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GATE Marine Engineering Applied Mechanics and Structures Syllabus
The Applied Mechanics and Structures syllabus for GATE Marine Engineering in full: 4 chapters and 18 topics, every one listed below, plus 53 Applied Mechanics and Structures flashcards with answers.
Applied Mechanics and Structures syllabus: all 4 chapters
Expand any chapter to see its topics and sub-topics.
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Engineering Mechanics
5 topics- Free-body diagrams and equilibrium
- Trusses and frames
- Virtual work
- Kinematics and dynamics of particles and rigid bodies in plane motion
- Impulse and momentum (linear and angular) and energy formulations
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Mechanics of Materials
8 topics- Stress and strain, elastic constants, Poisson’s ratio
- Mohr’s circle for plane stress and plane strain
- Shear force and bending moment diagrams
- Bending and shear stresses
- Torsion
- Euler’s theory of columns
- Energy methods
- Theories and failure, material testing methods
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Vibrations
2 topics- Free and forced vibration of damped and undamped systems
- Single and multi DOF systems
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Machine Design
3 topics- Design for static and dynamic loading
- Design of machine elements such as shafts, gears, rolling and sliding contact bearings
- Joining techniques such as bolting, riveting and welding
GATE Marine Engineering Applied Mechanics and Structures flashcards — 53 questions with answers
What two conditions must be satisfied for a rigid body in static equilibrium in a plane?
The vector sum of all forces and the sum of all moments must vanish: $\sum F_x = 0$, $\sum F_y = 0$, and $\sum M = 0$ (about any point).
What is a free-body diagram (FBD)?
A sketch of a single body isolated from its surroundings, showing all external forces and moments (applied loads, reactions, and weight) acting on it, used to apply the equilibrium equations.
State the condition for a planar truss to be statically determinate, with $m$ members, $r$ reactions, and $j$ joints.
$m + r = 2j$. If $m + r > 2j$ it is statically indeterminate; if $m + r < 2j$ it is a mechanism (unstable).
What is a zero-force member in a truss, and name one rule to identify it?
A member carrying no axial load. Example rule: at an unloaded two-member joint where the members are non-collinear, both members are zero-force; at a three-member joint with two collinear members and no external load, the third (non-collinear) member is zero-force.
Compare the method of joints and the method of sections for truss analysis.
Method of joints applies $\sum F_x=0,\ \sum F_y=0$ at each pin (only 2 equations per joint, good for finding all forces). Method of sections cuts through the truss and uses $\sum M=0$ to find a few specific member forces directly.
State the principle of virtual work for a system in equilibrium.
For a system in equilibrium, the total virtual work done by all external (and constraint-free) forces through any kinematically admissible virtual displacement is zero: $\delta W = \sum \vec{F}_i \cdot \delta \vec{r}_i = 0$.
How is the unit-load (virtual work) method used to find a deflection at a point in a structure?
Apply a unit virtual load at the point in the desired direction; the deflection is $\delta = \int \frac{M\,m}{EI}\,dx$ (bending) plus axial/shear terms, where $M$ is the real internal moment and $m$ the moment due to the unit load.
Write the kinematic equations for a particle under constant acceleration.
$v = u + at$, $\quad s = ut + \tfrac{1}{2}at^{2}$, $\quad v^{2} = u^{2} + 2as$.
For a rigid body in general plane motion, how is the velocity of point B related to point A?
$\vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$, i.e. translation of A plus rotation about A with angular velocity $\omega$.
What is the instantaneous centre of rotation (ICR) for a body in plane motion?
The point in the body (or its extension) that has zero velocity at a given instant; the body appears to rotate purely about it, so $v = \omega r$ for every point at perpendicular distance $r$ from the ICR.
State the acceleration of point B relative to A for a rigid body in plane motion.
$\vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} - \omega^{2}\vec{r}_{B/A}$, where $\vec{\alpha}\times\vec{r}$ is the tangential and $-\omega^{2}\vec{r}$ the centripetal (normal) component.
State the linear impulse–momentum principle for a particle.
$\int_{t_1}^{t_2}\vec{F}\,dt = m\vec{v}_2 - m\vec{v}_1$, i.e. the linear impulse equals the change in linear momentum.
State the angular impulse–momentum principle about a fixed point.
$\int_{t_1}^{t_2}\vec{M}\,dt = \vec{H}_2 - \vec{H}_1$, where the angular momentum $H = I\omega$ for a rigid body about its axis; angular momentum is conserved when net external moment is zero.
Define the coefficient of restitution $e$ for a direct central impact.
$e = \dfrac{\text{relative velocity of separation}}{\text{relative velocity of approach}} = \dfrac{v_2' - v_1'}{u_1 - u_2}$, with $e=1$ for perfectly elastic and $e=0$ for perfectly plastic impact.
State the work–energy principle for a rigid body in plane motion.
Work done by all external forces and moments equals the change in total kinetic energy: $W = \Delta KE = \left(\tfrac{1}{2}mv^{2} + \tfrac{1}{2}I\omega^{2}\right)_2 - \left(\tfrac{1}{2}mv^{2} + \tfrac{1}{2}I\omega^{2}\right)_1$.
Define normal (direct) stress and normal (longitudinal) strain.
Normal stress $\sigma = \dfrac{P}{A}$ (force per unit area). Normal strain $\varepsilon = \dfrac{\Delta L}{L}$ (change in length per unit original length).
State Hooke's law and define Young's modulus.
In the elastic region stress is proportional to strain: $\sigma = E\varepsilon$, where $E$ (Young's modulus) is the slope of the linear stress–strain curve, with units of Pa.
Define Poisson's ratio and give its typical range for metals.
$\nu = -\dfrac{\varepsilon_{lateral}}{\varepsilon_{longitudinal}}$, the ratio of lateral to longitudinal strain. For most metals $\nu \approx 0.25$–$0.35$; the theoretical limit is $-1 < \nu \le 0.5$.
Give the relationships among elastic constants $E$, $G$, $K$, and $\nu$.
$E = 2G(1+\nu)$ and $E = 3K(1-2\nu)$, and combining gives $E = \dfrac{9KG}{3K+G}$.
Define shear stress, shear strain, and the modulus of rigidity.
Shear stress $\tau = \dfrac{V}{A}$, shear strain $\gamma$ is the angular distortion (rad), and the modulus of rigidity $G = \dfrac{\tau}{\gamma}$.
Show the other 33 Applied Mechanics and Structures flashcards
What does Mohr's circle represent for plane stress, and what is its centre and radius?
It is a graphical representation of normal vs. shear stress on planes at all orientations. Centre $=\left(\dfrac{\sigma_x+\sigma_y}{2},\,0\right)$; radius $R = \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}$.
Write 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?
$\tau_{max} = \dfrac{\sigma_1 - \sigma_2}{2} = \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}$ (the radius of Mohr's circle).
On Mohr's circle, how does the angle between physical planes relate to the angle on the circle?
An angle $\theta$ rotation of the physical plane corresponds to a $2\theta$ rotation on Mohr's circle (in the same rotational sense).
State the relationships among load $w$, shear force $V$, and bending moment $M$ along a beam.
$\dfrac{dV}{dx} = -w$ and $\dfrac{dM}{dx} = V$, so $\dfrac{d^{2}M}{dx^{2}} = -w$. Bending moment is maximum where shear force is zero or changes sign.
For a simply supported beam of span $L$ with a central point load $W$, give the maximum shear and bending moment.
Maximum shear $V_{max} = \dfrac{W}{2}$ (at supports); maximum bending moment $M_{max} = \dfrac{WL}{4}$ (at midspan).
For a simply supported beam of span $L$ under uniformly distributed load $w$, give max shear and max moment.
$V_{max} = \dfrac{wL}{2}$ at the supports; $M_{max} = \dfrac{wL^{2}}{8}$ at midspan.
State the flexure (bending) formula for beams.
$\dfrac{\sigma}{y} = \dfrac{M}{I} = \dfrac{E}{R}$, where $\sigma$ is bending stress at distance $y$ from the neutral axis, $M$ the moment, $I$ the second moment of area, and $R$ the radius of curvature.
Write the transverse shear stress distribution formula in a beam cross-section.
$\tau = \dfrac{VQ}{Ib}$, where $V$ is shear force, $Q$ the first moment of area above the level considered, $I$ the second moment of area, and $b$ the width at that level.
For a solid rectangular beam, where is the transverse shear stress maximum and what is its value?
Maximum at the neutral axis: $\tau_{max} = \dfrac{3}{2}\,\dfrac{V}{A} = 1.5\,\tau_{avg}$. It is zero at the top and bottom fibres.
State the torsion equation for a circular shaft.
$\dfrac{T}{J} = \dfrac{\tau}{r} = \dfrac{G\theta}{L}$, where $T$ is torque, $J$ the polar moment of area, $\tau$ shear stress at radius $r$, $G$ modulus of rigidity, $\theta$ angle of twist over length $L$.
Give the polar moment of area $J$ for solid and hollow circular shafts.
Solid: $J = \dfrac{\pi d^{4}}{32}$. Hollow: $J = \dfrac{\pi (d_o^{4} - d_i^{4})}{32}$.
Define the torsional stiffness of a shaft.
Torsional stiffness $k_t = \dfrac{T}{\theta} = \dfrac{GJ}{L}$ (torque required per unit angle of twist).
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 $I$ the least second moment of area.
Give the effective length $L_e$ for the four standard column end conditions.
Both ends pinned: $L_e = L$. Both ends fixed: $L_e = 0.5L$. One end fixed, one free: $L_e = 2L$. One end fixed, one pinned: $L_e \approx 0.7L$.
Define slenderness ratio and the validity limit of Euler's theory.
Slenderness ratio $\lambda = \dfrac{L_e}{k}$, 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.
State Castigliano's second theorem (energy method for deflection).
The partial derivative of total strain energy $U$ with respect to a load gives the displacement at that load in its direction: $\delta_i = \dfrac{\partial U}{\partial P_i}$, and $\theta_i = \dfrac{\partial U}{\partial M_i}$ for rotation.
Write expressions for strain energy stored in axial, bending, and torsional loading.
Axial: $U = \dfrac{P^{2}L}{2AE}$. Bending: $U = \displaystyle\int \dfrac{M^{2}}{2EI}\,dx$. Torsion: $U = \dfrac{T^{2}L}{2GJ}$.
Name the principal theories of elastic failure used in machine design.
Maximum principal stress (Rankine), maximum shear stress (Tresca/Guest), maximum principal strain (St. Venant), maximum strain energy (Haigh), and maximum distortion (shear strain) energy (von Mises–Hencky).
State the maximum shear stress (Tresca) failure criterion.
Failure occurs when the maximum shear stress reaches the shear yield value: $\dfrac{\sigma_1 - \sigma_3}{2} = \dfrac{\sigma_y}{2}$, i.e. $\sigma_1 - \sigma_3 = \sigma_y$. Best for ductile materials.
State the distortion energy (von Mises) failure criterion for plane stress.
Yielding begins when $\sigma_v = \sqrt{\sigma_1^{2} - \sigma_1\sigma_2 + \sigma_2^{2}} = \sigma_y$. It is the most accurate criterion for ductile materials.
Distinguish hardness, toughness, and a fatigue (endurance) test.
Hardness: resistance to indentation (Brinell, Vickers, Rockwell). Toughness: energy absorbed before fracture, often from an impact test (Izod/Charpy). Fatigue test: cyclic loading (e.g. rotating beam) to find the endurance limit/S–N curve.
Write the differential equation of free undamped single-degree-of-freedom vibration and its natural frequency.
$m\ddot{x} + kx = 0$, with natural frequency $\omega_n = \sqrt{\dfrac{k}{m}}$ (rad/s) and $f_n = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}$.
Define the damping ratio and classify systems as under-, critically, and over-damped.
$\zeta = \dfrac{c}{c_c} = \dfrac{c}{2\sqrt{km}}$. Under-damped: $\zeta<1$ (oscillatory decay); critically damped: $\zeta=1$ (fastest non-oscillatory return); over-damped: $\zeta>1$ (slow non-oscillatory).
Give the damped natural frequency in terms of $\omega_n$ and $\zeta$.
$\omega_d = \omega_n\sqrt{1-\zeta^{2}}$ (valid for under-damped systems, $\zeta<1$).
Define the logarithmic decrement and relate it to the damping ratio.
$\delta = \ln\dfrac{x_1}{x_2} = \dfrac{2\pi\zeta}{\sqrt{1-\zeta^{2}}}$; for small damping $\delta \approx 2\pi\zeta$.
What is resonance, and what governs the amplitude at resonance in forced vibration?
Resonance occurs when the forcing frequency equals the natural frequency ($\omega = \omega_n$), producing maximum amplitude. The peak amplitude is limited only by damping; the magnification factor at resonance is $\dfrac{1}{2\zeta}$.
How many natural frequencies and mode shapes does an $n$-DOF undamped system have?
It has $n$ natural frequencies and $n$ corresponding mode shapes (eigenvalues and eigenvectors of $[K]\{\phi\} = \omega^{2}[M]\{\phi\}$).
State Soderberg and Goodman criteria for design under fluctuating (dynamic) loads.
Soderberg: $\dfrac{\sigma_m}{\sigma_y} + \dfrac{\sigma_a}{\sigma_e} = \dfrac{1}{N}$. Goodman: $\dfrac{\sigma_m}{\sigma_u} + \dfrac{\sigma_a}{\sigma_e} = \dfrac{1}{N}$, where $\sigma_m,\sigma_a$ are mean and alternating stress, $\sigma_e$ endurance limit, $N$ factor of safety.
Write the ASME design equation for a shaft under combined bending and torsion.
Equivalent twisting moment $T_e = \sqrt{M^{2}+T^{2}}$ and equivalent bending moment $M_e = \tfrac{1}{2}\left(M + \sqrt{M^{2}+T^{2}}\right)$; diameter from $T_e = \dfrac{\pi}{16}\tau\,d^{3}$ or $M_e = \dfrac{\pi}{32}\sigma_b\,d^{3}$.
For a pair of meshing spur gears, give the velocity ratio and the law of gearing.
Velocity ratio $\dfrac{N_1}{N_2} = \dfrac{T_2}{T_1} = \dfrac{d_2}{d_1}$ (speeds inverse to teeth/diameters). Law of gearing: for constant velocity ratio the common normal at the point of contact must always pass through the fixed pitch point.
Compare rolling-contact and sliding-contact (journal) bearings.
Rolling (ball/roller) bearings: low starting and running friction, carry combined loads, need little lubricant, but are noisier and fail by fatigue. Sliding (journal) bearings: higher friction, quieter, better shock/vibration damping and high-speed capacity, and rely on a hydrodynamic oil film.
Compare bolted, riveted, and welded joints.
Bolted: non-permanent, allows disassembly, needs locking against loosening. Riveted: permanent, good for shear and fatigue (e.g. ships/structures), assembled hot or cold. Welded: permanent, strong and lightweight (no holes weakening the section), but introduces residual stresses and a heat-affected zone.
GATE Marine Engineering Applied Mechanics and Structures FAQ
What is in the GATE Marine Engineering Applied Mechanics and Structures syllabus?
GATE Marine Engineering Applied Mechanics and Structures is split into 4 chapters: Engineering Mechanics, Mechanics of Materials, Vibrations and Machine Design. Together they contain 18 topics.
How many chapters are in GATE Marine Engineering Applied Mechanics and Structures?
4 chapters, containing 18 topics. The largest chapter is Mechanics of Materials, with 8 topics.
Are there flashcards for GATE Marine Engineering Applied Mechanics and Structures?
Yes — a 53-card Applied Mechanics and Structures deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.