Question

In an elastic material, the direct stresses of 120 MN/m² and 90 MN/m² are applied at a certain point on planes at right angles to each other in tension and compressive respectively. Estimate the shear stress to which material could be subjected, if the maximum principal stress is 150 MN/m². Also find the magnitude of other principal stress and its inclination to 120 MN/m².

Answer 1

Given that 

σ_x = 120MN/m² (tensile i.e + ive) 

σ_y = 90MN/m² (Compressive i.e. – ive) 

τ_xy = ? 

σ₁ = 150MN/m² 

Since we have

150 = ½[(120 – 90) + {(120 – (– 90))² + 4(τ_xy)²}^1/2]

τ_xy = 84.85MN/m²  

Now the magnitude of other principal stress σ₂

σ₂ = ½[(120 – 90) – {(120 – (– 90))² + 4(84.85)² }^1/2] 

σ₂ = – 120MN/m² 

The direction of principal planes is: 

tan 2θ = 2 τ_xy / (σ_x −σ_y) 

tan 2θ = (2 × 84.85)/(120 – (– 90)) 

2θ = 38.94º or 2218.94º 

θ = 19.47º or 109.47º