The plastic collapse load $${{W_P}}$$ for the propped cantilever supporting two point loads as shown in the figure in terms of plastic moment capacity, $${{M_P}}$$ is given by

When the triangular section of a beam as shown below becomes a plastic hinge, the compressive force acting on the section (with $${\sigma _y}$$ denoting the yield stress) becomes

A cantilever beam of length $$l,$$ width $$b$$ and depth $$d$$ is loaded with a concentrated vertical load at the tip. If yielding starts at a load $$P,$$ the collapse load shall be

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