Question

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.

• A. \([Q][\sigma][Q]^{T}\)
• B. \([Q][\sigma][Q]^{-1}\)
• C. \(([Q]^{-1})^{T}[\sigma][Q]^{T}\)
• D. \([Q]^{-1}[\sigma][Q]\)

Hint:

Stress transforms as \([\sigma']=[Q][\sigma][Q]^{T}\); since \([Q]\) is orthogonal, \([Q]^{-1}=[Q]^{T}\), so any option that algebraically reduces to this form is also correct.

Answer 1

The Correct Option is A, B, C

Step 1: Recall the Stress Transformation Rule.
For a second order tensor like stress, the standard rule for transforming its matrix representation from an old \((x,y)\) axis system to a new, rotated \((x',y')\) axis system is
\[ [\sigma'] = [Q][\sigma][Q]^{T} \]
where \([Q]\) is the rotation (direction cosine) matrix taking the old axes to the new axes. This makes option (A) correct by definition.

Step 2: Use the Key Property of a Rotation Matrix.
A rotation or transformation matrix between two Cartesian coordinate systems is always orthogonal, meaning
\[ [Q]^{T} = [Q]^{-1} \]
This is because a rotation preserves lengths and angles, so \([Q]^{T}[Q] = [I]\).

Step 3: Check Option (B).
Option (B) is \([Q][\sigma][Q]^{-1}\). Substituting \([Q]^{-1} = [Q]^{T}\) from Step 2 turns this expression directly into \([Q][\sigma][Q]^{T}\), which is identical to option (A). So option (B) is just another way of writing the same correct transformation, and it is also correct.

Step 4: Check Option (C).
Option (C) is \(([Q]^{-1})^{T}[\sigma][Q]^{T}\). Using \([Q]^{-1} = [Q]^{T}\) again, \(([Q]^{-1})^{T} = ([Q]^{T})^{T} = [Q]\), the transpose of a transpose returns the original matrix. Substituting this back gives \([Q][\sigma][Q]^{T}\), again exactly option (A). So option (C) is also correct, just written in a disguised form.

Step 5: Check Option (D).
Option (D) is \([Q]^{-1}[\sigma][Q]\). Substituting \([Q]^{-1}=[Q]^{T}\) gives \([Q]^{T}[\sigma][Q]\). This is generally NOT the same as \([Q][\sigma][Q]^{T}\) from Step 1, the two rotation matrices are on the wrong sides, unless \([Q]\) happens to be symmetric, which a general rotation matrix is not. So option (D) is the wrong transformation and does not represent the stress in the new coordinate system in general.

Final Answer:
Options (A), (B) and (C) all reduce to the same correct transformation once \([Q]^{-1}=[Q]^{T}\) is used; only (D) is wrong. \[ \boxed{\text{(A), (B), (C)}} \]