Numerical Solution to the Propagation of a Waveform Caused by

A finite element model of the half-space is developed using the frequency domain computer code SASSI, “a System for Analysis of Soil-Structure Interaction” [Lysmer et al., 1981]. Analyses using this model can predict wave forms and travel velocities that incorporate effects not considered in the previous analytical and numerical pre- diction models. The use of finite element analyses permits assessment of changes in

velocity, Vs, of 200 fps and Poisson’s ratio of 0.25 is described below. The surface of the half-space is loaded over a uniform circular area having a radius of 3 in. and, separately, a rigid disk having the same diameter. The disk is meshed using trian- gular plate elements sized to provide a model of the plate consisting of three 1 in. wide rings. The meshing of the 6 in. diameter plate is shown in figure 2.27. Nodes are located on the surface of the half-space starting at a distance of 6 in. from the center of the loaded region and extending to a distance of 78 in. in 6 in. increments. This model is shown schematically in figure 2.28 where the black disk in the center of the block represents the loading source, the larger orbs represent the locations of ac- celerometers in the experimental program, and the smaller orbs represent “receiver” locations in the finite element model where responses are computed. The impulse load (a pulse with a duration of 1.5 msec) and soil properties are the same as those used in Section 2.4 to compute the waveform shape and travel velocity using numer- ical and analytical approaches.

The solution method used in SASSI is described briefly, followed by a comparison of the displacements computed using the SASSI model to the solutions developed in Section 2.4 for the cases without mass. The change in response due to addition of mass is developed and compared to the results for the response computed neglecting mass. Finally, the computed displacements at each of the “receiver locations” from the computer analyses are plotted for all distances demonstrating the propagation of the surface wave with time.

The SASSI computer program was developed at the University of California, Berkeley. The theory and analytical methods are described in detail in four doc- torial disserations [Ostadon, 1983, Tabatabaie-Raissi, 1982, Tajirian, 1981, Vahdani, 1981]. The program uses a sub-structuring approach known as the flexible volume method wherein the problem is divided into a series of sub-problems. A frequency do- main solution is implemented, thereby imposing the assumptions of elastic behavior.

The flexible volume method for the foundation vibration problem involves computing the Green’s function for each node located at the interfaces between the structure and the surrounding soil (site) and for the nodes located within the volume enclosed by the structure to soil interfaces. A compliance matrix is assembled using the Green’s functions, which is then inverted to obtain and impedance matrix (dynamic stiffness matrix). The dynamic matrix associated with the impedances is combined with the dynamic matrix of the structure(s) to describe the entire system. This system defines the [C] part of the matrix equation,

[C]{U}={Q}, (2.45)

where {U} are the nodal displacements and{Q} is the vector of applied forces. The

details of this methodology are described in Appendix D.

The results from the SASSI analyses for the flexible and rigid solutions of the disk subjected to a 1.5 msec long cosine load function pulse are compared to the displacement computed analytically from the flexible disk in figures 2.29, 2.30, and 2.31 and to the displacement from the rigid disk in figures 2.32, 2.33, and 2.34. As shown, the general shape of the wave forms are similar to the analytic solution although considerable smoothing of the response occurs. This smoothing is a result of the use of Fourier transform representation of the loading function (also seen in figures 2.14, 2.15, and 2.16) but mostly from the use of finite elements, which impose a linear relationship on the displacement regime between nodes, to model the loaded region and the local zone around the reciever nodes where the response is computed.

figures 2.33 and 2.34, the finite element solution to analytic comparison improves as distance increases indicating that, as separation distance from the source increases, the finite element approximation becomes better at capturing response.

The effect of mass on the results for the rigid disk on the surface is shown and compared to the no-mass solutions in figures 2.35, 2.36, and 2.37. As expected, the inclusion of mass adds inertial effects to the system, lengthening the arrival times for both the first and second peaks, further smoothing the response, and adding some oscillatory behavior to the response.

The response of the surface of the sand half-space is plotted as displacement amplitude vs. time, for increasing distance from the source (figure 2.38). As shown, the waveforms are similar in character to the analytic forms and arrival times are well-reproduced by the finite element solutions. Since changes in the propagating wave are of primary interest in this study, the results from the finite element model are adequate for assessing changes to the wave’s response caused by embedded objects in its path.