Question

The state of stress at a point in a loaded component principal stresses is found to be as given below : σ_x =50 GN/m²; σ_y =150 GN/m²; τ_xy =100 GN/m²;Determine the principal stresses and maximum shearing stress. Find the orientations of the planes on which they act.

Answer 1

Principal stresses is given by the equation:

σ_1,2 = ½ [(50 + 150) ± {(50 – 150)² + 4 x (100)²}^1/2] 

σ_1,2 = 100 ± 111.8 

σ₁ = 211.8 GN/m²

σ₂ = – 11.8GN/m² 

Now Maximum shear stress is given by the equation:

t = 1/ 2 (σ₁ +σ₂)sin 2θ

(Since Maximum Shear stress is at θ = 45º) 

= ½ (211.8 + 11.8) sin 90º 

= 111.8GN/m² 

The orientation of the planes on which they act is given by the equation:

2θ = 2τ_xy/(σ_x − σ_y)

tan 2θ = (2 × 100)/(50 – 150)) 

2θ = – 63.43º 

θ = – 31.72º

Major principal plane = θ = – 31.72º

Miner principal plane = θ + 90º = 58.28º