 1“,. qty)

1 2.4048 1.39 1.602 10(0.00874y)
2 5.5201 3.19 -l .065 JO(O.O2007y)
3 8.6537 5.01 0.851 10(0.O3 l47y)
4 11.7915 6.82 -0.730 JO(O.O4288y)

 

 

 

 

 

 

l6

 

 

 

 

 

 

 

0

— First mode
''''' Second mode
--- Third mode
‘-' Fourth mode

8

5

Q.

a)

Q

275

Figure 2.3 One-dimensional shear beam vibration modes

high confining pressure. As a result, the upper portion of the dam is considerably more

ﬂexible, and the vibration modes can be rather different from those shown in Fig. 2.3.

Sawada and Takahashi (1975) investigated the velocity of seismic waves in the
bodies of existing rockfill dams and obtained important data on the distribution of the
rigidity in dam bodies. According to their tests, the velocity of transverse wave at any depth

can be represented by the formula

v = cry“ (2.13)

S
where VS = velocity of the shear wave, )2 = depth from the surface of the dam, and the two
parameters a and B are determined by the geometry and material properties of the dam.

Table 2.2 shows the typical velocity of a transverse wave and the Poisson’s ratio.

17

Further refinement can be performed by adjusting the shear modulus based on the
level of stresses and strains during the response to earthquake motions. This would lead to
an iterative analysis. A simpler method of obtaining equivalent dynamic shear modulus that
accounts for stresses and strains due to the ﬁrst mode only, has been proposed for 1-D and
2-D shear beam models by Abdel-Ghaffar and Scott (1979a). In their studies, a formula for
the variable shear modulus used in the analysis of the Santa Felicia dam response to

earthquakes was

WI—

0 (y) = only?) (2.14)

where p = mass density; 7 = speciﬁc weight; and v30 = parameter defined by Price (1956)
to determine G as a function of depth y. Furthermore, Abdel-Ghaffar and Koh (1982) used
the shear modulus as functions of position within the dam to investigate 3-D dynamic

behaviors. It has the form

our-—

 

) 1- B x (2.15)

Guy) = G.(
5—2.2(H—))

IN

where Go is a constant parameter.

It should be noted that the damping ratio increases with increasing shear strain while
the shear modulus decreases with increasing shear strain. More comprehensive
investigations from laboratory and field tests suggest that the main factors affecting shear

moduli and damping factors for soils are strain amplitude, confining pressure, void ratio,

TABLE 2.2 Velocity of Transverse Wave (m/sec)

 

 

 

 

 

 

 

Zone Depth (m) Shell Core
0-5 Vs=245 Vs=210
5-30 VS : 250y0-20 Vs = 180y0-35
30 < Vs : 200y0-315 Vs : 180y0'35
Poisson’s ratio v = 0,49-()_()()1y0.95 V = 0_45_0.006yo.oo

 

18

number of cycles of loading, and the degree of saturation for cohesive soils. For sand, the
modulus values are strongly influenced by the confining pressure, the strain amplitude, and
the relative density but not significantly by variation in grain-size characteristics. In case a
dam is constructed mainly of cohesionless soils, the shear modulus G (at low strain) and

confining pressure p’ are related by

G = 1,000K2(p') ”2 Ib/f12 (2.16)
so that the inﬂuence of void ratio and strain amplitude can be expressed through their

inﬂuence on the soil parameter K 2.
2.2.2.3 Forced vibration analysis

Supposing the dam shown in Fig. 2.2 is subjected to a horizontal upstream-

downstream ground motion, ug(t), the equation (2.6) will become

 

 

__._____—‘_.'. : —~___." (2.17)
03:2 a 2 23> all
The solution to equation (2.17) can be expressed
°° 2] [Z (v/H)] ..
_ 0 n ' - I I
u(y,t) — 2 wnZn‘ll (Z) Lllg81n[wn(t—t)ldt (2.18)
n = l It
If viscous damping is included, then equation (2.18) will be modified to the form
0° 2] [Z (y/H)] .. —Z;u) (r-r’)
_ 0 n n n - _ I I
u(y,t) — 21 wnzn‘ll (Zn) Ouge sm [(udn(r t)ldt (2.19)
n:

where C" = damping factor in the nth mode; and

(odn = (on ,H 1 — C: i = damped natural angular frequency in the nth mode.

For small values of damping, cod" = (on. Thus, from equation (2.19), the absolute

acceleration response is

19

.. (Zn y/H) .. — a) [—1’
ua(y,t)= 21,120) [Hg—1(2) ]ruge 8C" "( )sin[(r)n(t—t')]dt’ (2.20)

Equation (2.20) can be rewritten as

 

(1,6,0 = 2 w,v,,(y>vn(t) (2.21)
n=l
where
210(Zny/H)
wntv) — 2.11%) and (2.22)
-- —c.w.<r-t'> .
vnu) = fouge srn[(on(t—t’)]dt’ (2.23)

For a given ground acceleration record, ﬁg (1) , and dam, equation (2.20) can be evaluated

numerically to obtain the absolute acceleration with depth.

Regarding the average value of kh in the Section 2.2.1, it is worth noting that a
theoretical derivation has been recommended by Seed and Martin (1966). Using equation

Bu (.v, t)
3y

(2.19), the distribution of shear stress, I (y, t) = G , can be derived. Considering

the shear force, F (y,t), induced by inertia force at the base of an arbitrary soil wedge, i.e.
F (v. t) = r (y. r) b = (épbyju'a, (r) (2.24)
the average lateral acceleration rim, (r) is

.. °° 4011 (Zny/H)
“mm — glpwnHyJMZ.)

 

V" (t) (2.25)

.k —la° (I) _ i 4GJ1(Zny/H)
” " g a “V _ gownHy11(Z,,)
n—l

 

(2.26)

Note that the value of kh is a function of time. Since the average acceleration varies with

the depth y, the magnitude of kh also varies with y.

20

2.2.2.4 Two-dimensional shear beam theory
The schematic of a typical dam is shown in Fig. 2.4(a). Fig. 2.4(b) shows the shear
forces on the four internal faces of an infinitesimal element of a dam with sides dy and dz
and width b. The forces acting on the element during free vibrations include the inertia
force, and the shear forces, Q” and Q a. Equilibrium of the forces acting on the element in

the x-direction yields the following equation of motion for a constant shear modulus

(Martin 1965):

+ -— + — (227)

For earth dams consisting mainly of cohesive materials, it is acceptable as a first

approximation to assume a constant shear modulus. However, for dams constructed with

(a)

 

 

 

(b)

 

 

 

 

 

 

Figure 2.4 Shear wedge in a rectangular canyon

21

non-cohesive materials, it is more appropriate to assume that the average shear modulus
varies with the depth from the crest in the form (Abdel-Ghaffar and Scott 1978, Gazetas
1980)

y B
G, = C(71) (2.28)

where G is a nominal shear modulus equal to the shear modulus at the base. Substitution of

equation (2.28) into equation (2.27) yields

2 l3 2 2
2;: 0W”) [agﬂwngmig] (2.29)
a: D By y y dz
where u is the lateral displacement of the dam and p is the mass density of the dam material.

It is assumed that the displacements of the dam at its base and at its banks are zero, whereas

at the crest the lateral shear force vanishes.

Using the separation of variables method, equation (2.27) can be uncoupled. By
enforcing the boundary conditions, the dynamic characteristics of the dam model can be
obtained (Ambraseys and Sarma 1967, Abdel-Ghaffar and Scott 1978, Martinez and Bielak
1980). The natural frequencies of vibrations are given by

V5 2 TC” 2 “2
mm" = ﬁ(zm+("T) ) (2.30)

where tom” is the frequencies of vibration modes, Vs = ./G/ p = is the nominal velocity of
propagation of shear waves within the dam, and Zm are roots of the characteristic equation
10(Zm)=0. The corresponding modes of vibrations are expressed as

Z y
(Dnm(y, z) = J0(—}m7)sin(£:—Z) (2.31)

Fig. 2.5, Fig. 2.6 and Fig. 2.7 show the vibration modes for m=1 and n=l, m=l and n=2,

and m=2 and n=2, respectively.

22

 

 

 

Figure 2.5 2-D model vibration mode for n=1 and n=1

 

 

 

Figure 2.6 2-D model vibration mode for m=1 and 11:2

 

 

 

 

Figure 2.7 2-D model vibration mode for m=2 and n=2

 

 

2.2.2.5 Relevant research

Although the shear beam model treats earth dams as deformable structures, the
analysis of the 1-D model, in general, overestimates the fundamental frequency and
overestimates the frequencies of higher modes even more. The assumption of pure shear
deformation is reasonable near the vertical axis of the cross section, but results in
significant error near the upstream and downstream faces. The displacement near the faces
of the dam may be significantly different form the center-line displacement, which is
neglected in the theory. Apparently, the analysis does not give an adequate prediction of
the dam behavior because the design criteria for earth dams are based on slope stability
which is controlled by conditions near the free surfaces. Consequently, further work has
been done in order to overcome the shortcomings. A summary of the most significant

developments and studies is presented here.

0 Hatanaka (1952) performed a 2-D shear beam analysis for dams in rectangular
canyons and showed that shearing deformations are predominant. In addition, a

rational design procedure using response spectra was developed.

' Martin (1965) conducted parametric studies using the l-D shear beam model, and
proposed a non—linear variation of the shear modulus due to conﬁning pressure, in
which the modulus was assumed to be proportional to the cube root of depth from
the crest. The results differ from those using homogeneous shear beam models

which was shown to be inadequate when dealing with the vibrations of earth dams.

° Ambraseys and Sarma (1967) summarized that the magnitude and time distribution
of inertia force, geometric configuration, damping and strain-rate are the principal
factors that control response characteristics. They used the l-D shear beam model
to obtain seismic coefficients by performing stability analysis on potential sliding

blocks.

24

° Gazetas (1981) studied the response of embankment dams to longitudinal
vibrations using a triangular elastic wedge model. Transverse cracks that can induce
uncontrolled seepage through the dam and thereby result in piping failure, are
mainly due to tensile stresses created by longitudinal vibrations of the dam. The
numerical results indicate that, for a narrow dam (L/ H S 1.5 ), the fundamental
natural period is independent of the dam height and is only a function of the length
and shear wave velocity. In contrast, the period depends only on the height for long
dams. It was also found that axial deformations are more significant than shear
deformations in narrow dams, while shear deformations are more important than

axial deformations in relatively long dams.

° Oner (1984) studied the shear vibration of inhomogeneous earth dams in
rectangular canyons using a shear wedge model. The effect of the restraint provided
by lateral valley was considered in the analysis. The solutions indicated that the
effect of the valley width to depth ratio and the modulus inhomogeneity are
coupled. This coupling affects the mode shapes, frequencies, and modal
participation factors. The fundamental period can be predicted, within 6-9% error,

by using an average modulus value in the homogeneous shear wedge solution.

- Prato and Delmastro (1987) used a 1-D shear beam model coupled with an
equivalent linear method for approximating the dynamic soil properties to analyze
a gravel shell dam. The results compared favorably the more precise 2-D analyses.
Crest accelerations were predicted within 5% of those given by 2-D analyses. Shear

stress patterns and values within shells exhibited differences between 5 and 12%.

° Emesto-Motta and Zeng (1993) studied the effect of travelling seismic waves on the
dynamic response of earth dams, using a truncated 2-D elastic wedge. For
comparison, 1-D and 2-D wedge models were also used without considering the
travelling wave effect. The results showed that: (l) for the longitudinal mode of

vibration, the greater the ratio of the height to the length of the complete wedge, the

-ﬁ)

25

more the natural transverse period of vibration of the 2-D wedge is less than that of
the 1-D wedge; (2) when the phase difference of ground motion at both ends of the
dam equals 1:, the amplification for a 2-D wedge is less than that for a 1-D wedge;
and (3) in general, the displacement caused by an earthquake is greater for a 1-D
model than for a 2-D model when considering travelling waves, but the acceleration
response of a long 2-D wedge to travelling waves with a long dominant wave length

is greater than that of a l-D wedge.

2.2.3 Plane strain finite element model

The FEM for 2-D plane—strain analysis in earth dams was first presented by Clough
and Chopra (1966). The structure was assumed to be of infinite length and loaded
uniformly along the length so as to produced plane strain behavior of the cross section. The
method can account for any arbitrary geometry and material property variations and can be
applicable to the analysis of any elastic system subjected to any type of dynamic loading.
Another significant advantage is that the method can be adapted to take account of non-
uniform base motions in the width direction; that is, of the effects of seismic waves passing
under the dam in which the wave lengths are short compared to the cross-section width of

the dam base.

Concerning the finite element (FE) plane strain analysis procedure in general, it may
be noted that: (1) compatibility is satisfied everywhere in the system; (2) equilibrium is
satisﬁed within each element; and (3) equilibrium of stresses is not satisfied along the
element boundaries, in general, but the nodal force resultants are in equilibrium. This local

discrepancy in stress equilibrium represents the type and extent of the approximation

involved in the FEM.

It has been the limitations of computer storage capacity and high costs that have
generally restricted the use of FE models to the plane-strain case. The following lists the

evolution of developments related to the 2-D FE model.

26

° Chopra (1967) compared the response of a shear wedge model with that of the 2-D
FE model to a prescribed horizontal base acceleration (El Centro earthquake 1940).
The important conclusion from this comparison was that the natural mode shapes
obtained by the FEM differed significantly from those obtained with the shear
wedge model. Only the fundamental mode resembles the shear wedge behavior, but
even in this case the 2-D deformations are much more complex than those assumed
in the shear wedge analysis. On the other hand, stresses near the faces of the dam
computed by the FEM were found to depend significantly on the slope of the sides,
with steeper side slopes resulting in greater stresses; this is in contrast to the shear

wedge approach which yields stresses independent of the side slopes.

° Dibaj and Penzien (1967) studied the response of a 2-D FE model of a dam to
horizontally travelling waves. It was concluded that the time variation of the ground
motion must be taken into account especially for seismic waves with low velocity.
They also concluded that the spatial variation of ground motion was negligible only

when the ratio of base width to wave velocity is less than 0.1 second.

0 Ghaboussi and Wilson (1973) modeled the soil as a porous elastic medium and
studied the dam-reservoir interaction. The upstream shell and the central core were
considered as a coupled two-phase ﬂuid-solid zone. This was the first attempt
determining the pore pressures and the intergranular stresses under earthquake
loading conditions. It should be noted that, while in a local shear failure the soil
regains its strength when the load reverses, once local liquefaction develops, the
soil in that zone is likely to remain liquefied for the rest of the duration of the
earthquake. It was also concluded that an important source of the pore pressure built
up in the earth dams is due to the dilatancy of soil under shear deformation, which
cannot be simulated by linearly elastic material models. Therefore, it was

concluded that nonlinear material properties should be adopted.

27

- Martinez and Bielak (1980) combined a 2-D FE discretization of a dam’s cross

section with Fourier expansion in the longitudinal direction to obtain the response

of dams in symmetric canyons.

Mansuri et al. (1983) studied the liquefaction of earth dams due to shear straining,
using a 2-D FE mode] with soil treated as a poroelastic material. The effect of
elastic wave propagation and the volumetric strain resulting from an applied shear
strain were considered. The results of an analysis of a hypothetical dam were
observed to agree well with the observed liquefaction in the lower San Fernando

Dam during the San Fernando Earthquake of February 8, 1971.

Comprehensive comparative studies of the linear viscoelastic predictions by the shear

beam model and plane strain model has been reported by Tsiatas and Gazetas (1982) and

Dakoulas and Gazetas (1985). The following are the main conclusion from these studies:

1.

The two models yield very similar fundamental natural frequencies and
fundamental mode shape. Specifically, the shear beam invariably under-predicts

the natural period by about 10% in most cases.

The significant higher lateral modes computed with finite elements exhibit an
increasingly stronger component of vertical deformation. The shear beam neglects

this component and thereby leads to higher natural frequencies.

. Peak values and time histories of horizontal relative displacements computed with

the two models are in very close agreement, the discrepancies being within 10% for

all dams and all seismic excitations studied.

The most significant disagreement of the two models involves absolute
accelerations of relatively ﬂexible dams, for which the shear-beam frequently

overestimates the crest peak acceleration by as much as 50%.

In general, the shear beam also tends to overestimate slightly the largest peak values

of strain, which usually occur within the upper third of the dam. On the other hand,

28

shear stresses are in very good accord; the shear beam predictions are

approximately equal to the average-across-the-width shear stress computed using

the FEM.

2.2.4 Three-dimensional model

It has been conventional practice in the past to evaluate the dynamic response of an
earth dam by making plane-strain analyses of several representative sections through the
dam. The assumption of plane-strain conditions is valid for infinitely long dams subjected
to synchronous lateral motion on longitudinal lines along the base. However, for dams built
in narrow valleys, which is the frequent case in mountainous regions, plane strain analyses
can give grossly misleading results. The presence of relatively rigid abutments creates a
3-D stiffening effect. As the canyon becomes narrower, natural frequencies increase and
displacement mode shapes tend to become sharper. Thus while much effort is always
directed towards simplification of the evaluation procedure, it is sometimes necessary to

use more sophisticated analyses in the interest of improved understanding and accuracy.

A number of studies using 3-D model have been performed in the 1980s. For earth
dams with a plane of symmetry perpendicular to the longitudinal axis, Martinez and Bielak
(1980) developed a numerical procedure for 3—D FE analysis, and concluded that the 3-D
deformation behavior yields higher stiffness characteristics than does the 2-D idealization.
Ohmachi (1981) extended the shear wedge model into a simplified finite element model,
and concluded that three dimensionality has an important effect on the earthquake response
of dams located in narrow canyons. Abdel-Ghaffar and Koh (1982) presented a more
rigorous although computationally quite efficient 3-D semi-analytical solution for dams
built in canyons of any shape but having a plane of symmetry. This solution is based on the
Rayleigh-Ritz method with the shear-beam shapes or-even simple sinusoids as "basis
function", and involves an appropriate transformation of the dam geometry into a cuboid.

An attempt was made to reproduce the recorded seismic response of the Santa Felicia dam

29

during the 1971 San Fernando earthquake. Of particular significance is the versatility of the
method which has been extended to formulate a realistic approximate solution for the non-
linear inelastic response of dams in non-rectangular canyons (Elgamal et al. 1984). Makdisi
et al. (1982), Mejia et al. (1982) and Mejia and Seed (1983) developed relatively efficient
3-D FE formulations to obtain the steady state and transient responses of homogeneous
dams in triangular canyons. The results show that the presence of the rigid triangular
boundary increases the fundamental frequencies, and seems to substantially increase the

seismic acceleration while decreasing the seismic strains.

More sophisticated deterministic analyses including non-linear effects and differential

ground motion are summarized below:

0 Prevost et a1. (1985) performed 2-D and 3-D non-linear finite element analyses of
the Santa Felicia earth dam utilizing a multi-surface plasticity theory. In the study,
all three orthogonal components of the input ground motions were considered,
rather than only base excitation in the upstream-downstream direction. The results
are more reliable because all possible vibration modes of the dam were taken into
account. Comparing the responses from the 2-D model with that of the 3-D model,
it was found that the 2-D model predicted the permanent shear strain to be 30%
lower when the dam was subjected to strong shaking. The three-dimensionality of

the dynamic earthquake responses of earth dams was shown to be important.

0 Abdel-Ghaffar and Elgamal (1987) developed a simplified analytical-numerical
procedure based on a Galerkin formulation of the 3-D equations of motion. A
simpliﬁed elastic-plastic technique was applied to compute the non—linear seismic
3-D response of the Santa Felicia dam. The linear mode shapes were obtained using
low-strain elastic moduli and a variational energy approach utilizing both
Hamilton’s principle and the Rayleigh-Ritz method. The results of the simplified

analysis were found to compare favorably with those of an elaborate 3-D FE model.

30

0 Lacy and Prevost (1987) modeled soil as a non-linear two-phase system, and used
FE discretization of the field equations followed by time integration to obtain the
resulting non-linear semi-discrete FE equations. Since earth dams are susceptible to
liquefaction failure, an integrated, coupled treatment of the dynamic interaction
between the soil skeleton and the water ﬂuid phase was used in the analysis. The
results of the numerical calculations were in good agreement with the recorded

response of the dam.

- Haroun and Abdel-Hafiz (1987) performed a deterministic study of the effect of
differential ground motion on the displacement response of earth dams using a 2-D
shear beam model and the FE model. The effective force vector was varied at each
time step to reﬂect both the temporal and spatial variations of the input ground
motion. The results for spatially uniform ground motion was shown to be

conservative.

2.2.5 Model testing
2.2.5.1 Elastic vibration test

The purpose of performing elastic vibration tests on a model dam is to experimentally
determine the natural period and mode shape. The similarity between the prototype and the
model is determined by the law of similitude. According to elastic vibration theory of the
shear beam model presented in Section 2.2.2.1, the natural period is proportional to
H ./p7G Therefore, if subscript p and m are added to express the quantities relating to the

prototype and the model, respectively,

:11 = :11 _pme (2.32)
Tm Hm pmGp

If the natural period Tm of the model is measured in the model test, the natural period Tp of
the prototype may be calculated from equation (2.32). The elastic modulus of a real dam

will vary depending on the soil, construction process and the length of time which has

31

elapsed since construction. However, its average value ranges from 50 to 270 kg/cmz. This
was calculated from measurements of propagation velocities of elastic waves in several
existing earth dams (Severn, 1979). The propagation velocity of transverse waves and the

Poisson’s ratio for rockﬁll dams are given in Table 2.2.
2.2.5.2 Dynamic failure test

In an earth dam failure test, a valid similarity between the displacements, the major
external forces and resisting forces of the prototype and the model is required. In this case
major external forces consist of the weight of the dam and the earthquake-induced inertia
force, and the major resistance force comes from the frictional and cohesive forces. The

following symbols are used: (Okamoto 1984)
L = length; 7» = length scale factor; it = displacement; f = force; I = stress;

5 = strength of materials; and 0 = angle of internal friction.

The similitude of displacement is the same as the similarity of length. Thus,

= _’" = A (2.33)
The ratio between the weight of the model and that of the prototype will be

3
L
pmg ’" = p—"Wf (2.34)

3
ppg LP pp

 

The same ratio must be valid for force f. As stress is force per unit area,

2
1' /L
- f“ ’" = p_,,, (2.35)

m
1:— _ 2 p
p fp/Lp p

 

Since the strength of the material is of the same order as stresses,

_ = —>t (2.36)

32

The strength of granular material is given by the product of the normal stress and
coefficient of friction. However, since the similitude of normal stress satisfies equation

(2.35), the coefﬁcient of friction must be identical for the prototype and model. Namely,

tan em = tan 0,, (2.37)

The objective of a vibration test is to determine the displacement and surface strain

of the dam during earthquakes taking the detailed configuration of the valley into
consideration. The type of damage may be observed during the dynamic failure test.
Regarding the material modeling, plaster or dense gelatin is generally used for the valley
and gelatin for the dams, since these materials are convenient for reproduction of the shapes

of valleys and dams.

2.2.6 Field measurement

The seismic behavior of earth dams can be learned by installing instruments to carry
out observations during earthquakes. Measurements have been made on and within dams,
including dynamic earth pressures, dynamic pore-water pressures and residual
deformations after an earthquake. There is no difficulty when instruments are installed on
the dam or on the ground, but when they are set up within the dam, there is liable to be
problems with moisture entering the instruments, tilting of instruments, and breaking of
lead wires due to dam settlement or heavy equipment used during construction. Therefore,

care must be taken on the installation of instruments.

Abdel-Ghaffar and Scott (1981a) performed full-scale dynamic tests, including
ambient vibration tests, forced vibration tests and popper tests, which were carried out on
the Santa Felicia dam. The measuring equipment consisted of: (1) motion sensing
transducers (SS-1 seismometer by Kinemetrics, Inc.); (2) recording instruments including
signal conditioners (SS-1 also by Kinemetrics, Inc.), and an oscillograph recorder (7418A
by Hewlett-Packard), and tape recorders (3460A by Hewlett-Packard); (3) force generating
systems or shaking machines capable of producing a force up to 44,500 N (10,0001bs); and

33

(4) data processing instruments including an electronic analog-digital converter (DDS-
1103 by Kinemetrics) and a digital signal processor (SD36O manufactured by Spectral
Dynamics Corp.).

According to the test results, it was found that existing shear-beam theories do not
predict all the upstream-downstream modes, indicating the inadequacy of these theories for
comprehensive earthquake response computation. The correspondence between resonant
frequencies from full-scale tests and those estimated from the spectral analysis of two
earthquake records (San Fernando earthquake 1971 and Southern California earthquake
1976) is reasonably good over the first few frequencies, but higher modes could not be
reliably matched. It should be noted that full-scale tests at levels of excitation much lower
than those experienced during the earthquakes produced substantial changes in the dynamic
properties of the dam. Thus it was concluded that the behavior under real earthquakes was

significantly non-linear.

Several actual measurements of earth dam responses during earthquakes has been
conducted in the past. Attention has generally been paid to three major concerns: (1)
earthquake acceleration; (2) magnification of the maximum acceleration at the tops of
dams; and (3) stresses developed in the embankment. The accelerations of dams induced
by earthquakes is important in determining the earthquake inertia forces acting on the dam.
The ratio of the maximum acceleration at the ground surface and the maximum acceleration
at the crest of a dam is also of interest because most failures occur near the crest. In
addition, the stresses developed in the embankment are mainly due to earth pressure and
pore water pressure. Their magnitude has a signiﬁcant effect on the seismic stability of
dams. Although the measured responses can give direct information for design, the cost of

testing prohibits its widespread use.

34

2.3 Recent Studies

Since most structures are quite sensitive to the details of ground excitation, which are
of an unpredictable nature, little conﬁdence can be achieved form the results of a single
deterministic dynamic analysis using a recorded or artificially generated ground motion. To
avoid the expense and effort required for multiple deterministic analyses, random vibration
methods have been developed. In the 19805, the application of probabilistic methods to the

seismic analysis of earth dams increased notably:

0 Gazetas, DebChaudhury and Gasparini (1981 & 1982) used the random vibration
technique to study the seismic response of earth dams. The Kanai-Tajimi spectral
density function was used to characterize the ground motion, and a shear beam
model with a variable cross section was used to characterize the dam. However, due
to the limitations of the idealized model of the dam, wave passage effects were

neglected.

° Gasparini and Sun (1982) used the FEM in conjunction with random vibration
analysis. They studied the effect of correlation between vertical and horizontal
ground motion components, but did not account for spatially varying earthquake

ground motions (SVEGM).

° Luco and Wong ( 1986), and Harichandran (1987) studied the response of rigid mat
foundations to SVEGM. However, it is rather unreasonable to consider an earth
dam as being located on a rigid mat foundation. If the base of a dam is considered
as rigid, then the averaging effect over its large base dimension would reduce the

earthquake excitation drastically, severely under—predicting the dam response.

0 Novak and Suen (1987) are the first to include SVEGM in the random vibration
analysis of dams. Their study was a preliminary one, in which the darn was modeled
as a prismatic beam, wave propagation effects were neglected, and only vertical

excitation was considered. They did include dam-foundation interaction effects,

35

and found that while this was important for concrete dams, it may be neglected for
earth dams. In spite of the rather simplistic model used, their study confirmed that
bending distortions produced by spatially correlated ground motion substantially

increased the stresses within the dams.

° Dakoulas and Hashmi (1992) presented an analytical model for the steady-state
lateral response of earthfill and rockfill dams in canyons subjected to a steady train
of harmonic SH waves incident at an arbitrary angle. The dam was modeled as a 2-
D shear beam with a triangular cross-section, while the canyon was considered to
be rectangular and consisting of elastic rock. The result showed that obliquely
incident waves traveling at an angle of about 550-600 to the horizontal yielded the

maximum response.

0 Ramadan and Novak (1992) addressed dam-reservoir-foundation interaction on a
long gravity dam under SVEGM. A simpliﬁed mathematical model which accounts
for spatial coherence effects as well as wave passage effects was formulated in the
study. However, the analysis is adequate only for long gravity dams and may not be

suitable for the analyzing the behavior of earth dams under earthquake conditions.

Based on the above review, it is apparent that although a substantial body of work
exists on the seismic response of earth dams, only marginal attention has been paid to
include the effects of SVEGM. This is due to the fact that spatial variation of strong ground
motion has been quantified only very recently, based on actual observations from dense
seismograph arrays (Loh 1985, Harichandran and Vanmarcke 1986, Harichandran 1987,
Abrahamson et al. 1990, Schneider et a1. 1990). A new thrust is therefore necessary to

include the effect of SVEGM on the seismic analysis of earth dams.

3. DETERMINISTIC RESPONSE OF THE SANTA
F ELICIA EARTH DAM

3.1 Description of Santa Felicia Dam

The Santa Felicia earth dam is a modern rolled-fill earth embankment located on Piru
Creek, 65 km northwest of Los Angeles. It is owned by the United Water Conservation
District of Ventura County and was completed in December 1955. The dam, shown in
Fig. 3.1, is 83.8 m (275 ft) high above its rock foundation and 137.2 m (450 ft) long across
the valley at the base. The crest has a width of 9.14 m (30 ft) and a maximum length of
388.6 m (1,275 ft). The dam is made of a central impervious core and pervious shell
upstream and downstream. The core and shell materials are basically alluvial, consisting of
clay, sand, gravel and boulders. The core rises from bedrock with a slope of 0.33:1. The
shells rest on existing stream gravel, 22.9 m (75 ft) in height. Upstream and downstream

slopes have slopes of 2.25:] and 2: 1, respectively.

W W

70' 0‘ MI EL 1075

IH'

279‘
KQVIOUS ‘ ’00
SNILL .. . .- 7 m 29'

unsrmc sun. GRAVELS
ROCK

:4)—

 

 

 

PLAN VIEW

Figure 3.1 Structural Details of Santa Felicia Earth Dam

36

37

Fig. 3.2 is the side view of the Santa Felicia dam model, which shows an impervious
clay core covered by a sand and gravel shell resting on a stiff layer of gravel and sand down
to bedrock. Due to conﬁning pressure, the elastic moduli of the soil varies with the depth
from the crest of the dam. The low strain elastic moduli for each material at each depth were
computed from the results of both shear wave velocity measurements and full-scale
dynamic tests performed at the site and have been reported by Abdel-Ghaffar and Scott
(1981). The elastic shear moduli back-calculated from those measurements and used in the
analysis are listed in Table 3.1 (adopted from Prevost et al., 1985). The material zones listed
in Table 3.1 are shown in Fig. 3.3. Poisson’s ratio is assumed equal to 0.45 for the clay core
and 0.30 for the remaining materials. Saturated unit weights of 20.59 kN/m3 (134 pcf) and

22.17 kN/m3 (141 pcf) are used for the clay and the remains, respectively.

Prevost et a1. (1985) simplified the model and assumed zone SS to be stream gravel.
This simpliﬁcation on author’s model (Case I) causes an error in the fundamental natural
frequency of about 3% compared with the model with pervious shell material at zone SS
(Case 11). The ﬁrst ten natural frequencies of the model (Case II) used in the study and of

the simplified model (Case I) are compared in Table 3.2.

 

 

upstream downstream
-/"'/ . 1' x x
.r'// o 'i‘ . x \
-/' pervrous ; pervnous \«..
//' V“. '. ‘\
shell she“
/"//‘———-/ 5;, o o ‘1‘". \VIIF\\\\T\M ““
\\ imperkus / l
e o \ j“. ,// o o
exrstmg stream gravels \\ core exrstmg stream gravels
\ . .‘ /,

 

 

////////////7/7///////>////)7////7////////////7//7///////77
bedrock

Figure 3.2 Santa Felicia earth dam model side view

38

 

 

 

IS
23
3s
_ -_ 4_S_ __ L
" 53
G G G G ‘ G G
l

 

 

 

 

Figure 3.3 Illustration for different material zones

TABLE 3.] Low strain shear moduli for Santa Felicia earth dam

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Depth below Zone Core Zone Shell Zone Gravel
crest (ft) number (lb/ftZ) number (lb/itz) number (lb/rtz)
0—50 1C 2,441,492 1 S 1,725,423
50-100 2C 3,755,302 28 2,939,447
100-150 3C 4,414,405 33 3,781,540
150-200 4C 4,763,929 43 3,972,660
200—275 5C 4,763,929 SS 4,062,370 G 39,726,600
TABLE 3.2 Natural frequencies (Hz) compared with simpliﬁed model
M0“ (the 51:3:Ililsed) (simplifift: Model) M°d° (the :22::Ilsed) (simplgifc: Model)
1 1.392 1.427 6 1.859 1.873
2 1.466 1.510 7 1.965 1.982
3 1.582 1.609 8 2.138 2.154
4 1.812 1.812 9 2.244 2.259
5 1.820 1.820 10 2.303 2.338

 

 

 

 

 

 

 

 

39

3.2 Model Assumptions
3.2.1 Geometry

The Santa Felicia earth dam is located on Piru Creek whose banks on both sides do
not exactly have the same shape. However, in this study it is assumed to be symmetric about
the middle of the dam axis. This simpliﬁcation does not have a significant effect on the
response, which was found by Elgamal and Abdel-Ghaffar (1987). Another simplification
made was to terminate the very long dimension of the existing stream gravel in the

upstream-downstream direction shortly after the extent of the pervious shell.

3.2.2 Soil properties

Soil is, in reality, an inhomogeneous, anisotropic, nonlinear and hysteretic material.
During an earthquake, its dynamic shear modulus decreases with increasing shear strain,
while its damping ratio increases with increasing shear strain. However, in this study the
emphasis was on obtaining better insight into the effect of SVEGM rather than on complex
material models and so linearly elastic, isotropic dam materials were assumed in the
analysis. In addition, low strain viscous damping is adopted to facilitate a simple ﬁnite
element dynamic analysis. In this study, the increase in the pore water pressure of the soil

during seismic response is neglected.

3.2.3 Hydrodynamic effect

It has been reported that the hydrodynamic effect on earth dams is not significant
compared to the response due to earthquakes. Assuming that water is incompressible and
that the ground motion is horizontal, Zangar (1955) reported that the water pressure acting
against an inclined upstream face decreases with decreasing angle of inclination. For the
case of the Santa Felicia dam, the water pressure is only one—fifth of that acting on a vertical
upstream face. As a result, the hydrodynamic effect may be important for concrete gravity
dams with a steep upstream surface (Chopra and Gupta 1981, Hall and Chopra 1982), but

it is negligible on earth dams.

40

3.2.4 Dam-foundation interaction

It has been concluded that soil-structure interaction is important for concrete gravity
dams but insignificant for earth dams built on rigid foundations (Chopra and
Perumalswami 1969, Idriss et. a1 1974, Novak, 1987). Because the material properties of
concrete gravity dams are close to those of rock foundations, resonant vibration is very
likely to occur during earthquake excitation. For the Santa Felicia dam, since the
magnitudes of the earth dam stiffnesses are much smaller than those of the foundation
stiffness, the vibration characteristics of the dam differ greatly from that of the bedrock.
Therefore, dam-foundation interaction can be neglected so that the free-field bedrock

motions may be applied at the base of the dam and the responses analyzed directly.

3.2.5 Radiation damping

When an earth dam subjected to ground motion vibrations, energy dissipation may
take place through feedback into the ground soil. Radiation damping is considered as the
energy loss through partial refraction of the waves in the soil back into bedrock. Finite
element methods have inherent difficulties in modeling the radiation damping effects (Seed
et a1. 1975). Dakoulas and Hashmi (1992) related the radiation damping of the dam-
foundation system to the effect of the impedance ratio. By observing acceleration
amplification corresponding to different impedance ratios between canyon and dam, they
concluded that the ﬂexibility of the canyon rock has a dramatic effect on the response of
earth dams, as it affects the amount of energy radiated back to the canyon. However, for a
dam built in a rigid canyon, the effect of radiation damping is not significant, and hence it

is neglected in this study.

3.3 Modeling and Analysis using I-DEAS VI.i

3.3.1 Introduction

Deterministic analysis of the Santa Felicia dam was performed using I-DEAS VI.i
ﬁnite element program. The simultaneous vector iteration technique was used to obtain the

dynamic modes of the dam. The I-DEAS computer software was developed and marketed

41

by Structural Dynamics Research Corporation of Milford, Ohio. I-DEAS provides a broad
array of analysis and pre- and post-processing tools that are suited for this study. The main
features of I-DEAS families used in modeling and analysis are brieﬂy presented in

Appendix A.1.

3.3.2 Finite element model

The three-dimensional finite element model, shown in Fig. 3.4, was used for all
analyses in this study. The model consists of 1,004 nodes and 4,140 tetrahedral elements.
Since the dam is located on a rigid rock base, 288 nodes on the boundary surface between
the dam and the valley are completely restrained. A brief description of the modeling

procedures used in I-DEAS VI.i is presented in Appendix A.2.

3.3.3 Free vibration analysis

The free vibration characteristics of a dam is fundamental to understanding the
behavior of the dam during an earthquake. The free vibration characteristics are represented
by natural frequencies and vibration modes and can be computed by performing modal

analysis in I—DEAS. The steps for free vibration analysis are described in Appendix A.3.

In the FE method, there is a trade-off between computational effort and numerical
accuracy. Progressively finer meshes were used to determine the number of elements

required for acceptable accuracy. Four cases using coarser to finer finite element meshes

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.4 Finite Element Mesh

42

are compared in Table 3.3. The elapsed time for the analyses listed in the table is for an HP
model 715/50 UNIX workstation. Ten modal frequencies are solved for in each case. The
results are listed in Table 3.4. Based on the convergence of the ﬁrst ten modal frequencies,
the third case was selected as being sufficiently accurate. The first eighty natural
frequencies for the third case are listed in Table 3.6, and these agree well with values
reported in previous studies (Abdel-Ghaffar and Scott 1979, Prevost et a1. 1985). The

vibration modes may be either symmetric or antisymmetric about the mid-plane of the dam

TABLE 3.3 Comparison of elapsed time for solving ten vibration modes

 

 

 

 

 

 

Case Average element Total number Total number CPU time
size (ft) of elements of nodes (sec)
I 100 1916 491 196
ll 85 2626 666 363
III 70 4140 1004 810
IV 55 5952 1446 874

 

 

 

 

 

 

 

TABLE 3.4 Comparison of natural frequencies (Hz) using different mesh sizes

 

 

 

 

 

 

 

 

 

 

 

 

Mode Case I Case 11 Case III Case IV
Mode 1 1.414 1.400 1.391 1.385
Mode 2 1.491 1.474 1.466 1.458
Mode 3 1.631 1.604 1.582 1.568
Mode 4 1.951 1.802 1.812 1.634
Mode 5 1.993 1.973 1.820 1.648
Mode 6 2.235 2.163 1.859 1.825
Mode 7 2.338 2.261 1.965 1.937
Mode 8 2.355 2.296 2.138 2.088
Mode 9 2.441 2.362 2.244 2.110
Mode 10 2.593 2.464 2.303 2.195

 

 

 

 

 

 

 

43

axis. The types of the first 20 vibration modes are shown in Table 3.5. And the first eight

mode shapes are shown in Figs. 3.5 and 3.6.

3.3.4 Response to ground motion

A convergence analysis indicated that about 80 vibration modes were required in
order to obtained an adequately accurate response to the earthquake excitation shown in
Fig. 3.7 which is similar to the El Centro accelerogram. The excitation was applied at all
nodes on the boundary in the upstream-downstream direction. A viscous damping ratio of
0.06 corresponding to low strain deformation was adopted for all modes. The steps used in

I-DEAS to perform deterministic dynamic analysis are brieﬂy described in Appendix A4.

The ground acceleration, velocity and displacement in the upstream-downstream
direction was speciﬁed at nodes along the dam and rock interface. Figs. 3.8 and 3.9 show
the velocity and displacement time histories obtained by integrating the acceleration time

history in Fig. 3.7. Fig. 3.8 shows that the final ground velocity is not zero as expected.

TABLE 3.5 Types of mode for the ﬁrst 20 modes

 

 

 

 

 

 

 

 

 

 

 

 

Mode no. Type of Mode Mode no. Type of Mode
1 Symmetric 1 1 Anti-symmetric
2 Anti-symmetric 12 Anti-symmetric
3 Anti—symmetric l 3 Symmetric
4 Anti-symmetric l4 Anti-symmetric
5 Symmetric 15 Symmetric
6 Symmetric 16 Symmetric
7 Symmetric 17 Symmetric
8 Anti-symmetric 18 Anti-symmetric
9 Anti-symmetric l9 Symmetric
10 Symmetric 20 Symmetric

 

 

 

 

 

 

 

44

TABLE 3.6 Natural Frequencies

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mm!" Fifi?» M°de 17:13:11) M°d° Fifﬁailz) M°d° F1? (:2)
l 1.392 21 2.807 41 3.356 61 3.876
2 1.466 22 2.814 42 3.371 62 3.880
3 1.582 23 2.837 43 3.384 63 3.937
4 1.812 24 2.889 44 3.411 64 3.940
5 1.820 25 2.911 45 3.446 65 3.958
6 1.859 26 2.929 46 3.499 66 3.974
7 1.965 27 2.950 47 3.507 67 4.008
8 2.138 28 2.964 48 3.590 68 4.029
9 2.244 29 2.966 49 3.597 69 4.065
10 2.303 30 2.980 50 3.659 70 4.081
11 2.341 31 3.041 51 3.694 71 4.099
12 2.430 32 3.069 52 3.709 72 4.105
13 2.455 33 3.093 53 3.724 73 4.142
14 2.461 34 3.143 54 3.745 74 4.167
15 2.493 35 3.167 55 3.752 75 4.179
16 2.511 36 3.184 56 3.802 76 4.192
17 2.587 37 3.251 57 3.803 77 4.210
18 2.609 38 3.258 58 3.814 78 4.212
19 2.640 39 3.298 59 3.831 79 4.247

20 2.756 40 3.325 60 3.849 80 4.292

 

 

 

 

 

 

 

 

 

 

 

 

45

 

 
   
      
     
       

 
 
   
 

  

 
 

   
      

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47

Typically, baseline correction procedures are employed to correct the acceleration time
history so that the final ground velocity is zero. However, for the convergence analysis

performed here, base line correction is not necessary and hence it was not performed.

Table 3.7 lists some nodal responses computed with an increasing number of
vibration modes. The maximum acceleration responses at four different locations were
computed in order to study the sensitivity of the response to the number of vibration modes
included in the analysis. The table indicates that using the first eighty modes yields

sufﬁcient accuracy in the computations.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Figure 3.7 Accelerogram similar to the El Centro, California, earthquake of May 18, 1940

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TABLE 3.7 Maximum acceleration (ft/secz) response to different number of modes

 

 

 

 

 

 

 

 

 

Number Of Middle point at Side point at the point on the Point on the
activated . .
the crest crest upstream s1de downstream srde
modes

20 45.1 42.3 11.0 12.0
40 45.0 38.6 11.0 12.0
60 46.0 41.0 11.3 12.3
70 48.1 41.4 11.3 12.0
80 47.1 40.7 11.0 12.0
90 48.1 41.8 11.0 11.7
100 47.6 40.6 11.0 12.0

 

 

 

 

 

 

 

3.4 Types of Failures in Earth Dams

Over the years several types of earthquake damage have been observed in earth dams
and embankments (Seed et al. 1978). Three major factors affect the stability and
performance of an embankment during an earthquake: (a) section geometry (upstream and
downstream slopes); (b) construction method and compaction procedure; and (c) type of
embankment and foundation material. The possible ways in which an earth darn might fail

during an earthquake include (Sherard et al. 1963):
1. Failure due to disruption of the dam by major fault movement in the foundation.
2. Slope failures induced by ground motions.
3. Loss of freeboard due to differential crest settlement.
4. Piping failure through cracks induced by the ground movements.
5. Overtopping of the dam due to failure of the spillway or outlet works.

In general, the principal types of damage can be classified as sliding failure, liquefaction

failure, longitudinal cracks, transverse cracks, and piping failure.

50

3.4.1 Sliding failure

Sliding, as shown in Fig. 3.10, is a major type of damage that can occur in earth dams
subjected to earthquake ground motions. Sliding initially causes settlement and subsequently
leads to dam failure. The slope stability of earth dams is usually evaluated in terms of the
shear strength of soils and the Mohr-Coulomb strength criterion is often used to characterize
local failure. According to the Mohr-Coulomb criterion, the shear strength of the soil, 1'], is

expressed as

If = c+t'tan¢ (3.1)
where t' = effective normal stress on the failure surface, and c and (1) are the cohesion and
angle of internal friction, respectively. When earthquake—induced maximum shear stress
exceeds the shear strength of soils, local yielding is expected to occur. Consequently, if the
shear strength along a trial sliding surface cannot resist destabilizing seismic forces, sliding

failure may occur.

3.4.2 Liquefaction failure

Under earthquake conditions, due to rapid cyclic straining, gravity loading is
transferred from soil solids to the pore water. This results in an increase of pore water

pressure with a reduction in the capacity of the soil to resist loading. This process by which

 
 
    
       

Upstream

V )
Sliding
surface

  

'0'.'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O
H a
8

Figure 3.10 Sliding failure of earth dam subjected to earthquake motions

51

loss of strength occurs in soil is called liquefaction. The phenomenon of soil liquefaction
is primarily associated with medium- to ﬁne—grained saturated cohesionless soils. Sliding
of a slope is attributed to the dam losing stability due to variation in stress, and the soil
losing resistance due to vibration and the rise of pore-water pressure. Such damage induced
by ground motions occurs most often at the upstream slope as shown in Fig. 3.11. If the
earth dam is saturated at the upstream side, it may liquefy when subjected to vibration

exceeding a certain limit.

The occurrence of liquefaction failure is usually related to the volumetric strain in the

soil which is given by

e = 21 +82+€3 (3.2)

V

where 81, £2 and 83 are the principal strains for 3-D problems.

Another method to evaluate possible liquefaction is the use of a threshold strain. If
the cyclic shear strain in soil as a result of an earthquake does not exceed a certain threshold
level, liquefaction should not occur. The peak shear strain caused by an earthquake ground

motion can be estimated by (Commission on Engineering and Technical Systems 1985)
Ymax = . 3“” (33)

U t
V ps ream

Liquefaction zone

J

Figure 3.11 Liquefaction at the upstream side

52

in which am = peak acceleration of earthquake motions, h = depth from the crest and vs =

shear wave velocity in the soil.

3.4.3 Longitudinal cracks

Longitudinal cracks take place mostly in the crest area as shown in Fig. 3.12. The
formation of wide longitudinal cracks is considered to be due to tensile stresses produced
at the surface. Shear sliding deformation may also contribute to such failure. Another cause
of longitudinal cracks is the uneven settlement of the core or foundation. Uneven settlement
usually occurs when the strength of the foundation is not uniform or when loose river
deposits are left unexcavated. It should be noted that longitudinal cracks are sometimes
concealed. The internal cracks formed in the Hachi Dam in Niigata, Japan were discovered
when the dam was excavated to repair sluiceways damaged in the Niigata earthquake.
Therefore, careful investigation is necessary even when no damage is visible (Okamoto

1984).

3.4.4 Transverse cracks

Transverse cracks consists of four types: (a) those formed due to violent vibration in

the direction parallel to dam axis, (b) those formed near both ends of an embankment

Longitudinal cracks

‘
/
core \
/ \

Exaggerated» /

settlement Dumped rock
/ shells
/ .
/ \
\
\
/ \
/ \

 

Figure 3.12 Cracking caused by differential settlement between embankment sections of
dumped rock and rolled earth

53

because of the difference in the vibration characteristics of the embankment and the banks,
(c) those formed due to nonuniforrnity of consolidation within the dam when repairs were
made on sluiceways, and (d) those due to uneven settlement of the foundation. Typical
transverse cracks due to differential settlement are shown in Fig. 3.13. Generally,
transverse cracks are fewer than the number of longitudinal cracks, but these can serve as

water channels, causing breaking, so early repair is necessary.

3.4.5 Embankment and foundation piping

Piping, or progressive erosion of concentrated leaks, has caused a number of
catastrophic failures. As water seeps through the compacted soil of an embankment or the
natural soil of a foundation, the pressure head is dissipated in overcoming the viscous drag
forces which resist the ﬂow through the small soil pores. Conversely, the seeping water
generates erosive forces which tend to pull the soil particles with it in its travel through and
under the dam. If the forces resisting erosion are less than those which tend to cause it, the
soil particles are washed away and piping occurs. The resisting forces depend on the
cohesion and the weight of the soil particles, as well as on the action of the downstream

ﬁlter.

Longitudinal section

Exaggerated crest settlement *—

A
\ Open cracks/

     
    

 
 

 

Figure 3.13 Typical transverse differential settlement cracks

4. STOCHASTIC RESPONSE ANALYSIS USING A
FINITE ELEMENT MODEL — THEORY

4.1 Characteristics of Earthquakes

An earthquake is a phenomenon of strong vibration occurring on the ground due to
the release of a large amount of energy within a short period of time through a sudden
disturbance in the earth’s crust or in the upper part of the mantle. Energy released
underground moves through the earth in the form of wave motions to reach the surface.
Two types of wave motion of different nature, longitudinal waves (P waves) and transverse
waves (S waves), may be transmitted through the ground. In the case of P waves, the
direction of motion of particles of matter coincides with the direction of advance of the
waves. For 8 waves, the direction of motion of particles of matter is orthogonal to the
direction of advance. In particular, the transverse wave is called an SV wave when it is

vertically polarized or an SH wave when it is horizontally polarized.

For any specified ground motion, the responses of any structure can be computed
deterministically. However, earthquake motions are irregular and each one differs from
another in the fine details. It is not possible to predict the precise details of a future
earthquake ground motion. Owing to the variation in ground motion details, stochastic
modeling of strong ground motions is more appropriate and is finding increasing use. In
addition, recent studies of seismograms recorded by the SMART-1 array in Lotung,
Taiwan and other arrays indicate that earthquake ground motions exhibit travelling wave
effects as well as spatial correlation effects. Spatially varying earthquake ground motion
(SVEGM) can only be effectively characterized in terms of stochastic models. Therefore,

the effect of SVEGM on the Santa Felicia dam is studied using random vibration analysis.

4.2 Engineering Interpretation of Stochastic Responses

The trend in civil engineering today, more than ever before, is toward providing

economical designs at speciﬁed levels of safety. Earthquake ground motion is one of the

54

55

main factors which significantly affects the safety of structures. Current earthquake-
resistant design procedures, which generally have been determined only from a specific
earthquake record, often fall short of expectations in future situations. Little confidence can
be achieved from the results of a single deterministic dynamic analysis. Therefore, to avoid
the expense and effort required for multiple deterministic analyses, probabilistic methods

are employed.

In the simplest case, the time history of earthquake-induced ground motion can be
assumed to be ergodic random process. The ergodic process is characterized by: (a) all the
ensemble averages being stationary with respect to the time scale; and (b) the averages
taken along any single sample function being the same as the ensemble averages. In
practical terms, each sample function is completely representative of the ensemble that
constitutes the random process. Based on this assumption, some statistical properties of
interest, as described in the following sections, can be obtained from a single earthquake

record.

Random vibration is rooted in the theory of probability and random variables. The
behavior of a random variable is characterized by its probability distribution. The most
important probability distribution in theory as well as in application is the Gaussian
distribution. The earthquake acceleration excitation is assumed to be a zero-mean Gaussian
process throughout this study. It follows that most stochastic responses must be zero-mean
Gaussian random processes for a linear system. Two of the most important moments of the
response process are the mean and the variance. The ﬁrst moment, i.e., the mean, is the
average value of the response. The second moment, i.e., the variance, measures the
dispersion of the response about its mean. The positive square root of the variance is called
the standard deviation which has the same unit as the mean and for a zero-mean, this is
identical to the root-mean-square (r.m.s.). The standard deviation measures the spread of

the distribution.

56

As mentioned in the beginning, economics as well as safety is the goal of structural
design. Usually, a structure designed to be almost 100% safe over a certain period of time
will be uneconomical. In general, the expected reliability of most civil engineering systems
is in the range of 0.95 to 0.999. The reliability varies depending on the importance of the
structure. For earth dam structures, a reliability of more than 0.997 is typically required.
Typically, responses such as displacement, maximum shear stress and maximum shear
strain are of primary interest. Mean plus three standard deviation (n+30) values of the

responses are computed in this study.

Fig. 4.1 shows a typical sample of a random response. In deterministic problems, one
is usually interested in designing for the peak response. However, for stochastic loading,
the peak response is itself a random variable. For a stationary response duration T, the

response threshold that will not be exceeded the probability P can be expressed as

 

 

ypeak = ”+TT,PO (4-1)
Response Upcrossing
f point \
ypeak r— ———————————————————— —
u, __ ‘ A I _ I' _ 1 _ _ L t A ' ‘ ‘l __ _
4’
« > Time
T

 

Figure 4.1 First upcrossing of the level ypeak in time history of response

57

where u. and 0' are the mean and standard deviation of the response and TT’ P is a peak factor
that is dependent on the duration T and probability of non-exceedance P. Expression for
TT‘ P have been proposed for Gaussian process (Der Kiureghian 1980). In order to compare
peak responses due to different excitations, a value of YT. P = 3 should suffice, and this is

used herein.

4.3 Ground Motion Model

In order to study the response of the Santa Felicia earth dam to SVEGM, the ground
motion model proposed by Harichandran and Vanmarcke (1986) is adopted. The model
which accounts for both spatial correlation and wave passage effects is used to specify the
base motions in the upstream-downstream direction of the dam. In the model the ground
accelerations are assumed to constitute a homogeneous random field. The auto spectral
density function (SDF) of the ground acceleration, $ng ((0), is therefore assumed to be the
same at all spatial locations. The correlation between the accelerations at two different
points is characterized by a coherency function, I'Y(V,(D)l, and all waves are assumed to have
a constant apparent propagation velocity V. The cross SDF between the accelerations at two
locations A and B is then expressed as the product of the point SDF, coherency function

and phase delay

—i(ov/V

511,113 ((1)) = Sag (03) 170% (9)18 (4.2)
Based on the analysis of some events recorded by the SMART-1 array, Harichandran and

Vanmarcke ( 1986) proposed the following empirical coherency function:

2v
(19 ((0)

(1 —A+0tA)] + (1 —A)exp[—6%Z)—)(l —A+0tA)](4.3)

 

1m. (9)1 = Aexp[-
where

9(0)) = k[1+ ((1)/(1)0)b]_l/2 (4.4)

58

and A, 0t, k, (00 and b are empirical model parameters; v = separation between locations A
and B; V = apparent wave propagation velocity from A to B; and S a. (00) 2 auto SDF of

the ground acceleration.

A commonly-used form of the auto SDF is (Clough and Penzien, 1975)

604 + 409091;? 4
Sﬁg (0)) = [(002 —80)2) 2 + 450210213] [(0)2 — (02):: 4(07-(02C2:l 5" (4'5)
8 8 8 f f f

 

 

in which the five parameters cog, Cg, (0], Cf and S O can be estimated by fitting the above
function to observed acceleration spectra. This ground acceleration spectrum, is a modiﬁed

version of the popular Kanai-Tajimi spectrum.

4.4 Finite Element Formulation of the Equations of Motion

The ﬁnite element method has been widely used in many applications. For analyzing
the vibration of earth dams, the finite element method successfully overcomes geometric
irregularity and material inhomogeneity. It is therefore expedient to resort to the finite
element technique for the analysis of the Santa Felicia dam. Tetrahedral elements in which
each node has three translation components, x, y and 2, were used in the analysis. The
equation of motion of the nodal points in the finite element system may be expressed in

matrix form as

[M] {ii} + [Cl {11} + [K] {H} = 0 (4.6)

in which [M] = the mass matrix, [C] = a viscous damping matrix, and [K] = the stiffness
matrix. The vector {u} represents the nodal displacements, and {a} and {131' } are the nodal

velocity and acceleration vectors, respectively.

4.4.1 Finite element discretization

From equation (4.6), by reordering and partitioning [M], [C] and [K], to correspond
to the partitioning of the nodal displacements {u} into free and restrained displacements

{ uF} and {u R}, the dynamic equations of motion become

S9

[IMFFI [W1] {uF} +[[CFF] [le “3”
[Mn] [Mme] {tip} [CRF] [CRR] {up}
(4.7)

+ [KFF] [KFR] {HF} =({0})
[KRF] [KRR] {uR} {O}

in which {14F} are the free (unrestrained) degrees-of—freedom (DOF) and {uR} are the
restrained (support) DOF.

The free nodal displacements {uF} can be decomposed into pseudo-static and

dynamic parts, {“5} and {u d}, respectively:

{“F} = {“5} + {“d} (4.8)

The pseudo-static displacements are obtained from the support displacements, { u R}, using

the static equilibrium equations (with no external loading)

IKFFI {Us} + [KFR] {uR} = {O}

yielding

{14,} = —[KFF1"‘[KFR1 {uR} = [A1 {14R} (4.9)

. . -1 . .
in which [A] = -[KFF] [KFR] . These displacements are the 1nstantaneous free
displacements of the structure due to support movement {u R} at a given time t. Substituting

equation (4.8) and equation (4.9) into equation (4.7) yields

[MFFI {ad} + [CFF] {dd} '1" IKFFI {"d}
(4.10)
= -(1Mppl 1A1+1MFR1){17R}-(1CFF1 1A1+1CFRI){L3R}

60

For light damping, the second term on the right-hand-side of the above equation may be

neglected. Thus the final equations of motion for the free DOF become

[Mfr] {ﬁd}+[Cppl {lid} +1KFF] {14d} =—(1MFF1 [41+1MFR1) {iip} (4-11)

4.4.2 Modal analysis

In order to simplify the damped equations of motion of the multi-DOF system, the
orthogonality properties of the normal coordinates may be used to decouple equation

(4.11). For undamped free vibration

[MN] {lid} + [KFF] {11,} = 0 (4.12)

Assuming harmonic vibration in the form

04,} = {4,} e...) (4.13)

results in the generalized eigenvalue problem

I
O

[[KFFI —wf [Mppl] {4),} - (4.14)

The solutions of these equations yield the natural frequencies of vibration (0]., and the mode
shapes, {$1.}, of the structure. For general vibration, the nodal displacements may be

expanded in terms of the mode shapes as

{ad} = 2 {mil/((0 (4.15)

k=1

where U k(t) are generalized modal responses. Substituting equation (4.15) into equation
(4.11), premultiplying by the transpose of mode-shape vector {(1)1}T and assuming that the

mode shapes are orthogonal to the damping matrix, i.e.

T
{4,} [Cl {4.} = 0,j¢k

results in the uncoupled modal equations

61

Uj+2§jijj+waj = G}, j: l, ...,n (4.16)
where
-1 T T
G}. = ﬁjo-l (IMFFI [141+ [Mppll {17R} = {Fj} {HR} (4-17)
{1]} = X7111Mppl [A] + [MFR]]T{0J.} (4.18)
J
T

in which {Fj} is the modal participation factor, M]. is the normal-coordinate generalized
mass (modal mass), and C} represents the modal damping ratio. In practice, it is common to
assume the damping ratio for each mode than it is to try to evaluate the coefficients of the
damping matrix [C]. It is convenient to collect the excitations Gj(t) into a vector {G(t)} and

the modal participation factors { Fj} into a matrix

in = {{1}} {r2}...{r,,}]
in which case equation (4.17) may be written as
{G} = lFlTUiR} (4.20)

In addition, the mode shapes may be collected into a matrix

l—

[(1)] = [{4.} {42}...{¢,,}]

and equation (4.15) may be written as

{ud} = [(1)] {11(1)} (421)

62

4.5 Random Vibration Analysis

4.5.1 Introduction

Assuming that the earthquake-induced ground acceleration is a stationary random
process, it can be shown that the displacement response is also a stationary process.
Consequently, the autocorrelation matrix of the random process {14],} will be independent

of time t and is defined as

{16,501 = E[{u.<r)}{u.(t+r>}1] (4.22)

where E[ ] denotes the expectation of the random variable. Substituting equation (4.8) into

equation (4.22),

T T
[R,,F(I)l =E[{u.<t)+ud<z>}{{u,<r+r)} +{u,,<r+r)} }]

= [R4 (01 + “in. (r) l + 18,,” (r) l + IR... (0 l (4.23)
in which [Ru (1)] and [Rud(t)] are the autocorrelation matrices of the pseudo-static
and dynamic displacements, respectively, and [R

1(0] and [R (1)] are the cross

11,14! udu‘

correlation matrices. For stationary response

1R...” (01 = {wa (4) l
The Fourier Transform of equation (4.23) yields the spectral density matrix of the free

displacements:

[Sur(w)l = [Sudan] + [Sud(w)l + [Su",,d(w)l + [541(4me (424}

For stationary response, the cross spectral density matrices are related through
{5,3,4 ((0) l = [514.111. (40)] (4.25)
where the overbar denotes the complex conjugate.

The variance of the ith free displacement can be obtained from the ith diagonal terms

of the spectral density matrices

63

0
II

J SH!" (0)) dm + j Suit. (0)) do) + 2Re [ of Sta—jun, ((0) dw]

(4.26)

_oo —00

2 2
= out + oud1+ 2C0v (”s.’ “(1)

l

2 2 .
where Re [ ] denotes the real part of the argument. Cu and 0'“ are the variances of the
d

S .
I l

pseudo-static and dynamic ith free displacement, and Cov (us, “(1) is the covariance

between the static and dynamic displacements.

4.5.2 Variance of dynamic displacements

Let h j(t) be the impulse response function of the jth mode corresponding to equation
(4.16), i.e. the response to Gj(t) = 5(t) is Uj(t)=hj(t). For linear response, applying

superposition for any general excitation, the response can be expressed by

Uj(t) = jhj(0)Gj(r—0)de (4.27)

—00

In matrix notation, the modal response vector may be expressed as

00

{U,<r)} = jlhj(9)l{Gj(t-9)1a'9 (4.28)

—oo

in which the diagonal n x n modal impulse response matrix is

lh(9>l = [diag(hj(9))l

The autocorrelation matrix of the dynamic displacements { “(1} is

[Rud(I)1 5[{ud(()}{ud(r+r)}’]

4.29
E1191 {U(t)}{U(t+0}Tl<I>l’] ( ’

l<blE[{U(t)} {U(t+r)}’] 191T: {<91 {Rum} [(D1T
Using equation (4.28)

64

18,/(01 = E[{U(r>} {U(t+r>}’l

WOO

jj[h(0,)]E[{G(i—0,)} {G(t+T—02)}]][h(02)]Td01d02

—°O—OO

(4.30)

0000

j j [h(91)I[RG(T+91—92)][h(92)]T(191d82

—OO—OO

Since {G} = [F]T{iiR},

[RG(r+01—02)] = E {G(t—01)}{G(t+T—02)}]:|

 

= E—[F]T{iiR(t—01)}{l'iR(t+T—02)}T[F]] (4.31)
L

[FlTlR,-,R(I+9,—92)l lrl

Finally, combining equation (4.29), equation (4.30) and equation (4.31),
”in. (r) 1 =

°° °° (4.32)
1¢1[jj[h(9,)l[FlTR,.,-R(I+9,—92)[F1[h(92)1Td9,d02]1¢1T

—-OO —00

The Fourier Transform of the above equation yields the spectral density matrix of the

dynamic displacements:

Sudan) = {<91 [IN—(9)1 {rlTlS,R(w)l {r1 [H(w)lTl<l>lT (4.33)
where the diagonal modal frequency response matrix [H((D)] is
{Hum = jlh<t)1e“"”dr = ldiag<Hj<w))l (4.34)

--00

and S u" (00) is the spectral density matrix of the support accelerations.
R

The diagonal terms of [Sud (0)) ] may be written in scalar form as

65

Ste/1m): 2 2 2 2 q)ijcblkr lj mkHj(_ _m)Hk (0‘)) 511,, [HR ((0) (4'35)

j=lk=1l=1m=l R“

where n is the number of free degree-of-freedom and r is the number of restrained degree-

of—freedom. The modal frequency response function has the form

1
H}. ((b) = 2 (4.36)

2 .
(al.-0) +21%.ij

 

The variance of the ith free dynamic displacement is

=j 5“,) ((1))d0)= X Z oijoikl 1k (4.37)
j— = 1k =1
where
:2 2 ﬁrm). j H (— —0))Hk((l))Su. I“. m(00)d0) (4.38)
1: lm =1 _oo

4.5.3 Variances of pseudo-static displacements

From equation (4.9), {“5} = [A] {uR} , thus the autocorrelation matrix of the

pseudo-static free displacements {“5} is

[Rpm] E {u,<t)}{u,<r+r)}?

 

= ELI/110mm}{uR(t+I)}T1A1T] (439)

T
[A1 [Rukml [A]
The Fourier Transform of the above equation yields the spectral density matrix

[Sutton = [A1 15,,R((9)l lAlT = i,( [M 15,-,R(w)l 141’) (4.40)
(t)

The diagonal elements of this matrix may be written in scalar form as

66

S“.(.((D)= 142 Z AilAimS u'Rl'l'Rm ((1)) (4.41)
' (D 1: 1m -1

The variances of the pseudo-static displacements are therefore

oo

:2 XAilAimJ1'4SliRliRm(w)dw (4-42)
(0 R1

l=lm=l —00

4.5.4 Covariance of pseudo-static and dynamic displacements

The cross correlation matrix of the pseudo-static and dynamic displacement is

defined as

lR,,_I,,I<r)l = E[{u,(r)} {loom}?

= E-[A] {uR(t)} {U(t+t)}Tl<DlT]

°° T T I (4.43)
= E [A]{uR(t)} [{uR<r+r—9)} lrl [11(0)] 49 [<91]
L -..

 

[A] {j {R,,I,.,I(r-9)} lrl {12(9)le9} [(DlT

Again, the Fourier Transform of the above equation yields the cross spectral density matrix

[SuI,,I(w)1 = [A1 lS,,I,.,.I(m)l lrl [H(9)1T[<I>1T (4.44)

There exists a direct relationship between [S uRiiR ((1)) l and [SIIR ((1)) ] . Consider

2

[R",,II(I)1 = —d—2—E[{uR(t)} {uR(t+t)}7]
d1 .1
=E[{uR(t)} {uR(t+r)}7]_ — [RII II. (1)] (4.45)

The Fourier Transform of [R"IIR(t)] is —(02 [SIIR((1))] , and therefore, the Fourier

Transform of the equation (4.45) becomes

67
[5,,II,-,-II(<9)1 = —w2[S,,II(w)] = —iI,_lS,,.I(m)l (4.46)
(I)

As a result, equation (4.44) may also be written as

[5,,I,,I(w)1 = «‘3141 lS,.,I(co)l {r1 [H(9)1T[<I>1T (4.47)
(D

The diagonal elements of [SII III (0)) ] in scalar form are

S“.(~."d.(w) = _

ZI I2 2 ¢,I-A ,,r,,,I-HI(w)S,I,. III (4)) (4.48)

lm=1

L
(1)2}
The covariance of the ith free pseudo-static and dynamic displacements is therefore

00

Cov(us, u II) = Re[ I sII IIIII (6))de (4.49)

—00

4.5.5 Calculation of covariance matrix of nodal stresses

As with the displacement response, any Cartesian stress can also be decomposed into
pseudo-static and dynamic parts, {IS} and {Id}, respectively. Consider two Cartesian
stresses, 1'1 and 12, whlch can be expressed as II = 115 + III] and 12 = 125+12d,

respectively. Thus,
E [1112] = EH1” + I1d) (125 + T2(1)]

(4.50)
= E [TldTZd + I15123 + TldTZS + Tutu]

Since all Cartesian stresses are zero-mean Gaussian processes, equation (4.50) becomes

Cov(tl,tz) = Cov(tIII,12II) +Cov(tIII, 125) +Cov(tIII, 125) +Cov(tls,12d) (4.51)

Therefore, the total covariance matrix of the six Cartesian stresses at the ith node may be

expressed as

IIIIII)] + [Cov(t

[Cov(1:II,tII)]I, = [Cov(1: 1’3,qu

pd’

)1I.

(4.52)
+ [Cov(tIIII,tIIII)]I+ [Cov(tIIII,tIIII)]I,

68

where [Cov (t qu) ] I, and [Cov (t P d, tqd) ] I. are the covariances matrix of the pseudo-

[’5’
static and dynamic stresses at the 1th node; [Cov (tpd, IIIS) ]I, and [Cov (’th, IIIII) ]I are
the covariance matrix between the static and dynamic stresses, and p=l, .6 and q=1,. .,6

are subscripts denoting the stress components, in which the six stress components III, III, 1,,
Ix)" 'tIIIz and 'tII,z are denoted by 1'], 1:2, 13, I4, 15 and “to, respectively. In expanded form, the

covariance matrix is

C0v(tI,tI) COV(I],TZ) C0v(Tl,T6)

Cov (12, 12) Cov (12, 1:6)

[Cov (III, Tq) ]I = (4.53)

 

 

_ Symmetric Cov (T6, 16)‘ I

4.5.5.1 Dynamic response

As with the dynamic displacement response in equation (4. 15), the nodal stresses may

also be expanded in terms of the stress mode shapes, [‘1’], as

{far} = [‘P1{U(t)} (4-54)

Consequently, following the procedures in Section 4.5.2, the covariance matrix of the

dynamic stresses at each node can be obtained as

n ’1

6 6
1 .
[Cov(rIIII,rIIII)]I= z 2 IIIz III”); III. pIII+uIIwIII)II.I 1=1,...,NII (4.55)
k=1j=l = =

where subscripts u = 6 (i — 1) +p and v = 6 (i — 1) + q,NII is the total number of nodes,

n is the number of modes used and Ijk was deﬁned in equation (4.38).
4.5.5.2 Pseudo-static response

The pseudo-static stresses may be expressed as

{1,} = in] {up} (4.56)

69

where the jth column of [n] are the stresses due to a unit displacement along the jth
restrained DOF. Equation (4.56) is similar to the pseudo-static displacement representation

in equation (4.9). Following the procedures in Section 4.5.3, the covariance matrix can be

obtained as
[Cov (IIII,1:IIII 5)] ,
r r 6 1( 0° 1 i: [I ..., N" (4.57)
= 2 21p 61‘]; §( TLilr‘vm + nvlnum) J jsl'iR l'iR ((0) (1(1)

4.5.5.3 Covariance of pseudo-static and dynamic responses

The calculations of the covariance matrix between the pseudo-static and dynamic
stress responses are similar to those presented in Section 4.5.4. It can be shown that the

covariance matrix of the stresses at the ith node may be expressed as

[Cov (t IIII) + Cov (1

pr!’ ps’ chI) l i

= RC 2 z 2 2 z (Wujnvl+ wvjnul):J:r mj m—zsl‘iR IliR m((1))d(1)

=ll=lm=1p=lq=l

(4.58)

The covariance matrices of the three strain response components can be obtained
using equations (4.55), (4.57) and (4.58) by replacing w and n with strain mode shapes and
strain responses due to a unit displacement at each restrained DOF, respectively. Finally,
the total covariance matrix of Cartesian strains at ith node can be obtained analogous to

equation (4.52).

4.6 Viscous Damping vs. Hysteretic Damping

It is customary in dynamic analysis to use viscous damping to account for the loss of
energy during vibration. The main shortcoming of viscous damping is that the energy
dissipated is frequency dependent, with more energy being dissipated at higher frequencies.
Experimental results indicate that earth dams exhibit frequency-independent energy
dissipation due to hysteretic damping. However, the equation of motion with exact

hysteretic damping is non-linear. The most common method of accounting for hysteretic

70

damping is to linearize the equations of motion by replacing the stiffness of the system, k,
by the complex-valued stiffness I} = k (1 + 2iC) where C is the hysteretic damping
parameter. The effect of viscous and hysteretic damping on the response of earth dams is

examined in the next sections using a simple one-dimensional shear beam model.

4.6.1 Shear beam with viscous damping

The one-dimensional shear beam model was reviewed in Section 2.2.2. 1. It is known
that stress can be expressed as a function of strain and rate of change of strain with time for

viscous behavior. Therefore (Okamoto 1984),

= (G + 6I%)(g—:) (4.59)

where G and G I are material constants. Substituting this into equation (2.4),

2
pa—2”— (G+GI8)(a—2“+1a“] :0 (4.60)
3,2 at ayz yBy
For each mode of vibration, letting u = (DI. U} , the above equation becomes
2 2
(Djd—U (GUI. G dU _)[d—(I:+ ld—(Dj—J O ' 1 (461)
_ + - , = n .
p dIz d (1)2 y dt 1

Substituting equation (2.7) into equation (4.61),

d2 U} G Zde
—7+—6mjd—+mI2IU =0, j=1,. (4.62)
GI dt
Letting CI. = 2G0)!" ,the familiar 1 -D modal equation of motion is obtained:
2
d U de
1+2§..0)—+(0I.2I.U =0 (4.63)

dt2 1 J d).

For a base acceleration of ﬁg, the modal equation of motion becomes

U J+2ijjUj+mj2 Uj - —-1“I.x'g (4.64)

where 1"}. is given in equation (2.12). The frequency response function is therefore

7l

1
H]. (0)) = 2 (4.65)

2 .
(1)]. —a) +21%ij

 

so that

 

 

IH. ((1)) '2 = 1 (4.66)
J ( 2 2)2 + 4C2 2 2
Using the Kanai-Tajimi spectrum for jc'g,
m4 + 4602612§2
5.. (1) = g g g 4.67
"x( ) [(wg-w2)2+4w2w&?§§] " ( )

The two parameters (0g and Cg are taken to be 15.0 rad/s and 0.55, respectively. The
intensity parameter So is adjusted such that the standard deviation of the ground

acceleration is 0.09g.

4.6.1.1 Displacement response
It

Using random vibration analysis with n vibration modes (i.e., u = 2 (DjUj), the

displacement variance for widely separated modes can be approximated by}

n
of; z ofrffwmjmuzsmmww (4.68)
j: 1

in which modal cross correlation have been neglected. The mode shape of the 1-D shear
beam model was shown to be a Bessel’s function in Section 2.2.2.1. Therefore, the

displacement response at the top (y = O) is

2 n 2 2 .2
614(0): 210(0)rj£!11j(m)( ng((1))dw (4.69)
.=1

4.6.1.2 Shear force
For the generalized single-degree-of—freedom system, using the principle of virtual
work, the elastic forces f3 for each mode developed during an earthquake response can be

estimated by

72

f, (.v. t) = (02m (2') u (y. I) = wzm (y) <I>j (1’) U,- (I) . j= 1. n (4.70)
where m (y) = pyB/ H for a unit width of the dam. The same notation was used in the
1-D shear beam theory in Section 2.2.2.1. Consequently, the base shear f3 for each mode is

given by

fB(t)= ﬁpyHCDj (_\)th02 U]. (I), j=l,. (4.71)

Taking the Fourier Transform, the spectral density function of the base shear is then

SfB((1))= (rpv—<D(\)d)) fIHj(0))|sz'(0)) (4.72)
j_ _ .
Therefore, the variance of the shear force at the base ()7 = H) is

n
2 B Z')’ 2 4 2 2
of8 = 2(pryl—110(#)dy) to]. 1‘]. _°o:Hj(0))| ng((t))d(1) (4.73)

1' = 1
4.6.1.3 Bending moment

Similarly, from the above result, the base moment for each mode can be expressed by

28 2 .

The spectral density function of the base moment is then

SMB((1)) = z(ﬁpyzgcb(_v)dy)“mjrf|Hj(m)lzsxcm) (4.75)
j=1 L

Therefore, the variance of the bending moment at the base of the dam is given by

ZUJO’pyzg 102% )d)’)2(11j41“f mlHj(u))|ZS_.r.g(w)dw (4.76)

j- — 21
4.6.2 Shear beam with hysteretic damping

For an analysis with hysteretic damping, a complex modulus is used instead of an

elastic modulus. For mathematical convenience, a complex-valued stress E and a complex—

73

valued displacement E are used. The stress and the corresponding displacement must satisfy

an equation similar in form to equation (2.4), i.e.

Substituting 1: = (G + iGz)Ea—u into the above equation,
)1

2 2
p-a—zl—l— (64-102) [%u+-lz-a§-u] : O
at 8y y y

For each mode of vibration, letting it = (bk-(7k , the above equation becomes
2_
d . G 2_
—(:"+(l +i——2)w Uk = 0, k=1.....n
dt G k

Letting C = GZ/G , equation (4.79) becomes

Uk+ (1 +2iC)w:L—/k = O, k=1,...,n
and the modal equation for base excitation becomes,

4' . 2— ..
Uk+ (l +21§)(1)kU,< = 4“ng

Therefore, the frequency response function is

l

Hk((0) = 2
(1 +2i§)wk—w

 

2
so that

1

2 2 2 2 2
(wk—co) +4C wk

 

IHk("’)|2 =

(4.77)

(4.78)

(4.79)

(4.80)

(4.81)

(4.82)

(4.83)

The variance of the displacement, base shear and base moment are obtained by using these

frequency response functions in equations (4.69), (4.73) and (4.76).

74

4.6.3 Discussion

The following values were used for the dimension and material properties of the
Santa Felicia dam: dam height H = 275 ft, dam width B = 1,050 ft, soil mass density p =
4.02 lb-sz/ft4 and damping ratio C = 0.06. Four vibration modes were used in the analysis.
The results of three cases: constant modal viscous damping ratios, modal viscous damping
ratios varying with frequency and hysteretic damping, are compared in Table 4.1 and Table
4.2. For the case with varying modal viscous damping ratios, the damping ratio of the ﬁrst
mode was set at 0.06 and the ratios for higher modes increased with the natural frequency

so that they were consistent with a constant damping c. The values listed in the Tables are

TABLE 4.1 Comparison of responses using hysteretic damping and varying viscous damping

 

 

 

 

 

 

 

 

 

R.m.s. R.m.s. R.m.s.
Case displacement shear force at the base bending moment at the
at the top (ft) (kips/ft) base (kips-ft/ft)
Hys‘eiet‘c 0.19111 7.41576x108 1.86148x10”
dampmg
vary” Y‘scous 0.18993 4.80325x108 1.20568x10“
damping
Difference -0.6% -35.2% -35.2%

 

TABLE 4.2 Comparison of responses using hysteretic damping and constant viscous damping

 

 

 

 

 

 

 

 

 

R m 8 dis Iacement R.m.s. R.m.s.
Case ' a; t. the t: (ft) shear force at the base bending moment at the
" (Rips/ft) base (kips-ft/ft)
Hys‘eiet'c 0.19111 7.41576x108 1 .86148x10ll
dampmg
“mm“ .V'SCOUS 0.19143 7.46685x103 1.8743lx10ll
dampmg
Difference 0.17% 0.69% 0.69%

 

 

 

75

the root-mean-square (r.m.s.) values of the responses. Table 4.1 indicates that for the case
of varying viscous damping the r.m.s. values of the base shear and base moment are
significantly lower than those for the case of hysteretic damping due to frequency-

dependent energy dissipation. At higher frequencies, the loss of energy is higher.

On the other hand, if the viscous damping ratio for each mode is kept constant, as was
done in the 3-D finite element model Table 4.2 indicates that the responses are very similar
to those obtained from a hysteretic damping model. This indicates that although soil
displays hysteretic behavior, the simpler viscous damping model with constant modal

damping ratios is adequate for evaluating the seismic responses.

4.7 Statistical Moments of Maximum Shear Stress

4.7.1 Introduction

The random vibration analysis described in Section 4.5.5 yields the statistical
moments of Cartesian stress components. However, an important quantity of interest in the
seismic analysis of earth dams is the maximum shear stress 1mm. Several approximate
methods have been developed to estimate the moments of functions of random variables.
The ﬁrst-order second-moment (FOSM) method is one of the simplest and most popular.
This method is based on truncating the Taylor series expansion of a function to retain only
linear terms. Since the relationship between maximum shear stress {max and Cartesian
stresses is nonlinear, the FOSM method may be employed to approximately estimate the

mean and variance of 1mm.

As an example, consider a function of two variable F (x, )1). Its Taylor series expansion

about the point (E3), retaining only linear terms, is

- 25 - a_F .,
F(x.1) ~F(x..)) + ax (x—X) + ay(.v-.1) (4.84)

76

Commonly, E and y are taken to be the mean values (in which case the method is called the
mean-value FOSM method). As a result, the mean and variance of function F (x, y) can be

expressed as

E [F(x, '11)] 2 F (X, 3'1) (4.85)

and

2 2
Var[F(x,_v)] =(g—fl ) Var[x] +(g—Ifl ) Var[_v]

+20” )(a—F
1,9 8y

8?
where all derivatives are evaluated at the expected values of the variates.

(4.86)
)Cov [x, y]

 

 

i, 5'

4.7.2 Calculation of maximum shear stress for special cases

It is well known that for 2-D case 1mm can be determined by

1' — 1' 2
x y 2
Irma: ,J( 2 ) + Txy (487)

The exact moments of 1mm can be computed for certain specified cases, and these cases are

 

 

ﬁrst used to gain insight into the adequacy of the FOSM method.

4.7.2.1 Uniaxial loading I},

In this case,

TX 2 '1‘!
Tm? F (1..) = (3) = -2'— (4.88)

Assumin that r is a zero-mean Gaussian random variable with a robabilit densit
g x P Y y
T2
. 1 .
function fI = exp ———x7 , the exact first two moments of 1mm are
. ./21rotx 20

I

X

T T .
518....-.1 = E['—;—'] = £984..

6

l
= i( J: Trftrdt.r+foo_1xftxdt.r) = 2

1‘.
TC

 

(4.89)

 

77

and

 

 

 

2 2
2 2 T, or,
var [THICK] = E[TIN(II] — (E [Tmax] ) = r jft dTX —[ ‘ J
-°° " 27! (4.90)
_ l 2 1 2 _ 2
— 401,. — 2176‘: — 0090845611
Therefore, the standard deviation of the maximum shear stress is 0.30146I .
Using Ix as the basic variable, the mean-value FOSM method yields
_ |E [all
E[‘tmm_] =F Tx 2 —2— = 0 (4.91)
_ . aF(tr) lalt‘i -
and the variance of 1mm. unable to obtain because at ‘ = :2- 81' 1s undefined.
x _ X
I 0

 

 

These are clearly not good approximation. However, if I‘txl is’used as the basic variable,

then the mean-value FOSM method yields

 

 

Emmi :4m) = 2" = J21: (4.92)

8F (ltxl)
varltmaxl ‘7 WIT—I—

l
A
NI—

 

.K

_1 2 2 2
‘2. 02-59.,

which are exact results for E [T

2 2 [- 2 2
)Varllfxll - ) X(ELTx]_ (EHTle) )
m (4.93)

0.0908450:

and (SI . The reason for this is as follow:

M (II

max]
0 With II as the basic variable, Fax) is bilinear as shown in Fig. 4.2(a).
Linearization of F (1)) at any value of It yields a poor approximation for half the

range that Tx can assume.
0 With ltxl as the basic variable, F (ltxl) is linear as shown in Fig. 4.2(b), and hence

the FOSM method yields exact results.

If consider I: as the basic variable, rm“: F( 1:) = alt}. Using the FOSM

method,

78

(a)

 

 

  
 

 

 

 

 

  
 

‘ F (Ix)
TI
r +
/ \
/ \
\
/ / \ \ <\
ﬂ , \
linearization of F (1‘) V fI linearization 0f F(t_,)
at any Ix > 0 ' " at any 12x < 0
Probability density of “Ex showing
the range of values Ix can assume
(b) +
4 F (l'cxl)
Density function of 1mm.
Itxl
% n
f '
Density function of III!

 

11:1

Figure 4.2 Illustration of probability distribution of maximum shear stress for 1-D case

 

 

‘2 1 2 1
E [Tmax] “(1.) = 5 Elk] = 50. (494)
2 7 2
BF TX 2 l 2
Varhmm] z 2 Var [Ix :l = 2 x 2 (Var [3])
81x —. 4 It
I: ' :7. = O: (4.95)
= 2 X20: = 0.]250':
I60T " "

Consequently, the standard deviation of 1mm is 0.353601‘ . Table 4.3 shows the differences
of the results using different basic variables, and indicates that using FOSM method with
I: as the basic variable has 25.3% error for the mean value and 17.3% error for the standard
deviation of 1mm. Note that the correction factors (CF) defined as the ratio of the exact
mean and standard deviation to the approximate mean and standard deviation are CF11 =

0.80 and CFO = 0.85, respectively.

4.7.2.2 Pure shear loading In

. 2 . . . . .
For th1s case, 1' F (In) — ,l‘cxy = '14,)" . As 1n the preV1ous sect1on, 1f It”! 1s

max—

taken as the basic variable, then 1mm is linearly related to l‘tnl and the FOSM method yields
exact results. If ‘13:, is used as the basic variable, the FOSM method yields the same

percentage error percentage as for uniaxial loading.

4.7.2.3 2-D beam bending problem with stresses Ix and Txy

For this case, since Iv = 0, equation (4.87) reduces to

TABLE 4.3 Comparison of statistical moments of 1mm using different approaches

 

 

 

 

Maximum Basic variable used in FOSM method
shear stress Exact value 2
T x lTxl Ix
Mean Value (3'T /x/21[ 0 (31'T /./21t 0'1 /2 (CFp = 0.80)
5‘31“?“ 0.30146 undeﬁned 0.30146 0.35366 (c1: = 0.85)
dev1at1on 1'. T... T. 9

 

 

 

 

 

 

 

2
T 2
rmax = F (TX, I”) = I" +1”, (4.96)
If Ix and In, are used as the basic variables, the FOSM method yields zero for the
mean value of 1mm and an undefined variance for 12mm since F (I x, I“) is not differentiable

at the origin. Fig. 4.3 shows F (TX, 1:“) with the cusp at the origin.

Alternatively, if ITxl and I1: ”I are used as the basic variables, for all positive values, the

function F can be rewritten as

 

Tmax = F(|Txl’lt.r1'l) = (4-97)
and the FOSM method yields
- — — _ 1( 2
El‘mwl~Fllr.1’11..1l - J f
(4.98)

 

 

 

 

 

 

 

2 2
Figure 4.3 Display of function F (TX, I”) = .11 x/ 4 + In

81

 

 

 

2 2
Varnnm] z[% ] Var[|rx|] +[élaTF—l ] Var[|1x).|]
.. 122—112.1 " 127121271 (4.99)
BF 8F
[211 _ 1121—21 _ 1... ”‘1’ ""‘-"“

 

 

 

 

IBI’ I‘m-l IRI‘ Ital

in which the variances of l‘ttl and I‘tnl are

Var[|rxl] E[’C:] - (EMT-Al)2

2 2 (4.100)

2 2
— E61} = 0.363401!

I

I

and Var [ITxyl] = 0.363403“, respectively; the covariance is (Papoulis 1991, p. 179)
Cov ['14, Irwl] = E [ ltxlltxxvl] - E [Itxl] E [ Iti‘yl]

‘ , -1 . —l 2 2
261x614 cos(s1n (P) j + PSI“ (P) L“ — #GTA/TY‘IGTH (4,101)

20T (3T

1’; “(cos(sin_l (p) ) + psin_I (p)_1)

 

where p is the correlation coefficient between It and In; and the derivative terms are

 

 

 

 

 

 

 

 

 

3F _ l 117/2 - l G“ (4102)
atx|__—2J_2 _21 _4(/2/4 2 '
it!" lrn'l ITII /4 + ITXVI — — 611 + 01-0
' ' ITJ‘ I‘m-l
and
——aF - GT“ (4103)
612.1 _ ’ J 2 4 2 '
- l1!" ltul 011/ + Ctr.)-

. . 2 2 . .
F1nally, 1f TX and I“, are used as the bas1c variables,

2 2 j 2 2
1”!th = F(Tx’ T1"): Tr/4 +11.) (4.104)

and the FOSM method yields

 

E[r"m1=F(r I) = (Tr/4+5: (4.105)

82

 

 

 

 

 

 

2 2
BF 2 BF 2
Var [Imax] 7 V3.1‘[:Tx] + —_2_ Var [In]
an _ _ at“ _ _ '
3, I; ' tit?)
- (4.106)
+2 9% ‘a_}:' C V[I:’T:v
at, _ _ 31x, _ _ .
If” I: i I: Til
in which
2 4 2 2
Var 13 = E T - (E T J
I: x] I: x3 [ x] 2 2 (4.107)
= 3x (Var[1'x]) — (Varltxl) = 2X Warn.“
2 2 2 —2 2 T 2 2 2 2
COVEI: T”. = Ell 1; —:x)(:3)'2- 14.1)] = E[Txtxy] _ (:t-‘o-T“ (4.108)
___ 611011." + 2(511x1’nj) — 61‘s,”: 2 x (Cov [1,, 1,1,1)
81:2 = '8 2 1 2 (4.109)
8‘;ro lo"T /4+0'T
Ir’ 1;)" x U
and
a): = 1 1 (4.110)

 

2 2 2
. _2 _2 61/ 4 + or"
,T ' "

As an example, consider a cantilever beam supporting a concentrated load P at its free
end. Distributions of the normal stress TX and shear stress I)“, along the depth of the beam
are illustrated in Fig. 4.4. The normal stress 15x at any distance y from the neutral axis is

determined by

r = —- (4.111)

where M is the bending moment at the cross section under consideration, and I is the
moment of inertia of the cross section with respect to the neutral axis. The shear stress Txv

is determined by

83

‘p

T T ,
. .ir . ' ..X) t
distribution distribution

.... ?
[é

 

 

 

 

 

 

V

\\\\\\ \\\\\\\\\\\\\\\\

 

Figure 4.4 Illustration for stress distributions of beam bending problem

1:193

.ry I, (4.112)

in which t is the width of the beam and Q is the first moment with respect to the neutral axis.

For this problem, Fig. 4.4 shows that the sign of the shear stress may be the same as
or opposite to that of the normal stress depending on whether the point under consideration
is above or below the neutral axis. This implies that the correlation coefﬁcient between Ix
and In could be positive or negative. For some specified cases given in Tables 4.4, the

. . . . 2
mean and standard deViation of 1m were computed usmg the FOSM method With 1x and

at

2 . . .
In, as well as I‘txl and Itxvl as the ba51c variables. These results are compared With those

computed by using a sufﬁcient number of simulated samples. The details for simulation are
described in Section 4.8. Two correction factors CF11 and CFO for the mean and standard

devration of 1mm,

respectively, are deﬁned by
_ it based on simulation

CF —
1‘ tt based on FOSM

 

(4.113)

and

0' based on simulation
CF = 4.
° 0' based on FOSM ( “4)

 

84

which can be used to improve the accuracy of the FOSM method.

Table 4.4 shows that using ltxl and I'cxvl as the basic variables in the FOSM method
yields estimates for Ehmax] that are quite close to the simulation results for all cases (i.e.,

fluctuate and are not uniformly accurate. On the other

max

CFu z 1 ), but estimates of 0’I

hand, if I: and 1:, are used as the basic variables, then the FOSM method yields estimates

of 0'T and E[t

"I01

mar] that are consistently on the low side. Therefore, correction factors of

0.82 and 0.85 for the mean and standard deviation of 1mm, respectively, may be used for

this case to correct the FOSM estimates so that they are close to the simulation results. The

TABLE 4.4 Simulation results and correction factors (CF) for the mean and standard deviation of
tmx estimated by FOSM methods using different basic variables

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 2 Simulation results Barrie Zraaiafles: Basizc variabges:
Case 611 0'!" p I x." 1:x and 1:xy

' Ehmr] 01...... Ci:Ll CFO C1:L1 CFo
1 1 1 1.0 0.8896 0.6725 0.9972 0.9979 0.7957 0.8508
2 1 1 0.5 0.9469 0.5818 1.0615 0.9957 0.8470 0.8441
3 1 1 0.0 0.9735 0.5621 1.0913 1.0114 0.8707 0.8600
4 1 1 0.5 0.9577 0.5904 1.0736 1.0103 0.8566 0.8533
5 1 l —1.0 0.8924 0.6690 1.0004 0.9928 0.7982 0.8507
6 10 1 1.0 1.4905 1.1253 0.9985 0.6168 0.7967 0.8479
7 10 1 0.5 1.6200 0.9493 1.0853 0.6623 0.8659 0.8599
8 10 1 0.0 1.6237 0.8904 1.0878 0.6855 0.8679 0.8829
9 10 1 0.5 1.6125 0.9436 1.0803 0.6583 0.8619 0.8573
10 10 1 -1.0 1.4843 1.1231 0.9944 0.6156 0.7934 0.8549
11 l 10 1.0 2.5489 1.9195 0.9978 2.9877 0.7961 0.8476
12 1 10 0.5 2.5953 1.8582 1.0160 3.0578 0.8106 0.8466
13 l 10 0.0 2.6137 1.8520 1.0232 3.1008 0.8163 0.8353
14 1 10 0.5 2.5946 1.8682 1.0157 3.0743 0.8104 0.8322
15 1 10 -1.0 2.5558 1.9276 1.0005 3.0002 0.7983 0.8589

 

 

85

2 2 . .
reason that the use of Ix and In, as the base variables produces good results is that the

relationship between 1

mm and these variables is flatter than when 1111 and 11“] are used as

the basic variables.

To sum up, using I? as basic variables in the FOSM method does not yield exact
results for 1-D problems while the use of 1111 does yield exact results. However, for 1-D and
2-D problems, the use of 11.2 as basic variables yields estimates that are under-predicted by

consistent factors.

4.7.3 F OSM method for 3-D problems

For 3-D problems, the maximum shear stress, 1mm, is a function of all six Cartesian
stresses. The Six stress components Ix, I), 12, TX)" In and I): are denoted by I 1’ 12, 13, T4, 15

and 1:6, respectively. The closed-form solution for 1: is too complicated to work with.

max

However, I

. may be expressed as
”“11

T —T
—"‘ ”3 (4.115)

Tmax — 2

where “cpl and Tp3 are the major and minor principal stresses, respectively, and are the

largest and smallest eigenvalues of the matrix

r-

7
T1 T4 T5

.14 12 16 (4.116)
115 T6 13—

It should be noted that since the input earthquake excitation is assumed to be a zero-

 

 

me an Gaussian process, the expected values of all Cartesian stresses are zero. However, as

with the 1-D and 2-D cases, the expected value of 1mm is not zero.

4-7 -3.1 Using Itil as basic variables

For this case, the maximum shear stress is expressed as

Ipl_Tp3
1mm = F(|‘C]|, |12|, |r3|, |t4|, |15|, [16]) = —2— (4.117)

86

The FOSM method yields the expected value of I as

max

E[ ]~F( —————— j—TIL—_I—'_’3 4118
Tmax ‘~ 111117211131114111511161 " 2 ( ° )

where Ip1 and I p3 are the largest and smallest eigenvalues of

 

 

 

 

 

”_ _ _T
111111411151
M [I] lit—6| (4.119)
JISll‘bllt3 L
The FOSM estimate of the variance of Immis
6 31? 2
Var ”mm-1 = 2,1411-) Var [IE-ll
1:
6 6 i I a (4.120)
8F F 1
'11]; 21'? _aT-J]_COV [ITiI’ ‘ITJ-l]
‘ ,2; 111 17,1
in which the variance is (see Section 4.7.2.3)
Var[|Ii|] = E[1:f]— (E[|Ii|])2 = (xii—£0 f _ 0.363403 (4.121)

and the covariance is (see equation (4.101))

COVHTil’ l’tjll E[|Ti|ltj|] —E[ltii]E[lTJ-|]

. -1 . —1 2 2
201,01,(C05(5'“ (pij))+P,-,-Sln (Pull/“nﬂgcmﬂzcg (4.122)

20 6 . —1 . —1 )
7t ’(cos(sm (pij))+pijs1n (pip—1

 

where pi]. is the correlation coefﬁcient between Ti and 15].; and the derivative terms can be
de‘ie mined by
8F _ 1(81121 81.123

am _ 2 a—lﬁ—a—li-l] fori=1,...,6 (4.123)

87

The derivatives of “CPI and Tp3 with respect to W can be determined from the characteristic

equation governing the eigenvalue problem. II) I and TM are the largest and smallest roots of

 

rIl -}. 1'4 1'5 1
I4 IZ—k T6 = (II —2.) (132—2t) (13—71.) +2141516
(4.124)
L 1'5 I6 T3—2:

 

 

 

4: (1'2— X) 42(11 —— k) 4: (I3 — 2») = O

Denoting 11,1 by al., i = 1, 6, for all ‘ti 2 0, equation (4.124) becomes

3 2
g(2\.,(11,(12,(13,a4,(15,(16) = 2» —(0t1+0t2+0t3)l

+(0t10t2+0t10t3+a2a3—aé—az—ai)?» (4.125)

2 2 2
— ala2a3 — 2040(5a6 + 0150:2 + a6a1+ a40i3 = 0
Equations (4.124) and (4.125) are equivalent only for all 11.2 0 , but this is sufficient since

the partial derivative in equation (4.120) are always evaluated at positive values of the

BI
variables. —pl and 5L3 required in equation (4.123) can be computed by treating k as a

809. oci
function of (1,, (12, ..., (16, taking the derivative of equation (4.125) with respect to al.,

solving for i and evaluating it at A = Ipl and 2t. = IP3, respectively.

8a.

1

The total derivative of equation (4.125) is

6
dg = 2 (fﬁingajﬂaa. = 0 (4.126)

Since equation (4.126) must be true for arbitrary aai, this yields

53% Eli _ -_
828013800. — 0, l— l, ..., 6 (4.127)
ax dg/dai

Z) a—(XI — —m, 1:], ..., 6 (4-128)

Thus,

88

 

BI a /80t.
P1__g__' ,1=I,...,6
8a,. Bg/dl k:
and
GI a /80t.
...L3 = _;g_—l , i=1, ..., 6
Bat. ag/dk k=1

 

The partial derivatives in equations (4.129) and (4.130) are:

2 2 2 2
31;: = 3?» —22\.((1]+(12+a3) +(Otlaz+0t10t3+(12a3—OtS—Otb—Ot4)

3_8
8a]

—(22— (0t2 + 0:3) 2+ (120t3 — 01(2))

L8
as,

2 2
—(7s —((11+(13)2\.+a1(13—(15)

is;

2 2
8&3 —(}\. — (0tl +0t2)}t+0110t2—oc4)

— - —2 (0%)» + a5a6 — Ot4Ot3)

4 - —2 ((152 + a40t6 — a5012)

and

—2 (016?» + was — (160(1)
437-32 Using 1,2 as basic variables

For this case, Tmax can be expressed as

-F 2 2 2 2 2 2
t — I], 12, T3, T4, Ts, T6

max

The FOSM method yields the expected value of I as

mar

‘2‘2‘2‘2‘2—2 Tpi—Tp3
E[Tmax] =F(T],TZ,T3,I4,TS,16) —

 

(4.129)

(4.130)

(4.131)

(4.132)

(4.133)

(4.134)

(4.135)

(4.136)

(4.137)

(4.138)

(4.139)

89

where I]; and 173 are the largest and smallest eigenvalues of

REE (4.140)

The FOSM estimate of the variance of I , is

"1th

 

 

7

6 -
Varleaxl z 2 —2

6 6
Varlrfl 2 2:;
i=1 317,-_ i: =

1J
I; i¢j

Cov [I33 I12] (4.141)

“'SJIN
l-NI:

 

 

 

T: T;

where Var[If] and Cov [112,112] were shown in equations (4.107) and (4.108),

respectively, and the derivative terms can be determined by

as? 2

l

 

BI 3:
BF— ][_”2_‘_ ”3] fori=l,...,6 (4-142)

2
31". 81:1.
The derivatives of IN and 1,73 with respect to 1".2 can be determined from the characteristic
e(ll—lattion governing the eigenvalue problem. Ip, and Tp3 are the largest and smallest roots

0f equation (4.124). Denoting It? by B 1., i = 1. 6, for all I12 0 , equation (4.124) becomes

“1,131,132, [33’ [34,135, Bo) = A3‘ijl3—I+a/B_2+(/B—3J7xz
W“ 82 +34 +BB2 3 --B5 136 134) (4.143)
—VBIBZB3_21::B566+B5@+66JB—1+B4JE=0

Again, equations (4. 143) and (4.124) are equivalent since the partial derivatives in equation

BIN
(4- l 41) are always evaluated at positive values of the variables. and — 3required in

3B 3B,-

e(ll—lemon (4.142) can be computed by treating K as a function of B 1’ [32, [36, taking the

derivative of equation (4.143) with respect to Bi, solving for g}: and evaluating it at

B.-

)k 2 — '
Ipl and A — Ip3, respectively.

90

The total derivative of equation (4.143) is

8h 81 811
d” E (@7813- BB,- lab“
Since equation (4.144) must be true for arbitrary 881., this yields

8h ax 8h

 

     

 

     

a—Xa—B—‘l'a—E =0,i=l,...,6

BA Bh/BBi . l 6

2) TB, — —ah/a}\,, l— ,...,
Thus,
ail. : —— l— l 6
913,. ail/ax L
and

.aig : —— i: 1 6
1913,. 311/31 ,_ ’

 

The partial derivatives in equations (4.147) and (4.148) are:

§¥,=312—2B(B+B2+JB3)
+(J7+B+J7+ B2133-B5—B6—B4)
313—1: Tﬂl HUB—2 J57)“ 5253-36)
53—52 37,—} -(JBT Mum B.)
59—53 -2—}_3( -+(JBT Mum—134)

J—sJB;
%;=_;,_ f 3

 

(4.144)

(4.145)

(4.146)

(4.147)

(4.148)

(4.149)

(4.150)

(4.151)

(4.152)

(4.153)

91

ah _ _ JEJE
53;— x J13: +ﬁ‘2 (4.154)
and
(4.155)

ah _ JBlJB—s
aB.-"“W*B

4.8 Simulation
The use of higher order terms in the Taylor series expansion in equation (4.84) would

be expected to increase the accuracy in the moment estimates of 1mm for the 3-D case.

However, such an analysis becomes intractable in practice. An alternative and more
reliable approach is to use Monte Carlo simulation. Numerous samples of Gaussian
Cartesian stresses consistent with a speciﬁed covariance matrix can be simulated, samples

of 1mm can be computed for each Cartesian sample set, and the mean and variance of the

samples can be computed. The Naval Surface Warfare Center mathematics

Tmax
SUbroutines library (NSWC 1993) provides a routine to generate multivariate normal

ran dom samples with a specified mean and covariance matrix. The density function of the
x ]T is ' b
., g1ven y

”I

mLIItivariate Gaussian distribution for {x} = [x,,

Jun—{111) [41"(1x1- {111)T) (4.156)

f(x) = 1 exp( 2
J(2n>”’der(1A1)

in Which [A] is the covariance matrix of the variates {x} and {u} is the mean vector. lln the

 

 

Case studied, {u}={0}.
A sufficient number of samples must be generated to obtain sufficient accuracy in the

mOl‘nent estimates. Figs. 4.5 and 4.6 show typical estimates of the mean and standard

deviation of 1mm, respectively, as the number of samples is increased. The figures indicate

that 8,000—10,000 samples are required to accurately estimate the first two moments of

T
ma.x‘

92

 

 

 

 

 

 

780 f 1 ' r ' 1 1 l
770
NA
£
:1
b 760
X
E
(a
“5
g 750
a)
2
740 ~ -
730 1 l 1 l 1 l 1 l 1 l
0 2000 4000 6000 8000 1 0000
Number of samples
Figure 4.5 Estimate of mean of Tm: for different number of samples
350 . 1 1 1 1 1 1 T r 4

340

l

 

Standard Deviation 0171111 (ft/lbz)

 

33o — \/\«

 

 

320 ~ ~
300 1 l 1 l 1 1 1 1 1 l
0 2000 4000 6000 8000 10000

Number of samples

Figure 4.6 Estimate of standard deviation of Tm“ for different number of samples

12000

Probability Density Distribution

93

A typical histogram of 1mm based on 10,000 sample is shown in Fig. 4.7. Although
the probability density function of {max is a function of the covariance matrix of the

Cartesian stresses It at the point under consideration, typically about 99.55% of the samples

lie below the mean plus three standard deviations of 1mm.

0-25 1 1

 

2
0.2 . 517.11.]: 745 mm

a,” = 328 mm2

0-15 *

(11*

0.05 *

 

 

0.0

 

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400
2
TM (ft/lb )

Figure 4.7 Probability density distribution of maximum shear stress

5. STOCHASTIC RESPONSE OF THE SANTA
FELICIA DAM — CASE STUDY

5.1 Ground Motion Model

The ground motion model shown in equation (4.2) is based on the analysis of
recordings made by the SMART-1 seismograph array in Lotung, Taiwan. The model which
considers the spatial as well as the temporal variation of earthquake ground motion is used
in this study. The parameters of the auto spectral density function (SDF) S“.g were
estimated from the El Centro earthquake record by fitting the function expressed in
equation (4.5) to observed acceleration spectra. Fig. 5.1 shows the normalized

autospectrum and the fitted model. The parameters (0g, Cg, (of and wahich control the shape

TABLE 5.1 Double-ﬁlter Autospectrum Parameters

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

wg(radls) Cg cof(rad/s) Cf
15.0 0.55 3.0 0.60
100 . T I r 1 1 1 I
; — Estimated
5 : —- Fitted
LL. 2 \
a 10" - I \ -
o E l \
g 5 I \
1: Z l ‘ \
0 l
.E 2 . l \ v
-2 ,_ ..
O 10 :l \\\\
Z i, \ ‘
5 " \\4
2" 3
2 u-
1
10-3 1 L l 1 l 1 I 1 l 1 l 1 l 1 l
O 2 4 6 8 10 12 14
Frequency (Hz)

Figure 5.1 Fitted Autospectra for Ground Motion Accelerogram

94

95

of the spectra are listed in Table 5.1. In the auto SDF, the intensity parameter So was
adjusted such that the standard deviation of the ground acceleration is 0.09g corresponding

to a peak acceleration of about 0.27g.

For the coherency function l'y(v,(o)l in equation (4.3), the five parameters used in the
analysis are listed in Table 5.2. These parameters correspond to Event 20 recorded by the
SMART-l array. In general, the coherency decreases with increasing separation and
increasing frequency, as shown in Fig. 5.2 and Fig. 5.3. To account for the wave passage
effect, an exponential function, exp(-i(1)v/V), is used, in which v is the separation distance
and V is the apparent wave velocity. Since the Santa Felicia dam is located on bedrock, an
apparent wave velocity of 4,267 m/s corresponding to waves in the bedrock propagating at
2,134 m/s with an incident angle of 60° to the horizontal was assumed. An incident angle
of 60° was found to be critical by Dakoulas and Hashmi (1992). This angle is used to
determine the delay due to wave propagation for the nodes along inclined base surfaces.
Unless noted otherwise, the waves are assumed to propagate in the upstream/downstream
direction (x-direction). Consequently, the delay time between two locations was computed
as (vxcose + vysinB) /Ve, in which V9 is the actual wave propagation velocity in the
bedrock, 0 is the incident angle to the horizontal, and vx and vy are separations between two

nodes in the x and y directions, respectively.

In order to obtain a physical feel for accelerograms that are consistent with the
prescribed SDF, coherency function and propagation velocity, stationary accelerogram
segments at the extreme downstream and upstream comers and the middle of the base along

the axis of symmetry were simulated. These three accelerograms are shown in Fig. 5.4.

TABLE 5.2 Coherency Function Parameters

 

A a k (m) 0),, (rad/S) b

 

 

0.636 0.0186 31,200 9.49 2.95

 

 

 

 

 

 

 

Coherency

Coherency

0.1

0.0

0.9

0.8

0.7

0.5

0.4

0.3

0.1

0.0

96

 

0.2

 

.—

l 1 l 1 l

 

 

 

 

 

4 5 6
Frequency(Hz)

Figure 5.2 Coherency vs. frequency

 

0.6

0.2

I

 

1

~

—- f=lI-Iz
------ f=3Hz
f=SHz
°—° f=7Hz
—- f=9Hz

 

 

'-
o
.......
o
n—
'—o—.._--

 

 

100

Figure 5.3

200
Separation (m)

Coherency vs. separation

300

400

Acceleration (g)

Acceleration (g)

-0.3
-0.4

'05 1 l 1 l J;
O

Acceleration (g)

97

 

0.5 V r V r v I *7

0.4 1-

I

0.3

0.2

3.11 1.11 11.11

I I

I '7‘

 

l

r ' 1 ' , T
Downstream Side

11111111111111

*"111' 111 1111 1111 111 1‘11‘11‘1'1'11

-0.2 -

p—
I-

4 1 I

 

 

"’1'

4

S 6 7 8

Time (second)

 

0.5 7 I V I v r T

I F

0.4

0.3

1.1111111. 11

I I V

I

l

-0.3

I

-O.4 r-

 

)—

' Mi'ddleTof 111}: bas'e

b
b

l 1 l

. , , , 1 11..“11l11.111 ‘11.)
-01. 11'1”“1wuw'l'nwuql‘n' TIT!

-0.2

 

 

-o.5-111- 41
012 3 4

5 6 7 8

Time (second)

10

 

0.5 ' I V I V I r

0.4

0.3

0.2

fIﬁIrI'

0.1

 

7"

0.0

a

-0.1

I

-02 ..

-0.3

V I I

-0.4

' I

 

T ' 1 . Y
Upstream s1de

1111 1111111111 .1111
“1111‘111‘11'1 M1111H111H'1111N'9'11

1

 

{1 Ln

 

 

 

 

_05 1 1 1 1 1 l 1 _L
0 1 2 3 4

1 1 1 1 4 1 1
7

5 6 8

Time (second)

Figure 5.4 Accelerograms at downstream side, middle of the base and upstream side

98

5.2 Computation Steps

A series of computer programs were written to perform the stochastic response

analysis of the earth dam. The basic steps of the analysis are summarized below:

1.

3.

In equation (4.17), mode shapes {(1)} and modal mass Mj as well as the mass matrix
[M] and the static constraint matrix [A] are required in computing the participation
factor matrix [F]. {(1)} and Mj can be obtained directly from regular output in
I-DEAS, but [M] and [A] need to be generated by special techniques which are
described in Appendix A.5.

Using equation (4.18), the participation factor matrix [F] was computed. [1"] is of
order r x n , where r = number of restrained DOFs with excitation and n = number
of modes (for the model used, r = 288 and n = 80). The static constraint matrix [A]
has a size of N x r, in which each column represents the deformation shape due to
unit displacement at a restrained node, and N = total number of DOFs in the model

(for the model used, N = 3012 and r = 288).

The lower triangular part of the integration matrix of equation (4.38) was computed
by using the numerical subroutine DQAGI in the NSWC mathematics library, and

stored in a 1-D array oforder n (n + 1) /2.

. l . .
The integrals F 71512 11 ((1)) do) were computed and stored 1n an array of srze
.01; RI Rm

r(r+1)/2. The integrals r

—00

15H . ( (11)) S .. ._ ((0)610) were computed and stored in
J uRluRm

an array of size n x r. It should be mentioned that the calculations of the

integrations using the adaptive algorithms in DQAGI are extremely time-

consuming. The approximate elapsed time using an HP Model 715/50 workstation

is shown in Table 5.3.

99

TABLE 5.3 Approximate elapsed time for calculation of integration

 

Static Dynamic Static-Dynamic

 

 

15 minutes 21 days 16 hours

 

 

 

 

 

. The variance of the total displacement responses were computed using equations
(4.37), (4.42) and (4.49) for each degree of freedom. The three displacement

components in the x, y and 2 directions were computed at each node.

. For 3-D problem, six Cartesian stresses and strains were computed and the
covariance matrix for each node is of order 6 by 6. Because of symmetry, only 21
components in the matrix were calculated. The total covariance matrix of stresses
(strains) was computed using equations (4.55), (4.57) and (4.58) for each node, in
which the integration values are identical to the results computed in step 4. The
stress mode shape [‘P] is of order N X n (for the model used, N = 6024 and n = 80).
The static constraint matrix [n] has a size of N X r (for the model used, N = 6024

and r = 288).

. Using the covariance matrices for stresses and strains computed in step 6, 10,000
normal random samples were simulated for the six Cartesian stresses and strains at
each node. 10,000 values of the maximum shear stress and strain were computed at
each node using the Cartesian stresses and strains, and the u+30’ values were

estimated from the simulated data.

. The “+30 of maximum shear stress were also estimated using the FOSM method

w1th Ti 1n add1t10n to With 11:11 as the ba51c var1ables (see Sectlon 4.7.3).

. Mohr-Coulomb strength criterion is employed to determine the shear strength of
soils. By considering gravity and earthquake loads, reliability analysis was

performed.

100

5.3 Numerical Results

Finite element analysis yields a vast amount of results, and a good post—processor is
essential to suitably display relevant quantities. Fortunately, the excellent post-processor in
I—DEAS can be utilized to display response quantities computed with customized programs
for random vibration analysis. The techniques to display response contours are described
in Appendix A.6. Shaded image contours of the displacement, stress and strain responses
are displayed over the entire dam in this section. All responses should be symmetric about
the middle axis of the dam and any observable asymmetry is due to the variability in the
simulated data on either side of the axis of symmetry. The origin of the coordinate system

used in the study is located at the mid—point of the base in the core clay region.

5.3.1 Responses to general excitation

This section shows the dam responses to general SVEGM. Both correlation and wave

propagation effects were included for the base excitations.
5.3.1.1 Displacement

Since the excitations are assumed to be zero-mean Gaussian processes, the mean
value of the displacement response must be zero. The variance of the total x-, y- and
z-displacements at each node was computed using equation (4.26). The standard deviation
(0) is taken to be the positive square root of the variance. Fig. 5.5 shows four different
views of the 30’ x-displacement response contours so that the displacements on the dam
surface and within the dam can be easily visualized. The scale for the contour is shown on
the right edge of each picture. The darker contour shades signify larger response values.
The figures show that the maximum x-displacement occurs at the middle of the crest, with
a 30' value of 191.8 mm (0.630 ft). The displacements decrease with increasing distance

from the mid-point of the crest.

In 3-D analysis, the total displacement response consists of the x, y and 2 components.

Fig. 5.6 shows the 3G contours for each displacement component. As expected, the

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103

x-displacement dominates the total displacement response. Since the magnitudes of the y
and 2 components are very small in comparison to the x-displacement, it is reasonable to

neglect them.
5.3.1.2 Stress

As with the displacement, for zero-mean excitations, the mean of the Cartesian
stresses is zero. Figs. 5.7 (a), (b) and (c) show the 30' contours of the stresses, Ix, Txy and
In, respectively at the base where they are critical. These three components are the most
significant. The stress component Try is mainly caused by bending effects. In at the base is
primarily caused by the spatial variation of the base excitation. According to the Mohr-
Column failure criterion, the failure of soil material is related to the maximum shear stress,

I and the soil properties. The mean plus three standard deviation (u+3o) contours of

max’
Im at the base are shown in Fig. 5.7(d). It should be noted that the high values of stresses
occurring on the upstream and downstream sides are due to the artiﬁcial cut-off boundary
and should be disregarded. Excluding responses near the boundary, Table 5.4 lists the

maximum u+3o nodal stress for each component. A maximum shear stress of 1,670 kPa

(34,869 lb/ftz) occurs at nodes 138 and 274 where Txy is maximum.

TABLE 5.4 Maximum u+30 nodal stress response for each component

 

 

 

 

 

 

 

 

 

Component Nodes Node Location (x,y,z) (ft) Stress (kPa)

Ix 102, 241 247.7, 0.0, i150.0 3,222

Iv 132, 268 289.5, 0.0, i172.5 772

IZ 102, 241 247.7, 0.0, 11500 1,005

In, 138, 274 —-l60.2, 0.0, i121.8 1,467

In 105, 244 176.3, 0.0, i225.0 1,187

IY2 138, 274 -160.2, 0.0, i121.8 394
Tmax 138, 274 —160.2, 0.0, i121.8 1,670

 

 

 

 

 

 

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1:. . 1:... , y. .2 . 2x
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2 2 ...: .2... 2 2 2 . 2.; 1 _. .2
2122 . 1 2.. 11 21.2.. 1.1 1 1.. 1.
.11 2 12 x as, (2 11.1 1 x p. ..2

 

 

 

l 05
5.3.1.3 Strain

Although stresses are critical at the base, strains are critical at the top free surface of
the dam because the material at the top is not as stiff as the material of the base. The “+30
values of the three dominant strain components, 7x, ny and sz’ and the maximum shear
strain 7mm at the top surface of the dam are shown in Figs. 5.8 (a), (b), (c) and (d),
respectively. In Fig. 5.8(a), the high values of 7x along the side boundaries are due to ground
straining. The contour pattern of 7mm is similar to that of ny’ which indicates that y”
contributes most to 7mm. The critical shear strain responses occur on both surfaces of the
upstream and downstream sides. Table 5.4 lists the maximum u+3o nodal strain for each
component. It is shown that ny dominates the strain response, which gives most

contributions to 7mm.

5.3.2 Modal contributions

The dynamic response variances consist of individual modal response variances and
covariances between pairs of modal responses as indicated in equation (4.37). The relative
modal contributions to the x-displacement are examined in detail at the middle of the dam

crest (which has the maximum displacement). Table 5.6 shows the percent contribution of

TABLE 5.5 Maximum u+3o nodal strain response for each component

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Component Nodes Node Location (x,y,z) (ft) Strain
yx 336 —212.5, 187.5, 0.0 9.4008x10‘4
y), 644, 977 147.7, 125.0, :64] 4.9636x10‘4
y, 438 195.0, 1850,00 5.1313x10‘4
yr), 591, 924 129.1, 176.1, i392 2.2613x10‘3
7,, 504, 837 —125.3, 226.1, :3490 9.8622x10“4
7),, 627,960 —203.3, 180.8, i187.0 6.2395x10'4
7m 591, 924 129.1, 176.1, i392 1.9340x10‘3

 

106

 

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107

the model variances (diagonal terms) and covariance (off-diagonal terms). The variances
(diagonal elements in Table 5.6) of modes 1, 6 and 16 and the covariances between modes
1 & 6 and 15 & 16 contribute more than 1% to the dynamic displacement. However, the

mode 1 variance dominates, with a contribution of 96%.

The relative modal contributions to the x-displacement are also examined at node 329
(x=-15 ft, y=275 ft, 2:354 ft) which is located approximately at quarter length along the
crest. Table 5.6 shows the percent contribution of the modal variances and covariances. The
variances of modes 1, 3, 6, 9, 13 and 19 and the covariances between modes 1 & 6 and 6 &
l9 contribute more than 1% to the dynamic displacement. At this node, modes 1 and 6
dominate with contributions of 34% and 40%, respectively. While only symmetric modes
contributed to the displacement at the middle of the crest, but symmetric and antisymmetric

modes contribute to the displacement at the quarter-length locations.

The In, component at node 275 (x=-167 ft, y=O ft, z=-57.9 ft), which has the
maximum dynamic variance, is selected for a detailed examination of the relative modal
contributions. Table 5.6 shows the percent contribution of the modal variance and
covariance. The variances of modes 1, 6, 10 and 16 and the covariances between modes 1
& 6, 1 & 10 and 10 & 16 contribute more than 1% to the dynamic TX), at this node. As with
the displacement at the mid-crest, mode 1 dominates with a contribution of 94%. Basically,
symmetric modes give some contribution to the dynamic Ix), while anti-symmetric modes
do not contribute at all at this node. Anti—symmetric modes may of course have larger

contributions at other nodes.

The largest dynamic variance of 7n, occurs at node 441 (x=181 ft, y=108 ft, 2:0 ft).
The percent modal contributions to the total dynamic response at this node are listed in
Table 5.6. The variances of modes 1, 6 and 10 and the covariances between modes 1 & 6,

1 & 10 and 6 & 10 contribute more than 1% to the response. Again, mode 1 dominates with

108

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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a contribution of 98%. Detailed examination of modal contributions indicates that a

dynamic analysis using only the ﬁrst mode should be sufficient in preliminary design.

5.3.3 Responses to simplified excitations

In conventional seismic analysis of earth dams, simplified ground motion models are
typically used. In order to assess their adequacy, three simplified ground motion models are

considered in addition to general SVEGM model.

0 Case I: Propagation only without coherency decay. Correlation effects between
support excitations is neglected in the case, i.e. Iy(v, (1))I = 1 . However, the time

delay for seismic waves to propagate from one support to another is considered.

0 Case II: Identical ground motion. This case considers the earthquake excitation to
be identical at all support points, an assumption often used in practice. For this case,

—'(0v/V .
17W, (1)) Ie ' 1s equal to one.

0 Case III: incoherency only without wave passage effect. For this case the wave

—i(0v/V _

propagation effect is neglected and hence e 1 .

For cases I and II, Harichandran (1991) developed closed-form solutions for the
integrals in equations (4.38), (4.42) and (4.49). Use of closed-form solutions instead of
numerical integration result in a tremendous saving of computation time, especially for the
dynamic response. Computation of the dynamic response using the closed-form solutions
took only about 6 hours on a HP 715/50 workstation compared to about 21 days when

numerical integration was used.

5.3.3.1 Displacement

Fig. 5.9(a) shows 36 x-displacement contours for the general SVEGM model, while
Figs. 5.9 (b), (c) and (d) show the contours for the simpliﬁed excitation cases: propagation
only, identical and incoherency only, respectively. The x—displacement contours for the
four types of excitation are similar to one another. However, the maximum x-displacement

response to identical excitation is 8.1% larger than that due to general SVEGM. The critical

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responses for these four cases are listed in Table 5.10. The figure indicates that for the
assumed seismic wave speed, the wave propagation effect is not very signiﬁcant. The effect
of incoherent ground motion slightly decreases the displacement response. The use of
identical excitations (typical practice) appears to be adequate and conservative as far as

predicting displacements.
5.3.3.2 Maximum shear strain

The 114-30' maximum shear strain responses are depicted in Figs. 5.10. (a), (b), (c) and
(d) due to cases of general excitation, wave propagation only, identical excitation and
incoherency only, respectively. There is no significant difference among these contour
patterns, and the critical values are listed in Table 5.10. The effect of SVEGM slightly
lowers the maximum shear strain response. The wave passage effect does not affect 7mm
significantly.

5.3.3.3 Maximum shear stress

The u+3o contours of 1mm response are shown in Fig. 5.11 in which figures (a), (b),
(c) and (d) are for the cases of general, propagation only, identical and incoherency only
excitations, respectively. A comparison of figures (a) & (d) with (b) & (c) indicates that
incoherency of the base excitations has a very significant effect on the magnitude and
distribution of 1mm at the base. The critical response for each case, excluding that on the

cut-off boundaries, is also listed in Table 5.10 and indicates that the traditional assumption

TABLE 5.10 Comparison of maximum “+36 response for displacement, maximum shear stress
and maximum shear strain for four different excitation models

 

 

 

 

 

 

 

 

 

 

 

Case x-displacement (mm) 1m (kPa) 7m“
General Excitation 191.8 1,670 1.934x10"3
Propagation Only 207.2 51 l 2. 127x 1 0—3
Identical Excitation 207.4 498 2. l 33 X10—3
Incoherency Only 195.0 1669.9 1 .975X10—3

 

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117

of uniform earthquake ground motion significantly underestimates the stress response. The
response of rm“ to general SVEGM is 3.35 times larger than that due to identical
excitation. This is because the stress component, Ty, signiﬁcantly increases due to the base

shearing effect caused by SVEGM.

The contours in Fig. 5.1 l were generated using 10,000 simulations of 1mm at each
node consistent with the covariance matrix of the Cartesian stresses computed by random
vibration analysis. In addition, the mean-value FOSM method was also employed to
estimate the p+3o contours of 1mm. The main advantage of using the FOSM method is that
the elapsed computing time was only about 4 seconds on a HP Model 715/50 workstation,
while simulating 10,000 samples and computing moments required about 21 minutes. The
FOSM estimates using l'cil as the basic variables (see Section 4.7.3.1) for the four excitation
cases are displayed in Fig. 5.12. A comparison with Fig. 5.11 indicates that the FOSM
estimates are clearly unacceptable. The poor estimates are due to the derivative terms
evaluated at the mean values of all variables in equation (4.120) being unexpectedly high
at some nodes, as a result of which the variances of tmax at these nodes deviate excessively
from simulation results. A similar observation was made for the illustrative 2-D beam
bending problem in Section 4.7.2.3. Therefore, it may be concluded that the accuracy of the
FOSM estimate for the standard deviations of T can be poor and ﬂuctuating if 1111 are

mar

used as the basic variables.

The FOSM approach with If as the basic variables was presented in Section 4.7.3.2.
The u+3o 1mm responses computed for this case are shown in Fig. 5.13, in which ﬁgures
(a), (b), (c) and (d) are contours for the general, delayed, identical and incoherency only
excitations, respectively. The contour patterns are now closer to those in Fig. 5.11,
especially for cases yielding higher stresses. Thus, the use of 11.2 as the basic variables in
the FOSM approach is better than using Itil as the basic variables. The differences between

critical u+3o estimates of 1mm using the simulation and FOSM approach with 1".2 as the

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TABLE 5.1] Comparison of critical p.+3o estimates of rm (kPa) using simulation and FOSM
approaches with If as the basic variables

 

 

 

 

 

 

Case Simulation FOSM Difference
General Excitation 1,670 1,812 8.5%
Propagation Only 51 1 623 22%
Identical Excitation 498 605 2 I %
Incoherency Only 1,670 1,812 8.5%

 

 

 

 

 

 

basic variables are compared in Table 5.1 l. A difference of 8.5% occurs for the general

excitation case while a larger error of 21% occurs for the identical excitation case.

Figs. 5.14 (a) and (b) show the u+3o contours of 7mm at the top free surface using the
FOSM approach with Itil as the basic variables for the general and identical excitations,
respectively. As with the 1mm response, the FOSM estimates are clearly unacceptable.
However, Figs. 5.14 (c) and (d) show the 7mm response using the FOSM approach with 11.2
as the basic variables for the general and identical excitations, respectively. The contour
patterns are quite similar to those using simulation samples as shown in Figs. 5.10 (a) and
(c). Therefore, the FOSM method with 11.2 as the basic variables can be used to estimate

statistical moments of maximum shear stress and strain.

5.3.4 Correction factors for the FOSM method

The linearization of the non-linear function of Tmax using the mean-value FOSM
method can be efficient and have acceptable accuracy if calibrated correction factors are
used. By comparing the 1mm responses over the entire dam estimated by the FOSM method
with those obtained using simulation, correction factors CF“ and CFO as defined in
equations (4.113) and (4.114) can be determined for the mean and standard deviation of

T respectively, at each node. The statistics of these correction factors are shown in

max’

Table 5.12. The two cases of general and identical excitations are selected to illustrate the

accuracy of the FOSM method using 11,1 as well as If as the basic variables. In addition to

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the mean and standard deviation of the CFs, the coefficients of variation (c.o.v.) are also

shown in Table 5.12. The c.o.v. of a random variable x is defined as

- gmx 100 (%)

C.O.V. — ELY]

(5.1)

The higher the coefficient of variation, the greater the scatter. The maximum and minimum

values of the CFs for each case are also given in Table 5.12.

Table 5.12 shows that CFO computed by the FOSM method with I} as the basic
variables are more consistent (i.e., have less scatter) than those obtained by the method with
nil as the basic variables for both excitation cases. For the identical excitation case, the
FOSM method using I} as the basic variables yields a c.o.v. for CFO only 3.3%. However,
the FOSM method using Itil as the basic variables yields a c.o.v. as large as 47% for the
general excitation case. For CF“, however, the two FOSM approaches yield the same c.o.v.
for both excitation cases. Therefore, if 1.".2 are used as the basic variables, then it is possible
to arrive at a CF11 and CFO that can be used to correct estimates of the mean and standard

deviation, respectively, computed using the FOSM method.

 

TABLE 5.12 Statistics of the correction factors CF11 and CF<I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

General Excitation Identical Excitation
Statistics Basic variables Basic variables Basic variables Basic variables
of I‘L'il 1,2 1131.1 1,2
CFs

CFl1 CFO CF“ CFO CFLl CFO CF" CFO
Mean 1.223 0.526 0.976 0.856 1.076 0.644 0.858 0.826
Std. Dev. 0.075 0.247 0.060 0.072 0.052 0.191 0.041 0.027

Coefﬁcient of 6.1 47.0 6.1 8.4 4.8 29.7 4.8 3.3

Variation (%)

Maximum 1.476 1.337 1.178 1.095 1.408 2.289 1.124 1.158
Minimum 1.059 0.006 0.845 0.757 0.994 0.006 0.793 0.777

 

 

 

123

Fig. 5.15 show the u+3o 1mm contours at the base using the corrected FOSM method
with It? as the basic variables compared with those using simulation results. Figs. 5.15(a)
and (b) show the 1mm responses due to general excitation for the simulation and corrected
FOSM methods, respectively. Correction factors of CFH=1.04 and CFO=0.91 were used to
correct (multiply) the estimated mean and standard deviation of 1mm at each node. The
difference between the maximum 1' using the corrected FOSM method and simulation

mat

samples is only 4%.

Figs. 5.15(c) and ((1) show the 1mm responses due to identical excitation for the
simulation and corrected FOSM methods, respectively. Correction factors of CFH=0.86 and
CFO=0.83 were used. The 1mm contours using the corrected FOSM method and simulation

results are almost identical, and the difference in the maximum 1mm value is less than 0.5%.

In summary, the use of approximate correction factors for the mean and standard
deviation estimated by the FOSM method with If as the basic variables can yield fairly
accurate results. The corrected results are considerably more accurate for the identical
excitation model than for the general excitation model. The correction factors must be
arrived at through a calibration process which uses simulation, which partially offsets the
computational advantage of using the FOSM method. However, if several analyses are to
be performed, then the calibration process and the subsequent use of the corrected FOSM
method may be justified. Owing to its better accuracy, the simulation approach was used to

generate all contour presented in this study hereafter.

5.3.5 Response details over the dam

Although the critical values of 1mm occur at the base and the critical values of 'ymax
occur at the top free surface as shown in previous sections, in order to have global insight
on the responses of the dam, Tmar at the top free surface and on the cutting plane at z = 0

and y at the base are also depicted in this section.
mar

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Figs. 5.16 (a), (b), (c) and ((1) display the u+3o contours of 1mm at the top free surface
for the cases of general, delayed, identical and incoherency only excitations, respectively.
All ﬁgures are shown on the same scale. The dark regions which signify high “cm“ on both
upstream and downstream sides are due to the artificial cut-off boundaries and should be

disregarded. At the top free surface, the 1: response due to identical and delayed

m at

excitations are larger than those due to the other two cases.

Figs. 5.17(a), (b), (c) and ((1) display the u+30 1mm contours on the cutting XY plane
at z = 0 for the cases of general, identical, delayed and incoherency only excitations,
respectively. Identical excitation case yields the critical 1mm in the core, while the general
excitation yields the critical 1mm in the gravel streambed. Table 5.13 shows the critical
u+3o values of 1mm in the core, shell and gravel material for all these cases and indicates
that the traditional assumption of identical earthquake excitation overestimates the 1mm,
response in the core and shells by about 10% and 8%, respectively. The wave passage effect
slightly lowers the 1mm response in the core, while it does not affect the Tmax response in

the shell signiﬁcantly.

Figs. 5.18 (a), (b), (c) and ((1) display the u+3o contours of ymax at the base for the
cases of general, delayed, identical and incoherency only excitations, respectively. The
contour patterns have no significant difference among these cases. High 7mm values occur

in ﬂexible material regions, i.e., core and shell. However, shear stresses are related to shear

TABLE 5.13 Critical u+3o values of “cm (kPa) in different material zones

 

 

 

 

 

 

Case Core Shell Gravel
General Excitation 373.8 161.9 1,670
Delayed Excitation 405.5 174.6 511
Identical Excitation 410.6 174.1 498
Incoherency Only 380.5 164.2 1,670

 

 

 

 

 

 

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; all. : ....x 1 .rl... .1 a ....A. .....x .. ...l‘ L. .
. .. .. ... .. .. . . ..:...

 

 

129

strains through the shear the shear moduli of the materials, and the extremely high values

of 13mm that occur at the base is due to the very high shear modulus of the gravel.

5.3.6 Contribution of response components to total response

As indicated in equation (4.26), the variance of the total response consists of three
components: the variance of the pseudo-static response, the variance of the dynamic
response, and the covariance between the pseudo-static and dynamic responses. It is

instructive to deduce which component gives most contributions to the total response.

5.3.6.1 Displacement

The maximum variance of 15 displacement responses in the upstream-downstream
direction are extracted to show the contributions of the response components. The 15 nodes
having the largest responses are the same for the three cases: general excitation, identical
excitation and delayed excitation. The coordinates of these nodes are listed in Table 5.14.
The percentage contribution of the three response components to the total variance at these
nodes are shown in Table 5.15. Clearly, the dynamic response component contributes the
most for all the cases but the other components are also significant. It should be noted that
for identical excitation, the pseudo-static response consists of a pure rigid body translation
of the dam and the static displacement component at all nodes is identical to the ground

displacement.

TABLE 5.14 Coordinates of the nodes (ft) at which the 15 largest
x-displacement responses occur

 

Node x y 2 Node x y 2 Node x y z

 

 

29] -15 275 0 333 -15 275 71 692 -15 275 -l42

 

299 15 275 0 338 -81 246 O 332 -15 275 142

 

732 15 275 -7l 436 75 245 O 963 67 249 -95

 

382 15 275 71 73] 15 275 - l 42 630 67 249 95

 

693 ~15 275 -71 381 15 275 142 816 -61 254 -120

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

130

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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5.3.6.2 Strain

The strain component 7”, is selected to show the contribution of the response
components to the total variance. The coordinates of the nodes at which the 15 largest ny
response occur for the two cases of general and delayed excitations are listed in Table 5.16.
The maximum responses occur near the top free surface of the dam for both cases. The
percentage contribution of the response components are shown in Table 5.17 and indicate
that the dynamic response is dominant for both cases. The pseudo-static and covariance

components can be neglected when estimating y” at the free surface.
5.3.6.3 Stress

The stress component TX), is selected to show the contribution of response components
to the total variance. Since the largest variance of TX), occurs at different locations for
different excitation models, the nodes at which the 15 largest 13x), responses for the two cases
of general and delayed excitations are listed separately in Table 5.18. The percentage
contribution of the three response components at these nodes are listed in Table 5.19 for
each excitation case. For the general excitation the static component contributes the most
to the total response, although the dynamic component is also signiﬁcant at some nodes.
For delayed excitations with an apparent wave speed of 4.27 km/s, the static response is not
significant, but the covariance between the static and dynamic responses becomes
significant. It should be noted that for identical excitation the static stress component is zero
since this consists of a pure rigid body translation of the dam. As a result, the covariance
between the static and dynamic components is also zero, and the total stress response

consists of only the dynamic component.

In addition, it is of interest to examine the contribution of the response components
to the total response at the critical point in the core. Node 396 (0 ft, 125 ft, 0 ft) was found
to have the maximum u+3o 1mm value. Table 5.20 shows the percentage contribution of

response components to the total variance of the shear stress In, at node 396 and indicates

132

TABLE 5.16 Nodal coordinates (ft) for the largest 15 variances of ny

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sorted General Excitation Delayed Excitation
Order Node x y 2 Node x y z
1 591 129 176 39 352 -104 150 0
2 924 129 176 -39 353 — 161 125 0
3 441 181 108 441 181 108 O
4 440 114 142 440 114 142 0
5 352 -104 150 311 -99 195 0
6 353 -161 125 295 -48 175 0
7 977 148 125 -65 591 129 176 39
8 644 148 125 65 924 129 176 -39
9 437 135 215 857 -185 114 ~35
10 446 114 100 524 -185 114 35
11 418 91 195 446 114 100 0
12 857 -185 114 -35 856 -153 128 -57
13 524 -185 114 35 523 -153 128 57
14 417 151 165 977 148 125 -65
15 295 -48 175 644 148 125 65
TABLE 5.17 Percentage contribution of response components to the total variance
of the shear strain 7:9
Sorted General Excitation Delayed Excitation
Order Static Dynamic Covariance Static Dynamic Covariance
1 f 0.1 100.0 -0.1 jFPOD 100.1 -0.1
2 0.1 100.0 -0.1 0.0 100.5 -0.5
3 0.3 100.4 -0.8 0.0 99.4 0.6
4 0.1 100.1 -0.1 0.0 99.9 0.1
5 0.1 99.8 0.1 0.0 100.1 -0.1
6 0.3 99.4 0.3 0.0 100.0 0.0
7 0.2 100.3 -O.5 0.0 99.9 0.1
8 0.2 100.3 -0.5 0.0 99.9 0.1
9 0. 1 100.0 -0.1 0.0 100.8 -0.8
10 0.2 100.2 0.3 0.0 100.8 -0.8
1 l 0.0 100.0 0.0 0.0 99.8 0.2
L 12 0.4 99.1 0.5 0.0 100.4 -05
L 13 1.9 97.5 0.6 0.0 100.4 -0.5
l 14 0.1 100.1 -03 0.0 99.7 0.3
15 0.1 99.9 0.0 0.0 99.7 0.3

 

 

 

 

 

 

 

 

 

 

 

 

133

TABLE 5.18 Nodal coordinates (ft) for the largest 15 variances of ‘ny

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sorted General Excitation Delayed Excitation

order Node x y 2 Node x y z
1 274 - 160 0 -122 275 -167 0 -58
2 138 -160 0 122 139 -167 0 58
3 176 595 75 ~338 274 - 160 0 -122
4 26 595 75 338 138 -l60 0 122
5 284 -322 25 -263 110 -189 0 0
6 148 -322 25 263 276 -224 0 -47
7 177 595 0 —225 140 -224 0 47
8 28 595 0 225 270 176 O -44
9 150 -452 0 112.5 134 176 0 44
10 286 -452 0 -112.5 396 0 125 O
11 110 -189 0 0 273 -218 0 -103
12 40 -535 0 0 137 -218 0 103
13 245 -189 O -225 269 176 0 -113
14 107 -189 0 225 133 176 0 113
15 285 -452 50 -300 662 -228 38 0

 

 

 

 

 

 

 

 

 

 

TABLE 5.19 Percentage contribution of response components to the total variance
of the shear stress Txy

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sorted General Excitation Delayed Excitation

Order Static Dynamic Covariance Static Dynamic Covariance
1 89.8 12.3 -2.1 1.55 91.44 7.01
2 89.8 12.3 —2.1 1.55 91.44 7.01
3 100.0 0.0 0.0 2.82 87.40 9.78
4 100.0 0.0 0.0 2.82 87.40 9.78
5 98.0 1.9 0.2 1.03 93.51 5.46
6 98.0 1.9 0.2 0.72 94.62 4.66
7 99.9 0.1 0.0 0.72 94.62 4.66
8 99.9 0.1 0.0 1.73 106.20 -7.92
9 98.8 0.7 0.5 1.73 106.20 -7.92
10 98.8 0.7 0.5 0.00 99.92 0.08
11 74.1 28.9 -3.0 0.82 93.92 5.26
12 99.1 0.3 0.6 0.82 93.92 5.26
13 93.5 6.4 0.1 2.09 106.98 —9.08
14 93.5 6.4 0.1 2.09 106.98 -9.08
15 99.5 0.3 0.2 0.61 95.11 4.28

 

 

 

 

 

 

 

 

134

TABLE 5.20 Percentage contribution of response components to the total
variance of the shear stress In at the critical node in the core

 

 

 

General Excitation Delayed Excitation
Node
Static Dynamic Covariance Static Dynamic Covariance
396 0.4 99.7 -0.1 0.0 100.0 0.0

 

 

 

 

 

 

 

 

 

 

 

that for the general excitation the dynamic component contributes the most to the total
response, which is in contrast to the response at the base. However, for the delayed
excitation, the dynamic component is dominant for the critical response in the core as well

as at the base.

It should be noted that the contribution of the stress response components to the total
response also depends on the shear modulus of the material. Fig. 5.19 shows u+3o contours
of 1mm at the base. Figures (a) and (c) show the total response and (b) and (d) show the
static component, both due to general excitation. The difference between the total response
and static component would indicate the contribution of the dynamic component. The
contour patterns in ﬁgures (a) and (b) are very similar to each other, which indicates the
static component dominates the total response in the base gravel region. For figures (c) and
(d), the scales were adjusted so that the 1mm response in the core can be displayed. The
ﬁgures show that signiﬁcantly higher Tmax values occur in the clay core and shell for the
total response, which indicates the dynamic component is significant for the stress response

at the base for flexible material.

In addition, to examine the contribution of the stress response components inside the
dam, the 1m contours on the horizontal cutting plane at a height of 21.3 m (70 ft) (i.e., near
the top of the gravel streambed) are displayed in Fig. 5.20. Figures (a) and (c) show the total

response and (b) and ((1) show the static component. Figures (a) and (b) are similar in the

gravel region, which indicates that the static component dominates the total response

135

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137

through the gravel layer. However, in the core region the dynamic component is signiﬁcant

since ﬁgures (c) and ((1) show difference in this region.

The u+3o 1mm contours at a height of 38.1 m (125 ft) (figures (a) and (b)) and 61 m
(200 ft) (figures (c) and (d)) are shown in Fig. 5.21. Figures (a) and (b) are the total
responses and (b) and (d) are the static components. Clearly, at these layer heights, the
dynamic component is significant in the shell as well as in the core since the static

responses are much lower than the total responses.

To sum up, ground straining at the base due to SVEGM causes higher stresses in rigid
material like gravel, and the static component is dominant for these material. On the other
hand, flexible materials are more sensitive to the ground vibration, and hence the dynamic

component is more signiﬁcant for these material.

5.4 Reliability against Local Failure under Gravity and Earthquake Loads

5.4.1 Introduction

Since the main motivation of this study is to assess the effect of SVEGM, gravity
effects are generally neglected and only the seismic response of the dam is considered. This
section illustrate how gravity effects can be included in a more complete investigation.
Only one failure mode, local yielding governed by the Mohr-Coulomb criterion, is

considered in this illustration.

The Mohr-Coulomb failure criterion is usually expressed as

Is = c + I'tanq) (5.2)
in which c = cohesion, ¢ = friction angle, 1' = effective normal stress and '1:5 = shear strength
of the soil. If the shear stress due to loading exceeds the shear strength given by equation

(5.2), then the soil would fail locally. In terms of principal stresses, equation (5.2) can be

written as

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intermediate principal stress in 3-D problems is neglected by the Mohr—Coulomb criterion.

When the stresses 1] and T3 and/or the material parameters c and 4) are random, safety

against failure may be expressed by the safety margin failure criterion

T1+T Tl-

Y= 2 3sin¢+ccos¢— 2

 

 

(5.4)

Failure occurs when Y < O, and hence the probability of failure is pf = P [Y< O] . Rather
than using Pp a popular measure that is used to assess the reliability of engineering
structures is the reliability index deﬁned as

B = sly (5.5)
where Y and CY are the mean and standard deviation of Y, respectively. For a non-linear
failure criterion the magnitude of the reliability index is dependent on the form of the
failure criterion. For example, a failure criterion written in terms of the factor of safety can
yield a different B-value than a criterion written in terms of the safety margin. A
comprehensive reliability analysis should use the Hasofer-Lind reliability index (Madsen
et al. 1986), which is invariant with respect to different but mechanically equivalent
formulations of the failure criterion. However, to keep this illustration simple, the
reliability index based on the safety margin is used. For most civil engineering structures,

a reliability index of 3 is considered acceptable.

One method of assessing the safety against local failure in a finite element analysis of
earth dams is to compute the B-value at each node. The principal stresses 1'] and T3 should
include the effects of both gravity and seismic loads. In this illustration, the material
parameters c and q; are assumed to be deterministic (i.e., constant), and the values used for

the three material types are given in Table 5.21.

140

TABLE 5.21 Physical properties of the core, shell and gravel

 

 

 

 

Physical property Core Shell Gravel
Friction angle 20 36 42
Cohesion (kN/mz) 100 0.72 0.72

 

 

 

 

 

 

A high degree of compaction on the dam materials is assumed and thus a small
cohesion of 0.72 kN/m2 for the shell and grave] is used. Typical friction angles of 36° and
42° are adopted for the shell and gravel, respectively. The determination of the
consolidated-undrained shear strength of clay is described in Lowe (1967) and Johnson
(1975). A typical friction angle of 20° and cohesion of 100 kN/m2 are used for the clay core.
Those nodes at the interface between the core and shell or between the core and gravel were
assigned to have the properties of the core, and those nodes between shell and gravel were
assigned to have properties of the gravel material. For simplicity, these properties are
assumed to be homogeneous for each material region. The excess dynamic pore water

pressure induced during earthquakes is neglected.

5.4.2 Gravity load

Soil strength depends on effective stresses. The Santa Felicia dam has a free board of
20 ft as shown in Fig. 3.1. In this study, the situation of full water level is considered so that

on the upstream side the effective vertical normal stress due to gravity load is determined

by

'c’ = (height of the soil column) x Y (5.6)

where y’ = — yw equals the submerged unit weight of soil.

sat

The gravity-induced stresses can be obtained using I-DEAS by performing a static
analysis of the Santa Felicia dam subjected to an acceleration of Ig in the vertical direction.
The steps of the analysis are described in Appendix A7. The computed horizontal normal

stresses were about 50%~60% of the vertical normal stresses 1'.

141

The values of the safety margin, Y, due to gravity induced stresses are shown in
Fig. 5.22. Figures (a), (b), (c) and (d) are the contours shown at the base, at the top, on the
XY cutting plane at z = O and on the YZ cutting plane at x = 0, respectively. Note that the
lighter contours signify lower safety. The lightest color regions have negative Y values,
where local failure is predicted. It is unreasonable that the gravity-induced stresses at some
locations near the uppermost layer of elements indicate local failure as shown in
Fig. 5.22(c). The reason for this is that the ﬁnite element analysis yields tensile principle
stresses at these nodes. It is well known that finite element analysis is an approximate
procedure, and computed stresses usually have greater error than displacement. The order
of the error in nodal stresses arising from finite element analysis is close to the magnitude
of the gravity-induced stresses near the top free surface. Yet the Mohr-Coulomb criterion
predicts a very small shear strength near the free surface for the cohesionless shell since
effective normal stresses are very small. Thus, the results of the analysis in material zone
IS (see Fig. 3.3) is not reliable. For more acceptable and accurate analysis, a finer element

mesh near the top free surface should be used.

5.4.3 Results

The reliability analysis was performed considering gravity and earthquake loads for
the cases of identical and general excitations. Again, the root-mean-square ground
acceleration was set to 0.09g, which corresponds to a peak ground acceleration of about
0.27g. The computed reliability index is shown as shaded image contours over the entire
dam. Fig. 5.23 shows the contours for the identical excitation case, in which ﬁgures (a), (b),
(c) and (d) show the base view, plan view, XY cutting plane at z = O and YZ cutting plane
at x = 0, respectively. The area in which the reliability index is less than zero is the potential
failure zone. It is seen that the material in the bottom half of the dam has high reliability
against local failure. However, the soils at about two-thirds the height of the dam in the
shell as well as in the core show local failure. When the potential failure zones are narrow

and isolated, sliding failure does not develop. However, if the potential failure zones

142

 

 

(b)

 

 

’532

(d)

 

 

 

 

 

 

 

 

=0.

0 (d) YZ cutting plane at x

Figure 5.22 Safety State under gravity load (a) base view (b) top view (c) XY cutting plane at z

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expand to some extent and are connected with each other, sliding of the slope as shown in
Fig. 5.23(c) may occur (Okamato 1984). Note that the zigzag shape of contour boundaries

is due to the coarse resolution of the ﬁnite element mesh.

Figs. 5.24 (a), (b), (c) and ((1) show the reliability index contours at the base, at the
top, on the XY cutting plane at z = 0 and on the YZ cutting plane, respectively, for the
general excitation case. The figures show larger regions of local failure in the dam due to
the general excitation than due to identical excitation, especially in the gravel streambed.
As shown in Fig. 5.24(c), it would appear that a greater variety of sliding failures could
occur due to the SVEGM than due to identical excitation. In particular, sliding masses

could include parts of the gravel streambed.

The Santa Felicia earth dam located in southern California experienced the
Northridge earthquake of January 17, 1994. The ground acceleration of the Northridge
earthquake had a peak value of about 0.27g in the upstream—downstream direction, which
is close to the peak acceleration of the simulated El Centro earthquake used in this analysis.
However, no severe damage except a 1/8 inch transverse crack in fill placed to connect an
access road to the crest of the dam at the abutment was reported due to the Northridge
earthquake (Stewart 1994). The analysis results therefore appear to be unrealistic. There are

several reasons why the results of the analysis may be over-conservative:

0 Linear analysis of the dam was performed by assuming that the entire construction
took place in a single operation. In reality, earth dams are constructed in a definite
sequence of operations. In embankments, the behavior of soil at a particular stage
of loading is dependent upon the state of stress and stress history. Thus the stresses
in the ﬁnal conﬁguration depend on the sequence of intermediate configurations
and loadings (Desai and Abel 1972). Nonlinear finite element analysis can be used
to compute the stresses in the final conﬁguration of the slope under the effect of

gravity (Dunlop and Duncan 1970). The computed horizontal normal stresses in

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this analysis were typically about one-half of the vertical stress. However, for over-
consolidated dam soil, the lateral earth pressure may be larger than the vertical
stress, i.e., the lateral earth pressure coefﬁcient can be greater than unity. These
under-estimated horizontal stresses could signiﬁcantly affect the shear strength of
soil against seismic force. However, the effect of sequential construction and

compaction was not considered in this study.

To determine sliding failure, the number of cycles of peak shear stresses required
to cause failure at points along a potential sliding surface should be taken into
account (Peakcock and Seed 1968). The Mohr-Coulomb criterion assumes that
failure occurs even under a single excursion of the shear strength during dynamic
response. However, in reality, several excursions of the shear strength are required
during dynamic response to cause sliding failure; the required number of excursions
is a function of confining pressure, void ratio, relative density, etc. Thus, a more
realistic failure criterion should include not only the shear strength of soil but also
the number of excursions of the shear strength by the dynamic load induced shear

SII'CSS.

Earthquake induced sliding failure in the shell has been found to be related to the
gradient of the slope (Okamoto I984). The intensity of vibration necessary to cause
sliding failure is lower for steeper slopes. The local yielding failure criterion
governed by the Mohr-Coulomb relationship does not account for this effect. For
global failure, the effect of slope gradient on the sliding of the slope should be taken

into account.

Earthquakes, in reality, are non-stationary random processes. Fig. 5.25 shows the
accelerogram of the Northridge earthquake at the base of the Santa Felicia dam in
the upstream-downstream direction. High accelerations occurred only for the short
duration from about 3 to 7 seconds. In this study, stationary earthquake excitation

(see Fig. 5.4 for a typical stationary accelerogram) and response were assumed.

I47

 

0.5

0.4

0.3

0.2

Acceleration (g)
C
O

0.1

 

Figure 5.25 Accelerogram at the base of the Santa Felicia dam in the upstream-downstream

1 ' r ' l T I ' T

upstream-downstream direction

”J l‘l‘ In 3"- ‘ .... ‘7‘ __-... AA

lﬂll‘” w... .

l 1 4 1 l 1 1 n l

1 I J

 

 

5 10 15 20 25
Time (second)

direction due to the Northridge earthquake of January 17, 1994

30

Clearly, the assumption of stationary response is conservative; a dam that would fail

under stationary response may not fail under non-stationary, transient response.

Consequently, the reliability analysis based on stationary response is conservative.

° The dynamic analysis of the Santa Felicia dam subjected to the Northridge

earthquake is described in Appendix B. Comparison of the computed results with

seismograph records indicates that a modal damping ratio as high as 0.45 is

appropriate. A higher modal damping ratio would reduce the magnitude of the

computed stresses and hence the analysis based on a modal damping ratio of 0.06

is conservative.

This section merely illustrates how the computed earth dam responses can be used to

assess reliability against local failure. The reasons given above indicate why the simplistic

approach used in this section may yield over-conservative results. Nevertheless, this

illustration shows how SVEGM can aggravate sliding failures in earth dams. More accurate

148

global failure criteria need to be developed in order for a more realistic assessment of

reliability against sliding failure.

5.5 Summary

Due to SVEGM, a maximum 30' displacement of 193 mm occurs at the middle of the
crest with the upstream-downstream displacement dominating the total response. A u+3o
maximum 1mm of 1,670 kPa occurs at the base with the TX), component being dominant. The
critical 7mm of 0.193% occurs at the top free surface on the downstream side with the ny

component contributing the most.

The conventional identical ground motion model slightly over-estimates the
displacement and ymax responses, and significantly under—estimates the 1mm response at the
base. However, the critical 1mm response in the core as well as in the shell is over-estimated
by the identical excitation model. In addition, the contour patterns of the displacement and
7mm responses due to identical excitation are similar to those due to general excitation,
while the contour patterns for the 1mm, responses at the base are dissimilar for the two types

of excitations.

Modes 1 and 6 contribute most to the dynamic component of the displacement, strain
and stress responses. The dynamic component dominates the displacement and strain
responses for all types of excitations. However, the pseudo—static component dominates the
stress responses in the gravel near the base for general excitation, while the dynamic
component dominates for identical and delayed excitations. On the other hand, for the

critical stress response in the core, the dynamic component dominates for all cases.

The stochastic moments of the maximum shear stress response were generally
computed using Monte Carlo simulation. However, the use of mean-value FOSM methods
were also investigated and found to be fast and acceptable if appropriately calibrated
correction factors were employed. Finally, reliability against local failure was investigated

for the identical and general excitations using the very simple Mohr-Coulomb criterion.

149

Soils on the upstream side are not as strong as those on the downstream side due to the static
pore water pressure. The critical regions susceptible to local failure under identical
excitations are typically within the upper one-third height of the dam. However, under
SVEGM, the gravel streambed is also susceptible to local failure, and a larger variety of

sliding blocks appear feasible.

6. RESPONSE TO DIFFERENT GROUND MOTIONS
— PARAMETRIC STUDIES

6.1 Effect of Propagation Speed and Direction for SV/P Waves
Earthquakes, in general, release two types of body waves: P waves and S waves, in
which S waves may consist of SV waves and SH waves. P waves are compression waves
that cause motions along the direction of wave propagation. S waves are shear waves that
cause motions perpendicular to the direction of wave propagation. SV waves are polarized
vertically while SH waves are polarized horizontally. P and SV waves are coupled, i.e., one
type generates the other upon reflections and refractions. SH waves, on the other hand,
remain pure SH waves. For seismic waves propagating in the upstream-downstream

direction, SV/P waves would cause critical responses of the earth dam.

According to Abrahamson’s investigation (1992), the lowest apparent velocity of
S-waves in California is about 2.0 km/s. Therefore, in addition to the apparent speed of
4.3 km/s used in Chapter 5, a wave speed of 1.0 km/s (3,300 ft/s) with an incident angle of
60°, as shown in Fig. 6. l (a), was also used so that the apparent wave velocity (Va) is close
to 2.0 km/s. Further, since the location of earthquake source is usually uncertain and the
incident angle of seismic waves approaching the dam will depend on the hypocentral
distance and depth, the effect of a shallow incidence angle of 30°, as shown in Fig. 6.1(b),

is also investigated. In this case, the apparent velocity is V a = 2.1 km/s.

60°; 30):”
V = 1.0 km/s (3,300 ft/s) V = 2-1 km/s (7,000 W5)

Figure 6.1 (a) SV/P waves at lower speed, (b) SV/P waves at shallow incident angle

150

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TABLE 6.1 Maximum [1+30' responses to different apparent wave velocities

 

 

 

 

Response 4.27 km/s (60°) Infinity (90°) 2.0 km/s (60°) 2.47 km/s (30°)
x-displacement (mm) 207.4 207.6 202.8 203.2
{mm (kPa) 511.0 498.0 595.3 567.8

 

 

 

 

 

 

 

The 30' responses of displacement in the x direction due to apparent wave velocities
of Va = 4.27 km/s (60°), Va = infinity (90°), Va = 2.0 km/s (60°) and V2‘ = 2.47 km/s (30°)
are shown in Figs. 6.2 (a), (b), (c) and (d), respectively. The displacement responses for the
four cases are plotted on the same scale. These contour patterns are very similar to one
another and the maximum response in each figure is given in Table 6.1. A higher apparent
velocity results in a larger x-displacement response, but the increases are not very

significant.

In addition, the u+3o maximum shear stress responses at the base are also compared
for the four wave speeds, and the corresponding contours shown in Fig. 6.3 arranged in the
same order as the displacement contours. These patterns of the stress distribution are
significantly different from each other and indicate that lower apparent wave speeds can
cause significant stress increases, especially in the base gravel region. The critical response
value for each case is given in Table 6.1. The 1mm due to the lowest wave speed of 2.0 km/s
is about 20% higher than that due to identical excitation (i.e., infinity apparent wave
velocity). The reason for the stress increase is that waves with lower speed have shorter
wave lengths, and hence the ground movement at the dam base is less in phase giving rise

to greater spatial variation.

6.2 Effect of Propagation Speed for SH Waves

Since all particle motion for SH waves takes place in a direction horizontally
perpendicular to the direction of wave propagation, SH waves traveling in the longitudinal

direction of the earth dam, as shown in Fig. 6.4, will give rise to upstream-downstream

153

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\ /\ 4

V = 2.1 km/s (7,000 ft/s) V = 1.0 km/s (3,300 ft/s)

 

 

Figure 6.4 SH waves propagating in the longitudinal direction of Santa Felicia dam

motion and hence the critical response of the dam. The two cases illustrated in Fig. 6.4(a)
and (b) were used to study the effect of SH wave excitations with high and low propagation

speed, respectively.

Displacement response due to SV/P and SH waves, both having a high speed of
2.134 km/s and an incident angle of 60° are compared in Fig. 6.5. Figures (a) and (b) show
x-displacements at the top free surface for SV/P and SH waves, respectively, while Figures
(c) and (d) show x-displacements on the vertical YZ plane at x = 0 for SV/P and SH waves,
respectively. The displacement responses are very similar for these two kinds of waves
with the same high speed. However, if SH waves propagate at the lower speed of 1.0 km/s,
the x-displacement contours have a signiﬁcant shift in the propagation direction as shown
in Fig. 6.7(a) and (b). Table 6.2 lists the critical response values for all these cases and
indicates that at the same speed the difference between the maximum x-displacement
response to SV/P and SH waves is negligible. The lower wave speed yields a slightly
smaller displacement response. However, it should be noted that for SH waves the location

of the maximum response is no longer at the middle of the crest.

The comparison of the u+3o maximum shear stress responses are shown in Fig. 6.6,
in which ﬁgures (a) and (b) are 1m ax responses at the base, and (c) and (d) are Tmax on the
vertical YZ plane at x = 0 for SV/P and SH waves, respectively. At a wave speed of

2.1 km/s, the distribution of 17mm along the YZ plane has a slight shift in the negative

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156

TABLE 6.2 Maximum ”+36 responses due to different wave types and speeds

 

 

 

 

 

Response SV/P waves (2.1 km/s) SH waves (2.1 km/s) SH waves (1 km/s)
x-Displacement (mm) 207.41 207.36 204.55
1mm (kPa) 510.97 499.43 519.40

 

 

 

 

 

z—direction for SH waves, compared to SV/P waves. The shift is more significant for the
response corresponding to the lower SH wave speed of 1.0 km/s as shown in Fig. 6.7(d).
On the other hand, for 1mm along the base (see Fig. 6.7(c)), the higher stresses shift to the
positive 2 direction, and the critical responses occur at the comer between the base and

inclined boundary.

The maximum 1mm value for each case is also given in Table 6.2. The critical 1mm
value due to SH waves is about 2% lower than that due to SV/P waves at the same speed.
However, as mentioned earlier, waves with lower speed have smaller wave lengths and
cause the stress responses to increase due to out-of-phase motions. The maximum Tm for
SH waves with a low velocity of 1.0 km/s is about 4% larger than that for waves with a
higher speed of 2.1 km/s. It should be noted that although the magnitude of the stress
response does not change significantly, the critical response regions due to SH waves differ

considerably from those due to SV/P waves.

6.3 Effect of Different Coherency Models

The effects of spatial variation of seismic ground motion on large structures has been
studied by several investigators in the last decade. A number of empirical models for the
coherency function have been proposed based on strong motion records obtained at dense
seismography arrays. The discrepancy between these models can be quite significant. In
general, the ground motions is considered to be a homogeneous random field and the
coherency function is assumed to be direction-independent. In this section, the effects of

different coherency models on the responses of the Santa Felicia dam are investigated.

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6.3.1 Coherency models
6.3.1.1 Abrahamson’s model

Abrahamson (1993) investigated a number of earthquake events in California and
Taiwan. His coherency model was determined empirically based on from several

seismography arrays data (Abrahamson 1992), and is

 

Ivtﬁv)| = lvltﬂv)|h(f.v) (6.1)
where
[y] (f,v)| = tanh[ C3(v) + (4.8—c3 (V))exp (c6(v)f) +0.35] (6.2)
1+c4(v)f+c7(v)j2

f is the frequency in Hertz, and v is the separation distance in meters. The CI. functions in

equation (6.2) are given by

c3(v) — , 3°95 \ +0.85exp(—0.00013v)

[I + 0.0077v + 0.000023%)

0.4[1——1—3]
1+ (v/S)

[1+ (v/190)8][1+ (v/180)3]
C6(V) 3(exp(—v/20)-l)-0.0018v
67 (v) = — 0.598 + 0.1061n (v + 325) — 0.0151 exp (—O.6v)

 

 

C4 (V) =

In addition, the plane wave factor h (f, v) is

h (f, v) = 1 (6.3)

+ (cgij) )6

 

 

where

c8 (v) = exp (8.54- 1.07ln (v + 200)) +100exp(—v)
This function has smaller coherency loss at lower frequency as shown in Fig. 6.8
compared to Harichandran’s model. The relationship between the coherency and

separation distance is shown in Fig. 6.9. Abrahamson’s model is not based on a single

Coherency

Coherency

0.8 r

0.7

0.6 -

0.5 r

0.4 .

0.3 -

0.2 -

0.1 r

0.0

1.0

0.9

0.8

0.7

_o
o

0.4

0.3

0.2

0.0

160

 

 

 

 

 

l A L 4 l A 1 1 l 1 l 1 l 1 l

 

 

0.5

0.1

 

1 2 3 4 5 6 7 8
Frequency (Hz)

Figure 6.8 Coherency vs. frequency for Abrahamson’s model

 

I.
-.
‘0
o
'-
.
o
...-
.
o
u
-
u
o
u
a
-.
.

 

1----f=5Hz

—- f=le
----- f=3Hz

' f=7Hz
f=9I-Iz

 

 

 

l 00 200

Separation (m)

Figure 6.9 Coherency vs. separation for Abrahamson’s model

161
event, but in the average coherency for several events and sites. Abrahamson claimed that
the event to event and site to site coherency variations are not statistically significant and

hence justified the averaging.
6.3.1.2 Novak’s model

Novak (1987) proposed the following single exponential coherency function for

SVEGM

 

Mm, v)! = expf—a(2",§"v)ﬁj (6.4)

where 0t and B are empirical constants and V is a wave velocity.

In order to compare Novak’s and Harichandran’s models, the constants 0t and B were
adjusted so that the two coherency functions matched to some degree. Using the
non-dimensional parameter x = ccov, where c = 21t/V, Novak’s coherency model
becomes expL—OLXB). Since v = x/ (cm) ,Harichandran’s model, equation (4.3) can be

rewritten as

|y(v,m)| = Aexpl:—%g%(;l)—(l —A+0tA)]
(6.5)
+ (1—A)exp[—2—X-e/(—((§)(—9-)—(l—A+aA)]

It is not possible to match the two coherency models closely over all separations. Two

types of fits are considered:

Case I — The coherency curves are matched at lower normalized separations as shown in
Fig. 6.10. In this case the values of the three parameters, 0t, B and c, are 1.1, 0.9 and
0.00069, respectively. This coherency function emf—0.001572 (fv) 0'9) is shown in
Fig. 6.11.

Case 11 — The coherency curves are matched at higher normalized separations as shown in
Fig. 6.12. In this case the values of the three parameters, 0t, B and c, are 2.0, 0.31 and

0.00002827, respectively. The coherency function expL —0.077797 (fv) 0'31 J is shown in

Coherency

Coherency

l62

 

1.0

 

 

 

 

 

 

 

 

 

 

 

' I I I I I I T I
- \ -— Novak’s model
0.9 i- ‘\_\\ ------ Harichandran’s, f=l Hz
\~ \\\ ---- Harichandran's, f=3 Hz
0.8 ” \.§ \ \ - - - - Harichandran’s, f=5 Hz
’ ‘§‘\ \~\\ -- Harichandran’s, f=7 Hz
0-7 “' \.\\\ ‘~:\~\\ - — - Harichandran’s, f=9 Hz
06? '\\~\\;"“':7 ~~~~
' — '\ iii?“ 573:? ~~~~~~~
0.5 - \'-~\.:.-~E"""~--.-...7:ZT.‘. ~~~~~~~~ ‘
0.4 — T \ :TL‘*3E€
0.3 - "
0.2 ’- ..
0.] - ‘
00 L I l _A_ l l l l. I l l 1 l
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Normalized Separation
Figure 6.10 Match of Novak’s and Harichandran’s models (Case I)
10 ' I I ' I I I f I ' I ' I
(3“ — V: 100 m
0.9 " \.\' """ W 200 m
. . .\" — ' W 300 m
0.3 " \ .\ —- V: 400 m
0.7 — ‘\. .' _
_ \\ \. .
\
0.6 e \‘ -\. -
.. \ \.
0.5 - ‘ \.\ _
. \\ -\_
0.4 i- \ ‘ \ ' \ . ﬂ
- \ ‘ \ ‘ \ . . .
0.3 " \ ‘ \ ' \ , ............ -
r- \‘ \.\.\ ........"'-..‘
02 b \‘ \_ \\\\\ A
0.1— \' “ E- E_\;
00 b l l 1 l I l l I l l l l 1
0 l 2 3 4 5 6 7 8 9 10
Frequency(Hz)

Figure 6.11 Coherency vs. frequency for Novak’s model (Case I)

Coherency

Coherency

1.0'

163

 

0.9

0.8

0.7

 

-— Novak’s model

°°°°°° Harichandran’s, f=l Hz
---' Harichandran’s, f=3 Hz
- ° ' ' Harichandran’s, f=5 Hz
-- Harichandran’s. f=7 Hz
-—- Harichandran’s, f=9 Hz

 

 

 

 

 

' 0.0

0. l 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Normalized Separation

Figure 6.12 Match of Novak’s and Harichandran’s model (Case 11)

 

0.2 -

0.1 -

 

 

 

 

l 4 l 4 l 1 l 1 l 1 l L l 1 4 1 l 1

 

 

0.0

l 2 3 4 5 6 7 8 9
Frequency(Hz)

Figure 6.13 Coherency vs. frequency for Novak’s model (Case II)

164
Fig. 6.13. Case 11 has lower coherencies at lower frequencies compared to Case I, while at

higher frequencies Case I has lower coherencies.

6.3.1.3 Luco and Wong’s model

Luco and Wong (1986) used the coherency function

2 2 2
Iy(v,00)l = expi—a 00 v ) (6.6)
where a is an empirical constant, 00 is the frequency in rad/s and v is the separation distance

between two stations. In order to assess the effects of this coherency function on earth dam

responses, two rates of coherency decay are considered:

1. a = 2 x 10'4 s/m is used to represent low coherency decay. Iy(v, (0)] for this
value of a is shown in Figs. 6.14 and 6.15, and is somewhat similar to
Abrahamson’s model. The relatively ﬂat curve at a frequency of 1 Hz indicates

nearly fully correlated ground motions at low frequencies.

2. a = 1 x 10—3 s/m is adopted to represent high coherency decay, and the
corresponding coherency function is plotted in Figs. 6.16 and 6.17. The abrupt drop
of the coherency at low frequencies implies highly incoherent ground movement.

6.3.1.4 Hao’s model

Hao et al. (1989) proposed a coherency function based on the study of 17 events

recorded by the SMART-1 array. This function is anisotropic and has the form

y(v€vT,0)) = exp(—Ble—BZVT)Xexp(—(al«/\7“+a2«/\7)(%)2) (6.7)

where vL and VT are the projected distances between two points in the longitudinal or radial
(i.e., wave propagation) and transverse directions, respectively; B1 and B2 are empirical

constants, and 0tl and a2 are defined as

21m b0)
(1 = —+—+
' (0 2
0.314 S (0 S 62.83 (rad/s)
21rd ea)
(12 - —+—+

Coherency

Coherency

 

165

 

 

 

 

 

 

 

Frequency (Hz)

Figure 6.14 Luco and Wong’s low coherency decay model plotted against frequency

 

 

 

 

 

 

 

—- f= 1 Hz
‘‘‘‘‘ f=3Hz
f=5Hz
‘—' f=7Hz
—‘ f=9Hz
. \ .1
0.6 '— \\ ‘\ J
0.5 ’- \\ \'\ . ... A
0.4 - \. .\. —
I- \ i'. “2.. .
0.3 ~ \ \. -
h \ ... .-~.... ‘
0.2 — \. \. -
. \‘ \\ .
0.1 - ~ "-‘
. \‘ \.\-\ ... ..
00 I l l \‘ -L \M‘r—_ __'L.""°--;-4.#-
0 WC 200 300 400

Separation (m)

Figure 6.15 Luco and Wong’s low coherency decay model plotted against separation

Coherency

Coherency

166

 

 

l I l

—

—— 11:100m
""" v=200m
- u=300m

 

11:400m

 

 

 

 

4 5

Frequency (Hz)

Figure 6.16 Luco and Wong’s high coherency decay model plotted against frequency

 

 

 

 

 

 

1.0 .- l r I

“2“: i _ f: l HZ
0.9 —\\ ------ f: 3 Hz

. |\ ' ' ' ‘ f: 5 HZ
0.8 - \\ z 1.. ' f: 7 HZ

~ ‘. _ —- f= 9 Hz
0.7 — I: -.

L \\ I
0.6 L "\ -. _
0.5 - -\ 1 ~

\\ '.

0.4 — .'\ 1 -
0.3 - \l —
0.2 — \\ '. -
0.1 — \\.\ '. _
0.0 - \.\. l 1 1

0 100 200 300 400

Separation (m)

Figure 6.17 Luco and Wong’s high coherency decay model plotted against separation

167

TABLE 6.3 Parameters in coherency function ( X 10—4 )

 

 

 

 

Event B, B2 a b c d e g
45 1.109 0.673 35.83 -0. 181 1.177 51.63 -0.076 -1.905
30 2.250 5.100 106.6 0.265 —0.999 66.55 0.588 -1 1.18

 

 

 

 

 

 

 

 

 

 

 

where a, b, c, d, e and g are constants. For (0 > 62.83 rad/s, the coherency is assumed to be

constant and equal to the value at (0 = 62.83 rad/s.

The constants computed by Hao for the SMART-l Events 45 and 30 are given in
Table 6.3 (Hao 1993) and are plotted in Fig. 6.18 to Fig. 6.21. It is apparent that the
coherency decay for Event 30 is larger than that for Event 45. Fig. 6.19 indicates that at
f = 5 Hz, the coherency decay along the longitudinal and transverse directions is similar for
Event 45. However, Fig. 6.21 indicates greater anisotropy for Event 30, with coherency

decaying faster in the longitudinal direction than in the transverse direction.

6.3.2 Responses due to different coherency models

6.3.2.1 Displacement and strain

Displacement and strain responses using different coherency functions are presented
in this section. The 30' x-displacement responses due to four coherency models are shown
in Fig. 6.22. The four coherency models are: (a) Harichandran’s, (b) Abrahamson’s, (c)
Hao’s (Event 30) and (d) Hao’s (Event 45) models, respectively. The figures indicate that
displacement response is not very sensitive to the coherency model. The maximum 30'
x-displacements are given in Table 6.4 for each coherency model. Basically, higher
coherency decay models result in smaller displacements. The largest difference between
the critical 3o x-displacements is about 10.3% (between Abrahamson’s and Hoa’s

Event 30 models).

Coherency

 

— L= 100m 11 =1m
------ VL=1m,1/ =100m
-—- VL=150m,l/T=150m
—- uL=300m,I/T=300m
\
.
\‘ \~
0.6» \ \.\\_
\~ \‘\-\ §__‘
0.5 > \.\~ .........
0.4- \‘ \_ E_ __ __ __
0.3-
0.2 -
0.1 -
01) l . l . l . l . 1 L L JEAAAL 1 4L4431 .L
0 l 2 3 4 5 6 7 8 9
Frequency (Hz)

168

 

 

 

 

 

 

Figure 6.18 Hao’s coherency model for SMART-l Event 45

 

 

    

150

100
Transverse Longitudinal

 

 

Figure 6.19 Hao’s coherency model for SMART-1 Event 45 (at f = 5 Hz)

Coherency

I I I I I I
— L=100m,r/ =1m
09-. -------- VL=lm,1/T=100m
..................... —- VL=150m,r/T=150m
0'8 -\ .\ -------------------------- —- VL=300 m,uT=300m
\ '\ I
0.7 ’ \ '\ I
\ .\-
0.6 h \\ \‘\'
\ \-
0.5 ’ \‘
0.4 \~
0.3 \. ~\.
\~ \.\-\
0.2- \~ \~ -\.\ \
OJ ’ - §§' ‘~. ‘~‘
01) I AAJ I | I 1 I 1 I l I 1 I l I l I l L
0 1 2 3 4 5 6 7 8 9 l

169

 

   
   

 

 

 

 

 

Frequency (Hz)

Figure 6.20 Hao’s coherency model for SMART-1 Event 30

 

 

 

200

150
Transverse

100
Longitudinal

 

 

Figure 6.21 Hao’s coherency model for SMART-1 Event 30 (at f = 5 Hz)

170

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TABLE 6.4 Maximum responses (n+30') corresponding to different coherency models

 

 

 

 

Response Harichandran’s Abrahamson’s Hao’s (Event 30) Hao’s (Event 45)
x-Displacement 191.8 mm 202.0 mm 183.2 mm 193.2 mm
7mm 0.001934 0.002060 0.001839 0.002050

 

 

 

 

 

 

 

The u+3o contours of the maximum shear strain viewed from the top using these four
coherency models are shown in Fig. 6.23, all plotted to the same scale. There is some
difference among the contours, especially for Abrahamson’s model. However, the critical
responses for all cases are similar, occur around both the upstream and downstream surface
regions, and are also given in Table 6.4. Basically, higher coherency decay models give rise
to lower strain. The largest difference between the critical u+3o 7mm, values is about 12%

(between Abrahamson’s and Hao’s Event 30 models).
6.3.2.2 Maximum shear stress

It was found in Section 5.3.3.3 that stress responses at the base have a significant
increase due to SVEGM and that the static response component dominates the total
maximum shear stress response for the general excitation case. Since 1mm is much more
sensitive to the coherency model than displacement and strain, eight different coherency
functions are used to evaluate their impacts on the maximum shear stress. The total 1mm
responses corresponding to these models are compared. In addition, the static 15mm
component is also shown so that the contributions of response components can be

recognized.

Fig. 6.24 shows the u+3o total 1mm response along the base computed using (a)
Harichandran’s, (b) Abrahamson’s, (c) Novak’s Case II and (d) Novak’s Case I models,
respectively. The corresponding u+3o static response contours of are shown in Fig. 6.25.
Note that the magnitudes and distributions of 1mm are significantly different for each case.

The contours are not drawn on the same scale because of the large differences in the

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175
response magnitudes. The total and static tmax responses are quite different for
Abrahamson’s coherency function, indicating that the static component does not dominate
for this low coherency decay model. For the other three high coherency decay functions,

however, the static response dominates the total response.

The total u+3o 1mm response for (a) Luco & Wong’s low coherency decay model, (b)
Luco & Wong’s high coherency decay model, (c) Hao’s model for SMART—l Event 45,
and (d) Hao’s model for SMART-l Event 30, respectively, are shown in Fig. 6.26. The
corresponding static responses are shown in Fig. 6.27. The tmax response due to Luco &
Wong’s low coherency decay model (Fig. 6.26(a)) is similar in magnitude and distribution
to the response due to Abrahamson’s model (Fig. 6.24(a)), and is also not dominated by the
static component (see Figs. 626(3) and 6.27(a)). The other three responses in Fig. 6.26,
however, are dominated by the static component. Note that although the magnitude of the
TM a response due to Hao’s two models is significantly different, their distributions are
similar (see Figs. 6.26(c) and (d)).

Excluding responses around the artificial upstream and downstream boundaries, the
maximum u+3o total 1mm, responses are given in Table 6.5 for each coherency function,
and the numbers listed in parentheses are the static components. There is a wide variation

in the maximum 1mm response, with Novak’s model (Case 11) being ten times higher than

that due to Luco and Wong’s low coherency decay model. For high coherency decay cases,

TABLE 6.5 Maximum n+30' Tm response (kPa) to different coherency models

 

Harichandran’s Abrahamson’s Novak’s (case I) Novak’s (case 11)

 

 

1,670 (1,636) 523 (342) 2.650 (2,639) 4,866 (4,859)

 

 

 

 

 

 

 

Luco & Wong’s (low Luco & Wong’s (high

decay) decay) H30 5 (Event 45) Hao 5 (Event 30)

 

 

476 (231) 1,083 (1,074) 1,368 (1,319) 2,026 (2,000)

 

 

 

 

 

 

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178
the static stress component alone is a very good approximation of the total stress response.
Surpnsmgly, the critical 13mm responses occur at the same location for all these high

coherency decay models (x = -160 ft, y = 0 ft, 2 = i 122 ft).

Note that for the low coherency decay models (Abrahamson’s and Luco & Wong’s),
the 1mm responses are very similar to that due to identical ground motion (see Fig. 5.1 l(c)),

and hence the effect of SVEGM can be neglected for these cases.

6.3.3 Discussion

The pseudo-static stress component contributes the most to the total responses. This
component is based on the calculation of the integral of the ground displacement spectrum
as shown in equation (4.42). Fig. 6.28 shows plots of the integrand Sag (00) Iy(v, 01))I/(104
against frequency for various separations for Harichandran’s model, and indicates that only
low frequencies up to about 10 rad/s contributes to the area and hence the static stress

component. The fact that the curves shown in Fig. 6.28 for different separations are distinct

 

0.002 T r r I . r ' l . I
' _ V: 1 m
---‘ V=100m

—- VzZOOm
------ V—-—400m

0.0018

 

0.0016

 

0.0014
0.0012 "
0.001

0.0008

Sg((.u)l')r(z/,w)llw4

0.0006
0.0004

0.0002

 

 

 

0.0

 

0 2 4 6 8 10 12 14
Frequency (rad/s)

Figure 6.28 Plot of static response integrand for Harichandran’s ground motion model

179
implies that the coherency loss is significant at the low frequencies that contribute to the
static stresses. Thus, a plot of the static response in integrand, as in Fig. 6.28, can be used
to assess whether or not the static stress response will be significant for the coherency

model used.

To illustrate this further, the static response integrand is plotted in Figs. 6.29 and 6.30
for Novak’s Case I and Case 11 models, respectively. The integrand values decay much
faster than for Harichandran’s model, and so it can be concluded that stress responses at the

base would be more critical for these models than for Harichandran’s model.

Another two coherency models proposed by Hao were developed according to Events
30 and 45 from SMART-l array. Plots of the pseudo-static response integrand for the
coherency models proposed by Hao for the SMART-1 Events 30 and 45 are shown in Figs.
6.31 and 6.32, respectively. Six combinations of the longitudinal and transverse separations
are used in these plots. Based on these two figures alone, it can be concluded that the static
stress responses at the base will be greater for Event 30 than for Event 45, since the family
of curves are more distinct for the former event. A comparison of Harichandran’s model
used in the study, which corresponds to Event 20 recorded by the SMART-l array, and
Hao’s model for Event 30 & 45 reveals that coherency loss and hence the stress response

may change significantly from one earthquake ground motion to another.

Finally, according to Abrahamson’s coherency model for horizontal ground motions,
coherency loss results in only 5% increase of Tmax' The curves for the static response
integrand at different separations almost overlap each other as shown in Fig. 6.33. Within
the dimensions of the Santa Felicia darn, the effect of SVEGM is not significant for
Abrahamson’s model. One possible reason is that Abrahamson averaged the coherency
curves for a number of earthquake events and sites which may have resulted in high

coherencies at low frequencies.

0.002 . . . , . ,
0.0018
0.0016 ‘

0.0014

0.0012

0.001

Sg(w)l'y(u,w)l/w4

' é

0.0004

0.0002

0.0

0.0008

180

 

 

 

Frequency (rad/s)

 

10

 

—- ur—lm

---'V=100m
-- V:200m
""" u=400m

 

L4

 

 

12 14

Figure 6.29 Plot of the static response integrand for Novak’s Case I model

0.002

0.0018

0.0016

0.001

Sg(w)l'y(u,w)I/w4

é

0.0004 -
0.0002

0.0

0.0014 ._

0.0012 "\

0.0008

 

 

 

Frequency (rad/s)

 

10

 

—— Vzlm

----v=100m
—- u=200m
------ u=400m

 

 

12 14

Figure 6.30 Plot of the static response integrand for Novak’s Case II model

4

Sg(w)|'y(u,w)l/w

346176.612)“

0.002

0.0018

0.0016

0.0014

0.0012

0.001

0.0008

0.0006

0.0004

0.0002

0.0

181

 

I

I

I

I

T

I

 

Frequency (rad/s)

 

— u‘=lm,I/T=lm
------ u =1m,VT=150m
-—- u =150m,1/T=lm
---' V =150m,uT=150m
—- u =300m,z/ =150m
~u =300m,u =300m

PVT—1""

T

r

T

 

 

 

Figure 6.31 Plot of the static response integrand for Hao’s model—Event 30

 

 

 

 

 

 

 

0.002 r r T T
.. L T
. —- u =1 m,u =1 m
0.0018 - ...... ”L: 1 m,uT=150m
' -—- L=150m,uT=1m
0.0016 . ---. VL=150 m, VT=150m
. —- VL =300 m, VT=150m
0.0014 F -- 1} =300 m, VT=3OOm
0.0012 — 4
0.001 ~ _
0.0008 - ‘
0.0006 - ‘
0.0004 — ‘
0.0002 b 4
00 4 1 l I 1 i 1 A
0 2 6 8 10 12 14
Frequency (rad/s)

Figure 6.32 Plot of the static response integrand for Hao’s model-Event 45

182

 

0.002.,.,...l

— v=1m

---‘ u=100m
—- V:200m
"""" 11:400111

0.0018

0.0016

 

 

0.0014
0.0012
0.001

0.0008

S g(LIJ)|’y(1/,<.IJ)I/LIJ4

0.0006
0.0004

0.0002

 

 

0.0

 

 

Frequency (rad/s)

Figure 6.33 Plot of the static response integrand for Abrahamson’s horizontal ground motion

To sum up, the stress response at the base of earth dams are sensitive to the specific
coherency model used. The stress response at the base is governed by low frequency
excitations while displacement and strain responses are dominated by medium frequency
motions. At frequencies less than 0.5 Hz, Abrahamson’s model is essentially unity for most
separations as a result of which the stress responses at the base are essentially the same as

those due to identical excitation.

In fact, coherencies in the frequency range from 0 to 0.5 Hz cannot be reliably
estimated from data. The reason for this is that estimation of the coherency requires
adequate smoothing of the auto and cross spectral estimates (Jenkins and Watts 1969).
However, smoothing results in a loss of resolution. For limited duration accelerograms (e. g.
10 second long records sampled every 0.01 second), the bandwidth of the smoothing
window used in spectral estimation is typically about 0.5 Hz. If at any frequencies the

spectra vary very rapidly within the bandwidth of the smoothing window centered at those

183
frequencies, then the spectral estimates and hence the coherency at those frequencies will
be highly biased and unreliable. For frequencies approaching zero, physical considerations
require that the auto and cross spectra of accelerograms must rapidly decay to zero. Since
this rapid decay usually occurs within the frequency range of 0 to 0.5 Hz (see Fig. 5.1 El
Centro spectrum) which corresponds to the bandwidth of the smoothing window, the

coherency estimates in this frequency range are unreliable.

Thus, most coherency models that are developed are reliable only for frequencies
above 0.5 Hz. The extrapolation of the coherency from 0.5 Hz to 0 Hz is strongly
influenced by the functional form chosen for the empirical coherency model, and is
somewhat arbitrary. However, since ground displacements are dominated by low
frequency components, the coherency in the range of 0 to 0.5 Hz is extremely important for

characterizing the correlation between ground displacements at different spatial locations.

Although a few coherency functions have been proposed in the last decade, the
investigation of SVEGM is still at an early stage. It has been shown that the coherency
models presented in this study are rather inconsistent. It appears important to address this

incoherency in future efforts at processing seismograph array data.

7. Simpliﬁed Modeling and Analysis Techniques

7.1 Introduction

A finite element-based random vibration analysis of a 3-D earth dam model subjected
to earthquake ground motion is time—consumin g and costly. From a practical point of view,
it is of interest to explore simplified modeling and analysis that may be capable of yielding
acceptable results. In other studies, simplified earth dam models such as 2-D shear beam
models and plane strain ﬁnite element models have been used. As mentioned previously,
SVEGM cannot be represented exactly for such models in reduced dimensions. Techniques
to simplify the SVEGM at the base, so that a 2-D shear beam model can be used for

approximate analysis, are explored in this chapter.

7 .2 One-dimensional Excitation Model

For a 2-D shear beam model of an earth darn, as shown in Fig. 7.1, the base consists
of a finite number of truncated wedges. The base of each wedge is a strip along the
upstream to downstream width of the dam. The strips where ground motion excitation is

applied each have a single degree of freedom. As a result, all points within a strip have

 

 

Figure 7.1 Illustration of truncated wedges in 2-D shear beam model

184

185

identical excitations. However, the ground motion at two points on different strips may be
incoherent. One method of obtaining “equivalent” excitations for each strip is to assume
that it is the average of the correlated excitations at each point within the strip. The
coherency between the equivalent excitations at different strips would then be the
coherency between each average excitation. However, the calculations required to obtain
the SDFs of the average excitations and the coherency between them is tedious. Therefore,
as a ﬁrst step, the coherency decay along the width of the dam is neglected, and the SDF of
the equivalent excitation for each strip is taken to be the point SDF, 5115(0)) . Further, the
coherency between the excitations at two different strips is taken to be the coherency
evaluated at the average separation between all pairs of points in the two strips. For any pair
of strips i and j as shown in Fig. 7.2, the average separation between a point on strip i and

a point on strip j is

 

1 IA“ '1
V.. :
U AXIAZIAXZAZZ .t‘|=0 21:0

7.1
l+At2 2+Az2 2 2 ( )
sz=Ll K2=L2 ’\/(x2_x1) '1' (ZZ—zl) dxldzldxzdzz

 

 

 

 

 

 

 

 

 

 

 

Figure 7.2 Illustration for calculation of average separation between two rectangular areas

186

where Ax] and Az, are the width and length of strip i; sz and A22 are the width and length
of strip j; and L1 and L2 are separations between the lower left comers of these two strips

in the x and z directions, respectively.

The analytical expression in equation (7.1) cannot be evaluated in closed-form.
However, Vi]. may be estimated by taking the average separation of a sufﬁcient number of
simulated pairs of points on strips i and j. The average separation is used to determine the
coherency loss in the simplified excitation model, which is called a l-D excitation model
because only separation in the longitudinal direction of the dam (perpendicular to the length
direction of strips) is used to account for the incoherency effect. For the general excitation
model, coherency loss in the upstream-downstream direction as well as the longitudinal
direction at the base was considered. The base of the dam was divided into 15 strips for the
1-D excitation model. In addition, the effects of using a different number of strips at the
base on the earth dam responses were investigated by considering 9 and 21 strips in

addition to the 15-strip case.

It should be noted that a 2-D shear beam model of the darn was not actually used. Both
the 1—D and general excitation models were used with the 3-D finite element model of the
darn. For the 1-D excitation model, all base nodes located within a strip were assumed to
have identical excitations. The intent of the analysis was to assess the effect of simplifying
the excitation model. If the simplified excitation model is acceptable, then it could be used

with a 2-D shear beam model of the dam.

7.2.1 Fifteen strip subdivision of the base
7.2.1.1 Introduction

It is convenient to use a new Cartesian coordinate system as shown in Fig. 7.3. The
origin is located on the left-hand side of the base instead of at the middle of the base so that
the coordinates of every point of the darn are always positive. In Fig. 7.4, the boundary of

the dam was divided into 15 strips. The ith strip is denoted by Si. Each strip has a reference

187

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275 nI ’
' z (0,412.5)
‘

1,275 ft

Figure 7.3 YZ Cutting Plane at x=0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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P.
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l
s m
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Illustration of lS-strip division

Figure 7.4

188

point which is denoted by Pi. The point closest to the origin of each strip was selected to be
the reference point. The length Ax and width Az of each strip and the x and z coordinates of
each reference point are given in Table 7.1. Because the length Ax is variable along the

width on the inclined boundary, an average value of Ax was adopted as shown in Fig. 7.4.

In order to randomly generate points on strip i, n pairs of independent random
numbers were simulated in the range between 0 and 1 using a uniform distribution function
U(0,1). If the pairs of random number sequences are denoted by ax. (n) and a, (n) , the x

and z coordinates for each point at the ith strip can then be determined by

TABLE 7.1 Strip dimensions and reference point coordinates (ft)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Strip Strip Width Strip Length Reference

Number (Az) (average Ax) Point 2 x
S] 75 130 P1 0 472.75
32 75 330 P2 75 378.25
S3 75 530 P3 150 283.75
S4 75 730 P4 225 189.25
SS 1 12.5 980 P5 300 71
S6 90 1130 P6 412.5 0
S7 90 1 130 P7 502.5 0
88 90 1 130 P8 592.5 0
S9 90 1 130 P9 682.5 0
510 90 1130 P10 772.5 0
811 112.5 980 P11 862.5 71
512 75 730 P12 975 189.25
813 75 530 P13 1050 283.75
514 75 330 P14 1125 378.25
515 75 130 P15 1200 472.75

 

 

 

 

 

 

 

 

 

189

x . = xPi+AxSiXaxi(n)

III

(7.2)
2 . = zPi+AzSiXazi(n)

IN

The y coordinate has the following relationship with the 2: coordinate:

(412,5 _ Zni) tang if 0.0 S zm. < 412.5 ft
Ym' = 0 if 412.5 S 2,". S 862.5 ft
(Z,,,- - 862.5) tanG if 862.5 < zm. S 1275 ft

275
412.5

(7.3)

where 0 is the angle of inclination shown in Fig. 7.3 (tan0 = = 0.66667).

A total of m points on strip j can be generated in a similar manner using two other
pairs of random number sequences ax. (m) and a,_ (m) . Consequently, the average

separation between strips i and j can be calculated by

 

n m
1 2 2 2
Va- = akXHZIJUki-xu) +<.v..-—.v,,-> +<z,..-—z1,> (7.4)

For the two strips 81 and S1 1, the average separation corresponding to different numbers

of samples are plotted in Fig. 7.5, and the figure indicates that using one hundred samples

 

970'1'1'1'1

 

 

 

I ' n ' I ' l T l
960 - ..
A
‘5
1:
.2 J
*5 950 ~
5
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930 - -
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0 20 40 60 80 100 120 140 160 1 80 200
Number of samples

Figure 7.5 Convergence of average separation between S] and 811

190

is adequate to estimate 9. The average separation between each pair of strips using 100
randomly generated points in each strip is shown in Table 7.2. The responses due to the
simplified 1-D earthquake excitation were obtained using these values for V in equation

(4.3).
7.2.1.2 Displacement and maximum shear strain responses

The 30' x-displacement responses for the 1-D and general excitation models are
shown in Fig. 7.6, in which ﬁgures (a) and (b) are plan views of the response contours, and
figures (c) and (d) are contours on the XY cutting plane at z = 0. All contours are set to the
same scale. It is apparent that x-displacement responses inside the dam as well as on the top
surface due to the 1-D excitation model agree well with those due to the general excitation
model. Using the 1—D excitation model, the maximum 30' x-displacement response is only
1.3% higher than that using the general excitation model. Note that the maximum 36
displacement response due to the identical excitation model is 8.1% larger than that due to

the general excitation model.

The u+3o maximum shear strain responses using these two excitation models are
shown in Fig. 7.7. Figures (a) and (b) are plan views of the ymax contours, and figures (c)
and (d) are the ’ymax contours displayed on the XY cutting plane at z = 0. Although ymax is
slightly different near the side boundaries between the two excitation models, the critical
response regions are quite similar to each other. The maximum u+3o 'ymax response is about
1.9% larger for the 1-D simplified model compared to the general model. Note that for the
identical excitation model, the maximum 114-30' 'ymax response is 10.3% higher than that

using the general excitation model.
7.2.1.3 Maximum shear stress response

The u+3o maximum shear stress responses at the base are shown in Figs. 7.8(a) and
(b) for the 1-D and general excitation models, respectively, and these are quite dissimilar.

The maximum 13mm response is 50% larger for the l-D excitation model compared to that

191

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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due to the general model. Figs. 7.8(c) and ((1) show the “+30 1mm contours on the XY
cutting plane at z = 0. Although the l-D excitation model does not predict 1mm well near
the base, the predictions at locations away from the boundary are reasonable. Figs. 7.9(a)
and (b) display the 1mm contours on the horizontal cutting plane at the mid-height of the
dam, and show that the two contour patterns match each other very well. The Tmax responses
in the core clay region are shown in Figs. 7.9(c) and (d) which display the YZ cutting planes
at x = O. The “(max response due to the l-D simpliﬁed excitation is very similar to that due

to general excitation over the entire dam except near the base.

7.2.2 Effect of different number of strips on the boundary
7.2.2.1 Introduction

As the number of strips on the base is reduced, the accuracy of the 1-D excitation
model would be expected to become poorer. Analyses with 9 and 21 strips on the base were
performed to assess this effect. The horizontal and two inclined portions of the base were
divided into 3 strips each for the 9-strip case, and 7 strips each for the 21-strip case,
respectively.
7.2.2.2 Responses

The 30' x-displacement response contours are shown in Fig. 7.10, in which figures (a)
and (b) are plan views of the contours for the 9- and 21-strip cases, respectively, and ﬁgures
(c) and (d) show the x-displacements on the XY cutting plane at z = 0 for the 9- and 21-strip
cases, respectively. Generally speaking, the displacements do not vary significantly
between the 9-, 15- and 21-strip cases. The values of the critical x-displacement response
for all excitation cases used in this study are given in Table 7.3, which indicates that as
more strips are used on the base, the maximum x-displacement response becomes closer to
that due to the general excitation. However, the use of 9 strips, which appears minimal,

seems adequate, and the improvement with 15 and 21 strips is marginal. In the limit, as

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TABLE 7.3 Maximum 30' displacement responses using simplified and general excitation models

 

 

 

 

Identical l-D excitation model General
Response excitation excitation
model 9 strips 15 strips 21 strips model
x-displacement (mm) 207.4 194.93 194.26 194.02 191.80

 

 

 

 

 

 

 

 

 

 

 

TABLE 7.4 Critical [1+30' maximum shear strain responses using simplified and general
excitation models

 

 

 

 

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model 9 strips 15 strips 21 strips model
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fewer and fewer strips are used on the base, the 1—D excitation model approaches the

identical excitation model.

The u+3o maximum shear strain responses are shown in Fig. 7.11, in which the
responses for all cases are set to the same scale. Figs. 7.] 1(a) and (b) show 'ymax responses
at the top free surface for the 1-D excitation model with 9 and 21 strips, respectively; and
Figs. 7.11(c) and ((1) show 'ymax responses on the XY cutting plane at z = 0 for the 9- and
21-strip cases, respectively. The use of 21 strips on the base yields ymax response contours
very similar to those shown in Fig. 7.7(b) for the general excitation model. The maximum
p.+3o ymax responses for the cases of identical, l-D and general excitations are listed in
Table 7.4 and indicates that the 1-D excitation model yields better 7mm responses than the
identical excitation model. As with the displacement response, the accuracy of the 1-D
model becomes better as the number of strips is increased, but the improvement beyond 9

strips is marginal.

The u+3o maximum shear stress response contours at the base are shown in

Figs. 7.12(a) and (b) for the 1-D excitation model with 9 and 2] strips, respectively.

199

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Fig. 7.12(a) shows that high stresses occur near the edge of each strip in the gravel material
regions due to the relative displacement between adjacent strips. For this 9-strip case, each
strip is quite wide and since the excitations at all nodes in a strip are fully—correlated, the
1mm responses are small within the strip and are relatively high on the boundary between
adjacent strips. In Fig. 7.12(b), even larger 1mm values occurs over the gravel region.
Instead of becoming closer to the results obtained with the genera] excitation model the
“cm response at the base becomes worse as the number of strips is increased. The contour
pattern is also quite different from that due to the general case. However, for the core and
shell regions, the 21-strip case is better than the other two. Figs. 7.13 (a), (b), (c) and ((1)
show the u+3o 1mm response contours at the base for the shell and core using the general,
9-strip, 15-strip and 21-strip models, respectively. The scale has been adjusted so that
details within the core and shell are displayed. The contour pattern for the 21-strip model
is similar to that for the general model, except around the central part of the base where the

response is overestimated.

In order to assess the stress distribution inside the dam, some cutting planes were
taken. Figs. 7.12(c) and ((1) show the u+3o 1mm response on the XY cutting plane at z = 0
for the 9- and 2l-strip models, respectively. These figures, along with Fig. 7.8(c), indicate
that the stress response near the base is sensitive to the number of strips used. However, at
locations distant from the base the stress responses is not very sensitive to the number of

strips used.

This is investigated further by plotting the contours of 1mm response on the horizontal
plane at the mid-height (137.5 ft) as shown in Figs. 7.14(a) and (b) for the 9- and 2l-strip
cases, respectively. In addition, the responses on the YZ cutting plane at x = 0 are shown
in Figs. 7.14(c) and (d) for the 9- and 21-strip cases, respectively. It is seen that the stress
contours on these cutting planes are very similar to each other. The contours on these two

cutting planes for the identical excitation case are also shown in Figs. 7.15(a) and (b),

201

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which indicate that higher 1m ax responses occur in the core clay region due to identical

excitation than for the l—D and general excitations.

Various horizontal cutting planes were used to assess the minimum height at which
stresses due to the simpliﬁed l-D excitation were acceptable, and it was found that this
minimum height was about 120 ft, which is just below the mid-height. Figs. 7.16 (c) and
((1) show 1mm contours at a height of 120 ft, and these are reasonably similar. Figs. 7.16 (a)
and (b) show Tm contours at a height of 75 ft, and these are quite dissimilar. Figs. 7.15 (c)
and ((1) show Tmax contours on horizontal cutting planes at 75 ft and 120 ft heights for
identical excitation. Clearly at 75 ft, the identical excitation model significantly under-

estimates 1mm response in the gravel region, while at 120 ft 1t overestimate 1mm in the core.

The maximum tmax responses are given in Table 7.5 for each excitation model. The
1-D excitation model yields stress responses that are only 1.5% to 2% larger than those due
to general excitation at the mid-height of the dam, with marginal improvement as the
number of strips is increased from 9 to 21. However, stresses predicted at the base using
the l-D excitation model is poor (over-conservative) and becomes worse as more strips are

used.

To sum up, for the critical displacement and 7mm responses and for the tmax response

away from the base, the use of 9 strips is sufﬁcient and little improvement is obtained by

TABLE 7.5 Critical 114-30' Tm responses (kPa) using simplified and general excitation models

 

 

 

 

 

 

 

 

 

 

 

 

 

Identical l-D excitation model General
Cutting plane excitation excitation
model 9 strips 15 strips 21 strips model
At the base (excluding

response on the 498 2,607 2,749 3,212 1,670
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Horizontal plane at

mid-height (same as 410.57 379.1 378.2 378.0 373.8
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207

increasing the number of strips. However, when using a 2-D shear beam model, the number
of strips at the base would equal the number of wedge-type finite elements used along the
length of the dam, and the analyst may as well consider correlation effects between all such

strips.

7 .3 Conclusion and Discussion

For the dynamic analysis of earth dams, the 2-D shear beam model exhibits slightly
higher natural frequencies and has larger responses than the 3-D model. Similarly, for
earthquake ground motion models, using the simplified l-D excitation model discussed in
this chapter gives rise to slightly larger displacement, stress and strain responses than using
the general excitation model. As a consequence, the use of the 2-D shear beam model in
conjunction with the simplified l-D excitation model would be expected to yield

reasonable and slightly conservative results at locations distant from the base.

For the prediction of stresses in the bottom half of the dam, especially in the stiff
gravel, the simplified l-D excitation significantly overestimates the response and is
unacceptable. On the other hand, the use of identical excitation significantly under-
estimates the stress response when ground displacements are not highly coherent.
Therefore, a 3-D model is required in order to predict stresses in the lower half of the dam
accurately under SVEGM. However, it was shown in Section 5.3.6.3 that the use of static
response component can approximate the stress response in the stiff gravel region well such

that the time-consuming 3-D dynamic analysis can be avoided.

8. Conclusions and Recommendations

8.1 Summary and Conclusions

This research was conducted to study the effect of spatially varying earthquake
ground motions (SVEGM) on the response of earth dams. The Santa Felicia earth dam was
selected for analysis because it has been studied extensively and data was easily available
for it. The Santa Felicia earth dam located on southern California is a modern rolled-fill
embankment with a central impervious core and pervious shell upstream and downstream
resting on a rigid bedrock. The dam is 83.8 m (275 ft) high above its rock foundation,
388.6 m (1,275 ft) long at the crest and 137.2 m (450 ft) long across the valley at the base.
A three-dimensional inhomogeneous finite element model consisting of 1,004 nodes and
4,140 tetrahedral elements was used for the dam. The model was assumed to be symmetric
about the middle of the dam axis. Shear modulus increasing with depth from the crest was
used to account for the effect of confining pressure. A response convergence analysis
indicated that eighty vibration modes were required to obtain accurate responses to
earthquake excitation. A viscous damping ratio of 0.06 corresponding to low strain

deformation was adopted for all modes.

In most cases, the SVEGM model proposed by Harichandran and Vanmarcke (1986),
which accounts for both spatial correlation and wave propagation effects, was used to
specify the base motions in the upstream/downstream direction. However, the effect of
using coherency models proposed by other researchers was also investigated. The auto
spectral density function of the ground acceleration estimated from the El Centro record
with a root-mean-square intensity of 0.09g was used. The coherency decay was
characterized by a double exponential function based on Event 20 recorded by the
SMART-1 array in Lotung, Taiwan. An apparent wave velocity of 4,267 m/s corresponding
to waves in the bedrock propagating at 2,134 m/s with an incident angle of 60° to the

horizontal was assumed in the study. In addition, conventional simplified ground motion

208

209

models such as identical excitation and a simple propagating wave were also used so that
the effect of SVEGM on earth dam responses could be evaluated. The effects of soil-
structure interaction, hydrodynamic impact, radiation damping and topography were

neglected in this study.

Stochastic moments of the displacement, stress and strain responses were computed
and displayed as shaded image contours over the entire dam. SVEGM, characterized by
wave propagation as well as by coherency loss, was found to have a significant impact on
the responses of the dam, especially on the maximum shear stress in the stiff gravel in the
streambed. A study of the reliability analysis against local failure characterized by the
Mohr-Coulomb criterion further highlighted the significance of SVEGM. SVEGM can
cause local failure in the granular streambed In addition to the core and shell material,
thereby making a larger variety of sliding failure possible. However, the stress response due
to SVEGM is very sensitive to the low frequency decay of the coherency used. The
coherency model proposed by Abrahamson (1992), which is highly coherent at low
frequencies, has very little effect on the stress response. It is therefore imperative that the
current discrepancy in the low frequency coherency decay in different coherency models
be investigated further and resolved. Finally, method of simplifying SVEGM excitation so
that it can be used with a 2-D shear beam model of the dam is described. More detailed
summaries are given below. Except for Section 8.1.6, in all other sections the Harichandran

and Vanmarcke’s coherency model was used to characterize the SVEGM.

8.1.1 Responses
Compared to SVEGM, the identical and delayed (wave propagation only) excitation

models both overestimate the maximum three standard deviations (30') x-displacement
response at the middle of the crest by about 8%. However, the contour patterns of the
displacement responses are quite similar to each other for all the excitation cases. The two

simplified excitation models both over-predict the critical mean plus three standard

210

deviations (u+3o) maximum shear strain (7mm) response by about 10%. As with the

displacement response, the contour patterns are similar for all excitation models.

The critical u+3o maximum shear stress (tmax) response in the stiff granular material
at the base due to SVEGM is about 3.5 times larger than those due to identical and delayed
excitations. In addition, the contour pattern of the 1mm response due to SVEGM differs
greatly from those due to identical and delayed excitations. However, the effect of SVEGM
is not very significant on ﬂexible materials such as the core and shell. For these regions, the
identical excitation model slightly overestimates the critical 1mm response. In addition, the
tmax response at the top free surface and the ymax response at the base both are larger due to
identical excitation compared to SVEGM. Although higher strains occur in flexible
materials at the base, the larger stresses occur in the stiff gravel material because of the

much higher shear modulus of the gravel.

8.1.2 Modal contributions

The dynamic response variances consist of individual modal response variances and
covariance between pairs of modal responses. The relative modal contributions to the
upstream-downstream displacement (x-displacement) at the middle point of the crest were
examined. Basically, symmetric vibration modes contribute much more than anti-
symmetric modes under SVEGM. Mode 1 dominates the dynamic variances with a
contribution of more than 90% for all the critical displacement, stress and strain responses.

Identical support motions on the other hand do not excite anti-symmetry modes at all.

8.1.3 Contribution of response components

The variance of the total response consists of three components: the variance of the
pseudo-static response, the variance of the dynamic response, and the covariance between
the pseudo-static and dynamic responses. For the 15 largest x-displacements, the dynamic
response component contributes the most for all the excitation cases: general, identical and

delayed excitations, but the other response components are also significant. For the 15

211

largest ny responses Wthl‘l occur at the top free surface, the dynamic response component
is dominant for all excitation cases and the pseudo-static and covariance components can

be neglected.

The largest variance of 1: )0, occurs at different locations for different excitation
models. For the general excitation, the static component contributes the most to the largest
In response. For delayed excitations with an apparent wave speed of 4.27 km/s, the static
response is not significant, but the covariance between the static and dynamic responses
becomes signiﬁcant. Note that for identical excitation the static stress component is zero
since this consists of a pure rigid body translation of the dam. Consequently, the covariance
between the static and dynamic components is also zero, and the total stress response

consists of only the dynamic component.

However, for the critical Txv response in the core region, the dynamic component
overwhelms the static component for all excitation cases. On the other hand, for the 1mm
responses in the base gravel layer due to general excitation, the pseudo-static component is

dominant and the other two components can be neglected.

8.1.4 Reliability against local failure

As a simple illustration of the failure analysis of earth dams subjected to a
combination of gravity loads and SVEGM, the reliability against local shear failure is
assessed. The effective shear strength of soils is determined by the Mohr-Coulomb criterion
and the static pore water pressure on the upstream side is accounted for. The reliability
index against local failure was computed at each node and shown as shaded image contours
over the entire dam for the identical and general excitations. The soils on the upstream and
downstream surfaces and at around two-thirds the height of the dam violate the Mohr-
Coulomb criterion due to identical excitation having a r.m.s. acceleration of 0.09g. For the

SVEGM, the potential failure zones become larger, and include the base gravel material

212

because higher stresses occur in this region. Reasons why this simplistic illustration is

likely to be over—conservative are discussed.

8.1.5 Effect of wave propagation

The wave passage effect is not so significant as coherency decay if the ground motion
is signiﬁcantly incoherent. In general, seismic waves with lower apparent velocities cause
smaller displacements but result in larger stresses at the base due to out-of—phase motions.
For upstream-downstream excitation, the responses to SV/P waves propagating in the
upstream-downstream direction are more critical than for SH waves propagating in the
longitudinal direction at the same apparent velocity. However, the critical responses due to
these two kinds of waves do not occur at the same locations. SH waves cause the critical
response locations to shift in the longitudinal direction. The shift is more signiﬁcant for

responses corresponding to lower SH wave speeds.

Variation of the incidence angle of the seismic waves within realistic limits (30° to

600 to the horizontal) did not significantly affect the responses in this study.

8.1.6 Effect of coherency function

In the last decade, a number of empirical coherency functions have been proposed
based on strong motion records. In this study, in addition to Harichandran’s model (1986),
coherency functions proposed by Abrahamson (1993), Novak (1987), Luco & Wong
(1986) and Hao (1989) models were also used to investigate the effect of the different

models on the Santa Felicia dam.

The x-displacement response is not very sensitive to the coherency model. Higher
coherency decay models give rise to smaller x-displacements. The largest difference
between the critical 3o x-displacements is about 10% (between Abrahamson’s and Hao’s
Event 30 models). For the u+3o 7m response, higher coherency decay models result in
lower 7mm, but the largest difference between the critical u+3o ynm values is about 12%

(between Abrahamson’s and Hao’s Event 30 models).

213

Stress responses are very sensitive to the coherency functions. The Tmax response
using Novak’s model (which was fitted to Harichandran’s model in this study) has the
largest value among the models investigated. Hao’s model calibrated to SMART-l Events
45 and 30, respectively, yields 1mm responses that are closer to Harichandran’s model
which is calibrated to SMART-1 Event 20. Surprisingly, the tmax response corresponding
to Abrahamson’s coherency model is very similar to that due to identical excitation. This
is because the stress response at the base is dominated by low frequency excitations (i.e.,
ground displacements) while displacements and strains are governed by medium frequency
motions. At frequencies less than 0.5 Hz, Abrahamson’s model yields essentially unit
coherency for most separations. Consequently, the u+3o 1mm response at the base are
essentially the same as those due to identical excitation. Harichandran’s and Hao’s models
yield significantly lower coherencies at lower frequencies than Abrahamson’s model (i.e.,
ground displacements that are more incoherent), which is why they yield large stresses at
the base. However, coherencies in the frequencies range from O to 0.5 Hz cannot be reliably
estimated from data using conventional spectral analysis, and yet the coherency in this
frequency range strongly affects the stresses in stiff material at the base. It is therefore
extremely important to focus future research on accurately describing the coherency at low

frequencies.

8.1.7 FOSM method for estimating the statistics of the maximum shear stress

The 114-30' 1mm response at each node is non-linearly related to the Cartesian stresses,
and its statistics were computed based on 10,000 Monte Carlo simulations. However, for
more efficient computation, the first-order second-order (FOSM) method was also
employed to approximately estimate the statistics of 1mm. The FOSM method using the
square of each Cartesian stress component as the basic variable was found to be most
suitable. The use of correction factors for the mean and standard deviation of 12mm enabled
the FOSM estimates to be sufficiently accurate, but the correction factors needed to be

calibrated using simulation results, which partially offset the computational saving of the

214

FOSM approach. Correction factors for the mean and standard deviation of Im a x of 1.04 and
0.95, respectively, for general excitation and 0.86 and 0.83, respectively, for identical
excitation produced good results, but the optimal correction factors for the general

excitation may change for different coherency parameters than the ones used.

8.1.8 Simplified ground motion excitation model

A finite element—based random vibration analysis of a 3-D earth dam model subjected
to SVEGM is time-consuming and costly. Therefore, simpliﬁed modeling and analysis that

is capable of yielding acceptable results were explored.

Attention was focussed on simplifying the excitation for use with a 2—D shear beam
model which could at least represent the excitation incoherence along the length of the dam.
The base of the dam was divided into several strips along the length of the dam to simulate
the bases of truncated wedge elements of the 2-D shear beam model. To be consistent with
each wedge element of a 2-D shear beam model having a single DOF, the excitation within
each strip along the base was taken to be fully correlated. However, the ground motion
between two different strips could be incoherent. The coherency between the excitations at
two different strips was taken to be the coherency evaluated at the average separation
between all pairs of points in the two strips. Since only separation in the longitudinal
direction of the dam (perpendicular to the length direction of strips) was used to account

for the incoherency effect, the simplified excitation model is called a l-D excitation model.

Using the 1-D excitation model, the maximum 36 x-displacement response is only
1.3% higher than that using the general SVEGM model, while the response due to identical
excitation is 7.6% higher than that due to SVEGM. The maximum u+30 7mm response is
about 1.9% larger for the l-D simplified model compared to the general model, while it is
10.3% higher for the identical excitation model. Apparently, the improvement on the

displacement and 'ymmr responses using the ID excitation model is signiﬁcant.

215

For the maximum “+30 tmax response, the 1-D excitation model does not yield good
results in the gravel material since it is very sensitive to the relative displacement between
each pair of strips, but the predictions at locations away from the boundary are reasonable.
The 1-D excitation model yields critical u+3o rm responses in the core as well as in the
shell that are only 2% larger than that due to general excitation. Therefore, the use of a 2-D
shear beam model in conjunction with the simplified l-D excitation model would be
expected to yield reasonable and slightly conservative results at locations distant from the
base. To predict stress responses at the base in the gravel streambed, the pseudo-static
component from a 3-D analysis can be used as long as ground displacements are not highly
coherent because this component dominates the total response. As a result, in many cases

a costly 3-D dynamic analysis can be avoided.

The effect of using a different number of strips on the base for the l-D excitation
model was evaluated. Models with 9, 15 and 21 strips were used in this study. It was found
that the displacement, 7mm and 1mm responses using the 2 l —strip model, were all closer to
those due to the general excitation than using the 9- and 15-strip models, except for the 1mm
response in the gravel streambed. However, the improvement in the responses from the 9-
to 21-strip models was not very significantly. In general, when using a 2-D shear beam
model, accuracy considerations would govern the discretization along the length of the dam

and more than 9 elements would be expected at the base.

8.2 Recommendations for future research

8.2.1 Topographic effects

This research is restricted to assessing the effect of spatial variation due to
incoherence and global wave propagation effects on earth dam responses. It is well known
that SVEGM can also occur due to ampliﬁcation of ground motions that is dependent on
the canyon shape, the relative rigidities of the canyon and the dam, and the type of

excitation wave (SH, SV, P, Rayleigh, etc.). In general, intensity and phase variations due

216

to topographic effects should be combined with the spatial variation due to incoherence
effects when specifying the input motions at the base of the dam.

8.2.2 2-D shear beam model and SVEGM
In Chapter 7, techniques to simplify the general SVEGM model to a 1-D excitation

model that can be used with a 2-D shear beam model of the dam are proposed. The
simplified excitation model was used with the 3-D ﬁnite element model of the dam, and the
results were found to be slightly conservative but acceptable for most responses. Although
the proposed excitation model should be applicable to a 2-D shear beam model, a 2-D shear
beam model of the dam was not actually used. The use of 2-D shear beam models for earth
dams subjected to conventional excitation models has been found to cause slightly higher
natural frequencies and larger responses than 3-D models and hence conservative results
are expected. However, this postulate should be veriﬁed by investigating the responses of

a 2-D shear beam model excited by the simplified l-D SVEGM model.

8.2.3 Nonlinear analysis

In reality, soil exhibits nonlinear hysteretic behavior. When large responses occur
under strong ground motion, linear random vibration analysis might not be adequate for
predicting the dam response. The restoring force of a hysteretic system depends not only
on the instantaneous displacement but also on its past history. For lightly damped systems,
the method of equivalent linearization including the effects of strain-dependent shear

modulus may be used for an approximate solution by performing iterative analysis.

8.2.4 Dynamic pore water pressure

Increasing pore water pressure during an earthquake may affect the stability of earth
dams significantly. More realistic models of the saturated granular soil, account for the
two-phase fluid-solid mixture, can be used to account for the build-up of pore water
pressure during the dynamic response. The use of such advanced models with SVEGM

input may yield insight into the effect of SVEGM on liquefaction failure.

APPENDICES

APPENDIX A
Outline of the Use of I-DEAS VI.i Computer Software

A.1 Introduction
The I-DEAS families including Solid_M0deling, FE_Modeling_&_Analysis and
System_Dynamics_Analysis were used in the modeling and analysis of the Santa Felicia

dam. Each one offers a set of interactive, graphically-oriented tasks.

0 Solid_M0deling: The Solid Modeling application offers a full range of tools needed
to develop geometric concepts and detailed mechanical designs for parts,

assemblies, and mechanism.

0 FE_M0deling_&_Analysis: I-DEAS Finite Element Modeling (FEM) is a
mechanical design automation tool which can be used to build a complete finite
element model, including physical and material properties, loads, and boundary
conditions. With FEM, the linear statics and dynamics behavior of mechanical

component and structures can be modeled and analyzed.

- System_Dynamics_Analysis: The System Dynamics Analysis family is an interac-
tive, graphically-oriented system modeling and analysis module which can be used
to simulate the dynamic response of complex mechanical systems. This application
evaluates the dynamic performance in terms of modal, frequency, and transient re-

sponse of interconnected linear systems.

A.2 Finite Element Modeling Procedures
The Santa Felicia earth dam was modeled as a three-dimensional inhomogeneous
finite element model using I—DEAS VI.i. The following is a brief description of the

modeling procedures:

1. The Solid_M0deling/0bject_M0deling/Credte command was used to generate a

block for each material zone, and Construct/Cut was used to trim its dimension to

217

2.

3.

218
match the geometry of the dam. Twenty-two objects are needed to satisfy the
discretization requirement for elements with different material properties. Since the
dam is assumed to be symmetric about the mid-plane of the dam axis, only half of

the dam is generated as shown in Fig. A. 1.

The Solid_M0deling/Assembly_M0deIing/Create command was used to put
together parts which were modeled and stored as objects into a system. Because
each object was created according to its own coordinate system, some orientations
to desired location need to be done in the assemblage of these parts to an entire dam

model.

The FE_M0deling_&_Analysis/Geometry_Modeling/Su(face/Create command
was used to import the dam system from the solid modeling module to the FE
module. In this step, the default for the Model _ﬁle/I’ olerances/Point_C0incidence

needed adjustment to avoid dangerously large or unnecessarily tight tolerances.

 

 

 

 

 

 

 

Figure A.1 Display of half of the Santa Felicia dam model consisting of twenty-two objects

219

The Mesh_Creati0n/Mesh_Areas and Mesh_Creati0n/Mesh_Volume commands
were used to create mesh areas and mesh volumes, respectively. The
Mesh_Creati0n task can automatically apply free meshing to the geometry as it is
extracted from solid objects or assemblies. Material properties were defined for

each volume at this stage.

The Mesh_Size/Free_Meshing/Settings command was used to specify global
element sizes, and the Generate_Mesh/Solids command was used to generate the

finite element mesh. Linear tetrahedral solid elements were used throughout.

Coincident nodes lying on the interfaces of the various assembled objects were
merged to obtain a single monolithic model. Nodes were renumbered for
convenience. After these steps it was necessary to reload the graphics database

using the Model_FiIe/Reload_Graphics_Db command.

The F E_M0deling_&_Analysis/Boundary_C0nditi0ns/Restraints command was
used to specify boundary conditions. The nodal degrees of freedom (DOF) for solid
elements consist of the three translation components (X, Y and Z translations) and
these were all restrained at the nodes on the boundary between the dam and the

canyon.

A.3 Steps for Free Vibration Analysis

Free vibration analysis of the Santa Felicia dam model was performed using I-DEAS

so that its natural frequencies and vibration modes were obtained. Subsequent to the ﬁnite

element modeling procedures described in the last section, the following steps were used

to perform a free vibration analysis:

1.

The Case_Management/Create command was used to generate a case set. For free

vibration analysis, the case set includes only restraint conditions.

The Model_Solution/Normal_M0de_Dynamics/Case_Set command was used to

specify the case set to be used.

220
The Method/Simult_Vect0r_IteraIi/Solution_C0ntrol command was used to define

the number of ﬂexible modes to be solved.

Modes, Stresses and Strains were selected for output. This yielded the stresses and

strains corresponding to each mode.

A.4 Deterministic Dynamic Analysis

The steps using I-DEAS VI.i to conduct deterministic dynamic analysis are briefly

described below.

1.

The FE_M0deling_&_Analysis/Boundary_C0nditi0ns/Restraints command was to
set the upstream-downstream DOF at the nodes on the boundary to be free, and the
D0f_Sets command was used to deﬁne these as kinematic DOF. In I-DEAS VI.i,

enforced motion can only be applied along kinematic DOF.

The Solution/Control command was used to specify the case set to be used and the
number of rigid body modes. The N0rmal_M0de_Dynamics command was used to

perform modal analysis.

The C0mp0nent_Deﬁnition/Create__C0mp0nent/Analytical_m0dal command in the
System_Dynamics_Analysis module was used to import analytical modes from the

F E_Modeling_&_Analysis module. Damping ratios were specified at this step.

The System_Deﬁnition/Components/Degrees_0f_Freedom/Physical command was

used to restrain all kinematic DOFs.

The System_Soluti0n/I'ransfonnation_Meth0d command was used to solve damped

vibration modes.

The Excitati0n_Deﬁnition/Functions command was used to create an acceleration
excitation function at the restrained nodes. The time history of velocity and
displacement were obtained from the acceleration function using the Integrate

command.

221

7. The L0ad_Cases command was used to combine acceleration, velocity and

displacement functions into a load case.

8. The Response/Madal_Properties/M0de_Shapes command was used to activate the
number of vibration modes used in the enforced motion analysis. The load case in

step 7 was set to be active using Response/Load_Set.

9. The Response_DOF_Sets command was used to select component entities and

physical DOF at nodes where responses were to be evaluated.

10. The Evaluate/Options/Limitsﬂ‘ime command was used to specify the desired length
of the responses. The types of responses were defined using the Response_Form

command and the analysis was performed using the Evaluate/Response command.

A.5 Generating [M] and [A] Matrices

In I-DEAS, the mass matrix [MFF] in equation (4.17) corresponding to DOF is
identical to the matrix [Mu] corresponding to independent DOF, and [MFR] is identical to
[M‘s]. Program files written in a special language are used in I-DEAS to drive each type of
computation. By modifying these program files, intermediate or specialized results can be

output. The program ﬁles and modification are brieﬂy described below:

1. Copy files nbb42.prg and svi.prg from I-DEAS programs library directory to the

directory in which normal mode analysis will be performed.

2. Edit the file nbb42.prg so that it calls the local copy of svi.prg, instead of

svnores.prg. Change the lines

 

C .' TO RUN A USER DEFINE PROGRAM FILE CHANGE “HMPACK” TO
“MODEL_FILE ”

C .' IN THE FOLLOWING LINE

K : /I-IMPACK PROGRAM_FILE RUN

K : "SVNORES.PRG"

 

 

 

To:

222

 

 

K : /MODEL_FILE PROGRAM_FILE RUN
K : "SVI.PRG"

 

 

Edit the file svi.prg to write the mass matrix MII immediately after
EIGENRATPRG is called, because in the process of checking orthogonality ratios
and eigenratios M11 is deleted. After the ﬁrst line shown, add the remaining lines

that follow in the boxed section below:

 

 

C .' CALCULATE AND LOG ORTHOGONALITY RATIOS AND EIGENRATIOS

.' WRITE MASS MATRIX MII
TO UNIV FILE MII. UNV
: /LOG MESSAGE "Writing mass matrix M1] to ﬁle "
: /HMPACK UNIVERSAL_FILE WRITE_READABLE MII
' " MII.UNV "

 

ENWGOGG

 

 

Edit the file svi.prg to write MIS immediately after SREACTPRG is called. In the
process of computing reaction forces, M18 is deleted. After the ﬁrst line shown, add

the remaining lines that follow in the boxed section below:

 

 

C: CALCULATE REACTION FORCES

.' WRITE MASS MATRIX MIS

TO UNIV FILE MIS. UNV

.' /LOG MESSAGE "Writing mass matrix MIS to ﬁle "

: /HMPA CK UNI VERSAL_FILE WRITE_READABLE MIS
: " MIS. UN V "

 

NRKGCGQ

 

 

Run the normal mode solution. Use the Executi0n_0pti0ns command, turn off

proﬁle reduction so that the mass matrix components will be sequenced using the

223

user-specified node numbers. By default I-DEAS internally renumbers nodes to
minimize the bandwidth of the stiffness matrix and this shufﬂes the elements of the
mass matrix. Since the mode shape matrix and displacement vector are output
according to the user-specified node number sequence, the elements of the mass

matrix must be consistent with these.

Use the simultaneous vector iteration method and select the User_Deﬁned method.
The program will prompt for the name of the program ﬁle to run. Enter the name

given to the copy of nbb42.prg.

Use the 0utput_Selection command to select reaction forces as well as vibration
modes, because M13 is created only when the reaction forces are selected for

output.

Watch the solution log to ensure that the proper version of the program runs, and
that the mass matrix is written. If the files are edited at the wrong locations, a
message saying that the M11 and/or MIS matrix does not exist will usually be

reported.

The universal files MII.UNV and MIS.UNV contain the mass matrices. The format
of these files is documented in the Model Solution Manual, in the Open
Architecture section. It should be noted that SI units are used in these ﬁles. If the

model file is based on other units, appropriate conversion of units is necessary.

The [A] matrix is expressed as [A] = — [KFF] ‘1 [Km] in equation (4.9). Each

column of [A] consists of the deformed shapes of the model subjected to a unit

displacement at each restrained node. The [A] matrix can be obtained directly in I-DEAS

using static constraint analysis, and hence there is no need to obtain the stiffness matrices

and invert [K FF]. The following steps are used to obtain the [A] matrix:

Since only upstream-downstream excitation (x-direction) is considered, the unit

displacements are applied only in this direction at the boundary nodes. All

224
x-translation DOFs of restrained nodes were set to be active using the
Boundary_Conditions/Dof_Sets command, and then deﬁned to be connection DOF
using the Case_Management command. The other two translation components (y

and 2 components) are kept restrained.

2. The ModeI_Solution/Constraint_Mode_Dynamics command was used to select the
case set specified in step 1. The other procedures are similar to those in normal
mode analysis. Since displacement, stress and strain responses are of interest, all of
these were selected for output. It is of interest to note that the stress or strain output

file occupied about 500 MB of storage space.

3. Stress and strain constraint modes are computed for each element. The average of
the element stresses and strains at nodes from adjacent elements was used to
represent the nodal stresses and strains. A computer program was written to extract

the nodal values from the I-DEAS output ﬁles.

4. It should be noted that the static constraint modes are derived corresponding to a
one meter displacement at connection DOF. If the model uses other units, the output

should be converted appropriately.

A.6 Techniques to Display Response Contours

The Manage_Models command in the I-DEAS Finite Element Modeling family can
be used to read and write universal files. Universal ﬁles provide a means of passing data in
and out of I-DEAS model files. An I-DEAS universal file is a sequential formatted file
consisting of one or more I-DEAS universal datasets. An universal dataset is a group of
records that define the properties of an I-DEAS entity such as a proﬁle or a point. The I-
DEAS Core Utilities User’s Guide documents the record format for each type of universal
dataset. In order to display response contours through I-DEAS, the first step is to write
nodal response values in the format required for the universal file. Fig. A.2 shows a partial

dataset of a typical displacement ﬁle.

225

-1
151
lusr/hpideas_tmp/chenm/con/contour
Santa Felicia Earth Dam
SDRC I-DEAS VI.i: Monitor
05-DEC-94 11:04:49 6 0
Never Never
SDRC I-DEAS VI.i: FE_Modcling_&_Analysis
05-DEC-94 12:46:50
-1
-l
164
2British grav 2
3.28083989501 312330E+00 2248089430997 10450E-0 l 1 .8()000000000000000E+00
4.59670000000000000E+02
-l
-l
800
l
WORKING_SET1
-1
-l
770
l 0
MAIN
-1
-1
771
1 1 1
FE MODEL]
-1
-1
55
General Excitation (38D)
Model Solution
NONE
ThetimeofAnalysis was 05-DEC-94 11:33:41
NONE
1 2 3 8 2 6
2 3 0 O
0.00000E+00 3.43029E+08 0.00000E+00
1
.28800E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00
2
.28800E+00 .00000E+00 .00000E+00 .OOOOOE+00 .00000E+00 .00000E+00
3
.28800E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00

Figure A.2 Partial dataset of an universal file

226

The upper part (the first line down to the seventh line from the bottom shown) can be
reproduced by copying it from any contour universal ﬁle written from within I-DEAS. The
remaining part specifies nodal numbers and nodal values, which have been replaced by the
values of the mean plus three standard deviations of the displacement, stress or strain
responses in order to display contours of these quantities. Using I-DEAS, the modified
universal file is read back to create a new analysis dataset. Subsequently, contours of the
new data component can be displayed using the Post_Processing command. I-DEAS
assumes that the values read from the universal file are the six displacements at each node,

but this is of no real signiﬁcance.

A.7 Gravity-induced Stresses

The stresses due to gravity were obtained using the following steps:

1. The finite element model and restraint boundary condition used for the free vibra-

tion analysis were used.

2. The Boundary_Conditions/Structural_Loads/Acceleration_Load/Create command
was used to apply an acceleration vector of Ig in the vertical direction. An
acceleration vector represents the loading effects from the acceleration on the entire

model.

3. The Case_Management/Modtfv command was used to add the acceleration load to

the case set.

4. The Linear_Statics command was used to solve the static loading problem and the

stress response was selected for output.

APPENDIX B

Comparison between Seismograph Records and Computed Responses
for the Northridge Earthquake of January 17, 1994

The Northridge earthquake having a magnitude of M = 6.7 occurred at 4:30 am (PST)
on January 17, 1994 under the north-western end of the San Fernando Valley, Los Angeles.
The epicenter location determined by the US. Geological Survey is 34.213 N and
1 18.537 W, with a focal depth of 18.4 km. The Northridge earthquake was recorded by two
seismographs located on the Santa Felicia dam; one on the downstream abutment and the
other on the crest of the dam (Stewart et al. 1994). Fig. B] shows the location of the
seismographs and directions in which the ground accelerations were recorded. The
instrument and baseline-corrected acceleration, velocity and displacement in the upstream-
downstream (U-D) direction at the downstream station are shown in Figs B2, 3.3 and 3.4,

respectively. The peak U-D acceleration is 0.27g.

The U-D ground motion recorded at the downstream station was used as input to the
3-D finite element model of the dam and the response at the node (x = 15 ft, y = 275 ft,
2 = -71 ft) closet to the crest station was computed. Fig. B.5 shows the recorded and
computed displacement responses due to U-D excitation only. The station at the crest
triggered 1.18 seconds before the downstream station, and hence the recorded motions
were translated to the left by 1.18 seconds. Although the peak displacements are in
reasonable agreement, the computed response has many more oscillations than the

recorded response indicating that the assumed modal damping ratio of 0.06 is too low.

In reality, the Northridge earthquake ground motion contains three components in the
upstream-downstream, vertical and longitudinal directions. To examine the adequacy of
the earth dam model, all three recorded components at the downstream station were applied

as excitations at those nodes on the boundary between the dam and the bedrock. Fig. B6

227

228

 

 

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-300 — -
_40011111111111111111111111[11111111111111.
0123456789101112131415161718192(
Time (second)
Figure B.2 Ground accelerations of Northridge earthquake in the upstream-downstream direction
60 I I I I Ii I I I l I I f I I I I I I I I I
40 - -

20

Velocity (cm/sec)
O

 

I

 

..:”: M [\LAVJl AL rA'\/\ AVAVA /\
W V W V “v \/

mlrlnlnlilllnlllil11.1111

l 1

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123456789101112
Time (second)

l3

14

15

16

17

18

19 20

Figure B.3 Ground velocities of Northridge earthquake in the upstream-downstream direction

30

20

-10

Displacement (cm)
O

230

 

I

T

 

I

 

 

 

 

-20 - -<
_30.11111111111.111111111111L4111111111.1.
0 1 2 3 4 S 6 7 8 9 10 ll 12 13 14 15 16 17 18 19 20
Time (second)
Figure B.4 Ground displacements of Northridge earthquake in the upstream-downstream direction
30 I I I I I I I I I I I I I I—r
— Recorded
------ Computed (C: 0.06)
20 - -

10

-10

Displacement (cm)
0

 

 

 

I

 

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1.1.1.111.

 

123456789101112 1920

Time (second)

l3 14 15 16 17 18

Figure 8.5 Comparison of upstream-downstream displacement records at the crest

with computed response due to upstream-downstream excitation

231

shows the time history of computed U-D displacement at the crest corresponding to modal
damping ratios of 0.06 and 0.25. It is seen that the displacement corresponding to a
damping ratio of 0.06 is higher for each vibration cycle than that corresponding to a
damping ratio of 0.25, and both responses are higher than the recorded response shown in
Fig. B.5. Fig. B.7 shows the response computed using a modal damping ratio of 0.45 as well
as the recorded displacement, and the two are in reasonable agreement although the
computed peak response is a little higher. Figs. B8 and B9 show the computed and
recorded velocity and acceleration traces, which also agree reasonably well. It appears
therefore that a modal damping ratio of 0.45 is more appropriate for the level of excitation
imparted by the Northridge earthquake. The Northridge data became available after most
of the research described in this study (which is based on a modal damping ratio of 0.06)

was completed.

 

 

 

 

 

30'l'l'l'I'l‘I’TTI'I'I'I'E‘TT
— Computed (C: 0.25)
------ Computed (C: 0.06)
20- ~
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'30_11141%1:1111 L11111111.L.l.l.l.1111"

 

 

 

0112 3 4 5 6 7 8 91011121314151617181920
Time (second)

Figure 8.6 Comparison between the computed upstream-downstream displacement responses
at the crest of the dam having damping ratios of 0.25 and 0.06.

232

 

30

20

10

-10

Displacement (cm)
O

I

I

IIII

T I r r Tj V

— Recorded
:::::: Computed (C: 0.45)

 

 

_

 

 

 

 

111111111111111111111111111111141

 

3456789

10 ll 12 13 14 15 l6 17 18 19 20

Time (second)

Figure 8.7 Comparison between computed displacement response of Santa Felicia dam with

60

40

20

Velocity (cm/sec)
O

a damping ratio of 0.45 and seismograph records

 

I

I

 

11.14I11111111111111111

— Recorded
------ Computed (C: 0.45)

 

 

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ll" WV‘WW

 

 

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3456789

10 11 12 13 14 15 16 17 18 19 20

Time (second)

Figure 8.8 Comparison between computed velocity response of Santa Felicia dam with

a damping ratio of 0.45 and seismograph records

233

400 r I I I t Iﬁ' I I I v I v I r I v I l

P — Recorded
------ Computed (C: 0.45)

 

 

 

 

 

 

. 2
Acceleration (cm/sec )

 

 

 

 

_4(X) 4L 1 1 141 l 1 l 1 l 1 l. 1 l 1 l 1 l 1 l 1 l 1 l 1 I 1 l 1 l 1 l 1 l 1 l 1 I 1

012 3 4 5 6 7 8 91011121314151617181920
Time (second)

 

Figure B.9 Comparison between computed displacement response of Santa Felicia dam
with a damping ratio of 0.45 and seismograph records

The maximum U-D displacement, velocity and acceleration for all the cases
investigated in this study are given in Table B]. The best agreement is obtained for the

three-component excitation and a modal damping ratio of 0.45.

234

TABLE B.1 Critical computed and recorded responses of the Santa Felicia dam
at the crest to the Northridge earthquake

 

 

 

 

 

 

 

Upstream] Three-component excitations
Response type downstream
(Upstream! . Recorded
d t excitation only
owns ream) (C =0.06) Q: 0,05 C: 0.25 C = 0.45
Displacement 16.2 30.4 22.6 19.3 15,6
(cm)

Velocity

(cm/sec) 125 155 82.3 57.9 38.]
Acceleration 1024 1039 485 326 243.8

(cm/secz)

 

 

 

 

 

 

 

 

 

 

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