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M ISCELLA N EO U S PAPER GL-79-13
RESPONSE AND STABILITY OF EARTH DAMS DURING STRONG EARTHQUAKES
by
S. K. Sarma Department of Civil Engineering Imperial College of Science and Technology, London, U. K.
June 1979 Final Report
Approved For Public Release; Distribution Unlimited
Prepared for Office, Chief of Engineers, U. S. Army Washington, D. C. 20314
Under Grant Agreement No. DAERO-75-G-OIO
Monitored by Geotechnical Laboratory U. S. Army Engineer Waterways Experiment Station P. O. Box 631, Vicksburg, Miss. 39180 i
library NOV 2 3 1979
Bureau of Reclamation penver, Colorado
Destroy this report when no longer needed. Do not return it to the originator.
The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents.
V BUREAU OF RECLAMATION DENVER LIBRARY /A 92067231 dr 92067231 Unni aggi -P-j SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER Miscellaneous Paper GL-79-^3
4. T IT L E (and Subtitle) 5. TYPE OF REPORT ft PERIOD COVERED
' RESPONSE AND STABILITY OF EARTH DAMS Final report DURING STRONG EARTHQUAKES 6. PERFORMING ORG. REPORT NUMBER
7. AUTHORfa; 8. CONTRACT OR GRANT NUMBERfs) S. K. Sarnia Grant Agreement Wo. D A E R O - 75 - G - 010
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK AREA ft WORK UNIT NUMBERS Department of Civil Engineering Imperial Coll'ege of Science and Technology CWIS Work Unit 311*+*+ London, United Kingdom 11. CONTROLLING OFFICE NAME AND ADDRESS REPORT DATE June 1979 / Office, Chief of Engineers, U. S. Army 13. NUMBER OF PAGES Washington, D. C. 2031*+ 269 14. MONITORING AGENCY NAME ft ADDRESS^// different from Controlling Office) 15. S E C U R IT Y CLASS, (of thia report) U. S. Army Engineer Waterways Experiment Station Unclassified Geotechnical Laboratory 15«. D E C L ASSI FI C A T IO N / DO W NG RADING P. 0. Box 631, Vicksburg, Miss. 39180 SCHEDULE
16. DISTRIBUTION STATEMENT (of thie Report)
Approved for public release; distribution unlimited.
17. DISTRIBUTION STATEMENT (of the abatract mitered in Block 20, if different from Report)
18. SUPPLEMENTARY NOTES
19. KEY WORDS (Continue on reverae aida if neceaaary and identify by block number) Earth dams Earthquake resistant structures Earthquakes Earthquake engineering Dam stability
20, A B S T R A C T fConCfisu« a m reverwm f t naceaeaty and identify by block number) This earth dam study considers the response of an earth dam resting on a layer underlain by a rigid base and subjected to a strong ground motion. The dam and the layer are assumed to be elastic, homogeneous, and of simple geo metric form. The properties of the dam and the layer may be different. Several combinations of the properties of the dam and the layer, as well as. several ratios of the height of the dam to the depth of the layer, are studied. (Continued)
OD ,;s rw M73 EDITION OF f NOV 65 IS OBSOLETE ______Unclassified______SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) Unni assi fi prl SECURITY CLASSIFICATION OF THIS PAGEfBTian Data Entered)
20. ABSTRACT (Continued).
All of these are subjected to nine strong motion acceleration records, and their response is calculated. At the same time, seismic coefficients for various sizes of sliding wedges are computed. The results of individual cases and their envelopes are presented in the spectral form for design purposes. Methods of stability analysis for static and pseudostatic conditions are examined, and two new methods are presented. One of these methods is suitable for analyzing existing slip surfaces, and the other for quick compu tation of the critical acceleration for a given surface. Since pore water pressure developed in the dam during an earthquake is an important factor, a method of stability analysis is developed that takes the excess dynamic pore water pressure into account. The method is based on limit equilibrium principle. Pore water pressures are introduced in the form of dynamic parameter An . The result is obtained in the form of a critical acceleration required to cause failure as a function of the number of cycles of earthquake load. Also shown is the computation of displacements of sliding wedges when the earthquake load is greater than the critical acceleration. Then the study is concluded with an example of the analysis of the effects of the San Fernando earthquake of 9 February 1971 on the Lopez Dam in California.
Unclassified
SECURITY CLASSIFICATION OF THIS PAGEfWTien Data Entered) PREFACE
The work reported herein was performed by Dr. S. K. Sarma, of the Department of Civil Engineering, Imperial College of Science and Technology, London, United Kingdom, under Grant Agreement No. DAER0-75-G-010, dated 6 November 197*+ • This grant was sponsored by the Civil Works Directorate, Office, Chief of Engineers, U. S. Army, under CWIS Work Unit 311*+*+» and administered by the U. S. Army Procurement Agency (Europe). The research was conducted during the period from November 197*+ to November 1976 under the direction of Drs. F. G. McLean, former Chief, and A. G. Franklin, Research Civil Engineer, Earthquake Engineering and Vibrations Division, Soils and Pavements Laboratory (now designated Geo technical Laboratory (GL)). General guidance was provided by Mr. J. P. Sale, Chief, and Mr. S. J. Johnson (retired), Special Assistant, GL. The research in this document was made possible through the support and sponsorship of the U. S. Government through its European Research Office. The various computer programs were developed on the CDC 6*+00 computer at Imperial College. Special acknowledgement is due to Professor N. N. Ambraseys for several discussions during the course of the project. Directors of the WES during the performance of this work and preparation of the report were COL G. H. Hilt, CE, and COL J. L. Cannon, CE. The Technical Director was Mr. F. R. Brown.
1 CONTENTS
Page PREFACE ...... 1 CONVERSION FACTORS, U. S. CUSTOMARY TO METRIC (Si) UNITS OF MEASUREMENT ...... 3 PART I: INTRODUCTION...... 1+ Background ...... 4 Purpose ...... 5 PART II: INERTIA FORCES AND SEISMIC COEFFICIENTS IN EARTH DAMS . 6 Inertia Forces ...... 6 Seismic Coefficient ...... 19 Discussions and Design Curves ...... 25 PART III: STABILITY ANALYSIS...... 3b Critical Acceleration of Slip Surfaces ...... 3^ Existing Methods of Stability Analysis ...... 35 New Methods of Stability Analysis ...... 39 Stability of Slopes Under Earthquake Conditions with Dynamic Pore Pressure ...... 1+7 PART IV: DISPLACEMENT CRITERION OF D E S I G N ...... 58 PART V: ANALYSIS OF THE LOPEZ D A M ...... 66 Seismic Coefficients ...... 66 Critical Accelerations ...... 73 Displacements ...... 73 PART VI: CONCLUSIONS...... 79 R E F E R E N C E S ...... 80 APPENDIX A: SOLUTION OF THE GOVERNING EQUATIONS FOR A DAM-FOUNDATION SYSTEM ...... A1 APPENDIX B: SEISMIC COEFFICIENTS ...... B1 APPENDIX C: ROOTS OF THE EQUATION m tan (qa ) = J (a )/J, (a ) . Cl n o n 1 n APPENDIX D: ACCELERATION RECORDS OF FIVE EARTHQUAKES ...... D1 APPENDIX E: SEISMIC COEFFICIENTS AND POINT ACCELERATION SPECTRA ...... El APPENDIX F: ENVELOPE OF SEISMIC COEFFICIENTS AND POINT ACCELERATION SPECTRA ...... FI APPENDIX G: NOTATION...... G1
2 CONVERSION FACTORS, U. S. CUSTOMARY TO METRIC (Si) UNITS OF MEASUREMENT
U. S. customary units of measurement used in this report can he con verted to metric (Si) units as follows: Multiply JJL To Obtain inches 2 .5 1* centimetres feet 0.301+8 metres pounds (mass) per 16.018U6 kilograms per cubic cubic foot metre pounds (force) 1+. 1+1+8222 newtons pounds (force) per 689^.757 pascals square inch pounds (force) per U7•88026 pascals square foot feet per second 0.301+8 metres per second degrees (angle) 0.017U5379 radians
3 RESPONSE AND STABILITY OF EARTH DAMS DURING STRONG EARTHQUAKES
PART I: INTRODUCTION
Background
1. With the failure of the Lover San Fernando Dam during the San Fernando earthquake of 9 February 1971, the attention is directed towards the problem of rational design of earth and rockfill dams against strong earthquake ground motion. With the aid of a high-speed computer and a very sophisticated method of finite element analysis using the nonlinear properties of the materials at the time of failure and knowing the ground motion characteristics of the San Fernando earthquake, it is possible to back-analyze and give the cause of this failure, Seed et al. (1973). But for the designer, who has yet to fix a dam section, to satisfy himself about the soil properties obtained from laboratory tests and assume the ground motion characteristics of a future event, the use of the finite element method is not economical. Moreover, the results obtained by a sophisticated method with doubtful input data are equally doubtful. Therefore, the use of a sophisticated method with doubtful input cannot be justified unless, of course, the method is equally costly and can be used with equal ease— not the case with the finite element method. On the other hand, simplified methods, such as those based on linear analysis of response and limit equilibrium analysis of stability, have produced results comparable to those of the finite element method. What is needed, therefore, is further refinement of the simplified approach without adding extra cost or time to the designer but with a more extensive use of information about the process of analysis. Thus, within the bounds of present-day knowl edge regarding the soil parameters and the ground motion input, a simplified method of approach to the design of earth dams is desirable.
k Purpose
2. The purpose is to provide a comprehensive method for such an analysis. In order to analyze the stability of a dam during an earth quake, the following information is needed: a.. The inertia forces that are being generated in the dam during an earthquake. b_. The resistance of the dam against these forces, along with the preexisting static forces. c. The possible consequences when the resistance is not sufficient to withstand these forces temporarily. These three factors are discussed in Parts II, III, and IV, respectively. In Part V, the proposed method is used to study the effect of the San Fernando earthquake on the Lopez Dam in California, United States of America.
5 PART II: INERTIA FORCES AND SEISMIC COEFFICIENTS IN EARTH DAMS
Inertia Forces
3. Earth dams are not absolutely rigid, and when excited into oscillations by strong earthquakes, they will respond in a manner that will be dictated by (a) the geometry of the dam, (b) the material properties of the dam and its foundation, and (c) the nature of the ground movements. Depending on circumstances, the ground movements may cause accelerations to develop in the body of the dam with associated fluctuating stresses, which may be larger or smaller than those of the ground.
k. During a strong earthquake, which may last from a few seconds to a few minutes, the ground accelerations will fluctuate with time in magnitude and direction. A dam to which this excitation is applied will respond in a manner that is determined by its resilience and its capacity to dissipate energy. The lower its capacity to dissipate energy, the greater will be its response. Moreover, the dam will seek out the periods of ground movements and respond strongly to those with which it can resonate. Consequently, an earthquake cannot be specified by an acceleration alone, nor can a dam be assumed to be rigid. 5. To make the problem amenable to analysis, it is necessary to make assumptions to obtain such solutions as various authors have done. For instance, Ambraseys and Sarma (1967) give a table of these solutions, which is reproduced here for easy reference (Table l). Ambraseys and Sarma (1967) and Sarma (1968) deal with the problem of the dam on the rigid foundation and present design curves. Since it is not always possible to assume the foundation to be rigid, it is thought better to produce another set of curves which will take the depth and the shear modulus of the foundation into consideration. 6. Thus, for the sake of analysis, the following assumptions are made regarding the geometric configuration of the dam. We define the geometry by Figure 1:
6 Table 1 Solutions for Dam Response Problem
One-Dimensional Solutions: Width of Thickness of Ground Shear Length of Dam Crest Side Slopes Overburden Acceleration Modulus L b 4> d g G(y) References L b = 0 g(t) constant Ambraseys (1960a, b, c) *1 *2 1960 1960 L 0 = 0 h *2 g(t, x) constant Hatanaka (1952, 1955) L 0 0 h * *2 sine constant Minami (i960) 00 0 - d g(t) constant h +2 Ambraseys (1960a) 00 0 0 h = *2 g(t) y 1/3 Rashid (1961) 2 L b = C\J 0 h g(t, x) y Ambraseys (1963)* * 00 n 1 b — = 0 g(t, x) Ambraseys (1963)* (n ^ 2 ) 1 *2 y
Two-dimensional solutions: Yokoo et al. (1957) Ishizaki and Hatakeyma (1962) Medvedev and Sinitsym (1965) Clough and Chopra (1965) 2 * Unpublished reports. G(y) = s(y) p. V 75. 777?:------777*------7-77*------7~77*------/S/S----- =------7 7 X \------7 7 ^ 903" rocK
Figure 1 . Geometry of dam-foundation system
a1. A triangular untruncated wedge is resting on a layer underlain by base rock. b. The length of the dam is much greater than its height. c. The dam is symmetrical about a vertical axis. <1. The slopes are flat. e. It vibrates in shear only.
7. Under these assumptions, a one-dimensional shear beam analysis can he applied. Slight deviations from these assumptions do not affect the result. For example, Ambraseys (i960) has shown that when the top width of the dam is less than 10 percent of the base width of the dam, the wedge behaves as an untruncated wedge. The same author has also shown that when the length of the dam is larger than four times the height, a one-dimensional solution applies. Two-dimensional solutions by Clough and Chopra (1965) and Hatanaka (1955) have shown that for slopes flatter than 1:1.5 a shear beam solution applies.
8 8. Regarding the properties of the material, we assume that the material for both the dam and its foundations are linearly elastic, homogeneous, and isotropic but that the properties of the dam and the foundations are different. Also, the material has an energy dissipating capacity or damping. 9. For small values of inertia forces, the dam behaves elasti cally, but the stress-strain behaviour of the material over the stress range in question is not elastic. However, with proper use of damping values, an average linear relationship may be used. The elastic proper ties of the dam are most conveniently introduced in terms of the shear wave velocity of the material. For zoned dams, when more than one type of soil is used, a weighted average value may be used. Elastic proper ties of soil generally vary with depth, but the actual variation is not * exactly known; therefore, its use cannot be justified in a simplified solution. Energy dissipating mechanisms 10. The various kinds of energy dissipating mechanisms present in the dam foundation system are as follows: a, . Viscous damping. In this case, energy loss is propor tional to the velocity. With pure viscous damping in elastic material, there is no residual displacement at the end of vibrations. In the vibration problems for analysis, this is the most convenient form of damping. b. Hysteretic damping due to nonlinearity. This is the energy loss due to nonrecoverable work done when the material is strained into an inelastic zone. In this case, there may be permanent deformations of the struc ture at the end of vibrations. This kind of damping varies with the level of strain. c_. Radiation damping. This is the energy loss due to non rigidity of the base rock when some of the vibrational energy is filtered back into the base rock. This depends on the relative properties of the structure and the base rock.
In reality, all three energy-dissipating mechanisms are present. Since viscous damping provides an easy method of solution, we lump all of these into one equivalent viscous damping term. As a result, the dis placement that is computed is much less than the real one. On the other
9 hand, the stresses and, therefore, the inertia forces that are com puted with equivalent viscous damping become realistic. Even though the pure viscous damping is a frequency-dependent characteristic, in vibration problems this damping is assumed to be constant for all frequencies. 11. There is no direct way to estimate the equivalent viscous damping. The only possible way is vibrating real structures and re calculating the damping from the response of the structure. Alterna tively, it may be possible to separate the radiation damping from the other two. In this case, the radiation is considered to be part of the problem, but the solution to this problem does not exist at present. 12. The nature of ground movements depends on various factors, such as the source mechanism, the distance of the site from the source, and the geological conditions of the wave path from the source to the site. The radiation of energy from the structure to the base rock also affects the input ground motion. 13. The earthquake parameter that can be predicted with some confidence is the maximum ground velocity. Otherwise, there is no way of predicting a future ground motion. One possible method is to look through the library of existing records of strong motion earthquakes and find the one that was produced under similar conditions of source, distance, and geology. An alternative is to select a few records that have different frequency characteristics and normalize the amplitudes to obtain the expected ground velocity. For the purpose of this analysis, we assume that the earthquake ground motion is an arbitrary disturbance,
Derivation of governing equations and solutions Ik. Let us consider the response of an elastic untruncated wedge, resting on an elastic foundation layer underlain by a rigid base (Figure l). We consider the vibrations to be in shear in one dimension only, i.e., perpendicular to the y-axis and in the plane of the paper. It can be shown that the equations governing the vibra tions of the dam foundation system when the base of the system is
10 subjected to a ground acceleration g(t)* are: a. For the dam
1 9ul 1 n- •• 1 (l) 7 7 + 7 3 r = 72 [ ui tcii tBlt)] •••
b. For the foundation layer
a2 3 U r 2 = ~ 2 [ “2 + 0Û2 + *(t)] ••• ( 2) 9y S
where displacement in the dam and the layer, respectively, U1U2 measured relative to the base
y vertical coordinate shear wave velocity in the dam and the layer, respectively S1’S2 ul5u2 absolute acceleration of a point on the dam and the layer, respectively c = damping coefficient for both dam and layer ux,u2 velocity of a point in the dam and the layer relative to the rock, respectively
15* These assume that the damping coefficient for both the dam and the foundation layer is the same. The boundary conditions are
1^ ( 11,t) = u2 (h,t) (3)
u2 (H,t) = 0 (b)
= 0 at y = 0 (5) 3y
For convenience, symbols and unusual abbreviations are listed and defined in the Notation (Appendix G).
11 3u 3u (6) aiW sa2W a t y = h where h = height of dam t = time H = total height from crest of dam to rock level = shear modulus of the material of the dam and the layer, respectively
The initial conditions are at rest, i.e.
U-^yjO) = u2 (y,0) = û-^y-jO) = ù2 (y,0) = 0 (T)
l6. The solutions of these équations give (Ambraseys (1960b) and also Appendix A):
2Jo(any/h) u. (j,t) = ^ - Sdn ; 0 < y < h (8) a P (q,m,n) n=l n o
co 2M ( y ) (9) U, ,(y,t) = y - — ° - ... , Sdn ; h i y - H - Z j a P (q,m,n) n=l where - Bessel function of the first kind of order zero and the first kind of order one, respectively
a^ = n^*1 root of the equation m tan q|(an)/Jl(a„) (10)
i M 2 2/ — v 2/_ \ P (q,m,n) = mq + m sin y^anj + COS COS
- Z T s i n (qaj cos (qaj (id an
12 MQ(y) (12)
q = m(H-h)P2y , (hPp (13)
(lU) m s 1P i/S2P 2
J g(x)e sin ^ü)n(t_T)J dT (15) dn 0) n o
0) damped and undamped circular frequency, n respectively, of the dam foundation system (16)
0) ——— a = undamped circular frequency of the on h n n^h mode of the dam foundation system (IT)
C X damping as a factor of critical, 2w on assumed the same for all modes (18)
t = variable of integration
17. There is no way of proving, at least not to the knowledge of the author, that the solutions ux(y,t) and u2(y,t) obtained by applying the Laplace transform technique satisfy the governing equa tions. Therefore, we apply the limit tests. For H=h , the solution ^ (y,t) converges to the well-known case of the dam on the rigid base which is
2J (a y/h) u \ O n g (19) 1 Z# anJl (an ) ^
where a^ represents root of the equation JQ(an) = ^ • ^or ^-0 5 the solution u^ converges to the well-known case of the layer on a
rigid base, which is
13
A ex ** cos \ (2n-l)ir/2 y/Hl U 2 I <-i>n (2n-l) ir bdn (2 0 ) n=l
1 8 . There is another indirect test that can be applied to test the solution, which is based upon the transient part of the solution. The solution obtained through the Laplace transform technique includes the transient part of the solution. For the undamped solution, the relative displacements u^ and u2 should equal the ground displace ment at any time before the first arrival of the wave to the point in question. Solution u^ is obtained as convolutions of-two functions F^ and g , and u^ is obtained as convolutions of F^ and g , where for the undamped case
2Jo (any /h) sin ('s1an fc/h ) Fx (y,t) ; h _> y >_ 0 (21) I a P (q,m,n) S.a /h n o 1 n
2Mo(y) sin ( Slant/h) F2 (y,t) ; H > y >_ h (22 ) x a P (q,m,n) S a /h n o ’ 1 n
If the solution is true, then
F2 (y,t) = t for 0 <_ t <_ (H-y)/S (23)
and
F1 (y,t) = t for 0 < t < [(H-h)/S2 + (h-y)/sj (2k)
19. For a set of typical values of m , q , and y , these two functions are evaluated numerically for t increasing from zero. Func tion F2 was found to converge rapidly to t during the period con cerned. For values of y/h > 0.2 , the function F^ also converged to
lU
k. a. Untruncated wedge on a layer t (Figure 2); for smaller values of y/h , particularly for y = 0 , the convergence is very poor, and it is assumed that it will converge to t with very large n . 20. If the two solutions are correct, then
2Jo (any/h) (25) a P (q,m,n) 2n o
2MQ (y) 1 (26) a P (q,m,n) I*n = 2n o where d> and il; are the n^*1 mode shapes for the dam and the layer, n n respectively. 21. For the set of typical values of m , q , and y , the con vergence of £<|> and is tested and found to he not uniform at all points of the dam (Figure 3). However, convergence can he achieved with very large numbers of modes except at the top part of the dam. 22. Double differentiation of functions and u^ now yields
u^abs) = + g = (27) *->y
u2 (abs) = u2 + g = y * s (28) and L* Tn an
2 2 2 CO — À Cü t.. -v / , \ d x where S = — ------— f gi^)e~ “on sin [“ n (t " T)] an Jo
t .. _ + 2Ao) f g ) t - x ) co s Ito ( t —t )J dx (29) on %/ l n
From here on, we shall drop the term (abs) from u^ab s ) and u2 (abs) , and simply write u and u2 to refer to absolute accelerations, but
16 a. Test for E<|> at y/h = 0 b. Test for 2
c. Test for Figure 3. Convergence tests for £d> and Eili n : u , u2 an<^ ^1 5 an(i ^2 w^-11 still refer to relative displacement and velocity, respectively, 23. In view of the nonuniform convergence, the numerical values of and u^ are obtained in the following way:
oc
U^ ip S + ip S (31) 2 n an 2 n an VI = 1
If we assume that the response acceleration S (t) of a pendulum of an period T , such that T equals 0.05 sec or less, is equal to the ground acceleration, i.e., the pendulum behaves as a rigid one, then
U u cj> S + (32) n an >(t) -2 n n=l n+1
(33) V ^ *nSan+ «(t)(1 - J, n=l n=l
and j is so chosen that
T 0.05 sec
2 k . Thus, for a given ground acceleration g(t) , the complete response time history can be computed at various heights of the dam. From these time histories, the maximum value of un and u_ 1 max 2 max can be determined at different heights of the dam and the layer. We can determine the magnification Ma of ground acceleration (smax) a point
y = ah given by
18 un un 1 max 2 max M = ------or ------a g g max max
Seismic Coefficients
25. From the solutions of and u^ , it is seen that the accelerations induced in the dam and in its foundations are not constant with respect to height or in time. However, most of the stability analysis techniques prefer to have the accelerations constant with height. According to Seed and Martin (1966), Ambraseys and Sarma (1967), and Sarma (1968), if the potential sliding surface can be defined in advance, then an average seismic acceleration can be computed for that surface that will be a function of the ground vibrations and periods of the dam. 26. The "average seismic coefficient" is defined as that factor which, when multiplied by the weight of the sliding mass, gives the maximum inertia force that will act on the mass during a particular earthquake. Thus
K = (3U) where K = average seismic coefficient F = maxiumum inertia force on the potential sliding mass max ¥ = weight of the sliding mass
For convenience of comparison, we shall express the seismic coefficient in terms of the maximum ground acceleration instead of the acceleration due to gravity as
max K = Mg (35) ®max max where M represents total mass of the sliding body. In order to
19 compute K , therefore, we have to determine the total inertia force F and the total mass M of the sliding body. 27. In Figure Ua, consider an element of the sliding mass of thickness dy and width b(y) at a depth y from the crest. The mass of the elemental slice is
dm = pb(y) dy (36) where p is the mass density. The corresponding absolute acceleration at the depth y at an instant t is ii(y,t) where ii = u^ if y <_ h, and u = iig if y > h . Then the inertia force on this elemental slice is
dF = ü dm
= pb(y)ü(y,t) dy (37)
Then, the total mass M of the sliding hody is
ah M = f Pb(y) dy (38) o and the total force F is
ah F = J pb(y)U(y,t) dy (39) o
As can he seen from Equation 39, F is a function of both a and t . We can evaluate the complete time history of the function F and find the absolute maximum value of it at some instant that will give F max for a given base motion. Thus, knowing F , M , and the maximum max value of the ground acceleration g , we can determine K , which max a will be a function of a . For convenience of computation, we may determine a quantity
A(a,t) = ^ (lO)
20 U,
JLl^
21 and then compute
A(a,t) max K a (M) max
One-parameter sliding wedge 2 8 . In order to compute the average seismic coefficient, it is essential to define the potential sliding surface in advance. The most convenient is the one suggested by Seed and Martin (1966), Ambraseys and Sarnia (1967), as shown in Figure Vb. These sliding wedges have a horizontal base, and the slip surfaces pass through the apex of the dam. The seismic coefficients computed for these surfaces become independent of the base width of the sliding wedge, and these are also independent of the slopes of the dam. The only parameter that is required to de scribe such a surface is a the depth of the base of the sliding surface from the crest. As can be seen, these surfaces can be defined only in the dam and the foundation layer cannot be included. For such a surface,
M y ) = By (1*2 ) where B is a constant and
a(y,t) = y y . t ) = y V a n ♦ g(t) (l - £ +n (U3 ) n=l \ n=l ,
Then
“h J / J \ / y 1 +nS*m + 8(t) l1 " I O dy A(a,t) = o_____ nfl______\ n=l / ah / y dy
ah s an f y
22 Then
j A(a,t) = V c()l (a J-* n )San + 8 ( t) I 1 l +!„(«) (^) n=l [ n=l where ah / dy (j)l (a ) = °------n ' «h (U5) / y dy o
As shown in the Appendix B
(aa^) c|>ln(a) — 2 (U6) aa P (q,m,n) no
Four-parameter sliding wedge 29. Because of the inability of the one-parameter sliding wedge to include a part of the foundation layer, Ambraseys and Sarma (1967) devised the four-parameter sliding wedge as shown in Figure be. In these wedges, the four parameters are the depth of the point C from the crest (a), the slope of the dam, and the two angles 0 and i|It was assumed that the position of the point C is fixed. But in reality, the position of the point C can also be changed, which of course increases one parameter and seems unnecessary. 30. For such a surface, assuming = p
ah F = f pbCy)^ dy 0 £ a £ 1 (1+7) o or h ah F = J pb(y)u^ dy + J pb(y)u2 dy if a > 1 (U8) and ah M = J pb(y) dy (U9) o
23 Therefore, using Equations 32 and 33 for 0 a <_ 1 ,
ah j / j \ dy / b
This yields
[x - ♦ ■,„<•>] (51) Aa ’ I + V ^ a n + *(t) I n=l where ah J' i(y)4>n dy
Similarly, when a > 1 ,
A = a
h J / J \ ah j / J \ /b(y) dy+ b(y) dy I * n San+'«(t) b- I / n ) f o _n=l \ n=l /_ (53) ah f b(y) dy
This also yields,
A 4>ltn(a)S + §(t) <()Un(a) (5U) a an ■] where h ah f b(y) 1 (55) n ah f b(y) dy o
See Appendix B for <|>An(a) . 2h Discussions and Design Curves
31. The single parameter by which a dam-layer system can be represented is its fundamental period. The fundamental period and the higher modes are functions of four factors - , h , m , q . These represent (a) The shear wave velocity of the material of the dam = . (b) The height of the dam = h . (c) The shear wave velocity of the foundation pisi layer = ---- m .
(d) The height of the layer on which the dam is situated
h . m The fundamental period of the dam-layer system is then given by
h T = 2tt (56) aisi where a^ is the 1st root of the equation
Jo(an> m tan(qa ) = (57) n V a„> 32. The higher roots of the equation give the higher mode periods of the system. These periods are given by
T (58) n W an
Figure 5 gives the values of a^ for a combination of values of m and q . These curves can be used to determine the fundamental period of the dam-layer system. Some higher roots of the equation are given in a tabular form in the Appendix C. 33. The following earthquake acceleration records are chosen in order to determine the maximum accelerations at various heights of the dam. These records were chosen mainly because of their differences in frequency characteristics. The ground motions, acceleration, velocity,
25 9 Apni ) A xml (the shownin isexampleHuenemerecord) Port An (Appendix D).D9 nte eod hc.sue o h studyisforofLopez added theDam record, the used Anotherwhich.is and displacement spectrashowninandofareD1throughrecordsFigures displacementthese ae. Thisis record later. 0-5 IO 15 20 2-5 Figure 5. Fundamental period of dam-foundation systemofdam-foundation Fundamental period 5.Figure jC. b. The d. a. . e f_. San Fernando Earthquake 1971 - Deconvoluted and1971 San- FernandoEarthquakeDeconvoluted records arerecords on (India) Earthquake Koyna Koyna Dam Dam Record Koyna El-CentroRecord San Fernando Earthquake 1971San-FernandoEarthquakePacoimaDam modified version.modified Cholame,-Earthquake1966 Parkfield Port Hueneme Record 1957Port Record Hueneme Imperial19^0- EarthquakeValley hno ra o 2 ShandonNo. Array 2
6 1967 -
1 1 2 2 3
components components component component components
PHN/N /SS/C N 5 PORTHUENEME 18 /3/57
Figure 6. Ground motion records and spectra: Port Hueneme record of 1957 (sheet 1 of 2) ACCELERATION SPECTRA H/ /SC S OTUNM 1//7 '/• 18/3/57 PORTHUENEME S N /SS/C PHN/N * / . ' / . " / . ' / . ‘ / . 7 . 7 . ’ / . Y iue6 (sheet6 22)Figureof . / ' 7.7.’/.7.7.7.7.’/.Y "/."/.7.*/.7.7.777Y 3k. In Figures 7 and 8 the effects of m and q are shown. It appears that for m less than 0.2, there may he some convergence problems, particularly in the upper part of the dam, when the predomi nant period of the earthquake ground motion nearly coincides with the fundamental period of the dam-layer system. For values of m less than 0.7, there is some magnification in the response of the dam, compared with the response of the dam on the rigid base. The magnification is largest for values of m about 0.25. The magnification is also af fected by the value of q/m as shown in Figure 8. The magnification is different at different heights of the dam and at different periods of the dam-layer system. 35. Figure 9 is an explanation of Figures E-2 through E-127 (Ap pendix E) that show, in spectra form, the maximum response of dams of various fundamental periods, and m and q factors subjected to the first nine of the earthquake records listed above. The responses are calculated for a damping value of 20 percent critical for all modes. A maximum of 20 modes was used. The values of m and q that are chosen
are m = 0 0.5 1.0 q = 0 0.125 0.25 0.25 0.50 0.375 0.75 0.50 1.00 0.75 1.50 1.0 2.00 36. These sets of q values correspond to depth of layer to height of dam ratios of
H ~ — = 1/U, 1/2, 3/k, 1, 1-1/2 and 2 for p_L = p 2
At the same time, the seismic coefficients for one-parameter and four- parameter sliding wedges are computed and shown in Figure 9 in spectra form. In this figure, Figure 9a gives the values of the amplication factor M (the maximum acceleration at a point divided by the maximum a ground acceleration), for values of cn = 0, 0.2, 0.U, 0.6, 0.8, 1.0, 1.2, 1.It, 1.6, 1.8 if the depth of the layer is greater than, or equal to
29 AMPLIFICATION FACTOR Ma AMPLIFICATION FACTOR, M 30 MfiG.OF GROUNO «CClN. flV.SEISMIC COEF. l 1) SO 1-00 I.SO 2.00 2 .SO .1^00 0-SO I-00 I SO G TUNM ERHUK 1//7 S COMP. S N 18/3/57PGRTHUENEME EARTHQUAKE MAXIMUM ACCELERATION AT A POINTA AT MAXIMUM ACCELERATION iue9 Explanations forthrough FiguresE-2E-127 Figure9* 5 Q = 0.75 M = 0 -50 SEISMIC COEFFICIENTS FOR FOUR-PARAMETER SLIDING WEDGESLIDING FOUR-PARAMETER FOR COEFFICIENTS SEISMIC MAX. R. CL.01 G GR ACCLN.=0.17 . the height of the dam. For smaller depths of layer, the number of curves is curtailed at the corresponding point. For example, q/m =0.5 represents a layer depth which is half the height of the dam. Therefore, a = l.U is the last curve as the total depth from the crest of the dam to the base rock if y = 1.5h . 37* Figure 9t> gives the seismic coefficients for one- parameter sliding wedges (seis. coef. l) for values of a = 0.2, O.U. 0.6, 0.8, and 1.0 . 38. Figure 9c gives the seismic coefficients for four- parameter sliding wedges (seis. coef. k) for values of a = 0.2, 0.U, 0.6, 0.8, 1.0, 1.2, l.U, 1.6, 1.8, 2.0 when the layer is greater than, or equal to, the height of the dam. For smaller layer depths, the number of curves is curtailed at the corresponding a level, the last curve being an odd one representing a wedge touching the base rock. For example, for q/m = 0.5 5 the next to last curve is for a = l.k , and the last curve is for a = 1.5 . The other parameters for the four-parameter wedges are 0 = ^5 deg, ^ = 80 deg, slope = 1:2.5 . 39- Figure 10 is an explanation of Figures F-2 through F-15 (Ap pendix F) that show the envelopes of the nine earthquake acceleration records. These envelopes can be used to determine the maximum values of seismic coefficient when the frequency characteristics of probable ground motion are unknown.
32 MAXIMUM OF 9 RECORDS M =0.5 Q =0.75
oo oo
SEISMIC COEFFICIENTS FOR FOUR-PARAMETER SLIDING WEDGE
Figure 10. Explanation of Figures F-2 through F-15 PART III: STABILITY ANALYSIS
Critical Acceleration of Slip Surfaces
1+0. In Part II, the inertia forces and the seismic coefficients are determined for the dam and its foundations. The problem then is to determine whether the dam section will be able to withstand these forces. Ul. A very convenient term to describe whether a given slip surface will be able to withstand these forces is the critical accelera tion. The critical acceleration is defined as that horizontal acceler ation which is required to bring a slip surface to a state of limiting equilibrium, and this term is therefore analogous to the factor of safety. If the critical acceleration for the slip surface is larger than the corresponding seismic coefficient, then the surface has a factor of safety greater than one and is therefore safe. In contrast to the static problem of determining the minimum factor of safety for the entire section, a minimum critical acceleration for the section for the earthquake problem is not meaningful. As shown in Part II, the seismic coefficients for slip surfaces at different heights are different. The surface that produces the minimum critical acceleration could be safe against the corresponding seismic coefficient, but another surface with a higher critical acceleration may not be safe, particularly in the upper part of the dam. Therefore, it will be necessary to determine the critical acceleration for slip surfaces at various heights of the dam. 1+2. According to Sarma and Bhave (197*0 > without the change of strength characteristics and change of pore pressures, the critical acceleration for any given slip surface bears a relationship with the factor of safety; therefore, the critical acceleration can be used as a measure of the static factor of safety. However, any change in these conditions, such as the increase of pore pressures due to earthquake loading, removes these relationships. It has been shown (Sarma, 1973) that for a given dam section and for static condition or pseudo-static condition of earthquake loading with horizontal accelerations, the de termination of the critical acceleration is easier than the determination
3*+ of the factor of safety. The advantages of using a critical accelera tion solution instead of a direct factor of safety solution are men tioned in Sarma and Bhave (197^)• 1+3. In this analysis, it is assumed that the earthquake accelera tion is constant along the height. It is known that the earthquake acceleration is not constant along the height, and that is why the seismic coefficient was defined in Part II. Therefore, an analysis is made to study the effect of the acceleration distribution on the critical earthquake load (Valenzuela, 1975). From this limited study, it is found that the critical earthquake load for a given slip surface is minimum when the acceleration distribution is constant and is not minimum when the accelerations are increasing towards the crest. Thus, the use of a constant acceleration with the seismic coefficients for that surface produces conservative results. But the use of the critical acceleration with constant distribution cannot produce the critical surface, whereas the use of varying acceleration can. However, the use of varying acceleration for design purposes is not suggested, for the variation is not known in advance.
Existing Methods of Stability. Analysis
kk. It is accepted that the pore water pressures in a dam increase with cyclic loading induced by earthquakes. Since increase in pore pressures reduces the total strength, this plays an important part in the stability of the dam section. But before going into the solution of the problem for earthquake loading, it is necessary to examine the problem of the stability analysis both for static and earthquake condi tions. What follows below is an assessment of this problem. From this we will see that different techniques can be developed that will be suitable for specific types of problems. For example, to analyze an existing slip surface in a dam or a slope, the technique of solution should be different from that required for designing a dam. Since in the design of dams, one can reject a slip surface on the basis that the surface is not kinematically admissible, the existing slip surface
35 must produce a kinematically admissible solution. Thus, the assumptions may be varied to suit the problem. But whatever the technique, the so lution is based on the limit equilibrium principle. *+5. There are several methods of stability analysis in existence. All of these have certain things in common: a. These methods are based on the limit equilibrium principle which defines the factor of safety or the critical acceleration. b. They employ the Mohr-Coulomb failure criterion with rigid-plastic models. c. All of these apply a method of slices. d. They are based on assumed slip surfaces.
These methods differ in e. The shape of the assumed slip surfaces. f. The handling of the indeterminancy of the problem as shown later.
U6. Before proceeding further, we need to define a few terms. a. Simplified solution. A solution which does not satisfy all the static equilibrium conditions. Assumptions are made to obtain the solution in a simple form. Most of the available solutions, including Bishop’s simplified method (1955)* Kenney's method (1956)5 and even Janbu's generalized method of slices (1957), fall into this category. b. Rigorous solution. A solution which satisfies the complete equilibrium condition of statics. The implied forces obtained from the solution may not be acceptable. c. Acceptable solution. A rigorous solution which has the added criterion of the implied forces satisfying the criterion of acceptability. The basic criteria are: the forces must not violate the Mohr-Coulomb failure law, no tension is implied, and the direction of forces must be kinematically admissible. Our aim is to obtain an acceptable solution with the least possible effort. 1+7. In the method of slices, the possible failure mass is divided into n slices (Figure ll). For each slice, the known and unknown forces and their points of applications are shown. For n slices, we have the following (6n-2) unknowns:
36 n numbers of the normal force N n numbers of the shear force T (n-l) numbers of the body force E (n-l) numbers of the body force X (N-l) numbers of the point of application of E n numbers of the point of application of N 1 in the form of factor of safety or critical accelerations
+ c — o C — #«J3
Figure 11. Stability analysis by method of slices
k8. From the static equilibrium conditions of each slice, ¥e have the following equations:
£ Moment = 0 £ Vertical forces = 0 £ Horizontal forces = 0
The Mohr-Coulomb failure criterion gives T = f(N) ; hence, for n slices we have kn equations. In order to obtain a rigorous- solution, therefore, one has to make 2n-2 assumptions. If more assumptions are made, more unknowns must be introduced. Now let us look into the various
3T available solutions and see how these assumptions are made. a. Bishop’s simplified method: In this method, the assump tions are n numbers of points of application of N , and (n-l) numbers of the magnitude of X (X = 0). Since one more assumption is made than required and no unknown is introduced, one known condition must be sacrificed. In this case, the condition that is not satisfied is En+q i 0 • For circular slip surfaces this error is small, and the result is therefore mostly acceptable. b. Janbu’s generalized method of slices: The assumptions are n numbers of points of application of N , and (n-l) numbers of points of application of E . Again there is one more assumption than is required. In this case, the condition that is not satisfied is M ^ 0 for n the last slice. c. Kenney’s method: The same assumption as Janbu’s is made, and the equation that is not satisfied is En+1 ^ 0 • d. Morgenstern and Price method (1966): The assumptions are n numbers of point of application of N , and (n-l) numbers of the relationship between X and E . Since they have made one assumption more than is required, an extra unknown X is introduced. This method, there fore, satisfies static equilibrium conditions rigorously. e. Sarma’s method (1973): The assumptions are n numbers of points of application of N , and (n-l) numbers of relative magnitudes of X . As in Morgenstern and Price’s method, an extra unknown X is introduced; therefore, it is a rigorous solution.
^■9* Tbe necessary and sufficient number of independent assumptions that are required are (2n-2). If one can make these assumptions satis factorily, a rigorous solution can be obtained. Some possible combina tions of assumptions that one can make are given below. a. (n-l) numbers of relationship between X and E , and (n-l) numbers of point of application of E or N . A method of solution based on these assumptions is provided herein. b. (n-l) numbers of point of application of E , and (n-l) numbers of point of application of N . This means that Janbu’s method can be a rigorous one by assuming that the point of application of the last N is unknown and then determining it as part of the solution. c. (n-l) numbers of magnitudes of X , and (n-l) numbers of point of application of E or N . This method is not
38 recommended, as the solution depends on the absolute magnitude of X , which may be very difficult to guess. d. n numbers of relative magnitude of N and (n-l) numbers of point of applications of E or N introduce an extra unknown in the form of X . Since this is a very simple solution, it is also provided here.
New Methods of Stability Analysis
50. It may be possible to find other sets of assumptions, but whether any one of these assumptions will provide acceptable solutions is debatable. What follows below are two new methods of solutions of stability of slopes. Method 1 51. Figure 12 shows a slip surface in a given slope. The slip surface is formed of segments of n straight lines. We subdivide the sliding mass into n slices, but the slices need not be vertical and the edges of the slices need not be parallel. The force on the slice i
39 is shown in the inset. We assume that under the action of the horizontal] acceleration k^g , the slip surface is in limiting equilibrium. 52. From the vertical and horizontal equilibrium of the slice,
N. cos a. + T. sin a. = W. + X . c o s 6. - X. cos 6. l l i i i l+l l+l l l
- E. ,. sin 6.,_ + E. sin 6. (59) l+l l+l i l where a = the slope of the base of the slice with respect to the hori zontal 6 = slope of the side of the slice with respect to the vertical
T. cos a. - N. sin a. = k W. + X.sin 6.,n - X. sin 6. 1 1 1 1 c 1 1+1 1+1 1 1
+ E.cos 6. _ -E. cos 6. (60) l+l l+l l l where k is the critical acceleration required to bring the stresses c on the probable failure surface to a state of limiting equilibrium. The limiting equilibrium condition of the slip surface gives
T. = (H. - U.) tan cf)! + c!b. sec a. (6l) i i i i ii i where
X = (E - PW) tan + F fd (62)
hO where
and c T = the average strength parameters at the slice edges PW = the force exerted by the pore water pressure on the side of the slice Putting Equations 6l and 62 in Equations 59 and 60 and eliminating I\L , T. , X. , and X. _ , we obtain 1 ’ i i +I 5
E = a. - b.k + E.e. (63) 1+1 1 1C 11 where
a. W^sin(<|>^-a^ )+R_^cos
W. cos (d) ! - a. ) l Yi l b. l (65) cos ( q - a. + COS (<|>^ - a . + d)! - <5. ) sec d>! l l l l e. ( 66) l cos (d>! - a. l l + *l+i • ôi+i) sec *î+i R. = c!b. sec a. - U. tan d>! 6 ill il l ( ?) S. = (c!d. - PW. tan cj>! ) ( 68 ) l li l Ti Thus E a - b k + E e = (a + a _e ) - (b + b _e )k n+1 n n c n n n n-1 n n n-1 n c + E e e n-1 n n-1 Therefore, proceeding further with the recurrence relation hi E n=(a +a ne +a ^ee ,, + ... to n terms) n+1 n n-1 n n-2 n n-1 - k (b + b .,e + b .e e ,. + ... to n terms) c n n+1 n n-2 n n+1 + E, • e . e , . e ^ ... e. 1 n n-1 n-2 1 Since in the absence of external forces E = E =0 1 n+1 , a+a..e+a_ee..+ , . to n terms k = n n-1 n n-2 n n-1 (69) c ------b + b ne + b „ e e .,+... to n terms n n-1 n n-2 n n-1 Once the value of k is determined, all E forces can be determined c from the recurrence relation (Equation 63) and then Equation 62 gives all the X forces. Equations 59 and 6l -will give N. = (W. + X. ... cos <5. - X. cos <5. - E. sin <5 i i i+I l+l l l l+l i+l cos and then Equation 6l will give all values of Tk . 53. In order to complete the solution, we have to obtain the points of applications of N and E forces. For this we make the second set of assumptions. Let us assume the point of application of all E forces is known. Then taking moment about the corner of the slice, we obtain !i = jw.(xg. - x.) - kcW.(ygi - y.) + E.z. - E.+1 [z.+1 + b., sec a. sin (a. + <5. _ )1 + X.^.b. sec a. cos (a. + 6 (71) v l i+l J i+l l l i i+l )/ 1 b2 where (xg^, yg^) are the coordinates of the center of gravity of the slice. 5U. Alternatively, we may determine by assuming 1^ as known, in which case the last 1 should be determined as part of the solution. 55- For acceptability, all forces (i.e., flL , , and E^) must have a positive result, and 1^ must be within the slice, preferably within the middle third. 56. In obtaining the solution, it is assumed that shearing exists between the slices. The assumed angle 6^ may not be the critical one, and it is therefore necessary to try various angles of 6^ . The set of 6. that will produce minimum k will be the critical set. The tech- 1 c nique of achieving this minimum value will be: fix all but one 6^ , and change this 6^ gradually to obtain the minimum. Then, fixing this angle S_^ , proceed to another point until all the 6_^!s are tried and the smallest value of k is reached. It is assumed that the c changing of the order of 6^ will not produce a different result. How ever, this point is not checked. It is seen that the assumption of 6^ = 0 for all slices produces a nonconservative result, but only slightly. Hence, for routine procedure, 6 = 0 can be used. 57. This solution can be converted to another form: fix 6^ = 0 and introduce a local factor of safety in * Sarma's (1973) method determines the shape of the X force distribu tion on the basis of the internal strength. Because of the factor X involved in the solution, the exact effect is not deterministic. 1-304 Figure 13. Comparison of method 1 with Morgenstern-Price and Sarma methods 63. The method is based on the assumption that the shape of the normal force distribution on the slip surface is known, but the magni tude is not known. The slip surface is formed of segments of straight lines, and the forces on the base of the slip surface are as shown in Figure lU. From the vertical and horizontal equilibrium of the whole sliding body, (N! + U. ) cos a. + VT. sin a. = Y^. (72) Zr 1 1 1 . a* 1 1 ¿*1 YT. cos a. - Y(N! + U.) sin a. = k Y w - (73) ¿*1 1 ¿*1 1 1 c^i The Mohr-Coulomb failure criterion gives T. = N! tan 6! + c!b. sec a. 11 Ti 1 1 1 m ) ^5 Figure lk. Stability analysis - method 2 We assume that N» = AP. (75) where is known and A is to be determined. Eliminating N! and from Equations 72, 73, 7^* and 75, we obtain c!b. tan a. - U. cos a.) li ii l A (76) cos (d>! - a . ) sec d)! i i i V(c!b. - U. sin a.) + A Y P . sin ( 6k. Since the solution depends on P^ , the following expression for P^ gives acceptable results in many cases. If and a_!+^ are the normal stresses on the two edges of the slip line segment, then a! + a i i+1 P. b. sec a. (78) l 2 i l where a f = fcos h6 and R u (80) u yh u = pore pressure at the point under consideration g = 2a - (j)1 (81) 65. In order to compare the result with that obtained from the other method of slices, it is necessary to divide the sliding body into vertical slices as in Figure 11 and then to determine the interslice forces and their points of application. From the vertical and hori zontal equilibrium of the slice X = (N! + U.) cos a. + T. sin a. - W. + X. (82) l+l 1 1 1 1 1 1 1 v ' E = T. cos a. - (N! + U.) sin a. - k W. + E. (83) l+l 1 1 1 1 1 ci 1 Since A is known from the solution, RL from Equation 75 * and from Equation 7^> starting from known initial condition of = E^ = 0 , all X and E forces can be calculated. 66. After determining the X and E forces, the points of applications of E and N T forces can be determined by assuming the position of one of these quantities as in method 1. 67. This method appears to be suitable for quick computation of kc for a slip surface and should give acceptable results for existing slip surfaces. However, it is not easy to see whether it will give acceptable results for any assumed slip surface. Thus, it appears that for design of dams where one has to start with assumed failure surfaces, Sarmafs method (1973) is most preferable and that for the analysis of existing slips, the first method presented herein is preferable. Stability of Slopes Under Earthquake Conditions with Dynamic Pore Pressure 68. Under the earthquake loading conditions, the inertia force ^7 as well as the excess pore pressure generated by the earthquake play an important part in the stability of the dam and its foundations. Until now, this excess pore pressure was introduced into the analysis as equivalent static pressure. Seed and Martin (1966) proposed using the total strength determined under appropriate cyclic loading conditions for the stability analysis. Seed et al. (1969) extended this idea to develop the finite element method for studying the liquefaction poten tial of the dam during an earthquake, and Seed et al. (1973) used this method for the analysis of the Lower San Fernando Dam in California, which was affected by the San Fernando earthquake of 1971* However, since this is a finite element method, it is outside the scope of this report. 69. The method presented herein is an effective stress method which can take into account the excess pore pressures developed during cyclic loading. This method uses the concept of the pore pressure parameters which was originally defined by Skempton (195*0 Tor static loading conditions. Dynamic pore pressure parameter 70. Cyclic loading tests on saturated samples of cohesionless soil show that the pore pressures increase with every cycle of loading until failure by large strain or liquefaction occurs (Lee and Seed, 1967a; Lee and Seed, 1967b; Peacock and Seed, 1968; Martin et al., 1975)* This pore pressure depends primarily on the applied cyclic shear stress and the number of cycles. Martin et al. (1975) proposed a set of formulas to determine the change of pore pressure ratio Au /o ^q against the number of cycles for a given stress ratio At /o ^ in a simple shear test, where Au is the pore pressure, a^ is the effective consoli dation pressure, and At is the applied shear stress. The same relationships are replotted in Figure 15 to show the variation of Au/a^o against At /o^ for a given number of cycles. From these curves we see that At (8H) n a vo U8 Figure 15. Pore pressure response in simple shear test (data from Martin et al., 1975) where is the pore pressure parameter that is a function of the num ber of cycles and the level of stress ratio. However, for the initial part of the curve, the curves appear linear; and is a function of the number of cycles only. Higher order equations for these curves can easily be found if desired. 71. Initial failure is defined as that state in which the Mohr’s circle of stress first touches the failure envelope. From the geometry of the Mohr’s circle, Figure 16, we obtain 1_ At a' R a ’ (85) vo vo where + ko >2 . 2 d - V 2 R = sin 86 2 72. This relationship is shown dotted in Figure 15 . Equations U9 (assume k0 co n stan t) Figure 16. Initial failure in cyclic simple shear test Equations Qk, 85, and 86 are derived for simple shear tests where the ini tial stress conditions are represented by a f and k o1 in which k vo o vo o is the coefficient of earth pressure at rest. Equations 85 and 86 are derived assuming that kQ remains unchanged until the initial failure. 73. In the slope stability problem involving dams and slopes, simple shear conditions do not represent the true state. Neither does the triaxial test condition where either no rotation of the principal stresses are allowed or they are rotated through 90 deg.* However, in the dam during earthquakes, the rotation will be small (Seed, 1968). Therefore, the consolidated, undrained, triaxial test with anisotropic consolidation appears to be nearer the truth, and we may assume that the pore pressure parameters will not be affected by the rotation of the principal stresses. From the limited test data available in the literature on triaxial cyclic test results, utilizing both isotropically and anisotropically consolidated samples, we see a similar pattern to the simple shear tests; and we may write for the initial part * A table of factors for converting U. S. customary to metric (Si) units of measurement is presented on page 3. 50 A A O . —7 — = A - ± (87) °3e n °3c and for the anisotropic consolidations, perhaps Au = A ( ) a 88 CT3c(l+kc ) 3c(1+k ) c where a^c = confining cell pressure for consolidation A = change in axial stress k = a ( ) c l(/a3c 88 However, Equation 88 -will need extensive laboratory verification to see whether or not A is independent of k . The parameters A in n c n triaxial tests are different from those of the simple shear tests. 7^+. Equations 87 and 88 suggest that the Skempton (195^-) equa tion can be adopted for dynamic pore pressure as well, and we may write Au = B[4o3 + V 4ol (90) where B is a pore pressure parameter. The analysis 75« In this analysis, the suffix o denotes the preearthquake conditions, the suffix d for dynamic loading conditions. The method is based on Sarma!s (1973) technique of determining k^ , the critical acceleration factor. 76. Let us consider a possible slip surface in a slope as shown in Figure 17- We subdivide the sliding body into n vertical slices. Before the earthquake the slice is acted upon by forces as shown in the figure. Let us assume that there is a factor of safety F q under static conditions for this surface. We may find this value of Fq , the interslice body forces E and X , and the normal and shear J o o forces at the base of the slices N q and T^ by any of the rigorous analysis methods. 51 during earthquake critical acceleration Kc « ^tj/bseefct t * ^ ./b S«e«( 7^ % Tci/bsec*. Figure 17. Stability analysis by Sarma method 77. Let us now assume that a horizontal earthquake load kW is applied, under the influence of which limiting equilibrium is achieved. In that case, k is equal to kc . From Figure 17c, we may then write N., cos a + T, sin a = W - DX, (9l) d d d T, cos a - .N, sin a = k W + DE, (92) d d c d T, = (N, - U - DU,) tan cf>! + c !b sec a (93) d d o d' where DU, = the dynamic pore pressure developed under the influence of d K W c ¥ = weight of the slice DX and DE are as shown in Figure 17. 78. At this stage, let us introduce the hypothesis proposed by Sarma (1975). This hypothesis states that if a possible failure plane is acted upon by a normal stress o' and a shear stress t and the shear stress has a factor of safety F , then the state of stress in all planes will be the same as that in which the material has a strength property given by i|^! = tan “^(tan cj>’/F) and c ! = c!/F . Under this hypothesis, it is possible to draw the Mohr’s circle of stress for static conditions, Figure 17b. From the geometry of the circle, o' . = o' + t (tan tJj1 + sec i|>!) (9^-) 1.0 o o ro o o' „ = o' + t (tan - sec (95) 3.0 o o v Yo ro where c ’ c ’/F o o tan" (tan cj>1 /F ). Adding u (u = pore pressure at the base of the slice) to both sides 53 a, _ = a + t (tan ip' + sec ip’) 96 1.0 o o ro ro ( ) o = o + t (tan - sec <|U ) (97) 3.0 o o o o 1 ' o = stress component Similarly, for the dynamic loading conditions, we shall have II Q 0 , + T,(tan ^ ’ + sec iH) (98) H d d d Td a, + T..(tan - sec ip') °3,d d d rd rd (99) We define (100) 40i = ci,a - °i,o and (101) 4 o 3 = 0 3,a - °3,0 and therefore Equation 90 gives Au = b [a o 3 + An(Aa1 - Aa3)] Multiplying both sides by b sec a , we have, with = DUd = B|(Ba - No ) + Td [tan - sec ip^(l - 2A)]j[(102) 79. Eliminating and from Equation 102, with Equations 91 and 93, we obtain DU, = a - b DX d d (103) where E Ü* h O _ B(a - N ) + dq - O 1 - Bb - eq. (lOll) 0 51* b = B(C + fq)/l - B(bQ - eq) (105) a = (W + Uq sin a tan c|)f - c!b tan a cos (|>f sec(<|>! - a) (106) b = tan a tan (|>T/(l + tan a tan cf)!) (107) o C = cos (|>? sec(i))1 - a) (108) d = (cfb + W tan (j)! - cos a tan c()T) cos c|>! sec( e = tan cf>f /(1 + tan a tan <|>f) (110) f = sin (f)1 sec((|>! - a) (ill) q = tan q = tan i|;! - sec ip!(l - 2A ) (113) o o o n (W + U sin a tan if;1 c'b tan a) cos sec (il»! - a) N = o o o o o - DX cos sec (ip' - a) (llh) o o o T = (c!b + W tan ibf — U cos a tan ib1) cos sec (1p1 - a) 0 0 ro o o o o - DX sin to1 sec (iJj! - a) o o o (115) -1 = tan K (tan 6 (116) 'o F (117) Now eliminating N^ and from Equations 91» 92, and 93, we obtain DX, tan (d>’ - a) + DE, = W tan («j)' - a) + (c'b sec a cos ((>' a a - UJQ sin 55 and rearranging terms DX^ £t'an (<(>' - a) - b sin ((j)' - a) - a sin ^ D X ^ t a n (cj)' - a) - b sin 80. Let us assume that the point of application of forces are known and that the coordinate of this point is xm >ym which, for practical purposes, can be the middle point of the base of the slice. Then taking moment about the centre of gravity of the whole sliding body (x ,y ) and remembering that the net moment of W and kW about s § the centre of gravity is zero, we have (N, cos a + T, sin a)(x - x ) + Y (T, cos a Z d d m g L* d - Nd sin a) (ym - y ) = 0 (121) and using Equations 91» 92, and 118 in Equation 121, b sin c|>f sec ((j)1 - I DXd { U m " Xg } + (ym ' yg } tta n ( = 5>(xm - xg) +2^ ym - yg) (122) The problem now is to find a set of forces which will satisfy Equation 122. 8l. Let us assume that the shape of the force distribution 56 is known but not the magnitude. Then we can say that xd= XQ (123) where Q is known but A is unknown. Then DX., = A(Q. . - Q.) for the th ^ i+l 1 i slice = AP Then Equations 120 and 122 become A^Pj^tan (<}>* - a) - b sin A x x ) + (y - y ) £tan ( = 2 > ( * n - xg ) +jD(ym - yg ) (125) Therefore, X and k can be solved. The rest of the solution will c v follow the same pattern as in Sarma (1973) and is not shown here. Procedure 82. A procedure for applying the above method of analysis can be summarized as follows: a. Determine An parameters for various cycles, say 1 cycle, 5 cycles, 10 cycles, 20 cycles, etc., from anisotropically consolidated, undrained tests for different materials of the dam. b. For each set of An values, determine the k values — c for the possible slip surface. Therefore, we obtain the critical accelerations for a corresponding number of cycles. c_. From the strong motion record and the average seismic coefficient for the sliding mass, find the equivalent number of uniform cyclic seismic coefficient of kc (Seed et al., 1969). If the available number of cycles is more than the number of cycles required for failure, the factor of safety will drop below one. Since the avai-lable number of cycles will depend primarily on the choice of the record, it is better to consider the critical acceleration for 10-20 cycles as the most crit ical value and to compute the probable displacement on the basis of it. 57 PART IV: DISPLACEMENT CRITERION OF DESIGN 83. Within the limit equilibrium principle, a factor of safety smaller than one represents failure. This signifies that when the factor of safety is less than one, the mean strength along the failure surface is less than that required to maintain equilibrium. Therefore, a section of the dam or embankment will slide along the failure surface and will come to rest again at a time and place when the new mean stresses do not exceed the strength. Depending on the amount of rela tive displacement, which may vary from a fraction of an inch to a few yards, the structure will be said to be safe or to have failed. It is therefore obvious that a factor of safety less than one cannot be per mitted under the static conditions, as the stresses producing this stage will exist until large displacements change the geometry of the structure. 8k. However, under earthquake conditions, it may be possible to allow the factor of safety to drop below one, as this state will exist only for a very short time. Nevertheless, the consequences of allowing the factor of safety to drop below one must be known, and this can be measured in terms of relative displacements of the sliding mass relative to the main body of the structure. Newmark (1965) proposed the sliding block technique to measure this displacement, later Ambraseys (1973) proposed the upper bound formula, and Sarma (1975) showed that displace ments can be computed on the basis of simple pulses once the critical acceleration is known. 85. Sarma*s (1975) work is based on the effective stress prin ciple. The displacements were computed on the basis of the assumption that the pore pressures at the base of the sliding block remain constant during movement. In contrast to laboratory testing, where pore pres sures continue to rise after the factor of safety has become equal to one in cyclic loading, any attempt to increase the earthquake load only increases the displacement, creating dilatency in the slip zone and thereby reducing the pore pressure. In the case of slope failures, therefore, the assumption that pore pressures remain constant during movement is rather conservative. 58 86, It is also assumed that the critical slip surfaces can he converted into plane slip surfaces. The shear strength properties of this plane surface which separates the sliding block from the main body of the structure are the average <|>f and c! . The pore pressure is equal to that at incipient failure (Figure 18). Figure 18. Sliding block model 87. The sliding block is subjected to a horizontal earthquake acceleration kg , which imparts a horizontal load kW on the sliding block, as shown in the figure. The earthquake acceleration is greater than the critical acceleration kcg , which was determined for the original slip surface. 88. The driving force D on the sliding block is D = W sin 3 + kW cos 3 (126) where W = the weight of the sliding block 3 = the slope of the sliding plane and the resisting force R is R = (W cos 3 - kW sin 3 - U_p) tan (f>1 + c!L (127) where L is the length of the sliding block. When the earthquake acceleration is kcg , the driving and the resisting forces are equal, which gives 59 W sin 3 + k^W cos 3 = (W cos 3 - k^W sin 3 - U^,) tan Using Equation 128 in Equation 127 R = W sin 3 + k^W cos 3 - (k - k^)W sin 3 tan W - x = D - R g = W r . 6 ). (k . t ) (130) cos d>* c or x _ S cos («l»' - 3) (k - kc ) cos Since the earthquake acceleration kg is a function of time, Equa tion 131 can be evaluated to determine the total displacement x at the end of the earthquake. 89. Eor a rectangular pulse of duration T/2 and amplitude k^g , Figure 19a, the solution of Equation 131 gives Ux m cos (132) cos ( 60 Figure 19. Forms of acceleration pulses used for computation of displacement For a triangular pulse of duration T/2 and amplitude k^g , with a peak at time AT/2 , Figure 19"b, 1+x m ^ cosij) ' 2 cos( fo r 0 < k /k < \l - V( 1 - *)"| A (133) = |[(1 - k/km)3 (2+2V1 - A - x) for (l -VITx) /X < k /k < 1 (13U) For a half/sine pulse of duration T/2 and amplitude k^g , Figure 19c, 61 Ux ^9„2 * Po'qCf'A^' cos (q> - 'bfs;V = ^Kc^km c m " Sin for 1 > k /k > 0.725 — c m = (k^/k^ + a - 7T + cos^a/2 cot a/2)/7i^ (135) 0 < k /k < .T (136) — c m — 0 25 where q = a + k /k (cos a - cos q) c m ^ (137) and a = sin ^(k /k ) (138) c m 90. The solution for earthquake records is obtained by assuming piece-wise linear acceleration. Since the problem is that of a slope failure, the critical acceleration required to move the body up the slope is assumed to be very large, and no reversal of displacements occurs. 91. Between the times t. and t. , the acceleration k can l-l 1 g be written as ( ) k(t) = ki-i♦ 1 3 9 1 1-1 then the solution of Equation 131 gives , , . k.-k (t-t. n ) cos4>' x , Wj_ ^ . 1 1 -1 l-l . . /.; „■> • — = (k. n-k ) (t-t. n ) + t— 7--- • ------+ x. (lb0) coss(d>'-3) g l-l c l-l t.-t. . 2 ' “i-l r & 1 l-l 3 cost*__ . 2L = _v ) ^ h - i * . ki-ki-i ^ - h - i ) cos( * * V i (Ha) 62 where x is the downslope velocity of the sliding block relative to its original position. 92. The procedure for determining the displacement for earthquake record will then he to a. Scan the earthquake record from the beginning until the pulse is bigger than k . This will give the initial conditions. Determine ?he velocity and displacement at the end of the linear pulse, which provides the initial conditions for the next pulse. b. It is possible that the movements may stop in the middle of a decreasing pulse. This will be indicated by the change of the sign of the velocity; in which case, the time for zero velocity is interpolated, and the dis placement is computed up to that time only. c_. Once the movement is stopped, it will not start unless the pulse is bigger than k . Therefore, we are back in step a. 93. The total displacement at the end of the record gives x m The maximum acceleration in the record gives k^ , and the predominant period of the acceleration spectrum gives T . We can therefore compute ^Xn/kmsT ’ The ear-th(luak:e acceleration time history is not symmetric about the zero axis, and it is therefore necessary to compute the dis placement xm by changing the signs of the accelerations. 9b. Figiure 20 shows the quantity (l/C)(Uxm /kmgT ) computed against k /k for the four cases mentioned above. This figure shows that for kc/km > 0.5 , the triangular pulse and for k /k < 0.5 , the rectangular pulse effectively give the displacement from an earthquake record, where k for the pulse is the maximum of the acceleration record and m T for the pulse is the predominant period of the record. Also, it is seen that the absolute displacement for k^/k^ >0.5 is very small for real earthquake records. C represents cos (<|)! - g)/cos 63 Figure 20. Computed permanent displacements 6k 96. The methods presented in Part III will give the critical ac celeration kcg . From this solution we can draw the force-polygon for the entire sliding surface, as shown in Figure 21. The slope of the shear force vector can he used as the slope angle B . From the same solution, we can obtain the average shear stress x and the average effective normal stress n' on the slip surface. Since we can compute c' , the average angle T - C -1 av = tan (1U2) n f W S = Shear force vector KW N = Normal » » Figure 21. Vector diagram of forces 65 PART V: ANALYSIS OF THE LOPEZ DAM 97* In this part, the Lopez Dam of California is analyzed by the method mentioned in the previous chapters, and the stability of the dam is checked against the San Fernando Earthquake of 9 February 1971« The probable ground motion at the Lopez damsite during this earthquake is supplied by the U. S. Army Engineer Waterways Experiment Station (WES). This record is a modified version of the Pacoima Dam record. The record is shown in Figure 22, along with the integrated velocity and displacement records. The three spectra are shown in Figure 23. The cross section of the Lopez Dam is shown in Figure 2k. The material properties, as supplied by the WES, are shown in the Table 2. Seismic Coefficients 98. The presence of the debris in the upstream side creates an asymmetric section. However, for the sake of analysis, the debris will be neglected in finding the inertia forces and the seismic coefficients. The average shear wave velocity for a nonhomogeneous section can be computed in the following way: b b. dy si ’pi S2’P2 S3 ,P3 99* Consider an elemental slice of thickness dy . Assuming constant strain along the entire width, the net shear force T across the section is T = (G1b1 + G , b J f 1 a2h2 3 3 dy du S2p.b. (1U3) dy j 1 1 1 The average shear stress is therefore 66 CLOUGH'S'DECONVOLVED PflCOIMfi S 16 E RECORD E N R I 7.7. V. 7. V. X Y */. 7. Z X 7.7. X X 7. Y 7. X 7. Z 7. X 7. Z 7. Y Figure 22. Ground motion record used in analysis of Lopez Dam VELOCITY SPECTRA > ACCELERATION SPECTRA CLOUGH'S DECONVOLUTED RECORD 16 E S PACOIMA ENR iue2. pcrlcre forcurves Spectralgroundof Figurerecord motion 23.Figure *— NATURAL PERIOD (SEC)-» 17.7.7. . Z .777Y7.7.7.7.7.7.7.7. 7.77.7.7.Y 7 Y . /. . .7 '/ 7.7.7.7.7.77Y 22 ) y y v ZONE moist sat (pcf) (pcf) (fps DEBRIS 1070 130-0 700 SLOPE 1330 H6-8 750 CORE 1230 I— 20— i 139-3 850 FOUND. 127 5 H2-A 1000 ~T77^----- 7 * ? ------ZT? ----- ^ ------7*3=— 777$----- '/jfëTTBS------WJ> i Figure 2k. Lopez Dam t (ill) T av = r- b and therefore the average shear modulus is G av ìysV.b.b ^ l l l (ll5) The average shear wave velocity is therefore , p \1/2 G \ 1 / 2 av \ I 4 i h \ S (116) av av ,1 pih / However, for all practical purposes, a linear average can be assumed, so that S dvr) av Thus for the Lopez Dam section, considering the moist densities, the 69 Table 2 Lopez Dam - Material Properties Zone 1 Zone 2 Zone 3 Zone k Properties Debris Slope Core Foundation c' , psf 0 0 560 22k 1+0.8° In. 8° 27.5' 38.5° 130 Tsatpof 1I+6.8 139.3 lk2.k ^moist^c^ 107 133 123 127.5 S ft/sec 700 750 850 1000 A * A1 0.3 0.3 0.6 0.2 A * 1.0 1.0 2.0 5 0.9 A * Aio 1.9 1.9 2.2 1.7 A * 2.2 2.2 20 2.5 2.0 An = pore pressure parameter A at n cycles. average shear wave velocity is 795 ft/sec; and the average density is 128 lb/ft . The shear wave velocity of the foundation layer is 1000 ft/ sec, and the density is 127.5 lb/ft3. Therefore, m = S1P1/ S2P2 = 0,3 and q = m(H - h)p2/hp1 = 0.71 . For this pair of values of m and q , the first root of the transcendental equation J (a) n m tan (qa ) = n (ikQ) J.,(a ) 1 n IS a-L = 1.3U , which is obtained from Figure 5 of Part II (also com puted numerically in this case). 100. Thus, the fundamental period of the dam-layer system is 27ih T1 = 3 “ = 0.27 sec '(Ifc9) S an av 1 Figure 25 shows the seismic coefficient spectra for the period to range from 0.2 to 0.1+ sec for the modified version of the Pacoima Dam record for the pair of values of m and q mentioned above. 101. If, on the other hand, the debris is not neglected but is 70 iue2. esi ofiins ifdebris iscoefficients Seismic Figure26. Seismic Coefficient {4) V Seismic^Coefficient (4) 5 Simc ofiins ifisdebris neglected coefficients Seismic e25- considered part of foundationconsideredofpart oiid aom Dm Record Dam Pacoima Modified OIID AOM DAM PACOIMA MODIFIED sIO m =2-4 q g =0-65 g m 5:89i RECORD considered as part of the foundation, so that the dam height is reduced by 20 ft and the layer thickness is increased by the same amount, then the values of the two parameters become m = 1 and q = 2.k . For this pair of values a^ = 0.5*+ , and the fundamental period of the dam-layer system becomes T = 0.37* sec (150) 102. Figure 26 shows the seismic coefficient spectra for this case for the period to range from 0.2 to 0.*+ sec. From Figures 25 and 26, we can replot the seismic coefficients for the two periods in question (Fig ure 27). The maxima of the two will give the design seismic coefficients In this analysis, this is obtained entirely from the first case. The point at the crest of the dam is obtained from the point response. Figure 27- Seismic coefficient versus depth * The assumption of an untruncated wedge is not strictly applicable in this case. However, it is considered to be a reasonable approximation. 72 Critical Accelerations 103. The Lopez Dam section is analyzed by the limit equilibrium principle using the Sarma (1973) method, with the computer program written by the author for the CDC 6U00-6600 computer at the Imperial College of Science and Technology, London. The various probable slip surfaces and their critical accelerations are shown in Figure 28. IOU. WES supplied cyclic triaxial test data indicating the prob able values of the pore pressure parameter for various cycles, shown in Figure 29; and the results are tabulated in Table 2. These A^ values are used to determine the critical accelerations for a few upstream surfaces which were considered critical. As far as the dy namic pore pressures were concerned, only the material below the water table is assumed to be affected by the earthquake loading. The results of the analysis are also tabulated in Table 3. 105. From the results we can see the following: aL. There will not be any post earthquake slope failure due to liquefaction. Even the worst case can withstand a critical acceleration of 0.2 g with 20 cycles of pore pressure development, which means, in other words, that without an earthquake, there will be a factor of safety greater than one (a critical acceleration of 0.2 g corresponds to a static factor of safety of approxi mately 1.6 without earthquake). b. Even without any dynamic pore pressure development, the critical accelerations of some surfaces are smaller than the design seismic coefficients (Figure 30). Therefore, during the earthquake, the factor of safety of the structure will drop below one; and some slip surfaces will develop. Displacements 106. Figure 31 shows the dimensionless displacement graph for the modified version of the Pacoima Dam record. The assumed predominate period is 0.^- sec. There are two peaks in the acceleration spectra, one at 0.2 sec and the other at 0.U sec; however, the velocity spectra shows that O.k sec is perhaps more important than 0.2 sec. Table U 73 LOPEZ DAM a. LOPEZ DAM downstream section b. Figure 28. Trial slip surfaces and critical accelerations 2*5 r © O 5 IO 15 20 25 30 Cycles of stress a. b . Figure 29. Probable values of pore pressure parameter A Table 3 Results of Stability Analysis k k k k k Surface c c c c c No. No DPP* PP** = 1 cycle 5 cycle 10 cycle 20 cycle 6 0.79 0.59 0.55 0.53 0.53 7 0.71 0.51+ 0.1+1 0.36 0.35 8 0.66 0.50 0.31+ 0.28 0.27 9 0.67 0.U8 0.33 0.27 0.26 10 0.85 0.52 0.37 0.32 0.30 11 0.70 0.50 0.1+6 0.39 0.37 12 0.73 0.51 0.30 0 .21+ 0.22 * Dynamic pore pressure. ** PP = pore pressure. Figure 30. Comparison of criti cal accelerations and design seismic coefficients 76 Figure 31. Sliding block displacement from modified Pacoima Record 77 Table h Computed Displacements k k Displacement = ah c c k No DPP* 3 ft m No DPP* * 20 cycle c** in. 20 cycle deg 10 1.25 o.Uo _ 1.0 7.2 ____ 20 20 1.15 0.1t5 — 1.0 2.8 — 17 30 1.05 0.66 0.25 1.0 0.2 ll+ 7 ko 0.9^ 0.70 0.30 1.0 0.06 5.2 2 50 0.80 0.73 0.22 1.0 — 7.3 0 * Dynamic pore pressure. _ cos(61 - 8) C = ----*— -- is assumed 1.0. cos 4>! shows the various values needed for computing the displacements at dif ferent levels and also gives the computed displacements from Figure 31. * From the table, it can be seen that the maximum relative displacement at the top of the dam will be about 7 in., which will mean a vertical drop of about 2.5 in. and horizontal spread of about 6.5 in. Deeper slides will have negligible displacement without the dynamic pore pres sure considerations. 107* With a dynamic pore pressure equivalent to 20 cycles of loading, the deeper slip surfaces are affected. But the displacements are still small, the maximum being about lU in. This displacement does not show any appreciable drop of the crest level, but it will show as a crack in the debris at about 100 ft from the toe of the dam. 108. Thus, if the assumed ground motions were correct, the simpli fied analysis shows that the Lopez Dam of California should withstand the San Fernando earthquake of 9 February 1971 with a very minor amount of damage. There would be about a 2.5-in. slumping of the crest with a cor responding bulge on the side near the top 10 ft of the dam. There should be some cracking of debris about 100 ft away from the dam, which would be caused by a small movement of a deeper slide. 78 PART VI: CONCLUSIONS 109. The method of analysis and design of earth dams under earth quake loading conditions, which is developed in this report, is a simple one. The work in the second part is an extension of the original work done by Ambraseys and Sarma (1967)» which now includes the presence of a foundation layer. The presence of this layer does seem to affect the response instead of merely changing the fundamental period. The rest of the conclusions that can be drawn are the same as those in the original paper mentioned above. For example, the upper part of the dam is af fected more than the lower part, and a deeper slide has a smaller seismic coefficient than a slide on top of the dam. The various seismic coef ficient spectra will help the designer in computing the seismic coef ficients for his dam. 110. The stability analysis method with the dynamic pore pressure parameters is a simple method and is appropriate for design purposes. However, the dynamic pore pressure parameters need further study. Per haps a high order curve would be more appropriate than a linear one; but with the present available data, this is superfluous. 111. The displacement criterion of design is still a matter of opinion. However, the displacements computed for the Lopez Dam do not seem to disagree with the field behaviour. 112. The analysis of the Lopez Dam is used more as an example of the concept of this method of analysis than as an attempt to analyze it in detail. For example, the response of the dam and the seismic coef ficients are not shown in the time history sequence; only the maximum values are picked up. The seismic coefficients for particular slip surfaces are not computed but are taken from appropriate graphs. Thus, the analysis shows that the concept is a simple one. 79 REFERENCES Ambraseys, N. 1960a. "The Seismic Stability of Earth-Dams," Proceed ings, Second World Conference on Earthquake Engineering, Vol 2, pp 131+5-1363. ______1960b. "On the Shear Response of Two-Dimensional Truncated Wedge Subjected to an Arbitrary Disturbance," Bulletin, Seismological Society of America, Vol 50, pp 1+5-1+6. ____ _ 1960c. "Uber die Berechnung von erdbebensicheren Erddammen," Ver Deutsch Ing. Zeitsch, Vol 102, pp 121+1-121+3. ______. 1973. "Dynamics and Response of Foundation Materials in Epicentral Region of Strong Earthquakes," Proceedings, Fifth World Conference on Earthquake Engineering, Vol 1 , pp 12l+-ll+l. Ambraseys, N. and Sarma, S. K. 1967. "The Response of Earth Dams to Strong Earthquakes," Geotechnique, Vol 17, PP 181-213. Bishop, A. W. 1955* "The Use of the Slip Circle in the Stability Analysis of Slopes," Geotechnique, Vol 5, No. 1 , pp 7-17» Clough, R. and Chopra, A. 1965. "Earthquake Stress Analysis in Earth Dams, Report No. 65-68, Structural Engineering Laboratory, University of California, Berkeley, Calif. Hatanaka, M. 1952. "Three Dimensional Consideration on the Vibration of Earth Dams," Transactions, Japan Society of Civil Engineers, Vol 37, No. 10, pp 1+23-1+28. ______. 1955. "Fundamental Considerations on the Earthquake Resistant Properties of the Earth Dam," Bulletin, Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan, Vol 11. Ishizaki, H. and Hatakeyama, N. 1962. "Considerations on the Vibra tional Behaviour of Earth Dams," Bulletin, Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan, Vol 52. Janbu, N. 1957. "Earth Pressures and Bearing Capacity Calculations by Generalized Procedure of Slices," Proceedings, Fourth International Conference, Soil Mechanics and Foundation Engineering, Vol 2, pp 207-212. Kenney, T. C. 1956. "An Examination of the Methods of Calculating the Stability of Slopes," M.S. Thesis, University of London, London, England. 80 Lee, K. L. and Seed, H. B. 1967a- "Cyclic Stress Conditions Causing Liquefaction of Sand," Journal, Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol 93, No. SMI, pp 1+7-70. ______. 1967b. "Dynamic Strength of Anisotropically Consolidated Sand," Journal, Soil Mechanics and Foundations Division, American Soxiety of Civil Engineers, Vol 93, No. SM5, pp 169-190. Martin, G., Finn, W. D. L., and Seed, H. B. 1975* "Fundamental of Liquefaction Under Cyclic Loading," Journal, Geological Engineering Division, American Society of Civil Engineers, Vol 101, No. GT5, pp 1+23-1+38. Medvedev, S. and Sinitsym, A. 1965. "Seismic Effects on Earth Fill Dams," Proceedings, Third World Conference on Earthquake Engineering, Paper IV/M/18. Minami, I. i960. "A Consideration of Earthquake-Proof Design Method of Earth Dams," Proceedings, Second World Conference on Earthquake Engineering, Vol 3, pp 2061-207^+. Morgenstern, N. R. and Price, V. E. 1966. "The Analysis of the Sta bility of General Slip Surfaces," Geotechnique, Vol 15 , pp 79-93. Newmark, N. 1965- "Effects of Earthquakes on Dams and Embankments," Geotechnique, Vol 15, No. 2, pp 139-160. Peacock, W. H. and Seed, H. B. 1968. "Sand Liquefaction Under Cyclic Loading Simple Shear Conditions," Journal, Soil Mechanics and Founda tions Division, American Society of Civil Engineers, Vol 9I+» No. SM3, pp 689-708. Rashid, Y. 1961. "Dynamic Response of Earth Dams to Earthquakes," Report Graduate Student Research, University of California, Berkeley, Calif. Sarma, S. K. 1968. Response Characteristics and Stability of Earth Dams During Strong Earthquakes, Ph. D. Dissertation, University of London, London, England. ______. 1973. "Stability Analysis of Embankments and Slopes," Geotechnique, Vol 23, No. 3, pp 1+23-U33. ______. 1975. "Seismic Stability of Earth Dams and Embankments," Geotechnique, Vol 25, No. 1+, pp 7^+3-761. Sarma, S. K. and Bhave, M. V. 197^- "Critical Acceleration Versus Static Factor of Safety in Stability Analysis of Earth Dams and Embank ments," Geotechnique, Vol 2U, No. U, pp 661-665. 81 Seed, H. B. 1968. "The Fourth Terzaghi Lecture: Landslides During Earth quakes Due to Liquefaction," Journal, Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol 9*+» No. SM5, pp 1053-1122. Seed, H. B. and Martin, G. 1966. "The Seismic Coefficient on Earth Dam Design," Journal, Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol 92, No. SM3. Seed, H. B., Lee, K. L., and Idriss, I. M. 1969« "Analysis of Sheffield Dam Failure," Journal, Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol 95» No. SM6, pp l*+53-l*+90. Seed, H. B. et al. 1973. "Analysis of the Slides in the San Fernando Dams During the Earthquake of February 9» 1971»" Report No. EERC 73-2, Earthquake Engineering Research Center, University of California, Berkeley, Calif. Skempton, A. W. 195*+• "Pore Pressure Parameters A and B," Geotechnique, Vol *+, No. U, 95*+, P l*+3. Valenzuela, L. 1975. "Influence of the Distribution of Seismic Acceleration on the Analysis of Stability of Earth Dams and Embankments," M.S. Thesis, Imperial College of Science and Technology, University of London, London, England. Yokoo et al. 1957« "Vibration Characteristics of Earth Dams," Transactions, Japan Society of Civil Engineers, No. U9 » PP 17-18. 82 APPENDIX : SOLUTION OF THE GOVERNING EQUATIONS FOR A DAM-FOUNDATION SYSTEM 1. The equations of motion of a dam-layer system subjected to a ground motion are: a2 d U n + 1 . 9U1 _ (A-l) y 3y U1 + CU1 + ay ?[ ,2 9 u2 _ i cu2 + g(t)] (A-2) a 2 s2 3y S2 The boundary conditions; areare: 9ui = 0 at y = o (A-3) 3y = 0 at (A-10 U2 y = h at y = h ui = U2 (A-5) , 9ui ^ 2 at y = h (A-6) rl 3y = G2 3y are at rest : 0 at t = 0 = u2 = = ^2 = (A-7) Applying Laplace transforms to Equations A-l and A-2 with the conditions in Equation A-7, d2^ du. 1 + i (A-8) 2 y dy (P2 + CP)U-| + dy d2u„ (p2 + cp)uu- + (A-9) dy S2 S2 A2 where the bar signifies the Laplace transform of the function. Let JL-i-ja = , ana 2i£l = g (A-10) si si Equations A-8 and A-9 then become d2"i + i dui 2— — k + g (A-ll) 2 y dy dy' d2^ (A-12) dy S2 The transformed boundary conditions are dun dy = 0 at y = 0 (A-13) u2 = 0 at y = H (A-11+) U1 = U2 at y = h (A-15) du du G. -r- = G0 at y = h (A-16) 1 dy 2 dy Solution of Equation A-ll with Equation A-13 gives U1 = CIo(ky) - h (A-17) k and solution of Equation A-12 with Equation A-lk gives A3 B sinhinh k(y - H)J c°sh[ g-kyl - 1 (A-18) U2 = c o s h f - k H c°shl— kH where B and C are unknown constants and I () is the modified Bessel o function of order 0 • Equation A-15 then gives sinh cosh I — kH B \ 2 / C = f t k Equation A-l6 gives „2 dul „2 du2 . S1P1 1 7 = S2P2 1 7 at y ‘ h 2 — — S1 P1 dUl dU2 (A-20) g2 p 2 dy dy cosh — k(h - H) sinh [ — khl S p \ S2 / i (A-21) c Sfl Ii (Hl) ■ B cosh ig i kHj coshf — kH where I^() Bessel function of order 1. Put V i = m (A-22) S2P2 S1 (h - H) = a (A-23) Ak (A-2U) Equations A-19 and A-21 therefore give — |"IQ(kh) sinh (bk) - ml^(kh) cosh (bk) B k^ Lml^(kh) sinh (ak) - IQ (kh) cosh (ak) (A-25 ) Substituting Equation A-25 into Equation A-19 yields n = S_ • (A-26) ^2 ml^(kh) sinh (ak) - I (kh) cosh (ak) Putting Equation A-26 back into Equation A-IT yields Io (ky) u„ = 1 .2 1 (h) cosh (ak) - ml., (kh) sinh (ak) (A-27) k I[- o 1 “] Similarly, substituting Equation A-25 into Equation A-l8 yields S1 S1 I (kh) cosh -3— k(y - h) + ml (kh) sinh — k(y - h) o b 1 - 1 (A-28) U2 = % K S1 I (kh) cosh -— k(h - H) - ml, (kh) sinh (ak) o S25 1 2. The inversion of Equation A-27 and Equation A-28 can be ob tained by applying Melin's inversion theorem. We can write Ui = s * Fl (A-29) and u2 = I • F2 (A-30) where F^ and F^ are Laplace transforms of F^ and F^, and therefore A5 / *2 - T ) dT u2 ~ f (A-32) O S Therefore, the problem is to obtain F^ and F^ from ^(ky) IQ (kh) cosh (ak) - ml^(kh) sinh (ak) (A-33) W[ ■ * ] S1 S1 I (kh) cosh — k(y - h) + ml (kh) sinh — k(y - h) _1 ° S2 1 S2 F„ = » 1 2 Io (kh) cosh (ak) - ml^kh) sinh (ak) (A-3U) From the inversion theorem pt S Y+ioo __ e Io S S I (rh)cosh — h(y-h) + ml (rh)sinh — r(y-h) S y+iœ pt ° S2 1 S2 / Io^rh)cosh(ar)-mI1 (rh)sinh(ar) -ST Y-i“ (p +cp) . » 1 dp (A-3’6) where 1/2 = h - (p2 + cp) A6 To apply the theory of residues, the poles are at p = 0 ; p = -c ; and at those values of p = p that make n I rh cosh ar - ml.rh sinh ar = 0 o 1 (A -37) The roots of Equation A-37 are all imaginary. Let rh ia be the n roots. Then — 2 2 a Sf n 1 Pn (A-38) h are the poles, and Equation A-37 becomes J a o n = m tan qa (A -39) J-. a 1 n where a 2 H - h q = - - = m (A-Uo) >, * h Put a S n = 0) h on (A-Ul) and c = 2A co (A-k2) n on then P = -A a) + n n on — (A-U3) 3. The residues at p = 0 and p = -c are zero, therefore p nt f£ > - S1 i Jo(any/h) (A-MO n "*■ Pn + c^n ^■jl0(rh)cosh(ar)-mI^(rh)sinh(ar)J P = A7 t e L F = we rh = a i = h r when p = p r o f 2 = ih = D = D = ) t ( h ro n = l p + cp cp + p l = n 2 dp S./ cp + 2S-.\/P S^a/p2+cp S^a/p2+cp ( h i W \ 1 n n *n Q oh ^(h) nh (ar)J ) r a ( h in s ) (rh l^ m - ) r a ( cosh ) h r IQ( ^ m n ( ) a (q in s m + n n 2 2S. — — - ) h r Q( I ) r a ( h in s mh - fn n n f pn+c) t J J t p ° -(rh) oh al (rh) si ) r a ( h in s ) h r ( l a + ) r a ( cosh ) h r ( l- h < — ° + 2 + c) + (2p n CPn + Pn V . o n 2 / ■n o (a )|^ (qa oos n [ W a c°^ - y- j -mJ )j -h (y -^ °S^ )c (an 1 o _ d dp a ) - ) (a J I-^rh) r n o 1 n n u o n ^ n ^ n o u n n 1 I ^(rh)si )J r a ( h in s ) h r ( I^ m - ) r a ( h s o c ) h r ( o j [ - ( cs qa + ( si qa ) a (q in s ) (a J q + ) a (q cos ) (a J-, q m n q^ cs qa^ cos + qa^ in s m + mq a ) (a J 2 2 - 2 — 2 . 2 . A n 1 8 n - s in (q aß ) cos (q aQ) aQ) (q cos ) aß (q in s - J a 2S. n ( h i V L -J -J •v/p + cp cp + •v/p mqJ + 2 I 2 + ) c + p mal^ - 2 n n 1 ----- p q, n) ,n ,m (q p • ------a sn^i £ (y-h) -£ ■ )sin|^gi (an 1 an) o (Q.an )! ( cos ) n (a oh ar)/ r (a cosh ) h r ( P=P, n (A-U6) (A-1^5) where / \ JJ aJ ( 2 . 2 - p (q,m,n) = ----- — — I mq + m sin qa cos (qa n ) \ N 2 •— — — \ + cos qa - — sin qa cos qa i (A-UT) n — n n a n At p = -X a) + io) . /I - X2 n n on — o n \ n D = + i — p (q,m,n)x/l - X2 — s_ o \ n Therefore -X a) t ico 1 - X t e non e °n n J^y/h) + S^â2 r /----—■ -X o) t __ e " 0n J0(any/h) /!»on Jl - X2t -iu Jl - X2t \ n on X n - e o) îaianp 0(q,m,n)^l - A2 on — X 03 t e n °n J0 (any/h) = Sn ------— ------Sln (u) Jl - 2 \ on \ x n t) / (A-U8) anP0 (l.” .° ) - 0)o J l - X2 on y n Similarly, A9 -r f — \ . / — H - y\ -X a) t / 2mJn (a ; s m ( qa ---- M n on I:sin co 1 - x2t) 1 n \ n i ■ h/ e______V on n / (A-U9) F2 (t) = S 1 cos (qa )a p (q,m,n) n no -U) on V1 - Xn Equations A-31 and A-32 then yield the solutions for u^(y,t) and Ug(y,t) given in the main text. A10 APPENDIX B: SEISMIC COEFFICIENTS 1. For the one-parameter sliding wedge ah / y*n dy o______ rh 2Jo (any/h) y a p (q.,m,n) dy oJ n o UJn (aa ) 1 n (B-l) aanPo(q,m,n) 2. For the four-parameter sliding wedge, no such simple form exists. In computing ^^(a) , there are three possibilities, depending on the geometry of the sliding wedge. Case 1: The entire sliding surface is in the dam: a <_ 1 . Case 2: Part of the sliding surface is in the foundation, but the exit points are in the dam: a > 1 . Case 3: Part of the sliding surface is in the foundation, and the lower exit point is in the foundation: a > 1 . 3. Sarma (1968) determined these seismic coefficients by comput ing the accelerations in the dam first and then numerically determining the seismic coefficient. But it is possible to formulate the seismic coefficients analytically for the three cases separately. Case 1: ah f t>(y) B2 ah Denominator = / b(y) dy o^h ah ah = dy + b2^ dy + M y ) dy fo \ {*] S a^h S ah J where B1 (B? " Bi} V y) = ^ y ’ V y) = Bih + ~a~ r ^ (y - aih) ; B2(ah - y) B3 (y) (a - a ) T h erefo re, h2 1 Denominator = — + B2(a - a^J a^h a2h ah Numerator = r bl(y )<(>n ^ + f b2^y ^ n dy + f b3^y ^ n dy a^h a2h B1 rl h (a2Bl ■ aiB2)h ra2h B2 ■ Bi ? 2 h = - / y^ndy^ Si y^ndy 1 o <0,2 - “l* 2 1 a h aB^h ah B^ ah a~rS72 ~ a2h/„ ^ +n 37 - (a - a,) 2 a^h / y+n dy 2h ®1 . ^1 T / - \ E2 a, — 1 al an (a - a ) = - Jl(a an> - ~ Jl(“2an a p (q,m,n) I I a a n o \ n L n n B2 - B1 a2 - “l - — J-Jo^n) - IT J!(aian) «2 ~ ai a a l n n B3 + ~ ^ 2 2' [BESIBT (an* “2> - BESIBT < V “l>] + — LBESINT (ania) " BESINT (ana 2 }] wh er e a B E SI NT (a ,a) = f J (a y) dy n J o rr J o T h u s , Case 2 : o^h ah h Numerator = J t>^(y)n dy + / ■b2 (y) ah B a^h (a 2B i “ y b2 " B! “2h aB2h h b2 h + ^ 7 L dy + ^ 7 ¿** 47 ' ^ b U B ah + 7------r / (ah - y)i|> dy = — — — / --- \ [ — a. • — — J 1(a 1a n) (a “ a2 h n —a p (q,m9n) V I a n o ^ \ n B2 - Bl “? a1 ^ V \ > a2 " “l “a W ' - V a Y r i 1 (a - a2) a ■ n n L n _ (a0B - a B ) — a ^ 2 [BESI" ' W - BESIm (an*al n aB^ _ (a ) ' BESINT (a ,l) - BESINT 1 n a - a^ | n '“„•“s ’] D cos (qa ) n B2 ( (H/h - l)2 [sin (H/h - 1 ) (a - 1) cos qa qa Case 3: h ah f t(y) o^h h ah Denominator = r ^(y) dy + / b2 (y) dy + f "b (y) dy o a h h i where bi(y) = ^ y B5 (B2 ■ t>2(y) = Bxh + J _ a---V (y - ot-jh) and B,(ah - y) ■ 'I -1 h2 r -i Denominator = — + B^Cl - a^) + B^(a - l)J \ b2 - B h Numerator = — / yd) dy + — ------f y (“l - alB 2) h 5 ---- — f B ah + irry / (ah - y)+n B a _ B - B 2h i _ ai _ — • — J,(ana ) + -f---- ± — J , (a ) - — J . (a, a ) / % i a — lv 1 n 1 - a — I n — 11 n anP0(q,m,n) \ 1 an 1 a a L n n (Bl'_ aiBp^ T — _ "I + (i _ a ).1 [BESINT (an ,l) - BESIÏÏT ( a ^ a ^ J B 6 APPENDIX C: ROOTS OF THE EQUATION m tan (qan) “ J0(an)/Ji(an ) ROOTS OF EQUATION M.TAN(Q.AN)=J0(AN)/J1(AN) ********************************************* M= 1 .0 0 0 0 Lf\ 0 O r .250 .750 1.000 I. 5OO 2.000 1 1 . 8771* 1 . U93U 1 .2 2 3 8 1.0 312 .7803 .6262 2 1*. 31*1*1* 3. 6O63 3 . 12I+I 2.7612 2.2156 I.8256 3 6.8703 5. 77O8 *+.9329 1+.2832 3. 1+266 2 . 8821* k 9 . 1*152 7.8*123 6.7 3 5 6 5.8958 I+.7OO3 3 . 8951» 5 n . 9^57 9.9 2 5 9 8 . 53*+9 7.1+1+05 5.9713 lt.9731 6 i i * .1*523 1 2 . 0*180 10.3019 9 .0 3 5 3 7.2055 6.0221 7 16 .9 515 1*1.1309 1 2 .1211 IO .5883 8.1+87*+ 7 . O5O9 8 19.1*631 1 6 .2177 13.9030 12.1759 9 . 728I 8.1151» 9 21.9868 18.329*+ 15.7036 13.7332 IO .9907 9. I633 10 2I*. 5081 20. *+l6l 17.5025 15 .3 172 12.2530 IO.I98O 11 27.0183 22. 50I+7 19.2890 16 .8 771 13.1+981 II.2572 12 29.5235 2*1.6117 21.0967 18.1+58*+ 1 I+.768I+ 12.3¡»07 13 32.0352 26. 7OO3 22.8796 28.0200 16.015*+ 13.31*21* Ik 31*. 551*7 28. 79OI 2*+. 6856 21.5997 17.2758 il*. 3988 15 37.0732 30.89*+*+ 26. *+7*+0 23.1626 18.5360 15.1*1*61* 16 39.581*5 3 2 .98*+l 28.2716 2*+.7*+H 1 9 .781+9 16.1*858 17 1*2.0921 35.07*+7 30.0692 26.3852 21.0508 17.51*05 18 l*l*.6ol*l 3 7 .17 7 1 31.8591 27.8827 22.3005 18.5877 19 1*7.1218 39.2677 33.6619 29.1+1+73 23.5597 19.6286 20 1*9.6390 *+1.3589 35. *+*+92 3 1 . 02I+I 2I+.8190 20.6821 M= .500 O r .125 .250 .375 .500 .750 1 .0 0 0 1 2 .2 5 6 3 2 .1032 1.9*+07 I .7727 1.1+591 1.2109 2 5.11*92 *+.6135 3.9859 3.1+996 2.9*+65 2.6526 3 7.9720 6.7769 6.0396 5.6739 5.1122 1*. 1*157 k 1 0 .6397 9.159*+ 8.6100 8.0771 6.5901 5.7817 5 13 .173 6 11.8779 11.0926 9.8089 8.5779 7.5670 6 15.7793 l*+ .6*+92 13.006*+ 11.9*+87 10.3526 8.9191» 7 18.5585 17.2027 15.2375 1 I+.3616 1 2 . 008*+ 10.7126 8 21.1*531 19.36Í+2 1 7 . 85I+5 16.0986 l*+.0852 12.0591* 9 2I+. 3981* 21.7250 20. 16I+7 18.2293 I 5 . 513O 13.8561* 10 27.3519 2*+.*+396 22. 0I+79 20.61+1+8 17.675*+ 15.2000 11 30.271*1* 27.2133 2I+.U770 22.381+1 19.1678 16.9992 12 33. m l * 29.7715 27.0719 2*+. 5II 2 21.1383 18.31*06 13 35.7882 31.9362 29. 16*15 26.9282 22.9269 2 0 . 11*18 Ik 38.3206 3*+. 2913 3 1.17 15 28.6689 2I+.5716 2 1 . 1*821 15 1*0 .9 171 3 7 .00*+7 33.7338 30.7936 26.65*+3 23.281*3 16 1*3.6907 39.7788 36.2293 33.2120 28.0797 2l*.623l* 17 1*6.5835 *+2.3391 38.11+83 3*+.9530 30.21+11 26.1*261 18 1*9.5287 *+*+.5038 *+0. 367*+ 37.0763 31.7365 27.761*1* 19 52.1*821 *+6.8576 *12. 98*+*+ 39.*+9*+8 33.7031 29.5681* 20 55.1*061* i+9.5701 *+5.3016 *+1.2369 35.1+950 30.9058 M= 0 Q= 0 1 2.1*01*8 2 5 . 52OI 3 8 .6 5 3 8 1+ II .7915 5 1 U.9309 6 I 8.O7 1 O 7 21.2163 8 2¡*.3525 9 27.1*933 10 30.631*6 11 33.7762 12 36.9171 13 1*0.0581* 111 1*3.1998 15 1*6.31*12 16 1*9.1*820 17 52.621*1 18 55.7657 19 58.9070 20 62.01*85 C 2 APPENDIX D: ACCELERATION RECORDS OF FIVE EARTHQUAKES In each of the following figures (D1-D9), represents accelera tion, velocity, displacement records, and energy flux plots; and b rep resents acceleration, velocity, and displacement spectra of strong motion records. D3 GROUND VELOC i CM/SEC) —^ ►— GROUND ACCEL (PERG) H/ /SC S OTUNM 18/3/57 8 1 PORTHUENEME S N /SS/C PHN/N ...... /77Y'...... 7.Y X 7.7.7.7.7.7.X '/.7.7.7.7.7.7.7.7.Y 7.7.7.7.7.7.'/.7.7.Y FigureDia ►-VELOCITY SPECTRA—» »— ACCELERATION SPECTRA H/ /SC S OTUNM 1//7 ’/. 18/3/57 PORTHUENEME S N /SS/C PHN/N V.V.7 '.X77Y7.'/.‘/.7.7.7. 7.7.Y .'/./X / iue Dlb Figure 7.7.Y Z./ 7.7. / 7.7.7.7.Y "1 • P F 2 / N E / D P / C N 6 5 E P O R K F I E L D - 2 2 7 / 6 / 6 6 ‘/. '/. 7. ’/. '/. ’/. ’/. ’/. V. Y '/. '/. V. '/. 7. ’/. /. '/. /. Y Z. 7. V. */. 7. /. V. 7. Y Figure D2a PF2/NE/DP/C NS5E PflRKF I ELD-2 27/6/66 7.’/.7.7.7.7.'/.7.7.Y 7.7.7.7.7.'/.7.7.7.Y 7.7./ 7.X X 7.X 7.Y ► — NRTURRL PERIOD £ SEC ) — > Figure D2b ECR/N /Nfl/C N S ELCENTRO 18/5/40 PflR.CORR. IC 07- TT Figure D3a Figure D3"b Figure Figure DUa DIO Figure Dlrt> Figure LIN.CORR. IC ECR/E /Nfl/C ECR/E /Nfl/C E W ELCENTRO 18/5/40 Dll Figure D5a KNfl-S/L/SS/C LONG KOYNfl-SAINI 10/12/67 10/12/67 KOYNfl-SAINI LONG KNfl-S/L/SS/C 48- IC PflR.CORR. D12 F i g u r e D5"b D13 KNP-S/T/SS/C TRAN KOYNA-SAINI 10/12/67 PAR.CORR. 1C 47- Figure D6a VELOCITY SPECTRA—» »— ACCELERATION SPECTRA N-//SC RN ON-AN 1/26 PAR.CORR. 47- IC TRAN KOYNA-SAINI 10/12/67 KNA-S/T/SS/C Figure D6b Figure Figure DTa VELOCITY SPECTRA > | »— ACCELERATION SPECTRA 1 E OP F PN ENNO ATQAE //1 S 1C SS SPIN 9/2/71 EARTHQUAKE FERNANDO OF COMP E 16 S Figure D7b Figure I GROUNO VELOC (CM/SEC1— ► *— GROUND ACCEL (PERG) 7 W OP F R FRAD ERHUK 927 S IC SS 9/2/71 EARTHQUAKE FERNANDO SRN OF COMP W 74 S FigureD8a Figure D&b Figure S S 74 W COMP OF FERNANDO SflN EARTHQUAKE 9/2/71 SS IC D19 Figure D9a VERT COMP OF SAN FERNANDO EARTHQUAKE 9/2/71 VERT EARTHQUAKE OF SAN FERNANDO COMP SS IC D20 Figure D9b Figure VERT COMP OF SAN FERNANDO EARTHQUAKE 9/2 /7 1 SS 1C D21 APPENDIX E: SEISMIC COEFFICIENTS AND POINT ACCELERATION SPECTRA MAC-OF GROUND ACClN. OTUNM ERHUK 1//7 S COMP. S N 18/3/57 PORTHUENEME EARTHQUAKE MAXIMUM ACCELERATION AT A POINTA AT MAXIMUM ACCELERATION Figure El. Explanations for Figures E2 through E127 through E2 Figures for Explanations El. Figure r.0 Q = 0 -75 Mr0.50 SEISMIC COEFFICIENTS FOR FOUR-PARAMETER SLIDING WEDGESLIDING FOUR-PARAMETER FOR COEFFICIENTS SEISMIC MAX. GR• ACCLN. rO. G 17 Figure £2 PRRKFI ELD ERRTHQURKE 27/6/66 ST. 2 N65 ECOMP. MRX.GR.RCCLN.=0.52 Figure E3 Figure E h Figure E5 Figure E7 Figure E8 Figure E9 Figure Figure E10 Ell Figure Ell E12 .,«?6•OF0.88OUND,.%CLW• ,,M 2.oo . 00 'SEISMIC COEF.CH FigureE12 riPG.OF GROUNO PCCLN. RV. SE ISM1C COEF.ll Figure E13 .,?.fl0-|)F. . g OU,IO,.BaSi:LW-,.H, 2.00 J .00 0.?0V-SEI0S.?JC C0f.F20, n Figure El4 Figure Figure El6 HAG «OF GROUND R£CLN • „flV.SE.SHJC COEF.,1, Figure E17 Figure El8 E19 Figure E19 Figure E20 PORTHUENEME ERRTHQURKE ERRTHQURKE PORTHUENEME 18/3/57 N S COMP. . . RCCLN.=0.16 GR MRX G E21 Figure E21 Figure E22 Figure E23 Figure E2k F igu re E25 E26 SRN FERNANDO EARTHQUAKE 9/2/71 S 16 E COMP. MAX.OR-ACCLN.=1.03 G M = 0.50 Q = 0.25 Figure E26 Figure E27 F igu re E28 F igu re E29 F igu re E30 Figure E31 Figure E32 Figure E33 Figure E3^ 40ioG' 0 FnSn0UND.0.8 0 1-20 ^ CLN-,1.60 cn 2 , .00 _0.00 « 0.40 RV.SEISNICCOEF.il) O.HO i .90 iue E35 Figure Figure E36 E37 Figure E37 Figure E38 Figu re E39 Figure E^O Figure EUl Figure EU2 Figure E^3 Figure EUU SAN FERNANDO EARTHQUAKE 9/2/71 S 74 W COUP . MAX . GR • ACCLN•=0•86G Figure E^5 Figure E46 Figure Ehi Figure EU8 9 Figure EU E50 Figure E50 E51 Figure E51 KOYNR EARTHQUAKE 10/12/67 TRAN. COMP MAX. GR. ACCLN .=0.46 G M = 0.50 Q = 0.75 1.80 2.0 Figure E52 MRG.OF GROUND RCCLN. RV.SEISMIC COEF.(l) Figure E53 Figure SAN FERNANDO EARTHQUAKE 9/2/71 S 74 N COMP . MAX.GR.ACCLN . = 0.86G Figure Figure E55 Figure E56 E57 Figure E57 Figure E58 E59 F igu re E59 F igu re E60 Figure E6l SAN FERNANDO EARTHQUAKE 9/2/71 S 16 E COMP MAX - GR.ACCLN. = 1.03 G Figure EÔ2 Figure EÓ3 Figure E6U Figure E65 Figure E66 ELCENTRO ERRTHQURKE 18/5/40 N S COMP MRX . GR • RCCLN . = 0.31 G Figure EÓ7 ELCENTRO ERRTHQURKE 18/5/40 E N COMP. MAX. GR. RCCLN .=0.22 G Figure E68 F igu re E69 F igu re E70 Figure E71 F igu re E72 SRN FERNRNDG ERRTHQURKE 9/2/71 VERT COMP. MRX . GR. RCCLN .=0 -72G Figure E73 Figure E7^ Figu re E75 Figure E76 Figure E77 KOYNfl ERRTHQURKE 10/12/67 LONG COMP. MRX.GR. RCCLN 0-63 G M- 1 . 00 QzO . 50 Figure E78 Figure E79 SRN FERNANDO EARTHQUAKE 9/2/71 S 16 E COMP HAX .GR•ACCLN. = 1 .03 G CO H Figure E80 Figure E8l SñN FERNANDO EARTHQUAKE 9/2/71 VERT COMP. MAX. GR . ACCLN.zO.72G Figure e 82 F igu re E83 Figure E8U Figu re E85 Figu re E86 Figure E87 Figure E88 MAX .GR•ACCLN. = 1.03 1.03 = G .GR•ACCLN. MAX 89 Figure E SfiN FERNANDO EARTHQUAKE 9/2/71 EARTHQUAKE SfiN FERNANDO S 16 E COUP. E90 F igu re E90 Figure E91 F igu re E92 Figure E93 Figure E9k Figure E95 F igu re E96 Figure E97 SFIN FERNANDO EARTHQUAKE 9/2/71 S 16 E COMP. MAX .GR . ACCLN . = 1.03 G M = 1 .00 Q=1.00 Figure E98 Figure E99 Figure E100 Figure E101 Figure E102 Figure EIO3 ELCENTRO EARTHQUAKE 18/5/40 E W COMP. MAX. GR. RCCLN.=0 -22 n = i.oo q = i .so Figure ElOU Figure E105 Figure E106 E107 Figure E107 SAN FERNANDO EARTHQUAKE 9/2/71 S 74 N COUP MAX.GR.ACCLN .=0 -86G Figure E108 Figure E109 Figure E110 EHI F igu re E H I Figure E112 Figure E113 KOYNfl EARTHQUAKE 0/12/67 LONG COMP . MAX .GR • ACCLN . = 0.63 G M=1 .00 Q = 2.00 Figure EllU E115 Figure E116 Figure EIl6 SRN FERNANDO EARTHQUAKE 9/2/71 S 74 W COMP. MAX .GR . ACCLN . =0 -86G M=1 -00 Q = 2.00 Figure E117 SRN FERNANDO EARTHQUAKE 9/2/71 VERT COMP. MAX . GR. ACCLN.zQ.72G Figure E118 Figure E119 Figure E120 E121 Figure E121 Figure E122 ^>•00 8 « G-OF0 ,« OUNO, S2SCLM-,.« J I.0 0 0.?.V Sei«S"JC « f k " ' 0.20 KOYNfl ERRTHQURKE 10/12/67 0 . 40 . 0.60 0.80 ______PERIODI SEC) PERIODI 1.00 1.20 1.40 1.60 1.80 FigureE123 LONG COMP. 2.00 s MRX RCCLN .DR. G 0.63 .= 8 Q M= .80 0 = .800 E12k Figure E125 SRN FERNANDO EARTHQUAKE 9/2/71 S 16 E COMP. MAX .GR.ACCLN. = 1 .03 G M = 0.80 Q = 0 -80 Figure E125 PER 100(SEC) PER M = 0 • 80 80 • 0.80 0 = = Q M MAX.GR .ACCLN.=0.860 MAX.GR 0.80 1.00 1.20 Figure E126 PERIODI SEC PERIODI 0.80 1.00 SAN FERNANDO EARTHQUAKE 9/2/71 EARTHQUAKE SAN FERNANDO S 74 M COMP. E127 Figure E127 E128 APPENDIX F: ENVELOPE OF SEISMIC COEFFICIENTS AND POINT ACCELERATION SPECTRA • SEISMIC COEF .1 1 1 SEISMIC COEFFICIENTS FOR ONE-PARAMETER SLIDING WEDGE SLIDING ONE-PARAMETER FOR COEFFICIENTS SEISMIC ' 0 o'. o'. 20 40 AIU O RECORDS 9MAXIMUM OF ' 0 ' 0 To ^0 To ^0 2-00 0 8 U 1^60 TTio 1^20 ¡Too o'. 80 o'. 60 PERIOD!SEC PERIOD!SEC ! iue FI Figure 0 ONE-PARAMETER SLIDING WEDGE SLIDING ONE-PARAMETER 0 SEISMIC COEFFICIENTS FOR FOUR-PARAMETER SLIDING WEDGE SLIDING FOUR-PARAMETER FOR COEFFICIENTS SEISMIC 05 Q=0.75 M=0.5 @ FOUR-PARAMETER SLIDING WEDGE SLIDING FOUR-PARAMETER @ r ---- m=Si/°i/S2/,2 q=$j (H-h)/Sq=$j mm% ----- 2 h F ig u re F2 UP 3303 3 1 US I 3S* Ad * N333d ONflOdO 30*0dU Figure F3 Figure FU MflO.OF GROUNO «CCLN MAXIMUM OF9 RECORDS Figure F5 Figure Figure Fé 0 0 GROUND RCCLN AIU O 9 RECORDS 9 OF MAXIMUM iue F7 Figure 05 Q=0.75 M=0.5 HflG.OF GROUND flÇCLN...... o'. o'. 20 VTÔÔ flV .SEISMJC COEFcA( 1) MAXIMUM OF 9 RECORDS 40 40 o'. 0 *8 l.O ^0 ’4 l‘6 l. 00 -0 2 0 l'.8 ‘.60 l l’.40 1^20 l'.OO 0*.80 60 PERIOD(SEC ) iue F8 Figure M = 0.5 0 QM = = 1 - 0.5 0 0 0 Figure F9 MAXIMUM 9 OF RECORDS MAXIMUM FIO Figure FIO MAXIMUM 9 OF RECORDS MAXIMUM FU Figure Fil Figure MAXIMUM OF 9 RECORDS 9 OF MAXIMUM F12 Figure Figure F12 FI 3 tlTá F igure FI 3 FI igure F Figure FlU MAXIMUM OF 9 RECORDS M = 0.80 Q = 0 • 80 Figure Fl5 APPENDIX G NOTATION PART II a The root of the equation J (a ) = 0 n ^ o n a n^*1 root of the equation m tan q(a )/J_ (a ) n ^ ^ n 1 n h Width of crest B Unknown constant c Damping coefficient for both dam and layer C Unknown constant F Total inertia force F Maximum inertia force max F^ A function defined in the text F^ A function defined in the text g Acceleration due to gravity g(t) Base accelerations g Maximum base acceleration max G^ Shear modulus of the material of the dam G^ Shear modulus of the material of the layer h Height of dam H Total height from crest of dam to rock level i Square root of -1 I Modified Bessel function of order zero o 1^ Modified Bessel function of order one J Bessel function of the first kind of order zero o J_^ Bessel function of the first kind of order one K Average seismic coefficient expressed as a fraction of gravity K Average seismic coefficient expressed as a fraction of the maximum ground acceleration L Length of dam m The ratio Sn n/S0 _ lpl 2p2 M Total mass of the sliding body M A function defined in the text o M^ Amplification factor, magnitude of ground acceleration P A function defined in the text o q The ratio miH-hJp^/hp^) G2 s Response acceleration an Shear wave velocity in the dam and in the layer, respectively S1 ’S2 Time Horizontal displacement of a point in the dam and in the U1 ’ U2 layer, respectively, relative to the rock base Velocity of a point in the dam and in the layer, respectively, U1 ,U2 relative to the rock base ulSU2 Absolute acceleration of a point in the dam and in the layer, respectively W Weight of the sliding mass y Vertical coordinate a The depth of the base of the sliding surface from the crest A Damping as a factor of critical Mass density Mass density in the dam and in the layer, respectively P1’P2 Angle of slope with the horizontal *n(y) n ^*1 mode shape for the dam ♦ V ^ n Shape function for one-parameter and four-parameter sliding wedge, respectively *n(y) n^*1 mode shape for the layer ( J L ) , 0 ) Damped and undamped circular frequency, respectively, of the n on n ^*1 mode of the dam-foundation system PART III A Pore pressure parameter A^ Dynamic pore pressure parameter b Width of the slice B Pore pressure parameter c* Cohesion at the base of the slice in terms of effective stresses c f Average cohesion at the side of the slice c! Cohesion at the base of the slice i 1 DU^ Dynamic pore pressure developed under the influence of k^W E Total normal force on the side of the slice Eq Interslice body force F Factor of safety G3 Part III (Continued) F Static factor of safety 0 Local factor of safety in the side of the slice f l h Height of the section above the slip surface 1 Slice number k Seismic coefficient kW Horizontal earthquake load k Critical acceleration c k Coefficient of earth pressure at rest 0 k Coefficient of anisotropic consolidation c 1 Distance of the point of application of the normal force at the base of the slice from the edge of the slice N,N’ Normal force at the base of the slice in terms of total stresses and effective stresses, respectively N Normal force at the base of the slice o PW Force exerted by the pore water pressure on the side of the slice T Shear force at the base of the slice T Shear force at the base of the slice o u Pore pressure at the base of the slice U Force exerted by the pore water pressure on the base of the slice W Weight of the slice x Horizontal coordinate X Coordinate of the center of gravity of the whole mass g' X Coordinate of the point of application of the normal force m m X Shear force on the side of the slice section X Interslice body force 0 Z. Height of the point of application of E force above the 1 slip surface a Slope of the base of the slice with respect to the horizontal Y Density of the material 6 Slope of the base of the slice with respect to the horizontal Au Change of pore pressure due to cyclic loading Part III (Continued) Aa^ Change in axial stress Aa^ Change in cell pressure At Applied cyclic load in simple shear test X An unknown to he determined from the solution a Stress component a ! Normal stress at the base of the slice a f Vertical consolidation pressure vo Axial pressure for consolidaton a^c Confining cell pressure for consolidation t Shear stress PART IV c 1 Average cohesion in terms of effective stresses av D Driving force k The maximum value of the seismic coefficient m L Length of sliding block n ! Average effective normal stress on the slip surface R Resisting force T Predominant period of an accelerogram or twice the duration of the pulse Force due to pore pressure at failure W Weight of the sliding block x Instantaneous displacement of the sliding block relative to the base x Instantaneous velocity of the sliding block relative to the base x Instantaneous acceleration of the sliding block relative to the base x Permanent displacement of the block relative to the base m 3 Slope of the sliding plane t Average shear stress on the slip surface In accordance with letter from DAEN-RDC, DAEN-ASI dated 22 July 1977, Subject: Facsimile Catalog Cards for Laboratory Technical Publications, a facsimile catalog card in Library of Congress MARC format is reproduced below. Sarma, S K Response and stability of earth dams during strong earth quakes / by S. K. Sarma, Department of Civil Engineering, Imperial College of Science and Technology, London, U. K. Vicksburg, Miss. : U. S. Waterways Experiment Station ; Springfield, Va. : available from National Technical Informa^- tion Service, 1979. 82, [188] p. : ill. ; 27 cm. (Miscellaneous paper - U. S. Army Engineer Waterways Experiment Station ; GL-79-13) Prepared for Office, Chief of Engineers, U. S. Army, Washington, D. C., under Grant Agreement No. DAERO-75-G-010. References: p. 80-82. 1. Dam stability. 2. Earth dams. 3. Earthquake engineering. 4. Earthquake resistant structures. 5. Earthquakes. I. London. University. Imperial College of Science and Tech nology. Dept, of Civil Engineering. II. United States. Army. Corps of Engineers. III. Series: United States. Waterways Experiment Station, Vicksburg, Miss. Miscellaneous paper ; GL-79-13. TA7t,WS4m n°T GL-7jLJL3------