￥ relation is given by Recall that for a homogeneous,isotropic beam,the moment-curvature original ply,so that the total number of plies is now even. divide each ply into two identical plies having half the thickness of the Equation(7.6)can also be used for an odd number of plies if we simply (7.3)in equation (7.4)gives defined by the xy plane. 巴 20 thickness z;=jh/N and equation (7.5)becomes surface to the outside of the jth ply.For an even number of plies of uniform where N is the total number of plies and z is the distance from the neutral where the symmetry assumption 2 has been used.Substitution of equation be related to the longitudinal stresses by Static equilibrium requires that the applied bending moment M must equation(7.1)and equation(7.2),the longitudinal stress is seen to be (e,);is the longitudinal strain in the jth ply along the x direction.From where(E.);is the Young's modulus of ith ply along the x direction and From assumption 3,the longitudinal stress in the jth ply is given by the angle defined in figure 7.2,and z the distance from neutral surface where p is the radius of curvature of the neutral surface during flexure, Principles of Composite Material Mechanics 园 金 8 8 (7.7)to get buckling load,Pfor a laminated beam can be estimated by the formula where p is the applied tip load and L is the beam length.The Euler 三 in figure 7.3 would be given by the familiar equation and the maximum deflection at the tip of the laminated cantilever beam would be of the form Young's modulus. ferential equation for the transverse deflection,w,of a laminated beam equations from elementary mechanics of materials.For example,the dif- flexural modulus in place of the Young's modulus in the beam deflection The deflections of laminated beams can now be calculated by using the ing sequence and the ply moduli.That is,if the properties do not change through the thickness of a beam,the flexural modulus is the same as the modulus of the homogeneous isotropic beam,depends on the ply-stack- Thus,the flexural modulus of the laminated beam,unlike the Young's 冬 or for an even number of plies we can combine equation(7.6)and equation flexural modulus of the laminated beam can be expressed as Combining equation(7.5)and equation(7.7),we find that the effective effective flexural modulus of the beam(which is same as Young's modulus of the beam material for a homogeneous,isotropic beam). whereIw=zdA=/12 is the moment of inertia of the cross section about the neutral axis (y axis),A the cross-sectional area,and Ef the Analysis of Laminates 是 G10 8 品 务