SECTION A (2 × 10 = 20 Marks)
Attempt all questions in brief
(a) Hooke’s Law
Hooke’s law states that within the elastic limit of a material, stress is directly proportional to strain. Mathematically, it is expressed as stress = E × strain, where E is Young’s modulus of elasticity. This law is valid only up to the elastic limit, beyond which permanent deformation occurs.
(b) Thermal stress and thermal strain
Thermal strain is the strain developed in a material due to change in temperature and is given by the product of coefficient of thermal expansion and temperature change. Thermal stress develops when the free expansion or contraction of a material is restrained, resulting in internal stresses.
(c) Spring and its types
A spring is an elastic mechanical element that stores energy when deformed and releases it upon unloading. Springs are classified as helical springs, leaf springs, spiral springs, and volute springs depending on their shape and application.
(d) Thin and thick shells
Thin shells are those in which the thickness is very small compared to the diameter, and stress distribution across thickness is assumed uniform. Thick shells have significant thickness, and stress varies across the section, requiring Lame’s equations for analysis.
(e) Deflection of simply supported beam under UDL
Using the area-moment method, the maximum deflection occurs at the mid-span of the beam. The slope and deflection are obtained by evaluating the area of the bending moment diagram and its moment about the point where deflection is required.
(f) Principal stresses and principal planes
Principal stresses are the maximum and minimum normal stresses acting on a plane where shear stress is zero. The planes on which these stresses act are known as principal planes.
(g) Castigliano’s theorem
Castigliano’s theorem states that the partial derivative of total strain energy with respect to an applied load gives the deflection in the direction of that load. It is widely used for deflection analysis in complex structures.
(h) Shear centre
The shear centre is the point through which transverse load must pass so that the member bends without twisting. It is particularly important in unsymmetrical sections like channels and angles.
(i) Open and closed coiled helical springs
Open-coiled springs have a large helix angle and are subjected to bending and torsion, while closed-coiled springs have a small helix angle and are primarily subjected to torsional shear stress.
(j) Section modulus and slenderness ratio
Section modulus is the ratio of moment of inertia to the distance of extreme fiber and indicates bending strength. Slenderness ratio is the ratio of effective length of a column to its least radius of gyration and determines buckling behavior.
SECTION B (10 × 3 = 30 Marks)
(a) Gradual and sudden loading on a bar
When a tensile load is applied gradually, stress, strain, and elongation are calculated using basic axial load formulas. Strain energy stored equals half the product of load and elongation. When the same load is applied suddenly, the maximum stress and elongation become twice those of gradual loading, and strain energy increases significantly.
(b) Compound cylinder under internal pressure
In compound cylinders, residual stresses are induced due to shrink fitting. When internal pressure is applied, stresses at inner and outer surfaces are calculated using Lame’s equations. The resultant radial pressure at the common surface is obtained by superposition of shrinkage and pressure stresses.
(c) Hollow shaft vs solid shaft
For the same material and weight, a hollow shaft is stronger in torsion compared to a solid shaft. Hollow shafts also save material and reduce weight while providing higher torque transmission capacity.
(d) Rankine’s crippling load of column
Rankine’s formula combines crushing strength and Euler’s buckling theory to calculate crippling load. For a fixed-end column, effective length is reduced, increasing load carrying capacity.
(e) Shear centre of I-section
For symmetrical I-sections, the shear centre coincides with the centroid. It is determined by equating the moments of shear flow in flanges and web.
SECTION C
Q3 (a) Principal stresses and maximum shear stress
Using stress transformation equations or Mohr’s circle, principal stresses are calculated along with their directions. Maximum shear stress acts on planes inclined at 45° to principal planes, and corresponding normal and shear stresses are evaluated.
Q4 (a) Deflection of simply supported beam
Deflection under point loads is calculated using double integration or moment-area method. Maximum deflection occurs where slope is zero, usually near mid-span.
Q5 (a) Closed-coiled helical spring
Axial elongation of a closed-coiled spring depends on applied load, coil diameter, number of turns, and modulus of rigidity. Maximum shear stress occurs at inner fiber, and Wahl’s factor accounts for stress concentration due to curvature.
Q6 (a) Lame’s equations for thick cylinders
Lame’s equations express radial and hoop stresses as functions of radius. Assumptions include elastic material, axisymmetric loading, and negligible end effects. These equations are essential for thick pressure vessels.
Q7 (a) Beam under inclined load
An inclined load produces bending in two perpendicular planes. Stresses at corners are obtained by combining bending stresses due to vertical and horizontal components of the load.
