Copyrighted Materials 0CpUyPress o CHAPTER FIVE Sandwich Plates Sandwich plates,consisting of a core covered by facesheets,are frequently used instead of solid plates because of their high bending stiffness-to-weight ratio.The high bending stiffness is the result of the distance between the facesheets,which carry the load,and the light weight is due to the light weight of the core. Here,we consider rectangular sandwich plates with facesheets on both sides of the core(Figs.5.1 and 5.2).Each facesheet may be an isotropic material or a fiber- reinforced composite laminate but must be thin compared with the core.The core may be foam or honeycomb(Fig.5.1)and must have a material symmetry plane parallel to its midplane;the core's in-plane stiffnesses must be small compared with the in-plane stiffnesses of the facesheets. The behavior of thin plates undergoing small deformations may be analyzed by the Kirchhoff hypothesis,namely,by the assumptions that normals remain straight and perpendicular to the deformed reference plane.For a sandwich plate, consisting of a core covered on both sides by facesheets,the first assumption (normals remain straight)is reasonable.However,the second assumption may no longer be valid,because normals do not necessarily remain perpendicular to the reference plane (Fig.5.3).In this case the x and y displacements of a point located at a distance z from an arbitrarily chosen reference plane are u=°-zXx:v=u°-ZXyz (5.1) where uo and vo are the x and y displacements at the reference plane (where =0)and xx,Xyz are the rotations of the normal in the x-z and y-z planes.The angle xr is illustrated in Figure 5.3. As shown in Figure 5.3,the first derivative of the deflection w of the reference plane with respect to x is 8w° =Xxz+yxz. (5.2) ax 169
CHAPTER FIVE Sandwich Plates Sandwich plates, consisting of a core covered by facesheets, are frequently used instead of solid plates because of their high bending stiffness-to-weight ratio. The high bending stiffness is the result of the distance between the facesheets, which carry the load, and the light weight is due to the light weight of the core. Here, we consider rectangular sandwich plates with facesheets on both sides of the core (Figs. 5.1 and 5.2). Each facesheet may be an isotropic material or a fiberreinforced composite laminate but must be thin compared with the core. The core may be foam or honeycomb (Fig. 5.1) and must have a material symmetry plane parallel to its midplane; the core’s in-plane stiffnesses must be small compared with the in-plane stiffnesses of the facesheets. The behavior of thin plates undergoing small deformations may be analyzed by the Kirchhoff hypothesis, namely, by the assumptions that normals remain straight and perpendicular to the deformed reference plane. For a sandwich plate, consisting of a core covered on both sides by facesheets, the first assumption (normals remain straight) is reasonable. However, the second assumption may no longer be valid, because normals do not necessarily remain perpendicular to the reference plane (Fig. 5.3). In this case the x and y displacements of a point located at a distance z from an arbitrarily chosen reference plane are u = uo − zχxz v = vo − zχyz, (5.1) where uo and vo are the x and y displacements at the reference plane (where z = 0) and χxz, χyz are the rotations of the normal in the x–z and y–z planes. The angle χxz is illustrated in Figure 5.3. As shown in Figure 5.3, the first derivative of the deflection wo of the reference plane with respect to x is ∂wo ∂x = χxz + γxz. (5.2) 169
170 SANDWICH PLATES Core Facesheets Figure 5.1:Illustration of the sandwich plate and the honeycomb core. Similarly,the first derivative of the deflection wo of the reference plane with respect to y is dwo ay Xyz+Yyz. (5.3) 5.1 Governing Equations The strains at the reference plane are (Eq.4.2) au° 8v° 0°.auo e= ax ay y8= av ax (5.4) The transverse shear strains are(Egs.5.2 and 5.3) aw° 8w0 Yxz= ax -Xxz Yy:=ay -Xyz. (5.5) For convenience we define Kx,Ky,and Kxy as Kx=一 Xxz Ky=- aXyz Koy =-dxe aXy. (5.6) ax ay ay ax We note that K,K,and Ky are not the curvatures of the reference plane. They are the reference plane's curvatures only in the absence of shear deforma- tion. The three equations above represent the strain-displacement relationships for a sandwich plate. h, Reference plane Figure 5.2:Sandwich-plate geometry
170 SANDWICH PLATES x z y Core Facesheets Figure 5.1: Illustration of the sandwich plate and the honeycomb core. Similarly, the first derivative of the deflection wo of the reference plane with respect to y is ∂wo ∂y = χyz + γyz. (5.3) 5.1 Governing Equations The strains at the reference plane are (Eq. 4.2) o x = ∂uo ∂x o y = ∂vo ∂y γ o xy = ∂uo ∂y + ∂vo ∂x . (5.4) The transverse shear strains are (Eqs. 5.2 and 5.3) γxz = ∂wo ∂x − χxz γyz = ∂wo ∂y − χyz. (5.5) For convenience we define κx, κy, and κxy as κx = −∂χxz ∂x κy = −∂χyz ∂y κxy = −∂χxz ∂y − ∂χyz ∂x . (5.6) We note that κx, κy, and κxy are not the curvatures of the reference plane. They are the reference plane’s curvatures only in the absence of shear deformation. The three equations above represent the strain–displacement relationships for a sandwich plate. t b t t c hb ht d b d t h d Reference plane Figure 5.2: Sandwich-plate geometry
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 distance 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
172 SANDWICH PLATES By combining Eqs.(5.4),(5.7),(5.9),and(5.10)and by utilizing the definitions of the [A],[B],[D]matrices(Eq.3.18),we obtain ax [4 +[B aXy ay (5.11) 8Xx ay ax =[B +[D (5.12) With the definitions in Eq.(5.6),these equations may be written as N Kx [4 +[B (5.13) Kxy M M [B +[D Ky (5.14) Mxy 8 Kxy In addition we need the relationships between the transverse shear forces and the transverse shear strains.The relevant expressions are derived in Section 5.1.3. Here we quote the resulting expression,which is -[原 (5.15) where [S]is the sandwich plate's shear stiffness matrix. In the analyses we may employ either the equilibrium equations or the strain energy.The equilibrium equations are identical to those given for a thin plate (Eqs.4.4and4.5). 5.1.1 Boundary Conditions In order to determine the deflection,the conditions along the four edges of the plate must be specified.An edge may be built-in,free,or simply supported. Boundary conditions for an edge parallel with the y-axis (Fig.5.4)are given below. Along a built-in edge,the deflection w°，the in-plane displacements.°，v°，and the rotations of normals xr,Xyz are zero: w°=0°=v°=0Xr:=Xz=0. (5.16) Along a free edge,where no external loads are applied,the bending M and twist Mry moments,the transverse shear force V,and the in-plane forces Mr
172 SANDWICH PLATES By combining Eqs. (5.4), (5.7), (5.9), and (5.10) and by utilizing the definitions of the [A], [B], [D] matrices (Eq. 3.18), we obtain Nx Ny Nxy = [A] o x o y γ o xy + [B] −∂χxz ∂x −∂χyz ∂y −∂χxz ∂y − ∂χyz ∂x (5.11) Mx My Mxy = [B] o x o y γ o xy + [D] −∂χxz ∂x −∂χyz ∂y −∂χxz ∂y − ∂χyz ∂x . (5.12) With the definitions in Eq. (5.6), these equations may be written as Nx Ny Nxy = [A] o x o y γ o xy + [B] κx κy κxy (5.13) Mx My Mxy = [B] o x o y γ o xy + [D] κx κy κxy . (5.14) In addition we need the relationships between the transverse shear forces and the transverse shear strains. The relevant expressions are derived in Section 5.1.3. Here we quote the resulting expression, which is Vx Vy = S 11 S 12 S 12 S 22! γxz γyz , (5.15) where [S ] is the sandwich plate’s shear stiffness matrix. In the analyses we may employ either the equilibrium equations or the strain energy. The equilibrium equations are identical to those given for a thin plate (Eqs. 4.4 and 4.5). 5.1.1 Boundary Conditions In order to determine the deflection, the conditions along the four edges of the plate must be specified. An edge may be built-in, free, or simply supported. Boundary conditions for an edge parallel with the y-axis (Fig. 5.4) are given below. Along a built-in edge, the deflection wo, the in-plane displacements uo, vo, and the rotations of normals χxz, χyz are zero: wo = 0 uo = vo = 0 χxz = χyz = 0. (5.16) Along a free edge, where no external loads are applied, the bending Mx and twist Mxy moments, the transverse shear force Vx, and the in-plane forces Nx,
5.1 GOVERNING EQUATIONS 173 Simply supported Built-in Free Without With side plate side plate Figure 5.4:Boundary conditions for an edge parallel to the y-axis. Nry are zero: M=Mry=0 Vt=0 Nt=Nty=0. (5.17) Along a simply supported edge,the deflection wo,the bending Mr and twist Mry moments,and the in-plane forces N,Nry are zero: w°=0M=My=0N=Nxy=0. (5.18) When in-plane motions are prevented by the support,the in-plane forces are not zero(N 0,Ny0),whereas the in-plane displacements are zero: °=0v°=0. (5.19) When there is a rigid plate covering the side of the sandwich plate the normal cannot rotate in the y-z plane,and we have Xz=0. (5.20) However,the twist moment is not zero (My0). For an edge parallel with the x-axis,the equations above hold with x and y interchanged. 5.1.2 Strain Energy As we noted previously,solutions to plate problems may be obtained by the equa- tions described above or via energy methods.The strain energy(for a linearly elastic material)is given by Eq.(2.200).The thickness of the sandwich plate is assumed to remain unchanged and,accordingly,e=0.The expression for the strain energy(Eq.2.200)simplifies to L Ly h (oxex +oyey+txyrxy +txzYxz+tyzryz)dzdydx. (5.21)
5.1 GOVERNING EQUATIONS 173 Built-in Free Simply supported Without With side plate side plate z x Figure 5.4: Boundary conditions for an edge parallel to the y-axis. Nxy are zero: Mx = Mxy = 0 Vx = 0 Nx = Nxy = 0. (5.17) Along a simply supported edge, the deflection wo, the bending Mx and twist Mxy moments, and the in-plane forces Nx, Nxy are zero: wo = 0 Mx = Mxy = 0 Nx = Nxy = 0. (5.18) When in-plane motions are prevented by the support, the in-plane forces are not zero (Nx = 0, Nxy = 0), whereas the in-plane displacements are zero: uo = 0 vo = 0. (5.19) When there is a rigid plate covering the side of the sandwich plate the normal cannot rotate in the y–z plane, and we have χyz = 0. (5.20) However, the twist moment is not zero (Mxy = 0). For an edge parallel with the x-axis, the equations above hold with x and y interchanged. 5.1.2 Strain Energy As we noted previously, solutions to plate problems may be obtained by the equations described above or via energy methods. The strain energy (for a linearly elastic material) is given by Eq. (2.200). The thickness of the sandwich plate is assumed to remain unchanged and, accordingly, z = 0. The expression for the strain energy (Eq. 2.200) simplifies to U = 1 2 ) Lx 0 ) Ly 0 ) ht −hb (σxx + σyy + τxyγxy + τxzγxz + τyzγyz) dzdydx. (5.21)
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