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NASA Reference Publication 1351 Basic Mechanics of Laminated Composite Plates A.T. Nettles (NASA-RP-1351) BASIC MECHANICS OF LAMINATED COMPOSITE PLATES (NASA. Marshall Space Flight Center) 103 p Uncl as October 1994

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Page 1: Nasa Laminates

NASA Reference Publication 1351

Basic Mechanics of Laminated Composite Plates A.T. Nettles

(NASA-RP-1351) BASIC MECHANICS OF LAMINATED COMPOSITE PLATES (NASA. Marshall Space Flight Center) 103 p Uncl as

October 1994

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TABLE OF CONTENTS

Page

I . INTRODUCTION ..................................................... ....., ............................................... I

A . Intent and Scope ..................................................................................................... I B . Terminology and Notation ..................................................................................... B C . Summary of Sections ............................................................................................. 2

I1 . GENERALIZED HOOKE'S LAW FOR NONISOTROPIC MATERIALS ................. 3

A . Norn~al Stress and Strain. Uniaxially Applied Force .................................... ... ...... 3 B . Stress and Strain. Plane Stress for Specially Orthotropic Plates ......................... ... 4 C . Stress and Strain, Plane Stress for Generally Orthotropic Plates ........................... 6 D Invariant S tiffnesses ............................................................................................... 10 .

111. MECHANICS OF LAMINATED COMPOSITES ........................................................ 11

A . Assumptions ........................................................................................................... 11 B . Definitions of Strains and Displacements .............................................................. 11 C . Definitions of Stress and Moment Resultants ........................................................ 15 D . Constitutive Equations for a Laminate ................................................................... 17 E . Physical Meanings of the [A] , [B], and [Dl Matrices ............................................* 23

IV . NOMENCLATURE FOR DEFINING STACKING SEQUENCES .............................. 25

A . Coordinate System ................................................................................................. 25 B . Nomenclature ......................................................................................................... 25

V . IN-PLANE ENGINEERING CONSTANTS FOR THE LAMINATE .......................... 28

A . Orientation of the Laminate ................................................................................... 28 B . Symn~etric Laminates ............................................................................................ 28 C . Nonsymmetric Laminates ...................................................................................... 33 D . Summary ................................................................................................................ 40

VI . ENVIRONMENTAL EFFECTS .................................................................................... 43

A . Importance ............................................................................................................. 43 B . Coefficients of Thermal Expansion ....................................................................... 45 C . Moisture Effects ..................................................................................................... 45

VII . STRESSES AND STRAINS WlTHIN LAMINAE OF A SYMMETRIC LAMINATE .................................................................................................................... 46

A . Strains Within the Laminae .................................................................................... 46 B . Stresses Within the Laminae .................................................................................. 50

iii

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LIST OF ILLUSTWTIONS

Figure Title

Difference between an isotropic and an orthotropic plate ...........................................

Definition of shearing strains .......................................................................................

Generally orthotropic lamina .......................................................................................

Displacements of a plate ..............................................................................................

Total displacements in a plate ......................................................................................

Definitions of plate curvatures ....................................................................................

Definition of stress resultant ........................................................................................

Direc~ion of stress and moment resultants ...................................................................

Cross section of a laminate ..........................................................................................

Displacements in an unsymmetrical plate ....................................................................

Coordinate system l'or a typical laminate .....................................................................

Some laminate stacking sequences and their notation .................................................

Page

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REFERENCE PUBLICATION

BASIC MECHANICS OF LAMINATED COMPOSITE PLATES

I. INTRODUCTION

A. Intent and Scope

This report is intended only to be used as a quick reference guide on the mechanics of continuous fiber-reinforced laminates. By continuous fiber-reinforced laminates, the following is assumed:

(1) The material to be examined is made up of one or more plies (layers), each ply consisting of fibers that are all uniformly parallel and continuous across the material. The plies do not have to be of the same thickness or the same material.

(2) The material to be examined is in a state of plane stress, i-e., the stresses and s e r ~ n s in the through-the-thickness direction are ignored.

(3) The thickness dimension is much smaller than the length and width dimensions.

An attempt is made in this report to develop a practical guide that can be easily referenced by the engineer who is not familiar with composite materials, or to aid those who have seen this subject matter before.

The scope of the report will be limited to the elastic response of the above-mentioned class of material. Strength-of-laminated composites will not be covered. General composite material mechanics and strength are developed in more detail in texts such as Jones1 and Walpin.2 It. is assumed that the reader has a general knowledge of elastic stress-strain behavior.

B. Terminology and Notation

Some terminology important to composite materials follows:

Isotropic-Possessing the same mechanical properties in all directions. Composite laminates are never isotropic.

Laminate-A material consisting of layers (laminae) bonded together.

Transversely Isotropic-Possessing one plane that has the same mechanical properties at any direction in that plane, i.e., the laminate will have the same stress-strain behavior at any direction in the plane of the nlaterial (son~etinles called quasi-isotropic).

Orthotropic-A material that has different mechanical properties in three mutually perpendicular planes. Note that the properties of the material are direction specific in this case. All unidirectional laminae are individually orthotropic. Most laminated composites fall into this categoq.

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Section VI introduces the effects of temperature and moisture on the strain of a composite laminate. These effects are often neglected, but are very important in determining the skesses md strains within each ply of the laminate. Determining these ply stresses and strains for symmetric laminates is presented in section VII and in section VIII for unsymmetric laminates. These sections provide information necessary to study the strength of composite laminates.

Section IX is a summary of the most important equations presented in this paper.

II. GENERALIZED WOOKE'S LAW FOR NONISOTROPIC MATEWICALS

A. Normal Stress and Strain, Uniaxially Applied Force

Normal stress is defined as the force per unit area acting perpendicular to the surface of the area. The corresponding strain is defined as the elongation (or stretch) per unit length of material in the direction of the applied force. For isotropic materials, the relationship between slress and ser~ne! is independent of the direction of force, thus only one elastic constant (Young's modulus) is required to describe the stress-strain relationship for a uniaxially applied force. For a nonisotrspic material, at least two elastic constants are needed to describe the stress-strain behavior of the material.

Figure 1 is a schematic of an isotropic and a unidirectional fiber-reinforced material, The stiffness of the isotropic plate can be described by one value, the modulus, E, of the material, regardless of direction of load. The stiffness of the orthotropic plate must be described by two vdues, one along the longitudinal direction of the fibers, commonly referred to as EL, and one wansverse to the direction of fibers, usually denoted by ET. Subscripts 1 and 2 will be used such that EL = El and ET = E2. Thus, indices must be added to the stress, strain, and modulus values to describe the direction of the applied force. For example, for an isotropic material, the stresslstrain relationship is written:

Isotropic Plate , Orthotropic Plate

1 1 Reinforcing fibers aligned in 1-direcuon

stiffness in 1-direction = stiffness in 2-direction stiffness in 1-direction >> stiffness in = stiffness in any direction 2-direction # stiffness in other directions

Figure 1. Difference between an isotropic and an orthotropic plate.

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For the orthotropic system, the direction must be specified. For example:

If the applied load acts either parallel or perpendicular to the fibers, then the plate is considered specially orehohropic.

IB. Stress and Strain, Plane Stress for Specially Orthotropic Plates

The previous section dealt with an extremely simple type of stress state, uniaxial. In general, plates will experience stresses in more than one direction within the plane. This is referred to as plane s&ess. In addition, Poisson's ratio now becomes important. Poisson's ratio is the ratio of the swain perpendicular to a given loading direction, to the strain parallel to this given loading direction:

&T &2 Poisson's ratio = v12 = - = - &L &1 or v ~ ~ = - = - . (3)

&L &1 &T &2 For loading along For loading the fibers perpendicular to the fibers

The strain components are now stretch due to an applied force, minus the contraction of Poisson's effect due to another force perpendicular to this applied force. Thus:

01 0 2 &I=-- v ~ ~ E ~ and ~ z = - -

E l E 2 v12E1 .

Using equation (2):

01 O2 O2 O1 & I = % - "21% and E ~ = - - v E2 1 2 ~ 1 - .

Shear forces can also be present. Shear stress and shear strain are related by a constant, like the normal stresses and strains. This constant is called the shear modulus and is usually denoted by C. Thus:

Where 712 is the shear stress (the 1 and the 2 indices indicating shear in the 1-2 plane), and 3'i2 is the shear skaln. Figure 2 gives a definition of shear strain.

Since it is known that a relationship exists between Poisson's ratios and the modulii in each of the two axes directions, namely:

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Equa~ons (4b) and (5) can be written in matrix form as:

where,

Note fiat at the 3,3 position in this 3x3 matrix (called the compliance matrix), the subscripts are 6,6. This evolves from a detailed treatment of arriving at a constitutive equation for an orthotropic materid from an anisotropic one.

By invehting the compliance matrix, one can get stress as a function of strain. This turns out to be:

where:

The Q9s ulre rekrred to as the reduced stiffnesses and the matrix is abbreviated as [Q].

C. Stress and Strain, Plane Stress for Generally Orthotropic Plates

Now suppose that the unidirectional lamina in figure 1 is loaded at some angle other than 0' or 90°, The lmina is now referred to as generally orthotropic, since, in general, the loading direction does not coincide with the principal material directions. The stresses and strains must now be uanshmed into coordinates that do coincide with the principal material directions. This can be accomplished using the free-body diagram in figure 3. From free body diagram (a), and summing forces in &e I-direction:

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Simplifying equations (1 I), (12), and (13);

2 2 0, = g cos e + o sin 8 +27 sin 0 cos ,

Y XY

2 2 4 = g sin e + o cos 8-27 sin gcos g , Y xy

2 2 ~ , ~ = - g s i n B cos8+0 s inBcos8+~ (cos &sin 8) . Y xy

Equation (14) can be written in matrix form as;

The 3x3 matrix in equation (15) is called the transformation matrix and is denoted by [a. The same matrix is used to transform strains. Note that the tensorial shear strain must be used, not the engineering shear strain, when transforming strains. This arises from the geometrical considerations that the amount of shear must be equivalent with respect to both the x- and y-axes, since these axes will be transformed into new ones (fig. 2(b)).

7

2 cos 8 sin2 8 2 sin ecos 8

sin2 2 cos 8 -2 sin ecos 8

-sin 8cos 8 sin 0 cos 8 (cos2 8- sin2 8)

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- 3 Qz6 = (Q 11-Q 1 2 - 2 ~ 6 6 ) n3m + (Q 1 2 - ~ 2 2 + 2 ~ 6 6 )

- (22) (cont.) Qs6 = (Q 1 1 + ~ 2 2 - 2 ~ 1 2 - 2 ~ 66) m 2 n 2 + ~ 66(m4+n4) -

Note that if 8 is any angle other than zero, there will be nonzero Q16 and Q26 terms. Putting this into equation (20):

it can be seen that a shear strain will produce normal stresses, and normal strains will contribute to a shear stress. This is referred to as extension-shear coupling and will take place in a lamina that is loaded at an mgle to the fibers (other than 0" and 90"). That is, there will be coupling if the Ql6 anuor QZ6 terms in the lamina stiffness matrix are nonzero.

D. Invariant Stiffnesses

The "Q-Bar" terms can be written as:

- Q l l = U ~ + U ~ C O S (28)+u3 cos (48) , - Qu = U l - 4 cos (26)+u3 COS (48) ,

- Q 12 = (14-U3 COS (40) , - QS6= u5-u3 cos (40) 7

- 1 - - u2 sin (28)+u3 sin (48) . Q 1 6 - ,

1 = - u2 sin (28)-u3 sin (48) , 2

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where,

Note that only U2 and U3 are coefficients to the sine or cosine terms in equation (24a). This implies that when calculating the Q-Bar values, U1, U4, and US are independent or invariant to the ply orientation 8. This concept of "invariant" quantities can make some calculations easier. This paper will not go into detail on this subject since only the basics are being presented.

1.1. MECHANICS OF LAMINATED COMPOSITES

A. Assumptions

The following assumptions are made for the remainder of this paper:

(1) The laminate thickness is very small compared to its other dimensions.

(2) The lamina (layers) of the laminate are perfectly bonded.

(3) Lines perpendicular to the surface of the laminate remain straight and perpendicul~ to the surface after deformation.

(4) The laminae and laminate are linear elastic.

(5) The through-the-thickness stresses and strains are negligible.

These assumptions are good ones as long as the laminate is not damaged anid undergoes small deflections.

B. Definitions of Strains and Displacemenb

A displacement of the plate in the x-direction is designated as u. For the y-direction, lit is designated as v and for the z-direction w. Figure 4 shows these displacements. The slrains axare now defined as:

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Deformed

---------- J """ I

Normal displacement Z

Shear displacement

Bending displacement

Figure 4. Displacements of a plate.

The slope of the plate if it is bending is given as:

aw - along the xdirection , ax dw - along the ydirection . ay

The total in-plane displacement at any point in the plate is the sum of the normal displacements plus the displacements introduced by bending. Denoting the displacements of the midplane of the plate for the x and y directions as u, and v, respectively, with the help of figure 5 the total displacements are:

Note that for figure 5, x can be replaced by y when u is replaced by v (i.e., the view could be from any side of the plate). It is assumed that there is no strain in the thickness direction, only a displace- ment.

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From equation (23), the stresses in each ply of the laminate can be determined with equation (30):

6. Defi~tions of Stress and Moment Resultants

Since the stress in each ply varies through the thickness of the laminate, it will be convenient to define stresses in terms of equivalent forces acting at the middle surface. Referring to figure 7 , it can be seen that the stresses acting on an edge can be broken into increments and summed. The resulting integral is defined as the stress resultant and is denoted by Ni, where the E' subscript denotes direction. This stress resultant has units of force per length and acts in the same dkection

Total force in x-direction = ~x(dz)(y)

Nx = ox dz

Figure 7. Definition of stress resultant.

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as the suess state it represents. Figure 7 could also be drawn for the y-direction stress and shear stress, The three stress resultants are therefore:

As can be seen from figure 7, the stress acting on an edge produces a moment about the midplane. The force is ox (dz)(y) as denoted in figure 7. The moment arm is at a distance z from the midplane. Following the same procedure as for the stress resultants, the moment resultants can be defined as:

M y = o zdz , -W2 y

These moment resultants have units of torque per unit length.

The directions for all of the stress and moment resultants are shown in figure 8 for clarity. The double-headed arrow indicates torque in a direction determined by the right-hand-rule (i.e., point the thumb of your right hand in the direction of the double-headed arrows and the direction of rotation of the torque is in the direction that your four fingers are pointing). Note that Mx and My will cause the plate to bend and Mxy will cause the plate to twist.

As an example of the relationship between stress and stress resultants, if a tensile test specimen is 2.54-cm (1-in) wide and 2-mm (0.08-in) thick, and is pulled on with a force of 4,500 N (1,000 Ib), then the average stress on the cross section is:

- 4,500 N lb ox = = 88.6 MPa = 12,500 7 .

(0.0254 m) (0.002 m) in

The stress resultant would be the average stress multiplied by the specimen thickness:

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Stress Resultants

X Y

Moment Resultants

X Y

Figure 8. Direction of stress and moment resultants.

Since 1,000 lb was applied over 1 inch of specimen length, the expected result of 1,000 Ib/in is obtained. Each individual ply of the specimen may have a stress other than 88.6 MPa (12,500 IWiar"), but the average stress will be 88.6 MPa (12,500 lblinz).

D. Constitutive Equations for a Laminate

Putting equation (31) in matrix form:

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ww

w

16 lG 1s

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Since the middle surface strains and curvatures (the 8 ' s and K's) are not a function of z (because these values are always at the middle surface z = 0), they need not be included in the integration. Also, the laminate stiffness matrix is constant for a given ply, so it too will be a constant over the integration of a lamina thickness. Pulling these constants to the front of the integral in equations (39) and (40) gives:

and

zdz

Performing the simple integrations gives:

and

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Since the middle surface strains and curvatures are not a part of the summations, the laminate stiffness matrix and the hk terms can be combined to form new matrices. From equations (43) and (44), these can be defined as:

In matrix form, the constitutive equations can easily be written as:

Written in contracted form, equation (48) becomes:

A I B

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This can be p;ar~ally inverted to give:

where,

The fully inverted form is given by:

where,

[A1 = [A*] - [B*] [D*]"[c*

[B'] = [B*] [o*]-' ,

[D'] = [D*Id .

The fuFuQBy inverted form is the most often used form of the laminate constitutive equations.

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For symmetric laminates (laminates that are configured such that the geometric midplme is a mirror image of the ply configurations above and below the midplane), the geometric midplane is also the neutral plane of the plate, and the [B] matrix will have all elements equal to zero (as will be shown later). However, if the laminate is unsymmetric, i.e., if the plies near the bottom of the plate are much stiffer in the x-direction, then the geometric midplane will not be the neutral plane of the plate; and the neutral plane will be closer to the bottom of the plate for x-direction beading as shown in figure 10. This is accounted for in the constitutive equations, since the [B] matrix will have some nonzero elements (as will be shown later), implying that a bending strain (plate curvature) will cause a midplane strain as depicted in figure 10. Likewise, a midplane strain will cause a bending moment. A method to find the neutral axis of the plate will be discussed in a later section about stresses within the plies of a laminate.

Z Undeformed edge of plate k TOD surface of olale

1 -

I Very stiff plies

Bottom surface of plate X

Z

Pure bending causes axial compression of geometrical midplane

Figure 10. Displacements in an unsymmetrical plate.

E. Physical Meanings of the [A], [B], and [Dl Matrices

Recalling the definitions of the [A], [B], and [Dl matrices,

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and referring to figure 9, it can be seen that the last term in equation (45) is the kth lamina thickness which will be denoted by tk. Thus, equation (45) can be written as:

This matrix is called the extensional stiffness matrix. From the constitutive equation (48), it can be seen that these terms relate the normal stresses and strains (much like the modulii of elasticity), except for the A16 and terms which relate shear strains to normal stresses and normal suains to shear stresses. Thus, when A16 and are nonzero, and the laminate has a shear strain applied to it, normal stresses will result and vice-versa. These terms are analogous to the Q16 and terns mentioned in the final part of section 11.

Equation (46) can be written as:

where tk is the thickness of the kth ply, and (hk+hk-l)/:! is the distance from the geometric midplane to the center of the kth ply. This matrix is called the coupling stiffness matrix. From the constitutive equation (481, it can be seen that these terms relate bending strains (plate curvatures) with normal stresses and vice-versa. The B16 and B26 terms relate twisting strains to normal stresses and shear strains to bending stresses.

If the laminate is symn~etric, then the Bq terms will be the same for each mirrored ply above and below the midplane, with the exception of the sign of the (hk+hk-1)/2 term being negative if it is below the midplane (-2) and positive if it is above the midplane (+z). Thus, when summed, the result will be zero for all Bij. Now define:

Part of equation (47) can be written as:

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Therefore, equation (47) can be written as:

It can be seen that the last term is the second moment of the kth ply with respect to the geometric midplane. Dij is called the bending stiffness matrix and relates the amount of plate cmvatures tvih the bending moments.

IV. NOMENCLATURE FOR DEFINING STACKING SEQUENCES

A. Coordinate System

The choice of coordinate system used for the laminate will determine its stacking sequence. For example, a unidirectional composite can be said to be made up of all 0' plies in the x-direction, and all 90' plies in the y-direction. Alternatively, the same composite can be said to be made up of all 0' plies in the y-direction, and all 90' plies in the x-direction. The composite can also be referenced by any other x-y coordinate system in the plane of the plate. The choice of coordinate system is totally arbitrary, but some general procedures are usually followed to make calculations or communication about the laminate easier for others to understand.

A coordinate system is almost always chosen such that one of the axes runs in the dkection of fibers of one of the plies of the laminate. This will make analysis much easier. The x-=is is usually chosen as the "longitudinal" axis, with the corresponding y-axis being the "&ms~erse '~ direction. This is similar to what was defined earlier for laminae. The main load bearing fibers (if these are known) are usually called the 0' fibers, longitudinal fibers, or x-direction fibers. The other ply orientations will then be defined with this coordinate system. An example of a typical coordinate system is given in figure 11.

B. Nomenclature

There is more than one way to denote the stacking sequence of laminates. However, once one method is learned, any other is easy to interpret, even though it may not be in the form that the user is accustomed to.

Once the 0' fiber direction has been defined (and thus the x-axis), the plies &at are not at 0' must be assigned an angle. To do this, start from the x-axis and rotate to the fiber d ~ e e t i o n of the ply being defined. Clockwise rotations are positive angles, and counterclockwise rotations are negative angles, although the reverse can also be used since only plane-stress is being examined for plates and the material is the same whether viewed from one surface or the other sudaee, Now that all plies have an angle associated with them, a method of presenting the stacking sequence folllows.

If the laminate is symmetric, then start with the angle of the outermost ply and wite the ply angles, separated by a comma, until the midplane is reached. Enclose this string of angles in brackets or parentheses and subscript the brackets or parentheses with an " S ' t o denote "symmetric." If the laminate is not symmetric, then proceed as above until the bottom ply is

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reached. Subscript the brackets or parentheses with a "2" to denote "total" laminate, Referring to figure 10, this laminate is denoted as [O, +45, -45, 901s . This is more convenient than wfiting [O, +45, -45,90, 90, -45, 4 5 , OIT.

Further simplifications can be made when two or more plies of the same o~entation me grouped together. The angle of these plies need only be written once with a subscripted number denoting the number of plies in the group. For example, [O, 90, 90,90,90, OJT can be written as [o, 904, OIT. Since this laminate is symmetric, further simplifications can result, and this laminate could be described by [0, 9O2ls. If a symmetric laminate consists of an odd number of plies, then the geometrical midplane of the laminate will lie at the midplane of the center ply. In this case, ;a bar is placed over the angle of this ply to denote that half of it resides in the top half of the Bminate and the other half resides in the bottom half of the laminate. For example, a laminate with stacfing sequence [O, 90, 90, 90, 0IT can be written as, [O, 90, wls . Any repeating units within the lanlinate cu.1 be placed in parentheses with a subscripted number representing the number of repeats. For example, a [O, 90, 0, 90, 0, 90, 0, 90Is laminate can be written as [(0, 90)4Is. If adjacent plies are of the same angle, but with different signs, then a plus-minus sign is usually placed in front of the angle of the plies. For example, a [o, +45, -45, 90, +30, -30IT laminate can be written as [O, k45, 90, -b. 30IT. Some examples of stacking sequences and how they can be denoted are given in figure 12,

1

Figure 12. Some laminate stacking sequences and their notation.

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V, m-PLANE ENGINEERING CONSTANTS FOR THE LAMINATE

A. Orientation of the Laminate

For a given stacking sequence of laminae whose engineering properties are known, it is possible to determine the in-plane engineering constants of the laminate from the Aij matrix for symmeuic lminates, and the AV, Bii, and Dii matrices for unsymmetric laminates. The choice of coordinates will determine the directions of the laminate engineering constants being evaluated. These dkections are arbitrary, but are usually chosen as described in section IV. A.

B. Symmetric Laminates

Recdl h a t for symmetric laminates, the Bii matrix consists of all elements being zero. This greatly simplifies finding the in-plane engineering constants of the laminate. To find the x-direction modulus, the vdue of the x-direction stress to the x-direction strain must be calculated. In equation form:

where h PS the thickness of the laminate. Since the Be's are zero, the constitutive equations are:

Since a relationship between NX and Ex are being sought, when a load is applied in the x-direction, from equation (58):

From equations (60) and (61):

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Page 40: Nasa Laminates
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Page 42: Nasa Laminates

To find sought.

L

Ex, only the x-direction in-plane load is applied The cons~tut ive equation now becomes:

-

and a relationship

(48)

between Nx and E! is

Using Crmerys rule to solve for E: :

Page 43: Nasa Laminates

Determinants for two 6x6 matrices must be found.

At this point, it should be clear why a computer (or calculator) program makes calculadons much easier. To write out the solution for equation (82) would require 6! = 720 terms for the numerator and the denominator. Cofactor expansion can be used in the numerator for some simplification:

From equation (57), Ex can be found by evaluating:

Appendix A shows a Fortran program to calculate the determinant of a 6x6 and 5x5 matrix,

Page 44: Nasa Laminates

Ey can be found in a similar manner. The denominator will be different since equation (81) is being solved for E: .

Gjcy will be given by:

Page 45: Nasa Laminates

Poisson's ratio will be determined as it was for symmetric laminates. For vq, the contraction in the y-direction upon an applied stress in the x-direction must be obtained. This is given by:

Thus, v,, will be given by:

vyx is given by:

Page 46: Nasa Laminates

Example 2:

Given a %ply laminate of stacking sequence [0,+45IT, Ex will be calculated. From example 1, the Aq9s .are known. The Bu's and Du's must be calculated.

To obtain the B y use equation (54) with the already determined values in equations (E1.2) and (E1,3),

BI1 = 20,130,755 lblin' (0.005 in)(0.0025 in)

+ 6,557,237 lb/in2 (0.005 in)(-0.0025 in) = 170 lb ,

B 12 = 392,656 lb/in2 (0.005 in)(0.0025 in)

+ 4,555,238 lb/in2 (0.005 in)(0.0025 in) = -52 lb ,

Bu = 1,308,853 1b/in2 (0.005 in)(0.0025 in)

+ 6,557,237 lb/in2 (0.005 in)(-0.0025 in) = -66 lb

B 16 = 0+4,705,483 lb/in2 (0.005 in)(-0.0025 in) = -59 lb ,

826 = 0+4,705,483 lb/in2 (0.005 in)(-0.0025 in) = -59 lb ,

B 66 = 1,00 1,000 lb/in2 (0.005 in)(0.0025 in)

+ 5,163,582 lb/in2 (0.005 in)(-0.0025 in) = -52 1b

Use equafion (56) to find the Dij's:

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For unsymmetric laminates, the engineering constants are given by:

B B B D D D 1 16 26 66 16 26 6 6 1 ,

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Page 51: Nasa Laminates

VI. ENVIRONMENTAL EFFECTS

A. Importance

Like all engineering materials, laminates will contain residual stresses from processing, or will experience strains due to thermal effects or moisture absorption. However, because of the anisotropic nature of composite laminates, these effects become much more important. I& is well known that most laminates are processed or "cured" at elevated temperatures. It is at this elevated temperature that the molecular structure of the material is set. Upon cooling, the laminate will experience many internal stresses as each ply contracts a different amount in different directions. For unsymmetric laminates, this can be seen by warpage of a flat plate as it is removed from its molding platens. In some cases, this warpage is desirable to achieve a natural "twist" in the material (such as for helicopter rotors). In general, the more anisotropic the laminate, the more important the residual thermal and moisture absorption stresses become. The thermal effects will be quite important in the next section on ply stresses.

B. Coefficients of Thermal Expansion

Just like any other material, fiberfresin systems will experience a change in s t r ~ w with a change in temperature. These strains are defined by the coefficients of thermal expansion. These values are material constants in each principal material direction. Thus, two constants will describe the thermal expansion coefficients for any lamina. These are defined as:

a1 - coefficient of thermal expansion in the fiber direction

and

a2 - coefficient of thermal expansion in the direction perpendicular to the fibers.

Page 52: Nasa Laminates

These values have dimensions of inch/incN°F or incNinch/OC, depending on the temperature scale that is being used.

Just as with the mechanical strains, the thermal strains must be transformed into the laminate coordinate system:

cos2 e sin2 e

sin2 e 2 cos e

sin 8 cos 8 -sin 8 cos 0

Note that a coefficient of thermal expansion for shear is formulated if the lamina being examined does not have its material axes as principal material axes (i.e., 8 # O0 or 90'). The amount of thermal swain induced in each lamina is given by:

T Ex = axAT 7

where the superscript T denotes "thermal," and AT denotes the change in temperature from cure to opera~ng temperature.

These thermal strains are now treated just like the mechanical strains considered earlier. Therefore, from equation (43), it can be seen that when these thermal strains are combined for each layer of the laminate, thermal stress resultants are present:

From equation (44), it can be seen that thermal moment resultants also develop:

Page 53: Nasa Laminates

These thermal stress and moment resultants can be added to the mechanical stress and moment resultants to arrive at the total stress and moment resultants:

where the superscript Tot denotes "total".

Typical values of the thermal expansion coefficients for carbodepoxy is a1 = -0.072~l0-~ in/id°F and a2 = 32.4~10-6 id id°F. Note that in the fiber direction, the material actually eon&acts upon heating (or expands upon cooling). Also, note the large differences in the two coefficients, For this reason, composite materials have been used in many applications where a certain itlhermal expansion is desired, since plies can be combined to give a wide range of values (including near zero which is very convenient for optical benches).

C. Moisture Effects

Swelling of a composite material due to moisture absorption is handled in the exact s m e manner as expansion due to temperature differences. The moisture swelling coefficients in each principal material direction must be known. They are designated as Dl, along the fiber dbection, md P2, perpendicular to the fiber direction. The strain due to moisture absorption is given by:

P

where the Pi's are the transformed moisture expansion coefficients, Am is the moisture concenbation in weight moisturelweight material, and the superscript M denotes "moisture."

Stress and moment resultants can be determined due to the effects of moisture absorption.

Typical values of moisture expansion coefficients for carbodepoxy are, Dl = 0.01 irn/idg/g and f l = 0.35 in/in/g/g. The moisture concentration is usually a very low number under normal operating circumstances -0.0005 glg. However, in humid environments this number may be much higher.

Page 54: Nasa Laminates
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These are the midplane strains and curvatures. Using equation (30), it can be seen that since the midplane curvatures are zero, then the strains in each ply will be equal to the midplane swains. In other words, the strains are constant through the thickness in any given direction since there is no bending and therefore z does not enter into the calculation.

12.3 0.609 0.392 0.609 1.31 0.392 in-lb 0.392 0392 1.014

To transform these strains into the principal material directions for the +4S0 ply:

Therefore, the constitutive equation is:

--1 -

= 0.0836 -0.033 -0.0195 -0.033 0.876 -0.326 -0.0195 -0.326 1.12

-

Page 58: Nasa Laminates

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Page 62: Nasa Laminates
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Transforming these strains into principal material direction strains for the +45" plies;

cos2 45O sin2 45" 2 sin 45" cos 45" - - sin2 45" cos2 45" -2 sin 45" cos 45"

- sin 45" cos 45' sin 45" cos 45" (cos2 45O - sin2 45"

The stresses in the 0" plies are given by:

Page 64: Nasa Laminates

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Page 65: Nasa Laminates

The constitutive equation is:

K) = D;$, = -0.013 - (5 T) = - 0 165 - in-lb in

Using equation (30), it can be seen that the strains in each ply will vary across the thichess of the ply. Thus, the distance the ply is away from the geometric midplane must be t&en into account, as well as the direction of the fibers as in the previous examples. For clarity, assign the four plies of the laminate numbers as shown in the following:

.005 in. I Ply # 1 0"

.005 in. I # +450

.005 in. Ply # 4 0"

Since this laminate is symmetric, the geometric midplane is also the neutral plme of the plate. Thus, the strains will be zero at the bottom of ply No. 2 and at the top of ply No. 3, The following are calculations for other planes within the plate.

Page 66: Nasa Laminates

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Page 67: Nasa Laminates

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Page 68: Nasa Laminates

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Page 69: Nasa Laminates

The stresses in the bottom of ply No. 3 will be the mirror of the stresses in the top of ply No. 2, thus, for the bottom of ply No. 3:

All of the information in example No. 6 can be tabulated:

Strains Plane I &Y

Stresses

Since the stresses and strains vary linearly through the thickness of each ply, a diagram of the stresses and strains through the laminate can be made. A few examples follow.

Page 70: Nasa Laminates
Page 71: Nasa Laminates

VIII. STRESSES AND STRAINS WITHIN LAMINAE OF AN UNSUMMETRIC LAMINATE

A. Difference From Symmet~e Ladnates

If the laminate being examined is not symmetric, then additional complexities arise in the material's behavior from that examined in the previous section. The neutral plane of the plate will not coincide with the geometrical midplane of the plate as it did for symmetric plates. Thus, there will be less simplifications, such as knowing the bottom half of a symmetric laminate will consist of a negative mirror image of stresses and strains from the top half due to bending moments. Also, environmental effects may become critical since the plate can warp due to these effects. Coupling now becomes extremely important and can cause very unique mechanical behavior chuacteristic of anisotropic plate!,

B. Example (0/+45)T Laminate

Example 7:

Consider a 2-ply laminate of AS413501-6 with a (0/+45)T stacking sequence. The Barnina properties were given in example 1 as:

Ply thickness = 0.005 in ,

and the resulting values for Gij for the 0' and the +45O plies were given as:

Page 72: Nasa Laminates
Page 73: Nasa Laminates

Find the stresses and strains in each of the two plies given a stress resultant in the X-direction of 250 Iblin. The primed matrices must be obtained from equations (51) and (53):

Therefore,

Page 74: Nasa Laminates
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Page 76: Nasa Laminates

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Page 77: Nasa Laminates

Note that although only a tensile stress is applied to the plate, a small amount of compaessive strain is produced in the outer fibers of part of the 0' ply. This is due to the Bii terms which relate midplane strains to plate curvatures (i.e., due to the nonsymmetry, the plate is bending even though no bending moment is applied). As can be seen from the diagram, the plate is curving into the Oa fiber direction.

Example 8:

Consider an applied moment of 2.5 inch-lblinch in the 0' fiber direction. The constitudve equation is:

, From this the midplane strains and plate curvatures can be found.

I

K,=D,,Mx=2.05- in-lb

, K, = D,,M, = -0.366 -

in-lb in

K, = D ; ~ M , = -1.37 - (2.5 y) = -3.43 : in-lb m

Page 78: Nasa Laminates
Page 79: Nasa Laminates

Putting the above values of strain into tabular form:

Strains

A diagram of the through-the-thickness strain in the x-direction follows:

Z (in.)

Ex (percent)

Note that at the geometric midplane, the strain is not zero but -1.02 percent, even though only a bending moment has been applied. This is due to the bending-extension coupllixag that is present in this unsymmetric laminate. The plane of zero strain in the x-direction-can be determined by setting Ex equal to zero in equation(30) and solving for z:

Thus, at z = 0.002 in, the x-direction strain is zero for this example. A diagram of the though-the- thickness strain along the fibers in each ply follows:

Page 80: Nasa Laminates
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Page 82: Nasa Laminates

The lamina properties are:

E l = 20,010,000 lb/in2 , 2

E2 = 1,30 1,000 lb/in ,

G12 = 1,001,000 lb/inL ,

v12 = 0.3 ,

v21 = 0.02 ,

a, = -0.072x10-~ - FO '

a, = 32.4x10-, - , FO

Ply thickness = 0.005 in

From equation (lo), the reduced stiffnesses can be calculated:

From equation (22); for the 0' plies:

Page 83: Nasa Laminates

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Page 84: Nasa Laminates

From equation(53):

All = [20,130,785+6,557,237+1,308,853] lb/in2 (0.005 in)= 139,984 lbin '

A12=[+4,,,238+3926,611_,in2/000,in/=267031___in

A22=[13088,3+6,,7237+2013078,]_b,in2l000,_/=139,984ml--_a1o=[4,705,483]l_,in2(0.005in)--23,5271±

in

A26=[4,70,,48311_,in2(o.oo5in)=23,5271±in

A66[1001000+,163,82+100100011_,in2/0005in/=35828l__in

(E9.5)

From equation (54); the Bij's will be:

B 11 = [6,557,237(0.005)- 1,308,853(0.005)] .1---b(0.005 in) = 131.2 lbin

lbB 12 = [4,555,238(0.005)-392,656(0.005)] ]-ff (0.005 in) = 104 lb,

lbB22 = [6,557,237(0.005)-20,130,785(0.005)] _-_ (0.005 in) = -339.4 lb ,

B16 = [4,705,483] lb/in 2 (0.005 in) (0.005 in) = 117.6 lb ,

B26 = [4,705,483] lb/in 2 (0.005 in) (0.005 in) = 117.6 lb ,

lbB66 = [5,163,582(0.005)-1,001,000(0.005)] ]-ff (0.005 in) = 104 lb.

(E9.6)

From equation (56); the Dij's will be:

76

Page 85: Nasa Laminates
Page 86: Nasa Laminates

The extensional stiffness [ A ] , coupling stiffness [ B ] , and bending stiffness [Dl matrices are:

The .Fully invested form of the constitutive equations is needed. First, the partially inverted form is given by equafon (51):

in-lb . =

-

1.275 0.674 0.637 0.674 3.628 0.637 0.637 0.637 0.845

- -

Page 87: Nasa Laminates

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Page 88: Nasa Laminates
Page 89: Nasa Laminates

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0.00002016 -0.009072 + x106 (0.005 in) lb/in2

0 d

The thermal moment resultant is given by equation (93):

- 20.131 0.393 0 0.00002016 0.393 1.309 0 -0.009072 (0 in) +

0 0 1.001 0 -

P

1.309 0.393 0 -0.009072 0.393 20.13 1 0 0.00002016 (-0.005 in)

0 0 1,001 0 - i

x106 (0.005 in) 1b/in2

Page 90: Nasa Laminates
Page 91: Nasa Laminates

Now equation (30) can be used to find the strains within the laminate, and once these are found, equation (23) can be used to find the stresses.

At the top of the +4S0 ply, z = 0.0075 in:

At the bottom of the +45' ply (the top of the 0" ply, z = 0.0025 in:

At the bottom of the 0' ply (the top of the 90' ply), z = -0.0025 in:

At the bottom of the 90" ply, z = -0.0075 in:

Page 92: Nasa Laminates

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Page 94: Nasa Laminates
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Page 104: Nasa Laminates

APPENDIX

A Foruan program to find the determinant of a 6x6 matrix is given below:

@ THIS PROGRAM CALCULATES THE DETERMINANT C OF A 6 X 6 MAT= WITH ELEMENTS "E" C

REAL E(4,6), ES(6,6), M IIWEGER TAG,I,J.N M= l N= l DO 10 Ih=l,6 PRINT *, 'ENTER ROW #', I, 'OF THE MATRIX'

READ *, E(I,1),E(I,2)7E(173)7E(I,4),E(I,5)7E(I,6) BO CONTINUE 40 CONnNUE

TAG=N DO 75 I=N,5

IF (ABS(E(TAG,N)) .GT. E(I+l ,N)) THEN TAG=TAG ELSE TAG=li+ 1 EMDIF

75 CONTINUE DO 80 J=N,6 ES(N,J)=E(TAG,J) ES(TAC,Y)=E(N,J)

80 CONTINUE DO 90 J=N,4 E(N,J)=ES(N,JO E(TAG,J)=ES(TAG,J)

96 CONTINUE M=M"E(N,N) DO 91 J=N,6

ES(N,J>=E(N,J)/E(N,N) 91 CONTINUE

DO 95 J=N,6 E(N,J)=ESQN,J)

95 CONTINUE DO 200 I=N+1,6 DO 150 J=N,6

Es(a,J)=(-:i *E(I,N)*E(N,J))+E(I,J) 150 CONTINUE 200 CONTINUE

DO 300 I=N+1,6 DO 250 J=N,6 EQI,J)=ES(I,J)

250 CONTINUE 300 CONTINUE

Page 105: Nasa Laminates

N=N+l IF (N .LT. 6) THEN

GO TO 40 ELSE CONTINUE ENDIF M=M*E(6,6) PRINT *, 'DETERMINANT =', M STOP END

The determinant of a 5x5 matrix can be determined in a similar manner, substituting 5 for 6 in the appropriate sections of the above program.

*U.S. GOVERNMENT PRINTING OFFICE 1994-633-109100124

Page 106: Nasa Laminates

Form Approved

OM8 No. 0704-0188

7. PERFORMING ORGAFJlZATlON NAME(S) AND ADDRESS(ES)

Geolr,oe C. Mashall Space Flight Center M a s h a Space Flight Center, Alabama 35812

9. SPONSORlbgC I MONlTORlNG AGENCY NAME(S) AND ADDRESS(ES)

N a ~ o n d Aeronautics and Space Administration Wuasllnington, DC 20546 NASA RP-1351

Prepxed by Materials and Processes Laboratory, Science and Engineering Directorate.

AILABILITY STATEMENT

Subject Category: 24 hincl~sified-unlimited

63. ABSTRACT (EAsJximwm ZOO words)

The mechanics of laminated composite materials is presented in a clear manner with only essenfid derjiva~ons included. The constitutive equations in all of their forms are developed and then summzized in a separate section. The effects of hygrothermal effects are included. The prediction of the engineering constants for a laminate are derived. Strength of laminated composites is not covered.

NSN 7548-01-280-5500 Standard ~orrn-298 (Rev 2-89)

Page 107: Nasa Laminates

National Aeronautics and Space Administration Code JTT Washington, DC 205464001

official Business Penalbyfor Private Use, $300

Postmastee y Undeliverable (Section 158 Postal Manual), Do Not Return