Analytical Solutions

We have developed closed-form analytical solutions for the pile KSSI problem for soil profiles with constant stiffness for both vertically propagating shear waves and planar Rayleigh waves. Although the assumption of constant soil stiffness over the length of a deep foundation is unlikely to match typical field conditions, the analytical solutions provide a convenient reference point for comparison to the numerical solutions and field measurements. For example, we use the analytical solutions to help us understand over what ranges of soil stiffness, foundations stiffness, and ground motion frequency the KSSI effects are most significant, then use this information to set the ranges of parameters we will study with the numerical simulations.

We have also developed solutions for soil profiles with linearly or parabolically increasing soil stiffness with depth, which is more realistic. However, a closed form solution to the governing differential equation for a laterally loaded pile is not available for this case. Instead, we can solve the differential equation numerically. This solution approach is more complicated and time consuming than the closed-form approach for constant soil stiffness, but still less involved than the numerical procedure involving 1-D site response and a BNWF model. Ultimately we plan to develop tools that utilize a variety of solution approaches to be utilized at different phases of a project, from preliminary planning when only basic soil and foundation properties are known, to the final design where more robust modeling is possible.