Mohr’s circle & stress transformation worksheet with answers

A point in a loaded part has σₓ = 140 MPa, σᵧ = 120 MPa and τₓᵧ = −30 MPa (tension positive). Find the smallest principal stress σ₂.

A point in a loaded part has σₓ = 20 MPa, σᵧ = −40 MPa and τₓᵧ = −40 MPa (tension positive). Find the largest principal stress σ₁.

A point has σₓ = 140 MPa, σᵧ = 10 MPa and τₓᵧ = −60 MPa (tension positive). The element is rotated 20° counter-clockwise. Find the normal stress σ_y′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

A point has σₓ = 160 MPa, σᵧ = 100 MPa and τₓᵧ = 20 MPa (tension positive). Find the angle θ_s, between −90° and 90°, from the x-axis to the normal of the plane on which the shear stress has its maximum positive value +R (shear positive in the same sense as τₓᵧ).

A point in a loaded part has σₓ = 0 MPa, σᵧ = 110 MPa and τₓᵧ = 50 MPa (tension positive). Find the shear stress τ_θ on a plane whose normal is rotated 30° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

A point in a thin plate is in plane stress: σₓ = 120 MPa, σᵧ = 70 MPa and τₓᵧ = 30 MPa (σ_z = 0). The material has a yield strength of 420 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.