GATE Mechanical Engineering (ME) Syllabus

Give the principal stresses for a plane-stress state in terms of $\sigma_x$, $\sigma_y$, $\tau_{xy}$.

$\sigma_{1,2} = \dfrac{\sigma_x+\sigma_y}{2} \pm \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}$.

What is the maximum in-plane shear stress in terms of the principal stresses?

$\tau_{max} = \dfrac{\sigma_1 - \sigma_2}{2}$, equal to the radius $R$ of Mohr's circle. The planes of maximum shear are oriented $45^{\circ}$ from the principal planes.

On Mohr's circle, what is the angular relationship between the physical plane and the circle?

Angles on Mohr's circle are twice the physical angles: rotating a plane by $\theta$ in the body corresponds to rotating $2\theta$ on the circle (in the same sense).

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

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

Where do the maximum bending stress and the neutral axis lie in a symmetric beam section?

Maximum bending stress occurs at the extreme fibre (largest $y$): $\sigma_{max} = \dfrac{M\,y_{max}}{I} = \dfrac{M}{Z}$. The neutral axis passes through the centroid where bending stress is zero.

Define section modulus $Z$ and give it for a rectangular section $b \times h$.

$Z = \dfrac{I}{y_{max}}$. For a rectangle, $I = \dfrac{bh^{3}}{12}$ and $y_{max} = \dfrac{h}{2}$, so $Z = \dfrac{bh^{2}}{6}$.

Write the transverse shear stress distribution formula in a beam.

$\tau = \dfrac{VQ}{Ib}$, where $V$ is shear force, $Q$ the first moment of area above the level considered, $I$ the second moment of area and $b$ the section width. For a rectangle $\tau_{max} = 1.5\,\tau_{avg}$ at the neutral axis.

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

$\dfrac{dV}{dx} = -w$ and $\dfrac{dM}{dx} = V$. Thus the slope of the SFD equals minus the load intensity, and the slope of the BMD equals the shear force.

On a bending moment diagram, what occurs where the shear force is zero (and changes sign)?

The bending moment is a local maximum or minimum, since $\dfrac{dM}{dx}=V=0$ there. This locates the point of maximum bending moment.

What is a point of contraflexure (inflection) in a loaded beam?

A point where the bending moment changes sign (passes through zero) and the beam curvature reverses. It occurs in beams with overhangs or continuous spans.

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

$\dfrac{T}{J} = \dfrac{\tau}{r} = \dfrac{G\theta}{L}$, where $T$ is torque, $J$ the polar moment of area, $\tau$ shear stress at radius $r$, $G$ the shear modulus, $\theta$ the angle of twist and $L$ the shaft length.

Give the polar moment of inertia $J$ for a solid and a hollow circular shaft.

Solid: $J = \dfrac{\pi d^{4}}{32}$. Hollow: $J = \dfrac{\pi (d_o^{4} - d_i^{4})}{32}$, where $d_o$ and $d_i$ are outer and inner diameters.

Where is shear stress maximum in a shaft under torsion, and what is the power-torque relation?

Shear stress is maximum at the outer surface (largest $r$). Power transmitted: $P = T\omega = \dfrac{2\pi N T}{60}$, with $N$ in rpm and $\omega$ in rad/s.

What is the torsional stiffness of a shaft?

$k_t = \dfrac{T}{\theta} = \dfrac{GJ}{L}$, the torque required per unit angle of twist.

Define beam deflection and write the governing Euler-Bernoulli differential equation.

Deflection is the transverse displacement $y$ of the beam axis. Governing equation: $EI\dfrac{d^{2}y}{dx^{2}} = M(x)$, where $EI$ is the flexural rigidity.

Give the maximum deflection for a simply supported beam with a central point load $W$.

$\delta_{max} = \dfrac{W L^{3}}{48 EI}$, occurring at mid-span.

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

End load: $\delta_{max} = \dfrac{W L^{3}}{3EI}$. Uniformly distributed load: $\delta_{max} = \dfrac{w L^{4}}{8EI}$, both at the free end.

State the maximum principal stress (Rankine) theory of failure.

Failure occurs when the maximum principal stress reaches the uniaxial yield (or ultimate) strength: $\sigma_1 = \sigma_y$. Best suited to brittle materials.

State the maximum shear stress (Tresca / Guest) theory of failure.

Yielding begins when the maximum shear stress equals that at yield in a tensile test: $\dfrac{\sigma_1 - \sigma_3}{2} = \dfrac{\sigma_y}{2}$, i.e. $\sigma_1 - \sigma_3 = \sigma_y$. Suited to ductile materials.

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

Yielding occurs when the von Mises stress equals the yield stress: $\sqrt{\dfrac{(\sigma_1-\sigma_2)^{2}+(\sigma_2-\sigma_3)^{2}+(\sigma_3-\sigma_1)^{2}}{2}} = \sigma_y$. Most accurate for ductile metals.

State Euler's formula for the critical buckling load of a column.

$P_{cr} = \dfrac{\pi^{2} EI}{L_e^{2}}$, where $L_e$ is the effective length and $EI$ the flexural rigidity. Buckling occurs about the axis of least $I$.

Give the effective length $L_e$ for the four standard end conditions of a column.

Both ends pinned: $L_e = L$; both fixed: $L_e = 0.5L$; one fixed–one free: $L_e = 2L$; one fixed–one pinned: $L_e = 0.707L$.

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

Slenderness ratio $= \dfrac{L_e}{k}$, where $k = \sqrt{I/A}$ is the radius of gyration. Euler's formula is valid only for long (slender) columns where the critical stress stays below the proportional limit; short columns fail by crushing.

Distinguish a mechanism, a machine and a kinematic chain.

A kinematic chain is an assembly of links connected by joints with relative motion possible. A mechanism is a kinematic chain with one link fixed. A machine is a mechanism (or set) that transmits forces to do useful work.

State Kutzbach's (Grübler's) equation for the degrees of freedom of a planar mechanism.

$F = 3(n-1) - 2j_1 - j_2$, where $n$ is the number of links, $j_1$ the number of lower (1-DOF) pairs and $j_2$ the number of higher (2-DOF) pairs.

State Grashof's law for a four-bar linkage.

For at least one link to make a full revolution, the sum of the shortest ($s$) and longest ($l$) links must be less than or equal to the sum of the other two ($p$, $q$): $s + l \leq p + q$.

Differentiate a lower pair from a higher pair, giving examples.

A lower pair has surface (area) contact between links (e.g. revolute/pin, prismatic/slider, screw). A higher pair has point or line contact (e.g. cam-follower, meshing gear teeth, ball bearing).

Define the base circle, pressure angle and pitch curve of a cam.

Base circle: smallest circle tangent to the cam profile (defines minimum radius). Pressure angle: angle between the direction of follower motion and the normal to the cam profile at the contact point. Pitch curve: locus of the trace point of the follower's reference.

State the law of gearing and define the module of a gear.

Law of gearing: the common normal to the tooth profiles at the point of contact must always pass through the fixed pitch point, giving a constant velocity ratio. Module $m = \dfrac{d}{T}$, the pitch-circle diameter $d$ per number of teeth $T$ (in mm).

For a pair of meshing gears, write the velocity (gear) ratio.

$\dfrac{N_1}{N_2} = \dfrac{T_2}{T_1} = \dfrac{d_2}{d_1}$, i.e. speed is inversely proportional to the number of teeth (and pitch diameter).

What is the function of a flywheel, and what is the coefficient of fluctuation of speed?

A flywheel stores and releases kinetic energy to smooth out speed fluctuations caused by varying torque, keeping speed nearly constant. Coefficient of fluctuation of speed $C_s = \dfrac{N_{max}-N_{min}}{N} = \dfrac{\omega_{max}-\omega_{min}}{\omega}$.

Relate the maximum fluctuation of energy of a flywheel to its mass moment of inertia.

$\Delta E = I\,\omega^{2}\,C_s = m k^{2}\,\omega^{2}\,C_s$, where $I$ is the mass moment of inertia, $\omega$ the mean angular speed and $C_s$ the coefficient of fluctuation of speed.

What is the function of a governor, and how does it differ from a flywheel?

A governor controls the mean speed over time by regulating fuel/working-fluid supply as the load varies (works between cycles). A flywheel limits speed fluctuation within a single cycle; it does not control the mean speed.

State the conditions for complete static and dynamic balancing of rotating masses.

Static (force) balance: $\sum m r = 0$ — the resultant of centrifugal forces is zero. Dynamic balance also requires $\sum m r l = 0$ — the resultant couple about any reference plane is zero. Both must hold for complete balance.