Calculate critical force for structure.

A continuous beam with infinitely stiff column under compression force $P$ is considered.

a$)$ Derive expression for bending moment and rotation angle at point of support in the middle.

b$)$ Derive equilibrium equation.

c$)$ Assume that rotations are small and compute critical force.

d$)$ Sketch post$-$buckling displacement for structure.

e$)$ Write assumptions, which are used to derive critical force.