Mohr’s circle & stress transformation worksheet with answers

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = −110 MPa (σₓ = σᵧ = 0). Find the normal stress σ_θ on a plane whose normal is rotated 35° counter-clockwise from the x-axis.

A bar with cross-sectional area 400 mm² carries an axial tensile force of 100 kN. Find the magnitude of the shear stress |τ_θ| on a section whose normal makes an angle of 35° with the bar axis.

A point in a loaded part has σₓ = 10 MPa, σᵧ = 50 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 has σₓ = 70 MPa, σᵧ = 20 MPa and τₓᵧ = 50 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 = −30 MPa, σᵧ = 100 MPa and τₓᵧ = 40 MPa (tension positive). Find the normal stress σ_θ on a plane whose normal is rotated 20° clockwise from the x-axis.

A point in a loaded part has σₓ = 110 MPa, σᵧ = 50 MPa and τₓᵧ = −60 MPa (tension positive). Find the shear stress τ_θ on a plane whose normal is rotated 50° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).