Principal Plane Angle from Stress State
MediumMohr's circleprincipal angle
At a point, σx = 80 MPa, σy = 20 MPa and τxy = 30 MPa. The angle from the x-axis to the principal plane is most nearly:
- A11.25°
- B45°
- C67.5°
- D22.5°
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Answer: D. 22.5°
Equation: the principal planes are at θp where tan 2θp = 2τxy / (σx − σy). Values: σx = 80 MPa and σy = 20 MPa are positive because the arrows point outward (tension). τxy = +30 MPa is positive because on the right (+x) face it points up (+y). Step 1: tan 2θp = 2(30)/(80 − 20) = 60/60 = 1.0. Step 2: 2θp = tan⁻¹(1.0) = 45°. Step 3: θp = 45°/2 = 22.5°. Reporting 45° forgets that the formula gives 2θp, because angles on Mohr's circle are doubled.
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