Principal Stresses at a Point

Equations: Mohr's circle has centre C = (σx + σy)/2 and radius R = √[((σx − σy)/2)² + τxy²]. The principal stresses are σ1,2 = C ± R. 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: C = (80 + 20)/2 = 50 MPa. Step 2: R = √[((80 − 20)/2)² + 30²] = √(900 + 900) = 42.4 MPa. Step 3: σ1 = 50 + 42.4 = 92.4 MPa and σ2 = 50 − 42.4 = 7.6 MPa. Answering 80 and 20 MPa ignores the shear stress; σx and σy are only principal when τxy = 0.