Mohr’s circle & stress transformation worksheet with answers

A point in a loaded part has σₓ = 30 MPa, σᵧ = −40 MPa and τₓᵧ = 50 MPa (tension positive). Find the smallest principal stress σ₂.

A point has σₓ = 20 MPa and σᵧ = 20 MPa, plus some shear stress. On an element rotated by 25° the normal stress in the rotated x′ direction is 58.3 MPa. Find the normal stress σ_y′ in the rotated y′ direction.

A point has σₓ = 130 MPa, σᵧ = 100 MPa and τₓᵧ = −40 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 thin plate is in plane stress: σₓ = 90 MPa, σᵧ = 80 MPa and τₓᵧ = 40 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 = −10 MPa, σᵧ = −20 MPa and τₓᵧ = −20 MPa (tension positive). Find the shear stress τ_θ on a plane whose normal is rotated 35° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

A point in a loaded part has σₓ = 0 MPa, σᵧ = 0 MPa and τₓᵧ = −10 MPa (tension positive). Find the normal stress σ_θ on a plane whose normal is rotated 15° counter-clockwise from the x-axis.