A steel portal frame has dimensions, plastic moment capacitance and applied loads as shown in the figure. The vertical load is always twice of the horizontal load. The collapse load $$P$$ required for the development of a beam mechanism is

A steel beam (with a constant $$EI,$$ and span $$L$$) is fixed at both ends and carries a uniformly distributed load ($$w$$ $$kN/m$$), which is gradually increased till the beam reaches the stage of plastic collapse (refer to the following figure). Assuming $$'B'$$ to be at mid-span, which of the following is true.

A cantilever beam of length $$L$$ and a cross section with shape factor $$'f'$$ supports a concentrated load $$P$$ as shown below:

The length $${L_P}$$ of the plastic zone, when the maximum bending moment, equals the plastic moment $${M_P}$$, given by

The four cross sections shown below are required to be ordered in the increasing order of their respective shape factors.