Question

A thin plate is loaded in plane stress condition with \( \sigma_{xx} = 110 \) MPa, \( \sigma_{yy} = -50 \) MPa, \( \tau_{xy} = -70 \) MPa. The maximum principal stress in MPa is ................... (round off to nearest integer).

Hint:

When calculating principal stresses, look for numbers that might form a Pythagorean triple (like 3-4-5, 5-12-13, 6-8-10) to simplify the square root calculation. This is a common feature in exam problems to make the arithmetic cleaner. Here, 80 and 60 form a 6-8-10 triple scaled by 10.

Answer 1

Step 1: Understanding the Concept:
Principal stresses are the maximum and minimum normal stresses acting on a body at a point. For a 2D plane stress state, they can be calculated from the stress components (\(\sigma_{xx}, \sigma_{yy}, \tau_{xy}\)) using a standard formula, which is derived from Mohr's circle analysis.
Step 2: Key Formula or Approach:
The principal stresses (\(\sigma_1\) and \(\sigma_2\)) for a plane stress condition are given by the formula: \[ \sigma_{1,2} = \frac{\sigma_{xx} + \sigma_{yy}}{2} \pm \sqrt{\left(\frac{\sigma_{xx} - \sigma_{yy}}{2}\right)^2 + \tau_{xy}^2} \] The maximum principal stress (\(\sigma_1\)) corresponds to the '+' sign in the formula.
Step 3: Detailed Calculation:
Given stress components:
- \( \sigma_{xx} = 110 \) MPa
- \( \sigma_{yy} = -50 \) MPa
- \( \tau_{xy} = -70 \) MPa
First, calculate the average stress (the center of Mohr's circle): \[ \frac{\sigma_{xx} + \sigma_{yy}}{2} = \frac{110 + (-50)}{2} = \frac{60}{2} = 30 \text{ MPa} \] Next, calculate the term inside the square root (the radius of Mohr's circle): \[ \frac{\sigma_{xx} - \sigma_{yy}}{2} = \frac{110 - (-50)}{2} = \frac{160}{2} = 80 \text{ MPa} \] Radius \(R = \sqrt{\left(\frac{\sigma_{xx} - \sigma_{yy}}{2}\right)^2 + \tau_{xy}^2} = \sqrt{(80)^2 + (-70)^2} \) \[ R = \sqrt{6400 + 4900} = \sqrt{11300} \approx 106.3 \text{ MPa} \] Now, calculate the maximum principal stress (\(\sigma_1\)): \[ \sigma_1 = \text{Center} + \text{Radius} \] \[ \sigma_1 = 30 + 106.3 = 136.3 \text{ MPa} \] Step 4: Final Answer:
The maximum principal stress is 136.3 MPa.