A solid circular shaft of diameter d and length $$L$$ is fixed at one end and free at the other end. A torque t is applied at the free end. The shear modulus of the material is $$G.$$ The angle of twist at three free ends is

For the simply supported beam of length $$L,$$ subjected to a uniformly distributed moment $$(M)$$ $$kN$$-$$m$$ per unit length as shown in the figure, the bending moment (in $$kN$$-$$m$$) at the mid-span of the beam is

Two people weighing W each are sitting on a plank of length $$L$$ floating on water at $$L/4$$ from either end. Neglecting the weight of the plank, the bending moment at the center of the plank is

The major and minor principal stresses at a point are $$3MPa$$ and $$-3MPa$$ respectively. The maximum shear stress at the point is