Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Q.1 Choose the correct answer of the following (any seven):
Two parallel, equal and opposite forces acting tangentially to the surface of the body is called
Maximum shear stress is
What will be the radius of gyration of a circular plate of diameter $ 10 \text{ cm} $?
Which of the following are statically determinate beams?
In a cantilever carrying uniformly varying load starting from zero at the free end, the bending moment diagram
Calculate the deflection if the slope is $ 0.0225 \text{ radians} $. Take the distance of centre of gravity of bending moment to free end as 2 metres.
Hoop stress in a thin vessel is
Hoop shrinking in thick cylinders is done to achieve
Two shafts in torsion will have equal strength if
What is the maximum principal stress induced in a solid shaft of $ 40 \text{ mm} $ diameter which is subjected to both bending moment and torque of $ 300 \text{ kN-mm} $ and $ 150 \text{ kN-mm} $ respectively?
Q.2 Solve both questions :
Derive the relation between $ E $ and $ K $. A bar of $ 30 \text{ mm} $ diameter is subjected to a pull of $ 60 \text{ kN} $. The measurement extension on gauge length of $ 200 \text{ mm} $ is $ 0.1 \text{ mm} $ and change in diameter is $ 0.004 \text{ mm} $. Calculate $ E $, Poisson ratio and $ K $.
Three bars made of copper, zinc and aluminium are of equal length and have cross-section $ 500 $, $ 700 $ and $ 1000 \text{ mm}^2 $ respectively. They are rigidly connected at their ends. If this compound member is subjected to a longitudinal pull of $ 250 \text{ kN} $. Estimate the proportional of the load carried on each rod and the induced stresses. Take the values of $ E $ for copper $ = 1.3 \times 10^5 \text{ N/mm}^2 $ and for zinc $ 1.0 \times 10^5 \text{ N/mm}^2 $ and for aluminium $ = 0.8 \times 10^5 \text{ N/mm}^2 $.
Q.3 Solve this question :
A material is subjected to two mutually perpendicular tensile direct stresses of $ 40 \text{
MPa} $ and $ 30 \text{ MPa} $ together with a shear stress of $ 20 \text{ MPa} $,
shear stress being clock-wise on the face carrying the $ 40 \text{ MPa} $ tensile stress.
Determine-
(a) the stresses on a plane making an angle of $ 40^{circ} $ counter-clockwise to the plane
of the $ 40 \text{ MPa} $ stress;
(b) the principal stresses and their planes;
(c) the maximum shear stress and its plane.
Q.4 Solve both questions :
With the help of mathematical proof, show that the torque transmitted by the hollow shaft is greater than the solid shaft.
A hollow shaft with diameter ratio $ 3/5 $ is required to transmit $ 450 \text{ kW} $ at $ 120 \text{ r.p.m.} $ The shearing stress in the shaft must not exceed $ 60 \text{ N/mm}^2 $ and the twist in a length of $ 2.5 \text{ m} $ is not to exceed $ 1^{circ} $. Calculate the minimum external diameter of the shaft. Take $ C = 80 \text{ kN/mm}^2 $.
Q.5 Solve this question :
A simply supported beam is subjected to a combination of loads as shown in Fig. 1. Sketch the shear force and bending moment diagrams and find the position and magnitude of maximum bending moment.
Q.6 Solve both questions :
Derive expression for moment of inertia for circular lamina and thin ring.
Determine the moment of inertia of the beam cross-section about the centroidal axis shown in Fig. 2.
Q.7 Solve both questions :
Derive the expression for the change in diameter and for the change in volume of a thin spherical shell when it is subjected to an internal pressure.
A thin cylinder is $ 3.5 \text{ m} $ long, $ 90 \text{ cm} $ in diameter, and the thickness of the metal is $ 12 \text{ mm} $. It is subjected to an internal pressure of $ 2.8 \text{ N/mm}^2 $. Calculate the change in dimensions of the cylinder and the maximum intensity of shear stress induced. $ E = 200 \text{ GPa} $ and Poisson's ratio $ = 0.3 $.
Q.8 Solve both questions :
Calculate circumferential and radial stress in a thick cylinder assuming internal pressure $ p_i $ and internal pressure = zero.
An external pressure of $ 10 \text{ MN/m}^2 $ is applied to a thick cylinder of internal diameter $ 160 \text{ mm} $ and external diameter $ 320 \text{ mm} $. If the maximum hoop stress permitted on the inside wall of the cylinder is limited to $ 30 \text{ MN/m}^2 $, what maximum internal pressure can be applied assuming the cylinder has closed ends? What will be the change in outside diameter when this pressure is applied? $ E = 207 \text{ GN/m}^2 $, $ v = 0.29 $.
Q.9 Solve both questions :
Deduce the expression for bending equation. What is section moduli of hollow circular section and solid circular section?
A simply supported beam is subjected to uniformly distributed load in combination with couple $ M $. It is required to determine the deflection shown in Fig. 3.
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Q.1 Choose the correct answer of the following (any seven):
Two parallel, equal and opposite forces acting tangentially to the surface of the body is called
Maximum shear stress is
What will be the radius of gyration of a circular plate of diameter $ 10 \text{ cm} $?
Which of the following are statically determinate beams?
In a cantilever carrying uniformly varying load starting from zero at the free end, the bending moment diagram
Calculate the deflection if the slope is $ 0.0225 \text{ radians} $. Take the distance of centre of gravity of bending moment to free end as 2 metres.
Hoop stress in a thin vessel is
Hoop shrinking in thick cylinders is done to achieve
Two shafts in torsion will have equal strength if
What is the maximum principal stress induced in a solid shaft of $ 40 \text{ mm} $ diameter which is subjected to both bending moment and torque of $ 300 \text{ kN-mm} $ and $ 150 \text{ kN-mm} $ respectively?
Q.2 Solve both questions :
Derive the relation between $ E $ and $ K $. A bar of $ 30 \text{ mm} $ diameter is subjected to a pull of $ 60 \text{ kN} $. The measurement extension on gauge length of $ 200 \text{ mm} $ is $ 0.1 \text{ mm} $ and change in diameter is $ 0.004 \text{ mm} $. Calculate $ E $, Poisson ratio and $ K $.
Three bars made of copper, zinc and aluminium are of equal length and have cross-section $ 500 $, $ 700 $ and $ 1000 \text{ mm}^2 $ respectively. They are rigidly connected at their ends. If this compound member is subjected to a longitudinal pull of $ 250 \text{ kN} $. Estimate the proportional of the load carried on each rod and the induced stresses. Take the values of $ E $ for copper $ = 1.3 \times 10^5 \text{ N/mm}^2 $ and for zinc $ 1.0 \times 10^5 \text{ N/mm}^2 $ and for aluminium $ = 0.8 \times 10^5 \text{ N/mm}^2 $.
Q.3 Solve this question :
A material is subjected to two mutually perpendicular tensile direct stresses of $ 40 \text{
MPa} $ and $ 30 \text{ MPa} $ together with a shear stress of $ 20 \text{ MPa} $,
shear stress being clock-wise on the face carrying the $ 40 \text{ MPa} $ tensile stress.
Determine-
(a) the stresses on a plane making an angle of $ 40^{circ} $ counter-clockwise to the plane
of the $ 40 \text{ MPa} $ stress;
(b) the principal stresses and their planes;
(c) the maximum shear stress and its plane.
Q.4 Solve both questions :
With the help of mathematical proof, show that the torque transmitted by the hollow shaft is greater than the solid shaft.
A hollow shaft with diameter ratio $ 3/5 $ is required to transmit $ 450 \text{ kW} $ at $ 120 \text{ r.p.m.} $ The shearing stress in the shaft must not exceed $ 60 \text{ N/mm}^2 $ and the twist in a length of $ 2.5 \text{ m} $ is not to exceed $ 1^{circ} $. Calculate the minimum external diameter of the shaft. Take $ C = 80 \text{ kN/mm}^2 $.
Q.5 Solve this question :
A simply supported beam is subjected to a combination of loads as shown in Fig. 1. Sketch the shear force and bending moment diagrams and find the position and magnitude of maximum bending moment.
Q.6 Solve both questions :
Derive expression for moment of inertia for circular lamina and thin ring.
Determine the moment of inertia of the beam cross-section about the centroidal axis shown in Fig. 2.
Q.7 Solve both questions :
Derive the expression for the change in diameter and for the change in volume of a thin spherical shell when it is subjected to an internal pressure.
A thin cylinder is $ 3.5 \text{ m} $ long, $ 90 \text{ cm} $ in diameter, and the thickness of the metal is $ 12 \text{ mm} $. It is subjected to an internal pressure of $ 2.8 \text{ N/mm}^2 $. Calculate the change in dimensions of the cylinder and the maximum intensity of shear stress induced. $ E = 200 \text{ GPa} $ and Poisson's ratio $ = 0.3 $.
Q.8 Solve both questions :
Calculate circumferential and radial stress in a thick cylinder assuming internal pressure $ p_i $ and internal pressure = zero.
An external pressure of $ 10 \text{ MN/m}^2 $ is applied to a thick cylinder of internal diameter $ 160 \text{ mm} $ and external diameter $ 320 \text{ mm} $. If the maximum hoop stress permitted on the inside wall of the cylinder is limited to $ 30 \text{ MN/m}^2 $, what maximum internal pressure can be applied assuming the cylinder has closed ends? What will be the change in outside diameter when this pressure is applied? $ E = 207 \text{ GN/m}^2 $, $ v = 0.29 $.
Q.9 Solve both questions :
Deduce the expression for bending equation. What is section moduli of hollow circular section and solid circular section?
A simply supported beam is subjected to uniformly distributed load in combination with couple $ M $. It is required to determine the deflection shown in Fig. 3.