## Mechanics of Materials - Jun 2014

### Mechanical Engg. (Semester 4)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1 (a)** Define i) Stress ii) Hoke's law iii)Elasticity iv) Lateral strain(4 marks)
**1 (b)** Explain stress-strain relationship showing salient points on the diagram.(6 marks)
**1 (c)** A stepped bar is subjected to an external loading shown in fig Q1(c). calculated the change in the length of bar. Take E= 200 Gpa for steel, E=70 Gpa for aluminium and E=100GPa for copper.

img(10 marks)
**2 (a)** Define i) Poisson's ratio ii) Bulk modulus.(2 marks)
**2 (b)** Derive an expression for establishing the relationship between Young's modulus and modulus of rigidity.(6 marks)
**2 (c)** A 25 mm diameter steel rod passes concentrically through a bronze tube 400mm long and provided with nuts and washers which are adjusted initially so that there is no end play at 20°C. Assuming that there is no bronze when one of the nuts its tight end by giving at one-tenth of a turn, the pitch of the thered being 2.5mm, take E for steel =200 kN/mm^{2} and E for bronze=100 kN/mm^{2}(12 marks)
**3 (a)** Define the principle planes and principal stresses.(4 marks)
**3 (b)** Explain procedure for constructing Mohr's circle, for an element acted upon by two tensile stresses and sher stresses.(6 marks)
**3 (c)** The state of stress in two dimensionally stressed body is as shown fig Q3(c). determine the principle planes, principle stresses, maximum shear stress and their planes.(10 marks)
**4 (a)** Define i) Strain energy ii)Work(3 marks)
**4 (b)** Prove that volumetric; strain in thin cylinder is given by \frac{Pd}{4tE}(5-4\mu ), with usual notations(7 marks)
**4 (c)** A.C.I pipe has 200 mm internal diameter and 50 mm metal thickness and carries water under a pressure of 5 N/mm^{2}. Calculate the maximum and minimum intensities of circumferential stress and sketch the distribution of circumferential stress radial pressure across the section.(10 marks)
**5 (a)** Derive the relationship between load, shear and bending moment.(5 marks)
**5 (b)** Briefly explain the different types of loads.(3 marks)
**5 (c)** Draw the SFD and BMD for the loading pattern on the beam in FigQ5(c). indicate the point of contraflexure. Also locate the maximum BM with its magnitude

img(12 marks)
**6 (a)** What are the assumptions made in simple theory of bending?(4 marks)
**6 (b)** Prove that the maximum shear stress is 1.5 times the average shear stress in beam of rectangular cross- section.(6 marks)
**6 (c)** At a given position in a beam of uniform I-section is subjected to a bending moment of 100kN-m. Plot the variation of bending stress across the section.(Refer FigQ6(c))

img(10 marks)
**7 (a)** Derive the deflection equation for the beam in the standard form

EI\frac{d^{2}y}{dx^{2}}=M(x)(6 marks)
**7 (b)** For the beam loaded as shown in fig Q7(b), find the position and magnitude of maximum, deflection. Take I=4.3×10^{8} and E=200kN/mm^{2}.

img(14 marks)
**8 (a)** What are the assumptions made in simple theory of columns?(3 marks)
**8 (b)** Derive an expression for the critical load in a column subjected to compressive load, when one end is fixed and other end is free.(7 marks)
**8 (c)** Find the diameter of the shaft required to transmit 60kW at 150 RPM if the minimum torque is 25% more than the mean torque for a maximum shear stress of 60MPa. Find also the angle of twist in a length of 4m. Take G=80 Gpa.(10 marks)