GATE Aerospace Engineering Structures Syllabus

In plane strain, what is the out-of-plane stress $\sigma_z$?

$$\sigma_z = \nu(\sigma_x+\sigma_y)$$ This stress is required to enforce $\varepsilon_z=0$.

State the Maximum Principal (Normal) Stress theory (Rankine) of failure.

Failure occurs when the maximum principal stress reaches the material's yield (or ultimate) strength: $$\sigma_1 = \sigma_y$$ Best suited to brittle materials; unsafe for ductile materials under shear.

State the Maximum Shear Stress (Tresca) yield criterion.

Yielding begins when the maximum shear stress equals that at yield in uniaxial tension: $$\tau_{max}=\frac{\sigma_1-\sigma_3}{2}=\frac{\sigma_y}{2}\quad\Rightarrow\quad \sigma_1-\sigma_3=\sigma_y$$

State the Maximum Distortion Energy (von Mises) yield criterion in terms of principal stresses.

Yielding occurs when $$\frac{1}{2}\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right]=\sigma_y^2$$

Write the von Mises equivalent stress for a general 3D stress state in Cartesian components.

$$\sigma_{vm}=\sqrt{\tfrac{1}{2}\left[(\sigma_x-\sigma_y)^2+(\sigma_y-\sigma_z)^2+(\sigma_z-\sigma_x)^2\right]+3(\tau_{xy}^2+\tau_{yz}^2+\tau_{zx}^2)}$$ Yield occurs when $\sigma_{vm}=\sigma_y$.

Compare Tresca and von Mises predictions in pure shear.

In pure shear, Tresca predicts yielding at $\tau_y=0.5\,\sigma_y$ while von Mises predicts $\tau_y=\dfrac{\sigma_y}{\sqrt{3}}\approx 0.577\,\sigma_y$. Tresca is more conservative; von Mises agrees better with experiment for ductile metals.

Geometrically, how do the Tresca and von Mises yield surfaces appear in 2D principal-stress space?

Von Mises is an ellipse; Tresca is a hexagon inscribed within (touching) the von Mises ellipse at six points. Tresca always lies inside or on the von Mises locus, so it predicts yield at equal or lower loads (more conservative).

Give the general expression for strain energy stored in a linearly elastic 3D body.

$$U=\frac{1}{2}\int_V (\sigma_x\varepsilon_x+\sigma_y\varepsilon_y+\sigma_z\varepsilon_z+\tau_{xy}\gamma_{xy}+\tau_{yz}\gamma_{yz}+\tau_{zx}\gamma_{zx})\,dV$$

Write the strain energy stored in an axially loaded bar, in a beam under bending, and in a shaft under torsion.

Axial: $U=\displaystyle\int \frac{N^2}{2AE}\,dx$. Bending: $U=\displaystyle\int \frac{M^2}{2EI}\,dx$. Torsion: $U=\displaystyle\int \frac{T^2}{2GJ}\,dx$.

What is the strain energy density (per unit volume) for a uniaxial stress state?

$$u=\frac{1}{2}\sigma\varepsilon=\frac{\sigma^2}{2E}=\frac{1}{2}E\varepsilon^2$$ This is the area under the linear stress–strain curve.

State Castigliano's second theorem and its use.

For a linearly elastic structure, the deflection at the point and direction of an applied load $P_i$ equals the partial derivative of total strain energy with respect to that load: $$\delta_i=\frac{\partial U}{\partial P_i}$$ Likewise rotation $\theta_i=\dfrac{\partial U}{\partial M_i}$.

What is the 'dummy (fictitious) load' technique in Castigliano's method?

To find the deflection where no real load acts, apply a fictitious load $Q$ at that point, compute $U$ including $Q$, take $\delta=\dfrac{\partial U}{\partial Q}$, then set $Q=0$. It gives the deflection in the direction of $Q$.

State Castigliano's first theorem.

For an elastic structure expressed in terms of displacements, the force at a point equals the partial derivative of strain energy with respect to the corresponding displacement: $$P_i=\frac{\partial U}{\partial \delta_i}$$

State the condition of static determinacy for a plane (2D) pin-jointed truss.

With $m$ members, $j$ joints, and $r$ reaction components: determinate if $m+r=2j$. If $m+r>2j$ it is statically indeterminate (redundant); if $m+r<2j$ it is a mechanism (unstable).

State the static determinacy condition for a space (3D) truss.

For a space truss: $m+r=3j$ for static determinacy, where $m$ = members, $r$ = reactions, $j$ = joints. $m+r>3j$ indicates indeterminacy.

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).

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.

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).

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).

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).

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.

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.

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.

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.

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 $L$.

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.

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.

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.

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.

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 $T=\dfrac{2\pi}{\omega_n}$.

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).

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.

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

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.

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.

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 $\omega_1,\omega_2$.

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.

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.

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 $\tau_{max}=1.5\,V/A$).