GATE AE 2026 • Q60 NAT • 2 Marks Aerospace Structures

Aerospace Structures

Strength of Materials & Elasticity

Subtopic: Stress and Strain Tensors & Generalized Hooke's Law

Question Statement:

A stepped cantilever beam, made of a material having Young's modulus E = 200 GPa, is shown in the figure below. The length and the moment of inertia of the beam from point A to B are L1 = 100 mm and I1 = 100 mm4, respectively. The length and the moment of inertia of the beam from point B to C are L^2 = 100 mm and I^2 = 700 mm4, respectively. A shear force P = 30 N is applied at point A of the beam. The magnitude of the deflection of the beam at point A is ________ mm (rounded off to 1 decimal place).
Question figure for GATE AE 2026 Q60

Scientific schematic diagram reproduced from official IIT paper

Official Answer Key:

1.0

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2026 Q.60)** **1. Intuitive Physical Setup & Given Parameters**: Stepped cantilever beam fixed at $C$: - Material Young's modulus: $E = 200\text{ GPa} = 200 \times 10^3\text{ N/mm}^2$ - Segment $AB$ (tip to step): $L_1 = 100\text{ mm}$, $I_1 = 100\text{ mm}^4$ - Segment $BC$ (step to wall): $L_2 = 100\text{ mm}$, $I_2 = 700\text{ mm}^4$ - Point load applied at tip $A$: $P = 30\text{ N}$ downwards. We need to find the magnitude of the tip deflection $\delta_A$ in $\text{mm}$. **2. Step-by-Step Structural Mechanics Derivation**: 1. **Castigliano's Theorem / Moment-Area Method**: Bending moment at distance $x$ from free tip $A$: M(x) = P x 2. **Deflection via Strain Energy**: \delta_A = \int_0^{L_1 + L_2} \frac{M(x)}{E I(x)} \frac{\partial M}{\partial P} dx = \frac{P}{E} \left[ \int_0^{L_1} \frac{x^2}{I_1} dx + \int_{L_1}^{L_1 + L_2} \frac{x^2}{I_2} dx \right] 3. **Evaluate the Integrals**: - First integral over segment $AB$ ($0 \le x \le 100\text{ mm}$): \int_0^{100} \frac{x^2}{100} dx = \frac{1}{100} \left[ \frac{100^3}{3} \right] = \frac{10^6}{300} = \frac{10000}{3}\text{ mm}^3 - Second integral over segment $BC$ ($100 \le x \le 200\text{ mm}$): \int_{100}^{200} \frac{x^2}{700} dx = \frac{1}{700} \left[ \frac{200^3 - 100^3}{3} \right] = \frac{8 \times 10^6 - 1 \times 10^6}{2100} = \frac{7 \times 10^6}{2100} = \frac{10000}{3}\text{ mm}^3 4. **Sum of Integrals**: \text{Total Bracket} = \frac{10000}{3} + \frac{10000}{3} = \frac{20000}{3}\text{ mm}^3 5. **Compute Tip Deflection $\delta_A$**: \delta_A = \frac{P}{E} \times \frac{20000}{3} = \frac{30\text{ N}}{200 \times 10^3\text{ N/mm}^2} \times \frac{20000}{3}\text{ mm}^3 \delta_A = (1.5 \times 10^{-4}) \times \frac{20000}{3} = 1.5 \times 10^{-4} \times 6666.67 = 1.0\text{ mm} **3. Comprehensive Options & Distractor Analysis**: - **Evaluation Criteria**: Numerical Answer Type (NAT). - **Exact Analytical Result**: $\delta_A = 1.0\text{ mm}$. - **Rounding Requirement**: One decimal place $\implies \mathbf{1.0}$. **4. Official Master Key & Exam Takeaway**: - **Official Answer Key**: **1.0** (Acceptable Range: **0.9 to 1.1**) - **Exam Takeaway**: For stepped beams, integrate $\int \frac{M^2}{2EI} dx$ piecewise. Notice the elegant symmetry where $\frac{L_1^3}{3I_1} = \frac{10^4}{3}$ and $\frac{L_2^3 + 3L_1 L_2^2 + 3L_1^2 L_2}{3I_2} = \frac{10^4}{3}$, summing to exactly $1.0\text{ mm}$.
← Previous (Q59) All 2026 Questions Next (Q61) →

Related Strength of Materials & Elasticity Solved Questions

View All Strength of Materials & Elasticity PYQs →
GATE AE 2026 Q12 MCQ • 1M

The Fourier series representation of a square wave is shown in the figure below. The fluctuations seen near x = ±1 are named after which one of the following scientists?
GATE AE 2026 Q40 MCQ • 2M

Consider a unit square body as shown in the figure below. The body is subjected to the deformation field u = -ay and v = ax, where 'a' is a constant. Due to the application of this deformation field, the body undergoes _______ in the x-y plane.
GATE AE 2026 Q45 MSQ • 2M

The deformation of an open-section bar subjected to pure torsion can be solved by choosing an appropriate Prandtl stress function. Which of the following statements is/are true about the Prandtl stress function?
GATE AE 2026 Q57 NAT • 2M

WA and WB are the respective maximum take-off weights of an aircraft for two ambient air conditions given below. Condition A: p = 1 bar, T = 50 oC; Condition B: p = 0.66 bar, T = -30 oC If all other parameters relevant for take-off are kept the same in these two conditions, the ratio WB/WA is ________ (rounded off to 3 decimal places).

Practice This in Real CBT Exam Interface

Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.

Launch CBT Mock Test