## Mechanics of Materials - Dec 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)** State Hooke's law. Sketch and explain typical stress strain curve for aluminium.(4 marks)
**1 (b)** The tensile test was conducted on a mild steel bar. The following data was obtained from the test:Diameter of steel bar: 16mm

Load at proportionality limit : 72kN

Load at failure: 80kN

Diameter of the rod at failure : I 2mm

Gauge length of the bar: 80mm

Extension at a load of 60kN : 0.115mm

Final gauge length of bar : 104mm

Determine: i) Young's modulus; ii) Proportionality limit; iii) True breaking stress; iv) Percentage elongate(8 marks)
**1 (c)** Determine the magnitude of the load P necessary to produce zero net change in the length of the straight bar shown in Fig.Q.l(c). Area of cross section : 400mm^{2}.

img(8 marks)
**2 (a)** Explain volumetric strain and obtain the expression for volumetric strain for a circular bar.(5 marks)
**2 (b)** Establish a relationship between the modulus of elasticity and modulus of rigidity.(7 marks)
**2 (c)** A compound bar is made up of a central steel plate 50mm wide and 10mm thick to which copper plate 50mm wide and 5mm thick are connected rigidly on each side. The length of the compound bar at room temperature is 1000mm. If the temperature is raised by 100°C,determine the stress in each material and change in length of the compound bar. Assume E_{s}=200GPa, E_{c}=1..Gpa, ?_{s}=12 × 10^{-6}/°C and ?_{c}=18× 10^{-6}/°C.(8 marks)
**3 (a)** A point in a plate grider is subjected to a horizontal tensile stress of 100N/mm^{2} and vertical shear stress of 60 N/mm^{2}. Find the magnitude of principle stresses and its location.(10 marks)
**3 (b)** An element with the stresses acting on it, is as shown in Fig.Q.3(b) by Mohr's circle method ,determine: i) Normal and shear stress acting on a plane whose normal is at an angle of 11° with respect to X-axis; ii) Principal stresses and its locations; iii) Maximum shear stresses

img(10 marks)
**4 (a)** The maximum stress produced by a pull in a bar of length 100mm is 100N/mm^{2}. The area of cross sections and length are as shown in Fig.Q.4(a). Calculate the strain energy stored in the bar if E=2×10^{5}N/mm^{2}.

img(10 marks)
**4 (b)** A thick cylinder with internal diameter 80mm and extemal diameter 120mm is subjected to an external pressure of 40N/mm^{2}, when the internal pressure is 120N/mm^{2}, calculate circumferential stress at external and internal surfaces of the cylinder. Plot the variation of circumferential stress and radial pressure on the thickness of the cylinder(10 marks)
**5 (a)** Derive expressions relating load, shear force and bending moment with usual notations.(5 marks)
**5 (b)** Draw the SFD and BMD for the over hanging beam shown in Fig.Q.5(b).Indicate all significant values including point of contra flexure.

img(15 marks)
**6 (a)** What are the assumptions made in simple theory of bending?(4 marks)
**6 (b)** Derive an expression for relationship between bending stress and radius of curvature.(6 marks)
**6 (c)** An 1 section has the following dimensions. Flanges 200mm × 10mm; web 380mm × 8mm.The maximum shear stress developed in the beam is 20N/mm^{2}. Find the sheer force to which the beam is subjected.(10 marks)
**7 (a)** Derive an expression $$EI\dfrac{d^{2}y}{dx^{2}}=M$$ with usual notations.(10 marks)
**7 (b)** A beam of length 5m and of uniform rectangular section is simply supported at its ends. It carries a uniformly distributed load of 9kN/m run over the entire length. Calculate the width and depth of the beam if permissible bending stress is 7N/mm^{2} and central deflection is not to exceed 1cm. Take E for beam material = 1× 10^{4}N/mm^{2}.(10 marks)
**8 (a)** A hollow shaft having an inside diameter 60% of its outer diameter, is to replace a solid shaft transmitting the same power at the same speed. Calculate the percentage saving in material, if the material to be used is also the same.(10 marks)
**8 (b)** A hollow C.I. column whose outside diameter is 200mm has a thickness of 20mm. It is 4.5mlong and is fixed at both ends. Calculate the safe load by Rankine's formula using a factor of safety of 4. Calculate the slenderness ratio and the ratio of Euler's and Rankine's critical loads. Take $$f_c =550 \ N/mm^2 , \ a=\dfrac{1}{1600}$$ in Rankine's formula and E=9.4 × 10^{4} N/mm^{2}.(10 marks)