Mohr’s circle & stress transformation worksheet with answers

A point in a loaded part has σₓ = −40 MPa, σᵧ = 70 MPa and τₓᵧ = 30 MPa (tension positive). Find the largest principal stress σ₁.

A point has σₓ = 90 MPa and σᵧ = 10 MPa, plus some shear stress. On an element rotated by 40° the normal stress in the rotated x′ direction is 76.6 MPa. Find the normal stress σ_y′ in the rotated y′ direction.

A point has σₓ = 30 MPa, σᵧ = −20 MPa and τₓᵧ = −60 MPa (tension positive). The element is rotated 45° clockwise. Find the normal stress σ_y′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

A point in a loaded part has σₓ = 150 MPa, σᵧ = 100 MPa and τₓᵧ = 50 MPa (tension positive). Find the maximum in-plane shear stress τ_max.

A point in a thin plate is in plane stress: σₓ = 150 MPa, σᵧ = −70 MPa and τₓᵧ = 40 MPa (σ_z = 0). The material has a yield strength of 275 MPa. Find the factor of safety n against yielding by the von Mises (distortion energy) criterion.

A point in a loaded part has σₓ = 60 MPa, σᵧ = 70 MPa and τₓᵧ = −60 MPa (tension positive). Find the shear stress τ_θ on a plane whose normal is rotated 60° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).