The complexity of modern seismically isolated structures requires the analysis of the structural system and the isolation system in its entirety and the ability to capture potential discontinuous phenomena such as isolator uplift and their effects on the superstructures and the isolation hardware. In this paper, an analytical model is developed and a computational algorithm is formulated to analyze complex seismically isolated superstructures even when undergoing highly-nonlinear phenomena such as uplift. The computational model has the capability of modeling various types of isolation devices with strong nonlinearities, analyzing multiple superstructures (up to five separate superstructures) on multiple bases (up to five bases), and capturing the effects of lateral loads on bearing axial forces, including bearing uplift. The model developed herein has been utilized to form the software platform 3D-BASIS-ME-MB, which provides the practicing engineering community with a versatile tool for analysis and design of complex structures with modem isolation systems.