Nonlinear seismic response analysis of earth dams Sara J. Lacy and Jean H. Prevost Department of Civil Engineerin9, Princeton University, Princeton, NJ 08544, USA The objective of this paper is to propose a general and efficient numerical procedure for analysing the dynamic response of geotechnical structures, which are considered as both nonlinear and two phase systems. In Section 2, the appropriate coupled dynamic field equations for the response of a two-phase soil system are briefly reviewed. The finite element spatial discretization of the field equations is described and time integration for the resulting nonlinear semi-discrete finite element equations is discussed. In Section 3, iterative techniques are examined for the solution of the global nonlinear system of finite element equations. A large amount of computational effort is expended in the iterative phase of the solution and so the iterative procedure used must be both reliable and efficient. The performance of three iterative procedures is examined: Newton Raphson, Modified Newton Raphson and Quasi-Newton methods, including BFGS and Broyden updates. Finally, in Section 4, the elasto-plastic earthquake response analysis of a two phase nonhomogeneous earth dam is presented. Extensive documentation exists ~ for the particular problem selected including recorded earthquake motions at the base and crest of the dam. The results of the numerical calculations are compared to the recorded response of the dam. 1. INTRODUCTION The solution of geotechnical engineering problems is complicated for several reasons. First, geotechnical problems generally involve large nonhomogeneous structures. Second, the solution of dynamic problems is complicated because soil exists in general as a multi-phase solid-fluid system. Further, the behaviour of the soil skeleton is highly nonlinear, anisotropic and hysteretic. Under these conditions, the analytical solutions of geotechnical problems have necessarily involved many simplifying assumptions. However, the development of computing facilities has now made it possible to solve these complicated problems more exactly using numerical methods. As larger and more powerful computing facilities become available, the need grows for numerical techniques which efficiently and accurately model the complex behaviour of soil systems. One problem which has received considerable attention in geotechnical engineering is the dynamic response of earth dams to earthquakes. The disastrous consequences of a dam failure serves as a motive for making as accurate an analysis as possible of the problem. This exemplifies the kind of geotechnical problem in which there are many complexities, viz., the material is saturated in part, although the extent of the saturated region is a priori unknown. Further, the dam material is subjected to high amplitude cyclic loading, and in this circumstances develops inelastic, hysteretic behaviour. The stress-strain behaviour of the soil skeleton is complex and its modelling must be accurate and numerically efficient. In this paper numerical methods are presented which are applicable to the solution of nonlinear two phase geotechnical problems in general. The methods are then ReceivedOctober 1986.Discussioncloses March 1987. 0261 7277/87/010048 16 $2.00 © 1987ComputationalMechanicsPublications 48 applied to analyse the response of an earth dam to an earthquake. Dynamic analyses of the response of soil systems to earth- quakes have until recently been solved by very simple numerical models. Traditionally, the finite element methods is used assuming elastic soil behaviour with viscous damping as the dissipation mechanism. More recently equivalent linear methods have been developed 22,39 in which the nonlinear characteristics of the soil skeleton are accounted for by use of equivalent linear soil properties, and an iterative procedure is used to obtain modulus and damping values compatible with the effective strains computed in each zone of the soil mass. Although this approximate procedure has been used for many practical problems (see e.g., Refs 24, 28, 29, 43 for its application to earth dams) the availability of computing facilities has now made it possible to develop more rigorous nonlinear dynamic analysis procedures. Nonlinear hysteretic finite element analyses of the earthquake response of an earth dam were reported in Refs 36 and 37. The dam was modelled as a single phase soil structure. The stress-strain behaviour of the soil was approximated by a simple multi-surface yon Mises plasticity model. The only damping introduced was the hysteretic damping resulting from the material nonlinearities. Figure 1 compares computed acceleration for the one phase model with the recorded acceleration of the dam crest. Comparisons are shown for the motions in the upstream-downstream direction along with their corresponding Fourier transforms. The nonlinear model produces an accurate time history and in particular it captures the correct range of frequencies. However, the magnitude of the computed acceleration exceeds that recorded. Clearly the model used did not introduce enough damping. A more sophisticated two phase model Soil Dynamics and Earthquake Engineering, 1987, Vol. 6, No. 1