Stability of columns subjected to an intermediate concentrated load with ends restrained to move axially

Finite element formulation is employed to study the stability behaviour of columns subjected to an intermediate concentrated load with ends restrained to move axially. It is seen that for the case of a simply supported column there is a discontinuity point for the critical load at the centre of the column. For the clamped column the stability behaviour is more complex, with five discontinuity points of critical load