What are the key principles and methodologies involved in the analysis and design of geotechnical structures? Discuss the factors influencing soil behavior, including soil mechanics principles, soil composition, and environmental factors, and their significance in the design of foundations, retaining walls, and earthworks. Explore the various analytical methods used in geotechnical engineering, such as limit equilibrium analysis, numerical modeling techniques, and geophysical investigations, and their applications in assessing slope stability, bearing capacity, and settlement analysis. Furthermore, examine the role of geotechnical considerations in mitigating natural hazards, such as earthquakes, landslides, and liquefaction, and propose integrated approaches for designing resilient geotechnical structures in challenging environments.