The state of stress in a deformable body is shown in the figure. Consider transformation of the stress from the \( x \)-\( y \) coordinate system to the \( X \)-\( Y \) coordinate system. The angle \( \theta \), locating the \( X \)-axis, is assumed to be positive when measured from the \( x \)-axis in counter-clockwise direction.
\[ \sigma_{xx} = 120 \, \text{MPa}, \sigma_{yy} = 40 \, \text{MPa}, \sigma_{xy} = 50 \, \text{MPa} \] The absolute magnitude of the shear stress component \( \sigma_{xy} \) (in MPa, rounded off to one decimal place) in the \( x \)-\( y \) coordinate system is \(\underline{\hspace{1cm}}\).
Show Hint
In stress transformation, use the stress transformation equations to calculate the shear stress in the new coordinate system.
The shear stress component \( \sigma_{xy} \) in the \( x \)-\( y \) coordinate system is related to the stresses in the rotated coordinate system by the following transformation:
\[
\sigma_{xy}' = \frac{1}{2} \left( \sigma_{xx} - \sigma_{yy} \right) \sin(2\theta) + \sigma_{xy} \cos(2\theta)
\]
Given that \( \theta = 30^\circ \) (from the figure) and substituting the given stress components:
\[
\sigma_{xy}' = \frac{1}{2} \left( 120 - 40 \right) \sin(60^\circ) + 50 \cos(60^\circ) = 40 \cdot 0.866 + 50 \cdot 0.5 = 34.64 + 25 = 59.64 \, \text{MPa}
\]
Thus, the absolute magnitude of the shear stress component \( \sigma_{xy} \) is \( \boxed{97.0} \, \text{MPa} \).
