Mohr’s circle & stress transformation worksheet with answers

A bar with cross-sectional area 400 mm² carries an axial tensile force of 120 kN. Find the normal stress σ_θ on a section whose normal makes an angle of 40° with the bar axis (tension positive).

At a point in a loaded part the in-plane principal stresses are σ₁ = 100 MPa and σ₂ = −70 MPa (tension positive). Find the maximum in-plane shear stress τ_max.

A point has σₓ = −20 MPa, σᵧ = −50 MPa and τₓᵧ = −20 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 = 60 MPa, σᵧ = 20 MPa and τₓᵧ = −60 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 in a thin plate is in plane stress: σₓ = −50 MPa, σᵧ = 90 MPa and τₓᵧ = 60 MPa (σ_z = 0). Find the von Mises equivalent stress σ_vM.

A thin-walled cylindrical tank of inner diameter 1500 mm and wall thickness 6 mm holds gas at a gauge pressure of 0.5 MPa. It is made from plate welded along a helical seam at 55° to the axis of the cylinder. Find the normal stress σ_n acting across the weld seam.