During a big earthquake along the San Andreas fault in southern California, high excitation and low attenuation of long-period (3 to 10 sec) strong ground motions will cause wave motions to propagate efficiently far from the epicentral area. These ground motions could potentially be destructive to large-dimension structures in the Los Angeles basin. We performed calculations using the surface-wave Gaussian beam method for a 3D southern California crustal structure. Displacement field as well as the associated dynamic strain field produced by large propagating ruptures along the San Andreas fault are evaluated. Results indicate that in the presence of lateral heterogeneity, focusing and multipathing interference contribute significantly to a complex pattern of the displacement field and the associated dynamic strain field. For a big event on the San Andreas fault with a seismic moment of 1.8 x 1028 dyne-cm, long-period displacement in the Los Angeles basin could reach a maximum amplitude of meters in places. Since this calculation is fast, we have evaluated the displacement field for a dense grid of points; a differentiation gives the corresponding effective horizontal dynamic strain field. At times, the maximum effective dynamic strains may reach mid-10-3 to even 10-2--high enough to be of engineering concern. This computational result probably gives the upper bound values due to the large source assumed. For events of smaller seismic moment release along less extensive ruptures, these results can easily be scaled down proportionally. Different scenarios are considered in this study with different slip distributions. It is found that with a given seismic moment, a more evenly distributed fault slip over the rupture surface will result in lower peak values on both displacements and dynamic strains. Our displacement results give similar values to those obtained by Kanamori using empirical Green's functions but substantially higher than Bouchon and Aki's results.