A beams $$PQRS$$ is $$18$$ $$m$$ long and is simply supported at points $$Q$$ and $$R$$ $$10$$ $$m$$ apart. overhangs $$PQ$$ and $$RS$$ are $$3$$ $$m$$ and $$5$$ $$m$$ respectively. A train of two point loads of $$150$$ $$kN$$ and $$100$$ $$kN$$, $$5$$ $$m$$ apart, crosses this beam from left to right with $$100$$ $$kN$$ load leading.

The maximum hogging moment in the beam anywhere is

In a redundant joint model, three bar members are pin connected at $$Q$$ as shown in the figure. Under some load placed at $$Q$$, the elongation of the members $$MQ$$ and $$OQ$$ are found to be $$48$$ $$mm$$ and $$35$$ $$mm$$ respectively. Then the horizontal displacement $$'u'$$ and the vertical displacement $$'v'$$ of the node $$Q,$$ in $$mm,$$ will be respectively,

The stiffness $$K$$ of a beam deflecting in a symmetric mode, as shown in the figure, is _______

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