A frame $$ABCD$$ is supported by a roller at $$A$$ and is on a hinge at $$C$$ as shown below:

The reaction at the roller end $$A$$ is given by

A two span beam with an internal hinge is shown below.

Conjugate beam corresponding to this beam is

For the structure shown below, the vertical deflection at point $$A$$ is given by

Four column of the same material having identical geometric properties are supported in different ways as shown below

It is required to order these four beams in the increasing order of their respective first buckling loads. The correct order is given by