4.2 DEFLECTION OF RECTANGULAR PLATES 93 The stresses and the strains at a point are related by (Eq.3.13) (4.12) By substituting Eqs.(4.11)and (4.12)into Eq.(4.10)and by utilizing the definitions of the [A,[B],[D]matrices (Eq.3.18),we obtain the following expression for the strain energy: 公 「A1 A12 A16 B11 B12 B16 e L 内2 A22 426 B12 B22 B26 1 U= 46 6 A66 B16 B26 B66 dydx. Kx B11 B12 B16 D11 Di2 D16 Kx 0 0 Ky B12 B2 B26 D12 D22 D26 Ky Kxy B16 B26 B66 D16 D26 D66」 Kxy (4.13) The superscript T denotes the transpose of the vector. 4.2 Deflection of Rectangular Plates 4.2.1 Pure Bending and In-Plane Loads We consider an unsupported rectangular plate subjected to pure bending and to in-plane loads(Fig.4.2).The in-plane forces and moments are related to the reference plane's strains and curvatures by Eq.(4.3).Six of the twelve quantities appearing in this equation must be specified as follows: N or e Mx or Kx Ny or e My or Ky (4.14) Nry or esy Mry or Kxy. With six of the quantities chosen(Eq.4.14),the remaining six may be obtained by solving the six simultaneous equations given by Eq.(4.3).Once the curvatures Figure 4.2:Rectangular plate subjected to bending and in-plane loads.4.2 DEFLECTION OF RECTANGULAR PLATES 93 The stresses and the strains at a point are related by (Eq. 3.13)    σx σy τxy    = [Q]    x y γxy    . (4.12) By substituting Eqs. (4.11) and (4.12) into Eq. (4.10) and by utilizing the definitions of the [A], [B], [D] matrices (Eq. 3.18), we obtain the following expression for the strain energy: U = 1 2 ) Lx 0 ) Ly 0    o x o y γ o xy κx κy κxy    T          A11 A12 A16 B11 B12 B16 A12 A22 A26 B12 B22 B26 A16 A26 A66 B16 B26 B66 B11 B12 B16 D11 D12 D16 B12 B22 B26 D12 D22 D26 B16 B26 B66 D16 D26 D66             o x o y γ o xy κx κy κxy    dydx. (4.13) The superscript T denotes the transpose of the vector. 4.2 Deflection of Rectangular Plates 4.2.1 Pure Bending and In-Plane Loads We consider an unsupported rectangular plate subjected to pure bending and to in-plane loads (Fig. 4.2). The in-plane forces and moments are related to the reference plane’s strains and curvatures by Eq. (4.3). Six of the twelve quantities appearing in this equation must be specified as follows: Nx or o x Mx or κx Ny or o y My or κy Nxy or o xy Mxy or κxy. (4.14) With six of the quantities chosen (Eq. 4.14), the remaining six may be obtained by solving the six simultaneous equations given by Eq. (4.3). Once the curvatures Nx x Mx Mxy Nxy Nxy My Mxy Ny y z Figure 4.2: Rectangular plate subjected to bending and in-plane loads