Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.45)**
**1. Intuitive Physical Setup & Given Parameters**:
The problem concerns the Prandtl stress function $\Phi(x, y)$ used in the formulation of the Saint-Venant torsion of prismatic bars with open or arbitrary cross-sections.
We are asked to select all true statements about $\Phi(x, y)$ (MSQ).
**2. Step-by-Step Theory of Elasticity Derivation**:
1. **Definition of Prandtl Stress Function**:
- For pure torsion of a cylinder with axis along $z$, the only non-zero stress components are shear stresses $\tau_{zx}$ and $\tau_{zy}$.
- The Prandtl stress function $\Phi(x, y)$ is defined such that:
\tau_{zx} = \frac{\partial \Phi}{\partial y}, \quad \tau_{zy} = -\frac{\partial \Phi}{\partial x}
2. **Equilibrium Equations**:
- In the absence of body forces, the 3D equilibrium equations reduce to:
\frac{\partial \tau_{zx}}{\partial z} = 0, \quad \frac{\partial \tau_{zy}}{\partial z} = 0
\frac{\partial \tau_{zx}}{\partial x} + \frac{\partial \tau_{zy}}{\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial \Phi}{\partial y}\right) + \frac{\partial}{\partial y}\left(-\frac{\partial \Phi}{\partial x}\right) = \frac{\partial^2 \Phi}{\partial x \partial y} - \frac{\partial^2 \Phi}{\partial y \partial x} = 0
- The equilibrium equation is **identically satisfied** by the definition of $\Phi$. Statement A is **TRUE**, Statement D is **FALSE**.
3. **Compatibility Equation**:
- Compatibility of the shear strains gives the governing Poisson equation:
\nabla^2 \Phi = \frac{\partial^2 \Phi}{\partial x^2} + \frac{\partial^2 \Phi}{\partial y^2} = -2G\theta
where $G$ is shear modulus and $\theta$ is angle of twist per unit length. Statement C is **TRUE**.
4. **Boundary Condition on Lateral Surface**:
- The lateral surface of the cylinder is traction-free ($T_i = 0$):
\tau_{zx} n_x + \tau_{zy} n_y = 0 \implies \frac{\partial \Phi}{\partial y}\frac{dy}{ds} + \left(-\frac{\partial \Phi}{\partial x}\right)\left(-\frac{dx}{ds}\right) = \frac{d\Phi}{ds} = 0
- Thus, $\Phi$ is constant along the boundary curve. For simply connected cross-sections or open sections, this constant is arbitrarily set to zero: $\Phi = 0$ on the boundary. Statement B is **TRUE**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) It satisfies the equilibrium equation**: ✅ **CORRECT**. By definition of stress components via partial derivatives.
- **(B) It is zero on the lateral surfaces of the bar**: ✅ **CORRECT**. Required boundary condition for traction-free lateral surface.
- **(C) It satisfies the compatibility equation**: ✅ **CORRECT**. Leads to $\nabla^2 \Phi = -2G\theta$.
- **(D) It does not satisfy the equilibrium equation**: ❌ **INCORRECT**. Directly contradicts Option A.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A, B, C**
- **Exam Takeaway**: Prandtl's stress function satisfies equilibrium identically, compatibility via $\nabla^2\Phi = -2G\theta$, and traction-free lateral boundaries via $\Phi_{\text{boundary}} = 0$.