Calculation of Nonlinear Ground Response in Earthquakes by William B. Joyner and Albert

A method is presented for calculating the seismic response of a system of horizontal soil layers. The essential element of the method is a rheological model suggested by Iwan which takes account of the nonlinear hysteretic behavior of soils and has considerable flexibility for incorporating laboratory results on the dynamic behavior of soils. Finite rigidity is allowed in the underlying elastic medium, permitting energy to be radiated back into the underlying medium. Three alternate ways of integrating the equations of motion are compared, an implicit technique, an explicit technique, and integration along characteristics. An example is set up for comparing the different methods of integration and for comparing the nonlinear solution with a solution based on the widely used equivalent linear assumption. The example consists of a 200-m section of firm alluvium excited at its base by the N21E component of the Taft accelerogram multiplied by four to produce a peak acceleration of 0.7 g and a peak velocity of 67 cm/sec. The three techniques of integration give very similar results, but integration along characteristics has the advantage of avoiding spurious highfrequency oscillations in the acceleration time history at the surface. For the chosen example, which has a thick soil column and a strong input motion, the equivalent linear solution underestimates the intensity of surface motion for periods between 0.1 and 0.6 sec by factors exceeding two. The discrepancies, however, wonld probably be less for input motion of lower intensity. At longer periods the equivalent linear solution is in essential agreement with the nonlinear solution. For the same example both solutions show that, compared to a site with rock at the surface, motion at the surface of the soil is amplified for periods longer than 1.5 sec by as much as a factor of two. At shorter periods the amplitude is reduced. I N T R O D U C T I O N The problem we are concerned with is basically a very simple one. We postulate a system of horizontal soil layers bounded above by the free surface and below by a semiinfinite elastic medium representing the bedrock. We further postulate a vertically incident shear wave in the underlying medium, and we ask the question, "How will the overlying layers respond and in particular what will be the motion of a point on the free surface ?" This is a classical problem in engineering seismology. There is some disagreement concerning the range of applicability of the solution (Hudson, 1972; Newmark et al. 1972), but no one would deny the importance of solving this relatively simple problem correctly. The problem was solved by Kanai (1952) some years ago for the case of layers with linear viscoelasticity of the Voigt type. When we are dealing with input motion sufficiently intense to cause severe damage to structures, however, we cannot assume linear behavior over the entire range of strain. To do so would imply stresses many times greater than the strength of typical materials as measured in the laboratory. To circumvent this difficulty the method in common use currently is what we shall