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LESSON 9:

LAMINATE THEORY

I n t r o d u c t i o nPractical structures made of composite materials have fibers

placed in more than one direction. Because most composites,

for example, those with polymeric matrices, are extremely weak

in directions transverse to the fibers, fibers must be placed in

more than one direction. Otherwise, even secondary loads in

transverse directions could cause failure of the structure. The

proper selection of the placement angles of the different layers

of fibers is a key feature in the design of composite structures.

 The foundation for the analysis techniques necessary to

accomplish this design are explained in this lesson.

 The situation with composite materials is somewhat similar to

that with reinforced concrete, in that even if the overall loads aresimple, such as a uniaxial load, for example, the internal stressdistribution is not uniform. Rather the individual plies carry theload according to their relative stiffnesses. The situation iscomplicated in fiber composites, because the stiffnesses of the

individual plies depend on the angles of fiber placement withrespect to the loads.

 The material developed in this chapter has direct application tothe case in which the final composite form has distinct layers,such as the layers of a prepreg laminate or a filament- wound

structure. However, other material forms, such as the varioustextile forms (e.g., cloth or braids), can often be analyzed usingsimilar techniques. Thus, the material to be developed here isapplicable when the material has fibers aligned within a layer.Materials such as random chopped-fiber or random mat

composites have much less directionality to the various layers,and can be analyzed as if they were isotropic.

 This chapter proceeds in what perhaps seems to be a round-about way. The basic geometry to be considered is that of a flatplate under both tension (or compression) and bending, and

the basic problem is to relate the applied in-plane loads andbending moments to the stresses within the individual layers.Subsequent chapters show how to apply the proceduresdeveloped here to other structural geometries. The basicapproach is to first describe the geometry characterizing

extension and bending of a flat plate. Strain distributions arethen obtained, and stresses related to these strains by means of the stress-strain laws. Integration of the stresses gives theextension forces and bending moments

as a function of the strains. Finally, these extension forces and

bending moments re related back to the stresses in the indi-vidual plies.

De f o r m a t io n D u e t o Ex t e n s io n a n dBend ingConsider a flat plate made up of a number of individual layers

with an x,y coordinate system n the plane of the plate. The basic

assumption is that the individual layers are perfectly bonded

together, so that in terms of displacements, it is not necessary at

this point to consider the individual layers, as they all displace

together under the action of the applied loads.

In the usual (simplest) theory, which is now described, theKirchhoff-Love hypothesis is invoked by assuming thatnormals to the centerline remain normal after deformation.

 This assumption neglects through-the-thickness shear deforma-tion. Further discussion of shear deformation is givensubsequently. The displacements in the x,z plane that character-ize uniform extension are illustrated in Figure, along with thosethat characterize bending. A similar situation will exist in the y,z

plane for displacements in that direction. The displacements inthe x, y, and z directions are called u, v, and w, respectively, andare assumed to be described by continuous functions of x, y,and z. I t is assumed that the plate displaces in the z directiononly because of the bending motion, and that no variation of 

w through the thickness takes place. This is a usual assumptionof thin-plate (and beam) analysis.

As illustrated in the figure, the in-plane displacements u and vcan be related to the centerline displacements Uo and Vo andthe slopes by

Figure : Illustration of extensionand bending-plate deformation

 The displacements can then be used to relate to the strains byusing the usual strain

displacement relationships of linear elasticity. These relation-

ships are developed in most

texts on mechanics of materials or elasticity, and can be easilyobtained from Figure by using the usual definitions for strain,that is, that normal strain is the change in length per unitlength, and that engineering shear strain is the change in the

 

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angle between two initially perpendicular sides. These ideas canbe written mathematically from the deformations shown in Fig.

Figure : The relationship between strains and displacementgradients: (a) normal strains and (b) shear strains

for normal strain, and the relationship

 

A similar relationship holds for normal strain in the y direction,so that the strain displacement relations needed are given by

Substituting the displacements gives

and

Fo r ce a n d M o m e n t Re s u l t a n t s The next step in the development is to relate the internal

stresses (in overall x,y coordinates) to the applied loading,

expressed in terms of stress resultants {N} and moment

resultants {M}. The term stress resultant refers to the stress

integrated over the thickness of the laminate, and is thus the

applied force per unit width. A similar interpretation can be

given to the moment resultant, which is thus the applied

moment per unit width. Using equilibrium, we equate the force

and moment per unit width to the integral of the stress and

stress times distance from the center lint to get, for example,

or in compact form,

and for the moment resultants,

where the integral is taken over the total laminate thickness bysumming the integrals over each ply. That is, we take theintegrals to be

Figure : Notation for location of ply interfaces.Note that the h

k have the value of z

and are negative below the centerline

 

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where the positions of the limits for each ply are illustrated in

Figure Coordinates hkare the values of z at the interfaces and

they have the sign convention of the z coordinate. Note thatthe integral of a matrix is just the integral of each term in thematrix.

A,B a nd D Mat r ices The force and moment resultants can be related to the strains in

the laminate through the material properties for each ply group. This relationship can be expressed in terms of matrices that arelabeled A, B, and D by custom. These matrices are developed in

what follows.

 The first task in the development is to relate the stresses in eachply to the strains in the ply. The strains are given in terms of position z. This relationship between stresses and strains mustbe expressed in terms of the overall x, y coordinate system,

because the strains are given in terms of x, y coordinates.However, this can be done using the stress strain relationshipsfor a layer oriented at an arbitrary angle that were developed inEqs.

where

Substituting the stress-strain relationship in x,y coordinates intothe preceding equations gives

and

 The integrals are easy to carry out, because the material proper-ties are constant over each individual ply (or ply group) and theonly variable is z. Thus, for example,

and

By using these integrals, the equations can be written in theclassic relation between stress resultants, moment resultants,centerline strains, and curvatures in the following form:

where the A, B, and D matrices are each 3-by-3 matrices definedas

where the positions of the ply surfaces are denoted by hk, N is

the number of plies (or groups of plies), and the Q matrix isthe stiffness in the x, y coordinate system of each ply.

In the general there is a coupling between the in plane behavior

and the bending behavior because of the presence of the Bmatrix. This coupling leads to effects that are not present inisotropic materials; they are discussed in more detail subse-quently. It will also be shown that the B matrix vanishes forsymmetric laminates, that is, laminates that are symmetric with

respect to the midplane. When the B matrix vanishes, thecoupling between in-plane behavior and the bending behaviordoes not occur. For this case, Eq. reduces to

 The preceding relationships between the stress and momentresultants and the centerline strains and curvatures arewel1known, and are the heart of laminate analysis. Because alaminate is often made up of a number of ply groups, a certainamount of bookkeeping is involved in calculating the A, B, and

D matrices. However, this can be readily implemented on adesktop computer, either in a special program or in a spread-sheet.

 

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It can be seen that the A matrix is the sum of the moduli foreach layer in the overall coordinate system multiplied by thethickness of that layer. Thus, the A matrix divided by the total

laminate thickness represents an average in-plane stress-strainrelationship for the laminate, expressed as

It will be shown in later chapter~ that the average stress-strainproperties for a laminate can be useful in a numberof’calculations.

 The notation used in the A matrix, that of calling the term inthe first row and third column A

16rather than A

13should be

noted. The rationale for this is that these matrices involvedintegrals of the Q stress-strain matrix, and it is customary toidentify the position relative to the full three-dimensional

stress-strain matrix with a size of 6 by 6, before it is reduced tothe 3-by-3 matrix for plane stress. Although perhaps confusing,this notation seems well established in the composites litera-ture. Similar notation is used for the D-matrix terms.

 The inverses of the previous matrix equations are required for a

number of purposes, including calculating the stresses withinthe various plies of the laminate. In the general the entire 6-by-6ABD matrix must be inverted to get

 The inverse of the ABD matrix is sometimes called the A*B*D* matrix; in this text it is called the F matrix to simplifynotation. Thus,

For symmetric laminates, which thus have a zero B matrix, thetwo 3-by-3 matrices can be inverted to get the following:

If 

then

and

Lam ina t e Cod eBefore discussion of particular laminates, it is convenient to

explore a shorthand code used to specify the layup. Laminates

are specified by the plies or ply groups that constitute the layup.

Some examples will illustrate the procedure. A laminate

consisting of a series of layers (made up of individual plies in a

prepreg laminate) of one material in unidirectional form can be

specified by the angles and numbers of plies in each ply group,

such as [02/ 90/ 90/ 0

2], where the subscript refers to the number

of plies in the ply group. A [0/45/-45/ 90]5laminate is symmet-

ric about the midplane, and is thus equivalent to a [0/45/ -45/

90/ 90/ -45/45/ 0] layup. Another convention is that of repeated

groups of plies, such as [(0/ -60/60)2]5which is equivalent to

[0/ -60/ 60/ 0/ -60/ 60]5Another convention is that of an

overbar on a ply designation adjacent to the symmetry axis,

which means a half ply, for example, [0/ 90]5means [0/90/ 0].

 The symbol T is sometimes added for clarity to show that the

total stack is indicated. For example, [010] Tmeans 10 plies of 0°

orientation. A hybrid laminate contains more than one material,

such as mixing glass fiber plies and carbon fiber plies. In this

case, the notation must also identify the material, usually by

using a material designation as a subscript for the ply group.

Manufacturing techniques other than prepreg layup do not have

layers composed of a discrete number of plies. For example, in

filament winding, the material is applied in layers of fibers with

a common angle, but the thickness of each layer depends on the

processing variables. Here the amount of fiber is identified by

the thickness of the layer, given as a subscript.

Example

Calculation of A and D matrices for a unidirectional laminate

As a first example, consider the calculation of the A and Dmatrices for a unidirectional laminate with 10 plies of AS4/3501-6 carbon epoxy. The layup is thus designated as [0

10]

Because the laminate is symmetric about its midplane, that is, itcould equally well be designated as a [0

5]s. laminate, all terms of 

the B matrix are identically zero. The stress-strain matrix for thismaterial is calculated. Because the orientation of the pliescoincides with the overall x,y coordinate system, the coordinate-transformation matrix is just the unit matrix, so that theproperties are unchanged and noting that the sum is over the

single ply group, gives

and

Where t is the total laminate thickness. It is easy to see that forthis case, the A matrix is just the stress-strain matrix multipliedby the total thickness. The D matrix represents the bendingstiffness properties, and, noting that it is defined on a per unit

width basis, can be seen to be a two-dimensional version of thefamiliar EI stiffness of beam theory for a unit width beam.