4.1 GOVERNING EQUATIONS 91 Figure 4.1:Forces and loads acting on an element of the plate. where uo and vo are the displacements of the reference plane in the x and y directions,and w is the out-of-plane displacement (deflection)of this plane.The force-strain relationships are (Eq.3.21) [A1 A12 A16 B11 B12 B16 N A2 A 6 B12 Br B26 A6 6 A66 B16 B26 B66 M Bu B12 B16 D11 D12 D16 (4.3) Kx M B12 B22 B26 D12 D22 D26 Ky xy B16 B26 B66 D16 D26 D66 Kxy In the analyses we may employ either the equilibrium equations or the strain energy. The equilibrium equations are5 aNs aNg.=-px ax ay aNy +x」 (4.4) ay aNy=一Py 业+业 ax ay =一P V= OM:aMy ax V= aMy aMey ay ay ax (4.5) where pr,py,and p:are the components of the distributed surface load (per unit area);N,N,and Ny are the in-plane forces (per unit length);Vr and Vy are the transverse shear forces (per unit length);Mr.My and Mry are,respectively,the bending moments and the twist moment(per unit length)(Fig.4.1). 6 S.P.Timoshenko and S.Woinowsky-Krieger,Theory of Plates and Shells.2nd edition.McGraw-Hill, New York,1959,p.80.4.1 GOVERNING EQUATIONS 91 x y x y x y Nx Nxy Nyx Ny Mxy Mx My Myx Vy Vx px py pz Figure 4.1: Forces and loads acting on an element of the plate. where uo and vo are the displacements of the reference plane in the x and y directions, and wo is the out-of-plane displacement (deflection) of this plane. The force–strain relationships are (Eq. 3.21)    Nx Ny Nxy Mx My Mxy    =          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    . (4.3) In the analyses we may employ either the equilibrium equations or the strain energy. The equilibrium equations are6 ∂Nx ∂x + ∂Nxy ∂y = −px ∂Ny ∂y + ∂Nxy ∂x = −py (4.4) ∂Vx ∂x + ∂Vy ∂y = −pz Vx = ∂Mx ∂x + ∂Mxy ∂y Vy = ∂My ∂y + ∂Mxy ∂x , (4.5) where px, py, and pz are the components of the distributed surface load (per unit area); Nx, Ny, and Nxy are the in-plane forces (per unit length); Vx and Vy are the transverse shear forces (per unit length); Mx, My and Mxy are, respectively, the bending moments and the twist moment (per unit length) (Fig. 4.1). 6 S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells. 2nd edition. McGraw-Hill, New York, 1959, p. 80