Question

The two-dimensional state of stress in an infinitesimal element is given by: \[ \sigma_{xx} = 800 \, \text{MPa}, \sigma_{yy} = 0 \, \text{MPa}, \sigma_{xy} = 300 \, \text{MPa} \] 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:

Maximum shear stress in 2D = radius of Mohr’s circle = \(\sqrt{((\sigma_x - \sigma_y)/2)^2 + \tau_{xy}^2}\).

Answer 1

The Correct Option is B

Step 1: Principal stresses formula.
The principal stresses are: \[ \sigma_{1,2} = \frac{\sigma_{xx} + \sigma_{yy}}{2} \pm \sqrt{\left(\frac{\sigma_{xx} - \sigma_{yy}}{2}\right)^2 + \sigma_{xy}^2} \]

Step 2: Substitute values.
\[ \sigma_{avg} = \frac{800 + 0}{2} = 400 \] \[ R = \sqrt{(400)^2 + (300)^2} = \sqrt{160000 + 90000} = \sqrt{250000} = 500 \] \[ \sigma_1 = 400 + 500 = 900, \sigma_2 = 400 - 500 = -100 \]

Step 3: Maximum shear stress.
\[ \tau_{max} = \frac{\sigma_1 - \sigma_2}{2} = \frac{900 - (-100)}{2} = \frac{1000}{2} = 500 \] Wait — note maximum shear is the radius of Mohr’s circle: \[ \tau_{max} = R = 500 \, \text{MPa} \] But since the question asks among options, and correct calculation shows **500 MPa**, the right option is (A). Final Answer:
\[ \boxed{500 \, \text{MPa}} \]