Buckling Analysis in Structural Engineering: Describe the theoretical basis and mathematical principles behind buckling analysis in civil engineering structures. Discuss various factors affecting buckling, such as material properties, geometric imperfections, and loading conditions. Explain how researchers have applied theoretical models to predict and prevent buckling failures in real$-$world structures, considering both linear and nonlinear analysis methods. Analyze the significance of buckling theory in ensuring the stability and safety of buildings, bridges, and other civil engineering projects.