174 SANDWICH PLATES Substitution of Eqs.(5.4)-(5.15)and Egs.(5.26)-(5.32)(derived on pages 175- 176)into Eq.(5.21)gives e Au 42 A6 B11 B12 B16 42 A26 B12 B22 B26 A16 6 As6 B26 B B1 B12 B16 D D16 B2 B2 B26 D2 D26 Ky KxY B26 B66 D16 26 D66 +{ dydx, (5.22) where the superscript T denotes transpose of the vector. 5.1.3 Stiffness Matrices of Sandwich Plates The stiffness matrices are evaluated by assuming that the thickness of the core remains constant under loading and the in-plane stiffnesses of the core are negligi- ble.Under these assumptions the [A],[B],and D]stiffness matrices of a sandwich plate are governed by the stiffnesses of the facesheets and may be obtained by the parallel axes theorem(Eq.3.47,page 80).The resulting expressions are given in Table 5.1.In this table the [A],[B]',[D]'and [A],[B],[D]are to be eval- uated in a coordinate system whose origin is at each facesheet's reference plane. When the top and bottom facesheets are identical and their layup is symmetri- cal with respect to each facesheet's midplane,the B]matrix is zero and the [A], [D]matrices simplify,as shown in Table 5.1.(When the layup of each facesheet is symmetrical,the reference plane may conveniently be taken at the facesheets' Table 5.1.The [A],[B],[D]stiffness matrices of sandwich plates.The supersripts t and b refer to the top and bottom facesheets.The distances d,d,and d are shown in Figure 5.2. Layup of each facesheet with respect to the facesheet's midplane Symmetrical Unsymmetrical (identical facesheets) [4 [A+[A 2[A (B] d'[4-P[A+[B+[B 0 [D] (d2[4+(d)2[4+[D+[DP +2d [B]-2db [B] P[A'+2[D
174 SANDWICH PLATES Substitution of Eqs. (5.4)–(5.15) and Eqs. (5.26)–(5.32) (derived on pages 175– 176) into Eq. (5.21) gives 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 + {γxz γyz} S 11 S 12 S 12 S 22! γxz γyz dydx, (5.22) where the superscript T denotes transpose of the vector. 5.1.3 Stiffness Matrices of Sandwich Plates The stiffness matrices are evaluated by assuming that the thickness of the core remains constant under loading and the in-plane stiffnesses of the core are negligible. Under these assumptions the [A], [B], and [D]stiffness matrices of a sandwich plate are governed by the stiffnesses of the facesheets and may be obtained by the parallel axes theorem (Eq. 3.47, page 80). The resulting expressions are given in Table 5.1. In this table the [A] t , [B] t , [D] t and [A] b , [B] b , [D] b are to be evaluated in a coordinate system whose origin is at each facesheet’s reference plane. When the top and bottom facesheets are identical and their layup is symmetrical with respect to each facesheet’s midplane, the [B] matrix is zero and the [A], [D] matrices simplify, as shown in Table 5.1. (When the layup of each facesheet is symmetrical, the reference plane may conveniently be taken at the facesheets’ Table 5.1. The [A], [B], [D] stiffness matrices of sandwich plates. The supersripts t and b refer to the top and bottom facesheets. The distances d, dt , and db are shown in Figure 5.2. Layup of each facesheet with respect to the facesheet’s midplane Symmetrical Unsymmetrical (identical facesheets) [A] [A] t + [A] b 2 [A] t [B] dt [A] t − db [A] b + [B] t + [B] b 0 [D] (dt ) 2 [A] t + db 2 [A] b + [D] t + [D] b + 2dt [B] t − 2db [B] b 1 2d2 [A] t + 2 [D] t
5.1 GOVERNING EQUATIONS 175 Figure 5.5:Shear stress distribution x(left) in a sandwich plate and the approximate dis- tribution (right). midplane.)When the top and bottom facesheets are unsymmetrical with respect to the facesheets'midplane but are symmetrical with respect to the midplane of the sandwich plate,then [A]=[A],[B]=-[B],[D]=[D],and the [A],[B], [D]matrices of the sandwich plate become [A=2[A (5.23) [B=0 (5.24) [D]=2[4+2[D+2d[B. (5.25) The shear stiffness matrix [S]is determined as follows.In the core,as a conse- quence of the assumption that the in-plane stiffnesses are negligible,the transverse shear stress tx is uniform.In general,in the facesheets the shear stress distribu- tion is as shown in Figure 5.5(left).We approximate this distribution by the linear shear stress distribution shown in Figure 5.5(right).Accordingly,the transverse shear force V is Vx= :dk=+2+2=4 (5.26) -h where the superscripts c,t,and b refer to the core,the top,and the bottom facesheets,respectively.The distance d=c+t2+t/2 is shown in Figure 5.5. Similarly,we have Vy =t d. (5.27) The stress-strain relationship for the core material is given by Eqs.(2.20)and (2.27).With the superscript c identifying the core,these equations give 骨-[] (5.28) where C are the elements of the core stiffnesses matrix. We neglect the shear deformation of the thin facesheets.With this approxima- tion the shear deformation ye of the cross section is as shown in Figure 5.6(left). We approximate this deformation by the average shear deformation ys:shown in
5.1 GOVERNING EQUATIONS 175 τxz τxz t b t t c d c xz τ c xz τ Figure 5.5: Shear stress distribution τxz (left) in a sandwich plate and the approximate distribution (right). midplane.) When the top and bottom facesheets are unsymmetrical with respect to the facesheets’ midplane but are symmetrical with respect to the midplane of the sandwich plate, then [A] t = [A] b , [B] t = − [B] b , [D] t = [D] b , and the [A], [B], [D] matrices of the sandwich plate become [A] = 2 [A] t (5.23) [B] = 0 (5.24) [D] = 1 2 d2 [A] t + 2 [D] t + 2d [B] t . (5.25) The shear stiffness matrix [S ] is determined as follows. In the core, as a consequence of the assumption that the in-plane stiffnesses are negligible, the transverse shear stress τxz is uniform. In general, in the facesheets the shear stress distribution is as shown in Figure 5.5 (left). We approximate this distribution by the linear shear stress distribution shown in Figure 5.5 (right). Accordingly, the transverse shear force Vx is Vx = ) ht −hb τxzdz = τ c xzc + τ c xz tt 2 + τ c xz tb 2 = τ c xzd, (5.26) where the superscripts c, t, and b refer to the core, the top, and the bottom facesheets, respectively. The distance d = c + tt /2 + tb/2 is shown in Figure 5.5. Similarly, we have Vy = τ c yzd. (5.27) The stress–strain relationship for the core material is given by Eqs. (2.20) and (2.27). With the superscript c identifying the core, these equations give τ c xz τ c yz = Cc 55 Cc 45 Cc 45 Cc 44 ! γ c xz γ c yz , (5.28) where Cc i j are the elements of the core stiffnesses matrix. We neglect the shear deformation of the thin facesheets. With this approximation the shear deformation γ c xz of the cross section is as shown in Figure 5.6 (left). We approximate this deformation by the average shear deformation γxz shown in
176 SANDWICH PLATES Figure 5.6:Shear deformation of a sandwich plate. Figure 5.6(middle).The relationship between this average shear deformation and the core deformation is given by (see Fig.5.6,right) d Y:=cYaz (5.29) Similarly,we have d i=cYys (5.30) Equations(5.26)-(5.30)yield the relationship between the transverse shear forces and the average shear deformation: (5.31) By comparing this equation with Eq.(5.15),we obtain [-[剧 (5.32) The preceding four elements of the matrix [Ce]characterize the core material, whereas [is the shear stiffness matrix of the sandwich plate.We point out that [S]is not the inverse of the [C]matrix. Orthotropic sandwich plate.A sandwich plate is orthotropic when both face- sheets as well as the core are orthotropic and the orthotropy directions are parallel to the edges.The facesheets may be different,and their layups may be unsym- metrical.For such an orthotropic sandwich plate there are no extension-shear, bending-twist,and extension-twist couplings.Accordingly,the following elements of the stiffness matrices are zero: A6=A26=B16=B26=D16=D26=0. (5.33) Furthermore,for an orthotropic sandwich plate the transverse shear force V acting in the x-z plane does not cause a shear strain yy in the y-z plane.This condition gives 5i2=0. (5.34) Isotropic sandwich plate.A sandwich plate is isotropic when the core of the sandwich plate is made of an isotropic (such as foam)or transversely isotropic (such as honeycomb)material and the top and bottom facesheets are made of
176 SANDWICH PLATES γxz d c γxz γ c xz γ c xz Figure 5.6: Shear deformation of a sandwich plate. Figure 5.6 (middle). The relationship between this average shear deformation and the core deformation is given by (see Fig. 5.6, right) γ c xz = d c γxz. (5.29) Similarly, we have γ c yz = d c γyz. (5.30) Equations (5.26)–(5.30) yield the relationship between the transverse shear forces and the average shear deformation: Vx Vy = d2 c Cc 55 Cc 45 Cc 45 Cc 44 ! γxz γyz . (5.31) By comparing this equation with Eq. (5.15), we obtain S 11 S 12 S 12 S 22! = d2 c Cc 55 Cc 45 Cc 45 Cc 44 ! . (5.32) The preceding four elements of the matrix [Cc ] characterize the core material, whereas [S ] is the shear stiffness matrix of the sandwich plate. We point out that [S ] is not the inverse of the [C] matrix. Orthotropic sandwich plate. A sandwich plate is orthotropic when both facesheets as well as the core are orthotropic and the orthotropy directions are parallel to the edges. The facesheets may be different, and their layups may be unsymmetrical. For such an orthotropic sandwich plate there are no extension–shear, bending–twist, and extension–twist couplings. Accordingly, the following elements of the stiffness matrices are zero: A16 = A26 = B16 = B26 = D16 = D26 = 0. (5.33) Furthermore, for an orthotropic sandwich plate the transverse shear force Vx acting in the x–z plane does not cause a shear strain γyz in the y–z plane. This condition gives S 12 = 0. (5.34) Isotropic sandwich plate. A sandwich plate is isotropic when the core of the sandwich plate is made of an isotropic (such as foam) or transversely isotropic (such as honeycomb) material and the top and bottom facesheets are made of
5.1 GOVERNING EQUATIONS 177 Reference plane Neutral plane 不 d 米 e c/2 Midplane Figure 5.7:Neutral plane of an isotropic sandwich plate. identical isotropic materials or are identical quasi-isotropic laminates.The thick- nesses of the top and bottom facesheets may be different. For isotropic facesheets the [B]matrix is zero ([B]=0).The [A]and [D] matrices for the isotropic facesheets are (Egs.3.41 and 3.42) 1 0 0 A= 1 1-(v02 00 00号 (5.35) where the superscript i refers to the top (i =t)or to the bottom (i b)facesheet (Fig.5.7)and Ef and v are the Young modulus and the Poisson ratio of the facesheets. We now proceed to evaluate the [A],[B],[D]matrices for the entire sandwich plate.To this end,we choose a reference plane located at the center of gravity of the two facesheets.The distance o from the midplane of the core to the center of gravity is (Fig.5.7) 1(c+1)-1b(c+1b) (5.36) 2(+1) The distances dt and db between the reference plane (passing through the center of gravity)and the midplanes of the facesheets are t=5+写-e=+5+e (5.37) By substituting Eqs.(5.35)-(5.37)into the expression for the B]matrix given in Table 5.1(page 174)we obtain that for the entire sandwich plate the [B]matrix is zero with reference to the o reference plane.This means that for a sandwich plate with isotropic core and isotropic facesheets bending does not cause strains in this plane.Therefore,this reference plane is a"neutral plane." By substituting the expressions of d and db(Eq.5.37)into the expressions given in Table 5.1,we obtain the following [A]and [D]matrices for the sandwich
5.1 GOVERNING EQUATIONS 177 Reference plane Neutral plane ≡ t b t t d b d t c/2 c/2 Midplane Figure 5.7: Neutral plane of an isotropic sandwich plate. identical isotropic materials or are identical quasi-isotropic laminates. The thicknesses of the top and bottom facesheets may be different. For isotropic facesheets the [B] matrix is zero ([B] i = 0). The [A] and [D] matrices for the isotropic facesheets are (Eqs. 3.41 and 3.42) [A] i = ti Ef 1 − (νf )2 1 νf 0 νf 1 0 0 0 1−νf 2 [D] i = (ti )3Ef 12(1 − (νf )2) 1 νf 0 νf 1 0 0 0 1−νf 2 , (5.35) where the superscript i refers to the top (i = t) or to the bottom (i = b) facesheet (Fig. 5.7) and Ef and νf are the Young modulus and the Poisson ratio of the facesheets. We now proceed to evaluate the [A], [B], [D] matrices for the entire sandwich plate. To this end, we choose a reference plane located at the center of gravity of the two facesheets. The distance from the midplane of the core to the center of gravity is (Fig. 5.7) = tt (c + tt ) − tb(c + tb) 2(tt + tb) . (5.36) The distances dt and db between the reference plane (passing through the center of gravity) and the midplanes of the facesheets are dt = c 2 + tt 2 − db = c 2 + tb 2 + . (5.37) By substituting Eqs. (5.35)–(5.37) into the expression for the [B] matrix given in Table 5.1 (page 174) we obtain that for the entire sandwich plate the [B] matrix is zero with reference to the reference plane. This means that for a sandwich plate with isotropic core and isotropic facesheets bending does not cause strains in this plane. Therefore, this reference plane is a “neutral plane.” By substituting the expressions of dt and db (Eq. 5.37) into the expressions given in Table 5.1, we obtain the following [A] and [D] matrices for the sandwich
178 SANDWICH PLATES Table 5.2.The stiffnesses and the Poisson ratios of isotropic solid plates and isotropic sandwich plates;R is defined in Eq.(3.46). Isotropic sandwich plate Isotropic Isotropic Quasi-isotropic solid plate facesheets facesheets Aiso 品 +内海 (+1b)R Diso El (P+(d2b++ 121-2 1-( E [t(d)+(d]R Q11+02+602-40 8R plate: [A]= (5.38) where Aiso and Diso are defined in Table 5.2. When the core is isotropic in the plane parallel to the facesheets from Eq.(2.40) we have C4s =0,C44=(C11-C12)/2,and the shear stiffnesses are(Eq.5.32) 1=2=5=CGi,c金 c 2 32=0. (5.39) The sandwich plate may also be treated as isotropic when the top and bottom facesheets are quasi-isotropic laminates(page 79)consisting of unidirectional plies made of the same material.For such sandwich plates the [B]matrix is negligible, the [A]and [D]matrices are approximated by Eq.(5.38)(with the terms Aiso and Diso defined in Table 5.2),and the elements of the shear stiffness matrix are given byEq.(5.39). 5.2 Deflection of Rectangular Sandwich Plates 5.2.1 Long Plates We consider a long rectangular sandwich plate whose length is large compared with its width (Ly>Lx).The long edges may be built-in,simply supported,or free,as shown in Figure 5.8.The sandwich plate is subjected to a transverse load p(per unit area).This load,as well as the edge supports,does not vary along the longitudinal y direction. The deflected surface of the sandwich plate may be assumed to be cylindrical at a considerable distance from the short ends (Fig.4.4).The generator of this cylindrical surface is parallel to the longitudinal y-axis of the plate,and hence the
178 SANDWICH PLATES Table 5.2. The stiffnesses and the Poisson ratios of isotropic solid plates and isotropic sandwich plates; R is defined in Eq. (3.46). Isotropic sandwich plate Isotropic Isotropic Quasi-isotropic solid plate facesheets facesheets Aiso Eh 1−ν2 (tt + tb) Ef 1−(νf)2 (tt + tb)R Diso Eh3 12(1−ν2 ) (dt )2tt + (db)2tb + (tt)3 + (tb)3 12 1−(νf)2 Ef " tt (dt ) 2 + tb(db)2 # R νiso ν νf Q11 + Q22 + 6Q12 − 4Q66 8R plate: [A] = Aiso 1 νf νf 1 1−νf 2 [D] = Diso 1 νf νf 1 1−νf 2 , (5.38) where Aiso and Diso are defined in Table 5.2. When the core is isotropic in the plane parallel to the facesheets from Eq. (2.40) we have C45 = 0, C44 = (C11 − C12)/2, and the shear stiffnesses are (Eq. 5.32) S 11 = S 22 = S = d2 c Cc 11 − Cc 12 2 S 12 = 0. (5.39) The sandwich plate may also be treated as isotropic when the top and bottom facesheets are quasi-isotropic laminates (page 79) consisting of unidirectional plies made of the same material. For such sandwich plates the [B] matrix is negligible, the [A] and [D] matrices are approximated by Eq. (5.38) (with the terms Aiso and Diso defined in Table 5.2), and the elements of the shear stiffness matrix are given by Eq. (5.39). 5.2 Deflection of Rectangular Sandwich Plates 5.2.1 Long Plates We consider a long rectangular sandwich plate whose length is large compared with its width (Ly Lx). The long edges may be built-in, simply supported, or free, as shown in Figure 5.8. The sandwich plate is subjected to a transverse load p (per unit area). This load, as well as the edge supports, does not vary along the longitudinal y direction. The deflected surface of the sandwich plate may be assumed to be cylindrical at a considerable distance from the short ends (Fig. 4.4). The generator of this cylindrical surface is parallel to the longitudinal y-axis of the plate, and hence the