and that 之 ment relations for the in-plane strains(Appendix B),we find that Substituting equations(7.24)and equations(7.27)in the strain-displace- functions of the z coordinate as follows: 是 笔 P0.0 20010 verse shear strains and using assumption 5,we find that (7.24)in the strain-displacement equations(Appendix B)for the trans- only develop the CLT based on equation (7.24).Substituting equations Such a theory is beyond the scope of this book,however,and we will which is based on the assumption that the displacements are nonlinear lamination theory.For example,Christensen [11]describes one such theory, transverse shear deformations,it is necessary to use a so-called higher-order for possible warping of the cross section of the laminate and resulting =w(x,y).At this point it is appropriate to mention that in order to account displacement of any point having the samex and y coordinates,so w(x,y) displacement at the middle surface,w(x,y),is the same as the transverse Principles of Composite Material Mechanics G28 国 G25 equations.Similarly,in the laminated plate analvsis the midlane related to the loads on the structure by additional static equilibrium stress,equation (7.13),was developed.The bending moment.can be ship in equation (7.4).The result was that a more useful equation for to the applied bending moment by using the static equilibrium relation- known and is difficult to measure.Thus,the lamina stress was related to be of limited practical use because the curvature is not generally In the laminated beam analysis,equation(7.3)for lamina stress is seen sion and shear.In addition,the laminated plate analysis includes the stresses due to shear coupling,as discussed in chapter 2. the laminated plate analysis gives the 2-D lamina stresses y and due to bending and twisting curvatures and to the midplane biaxial exten- only gives the uniaxial stress,ox,due to the bending curvature,whereas equation(7.3),we notice several differences.The laminated beam analysis plate stresses in equations(7.31)with the laminated beam stress given by where the subscript k refers to the kth lamina.Comparing the laminated before deformation. Analysis of Laminates 9 g relationships from equations (2.35)as follows: 9 第十 樂十茶 may be found by substituting equations(7.28)into the lamina stress-strain surface,the stresses along arbitrary xy axes in the kth lamina of a laminate Since equations(7.28)give the strains at any distance z from the middle with out-of-plane twisting of the middle surface,which lies in the xy plane the middle surface in the yz plane.K is a twisting curvature associated in the xz plane and K,is a bending curvature associated with bending of K,is a bending curvature associated with bending of the middle surface 1 and the curvatures of the middle surface are where the strains on the middle surface are 者 G50 G.29