A variational method is outlined for constructing envelopes of the interaction yield hypersurfaces for beams subjected to a combined bending moment, axial tensile force, transverse shear force and torsion. The lower bounds of the envelopes for the interaction yield curves relating bending moment-torsion and axial tension-torsion are obtained by using a numerical scheme for beams having a rectangular-shaped cross-section.NOTATIONk 4 max (T)/(τ 0 /BH 2 )p i (s) distributed load vectorx, y, z coordinates defined in Fig. 1y * , z * y/(H/2), z/(B/2)B breadth of beamC i j , D m n coefficientsH depth of beamM y , M z bending moments about the y- and z-axes, respectivelyM 0 y , M 0 z fully plastic bending momentsM * y ,M * z M y M 0 y ,M z M 0 z N tensile forceN 0 fully plastic tensile forceN * N/N 0 P external load vectorQ y , Q z transverse shear forces in the y and z directions, respectivelyQ 0 fully plastic transverse shear forceQ * y ,Q * z Q y Q 0 ,Q z Q 0 T twisting momentT 0 fully plastic twisting momentT 0 1 lower bound of T 0 T e pure plastic twisting moment without warpingT * TT 0 T * * T * T 0 /T e α B/Hσ x , σ y , σ z , τ x y , τ y z , τ x z stresses defined in Fig. 1σ * x , τ * x y , τ * x z σ x σ 0 ,τ x y τ 0 and τ x z τ 0 σ 0 uniaxial yield stressτ 0 shear yield stressτ * x y i j shear stress defined by Eqn (15)τ * x z m m shear stress defined by Eqn (16) i external load factorΛ function defined by Eqn (18)