A solid cylindrical shaft with a diameter of $50$ mm is subjected to a combined loading of a bending moment of $600$ Nm and a torque of $500$ Nm$.$ Determine the principal stresses and the maximum shear stress at a point on the surface of the shaft. A thin$-$walled cylindrical pressure vessel with an outer radius of $0\mathrm{.}4$ meters and a wall thickness of $12$ mm is subjected to an internal pressure of $2$ MPa. Calculate the hoop stress, longitudinal stress, and radial stress in the wall of the vessel. A rectangular beam is simply supported at both ends and subjected to a uniformly distributed load. Determine the maximum bending stress and the deflection at the midpoint of the beam if the span of the beam is $4$ meters, the width is $100$ mm$,$ and the height is $200$ mm$.$ A steel rod of $20$ mm diameter and $2$ meters length is subjected to an axial tensile load of $30$ kN$.$ Calculate the elongation of the rod if the modulus of elasticity of steel is $210$ GPa. A cantilever beam with a length of $1\mathrm{.}2$ meters is subjected to a point load at its free end. Determine the slope and deflection at the free end of the beam. A circular plate with a radius of $0\mathrm{.}3$ meters is rotating at $1200$ rpm$.$ Calculate the centrifugal force acting on a mass of $0\mathrm{.}2$ kg located at the edge of the plate. A rectangular plate with a width of $250$ mm and a length of $500$ mm is subjected to an axial tensile load of $80$ kN$.$ Determine the stress and strain in the plate if the thickness is $10$ mm$.$ A cylindrical shaft with a diameter of $60$ mm is subjected to a torque of $1500$ Nm$.$ Calculate the shear stress and the angle of twist per meter length of the shaft. A simply supported beam with a span of $5$ meters is subjected to a point load of $10$ kN at the midpoint. Determine the maximum bending moment, shear force, and deflection at the midpoint of the beam. A helical spring made of steel wire with a diameter of $5$ mm and a mean coil diameter of $50$ mm has $12$ active coils. Calculate the stiffness of the spring and the maximum shear stress in the wire when a load of $150$ N is applied. A rectangular beam with a width of $80$ mm and a height of $160$ mm is subjected to a bending moment of $4000$ Nm$.$ Determine the maximum bending stress and the location of the neutral axis. A steel column with a length of $3$ meters is pinned at both ends and subjected to a compressive load. Determine the critical load and the factor of safety against buckling if the column has a rectangular cross$-$section of $100$ mm by $50$ mm and the modulus of elasticity of steel is $210$ GPa. What is Thermodynamics