$${U_1}$$ and $${U_2}$$ are the strain energies stored in a prismatic bar due to axial tensile forces $${P_1}$$ and $${P_2},$$ respectively. The strain energy $$U$$ stored in the same bar due to combined action of $${P_1}$$ and $${P_2}$$ will be

If principle stresses in two-dimensional case are $$\left( - \right)\,\,10\,MPa$$ and $$20$$ $$MPa$$ respectively, then maximum shear stress at the point is

A three-span continuous beam has an internal hinge at $$B.$$ Section $$B$$ is at the mid-span of $$A.C.$$ Section $$E$$ is at the mid-span of $$CG$$. The $$20$$ $$kN$$ load is applied at section $$B$$ whereas $$10$$ $$kN$$ loads are applied at sections $$D$$ and $$F$$ as shown in the figure. Span $$GH$$ is subjected to uniformly distributed load of magnitude $$5$$ $$kN/m$$. For the loading shown, shear force immediate to the right of section $$E$$ is $$9.84$$ $$kN$$ upwards and the sagging moment at section $$E$$ is $$10.31$$ $$kN$$-$$m.$$

The vertical reaction at support $$H$$ is

A homogeneous, simply supported prismatic beam of width $$B,$$ depth $$D$$ and span $$L$$ is subjected to a concentrated load of magnitude $$P.$$ The load can be placed anywhere along the span of the beams. The maximum flexural stress developed in beam is