Strength of Materials

(a) Derive the relation between Young’s Modulus ($E$) and Shear Modulus ($G$). [4] (b) A C.I. flat, $300\text{ mm}$ long and of $30\text{ mm}\times 50\text{ mm}$ uniform section, is acted upon by the following forces uniformly distributed over the respective cross-section; $25\text{ kN}$ in the direction of length (tensile); $350\text{ kN}$ in the direction of the width (compressive); and $200\text{ kN}$ in the direction of thickness (tensile). Determine the change in dimensions and volume of the flat. Take $E=140{\text{ GN/m}}^2,and$ Poisson’s ratio ($\nu$) = $1/4$. [6] (c) A rigid bar is supported by three rods at one end in the same vertical plane and equidistant. The other end of rods is rigidly fixed. The outer rods are of brass and of length $600\text{ mm}$ and diameter $30\text{ mm}$. The central rod is of steel of $900\text{ mm}$ length and of $37.5\text{ mm}$ diameter. Calculate the forces in the bars due to $a^n$ applied force $100\text{ kN},if$ the bar remains horizontal after the load has been applied. Take $E_s/E_b=2$.

Define hoop and longitudinal stress. Prove that longitudinal stress is half of the circumferential stress for the thin cylinder with neat sketch.

(a) A solid circular shaft is subjected to a torque $120\text{ Nm}$. Determine the diameter if the allowable shear stress is $100{\text{ N/mm}}^2$ and the allowable angle of twist is $3^{\circ }$ per $10$ diameter length of the shaft. $G=10^5{\text{ N/mm}}^2$. [8] (b) Prove that the torque transmitted by the hollow shaft is greater than the solid shaft of same weight, material and length.

a) Derive pure bending equation. [4] b) A cantilever beam $4\text{m}$ carries an uniformly distributed load of $50\text{kN/m}$ over the whole span. The beam is of T-section with web $45\text{cm}\times 3\text{cm}$ and flange is $35\text{cm}\times 5\text{cm}$ making overall depth of $50\text{cm}$. Find the maximum tensile and compressive stresses if the flange is at the bottom and horizontal. Also determine the stress and its nature at $3\text{ cm}$ from top.

Derive Euler's formula of critical load for a steel column with one end fixed and other hinged. Derive the condition of no tension in rectangular column.