Problem $1$

A simply supported beam$-$column with the length $L=2l$ is loaded with a point load $F$ at the middle of the beam and a concentric compressive force $P.$ The system is shown in figure $1.$ The cross section and the material properties are constant throughout the beam. Assume that Euler beam theory is valid.

Figure $1$: Beam$-$column with the length $L=2l,$ loaded with a point load $F$ at the middle of the beam and a concentric buckling load $P.$

a$.(1)$ Find the boundary conditions. $(2)$ What other conditions can you use to solve the problem?

b$.$ Show that the governing equation is

$y^{\text{'}\text{'}}+k^{2}y=-\frac{Fx}{2EI}$

c$.$ Derive the deflection function.

d$.$ Show that the buckling load is

$P_{cr}=\frac{{}^{2}EI}{4l^2}$