180 SANDWICH PLATES where ET and S are the bending and shear stiffnesses of the isotropic sandwich beam,respectively,and p'is the load per unit length. The equations describing the deflections of long sandwich plates and isotropic sandwich beams are identical when in Egs.(5.45)and (5.46),Du1,S11,and p are replaced,respectively,by EI,S,and p'.Therefore,the deflection of a long sandwich plate(symmetrical layup)may be obtained by substituting the values of Du,Su,and p for EI.S,and p'in the expression,given in Section 7.3,for the deflection of the corresponding isotropic beam. When the layup is unsymmetrical,the expression for the moment Mr can be derived analogously to the equation of a solid composite plate(Section 4.2.2). Here we only quote the result,which for sandwich plates is M (5.49) ax where xxz is shown in Figure 5.3.The term in parentheses is the bending stiffness parameter defined by Eq.(4.52).Equations(5.42),(5.43),(5.44,right),and (5.49), together with Eq.(5.2),give sandwich plate,unsymmetrical layup: -yd0+p=0 dx3 (5.50) dw dx2 dx-Xxz =0 (5.51) The preceding equations describing deflections of sandwich plates(unsymmet- rical layup)become identical to the equations of sandwich beams(Eqs.5.47 and 5.48)when Su,and p are replaced,respectively,by ET.S.and p'.Therefore, the deflection of a long sandwich plate(unsymmetrical layup)may be obtained by substituting the values of,Su,and p for El.S.and p'in the expression for the deflection of the corresponding isotropic beam. 5.1 Example.A 0.9-m-long and 0.2-m-wide rectangular sandwich plate is made of a 0.02-m-thick core covered on both sides by graphite epoxy facesheets.The ma- terial properties are given in Table 3.6 (page 81).The layup of each facesheet is [45/012/+453],and the thickness of each facesheet is 0.002 m (Fig.5.9).The 0-degree plies are parallel to the short edge of the plate.The plate is either simply supported or built-in along all four edges (Fig.5.10).The plate is subjected to a 下t=2mm d=22 mm c 20 mm t=2mm Figure 5.9:The cross section of the sandwich plate in Example 5.1.180 SANDWICH PLATES where EI and S
are the bending and shear stiffnesses of the isotropic sandwich beam, respectively, and p is the load per unit length. The equations describing the deflections of long sandwich plates and isotropic sandwich beams are identical when in Eqs. (5.45) and (5.46), D11, S 11, and p are replaced, respectively, by EI , S
, and p . Therefore, the deflection of a long sandwich plate (symmetrical layup) may be obtained by substituting the values of D11, S 11, and p for EI , S
, and p in the expression, given in Section 7.3, for the deflection of the corresponding isotropic beam. When the layup is unsymmetrical, the expression for the moment Mx can be derived analogously to the equation of a solid composite plate (Section 4.2.2). Here we only quote the result, which for sandwich plates is Mx = − - D 11 − B2 16 A 66 . % &' (  ∂χxz ∂x , (5.49) where χxz is shown in Figure 5.3. The term in parentheses is the bending stiffness parameter defined by Eq. (4.52). Equations (5.42), (5.43), (5.44, right), and (5.49), together with Eq. (5.2), give sandwich plate, unsymmetrical layup: − d3χxz dx3 0 + p = 0 (5.50)  d2χxz dx2 + S 11 dwo dx − χxz = 0. (5.51) The preceding equations describing deflections of sandwich plates (unsymmet￾rical layup) become identical to the equations of sandwich beams (Eqs. 5.47 and 5.48) when , S 11, and p are replaced, respectively, by EI , S
, and p . Therefore, the deflection of a long sandwich plate (unsymmetrical layup) may be obtained by substituting the values of , S 11, and p for EI , S
, and p in the expression for the deflection of the corresponding isotropic beam. 5.1 Example. A 0.9-m-long and 0.2-m-wide rectangular sandwich plate is made of a 0.02-m-thick core covered on both sides by graphite epoxy facesheets. The ma￾terial properties are given in Table 3.6 (page 81). The layup of each facesheet is [±45f 2/012/ ± 45f 2], and the thickness of each facesheet is 0.002 m (Fig. 5.9). The 0-degree plies are parallel to the short edge of the plate. The plate is either simply supported or built-in along all four edges (Fig. 5.10). The plate is subjected to a t = 2 mm c = 20 mm t = 2 mm d = 22 mm Figure 5.9: The cross section of the sandwich plate in Example 5.1.