% THEORY OF ELASTIC STABILITY BEAM-COLUMNS shown in Fig.1-2b.The lateral load may be considered as having con- stant intensity g over the distance da and will be assumed positive when beam-columns.If the axial force P equals zero,these equstions reduce in the direction of the positive y axis,which is downward in this case. to the usual equations for bending by lateral loads only. The shearing force V and bending moment Mf acting on the sides of the Instead of taking an element d=with sides perpendicular to the axis (Fig.1-2), element are assumed positive in the directions shown. we can cut an element with sides normal to the deflected axis of the beam (Fig.1-2c). Since the slope of the beam is small,the normal forees neting on the sides of the ele- 12, ment can be taken equal to the axial compressive foree P.The shearing force N in this cae is relsted to the ahearing force V in Fig.1-26 by the expresaion (b) N-r+P器 @) (a) hg.1-1 nd instead of Egs.(1-1)and (1-2)we obtain The relations among load,shearing foroe V,and bending moment are (-1a obtained from the equilibrium of the element in Fig.1-20.Summing g-器+P器 forces in the y direction gives and N- (1-2a) -v+qdz+(V+dv)=0 Equstions(1-1a)and(1-2)can also be derived by considering the equilibrium of the element in Fig.1-2c.Finally,combining Eq.(1-2a)with Eq.(1-3)yields the equstion or 9=- dt. (1-1) 数尝-N (1-4aj Taking moments about point n and assuming that the angle between the axis of the beam and the horizontal is small,we obtain Equstion (1-5)remains valid for the element in Fig.1-2c.Thus we have two seta of differential equations for s beam-column,depending on whether the shearing force is M+ga警+(v+am-(M+d0+P费-0 taken on s eross section normal to the deflected or the undeflected axis of the beam. 1.3.Beam-column with a Concentrated Lateral Load.As the first If terms of second order are neglected,this equation becomes example of the use of the beam-column equations,let us consider a beam of length I on two simple supports(Fig.1-3)and carrying a single lateral v-器-P器 (1-2) If the effects of shearing deformations and shortening of the beam axis are neglected,the expression for the curvature of the axis of the beam is 1碧=-M (1-3) The quantity BI represents the fexural rigidity of the beam in the plane of bending,that is,in the ry plane,which is assumed to be a plane of sym- metry.Combining Eq.(1-3)with Eqs.(1-1)and (1-2),we can express the differential equation of the axis of the beam in the following alternate forms: B路+P鼎=-V (1-4) and BI器+P器-g (1-5) (e) Equations(1-1)to (1-5)are the basie differential equations for bending of Fro.1-2