Mohr’s circle & stress transformation worksheet with answers

A point in a loaded part has σₓ = 20 MPa, σᵧ = 50 MPa and τₓᵧ = 20 MPa (tension positive). Find the largest principal stress σ₁.

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = 80 MPa (σₓ = σᵧ = 0). Find the normal stress σ_θ on a plane whose normal is rotated 25° counter-clockwise from the x-axis.

A point in a loaded part has σₓ = 160 MPa, σᵧ = −30 MPa and τₓᵧ = 30 MPa (tension positive). Find the angle θ_p from the x-axis to the direction of the largest principal stress, between −90° and 90°.

A point in a loaded part has σₓ = 100 MPa, σᵧ = −50 MPa and τₓᵧ = 60 MPa (tension positive). Find the maximum in-plane shear stress τ_max.

A point in a thin plate is in plane stress: σₓ = −50 MPa, σᵧ = 10 MPa and τₓᵧ = −60 MPa (σ_z = 0). The material has a yield strength of 350 MPa. Find the factor of safety n against yielding by the von Mises (distortion energy) criterion.

A point in a loaded part has σₓ = 140 MPa, σᵧ = 30 MPa and τₓᵧ = −50 MPa (tension positive). Find the shear stress τ_θ on a plane whose normal is rotated 45° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).