Copyrighted Materials 0CpUyPress o CHAPTER FOUR Thin Plates In practice we frequently encounter "thin"plates whose thickness is small com- pared with all other dimensions.Such a plate,undergoing small displacements, may be analyzed with the approximations that the strains vary linearly across the plate,(out-of-plane)shear deformations are negligible,and the out-of-plane normal stress o:and shear stresses tx,tyz are small compared with the in-plane normal ox,oy,and shear try stresses. Under certain conditions,solutions may be obtained for thin plates either by the solution of the differential equations representing equilibrium or by energy methods.Here we demonstrate the use of the first method via the example of long plates and the second method via examples of rectangular plates either with sym- metrical layup or with orthotropic and symmetrical layup.(For orthotropic plates the directions of orthotropy are parallel to the edges of the plate.)We chose these three types of problems because (i)they illustrate the analytical approaches and the use of the relevant equations,(ii)solutions can be obtained without extensive numerical algorithms,and last,but not least,(iii)they are of practical interest. Additionally,and importantly,these problems provide insights that are useful when analyzing plates by numerical methods. Although the specification of orthotropy may seem to be overly restrictive,in fact it does not unduly limit the applicability of the analyses.The reason for this is that plates are often made according to the 10-percent rule,and such plates behave similarly to orthotropic plates.2 Therefore,solutions for orthotropic plates provide good approximations of the deflections,maximum bending moments,buckling loads,and natural frequencies of nonorthotropic plates that have symmetrical layup and are constructed according to the 10-percent rule.The 10-percent rule 1 J.M.Whitney,Structural Analysis of Laminated Anisotropic Plates.Technomic,Lancaster, Pennsylvania,1987. 2 I.Veres and L.P.Kollar,Approximate Analysis of Mid-plane Symmetric Rectangular Composite Plates.Journal of Composite Materials,Vol.36,673-684,2002. 89
CHAPTER FOUR Thin Plates In practice we frequently encounter “thin” plates whose thickness is small compared with all other dimensions. Such a plate, undergoing small displacements, may be analyzed with the approximations that the strains vary linearly across the plate, (out-of-plane) shear deformations are negligible, and the out-of-plane normal stress σz and shear stresses τxz, τyz are small compared with the in-plane normal σx, σy, and shear τxy stresses. Under certain conditions, solutions may be obtained for thin plates either by the solution of the differential equations representing equilibrium or by energy methods.1 Here we demonstrate the use of the first method via the example of long plates and the second method via examples of rectangular plates either with symmetrical layup or with orthotropic and symmetrical layup. (For orthotropic plates the directions of orthotropy are parallel to the edges of the plate.) We chose these three types of problems because (i) they illustrate the analytical approaches and the use of the relevant equations, (ii) solutions can be obtained without extensive numerical algorithms, and last, but not least, (iii) they are of practical interest. Additionally, and importantly, these problems provide insights that are useful when analyzing plates by numerical methods. Although the specification of orthotropy may seem to be overly restrictive, in fact it does not unduly limit the applicability of the analyses. The reason for this is that plates are often made according to the 10-percent rule, and such plates behave similarly to orthotropic plates.2 Therefore, solutions for orthotropic plates provide good approximations of the deflections, maximum bending moments, buckling loads, and natural frequencies of nonorthotropic plates that have symmetrical layup and are constructed according to the 10-percent rule. The 10-percent rule 1 J. M. Whitney, Structural Analysis of Laminated Anisotropic Plates. Technomic, Lancaster, Pennsylvania, 1987. 2 I. Veres and L. P. Koll´ar, Approximate Analysis of Mid-plane Symmetric Rectangular Composite Plates. Journal of Composite Materials, Vol. 36, 673–684, 2002. 89
90 THIN PLATES requires that the plate satisfy the following conditions: The plate is made of unidirectional plies. There are at least three ply orientations. ·The angles between the fibers are at least 15° The number of plies in each fiber direction is at least 10 percent of the total number of plies. Plates conforming to the 10-percent rule have better load bearing capabilities than unidirectional or angle-ply laminates for the following reasons. Unidirectional plies are stiffer and stronger in the 0-degree fiber direction than in the 90-degree direction perpendicular to the fibers.Thus,laminates made of uni- directional plies are ill-suited to carry load in the 90-degree direction.Angle-ply laminates with only two fiber directions do not resist well tensile loads applied along the symmetry axis.Plates made by the 10-percent rule minimize these short- comings. The specification of symmetrical layup is less restrictive than it may appear because the analyses of symmetrical plates(for which([B]=0)can readily be extended to unsymmetrical plates ([B]0)with the use of the reduced bending stiffness [D]*,defined as 3.4.5 [D*=[D]-[BA-[B (4.1) The deflections,maximum bending moments,buckling loads,and natural fre- quencies of unsymmetrical plates can be approximated by replacing [D]by [D]* in the expressions derived for symmetrical plates. 4.1 Governing Equations In this section we summarize the equations used in analyzing thin plates.We employ the x,y,z coordinate system.The origin is at the midplane for plates with symmetrical layup and at a suitably chosen reference plane for plates with unsymmetrical layup. The strains and curvatures of the reference plane(Fig.3.10)are (Egs.3.1,3.8) au° e= avo au°，8v° ax 9= ay y,= ay ax (4.2) 02w° 82w0 282w° Kx三 8x2 ay2 Kxy=一 axay 3 J.M.Whitney,Structural Analysis of Laminated Anisotropic Plates.Technomic,Lancaster. Pennsylvania,1987,p.203. 4 E.Reissner and Y.Stavsky,Bending and Stretching of Certain Types of Heterogeneous Aelotropic Elastic Plates.Journal of Applied Mechanics,Vol.28,402-408,1961. 5 J.E.Ashton,Approximate Solutions for Unsymmetrically Laminated Plates.Joural of Composite Materials,Vol.3,189-191,1969
90 THIN PLATES requires that the plate satisfy the following conditions: The plate is made of unidirectional plies. There are at least three ply orientations. The angles between the fibers are at least 15◦. The number of plies in each fiber direction is at least 10 percent of the total number of plies. Plates conforming to the 10-percent rule have better load bearing capabilities than unidirectional or angle-ply laminates for the following reasons. Unidirectional plies are stiffer and stronger in the 0-degree fiber direction than in the 90-degree direction perpendicular to the fibers. Thus, laminates made of unidirectional plies are ill-suited to carry load in the 90-degree direction. Angle-ply laminates with only two fiber directions do not resist well tensile loads applied along the symmetry axis. Plates made by the 10-percent rule minimize these shortcomings. The specification of symmetrical layup is less restrictive than it may appear because the analyses of symmetrical plates (for which ([B] = 0) can readily be extended to unsymmetrical plates ([B] = 0) with the use of the reduced bending stiffness [D] ∗, defined as 3,4,5 [D] ∗ = [D] − [B][A] −1 [B]. (4.1) The deflections, maximum bending moments, buckling loads, and natural frequencies of unsymmetrical plates can be approximated by replacing [D] by [D] ∗ in the expressions derived for symmetrical plates. 4.1 Governing Equations In this section we summarize the equations used in analyzing thin plates. We employ the x, y, z coordinate system. The origin is at the midplane for plates with symmetrical layup and at a suitably chosen reference plane for plates with unsymmetrical layup. The strains and curvatures of the reference plane (Fig. 3.10) are (Eqs. 3.1, 3.8) o x = ∂uo ∂x o y = ∂vo ∂y γ o xy = ∂uo ∂y + ∂vo ∂x κx = −∂2wo ∂x2 κy = −∂2wo ∂y2 κxy = −2∂2wo ∂x∂y , (4.2) 3 J. M. Whitney, Structural Analysis of Laminated Anisotropic Plates. Technomic, Lancaster, Pennsylvania, 1987, p. 203. 4 E. Reissner and Y. Stavsky, Bending and Stretching of Certain Types of Heterogeneous Aelotropic Elastic Plates. Journal of Applied Mechanics, Vol. 28, 402–408, 1961. 5 J. E. Ashton, Approximate Solutions for Unsymmetrically Laminated Plates. Journal of Composite Materials, Vol. 3, 189–191, 1969
4.1 GOVERNING EQUATIONS 91 Figure 4.1:Forces and loads acting on an element of the plate. where uo and vo are the displacements of the reference plane in the x and y directions,and w is the out-of-plane displacement (deflection)of this plane.The force-strain relationships are (Eq.3.21) [A1 A12 A16 B11 B12 B16 N A2 A 6 B12 Br B26 A6 6 A66 B16 B26 B66 M Bu B12 B16 D11 D12 D16 (4.3) Kx M B12 B22 B26 D12 D22 D26 Ky xy B16 B26 B66 D16 D26 D66 Kxy In the analyses we may employ either the equilibrium equations or the strain energy. The equilibrium equations are5 aNs aNg.=-px ax ay aNy +x」 (4.4) ay aNy=一Py 业+业 ax ay =一P V= OM:aMy ax V= aMy aMey ay ay ax (4.5) where pr,py,and p:are the components of the distributed surface load (per unit area);N,N,and Ny are the in-plane forces (per unit length);Vr and Vy are the transverse shear forces (per unit length);Mr.My and Mry are,respectively,the bending moments and the twist moment(per unit length)(Fig.4.1). 6 S.P.Timoshenko and S.Woinowsky-Krieger,Theory of Plates and Shells.2nd edition.McGraw-Hill, New York,1959,p.80
4.1 GOVERNING EQUATIONS 91 x y x y x y Nx Nxy Nyx Ny Mxy Mx My Myx Vy Vx px py pz Figure 4.1: Forces and loads acting on an element of the plate. where uo and vo are the displacements of the reference plane in the x and y directions, and wo is the out-of-plane displacement (deflection) of this plane. The force–strain relationships are (Eq. 3.21) Nx Ny Nxy Mx My Mxy = 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 . (4.3) In the analyses we may employ either the equilibrium equations or the strain energy. The equilibrium equations are6 ∂Nx ∂x + ∂Nxy ∂y = −px ∂Ny ∂y + ∂Nxy ∂x = −py (4.4) ∂Vx ∂x + ∂Vy ∂y = −pz Vx = ∂Mx ∂x + ∂Mxy ∂y Vy = ∂My ∂y + ∂Mxy ∂x , (4.5) where px, py, and pz are the components of the distributed surface load (per unit area); Nx, Ny, and Nxy are the in-plane forces (per unit length); Vx and Vy are the transverse shear forces (per unit length); Mx, My and Mxy are, respectively, the bending moments and the twist moment (per unit length) (Fig. 4.1). 6 S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells. 2nd edition. McGraw-Hill, New York, 1959, p. 80
92 THIN PLATES 4.1.1 Boundary Conditions The conditions alongeach edge of the plate must be specified.Boundary conditions for an edge parallel with the y-axis are given below. Along a built-in edge,the deflection wo,the rotation of the edge aw/ax,and the in-plane°，v°displacements are zero: w°=0 aw -=0°=v°=0. (4.6 ax Along a free edge,where no external loads are applied,the bending moment Mr,the replacement shear force?V+a Mry/ax,and the in-plane forces N,Ny are zero: Mx=0 Vx+ 8My=0N:=Nw=0. ay (4.7) Along a simply supported edge,the deflection wo,the bending moment Mr, and the in-plane forces M,Nry are zero: w°=0Mx=0N=Nxy=0. (4.8) When in-plane motions are prevented by the support,the in-plane forces are not zero(N≠0，Nxy≠O)whereas the in-plane displacements are zero: °=0v°=0. (4.9) For an edge parallel with the x-axis,the preceding boundary conditions hold with x and y interchanged. 4.1.2 Strain Energy As we noted previously,solutions to plate problems may be obtained by energy methods that require knowledge of the strain energy.For a linearly elastic material the strain energy is given by Eq.(2.200).Under plane-stress condition the stress components o:,txz,and tyz are zero(Eq.2.121),and the expression for the strain energy simplifies to Ly (oxex +oyey+txyrxy)dzdydx, (4.10) where h and hp are the distances from the reference plane to the plate's surfaces (Fig.3.12).The strain components are (Eq.3.7) (4.11) 7bid,p.84
92 THIN PLATES 4.1.1 Boundary Conditions The conditions along each edge of the plate must be specified. Boundary conditions for an edge parallel with the y-axis are given below. Along a built-in edge, the deflection wo, the rotation of the edge ∂wo/∂x, and the in-plane uo, vo displacements are zero: wo = 0 ∂wo ∂x = 0 uo = vo = 0. (4.6) Along a free edge, where no external loads are applied, the bending moment Mx, the replacement shear force7 Vx + ∂Mxy/∂x, and the in-plane forces Nx, Nxy are zero: Mx = 0 Vx + ∂Mxy ∂y = 0 Nx = Nxy = 0. (4.7) Along a simply supported edge, the deflection wo, the bending moment Mx, and the in-plane forces Nx, Nxy are zero: wo = 0 Mx = 0 Nx = Nxy = 0. (4.8) 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. (4.9) For an edge parallel with the x-axis, the preceding boundary conditions hold with x and y interchanged. 4.1.2 Strain Energy As we noted previously, solutions to plate problems may be obtained by energy methods that require knowledge of the strain energy. For a linearly elastic material the strain energy is given by Eq. (2.200). Under plane-stress condition the stress components σz, τxz, and τyz are zero (Eq. 2.121), and the expression for the strain energy simplifies to U = 1 2 ) Lx 0 ) Ly 0 ) ht −hb (σxx + σyy + τxyγxy) dzdydx, (4.10) where ht and hb are the distances from the reference plane to the plate’s surfaces (Fig. 3.12). The strain components are (Eq. 3.7) x y γxy = o x o y γ o xy + z κx κy κxy . (4.11) 7 Ibid., p. 84
4.2 DEFLECTION OF RECTANGULAR PLATES 93 The stresses and the strains at a point are related by (Eq.3.13) (4.12) By substituting Eqs.(4.11)and (4.12)into Eq.(4.10)and by utilizing the definitions of the [A,[B],[D]matrices (Eq.3.18),we obtain the following expression for the strain energy: 公 「A1 A12 A16 B11 B12 B16 e L 内2 A22 426 B12 B22 B26 1 U= 46 6 A66 B16 B26 B66 dydx. Kx B11 B12 B16 D11 Di2 D16 Kx 0 0 Ky B12 B2 B26 D12 D22 D26 Ky Kxy B16 B26 B66 D16 D26 D66」 Kxy (4.13) The superscript T denotes the transpose of the vector. 4.2 Deflection of Rectangular Plates 4.2.1 Pure Bending and In-Plane Loads We consider an unsupported rectangular plate subjected to pure bending and to in-plane loads(Fig.4.2).The in-plane forces and moments are related to the reference plane's strains and curvatures by Eq.(4.3).Six of the twelve quantities appearing in this equation must be specified as follows: N or e Mx or Kx Ny or e My or Ky (4.14) Nry or esy Mry or Kxy. With six of the quantities chosen(Eq.4.14),the remaining six may be obtained by solving the six simultaneous equations given by Eq.(4.3).Once the curvatures Figure 4.2:Rectangular plate subjected to bending and in-plane loads
4.2 DEFLECTION OF RECTANGULAR PLATES 93 The stresses and the strains at a point are related by (Eq. 3.13) σx σy τxy = [Q] x y γxy . (4.12) By substituting Eqs. (4.11) and (4.12) into Eq. (4.10) and by utilizing the definitions of the [A], [B], [D] matrices (Eq. 3.18), we obtain the following expression for the strain energy: 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 dydx. (4.13) The superscript T denotes the transpose of the vector. 4.2 Deflection of Rectangular Plates 4.2.1 Pure Bending and In-Plane Loads We consider an unsupported rectangular plate subjected to pure bending and to in-plane loads (Fig. 4.2). The in-plane forces and moments are related to the reference plane’s strains and curvatures by Eq. (4.3). Six of the twelve quantities appearing in this equation must be specified as follows: Nx or o x Mx or κx Ny or o y My or κy Nxy or o xy Mxy or κxy. (4.14) With six of the quantities chosen (Eq. 4.14), the remaining six may be obtained by solving the six simultaneous equations given by Eq. (4.3). Once the curvatures Nx x Mx Mxy Nxy Nxy My Mxy Ny y z Figure 4.2: Rectangular plate subjected to bending and in-plane loads