Question

A point in a structural member is subjected to a plane stress state where \(\sigma_x=80\ \text{MPa}\), \(\sigma_y=20\ \text{MPa}\), and \(\tau_{xy}=40\ \text{MPa}\). What is the maximum principal stress?

• A. \(60\ \text{MPa}\)
• B. \(100\ \text{MPa}\)
• C. \(120\ \text{MPa}\)
• D. \(200\ \text{MPa}\)

Hint:

For plane stress, \[ \boxed{ \sigma_{1,2} = \frac{\sigma_x+\sigma_y}{2} \pm \sqrt{ \left(\frac{\sigma_x-\sigma_y}{2}\right)^2 +\tau_{xy}^{\,2} }. } \]

Answer 1

The Correct Option is B

Step 1: Recall the principal stress formula. The principal stresses are \[ \boxed{ \sigma_{1,2} = \frac{\sigma_x+\sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^{\,2}}. } \]

Step 2: Substitute the given values. Given, \[ \sigma_x=80\ \text{MPa}, \] \[ \sigma_y=20\ \text{MPa}, \] \[ \tau_{xy}=40\ \text{MPa}. \] Hence, \[ \frac{\sigma_x+\sigma_y}{2} = \frac{80+20}{2} = 50\ \text{MPa}, \] and \[ \sqrt{\left(\frac{80-20}{2}\right)^2+40^2} = \sqrt{30^2+40^2} = \sqrt{2500} = 50\ \text{MPa}. \] Therefore, \[ \sigma_1 = 50+50 = 100\ \text{MPa}. \] Hence, \[ \boxed{100\ \text{MPa}} \] is the correct answer. Thus, \[ \boxed{(B)} \] is the correct answer.