Answer
given,
σ x x = 100 MPa
σ y y = −50 MPa
σ x y = 30 MPa
σy = 160 MPa
using principal stress formula
[tex]\sigma = \dfrac{\sigma_x+\sigma_y}{2}\pm \sqrt{ (\dfrac{\sigma_x-\sigma_y}{2})^2+\tau^2}[/tex]
[tex]\sigma = \dfrac{100-50}{2}\pm \sqrt{ (\dfrac{100+50}{2})^2+30^2}[/tex]
[tex]\sigma = 25\pm80.77[/tex]
[tex]\sigma_{max} = 105.77 MPa[/tex]
[tex]\sigma_{min} = 55.77 MPa[/tex]
a) |σ₁ - σ₂| = σ f
|105.77 - 55.77| = σ f
σ_f = 50 MPa
b)[tex]\sigma_y=\sqrt{\sigma_1^2+\sigma^2_2-\sigma_1\times \sigma_2}[/tex]
[tex]\sigma_y=\sqrt{105.77^2+55.77^2-105.77\times 55.77}[/tex]
[tex]\sigma_y = 91.65 MPa[/tex]
