Mohr’s circle & stress transformation worksheet with answers

A point has σₓ = 160 MPa and σᵧ = 110 MPa, plus some shear stress. On an element rotated by 25° the normal stress in the rotated x′ direction is 135.7 MPa. Find the normal stress σ_y′ in the rotated y′ direction.

At a point in a loaded part the in-plane principal stresses are σ₁ = 120 MPa and σ₂ = −10 MPa (tension positive). Find the normal stress σ_avg that acts on the planes of maximum in-plane shear stress.

A point in a loaded part has σₓ = −50 MPa, σᵧ = −70 MPa and τₓᵧ = −30 MPa (tension positive). Find the maximum in-plane shear stress τ_max.

A point has σₓ = 100 MPa and σᵧ = −30 MPa. How large can the shear stress τ_xy (magnitude) become before the largest principal stress reaches 160 MPa?

A point in a loaded part has σₓ = 130 MPa, σᵧ = 100 MPa and τₓᵧ = −50 MPa (tension positive). Find the shear stress τ_θ on a plane whose normal is rotated 45° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

A point in a loaded part has σₓ = −20 MPa, σᵧ = 110 MPa and τₓᵧ = 30 MPa (tension positive). Find the normal stress σ_θ on a plane whose normal is rotated 60° counter-clockwise from the x-axis.