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

A truss, as shown in the figure, is carrying $$180$$ $$kN$$ load at node $${L_2}.$$ The force in the diagonal member $${M_2}{U_4}$$ will 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

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