ysis of Earth Dams," Journal of the Geotechnical Engineering Division, ASCE, Vol. 109, No. 11, 1983, pp. 1383-1398.

10. Ohmachi, T., and Tokimatsu, K., "Simplified Method for 3-Dimensional Dy-namic Analysis of Embankment Dams," Proceedings of Fourth International Conference on Numerical Methods in Geomechanics, Vol. 1, 1982, pp. 411-419.

11. Prevost, J. H., Abdel-Ghaffar, A. M., and Lacy, S. J., "Nonlinear Dynamic Analysis of Earth Dams: A Comparative Study," Journal of the Geotechnical Engineering Division, ASCE, Vol. I l l , No. 7, 1985, pp. 882-897.

12. Roesset, J., "Soil Amplification of Earthquakes," Numerical Methods in Geo-technical Engineering, C. S. Desai and J. T. Cristian, Eds., McGraw-Hill Pub-lishing Co., Inc., New York, N.Y., 1977, pp. 639-682.

13. Sarma, S. K., "Response and Stability of Earth Dams During Strong Earth-quakes," U.S. Army Corps of Engineers Waterways Experiment Station, Mis-cellaneous Paper, No. GL-79-13, 1979.

1-D SEISMIC ANALYSIS OF EMBANKMENT DAMS By Carlos A. Prato,1 A. M. ASCE, and Elver Delmastro2

INTRODUCTION

This technical note is concerned with a numerical integration scheme based on the shear beam model coupled with the linear equivalent method for dynamic soil properties of Seed, Idriss, and their co-workers (2, 3 -5, and 8). Solution to the dynamic equilibrium equation is performed in the frequency domain FFT. The proposed solution is, in effect, a com-bination of the shear wedge approach (1, 6, and 9) and Seed's material properties, which in a simplified manner , allows modeling of nonhomo-geneous cross section dams.

The practical attractiveness of a reasonably accurate 1-D procedure such as the one proposed here is obviously related to reducing the compu-tational effort associated with more elaborate 2-D and 3-D models (2-5). One must bear in mind, that even though the classical, triangular, ho-mogeneous, shear wedge modal solution (zero-order Bessel functions of the first kind, [9]) and the stratum-solution, modal shapes are very sim-ilar for the fundamental mode. However, they have the following short-comings for seismic design: (1) They can only be applied to homoge-neous sections; (2) modal shapes that are higher than the fundamental, differ considerably between these approaches; and (3) the fundamental period of a homogeneous wedge is T0 = 2,59 H/v,, and the stratum is

lProj. Engr., INCONAS Consulting Engrs., C6rdoba, Argentina; Prof., Natl. Univ. of C6rdoba, Cdrdoba, Argentina.

2Res. Asst., CONICET at Natl. Univ. of C6rdoba, C6rdoba, Argentina. Note.Discussion open until January 1, 1988. To extend the closing date one

month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on March 24, 1986. This paper is part of the Journal of Geotechnical Engineering, Vol. 113, No. 8, August, 1987. ASCE, ISSN 0733-9410/87/0008-0904/$01.00. Pa-per No. 21729.

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T0 = 4 H/vs. In this formula, H = height, and vs = shear wave velocity of the material. Although by using a fictitious value of vs it is possible to approximate the fundamental period of the wedge with the stratum expressions (i.e., v's = vs 4/2,59), the dynamic participation factors for seismic excitation differ by approximately 20% between these solutions. Therefore, one should not expect to obtain a reasonable representation of seismic dam response through stratum expressions in many typical cross sections used in practical cases.

To analyze a section such as Fig. 1(a) by means of a global 1-D model, the cross section is divided into a series of layers as indicated in Fig. 1(b). For each layer, the average shear stiffness, damping, and density is calculated numerically by subdividing the layers into horizontal seg-ments in which the mechanical properties are assumed constant. For granular materials, the dynamic shear modulus at small strains Gmax var-ies with the effective confinement stresses and is commonly assumed, as in Ref. 2, to depend on (tr^)1''2, where &m = the mean effective stress at a point under static conditions. For this purpose, v'm is approximated by two-thirds of the overburden effective stress, evaluated at the center point of each segment. A material constant, (K2)max, must be determined (2) for different granular materials in the cross section based on field and laboratory tests. For impervious materials, Gmax is commonly ex-pressed in terms of the consolidated, undrained shear strength, which is also a function of the confinement stresses. Successive iterations of the linear equivalent method lead to modifications of the average shear modulus and damping of each layer as a function of the effective shear strain. This is assumed constant within each layer in the shear beam theory. For the variation of the shear modulus with shear strain, ex-perimental results given in Ref. 5 were used for shell and core materials.

(a)

(b) FIG. 1.Embankment Cross Section: (a) Nonhomogeneous Section; (h) Layer Idealization

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INTEGRATION SCHEME

Dynamic equilibrium of a shear beam subject to base motion is gov-erned by:

[b(Gy + cy)] - pbua = 0 (i) in which (7 = du/dy) = the shear strain; u = the horizontal displacement relative to the base; y = the axial coordinate; (7 = dy/dt) = the shear strain rate; c = viscous damping coefficient; b = width of the shear beam as a function of y; p = mass density of the layer ( = tis + ii) = the absolute acceleration; and us = the acceleration at the base.

For a base acceleration of amplitude Usemt, Eq. 1 reduces to:

(bG*T) + pbw2U -pbUs = 0 (2) y in which G* = G[(l - 2g2) + i'2i;Vl - |2] is the complex shear modulus; T = the Fourier transform FT of the shear strain 7; = the damping ratio; U = the complex relative displacement response at frequency

tia=Ua-

FIG. 3.Maximum Stresses and Accelerations

suits are in good agreement in the range of frequencies from zero to approximately 1.3 times the fundamental frequency. However, they di-verge considerably for higher frequencies. QUAD 4 gives an average, overall damping ratio of 10.7%, while the proposed method gives 11.1%. The fundamental natural period (at level of shaking considered in the analysis) predicted by QUAD 4 is 0.89 s; by the proposed method, it is 0.91 s; and by SHAKE is 0.80 s. Fig. 2 contains the maximum shear stresses in the center of the bottom layer of the shell; those associated with QUAD 4 are given at element center point. In the proposed method they are calculated by multiplying the maximum shear strain -ymax at the layer by the secant elastic modulus G = /(cr^, K2max, 7max)/ a function of the initial static effective confinement stresses, of (K2)max values across the layer and of 7max. Difference between these results for the maximum stress at the upstream shell is 5.6% (referred to the QUAD 4 value), and for the downstream shell, is 12.1%. Maximum differences are observed at the transition zones between the core and shells, where the variation of the shear strains across the layer cannot be accurately represented by the shear beam theory.

CONCLUSIONS

A numerical scheme for seismic analysis of nonhomogeneous em-bankment dam sections has been presented and shown to provide re-sults that approximate reasonably well the more precise 2-D analyses in a gravel shell dam design. Crest accelerations are predicted within 5% of those given by QUAD 4. Shear stress patterns and values within shells exhibit differences between 5 and 12%, with the exception of those at the narrow transition zones between shells and core, where differences are larger. In spite of this shortcoming, the general good agreement found with the 2-D, QUAD 4 solution renders the proposed 1-D numerically integrated, shear beam model a potentially useful tool. It may be used to determine the dynamic response data required to calculate defor-mation potentials, dynamic pore water pressure increases, settlement, and stability analyses currently performed to assess the seismic behavior of embankment dams.

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APPENDIX.REFERENCES

1. Ambrasseys, N., and Sarma, J., "The Response of Earth Dams to Strong Earthquakes," Geotechniqae 17, 1967, pp. 181-213.

2. Idriss, I. M., et al., "QUAD 4: A Computer Program for Evaluating Seismic Response of Soils Structures by Variable Damping Finite Element Proce-dures," EERC Report, No. 73-16, Earthquake Engineering Research Center, University of California at Berkeley, Berkeley, Calif., Jul., 1974.

3. Lysmer, J., et al., "LUSH 2: A Computer Program for Complex Response Analysis of Soil-Structure Systems," EERC Report, No. 74-4, Earthquake En-gineering Research Center, University of California at Berkeley, Berkeley, Calif., Apr., 1974.

4. Lysmer, J., et al., "FLUSH: A Computer Program for 3-D Analysis of Soil-Structure Interaction Problems," EERC Report, No. 75-30, Earthquake Engi-neering Research Center, University of California at Berkeley, Berkeley, Calif., Nov., 1975.

5. Mejia, L. H., Seed, H. B., and Lysmer, J., "Dynamic Analysis of Earth Dams in Three Dimensions," Proceedings, Geotechnical Division, ASCE, Vol. 108, No. 12, Dec, 1982, pp. 1586-1604.

6. Prevost, J., Abdel-Ghaffar, A. M., and Elgamal, A. M., "Nonlinear Hysteretic Dynamic Response of Soil Systems," Proceedings, Engineering Mechanics Di-vision, ASCE, Vol. I l l , No. 5, May, 1985, pp. 696-713.

7. Schnabel, P. B., Lysmer, J., and Seed, H. B., "SHAKE: A Computer Program for Earthquake Response Analysis of Horizontally Layered Sites," EERC Re-port, No. 71-12, Earthquake Engineering Research Center, University of Cal-ifornia at Berkeley, Berkeley, Calif., Dec, 1972.

8. Seed, H. B., and Idriss, I. M., "Soil Moduli and Damping Factors for Dynamic Response Analysis," EERC 70-20, Earthquake Engineering Research Center, University of California at Berkeley, Berkeley, Calif., Dec, 1970.

9. Seed, H. B., and Martin, G. R., "The Seismic Coefficient in Earth Dam De-sign," Proceedings, Soil Mechanics and Foundations Division, Vol. 92, No. 3, Jan., 1966, pp. 25-58.

EFFECTS OF STRESS HISTORY ON BALLAST DEFORMATION

By Vishnu A. Diyaljee,1 M. ASCE

INTRODUCTION

The majority of published work on triaxial compression repeated load-ing of cohesionless soils has been concerned with evaluation the behav-

'Supervising Geotech. Design Engr., Alberta Transportation, 4999 98th Ave., Edmonton, Alberta, Canada, T6B 2X3.

Note.Discussion open until January 1, 1988. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on April 3, 1986. This paper is part of the Journal of Geotechnical Engineering, Vol. 113, No. 8, August, 1987. ASCE, ISSN 0733-9410/87/0008-0909/$01.00. Paper No. 21729.

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