STRUCTURAL ANALYSIS 181 for a stress in the x direction are then given by: E= Ey Ex Ex Proceeding in this way,it is found that: 号 Ex =As-Ay A E,=A (6.19) A Vyx A Gxy=Ais As illustrative examples of the above theory,consider a family of 24-ply laminates,symmetrical and orthotropic,and all made of the same material but with varying numbers of 0 and +45 plies.(For the present purposes,the precise ordering of the plies is immaterial as long as the symmetry requirement is maintained;however,to ensure orthotropy,there must be the same number of +45 as-45 plies.)The single-ply modulus data (representative of a carbon/ epoxy)are: E1=137.44GPaE2=11.71GPaG2=5.51GPa (6.20) v12=0.2500v21=0.0213 The lay-ups considered are shown in Table 6.1.The steps in the calculation are as follows: (1)Calculate the y(0)from equation (6.4). (2)For each of the ply orientations involved here0=0°，+45，and-45°， calculate the from equation (6.8).[Of course,here the (0)have already been obtained in step 1.] Table6.1 Moduli for Family of24-ply0°/±45°Laminates Constructed Using Unidirectional Tape Lay-up No.0° No.+45 No.-45 Ex E G.x Vry Vyr Plies Plies Plies GPa GPa GPa 24 0 0 137.4 11.7 5.51 0.250 0.021 16 4 4 99.4 21.1 15.7 0.578 0.123 12 6 6 79.5 24.5 20.8 0.647 0.199 8 8 8 59.6 26.4 25.8 0.693 0.307 0 12 12 19.3 19.3 36 0.752 0.752
STRUCTURAL ANALYSIS 181 for a stress in the x direction are then given by: O" x Ey Ex "= -- , Pxy ~ -- -- ~x ~'x Proceeding in this way, it is found that: * Ex = A* x , Ey = Ayy -- -- Ayy Axy A~y Pxy "= Ayy-'T Yyx ~ ~xx *2 A~ A~ G~y = As* ~ (6.19) As illustrative examples of the above theory, consider a family of 24-ply laminates, symmetrical and orthotropic, and all made of the same material but with varying numbers of 0 ° and _ 45 ° plies. (For the present purposes, the precise ordering of the plies is immaterial as long as the symmetry requirement is maintained; however, to ensure orthotropy, there must be the same number of + 45 ° as -45 ° plies.) The single-ply modulus data (representative of a carbon/ epoxy) are: El = 137.44GPa E2 = 11.71GPa Gt2 = 5.51GPa ]212 = 0.2500 V21 = 0.0213 (6.20) The lay-ups considered are shown in Table 6.1. The steps in the calculation are as follows: (1) Calculate the Qij(O) from equation (6.4). (2) For each of the ply orientations involved here 0 = 0 °, +45 °, and -45 °, calculate the Qij(O) from equation (6.8). [Of course, here the Qo(O) have already been obtained in step 1.] Table 6.1 Moduli for Family of 24-ply 0°/+_ 45 ° Laminates Constructed Using Unidirectional Tape Lay-up No. 0 ° No. + 45 ° No. -45 ° Ex, Ey, G~, l~xy ])yx Plies Plies Plies GPa GPa GPa 24 0 0 137.4 11.7 5.51 0.250 0.021 16 4 4 99.4 21.1 15.7 0.578 0.123 12 6 6 79.5 24.5 20.8 0.647 0.199 8 8 8 59.6 26.4 25.8 0.693 0.307 0 12 12 19.3 19.3 36 0.752 0.752
182 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES (3)Calculate the Ai from equation (6.16);in the present case,equation (6.16) becomes: A等=[n12(0)+n22(+45)+n32(-45]/24 where n is the number of0°plies,.n2of+45°plies and n3of-45°plies.. (4)Calculate the moduli from equation (6.19). The results of the calculations are shown in Table 6.1. The results in Table 6.1 have been presented primarily to exemplify the preceding theory;however,they also demonstrate some features that are impor- tant in design.The stiffness of a composite is overwhelmingly resident in the extensional stiffness of its fibers;hence,at least for simple loadings,if maximum stiffness is required,a laminate is constructed so that the fibers are aligned in the principal stress directions.Thus,for a member under uniaxial tension,a laminate comprising basically all 0 plies would be chosen;in other words,all fibers would be aligned parallel to the tension direction.As can be seen from Table 6.1,E decreases as the number of 0 plies decreases.On the other hand,consider a rectangular panel under shear,the sides of the panel being parallel to the laminate axes(Fig.6.7a).The principal stresses here are an equal tension and compression, oriented at +45 and-45 to the x-axis.Thus,maximum shear stiffness can be expected to be obtained using a laminate comprising equal numbers of +45 and -45 plies;this is reflected in the high shear modulus Gry for the all 45 laminate of Table 6.1. Fibre Fibre ◆45 a) (i)Shear panel (ii)Principal stresses (iii)Fibre directions x 68.9 MPa b) Fig.6.7 Ply orientations for example problems:a)fiber orientations for a shear panel:b)0°±45°laminate under uniaxial tension
182 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES (3) Calculate the A,~ from equation (6.16); in the present case, equation (6.16) becomes: Aij = [nlQo(0) + neQij( +45) + n3Q(i(-45)]124 where nl is the number of 0 ° plies, n2 of +45 ° plies and n3 of -45 ° plies. (4) Calculate the moduli from equation (6.19). The results of the calculations are shown in Table 6.1. The results in Table 6.1 have been presented primarily to exemplify the preceding theory; however, they also demonstrate some features that are important in design. The stiffness of a composite is overwhelmingly resident in the extensional stiffness of its fibers; hence, at least for simple loadings, if maximum stiffness is required, a laminate is constructed so that the fibers are aligned in the principal stress directions. Thus, for a member under uniaxial tension, a laminate comprising basically all 0 ° pries would be chosen; in other words, all fibers would be aligned parallel to the tension direction. As can be seen from Table 6.1, Ex decreases as the number of 0 ° plies decreases. On the other hand, consider a rectangular panel under shear, the sides of the panel being parallel to the laminate axes (Fig. 6.7a). The principal stresses here are an equal tension and compression, oriented at +45 ° and -45 ° to the x-axis. Thus, maximum shear stiffness can be expected to be obtained using a laminate comprising equal numbers of +45 ° and -45 ° plies; this is reflected in the high shear modulus G,:y for the all _+ 45 ° laminate of Table 6.1. 'L_ X a) (i) Shear panel ÷~5" Fibre Fibre \\ \\//.. /5"~ /'\ -~,s'/-'-7 iii \\4 (ii) Principal stresses (iii) Fibre directions b) L_ x " )~",. /\ ./",. IX./',,. /"y'l"""'*" ..__D ,,,_l -'~ --'-~'-'-2~C'~ "~'--~7 = :x = 68.9 MPa .,> I(-×.~X ,X.~ X ~,~.,~5< .NX-)I __,. ...~... v '~./ \/ ,J ~/ \/ V XJ~ Fig. 6.7 Ply orientations for example problems: a) fiber orientations for a shear panel: b) 0 ° _ 45 ° laminate under uniaxial tension
STRUCTURAL ANALYSIS 183 It should also be observed that,although for an isotropic material,Poisson's ratio cannot exceed 0.5,this is not the case for an anisotropic material. 6.2.3.5 Quasi-/sotropic Laminates.It is possible to construct laminates that are isotropic with regard to their in-plane elastic properties-in other words,they have the same Young's modulus E and same Poisson's ratio v in all in-plane directions and for which the shear modulus is given by G= E/2(1+v).One way of achieving this is to adopt a lay-up having an equal number of plies oriented parallel to the sides of an equilateral triangle.For example,a quasi-isotropic 24-ply laminate could be made with 8 plies oriented at each of0°，+60°，and-60°.Using the same materials data(and theory)as were used in deriving Table 6.1,it will be found that such a laminate has the following moduli: E=54.2Gpa,G=20.8Gpa,v=0.305 Another way of achieving a quasi-isotropic laminate is to use equal numbers of plies oriented at0°，+45°，-45°，and90°.A quasi--isotropic24-ply laminate (with,incidentally,the same values for the elastic constants as were just cited)could be made with6 plies at each of0°，+45°，-45°，and90°. (For comparison,recall that Young's modulus and the shear modulus for a typical aluminum alloy are of the order of 72 and 27 GPa,respectively,and that the specific gravity of carbon/epoxy is about 60%that of aluminum.) The term guasi-isotropic is used because,of course,such laminates have different properties in the out-of-plane direction.However,it is not usual practice to work with quasi-isotropic laminates;efficient design with composites generally requires that advantage be taken of their inherent anisotropy. 6.2.3.6 Stress Analysis of Orthotropic Laminates.The determination of the stresses,strains,and deformations experienced by symmetric laminates under plane stress loadings is carried out by procedures that are analogous to those used for isotropic materials.The laminate is treated as a homogeneous membrane having stiffness properties determined as described above.It should be noted, though,that while the strains and deformations so determined are the actual strains and deformations (within the limit of the assumptions),the stresses are only the average values over the laminate thickness. If an analytical procedure is used,then generally a stress function F is introduced,this being related to the (average)stresses by &F &F -82F x=a2,= 02, Try= axay (Eq.6.21)
STRUCTURAL ANALYSIS 183 It should also be observed that, although for an isotropic material, Poisson's ratio cannot exceed 0.5, this is not the case for an anisotropic material. 6.2.3.5 Quasi-lsotropic Laminates. It is possible to construct laminates that are isotropic with regard to their in-plane elastic properties---in other words, they have the same Young's modulus E and same Poisson's ratio v in all in-plane directions and for which the shear modulus is given by G = E/2(l+v). One way of achieving this is to adopt a lay-up having an equal number of plies oriented parallel to the sides of an equilateral triangle. For example, a quasi-isotropic 24-ply laminate could be made with 8 plies oriented at each of 0 °, + 60 °, and - 60 °. Using the same materials data (and theory) as were used in deriving Table 6.1, it will be found that such a laminate has the following moduli: E=54.2Gpa, G=20.8Gpa, v=0.305 Another way of achieving a quasi-isotropic laminate is to use equal numbers of plies oriented at 0 °, -t-45 °, -45 °, and 90 °. A quasi-isotropic 24-ply laminate (with, incidentally, the same values for the elastic constants as were just cited) could be made with 6 plies at each of 0 °, +45 °, -45 °, and 90 °. (For comparison, recall that Young's modulus and the shear modulus for a typical aluminum alloy are of the order of 72 and 27 GPa, respectively, and that the specific gravity of carbon/epoxy is about 60% that of aluminum.) The term quasi-isotropic is used because, of course, such laminates have different properties in the out-of-plane direction. However, it is not usual practice to work with quasi-isotropic laminates; efficient design with composites generally requires that advantage be taken of their inherent anisotropy. 6.2.3.6 Stress Analysis of Orthotropic Laminates. The determination of the stresses, strains, and deformations experienced by symmetric laminates under plane stress loadings is carried out by procedures that are analogous to those used for isotropic materials. The laminate is treated as a homogeneous membrane having stiffness properties determined as described above. It should be noted, though, that while the strains and deformations so determined are the actual strains and deformations (within the limit of the assumptions), the stresses are only the average values over the laminate thickness. If an analytical procedure is used, then generally a stress function F is introduced, this being related to the (average) stresses by 02 F OZ F - 02 F (Eq. 6.21) O" x -~- OX2, O'y = 0)22 , TrY- OxOy
184 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES It can be shown that F satisfies the following partial differential equation: F 8F E 4 =0 (6.22) Ex Solutions of equation(6.22)for several problems of interest can be found in Ref.10. Most structural analyses are now performed using finite element methods described in Chapter 16,and many general-purpose finite-element programs contain orthotropic membrane elements in their library.Once the laminate moduli are determined,they are used as input data for calculating the element stiffness matrix;the rest of the analysis proceeds as in the isotropic case. As has already been emphasized,the stresses obtained from the above procedures are only the average stresses.To determine the actual stresses in the individual plies,it is necessary only to substitute the calculated values of the strains in equation(6.10).An elementary example may clarify this.Consider a rectangular strip under uniaxial tension(Fig.6.7b)made of the 24-ply laminate considered earlier that had 12 plies at0°，6 plies at+45°，and6 plies at-45. Suppose the applied stress is =68.9 MPa.The average stress here is uniform in the xy plane and given by: Ox =68.9 MPa,y=Ts =0 Using the values of Ex and v given in Table 6.1,it follows that the associated strains are: ex=0.8667×10-3,ey=-0.5607×10-3,Y,=0 The stresses in the individual plies can now be obtained by substituting these values into equation(6.10),with the appropriate values of the ()The results of doing this are shown in Table 6.2. Thus,the actual stress distribution is very different from the average one.In particular,note that transverse direct stresses and shear stresses are developed,even though no such stresses are applied;naturally,these stresses are self-equilibrating over the thickness.It follows that there is some"boundary layer"around the edges of the strip where there is a rapid transition from the actual stress boundary values (namely,zero on the longitudinal edges)to the calculated values shown in the table. This boundary layer would be expected to extend in from the edges a distance of the order of the laminate thickness(from the Saint-Venant principle).In the boundary Table 6.2 Stresses in Individual Plies of a 24-Ply Laminate (12 at 0,6 at +45 and 6 at-45)under Uniaxial Stress of 68.9 MPa 0 (0)MPa (0)Mpa To(0)MPa 0 118.1 -4.1 0 +45 19.8 4.1 9.7 -45 19.8 4.1 -9.7
184 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES It can be shown that F satisfies the following partial differential equation: (1) 04F (1 2Vxy'~ 04F (1) O4F ~y ffx~-+ ~-y ~-)3x--5-~+ ~ -~--=0 (6.22) Solutions of equation (6.22) for several problems of interest can be found in Ref. 10. Most structural analyses are now performed using finite element methods described in Chapter 16, and many general-purpose finite-element programs contain orthotropic membrane elements in their library. Once the laminate moduli are determined, they are used as input data for calculating the element stiffness matrix; the rest of the analysis proceeds as in the isotropic case. As has already been emphasized, the stresses obtained from the above procedures are only the average stresses. To determine the actual stresses in the individual plies, it is necessary only to substitute the calculated values of the strains in equation (6.10). An elementary example may clarify this. Consider a rectangular strip under uniaxial tension (Fig. 6.7b) made of the 24-ply laminate considered earlier that had 12 plies at 0 °, 6 plies at +45 °, and 6 plies at -45 °. Suppose the applied stress is ~rx = 68.9 MPa. The average stress here is uniform in the xy plane and given by: o-x = 68.9 MPa, O'y = T s = 0 Using the values of Ex and Vxy given in Table 6.1, it follows that the associated strains are: ex = 0.8667 x 10 -3, 8y = -0.5607 x 10 -3, % = 0 The stresses in the individual plies can now be obtained by substituting these values into equation (6.10), with the appropriate values of the Qij(O). The results of doing this are shown in Table 6.2. Thus, the actual stress distribution is very different from the average one. In particular, note that transverse direct stresses and shear stresses are developed, even though no such stresses are applied; naturally, these stresses are self-equilibrating over the thickness. It follows that there is some "boundary layer" around the edges of the strip where there is a rapid transition from the actual stress boundary values (namely, zero on the longitudinal edges) to the calculated values shown in the table. This boundary layer would be expected to extend in from the edges a distance of the order of the laminate thickness (from the Saint-Venant principle). In the boundary Table 6.2 Stresses in Individual Plies of a 24-Ply Laminate (12 at 0% 6 at + 45 and 6 at - 45 °) under Uniaxial Stress of 68.9 MPa 0 ° Ox(O) MPa o~v (0) Mpa "rxy(O) MPa 0 118.1 -4.1 0 +45 19.8 4.1 9.7 - 45 19.8 4.1 - 9.7