GATE AE 2018 • Q62 NAT • 2 Marks Aerospace Structures

Aerospace Structures

Strength of Materials & Elasticity

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

Question Statement:

The components of stress in a body under plane stress condition, in the absence of body forces, is given by: $\sigma_{xx} = Ax^2; \sigma_{yy} = 12x^2 - 6y^2$ and $\sigma_{xy} = 12xy$. The coefficient, $A$, such that the body is in static equilibrium is _________.

Official Answer Key:

-6.0

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2018 Q.62 / AE Q.52)** **1. Intuitive Physical Setup & Given Parameters**: - Stress tensor components under plane stress without body forces: \sigma_{xx} = A x^2 \sigma_{yy} = 12x^2 - 6y^2 \tau_{xy} = \sigma_{xy} = 12xy **2. Step-by-Step Mathematical Derivation**: 1. **Differential Equations of Equilibrium (Plane Stress)**: In the absence of body forces ($F_x = 0, F_y = 0$): \frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \tau_{xy}}{\partial y} = 0 \quad \text{--- (Equation 1)} \frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_{yy}}{\partial y} = 0 \quad \text{--- (Equation 2)} 2. **Checking Equation 2**: \frac{\partial \tau_{xy}}{\partial x} = \frac{\partial}{\partial x}(12xy) = 12y \frac{\partial \sigma_{yy}}{\partial y} = \frac{\partial}{\partial y}(12x^2 - 6y^2) = -12y 12y + (-12y) = 0 \implies \text{Equation 2 is identically satisfied!} 3. **Evaluating Equation 1 for Coefficient $A$**: \frac{\partial \sigma_{xx}}{\partial x} = \frac{\partial}{\partial x}(A x^2) = 2A x \frac{\partial \tau_{xy}}{\partial y} = \frac{\partial}{\partial y}(12xy) = 12x Substituting into Equation 1: 2A x + 12x = 0 (2A + 12) x = 0 \quad \forall x 2A + 12 = 0 \implies 2A = -12 \implies A = -6.0 **3. Comprehensive Options & Distractor Analysis**: - **Question Type**: Numerical Answer Type (NAT). - **Exact Analytical Result**: **-6.0** (Acceptable Range: [-6.0, -6.0]). **4. Official Master Key & Exam Takeaway**: - **Official Answer**: **-6.0** - **Exam Rule**: In plane elasticity equilibrium, differentiate directly and set the divergence $\nabla \cdot \boldsymbol{\sigma} = 0$.
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