UPSC ESE Mechanical Engineering Strength of Materials Syllabus

Strength of Materials syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Strength of Materials in UPSC ESE Mechanical Engineering, not a summary of it.

1. 1

   Stress and Strain

   3 topics

   • Axial Loading
   • Shear Stress
   • Thermal Stresses
2. 2

   Bending and Torsion

   3 topics

   • Bending Moment and Shear Force
   • Torsion of Shafts
   • Bending Stress
3. 3

   Deflection of Beams

   3 topics

   • Double Integration Method
   • Macaulay's Method
   • Moment Area Method

Strength of Materials flashcards for UPSC ESE Mechanical Engineering

25 of 55 cards from the Strength of Materials deck — real questions with worked answers.

1. What is the formula for axial (normal) stress in a bar of cross-sectional area $A$ carrying axial load $P$?

   $$\sigma = \frac{P}{A}$$ where $\sigma$ is the normal stress, $P$ the axial force, and $A$ the cross-sectional area.

2. State the formula for the axial elongation of a prismatic bar of length $L$, area $A$, modulus $E$, under axial load $P$.

   $$\delta = \frac{PL}{AE}$$

3. Define Young's modulus (modulus of elasticity) $E$ in terms of stress and strain.

   It is the ratio of normal stress to normal strain within the elastic limit: $$E = \frac{\sigma}{\varepsilon}$$

4. What is normal (longitudinal) strain for a bar that elongates by $\delta$ over original length $L$?

   $$\varepsilon = \frac{\delta}{L}$$

5. Define Poisson's ratio $\nu$.

   The magnitude of the ratio of lateral strain to longitudinal strain: $$\nu = -\frac{\varepsilon_{lat}}{\varepsilon_{long}}$$ For most metals $0.25 \leq \nu \leq 0.35$.

6. For a bar with several segments of different loads, areas or materials, how is total axial elongation computed?

   By superposition, summing each segment's elongation: $$\delta = \sum_{i} \frac{P_i L_i}{A_i E_i}$$

7. What is the elongation of a vertical bar of length $L$, area $A$ due to its own self-weight (total weight $W$)?

   $$\delta = \frac{WL}{2AE}$$ It is half of what the same total load would produce if applied at the free end.

8. Give the relationships between the elastic constants $E$, $G$, and bulk modulus $K$ with Poisson's ratio $\nu$.

   $$E = 2G(1+\nu) \qquad E = 3K(1-2\nu) \qquad E = \frac{9KG}{3K+G}$$

9. Define shear stress $\tau$ on a section carrying tangential force $V$ over area $A$.

   $$\tau = \frac{V}{A}$$ It acts parallel (tangential) to the surface, unlike normal stress which acts perpendicular to it.

10. What is shear strain $\gamma$ and how does it relate to shear stress through the modulus of rigidity $G$?

    Shear strain is the angular distortion (in radians) of an element. Within the elastic limit: $$\tau = G\,\gamma$$

11. Distinguish single shear from double shear for a pin/rivet carrying load $P$.

    Single shear: one shear plane, $\tau = \dfrac{P}{A}$. Double shear: two shear planes, $\tau = \dfrac{P}{2A}$, so the stress is halved.

12. What is complementary shear stress?

    Shear stresses on perpendicular planes always occur in pairs of equal magnitude; the shear stress on one plane is accompanied by an equal complementary shear stress on the perpendicular plane to maintain rotational equilibrium.

13. For a circular shaft of diameter $d$ punched through a plate of thickness $t$, what is the punching shear area and force?

    Shear area $= \pi d t$ (cylindrical surface). Required punching force $$P = \tau_{ult}\,\pi d t$$

14. Write the formula for thermal (free) strain in a bar subjected to a temperature rise $\Delta T$ with coefficient $\alpha$.

    $$\varepsilon_{T} = \alpha\,\Delta T$$ and free elongation $\delta_T = \alpha\,\Delta T\,L$.

15. What thermal stress develops in a bar that is fully restrained (both ends fixed) under temperature rise $\Delta T$?

    $$\sigma = E\,\alpha\,\Delta T$$ It is compressive for a temperature rise and independent of the bar's length and area.

16. A bar between two supports has a gap $\Delta$ allowing some expansion before contact. What is the resulting thermal stress for temperature rise $\Delta T$?

    $$\sigma = \frac{E\,(\alpha\,\Delta T\,L - \Delta)}{L}$$ Stress is zero until free expansion exceeds the gap $\Delta$.

17. In a composite bar of two materials rigidly joined and heated, what governs the thermal stresses if $\alpha_1 > \alpha_2$?

    Both bars expand by the same final amount (compatibility). The material with higher $\alpha$ is put into compression, the lower-$\alpha$ material into tension, and equilibrium requires $P_1 = P_2$ (equal and opposite internal forces).

18. State the differential (sign-convention) relationships among load $w$, shear force $V$, and bending moment $M$.

    $$\frac{dV}{dx} = -w \qquad \frac{dM}{dx} = V \qquad \frac{d^{2}M}{dx^{2}} = -w$$

19. What does the area under the shear force diagram between two sections represent?

    The change in bending moment between those sections: $$\Delta M = \int V\,dx$$

20. Where does the maximum bending moment occur along a beam (in terms of the SFD)?

    At the section where the shear force is zero (or changes sign), since $\dfrac{dM}{dx} = V = 0$ there.

21. For a simply supported beam of span $L$ with a central point load $W$, give the maximum shear force and maximum bending moment.

    $$V_{max} = \frac{W}{2} \qquad M_{max} = \frac{WL}{4} \text{ (at midspan)}$$

22. For a simply supported beam of span $L$ under a uniformly distributed load $w$ per unit length, give $V_{max}$ and $M_{max}$.

    $$V_{max} = \frac{wL}{2} \text{ (at supports)} \qquad M_{max} = \frac{wL^{2}}{8} \text{ (at midspan)}$$

23. For a cantilever of length $L$ with a point load $W$ at the free end, give $M_{max}$ and its location.

    $$M_{max} = WL \text{ at the fixed end}$$ The bending moment is hogging (negative).

24. For a cantilever of length $L$ carrying a UDL $w$, give the maximum bending moment.

    $$M_{max} = \frac{wL^{2}}{2} \text{ at the fixed support}$$

25. What is a point of contraflexure (inflexion) in a beam?

    A section where the bending moment changes sign (passes through zero) and the beam curvature reverses from sagging to hogging or vice versa.

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