CE-305 Strength of Materials - Dec 2024 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aDeduce the relationship between the Modulus of Elasticity and Modulus of Rigidity.
   How to answer

   • Define Modulus of Elasticity (E), Modulus of Rigidity (G), and Poisson's ratio (\mu)
   • Consider a square element subjected to pure shear stress
   • Show tensile and compressive strains along the diagonals
   • Relate shear strain to normal strain on 45-degree planes
   • Conclude with relation: E = 2G(1 + \mu)
2. 7 Marks
   bA point is subjected to a tensile stress of 250 MPa in the horizontal direction and another tensile stress of 100 MPa in the vertical direction. The point is also subjected to a simple shear stress of 25 MPa. Such that when it is associated with the major tensile stress, it tends to rotate the element in the clockwise direction what is the magnitude of the normal and shear stresses on a section inclined at an angle of 20°, with the major tensile stress.
   How to answer

   • Identify stress components: \sigma_x = 250 MPa, \sigma_y = 100 MPa, \tau_{xy} = 25 MPa
   • Determine the angle of plane: \theta = 70° (or 20° relative to specified direction)
   • Apply formula for normal stress \sigma_n
   • Apply formula for shear stress \tau
   • Calculate and state magnitude of \sigma_n and \tau
3. 7 Marks
   aA rectangular beam 60 mm wide and 150 mm deep is simply supported over a span of 4 meters. If the beam is subjected to a uniformly distributed load of 4.5 kN/m. Find the maximum bending stress induced in the beam.
   How to answer

   • Calculate maximum bending moment: M = wL^2 / 8 = 4.5 * 4^2 / 8 = 9 kN.m = 9 * 10^6 N.mm
   • Calculate section modulus: Z = bd^2 / 6 = 60 * 150^2 / 6 = 225,000 mm^3
   • Compute maximum bending stress: \sigma = M / Z = 9 * 10^6 / 225,000 = 40 N/mm^2 (MPa)
4. 7 Marks
   bShow that for a rectangular section, the distribution of shearing stress is parabolic.
   How to answer

   • State standard shear stress formula: \tau = (V * A * y_bar) / (I * b)
   • Consider a fiber at distance y from the neutral axis of a rectangular section (b x d)
   • Determine area above the fiber and its centroidal distance
   • Substitute into formula to get quadratic equation in y: \tau = (V / 2I) * (d^2/4 - y^2)
   • Conclude that the equation represents a parabolic distribution with maximum at the neutral axis
5. 7 Marks
   aWrite the assumptions made in the theory of simple bending.
   How to answer

   • Material is homogeneous and isotropic
   • Beam is initially straight and has constant cross-section
   • Plane sections before bending remain plane after bending
   • Radius of curvature is large compared to dimensions of the cross-section
   • Modulus of elasticity is same in tension and compression
   • Hooke's law is obeyed
6. 7 Marks
   bDerive the relation between the rate of loading, shear force and bending moment.
   How to answer

   • Consider an elemental length dx of a beam carrying distributed load w
   • Apply vertical equilibrium: \sum F_y = 0 to get dV/dx = -w
   • Apply moment equilibrium: \sum M = 0 about a point to get dM/dx = V
   • Combine to show d^2M/dx^2 = -w
7. 7 Marks
   aA T-Section 150 mm × 120 mm × 20 mm is used as a strict of 4 m long with hinged at its both ends. Calculate the crippling load if modulus of elasticity for the material be $2.0 \times 10^5\text{ N/mm}^2$.
   How to answer

   • Calculate centroid and moments of inertia $I_{xx}$ and $I_{yy}$ of the T-section
   • Find minimum moment of inertia $I_{min} = \min(I_{xx}, I_{yy})$
   • Determine effective length $L_e = L = 4\text{ m} = 4000\text{ mm}$ (hinged-hinged)
   • Substitute into Euler's buckling formula: $P_{cr} = \frac{\pi^2 E I_{min}}{L_e^2}$
   • State final value in kN or N
8. 7 Marks
   bState the Euler's assumption in column theory and also explain the middle third rule.
   How to answer

   • List assumptions in Euler's column theory (straight column, axial load, buckling failure, long column, etc.)
   • Explain eccentricity and combined bending and axial stresses
   • Derive condition for no tension: $e \le d/6$
   • Explain core/kernel of rectangular section being middle third
9. 7 Marks
   aA solid steel shaft is to transmit a torque of 10 kN-m. If the shearing stress is not to exceed 45 MPa, find the minimum diameter of shaft.
   How to answer

   • State torsion equation: T / J = \tau / r
   • Write formula: T = (\pi / 16) * \tau * d^3
   • Substitute T = 10 * 10^6 N-mm and \tau = 45 N/mm^2
   • Solve: d^3 = (16 * 10 * 10^6) / (\pi * 45) \approx 1.13177 * 10^6
   • Compute d \approx 104.2 mm
10. 7 Marks
    bDerive an expression for the angle of twist in the case of a member of circular cross-section subjected to torsional moment.
    How to answer

    • Consider a cylindrical shaft of length L and radius R subjected to torque T
    • Relate shear strain \gamma at radius r to angle of twist \theta: \gamma = r \theta / L
    • Apply Hooke's law in shear: \tau = G \gamma = G r \theta / L
    • Integrate elemental torque over circular cross-section: T = \int r \tau dA = (G \theta / L) J
    • Rearrange to obtain \theta = T L / (G J)
11. 7 Marks
    aExplain the principle of super position.
    How to answer

    • Define principle of superposition (total response/displacement equals sum of responses due to individual loads acting independently)
    • State necessary conditions (linear elastic material, small displacements)
    • Provide mathematical representation: $\delta = \sum \delta_i$
    • Give an example of a bar with stepped cross-sections or multiple axial loads
12. 7 Marks
    bA I-section, with rectangular ends, has following dimensions: Flanges = 150 mm × 20 mm, Web = 300 mm × 10 mm. Find the maximum shearing stress developed in the beam for a shear force of 50 kW.
    How to answer

    • Note: '50 kW' in question is a typographical error for 50 kN
    • Total depth D = 300 + 2 * 20 = 340 mm
    • Calculate moment of inertia I about neutral axis: $I = \frac{150 \times 340^3}{12} - \frac{140 \times 300^3}{12}$
    • Calculate first moment of area $A \bar{y}$ about NA (flange contribution + web contribution above NA)
    • Apply $\tau_{max} = \frac{V (A \bar{y})_{NA}}{I \cdot t_w}$ with $t_w = 10\text{ mm}$ and calculate the stress
13. 7 Marks
    aWrite short notes on Theory of Failures.
    How to answer

    • Introduction to theories of failure for predicting yield/fracture under multiaxial stress states
    • Maximum Principal Stress Theory (Rankine's Theory) - statement and condition
    • Maximum Shear Stress Theory (Tresca's Theory) - statement and condition
    • Maximum Distortion Energy Theory (von Mises Theory) - statement and condition
    • Applications of theories to ductile versus brittle materials
14. 7 Marks
    bA circular alloy bar 2 m long uniformly tapers from 30 mm diameter to 20 mm diameter. Calculate the elongation of the rod under an axial force of 50 kN. Take E for the alloy as 140 GPa.
    How to answer

    • State formula for extension of uniformly tapering circular bar: $\Delta L = \frac{4 P L}{\pi E d_1 d_2}$
    • List given parameters: $P = 50 \times 10^3\text{ N}$, $L = 2000\text{ mm}$, $d_1 = 30\text{ mm}$, $d_2 = 20\text{ mm}$, $E = 140 \times 10^3\text{ N/mm}^2$
    • Substitute: $\Delta L = \frac{4 \times 50000 \times 2000}{\pi \times 140000 \times 30 \times 20}$
    • Compute: $\Delta L = \frac{400,000,000}{263,893,782.9} \approx 1.516\text{ mm}$
15. 7 Marks
    aWrite short notes on: Mohr's Circle
    How to answer

    • Definition of Mohr's circle as graphical method for determining normal and shear stresses on inclined planes
    • Construction steps: coordinates of center $(\frac{\sigma_x + \sigma_y}{2}, 0)$ and radius $R = \sqrt{(\frac{\sigma_x - \sigma_y}{2})^2 + \tau_{xy}^2}$
    • How to find principal stresses and orientation of principal planes
    • Sketch Mohr's circle with typical labels
16. 7 Marks
    bWrite short notes on: Longitudinal and lateral strain
    How to answer

    • Define longitudinal strain as deformation per unit length along the line of application of force: $\epsilon_L = \Delta L / L$
    • Define lateral strain as deformation per unit dimension perpendicular to the line of application of force: $\epsilon_d = \Delta d / d$
    • Explain sign difference (extension in axial direction causes contraction in lateral directions)
    • Define Poisson's ratio as ratio of lateral strain to longitudinal strain: $\mu = -\epsilon_{lateral} / \epsilon_{longitudinal}$