xvi NOTATIONB Thiekness,time,temperature Work,tenaile foree Axisl load factor for beam-columns (u/2) ,, Displacementa in多，，and z directions U Btrain energy 可，0 Displacements in tangential and radial directions CHAPTER 1 y Shearing foree in beam Warping displacement in beam BEAM-COLUMNS ,多 Rectangular coordinates Section modulus (2=I/e) 1.1.Introduction.In the elementary theory of bending,it is found that stresses and deflections in beams are directly proportional to the restraint applied loads.This condition requires that the change in shape of the y Shearing unit strain,weight per unit volume,spring conatant, beam due to bending must not affect the action of the applied loads. numerical factor For example,if the beam in Fig.1-la is subjected to only lateral loads, 8 Defection Unit normal strain,coefficient of thermal expansion such as Q amd Qa,the presence of the small deflections 61 and and slight , Unit normal strains in z,y,and s directionn changes in the vertical lines of action of the loads will have only an insig- Amplifcation factors for beam-columns nificant effect on the moments and shear forces.Thus it is possible to Angle,angulnr coordinate,angle of twist per unit length make calculations for deflections,stresses,moments,ete.,on the basis of Distance,numerical factor Poisson's ratio the initial configuration of the beam.Under these conditions,and also 名男下 Reetangular coordinates if Hooke's law holds for the material,the deflections are proportional to Radius of curvature the acting forces and the principle of superposition is valid;Le.,the final Unit normal streas deformation is obtained by summation of the deformations produced by Ua dn ds Unit normal stresses in z,y,and directions the individual forces. 4 Average compresaive unit atreas for columns Compreesive unit streas at critical load Conditions are entirely different when both axial and lateral loads act olt Unit streas at ultimate load simultaneously on the beam (Fig.1-16).The bending moments,shear Working unit streas forces,stresses,and deflections in the beam will not be proportional to Yield-point streas the magnitude of the axial load.Furthermore,their values will be Unit shear stres dependent upon the magnitude of the deflections produced and will be Te,T南Tu Unit shear stresses on planes perpendicular to the馬，斯，and&axes and sensitive to even alight eccentricities in the application of the axial load. parallel to the y,美，8od￥axw Angle,angular coordinate,angle of twist of bar Beams subjected to axial compression and simultaneously supporting X Change of curvature in shell lateral loads are known as beam-columns.In this first chapter,beam- Radian froquency of vibration columns of symmetrical cross section and with various conditions of Warping function support and loading will be analyzed. 1.2.Differential Equations for Beam-columns.The basic equations for the analyais of beam-columns can be derived by considering the beam in Fig.1-2a.The beam is subjected to an axial compressive force P and to a distributed lateral load of intensity g which varies with the dis- tance z along the beam.An element of length dz between two cross sections taken normal to the original (undeflected)axis of the beam is 1 For an analysis of beama subjected to axial tension see Timoshenko,"Strength of Materials,"3d od.,part II,p.41,D.Van Nostrand Company,Ine.,Princeton,N.J. 1966