Mohr’s circle & stress transformation worksheet with answers

A point in a loaded part has σₓ = −30 MPa, σᵧ = −80 MPa and τₓᵧ = −30 MPa (tension positive). Find the smallest principal stress σ₂.

A point has σₓ = 20 MPa and σᵧ = 80 MPa, plus some shear stress. On an element rotated by 30° the normal stress in the rotated x′ direction is −8.3 MPa. Find the normal stress σ_y′ in the rotated y′ direction.

A point has σₓ = 60 MPa, σᵧ = 70 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 point in a thin plate is in plane stress: σₓ = 140 MPa, σᵧ = 110 MPa and τₓᵧ = 50 MPa, with σ_z = 0. Find the absolute maximum shear stress τ_abs at the point (considering all planes in three dimensions).

A point in a loaded part has σₓ = 110 MPa, σᵧ = 40 MPa and τₓᵧ = −10 MPa (tension positive). Find the normal stress σ_θ on a plane whose normal is rotated 15° counter-clockwise from the x-axis.

A thin-walled cylindrical tank of inner diameter 400 mm and wall thickness 15 mm holds gas at a gauge pressure of 2 MPa. It is made from plate welded along a helical seam at 60° to the axis of the cylinder. Find the magnitude of the shear stress |τ_n| acting along the weld seam.