1.a. A material has young's modulus of 2x 10⁵ N/m² and poisson's Ratio of 0.32 . Calculate the modulus of rigidity and bulk modulus of the material.

1.b. Derive the relationship between the rate of loading, shear force and bending moment in a beam.

1.c. A simply supported beam of span 4 m with EI constant throughout the span is subjected to a load of 24kN at 3 m from left and support. Find the total strain energy of the beam in bending.

1.d. State the assumption made in the theory of the pure bending and derive the formula.

1.e. A short column of external diameter 400 mm and internal diameter 200 mm carries an eccentric load of 90kN. Find the greatest encentricity which the load can have without producing tension on the cross section.

2.a. For a beam loaded as shown in the figure, calculate the value of UDL w so that bending moment at C is 50 kNm. Draw the shear force and bending moment diagram for the beam for the calculated value of w. Locate the point of contra flexures, if any.

2.b. An elemental cube is subjected to tensile stresses of 30 N/mm² acting on two 08 mutually perpendicular planes and a shear stress of 10 N/mm² on these planes. Draw the Mohr's circle of stresses and hence or otherwise determine the magnittudes and directions ofprincipal stresses and also the greatest shear stress.

3.a. A box beam supports the loads asshown in figure. Compute the maximum value 12 of Pthat will not exceed bending stress $\sigma$=8 MPa or shear stress t1.2 MPa forsection between the supports. Also, draw the shear stress distribution diagram at a section where shear force is maximum.

3.b. Find the principal moment of inertia and directions of principal axes for the angle section shown. All dimensions are in cm.

4.a. A stepped round bar ABCD is fixed to unyielding support at section A&D as shown in figure. It is subjected to axial loads at section B and C. Determine stresses in each portion of the bar and deflection of section B and C. Take E= 200 GN/m2.

4.b. A cylindrical vessel of 1.5 m diameter and 4 m long is closed at ends by rigidplate. It is subjected to an internal pressure of 3 N/mm² If maximum circumferential stress is not to exceed 150 N/mm² find the thickness of the shell. Find change in diameter, length and volume of the shell.

Assume E- 2 x10 N/mm2 and Poisson's ratio0.25.

5.a. Determine the diameter of the solid steel shaft that will transmit 150kW at a speed of 3 rev/sec, if the allowable shearing stress is 85 MPa. Also, determine the diameter of a hollow steel shaft, whose inside diameter is 4th of its outside diameter for the same conditions. What is the ratio of angle of twist per unit length for these two shaft?

5.b. An overhanging beam ABC is loaded as shown in figure. Find the slopes over each support and the deflection at the right end. Take E=2x10⁵ mm⁴.

6.a. VIa. A steel tube of 30 mm external diameter and 20 mm internal diameter encloses a 10Sas 2.1x 103 N/mm2 and 1 x 10 N/mm2 respectively. The value of coefficient ofcopperrod of 15 mm diameter to which it is rigidly fixed at each end. If at a temperature of 10°C, there is no longitudinal stress, calculate the stresses in the rod and tube when the temperature is raised to 200 °C. Take E for steel and copper as 2.1x10⁵ n/mm² and 1x10⁵N/mm² respectively. the value of coefficient of linear expansion for steel and copper is given as 11 x 10C and 18 x 10PC Srespectively.

6.b. Form the following data , determine the thickness of cast iron column. Assume both the end of the column arr fixed.

Length of the column=3 m factor of safety = 5

External diameter=200mm Ultimate compressive stress = 570 N/mm²

Safe working load = 600kN Rankine constant= 1/1600.