Mohr’s circle & stress transformation worksheet with answers

A point in a loaded part has σₓ = 110 MPa, σᵧ = 40 MPa and τₓᵧ = 20 MPa (tension positive). Find the largest principal stress σ₁.

A bar with cross-sectional area 400 mm² carries an axial tensile force of 120 kN. Find the magnitude of the shear stress |τ_θ| on a section whose normal makes an angle of 20° with the bar axis.

A point has σₓ = −10 MPa, σᵧ = 60 MPa and τₓᵧ = −50 MPa (tension positive). The element is rotated 15° counter-clockwise. Find the normal stress σ_y′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

A point has σₓ = 30 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 solid shaft of diameter 30 mm carries a bending moment of 1000 N·m together with a torque of 200 N·m. At the top surface, where the bending stress is tensile and largest, find the maximum in-plane shear stress τ_max.

A thin-walled cylindrical tank of inner diameter 1500 mm and wall thickness 12 mm holds gas at a gauge pressure of 2 MPa. It is made from plate welded along a helical seam at 40° to the axis of the cylinder. Find the normal stress σ_n acting across the weld seam.