1.a. Define : i) Hooke's law ii) Poisson's ratio iii) Factor of safety iv) Bulk modulus v) Modulus of elasticity

1.b Draw and explain stress-strain diagram of a mild steel specimen subjected to a tension test.

1.c A circular rod of 100 mm diameter and 500 mm length is subjected to a tensile load of 1000 kN . Determine the i) Modulus of rigidity ii) Bulk modulus iii) change in volume. Take Poission's ratio = 0.30 and E = 200 GPa.

2.b Derive a relation between modulus of elasticity and bulk modulus.

2.c A bar of brass 25mm diameter is enclosed in a steel tube of 50 mm external diameter and 25 mm internal diameter. The bar and the tube fastened at the ends and are 1.5m long. Find the stresses in the two materials when the temperature raises from 30°C to 80°C.

Take :E_steel = 200 GPa; E_brass = 100 GPa.

α_steel = 11.6 x 10^-6/°C; α_brass = 18.7 x 10^-6/°C;

3.a Derive a relation between for normal stress, shear stress and resultant stress on an oblique plane inclined at an angle θ with vertical axis (x - plane) in a biaxial stress system subjected to σ_x, σ_y and τ_xy also find angle of obliquity Φ.

3.b Derive expression for hoop stress and longitudinal stress for a thin cylinder subjected to internal fluid pressure.

4.a A point in a strained in subjected to a tensile stress of 500 N/mm² and 300 N/mm² in two mutually perpendicular planes and also these carries a shear stress of 100 N/mm². Calculate the normal, tangential, resultant stresses $\left(\sigma_{\theta}, \tau_{0}, \sigma_{r}\right)$ on a plane making an angle of 30°C with the vertical axis(x-plane). also find principal stresses.

4.b A thin cylinder shell 1.2m in diameter and 3m long has a metal wall thickness of 12mm. It is subjected to an internal pressure of 3.2 MPa. Find the circumferential and longitudinal stress in the wall. Also determine change in volume of the cylinder. Assume E = 210 GPA and μ = 0.30.

5 For the beam shown in Fig.Q5. Draw shear force and bending moment diagrams. Locate the point of contraflexture if any.

6.a Derive the relationship between load shear force and bending moment for UDL.

6.b List the assumptions made in theory of pure bending. Write the bending equation with usual notations with their meanings.

6.c Derive an expression relating slope, deflection and radius curvature in a beam in terms of E,I and M with usual notations.

7.a State the assumption made in pure torsion and derive $\frac{T}{Jp}=\frac{G \theta}{L}=\frac{\tau}{R}$ with usual meanings.

7.b A 1.5m long column has circular cross section of 50mm diameter. One end of the column is fixed in direction and position and the other end is free. Taking the factor of factor of safety as 3 calculate:

i) Safe load according to Rankine's formula $\sigma_{c}=560 \mathrm{MPa}$ and $\alpha=\frac{1}{1600}$

ii) Safe load according to Euler's formula taking E = 120 GPa.

8.a State the assumptions made while deriving Euler's column formula. Also derive Euler's expression of buckling load for column with both ends hinged.

8.b A solid circular shaft has to transmit a power of 1000 kW at 120 rpm. Find the diameter of the shaft if the shear stress of the material must is replaced by hollow one whose internal diameter is 0.6 times its external diameter, find diameter of hollow shaft.

9.a Explain : i) Castigliano's first theorem ii) Castigliano's second theorem iii)

9.b Write a note on: i) maximum principal stress theory ii) Maximum shear stress theory.

10.a A hollow circular shaft of 2 m length has an external diameter of 100 mm and a thickness of 10mm. Take G=80GPa.

10.b The plane state of stress at a point is given $\sigma_{x}=70 \operatorname{MPa} ; \sigma_{y}=140 \mathrm{MPa} ; \tau_{x y}=-35 \mathrm{MPa}$. If the yielding stress in tension in 175 MPa, check whether there is failure according to

i) Maximum principal stress theory

ii) Maximum shear stress theory.

If the material is safe then find the factor of safety.