Mohr’s circle & stress transformation worksheet with answers

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = 30 MPa (σₓ = σᵧ = 0). Find the normal stress σ_θ on a plane whose normal is rotated 45° clockwise from the x-axis.

A point in a loaded part has σₓ = 150 MPa, σᵧ = 10 MPa and τₓᵧ = 50 MPa (tension positive). Find the smallest principal stress σ₂.

A point has σₓ = 140 MPa, σᵧ = 100 MPa and τₓᵧ = −40 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 has σₓ = −80 MPa, σᵧ = 80 MPa and τₓᵧ = 30 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 = −10 MPa, σᵧ = 60 MPa and τₓᵧ = 30 MPa (tension positive). Find the normal stress σ_θ on a plane whose normal is rotated 45° counter-clockwise from the x-axis.

A thin-walled cylindrical tank of inner diameter 400 mm and wall thickness 6 mm holds gas at a gauge pressure of 0.8 MPa. It is made from plate welded along a helical seam at 20° to the axis of the cylinder. Find the normal stress σ_n acting across the weld seam.