A shaft with a circular cross-section is subjected to pure twisting moment. The ratio of the maximum shear stress to the largest principal stress is

A rigid horizontal rod of length $$2L$$ is fixed to a circular cylinder of radius $$R$$ as shown in the figure. Vertical forces of magnitude $$P$$ are applied at the two ends as shown in the figure. The shear modulus for the cylinder is $$G$$ and the Young’s modulus is $$E.$$

The vertical deflection at point $$A$$ is

A simply supported beam of length $$2L$$ is subjected to a moment $$M$$ at the mid-point $$x = 0$$ as shown in the figure. The deflection in the domain $$0 \le x \le L$$ is given by $$$W = {{ - Mx} \over {12EIL}}\left( {L - x} \right)\left( {x + c} \right)$$$

Where $$E$$ is the Young’s modulus, $$I$$ is the area moment of inertia and $$c$$ is a constant (to be determined).

The slope at the center $$X=0$$ is

A thin cylindrical pressure vessel with closed-ends is subjected to internal pressure. The ratio of circumferential (hoop) stress to the longitudinal stress is