92 THIN PLATES 4.1.1 Boundary Conditions The conditions alongeach edge of the plate must be specified.Boundary conditions for an edge parallel with the y-axis are given below. Along a built-in edge,the deflection wo,the rotation of the edge aw/ax,and the in-plane°，v°displacements are zero: w°=0 aw -=0°=v°=0. (4.6 ax Along a free edge,where no external loads are applied,the bending moment Mr,the replacement shear force?V+a Mry/ax,and the in-plane forces N,Ny are zero: Mx=0 Vx+ 8My=0N:=Nw=0. ay (4.7) Along a simply supported edge,the deflection wo,the bending moment Mr, and the in-plane forces M,Nry are zero: w°=0Mx=0N=Nxy=0. (4.8) When in-plane motions are prevented by the support,the in-plane forces are not zero(N≠0，Nxy≠O)whereas the in-plane displacements are zero: °=0v°=0. (4.9) For an edge parallel with the x-axis,the preceding boundary conditions hold with x and y interchanged. 4.1.2 Strain Energy As we noted previously,solutions to plate problems may be obtained by energy methods that require knowledge of the strain energy.For a linearly elastic material the strain energy is given by Eq.(2.200).Under plane-stress condition the stress components o:,txz,and tyz are zero(Eq.2.121),and the expression for the strain energy simplifies to Ly (oxex +oyey+txyrxy)dzdydx, (4.10) where h and hp are the distances from the reference plane to the plate's surfaces (Fig.3.12).The strain components are (Eq.3.7) (4.11) 7bid,p.84.92 THIN PLATES 4.1.1 Boundary Conditions The conditions along each edge of the plate must be specified. Boundary conditions for an edge parallel with the y-axis are given below. Along a built-in edge, the deflection wo, the rotation of the edge ∂wo/∂x, and the in-plane uo, vo displacements are zero: wo = 0 ∂wo ∂x = 0 uo = vo = 0. (4.6) Along a free edge, where no external loads are applied, the bending moment Mx, the replacement shear force7 Vx + ∂Mxy/∂x, and the in-plane forces Nx, Nxy are zero: Mx = 0 Vx + ∂Mxy ∂y = 0 Nx = Nxy = 0. (4.7) Along a simply supported edge, the deflection wo, the bending moment Mx, and the in-plane forces Nx, Nxy are zero: wo = 0 Mx = 0 Nx = Nxy = 0. (4.8) When in-plane motions are prevented by the support, the in-plane forces are not zero (Nx = 0, Nxy = 0) whereas the in-plane displacements are zero: uo = 0 vo = 0. (4.9) For an edge parallel with the x-axis, the preceding boundary conditions hold with x and y interchanged. 4.1.2 Strain Energy As we noted previously, solutions to plate problems may be obtained by energy methods that require knowledge of the strain energy. For a linearly elastic material the strain energy is given by Eq. (2.200). Under plane-stress condition the stress components σz, τxz, and τyz are zero (Eq. 2.121), and the expression for the strain energy simplifies to U = 1 2 ) Lx 0 ) Ly 0 ) ht −hb (σxx + σyy + τxyγxy) dzdydx, (4.10) where ht and hb are the distances from the reference plane to the plate’s surfaces (Fig. 3.12). The strain components are (Eq. 3.7)    x y γxy    =    o x o y γ o xy    + z    κx κy κxy    . (4.11) 7 Ibid., p. 84