6.2 RECTANGULAR,SOLID BEAMS SUBJECTED TO AXIAL LOAD AND BENDING 211 Figure 6.7:Rectangular laminated beam. x-z plane (Fig.6.7).In this section,we develop expressions for calculating the displacements and the stresses. We treat the beam as a narrow plate and build the analysis on the results of laminate plate theory presented in Chapter 3.By convention,for plates along an edge parallel to the y-axis,the in-plane force per unit length is N,and the moment per unit length is M(Fig.6.8).For beams,the total force Nalong an edge parallel to y and the total moment in the x-plane My are specified.The total axial force in the beam Ncorresponds to bN,in the plate,and the total moment in the beam My corresponds to bM in the plate(where bis the width).Thus,we can apply the laminate plate theory expressions to beams by making the following substitutions: N Nx=b 应， Mr= b (6.20) 6.2.1 Displacements-Symmetrical Layup The layup of the beam is symmetrical.As noted in the preceding section,we analyze this beam by the laminate plate theory,according to which the midplane strain and curvature of the plate are (Eqs.3.31 and 3.32) e=anN plate. (6.21) Beam Plate Figure 6.8:Internal forces and curvatures in a beam and in the corresponding plate.6.2 RECTANGULAR, SOLID BEAMS SUBJECTED TO AXIAL LOAD AND BENDING 211 z y x b h N My Figure 6.7: Rectangular laminated beam. x–z plane (Fig. 6.7). In this section, we develop expressions for calculating the displacements and the stresses. We treat the beam as a narrow plate and build the analysis on the results of laminate plate theory presented in Chapter 3. By convention, for plates along an edge parallel to the y-axis, the in-plane force per unit length is Nx, and the moment per unit length is Mx (Fig. 6.8). For beams, the total force N
along an edge parallel to y and the total moment in the x–z plane M
y are specified. The total axial force in the beam N
corresponds to bNx in the plate, and the total moment in the beam M
y corresponds to bMx in the plate (where b is the width). Thus, we can apply the laminate plate theory expressions to beams by making the following substitutions: Nx = N
b Mx = M
y b . (6.20) 6.2.1 Displacements – Symmetrical Layup The layup of the beam is symmetrical. As noted in the preceding section, we analyze this beam by the laminate plate theory, according to which the midplane strain and curvature of the plate are (Eqs. 3.31 and 3.32) o x = a11Nx κx = d11Mx plate. (6.21) x Nx y x N My b y Mx κx 1 ρy y ∂y w∂ o y o w w ψ Beam Plate x y x y z z Figure 6.8: Internal forces and curvatures in a beam and in the corresponding plate