In the discussion on boundary conditions for buckling of columns, participants clarify the constraints for fixed-fixed and simply supported cases. For fixed-fixed conditions, both x and y displacements must be restrained, while for simply supported conditions, only y displacement is restrained. A key point of confusion arises regarding the application of loads to a fixed-fixed column while maintaining fixed ends. One participant suggests that only one end should be fixed in both x and y, allowing the other end to move and facilitate buckling. The conversation emphasizes the importance of correctly interpreting the boundary conditions when analyzing column behavior under load.