In a real beam, at an end, the boundary condition of zero slope and zero vertical displacement exists. In the corresponding conjugate beam, the boundary conditions at this end will be:

A cantilever beam of span, $$'L'$$ is subjected to a downward load of $$800$$ $$kN$$ uniformly distributed over its length and a concentrated upward load $$P$$ at its free end. For vertical displacement to be zero at the free end, the value of $$P$$ is:

The kern area (core) of a solid circular section column of diameter, $$D$$ is a concentric circle of diameter, $$'d'$$ is equal to

The axial load carrying capacity of a long column of given material. Cross-sectional area, $$A$$ and length, $$L$$ is governed by