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GATE 2026 AE – Question 45

Structures · Flight vehicle structures: torsion, bending and shear of thin-walled sections, loads on aircraft · 2 marks · Multiple select

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?

  1. It satisfies the equilibrium equation
  2. It is zero on the lateral surfaces of the bar
  3. It satisfies the compatibility equation
  4. It does not satisfy the equilibrium equation

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Show answer and explanation

Correct answer: (A) It satisfies the equilibrium equation; (B) It is zero on the lateral surfaces of the bar; (C) It satisfies the compatibility equation

Explanation

The shear stresses are defined as $\tau_{zx} = \frac{\partial \phi}{\partial y}$ and $\tau_{zy} = -\frac{\partial \phi}{\partial x}$, which satisfies the equilibrium equation identically. Compatibility then requires $\nabla^2\phi = -2G\theta$, which $\phi$ must satisfy, and a traction-free lateral surface requires $\phi$ to be constant (taken as zero) on the boundary.