A propped cantilever made of a prismatic steel beam is subjected to a concentrated load $$P$$ at mid span as shown.

If the magnitude of loadܲ is increased till collapse and the plastic moment carrying capacity of steel beam section is $$90$$ $$kN$$-$$m$$, ݇ܰ݉determine reaction ܴ$$R$$(in ݇ܰ$$kN$$)(correct to $$1$$-decimal place) using plastic analysis. ____________

As per $$IS$$ $$800:2007$$ the cross-section in which the extreme fiber can reach the yield stress, but cannot develop the plastic moment of resistance due to failure by local buckling is classified as

All members in the rigid-jointed frame shown are prismatic and have the same flexural stiffness $$EI$$. Find the magnitude of the bending moment at $$Q$$ (in ݇ܰ݉$$kN$$-$$m$$) due to the given loading _______________

A uniform beam ($$EI=$$ constant) $$PQ$$ in the form of a quarter-circle of radius $$R$$ is fixed at end $$P$$ and free at the end $$Q,$$ where a load $$W$$ is applied as shown. The vertical downward displacement, $${\delta _q},$$ at the loaded point $$Q$$ is given by : $${\delta _q} = \beta \left( {{{W{R^3}} \over {EI}}} \right).$$

Find the value of $$\beta $$(correct to $$4-$$decimal places). _____________________