Mohr’s circle & stress transformation worksheet with answers

A bar with cross-sectional area 150 mm² carries an axial tensile force of 10 kN. Find the normal stress σ_θ on a section whose normal makes an angle of 25° with the bar axis (tension positive).

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = 80 MPa (σₓ = σᵧ = 0). Find the normal stress σ_θ on a plane whose normal is rotated 15° counter-clockwise from the x-axis.

A point in a thin plate is in plane stress: σₓ = 90 MPa, σᵧ = 10 MPa and τₓᵧ = −20 MPa, with σ_z = 0. Find the absolute maximum shear stress τ_abs at the point (considering all planes in three dimensions).

A point has σₓ = −40 MPa and σᵧ = 70 MPa. How large can the shear stress τ_xy (magnitude) become before the largest principal stress reaches 85 MPa?

A point in a loaded part has σₓ = 80 MPa, σᵧ = −10 MPa and τₓᵧ = −60 MPa (tension positive). Find the normal stress σ_θ on a plane whose normal is rotated 35° counter-clockwise from the x-axis.

A thin-walled cylindrical tank of inner diameter 500 mm and wall thickness 12 mm holds gas at a gauge pressure of 1 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.