Explore the complexities surrounding "Soil$-$Structure Interaction $($SSI$)"$ in geotechnical engineering and its implications on the design and analysis of foundation systems. Elaborate on the dynamic interaction between soil and structural elements, considering factors such as soil properties, foundation types, and loading conditions. Discuss how SSI phenomena influence the behavior of structures subjected to static and dynamic loads, and evaluate the methodologies employed to incorporate SSI effects into engineering analyses and design procedures.