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GATE 2025 AE – Question 28

Structures · Special topics: Equilibrium and compatibility equations in two-dimensional elasticity · 1 mark · Multiple choice

A stress field is given by $\sigma_{xx} = \sigma_{zz} = C_1 y$; $\sigma_{yy} = C_2 y$; $\tau_{xy} = \tau_{yz} = \tau_{zx} = 0$, where $C_1$ and $C_2$ are non-zero constants. If the stress field satisfies equilibrium, which one of the following options is correct?

  1. There is no body force per unit volume.
  2. There is a constant body force per unit volume in the y-direction.
  3. The body force per unit volume varies linearly in the y-direction.
  4. The direction of the body force per unit volume depends on the value of $C_1$.

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Correct answer: (B) There is a constant body force per unit volume in the y-direction.

Explanation

The equilibrium equations are $\frac{\partial\sigma_{xx}}{\partial x} + \frac{\partial\tau_{xy}}{\partial y} + \frac{\partial\tau_{xz}}{\partial z} + b_x = 0$ and similar ones. All the derivatives with respect to $x$ and $z$ are zero, so $b_x = b_z = 0$. In the $y$ direction $\frac{\partial\sigma_{yy}}{\partial y} + b_y = 0$ gives $b_y = -C_2$, a constant.