Strength of Materials - Dec 2012

Mechanical Engineering (Semester 3)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1 (a) A uniformly tapering rod of length l and diameter d₁ and d₂ is subjected to an axial pull P. Prove that the total extension of the rod is
$$\delta l =\dfrac{4PL}{\pi Ed_{1}d_{2}} $$(5 marks) 1 (b) Write the assumptions in simple bending and hence derive the bending formula,
$$ \dfrac{M}{I}=\dfrac{\sigma}{y} = \dfrac{E}{R}$$(5 marks) 1 (c) Derive an expression for the maximum and the minimum stress at the base of a column of rectangular section, when it is subjected to a load which is eccentric to both axes.(5 marks) 1 (d) Find the maximum shear stress induced in a solid circular shaft of diameter 150 mm, when it transmits 150 kW power at 180 rpm.(5 marks) 1 (e) A solid round bar of 3 m long and 50 mm diameter is used as column with both ends hinged. Determine the crippling load, if E=2×10⁵ N/mm²(5 marks) 1 (f) A cantilever of length 4 m carries uniformly varying load of intensities zero at free end and 2 kN/m at fixed end. Draw shear force and bending moment diagrams for the beam.(5 marks) 2 (a) A compound tubes consists of a steel tubes of 140 mm internal diameter and 160 mm external diameter; and an outer brass tube of 160 mm internal diameter and 180 mm external diameter. Both the two tubes are of 1.5 m length. If the compound tube carrier an axial compressive load of 900 kN, find its reduction in length. also find the stress and the loads carried by each tube.
$$ E_{s}=2 \times 10^{5} N/mm^{2}, \ E_{b}=1\times 10^{5}N/mm^{2}$$(10 marks) 2 (b) A point load of 10 kN applied to a simply supported beam at mid-span produces a deflection of 6 mm and a maximum bending stress of 20 N/mm². Calculate the maximum value of the momentary stress produced, when a weight of 5 kN is allowed to fall through a height of 18 mm on the beam at the middle of the span.(10 marks) 3 (a) Two mutually perpendicular planes of an elemets of material are subjected to tensile stress of 105 N/mm², compressive stress of 35 N/mm² and shear stress of 70 M/mm². Find graphically or otherwise,
(i) Magnitude and the direction of principal stresses
(ii) Magnitude of the normal and the shear stresses on plane, on which the shear stress is maximum(10 marks) 3 (b) Draw axial force, shear force and bending moment diagrams for the beam loaded as shown in figure. Locate all important points.
(10 marks) 4 (a) A steel bar 120 mm is diameter is completely encased in an aluminium tube of 180 mm outer diameter and 120 mm inner diameter, so as to make a composite beam. The composite beam is subjected to a bending moment of 15 kN/m. Determine the maximum shear stress due to bending in each material. Assuming that Young's modulus of steel is three-times that of aluminium.(10 marks) 4 (b) A simply supported beam carrier a UDL of intensity 2.5 kN/m over a span of 5 m. The cross-section is T-section having flange 125 mm x 25 mm and web 175 mm x 25 mm. calculate maximum shear for the section of the beam.(10 marks) 5 (a) Determine the position and the amount of maximum deflection for the beam shown in the figure. Take, El=1.8 × 10⁴ kNm²
(10 marks) 5 (b) A square column of 400 mm x 400 mm size is subjected to an axial load of 400 kN. In addition to this a load of 40 kN is acting at an eccentricity of 20 mm about both x-x and y-y axes. Find the stresses at all four conrners.(10 marks) 6 (a) A hollow shaft, having an internal diameter 40% of its external diameter, transmits 562.5 kW power at 100 rpm. Determine extertnal diameter of the shaft, is shear stress is not to exceed 60 N/mm², and the twist in a length of 2.5 m should not exceed 1.3°. Assume that the maximum torque is 1.25 time the mean torque and G=9 × 10⁴ N/mm²(10 marks) 6 (b) A close cylindrical vassel made of steel plates 4 mm thick plane end carries fluid under a pressure of 3 N/mm². The diameter of the cylinder is 250 mm and the length is 750 mm. Calcualate the logitudinal and hoop stresses in the cylinder wall and deterine the changes in diameter, length and volume of the cylinder.
$$ E=2.1 \times 10^{5}N/mm^2, \ \dfrac {1}{m}=0.286 $$(10 marks) 7 (a) A hollow cast iron column of 200 mm external diameter, 150 mm internal diameter and 8 m long has both ends fixed. It is subjected to axial compressive load. Taking factor of safety as 6, $$ \sigma_{c}=560 N/mm^2, \ \alpha=\dfrac {1}{1600}, $$ determine the safe Rankine load.(8 marks) 7 (b) A simply supported beam AB of 6 m long is loaded with a UDL of 50 kN/m over the entire span. At a section 1.2 m end A, find SF and BM magnitude to be resisted and draw the shear stress and bending stress distribution diagrams. The corss section of the beam is symmetrical I-section with flanges 300 mm x 200 mm and web 560 mm x 12 mm.(12 marks)