5.1 GOVERNING EQUATIONS 171 B Reference plane Figure 5.3:Deformation of a sandwich plate in the x-z plane. Reference plane Next we derive the force-strain relationships.The starting point of the analysis is the expressions for the forces and moments given by Egs.(3.9)and (3.10) h h N= o,dz Ny= Ny= -h (5.7) h h Mx= My= Ztxydz h V= (5.8) where M,M,and V are the in-plane forces,the moments,and the transverse shear forces per unit length (Fig.3.11,page 68),respectively,and ht and hp are the distances from the arbitrarily chosen reference plane to the plate's surfaces (Fig.5.2).The stresses (plane-stress condition)are (Eq.2.126) Ox Q11 012 Q16 (5.9) 16 From Egs.(2.2),(2.3),and (2.11)together with Eq.(5.1)the strains at a dis- tance z from the reference plane are auau° Ex= Xxz 一 -Z- ax auav°，aXy Ey= (5.10) ayay ay w器++-（+） du,avau°av°5.1 GOVERNING EQUATIONS 171 x z B A u  w  γxz χxz Reference plane Reference plane x w ∂ ∂ o A′ B′ Figure 5.3: Deformation of a sandwich plate in the x–z plane. Next we derive the force–strain relationships. The starting point of the analysis is the expressions for the forces and moments given by Eqs. (3.9) and (3.10) Nx = ) ht −hb σxdz Ny = ) ht −hb σydz Nxy = ) ht −hb τxydz Mx = ) ht −hb zσxdz My = ) ht −hb zσydz Mxy = ) ht −hb zτxydz (5.7) Vx = ) ht −hb τxzdz Vy = ) ht −hb τyzdz, (5.8) where Ni , Mi , and Vi are the in-plane forces, the moments, and the transverse shear forces per unit length (Fig. 3.11, page 68), respectively, and ht and hb are the distances from the arbitrarily chosen reference plane to the plate’s surfaces (Fig. 5.2). The stresses (plane–stress condition) are (Eq. 2.126)    σx σy τxy    =    Q11 Q12 Q16 Q12 Q22 Q26 Q16 Q26 Q66       x y γxy    . (5.9) From Eqs. (2.2), (2.3), and (2.11) together with Eq. (5.1) the strains at a dis￾tance z from the reference plane are x = ∂u ∂x = ∂uo ∂x − z ∂χxz ∂x y = ∂v ∂y = ∂vo ∂y − z ∂χyz ∂y (5.10) γxy = ∂u ∂y + ∂v ∂x = ∂uo ∂y + ∂vo ∂x − z ∂χxz ∂y + ∂χyz ∂x