Principal stresses at a point in plane stressed element are σx = σy = 500 kg/cm². Normal stress on the plane inclined at 45° to x-axis will be
- A0
- B500 kg/cm²Correct
- C707 kg/cm²
- D1000 kg/cm²
Solution
Concept
Normal stress calculation under equal principal stresses
Formula
σn = (σx + σy)/2 + ((σx - σy)/2)cos(2θ)
Step-by-step solution
1. Normal stress on an inclined plane at angle θ is given by σθ = (σx + σy)/2 + ((σx - σy)/2)cos(2θ) + τxysin(2θ). 2. Given principal stresses σx = 500 kg/cm², σy = 500 kg/cm², and shear stress τxy = 0 kg/cm². 3. Substituting θ = 45°: σ45 = (500 + 500)/2 + ((500 - 500)/2)·cos(90°) = 500 kg/cm². Answer: B
Explanation
When equal normal stresses act along two perpendicular axes, the stress state represents a hydrostatic stress condition. Under this state, the normal stress on any plane remains constant regardless of the angle of inclination. Therefore, the normal stress on a plane inclined at 45 degrees to the x-axis is also equal to 500 kg/cm².
Why the other options are wrong
A) Normal stress is not zero because equal principal stresses produce a constant normal stress on all planes. C) 707 kg/cm² is incorrect as normal stress equals principal stress when principal stresses are equal. D) 1000 kg/cm² incorrectly sums the principal stresses rather than applying the plane stress formula.
Shortcut
When orthogonal principal stresses are equal (σx = σy), normal stress on any inclined plane remains equal to σx.
Common mistake
Incorrectly applying inclination angle transformation formulas for an equal biaxial stress state.
Correct answer: B) 500 kg/cm²