For a general two-dimensional stress system, the maximum principal stress σ₁ is (where σx and σy are direct stresses on mutually perpendicular planes and τxy is shear stress on planes.)
- A(σx+σy)/2 − √[((σx+σy)/2)² + τxy²]
- B(σx+σy)/2 + √[((σx−σy)/2)² − τxy²]
- C(σx+σy)/2 − √[((σx+σy)/2)² − τxy²]
- D(σx+σy)/2 + √[((σx−σy)/2)² + τxy²]Correct
Solution
Concept
Principal stresses in plane stress
Formula
σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]
Step-by-step solution
1. The centre of the Mohr circle is at (σx + σy)/2 and its radius is √[((σx − σy)/2)² + τxy²]. 2. The maximum principal stress is centre + radius.
Explanation
The major principal stress is the average normal stress plus the radius of the Mohr circle.
Why the other options are wrong
Options (a) and (c) give the minimum principal stress or use the wrong terms inside the root. Option (b) has a minus sign under the root.
Shortcut
σ₁ = centre + radius.
Common mistake
Using the minus sign (minor principal stress).
Correct answer: D) (σx+σy)/2 + √[((σx−σy)/2)² + τxy²]