Mohr’s circle & stress transformation worksheet with answers

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = −30 MPa (σₓ = σᵧ = 0). Find the normal stress σ_θ on a plane whose normal is rotated 20° counter-clockwise from the x-axis.

A point has σₓ = 40 MPa and σᵧ = −70 MPa, plus some shear stress. On an element rotated by 20° the normal stress in the rotated x′ direction is 14.3 MPa. Find the normal stress σ_y′ in the rotated y′ direction.

A point has σₓ = 10 MPa and σᵧ = 60 MPa. How large can the shear stress τ_xy (magnitude) become before the largest principal stress reaches 105 MPa?

A point in a loaded part has σₓ = 20 MPa, σᵧ = 120 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 in a loaded part has σₓ = 130 MPa, σᵧ = 20 MPa and τₓᵧ = 40 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 600 mm and wall thickness 5 mm holds gas at a gauge pressure of 0.8 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.