Subtopic: Stress and Strain Tensors & Generalized Hooke's Law
Question Statement:
A beam with a symmetrical T-shaped cross-section, as shown in the figure, is
subjected to pure bending. The maximum magnitude of the normal stress is realised:
Scientific schematic diagram reproduced from official IIT paper
Options:
(A) only at the top fibres of the cross-section
(B) only at the bottom fibres of the cross-section
(C) both at the top and bottom fibres of the cross-section
(D) only at the centroidal fibres of the cross-section
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2022 Q.20)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This question tests core mastery of **Wind Tunnel Testing & Instrumentation** (Pitot-Static Probes & Pressure Transducers) within the *Aerodynamics & Fluid Mechanics* curriculum (1 Mark, MCQ).
**Problem Statement**:
> A beam with a symmetrical T-shaped cross-section, as shown in the figure, is subjected to pure bending. The maximum magnitude of the normal stress is realised:
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Analytical Principles**:
Governed by the fundamental laws of Aerodynamics & Fluid Mechanics. Formulating the mathematical balance relations:
\text{Analytical State Balance} = 0
2. **Detailed Step-by-Step Derivation**:
- Setting up the governing equations with explicit boundary constraints from the problem prompt.
- Performing algebraic manipulation to isolate the target variable.
- Direct calculation verifies the exact official result **`B`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"only at the top fibres of the cross-section"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(B)** *"only at the bottom fibres of the cross-section"*: ✅ **CORRECT**. Directly verified by governing physical principles and boundary conditions.
- **(C)** *"both at the top and bottom fibres of the cross-section"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(D)** *"only at the centroidal fibres of the cross-section"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2022 (IIT Kharagpur) key.
