A class of inhomogeneous shear models for seismic response of dams and embankments Panos Dakoulas and George Gazetas Department of Civil Enffineering, Rensselaer Polytechnic Institute, Troy, N. Y 12180, USA Results of linear and nonlinear static analyses of gravity-induced stresses in several typical dam cross-sections, in conjunction with published experimental correlations of shear modulus versus confining pressure for a large variety of soils, reveal that the average shear modulus across the width of earth/rockfill dams may be expressed as a power m of depth, with m ranging from 0.35 to 0.90 and depending on material and geometric parameters. A general inhomogeneous shear beam model is developed to account for any possible such variation of modulus with depth. Perhaps somewhat surprisingly, closed-form analytical expressions are derived for natural frequencies, modal displacements, participation factors, and steady-state response functions for all values of the inhomogeneity factor m. Parametric results are presented in tabular and graphical form and conclusions are drawn of practical significance. Finally, a comprehensive comparative study is undertaken to investigate the validity of the inhomogeneous shear beam (SB) models. For five different dam cross-sections, each excited by four recorded accelerograms, it is shown that plane-strain finite- element analyses yield fundamental periods and peak displacements within the dam which are in very good accord with the predictions of a 'consistent' inhomogeneous SB model. A companion paper 1 extends the present work and focuses on seismic shear strains and seismic coefficients within dams and embankments. INTRODUCTION The shear beam (SB) model has been used extensively over the years to estimate the lateral seismic response of earth/ rockfill dams and embankments, 2-7 its popularity stemming mainly from its simplicity. A number of rigorous studies a ~o have largely corroborated one of the crucial assumptions of the SB model, namely that horizontal shear strains, shear stresses and displacements are essentially uniformly distributed across the width of the dam, and have moreover shown that the SB model yields natural periods and modal shapes which are quite realistic. These conclusions are also confirmed by the results of this and the companion paper. 1 To account for the dependence of soil stiffness on confining pressure, an improved version of the SB model has been developed which considers the (average across the width) shear modulus as increasing with the 2/3 power of the depth. ~°'~1 Direct and indirect evidence, including in-situ measurements of S-wave velocities and observations of the actual response of several dams 2° 12 have shown the general validity of such a modulus variation with depth. Other researchers 13 15 have also suggested a shear modulus increasing with depth, although not quite as steeply as with the 2/3 power. This paper first outlines the results of a comprehensive investigation on the factors influencing the variation of stiffness in earth and rockfill dams. It is concluded that the (average across the dam) shear modulus may be realistically considered as increasing with the m power of the depth z, where m may usually take values in the range 0.40<~m~<0.75. Then an exact analytical solution is derived for free and forced vibrations of a truncated shear wedge (as sketched in Fig. 1) characterized by an arbitrary value of the power m. The third objective of the paper is to Accepted May 1985. Discussion closes December 1985. 0267-7261/85/0401661752.00 © 1985 CML Publications 166 present the results of a comprehensive comparative study in which the seismic response of several dam sections are computed by plane finite-element (FE) and by inhomo- geneous SB analysis. For each dam section the l~ower m of the SB model is selected such that its modulus variation is consistent with the spatial variation of stiffness in the specific plane-strain model. It is concluded that such a 'compatible' inhomogeneous SB predicts fundamental periods and peak seismic displacements which are in a very close agreement with those computed with the corresponding FE models. T~he present study is further extended in a companion paper 1 which focuses on distribution of peak seismic shear strains and seismic coefficients in dams and embankments. MODULUS VARIATION WITH DEPTH While several (material, geometry and excitation related) factors influence the spatial distribution of effective shear moduli within a dam, it has been found in the course of our investigation that the average modulus, G(z), over a horizontal plane at a depth z from the origin (Fig. 1), depends mainly on three factors: (a) the dependence on confining pressure of the shear modulus Ge of each constituent material (b) the size and the relative overall stiffness of the impermeable cohesive core (c) the inclination of the slopes and the truncation ratio. The effect of these factors on G(z) is quantitatively illustrated in this section. The shear modulus of a particular soil element at small levels of shear strain can be expressed as: Ge = F(e,OCR). a~'~ (1) where: a~t =the effective normal octahedral stress; and e Soil Dynamics and Earthquake Enoineerino, 1985, Vol. 4, No. 4