A fixed-end beam is subjected to a concentrated load $$(P)$$ as shown in the figure. The beam has two different segments having different plastic moment capacities $$\left( {{M_P},\,2{M_P}} \right)$$ as shown.

The minimum value of load $$(P)$$ at which the beam would collapse (ultimate load) is

The portal frame shown in the figure is subjected to a uniformly distributed vertical load $$w$$ (per unit length).

The bending moment in the beam at the joint $$'Q'$$ is

Consider the structural system shown in the figure under the action of weight $$W.$$ All the joints are hinged. The properties of the members in terms of length $$(L),$$ area $$(A)$$ and the modulus of elasticity $$(E)$$ are also given in the figure. Let $$L,A$$ and $$E$$ be $$1$$ $$m,$$ $$0.05$$ $${m^2}$$ and $$30 \times {10^6}\,\,N/{m^2},\,\,$$ respectively, and $$W$$ be $$100$$ $$kN.$$

Which one of the following sets gives the correct values of the force, stress and change in length of the horizontal member $$QR$$?

The kinematic indeterminacy of the plane truss shown in the figure is