Mohr’s circle & stress transformation worksheet with answers

A point in a loaded part has σₓ = 110 MPa, σᵧ = −60 MPa and τₓᵧ = −30 MPa (tension positive). Find the largest principal stress σ₁.

A point has σₓ = 40 MPa and σᵧ = 70 MPa, plus some shear stress. On an element rotated by 20° the normal stress in the rotated x′ direction is 75.6 MPa. Find the normal stress σ_y′ in the rotated y′ direction.

A point in a loaded part has σₓ = 120 MPa, σᵧ = −40 MPa and τₓᵧ = 40 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 has σₓ = 10 MPa, σᵧ = 20 MPa and τₓᵧ = 30 MPa (tension positive). The element is rotated 30° 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 thin plate is in plane stress: σₓ = −10 MPa, σᵧ = −20 MPa and τₓᵧ = −20 MPa (σ_z = 0). The material has a yield strength of 355 MPa. Find the factor of safety n against yielding by the von Mises (distortion energy) criterion.

A solid shaft of diameter 25 mm carries a bending moment of 100 N·m together with a torque of 1000 N·m. At the top surface, where the bending stress is tensile and largest, find the maximum in-plane shear stress τ_max.