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dissertation entitled

INVESTIGATION OF THEORIES FOR LAMINATED
COMPOSITE PLATES

presented by

Xiaoyu Li
has been accepted towards fulfillment

of the requirements for

Ph.D. degree in Mechanics

Mot.

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Date January 16, 1995

 

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INVESTIGATION OF THEORIES FOR LAMINATED
COMPOSITE PLATES

By

Xiaoyu Li

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Materials Science and Mechanics

1994

ABSTRACT

INVESTIGATION OF THEORIES FOR LAMINATED COMPOSITE PLATES

By

Xiaoyu Li

High-order Shear Deformation Theories give good results for in-plane stresses but
poor results for interlaminar stresses. However, Layerwise Theories give excellent results
for both global and local distributions of displacement and stress (both in-plane and out-
of-plane). A compromising theory, the so-called Generalized Zigzag Theory, is presented.
It has two layer-dependent variables in the zeroth- and first-order terms. Due to its success
in laminate analysis, the feasibility of assigning the two layer-dependent variables in the
second- and third-order terms is examined, resulting in the Quasi-layerwise Theories.
Unfortunately, a physical absurdity — coordinate dependency, takes place. It then requires
a technique, the so-called Global—Local Superposition Technique, to formulate the
laminate theory to be coordinate-independent for the numerical advantage. The recursive
expressions presented in this study, though somewhat tedious, are necessary to achieve the
numerical advantage. By examining the results based on the Superposition Theories, it is
concluded that the completeness of the terms is meant two fold: not only can no low-order
term be skipped, but more high-order terms are preferred.

The objective of completeness seems to conflict with the fundamental of two

continuity conditions in each coordinate direction. In order to satisfy both aspects, a
special technique called the Hypothesis for Double Superposition is proposed. Several
three-term theories, the so-called Double Superposition Theories, are examined. They
give excellent values for in-plane displacement, in-plane stress, and transverse shear
stress. However, because w is considered as constant in the examples, both transverse
displacement and transverse normal stress are not as good as the remaining components.
Among all the theories examined in this thesis, it seems that the Generalized Zigzag
Theory, with up to seventh-order terms, and the third-order Double Superposition
Theories give the best agreement with Pagano’s solution in all ranges of layer number for
both symmetric and unsymmetrical laminates. Although they both are layer-number
independent theories, the former has seven degrees-of-freedom while the latter has only
three, provided w is considered to be constant through the laminate thickness. As a
consequence, the Double Superposition Theories are concluded as the best selection for

laminate analysis in this thesis.

TABLE OF CONTENTS

CHAPTER 1 INTRODUCTION ............................................................................... 1
CHAPTER 2 SHEAR DEFORMATION THEORIES ............................................ 7
2.1 Introduction ............................................................................................................ 7
2.2 High-order Shear Deformation Theories (HSDT) .............................................. 8
2.3 Higher-order Shear Deformation Theories (HrSDT) ......................................... 10
2.4 Influence of High-order w .................................................................................... 12
2.5 Verification Technique .......................................................................................... 13
2.6 Numerical Results and Discussion ........................................................................ 16
2.7 Summary .............................................................................................................. 30
CHAPTER 3 LAYERWISE THEORIES ................................................................ 31
3.1 Introduction ........................................................................................................... 31
3.2 Formulation of Generalized Zigzag Theory ....................................................... 35
3.3 Generalized Zigzag Theory with Higher-order Terms ...................................... 45
3.4 Numerical Results and Discussion ....................................................................... 47
3.5 Summary ............................................................................................................... 57
CHAPTER 4 QUASI—LAYERWISE THEORIES .................................................. 61
4.1 Introduction .......................................................................................................... 61

iv

4.2 Formulation of the 1-3 Quasi-layerwise Theory ................................................ 64

4.3 Numerical Results and Discussion ....................................................................... 70
4.4 Summary ............................................................................................................... 81
CHAPTER 5 GLOBAL-LOCAL SUPERPOSITION THEORIES ........................ 82
5.1 Introduction .......................................................................................................... 82
5.2 Global-Local Superposition Technique ............................................................... 83
5.3 Formulation of the 1-3 Superposition Theory .................................................... 85
5.4 Numerical Solution ............................................................................................... 92
5.5 Results and Discussion ............................................................................................ 95
5.6 Summary ............................................................................................................... 97
CHAPTER 6 APPLICATION OF DOUBLE-SUPERPOSITION ..................... 105
6.1 Introduction ........................................................................................................ 105
6.2 Formulation of the 1,2-3 Double-superposition Theory .................................. 108
6.3 Variational Equation .......................................................................................... 116
6.4 Numerical Results and Discussion ..................................................................... 119
6.5 Summary ............................................................................................................. 120
CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS ......................... 130
7.1 Conclusions .......................................................................................................... 130
7.2 Recommendations ............................................................................................... 132
LIST OF REFERENCES ......................................................................................... 133

LIST OF FIGURES

Figure 2.1 - Plane-strain problem: a laminated plate under cylindrical bending.

Figure 2.2 - Comparison of Ox from Shear Deformation Theories of various orders for a
[0/90/0] laminate.

Figure 2.3 - Comparison of tn from Shear Deformation Theories of various orders for a
[0/90/0] laminate.

Figure 2.4 - Comparison of 62 from Shear Deformation Theories of various orders for a
[0/90/0] laminate.

Figure 2.5 - Comparison of 11 from Shear Deformation Theories of various orders for a
[0/90/0] laminate.

Figure 2.6 - Comparison of W from Shear Deformation Theories of various orders for a
[0/90/0] laminate.

Figure 2.7 - Comparison of In from Shear Deformation Theories of various orders for a
[0/90] laminates.

Figure 2.8 - Comparison of 6x from various theories for a [0/90/0] laminate.
Figure 2.9 - Comparison of sz from various theories for a [0/90/0] laminate.
Figure 2.10 - Comparison of 62 from various theories for a [0/90/0] laminate.
Figure 2.11 - Comparison of u from various theories for a [0/90/0] laminate.
Figure 2.12 - Comparison of W from various theories for a [0/90/0] laminate.
Figure 3.1 - Coordinate system, layer order, and interface locations.

Figure 3.2 - Comparison of Ox from Zigzag Theories of various orders for a [0/90/0]
laminate.

Figure 3.3 - Comparison of In from Zigzag Theories of various orders for a [0/90/0]
laminate.

vi

Figure 3.4 - Comparison of Oz from Zigzag Theories of various orders for a [0/90/0]
laminate.

Figure 3.5 - Comparison of 1.4 from Zigzag Theories of various orders for a [0/90/0]
laminate.

Figure 3.6 - Comparison of 71’ from Zigzag Theories of various orders for a [0/90/0]
laminate.

Figure 3.7 - Comparison of In from Generalized Zigzag Theory of various orders for
both [0/90/0] and [0/90] laminates.

Figure 3.8 - Comparison of R from the fifth-order Generalized Zigzag Theory and
elasticity solutions for 2-, 6-, 14-, and 30-layer unsymmetrical laminates.

Figure 3.9 - Comparison of In from the fifth-order Generalized Zigzag Theory and
elasticity solutions for 2-, 6-, 14—, and 30-layer unsymmetrical laminates.
Figure 4.1 - Variation of Bx from the 1-3 Quasi-layerwise Theory due to various shift

ratios in a [0/90/0] laminate.

Figure 4.2 - Variation of tn from the 1-3 Quasi-layerwise Theory due to various shifts
ratios in a [0/90/0] laminate.

Figure 4.3 - Variation of 62 from the 1-3 Quasi-layerwise Theory due to various shifts
ratios in a [0/90/0] laminate.

Figure 4.4 - Variation of 12 from the 1-3 Quasi-layerwise Theory due to various shifts
ratios in a [0/90/0] laminate.

Figure 4.5 - Variation of W from the 1-3 Quasi-layerwise Theory due to various shifts
ratios in a [0/90/0] laminate.

Figure 4.6 - Results of Ox for a [0/90/0] laminate from various Quasi-layerwise Theories
obtained by adjusting the thickness coordinate.

Figure 4.7 - Results of ‘txz for a [0/90/0] laminate from various Quasi-layerwise Theories
obtained by adjusting the thickness coordinate.

Figure 4.8 - Results of Oz for a [0/90/0] laminate from various Quasi-layerwise Theories
obtained by adjusting the thickness coordinate.

Figure 4.9 - Results of 11 for a [0/90/0] laminate from various Quasi-layerwise Theories
obtained by adjusting the thickness coordinate.

vii

Figure 4.10 - Results of W for a [0/90/0] laminate from various Quasi-layerwise Theories
obtained by adjusting the thickness coordinate.

Figure 5.1 - Comparison of Ox from various Superposition Theories for a [0/90/0]
laminate.

Figure 5.2 - Comparison of In from various Superposition Theories for a [0/90/0]
laminate.

Figure 5.3 - Comparison of 62 from various Superposition Theories for a [0/90/0]
laminate.

Figure 5.4 - Comparison of 12 from various Superposition Theories for a [0/90/0]
laminate.

Figure 5.5 - Comparison of W from various Superposition Theories for a [0/90/0]
laminate.

Figure 5.6 - Comparison of u from the 1-3 Superposition Theory and elasticity
solutions for 7-, 15-, and 31-layer symmetric laminates.

Figure 5.7 - Comparison of tn from the 1-3 Superposition Theory and elasticity
solutions for 7-, 15-, and 3 l-layer symmetric laminates.

Figure 6.1 - Comparison of 6,, from the 1,2-3, 1,3-2, and 2,3-1 Double Superposition
Theories with the elasticity solutions for a [0/90/0] laminate.

Figure 6.2 - Comparison of In from the 1,2-3, 1,3-2, and 2,3-1 Double Superposition
Theories with the elasticity solutions for a [0/90/0] laminate.

Figure 6.3 - Comparison of 62 from the 1,2-3, 1,3-2, and 2,3-1 Double Superposition
Theories with the elasticity solutions for a [0/90/0] laminate.

Figure 6.4 - Comparison of 11 from the 1.2-3, 1,3-2, and 2,3-1 Double Superposition
Theories with the elasticity solutions for a [0/90/0] laminate.

Figure 6.5 - Comparison of W from the 1,2-3, 1,3-2, and 2,3-1 Double Superposition
Theories with the elasticity solutions for a [0/90/0] laminate.

Figure 6.6 - Comparison of 12 from the 1,2-3 Double Superposition Theory and elasticity
solutions for 2-, 6-, and l4-layer unsymmetrical laminates.

Figure 6.7 — Comparison of 1x2 from the 1,2-3 Double Superposition Theory and elasticity
solutions for 2-, 6-, and 14-layer unsymmetrical laminates.

viii

Figure 6.8 - Comparison of 12 from the 1.2-3 Double Superposition Theory and elasticity
solutions for 3-, 7-, and 15-layer symmetric laminates.

Figure 6.9 - Comparison of In from the 1.2-3 Double Superposition Theory and
elasticity solutions for 3-, 7-, and lS-layer symmetric laminates.

CHAPTER 1

INTRODUCTION

Because of their high stiffness and high strength with low density, fiber-reinforced
polymer matrix composite laminates have been widely used for high-performance
structures. When compared to conventional metals, the major characteristics of laminated
composite materials include the orthotropy in the laminate plane, the low modulus in
transverse shear, and the lamination through the thickness. In addition, it should be noted
that most laminated composite materials are originally used in thin structures. As the
technology of composite advances, laminated composites are also used for thick and
moderately thick structures.

In the early days of study, the Classical Plate Theory (CPT) used for isotropic
structures was directly applied to composite structures. In fact, in formulating the isotropic
and orthotropic plates, the only difference lies in the constitutive equations. Modifying the
CPT for orthotropic material applications does not create any further inaccuracy in the
composite analysis. However, when applying the CPT to materials with low modulus in
transverse shear and to thick plates, the results will be unsatisfactory because the
transverse shear effect which is not considered in the CPT should be considered in both
cases. As a consequence, the Shear Deformation Theories, being derived from the
Classical Plate Theory, have been presented to improve the deficiency.

In composite structure analysis, both a stress approach and a displacement approach

have been utilized by many investigators. An example of the stress approach is the hybrid-

stress finite element method presented by Mau, Tong, and Pian [1]. By a‘sfisgnlingmamstress'

""N-A. —»O. ., ..,. I“

field which satisfies the equilibrium equations, and a simple displacement interpolation,
\___.___.,.____ h” _

the governing equations'and associated boundary conditions can be obtained from a mixed A
variational principle [2]. Since the continuity conditions on the laminate interfaces are also
satisfied, the results from the hybrid—stress finite element methods are in close agreement
with the elasticity solution presented by Pagano [3]. However, the major disadvantage of
the stress approach is the large number of degrees-of-freedom involved in the numerical
solution.

Originating from Classical Plate Theory, the displacement approach, which includes
the High-order Shear Deformation Theories (HSDT) [4], is much more efficient in
numerical analysis than the stress approach. It is also recognized as being variationally
consistent and is thus much more popular in composite structure analysis. In the
displacement approach, the assumption of continuous functions is correct for out-of-plane
displacement components. However, this assumption is not accurate for in-plane
displacement components because the in-plane strains are discontinuous across the
laminate interfaces. In fact, owing to the abrupt change of material properties across the
laminate interfaces, the in-plane displacement components should be of zigzag
distributions through the laminate thickness.

The inaccurate in-plane displacement assumptions can result in erroneous transverse
stresses, especially on the laminate interfaces. Mth the continuous functions, their
derivatives are also continuous. As a consequence, single-valued strains on laminate

interfaces are created. Since different composite layers have different material properties,

the transverse stresses obtained from the constitutive equations will be mistakenly double-

valued. Most investigators tend to ignore and bypass this error. They utilize the
equilibrium equations to ‘recover’ the transverse stresses [5]. Although the results from
this post-processing technique seem to be promising, the technique itself is very
controversial.

Another category of displacement based techniques is the so-called Layerwise
Theories (LT) [6, 7]. They are developed from the High-order Shear Deformation
Theories. Instead of modeling the whole laminate by the HSDT, each composite layer is
described by individual displacement components. The displacement field of the whole
composite laminate is nothing but the assembly of the displacement components of
individual layers. The observation of the important roles of individual layers matches with
the aforementioned recognition of the composite characteristic that the composite
laminates are made of layers through their thicknesses. In addition, it should be noted that
in assembling the composite layers, the continuity conditions of both displacements and
transverse stresses are imposed on the laminate interfaces. As a consequence, the
Layerwise Theories are recognized as the most accurate approaches toward laminated
composite analysis.

Taking the properties of individual layers into consideration, the Layerwise Theories
can. satisfy all the major characteristics of laminated composite materials. Accordingly, it
is obvious that they are superior to High-order Shear Deformation Theories. However, just
because the LT are based on the assembly of individual layers, the total number of
degrees-of-freedom is dependent on the number of layers. As the layer number increases,
the computational requirement becomes very demanding. They then have the same

disadvantages as those of hybrid-stress finite element methods. This major disadvantage

may explain why the LT are rarely used for composite analysis, although they are the most
accurate displacement approaches available in the literature.

At the same time that the Layerwise Theories were being developed, another major
effort in the development of the composite laminate theories was being developed by
attempting to remove the deficiencies in the High-order Shear Deformation Theories.
These deficiencies were that the in-plane displacements and their derivatives are smoothly
continuous through the laminate thickness instead of being zigzag disuibutions. The
primary approach of this group of studies, the Zigzag Theories [8, 9], is based on assumed
displacement functions with limited layer-dependent variables for individual layers. Many
special functions aimed at presenting the zigzag distributions for in-plane displacements
are presented and they all seem to be very promising for in-plane displacement
predictions. However, since all the presented functions are specially selected and most of
them do not satisfy the stress continuity conditions on the laminate interfaces, the
predictions of the transverse stresses from the Zigzag Theories are no better than those
from the HSDT. The recovering technique, as mentioned before, seems to be the only
technique to bypass the errors.

By carefully examining the Zigzag Theories available in the literature, it can be found
that they are just special cases of the Layerwise Theories. In addition, by comparing the
advantages and disadvantages of the Layerwise Theories and the High-order Shear
Deformation Theories, it can be concluded that a compromising theory may be an
optimum choice for laminated composite analysis. In fact, a carefully selected Layerwise
Theory which resembles the idea of the Zigzag Theories may be a candidate to satisfy the

requirements of both computational efficiency and numerical accuracy. Thus, the

objectives of this thesis are listed below:

1. Since the Classical Plate Theory is a special case of the High-order Shear
Deformation Theories, and the Zigzag Theories and the High-order Shear Deformation
Theories are special cases of the Layerwise Theories, the first objective is to propose a
theorytounify all the laminate theories based on displacement assumption.

2. “With the proposed theory, it is possible to judge the computational efficiency and
numerical accuracy of various High-order Shear Deformation Theories and Zigzag
Theories in a more systematic, consistent way.

3. Based on the proposed theory, it is possible to develop an optimum theory or
various optimum theories for studying different laminated composite structures. Hence, a
comprehensive study on all possible theories derived from the proposed theory is another
goal.

“With the above objectives, this thesis is divided into the following chapters. Chapter 2
starts with the Classical Plate Theory and the High-order Shear Deformation Theories.
The evolution of HSDT is carefully documented. Eventually a generalized HSDT is
presented. This generalized theory can be extended to theories of a much higher order,
namely the Higher-order Shear Deformation Theories (HrSDT). Evaluations of HrSDT
with various orders are of primary concern. In Chapter 3, a Generalized Zigzag Theory is
presented. It is to demonstrate that the generalized theory envelops all the High-order
Shear Deformation Theories and Zigzag Theories available in the literature and allows a
systematic investigation for other potential theories. The Generalized Zigzag Theory is
then extended to various quasi-layer-dependent theories. The Quasi-layerwise Theories

are examined in Chapter 4. Comparisons among the various types of Quasi-layerwise

Theories are also made. However, unfortunately, the Quasi-layerwise Theories are found
to be coordinate dependent. In order to remove this complexity, a global-local
superposition technique is presented in Chapter 5. 'Wrth this technique, many more
theories, namely the Superposition Theories, are made available. They are later extended
to cover more layer-dependent terms based on a hypothesis for double superposition. This
hypothesis is proved to be very useful. As a result, many more composite theories are
presented in Chapter 6. The selection of an efficient and accurate theory is also discussed.
The last chapter of this thesis gives the conclusions of the investigation of composite
laminate theories based on the criteria of computational efficiency and numerical

accuracy. Recommendations for future studies are also presented in Chapter 7.

CHAPTER 2

SHEAR DEF ORMATION THEORIES

2.1 Introduction

In the early days of study, the technique used for studying conventional plates was
lent to analyze laminated composite plates. The Classical Plate Theory (CPT) was
naturally the first tool for the investigations. As can be expressed by the following

equations, the CPT is of a displacement approach:

u(x.y.2) -

|
E
O
A
3*
‘<
V
l
”I
N

V(x.y.2) = v0(x.y)— 217:2 (2.1)

w (x, y, 2) = WO ()6. y)

where u and v are in-plane displacements in the x and y directions, respectively, while w is
the displacement in the plate thickness, the z, direction. The in-plane displacement

components uo and v0 can be viewed as translational components at the laminate

midplane while — 8_w and - B_w

Bx 3y

deformation through the laminate thickness, w is assumed to be independent of the 2-

as rotational angles. In addition, considering the small

coordinate.
Because of its simplicity, the CPT is widely used in composite structure design and
analysis. Reasonable results of displacements and in-plane stresses can be obtained for

plates with aspect ratios (in-plane dimensions versus thickness) larger than twenty.

However, it should be recognized that the CPT is based on Kirchhoff ’s hypothesis which
assumes that the lines perpendicular to the midplane before loading remain perpendicular
after loading. The transverse shear effect is not considered in the analysis although it is
significant to thick plate analysis and should not be neglected in materials with low shear
moduli.

Reissner [10] and Mindlin [ll] recognized the important role of transverse shear
effect in plate bending. They replaced the rotational angles with more general variables to

account for the shear efiect, i.e.:

u (x. y. 2)
v (1,)” 2)

uo (x. y) + \le (x. y) 2
v0 (x. y) + W, (x. y) z (2.2)

w (x. y, 2) = wo (x. y)

By closely examining Equations (2.2), it can be concluded that the in-plane displacements
are expressed as first-order equations of 2. Based on their pioneering works, high-order
terms of 2 were gradually added to u, v, and even w. Many theories of this type were
proposed in the past three decades and were called High-order Shear Deformation

Theories (HSDT).

2.2 High-order Shear Deformation Theories (HSDT)

HSDT are also based on assumed displacement fields. Lo, Christensen, and Wu [5]
unified the notations used in expressing various HSDT. They documented the
development of HSDT according to the order of z of the polynomial equations for in-plane
displacements. In addition to those mentioned in Lo, et. a1. many more HSDT can be

found in the review articles by Noor and Burton [12], and Kapania and Raciti [13]. As a

summary, Lo, et. al. also presented a third-order shear deformation theory for laminated
composite plate analysis, i.e.

utx.y.z> = u0(x.y) +w.(x.y)z+C.(x.y)zz+¢,<x.y)23

v (x, y, z) = v0 (x. y) + W, (x. y) z + Cyix,y122 + «p, (x. y) z3 (2.3)

w(x.y, 2) = wo (Ly) +‘l'z (Jay) 2 + C, (Jay) 22

Apparently, all the HSDT available in the literature can be viewed as special cases of the
above third-order theory. In addition, with the use of common notations, it is clear that
Equations (2.3) are extended from Equations (2.1) and (2.2).

Although in their numerical studies, Cw was kept constant through the laminate
thickness, Lo, et. a1. verified that the third-order theory was superior to lower-order
theories in deformation and stress analysis, especially of composite laminates with
moderate thickness. With the assumed third-order equations for u and v as expressed in

Equations (2.3), the displacements and their first derivatives with respect to the z-

coordinate are continuous. However, it should be pointed out that although the in-plane’l r .6 ,f
displacements are continuous across the laminate interfaces when the interfaces are ; '
perfectly bonded, their first derivatives are discontinuous because of the abrupt change of

L.
t
material properties across the laminate interfaces. In other words, n and v should be of .3

zigzag distributions through the laminate thickness.

Inaddition to the in-plane displacements and stresses, the transverse stresses are also
very important in composite laminate analysis since interlaminar stresses take the primary
responsibility for delamination, which is a unique and critical damage mode in laminated
composite plates. Consequently, accurate interlaminar stresses are highly required for

composite analysis and design. Since the third-order assumption for in-plane displacement

10

interlaminar stresses are obtained on the laminate interfaces when single-valued strains on
the interfaces are multiplied by different material properties of different layers. To ignore
the error, a post-process, the so-called recovering technique, which is based on
equilibrium equations instead of constitutive equations was proposed (e.g. Lo,
Christensen, and Wu [5]). Although relatively good transverse stresses can be obtained
from this method, the approach itself is controversial. It then is desired to identify a
technique which can give correct displacements and stresses (both in-plane and
transverse) directly from the constitutive equations.

Despite the aforementioned fundamental defects, there is a primary advantage of
using HSDT in composite plate analysis. By examining Equations (2.3) it can be
concluded that eleven coefficients need to be determined regardless of the total number of
layers of the composite laminate. The feature of layer-number independency is critically
important to computational efficiency, especially when the layer number of the composite
laminate becomes very large. Another way to express the layer-number independency of
HSDT is to use the concept of “global sense.” Instead of considering the composite layers
individually, HSDT present each composite laminate as a whole piece. Since the

interlaminar stresses are of “local” information, the reason that they cannot be calculated

from global techniques such as HSDT is clear.

2.3 Higher-order Shear Deformation Theories (HrSDT)
As discussed in the previous section, almost all HSDT available in the literature are

third-order theories or lower. There is no discussion on why the third-order seems to

ll

become the upper limit in laminate theories. Although Reissner [14] presented laminate
theories of sixth-order, no numerical result was given to compare his higher-order theories
with other lower-order ones. For completeness, it is necessary to further investigate HSDT
of fourth-order and higher. Such higher-order theories will be called Higher-order Shear
Deformation Theories (HrSDT) in this study.

HrSDT can be obtained by simply extending the third-order HSDT of Equations (2.3)
to higher-orders. For convenience and uniformity, new notations are used in expressing

the generalized displacement field:

u(x,y,z) = E'rrl.(rr,y)zi
i=0
v(x,y,z) = 2vi(x,y) zi (2.4)
i=0
1 .
w(x,y,z) = Zwi(x,y)z'
i=0

In the above equations, m is the order of in-plane displacement components and is usually
used to name the order of the theory since w is frequently kept lower or constant. For m =
1, Equations (2.4) represent Reissner’s and Mindlin’s first-order theories while for m = 3,
the equations are associated with the third-order HSDT. As an example, HrSDT of

seventh-order can be written as follows:

2 3 ' 4
u(x,y.z) = uo(x,y)+u1(x,y)2+u2(x,y)z +u3(x.y)z +u4(x.y)z +

5 6 7
“5(an)z +u6(X,)’)Z +u7(x,y)z

v(x,y,z) = v0(x,y) +v1(x,y)z+v2(x,y)22+v3 (x,y) z3+v4 (Ly) 24+ (2.5)
5 6 7
v5(x,y)z +v6(x,y)z +v7(x,y)z
w(x, y, z) = w0 (x, y)

12

Apparently, as the order of the theory increases, the number of variables also increases,
resulting in a better possibility to improve the accuracy of laminate analysis. However, it

should also be recognized that the numerical effort will also become quite demanding.

2.4 Influence of High-order w

Because composite laminates are usually very thin, the displacement component in
the z-direction is frequently assumed constant through the laminate thickness. In other
words, the deformation of thickness is considered negligible when compared to the
deformations in the plane. However, high-order terms can be added to w as those in u and
v under the following conditions: (1) the thickness of a composite laminate is relatively
high, (2) the composite laminate has low transverse shear modulus, and (3) in the case a
more accurate prediction is highly desired.

As pointed out by Reissner [10], even-order terms of w play a more important role in
thickness deformation than odd-order terms. If the polynomial equation of w is not to

exceed the third-order of its in-plane counterparts, w can be assumed as follows:

w (x. y. z) = w, (x.y) + w, (x. y) z + w2 em 22 (2.6)

As a consequence, the following third-order HSDT (of u and v) is chosen for

investigations of the influence of high-order w on composite laminate performance:

2 3
u(x.y.2) = uo(x.y) +u1(x.y)2+u2(x.y)z +u3(x.y)z

vO (x, y) + v1 (x, y) z + v2 (1:, y) 22 + v3 (x, y) 23 (2.7)

v (x. y. 2)

2
w(x,y,z) =w0(x,y) +w1(x,y)z+w2(x,y)z

13

2.5 Verification Technique

In view of the potential small discrepancies among the various theories to be
presented, a high-sensitivity technique for verification is desired. Because the nature of the
present study is of numerical analysis, an analytical technique for verification is more
appropriate than an experimental one. Among the theoretical and numerical techniques
available, Pagano’s widely used elasticity solution [3] seems to be the most suitable, due
not only to the consideration of convenience but also accuracy. In addition, it is
recognized that Pagano’s solution has been used for verification in almost all composite
laminate theories presented in the literature. The comparisons made with Pagano’s
solution can be used to assess the theories presented in this thesis.

Pagano examined an infinitely long strip of laminated plate simply supported along

both longer edges. As shown in Figure 2.1, a sinusoidal load, q = q a sinpx , is exerted on

the top surface of the laminate, resulting in cylindrical bending across the width of the
laminate. As a consequence, this problem can be reduced to be a plane-strain type. By
assuming linear strain-displacement relations and orthotropic constitutive equations, and
imposing continuity conditions of displacements and transverse stresses on the laminate
interfaces, the elasticity solution of the problem was identified by Pagano in the following

forms:

l4

 

4
(k) _ . 2 mjkzk
Ox — srnprAJ-kmjke
j=l
(k) 2 4 m 2
. jk k
0'2 =—p srnpszjke
.=1
(16) _ 4 A "'1ka
In — —pcospx2 jkmjke (2.8)
i=1
(’0 4 (k) 2 (k) 2 m 2
_ COpr jkk
“ — ' p 2411(R13 1’ —R11mjk)e
i=1

(k) 4 (k) Rm 2 z
. 33 ”’1'
W = SlnprAjk R13 mjk—Fp e k I:

where Ajk, mjk, R13, R11, and R33 are constants associated with material and
geometric properties of the km layer. Details of A it and mjk can be found in Pagano’s
solution [3].

Pagano’s study is valuable to all displacement-based theories since close-form
solutions to all laminate theories can be obtained by assuming ui = U icospx and
w‘. = Wisinpx (i = 0, l, 2, ). Thus, the sirnply-supported boundary conditions can be
automatically satisfied while the coefficients U i and W2 need to be determined from

variational analysis.

15

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.1 - Plane-strain problem: a laminated plate under cylindrical bending.

2.6 Numerical Results and Discussion

In numerical analysis, the same material properties examined by Pagano as listed
below are also investigated in this thesis for verification of theories. Also presented in the
following equations are the definitions of normalized in-plane stress, Ox, at midspan,
normalized transverse shear stress, In , on the edge, normalized transverse normal stress,

6 at midspan, normalized in-plane displacement, a, on the edge, and normalized

z,

transverse normal displacement, W, at midspan:

Err = 1 x 106 psi, ELL = 25X106psi, G” = 0.5 x106 psi

 

 

 

 

(2.9)
0(1 l (i l
O = x 2’ 6 :02 2’2 1: =1:n(0,z)
x qo ’ z qo ’ xz qo
I
- EnXU(0,Z) _ lOOxEnxh3xw(§,z) _ z
u: r W: , Z: -
hqo qoxl“ h

Both a two-layer laminate with a stacking sequence of [0/90] and a three-layer
laminate of [0/90/0] are examined due to the availability of close-forth solutions and their
popularity in assessing various laminate theories. The aspect ratio U]: = 4 (length to
thickness ratio) is chosen because of its ability to show the feasibility of a laminate theory

in the analysis of both thin and thick composite laminates.

A. [0/90/0] Laminate
Results of the normalized stresses and displacement components of the [0/90/0]

laminate are shown from Figures 2.2 to 2.6. It is found in Figure 2.2 that the in-plane

l7

stress Ox is improved as the order of the theory increases. As can be seen from the
diagram, the results from the third-order and the fourth-order are identical. Similarly,
those from the fifth-order are the same as those from the sixth-order. These results may

indicate the importance of the odd-order terms in this symmetric laminate.
- -~V"\\\_ ~ “‘h-jutew - ......L.. . _ - -- . .. , , ‘ ..

‘— -‘—. u... _

As can be seen from Figure 2.3, the correct values of In are not obtained. It is
believed that this discrepancy is due to the fact that the continuity conditions 3f “transverse
shear stresses are not satisfied on the laminate interfaces.

Figure 2.4 shows the transverse normal stress, 62. Although no continuity condition
is imposed on the laminate interfaces, the discontinuity of Oz at the interfaces seems to be
very small. In addition, because no boundary condition of transverse normal stress is
enforced, the result on the top surface is not equal to negative one and that on the bottom
surface is not equal to zero. Moreover, it should be pointed out that although the results in
Figure 2.4 seem to be different from the exact solutions, they are small quantities when
compared with the dominant component, Ox.

Another important observation is focused on the in-plane displacement a (Figure
2.5). Although the fifth-order and above have significant efi‘ect on the transverse shear,
along with other lower-order theories, they fail to give distinct kinks on the laminate
interfaces. It is believed that the discrepancy cannot be improved simply by increasing the
order of the theory.

The prediction of transverse normal displacement, W, is shown in Figure 2.6. The

results reveal that the prediction is improved as the order of the theory increases.

18

 

 

 

 

 

 

0.5 . ....I XX 1 I I I r r
7 ° X “ “ eiasricity solution
0.4- °"‘, “xxxxx” second-orderSD‘l‘ .
' ..\ “ooooo” third-order SD‘I‘
0 _ xx “\0 ~ “ ..... " fath-orderSDT q
'3 x e “-.-.- " fifth-orderSDT
Xx °@\ “- - - - " sixth-orderSDT
0.2 - x e \ “.. .. .. ." seventh-order SD‘I’ -
x ..

0.1 - : ..
2 0" d
-0.1 - . A d

\ -@ xx
-0.2- e x .
'- 6 xx
-O.3 - . x ..
x
-0.4 " ‘ x o d
x o
_o 5 r r I 1 L X o - 7.
5 10 15 20 25

L25 -2o -15 -10 -5

Figure 2.2 - Comparison of Oz from Shear Deformation Theories of
various orders for a [0/90/0] laminate.

0.5
x
x
0.4 ~ § .
x
§
0.3 - x .
0.2 " .......... e n J
4:.9'—__.—.——- -—-——
:79 x
e x .
0.1 - / e x
.- e x
1’ 2
2 0 -. .,\ e x .
x: r
-0.1 "- .'...\ 9 § ct
"\‘9'_--T.'I_'——""- ;
-O.2 " ‘Q‘D‘ ........... X ..
ex ------- 3i
9 \
-0.3 " @\ x u
x
. x_
-0.4 - § -
...... .x
1 .............. x
'°'§3 —zs —2 -1.5 -1 -o.5 , o

19

 

“ “ elasticity solution
“x x x x x” second-order SDT
“o o o o o” third-order SDT

“ ..... " forth-order SDT
“-..- - " fifth-orderSDT

- - - " sixth-order SDT
“.. .. .. .." seventh-order SDT

 

 

 

 

 

 

 

 

0.5

Figure 2.3 - Comparison of In from Shear Deformation Theories of
various orders for a [0/90/0] laminate.

0.5
0.4
0.3
0.2
0.1
2 0
-0.1
-0.2

-O.3

-O.5

20

 

 

 

“ “ elasticity solution
“x x x x x” second-order SDT
“ooooo” third-orderSD‘l‘

“ ..... " form-order SDT
..-.- " fifth-orderSDT

. . . - " sixth-order SDT

.. .. .” seventh-order SDT

 

3::

Figure 2.4 - Comparison of 52 from Shear Deformation Theories of
various orders for a [0/90/0] laminate.

  

 

2.1

 

0.5

—0.3 P

 

 

 

“ “ clasticitysolution
“xxxxx” second-orderSDT
“ooooo” third-orderSD‘l‘
“ ..... " forth-orderSD‘l'
“-.-.- " fifdr-orderSDT
“- - - - " sixth-orderSDT
“.. .. .. ." seventh-orderSD‘l‘

‘

 

 

-O.5
-i .5

0.5 1

 

Figure 2.5 - Comparison of 11 fi'om Shear Deformation Theories of
various orders for a [0/90/0] laminate.

1.5

0.5
0.4
0.3
0.2
0.1
2 O
-O.1
-O.2
-O.3

4.4

22

 

I

 

 

M

 

“ elasdcity solution
“x x x x x” second-order SDT
“ooooo” third-order SDT

“ ..... " forth-order SDT
-.-.- " tifth-orderSDT‘

- - - - " sixth-order SDT

. .. .. .." seventh-order SDT

8

.8:

‘v‘v‘v‘vvvvv‘v'vv

MVXMVVYIVMVMYVMY¥VMV§LMVVy

‘"V'"Vvv‘vvv‘vvvvv‘vavv‘v‘ V‘
- .

vvvvv'v‘vwvvvvv‘

yvyuvvuyvv

""

 

 

4:3.

-2.5

shim
.'..
U!
1.

Figure 2.6 - Comparison of W from Shear Deformation Theories of
various orders for a [0/90/0] laminate.

é:

23

B. [0/90] Laminate

The significance of the results from the investigation of a [0/90] laminate is
essentially the same as that mentioned in the study of the [0/90/0] laminate. That is, the
theories of the fourth-order and above have an important effect on 6x, 12, and W. The
results for 62 are fair since they are small when compared to Ox . Only In is presented in
Figure 2.7. It is found that even-order and odd-order terms both play important roles in the

[0/90] laminate while only odd-order terms are critical to the [0/90/0] laminate.

C. Influence of High-order w
In order to identify the influence of high-order w on composite analysis, comparisons
are made between a third-order HSDT with constant w and with second-order w. The

results for Ox, I 6

XZ’

2, 1'2 and W are shown in Figures 2.8, 2.9, 2.10, 2.11, and 2.12,
respectively.
As shown in Figures 2.8, 2.9, 2.11, and 2.12, the high-order terms of w seem to have

no significant effect on 6x, I

n, and 12. On the contrary, the values of Oz are clearly

improved with the addition of the high-order terms, as shown in Figure 2.10. However, it
should be pointed out that the values of Oz are small when compared with the dominant
components such as Ox and In. This may imply that the effort involved in adding the
high-order terms for w is not well rewarded. Hence, it may suggest that the use of
constant w in laminate theory is justified. Also, shown in Figures 2.8 to 2.12 are the

results from the Zigzag Theory. They will be discussed in a later section.

24

 

 

 

0.5 I l I I I .
I. 0 ’3‘
0.4 - 1,6.- - x
9" ' X
of” - x
0.3 ' 0.." . x
C {.1 ' X
0.2 - o 0. n. e x
o I I' ' X
o 51-, x
0.1 ' O on}! X
0 ': x
I.
2 o - o r;- ———-- - -~- xx
0 ‘ o . x
-o.1 - ’ ° x
o g x
,2” O X
4.2 ' {V4 0 x
e .[n . o x
-0.3 ' ."-.'.\ ° 0 x
._~ 0 X
n r O X
.0“ I. " \- ' O X A
‘ O x
-o.s . . . . .
-3 -2.5 -2 4.5 -1 _ -o.5 0

Figure 2.7 - Comparison of In fiom Shear Deformation Theories of various
orders for a [0/90] laminates.

 

0.5

25

 

 

 

 

 

0.5 1- \: fi I I l T I I I
\\~ .
'\
0.4 - d
0.3 - \~ -
. e ‘5,
X) \\
0.2 "’ n» \\ .-
.s.
0.1 - “ “ elasticity solution ‘
“x x x x x” third-order SDT(w°)
2 0 .. “o o o o o” third-order SDT(w°,w1,w7)
‘ ’“ ..... " third-order Z'I'(wo)
“ - . - . - " third-order Z'T(wo,w1,w7)

~O.1 - “- - - - ” third-order ZT(wo,w1,w2), B.C.‘

. - W». .
'0 2 \ no

\
\ O
-o.3 - \ .\ .
-O.4 - -
x
e \"
. \

.0.5 l I A l l L J ' l \ 1

-25 -20 -15 -10 -5 0 5 10 15 20

at

Figure 2.8 - Comparison of 6x from various theories for a [0/90/0] laminate

25

26

 

 

0.5

0.4

0.3

0.2

0.1

 

 

I
'3 “ elasticity solution ‘
a “x x x x x” third-order SDT(Wo) .
g “o o o o o” third-order SDT(wo.W1.W7) -
. O.
I
a
I
I

 

..... ” third-order ZT(wo)
“ ‘ . ‘ . " n mird'ordfl ”(woew19wfl
“- - - - ” third-order Z'T(w°.w1'.W2). B-C- '

 

I
r
l
l
l
I

 
  

 

 

 

Figure 2.9 - Comparison of In from various theories for a [0/90/0] laminate.

27

 

 

 

 
 
 
 
  

 

 

0.5 '6‘: I T X I I I
'Q - X
.0. . x
0.4 - v .x.x ‘1
x
X
0.3 - "' ‘
x e
x .
0.2 - )2: ' - ‘
x ee
§ e
0.1 '- x e '1
Xe
>0
7. 0 - x ‘
K
-x
on- I? ‘
x
X
-02 - .. 3‘. ‘
“ “ elasticity solution . X
0.3 r :x x x x x: third-order SDT(w°) ' i; -
o o o o o thud-order SDT(w°,w‘,w7) x
“ ..... " third-orderZ'Rwo) ”x,
‘0-4 ' “ - . - . - " third-order ZT(w°,wl,w7) ‘0‘ x- q
' - - - - " mini-order Z'1'(w..w1.wz). EC. 0‘0. x; -
-0.5 ‘ ‘ A 4 4 —°B ‘ x
-1.2 -1 ~0.8 -0.6 -0.4 ~02 - 0 0.2 0.4
32

Figure 2.10 - Comparison of Oz from various theories for a [0/90/0] laminate.

28

 

    

 

 

 

 

0.5 I I II
0.4 " «1
0.3 - ..
0.2 - / ' c?“ “ clasricity solution
8: “x x x x x” mid-order SDT(w,)
0.1 _ at “o o o o o. mini-order SDT‘(w..w|.wz)
cx ..... dud-order 21w.)
°‘ “~ . - . - " durd-ordaZT(w,.wl,w7)
z o- - - " mummw,w,,w,),n,c,.
-0.1 - . -
\O
-0.2 - I -
-0.3 +- -
-0.4 - -
O. l 1 L L
- .5 -1 -0.5 0 0.5 1 - 1.5
a

Figure 2.11 - Comparison of a from various theories for a [0/90/0] laminate.

29

 

 

 
  

 

 

 

‘8 x I I I r 1
0.5 ‘1‘ O x
i 8§
0.4- '1 -o x
oox
1.0x
0.3- 11%:
l- o<
02 h g
. - ’1: o:
k g « “ elasticity solution
0.1 - I “x x x x x” third-order SDT(w°)
“. 2 “00000” third-order SDT(Wo»WIvW2)
2 o . 3’ re “ ..... ” third-order ZT(wo)
:1 g “-,.,- ” third-orderZ'lIWo-Wlowr)
0‘ )O “- - - - ” Ulil’d-OIUCTZT(Worwl’w2)’ B'C'
-O.1- -| )0
:1 Q
. -l >0
-0.2 I | :2
.lxg
-0.3- : (£0
. lxo
-o.4- I '§8
. a...
I 0 X0 L i r ' l
'0: .5 -3 " -2.5 3' -1 5 '1 '0'5
W

Figure 2.12 - Comparison of W from various theories fora [0/90/0] laminate.

 

30

2.7 Summary

Based on the above numerical results, the following conclusions can be summarized:

A. HSDT are layer-number independent theories. They model a composite laminate
as a whole piece, disregarding the details of the laminate interfaces.

B. Equation (2.4) gives a generalized theory for HSDT. It covers all the available
HSDT and provides the potential for development in this area.

C. HSDT give reasonable in-plane stress and displacement components. However,
they fail to give zigzag distribution of in-plane displacement through the laminate
thickness.

D. With HrSDT, it is not possible to obtain satisfactory transverse shear stresses
directly from the constitutive equations because the continuity conditions of transverse
shear stresses on the laminate interfaces are not satisfied.

E. Although the distribution of the transverse normal stresses is not in close
agreement with the elasticity solution, the discrepancies seem to be acceptable, given the
fact that they are very small numbers when compared with the in-plane and transverse
shear stresses.

F. The assumption of a constant w in a displacement field seems to be acceptable due
to the small values of Oz and the insignificant difference in 172 between constant w and

second-order w.

CHAPTER 3

LAYERWISE THEORIES

3.1 Introduction

As mentioned in Chapter 2, the major drawback of Shear Deformation Theories arises
from the assumption of continuous functions for in-plane displacement components. This
poor assumption subsequently results in (1) the incapability of presenting zigzag
distribution of in-plane displacement through the laminate thickness and (2) the erroneous
double-valued interlaminar stresses on the laminate interfaces. Apparently, the overall
assumption of a displacement field for the entire composite laminate can always cause this
type of error, regardless of the order of Shear Deformation Theory. In order to remove this
fundamental defect, it is necessary to describe each composite laminate as an assembly of
individual layers. Theories based on this layer-assembly technique are called Layerwise

Theories.

A. Generalized Layerwise Theory (GLT)

In view of the important roles of the individual layers in overall performance of
composite laminates, a layerwise approach was presented by Barbero and Reddy [7]. By
assuming both translational and rotational displacement components for each composite
layer, the displacement of the whole composite laminate is nothing but the assembly of the
individual components. As a consequence, the total number of displacement variables is

dependent on the layer number of the composite laminate. In their study, Barbero and

31

32

Reddy only imposed displacement continuity conditions on the laminate interfaces. Lu
and Liu [15], and Lee and Liu [16] further employed the continuity conditions of
interlaminar shear stresses and interlaminar normal stress for composite layer assembly.
In their study, Lee and Liu [l6] assign two variables, atranslational component and
its derivative, for each of the displacement components at each surface. In order to model
the displacement field of a composite layer, a two-node element is required with each node
representing a surface of the layer. Consequently, as can be seen below, there are four
variables in each displacement component. Hermitian cubic shape functions of are

imposed to assemble these four variables in each composite layer, i.e.

.13“: 1...
k k k k k
u = U2k_2¢1+Tzk-2¢2+U2k—1¢3+T2k—1¢4
k k 'V k k k
V = Vzk—2¢1+Szk—2¢2+V2I—1¢3+Szk—1‘i’4 (3'1)

1: 1: «W135 k k k
W = Wzk-z‘l’l+R2k—2¢2+W2k-1¢3+R2k-1¢4

The above equations result in a total of 12n variables for an n-layer composite laminate.
The superscript k in the above Quations represents the order of layer, while the subscript
represents the order of interface in the thickness direction. In addition, U, V, and W denote
the translational displacement components in x, y, and 2 directions, respectively, while T,
S, and R denote the derivatives of U, V, and W with respect to x, y, and 2, respectively.

In assembling the individual layers together, the continuity conditions of both
displacement and transverse stress are employed at every laminate interface. Accordingly,
the total number of variables becomes 6n + 6. The imposition of the continuity conditions
through the laminate thickness not only largely reduces the total number of degrees-of-
freedom, but also greatly improves the accuracy of displacement and stress predictions.

The results from their Layerwise Theory are believed to be the most accurate among all

[Vt-w

,Varw

33

the laminate theories available in the literature. However, it should be noted that
computational efficiency is sacrificed in exchange for numerical accuracy. When
compared to the Shear Deformation Theories, which have total numbers of degrees-of-
freedom regardless of the number of layers, the computational efficiency in Layerwise
Theories seems to be totally neglected. A laminate theory which accounts for both
numerical accuracy and computational efficiency is highly desired.

The displacement field of Equations (3.1) is based on translational displacement
components and their derivatives, namely angles, and has a very distinct physical
meaning. However, this type of expression is not consistent with those used in the Shear
Deformation Theories. For comparison and unification purposes, the notations used in the
Shear Deformation Theories are utilized to express the Layerwise Theory. Since the
Hermitian shape functions are of third-order, with respect to the thickness direction, and
there are four variables in each displacement component, the following polynomial

equations are equivalent to Equations (3.1):

(3‘ .w (9" )

I I I 2 I' 3
u0(x,y) +u1(x.y)z+u2 (x,y)z +u3 (x,y)z

uk (x, y, 2)
vk (x. y, 2) = VS (x, y) + v: (x, y) z + V; (x. y) 22 + v3k (x, y) 23 (3-2)

k k k 2 k 3
W"(x,y,2) = w0(x,y) +w1(x,y)2+w2 (x,y)z +w3 (x,y)z

Equations (3.2) are of a general form, they are not limited to Equations (3.1) which
impose Hermitian cubic shape functions in layer assembly. Besides, they are based on a
general coordinate system. As a result, they are called the Generalized Layerwise Theory.
It can be seen that Equations (3.2) unifies the Shear Defamation Theories and all the

Layerwise Theories mentioned before. In fact, the Shear Deformation Theories are just

34

simplified cases of the Generalized Layerwise Theory.

B. Zigzag Theories (ZT)

In view of the incapability of presenting the zigzag distributions for in-plane
displacement components through the laminate thickness in the Shear Deformation
Theories, there were some efforts in improving the deficiency. Quite a few theories were
presented and named Zigzag Theories (ZT), owing to the zigzag distributions of in-plane
displacements through the laminate thickness. Di Sciuva [8] and Murakami [9] were
among the most recent to present their own Zigzag Theories. Their approaches were based
on assumed displacement components for individual layers. Similar to Barbero and Reddy
[7], only the displacement continuity conditions on the laminate interfaces were
considered. In their studies, the most important achievements of their theories were the
ability to show the zigzag shape of in-plane displacements in the thickness direction and
the associated improvement of the in-plane stress prediction. However, due to the low
order of their assumed displacement fields, the transverse shear stresses were constant
through the thickness in both cases. The aforementioned post-processing technique was
then required for finding the true values.

Although high-order terms were later added to their theories (Di Sciuva [17, 18],
Toledano and Murakami [19, 20]), their theories still suffered from the deficiency in
predicting the correct transverse shear stresses. Similar studies could also be found in the
articles by Lee, Senthilnathan, Lirn, and Chow [21] and Soldatos [22]. It was not until Cho
and Parmerter [23] presented a Zigzag Theory accounting for both the overall

performance and the interlaminar continuity conditions of transverse shear stresses, that

35

the direct, and correct, calculation for transverse shear stresses from constitutive equations
became possible. However, it should be pointed out that similar to all other Zigzag
Theories, their theory was also of a special case instead of a general form. Hence, it then is
the first objective of this study to present a Generalized Zigzag Theory (GZT) which can
not only give accurate displacement and stress efficiently, but can also be linked to the

Generalized Layerwise Theory.

3.2 Formulation of Generalized Zigzag Theory

In addition to the ability of describing the zigzag distribution of in-plane
displacement through the laminate thickness, the most important characteristic of the
Zigzag Theories is their layer-independency. This is very critical to computational

efficiency and should be considered as a requirement in developing a laminate theory.

A. Fundamental Equations

As mentioned above, all the Zigzag Theories available in the literature are of special
types since they all are based on specially assumed displacement functions. By examining
the Equations (3.2) of the Generalized Layerwise Theory, it can be found that if both
displacement and transverse shear stress are to be continuous across the laminate
interfaces, there should be two, and only two, layer-dependent variables in each in-plane
displacement component. After analyzing all the Zigzag Theories available in the
literature, it is concluded that they all designate the zeroth- and the first-order terms as

layer-dependent variables. It is then only natural to define the following displacement field

as a Generalized Zigzag Theory since it is of a general form instead of a special form:

36

I I I 2 3
u (x,y,z) u0(x,y) +u1(x,y)z+u2(x,y)z +u3(x,y)z

I 2 3
V0 (x, Y) + VI; (x, Y) Z + v2 (x, y) z + v3 (x, y) z (3.3)

vk(x,y1 2)

w" (x, y, z) = w0 (x, y)

The displacement field shown above represents the displacement components of the
W layer located between the interfaces 2 = zk and z = zk +1. Apparently, the
displacement components in both the x and the y directions are layer dependent, while in
the z direction it is layer independent. Details of the coordinate system, layer order, and
interface location are shown in Figure 3.1. In addition, the following linear strain-

displacement relations are employed in this study.

8k = 1“,“ 8" z: ivk' 8" = .a_wk'
’ 8x ’ Y By ’ 2 dz ’
a a a a k a k a k (3.4)
I = I _ I. I =_ k _ . = _ .
710' "‘“ay “‘axv’ 711 82v ”'aywk’ 7x2 "“32 +3xw'

In view of the subsequent close-form verification, only orthotropic laminates made of
cross-ply stacking sequences are considered. The constitutive equations for the k'h layer

are written below,

 

 

 

 

 

 

 

 

 

[ r - ' r ‘ f
01f 913 Qiz 95‘s 0 8f 85
< of 1 = Qiz 952 Q53 0 < e; 1 = [QI] t e; L
0: Qt. 951 Qt. 0 a: ’ a:
. Tfiy 1 _ O 0 0 Q65; L 7;? 1 ~ 7:? , (3.5)

 

 

 

 

 

\\ ‘

 

 

 

 

 

In 1
+ nth
z“
(nu-1)“ ,
2.4
M
II
M2
23 2th
" -_
11

Figure 3.1 -Coordinate system, layer order, and interface locations.

 

 

38

where Q; are stiffness components of the k‘h layer. Details of the definitions can be found

in Vinson and Sierakowski [24].

B. Continuity Conditions

In Equations (3.3), the number of layer-dependent variables in an n-layer composite
laminate is 4n. These variables can be replaced with layer-independent variables through
the enforcement of continuity conditions of displacement and transverse shear stress on

the laminate interfaces.

If the n-layer composite laminate has continuous displacement across the interfaces,

the following continuity conditions should be satisfied:

"I
z=z,_u 2:2,, (36)

I—t I
v I = |_ k=2,3,...,n

k—ll
u ;

2:2,

By substituting Equations (3.3) into Equations (3.6), the following equations can be

obtained:

It k—l _ k-~l k
uo—uo — (ul —u1)zk
(3.7)

I I-1_ k—l I _
vo—vo —(v1 -v1)zk k—2,3,...,n

. . . K K
In addition, if u; = u o and v; = v0 are defined, u o and v0 can be expressed as the sums

k k . .
of ”1 and v1 , rcspectrvely,1.e.:

j = 2 (3.8)

39
The number of layer-dependent variables is then reduced to Zn.
Enforcing the following continuity conditions of transverse shear stresses on laminate

interfaces, the remaining 2n layer-dependent variables can be eliminated:

k-l

 

 

 

 

1 -1k

xz Z=ZI xz 2:“ (3 9)
k—l I '
”[2 =12 k=2,3,4,...,n

y 2:2, y 2:2,,I

The displacement field shown by Equations (3.3) will then become a layer-number
independent theory.

By combining Equations (3.8) and (3.3), utilizing Equations (3.4) and (3.5), and then

substituting the transverse shear stresses into Equations (3.9), the following relations can
be achieved:

I I k—l I—1
Q55“ 11’st “1

cava—Qale-l

2
2flkzku2 + 352ka u3 + kao x

(3.10)
2(9kzkv2 + 39kzlfv3 + GkWO' y
k = 2, 3,4, ..., n

where,

_ I—1 I - I—r I
QI-st ‘st’ G‘)I-'Q44 'Q44-
Given the new definitions u} = u1 and v} = V], Equations (3.10) can be rewritten as:

I _ I I I I

u1 — F1u1+F2u2+F3u3+F4wax
I _.

v1—

It k k k
L1 v1 +sz2 + L3v3 +L4w

(3.11)
0.y

where,

4O

1
Ff: Q_.55 Lf=_Q_4_‘1
955 Q2;
2 k
k-
’02-‘72 (“j-1‘91"? = ‘TQ 22"” ,-_1
.. 2 2
F3--—3k_Q2(Qf-lflf)zf L§=Q3IX(Gj-lej)zj
Q55}- 2 Q44j= 2
F§=Ff—1 L§=Lf—1
k=2,3,4,...,n

It should be noted that F11=1, F5=F§=F3=O, L11=1, and

C. Boundary Conditions
The independent variables in Equations (3.11) can be further reduced by applying
shear-free conditions on both surfaces of the laminate because of their popularity in

laminate plate analysis, i.e.:

1 1
15x2(21) = o, efizum) =0, tyzul) =0, 1532”,...) =0 (3.12)
Utilizing Equations (3.4), (3.5), (3.11), and (3.12), the following equations can be

achieved:

It 2
an+22n+1 F3 +3zn+1[“2] = [47: _(pfi+1)[ul ]

221 32: “3 —l —l Wax
L2"+22n+1 L3 "+32 2:,,+1[2=_L]'1:1:.(L—+1)[v1 J (3.13)
221 32? v3 WO 0)’

 

 

41

Solutions of U2, u3. v2, and v3 can be expressed in the following manner:

= A1“1+A2Wo.x “3 = Blul +32%... (314)

“2
Dlv1+D2way

v2=C1v1+C2w0J v3 =

where the coefficients A i, Bi, C i, and Di can be easily identified from Equations

(3.13). However, it should be recognized that in order to have unique solutions to

Equations (3.13), it requires that,

(a) 21 $0, and
2F; + 6h2 2L; + 6h2 (3'15)
(b)Zl¢—n—-— or 21¢ n .
315‘2 — 6h 3L2 — 6h
Substituting Equations (3.14) into Equations (3.11) and (3.8), it yields:
uf = Rfu1+R§wM u]; = u0+Sful +S§wm (316)
vf = Oi‘v1 + 0§W0,y v5 = v0 +va1+P§woJ .

where,
R’; = F"; +A2F’; + 32F;

k

[If = F’,‘+A1F§+31F’3‘
0: = L2+C2Lg+DzL3

of = L: + CIL’; +1)ng

(=2

I I
5f = 2004-1302. 55‘ = 2 (RP-RIM
(=2 (=2
I I
Pi = 2 (0{'1-0{)z, P5 = 2 (Oi—1‘05”!
1:2

By substituting Equations (3.16) into Equations (3.3), a displacement field which is free of

layer-dependent variables but dependent on layer properties can be concluded:

42

u" = u0+¢f(z)ul+d>§(z)wo‘x
v" = v0+‘l’f (z) v1+‘P§(z) wo'y (3.17)
Wk = W0
where,
ofu) = Sf+sz+Alzz+Blz3 d>§(z) = s; +R§2+A222+Bzz3

2 3
I I
P2 + 022 + C22 + Dzz

wf (z) = Pf + ofz + C122 +0123 w; (2)

It should be pointed out that Rf, Sf, 0:, Pt, A i, B i, C ‘- and D i are only related to
material properties and layer thicknesses. The total number of independent variables then

becomes five.

D. Variational Formulation
In order to verify the Generalized Zigzag Theory, the cylindrical bending problem
mentioned in Chapter 2 is to be investigated. If normal forces are exerted on the surfaces
of the composite laminate, the governing equations can be obtained from the principle of

virtual displacement:

 

 

 

 

 

 

 

 

VT 7 T N
r f i of 5le N
n zI+r k k T
6 Se k k
I 2 I t Y H Y I +{ Ty? } { 87)? } dz—qi (5W1: ) dxdy =0
.0 k = l 2‘ 02k 88: 1:2 8 :2
‘K K ‘ tfiy J L 8710 A I /

 

(3.18)

For simplicity, the strains and displacements can be expressed in matrix forms, i.e.:

71:, = [~35] {X}. 71:, = [Nfz]{X}1 7;, = [~52] {X} (3.19)

where, {X} = {“0 v0 141 v1 w0}T

 

 

 

 

 

 

, - - 2
[v.51 = _§—, 58; @395,- wraa—x (¢5+W§)§:Ty]
111:4 =[o 0 %<1>f o (a‘éogngg
[N;Z]=[o o o gulf (:wgnaa]
[111; ]=[o o o o 1]
[Nw]=[o o o o 1]
Let [[Ng
N" k
[MEI]: [y] ’ [N91]: [NH] (3'20)
[Na [~11

 

 

 

 

 

44

the variational equation, Equation (3.19), can be rewritten as follows:

12 21+]

(12.1111. 2. ; (wreath 1+ [~rzo1,.[~a.)dz 111.1111 =
I{{t‘m} T( q+[ N; Ddxdy + I]; (1' [N;,] )dxdy

(3.21)

E. Close-form Solution

As stated in Section 2.2, it is possible to have a close-form solution to the
aforementioned cylindrical bending problem. If the composite laminate is of an infinitely
long strip and simply supported along the two long edges, the problem can be reduced to a
plane-strain type. In order to satisfy both the sinusoidal loading, q‘(x, g) = qgsir'rlit x,
on the top surface and the simply-supported boundary conditions, the following close-

form solution can be assumed:

uo = Uocospx u1 = Ulcospx wo = Wosinpx (3.22)

As a consequence, {X} can be simplified and {X} can be defined as follows:

{X} = {uo u1 w0}T and {X} = {U0 U1 wof (3.23)

Substituting Equations (3.22) into (3. 19), it yields:

N" = Nit sinpx = —p —p<I>" -p2<l>" sinpx
x , 1 2
[Nil = [mg] sinpx = [0 0 0] sinpx
(3.24)

1%] = [now [.1 II: ,(%.,1.)]...,..

 

W

[1.1:] = [NJ ]sinpx= [o o 1]sinpx

45

And the variational equation, Equation (3.21), then becomes:

zk+l

E. 1191191 ’191 911911991 9191191 1991191

2‘ (3.25)
+ Q; [Ni‘JTNL] )d2 {X} = 43[ ”J ]T

Equation (3.25) is a set of linear equations for three variables U 0, U 1 , and W 0 which can

be determined by solving the equations simultaneously.

3.3 Generalized Zigzag Theory with Higher-order Terms

HrSDT can be combined with the Generalized Zigzag Theory to improve the
accuracy of global response. This is especially efficient in composite laminates with lower
numbers of layers such as two-layer and three-layer laminates. In this investigation, a

Generalized Zigzag Theory of fifth-order is of interest, i.e.:

k k k 2 3 4 5
u (x1y12) = u0(x1y)+u1(x1y)2+u2(x.y)z +u3(x,y)z +u4(x.y)z +u5(x.y)z

k k k 2 3 4 5
v (x1y12) v0(x,y) +v1(x1y)2+v2(x1y)z +v3(x.y)z +v4(x1y)z +v5(x.y)z

w". (x, y, z) wo (x, y)

(3.26)

Mth a process similar to that shown in Section 3.2, the layer-dependent theory can be
converted to the following layer-independent theory:
u" = no + (bf (2) ul +<I>§ (2) a2 + (b; (2) u3 +<I>fi (z) wax

v" = vo+‘l’f (2) v1 +‘I’g (z) v2+‘l’§ (z) v3+‘Pfi (z) way (3.27)

k-
W—Wo

46

Furthermore, the matrices regarding the displacement components and their derivatives

can be found:

T
{X} = {“0 v0 141 v1 112 v2 u3 v3 W0}

 

 

 

2

N" = .9. La- 11 k.3_ I a
[I] .31 o (”131: 0 $231: 0 (D38): 0 ¢4ax2d
[9+]= .1 i .1 .191 .1 +191 .1 1.92. 19.63.
y _ 8y ray 23y 33y 4 Y.

 

 

- 1?. By 53; 13y ax 8y ax 8y ax axay
- k- = d I d I d I (d I a
9in [0 0 21-2“)1 0 1‘”2 0 #3 O #Nlax

d d d d 3
NR]: _ k _ k _ k (_ k )—
[11 [o o o dzwl o (1sz o dzsg d211,“ 8y]
[N;]=[000000001]
[N;]=[oooooooor]

(3.28)
The close-form solution can be obtained by utilizing Equations (3.28) and Equation

(3.25).

47

3.4 Numerical Results and Discussion

A. [0/90/0] Laminate

Results of the normalized stress and displacement components of the [0/90/0]
laminate are shown from Figures 3.2 to 3.6. It is found that the results are improved as the
order of the theory increases. As can be seen from the figures, the results from the third-
order and the fourth-order are identical. Similarly, those from the fifth-order are the same
as those from the sixth-order (see In in Figure 3.3). These results may indicate the
importance of the odd-order terms in this symmetric laminate.

As can be seen from Figure 3.2, the fifth-order and above seem to give exact in-plane
stress Ox for the case under examination. However, the exact values of 1” are more
difficult to obtain. Although the fifth-order and above have significant effect on the
transverse shear stress distribution, they fail to show the distinct kinks on the laminate
interfaces as given in the exact solution. It is believed that the deficiency cannot be
improved simply by increasing the order of the theory.

Figure 3.4 shows the transverse normal stress, 62. Although no continuity condition
is imposed on the laminate interfaces, the discrepancy at the interfaces seems to be very
small, especially in the higher-order theories. In addition, because no boundary condition
of transverse normal stress is enforced, the result on the top surface is not equal to
negative one and on the bottom surface is not equal to zero. Although the results in Figure
3.4 seem to be different from the exact solutions, it should be recognized that they are

small values when compared to the dominant component, Ox.

11111
cxceL
highs
comp

theth

48

Another excellent result takes place in the in-plane displacement u. The zigzag
distribution can be perfectly represented by the theories of the third-order and above. This
excellent agreement in conjunction with the excellent results in Bx seems to imply that the
higher-order (fifth and above) theories can give excellent in-plane displacement and stress
components. The prediction of transverse normal displacement, W, is very good for both

the third- and the fourth-orders and excellent for the fifth-order and above.

 

49

 

 

 

 

0.5—*— 1 xi 1 I r r r r
o x
o.4- ° .
.1
0.3- X 0 .
X 6
X 6
0.2- “,1 ° 0 1
“ “ elasticity solution X i o
01- “xxxxx” second-orderZT ’
° “ooooo” third-orderZT ‘
“ ..... " forth-orderZT _
50- “-.-.- ”fifth-orderZ'l‘ -- ‘
. “----”sixth-orderZT : -
-O.1- .. .. sevenm-orderl'l‘ I ..
o i x
~O.2- o x .
O X
0 X
.003. a 0 xx CI
“
-O.4- x. L -
xx 9
'0. l l l l l l r: l
- 5 -20 -15 ~10 -5 O 5 10 15 20
5
I

Figure 3.2 - Comparison of 3.: from Zigzag Theories of various orders

for a [0/90/0] laminate.

0.5

50

 

0.4

0.3

0.2

0.1

-O.1

-O.2

-O.3

-o.4

1

 

 

“ “ elasticity solution
“x x x x x” second-order 2T
“o 000 o” third-orderZT

“ ..... " forth-orderZT
“-.-.- " fifth-orderZT‘

“- - . . " sixth-order Z‘l‘

“ .. .." seventh-order 2T

0
9
9
0 2
05
@-
O
O
G

 

-O.

 

 

 

O XXXXXXXXXXXXXXXXXXXXXXXXXXXXXxxxXxX

Figure 3.3 - Comparison of ’ngz from Zigzag Theories of various orders

for a [0/90/0] laminate.

0.5

51

 

  
   
    

 

 

 

0.5% I ‘ I X ‘ t
R x
N x
O 4 ' xx
x
x
0.3 - x
x
x
0.2 - xx . . .
‘9 x “ “ elasucrty solutron
0 xx “xxxxx” second-orderZT
0.1- x “00000” third-orderZT
x " ..... " forth-orderZT
2 °'- x -.,.,. ”fifth-orderZT
' - - - " sixth-orderl‘l‘
xx “.. .. .. .." seventh-order Z'l'
-O.1 - x .
x
x
-O.2 - x
- x
x
-O.3 - xx
x
XX
-0.4 - x \
X ‘0
x
°O.5 l x L ' 1 ‘Q 1
.1 °0.8 0.6 -O.4 .0 2 o 02
5
2

Figure 3.4 - Comparison of Oz from Zigzag Theories of various orders
for a [0/90/0] laminate.

 

52

 

    

 

 

 

 

0 5 . x %
0.4-
0.3“-
0.2-
“ “ elasticity solution
“xxxxx” second-orderZ'I'
0.1' “00000” thitd-orderZT
“ ..... " forth-orderZT'
2 O- “..-,. ”fifdr-orderZT
“. - - - " sixth-orderZT
0.1L .. .. .. .. seventh-order“
G
-O.2- Q
-O.3-
-O.4-
-O. as? x 1 1
- .5 -1 90.5 0.5 1

RIO"

Figure 3.5 - Comparison of a from Zigzag Theories of various orders
for a [0/90/0] laminate.

53

 

 

 

 

 

 

0.5 If r r 1— r X
g x
x
0 4 - 9 x
'. 0 x
:1 1
0.3 - 0 x
a 1
o x
0.2 L- .. ,. . . .
g elastrcrty solution :3
o “x x x x x” second-orderZT X
0-1" 2 “00000" third-orderZ'l‘ i:
g “....."forth'-orderZT >3:
- “-.-.- ”fifth-orderZT
- o
z 0 e “- - - - " sixth-orderZT‘ 3:
g .. .. .." seventh-orderZT x
-o.1- x
o x
3 1:
~02 - 9 x
g x
x
o x
-O.3 ~ (9 x
g x
x
-o.4 - 9 X
0 X
o x
.0 l e l l l 15
1 .5 -3 -2.5 -2 -1.5 -1 -0.5

Figure 3.6 - Comparison of ‘7’ from Zigzag Theories of various orders

for a [0/90/0] laminate.

 

54

B. [0/90] Laminate

The significance of the results from the investigation of a [0/90] laminate is
essentially the same as that mentioned in the study of the [0/90/0] laminate. That is, the
theories of the fourth-order and above give excellent results for Ox , 12 , and W. The results

for Oz are fair since they are small when compared to Ox. Only 13 presented in Figure

x2 ’
3.7, shows the distinct difference of the unsymmetrical [0/90] laminate from the
symmetric [0/90/0] laminate. It is found that even-order and odd-order terms play equally
important roles in the [0/90] laminate while only odd-order terms are critical to the [0/90/

0] laminate.

C. Comparisons between GZT and HSDT

In order to further discuss the accuracy of the Generalized Zigzag Theory (GZT),
comparisons of the five different theories shown below are made:

(a) a third-order HSDT with constant w,

(b) a third-order HSDT with second-order w,

(c) a third-order GZT with constant w,

(d) a third-order GZT with second-order w, and

(e) a third-order GZT with second-order w plus imposed transverse normal stress
boundary conditions.

The results for normalized in-plane stress, transverse shear stress, transverse normal
stress, in-plane displacement, and transverse normal displacement are shown in Figures
2.8, 2.9, 2.10, 2.11 and 2.12, respectively.

As shown in Figure 2.8, the cases of HSDT give continuous in-plane stress

55

distribution in the thickness direction owing to the assumed continuous displacement field
through the laminate thickness. On the contrary, the cases of GZT give correct
discontinuous distribution. The superiority of GZT to HSDT is obvious.

None of the third-order theories give good results for the transverse shear stress
depicted in Figure 2.9. However, the results from GZT are continuous while those from
HSDT are not. As shown in Figure 3.3, it takes an order higher than three to show the two
bulging distributions in the 0° layers.

In Figure 2.10, the theories with second-order w give much better transverse normal
stresses than those with a constant w assumption. However, only the case with imposed
transverse normal stress boundary conditions gives the correct results at the surfaces
although the efforts involved in enforcing the boundary conditions should also be
recognized. In addition, it should be pointed out that in the cases of constant w, although
the transverse normal strains are zero, the transverse normal stresses are not zero due to
Poisson’s effect

Similar to the in-plane stresses, the in-plane displacements from GZT given in Figure
2.11 show excellent agreement with Pagano’s solutions. The “zigzag” distribution is very
distinct. This is believed to be the most distinguished contribution of the Zigzag Theories.
Generally speaking, the results from the GZT are better than those from HSDT in the

transverse normal displacement, as can be seen in Figure 2.12.

D. Multi-layered Laminates
As mentioned before, only cross—ply laminates are examined in this study due to the

availability of close-form solutions. In addition to the widely investigated [0/90/0] and

56

[0/90] laminates, it is also interesting to look into the applications of the fifth-order
Generalized Zigzag Theory for composite laminates with multiple 00-900 alternation. In
this study, layer numbers of 2, 6, l4, and 30 for unsymmetrical laminates and 3, 7, 15, and
31 for symmetrical laminates are examined. Details of the stacking sequence for the
composite laminates are shown below.

symmetric 3-layer: [0/90/0]

symmetric 7 -layer: [0/90/0/90/0/90/0]

symmetric lS-layer: [[0/90/]7/0]

symmetric 3l-layer: [[0/90]15/0]

unsymmetrical 2-layer: [0/90]

unsymmetrical 6-layer: [0/90/0/90/0/90]

unsymmetrical l4-1ayer: [[0/90]7]

unsymmetrical 30-layer: [[0/90]15]
Figures 3.8 and 3.9 are the results of u and in for even layer numbers. The elasticity
solutions are also shown along with the numerical predictions for comparison. Apparently,
the fifth-order Generalized Zigzag Theory can be used for composite laminates with all
ply ranges of 00-900 alternation as shown in Figure 3.8. However, the predictions of
transverse shear stresses are not as good as those of in-plane displacement. It can be
concluded from Figure 3.9 that the higher the layer number, the closer the agreement
between the elasticity solution and the Generalized Zigzag Theory prediction. It is
believed that this is due to the assumption that the Generalized Zigzag Theory initially has
layer-dependent terms for the zeroth-order and the first-order. In other words, the second-

order term, which is responsible for curvature of the composite laminate, is of global

57

sense. As the layer number increases, the composite laminate of 00-900 alternation is

becoming more homogeneous through the laminate.

3.5 Summary
«a

1. Both HSDT and GZT are layer-number independent theories. However, the former
is based on an averaging sense in assembling the composite layers through the laminate
thickness, while the latter considers the layer properties individually during the assembly
stage.

2. This study gives a generalized theory for composite laminate analysis. It unifies all
the available HSDT and ZT, and provides an efficient technique to examine various
theories. 3

3. Zigzag Theories are superior to High-order Shear Deformation Theories in giving
excellent in-plane stress and displacement. The zigzag shape of u is accurately predicted.
This is where the name “Zigzag” comes from and is a major contribution of the Zigzag
Theories to the composite laminate analysis.

4. With the Generalized Zigzag Theory of the fifth-order and above, it is possible to
obtain satisfactory transverse shear stresses directly from constitutive equations. Although
the distribution of the transverse normal stresses is not close, the discrepancy seems to be
acceptable, especially when considering that they are calculated directly from the
constitutive equations.

5. The fifth-order Generalized Zigzag Theory can be applied to laminates with a wide

range of 00-900 alternation. It seems that the higher the layer number, the better the

prediction as compared to the elasticity solution.

58

 

    
 
 
 
  
   
 

 

 

 

0.5 1 . .-
“ “ elasticity solution . g
04- “xxxxx” second-orderZT x
' “o o o o o" third-order Z'l‘ ' °§
“ ..... . fnnh-orderZT . x
O.3-“-.-.- "fifth-ordchT é
“- - - - " sixth-orderZT )K
0.2 __ .. .. .. seventh-orderZT :8 -

31‘
0.1 - g

2 0 - g .
-o.1 - 3"
E
-0.2 - z
-0.3 - 1 i
g
-O.4 - x
-0.5 i
-3 0

Figure 3.7 - Comparison of 1:“ from Generalized Zigzag Theory of various orders for
bath [0/90/0] and [0/90] laminates.

 

59

0.5 T I I .' I . I I

 

0.4 1-

0.3 ~ I

0.2 - '1

-0.1

-0.2 -'

.o.4 I

 

-O.5
_-2

 

81

Figure 3.8 - Comparison of a from the fifth-order Generalized Zigzag Theory and
elasticity solutions for 2+, 6-, l4-. and 30-layer unsymmetrical laminates.

 

60

 

0.5

0.4 -

0.3

0.2

0.1

,5

I
.0
U

l

 

 

 

 

Figure 3.9 - Comparison of tn from the fifth-order Generalized Zigzag Theory and
elasticity solutions for 2-, 6-, 14», and 30-layer unsymmetrical laminates.

CHAPTER 4

QUASI-LAYERWISE THEORIES

4.1 Introduction

Compared to the Generalized Layerwise Theory (GLT), Equations (3.2), the
Generalized Zigzag Theory (GZT) of Equations (3.3) is just a simplified case of GLT,
since the layer-dependent variables are designated to the zeroth-order and the first-order
terms only. Other possibilities of simplified cases with two layer-dependent variables are
those designated to the zeroth- and the second-order terms, the zeroth- and the third-order
terms, the first— and the second-order terms, the first— and the third-order terms, and the
second- and the third-order terms. Since initially each of these six theories has two layer-
dependent variables, they are of the form of the Layerwise Theories. With the use of
continuity conditions, they eventually all become layer-number independent theories.
They are thus called Quasi-layerwise Theories. For convenience, these six theories will be
named the 0-1, 0-2, 0-3, 1-2, 1-3, and 2-3 Quasi-layerwise Theories, respectively. Since
the 0-1 Quasi-layerwise Theory was previously introduced as the Generalized Zigzag

Theory in Equations (3.3), the remaining five theories are expressed below:

A. The 0-2 Quasi-layerwise Theory

I I I 2 3

u (x,y,z) = uo(x1y) +u1(x,y)z+u2(x,y)z +u3(x,y)z

vk (x, y, z) = v3 (x, y) -1-vl (x,y)z+v: (x, y) 22-1-v3 (x, y) 23 (4-1)
wk(x,y,z) = Wo(x1}’)

61

62

B. The 0-3 Quasi-layerwise Theory

1: k 2 k 3
u (x1y12) = u0(x.y) +u1(x1y)2+u2(x1y)z +u3(x1y)z

Vk (x, y, Z) VS (x, Y) + V1(X1 y) z + v2 (x, y) z2 + v; (x, y) 23 (4+2)

w" (x, y, 2) W0 (x, y)

C. The 1-2 Quasi-layerwise Theory

1: k k 2 3
u (x1y12) = u0(x1y) +u,(x1y)2+u2(x1y)z +u3(x1y)z
Vk (x, 1’1 2) = V0 (11 Y) + V: (I, y) z + v: (x, y) z2 + v3 (x, y) 23 (4+3)
99" (w. z) = WO (9, y)

D. The 1-3 Quasi-layerwise Theory

I I 2 I 3
u (x,y,z) u0(x,y)+u1(x,y)z+u2(x,y)z +u3(x,y)z

vk (x, y, 2) v0 (x, y) + v: (x, y) z + v2 (x, y) 22 + V; (x, y) 23 (4.4)

W"(x1y12) = we (x1 y)

E. The 2-3 Theory

I I 2 I 3

u (mm) = uo(x1y) +u1(x.y)2+u2(x1y)z +u3(x1y)z

vk (x, y, z) = v0 (x, y) +vl (x, y)z+v; (x, y) 224w; (x, y) 23 (4+5)
Wk(x,y,2) = W0(xay)

63

In a third-order displacement based theory, each of the four components has a
distinct physical meaning. The zeroth-order term represents the translational component,
while the first-order term represents the rotational component. The second-order term can
be viewed as curvature, while the third-order term can be viewed as the third-order
derivative of the displacement, or high-order rotation. Accordingly, different order terms
play different roles in laminate performance. The selection of different order terms can
strongly affect the foundation of a laminate theory. The objective of this chapter is to
investigate the feasibility of the individual theories for laminated plate analysis. Similar to
Section 3.3, in the present study, a constant w and a fifth-order u‘ will be employed for the
consideration of both computational efficiency and numerical accuracy. However, the
details of formulation will be presented below by taking the 1-3 Quasi-layerwise Theory
as an example, since there is a major difference between the 0-1 Quasi-layerwise Theory
and the remaining five theories. In fact, a special numerical technique is required in
formulating the remaining five Quasi-layerwise Theories.

By applying the continuity conditions, the layer—dependent variables can be reduced
to layer-independent variables. However, it will also lead to a set of constraint equations
which are generally coupled together. Consequently, implicit, instead of explicit,
expressions are obtained, and a huge matrix will be generated in the solution scheme. The
summation expression used in Chapter 3 is no longer feasible. In order to overcome this
drawback, a recursive technique will be necessary. By virtue of this technique, the coupled

equations can be resolved to be the desired explicit formulations.

64

4.2 Formulation of the 1-3 Quasi-layerwise Theory

Among the five theories given above from Equations (4.1) to (4.5), at least one layer-
dependent term is'higher than first-order. This implies that the curvature (the second
derivative and above) and the high-order rotation may be associated with the layer
properties. The analysis of these five theories may give an insight into the roles of
individual terms in laminate analysis. However, the high-order layer-dependent terms can
also create trouble in computation. For example, the numerical singularity may take place
if the global coordinate systemcoincides with any of the layer interfaces. In laminate
analysis, it is a general practice to choose the midplane of the laminate as the x-y plane. In
order to avoid the potential numerical singularity, it is proposed to move the coordinate
system out of the laminate before the numerical process, and move it back to the midplane
position after the process. As mentioned before, the 1-3 Quasi-layerwise Theory will be

taken as an example for numerical formulation:

k k 2 k 3
It Own) = u0(x,y) +u1(x,y)2+u2(x.y)z +u3(x1y)z

I 2 I 3
V0 (x, y) + v1 (x, y) z + v2 (x, y) z + v3 (x, y) z (4+6)

vk (x1y1 2)

wk (x,y, Z) = W0 (XIY)

A. Continuity Conditions

If all the interfaces of an n-layer composite laminate are perfectly bonded, the

following interlaminar displacement continuity conditions should be satisfied:

 

By 9

cone]

lithe

be rev

Them

10mm

Blsub

and (3.:

65

k—ll k
u -

k-ll
V

 

z=zk 2:2,,

k (1%
=vl k=Z3pmn

2:2, 2:21.

By substituting Equations (4.6) into (4.7), the two layer-dependent variables can be

correlated as follows:

k k-l

k k-l 3
It k—l k k

_ l) 3 (4.8)
zk 0
fly-11"]; '

If the coordinate zk is properly selected to avoid being equal to zero, Equations (4.8) can

 

berewrittenas: , «if?
k-l k (QM? .J n t-Jk) - .
I_ k—l “1 —ul ' 1l
2
I
4.9
I—1 I ( )
I_ I—1 V1 —v1
2!:

The interlaminar shear stress continuity conditions have to always be true for laminated

composite plates with any bonding conditions, thus:

 

 

 

k—l _ I
sz _sz

z=zk z—z,‘
k l k (4.10)
z'| =9 I=234,,n
yz z=zk ”2:2,,

By substituting Equations (4.6) into (4.9) and then (4.10), and utilizing Equations (3.4)

and (3.5), the following relations can be obtained:

 

where.

F1

Obvio

00th

B- Bo.

3130 he

11mm 1

66

 

 

I I—1 3 2 k—l 1
“i = 5(3‘0‘I)“1 +§(1'°‘I)ZI“3 ‘BIZI“2‘§BIW0.1I
(4.11)
1 k-l 3 2 k—l 1
Vi = §(3'CI)V1 +§(1 ‘CI)ZI"3 ‘nkzkvz’inkwo.y
where,
k-l k-l I
_ Q55 B _ Q55 ’st
I I k ‘ I
955 Q55
k—l I—1 I
C Q44 _ Q44 ’Q44
k - k k - I
Q44 Q44

For convenience, rewrite Equations (4.11) in another form:

1
k ._ It k k

1
u§ = Gj‘ul + 05123 + Gl‘wM
(4.12)

vf = 1.1111 +L5v; +1.11%.y
v; = val 1M9»;1114111110.y
Obviously, G: = G; = 0, G; = 1, M: =M; =0, and M; = l. The remaining

coefficients are to be determined later.

B. Boundary Conditions

. In this study, it is assumed that the bottom surface of the n-layer composite laminate

also has free shear tractions, i.e.:

1

I (z ) = 0
:2 1 (4.13)

Iyz(zl) = 0

It then results in,
u1+22 u +322u1+w - 0
2. 3 , ’
l l 1 0 x (414)

II
C

1 2 3 21
vl+ 21v2+ Z1V3+Wo,y

By utilizing Equations (4.14), the following coefficients

obtained:

—221 F2

1
= -—221 L2

67

of Equations (4.12) can be

—3zf -1

(4.15)

2
-321 = —1

It is of great advantage to express the higher-order coefficients by the following recursive

equations:

 

 

 

1 k-l 3 2 I— 1
Ff=-2-(3 or,‘)F1 +2(1—zork)sz —Bkzk
l k—l 3 2 011— 1
F§=§(3‘ k)F2 +§(1’°‘ZI)IG
1 k-l 3 2 I— 1 1
F; = 5(3—ak)F3 +§(1- (1k)Zk ZG3 -§.Bk
I 11.1 k—l (I)
01:01 +(Fl -F:) —2
\ZI)
I k—l I—1 1‘
02:02 (F2 ’ng'i
izk)
I I—1 k-l l
03:03 +(F3 —F§)[—2]
2]:
1 +3( 2 I- 1
Lf = 15(3-gk)L11 +2( ’29)]; 21M -lezk
I— 1 3( 2 1
2—(3 1;.)L +2(1—1;,.)zM*
1 3 2 1
=-2(3 C1)L§' +2(1- :1): M; -2n
I -1 I 1 1)
M1: MI; +(L1 —L'1‘-)[2
ZI)
-1 I—1 I 1h
M2=MI2 +(L2 ”L2)[‘i
2k)
—1 k—l I 1
.1; =9: +(9. 91(1)
ZI)
where, k — 2,3 4

(4.16)

 

 

 

 

it. t

equan

By'so

“hen

Subsh

lhflds

Where

[fihzn

68

The top surface of the composite laminate is also assumed to be free of shear traction,

i.e. 1:2 (2" +1) = 0 and 1320"”) = 0. It is necessary to satisfy the following

 

 

 

 

(4.17)

(4.18)

equations:
” 2 3 2 " - 0 " 2 3 2 " -o
. . 1 1
By solvrng Equatrons (4.17), 143 and 123 can be expressed as:
1 - A B 1 - A B
“3 — 1“2+ lwo.x V3 — 2V2+ 2Wo.y
where,
I"+3z2 G"+2z 11‘"+3z2 G”+1
A __ l n+1 l n+1 B _ 3 n+1 3
1 — " 1 ‘ ‘
A1 A1
L"+3z2 M"+22 L"+3z2 M"+l
l n+1 1 n+1 3 n+1 3
2 2
2 n

with A1 = F2+3z G2 A2 = L'2'+32:+1M;

n+1

Substituting Equations (4.18) into Equations (4.12) to further simplify the expressions, it

yields:
“1 = R1u2+R2wax “3 = Sfu2+sgwax
vf = 0{‘V2+02‘w0‘y v; = va2+P§w0J
wher€+ Rf = Ff+A1F§ sf = 011191110;
R2 = F§+BIF5 $2 = GB’f-t-BIG;
of = Lf +1421; Pf = Mf+A2M§
k.. k k k _
02 — L3 +B2L2 P2 — Mg +B2M§

(4. l 9)

Utilizing Equations (4.19) for Equations (4.6), a displacement field which is free of layer-

dept

whc

cquz

C1036

“We.

69

dependent variables, but dependent on layer properties can be concluded:

I k k '
u (x,y, Z) “0(x1)’) +¢1(Z)u2+¢2(z)wo,x

vk(x1y12) v0(x1y) + ‘1",‘(2)v2 + ‘1’: (2) WO‘ y (420)

Wk(x1)’1 Z) = W0 (113’)
where,

I

<1>f(z) = sz + z2 + 5:23 <1>’2(z) R22 + 5,223
1:

‘1’]:(2) = 0:2 + z2 + sz3 ‘1’];(2) 022 + Pfiz3

For numerical analysis, Equations (4.20) can be substituted into the variational

equation, Equation (3.21), with the matrices defined as follows:

T
{X} = {“0 v0 142 v2 W0}

l_'—I
Z
1111
ll
Q) 1
HIQJ
o
'9'
a?"
a1”
o
'9'
Ma-
cu
ale;
N N

 

 

d d
NI] = I ( I
[22 10 0 Ed)! 0 #24-

(4.21)
Close-form solutions based on the 1-3 Theory can be obtained by a similar manner as

expressed in Section 3.2.

stron'
Theo
funet
order
plane
deper
differ
result
direct
comp
thiekr
are In
Show;
reman
I The
Coordi:
2-3 011
one am

3 Quas

70

4.3 Numerical Results and Discussion

As mentioned before, the similar plane-strain problem is examined. In applying the
Quasi-layerwise Theories to the composite analysis, it is noted that the formulation is
strongly dependent- on the thickness coordinate, except for the 0-1 Quasi-layerwise

Theory, i.e. the Generalized Zigzag Theory. As expressed in Equations (4.16), G: are

functions of l2. Apparently, there will be numerical singularity if any zk equals zero. In
2 \_ , _

I
order to avoid the singularity problem, the thickness coordinate is shifted from the mid-

plane of the composite laminate. However, it is found that the numerical results are
dependent on the location of the thickness coordinate. In other words, with different 22 ,
different displacement and stress values will be obtained. Figures 4.1 to 4.5 show the
results of the 1-3 Quasi-layerwise Theory based on various coordinate shifts (along the z
direction). The\ shift ratio is defined as the shift of the z-coordinate to the thickness of the
composite laminate. As can be seen, the results are unreasonably dependent on the
thiCkness coordinate. Accordingly, only the results from the 0-1 Quasi-layerwise Theory
are true values since they are not dependent on the coordinate system. These results are

shown in Figures 4.6 to 4.10 for Ox, ‘1

xz’

Oz , u , and W , with the results from the
remaining five Quasi-layerwise Theories, which are adjusted to match with those of the 0-
1 Theory as closely as possible through the manipulation of shifting the thickness
coordinates. The shifting ratios are 1.2, 3.0, 1.6, 2.0, and 2.5, for the 0-2, 0-3, 1-2, 1-3, and
2-3 Quasi-layerwise Theories, respectively. Although all six theories can be very close to

one another, th fatal deficiency of coordinate dependence of the 0- 2, 0-3,1-,2 1-,3 and 2-

3 Quasi-layerwise Theories rs recognized.

 

”H

 

~12,

43

~04

~05-

~25

FINN

71

 

0.4

0.3

0.2

0.1

 

I

-0.2

-O.4

 

 

 

-O.5
-25

 

Figure 4.1 - Variation of 02 from the 1-3 Quasi-layerwise Theory due to various shift
ratios in a [0/90/0] laminate.

(J

C)

-0.1

~01

F1gun 4

72

 

0.5

0.4 r-

0.3 '-

 

0.2 -

0.1-

 

-0.1-

-0.2

-O.3

 

 

 

-O.S
-2.5

 

Figure 4.2 - Variation of 122 from the 1-3 Quasi-layerwise Theory due to various shifts
ratios in a [0/90/01 laminate. ‘

73

 

0.5

0.4 -

 

0.3 ~

0.2

0.1 -

Na
'0
I

-0.1

-0.2

-0.3

 

 

 

-0.5

 

Figure 4.3 - Variation of 62 from the 1-3 Quasi-layerwise Theory due to various shifts
ratios in a [0/90/0] laminate.

74

 

0.5

 

 

Figure 4.4 - Variation of a from the 1-3 Quasi-layerwise Theory due to various shifts

 

ratios in a [0/90/0] laminate.

 

 

75

 

0.5

0.4 '-

0.3

0.2

0.1

-O.2

-0.3

 

,._._._._._._._.-......._.__._......._._._._.2-.-.-.

 

 

-0.5
-3.5

 

Figure 4.5 - Variation of W from the 1-3 Quasi-layerwise Theory due to various shifts

ratios in a [0/90/0] laminate.

0.5

76

 

0.4 -

0.3

0.2

0.1 '

-O.1

U

-O.2

0.3

-O.4

1

 

-o.

 

 

-5

Figure 4.6 - Results of 62 for a [0/90/0] laminate from various Quasi-layerwise

 

Theories obtained by adjusting the thickness coordinate.

 

77

 

 

 

 

0.5 I I I T
0.4 " . - .1
0.3 - .
0.2 - .1
‘1'
0.1 " I ..
3
2 0 ' (f 4
-O.1 '1. ' a
1
-O.2 ‘- ( 4
-O.3 ' -
-O.4 - \ \ 2 «
-O. ‘ 1 4 L .
- .5 -2 -1.5 -1 -O.5 O O 5

Figure 4.7 - Results of 1:22 for a [0/90/0] laminate from various Quasi-layerwise
Theories obtained by adjusting the thickness coordinate.

0.5

0.4

0.3

0.2

0.1

2 O

-O.1

-O.2

-O.3

-O.4

-O.5

Figure 4.8 - Results of 62 for a [0/90/0] laminate from various Quasi-layerwise

78

 

I

 

 

 

Theories obtained by adjusting the thickness coordinate.

 

0.5

0.4

0.3

0.2

0.1

20

-O.1

0.2

90.3

-O.4

-0.

Figure 4.9 - Results of a for a [0/90/0] laminate from various Quasi-layerwise

79

 

I

 

 

 

 

Theories obtained by adjusting the thickness coordinate.

1.5

 

80

0.5

 

0.4

0.3 -

0.2

I

0.1

  

20.

0.1-

  
       

-y '-<‘--4«
O O O O O I O O C O D

32.:

-O.2

     

'-
-

 

-O.3

-O.4 -

575133;;5:

 

 

 

I
0. l t: I l l l l 9 L
. o
0

Figure 4.10 - Results of W for a [0/90/0] laminate from various Quasi-layerwise
Theories obtained by adjusting the thickness coordinate.

81

4.4 Summary

Of the six Quasi-layerwise Theories analyzed, only the 0-1 Quasi-layerwise Theory,
i.e. the Generalized Zigzag Theory, is correct. The other five theories are overly sensitive
to variations in the coordinate system. In other words, their results are dependent on the

thickness coordinate. This coordinate dependency is not reasonable and not acceptable.

CHAPTER 5

GLOBAL-LOCAL SUPERPOSITION THEORIES

5.1 Introduction

The Quasi-layerwise Theories are superior to the Shear Deformation Theories in
numerical accuracy if the thickness coordinate is properly selected. They are also superior
to the Layerwise Theories in computational efficiency. However, they suffer from a
serious defect —coordinate dependency. Only the 0-1 Quasi-layerwise Theory, i.e. the
Generalized Zigzag Theory, is not coordinate dependent. The remaining five theories are
sensitive to the thickness coordinate. If the coordinate system is not properly selected,
H singularity can take place. Although it is possible to extract reasonable results from
mamptnéfifig‘tfié’ufigkfiésé "coordinate for the aforementioned Pagano’s problem, the
procedure doesnot seem to be practical for general applications. As a consequence, it is
necessary to develop a new technique to avoid this numerical dilemma.

Based on a global displacement field, the Shear Deformation Theories give
reasonable results for the overall performance of composite laminates, such as in-plane
deformation and stress. On the contrary, due to their assumptions for individual layers, the
Layerwise Theories are excellent for local behaviors such as interlaminar stresses. The
Quasi-layerwise Theories of third-order, e. g. the 0-1 Quasi-layerwise Theory, have two
terms for layer independency and two terms for layer dependency. In fact, they are
mixtures of the third-order Shear Deformation Theory and the first-order Generalized

Layerwise Theory. As a consequence, these theories can be clearly divided into a global

82

83
component of shear deformation type and a local component of layerwise type, and

assembled by a superposition principle.

Taking the 0-1 Quasi-layerwise theory as an example, it can be rewritten as:

k I
u (x1y12) = 4(x1y12) +11 (x1y12)
Vk (x, y, z) = y (x, y, z) + yk (x, y, z) (5.1)

wk (x, y, z) = Iy(x1 y, 2)

where,

2 3
4(x1y12) = u2(x.y)z +u3 (x,y)z
13(x1y12) = 112(Jr1y)z2 + v3 (x, y) 23 (5'2)
Iy(x1y12) = W.) (Jr, y)
are global components of shear deformation type, and
I I I
Ll (x, y, z) = 140 (x,y) + u1 (x, y) z
(5.3)

k k k
l’ (x,y,z) = V0(x1)’) +V1(x:)’)z

are local components of layerwise type.

5.2 Global-Local Superposition Technique

By utilizing the technique of superposition for the global and local displacement
fields, a wide range of new laminate theories can be proposed. However, since only four
continuity conditions (two displacement components and two interlaminar shear stresses
on each laminate interface) need to be satisfied, only four layer-dependent terms (two for
uk and two for v") are allowed in a laminate theory if layer-number independency is a
primary concern. In addition, by superimposing the global and local components through

the composite laminate, two coordinate systems are required. For the global description, 2

84

is used, while for the local description, a linear coordinate £2 for the km layer is assumed.

As a consequence, the 0-1 Quasi-layerwise Theory can be rewritten as the 0-1
Superposition Theory:

I 2 3 I I
u (x,y,z) = u0(x,y)+ul(x,y)z+u2(x,y)z +u3(x,y)z +u0+u1§k
vk(x.y12)

2 3 I I
v0(x,y)+v1(x,y)z+v2(x,y)z +v3(x,y)z +v0+v1§k

W" (x1y, 2) w0(x1y)

(5.4)

Similarly, the remaining five Superposition Theories can be rewritten as follows:

A. The 0-2 Superposition Theory

uk (x, y, z) =

2 3 I I 2
u0(x,y) +u1(x,y)z+u2(x,y)z +u3(x,y)z +u0+u2§k
vk(x1y12)

v0 (x, y) + v1 (x, y) z + v2 (x, y) 22 + v3 (x, y) 23 + v3.1. pggi (5.5)
W"(x1y1z) = w0(x1y)

B. The 0-3 Superposition Theory

it" (any, 2)

2 3 I I 3
u0(x,y) +u1(x,y)z+u2(x,y)z +u3(x,y)z +uo+u3§k
vk(x1y12)

2 3 k k 3
v0(x,y)+V1(X1Y)Z+V2(X1)’)Z +v3(x.y)z +Vo+V3§I

(5.6)
w" (x, y, 2) “’0 (x, y)

C. the 1-2 Superposition Theory

I 2 3

u (x,y,z) = uo(x.y) +u1 (x,y)2+u2(x1y)z +u3(x1y)z +uf§k+u§§13
vk(x,y,z) = v0(x,y) +v1(x,y)z-1-v2(x,y)22+v3 (x,y)z3+vf§k+v2‘ If (5+7)
w" (x, y, z) = w0 (x, y)

85
D. The 1-3 Superposition Theory

11 2 3
u (x1y12) = u0(x1y) +u1(x1y)2+u2(x1y)z +u3(x1y)z +uf§k+u§§2

vk (x, y. z) = v0 (x, y) + v1 (x, y) z + v2 (x, y) 22 + v3 (x, y) Z3 + Vffik + V332 (5.8)

E. The 2-3 Superposition Theory

I 2 3
u (x,y,z) = u0(x,y)+u1(x,y)z+u2(x,y)z +u3(x,y)z +u§ f+u§§2

I 2 3
V (x, y, 2) = v0 (x, y) + vl (x, y) z + v2 (x, y) z + v3 (x, y) z + vggg + vgfif (59)

Wk (11% Z) = W0(x1)’)

The 0-1 Superposition Theory is actually identical to the 0-1 Quasi-layerwise Theory
since they are not coordinate dependent. However, the remaining five Superposition
Theories are different from their Quasi-layerwise counterparts because of the numerical
advantage. In fact, all the Superposition Theories are not layer dependent. As an example,
the 1-3 Superposition Theory is selected in the following section for numerical

formulation. Similar procedures can be applied to the remaining five cases.
5.3 Formulation of the 1-3 Superposition Theory
A. Continuity Conditions

For simplicity, the local coordinate Q of the k”l layer is defined as a linear function

of z, i.e.:

86

i. = akz+bk (5.10)
2 Z +2
where, ak = —— and bk = —L+—I-——k.
Zk+1—Zk ZI+1"ZI

It should be noted that §k(zk) = —1 and §k_1(zk) = 1 for the interface located
between the k‘h layer and the (k —1)"‘ layer. To meet the continuity conditions of

displacement, the following conditions should hold:

k _ k k-l k—l
“3 " —u1—u1 —u3
k _ It k—l k-l ,
k = 2,3,4,...,n

In this study, linear strain-displacement relations and orthotropic constitutive

equations are considered. Thus, the transverse shear stresses can be expressed in terms of

the displacement variables:

k
Iyz

I 2 I k 2
Q44(v1+ 2v2z + 3v3z +akv1+ 3akv3§k + way)

(5.12)
I _ I 2 I I 2
tn - Q55 (ul +2u2z+3u3z +aku1 +3aku3§k +w0.x)

e e e o k k e _ o a .
To meet the continuity condrtrons of tyz and In at rnterface z — zk , 1t yrelds.

akQva— —ak_ 1Q44' 1v"“1 + 3akQ44v3 —3ak_1Qfi4' v" 1 =
Gkv1 + 2922,32 + 3922,53 + 92w“ y
“Istur ‘ “II-1935 1“f1+ 3“IQ551‘3 3aI 1Q551-“3 l = (5‘13)
Qkul + 2kaku2 + 3ka,%u3 + kao’x
k = 2, 3, 4, ..., n

where, 92 = Q1121"l -Q§4 and DI = 935-1 ’Qis °

87

Substrtuting Equations (5.11) into Equations (5.13), the following relations can be
concluded:

m2»

3 l
’ — élfiku Bkzkuz ‘ 33/121375 " §BIW0.I

1
V3 1"§"1I"1 1lIzI"2 ‘ inkZIEVS ‘ 51%qu

 

(5.14)
O a " ‘1 9
Where a L—LQL,B = —"_’C = ._k_-_1Q_4_4_’andn = k
I I I I k 1:
“I955 “Ist “I944 “I944

B. Boundary Conditions

Imposrng the free shear traction condition on the bottom surface of the composite

laminate, i.e. 11 (21) = 0 and tylz(zl) = 0, and noting that §1(zl) = —1, the

following two equations are satisfied:

2
1 1 ._
u1 +221u2+3z1u3+w0'x+a1u1 -I-3a1u3 - 0

2 (5.15)
1 l _
v1+2z1v2+3zlv3-1-w0,y+a1v1+3alv3 - 0

By examining Equations (5.14) and (5.15), it can be concluded that u", vf, u3 , and

. 1 .
3 can be expressed as functrons of ul, u2, u3, u3, and wO x , i.e..

88

It- k k k 1:1 k

k _ k k k k 1 k

I I I I I 1 I (5'16)
v1: le1+L2v2 +L3v3 +L4v3 +L5w0'y

k _ k k k k 1 k

After comparing Equations (5.15) and (5.16), the coefficients for k = 1 can be identified:

1 1 1 221 1 32? 1 1 1
1'71=’—’ F2=‘—’ F3="—’ F4=‘3’ F5="—
a1 a] al ‘11
01:0, 05:0, 0.11:1, 01:0, G§=0
22 322 (5.17)
L: =—i, L;=-—1, L;=———1, L1—-3, L;=——1—

1 1 l l 1
M1=0, M2=O, M3=1, M4=O, M5:

Assuming k = l for the first two equations of Equations (5.16) and subsequently
substituting them into Equations (5.14) for k = 2, it can be revealed that F: is a function of
F: and G: (where i = I,2,.,5). By further examining the terms for k = 3 and above, the

following recursive equations can be achieved:
I' Mr

3 l _ 3 3 _ l
Ff=—(-+-ock)Ff 1-(§+§ak)Gf l’i’BI

F5=-(%+%ak)Fs-1-(%+%ak)G:-l—m
Fé‘=1%+éak)F§-l—(%+%ak)G§-*—%mz%
F:=--(%+%ak)Fs-l-(%+%«I)G:-l
Fé=1%+%«k)F§-1-(%+%«I)G§-l—%m

89

L1=—(%+%ck)Lf-‘—(%+%ck)Mf-l—%m
Ls“(i-+19%!—(%+%ck)Ms-l—mzk
LI=-(%+%ck)Lé-l-(%+%ck)Mé-l—aw%
L:=-(%+%ck)Lz-‘—(%+%ck)Ms-l
Ls=1%+%ck)Lé-‘-(%+3ck)Mé-l—%m

(5.18)

By virtue of Equations (5.11), it is possible to further establish the following relations:

{chm—1— -1” 01—1, Mf=—Lf-1-Lf-Mf'1
5=_Fk-1 pk- 05—1, Mg=_L§-1_L5_Mg-1
G§=—F§-1—F§—G§-1, M§=—L§-l—L§—M§-1 (5.19)
gum-1 pk cg—I, Mg=-Lg—I_L5_Mg-I
G§=-F§“- Fé-Gé‘“, M§=-L§“-L§-M§“

By imposing the free shear traction condition on the top surface of the n-layer

composite laminate, i.e. 1:2 (2M1) = 0 and I; (2“,) = O , another two equations can

be obtained with the recognition of E," (2" +1) = 1

n n _
ul+22n+1u2+32n+1u3+wo,x+anul +3anu3 — O

(5.20)

I
O

n n
v1+22n+1v2+3zi l0v3+w y+anvl+3anv3 -

These two equations can be rewritten by virtue of the notation defined in Equations (5.16),

i.e.:

90

(1+ anFi‘ + 3ant) u] + (22 + aan + BanGg) 142 + (321,1 + (1an + 3anGg)u3

n+1
+ (anFZ-t- 3anGg)u§+ (1+aan’+ 3anGg’) wax = 0

(l + (1an + 3aan) v1 + (22n+1+ anLg1 + 3anM5') v2 +(3zi+1+ anLgl + BanM§)v3
(0,114.? + 3anM2)v31+ (1+ anLg1 + 3anMg) W0, y = O

(5.21)

The above equations imply that two more dependent variables can also be eliminated from

. . 1 1 . . . .
the formulatron. Assume these two vanables are u3 and v3 , wrth followrng definitions:

1
u3 = A1u1+Blu2+ C1u3 +Dlw0.x

 

 

 

 

 

 

 

 

1 (5.22)
v3 = A2v1+ Bzv2 + sz3 + Dzwo,y
where,
A = l+aan+3ant B = 22n+1+aan+BanGg
1 Al 1 Al
C = 3211+ (1an + 3on0; D = 1 + (1an + BanGg'
1 Al 1 A1
A _ l + (1an + 3aan B _ 22" +1+ anLg + 3anM5’
2 2 '
A2 A2
C = 32,21+1 + anLg + BanMg D = 1+ anLgI + BanMg‘
2 A2 2 A2
A1 = aan + 30,102 A2 = 0an; + 3aan41
Utilizing Equations (5.22), Equation (5.12) can be simplified as follows:
uf = Ri‘u1+R§u2 + R§u3 + Rfiwofit ué‘ = Sfu1 + Sé‘u2 + Sé‘u3 + SfiwoJ (5 23)

I _ I I I I I - I I I I
v1 — 01v1+02v2+03v3+04wmy v3 — P1v1+P2v2+P3v3+P4way

91

Rf = Ff+A1Ff§ 5’; = Gf+AIGfi
R: = F§+31Ffi 5’; = G§+Blcfi
R’; = F§+C1Ffi 5’; = G§+Clofi
R: = F§+01Ffi S: = G§+DIGZ
0’; = Lf+A2L§ P’f = M’I‘+A2M’j
0’; = L§+32Lfi P: = M§+32Mfi
0; = L§+C2Lfi P: = Mg+C2Mj
0: = L§+02Lj P: = M§+02Mfi

Substituting Equations (5.23) into Equations (5.8), the assumed displacement field can be

expressed in terms of the layer-independent variables:

k
u =u0+d>f(z)u1+4);(z)u2+d>§(z)u3+<l>f(z)wo.x

v" = v0 + ‘Pf (2) v1 + ‘11; (2) v2 + w; (2) v3 + ~11]; (2) WO' y (5.24)
w" = wO

where,
of (z) = z +Rf (akz + bk) +sf (akz + bk) 3

<1>§(z) 22+R§ (akz-I-bk) +S§(akz +1293
(1);‘ (z) = z3 +R§ (akz + bk) + S; (akz + bk) 3
<1>§(z) = R]; (akz + bk) + $5 (akz + bk) 3

‘Pf (z) = z + 0’1‘ (akz + bk) +Pf (akz +bk) 3

115(2) 22 + 05 (akz + 1),) +P§ (akz + bk)3

wig (z) 23 + 0; (akz + bk) +P§ (akz + 12,93

‘11}; (z) = 0!; (akz-I-bk) +P§ (akz + bk) 3

92

It should be noted that all (bf and ‘l’f in the above equations are associated only with
material properties and thickness coordinates. As a consequence, Equations (5.24) have
eleven independent variables. They represent a layer-independent theory and have the
same computational efficiency as those of HSDT. In fact, the Superposition 1-3 Theory is
like a third-order Shear Deformation Theory, although the total number of independent
variables is higher by four than its Shear Deformation Theory counterpart. However,
being different from the HSDT, the coefficients of Equations (5.24) are dependent on the
individual layers, instead of just coefficients to be determined from a variational process.

The characteristic of layer-dependence of the Superposition Theories is different from
that of the Layerwise Theories, whose coefficients are also layer-dependent but need to be
determined from variational process. By examining Equations (5.8), it can be seen that not
only can the distributions of in-plane displacement and transverse shear stresses be zigzag
through the thickness, but also that the curvature of transverse shear stresses are dependent
on the individual layers. These characteristics are similar to those of the Layerwise
Theories. In summary, the Superposition Theories are like the Quasi-layerwise Theories

and have the advantage of coordinate independency and numerical accuracy.

5.4 Numerical Solution
The procedures to obtain the solution to a composite laminate under bending are
Similar to those stated in Section 4.4. First of all, the strain and displacement components

are expressed in matrix forms:

where,

93

8!: = [N13] {X}, 85‘ = [N5] {X}, sf = [Is/f] {X}
713 = [N3] {X }, 71:, = [N152] {X }. 7;, = [N52] {X} (5.25)

w*= [~;]{x1, w' = [M1] {X}

T
{X} = {“0 v0 u1 v1 u2 v2 u3 v3 Wu}

F 2
[Nfi] = i 0 (154:9. 0 (bk 3 0 (bk 3 0 (bk 3]
ax

 

_8x 25; 3% 43—x2-
[N§]= 0 afl. 0 ‘Pf—a— 0 1'53 0 ‘1’36 W512]
y 8y 3y ay ay2
[Nf]=[0 0 0 0 0 0 0]
[”13] = [ea-y 5%; «>153;- ‘Pfa‘i. ¢s§—. “'53—.
“4‘38; flea—x (“’5 VDaxaaz]

dz dz 8x

[Nyz] [0 O O dz‘Pr 0 5W2
d k (i k a]
0 d 3 dz‘P4+l 3y

94

By defining
N1:
N" N"
Nk = y Nk = XZ (5.26)
1 , N, [1,, N5:
N5

 

 

and substituting them into Equations (3.21), the variational equation becomes:

n 21+]

J, g] 1(MIMI[~1zra1,,[~1,,)dz {mm =
+ J10] + [NJ ])dxdy+ £(q' [N;])dxdy

(5.27)

If Pagano’s cylindrical bending problem of plane-strain type is of interest, both the
loading on the top surface of the composite laminate, q‘(x, 151:) qosinm —x , and

the displacement variables can be assumed as follows:

uo = Uocospx; u1 = Ulcospx; u2 = Uzcospx; u3 = U3cospx

= W0 sinpx

As a consequence, the simply supported boundary conditions are automatically satisfied

and the matrices expressed in Equations (5.25) can be further simplified as follows:

T
{X} = {“0 u1 u2 u3 W0}

{X} = {U0 U1 U2 U3 W0}T

95

[N15]: [Nflsinpx = [..p _pq>f ~pd>§ -—p<1>§ _p2<b§]sinpx

[Nfl=[NflSinpx=[0 0 0 0 0]sinpx

[”152] = [Niziwspx = [0 i‘bi fid’é‘ 5;“? (5.28)
p(%¢fi + 1)]cospx
[NS] =[ NJ]Si"Px= [0 0 0 o 1]sinpx
Substituting Equations (5.28) into (5.27), it yields:
’3 21+]
.2. I (9911 TN) 91911921 9191191 9191191 ...,
= 2* .

1+:1"1~:1)9 =qz1~:1’

Once the independent variables U0, U1, U2, U3, and W0 are identified by

simultaneously solving the equations, the displacement and stress components can be

obtained through the associated equations.

5.5 Results and Discussion

The results for the six Superposition Theories are shown from Figure 5.1 to Figure
5.5. As depicted in Figure 5.1, it seems that the Superposition Theories of 1-2, 1-3, and 2-
3 types are superior to the Superposition Theories of 0-1, 0-2, and 0-3 types in giving Ox.
For the displacement, a, illustrated in Figure 5.4, both the 1-2 and 1-3 Superposition

Theories give good results. However, only the 1-3 Superposition Theory gives excellent

96

results of I

n, as shown in Figure 5.2. Since the boundary condition of the transverse

normal stress oz is not satisfied and constant w is assumed through the composite
laminate, the differences of OZ and W as shown in Figure 5.3 and 5.5, respectively, are not
significant.

Based on the above results, it seems that the 1-3 Superposition Theory is the best of
the six meofi;_%ong thev’fo‘ur coefficients, it seems that the zeroth-order term, which is.
the midplane displacement of each layer, is probably the least important one because the
Superposition Theories of 0-1, 0-2, and 0-3 types give worse results when compared with
the remaining three theories. It is believed that a layer-independent zeroth-order term
combined with displacement continuity conditions and variational process should be
enough to give accurate midplane displacement for each layer.

Among the first-, second-, and third-order terms, it seems that the first-order term is
the most important one because the 1-2 Superposition Theory and the 1-3 Superposition
Theory are better than the 2-3 Superposition Theory in giving displacement, and the 1-3
Superposition Theory is the best one for transverse shear stress. A possible interpretation
is that the first-order term is the most fundamental one for both displacement and stress
components.

Between the second- and the third-order terms, the latter seems to be more important
since the 1-3 Superposition Theory gives excellent sz' It is recognized that the second-
order term becomes the first-order after being differentiated with respect to z , while the ‘
third-order becomes the second-order. Since the transverse shear stress is known to be of
parabolic distribution, a layer-dependent third-order term certainly is very important.

Unfortunately, the above interpretation is not validated by further numerical results.

97

Shown in Figures 5.6 and 5.7 are the results for composite laminates made of 7, 15, and 31
layers. Though 12 seems to be reasonable, non-negligible errors still exist in the transverse
shear stress. The oscillations of In is believed to be attributed to the absence of the
second-order layer-dependent term. The numerical error seems to become more distinct as
the number of composite layers increases. It is thought that the boundary conditions, the
free shear tractions on both surfaces, strongly constrain the final distribution of 1“ in the
[0/90/0] laminate, resulting in excellent agreement with the elasticity solution. However,
as the number of layer increases, the influence from the boundary conditions becomes
weaker, and the important role of the second-order term becomes very distinct. As can be
seen in Figure 5.2, comparing the 0-2 and 1-2 Superposition Theories with the 0-3 and 1-3
Superposition Theories, it is obvious that one is convex to the left and the other to the
right. If both terms are considered in a theory, there will be an opportunity of balancing the

extremities.

5.6 Summary

The zeroth—order term can be omitted from the local component of a displacement
field. The first-order is the most fundamental one and can not be neglected. For higher-
order terms, it is concluded that the completeness of a displacement field is very important

in general composite analysis.

98

 

 

 

0.5 ‘r: r I r I l l r r
0.4- " ‘.
0.3- “n
x
o 9 x
0.2- o '3 x
. x
0.1-
. “ “ elasticitysolution
2 0- -+ “xxxxx” theO-lTheory
' “ooooo” dreO-Z'I‘heory
-01- “ ..... " theO-3Theory
' . “-.-.- " tlrel-Z'lheory
xo-, - a...” thel-B‘l‘heory
4.2- X’.’ . “...... 00” (1102-3111001?
X0 o
x.
-O.3- . ..
-O.4- ‘ ‘_
.05 l I l L l l l :

 

L25 -20 -15 -10 -5

Figure 5.1 - Comparison of Ox from various Superposition
Theories for a [0/90/0] laminate.

 

99

 

0.5

0.2

0.1

20-

-O.1

t

-O.2

-O.3

4.4 ~

 

-O.5
-2.5

 

 

 

 

“ “ elasticitysolution
“xxxmr” theO—l‘l'heory
“ooooo” theO—Z‘Iheory

“ ..... "theO—3Theory
“-.-.- "thel-Z'lheory
“- - - - " thei-3Theory
.. .. .." meZ-3‘l'heory

 

 

 

Figure 5.2 - Comparison of tn from various Superposition

Theories for a [0/90/0] laminate.

0.5

100

 

 

    

 

 

0.5 l H 1 I 3K T T
0.41- 3
b
I
0.3 - E
o.2~ 9%
0.1 - :5“
2 0- I
.... 9?.-
“ “ elasticity solution °
“xxxxx” theO—lTheory
42" “ooooo” theO-2Theory
“ ..... " theO-STheory >Q'
.03.. “-.-.- ”thel-Z'lheory x
“- - - - ”‘thel-B‘Iheory
.. .. .." th 2-
_0.4_ e 3‘Iheory
.0-5 I l l l
-1 -O 8 -O 6 —O.4 -O 2 O 0.2

Figure 5.3 - Comparison of Oz from various Superposition
Theories for a [0/90/0] laminate.

 

101

 

 

 

 

 

 

Figure 5.4 - Comparison of a from various Superposition
Theories for a [0/90/0] laminate.

0.5 I 1 l 1 .0 3
0.4 - . ' ‘
0.3 - 7 ..
o 2 ° ’ . .
' x 3‘3 ' “ “ elasticity solution
. \-.. - “x x x x x” the 0-1 Theory
0.1 - \‘2. ', “ooooo” theO—2Theory “
~- ' . “ ..... " theO-3‘1heory
O- _ “-.-.- ”thel-Z‘I‘heory d
2 “- - - - " the l-3Theory
: .‘f .. .. .." the2-3 Theory
-o.1 '- . '.. \ «1
“AV
-x
-0.2 - . -..x X ..
o O x
-o.3 - p "
40.4 - , " -
.0. : l l l 1
-51.5 -1 -O.5 O 0.5 1 1.5
i

 

102

 

 

 

 

 

0.5 l T T x] I l
x
x
0.4 )- “ elasticity solution 5 "
“xxxxx” theO-l‘lheory K
031 “00000" theO-ZTheory : ‘
' “ ..... " the 0-3 Theory K
”thel-ZTheory I:
0.2- - - - " the l-3Theory K ‘
. .. .. .." the 2-3 Theory :2
.. x
0.1 x .
x
x
2 0 - x .
x
x
-o.1 - 1“ 9
x
x
-O.2 - K -
x
x
-o.3 - § .
x
x
-o.4 - g 9
x
x
.0.5 l L l l x] l L
-3.5 -3 4 -3.3 -3 2 -3 1 -2.8 -2.7 -2.6 -2.5

 

Figure 5.5 - Comparison of W from various Superpbsition
Theories for a [0/90/0] laminate.

103

 

0.5 T l I

   
    

0.4” “ “ elasticity solution
“ ..... '° 3-ply

- - - " 79y
0.3

 

I

0.2

0.1

I

20-

-0.1-

—0.2

I

-0.3

-0.4

 

 

 

—os

415 -1 -o.5 0.5 1

0"

II

Figure 5.6 - Comparison of a from the 1-3 Superposition Theory and elasticity solutions
for 7-, 15-, and 31-layer symmetric laminates.

104

0.5 l I I l‘

0.4 i” ..
0.3+- .,. .
0.2- ".‘- J .4

 

I

 

 

‘9»...
1 9+
0.1 ' .-.2i --’- " d
’3“... ‘ 7: “ “ elasticity solution
2 “N “ ..... 3-ply
0- .1: . .._ _ . . 7.”), .
(z... "I". u. . . w 15.ply
-Oo1 - .~.: "°‘. : q
I \
\
.0 2- 1"

-o.3 i I -

 

 

 

\4‘.“' " " ‘“
-O.5 I l 1 ~ ’
-2 -105 -1 .5 o
t.“

Figure 5.7 - Comparison of tn from the 1-3 Superposition Theory and elasticity solutions
for 7-. 15-. and 31-layer symmetric laminates.

 

CHAPTER 6

DOUBLE SUPERPOSITION THEORIES

6.1 Introduction

In Chapter 5, the global-local superposition technique is used to combine a global
component and a local component together. The global component is actually the third-
order Shear Deformation Theory and the local component resembles the Generalized
Layerwise Theory. However, due to the fact that only two continuity conditions need to be
satisfied in each coordinate-direction, the local component is allowed to have only two
layer-dependent terms. As shown in Chapter 5, the six possible combinations give
different results. It is believed that each local term has a distinct contribution to laminate
performance. The zeroth-order term is the midplane displacement of each composite layer.
It can «be omitted from layer-dependent displacement assumption since the continuity
conditions have been satisfied on the laminate interfaces, resulting in the uselessness of
assuming zeroth-order terms for individual layers. The first-order term is of rotational
angle. It is too fundamentally important to be ignored. The second-order term represents
the curvature of the displacement distribution of the composite laminate, while the third-
order term can be associated with the curvature of the transverse stress distribution. They
are equally important to composite performance, especially when a laminate has a large
number of layers. As a consequence, the numerical results are very sensitive to the various
combinations of the terms, and the selection of the terms is very critical to the success of a

theory.

105

106

In view of the fundamental roles of the individual terms, it is believed that not only
the completeness of the terms, but also the inclusion of as many terms as possible, is
important to a laminate theory. It then is the goal of this study to look into a theory which
can satisfy the requirement of completeness and include all the first-, second-, and third-
order layer-dependent terms in an assumed displacement field.

Since only two continuity conditions can be satisfied in composite layer assembly, the
number of variables associated with the local behavior are limited to two. If the
completeness of the local component is of concern, the only allowable global-local
combination is the 0-1 Superposition Theory. However, if more terms are to be included, a
special technique such as the following Hypothesis for Double Superposition should be
proposed:

The Global-Local Superposition Technique is applied to the three local terms twice,
once for grouping two local terms and the other for one local term.

The application of the Hypothesis for Double Superposition in proposing new
laminate theories is demonstrated in the following sections.

A laminate theory whose global component is of a third-order Shear Deformation

Theory is considered:
2 3
“0(x,)’,2) = u0(x,y) +u1(x,y)z+u2(x,y)z +u3(x,y)z
VG (x. y. 2) = v0 (x. y) + v1(x.y)z + v2 (11. y) 22 + v3 ()1, y) 23 (6.1)

wG (Ly, 2) = w0(x.y)

and the local component has two groups, one with two local terms and the other with only

107

one local term. The total displacement field can be summarized as follows:
1

2+-
I_ I I
u —uG+uL+uL
k_ k k 6.2
v —vG+v_L+vL ( )
w=wG

where uG, VG, and wG are global terms: all: and v: are of two-term local groups; and u:
and v: are of one-term local groups. Assuming the local component is also limited to the
third-order, the three possible combinations for grouping the three local terms twice,

resulting in three Double Superposition Theories, can be listed below:

A. The 1,2-3 Double Superposition Theory
In this theory, the local component can be divided into two groups: the first group
contains the first— and second-order terms while the second group contains the third-order

term, i.e.:

li(x,y, gk) “f (x,y) §k+u§ (LY) ékz

Vf (x, y) E), + vg (x. y) 5.)?

3:,(x’ y, 5k)

“.2 (x, Y9 5.11:) = “3’5 (x, Y) 52 (6.3)

v; (x, y, 1,) 9 (x,y) at

B. The 1,3-2 Double Superposition Theory

uifxg’, Ek) “f (x,_)’) Eek-f“; (x,)’) Ed]?

v_,’j (x. y. a.) v1 (x. y) a. + v1 (x. y) :2

108

112(99):.) u; (x,y) at (6.4)

V: (x, y, alt) V5 (x9 y) a]?

C. The 2,3-1 Double Superposition Theory

“—2 (x, y, gk) u§ (x,y) éi+u§ (x,y)éi’
2': (x, y, 9,) = v; (x,y) 12+ v1 (x. y) é)?

uf (x. y) ék (6.5)

u: (x. y. :1)

v: (x, y, ék) V’f (x, Y) g]:

6.2 Formulation of the 1,2-3 Double Superposition Theory

Taking the 1,2-3 Double-Superposition Theory as an example to demonstrate the
process of Hypothesis for Double Superposition, the global component is of a third-order
Shear Deformation Theory as given in Equations (6.1). The local component is divided
into two groups. The first group includes the first-order and the second-order terms while
the second group includes the third-order term. As a consequence, the displacement field

can be written as follows:

uk (x, y, z) = u0 (x,y) + u1 (x, y) z + u2 (x, y) 22 + u3 (x,y) z3 +
uf (x. y) Eh + 145 (x, y) i)? + ué‘ (x. y) E)?

vk (x, y, z) = v0 (x, y) + v1 (x, y) z + v2 (x, y) z2 + v3 ()1, y) 23 + (6-6)
Vf (x. y) 5,, + v; (x. y) i? + V5 (19 Y) 532

W" (x. y. 2) = wo (x, y)

where a linear relation for transformation between the local and the global coordinates is

109

assumed, i.e.:

ék = akz+bk (6.7)
and,
2 zk+1+zk
ak = ———-; bk = —-————
ZI+1—ZI ZI+1’ZI

As mentioned before, it should be noted that Ek (2,) = —1 and §k_l (zk) = 1 on the

laminate interfaces.

A. Continuity Conditions
The continuity conditions of displacement 'on the laminate interfaces should be
satisfied. By imposing the continuity condition for the two-term group, the following two

equations can be identified:
I _ I k-l k-l
u2 — ul + u + u2

l (6.8)

k k k—l k-l
V2 V1+V1 +V2

In addition, by enforcing the continuity conditions for the one-term group, another two

equations are obtained:

(6.9)

It should be pointed out that the above equations are valid for k = 2, 3, 4, . . ., n.
Assuming linear strain-displacement relations and utilizing three-dimensional
constitutive equations for cross-ply laminates, the transverse shear stresses for the k‘h

layer are:

110

I _ I 2 I I I 2
Tu — Q55(u1+2u22+3u3z +aku1+2aku2§k+3aku3§k +w0'x)

)

(6.10)
k _ k 2 k k k 2
In -— Q44(v1+2v2z+3v3z +akv1+2akv2§k+3akv3§k +w0‘y

In order to meet the continuity conditions of 1:2 and 11’“: on interface z = 2k, the

following two equations need to be satisfied:

I I I k-l k-l I I—1 I—1
’aIstul = (zakQSS + “I- 1955 )“1 + 21%st +‘II—rst )“2
+ 3(a,cQ’5‘5 + ak_1Q§5'1)u§-l + Qku] + 29%qu + 3kaiu3 + kao'x
(6.11)
I I I k-l I—1 I I-1 I-1
‘aIQ44V1 = (ZGIQM + ak-1Q44 )Vr + ziaIQu + ak-1Q44 )Vz
+ 3(akofi4 + ak_1Qfi4" 12:" + 9,121+ 29,,sz2 + 3oszfv3 + 9W0.)
where, 9k = Q5; 1 “ Q54, and 9I = 955-1“ Q55 '
Rearranging the terms, Equations (6.10) can be expressed in the explicit forms:
uf = —(2 +ozk)u’1"'1—2(1 +ozk)u§'1—3 (l +0tk)u§f'1
_Bku1" ZBkzkuZ ‘ 3BIZiu3 — BIW
°" (6.12)

vf = —(2+1;k)vf'1—2(1 -t-Ck)v§"1—3(1+Ck)v§‘1

_nkv1 " anzkvz ‘ 371IZEV3 ‘ TlIWo, y

 

k-l k-l

ak—IQSS QI “I-1Q44 9,,
where,ak=——k—-,Bk=-—k,Ck=——k+,andnk= k'
“IQ55 “I955 “IQ44 “IQ.“

B. Boundary Conditions
Free shear traction conditions are imposed in this analysis for both surfaces.
Therefore, on the bottom surface, both 1:2(21) = O and 1)}2 (21) = 0 should be

satisfied. Since 51 (21) = —l , these two boundary conditions become:

111

2
1 1 1 _
u1+ Zzlu2 + 321143 + wax + alul — 2a1u2 + 3a1u3 — 0
2 1 (6.13)
1_ 1 _
v1+221v2+3zlv3+w0'y+alv1 2a1v2+3alv3 — 0
By examining these equations with Equations (6.12), u", vf, ug, v5, ug‘ , and v§ can

be assumed as:

1 1
I - I I I I I I
u1 - F1u1+F2u2+F3ul+F4u2+F5u3+F6wax

1
I I I I I I
u Glu +qu2+G3ul+G4u2+05u3+G6w0J

1
I I I I I
+H2u2 +H3ul +H4u2 ~1-H5u3 +H6w0.x

u

wa- N7:-

1
1
1
I
Hrur
(6.14)
1 1
I _ I I I I I I
vl —L1v1-1-L2v2+L3v1+L4v2-i-L5v3+L6w0.y
1 1
I- I I I I I I
v2— M1v1+M2v2+M3vl +M4v2+M5v3+M6way
1 1
I _ I I I I I I
v3 — val+N2v2+N3vl+N4v2+N5v3+N6woJ

When compared to Equations (6.13), the coefficients for k = 1 can be easily identified:

1 1 1 1 1 1
1“1 =1, F2=F3=F4=F5=F6 0
1 1 1 1 1 1
02:1, Gl=G3=G4=GS=GG=O
1 1 1 1 1 1
L1 =1, L2=L3=L4=L5=L6=0
1 1 1 1 1 1
M2: 1, M1=M3=M4=M5=M6=O
1 1 1 2 1 1

H =—-, H =-, H =_—
1 3 2 3 3 3a1
22 z
H1___l, H;=——1, H1~-—1-
a a1 3al
1 1 1 2 1 1
Nl=_-’ N2=-—’ N3=__
3 3 3a1
2
22. z
1 1 1 1 1 1
N ———, N =——, N =—-—
4 Ba 5 a1 6 3a1

112

In a manner similar to that used in obtaining Equations (5.18) and (5.19), the following

recursive equations can be obtained:

Ff=— (2+01k)F{‘-1—2(1+ock)Gf‘1—3(l-1-ock)l-I{‘—1

F2" =— (2+ock)F§'1—2(l+ak)G§'1—3(l -I-01,()H§‘1

F; =— (2+ock)F§‘1-2(1 +ak)G§'1—3(l +ock)H§'1—Bk

Ffi =— (2+ak)F§-1—2(1+ozk)G§'1—3(l +ork)H§-1—213kzk
F§ = - (2+ork)F§-1—2(l +otk)G§-1—3(l +ak)H§'1—3Bkzi
F}; =- (2+ock)F§-1—2(l +ork)Gg‘1—3(l +ak)H§-1-Bk
Lf =- (2+§k)L’1‘-1—2(1+C,‘)M{‘-1-—3(1+QI‘)N{‘-1
Lg:—(2-1-§,c)L’2"1--2(1-1-1;l,‘)M§'1—3(1+1.:,,‘)N§‘1
Lg:—(2+gk)L§-1—2(1+§k)M§-1—3(1+1;k)N§-1—nk
L§=— (2+Qk)L§’1—2(1+§k)M[,‘—1—3(1+§k)Nfi-1—2nkzk

L; = -— (2+Ck)L§-1—2(1 -1-1;k)Mg‘-1—-3(1+Ck)Ng"1--3'r]kz,2c
L39 =— (2+1;k)Lg-1-2(1 +§k)Mg-1—3(1 +§k)N§‘1—nk

Gf=Ff‘1+Ff+Gf'1 Mf=Lf'1+Lf+Mf-1
G§=F§-1+F§+G§'1 M§=L§-1+L§+M§-1
G§=F§-1+F§+G§-1 M§=L§-1+L§+M§'1
G§=F§'1+F§+Gfi-1 Mfi=L§-1+L5+Mfi-1
G§=F§-1+F§+G§'1 M§=L§-1+L§+M§'1
Gg=F§-1+Fg+Gg'1 Mg=Lg-1+Lg+Mg-1

H’i =-Hf“ ~§=_~’;-‘
Hi =-H§" N§=-’£“
HS =-H§" ~;=_~;-1
Hf. =——Hfi" Nfi=-Nfi"
H? =-H’§“ N§=—N§"
H; =—H:-l Iv‘=..N‘“

6 6

I = 2, 3,4,...,n

(6.15)

113

Although the recursive equations seem to be somewhat complex, their advantage of
numerical efficiency should be recognized.

For free shear traction on the top surface of a composite laminate, another two
conditions need to be satisfied. By imposing 1:2 (2M1) = 0 and 1:2 (2“,) = 0 and

recognizing that in (2" H) = 1 , the following two equations can be achieved:

2
n n n _
“1+22n+1“2+ 32M1u3 + wow+anu1 -1-2anu2 + 3anu3 — 0
2 (6.16)
n n n _

Mth the use of recursive equations, these two boundary conditions become:

(aan + Zant + 3aan) u: + (aan + 2on0; + 3anH5’) u;
+ (aan’ + 2anG§1 + 3anH§+ l)u1+ (aan + 2on0}; + 3anH2 + 22” +1) u2

+(aan+2anGg+3aan+3zi+l)u3+ (aan+2anGg+3aan+ 1)w0.x = 0

((1an + 2(1an + 3aan) vi + (anLg + 2anMg + 3anN5') v;
+ (anLg’ + ZanMg' + 3anNg' + l)v1+ (anLg + ZanMg + 3anN2 + 22" +1) v2

2

n+1

(anLg’ + ZanMg + 3anNg’ + 32 )v3 + (anLg + ZanMg + 3anNg + 1) W0, y = O

(6.17)

These equations imply that another two independent variables can be further eliminated in

Equations (6.14), i.e.:

1 1
u2 = A u1+B u +C u2+D u3+E w.
1 11 l 1 101: (618)

1 1
v2 = A2v1+ 32V1+ sz2 + Dzv3 + Ezwo’ y

114

 

 

 

 

 

where,
A _ _ aan + 2onG;x + 3anH;1
1 A1
B = _ 1+ aan‘ + 2on0; + 3aan’
1 A1
C _ 22" +1+ aan; + ZanG",I + 3ant,’
1 ‘ ’ A
1
D _ 3zi+l+aan+2anGg+3anGg
1 ’ ’ A
1
E = _ 1+ aan + ZanGg + 3anGg
1 A1

with A1 = aan + 2on0; + 3anH3 ;

aan + Zaan + 3anN’l'
Az
l + anLg + ZanMg;1 + 3onN§1
Az

 

 

22" +1-1- anLl;l + 2anMg + 3anN",1
A2

 

32: +1 + anLg + ZanMg' + BanNg
A2
1 + anLg + ZanMg + BanNg
A2

 

 

A2 = anL’z' + ZanMg + 3anN5’ .

By utilizing Equations (6.18), Equations (6.14) can be rewritten as follows:

1
I _ I I I I I
u1 — Rlul+R2u1+R3u2+R4u3+R5wo,x

k-
“2-

It-
U3-

1
I I I I I
Slu1+ Szu1+S3u2 +S4u3 +stmr

1
I I I I I
T1u1+T2u1+ T3u2 + T4113 +T5wo’x

(6.19)

I .. I 1 I I I I
v1 - 01121+02v1+03v2+04v3+05way

I- I1 I I I I

o,y

1
I _ I I I I I
V3 ‘ Qrvr+92V1+Q3V2+Q4V3+sto.y

where,
R’,‘ = F’l‘mlr’; 5’; = Gf+AlG§
R’z‘ = F§+31F§ s’; = G§+BIG§
R; = Ffi+C1F§ 5’; = Gfi+Cng
R"; = F§+D1F: s: = G§+Dng
R: = FE+EIFE 5: = Gz+Eng

F. 5. 5.

+A1H:
+BIH§

4+C1H2

R

5’01”;

(#51112

mi #1 to?!» Ni Hi
II II
1,.

 

115

I I I I I I I
o, = L, +14sz P, = M, +A2M’; Q, = N, +A2N§

I I I I I
02=L3+32L2 P2=M§+32M§ Q2=N§+32N12

I I I I I I
03=L4+C2L2 P3=Mfi+C2M2 Q3=Nfi+C2NI2

0: = L§+02L§ Pi = M§+02M§ Q. = N§+Dz~§
o

I I I I I I
5=L6+E2L2 P5=M8+52M2 Qs'N6+EzA’2

It should be noted that Q? and Ef in Equations (6.18) and (6.19) are not the widely used

material constants Q; and E2. By virtue of Equations (6.19), the displacement field then

becomes:
uk = u0 + <1>f(z)ui + (D’z‘ (z) u, + d>§ (2) u2 + (I): (z) u3 +<I>§ (z) wax
vk = v0 + ‘I’f (2) v} + ‘1‘5‘ (2) v, + ‘I’g‘ (2) v2 + ‘1‘5‘ (2) v3 + ‘I’g‘ (2) way (620)
w" = wO

where,

¢f(z) =Rf(akz+bk) +51 (akz+bk)2+T’,‘(akz+bk)3
<r>§(z) = 2+R§(ak2+bk)+55(a,,2-1-b,,)2+1”,;(akz-t-bk)3
<I>§(z) = 22+R§(ak2+bk) +Sz’,‘(a,,z+b,,)3+T‘;(ak2+bk)3
ow) = 23+Rfi(a,z+b,) +S§(akz+bk)3+T:(akz+bk)3
<r>§ (2) = R; (a,z+b,,) +5; (akz wk)3 +r§ (akz+bk)3
91(2) = of (akz + bk) +19; (akz + bk) 2 + Q’,‘ (akz + bk) 3
115(2) = z+0§(ak2+bk) +195 (akz+bk)2+Q§(akz+bk)3
‘1’§(z) = 22+0§(ak2+bk) +P§(a,,z-1-b,,)2+Q’3‘(a,,z+b,,)3
alga) = 23+ofi(akz+b,,) +Pfi(akz+bk)2+Qfi(akz+bk)3

1919(2) = 0; (akz +bk) +19; (akz + bk)2 + Q; (akz +bk)3

116

In Equations (6.20), it can be seen that the number of total variables of the 1,2-3
Double Superposition Theory is independent of the number of layers. When compared
with the Shear Deformation Theory of the same order, there are six additional layer
independent variables. However, when compared with the Generalized Layerwise Theory,
coefficients of higher-order terms (of z) are related to the properties and the thickness
coordinates of the individual layers instead of unknown variables. In addition, based on
the results from the Superposition Theories discussed in Chapter 5, it is expected that the
through-the-thickness distributions of in-plane displacement will be kinky across laminate
interfaces and the curvature of the transverse shear stress distribution will be layer

dependent in the 1,2-3 Double Superposition Theory.

6.3 Variational Equation
The procedures to obtain the variational equation for the 1,2-3 Double Superposition
Theory is similar to those used in previous chapters. The corresponding matrices are

defined as follows:

H

e§=[~{|{X}. e;=[~;]{X}. e:=[~;]{X}
v:,=[~§y]{X}. vfi.=[N§,] {X}. v;,=[~;z] {X}
w+= [11;] {X}, w'= [M1] {X}

(6.21)

where,

 

117

1 1 T
{X}= {“0 v0 u, v, u, v, 112 v2 u3 v3 W0}

2
Nk = 1 [ti [ti [(1 ki [‘8
[I] [ax 0 (”lax 0 (D23): 0 ¢3ax 0 «>481, 0 ¢53x2

2
N"= .9. 19.3. La. 9.3- IE Ii
[, [0 3y 0 ‘1’,ay 0 +112,” 0 T3ay o ‘1’4ay +195,”
[Nf]=[o 0 0 0 o 0 0 0 0 o 0]
a a a a a a a a
k = _ _ k_ k_ k_ k_ Ic_ k—
1ny [ay ax ‘blay “flax d’zay W281: ¢3ay “'33):

By 3x8
[ij = [0 0 illef 0 H‘izog 0 fie; 0
—.9 9 (299“ .1]
[N52] = [0 0 0 $11; 0 $11129 0 % 3

By defining

k k
1~1= ”+ 1~1,,= [1;] <9

 

 

118

and substituting Equations (6.22) into (3.21), it yields:

n zI+1

1, g, 1(1111211991~+;1Q1,1~1,,)9z {999 =
:1! 4+[N;])dde+(12( q“[N;])dxdy

(6.23)

For a simply-supported, infinitely long, laminated strip under cylindrical bending, the

".1:

. B °
L y assurrung

loading can be expressed as qr+ (x, g) = q (j sinpx , where p =

uo = Uocospx u: = Uicospx u, = U, cospx u2 = Uzcospx (6.24)

u3 = U3cospx wo = Wosrnpx

the simply-supported boundary conditions are satisfied automatically. This is a plane-
strain problem with a; = yfiy = 7;: = 0. The six variables to be determined are of the

following matrix:
1 T
{X} = {“0 u, u, u2 u3 W0} (6.25)

By substituting Equations (6.25) into Equations (6.24), it is obvious that

{X} = {U0 U: U, U2 U3 W0}T (6.26)

and

 

119

[Nil = [Nfi] sinpx = [‘P "P‘bf -P¢§ —p¢§ —pd>§ —p2<b§]sinpx

[N5] = [1175:19an = [0 0 0 0 0 0] sinpx
d d d a
[”359] = [”591“pr = [0 a?“ 2715‘ #5 #1

p(-a(-1—<I>§ + 1)]cospx

Z
[NJ]=[NJ]sinpx=[o 0 0 0 0 1]sinpx

Substituting Equations (6.26) into Equations (3.25), it yields:

II”

E 1 (9919:) ’19:) 1991191 9191191 9191191

9191191 )9 9:191

(6.27)

By solving Equations (6.27), U0, U1, U,, U2, U3, and W0 can be obtained.

6.4 Numerical Results and Discussion
The results from all three theories turn out to be identical for a [0/90/0] laminate.
They are shown from Figure 6.1 to Figure 6.5. Excellent agreement with the exact

solution is obtained for Ox, I

x, , and 12. Since constant w is assumed and no boundary

condition for transverse normal stress is enforced in the analysis, W and 62 are not good,
although they both have small values.
The 1,2-3 Double Superposition Theory is also applied to both even-numbered and

odd-numbered laminates. The results for the former are shown in Figures 6.6 and 6.7 for

120

2, 6, and 14 layers, while in Figures 6.8 and 6.9 results for 3, 7, and 15 layers are shown.
Good results seem to be achieved. The reason that the 30-p1y and the 31-ply laminates are
not shown in corresponding figures is because of poor results attributed to numerical

round-off error.

6.5 Summary

The importance of the completeness and the roles of high-order terms in a
displacement field is demonstrated in this analysis. The application of Hypothesis for
Double Superposition, though not verified mathematically, seems to be effective in giving

a good laminate theory.

121

 

0.5 t

 

,1:

 

—— “ elastiCity solution
- - - - " thel.3-2Theory

-.-.- " the2.3-1Theory
_0.4_ .. .. .." the 1.2-3 Theory

1
.0
(d

l

 

‘0 5 l l l I L

 

 

 

7-25 -2o -15 -10 -5

Figure 6.1 - Comparison of 6,, from the 1,2-3, 1.3-2, and 2.3-1 Double Superposition
Theories with the elasticity solutions for a [0/90/0] laminate.

122

 

0.5 I

0.4

0.3 -

0.2 r

 

0.1-

00 q.

0 _ —- elasticity solution
2 .“- - - - " the 1.3-2‘lheory
“-.-.- ” “2.3-1.11130”
.0.1 - 'I “u .. .. u" "31.2-3qu

 

.5

 

 

 

 

Figure 6.2 - Comparison of Ixz from the 1,2-3, 1.3-2, and 2,3-l Double Superposition
Theories with the elasticity solutions for a [0/90/0] laminate.

123

 

     

 

 

 

 

0.5 t I
'\
0.49 \
\
\o
0.3 - \_
\
."
I
0.1 - /
2 0)- l/
I
l
-0.1 - /
-o.2- \
\e
\.
-o.3 - 2‘ . “ elasticity solution \
“- - - - " the 1.3-2T'heory \
_04_ " the2.3-1T'heory \\
' .. .." the 1.2-3 Theory \_
\
_O.s 1 1 g i 1 1\
-1 -O.8 -O.6 -O.4 -O.2 O 0.2

Figure 6.3 - Comparison of 6, from the 1,2-3, 1.3-2. and 2.3-1 Double Superposition
Theories with the elasticity solutions for a [0/90/0] laminate.

124

 

   
   

 

 

 

 

0.5 I 9 I r I

0.49 +
0.3 - .
0.2 - '1
(1.1- .
2 '0 F ‘
-o.1 - 9
-o.2- 9
.03 _ “ “ elasticity solution .

2. - - 0; thel.3-2T'heory

.0... «:32: 3:131:23 -
-O.5 ’ i L ’ i

-1.5 -1 -O.5 0 0.5 1

a

Figure 6.4 - Comparison of u from the 1,2-3, 1.3-2, and 2.3-l Double Superposition
Theories with the elasticity solutions for a [0/90/0] laminate.

125

 

 

 

0.5 I I f l I I
0.3 - .
0.2 -
0.1 - .
2 0-
-0.1 -
_ 1-
0'2 " “ elasticity solution
- - - " the 1.3-21heory
-039 " theZ,3-l Theory
.. .. ..” the 1.2-3 Theory
-0.4 -
_o.5 l l 1 1 1 1

 

 

 

Figure 6.5 - Comparison of W from the 1,2-3, 1.3-2, and 2,3-1 Double Superposition
Theories with the elasticity solutions for a [0/90/0] laminate.

126

 

 

 

 

005 I T I I I T
0.4 - 9
0.3 - ~
0.2 - ’ -
0.1 - ' ' .
Z 0 " ‘
-O.1 - -+
p i ' “ “ elasticity solution
.021- “-.-.- " 2-ply .
‘ : ..... : 6—plyl
.o.3 - ,9 ' ' ' ' 4'9 y -
.0..- / -
.005 I1 I l l l L
-2 -1 0 1 2 3 4 5

Figure 6.6 - Comparison of a from the 1,2-3 Double Superposition Theory and elasticity
solutions for 2-, 6-, and l4—1ayer unsymmetrical laminates.

 

127

0.5 I I I I I

 

0.4

I

0.1

.49

 

“ elasticity solution

a

I:

 

 

 

Figure 6.7 - Comparison of I u from the 1.2-3 Double Superposition Theory and elasticity
solutions for 2-, 6-, and l4—layer unsymmetrical laminates.

128

0.5 t r

 

—— elasticity solution

0-4" ~.-.- " 3-ply

0.3- ' ' ‘ ' 15'9”

0.2 r

0.1

20'-

-O.1

-0.2

-O.3

 

-0.4

I
J

 

 

4.5 l l L l
-1 .5 -1 -O.5 O 0.5 1 1.5

 

Figure 6.8 - Comparison of a from the 1.2-3 Double Superposition Theory and elasticity
solutions for 3-, 7-. and 15-layer symmetric laminates.

129

 

0.5

 

0.4

0.3

0.2

0.1

 

-O.1

,1:
'1’

e

is

 

 

 

 

 

 

Figure 6.9 - Comparison of I“ from the 1,2-3 Double Superposition Theory and elasticity
solutions for 3-. 7-, and lS-layer symmetric laminates.

CHAPTER 7

CONCLUSIONS AND RECOMMENDATIONS

7.1 Conclusions

The Shear Deformation Theories give good results for global distribution of in-plane
stresses but poor results for local distribution of interlaminar stresses. Layerwise Theories
give excellent results for both global and local distributions of displacement and stress
(both in-plane and out-of—plane). The former group of theories has the advantage of
numerical efficiency because the total number of variables is independent of the layer
number, while the latter group suffers from a numerical crisis if the layer number becomes
too large. A compromising theory, the so-called the Generalized Zigzag Theory, is
presented. Mth high-order terms, which are not necessary to be layer-dependent, both in-
plane stress and transverse shear stress distributions are greatly improved. Moreover, the
zigzag distribution of in-plane displacement can be accurately predicted.

The Generalized Zigzag Theory is a layer-independent theory, though having two
layer-dependent terms for both the zeroth- and the first-order terms initially. Due to its
success in laminate analysis, the feasibility of assigning the two layer-dependent variables
in high-order terms (i.e. the second- and the third-order terms) is examined, resulting in
the Quasi-layerwise Theories. Unfortunately, a physical impossibility — coordinate
dependency, takes place. It then requires the use of the Global-Local Superposition
Technique to formulate the laminate theory to be coordinate-independent. In addition, it is

necessary to express the displacement components in an explicit manner to have the

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advantage of numerical efficiency. The recursive expressions presented in this study,
though somewhat tedious, are aimed at this purpose. By examining the results based on
the Superposition Theories, an important question regarding the complete selection of the
high-order terms is raised. It is concluded that the completeness is two fold: not only can
no low-order terms be skipped, but more high-order terms are preferred.

The objective of completeness seems to conflict with the fundamental of two
continuity conditions in each coordinate direction. In order to satisfy both aspects, a
special technique, namely the Hypothesis for Double Superposition, is proposed. Several
three-term theories, called the Double Superposition Theories, are examined. Since these
are three layer-dependent terms in a displacement field instead of two, an extra continuity
condition of displacement is added twice to the formulation through the application of the
superposition principle. All the Double-Superposition Theories are shown to have
identical results for a [0/90/0] laminate. They give excellent values for in-plane
displacement, in-plane stress, and transverse shear stress. However, because w is
considered as constant in the example, both transverse displacement and normal stress are
not as good as the remaining components.

Among all the theories examined in this thesis, it seems that the Generalized Zigzag
Theory with up to the seventh-order term and the third-order Double Superposition
Theories give the best agreement with the Pagano’s solution in all ranges of layer number
for both symmetric and unsymmetrical laminates. Although they both are layer-number
independent theories, the former has seven degrees-of-freedom while the latter has only
three, provided w is considered to be constant through the laminate thickness. As a

consequence, the Double Superposition Theories are concluded to be the best selection for

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laminate analysis in this thesis.

7.2 Recommendations

This study presents three laminate theories for laminated composite analysis - the
1,2-3, 1,3-2, and 2,3-1 Double Superposition Theories. However, the success is primarily
referred to the dominant displacement and stress components, namely in-plane
displacement, in-plane stress, and transverse shear stress. Apparently, both the transverse
normal displacement and stress are not carefully considered in the analysis. It is suggested
that these two components should be included in the future development of laminate
theories.

In addition, it should be pointed out that the assembly of the composite layer through
the laminate thickness is based on the assumption that the interfacial bonding is perfectly
rigid. Due to the complexity involved in the composite fabrication, it is possible to have
non-rigid bonding such as shear slipping and normal separation on laminate interfaces. It
is then required to include special stress-displacement relations in the laminate
formulation. Besides, geometrical irregularities can also occur in composite fabrication,
e. g. wavy fibers in composite laminates. In order to model the geometrical irregularity,
special strain-displacement relation may be required. Certainly the application of the
laminate theories to composite structures with complex geometry such as shells of various

shapes, is another example.

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