210 BEAMS result is 0 0 0 0 且g 0 [W]= Elyy El -(Ely:) iwig-(i)月 Elyy orthotropic. 0 lwEx-(自P 0 0 0 1 (6.19) 6.1.4 Replacement Stiffnesses The parameters EA EI,GIt are referred to as replacement stiffnesses.By com- paring Eq.(6.5)with Eq.(6.6)and Egs.(6.4)with Egs.(6.7)and (6.8),we note the similarity in the force-strain relationships of isotropic and composite beams. Therefore,the strains(and consequently the displacements)of an orthotropic beam and of a composite beam with symmetrical cross section can be obtained by replacing EA EI,GI by EA,EI,GIt in the corresponding isotropic beam solution.For an orthotropic beam,the necessary substitutions are Isotropic beam Orthotropic beam EA→EA ELw,El,EL:→v,Eia,Ei GL→Gi For a composite beam with symmetrical cross section and loaded in the symmetry plane the following substitutions apply: Isotropic beam Composite beam symmetrical cross section symmetrical cross section EA.Elyy → EA.Elyy As the preceding discussion indicates,the displacements of orthotropic beams and of composite beams with symmetrical cross section are similar to the displacements of isotropic beams.However,the stresses are markedly different in isotropic and in composite beams.In an isotropic beam subjected to an axial load and pure bending,there is only axial stress.In a composite beam subjected to an axial load and pure bending,in addition to the axial stress,there are also transverse normal and shear stresses.Furthermore,in an isotropic beam subjected to pure torque,there is only shear stress,whereas in a composite beam there are also axial and transverse normal stresses. In this chapter the stresses are calculated by the laminate plate theory,which does not take into account the interlaminar stresses near free edges(page 166). 6.2 Rectangular,Solid Beams Subjected to Axial Load and Bending We consider rectangular laminated beams having solid cross sections with an axial force N acting at the centroid and a pure bending moment My acting in the210 BEAMS result is [W] =         1 EA 0 00 0 EI
zz EI
yyEI
zz − (EI yz)2 −EI yz EI yyEI zz − (EI yz)2 0 0 −EI yz EI yyEI zz − (EI yz)2 EI yy EI yyEI zz − (EI yz)2 0 00 0 1 GI t         orthotropic. (6.19) 6.1.4 Replacement Stiffnesses The parameters EA, EI , GI t are referred to as replacement stiffnesses. By com￾paring Eq. (6.5) with Eq. (6.6) and Eqs. (6.4) with Eqs. (6.7) and (6.8), we note the similarity in the force–strain relationships of isotropic and composite beams. Therefore, the strains (and consequently the displacements) of an orthotropic beam and of a composite beam with symmetrical cross section can be obtained by replacing EA, EI, GIt by EA, EI , GI t in the corresponding isotropic beam solution. For an orthotropic beam, the necessary substitutions are Isotropic beam Orthotropic beam EA =⇒ EA EIyy, EIzz, EIyz =⇒ EI yy, EI zz, EI yz GIt =⇒ GI t For a composite beam with symmetrical cross section and loaded in the symmetry plane the following substitutions apply: Isotropic beam Composite beam symmetrical cross section symmetrical cross section EA, EIyy =⇒ EA, EI yy As the preceding discussion indicates, the displacements of orthotropic beams and of composite beams with symmetrical cross section are similar to the displacements of isotropic beams. However, the stresses are markedly different in isotropic and in composite beams. In an isotropic beam subjected to an axial load and pure bending, there is only axial stress. In a composite beam subjected to an axial load and pure bending, in addition to the axial stress, there are also transverse normal and shear stresses. Furthermore, in an isotropic beam subjected to pure torque, there is only shear stress, whereas in a composite beam there are also axial and transverse normal stresses. In this chapter the stresses are calculated by the laminate plate theory, which does not take into account the interlaminar stresses near free edges (page 166). 6.2 Rectangular, Solid Beams Subjected to Axial Load and Bending We consider rectangular laminated beams having solid cross sections with an axial force N
acting at the centroid and a pure bending moment M
y acting in the