A ductile hot-rolled steel bar has a minimum yield strength in tension and compression of 350 MPa. Using the distortion-energy and maximum-shear-stress theories determine the factors of safety for the following plane stress states:

(a) σ_x = 100 MPa, σ_y = 100 MPa

(b) σ_x = 100 MPa, σ_y = 50 MPa

(c) σ_x = 100 MPa, τ_xy = −75 MPa

(d) σ_x = -50 MPa, σ_y = -75 MPa, τ_xy = -50 MPa

(e) σ_x = 100 MPa, σ_y = 20 MPa, τ_xy = -20 MPa