The GATE Grind

GATE 2026 AE – Question 26

Structures · Strength of materials: Transformation of stress and strain, Mohr's circle, combined loading and failure criteria · 1 mark · Multiple select

The state of stress at a point in a 2-D body, in the x-y Cartesian coordinate system, is represented in matrix form as $[\sigma]$. The transformation matrix $[Q]$ rotates the coordinate system to a new x'-y' Cartesian coordinate system. Select the CORRECT option(s) that represent(s) the state of stress in the new coordinate system.

  1. $[Q][\sigma][Q]^T$
  2. $[Q][\sigma][Q]^{-1}$
  3. $([Q]^{-1})^T[\sigma][Q]^T$
  4. $[Q]^{-1}[\sigma][Q]$

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Correct answer: (A) $[Q][\sigma][Q]^T$; (B) $[Q][\sigma][Q]^{-1}$; (C) $([Q]^{-1})^T[\sigma][Q]^T$

Explanation

A rotation matrix is orthogonal, so $Q^{-1} = Q^T$. The stress transforms as $\sigma' = Q\sigma Q^T$ (A), which equals $Q\sigma Q^{-1}$ (B) and also $(Q^{-1})^T \sigma Q^T$ (C), since $(Q^{-1})^T = Q$. Option D is the inverse transformation.