A ductile material has the properties S_yt = 60 kpsi and S_yc = 75 kpsi. Using the ductile Coulomb-Mohr theory, determine the factor of safety for the states of plane stress given in Prob. 5–3.

For a bar of AISI 1030 hot-rolled steel and:

(a) σ_x = 25 kpsi, σ_y = 15 kpsi 

(b) σ_x = 15 kpsi, σ_y = -15 kpsi 

(c) σ_x = 20 kpsi, σ_y = −10 kpsi 

(d) σ_x = -12 kpsi , σ_y = 15 kpsi , τ_xy = -9 kpsi 

(e) σ_x = -24 kpsi , σ_y = -24 kpsi , τ_xy = -15 kpsi