Question

The two-dimensional state of stress, in an infinitesimal element, is given by \[ \sigma_{xx} = 800 \, {MPa}, \sigma_{xy} = 300 \, {MPa}, \sigma_{yy} = 0 \, {MPa}. \] Which one of the following options is the maximum shear stress (in MPa) in the element?

• A. 500
• B. 400
• C. 800
• D. 300

Hint:

To calculate the maximum shear stress in a two-dimensional state of stress, use the formula: \[ \tau_{{max}} = \frac{1}{2} \sqrt{ (\sigma_{xx} - \sigma_{yy})^2 + 4\sigma_{xy}^2 } \] This formula is derived from Mohr's circle and can be applied directly to any state of stress.

Answer 1

The Correct Option is A

Step 1: The maximum shear stress in a two-dimensional stress state can be calculated using the following formula: \[ \tau_{{max}} = \frac{1}{2} \sqrt{ (\sigma_{xx} - \sigma_{yy})^2 + 4\sigma_{xy}^2 } \] where \( \sigma_{xx} \), \( \sigma_{yy} \), and \( \sigma_{xy} \) are the normal and shear stresses.
Step 2: Substituting the given values into the formula: \[ \tau_{{max}} = \frac{1}{2} \sqrt{ (800 - 0)^2 + 4(300)^2 } \] \[ \tau_{{max}} = \frac{1}{2} \sqrt{ 640000 + 360000 } \] \[ \tau_{{max}} = \frac{1}{2} \sqrt{ 1000000 } \] \[ \tau_{{max}} = \frac{1}{2} \times 1000 = 500 \, {MPa} \] Step 3: Therefore, the maximum shear stress is \( 500 \, {MPa} \), which corresponds to option (A).