12 MECHANICAL BEHAVIOR OF THIN LAMINATED PLATES The definition of a laminate was given in Chapter 5.In the same chapter the practical calculation methods for the laminate was also described.We propose here to justify these methods,meaning to study the behavior of the laminate when it is subjected to a combination of loadings.This study is necessary if one wants to have correct design with strains or stresses within their admissible values. 12.1 LAMINATE WITH MIDPLANE SYMMETRY 12.1.1 Membrane Behavior We consider in the following a laminate with midplane symmetry.The total thickness of the laminate is denoted as b.It consists of n plies.Ply number k has a thickness denoted as es(see Figure 12.1).Plane x,y is the plane of symmetry 12.1.1.1 Loadings The laminate is subjected to loadings in its plane.The stress resultants are denoted as NN T=T These are the membrane stress resultants.They are defined as: N:Stress resultant in the x direction over a unit width along the y direction. n中p Nx= c.dc=∑(o×es b/2 (12.1) k=1p时 M:Stress resultant along the y direction over a unit width along thex direction. =马，k=∑(o×e (12.2) =1"ply TSee Section 5.2. 2 The problem of buckling of the laminates is not the scope of this chapter.See Appendix 2. 3 See Section 5.2.3. 2003 by CRC Press LLC
12 MECHANICAL BEHAVIOR OF THIN LAMINATED PLATES The definition of a laminate was given in Chapter 5.1 In the same chapter the practical calculation methods for the laminate was also described. We propose here to justify these methods, meaning to study the behavior of the laminate when it is subjected to a combination of loadings. This study is necessary if one wants to have correct design with strains or stresses within their admissible values.2 12.1 LAMINATE WITH MIDPLANE SYMMETRY 12.1.1 Membrane Behavior We consider in the following a laminate with midplane symmetry. 3 The total thickness of the laminate is denoted as h. It consists of n plies. Ply number k has a thickness denoted as ek (see Figure 12.1). Plane x,y is the plane of symmetry. 12.1.1.1 Loadings The laminate is subjected to loadings in its plane. The stress resultants are denoted as Nx, Ny, Txy = Tyx. These are the membrane stress resultants.They are defined as: Nx: Stress resultant in the x direction over a unit width along the y direction. (12.1) Ny: Stress resultant along the y direction over a unit width along the x direction. (12.2) 1 See Section 5.2. 2 The problem of buckling of the laminates is not the scope of this chapter. See Appendix 2. 3 See Section 5.2.3. Nx sxdz –h/2 h/2 Ú sx ( )k ¥ ek k=1 stply nthply = = Â Ny sy dz –h/2 h/2 Ú sy ( )k ¥ ek k=1 stply nthply = = Â TX846_Frame_C12 Page 235 Monday, November 18, 2002 12:27 PM © 2003 by CRC Press LLC
dy dx Ny x dx Tydh Txyx dy Nx×dy h stresses ek displacement Figure 12.1 Definition of Laminate and Membrane Loading To(or T):Membrane shear stress resultant over a unit width along the y direction (or respectively along the x direction): n中p脚 =mgdk=∑(cgxe (12.3) k=1ply 12.1.1.2 Displacement Field The elastic displacement at each point of the laminate is assumed to be two dimensional,in the x,y plane of the laminate.It has the components:u,v.The nonzero strains can be written as: Eox duoldx Eoy =dvoldy Yoxy duoldy+duoldx It was shown in the previous chapter (Equation 11.8)that one can express, in a given coordinate system,the stresses in a ply as functions of the strains. Then the stress resultant N defined in Equation 12.1 can be written as follows: 中py N.=∑{疏eas+疏ew+疏ane =”p邮y then: Nx AuEox+A12Eoy+A13 Yoxy 2003 by CRC Press LLC
Txy (or Tyx ): Membrane shear stress resultant over a unit width along the y direction (or respectively along the x direction): (12.3) 12.1.1.2 Displacement Field The elastic displacement at each point of the laminate is assumed to be two dimensional, in the x,y plane of the laminate. It has the components: uo , vo. The nonzero strains can be written as: It was shown in the previous chapter (Equation 11.8) that one can express, in a given coordinate system, the stresses in a ply as functions of the strains. Then the stress resultant Nx defined in Equation 12.1 can be written as follows: then: Figure 12.1 Definition of Laminate and Membrane Loading Txy txy dz –h/2 h/2 Ú txy ( )k ¥ ek k=1 stply nthply = = Â e ox ∂u0 = /∂x e oy ∂v0 = /∂y g oxy ∂u0/∂y ∂v0 = + /∂x Nx E11 k e ox E12 k e oy E13 k { } + + g oxy ek k=1 stply nthply = Â Nx = A11e ox + A12e oy + A13g oxy TX846_Frame_C12 Page 236 Monday, November 18, 2002 12:27 PM © 2003 by CRC Press LLC
with: ply ply A11= 疏eA=∑疏e:Au=∑e。 =1"ply =1"ply =1ply In an analogous manner,one obtains for the Equation 12.2: Ny A21Eox+AzEoy+A2Yoxy with: np A2y= ∑e =1ply and for the shear stress resultant T one can write,starting from Equation 12.3: Txy A31Eox+A3Eoy+A33Yoxy with: ply A3y=】 =1ply Therefore,it is possible to express the stress resultants in the following matrix form: Nx A12 A A21 A23 Eoy A31 A32 (12.4④ with: nply 武ee=A =1ply Remarks: One observes from the above expressions that coefficients Ay are inde- pendent of the stacking order of the plies. One can also see that the normal stress resultants N or N,create angular distortions.This coupling will disappear if the laminate is balanced.This means that apart from the midplane symmetry,there are as many and identical plies that make with the x axis an angle +0 as those that make The developments of Ey are given in Equation 11.8. 2003 by CRC Press LLC
with: In an analogous manner, one obtains for the Equation 12.2: with: and for the shear stress resultant Txy one can write, starting from Equation 12.3: with: Therefore, it is possible to express the stress resultants in the following matrix form: (12.4)4 Remarks: One observes from the above expressions that coefficients Aij are independent of the stacking order of the plies. One can also see that the normal stress resultants Nx or Ny create angular distortions. This coupling will disappear if the laminate is balanced. This means that apart from the midplane symmetry, there are as many and identical plies that make with the x axis an angle +q as those that make 4 The developments of are given in Equation 11.8. A11 E11 k ek; A12 k=1 stply nthply  E12 k ek; A13 k=1 stply nthply  E13 k ek k=1 stply nthply ===  Ny = A21e ox + A22e oy + A23g oxy A2j E2j k ek k=1 stply nthply =  Txy = A31e ox + A32e oy + A33g oxy A3j E3j k ek k=1 stply nthply =  Nx Ny ÓTxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ A11 A12 A13 A21 A22 A23 A31 A32 A33 e ox e oy Óg oxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = with: Aij Eij k ek k=1 stply nthply = =  Aji Eij TX846_Frame_C12 Page 237 Monday, November 18, 2002 12:27 PM © 2003 by CRC Press LLC
with the x axis an angle-0.In effect,the coefficients E1s and E23 are antisymmetric in 0 and,therefore,cancel each other out for the pairs of plies at te when one calculates the terms A1s and A23.The result is then: A13=A23=0 and the stress-strain relation for the laminate is reduced to A12 Eox Ny A12 A2 0 (12.5) Tg) 0 0 It is possible to substitute the stress resultants NN T with the global average stresses (which are fictitious): Gox Nx/b Goy Nylb (12.6 Toxy Txylb One then deduces from Equation 12.4 the average membrane stress-strain behav- ior of a laminate as: An A12 A13 = A21 A22 A23 Eo时 (12.7) Toxy A31 A2 A3 Yoxy. One can also note that according to Equation 12.4 the terms of the matrix Al above can be written as: ply 1 =1ply Then the ratios e/b can be rearranged to obtain the proportions of plies having the same orientation.In case where these proportions have already been fixed-and therefore their numerical values are known-it becomes possible to calculate the termsAy without knowing the thickness.For example,if the selected orienta- tions are0°，90°，+45°，-45°，and by denoting p'(6)as the percentages of the plies along the different orientations,one has: 君x4y=可×p“+"×p+×p5+可×p8 (12.8) 5 See Figure 12.1 and figure in the Equation 11.8. 6 The expressions developed for Ey are given in Equation 11.8. 2003 by CRC Press LLC
with the x axis an angle -q. 5 In effect, the coefficients 13 and 23 are antisymmetric in q 6 and, therefore, cancel each other out for the pairs of plies at ±q when one calculates the terms A13 and A23. The result is then: and the stress–strain relation for the laminate is reduced to (12.5) It is possible to substitute the stress resultants Nx, Ny, Txy with the global average stresses (which are fictitious): (12.6) One then deduces from Equation 12.4 the average membrane stress–strain behavior of a laminate as: (12.7) One can also note that according to Equation 12.4 the terms of the matrix [A] above can be written as: Then the ratios ek /h can be rearranged to obtain the proportions of plies having the same orientation. In case where these proportions have already been fixed—and therefore their numerical values are known—it becomes possible to calculate the terms Aij without knowing the thickness. For example, if the selected orientations are 0∞, 90∞, +45∞, -45∞, and by denoting p k (%) as the percentages of the plies along the different orientations, one has: (12.8) 5 See Figure 12.1 and figure in the Equation 11.8. 6 The expressions developed for are given in Equation 11.8. E E Eij A13 = = A23 0 Nx Ny ÓTxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ A11 A12 0 A12 A22 0 0 0 A33 e ox e oy Óg oxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = sox Nx = /h soy Ny = /h t oxy Txy = /h sox soy Ót oxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ 1 h -- A11 A12 A13 A21 A22 A23 A31 A32 A33 e ox e oy Óg oxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = 1 h -- 1 h -- ¥ Aij Eij k ek h ¥ ---- k=1 stply nthply = Â 1 h -- 1 h -- ¥ Aij Eij 0∞ p0∞ Eij 90∞ p90∞ Eij +45∞ p+45∞ Eij -45∞ p-45∞ = ¥ + ¥ + ¥ + ¥ TX846_Frame_C12 Page 238 Monday, November 18, 2002 12:27 PM © 2003 by CRC Press LLC
given: thickness to stress resultants be determined proportions Figure 12.2 Practical Determination of a Laminate Subject to Membrane Loading 12.1.2 Apparent Moduli of the Laminate Inversion of Equation 12.7 above allows one to obtain what can be called as apparent moduli and coupling coefficients associated with the membrane behav- ior in the plane x,y.These coefficients appear in the following relation: Eox ox 匹 G对 ox Eoy =b[A] Ooy (12.9) 12.1.3 Consequence:Practical Determination of a Laminate Subject to Membrane Loading Given: The stress resultants are given and denoted as:NN T Using the values of these stress resultants,one can estimate the ply proportions in the four orientations.'Assume in Figure 12.2 that the plies are identical (same material and same thickness). The problem is to determine The apparent elastic moduli of the laminate and the associated coupling coefficients,in order to estimate strains under loading The minimum thickness for the laminate in order to avoid rupture of one of the plies in the laminate 7See Section 5.4.3. 2003 by CRC Press LLC
12.1.2 Apparent Moduli of the Laminate Inversion of Equation 12.7 above allows one to obtain what can be called as apparent moduli and coupling coefficients associated with the membrane behavior in the plane x,y. These coefficients appear in the following relation: (12.9) 12.1.3 Consequence: Practical Determination of a Laminate Subject to Membrane Loading Given: The stress resultants are given and denoted as: Nx, Ny, Txy. Using the values of these stress resultants, one can estimate the ply proportions in the four orientations. 7 Assume in Figure 12.2 that the plies are identical (same material and same thickness). The problem is to determine The apparent elastic moduli of the laminate and the associated coupling coefficients, in order to estimate strains under loading The minimum thickness for the laminate in order to avoid rupture of one of the plies in the laminate Figure 12.2 Practical Determination of a Laminate Subject to Membrane Loading 7 See Section 5.4.3. e ox e oy Óg oxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ h A[ ]–1 sox soy Ót oxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ 1 E x ----- nyx E y –------- h xy G xy -------- nxy E x –------- 1 E y ----- m xy G xy -------- h x E x ----- m y E y ----- 1 G xy -------- sox soy Ót oxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = = TX846_Frame_C12 Page 239 Monday, November 18, 2002 12:27 PM © 2003 by CRC Press LLC