CE-305 Strength of Materials - Jun 2023 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 total extension of a uniformly tapering rod of diameters $d$ and $D$ over a length of $L$, when the rod is subjected to an axial load $P$.
   How to answer

   • Consider an elemental length $dx$ at a distance $x$ from the smaller end.
   • Express diameter at section: $D_x = d + \frac{D-d}{L}x$.
   • Find elongation of element: $d(\delta) = \frac{P dx}{A_x E} = \frac{4 P dx}{\pi D_x^2 E}$.
   • Integrate from $x = 0$ to $x = L$.
   • Final expression: $\delta L = \frac{4 P L}{\pi E D d}$.
2. 7 Marks
   bA steel rod of $3\text{ cm}$ diameter and $5\text{ m}$ long is connected to two grips and the rod is maintained at a temperature of $95^\circ\text{C}$. Determine the stress and force pull exerted when temperature falls to $30^\circ\text{C}$, if i) The ends do not yield ii) The ends yield by $0.12\text{ cm}$. Take $E = 2 \times 10^5\text{ MN/m}^2$ and $\alpha = 12 \times 10^{-6}/^\circ\text{C}$.
   How to answer

   • Calculate temperature drop $\Delta T = 95 - 30 = 65^\circ\text{C}$ and cross-sectional area $A = \frac{\pi}{4}(0.03)^2\text{ m}^2$.
   • Case (i) Ends do not yield: Stress $\sigma = \alpha \Delta T E$, Force $P = \sigma \times A$.
   • Case (ii) Ends yield by $\delta = 0.0012\text{ m}$: Stress $\sigma' = \frac{L \alpha \Delta T - \delta}{L} E$, Force $P' = \sigma' \times A$.
3. 7 Marks
   aDraw the S.F and B.M diagrams of the beam shown in Figure 1.

   

   How to answer

   • Determine support reactions at A and B by applying static equilibrium conditions ($\sum F_y = 0$, $\sum M_A = 0$). Total span = $2 + 4 + 4 = 10\text{ m}$.
   • Calculate shear force at critical sections (A, at 2 m before and after 60 kN, along UDL, at 6 m before and after 40 kN, at B).
   • Calculate bending moments at key points (supports and load application points).
   • Plot SFD with straight lines for point loads and linear slopes for UDL.
   • Plot BMD with parabolic curve under UDL and straight linear variations elsewhere.
4. 7 Marks
   bFind the SF at left support, if a simply supported beam of span $6\text{ m}$ is subjected an eccentric point load of $9\text{ kN}$ at distance of $2\text{ m}$ from the left support.
   How to answer

   • Identify beam parameters: span $L = 6\text{ m}$, load $W = 9\text{ kN}$, $a = 2\text{ m}$, $b = 4\text{ m}$.
   • Take moment about right support B: $R_A \times 6 = 9 \times 4 \implies R_A = 6\text{ kN}$.
   • The shear force at the left support is equal to the reaction at the left support, i.e., $6\text{ kN}$.
5. 7 Marks
   aA simply supported beam of span $L$, carrying a point load $P$ at $0.3L$ from left support. Determine the mid-span displacements and slopes at the supports, using the method of integration.
   How to answer

   • Find reactions: $R_A = 0.7P$, $R_B = 0.3P$.
   • Set up differential equation using Macaulay's method: $EI \frac{d^2y}{dx^2} = 0.7P x - P [x - 0.3L]$.
   • Integrate once for slope: $EI \frac{dy}{dx} = \frac{0.7P x^2}{2} + C_1 - \frac{P [x - 0.3L]^2}{2}$.
   • Integrate twice for deflection and apply boundary conditions $y(0)=0$ and $y(L)=0$ to find constants $C_1$ and $C_2$.
   • Compute slopes at supports ($x=0$, $x=L$) and deflection at mid-span ($x=0.5L$).
6. 7 Marks
   bA compound cylinder is made by shrinking a cylindrical of external diameter $300\text{ mm}$ and internal diameter of $250\text{ mm}$ over another cylindrical of external diameter $250\text{ mm}$ and internal diameter $200\text{ mm}$. The radial pressure at the junction after shrinking is $8\text{ N/mm}^2$. Find the final stresses sent up across the section, when the compound cylinder is subjected to an internal fluid pressure of $84.5\text{ N/mm}^2$.
   How to answer

   • Determine Lame's constants due to shrinkage pressure alone ($p = 8\text{ N/mm}^2$ at $r = 125\text{ mm}$) for inner and outer cylinders.
   • Calculate hoop stresses at inner radius ($r=100$), junction ($r=125$), and outer radius ($r=150$) due to shrinkage.
   • Treat the compound cylinder as a single thick cylinder with $r_i = 100\text{ mm}$ and $r_o = 150\text{ mm}$ under internal pressure $84.5\text{ N/mm}^2$ and compute bursting hoop stresses.
   • Superimpose shrinkage stresses and internal pressure stresses to find final resultant stresses.
7. 7 Marks
   aDerive the shear stress formula from fundamentals.
   How to answer

   • Consider an elemental length $dx$ of a beam subjected to bending moments $M$ and $M + dM$.
   • Consider an area of cross-section above distance $y_1$ from the neutral axis.
   • Calculate unbalanced normal force $dF = \int \frac{dM \cdot y}{I} dA$.
   • Equate unbalanced force to shearing force on longitudinal area: $\tau b dx = dF$.
   • Deduce final formula: $\tau = \frac{dM}{dx} \frac{A \bar{y}}{I b} = \frac{V A \bar{y}}{I b}$.
8. 7 Marks
   bA $120\text{ mm} \times 50\text{ mm}$ I-Section is subjected to a shearing force of $10\text{ kN}$. Calculate the shear stress at the neutral axis and at the top of the web. Given $I = 220 \times 10^4\text{ mm}^4$, $\text{Area} = 9.4 \times 10^2\text{ mm}^2$, $\text{web thickness} = 3.5\text{ mm}$ and $\text{flange thickness} = 5.5\text{ mm}$.
   How to answer

   • Calculate first moment of area of the flange about neutral axis: $A_f \bar{y}_f = (50 \times 5.5) \times (60 - 2.75) = 275 \times 57.25\text{ mm}^3$.
   • Compute shear stress at the top of the web ($b = t_w = 3.5\text{ mm}$): $\tau = \frac{V (A_f \bar{y}_f)}{I \times t_w}$.
   • Calculate first moment of area of half web: $A_w \bar{y}_w = (3.5 \times (60 - 5.5)) \times \frac{54.5}{2}$.
   • Compute shear stress at neutral axis: $\tau_{\text{max}} = \frac{V (A_f \bar{y}_f + A_w \bar{y}_w)}{I \times t_w}$.
9. 7 Marks
   aA hollow steel shaft $3\text{ m}$ long must transmit a torque of $25\text{ kNm}$. The total angle of twist in this length is not to exceed $2.5^\circ$ and the allowable shearing stress is $90\text{ MPa}$. Determine the inside and outside diameter of the shaft if $G = 85\text{ GPa}$.
   How to answer

   • State torsion equations: $\frac{T}{J} = \frac{\tau}{D/2}$ and $\frac{T}{J} = \frac{G\theta}{L}$.
   • Convert $\theta$ into radians: $\theta = 2.5 \times \frac{\pi}{180}\text{ rad}$.
   • From strength criterion: $\frac{T}{J/R} \le \tau \implies \frac{16 T D}{\pi(D^4 - d^4)} \le \tau$.
   • From stiffness criterion: $\frac{T L}{G J} \le \theta \implies J = \frac{\pi}{32}(D^4 - d^4) \ge \frac{T L}{G \theta}$.
   • Assume diameter ratio $k = d/D$ or solve simultaneously for $D$ and $d$ satisfy both constraints.
10. 7 Marks
    bA thin spherical vessel $100\text{ mm}$ diameter and $12.5\text{ mm}$ thick is filled with water. More water is pumped in until the pressure reaches $4.2\text{ MPa}$. How much extra water is required to reach this pressure? Assume $E = 210\text{ GPa}$, $\gamma = 0.25$.
    How to answer

    • Calculate initial volume of spherical shell: $V = \frac{\pi}{6} d^3$.
    • Calculate volumetric strain of the spherical vessel: $\epsilon_v = \frac{3 p d}{4 t E} (1 - \nu)$ (taking $\gamma = \nu = 0.25$).
    • Determine the increase in volume of the sphere: $\Delta V = V \times \epsilon_v$.
    • Extra water required equals $\Delta V$ (assuming incompressible water, or including water compressibility if bulk modulus $K_w$ were given).
11. 7 Marks
    aConstruct shear and bending moment diagrams for the loaded beam shown below in the Figure 2.

    

    How to answer

    • Find reaction forces at supports A and B using $\sum M_A = 0$ and $\sum F_y = 0$.
    • Calculate SF at key sections: just right of A, just left of B, just right of B, and at overhang end C.
    • Locate point of zero shear to find maximum positive bending moment between A and B.
    • Calculate bending moments at A, at zero shear location, at support B, and at C.
    • Draw SFD with linear variation across AB and constant across BC; draw BMD with parabolic profile across AB and linear across BC.
12. 7 Marks
    bFind the maximum deflection and slopes at supports of a simply supported beam of span '$l$' subjected to a point load of '$W$' acting at a distance of '$a$' from the left hand support.
    How to answer

    • Set up bending moment equation: $M_x = \frac{W b}{l} x - W [x - a]$.
    • Integrate using Macaulay's method to get general expressions for slope and deflection.
    • Apply boundary conditions $y = 0$ at $x = 0$ and $x = l$ to evaluate constants $C_1$ and $C_2$.
    • Obtain slopes at supports: $\theta_A = \frac{W a b (l + b)}{6 E I l}$ and $\theta_B = \frac{W a b (l + a)}{6 E I l}$.
    • Determine position of zero slope (maximum deflection) and substitute back to find maximum deflection.
13. 7 Marks
    aA solid steel shaft has to transmit $75\text{ kW}$ power at $200\text{ rpm}$. Taking allowable shear stress $70\text{ N/mm}^2$. Find suitable diameter of shaft with the maximum torque transmitted on each revolution exceeds the mean by $30\%$.
    How to answer

    • Calculate mean torque: $T_{\text{mean}} = \frac{P \times 60}{2 \pi N} = \frac{75000 \times 60}{2 \pi \times 200}\text{ N}\cdot\text{m}$.
    • Calculate maximum torque: $T_{\text{max}} = 1.30 \times T_{\text{mean}}$.
    • Use torsion formula: $T_{\text{max}} = \frac{\pi}{16} \tau d^3$.
    • Solve for diameter $d$: $d = \left( \frac{16 T_{\text{max}}}{\pi \tau} \right)^{1/3}$.
14. 7 Marks
    bDerive torsion equation and give the assumptions made for the derivation of torsion equation.
    How to answer

    • List assumptions: homogeneous, isotropic material obeying Hooke's law, plane cross-sections remain plane and circular, twist is uniform along length.
    • Consider an element on shaft surface to establish relationship between shear strain and angle of twist: $\gamma = \frac{r \theta}{L} \implies \frac{\tau}{r} = \frac{G \theta}{L}$.
    • Express resisting torque of an elemental ring: $dT = \tau \cdot dA \cdot r = \frac{G \theta}{L} r^2 dA$.
    • Integrate over the entire section: $T = \frac{G \theta}{L} \int r^2 dA = \frac{G \theta}{L} J$.
    • Combine to obtain final torsion equation: $\frac{T}{J} = \frac{\tau}{r} = \frac{G \theta}{L}$.
15. 7 Marks
    aDefine three moduli of elasticity and state the relation between them.
    How to answer

    • Define Young's Modulus ($E = \frac{\text{Tensile/Compressive Stress}}{\text{Longitudinal Strain}}$).
    • Define Shear Modulus / Modulus of Rigidity ($G = \frac{\text{Shear Stress}}{\text{Shear Strain}}$).
    • Define Bulk Modulus ($K = \frac{\text{Direct Stress}}{\text{Volumetric Strain}}$).
    • State relationships connecting them: $E = 2G(1 + \nu)$, $E = 3K(1 - 2\nu)$, and $E = \frac{9KG}{3K + G}$.
16. 7 Marks
    bWrite a brief note on: i) Stress and strain ii) Mohr's circle
    How to answer

    • Part (i): Define normal and shear stress; define longitudinal, lateral, and volumetric strain; write Hooke's law.
    • Part (ii): Explain Mohr's circle as a graphical method to determine normal and shear stresses on inclined planes.
    • State center coordinates $\left(\frac{\sigma_x + \sigma_y}{2}, 0\right)$ and radius $R = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$.
    • Sketch representative Mohr's circle indicating principal stresses.