Mohr’s circle & stress transformation worksheet with answers

A bar with cross-sectional area 500 mm² carries an axial compressive force of 130 kN. Find the normal stress σ_θ on a section whose normal makes an angle of 35° with the bar axis (tension positive).

A point has σₓ = 40 MPa and σᵧ = 100 MPa, plus some shear stress. On an element rotated by 25° the normal stress in the rotated x′ direction is 20.1 MPa. Find the normal stress σ_y′ in the rotated y′ direction.

A point in a loaded part has σₓ = −80 MPa, σᵧ = −30 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 = 10 MPa and τₓᵧ = 60 MPa, with σ_z = 0. Find the absolute maximum shear stress τ_abs at the point (considering all planes in three dimensions).

A point in a loaded part has σₓ = 50 MPa, σᵧ = −60 MPa and τₓᵧ = 30 MPa (tension positive). Find the shear stress τ_θ on a plane whose normal is rotated 30° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

A point in a thin plate is in plane stress: σₓ = 90 MPa, σᵧ = 10 MPa and τₓᵧ = −10 MPa (σ_z = 0). The material has a yield strength of 275 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.