At a point in an elastic material, a direct tensile stress of 70 N/mm² and a direct compressive stress of 50 N/mm² are applied on planes at right angles to each other. If the maximum principal stress in the material is limited to 75 N/mm², what is the maximum shear stress ?
- A65 N/mm²Correct
- B55 N/mm²
- C75 N/mm²
- D25 N/mm²
Solution
Concept
Principal stress and maximum shear stress with an unknown shear
Formula
σ₁ = (σx + σy)/2 + √[((σx − σy)/2)² + τ²]; τmax = √[((σx − σy)/2)² + τ²]
Step-by-step solution
1. Mean stress = (70 − 50)/2 = 10; half difference = 60. 2. σ₁ = 10 + R = 75, so R = 65 N/mm². 3. Maximum shear stress = R = 65 N/mm² (this needs a shear τ = 25 N/mm² on the planes).
Explanation
The radius of the Mohr circle equals the maximum shear stress. Since σ₁ is limited to 75 and the centre of the circle is at 10, the radius must be 65.
Why the other options are wrong
25 N/mm² is the shear stress τ on the planes, not the maximum shear stress. 55 and 75 do not equal the circle radius.
Shortcut
R = σ₁ − centre = 75 − 10 = 65.
Common mistake
Giving τ = 25 instead of the radius.
Correct answer: A) 65 N/mm²