Question

For a plane strain problem in the \( xy \) plane, it is necessary that:

• A. normal stress \( \sigma_z \) is zero.
• B. normal strain \( \epsilon_z \) is zero.
• C. both the normal stresses \( \sigma_x \) and \( \sigma_y \) are zero.
• D. shear strain \( \gamma_{xy} \) is equal to \( \frac{\sigma_x - \sigma_y}{2} \).

Hint:

1. In plane strain problems, focus on strains in the out-of-plane direction (\( z \)-axis), which are always zero.
2. Stress components \( \sigma_z \), \( \sigma_x \), and \( \sigma_y \) can be nonzero depending on the boundary conditions.
3. Use compatibility equations and material constitutive relationships for solving.

Answer 1

The Correct Option is B

Step 1: Definition of plane strain. In a plane strain problem, deformation occurs in a two-dimensional plane (e.g., the \( xy \) plane), while the strain in the perpendicular direction (the \( z \)-axis) is assumed to be zero. This condition is expressed mathematically as: \[ \epsilon_z = 0. \] Step 2: Analyze the options. Option (A): \( \sigma_z = 0 \) — Incorrect. The stress in the \( z \)-direction (\( \sigma_z \)) can exist in a plane strain condition. Option (B): \( \epsilon_z = 0 \) — Correct. This is the defining characteristic of plane strain. Option (C): \( \sigma_x = 0 \) and \( \sigma_y = 0 \) — Incorrect. Plane strain does not require normal stresses to vanish. Option (D): \( \gamma_{xy} = \frac{\sigma_x - \sigma_y}{2} \) — Incorrect. Shear strain does not have this relationship in plane strain problems. Conclusion: For a plane strain problem in the \( xy \) plane, it is necessary that \( \mathbf{\epsilon_z = 0} \), corresponding to option \( \mathbf{(B)} \).