A variational method has been used to construct envelopes of the interaction yield surfaces for elastic, perfectly plastic beams subjected to combined bending moment, axial tension and transverse shear force. A lower bound of the envelope for the interaction yield curves relating bending moment and transverse shear force is obtained using a numerical scheme for beams having rectangular-shaped cross-sections. The result is recast into a simple equation with the aid of the least-squares method. A good upper bound of the envelope for the interaction yield curves which combine transverse shear force and axial tension is also derived. A formula for the interaction yield surfaces for combined bending moment, axial tension and transverse shear force is suggested.NOTATIONa stress defined in Fig. 3(a)b,c stresses defined in Fig. 3(b)e i (i = 1,2,3), e 0 distances defined in Fig. 4f,g,h,p,q,s functions defined in Appendix 2f a f a = 0, an approximate lower bound defined in Eqn (21)f c f c = 0, a conjectured lower boundf e f e = 0, a limit of elastic behaviourf s f s = 0, a lower bound found by the current calculationk i ,l i ,m i ,n i coefficientsp i (s) distributed load vectorx,y,z coordinates defined in Fig. 1x * , y * x/(H/2), y/(H/2)A cross-sectional areaA i (i = 1,2,3) areas defined in Fig. 4B breadth of beamH depth of beamI second moment of areaM bending momentM A , M A + B bending moments due to τ * A x y and τ * A x y plus τ * B x y , respectivelyM 0 fully plastic bending momentM * MM 0 N tensile forceN 0 fully plastic tensile forceN * NN 0 P external load vectorQ transverse shear forceQ 0 fully plastic transverse shear forceQ * QQ 0 Q * s maximum transverse shear force carried by web before shear plastic buckling of the web occursS(y) first moment of areaα i , α j coefficientsβ parameterγ i , δ j coefficientsσ x , σ y , τ x y stresses defined in Fig. 1σ * x , σ * y , τ * x y σ x /σ 0 , σ y /σ 0 and τ x y /τ 0 σ 0 uniaxial yield stressσ 0 stress defined in Fig. 3(b)τ 0 shear yield stressτ 0 stress defined in Fig. 3(a)τ * A x y symmetric shear stressτ * A x y i symmetric shear stress defined by Eqn (18a)τ * B x y antisymmetric shear stressτ * B x y i antisymmetric shear stress defined by Eqn (18b) i external load factorΔ c - bΔM M A + B - M A