CE-305 Strength of Materials - Jun 2025 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
   aAt a point in a shaft, the bending stress is $50\text{ MPa}$ (tensile) and torsional shear stress is $20\text{ MPa}$. Determine: i) Principal stresses ii) Maximum shear stress iii) Location of principal planes
   How to answer

   • State given values: $\sigma_x = 50\text{ MPa}$, $\sigma_y = 0$, $\tau = 20\text{ MPa}$.
   • Apply principal stress formula: $\sigma_{1,2} = \frac{\sigma_x}{2} \pm \sqrt{(\sigma_x/2)^2 + \tau^2}$.
   • Compute major and minor principal stresses.
   • Compute maximum shear stress: $\tau_{max} = \frac{\sigma_1 - \sigma_2}{2}$.
   • Find orientation of principal planes using $\tan(2\theta_p) = \frac{2\tau}{\sigma_x}$.
2. 7 Marks
   bWhat are principal planes and principal stresses and why are they important?
   How to answer

   • Define principal planes: planes carrying zero shear stress.
   • Define principal stresses: normal stresses acting on principal planes.
   • Explain importance in mechanical/structural design and predicting failure via failure criteria.
3. 14 Marks
   A simply supported beam $1\text{ m}$ long and $20\text{ mm} \times 20\text{ mm}$ in cross-section fails when a central load of $600\text{ N}$ is applied to it. What intensity of uniformly distributed load would cause failure of a cantilever beam $2\text{ m}$ long $40\text{ mm}$ wide $\times 80\text{ mm}$ deep made of the same material.
   How to answer

   • Find maximum bending moment for simply supported beam: $M_{1} = \frac{W L_1}{4} = \frac{600 \times 1}{4} = 150\text{ N}\cdot\text{m}$.
   • Calculate section modulus $Z_1 = \frac{b_1 d_1^2}{6}$.
   • Calculate bending stress at failure $\sigma_f = \frac{M_1}{Z_1}$.
   • Calculate section modulus for cantilever beam $Z_2 = \frac{b_2 d_2^2}{6}$.
   • Equate maximum bending moment $M_2 = \frac{w L_2^2}{2} = \sigma_f Z_2$ to determine UDL $w$.
4. 7 Marks
   aA cantilever beam having length "L" is subjected to UDL of intensity "W". Find the free end deflection using Macaulay's method.
   How to answer

   • Set up coordinate axis from fixed or free end.
   • Write the bending moment equation $M_x$ at section $x$.
   • Integrate once to obtain slope equation: $EI \frac{dy}{dx} = \int M_x dx + C_1$.
   • Apply boundary conditions at the fixed end to evaluate constants of integration.
   • Integrate again to get deflection equation and evaluate deflection at the free end ($x=L$ or $x=0$): $\delta = \frac{w L^4}{8EI}$.
5. 7 Marks
   bState and explain Mohr's first and second theorems related to the area moment method for determining slope and deflection.
   How to answer

   • State Mohr's First Theorem: change in slope between two points equals the area of the M/EI diagram between those points.
   • State Mohr's Second Theorem: tangential deviation of point B from the tangent at A equals the moment of the area of M/EI diagram between A and B about B.
   • Provide mathematical formulations and sign convention.
6. 14 Marks
   A hollow cast iron cylindrical column, having $4\text{ m}$ long with both ends firmly fixed, carries an axial load of $210\text{ kN}$. The internal diameter of the column is equal to $0.8$ times the external diameter. Determine the diameters of the column, taking $\sigma_c = 620\text{ MPa}$, $a = \frac{1}{1600}$ and a factor of safety of $5$.
   How to answer

   • Calculate crippling load $P_{cr} = 210 \times 5 = 1050\text{ kN}$.
   • Effective length for both ends fixed: $L_e = L / 2 = 2\text{ m} = 2000\text{ mm}$.
   • Express area $A = \frac{\pi}{4}(D^2 - (0.8D)^2) = 0.09\pi D^2$.
   • Express radius of gyration squared: $k^2 = \frac{D^2 + d^2}{16} = \frac{1.64 D^2}{16} = 0.1025 D^2$.
   • Substitute into Rankine's formula: $P_{cr} = \frac{\sigma_c A}{1 + a (L_e/k)^2}$ and solve for external diameter $D$ and internal diameter $d = 0.8 D$.
7. 14 Marks
   A hollow shaft of diameter ratio $3/8$ is required to transmit $600\text{ kW}$ at $110\text{ r.p.m.}$, the maximum torque being $20\%$ greater than the mean. The shear stress is not to exceed $63\text{ MN/m}^2$ and the twist in a length of $3\text{ m}$ not to exceed $1.4\text{ degrees}$. Calculate the maximum external diameter satisfying these conditions. Take $G = 80\text{ GPa}$.
   How to answer

   • Calculate mean torque $T_{mean} = \frac{60 P}{2\pi N}$ and maximum torque $T_{max} = 1.2 \times T_{mean}$.
   • Determine external diameter $D_1$ based on shear stress criteria: $T_{max} = \frac{\pi}{16} \tau D^3 (1 - k^4)$ where $k = 3/8$.
   • Determine external diameter $D_2$ based on rigidity/twist criteria: $\theta = \frac{T_{max} L}{G J} \le 1.4 \times \frac{\pi}{180}$.
   • Choose the larger of the two diameters as the required external diameter.
8. 14 Marks
   Explain the following: i) Middle Third Rule ii) Combined Bending and Torsion
   How to answer

   • Explain Middle Third Rule: condition for no tension in a rectangular column cross-section ($e \le d/6$).
   • Provide core/kern diagram for rectangular sections.
   • Explain combined bending and torsion in shafts subject to bending moment $M$ and torque $T$.
   • Give equivalent bending moment $M_e = \frac{1}{2}(M + \sqrt{M^2 + T^2})$ and equivalent twisting moment $T_e = \sqrt{M^2 + T^2}$.
9. 7 Marks
   aExplain any three theories of failure along with their graphical representation.
   How to answer

   • Select three theories (e.g., Maximum Principal Stress Theory, Maximum Shear Stress Theory, Maximum Distortion Energy Theory).
   • State criterion/equation for each theory.
   • Draw graphical representation (failure envelopes in $\sigma_1-\sigma_2$ plane: square, hexagon, ellipse).
   • State application suitability (brittle vs. ductile materials).
10. 7 Marks
    bDiscuss the Rankine's formula for an intermediate sized column.
    How to answer

    • Explain limitation of Euler's formula for intermediate columns where failure occurs due to both direct crushing and buckling.
    • State Rankine's empirical relation: $\frac{1}{P_R} = \frac{1}{P_c} + \frac{1}{P_E}$.
    • Derive standard form: $P_R = \frac{\sigma_c A}{1 + a (L_e/k)^2}$.
    • Explain behavior for very short and very long columns.
11. 14 Marks
    Write short notes on the following: a) Section Modulus b) Point of Contra-flexure c) Thermal Stresses d) Comparison of hollow and solid shafts
    How to answer

    • Define Section Modulus ($Z = I/y_{max}$) and its relation to bending strength.
    • Define Point of Contra-flexure as the location where bending moment changes sign and curvature changes.
    • Explain thermal stresses caused by temperature change with restrained expansion ($\sigma = E \alpha \Delta T$).
    • Compare hollow and solid shafts showing hollow shafts are stronger and stiffer per unit weight.