Question

At a critical point in a steel part of a machine, the stress components found were σxx = 100 MPa, σyy = −50 MPa and σxy = 30 MPa. Assuming the point is in state stress plane, and the material yield stress to be 160 MPa, determine whether the fails according to the criteria: (a) the maximum shear stress (Tresca). (b) the maximum deformation energy (von Mises).

Answer 1

given,

σ x x = 100 MPa

σ y y = −50 MPa

σ x y = 30 MPa

σy = 160 MPa

using principal stress formula

`\sigma = (\sigma_x+\sigma_y)/(2)\pm \sqrt{ ((\sigma_x-\sigma_y)/(2))^2+\tau^2}`

`\sigma = (100-50)/(2)\pm \sqrt{ ((100+50)/(2))^2+30^2}`

`\sigma = 25\pm80.77`

`\sigma_(max) = 105.77 MPa`

`\sigma_(min) = 55.77 MPa`

a) |σ₁ - σ₂| = σ f

|105.77 - 55.77| = σ f

σ_f = 50 MPa

b)
`\sigma_y=√(\sigma_1^2+\sigma^2_2-\sigma_1* \sigma_2)`

`\sigma_y=√(105.77^2+55.77^2-105.77* 55.77)`

`\sigma_y = 91.65 MPa`