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GATE 2023 ME – Question 26

Machine Design · Design for static and dynamic loading, failure theories · 1 mark · Multiple select

The principal stresses at a point P in a solid are 70 MPa, −70 MPa and 0. The yield stress of the material is 100 MPa. Which prediction(s) about material failure at P is/are CORRECT?

  1. Maximum normal stress theory predicts that the material fails
  2. Maximum shear stress theory predicts that the material fails
  3. Maximum normal stress theory predicts that the material does not fail
  4. Maximum shear stress theory predicts that the material does not fail

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Correct answer: (B) Maximum shear stress theory predicts that the material fails; (C) Maximum normal stress theory predicts that the material does not fail

Explanation

The maximum normal stress theory compares the largest principal stress, 70 MPa, with the yield stress of 100 MPa. As $70 < 100$, there is no failure (C). The maximum shear stress theory (Tresca) uses $\tau_{max} = \frac{\sigma_1 - \sigma_3}{2} = \frac{70 - (-70)}{2} = 70$ MPa and compares it with $\frac{\sigma_Y}{2} = 50$ MPa. Since $70 > 50$, the material fails (B).