GATE Marine Engineering Applied Mechanics and Structures Syllabus

Applied Mechanics and Structures syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Applied Mechanics and Structures in GATE Marine Engineering, not a summary of it.

1. 1

   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
2. 2

   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
3. 3

   Vibrations

   2 topics

   • Free and forced vibration of damped and undamped systems
   • Single and multi DOF systems
4. 4

   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

Applied Mechanics and Structures flashcards for GATE Marine Engineering

25 of 53 cards from the Applied Mechanics and Structures deck — real questions with worked answers.

1. 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).

2. 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.

3. 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).

4. 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.

5. 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.

6. 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: $\delta W = \sum \vec{F}_i \cdot \delta \vec{r}_i = 0$.

7. 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.

8. Write the kinematic equations for a particle under constant acceleration.

   $v = u + at$, $\quad s = ut + \tfrac{1}{2}at^{2}$, $\quad v^{2} = u^{2} + 2as$.

9. 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 $\omega$.

10. 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.

11. 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.

12. 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.

13. 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.

14. 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.

15. 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: $W = \Delta KE = \left(\tfrac{1}{2}mv^{2} + \tfrac{1}{2}I\omega^{2}\right)_2 - \left(\tfrac{1}{2}mv^{2} + \tfrac{1}{2}I\omega^{2}\right)_1$.

16. 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).

17. 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.

18. 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 $-1 < \nu \le 0.5$.

19. Give the relationships among elastic constants $E$, $G$, $K$, and $\nu$.

    $E = 2G(1+\nu)$ and $E = 3K(1-2\nu)$, and combining gives $E = \dfrac{9KG}{3K+G}$.

20. 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 $G = \dfrac{\tau}{\gamma}$.

21. 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 $R = \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}$.

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

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

23. 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).

24. 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).

25. 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.

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