Euler buckling instability

Bending of a rod with both ends constrained to be in the same straight line:

The force balance equation for the above situation can be written as,

(14.16)

We can integrate this equation using boundary condition that at , which yields , leading to

(14.17)

Notice that equation 14.16 is the analog of oscillation of a suspended pendulum for which the equation of motion can be written as

The critical load for buckling for the above problem can be obtained as,

(14.18)