SUMMARYNumerical codes which use a one-point quadrature integration rule to calculate stiffness matrices for the 2-D quadrilateral element and the 3-D hexahedral element, produce matrices which are singular with respect to a number of displacement patterns other than the rigid body patterns. In this paper an economical method is derived to remove this singularity and which also produces accurate flexural response. For rectilinear element geometry the method is equivalent to the incompatible model element of Wilson et al.' For non-rectilinear element geometry a slight modification of the scheme is required in order to assure that it passes the patch test. The method of this paper can also be used in finite difference codes which experience similar difficulties.

A new method is presented for computing the complete elastic response of a vertically heterogeneous half-space. The method utilizes a discrete wavenumber decomposition for the horizontal dependence of the wave motion in terms of a Fourier-Bessel series. The series representation is exact if summed to infinity and consequently eliminates the need to integrate a continuous Bessel transform numerically. In practice, a band-limited solution is obtained by truncating the series at large wavenumbers. The vertical and time dependence of the wave motion is obtained as the solution to a system of partial differential equations. These equations are solved numerically by a combination of finite element and finite difference methods which accommodate arbitrary vertical heterogeneities. By using a reciprocity relation, the wave motion is computed simultaneously for all source-observer combinations of interest so that the differential equations need only be solved once. A comparison is made, for layered media, between the solutions obtained by discrete wavenumber/finite element, wavenumber integration, axisymmetric finite element, and generalized rays.

A finite element variational method is described and applied to plane strain analysis of zero frequency seismic data. This technique presents a suitable tool for the analysis of permanent displacements, tilts, and strains caused by seismic events, since it can model variable fault offsets in heterogeneous media. The accuracy of the technique is demonstrated by detailed static field computations for vertical and dipping dislocations acting in plane strain, corresponding to a fault of infinite length in a homogeneous half space, by comparison with closed form analytical solutions. A parametric study of material inhomogeneities and variable fault offsets reveals that order of magnitude changes in the solutions can occur for both near‐ and far‐field displacements and strains. The technique was applied to the San Fernando earthquake. The best solution was obtained by separating the fault into two distinct parts, both parts having offsets near the surface a factor of 5 larger than the average slip. The seismic moment, defined in terms of the average displacement, is 6.2 × 1025 dyne cm, and the average stress drop is 24 bars for this fault system, although both stress drop and displacements vary by more than an order of magnitude along the fault plane, the maximum occurring at 1‐km depth. Several solutions are investigated for the hypocentral region, one solution giving as much as 5 meters of offset. This possible behavior of the fault characterized by large variations of slip as failure progressed implies that local geology controlled this thrust fault through its effect on spatial distribution of prestress.

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A method is presented for the computation of near-field particle displacements and particle velocities resulting from a dynamic propagating, stress relaxation occurring on a finite fault plane embedded within a three-dimensional semiinfinite medium. To check our numerical procedure we compare our results for a circular fault in a full space with Kostrov's (1964) analytic solution for a self-similar propagating stress relaxation. We have simulated two bilateral strike-slip earthquakes differing only in hypocentral location and examined the particle motion on the traction-free surface and on the rupture surface. Focusing of energy is evident in both ruptures. The static displacement on the rupture surface overshoots the theoretical static value by approximately 25 per cent. For the rupture that nucleated at depth the free surface almost doubled the particle velocities along the fault trace as compared with the rupture that nucleated at the free surface. Our numerical results indicate that for an earthquake occurring on a semi-circular fault with radius of 10 km, an effective stress of 100 bars and a rupture velocity of 0.9β in a medium characterized by β = 3 km/sec, α = α=3β and a density of 2.7 gm/cm3 particle velocities can reach 400 cm/sec and displacements 250 cm. We also compare our numerical results with the observations made by Archuleta and Brune (1975) for a spontaneous stress relaxation on a semi-circular crack in a prestressed foam rubber block.