Subtopic: Euler Buckling of Columns & Energy Methods
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 L2 = 100
mm and I2 = 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).
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 $1.0\text{ mm}$.
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