UPSC IES/ESE (Engineering Services) Engineering Mechanics, Strength of Materials and Theory of Machines Syllabus

Define normal stress and longitudinal (axial) strain.

Normal stress $\sigma = \frac{P}{A}$ (force per unit area). Longitudinal strain $\varepsilon = \frac{\delta L}{L}$ (change in length per original length); both are used in Hooke's law $\sigma = E\varepsilon$.

Write the formula for elongation of a prismatic bar under axial load $P$.

$\delta = \frac{PL}{AE}$, where $L$ is the length, $A$ the cross-sectional area, and $E$ Young's modulus.

Define the three elastic constants $E$, $G$, and $K$, and Poisson's ratio $\nu$.

$E$ = Young's modulus (axial stress/strain), $G$ = modulus of rigidity (shear stress/shear strain), $K$ = bulk modulus (volumetric stress/volumetric strain), $\nu$ = Poisson's ratio = $-\frac{\text{lateral strain}}{\text{longitudinal strain}}$.

State the relations among the elastic constants $E$, $G$, $K$ and $\nu$.

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

Write the normal and shear stress on an oblique plane at angle $\theta$ for a 2D state $(\sigma_x,\sigma_y,\tau_{xy})$.

$\sigma_\theta = \frac{\sigma_x+\sigma_y}{2} + \frac{\sigma_x-\sigma_y}{2}\cos 2\theta + \tau_{xy}\sin 2\theta$ and $\tau_\theta = -\frac{\sigma_x-\sigma_y}{2}\sin 2\theta + \tau_{xy}\cos 2\theta$.

Give the principal stresses for a 2D stress state $(\sigma_x,\sigma_y,\tau_{xy})$.

$\sigma_{1,2} = \frac{\sigma_x+\sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}$, and the maximum in-plane shear $\tau_{max} = \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}$.

What does Mohr's circle represent and what are its centre and radius?

A graphical representation of the 2D stress transformation, plotting $\sigma$ vs $\tau$. Centre at $\left(\frac{\sigma_x+\sigma_y}{2},\,0\right)$; radius $R = \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^{2}+\tau_{xy}^{2}} = \tau_{max}$. Principal stresses are the horizontal-axis intercepts.

Write the formula for thermal stress in a fully restrained bar.

$\sigma = E\,\alpha\,\Delta T$, where $\alpha$ is the coefficient of thermal expansion and $\Delta T$ the temperature change. If the supports yield by $\delta$, then $\sigma = E\left(\alpha\,\Delta T - \frac{\delta}{L}\right)$.

Define strain energy and give the strain energy stored in an axially loaded bar.

Strain energy is the energy stored due to elastic deformation (resilience). For an axial bar: $U = \frac{P^{2}L}{2AE} = \frac{\sigma^{2}}{2E} \times \text{Volume}$.

Define proof resilience and modulus of resilience.

Proof resilience is the maximum strain energy stored up to the elastic limit. Modulus of resilience is the proof resilience per unit volume, equal to $\frac{\sigma_e^{2}}{2E}$ where $\sigma_e$ is the elastic-limit stress.

Compare strain energy under gradual, sudden, and impact axial loading.

For the same final load $P$: stress under gradual load $= \frac{P}{A}$; under sudden load the stress is doubled $= \frac{2P}{A}$; under impact (load $P$ falling height $h$) the instantaneous stress is $\sigma = \frac{P}{A}\left(1 + \sqrt{1 + \frac{2AEh}{PL}}\right)$.

Name the principal theories of elastic failure.

(1) Maximum principal stress (Rankine), (2) Maximum shear stress (Tresca/Guest), (3) Maximum principal strain (St. Venant), (4) Total strain energy (Haigh/Beltrami), (5) Maximum distortion (shear strain) energy (von Mises-Hencky).

State the maximum shear stress (Tresca) theory failure criterion.

Failure occurs when the maximum shear stress reaches the shear at yield in uniaxial tension: $\frac{\sigma_1 - \sigma_3}{2} = \frac{\sigma_y}{2}$, i.e. $\sigma_1 - \sigma_3 = \sigma_y$. It is conservative and best suited to ductile materials.

State the distortion energy (von Mises) theory failure criterion.

Yielding begins when the distortion energy equals that at uniaxial yield: $\frac{1}{\sqrt{2}}\sqrt{(\sigma_1-\sigma_2)^{2}+(\sigma_2-\sigma_3)^{2}+(\sigma_3-\sigma_1)^{2}} = \sigma_y$. It is the most accurate for ductile materials.

Which failure theory is recommended for brittle materials and why?

The maximum principal stress (Rankine) theory, $\sigma_1 = \sigma_{ut}$, because brittle materials fail by fracture (separation) rather than yielding, and fracture is governed by the largest tensile principal stress.

State the differential relationships among load $w$, shear force $V$, and bending moment $M$.

$\frac{dV}{dx} = -w$ (load intensity) and $\frac{dM}{dx} = V$. Hence $\frac{d^{2}M}{dx^{2}} = -w$. Bending moment is maximum where shear force is zero (or changes sign).

What is the shape of the SFD and BMD for a simply supported beam under a central point load $W$?

SFD: constant $+\frac{W}{2}$ from the left support, dropping to $-\frac{W}{2}$ after midspan (rectangular steps). BMD: triangular, zero at supports and maximum $M_{max} = \frac{WL}{4}$ at the centre.

Give the maximum bending moment for a simply supported beam under a uniformly distributed load $w$ over span $L$.

$M_{max} = \frac{wL^{2}}{8}$ at midspan; the SFD is linear from $+\frac{wL}{2}$ to $-\frac{wL}{2}$ and the BMD is parabolic.

What is the significance of a point of contraflexure?

It is a point on the beam where the bending moment changes sign (passes through zero), i.e. the curvature reverses from sagging to hogging. The beam tends to be straight there.

State the bending (flexure) equation and define each term.

$\frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R}$, where $M$ = bending moment, $I$ = second moment of area, $\sigma$ = bending stress at distance $y$ from the neutral axis, $E$ = Young's modulus, $R$ = radius of curvature.

Define section modulus and give it for a rectangular section.

Section modulus $Z = \frac{I}{y_{max}}$ relates moment to maximum bending stress via $\sigma_{max} = \frac{M}{Z}$. For a rectangle $b\times h$: $Z = \frac{b h^{2}}{6}$.

Write the formula for transverse shear stress distribution in a beam.

$\tau = \frac{V Q}{I b}$, where $V$ = shear force, $Q = A\bar{y}$ = first moment of area above the level considered, $I$ = second moment of area, $b$ = width at that level. Shear stress is maximum at the neutral axis and zero at the extreme fibres.

Compare maximum shear stress to average shear stress for rectangular and circular sections.

Rectangular: $\tau_{max} = \frac{3}{2}\tau_{avg}$ (occurs at the neutral axis). Circular: $\tau_{max} = \frac{4}{3}\tau_{avg}$. In both, average $\tau_{avg} = \frac{V}{A}$.

State the torsion equation for a circular shaft and define the terms.

$\frac{T}{J} = \frac{\tau}{r} = \frac{G\theta}{L}$, where $T$ = torque, $J$ = polar moment of inertia, $\tau$ = shear stress at radius $r$, $G$ = modulus of rigidity, $\theta$ = angle of twist, $L$ = length.

Give the polar moment of inertia for solid and hollow circular shafts.

Solid shaft of diameter $d$: $J = \frac{\pi d^{4}}{32}$. Hollow shaft of outer/inner diameters $D,d$: $J = \frac{\pi (D^{4} - d^{4})}{32}$.

Write the relation between power, torque, and speed for a transmission shaft.

$P = \frac{2\pi N T}{60}$ (with $N$ in rpm, $T$ in N·m, $P$ in watts), or equivalently $P = T\omega$ where $\omega = \frac{2\pi N}{60}$ rad/s.

Define torsional rigidity and torsional stiffness of a shaft.

Torsional rigidity is $GJ$ (the product resisting twist). Torsional stiffness is the torque per unit angle of twist: $k_t = \frac{T}{\theta} = \frac{GJ}{L}$.

Write the governing differential equation of the elastic curve (deflection of beams).

$EI\frac{d^{2}y}{dx^{2}} = M(x)$, where $y$ is deflection, $M(x)$ the bending moment. Integrating once gives slope $\frac{dy}{dx}$ and twice gives deflection $y$, with constants found from boundary conditions.

Give the maximum deflection of a simply supported beam with central point load $W$ and with UDL $w$.

Central point load: $y_{max} = \frac{W L^{3}}{48 EI}$. Uniformly distributed load: $y_{max} = \frac{5 w L^{4}}{384 EI}$, both at midspan.

Give the maximum deflection of a cantilever with an end point load $W$ and with UDL $w$.

End point load: $y_{max} = \frac{W L^{3}}{3 EI}$. Uniformly distributed load over the whole span: $y_{max} = \frac{w L^{4}}{8 EI}$, both at the free end.

What is Macaulay's method and what is its key advantage?

A double-integration technique using bracket (step/Macaulay) functions $\langle x-a\rangle$ that are taken as zero when negative. Its advantage is that a single bending-moment expression covers the whole beam, so one pair of integration constants suffices even with several discrete loads.

State the two Mohr's moment-area theorems.

Theorem I: the change in slope between two points equals the area of the $\frac{M}{EI}$ diagram between them. Theorem II: the vertical deviation of a point from the tangent at another equals the moment of that $\frac{M}{EI}$ area about the first point.

Explain the conjugate beam method for finding beam deflections.

An imaginary 'conjugate' beam of the same length is loaded with the $\frac{M}{EI}$ diagram of the real beam. The shear in the conjugate beam equals the real beam's slope, and the bending moment in the conjugate beam equals the real beam's deflection. Supports are altered per conjugate-beam boundary rules.

State Euler's formula for buckling of a long column and define equivalent length.

Critical (crippling) load $P_{cr} = \frac{\pi^{2} E I}{L_e^{2}}$, where $L_e$ is the equivalent (effective) length depending on end conditions. It applies to long, slender columns failing by elastic buckling.

Give the effective length $L_e$ for the four standard column end conditions (actual length $L$).

Both ends hinged: $L_e = L$. Both ends fixed: $L_e = \frac{L}{2}$. One end fixed, other hinged: $L_e = \frac{L}{\sqrt{2}}$. One end fixed, other free: $L_e = 2L$.

Define slenderness ratio and the limitation of Euler's theory.

Slenderness ratio $= \frac{L_e}{k}$ where $k = \sqrt{\frac{I}{A}}$ is the least radius of gyration. Euler's theory is valid only for long columns (high slenderness); it fails for short columns because it can predict a buckling stress exceeding the crushing strength.

State Rankine's formula for columns and explain why it is preferred.

$\frac{1}{P_R} = \frac{1}{P_c} + \frac{1}{P_e}$, i.e. $P_R = \frac{\sigma_c A}{1 + a\left(\frac{L_e}{k}\right)^{2}}$, where $P_c$ is crushing load and $P_e$ Euler load. It is preferred because it applies to both short and long columns, approaching crushing for short and Euler for long columns.

Compare springs in series and in parallel for stiffness.

Series (same force, displacements add): $\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + \cdots$ (softer combination). Parallel (same displacement, forces add): $k_{eq} = k_1 + k_2 + \cdots$ (stiffer combination).

Write the deflection and stiffness formulas for a close-coiled helical spring under axial load $W$.

Deflection $\delta = \frac{64 W R^{3} n}{G d^{4}}$ and stiffness $k = \frac{W}{\delta} = \frac{G d^{4}}{64 R^{3} n}$, where $R$ = mean coil radius, $d$ = wire diameter, $n$ = number of active coils, $G$ = modulus of rigidity.

Contrast helical and leaf (laminated) springs in function.

A helical spring resists axial load primarily through torsion of the wire and is used for energy storage and shock absorption (e.g., vehicle suspensions, valves). A leaf spring is a flat semi-elliptical/cantilever beam resisting load by bending, used in vehicle suspensions to carry large transverse loads while distributing stress along its length.