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)