GATE Aerospace Engineering Structures Flashcards

24 sample cards from the Structures deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. What is the degree of static indeterminacy of a beam, and how is it computed?

   Degree of static indeterminacy $=$ (number of unknown reactions) $-$ (number of available equilibrium equations). For a 2D beam, equilibrium gives 3 equations, so $\text{DSI}=R-3$ (plus condition equations for internal hinges).

2. Compare statically determinate and indeterminate structures.

   Determinate: reactions/internal forces found from equilibrium alone; no stress from temperature change, settlement, or fabrication error; failure of one member can cause collapse. Indeterminate: needs compatibility + material law; develops self-stresses from settlement/thermal effects; has redundancy/load redistribution.

3. State Euler's critical buckling load for a pin-ended column.

   $$P_{cr}=\frac{\pi^2 EI}{L^2}$$ where $EI$ is the flexural rigidity and $L$ the length. Buckling is independent of yield strength (elastic flexural buckling).

4. Write Euler's buckling load using effective length, and give effective-length factors for common end conditions.

   $$P_{cr}=\frac{\pi^2 EI}{(KL)^2}$$ $K=1.0$ pinned–pinned; $K=0.5$ fixed–fixed; $K=0.7$ fixed–pinned; $K=2.0$ fixed–free (cantilever).

5. Define slenderness ratio and critical (buckling) stress for columns.

   Slenderness ratio $=\dfrac{KL}{r}$ with radius of gyration $r=\sqrt{I/A}$. Critical stress $$\sigma_{cr}=\frac{\pi^2 E}{(KL/r)^2}$$ Euler's formula is valid only when $\sigma_{cr}\le\sigma_y$ (slender columns).

6. List the four primary external load types acting on an aircraft.

   Aerodynamic loads (lift, drag, control surface), inertial/gravity loads (weight, maneuver/gust accelerations), propulsion loads (thrust), and ground/landing loads (impact, taxi). Also pressurization (cabin) and thermal loads.

7. What is a V-n (flight envelope) diagram and what are limit and ultimate load factors?

   A V-n diagram plots load factor $n=L/W$ versus airspeed, bounding the structural/aerodynamic operating envelope. The limit load is the max expected in service; the ultimate load $=1.5\times$ limit load (factor of safety 1.5), which the structure must withstand without failure.

8. Describe the characteristics of semi-monocoque aircraft structures.

   Skin carries shear and pressurization; longitudinal stringers/longerons carry axial (bending) loads; transverse frames/bulkheads maintain shape and distribute loads; ribs in wings maintain aerofoil shape and transfer skin loads to spars. Loads are shared between skin and stiffeners.

9. Name key desirable properties of aircraft structural materials and a typical example.

   High specific strength ($\sigma/\rho$) and specific stiffness ($E/\rho$), fatigue and corrosion resistance, fracture toughness, and damage tolerance. Examples: aluminium alloys (2024, 7075), titanium alloys, and carbon-fiber composites.

10. Give the torsion formula for a circular shaft.

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

11. State the Bredt–Batho formula for shear flow in a single-cell thin-walled closed section under torque.

    $$q=\frac{T}{2A_m}$$ where $q=\tau t$ is the (constant) shear flow and $A_m$ is the area enclosed by the wall midline. Shear stress $\tau=\dfrac{T}{2A_m t}$ is largest where thickness $t$ is smallest.

12. Write the angle of twist per unit length for a single-cell closed thin-walled tube (Bredt–Batho).

    $$\frac{d\theta}{dz}=\frac{T}{4A_m^2 G}\oint \frac{ds}{t}$$ where the line integral is taken around the wall midline.

13. Give the flexure formula and the transverse shear stress formula for beam bending.

    Bending stress: $\sigma=\dfrac{My}{I}$. Transverse shear stress: $\tau=\dfrac{VQ}{Ib}$, where $V$ = shear force, $Q$ = first moment of the area beyond the level, $I$ = second moment of area, $b$ = width at that level.

14. What is the shear center of a thin-walled open section?

    The shear center is the point through which a transverse load must act to produce bending without twisting. For open sections (e.g. channels) it generally lies outside the wall; for sections with two axes of symmetry it coincides with the centroid.

15. Write the equation of motion and natural frequency of an undamped SDOF system in free vibration.

    $$m\ddot{x}+kx=0\quad\Rightarrow\quad \omega_n=\sqrt{\frac{k}{m}}$$ The motion is simple harmonic, $x(t)=A\cos\omega_n t+B\sin\omega_n t$, with period $T=\dfrac{2\pi}{\omega_n}$.

16. Define the damping ratio and classify under-, critically-, and over-damped SDOF systems.

    $\zeta=\dfrac{c}{c_c}$ with critical damping $c_c=2\sqrt{km}=2m\omega_n$. Underdamped $\zeta<1$ (oscillatory decay); critically damped $\zeta=1$ (fastest non-oscillatory return); overdamped $\zeta>1$ (slow non-oscillatory return).

17. Give the damped natural frequency of an underdamped SDOF system.

    $$\omega_d=\omega_n\sqrt{1-\zeta^2}$$ Free response: $x(t)=e^{-\zeta\omega_n t}\left(A\cos\omega_d t+B\sin\omega_d t\right)$. The amplitude decays exponentially.

18. For a harmonically forced damped SDOF system, write the steady-state amplitude and dynamic magnification factor.

    With $F(t)=F_0\cos\omega t$ and $r=\omega/\omega_n$: $$X=\frac{F_0/k}{\sqrt{(1-r^2)^2+(2\zeta r)^2}},\qquad \text{DMF}=\frac{1}{\sqrt{(1-r^2)^2+(2\zeta r)^2}}$$

19. What is resonance, and at what frequency does the peak amplitude occur for a damped forced SDOF system?

    Resonance is the large-amplitude response when forcing frequency approaches $\omega_n$. Peak amplitude occurs at $\omega=\omega_n\sqrt{1-2\zeta^2}$ (for $\zeta<1/\sqrt{2}$). At resonance the phase angle is $90^\circ$ and amplitude is limited only by damping.

20. Define logarithmic decrement and relate it to the damping ratio.

    Logarithmic decrement $\delta=\ln\dfrac{x_n}{x_{n+1}}=\dfrac{2\pi\zeta}{\sqrt{1-\zeta^2}}$. For light damping, $\delta\approx 2\pi\zeta$. It is found experimentally from the decay of successive free-vibration peaks.

21. For an undamped 2-DOF system, write the matrix equation of motion and the eigenvalue problem for natural frequencies.

    $$[M]\{\ddot{x}\}+[K]\{x\}=0$$ Assuming $\{x\}=\{\phi\}e^{i\omega t}$ gives $$\left([K]-\omega^2[M]\right)\{\phi\}=0,\quad \det\!\left([K]-\omega^2[M]\right)=0$$ The frequency equation yields two natural frequencies $\omega_1,\omega_2$.

22. What are mode shapes and normal modes in a 2-DOF undamped system?

    Each natural frequency $\omega_i$ has an eigenvector $\{\phi\}_i$ (mode shape) giving the relative amplitudes of the masses. In a normal mode all coordinates oscillate harmonically at the same frequency, passing through equilibrium together. A general motion is a linear superposition of the two normal modes.

23. What is a node in a 2-DOF (or continuous) vibration mode shape?

    A node is a point that remains stationary during a particular mode of vibration. In the higher (out-of-phase) mode of a 2-DOF system the two masses move in opposite directions, and a node lies between them where the amplitude is zero.

24. Compare bending stress distribution and shear stress distribution across a beam cross-section.

    Bending stress $\sigma=My/I$ varies linearly with distance $y$ from the neutral axis (max at extreme fibers, zero at neutral axis). Transverse shear stress $\tau=VQ/Ib$ is zero at the extreme fibers and maximum at the neutral axis (parabolic for a rectangle, where $\tau_{max}=1.5\,V/A$).

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