A long structural column (length $$-L$$) with both ends hinged is acted upon by an axial compressive load $$P.$$ The differential equation governing the bending of column is given by: $$$EI{{{d^2}y} \over {d{x^2}}} = - Py$$$

Where $$y$$ is the structural lateral deflection and $$EI$$ is the flexural rigidity. The first critical load on column responsible for its buckling is given by

For linear elastic systems, the type of displacement function for the strain energy is

A $$''H''$$ Shaped frame of uniform flexural rigidity $$EI$$ is loaded as shown in the figure. The relative outward displacement between points $$K$$ and $$O$$ is

A simply supported beam of uniform rectangular cross-section of width $$b$$ and depth $$h$$ is subjected to linear temperature gradient, $${0^ \circ }$$ at the top $${T^ \circ }$$ at the bottom, as shown in the figure. The coefficient of linear expansion of the beam material is $$\alpha .$$ This resulting vertical deflection at the mid-span of the beam is