LAMINATE DESIGNJocelyn M . Seng

30

birch or spruce, laid over balsa core or fir stringers to form a sandwich structure1. This An early example of laminated composite successful production aircraft (7781 units built) materials is the de Havilland Mosquito was designed without the analytical techniques fighter/bomber used by the British Royal Air described in this chapter and without fancy Force during World War I1 (Fig. 30.1). This aircomputer tools. With current laminate design craft was built entirely out of wood because of and analysis techniques, todays higher-perforlimited metal supplies and the need for quick mance composite aircraft are made possible; delivery. The wings, for example, were made as with the increase in speed and accuracy of comthree-ply skins (each 1.5 mm (0.060 in) thick) of putation results, designer confidence in3 . INTRODUCTION 01

Handbook of Composites. Edited by S.T. Peters. Published i 1998 by Chapman & Hall, London. ISBN 0 412 54020 7 n

Fig. 30.1 The de Havilland Mosquito, an all-wood WWII Production aircraft.Printed With Permission, Zokeisha Publications, Inc.

Laminated plate theory 687composites structures is increased. This chapter presents the basic mathematical tools used to design laminates and provides insight on the many options for optimizing the material for particular needs. Three distinct levels of benefit can be derived when using composites and laminate design. With equal fiber distribution in multiple directions, rendering an effectively quasi-isotropic material, composites can approximate metals while providing a weight savings due to the difference in material densities. In addition, however, designers opting to use composite parts enjoy the advantage of being able to tailor the properties of their material by orienting load-carrying fibers in the directions that there are loads. The result is an anisotropic material, which by definition is a material with different properties in different directions. Ultimately, the composites industry is finally beginning to see the development of unique structures that have never before been attempted and with material behavior that is only possible with distinct laminate designs. This is a result of coupled behavior, for example, an extensional load on an anisotropic material can yield extension coupled with bending and twisting deformations. The objective of this chapter is to outline a method to design ply layouts which achieve structural design goals for composite parts. This method is based on laminated plate theory used with the quadratic failure criterion2. This general discussion, which assumes familiarity with undergraduate mechanical engineering fundamentals, shows how the principles for isotropic materials (such as metals) are extended to the analysis of advanced composites. The basic equations are presented and the analysis procedure is outlined. Simplifying concepts are introduced and discussed. Simple computer codes that embody these equations are now widely available, making it unnecessary to ever have to solve these equations by hand. Examples and sample problems are included to demonstrate concepts.30.2 LAMINATED PLATE THEORY

30.2.1 LAMINA

Advanced composite materials are typically supplied as a thin layer, called a ply or lamina, which is subsequently stacked into a thin plate, called a laminate. A unidirectional ply or lamina is a flat or curved layer of fibers oriented in one direction and held together by matrix material that serves to support the fibers. The stresses perpendicular to the planar surface are assumed to be zero. While the behavior of isotropic materials can be described with two elastic constants (typically the Youngs modulus and the Poissons ratio) and one strength value, a composite ply that is transversely isotropic is characterized by four elastic (stiffness) constants and five strength parameters in two-dimensional analysis. The material properties are defined along the fiber (x-direction ) and perpendicular to the fiber (y-direction ). For each unidirectional ply in its own axes, the four orthotropic elastic constants are the longitudinal tensile modulus, Ex; the transverse tensile modulus, EY; the major Poissons ratio, vx; and the shear modulus, Es. Only one Poissons ratio is necessary since Y,,= vx(,,/Ex). five strength parameters for each The unidirectional ply are the longitudinal tensile strength, X; the longitudinal compressive strength, X; the transverse tensile strength, Y; the transverse compressivestrength, Y; and the shear strength, S. The five initial coupon tests to experimentally determine the nine material constants are shown in Fig. 30.2. In the stress, o, versus strain, E , plot, the material is characterized by the slope of the line, which represents material stiffness, and by the failure point, which defines the maximum stress that the material can sustain, i.e. its strength, and by the corresponding maximum strain. Use of the strength parameters is discussed in the section on failure criteria. During three of the coupon tests, the nonzero strains are monitored and the relationships in eqn (30.1) are determined. The

688 Laminate designLongitudinal Transverse

E X

j

". . . . . .

=-- EY

Longitudinal Compression

Transverse Compression

Fig. 30.2 Coupon tests to determine the nine material constants used to characterize an anisotropic material.

material is assumed to be linear and elastic; thus, the stiffness of a material is the same in tension as in compression. Based on the four elastic constants, infinitely different laminates can be designed using laminated plate theory.

If a unidirectional specimen was simultaneously tested under the three load cases, longitudinal tension, transverse tension, and shear, then superposition of the strains results in

Longitudinal tension test3E E

= -0Y

1

= - 0

Ex

(30.1)

EY=

VX --0Y

Ex

1-0,E S

ES

Written conveniently in matrix notation,

Transverse tension test1Ey

= - 0

E,,

y

0

0

Shear testE

=-a,E,

1

(30.3)

Laminated plate theory 689Defining the plane stress stiffness matrix [QJ = [SI-', another form of eqn (30.3) is laminate, x, y, s and 1,2,6 are interchangeable. Material properties are specified with respect to the on-axis coordinates. The properties of an off-axis ply, anything other than 0 degrees, can be calculated by transforming the properties of the 0-degree ply. The angle of transformation, 8, is equal to the ply angle shown in Fig. 30.3, where 1 and 2 are the laminate axes and x and y are the rotated ply axes. 8 is positive counterclockwise from the 1-axis to 90, and negative clockwise to -90".

l o

0

E,

J

Y

This calculation of the plane stress stiffness matrix [Q]for a single ply is the starting point of laminated plate theory, once the engineering constants have been experimentally determined.30.2.2 COORDINATE TRANSFORMATIONS

1

Two coordinate systems are used in laminated plate theory. The local, or on-axis, coordinate frame is defined by x and y, also referred to as the ply axes. The x-axis is along the longitudinal direction of the ply (along the fiber); the y-axis is in the same plane, but in the transverse direction (perpendicular to the fiber direction). The subscripts is used with expressions for shear, and is a contraction for the subscript xy. The ply's material properties are defined in this axis system. Since not all plies are aligned in the same direction along the principal loading axis in a laminate, a second set of coordinates is necessary to analyze composite laminates. The global, or off-axis, coordinate frame is defined by 1 and 2, also referred to as the laminate axes. The 1 direction is along the principal orientation of the laminate; the 2 direction is perpendicular to it. The subscript 6 is used with expressions for shear, and is a contraction for the subscript 12. The loads on the laminate and the boundary conditions are usually defined in the global svstem. In the case of a 0-degree unidirectional , "

Fig. 30.3 Definition of ply axes (x,y) and laminate axes (1,2), where the lines indicate the fiber direction.

The laminate off-axis stiffness matrix is computed from the ply on-axis stiffness matrix by using the following ransformation relation:

Q1llQ22

m4 n4 m2n2 m2n2 m3n mn3

2rn2n' 4m2n2 2m2n2 4m2n2 m4+n4 -4m2n2 (m' -n2)' m2n2 --2m2n2 -mn3 mn3 - m3n 2(mn3- m3n: -m3n m3n- mn3 2(m3n- mn3:n4 m4 m2n2

(30.5)

690 Laminate design where rn = cos 8, n = sin 8. A summary of some other useful transformation relations is given in Section 30.7. It is clear from these relations that when 8 = 0, then Q,, = Q,, = 0, which means shear and extension are uncoupled, i.e. shear loading only causes shear deformation, and extensional loading causes extensional deformation with Poissons effect but no shear deformation. Using the off-axis plane stress stiffness coefficients, the constitutive relations of eqns (30.4) and (30.3) can be generalized to a ply of any orientation: with the [Q] matrix, the S12(and S terms J , reflect the deformation in the direction perpendicular to the direction of loading, and are commonly referred to as the component due to Poissons effect (implying the major Poissons ratio). The S,, and S2, (and S,, and S62) terms reflect the amount of shear deformation under extensional loading, and are commonly referred to as coupling terms. Unlike the off-axis unidirectional ply shown in Fig. 30.4@), the 0-degree unidirectional ply shown in Fig. 30.4(a) does not exhibit any shear deformation under extensional loading.30.2.3 KINEMATICS

and inversely

For example, a physical interpretation is shown in Fig. 30.4. In general, the terms on the diagonal (S,,, S,, S,) reflect the amount of deformation in the direction of loading. As

Kinematics is the study of movement and depends solely on geometry, not on material properties. Since composite laminates are often thin two-dimensional structures, plate theory is used to simplify the three-dimensional behavior. Plate theory tries to account for stretching and bending behavior relative to the midplane of the laminate. The key assumption of plate theory is that normals remain normal, straight and unstretched. In practical terms, the plies in the laminate are assumed to be completely bonded to each other, allowing no interlaminar shear. Some other assumptions are that the

Fig. 3 . Extensional loading of (a) a unidirectionalply and (b) an off-axis ply and their associated off-axis 04 stiffness matrix.

Laminated plate theory 691material exhibits perfectly linear elastic behavior and that there is a perfect bond between the fiber and matrix. The out-of-plane displacement, w, can be described by a function of the in-plane coordinates such that

E: -I-ZK1

w = wo(x,y)

(30.7)

Based on the Kirchhoff assumptions, Fig. 30.5 shows the deformation of a cross section of the plate in the x-z plane, relative to the x-direction

u

= uo-z-

aw0

ax

(30.8)

30.2.4 STRESS RESULTANTS

Similarly, the displacement along the y-axis isY = Y,-Z-

awo

aY

Based on the definition of strains,

au&1

=-=--z-

ax

auo azw, ax a x 2

Just as beam theory defines net tensile force, shear force, and moment, relating everything (30.9) to the neutral axis, plate theory defines stress resultants and moment resultants to eliminate any z-direction dependence and to relate everything to the midplane, as shown in Fig. (30.10) 30.6. Ply stresses along each loading direction are summed for the laminate: (30.12)

2

MUndeformed Cross Section Deformed Cross Section

Fig. 30.5 Extension and bending deformation.

Fig. 3 . Force and moment resultants acting on a 06 plate.

692 Laminate designThese resultants can be rewritten in terms of strain by substituting in the constitutive relations. Putting the two-part expression for strains, eqn (30.11), into the constitutive relations, eqn (30.6), and substituting the resulting stress expression into the definitions for the resultants, eqn (30.12),

{N] = [A] (E'}

All

1 '1

1 '2

1 '6

B216 '1

'2 2B62

B26'66

A,, indicates the relationship between longitudinal in-plane load, N,, and the longitudinal extension, E;; A,, indicates the coupling between longitudinal in-plane load, N,, and the extension in the transverse direction, E;, (the traditional Poisson's effect); A,, indicates the coupling between longitudi+ [B](K) (30.13) nal in-plane load, N,, and the in-plane shear, E;; A, indicates the relationship between inplane shear load, N,, and the in-plane Bll 12 ' B16 shear distortion, E;; B,, indicates the coupling between transverse 21 ' 2 '2 B26 in-plane load, N,, and the twist, K ~ ; '61 '62 '66 B, indicates the relationship between inDll D12 D16 plane shear load, N6, and the twist, K ~ ; D21 D26 D12 indicates the coupling between longitudinal bending load, MI, and the transverse D61 D62 D4 65 bending curvature, K,; (30.14) D, indicates the relationship between twisting moment load, M6, and the twist angle, K ~ .

The ply stacking sequence has no effect on the (30.15) A matrix coefficients, which reflect in-plane behavior. However, since the B and D matrix coefficients are a function of z, they are dependent on the stacking sequence. There are two unique physical situations that deserve mention. When the laminate is symmetric about its midplane, the B coefficients are zero, which means that there is no It can be shown that A,, = A,,, B,, = BZ1, D,,= coupling between in-plane loads and curvaDZ1,etc. Equation (30.14)represents the funda- tures, nor between bending loads and mental relationships in laminated plate theory. in-plane deformations. Another common sitThe 6 x 6 matrix is the laminate stiffness uation is when the A,, and A,, coefficients are matrix. zero (usually in the presence also of all the 3 A composite with unidirectional plies lami- coefficients being zero): this arises when a nated in different directions (a generally laminate is balanced, i.e. there are an equal anisotropic material) under an inplane load number of off-axis plies in the +8 and 4 may stretch, bend and twist, as a result of directions and they have equal thickness. In extensional/ shear coupling. By comparison, a this case, there is no coupling between extenmetal structure will stretch only under an sion loads and shear strain. If, in addition, inplane load, bend only under flexure, and those +8 and 4 plies are effectively the same twist only under torque. Each matrix coeffi- distance from the midplane, then the correcient in eqn (30.14) relates a particular sponding B and D matrix coefficients tend resultant to a strain expression. For example, toward zero.

[AI = IIQldz

Laminated plate theory 693Most laminates used today are symmetric to eliminate or reduce any tendency of the structure to warp unexpectedly. Most laminates are also balanced, often because it is erroneously thought to be necessary to prevent the structure from warping. A balanced laminate is really only necessary in situations with reversible shear loading conditions.30.2.5 RESULTING STRAIN STATE

a flexural contribution, then it must be added

The off-axisply strains can be transformed to on-axis ply strains for each ply and their significance can be evaluated per a failure criterion (refer to the relations given in Section 30.7).

Knowing the laminate stiffness matrix and the applied loads, the resulting strains can be computed. The strains are obtained by inverting the stiffness matrix and multiplying by the input load. Instead of inverting the 6 x 6 stiffness matrix, however, it is sometimes possible to simplify the analysis even further. If the laminate is symmetric about the midplane so that the B coefficients are identically zero, then the in-plane (described by N, [A], E ) and bending problems (described by M I [D], IC)become uncoupled. In this case, it is much easier to invert two 3 x 3 stiffness matrices to get the compliance matrices (see also Section 30.7 for the explicit terms to invert a matrix)

[EtEsI

m2 n2 =.i

-mn mn 2mn -2mn m2-n2

n2 m2

.,

f:

I

(30.19)

where, as before, m = cos 8, n = sin 8. One step further, the on-axis ply stresses can be obtained by multiplying the on-axis ply strains by the ply stiffness matrix [Q] as shown in eqn (30.6). The laminate engineering constants, which have meaning with symmetric laminates only, are calculated from the compliance matrix and are useful for comparison to the properties of other materials, such as metals

[a] = [AI- [d] = [Dl-(30.16)

Then, the compliance matrix is multiplied by the appropriate input load conditions to compute the laminate strainsEPE;\E:

=

I

1 1

12

6 1

IN,

(30.17),

where the compliance terms have been normalized to have the necessary units of [length2 force] /[a*] = [a]h

a12 aZ a26 N26 266. tN6

(30.21)

6 1

In summary, the mathematical process of analyzing composite laminates is indicated in Fig. 30.7. A laminates stiffness is calculated as the summation of its individual ply properties. Load on the laminate is described in terms of Since all the plies are bonded together, the the laminate coordinates. Calculated from the strains in each ply, in the laminate axes, (1,2), applied load and the known material stiffness are equivalent to the laminate strain. If there is properties, the response of the laminate is

694 Laminate design

expressed as laminate strain. In order to apply a failure criterion, the laminate strain is commonly transformed into ply strains and each ply is individually evaluated.30.3 ENHANCEMENTS TO THE BASIC

30.3.2 HYGROTHERMAL EFFECTS

LAMINATED PLATE EQUATIONS

30.3.1 SANDWICH CONSTRUCTION

In composite structures, sandwich constructions are commonly used. By increasing the distance between the load-carrying laminate skins, a core can provide increased bending stiffness without a significant weight penalty. The core is often idealized in laminate design: it is assumed that the core does not contribute to laminate strength or in-plane stiffness, and that the shear bonds between the skins and core are perfect. The parallel axis theorem can be used to account for the increased moment of inertia that the core creates by offsetting the laminate load-carrying skins from the midplane. Once the skin laminates have been sized, further calculations can be performed to confirm that the core assumptions are valid3.

Most structure is exposed to a variety of environmental effects. Of particular concern are heat and moisture. The design must account for the hygrothermal effects (hygrothermal means water and temperature). A laminate that is stress-free when curing at an elevated temperature will have residual stresses when brought back to room temperature. It has been thought that some of the apparent improvement in toughness of 250F resins over 350F resins is simply due to the reduction in residual stresses (AT = 75 - 250 = -175F versus AT = 75 - 350 = -275F). In addition, moisture is absorbed by the laminate, usually into the resin. The negative effect on the mechanical properties is particularly pronounced at both high temperature and high humidity. Assuming that the primary effect on residual stresses is due to different thermal expansions and moisture expansions along and transverse to the fiber direction, two additional strains on the laminate result:{E)

= {a)AT+@)Ac

(30.22)

Measured

PlyStiffnesses

Plane Stress Coeff.

Laminate Stiffness Matrix

Inverted Stiffness Matrix

Given Loads; Resulting Laminate Strains

Ply Strains

Fig. 30.7 Logical flow of calculations involved in analyzing a symmetric composite laminate that is loaded axially and/or in bending.

Failure criteria 695The summation process can be used to determine the effective laminate expansion coefficients. The hygrothermal load, sometimes called non-mechanical load, can be computed by multiplying the laminate stiffness by the hygrothermal strain (i.e. laminate thermal expansion coefficients multiplied by the change in temperature). The stress induced by moisture absorption can be accounted for similarly by using PAC in place of aAT. Thus, the non-mechanical loading in the laminate can be expressed asX' = transverse tensile strength, Y = longitudinal compressive strength, Y' = transverse compressive strength, S = shear strength. (30.24)

The maximum ply strain values can be interpreted by dividing the above strengths by the appropriate ply stiffness coefficient.cx* = E ~ ' *= E * = Eyf* = Y E,* =

max longitudinal tensile strain, max longitudinal compressive strength, max transverse tensile strength, max transverse compressive strength, max shear strength. (30.25)

The mechanical and non-mechanical loads, N and W , can be added together to determine the total load experienced by the laminate.30.4 FAILURE CRITERIA

The ultimate objective in any structural design is to create a structure able to withstand deflections or loads without failing. The initial concern is to remain below a prescribed deflection as part of stiffness criteria. Once these criteria are satisfied, the focus shifts to a strength criterion, such that applied stress must not exceed laminate strength. Composite materials normally possess different strengths when loaded in either tension or compression. The following represent the minimum number of strength properties necessary to characterize a unidirectional or fabric ply. They are determined using material coupon tests, previously outlined in Fig. 30.2.

Laminate strength is function of material (ply) strength and the constraints on the ply within the laminate. Thus, failure is best assessed at the ply level. The proper interpretation, however, of the significance of the applied stress relative to the material strength is still debated. Maximum stress and maximum strain failure criteria are common wherein the applied stress or strain value is compared directly to the strength value. A review of failure criteria has been published4. Early laminate failure theories fail to account for Poisson's effects and interaction between loads in orthogonal directions (a complex load condition). For example, the major weakness of both the maximum stress and the maximum strain failure criteria are their inability to couple stress, or strain, components in determining the ultimate failure of a ply. It is important to understand that the longitudinal tensile failure of a ply is affected not only by the longitudinal load, but also by the magnitude of applied transverse loads. As a result, stress interaction criteria are widely used throughout the industry to determine ply failure in a laminate.

30.4.1 QUADRATIC FAILURE CRITERION

X = longitudinal tensile strength,

Tsai developed a two-dimensional stress interaction failure criterion and predicted the strength of an orthotropic ply subjected to combined stresses or strains. This analysis

696 Laminate design

takes into account the effects of other stress components on the strength in any one direction. Tsai postulated a criterion in stress space consisting of the sum of linear and quadratic scalar products as follows:

F 1DOI1

+ Ftp1I 1 i, j

=

x, y, s

(30.26)

before failure occurs); R < l failure has already occurred (i.e. occurs prematurely at some point below the applied stress or strain) and the applied stress or strain level can not be attained (e.g. if R = 0.5, then only half the applied stress can be sustained). Equation (30.28) is substituted into eqn (30.26) and the solution of this quadratic equation can be obtained.

or, in expanded form,

+ ( F p x + Fyuy) 1 INote that Fxs = Fys= F, = 0, and the six strength parameters are interpreted from the ply strength values (reviewed in Section 30.8):

[ F I , ~ l ~ ~ ][F,u,] R - 1 = 0 + R

(30.29)

i, j

= x, y, s

F = -

1s 2

F =

1

1

-- -

x x

F =y

1

1

-- -

Y

Y

(30.27)

The stress interaction term, Fx,*, can have a value of -1 I Fx; < 1although is recommended to be -1/2. When F,; = -1/2, the quadratic failure criterion is a general case of the von Mises criterion (Section 30.8.2). Instead of simply evaluating the failure criterion to determine if the laminate failed, it is useful to consider a nondimensional ratio to provide a perspective of the significance of the applied stress relative to material strength. Tsai defines the strength ratio, R, such that

The positive and negative roots of the quadratic equation can be found and represent failure of the laminate in tension and compression (where the absolute value of the negative root is used), respectively. Failure envelopes can be plotted to show laminate strength for any combination of loads. Instead of the stress space representation, however, the examination of failure envelopes in strain space is a useful alternative. The representation of failure envelopes in strain space is preferred because strain is usually specified in laminated plate theory. Strain, unlike stress, is at most a linear function of the thickness. Furthermore, failure envelopes are fixed in strain space, and are independent of other plies with different angles which may exist in a laminate. Thus, they can be regarded as material properties. Another additional advantage of strain space is that the axes are dimensionless.30.4.2 STRENGTH OF LAMINATES

(30.28)

The strength ratio is always a positive number with the following physical implications:

X = 1 failure occurs (at the applied stress orstrain level); R > 1 failure has not occurred and R represents a factor of safety (e.g. if R = 2, then the applied stress can be safely doubled

Traditional failure criteria based on strength of materials are limited to the prediction of the FPF, the point beyond which the continuous and homogeneous material assumptions are no longer valid. The use of a simple method for modeling of degraded plies is recommended, from which the FPF can be estimated. The load-carrying capability of a laminate

Laminate design 697of the preselected orientations results in a quasi-isotropic laminate. This is the performance baseline, because load-carrying fiber is in effectively all directions. Laminate performance can only be improved beyond that of a quasi-isotropic laminate as fiber is biased into load directions, since, of course, fiber would never be put in unnecessary directions. Heretofore, quasi-isotropic laminates have been used because they give properties like those of metals, and predictable responses that are familiar, although they are not optimal in strength-to-weight or stiffness-to-weight ratios. Many laminates used today on aircraft structures tend to be of this type. In general, however, the more directional the loading, the bigger the payoff possible with anisotropic tailoring. To improve on the performance obtained with a quasi-isotropic laminate, the cost to design and analyze the anisotropic part (using the tools like those discussed in this chapter) is unfortunately often thought not to be worth the additional weight savings. This attitude is commonly rationalized by worry about holes, increase in work associated with more complicated fiber placement (preform assembly), etc. In practice, laminate designs, if not quasiisotropic, are certainly still symmetric about the midplane, balanced (equal quantity of -8 30.5 LAMINATE DESIGN and +8 plies), and orthotropic. Capitalizing on To simplify the analysis, it is commonly initially the benefits of anisotropy will probably occur specified that a laminate will be constructed of in other industries first before being adopted plies oriented with fibers in a few preselected by the more conservative aircraft industry. directions, where only the percentage distribuAn exception to traditional aircraft laminate tion in each orientation must then be design is the X-29 experimental aircraft, which determined. Laminates with plies distributed demonstrated a unique attribute of anisotropy every 45" are called n/4 laminates (plies can be (Fig. 30.8). The basis for this design lies in the in the 0, 45, 90 or 4 5 directions. Ply orienta- important assumption that the 1,2,6 axes are tions are usually specified as a value between usually the primary load directions for the -90 and 90". For example, instead of identifying laminate. With the coordinate system for loadthe orientation as 135, the laminate orientation ing changed to be 20" off a designated is more commonly called 45", although they laminate system, it can be shown that the lamare the same). Another class of laminates are inate behavior in flexure and torsion is called n/3, where plies are placed every 60" coupled. In fact, twisting will result with flex(plies can be in the 0, 60 or -60 directions). In ural loading, even though the material would both cases, an equal percentage of plies in each normally behave as most metals. This is the beyond the FPF can be formulated using a ply degradation model. Two possible methods are recommended: first, the simplified micromechanics model based on the modified rule-of-mixtures relations can be used. Plies with transverse cracks are replaced by plies with reduced matrix modulus, Em. Micromechanics translates the effect of the altered constituent material properties to the ply level, e.g. how a change in the matrix modulus affects the shear and transverse modulus of the unidirectional ply. Degraded plies are modeled by quasi-homogeneous plies so that laminated plate theory can be reapplied to determine the ply stresses and ply strains. Another approach for the prediction of post-FPF strength can be based on macromechanics, without resorting to micromechanics. The degradation factor (DF) is applied directly to the transverse and shear modulus, as well as the major Poisson's ratio. The exact value for the degradation factor must be determined empirically.A value between 0.1 and 0.3 is recommended. If the degradation factor is given a value close to zero, the quadratic failure criterion can be made to resemble the maximum strain criterion and results in a generally conservative estimation of laminate strength.

698 Laminate design

principle used on the X-296. The normal ten- the laminate. Composite materials are not dency for forward swept wings to diverge at merely a light-weight substitute for heavyincreasing speeds was counteracted by this weight metals. Structural performances which laminate design: the increase in lift creates a are not possible with metals are easily achievdecrease in angle of attack, as the laminate able. Examples of such unique properties twists in the direction opposing the forces. include Poissons ratios greater than unity or It is conceivable that in the future the even negative, bending-twisting coupling, and graphite golf shafts currently gaining in popu- zero or negative coefficients of thermal expanlarity could be tailored to the individual golfer. sion (CTE). The problems and examples below The same coupling principle could be applied. illustrate the engineering constants of angleA golfers tendency to consistently slice the ply and related laminates. Examples of large ball might allow the designer to customize a and negative Poissons ratios and examples of golf shaft which not only bends, but also bend-twist coupling are also given. twists slightly under the bending load of the bad swing.30.5.2 UNUSUAL POISSONS RATIOS 30.5.1 UMQUE BEHAVIOR

The most unique features of composite materials are the highly direction-dependent properties. Highly coupled deformation and load-carrying capability can be designed into

Personal computer software based on a computer spreadsheet allows rapid sensitivity studies and parametric analysis of the behavior of laminates. Laminated plate theory with micromechanics is programmed into MicMac/In-Plane2. A companion charting tool,

Fig. 30.8 Top view of the Grumman X-29 aircraft with wings that twist under flexure to counteract the detrimental aerodynamic effects.@ NASA)

Laminate design 699'Chart-quick', can be used to plot variation of CTE as a function of independent variables (0, E,,, E,, vf, etc.). For the following problems and examples, the carbon fiber reinforced polymer material data used are shown in Table 30.1. Figure 30.9(a) shows the engineering constants for a unidirectional laminate as it is rotated from the on-axis. The Poisson's ratio, vx, of a 0" laminate is approximately 0.3. With increasing angle of the off-axis laminate, the Poisson's ratio decreases. The Poisson's ratio of a 90" laminate is effectively zero, because contraction in the transverse direction is constrained by the fibers. Figure 30.9(b) shows the engineering constants for an angle-ply laminate. It is interesting to observe the very large Poisson's ratio of 1.32 for a [ S O ] laminate. A value of greater than one implies that the transverse dimensional change is more than in the dimensional change in the longitudinal direction of loading. When the ply angle is either 0 or 90", the laminates (and consequently the values for the engineering constants) in Figs. 30.9(a) and 30.9(b) are the same. In both Figs. 30.9(a) and 30.9(b), the transverse modulus, E,, is a 'mirrorTable 30.1 Material property data for three different carbon fiber systems: IM6/Epoxy, T300/5208 and M40J/F584lM6/ EpoxyLongitudinal tensile modulus, E x (Msi) Transverse modulus, EY(Msi) Poisson's ratio Shear modulus, Es (Msi) Longitudinal CTE, a1 Transverse CTE, a2 Volume fraction V,(%) 66 70 29.44 1.62 0.32 1.22

T300/

520826.27 1.49 0.28 1.04

M40J/ F85432.8 1.2 0.26 0.66 -0.14 15 62

image' of the longitudinal modulus, Ex. Figure 30.10(a) shows the engineering constants for cross-ply laminates. For any given laminate, the longitudinal modulus, Ex, and the transverse modulus, E , are equal. The Poisson's ratio, vx, of a [d/90] laminate is approximately zero, because of the presence of fibers in the transverse direction. The largest

Modulus (Msi) 30 0022.50 15.007.50

Modulus Poisson's (Msi) Ratio 30.00 21.5 22.50

Poisson's Ratio 21.5 1

10.5

15.007.50

0.00

0

0.00

Ply Angle, 8 (degrees)

f

-0.5

(4

(b)

'

0.5 0 15

30

45

60

75

Ply Angle, 8 (degrees)

Fig. 30.9 Engineering constants of IM6/epoxy laminates as a function of 6 for (a) off-axis unidirectional and (b) mgle-ply [+el,.

[e],;

700 Laminate design

LO7 (6 + 90>1, ,Modulus

L, O

f

6,IsPoisson's

(Msi) 30.0022.50 15.00 7.50

0.0015 30 45

60

75

15

30

45

60

75

Ply Angle, 8 (degrees)

Ply Angle, 8 (degrees)

(4

(b)

Fig. 30.10 Engineering constants of IMG/epoxy laminates as a function of 19for (a) cross-ply [I9,(0+ 90)],,;

and @) L, O,

* qs.

Poisson's ratio is 0.55 for a [*45] laminate. The shear modulus, E , is a maximum, of course, for the [*45] laminate. Figure 30.10(b) shows the engineering constants for laminates with 50% 0" plies and 50% angle-plies. With the exception of the transverse modulus, the results are similar to those for the angle-ply laminate shown in Fig. 30.9(b).When the ply angle is 90", the values for the engineering constants in Figs. 30.10(a) and 30.10(b)are the same. Figures 30.11(a) and 30.11(b)show the engineering constants for some unusual laminates. When the ply angle, 8, is 15", Fig. 30.11(a) shows an off-axisunidirectional laminate and Fig. 30.11(b)shows an angle-ply. For all other ply angles, the laminates are unbalanced. From Fig. 30.11(a),it can be observed that the [15/60Is laminate exhibits an extremely large negative Poisson's ratio of -0.32, meaning the laminate will expand in the transverse direction under longitudinal tension loading and compress in the transverse direction under longitudinal compressive loading. From Fig. 30.11(b), it can be observed that the [-15/30Is

laminate exhibits a very large Poisson's ratio of 1.32, when compared with that of an isotropic material (0.3). Besides the unique Poisson's ratio behavior, it is also important to examine the values of the other coupling coefficients.

EXAMPLETable 30.2 considers the resulting deformations on coupon specimens under load, and Fig. 30.12 indicates the relative magnitude of deformation due to large and negative Poisson's ratios.30.5.3 STIFFNESS AND COUPLING

It is useful to look at the A, B, D stiffness matrices of some simple laminates. For ease of comparison, the stiffness matrices can be normalized to have units of [force/length2]by defining

[A*] [ A ] / h , [B*] = 2[B]/h2, =

Laminate design 701Poissons Ratio T 2

30.00 T 22.50

15.00 7.50 0.00

,r1

Modulus (Msi) 3000

[-I 5/9 14s

Poissons Ratio

T1.5 1 0.5 0Ply Angle, 8 (degrees)

22.5015.00

0.50

7.500.00

1

Ply Angle, 6 (degrees)

1 -0.5(b)

-0.5

(4

Fig. 30.11 Engineering constants of IM6/epoxy laminates as a function of 8 for (a) [15/8],s; and (b) [-w~I,.

Table 30.2 Strains, deformations and strength ratio (based on first-ply-failure) of 10 in x 1 in x 0.1 in specimens under 1000 lb longitudinal load, N,Longitudinal strain Material40ksi Steel IM6/Epoxy IM6/Ep IM6/Ep 30ksi Aluminum IM6/Ep IM6/Ep E-glass/Ep IM6/Ep(1C3 in/in). -

Transverse strainE2

Longitudinal Transverse displacement displacement(1C3in)A1

El

in/in)

(10-3 in)-1.0 -1.1 -0.2 -12.9 -3.0 3.5 -17.7 -20.7 -1.13.9 50.0 7.9 6.8 2.9 3.1 2.7 4.5 0.8

A2

Strength ratio R

0.34 0.34 0.65 0.95 1.02 1.09 2.40 3.20 6.25

-0.10 -0.11 -0.02 -1.29 -0.30 0.35 -1.77 -2.07 -0.11

3.4 3.4 6.5 9.5 10.2 10.9 24.0 32.0 62.5

Fig. 30.12 Relative deformation of 10 x 1 x 0.1 in specimens under 1000 lb load along the centerline (laminates are IM6/epoxy, unless otherwise indicated).

702 Laminate design

A four-ply laminate consisting of two 0" and two 90" plies can be combined into four different laminates. From Table 30.3 it can be observed that while the A* matrix remains unchanged through varied ply stacking sequences, large differences arise in the B* and D*matrices. From Table 30.4 it can be observed that only the fourth laminate is unsymmetric and has a

B* matrix with nonzero terms. The first and fourth laminates are balanced and so the A*16 and A*26coefficients are zero. For the second and third laminates which differ by the sign of the off-axis plies, the stiffness behavior differs only in that the A*16,A*26, and D*26coeffiD*,6 cients are of opposite signs. Table 30.5 displays different quasi-isotropic laminates. Note that the normalized A* matrix

Table 30.3 Normalized stiffness coefficients for four IM6/epoxy laminates, in units of MsiLayup

10/90/90/01' 15.624

190/0/0/901

~0/90/0/901'

f 0/0/90/90115.624 0.525 0.525 15.6340

0.525

.15.624 0.525

[A*]~

0.525 35.634 0 0 0 1.2200 0 0 0

0 ' 0.525 15.634 0 0 0 1.220,00

15.624 0.525 0.525 15.634

,

0

0

0 1.2200 0 -

0

0 0-6.997

1.220

'

[B*]

0 ' 0

'

00

, o

0

o , ,0 0

0

0

26.129 0.525 0.525 5.138 0 0

.

:]I00 0 ' 01.220,,

3.499 0 0 -3.499

- 6.9970 0

0

o * ,0 0

0

0 ' 0 0 .

' 5.138

1.220.

.

0.525 0.525 26.129

' 15.634 0.525 0.525 15.634

. .15.634 0.525

0

0

0

0

1.220,

0 ' 0.525 15.634 0 0 0 1.220,

Table 30.4 Normalized stiffness coefficients for four IM6/epoxy laminates, in units of MsiLayup[0/0/+45/-451$ 10/0/+45/+451+ [0/0/-45/-45Is f+45,/45,1 9.297 6.859 6.859 9.297,

' 19.463 3.692[A*]3.692 5.469

0 ' 04.387.

, o* oIB"1

0 0 0 0

0

, o

:I0

' 19.463 3.692 3.499 3.692 5.469 3.499 , 3.499 3.499 4.387

I

19.463 3.692 -3.4993.692 5.469 -3.499 .-3.499 -3.499 4.387

.

0

0

0 0 7.555 *

0 0

0

J

o

0 0

:ll!0

00 0

: [:0

0 3.499 0 3.499 3.499 3.499 09.297 6.859 6.859 9.297 00 0 0

I

27.087 1.316 0.656 1.316 2.597 0.656 0.656 0.656 2.012

27.087 1.316 0.875 1.316 2.597 0.875 0.875 0.875 2.012

27.087 1.316 -0.875 1.316 2.597 -0.875 -0.875 -0.875 2.012

II

7.555

Laminate design 703Table 30.5 Normalized stiffness coefficients for four IM6/epoxy laminates, in units of Msi

0 12.466 3.692 3.692 12.466 0 0 0 4.387

20.932 3.098 1.312 3.098 5.188 1.312 1.312 1.312 3.793

I I

12.466 3.692 0 3.692 12.466 0 0 4.387 0

1[

12.466 3.692 0 3.692 12.466 0 0 0 4.387 18.194 2.782 3.735 2.782 8.558 2.208 3.735 2.208 3.477

12.466 3.692 0 3.692 12.466 0 I O 0 4.387 22.001 1.932 0.737 1.932 6.451 1.956 0.737 1.956 2.628

20.2115 2.504 1.968 2.504 7.094 1.968 1.968 1.968 3.200

terms are equivalent for all quasi-isotropic laminates. This means all have the same stiffness to weight ratio. The differences between these laminates thus manifests themselves only in how they behave in bending.

Multiplying out the strains,a12N2/ 6

= a , , ~ , , E2 =

(iii) Evaluate the displacements by integrating the strains. (Note that for the tube,E

= a16N1

=r-.)

dd) dx

PROBLEMFind the amount that an anisotropic 20-layer [0/30], T300/5208, 3 in diameter, 12 in tube tube will extend, change in circumference, and twist under an in-plane load, N, = 100 lb/in.

1

Eldx = l:$dx

= l:llNldx

-

u = allNIL

I

E2dy=$$dC

=[a12NldC

---t

z,

= ul2N12nR

SOLUTION(i) Compute the laminate stiffness matrix and invert to get the compliance matrix:21.159 2.567 3.935 2.567 2.476 1.458 , 3.935 1.458 3.195 62.563 -26.647 -64.893 [a*] = -26.647 563.836 -224.518 -64.893 -224.518 495.374 (ii) Evaluate the strains:@

-

1 R

L+@ = a , 6 ~ l R

I

(iv) Evaluate the displacements numerically: =-a*ll N L = - 62.563 x 100 x 12 h 0.1

I

I

= 0.75 x 10-3in

v = -Nl2nR a*,,h

= -___ x 100 x 2 x 3.14 x 1.5 26*M7

0.1

= -0.25 x 10-3 in

= -U*16 Nh

L = R

- x 100 x 12/1.5 L 64 893

= -0.52 x

0.1 radian = -0.03 degree

704 Laminate design30.5.4 CTE BEHAVIOR

the a1 is -2 to -2.5 (versus -0.14 for a1 of the The following four figures show the coefficient unidirectional tape, as shown in Fig. 30.13(a)). As the number of ply angles increases, the of thermal expansion (CTE) in two principal directions (referred to as a , and a,) for CTE behavior becomes less intuitive. Figure M40J/F584 carbon fiber laminates. Figure 30.14(a) shows the CTE of a laminate with 50% 30.13(a)shows the CTE of an off-axis unidirec- 0 plies and 50% angle-ply; Fig. 30.1303) shows tional ply; Fig. 30.1303) shows the CTE of an the CTE of a laminate with 25% 0 plies, 25% 90 angle-ply laminate. From Fig. 30.13@), it can plies and 50% angle-ply. Note that when & be observed that, due to the Poisson coupling equals 45", the resulting laminate is quasieffect, laminate CTE values less than that of a isotropic [0, 90, &45] as confirmed by a , unidirectional material are possible for specific equaling a*. Examination of the fundamental O ply angles. In Fig. 30.13(b), for 0 of G to do", trends in Figs. 30.13 and 30.14 indicates

1.0 60 1.0 4012.00

16.00

-r

1.0 40 1200 1.0 00

1.0 008.00

am600

c

6.00

40 .0 200 00 .0-200

4.00 200

a , ._0

0 C

p

0.00 1 0

20

30

40

SO

60

70

80

90-2 00-4.00

0

0

Ply Angle, 0 (degrees)

(b)

Fig. 30.13 Coefficient of thermal expansion of M40J/F584 laminates as a function of 8 for (a) off-axis undirectional [e,],; and (b) angle-ply [&J,.

40 .00c

c

200

--Ply Angle, 0 (degrees) Ply Angle, B (degrees)

(4

(b)

Fig. 30.14 Coefficient of thermal expansion M40J/F584 laminates as a function of 0 in the following laminates (a>[o,, and (b) IO, ,90,, 4 1 , .

Laminate design 705potential near-zero CTE laminates, particularly useful in spacecraft applications to minimize deformation due to the large cyclic thermal loading. dominantly in the longitudinal direction to accommodate flexural loading (like a mast or golf club), Fig. 30.14(a) indicates that a [O,, G0JS with a steel mandrel would be a problematic choice, resulting in a composite shaft locked on to the mandrel as shown in Fig. 30.15@).There is a need for sufficient fibers in the hoop direction (90") to result in a laminate CTE less than that of the mandrel material. The CTE of the metal materials given in Table 30.6 indicates that it is easier to remove a composite shaft from an aluminum mandrel than from a steel mandrel.

EXAMPLETo remove a composite shaft from a metal mandrel after elevated temperature cure, the laminate CTE in the hoop direction of the cylindrical section has to be less than that of the mandrel material to prevent lock-on. The composite is considered to be stress-free at cure-temperature, and thus the temperature loading is associated with the temperature decrease to room-temperature.

Table 3 . Average coefficients of linear thermal 06 expansion of selected materials

Figure 30.15(a) illustrates that it is preferable Aluminum alloy to have the metal mandrel contract more than Concrete the composite during cool-down, which Invar Steel means that the metal CTE must be more than Titanium alloy the composite CTE. For a shaft with fibers pre-

12.8 6.7 0.39 6.5 4.9

a composite ' aofmandrel(significantamount fibers in hoop direction)

a composite ' a mandrel(predominantly longitudinalfibers)

comDosite

AT