GATE Aerospace Engineering Structures Syllabus

Structures syllabus — full chapter and topic list

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

1. 1

   Strength of Materials

   6 topics

   • Stress and strain
     • Three-dimensional transformations
     • Mohr’s circle
     • Principal stresses
     • Three-dimensional Hooke's law
     • Plane stress and strain
   • Failure theories
     • Maximum stress
     • Tresca von Mises
   • Strain energy
   • Castigliano’s principles
   • Statically determinate and indeterminate trusses and beams
   • Elastic flexural buckling of columns
2. 2

   Flight Vehicle Structures

   3 topics

   • Characteristics of aircraft structures and materials
   • Torsion, bending and shear of thin-walled sections
   • Loads on aircraft
3. 3

   Structural Dynamics

   2 topics

   • Free and forced vibrations of undamped and damped SDOF systems
   • Free vibrations of undamped 2-DOF systems
4. 4

   Special Topics

   2 topics

   • Vibration of beams
   • Theory of elasticity
     • Equilibrium and compatibility equations
     • Airy’s stress function

Structures flashcards for GATE Aerospace Engineering

19 of 59 cards from the Structures deck — real questions with worked answers.

1. Define normal stress and shear stress on a plane.

   Normal stress $\sigma = \dfrac{F_n}{A}$ acts perpendicular to the plane (force component normal to area). Shear stress $\tau = \dfrac{F_t}{A}$ acts tangential (parallel) to the plane. Both have units of pressure (Pa).

2. Define engineering normal strain and shear strain.

   Normal strain $\varepsilon = \dfrac{\Delta L}{L_0}$ (change in length per unit original length). Engineering shear strain $\gamma = \dfrac{\partial u}{\partial y} + \dfrac{\partial v}{\partial x}$, the total change in angle between two originally perpendicular fibers.

3. State the relationship between the three elastic constants $E$, $G$, and $\nu$ for an isotropic material.

   $$G = \frac{E}{2(1+\nu)}$$ where $E$ is Young's modulus, $G$ the shear modulus, and $\nu$ Poisson's ratio.

4. What is the relationship between bulk modulus $K$, Young's modulus $E$, and Poisson's ratio $\nu$?

   $$K = \frac{E}{3(1-2\nu)}$$ Note that as $\nu \to 0.5$, $K \to \infty$ (incompressible material).

5. Define Poisson's ratio and give its typical range for engineering metals.

   Poisson's ratio $\nu = -\dfrac{\varepsilon_{lateral}}{\varepsilon_{axial}}$. For most metals $\nu \approx 0.25$–$0.35$. Thermodynamic limits for isotropic materials are $-1 < \nu \leq 0.5$.

6. Write the 2D stress transformation equation for normal stress $\sigma_{x'}$ on a plane rotated by angle $\theta$.

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

7. Write the 2D shear stress transformation equation $\tau_{x'y'}$ for a plane rotated by angle $\theta$.

   $$\tau_{x'y'} = -\frac{\sigma_x-\sigma_y}{2}\sin 2\theta + \tau_{xy}\cos 2\theta$$

8. For plane stress, give the formula for the in-plane principal stresses.

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

9. Give the formula for the maximum in-plane shear stress in terms of the applied stresses.

   $$\tau_{max} = \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2} = \frac{\sigma_1-\sigma_2}{2}$$

10. What is the orientation of the principal planes in terms of $\sigma_x,\sigma_y,\tau_{xy}$?

    $$\tan 2\theta_p = \frac{2\tau_{xy}}{\sigma_x-\sigma_y}$$ Principal planes carry zero shear stress and are oriented $45^\circ$ from the planes of maximum shear.

11. On which planes do the principal stresses act, and what is special about the shear stress there?

    Principal stresses act on the principal planes, which are mutually perpendicular planes on which the shear stress is zero. The principal stresses are the maximum and minimum normal stresses at the point.

12. How is Mohr's circle constructed, and what are its center and radius for plane stress?

    Plot $\sigma$ (x-axis) vs $\tau$ (y-axis). Center $C = \left(\dfrac{\sigma_x+\sigma_y}{2},\,0\right)$ and radius $R = \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}$. A rotation of $\theta$ in the physical plane corresponds to $2\theta$ on the circle.

13. On Mohr's circle, how do the principal stresses and maximum shear stress appear geometrically?

    Principal stresses are the points where the circle crosses the $\sigma$-axis: $\sigma_{1,2} = C \pm R$. The maximum shear stress equals the radius $R$ and occurs at the top/bottom of the circle, $90^\circ$ (i.e. $45^\circ$ physical) from the principal points.

14. For a 3D stress state with principal stresses $\sigma_1 \geq \sigma_2 \geq \sigma_3$, what is the absolute maximum shear stress?

    $$\tau_{abs,max} = \frac{\sigma_1-\sigma_3}{2}$$ It corresponds to the largest of the three Mohr's circles drawn between each pair of principal stresses.

15. What are the three principal stress invariants $I_1, I_2, I_3$ of the 3D stress tensor?

    $I_1=\sigma_x+\sigma_y+\sigma_z$; $I_2=\sigma_x\sigma_y+\sigma_y\sigma_z+\sigma_z\sigma_x-\tau_{xy}^2-\tau_{yz}^2-\tau_{zx}^2$; $I_3=\det(\boldsymbol{\sigma})$. They are independent of the chosen coordinate axes.

16. What is the characteristic equation whose roots are the principal stresses?

    $$\sigma^3 - I_1\sigma^2 + I_2\sigma - I_3 = 0$$ The three real roots are the principal stresses $\sigma_1,\sigma_2,\sigma_3$.

17. Write the 3D generalized Hooke's law for normal strain $\varepsilon_x$ in an isotropic material.

    $$\varepsilon_x = \frac{1}{E}\left[\sigma_x - \nu(\sigma_y+\sigma_z)\right]$$ with analogous expressions for $\varepsilon_y$ and $\varepsilon_z$ by cyclic permutation.

18. Write the 3D Hooke's law relations for the shear strains.

    $$\gamma_{xy}=\frac{\tau_{xy}}{G},\quad \gamma_{yz}=\frac{\tau_{yz}}{G},\quad \gamma_{zx}=\frac{\tau_{zx}}{G}$$ where $G=\dfrac{E}{2(1+\nu)}$.

19. Distinguish the conditions defining plane stress and plane strain.

    Plane stress: $\sigma_z=\tau_{xz}=\tau_{yz}=0$ (thin plate loaded in-plane); $\varepsilon_z \neq 0$. Plane strain: $\varepsilon_z=\gamma_{xz}=\gamma_{yz}=0$ (long body, constrained ends); $\sigma_z \neq 0$, with $\sigma_z=\nu(\sigma_x+\sigma_y)$.

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