Mohr’s circle & stress transformation worksheet with answers

A bar with cross-sectional area 800 mm² carries an axial compressive force of 140 kN. Find the normal stress σ_θ on a section whose normal makes an angle of 20° with the bar axis (tension positive).

A point has σₓ = 0 MPa and σᵧ = 80 MPa, plus some shear stress. On an element rotated by 15° the normal stress in the rotated x′ direction is 15.4 MPa. Find the normal stress σ_y′ in the rotated y′ direction.

A point in a loaded part has σₓ = −50 MPa, σᵧ = 80 MPa and τₓᵧ = −10 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 = 80 MPa, σᵧ = −10 MPa and τₓᵧ = −20 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 thin-walled cylindrical tank of inner diameter 600 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 40° to the axis of the cylinder. Find the normal stress σ_n acting across the weld seam.

A solid shaft of diameter 50 mm carries an axial compressive force of 200 kN together with a torque of 500 N·m. At the surface of the shaft, find the maximum in-plane shear stress τ_max.