Mohr’s circle & stress transformation worksheet with answers

A point in a loaded part has σₓ = 50 MPa, σᵧ = 30 MPa and τₓᵧ = −50 MPa (tension positive). Find the smallest principal stress σ₂.

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = 20 MPa (σₓ = σᵧ = 0). Find the normal stress σ_θ on a plane whose normal is rotated 60° counter-clockwise from the x-axis.

A point in a loaded part has σₓ = 10 MPa, σᵧ = −70 MPa and τₓᵧ = −20 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 = 20 MPa, σᵧ = 90 MPa and τₓᵧ = −10 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 solid shaft of diameter 25 mm carries an axial tensile force of 190 kN together with a torque of 500 N·m. At the surface of the shaft, find the maximum in-plane shear stress τ_max.

A point in a loaded part has σₓ = 70 MPa, σᵧ = −70 MPa and τₓᵧ = 20 MPa (tension positive). Find the shear stress τ_θ on a plane whose normal is rotated 35° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).