6 Structural Analysis 6.1 Overview In this chapter,the basic theory needed for the determination of the stresses, strains,and deformations in fiber composite structures is outlined.Attention is concentrated on structures made in the form of laminates because that is the way composite materials are generally used. From the viewpoint of structural mechanics,the novel features of composites (compared with conventional structural materials such as metals)are their marked anisotropy and,when used as laminates,their macroscopically hetero- geneous nature. There is a close analogy between the steps in developing laminate theory and the steps in fabricating a laminate.The building block both for theory and fabrication is the single ply,also referred to as the lamina.This is a thin layer of the material (typical thickness around 0.125 mm for unidirectional carbon/epoxy "tape"and 0.25 mm for a cross-ply fabric or"cloth")in which all of the fibers are aligned parallel to one another or in an orthogonal mesh.The starting point for the theory is the stress-strain law for the single ply referred to its axes of material symmetry,defined here as the 0-1,2,3 material axes.In constructing a laminate, each ply is laid-up so that its fibers make some prescribed angle with a reference axis fixed in the laminate.Here the laminate axes will be defined as the x-,y-,and z-axes. All later calculations are made using axes fixed in the structure (the structural axes).In a finite element model,the material properties are usually entered in the material axes.The lay-up of the laminate is defined in the laminate axes.The laminate theory described in this chapter will indicate how the properties of the laminate are derived.The transformation from the laminate axes to the global structural axes is then completed during the solution process.Because the designer can select his own lay-up pattern (because the laminate stress-strain law will depend on that pattern),it follows that the designer can design the material (as well as the structure). For more detailed discussions of the topics covered in this chapter,see Refs.1-7.For background material on the theory of anisotropic elasticity, see Refs.8-10. 171
6 Structural Analysis 6.1 Overview In this chapter, the basic theory needed for the determination of the stresses, strains, and deformations in fiber composite structures is outlined. Attention is concentrated on structures made in the form of laminates because that is the way composite materials are generally used. From the viewpoint of structural mechanics, the novel features of composites (compared with conventional structural materials such as metals) are their marked anisotropy and, when used as laminates, their macroscopically heterogeneous nature. There is a close analogy between the steps in developing laminate theory and the steps in fabricating a laminate. The building block both for theory and fabrication is the single ply, also referred to as the lamina. This is a thin layer of the material (typical thickness around 0.125 mm for unidirectional carbon/epoxy "tape" and 0.25 mm for a cross-ply fabric or "cloth") in which all of the fibers are aligned parallel to one another or in an orthogonal mesh. The starting point for the theory is the stress-strain law for the single ply referred to its axes of material symmetry, defined here as the 0-1, 2, 3 material axes. In constructing a laminate, each ply is laid-up so that its fibers make some prescribed angle with a reference axis fixed in the laminate. Here the laminate axes will be defined as the x-, y-, and z-axes. All later calculations are made using axes fixed in the structure (the structural axes). In a finite element model, the material properties are usually entered in the material axes. The lay-up of the laminate is defined in the laminate axes. The laminate theory described in this chapter will indicate how the properties of the laminate are derived. The transformation from the laminate axes to the global structural axes is then completed during the solution process. Because the designer can select his own lay-up pattern (because the laminate stress-strain law will depend on that pattern), it follows that the designer can design the material (as well as the structure). For more detailed discussions of the topics covered in this chapter, see Refs. 1-7. For background material on the theory of anisotropic elasticity, see Refs. 8-10. 171
172 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES 6.2 Laminate Theory Classical laminate theory defines the response of a laminate with the following assumptions: For two-dimensional plane stress analysis,the strain is constant through the thickness. .For bending,the strain varies linearly through the thickness. The laminate is thin compared with its in-plane dimensions. .Each layer is quasi-homogeneous and orthotropic. .Displacements are small compared with the thickness. The behavior remains linear. With these assumptions satisfied,the laminate theory allows the response of a laminate to be calculated,engineering constants to be determined to substitute into standard formulas for stresses and deflections,and material properties of the laminate to be defined for substitution into finite element analysis as described in Chapter 16. 6.2.1 Stress-Strain Law for a Single Ply in the Material Axes: Unidirectional Laminates Consider a rectangular element of a single ply with the sides of the element parallel and perpendicular to the fiber direction(Fig.6.1).Clearly,the direction of the fibers defines a preferred direction in the material;it is thus natural to introduce a cartesian set of material axes 0-1,2,3 with the /-axis in the fiber direction,the 2-axis perpendicular to the fibers of the ply plane,and the 3-axis perpendicular to the plane of the ply.Here,interest is in the behavior of the ply when subjected to stresses acting in its plane,in other words,under plane stress conditions.These stresses (also referred to the material axes)will be denoted by o1,2,T12 and the associated strains by s1,s2,and y12.(Note that in composite mechanics,it is standard practice to work with"engineering"rather than"tensor" shear strains.)Although a single ply is highly anisotropic,it is intuitively evident that the coordinate planes 012,023,and 031 are those of material symmetry,there being a mirror image symmetry about these planes. 02.∈2 T12.712 01,∈1 Fibres Fig.6.1 Material axes for a single ply
172 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES 6.2 Laminate Theory Classical laminate theory defines the response of a laminate with the following assumptions: • For two-dimensional plane stress analysis, the strain is constant through the thickness. • For bending, the strain varies linearly through the thickness. • The laminate is thin compared with its in-plane dimensions. • Each layer is quasi-homogeneous and orthotropic. • Displacements are small compared with the thickness. • The behavior remains linear. With these assumptions satisfied, the laminate theory allows the response of a laminate to be calculated, engineering constants to be determined to substitute into standard formulas for stresses and deflections, and material properties of the laminate to be defined for substitution into finite element analysis as described in Chapter 16. 6.2.1 Stress-Strain Law for a Single Ply in the Material Axes: Unidirectional Laminates Consider a rectangular element of a single ply with the sides of the element parallel and perpendicular to the fiber direction (Fig. 6.1). Clearly, the direction of the fibers defines a preferred direction in the material; it is thus natural to introduce a cartesian set of material axes 0-1, 2, 3 with the /-axis in the fiber direction, the 2-axis perpendicular to the fibers of the ply plane, and the 3-axis perpendicular to the plane of the ply. Here, interest is in the behavior of the ply when subjected to stresses acting in its plane, in other words, under plane stress conditions. These stresses (also referred to the material axes) will be denoted by trl, tr2, r12 and the associated strains by el,/32, and 712. (Note that in composite mechanics, it is standard practice to work with "engineering" rather than "tensor" shear strains.) Although a single ply is highly anisotropic, it is intuitively evident that the coordinate planes 012, 023, and 031 are those of material symmetry, there being a mirror image symmetry about these planes. I 02. ~2 0 L I r12, 712 ] ~-~ °I, ~I Fibres 3 Fig. 6.1 Material axes for a single ply
STRUCTURAL ANALYSIS 173 A material having three mutually orthogonal planes of symmetry is known as orthotropic.The stress-strain law for an orthotropic material under plane stress conditions,referred to the material axes,necessarily has the following form: 1 -V21 0 E E E1 12 1 1 E2 E 02 (6.1) Y12 T12 0 0 G12 where:E1,E2=Young's moduli in the I and 2 directions,respectively;v12= Poisson's ratio governing the contraction in the 2 direction for a tension in the I direction;v21=Poisson's ratio governing the contraction in the direction for a tension in the 2 direction;G12 =(in-plane)shear modulus. There are five material constants in equation(6.1),but only four of these are independent because of the following symmetry relation': = (6.2) E1 E2 For unidirectional tape of the type being considered here,E is much larger than either E2 or Gi2 because the former is a"fiber-dominated"property,while the latter are "matrix dominated".For a bi-directional cloth,E=E2 and both are much larger than G2.For tape,v12 is matrix dominated and is of the order of 0.3,whereas the contraction implied in v2 is resisted by the fibers and so is much smaller. The above equations are all related to a single ply but,because the ply thickness does not enter into the calculations,they also apply to a"unidirectional laminate"that is simply a laminate in which the fiber direction is the same in all of the plies.In fact,most of the material constants for a single ply are obtained from specimen tests on unidirectional laminates,a single ply being itself too thin to test conveniently. For much of the following analysis,it is more convenient to deal with the inverse form of equation (6.1),namely Q11(0) Q12(0) Q12(0) Q22(0) (6.3) 0 0 Q66(0)」 12 where the Qi(0),commonly termed the reduced stiffness coefficients,are given by E Q(0)=1-v22i Q22(0)= E2 1-V12V21 (6.4) V21E1 Q12(0)=1-v12v2i Q66(0)=G12
STRUCTURAL ANALYSIS 173 A material having three mutually orthogonal planes of symmetry is known as orthotropic. The stress-strain law for an orthotropic material under plane stress conditions, referred to the material axes, necessarily has the following form: t3 2 'Y12 1 -1)21 El E2 --1)12 1 E1 E2 0 0 0 0 o'2 T12 1 (6.1) where: El, E2 = Young's moduli in the 1 and 2 directions, respectively; v12 = Poisson's ratio governing the contraction in the 2 direction for a tension in the 1 direction; rE1 = Poisson's ratio governing the contraction in the 1 direction for a tension in the 2 direction; G12 = (in-plane) shear modulus. There are five material constants in equation (6.1), but only four of these are independent because of the following symmetry relation1: 1)12 1)21 -- = -- (6.2) E1 E2 For unidirectional tape of the type being considered here, E 1 is much larger than either E2 or G12 because the former is a "fiber-dominated" property, while the latter are "matrix dominated". For a bi-directional cloth, E 1 = E 2 and both are much larger than G12. For tape, 1)12 is matrix dominated and is of the order of 0.3, whereas the contraction implied in 1)21 is resisted by the fibers and so is much smaller. The above equations are all related to a single ply but, because the ply thickness does not enter into the calculations, they also apply to a "unidirectional laminate" that is simply a laminate in which the fiber direction is the same in all of the plies. In fact, most of the material constants for a single ply are obtained from specimen tests on unidirectional laminates, a single ply being itself too thin to test conveniently. For much of the following analysis, it is more convenient to deal with the inverse form of equation (6.1), namely I0-1 I IQll(0) Q12(0) 0 02 = Q12(0) Q22(0) 0 ~'12 0 0 Q66(0) ~32 "Y12 (6.3) where the Qij(O), commonly termed the reduced stiffness coefficients, are given by E1 E2 Qn(0) - Q22(0) -- 1 - 1)121)21 1 -- 1)121)21 1)21E1 Q12(0) Q66(0) = G12 1 - 1)121)21 (6.4)
174 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES It is conventional in composite mechanics to use the above subscript notation for the point of which becomes evident only when three-dimensional anisotropic problems are encountered.The subscript 6 is for the sixth component of stress or strain that includes three direct terms and three shear terms. 6.2.2 Stress-Strain Law for Single Ply in Laminate Axes: Off-Axis Laminates As already noted,when a ply is incorporated in a laminate,its fibers will make some prescribed angle 0 with a reference axis fixed in the laminate.Let this be the x-axis,and note that the angle 0 is measured from the x-axis to the 1-axis and is positive in the counterclockwise direction;the y-axis is perpendicular to the x- axis and in the plane of the ply (See Fig.6.2.).All subsequent calculations are made using the x-y,or"laminate"axes,therefore it is necessary to transform the stress-strain law from the material axes to the laminate axes.If the stresses in the laminate axes are denoted by ox and T,then these are related to the stresses referred to the material axes by the usual transformation equations, c2 231 6.5) where c denotes cos 6 and s denotes sin 6.Also,the strains in the material axes are related to those in the laminate axes,namely,ex,and yo,by what is essentially the strain transformation: 2-2 -0 (6.6) Txy.7xy Ox.Ex Fibres ---- Fig.6.2 Laminate axes for a single ply
174 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES It is conventional in composite mechanics to use the above subscript notation for Q, the point of which becomes evident only when three-dimensional anisotropic problems are encountered. The subscript 6 is for the sixth component of stress or strain that includes three direct terms and three shear terms. 6.2.2 Stress-Strain Law for Single Ply in Laminate Axes: Off-Axis Laminates As already noted, when a ply is incorporated in a laminate, its fibers will make some prescribed angle 0 with a reference axis fixed in the laminate. Let this be the x-axis, and note that the angle 0 is measured from the x-axis to the/-axis and is positive in the counterclockwise direction; the y-axis is perpendicular to the xaxis and in the plane of the ply (See Fig. 6.2.). All subsequent calculations are made using the x-y, or "laminate" axes, therefore it is necessary to transform the stress-strain law from the material axes to the laminate axes. If the stresses in the laminate axes are denoted by trx, try, and "l~y, then these are related to the stresses referred to the material axes by the usual transformation equations, Dry I = if S 2 C 2 csl[ l 2CS 0"2 1 (6.5) "lxy CS --CS C 2 -- S 2 "/'12 where c denotes cos 0 and s denotes sin 0. Also, the strains in the material axes are related to those in the laminate axes, namely, 8x, ey, and Yxy, by what is essentially the strain transformation: e2 1 = s 2 c s 2 -cs csl[ x ey 1 (6.6) 3/12 --2CS 2CS C2 -- S 2 %y I Oy, ey ......_..=, / . / / / / / /Yl / // / , / / ~/ ... / ./ / A /~'~ / e // /// ///" ~/// / / / / / /~ / / / / / / / / / / / i I f # / rxy, 7xy t ~ aXo ~X Fibres Fig. 6.2 Laminate axes for a single ply
STRUCTURAL ANALYSIS 175 Now,in equation (6.5),substitute for o,o2,and T12 their values as given by equation(6.3).Then,in the resultant equations,substitute for 1,82,and y12 their values as given by equation(6.6).After some routine manipulations,it is found that the stress-strain law in the laminate axes has the form x [2xx(0) 2() 2x(0) Ex 2(0) 2w() 2,() By (6.7) xs(0) Q() (0) where the (0)are related to the Q(0)by the following equations: Qx(0) c4 2c2s2 54 4c2s2 2() c22 c4+ c2s2 -4c2x2 2w() s4 2c2x2 c4 4c2s2 2x() es -cs(c2-s2) 一Cs3 -2cs(c2-s2) 2.() C cs(c2-s2) -c3s 2cs(c2-s2) 2sx()」 c2s2 -2c2s2 c2s2 (c2-s2)2 Q11(0)1 Q12(0) 222(0) (6.8) 266(0) Observe that,in equation(6.7),the direct stresses depend on the shear strains (as well as the direct strains),and the shear stress depends on the direct strains (as well as the shear strain).This complication arises because,for non-zero 0,the laminate axes are not axes of material symmetry and,with respect to these axes, the material is not orthotropic;it is evident that the absence of orthotropy leads to the presence of the sand ys terms in equation(6.7).Also,for future reference, note that the expressions for ),),v)and (contain only even powers of sin 0 and therefore these quantities are unchanged when 0 is replaced by-0.On the other hand,the expressions for s and ys contain odd powers of sin 6 and therefore they change sign when 0 is replaced by-0. Analogous to the previous section,the above discussion has been related to a single ply,but it is equally valid for a laminate in which the fiber direction is the same in all plies.A unidirectional laminate in which the fiber direction makes a non-zero angle with the x-laminate-axis is known as an"off-axis"laminate and is sometimes used for test purposes.Formulas for the elastic moduli of an off-axis laminate can be obtained by a procedure analogous to that used in deriving equation(6.7).Using equation(6.1)with the inverse forms of equations(6.5)and (6.6)leads to the inverse form of equation (6.7),in other words,with the strains expressed in terms of the stresses;from this result,the moduli can be written. Details can be found in most of the standard texts,for example page 54 of Ref.3
STRUCTURAL ANALYSIS 175 Now, in equation (6.5), substitute for oq, ~r 2, and T12 their values as given by equation (6.3). Then, in the resultant equations, substitute for 81, 82, and Y12 their values as given by equation (6.6). After some routine manipulations, it is found that the stress-strain law in the laminate axes has the form Eoxx o o o oxs o lE x 1 O'y = axy( O) Qyy( O) Qys( O) 8y "rxy Qx~( O) Qy~( O) Qss( O) yxy (6.7) where the Qij(O) are related to the Qij(O) by the following equations: iQxx o llc4 2c2s2 s4 4c2s2j Qxy( O) | C2S 2 C 4 -'~ S 4 C2S 2 --4C2S 2 Qyy(O) ] = s 4 2c2s 2 c 4 4c2s 2 Qxs(O) ] c3s --¢S(C 2 -- S 2) --CS 3 --2CS(C 2 -- S 2) Qys(O) I cs3 cs(c2 - $2) -c3s 2cs(c2 - s2) Qss(O) J c2s 2 _2c2s 2 c2s 2 (c 2 - $2) 2 F Q ll (0) 7 / Q~(o) / [_Q66(O) J (6.8) Observe that, in equation (6. 7), the direct stresses depend on the shear strains (as well as the direct strains), and the shear stress depends on the direct strains (as well as the shear strain). This complication arises because, for non-zero 0, the laminate axes are not axes of material symmetry and, with respect to these axes, the material is not orthotropic; it is evident that the absence of orthotropy leads to the presence of the Qx~ and Qy~ terms in equation (6. 7). Also, for future reference, note that the expressions for Qxx(O), Q,,y(O), Qyy(O) and Q~s(O) contain only even powers of sin 0 and therefore these quantities are unchanged when 0 is replaced by - 0. On the other hand, the expressions for Qx~ and Qy~ contain odd powers of sin 0 and therefore they change sign when 0 is replaced by - 0. Analogous to the previous section, the above discussion has been related to a single ply, but it is equally valid for a laminate in which the fiber direction is the same in all plies. A unidirectional laminate in which the fiber direction makes a non-zero angle with the x-laminate-axis is known as an "off-axis" laminate and is sometimes used for test purposes. Formulas for the elastic moduli of an off-axis laminate can be obtained by a procedure analogous to that used in deriving equation (6. 7). Using equation (6.1) with the inverse forms of equations (6.5) and (6.6) leads to the inverse form of equation (6.7), in other words, with the strains expressed in terms of the stresses; from this result, the moduli can be written. Details can be found in most of the standard texts, for example page 54 of Ref. 3
176 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES Only the result for the Young's modulus in the x direction,E will be cited here: =()+(+（） (6.9) The variation of Ex with for the case of a carbon/epoxy off-axis laminate is shown in Figure 6.3.The material constants of the single ply were taken to be E1=137.44GPa E2=11.71GPa G12=5.51GPa V12=0.25 21=0.0213 It can be seen that the modulus initially decreases quite rapidly as the off-axis angle increases from 0;this indicates the importance of the precise alignment of fibers in a laminate. 6.2.3 Plane Stress Problems for Symmetric Laminates One of the most common laminate forms for composites is a laminated sheet loaded in its own plane,in other words,under plane stress conditions.In order for out-of-plane bending to not occur,such a laminate is always made with a lay-up that is symmetric about the mid-thickness plane.Just to illustrate the type of symmetry meant,consider an eight-ply laminate comprising four plies that are to be oriented at 0 to the reference (x)axis,two plies at +45,and two plies at -45.An example of a symmetric laminate would be one with the following ply sequence: 0°/0°/+45°/-45°/-45°/+45°/0°/0° 150 100 50 0 0 30 60 90 Off-axis angle (degrees) Fg.6.3 Extensional modulus off-axis laminate
176 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES Only the result for the Young's modulus in the x direction, Ex, will be cited here: 1(~_.~) (1 2vle~c2s e(l~s 4 -- = c a + + (6.9) Ex 4: el / \E2/ The variation of Ex with 0 for the case of a carbon/epoxy off-axis laminate is shown in Figure 6.3. The material constants of the single ply were taken to be E1 = 137.44GPa E2 = 11.71GPa Gl2 = 5.51GPa 1)12 : 0.25 1)21 = 0.0213 It can be seen that the modulus initially decreases quite rapidly as the off-axis angle increases from 0°; this indicates the importance of the precise alignment of fibers in a laminate. 6.2.3 Plane Stress Problems for Symmetric Laminates One of the most common laminate forms for composites is a laminated sheet loaded in its own plane, in other words, under plane stress conditions. In order for out-of-plane bending to not occur, such a laminate is always made with a lay-up that is symmetric about the mid-thickness plane. Just to illustrate the type of symmetry meant, consider an eight-ply laminate comprising four plies that are to be oriented at 0 ° to the reference (x) axis, two plies at +45 °, and two plies at - 45 °. An example of a symmetric laminate would be one with the following ply sequence: 0°/0°/+45°/-45°/-45o/+45o/00/0 ° E X (GPa) i 50 0 J 0 90 1 J ]0 6O Off~is a~J!e (.rfi~-grlze~) Fig. 6.3 Extensional modulus off-axis laminate
STRUCTURAL ANALYSIS 177 On the other hand,an example of an unsymmetric arrangement of the same plies would be: 0°/0°/0°/0°/+45°/-45°/+45°/-45° These two cases are shown in Figure 6.4 where z denotes the coordinate in the thickness direction. 6.2.3.1 Laminate Stiffness Matrix.Consider now a laminate comprising n plies and denote the angle between the fiber direction in the kth ply and thex laminate axis by (with the convention defined in Fig.6.2).Subject only to the symmetry requirement,the ply orientation is arbitrary.It is assumed that,when the plies are molded into the laminate,a rigid bond (of infinitesimal thickness)is formed between adjacent plies.As a consequence of this assumption,it follows that under plane stress conditions the strains are the same at all points on a line through the thickness (i.e.,they are independent of z).Denoting these strains by ex,and y,it then follows from equation(6.7)that the stresses in the kth ply will be given by: x(k)=Qx(0x)Ex+Oxy(0x)Ey +Qxs(0x)Yxy y(k)=Qxy(0x)Ex +Qy()Ey +Qys(0x)Yy (6.10) Tx(k)=Qxs(0x)Ex+ys(0x)Ey+Qxs(0x)Yxy The laminate thickness is denoted by t and the thickness of the kth ply is h-h-1 with hi defined in Figure 6.5.Assuming all plies are of the same thickness (which is the usual situation),then the thickness of an individual ply is simply t/n.Now consider an element of the laminate with sides of unit length parallel to thex-and y-axes.The forces on this element will be denoted by NN 0 -45 0 +45 +45 -45 Mid-plane -45 +45 Mid-plane -45 0 +45 0 0 0 0 0 Fig.6.4 Symmetric (left)and non-symmetric (right)eight-ply laminates
STRUCTURAL ANALYSIS 177 On the other hand, an example of an unsymmetric arrangement of the same plies would be: O°lO°lO°lO° 1+45°1-45°1+45ol-45 ° These two cases are shown in Figure 6.4 where z denotes the coordinate in the thickness direction. 6.2.3. I Laminate Stiffness Matrix. Consider now a laminate comprising n plies and denote the angle between the fiber direction in the kth ply and the x laminate axis by Ok (with the convention defined in Fig. 6.2). Subject only to the symmetry requirement, the ply orientation is arbitrary. It is assumed that, when the plies are molded into the laminate, a rigid bond (of infinitesimal thickness) is formed between adjacent plies. As a consequence of this assumption, it follows that under plane stress conditions the strains are the same at all points on a line through the thickness (i.e., they are independent of z). Denoting these strains by ex, ey, and "Yxy, it then follows from equation (6. 7) that the stresses in the kth ply will be given by: O'x(k) = Q~( Ok)ex + axy( Ok)l?,y + Qxs( Ok)3'xy ~ry(k) = Qxy( ODex + ayy( Ok)f,y + Qys( Ok)'Yxy (6.10) "rxy(k) = Qxs( Ok)ex + ays( Ok)Sy + Qss( Ok)Yxy The laminate thickness is denoted by t and the thickness of the kth ply is hk -- hk-1 with hi defined in Figure 6.5. Assuming all plies are of the same thickness (which is the usual situation), then the thickness of an individual ply is simply t/n. Now consider an element of the laminate with sides of unit length parallel to the x- and y-axes. The forces on this element will be denoted by Nx, Ny, 0 0 *,kS Mid-plane - 45 -4.5 ",~.S 0 0 ~Z mw -45 *t.5 -45 * `k5 Mid-plane Fig. 6.4 Symmetric (left) and non-symmetric (right) eight-ply laminates
178 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES h n=t/2 h k Ply k h k-1 midplane Ply 2 Ply 1 h o=-t/2 Fig.6.5 Ply coordinates in the thickness direction,plies numbered from the bottom surface. and N,(Figure 6.6);the N are generally termed stress resultants and have the dimension"force per unit length."Elementary equilibrium considerations give M=2a-山N=2o物-山 心=2h-) (6.11) Ny Nx Fig.6.6 Stress resultants
178 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES h n=t/2 hk h k-I hi h 0=-if2 Ply k ,.- y midplane Ply 2 Ply 1 T t 1 Fig. 6.5 Ply coordinates in the thickness direction, plies numbered from the bottom surface. and Ns, (Figure 6.6); the N are generally termed stress resultants and have the dimension "force per unit length." Elementary equilibrium considerations give n Nx = ~ O'x(k)(hk - hk-1), k=l Ns = ~ Zxy(k)(hk - hk- 1) (6.11) k=l Ny = ~ O'y(k)(hk - hk-1), k=l 't_ X Ny l ~ Nxy -1 Fig. 6.6 Stress resultants. Nx
STRUCTURAL ANALYSIS 179 Substituting from equations(6.10)into(6.11),and remembering that the strains are the same in all plies,the following result is readily obtained: Nx AuEx +AxEy +Axs Yxy Ny =AxyEx +Ayyy +Ays Yo (6.12) N:AxsEx +Aysy +Ass Yxy where: Ag=>Qu(0x)(hk-hg-1) (6.13) k= The quantities Ay are the terms of the laminate "in-plane stiffness matrix." Given the single-ply moduli and the laminate lay-up details,they can be calculated routinely by using equations (6.4),(6.8),and (6.13).Equation (6.12)are generally taken as the starting point for any laminate structural analysis. 6.2.3.2 Laminate Stress-Strain Law.As was just implied,it seems to be the current fashion in laminate mechanics to work in terms of the stress resultants, rather than the stresses.However,for some purposes,the latter are more convenient.From the stress resultants,the average stresses(averaged through the thickness of the laminate)are easily obtained;writing these stresses simply as ox, dy,and Try then: 0x= t (6.14) Hence,in terms of these average stresses,the stress-strain law for the laminate becomes: Ox Atex Aryey Ars Yoy y=Atyex +Ayyey+Ays Yo (6.15) Tg=AtEx十A,Ey+AxY where: 2(0)hk-hk-i） (6.16) In some cases,equation (6.15)is more convenient than is equation (6.12). 6.2.3.3 Orthotropic Laminates.An orthotropic laminate,having the laminate axes as the axes of orthotropy,is one for which Axs=Ays=0;
STRUCTURAL ANALYSIS 179 Substituting from equations (6.10) into (6.11), and remembering that the strains are the same in all plies, the following result is readily obtained: Nx = Axxex + Axyey + AxsYxy Ny = Axyex + Ayyey "t- Ays Yxy (6.12) Ns = Axsex -t'-Ayse, y + Ass'Yxy where: Aij = XZ, Qij( Ok)(hk -- hk-1) (6.13) k=l The quantities A o are the terms of the laminate "in-plane stiffness matrix." Given the single-ply moduli and the laminate lay-up details, they can be calculated routinely by using equations (6.4), (6.8), and (6.13). Equation (6.12) are generally taken as the starting point for any laminate structural analysis. 6.2.3.2 Laminate Stress--Strain Law. As was just implied, it seems to be the current fashion in laminate mechanics to work in terms of the stress resultants, rather than the stresses. However, for some purposes, the latter are more convenient. From the stress resultants, the average stresses (averaged through the thickness of the laminate) are easily obtained; writing these stresses simply as o'x, try, and 7xy then: Nx Nx N, O'x=-- , O'y = --, "/'xy = -- (6.14) t t t Hence, in terms of these average stresses, the stress-strain law for the laminate becomes: O-x = Axxex + Axyey "l- Axs ]txy Or y = Axy 8 x -t- Ayy ~ y --I- Ays 'Yxy (6.15) where: "rxy = A*xsex + Aysey + Ass'Yxy AU 1 A~j =--= - ~ Qij(Ok)(hk -- hk-1) (6.16) t tk= 1 In some cases, equation (6.15) is more convenient than is equation (6.12). 6.2.3.30rthotropic Laminates. An orthotropic laminate, having the laminate axes as the axes of orthotropy, is one for which Axs = Ays = O;
180 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES clearly,this implies that: ∑2a()hk-hk-i)=0, >Qys(0k)(hg-hg-1)=0 (6.17) k=1 Thus,the stress-strain law for an orthotropic laminate reduces to: x=At8x+A刘Ey Oy Atyex +Ayyey (6.18) Txy =Ais Yo The coupling between the direct stresses and the shear strains and between the shear stresses and the direct strains,which is present for a general laminate, disappears for an orthotropic laminate.Most laminates currently in use are orthotropic. It can be readily seen that the following laminates will be orthotropic: (1)Those consisting only of plies for which 6=0 or 90;here it follows from equation (6.8)that in either case ()=vs(6)=0. (2)Those constructed such that for each ply oriented at an angle 6,there is another ply oriented at an angle -0;because,as already noted from the odd powers in equation(6.8), 2x(-)=-Qxx(),Qyx(-)=-2s(0 There is then a cancellation of all paired terms in the summation of equation (6.17). (3)Those consisting only of 0°，9o°，and matched pairs of±0 plies are also,.of course,orthotropic. An example of an orthotropic laminate would be one with the following ply pattern: 0°/+30°/-30°/-30°/+30°/0 On the other hand,the following laminate (while still symmetric)would not be orthotropic: 0°/+30°/90°/90°/+30°/0° 6.2.3.4 Moduli of Orthotropic Laminates.Expressions for the moduli of orthotropic laminates can easily be obtained by solving equation (6.18)for simple loadings.For example,on setting oy=Txy=0,Young's modulus in the x direction,E,and Poisson's ratio vxy governing the contraction in the y direction
180 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES clearly, this implies that: n Qxs(Ok)(hk -- hk-l) = O, k=l ~-~ Qys( Ok)(hk -- hk-1) = 0 (6.17) k=l Thus, the stress-strain law for an orthotropic laminate reduces to: O'x = Axxex + Axysy O'y = Axy ex + Ayy ~y (6.18) Txy = A,* Yxy The coupling between the direct stresses and the shear strains and between the shear stresses and the direct strains, which is present for a general laminate, disappears for an orthotropic laminate. Most laminates currently in use are orthotropic. It can be readily seen that the following laminates will be orthotropic: (1) Those consisting only of plies for which 0 = 0 ° or 90°; here it follows from equation (6.8) that in either case Qxs(O) = Qys(0) = 0. (2) Those constructed such that for each ply oriented at an angle 0, there is another ply oriented at an angle - 0; because, as already noted from the odd powers in equation (6.8), Qxs(-O)=-Qxs(O), Qys(-O)=-Qys(O) There is then a cancellation of all paired terms in the summation of equation (6.17). (3) Those consisting only of 0 °, 90 °, and matched pairs of + 0 plies are also, of course, orthotropic. An example of an orthotropic laminate would be one with the following ply pattern: 0°/+30°/-30°/-30°/+30°/0 ° On the other hand, the following laminate (while still symmetric) would not be orthotropic: 0°1+30°190°190°1+30°/0° 6.2.3.4 Modufi of Orthotropic Laminates. Expressions for the moduli of orthotropic laminates can easily be obtained by solving equation (6.18) for simple loadings. For example, on setting try = "rxy = 0, Young's modulus in the x direction, Ex, and Poisson's ratio Vxy governing the contraction in the y direction