MESH INDEPENDENT ANALYSIS OF 3D SHELL-LIKE LAMINATE STRUCTURES USING ABD-EQUIVALENT MATERIAL MODEL

By

LI LIANG

A THESIS PRESENTED TO THE GRADUATE SCHOOL

OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2016

© 2016 Li Liang

To my Mom, Dad and all of the teachers

4

ACKNOWLEDGMENTS

I would like to express the most sincere gratitude to my advisor, Dr. Ashok V.

Kumar. His support helps me not only for my thesis and research, but also for my entire

study life of the graduate school. It would be impossible for me to finish such a research

without him.

I extend my greatest thanks to the member of my supervisory committee, Prof.

Nam-Ho Kim for his guidance during my thesis. It is an honor for me to work in such a

team which developed my critical thinking and research ability.

I thank Prof. Peter G. Ifju and Prof. Bhavani V. Sankar for inciting interest in the

field of composites material and Finite Element Method during graduate study at the

University of Florida.

Finally, I would like to thank my Mom and Dad, who bore, raised me and taught

me the meanings of life.

5

TABLE OF CONTENTS page

ACKNOWLEDGMENTS .................................................................................................. 4

LIST OF TABLES ............................................................................................................ 7

LIST OF FIGURES .......................................................................................................... 8

LIST OF ABBREVIATIONS ............................................................................................. 9

ABSTRACT ................................................................................................................... 10

CHAPTER

1 INTRODUCTION .................................................................................................... 12

Goals and Objectives .............................................................................................. 13

Goals ................................................................................................................ 13 Objectives ......................................................................................................... 13

Outline .................................................................................................................... 14

2 MESH INDEPENDENT FINITE ELEMENT METHOD ............................................ 16

Overview ................................................................................................................. 16

Formulation of 3D Element (3D-Shell) .................................................................... 18

Stiffness Matrix for 3D-Shell Element ..................................................................... 20

Boundary Condition ................................................................................................ 24 Clamped ........................................................................................................... 24 Simply Supported ............................................................................................. 25

Symmetry Boundary Condition ......................................................................... 26

3 ANALYSIS OF COMPOSITE MATERIAL ............................................................... 27

Overview ................................................................................................................. 27 Stress-Strain Relations of a Composite Lamina ..................................................... 28 Classical Laminated Plate Theory (CLPT) .............................................................. 32

Shear Deformable Plate Theory (SDPT)................................................................. 40

4 ABD-EQUIVALENT MATERIAL MODEL OF COMPOSITE LAMINATE ................. 43

Overview ................................................................................................................. 43 The ABD-Equivalent Material Model of Laminate ................................................... 44

Local and Global Stiffness Matrix ........................................................................... 48

6

5 RESULTS AND DISCUSSION ............................................................................... 56

Overview ................................................................................................................. 56 Example of Square Plate ........................................................................................ 56

Example of Pressured Cylinder .............................................................................. 60 Example of Scordelis-Lo Roof ................................................................................ 62 Example of Doubly-Curved Shell ............................................................................ 65

6 CONCLUSION ........................................................................................................ 68

Summary ................................................................................................................ 68

Future Work ............................................................................................................ 69

LIST OF REFERENCES ............................................................................................... 70

BIOGRAPHICAL SKETCH ............................................................................................ 73

7

LIST OF TABLES

Table page

5-1 Maximum Displacement (×10-7 inch) 𝑟=1/10 ........................................................ 58

5-2 Maximum Displacement (×10-4 inch) 𝑟=1/100 ...................................................... 58

5-3 Maximum Displacement (×10-1 inch) 𝑟=1/1000 .................................................... 59

5-4 Maximum Radius Displacement of Cylinder Subjected to Internal Pressure (10-1 inch) ........................................................................................................... 62

5-5 Maximum Displacement of Scordelis-Lo Roof (inch) 𝑟=100 ............................... 64

5-6 Maximum Displacement of Scordelis-Lo Roof (10-1 inch) 𝑟=50 .......................... 64

5-7 Maximum Displacement of Scordelis-Lo Roof (10-2 inch) 𝑟=20 .......................... 65

5-8 Maximum Displacement (Non-dimensionalized) [0/90]T ..................................... 67

5-9 Maximum Displacement (Non-dimensionalized) [0/90]S ..................................... 67

8

LIST OF FIGURES

Figure page 2-1 Mesh Generation for A Circular .......................................................................... 17

2-2 A Shell-Like Structure in IBFEM ......................................................................... 19

4-1 Coordinate Systems ........................................................................................... 48

5-1 Geometry and Load of Clamped Square Plate ................................................... 57

5-2 Clamped Square Plate in IBFEM ........................................................................ 58

5-3 Converge Plot of Strain Energy .......................................................................... 60

5-4 Converge Plot of Maximum Displacement .......................................................... 60

5-5 Geometry and Load of Pressured Cylinder ........................................................ 61

5-6 Pressured Cylinder in IBFEM ............................................................................. 62

5-7 Geometry and Load of Scordelis-Lo Roof .......................................................... 63

5-8 Scordelis-Lo Roof in IBFEM ............................................................................... 64

5-9 Geometry and Load of Doubly-Curved Shell ...................................................... 66

5-10 Doubly-Curved Shell in IBFEM ........................................................................... 67

9

LIST OF ABBREVIATIONS

CLPT Classical Laminated Plate Theory

EBCs Essential Boundary Conditions

FEA Finite Element Analysis

FEM Finite Element Method

IBFEM Implicit Boundary Finite Element Method

SDPT Shear Deformable Plate Theory

SnS Scan and Solve

SW SolidWorks

10

Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science

MESH INDEPENDENT ANALYSIS OF 3D SHELL-LIKE LAMINATE STRUCTURES

USING ABD-EQUIVALENT MATERIAL MODEL

By

Li Liang

May 2016 Chair: Ashok V. Kumar Major: Mechanical Engineering

Mesh Independent Finite Element Analysis uses geometry imported from CAD

software to perform analysis without generating a conforming mesh to approximate the

geometry, as in the traditional Finite Element Method (FEA). The Implicit Boundary

Finite Element Method (IBFEM) is a numerical approach, where approximate step

functions are created to impose the boundary conditions by constraining the

displacement field in the prescribed manner. It uses an automatically generated

background mesh, which is independent of the geometry, to avoid the difficulty of mesh

generation and the error introduced by the traditional mesh that have distorted elements

to conform the geometry. For analysis of shell-like structures, it uses 3D stress-strain

relationship and the general principle of virtual work define a 3D-shell element. B-spline

basis functions are used to interpolate the displacement field within the shell so that

tangent (C1) continuity is guaranteed.

In this thesis, the 3D shell element in IBFEM is extend for modeling shell-like

structures that are made of composite laminate using ABD-equivalent material model

the for composite laminate. Laminate can be defined by specifying the sequence of

laminas in the laminate, the fiber orientations, the material properties and the thickness

11

of each lamina. The effective properties (ABD matrix) of the laminate can be determined

by combining properties of each lamina. For 3D shell elements, we need an equivalent

stress-strain relation for the laminate. An ABD-equivalent 3D stress-strain relation for

an equivalent 3-layer composite laminate (ABD-equivalent Material model for laminate)

can be built. This stress-strain relation for the laminate is then transformed into global

coordinate system. Some practical plate and shell examples with different geometry and

boundary conditions are analyzed and the results are compared with analytical

solutions, if available, as well as results obtained from commercial FEA software.

Finally, the advantages and limits of this method are discussed.

12

CHAPTER 1 INTRODUCTION

Traditionally, the Finite Element Method (FEM) [1, 2] uses a mesh to

approximate the geometry of the structure to be analyzed. Although this is an effective

method, it has some difficulties as well. For instance, automatically generating a mesh

for complex geometry is difficult. Regeneration of the mesh is needed for large

deformation analysis, because the original elements are distorted significantly when the

structure is loaded, and also analysis of crack propagation. Some methods including

meshless and mesh independent methods were invented to avoid mesh generation

process. Meshless methods use nodes scattered within the geometry to perform

analysis without connecting those nodes to form elements. Examples including Moving

Least Square method [3], Element-Free Galerkin Method [4], Meshless Local Petrov-

Galerkin Method [5] and so on. Those methods are effective in many cases but they

have their own difficulties as well.

Another alternative approach is Implicit Boundary Finite Element Method

(IBFEM) [6-9] which imposes the Essential Boundary Conditions (EBCs) by employing

step functions to construct test and trial functions. It generates the mesh automatically

and the mesh is independent of the geometry so that it is not necessary to conform to

the geometry. The accuracy can be improved because accurate geometry will be

directly used for the analysis. Meanwhile, the difficulties of traditional mesh can be

overcome since the mesh is independent of the geometry, especially for complex

geometry.

The composite material can provide high strength to weight and stiffness to

weight ratio along with lots of other advantages which give it wide usage in aerospace,

13

sports and other fields where high performance materials is needed [10, 11]. In general,

a composite laminate is difficult to analyze using 3D elements since it can be formed by

a large number of layers which could have different material properties. Therefore, each

layer in a laminate will need to be analyzed independently. All of these reasons will

make the numerical computation very computation expensive.

One way to overcome this difficulty is to employ ABD-equivalent model for

laminate where original multi-ply laminate is replaced by a new laminate that has a

fewer number of plies (3 plies) which will behave similar to the original multi-ply laminate

based on the effective laminate properties [12]. By doing that, only three laminae,

instead of the origin number of laminae, which can easily be over a hundred, need to be

analyzed. Thus, the total time taken for computation can be reduced dramatically and

the macro behavior of the laminate can be captured as well. Meanwhile, it can provide

stress strain relationship for 3D shell elements in IBFEM which use material stress

strain relationship directly, instead of the effective propriety (ABD matrix) of laminate.

Goals and Objectives

Goals

The main goal of this thesis is to extend the ABD-equivalent material model of

composite laminates and adapt it to 3D shell elements in Implicit Boundary Finite

Element Method and use it to analyze shell-like structures which are made of composite

laminate.

Objectives

Implement the ABD-equivalent material model of composite laminates for 3D shell elements to model flat composite laminates. The geometry is modeled as a flat surface which is made in CAD software.

14

Implement ABD-equivalent material model of composite laminates for 3D shell elements to model a curved shell that made of composite laminates. The geometry is modeled as a curved surface in CAD software.

Compare results with analytical solution, if available, and commercial FEA software where the geometry is approximated by mesh.

Outline

The remaining portion of the thesis is organized as follows:

In Chapter 2, the Implicit Boundary Finite Element Method is discussed,

especially how it is applied to the 3D shell-like structure. Details about how to derive the

weak form and construct the stiffness matrix for 3D shell are also discussed.

In Chapter 3, the properties and applications of composite structures are

discussed, followed by a detailed discussion of the Classical Laminated Plate Theory

(CLPT) or Classical Lamination Theory as well as the Shear Deformable Plate Theory

(SDPT) which are essential for setting up ABD-equivalent model of composite laminate.

In Chapter 4, the disadvantages of the traditional FEA laminate simulation and

the motivation for constructing ABD-equivalent model of composite laminate for 3D FEA

are discussed. Then the details of how the ABD-equivalent material properties of

composite laminate are computed and how it been used for 3D shell-like structure finite

element analysis is discussed. In addition, the transformation between material

coordinate system, element coordinate system and the global coordinate system are

discussed in detail.

In Chapter 5, examples of applying the ABD-equivalent material model of

composite laminates for 3D FEA in mesh independent finite element method are listed

and results have been compared and discussed. The first example is a square

composite plate which have all four edges fixed and a uniform pressure applied on the

15

top of the plate uniformly. The second example is a composite cylinder has been fixed

along its outer edge and subjected to an internal pressure. The third example is the

Barrel Vault Problem which is a vault been supported at its edges and loaded with a

vertical pressure. And the fourth example is a Doubly-Curved shell loaded with pressure

that is acting outward.

16

CHAPTER 2 MESH INDEPENDENT FINITE ELEMENT METHOD

Overview

The one of the most popular numerical method which is used to solve structural,

thermal and fluid problems is the Finite Element Method (FEM). It can solve problems

with arbitrary geometry combined with nonlinear, coupling and so on, for which it is

almost impossible to find an analytical solution. A mesh is generated to represent the

arbitrary geometry with simple shaped elements like triangles or tetrahedrons. By using

this approach, the arbitrary geometry can be split into pieces of simple geometry and

analyzed. However, there are some draw backs of this mothed. First of all, the geometry

will lose some accuracy when a mesh is used and it will introduce error. Secondly, for

complex geometry, it is very difficult and time consuming to generate a suitable mesh

and even a fine mesh will still introduce error because the mesh is always an

approximation of the geometry no matter how fine it is. Although the error cause by

geometry approximation will become smaller and smaller as the mesh goes finer and

finer, it is very computationally expensive to use a very dense mesh.

In order to avoid the difficulties caused by the mesh and mesh generation

process, numbers of meshless or mesh-free techniques have been developed.

Belytschko.T et al. [13] introduced Element Free Galerkin Method, a meshless method.

J.J. Monaghan [14] explained how mesh-free methods can be used to solve

astrophysical problems. In this approach, nodes are scattered all over the system which

needs to be analyzed and the system can be solved based on the nodes. (Figure 2-1)

17

Figure 2-1. Mesh Generation for A Circular. A) Traditional FEM; B) Meshless FEM; C) Mesh Independent FEM

Another alternate approach of the mesh-free method solves the system based on

a structured, non-conforming mesh, in other word, the mesh is independent of the

geometry. Because the mesh does not depend on the geometry of the structure, there

is no geometry approximation during meshing process and the error caused by using a

mesh to represent the geometry will goes to zero. Meanwhile, since the mesh does not

necessary need to conform the geometry, it is very easy to generate uniform mesh with

regular shape such as rectangle or cuboid, which will further reduce the error cause by

Jacobian transformation.

To satisfy the essential boundary conditions Hollig [15] constructed B-spline finite

elements by using a function shows as.

( ) ( ) ( )au x x U x a (2-1)

u is a field variable in the equation and u a must be satisfied at along a given

boundary a . By defining the function ( ) 0a x for any 𝑈(𝑥) when there is an essential

boundary condition present, the boundary conditions can be guaranteed to satisfy along

a certain boundary.

. .

. . .

. . .

. .

. .

. . . .

A B C

18

Formulation of 3D Element (3D-Shell)

The implicit boundary finite element method uses implicit equation of geometry in

its solution structure as follows.

hu Hu a (2-2)

The definition of the variables in the equation above are listed as below

𝑢 : the trial function

𝑢ℎ : the piecewise approximation of the element of the structured grid derived from the implicit equation of the boundary.

𝑎 : the boundary value function.

𝐻 : the step function which that has a unit value inside the domain of analysis and on any free boundaries, whereas it equal to 0 at the boundaries where an essential boundary condition is specified.

The function can be guaranteed to satisfy the boundary condition since the step

function value is set equal to zero at the boundary.

The main application here is to implement this method for shell-like structures.

For a shell-like structure, the mid-plane surface and the thickness of the shell are two

things that used to define the structure. A parametric surface can be used for

representing the mid-plane of a shell structure as ( , )X and its boundaries be defined

using a set of oriented edges ( )i that are defined parametrically as

( ( ), ( ))

( ) ( ( ), ( )) ( 1, , )

( ( ), ( ))

i i

i i i b

i i

x

y i n

z

(2-3)

Where bn is the number of boundaries. The ith boundary curve has a domain of

0 1[ , ]i i and its location in parameter space of the shell mid-surface is ( ( ), ( ))i i .

19

The boundaries of the mid-plane surfaces are the edges of the plane and the

vectors can be defined, and a typical shell can be shown as Figure 2-2

Figure 2-2. A Shell-Like Structure in IBFEM

The in is the normal on the boundary and it can be defined as

, ,

, ,

i

x xn

x x

(2-4)

The it is the tangent vector on the boundary which can be defined as

ii

dt

d

(2-5)

The ib is the binormal on the boundary which can be calculated as

i i ib n t (2-6)

So, any point in the shell-like structure are well defined, and the coordinate can

be denoted as

𝑦

𝑧

𝑥

( , , )X

( )i

20

( , , ) ( , )2

hX x n (2-7)

Where [ 1,1] and h is the thickness of the shell.

Under the edge coordinate system, points in the vicinity of oriented edges can be

expressed easily as the equation

ˆˆ( , , ) ( )i i iX n b (2-8)

Stiffness Matrix for 3D-Shell Element

The displacement field within the shell must satisfy the weak form for

elastostatics, which can be written in the following form for shell-like geometry,

1

0

1 1 1

1 1 12 2 2

t

t

T T T Ttb

dh h hd d u T d d u F d d u fd

d

(2-9)

Since the shell is defined as its mid-plane surface, the volume integration of the

weak form can be modified as integral through area of the mid-plane (domain ) and

integral through the thickness of the shell. is the virtual strain, u is the virtual

displacement, T is the traction acting on the edge t , bF is the body force and f is

pressure load per unit area acting normal to the shell.

The stresses and strains are separated into two parts, the homogeneous part h

and the boundary value part a , and they can be shown as

h a (2-10)

( )h a h aC C (2-11)

Where the homogeneous part and boundary value part strains can be calculated

based on the displacement field as

21

1

2

hhjh i

j i

uu

x x

(2-12)

1

2

ja i

j i

aa

x x

(2-13)

Submit the modified stresses and strains equation back to the weak form of the

elastostatics, equation (2-9) will become

1

0

1 1 1

1 1 1

1

1

2 2 2

2

t

t

T h T a T t

T T

b

dh h hC d d d d u T d d

d

hu F d d u fd

(2-14)

The traction T can be calculated as

2

2

2

6

3(1 )

2

6

t b

n

b t

M P

h h

VT

M P

h h

(2-15)

Where tension bP in the negative binormal direction, shear force nV in the normal

direction, bending moment tM about the tangent axis, and torque bM about the

binormal axis can be defined with respect to the edge coordinate system.

The displacement field trial and test functions within an element are

approximated using a B-spline shape functions, which can be shown as

ii i i

i i

u H N u N a HNu Na (2-16)

i i

i

u H N u HN u (2-17)

22

So the strains field can be derived by combining the definition of strains and

shape functions. The strain-displacement 𝐵 function can be split into two parts. They

are 𝐵1, which contains only the derivatives of the shape functions, and 𝐵2, which

contains derivatives of the approximate step functions, and it can be shown as

1 2 3 1 2 3 3

1 2 1 2

[ ]

[ ]

i i ii i i

i i i

i ii i

i i

B u B u B a B B u B a Bu B a

B u B u B B u B u

(2-18)

Plug in the terms inside and rewrite the 𝐵 matrix as

11 , 11 , 11 ,

1 22 , 22 , 22 ,

33 , 33 , 33 ,

11, 11, 11,

2 22, 22, 22,

33, 33, 33,

, , ,

3 , , ,

,

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

i x i y i z

T

i i y i x i z

i z i y i x

x y z

T

i i y x z

z y x

i x i y i z

T

i i y i x i z

i z

H N H N H N

B H N H N H N

H N H N H N

H H H

B N H H H

H H H

N N N

B N N N

N

, ,i y i xN N

(2-19)

Using these definitions of the strain-displacement 𝐵 function, the weak form

above (equation(2-14)) can be reformed as the standard discrete form as the traditional

finite element method as

1 1

( )T

ne neT T T

e e e e be fe ae e Te

e e e E

u K u u F F F u F

(2-20)

Where each terms can be computed as

1

12

e

T

e

hK B CB d d

(2-21)

23

1

3

12

e

T

ae

hF B CB a d d

(2-22)

1

0

1

12

e

e

T tTe

dhF N T d d

d

(2-23)

1

12

e

T

be b

hF N F d d

(2-24)

e

T

feF N fd

(2-25)

For the internal elements, in other words the element is completely inside and

there is no edges passing through the element, all the terms that relate to the boundary

will vanish and the strain-displacement matrix 𝐵 can be simplified as 𝐵3. Therefore,

equation used for computed the stiffness (equation (2-21)) can be simplified as

1 1

3 3 3 3

11 12 2

t

e i

nT T

e

i A

h hK B CB d d B CB d dA

(2-26)

For elements that contain edges with specified displacements boundary

conditions, the derivation of the approximate step functions are exist but only in a small

transition width. By making this transition width very narrow, say magnitude of 10-5, it is

reasonable to suppose that transition area is entirely inside the elements which edges

pass through. Because 𝐵2 only contain the derivation of the step functions so its value

will go to zero outside the transition width, the stiffness matrix computational equation

(equation(2-21)) can be simplified as

1

1 2 1 2 1 2 2 31

( ) ( )2

e

T T T

e e e e e

hK B B C B B d d K K K K

(2-27)

Where each term can be computed individually as

24

1

1 1 11

1 2

t

i

nT

e

i A

hK B CB d dA

(2-28)

1

0

1

2 1 2

1 02

e

e

T

e

hK B CB d d d

(2-29)

1

0

1

3 2 2

1 02

e

e

T

e

hK B CB d d d

(2-30)

Boundary Condition

In the implicit boundary method, the trial function and the test function are

defined as the equation (2-31)

h

h

u Hu a

u H u

(2-32)

𝑎 is the boundary value function and its value at the boundary is equal to the

essential boundary condition that applied at that boundary. The value of the step

function should goes zero at the boundaries where an essential boundary condition is

specified so that the boundary condition will be satisfied. The definition of approximate

step functions depends on the type of boundary conditions. Typical essential boundary

conditions for shells include three types: they are clamped, simply supported and

symmetric boundary condition.

Clamped

The edge-face of the shell can be defined as

( , ) ( )2

f

i i i

hn (2-33)

A point near the boundary can be defined based on the definition of the edge-

face coordinate system and its coordinate can be written as

25

( , , ) ( )i i iX n b (2-34)

Since the displacement field are equal to zero at edges where the fixed boundary

applied, the step function can be define as

11 22 33

1

2 0

0 0

H H H

(2-35)

Where is the binormal-component of the position vector of the point of interest

in the edge coordinate system and is a small distance. The is set to be small to

make that step-function can transition from 0 to 1 within a small distance. Various step

function can be used but the slop should not equal to zero around the edge to make

sure the derivation is existed.

Simply Supported

The essential boundary condition on a simply supported edge can be satisfied

with the following conditions: The displacement towards binormal direction equal to zero

at the edge and the shell is free to rotate about the edge. In the same time, there are no

external moments applied on the same edge. Define the radial distance from the edge

as

2 2 (2-36)

The step function for this kind of boundary condition can be defined as

1

2 0

0 0

iiH

(2-37)

26

Again, the is a small distance and set small to make that step-function can

transition from 0 to 1 within a small distance

Symmetry Boundary Condition

Symmetry boundary condition can be used to reduce the size of the mold

effectively, for instance, only half of a structure will need to be model if one symmetry

face (edge) exists in the structure. Since in the IBFEM, the boundary conditions on shell

are defined as displacement and rotation for each edge, the symmetry condition can be

easy to set up by setting displacement in specific direction or rotation around specific

axis equal to zero based on the symmetry type (symmetry or anti-symmetry) and the

face (edge) of symmetry. Take a shell structure with one symmetry edge for instance,

the nodes on edge should not have displacement along the binormal direction of the

symmetry edge. The rotation about the normal and tangent direction of that edge will

also need to be set equal to zero.

27

CHAPTER 3 ANALYSIS OF COMPOSITE MATERIAL

Overview

Composites, which consist of two or more separate materials combined in a

macroscopic structural unit, are made from various combinations of the other materials

such as metals, polymer and ceramic [10]. Although many man-made material have two

or more constituents, they are generally not considered as composites if the structural

unit is formed at the microscopic level rather than the macroscopic level. For example,

alloys and ceramics are made of many constituents, but they are no considered as

composite under this definition.

The composite material can provide a high strength to weight and stiffness to

weight ratio. In addition, they can provide a variety of other advantages such as

corrosion resistant, friction and wear resistant, vibration damping, fire resistant,

acoustical insulation, etc. The particle-reinforced composite, fiber-reinforced composite

and composite laminate are the most frequently used types of composites.

One important advantage that composites have over other materials is the

composite material itself is designable, in other words, the material properties of

composite can be designed according to the requirement. A deep understanding of the

material, however, is needed to design the composite properly. Experimental methods

can be used to determine the properties of composite by performing various tests on the

testing machine according to the standards. Because experimental methods can only

be applied on simple geometries, mostly a bar, subjected to simple load, the usage of

finite element method (FEM) [16-18], which can perform analysis of complex structures,

is increasing. Composite laminate with plate and shell form are the most common types

28

of composite laminate that are used in the industry. So it is important to derive the

equations of composite laminate that can be used for plate or shell. This chapter

contains the theory and equations that are commonly used for analyzing composite

laminate in a plate or a shell form.

Stress-Strain Relations of a Composite Lamina

From the generalized Hooke’s law, the Stress-Strain relationship of a material is

defined as:

11 12 13 14 15 161 1

21 22 23 24 25 262 2

31 32 33 34 35 363 3

41 42 43 44 45 4623 23

51 52 53 54 55 5631 31

61 62 63 64 65 6612 12

C C C C C C

C C C C C C

C C C C C C

C C C C C C

C C C C C C

C C C C C C

(3-1)

Where the ij and ij is the stress and strain vector respectively and ijC is

the material stiffness matrix. The equation above gives the Stress-Strain relationship for

an anisotropic material. Because of the symmetric of the stiffness matrix, there are only

21 independent material stiffness coefficients.

If there is one symmetric plane for material properties, some terms in the material

stiffness matrix will reduce to zero and the Stress-Strain relationship can be simplified

as

11 12 13 161 1

21 22 23 262 2

31 32 33 363 3

44 4523 23

54 5531 31

61 62 63 6612 12

0 0

0 0

0 0

0 0 0 0

0 0 0 0

0 0

C C C C

C C C C

C C C C

C C

C C

C C C C

(3-2)

29

The number of independent material stiffness coefficients will reduce to 13, and

this material is called monoclinic material.

If there are two symmetric planes exist in one material in the same time, the third

plane of material properties will become symmetric plane automatically. Four more

terms in the material stiffness matrix will reduce to zero and the Stress-Strain

relationship will become

11 12 131 1

21 22 232 2

31 32 333 3

4423 23

5531 31

6612 12

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

C C C

C C C

C C C

C

C

C

(3-3)

This type of material is called orthotropic material and it only has 9 independent

stiffness coefficients.

If there is a material such that the properties are equal in all the directions at any

point in a particular plane, which means the 2 and the 3 in the stiffness coefficients are

interchangeable, the Stress-Strain relation can be given by

11 12 121 1

21 22 232 2

21 32 223 3

22 2323 23

6631 31

6612 12

0 0 0

0 0 0

0 0 0

0 0 0 ( ) / 2 0 0

0 0 0 0 0

0 0 0 0 0

C C C

C C C

C C C

C C

C

C

(3-4)

The material is called transversely isotropic material and the number of

independent stiffness coefficients are 6.

30

Finally if all the planes are symmetry plane for the material, the Stress-Strain

relation becomes:

1 11 12 12 1

2 21 11 12 2

3 21 21 11 3

23 11 12 23

31 11 12 31

12 11 12 12

0 0 0

0 0 0

0 0 0

0 0 0 ( ) / 2 0 0

0 0 0 0 ( ) / 2 0

0 0 0 0 0 ( ) / 2

C C C

C C C

C C C

C C

C C

C C

(3-5)

The material is called isotropic material and there are only 2 independent

material stiffness coefficients.

The composite plate and shell that people are dealing with in most cases are

made of laminate which behalves as orthotropic material.

If the fibers are only aligned in the 1 and 2 direction, that is, no fibers are aligned

in the thickness direction, the discussion can be limited to rotation of the coordinate

system only about the 3-axis. Applying plane stress condition, the Strain-Stress

relationship will becomes

1 11 12 16 1

2 21 22 26 2

12 61 62 66 12

Q Q Q

Q Q Q

Q Q Q

(3-6)

Where the ijQ is the material constants and they can be determine by the

equations below

31

111

12 21

222

12 21

12 2 21 112 21

12 21 12 21

66 12

16 61

26 62

1

1

1 1

0

0

EQ

v v

EQ

v v

v E v EQ Q

v v v v

Q G

Q Q

Q Q

(3-7)

A transformed stiffness matrix is used to show the Strain-Stress relationship in

the global x-y coordinate system instead of the 1-2 material coordinate system. Strain-

Stress relationship in the global x-y coordinate system can be show as

11 12 16

21 22 26

61 62 66

x x

y y

xy xy

Q Q Q

Q Q Q

Q Q Q

(3-8)

By applying the transformation matrix for the stresses and strains, the

transformed stiffness matrix in the global x-y coordinate system can be obtained.

123

123

[ ]

[ ][ ]

[ ][ ][ ]

xyz

xyz

T

xyz xyz

xyz xyz

T

T Q

T Q T

Q

(3-9)

Where the transformation matrix is

2 2

2 2

2 2

cos sin 2sin cos

[ ] sin cos 2sin cos

sin cos sin cos cos sin

T

(3-10)

So, according to the equation (3-8), (3-9) and (3-10), the transformed stiffness

matrix can be obtained as

32

[ ][ ][ ]TQ T Q T (3-11)

Or each terms can be written explicitly as

4 4 2 2

11 22 12 6611

4 4 2 2

11 22 12 6622

2 2 4 4

11 22 66 1212

3 3

11 12 66 22 12 6616

11 12 6626

cos sin sin cos

sin cos sin cos

( 4 )sin cos (sin cos )

( 2 )cos sin ( 2 )

2( 2 )

2( 2 )

cos sin

( 2

2

Q Q Q Q Q

Q Q Q Q Q

Q Q Q Q Q

Q Q Q Q Q Q Q

Q Q Q Q

3 3

22 12 66

2 2 4 4

11 22 12 66 6666

) cos sin ( 2 )cos sin

( 2 2 )cos sin (cos sin )

Q Q Q

Q Q Q Q Q Q

(3-12)

Classical Laminated Plate Theory (CLPT)

Classical laminate theory (CLPT) was apparently developed in the 1950s and

1960s investigated by investigators such as Smith [19], Pister and Dong [20], Reissner

and Stavsky [21], Stavsky [22], Stavsky and Hoff [23], etc. The CLPT is used to analyze

thin plates by ignoring transverse shear stresses.

The basic assumptions of this theory are:

1. The x-y plane is the middle plane of the plate and the z-axis is in the thickness direction;

2. The plate contain several layers which bond perfectly and the 3rd-direction (thickness direction) of the material principal coordinate system coincides with the z-axis;

3. The thickness h of the plate is much smaller than the plate’s lateral dimensions in the x-y plane;

4. Displacements 𝑢, 𝑣, and 𝑤 are very small compared to the plate’s thickness ℎ;

5. The in-plane strains 𝜀𝑥, 𝜀𝑦 and 𝛾𝑥𝑦 are small compared to unity;

6. The transverse shear strains 𝛾𝑧𝑥 and 𝛾𝑦𝑧 are negligible and so is the transverse

normal strain 𝜀𝑧,

7. The transverse normal and shear stresses 𝜏𝑧𝑥, 𝜏𝑦𝑧 and 𝜎𝑧 are negligibly small

compared to the in-plane normal and shear stresses 𝜏𝑥𝑦, 𝜎𝑦 and 𝜎𝑥;

33

Using the assumptions above, the displacement fields ( , , )u x y z , ( , , )v x y z and

( , , )w x y z can be written as Taylor series expansion in terms of 𝑧 as

2

0 1 2( , , ) ( , ) ( , ) ( , )u x y z u x y z x y z x y (3-13)

2

0 1 2( , , ) ( , ) ( , ) ( , )v x y z v x y z x y z x y (3-14)

2

0 1 2( , , ) ( , ) ( , ) ( , )w x y z w x y z x y z x y (3-15)

Since the 𝑧 will very small because the ply is very thin according to the basic

assumption, the linear terms will dominate the equations(3-13), (3-14) and (3-15). So, if

only the linear terms are taken in to consideration, the equation (3-13) and (3-14) above

will reduce to

0 1( , , ) ( , ) ( , )u x y z u x y z x y (3-16)

0 1( , , ) ( , ) ( , )v x y z v x y z x y (3-17)

Where 0 ( , )u x y and 0 ( , )v x y is known as mid-plane displacement. It means the

displacements of a point located in the middle plane of the plate.

The transverse displacement 𝑤 will be independent of the z-coordinate which

means the 𝑤 displacement is a function only of 𝑥 and 𝑦 according to the assumption,

hence the equation (3-15) will reduce to

0( , , ) ( , )w x y z w x y (3-18)

However if the structure is loaded with large transverse load which make 𝜎𝑧

become non-negligible, then equation (3-18) will not be valid and other assumptions

must be made. But the in the most cases where the thin composite laminate are

employed, the transverse load will be small and the equation (3-18) will work.

From the definition of the strain,

34

1

2

jiij

j i

uu

x x

(3-19)

The transverse shear strains can be computed as

yz

zx

v w

z y

w u

x z

(3-20)

By plugging in the equations (3-16) and (3-17) into the equations (3-20), it will

become

1

1

( , )

( , )

yz

zx

wx y

y

wx y

x

(3-21)

Since the transverse shear stress are negligible in the plate based on the

assumptions, that is, both theyz and zx are equal to zero at any point, equations (3-21)

can be simplified as

1

1

( , )

( , )

wx y

y

wx y

x

(3-22)

Submitting the equation (3-22) into equation (3-16) and (3-17) to calculate the in-

plane displacement, the displacement can be computed as

0

0

( , , ) ( , )

( , , ) ( , )

wu x y z u x y z

x

wv x y z v x y z

y

(3-23)

By using the definition of the strain (equation (3-19)), the in-plate strains can be

obtained as

35

0

i i iz (3-24)

It can be shown explicitly as

0

0

0

x x x

y y y

xy xy xy

z

z

z

(3-25)

Where 0

x , 0

x and 0

xy are the mid-plane strain and each of them can be

computed as

0 0

0 0

0 0 0

x

y

xy

u

x

v

y

u v

y x

(3-26)

x , y and

xy are the mid-plane curvature and each of them can be computed

as

2

2

2

2

2

2

x

y

xy

w

x

w

y

w

x y

(3-27)

Equation (3-24) can be put into matrix form as

0

0

0

x x x

y y y

xy xy xy

z

(3-28)

36

The in-plane force resultants 𝑁𝑥, 𝑁𝑦 and 𝑁𝑥𝑦 are defined as the forces per unit

length along the edge. They can calculated by integrating the stresses on the edge

through the thickness direction as

/2

/2

/2

/2

/2

/2

h

x x

h

h

y y

h

h

xy xy

h

N dz

N dz

N dz

(3-29)

Equation (3-29) can be rewritten in matrix form as

/2

/2

x xh

y y

h

xy xy

N

N dz

N

(3-30)

Submitting the plane stress constitutive relationship (Stress-Strain relationship)

equation (3-8) in to the equation (3-30), the in-plane force resultants can be computed

as

11 12 16/2

21 22 26

/2

61 62 66

x xh

y y

h

xy xy

Q Q QN

N Q Q Q dz

N Q Q Q

(3-31)

By submitting the relationship between strains for any point and mid-plane strain

and curvature equation(3-28) , the in-plane force resultants can be determined by the

equation (3-32) show as

011 12 16

/2

0

21 22 26

0/2

61 62 66

x x xh

y y y

h

xy xy xy

Q Q QN

N Q Q Q z dz

N Q Q Q

(3-32)

37

Equation (3-32) can be rewritten in index notation as

/2

0

/2

h

i i iij

h

N Q z dz

(3-33)

Since the mid-plane strain and curvature are independent of the thickness, in

other words, they remain constant in the thickness direction, hence, they can be taken

out of the integral in the equation (3-33) as

/2 /2

0

/2 /2

h h

i i iij ij

h h

N Q dz Q zdz

(3-34)

And the equation (3-34) can be further simplified as

0

i i iN A B (3-35)

Where [𝐴] and [𝐵] are 3 by 3 matrix defined as

/2

/2

h

ij

h

A Q dz

(3-36)

/2

/2

h

ij

h

B Q zdz

(3-37)

Similarly to the in-plane force resultants, the in-plane moment resultants 𝑀𝑥, 𝑀𝑦

and 𝑀𝑥𝑦 are defined as the moments per unit length and they can be obtained by

/2

/2

/2

/2

/2

/2

h

x x

h

h

y y

h

h

xy xy

h

M zdz

M zdz

M zdz

(3-38)

Equations (3-38) can be put into matrix as

38

/2

/2

x xh

y y

h

xy xy

M

M zdz

M

(3-39)

Submitting the plane stress constitutive relationship equation (3-8), and the

relationship between strain at any point and in-plane strain and curvature

(equation(3-28)) the equation (3-39) above will become

11 12 16/2

21 22 26

/2

61 62 66

x xh

y y

h

xy xy

Q Q QM

M Q Q Q zdz

M Q Q Q

(3-40)

011 12 16

/2

0

21 22 26

0/2

61 62 66

x x xh

y y y

h

xy xy xy

Q Q QM

M Q Q Q z zdz

M Q Q Q

(3-41)

Equation (3-41) can be rewritten in index notation as

/2

0

/2

h

i i iij

h

M Q z zdz

(3-42)

And by taking out the in-plane strain and curvature which are independent of the

thickness, equation (3-42) will become

/2 /2

0 2

/2 /2

h h

i i iij ij

h h

M Q zdz Q z dz

(3-43)

The equation (3-43) can be rewritten as

0

i i iM B D (3-44)

In equation (3-44) [𝐵] matrix is the same as it derived from force resultants and

[𝐷] matrix is a 3 by 3 matrix defined as

39

/2

2

/2

h

ij

h

D Q z dz

(3-45)

To sum up, [𝐴], [𝐵] and [𝐷] are all 3 by 3 and called laminate stiffness matrices,

which known as follows:

[𝐴] : In-plane stiffness matrix

[𝐵] : Coupling stiffness matrix

[𝐷] : Bending Stiffness matrix

And the definition of those matrixes are

11 12 16/2

12 22 26

/2

16 26 66

h

ij

h

A A A

A Q dz A A A

A A A

(3-46)

11 12 16/2

12 22 26

/2

16 26 66

h

ij

h

B B B

B Q zdz B B B

B B B

(3-47)

11 12 16/2

2

12 22 26

/2

16 26 66

h

ij

h

D D D

D Q z dz D D D

D D D

(3-48)

Because both the force resultants and moment resultants are defined in a

similarly form so the equation (3-35) and equation (3-44) can be combined together and

rewritten as matrix form as

0N A B

M B D

(3-49)

It can be written explicitly as:

40

0

11 12 16 11 12 16

0

12 22 26 12 22 26

0

16 26 66 16 26 66

11 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

x x

y y

xy xy

x x

y y

xy xy

N A A A B B B

N A A A B B B

N A A A B B B

M B B B D D D

M B B B D D D

M B B B D D D

(3-50)

Shear Deformable Plate Theory (SDPT)

The CLPT is set up based on assumptions that transverse normal and shear

stresses are neglected. Although it is a good approach for a thin structure, the answers

will become inaccuracy when the structure goes thicker. In order capture the effects of

the transverse stresses, Mindlin [24] and Reissner [25] developed the Shear

Deformable Plate Theory (SDPT) for a thick plate, in which the transverse stresses are

also taken into consideration.

In the shear deformable plate theory, the out-of-plane shear strains are not be

ignored, that is yz and zx are no longer assumed to be zero. The displacement fields of

equations (3-16) and (3-17) will become:

0

0

( , , ) ( , ) ( , )

( , , ) ( , ) ( , )

x

y

u x y z u x y z x y

v x y z v x y z x y

(3-51)

Where the ( , )x x y and ( , )y x y are the rotation of the edge surface. The in-plane

strains will be the same as those in the CLPT but the equations in-plane curvature will

become

41

xx

y

y

yxxy

x

y

y x

(3-52)

In addition to the in-plane force and moment resultants which are same as those

in the CLPT, the shear force resultants also exist and can be defined similarly as

follows:

/2

/2

hy yz

x zxh

Qdz

Q

(3-53)

Transverse shear stresses can be computed as

44 45

45 55

yz yz

zx zx

Q Q

Q Q

(3-54)

Stiffness matrix for the out-of-plane shearing in the x-y-z global coordinate

system can be derived as

44 4544 45

45 5545 55

cos sin cos sin

sin cos sin cos

Q Q Q Q

Q QQ Q

(3-55)

Applying Stress-Strain relationship for the out-of-plane shearing, stiffness matrix

for the out-of-plane shearing in the x-y-z global coordinate system can calculated as

2344 45

3145 55

0cos sin cos sin

0sin cos sin cos

Q Q G

GQ Q

(3-56)

Submitting the equation (3-54) in to equation(3-53), the shear force resultants

can be obtained as

42

/244 45

/2 45 55

hy yz

x zxh

Q QQdz

Q Q Q

(3-57)

In the matrix form, equation(3-57) can be written as

y s sQ A (3-58)

Where

/2

44 4544 45

45 55/2 45 55

h

s

h

Q Q A AA dz

A AQ Q

(3-59)

Equation(3-35), (3-44) and (3-58) can be combined together to form constitutive

relation of the shear deformable laminate as

00

0

0 0s s s

N A B

M B D

Q A

(3-60)

Equation (3-60) can be written explicitly as

11 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

11 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

44 45

45 55

0 0

0 0

0 0

0 0

0 0

0 0

0 0 0 0 0 0

0 0 0 0 0 0

x

y

xy

x

y

xy

y

x

N A A A B B B

N A A A B B B

N A A A B B B

M B B B D D D

M B B B D D D

M B B B D D D

Q A A

A AQ

0

0

0

x

y

xy

x

y

xy

yz

zx

(3-61)

Instead of effective laminate properties (ABD matrix) in plate/shell theory, the

IBFEM uses 3D shell element formulation which requires an effective material stress-

strain relationship. So, the effective laminate properties need to be modified to a new

form that can be implemented into the IBFEM.

43

CHAPTER 4 ABD-EQUIVALENT MATERIAL MODEL OF COMPOSITE LAMINATE

Overview

In general, a composite laminate is difficult to analyze using 3D shell elements in

IBFEM since it can be formed by a large number of layers. When the structure is

loaded, different layer will perform differently because different layers of the composite

laminate have different material properties. Integration in the thickness direction is

needed for volume integration. That will make the numerical integration required for

volume integration present within each element extremely computationally expensive.

To fully integrate a linear brick element with constant material coefficients, 8 integration

points are needed for Gauss’s quadrature rule. [26] Say there is a laminate with 100

layers and 8 points are used for integration of the thickness direction for each layers, all

in all there are 800 integration points are used for the thickness direction for each

element. It is very computational expensive for a structure which involved large number

of elements [12].

Another approach is replacing the original multi-ply laminate by an equivalent

laminate that has fewer number of plies and behaves similar to the original laminate and

results in the same stiffness matrices. An ABD-equivalent material model for laminate

can be derived according to that assumption [12]. By doing that, the time taken for

integration of the thickness direction can be reduced dramatically. Although the laminate

will perform slightly differently due to the replacement and the stresses and strains

distribution in the laminate might change, differences will be small and acceptable. Also,

if a study is mainly focused on the behavior of the laminate in the global sense instead

of focusing on stresses and strains distribution within the laminate, this approach will be

44

good enough as long as the new laminate is guaranteed to have similar behavior in the

global sense as original laminate.

The ABD-Equivalent Material Model of Laminate

According to the classical laminate theory, for a give laminate, equation (3-49)

can be used to determine the force and moment resultants

N A B

M B D

(4-1)

Where [𝐴], [𝐵] and [𝐷] matrix are In-plane stiffness matrix, coupling stiffness

matrix and bending stiffness matrix defined as equation (3-46), (3-47) and (3-48),

respectively, and they can be calculated as equations below

1

1

1

0

1

0

1

0 2

1

k

k

k

k

k

k

n zk

ij ijz

k

n zk

ij ijz

k

n zk

ij ijz

k

A Q dz

B Q zdz

D Q z dz

(4-2)

One way to ensure that a new 3-ply laminate will have similar global behavior is

to assume that the new 3-ply laminate has the same effective properties or [𝐴], [𝐵] and

[𝐷] matrix as the original multi-ply laminate. By this definition, the equation (4-2) hold its

validity for the new 3-ply laminate,

1

1

1

3*

1

3*

1

3* 2

1

k

k

k

k

k

k

zk

ij ijz

k

zk

ij ijz

k

zk

ij ijz

k

A Q dz

B Q zdz

D Q z dz

(4-3)

45

In these equations (4-2)-(4-3), 𝑄0̅̅̅̅ denotes the properties of the original plies and

𝑄∗̅̅ ̅ denotes the properties of the 3 new plies.

Assuming that the 𝑄 matrix is constant through each ply, for both original multi-

ply and the new 3-ply laminate, the integration in the equations (4-2) and (4-3) will

become a summation. As there are only 3 plies in the new laminate, equations (4-3)

can be written explicitly as

*1 *2 *3

1 0 2 1 3 2

*1 2 2 *2 2 2 *3 2 2

1 0 2 1 3 2

*1 3 3 *2 3 3 *3 3 3

1 0 2 1 3 2

( ) ( ) ( )

1[ ( ) ( ) ( )]

2

1[ ( ) ( ) ( )]

3

ij ij ij ij

ij ij ij ij

ij ij ij ij

A Q z z Q z z Q z z

B Q z z Q z z Q z z

D Q z z Q z z Q z z

(4-4)

Equations (4-4) can be put into matrix format as

*11 0 2 1 3 2

2 2 2 2 2 2 *2

1 0 2 1 3 2

*3

3 3 3 3 3 3

1 0 2 1 3 2

1 1 1( ) ( ) ( )

2 2 2

1 1 1( ) ( ) ( )

3 3 3

ijij

ij ij

ijij

z z z z z z QA

B z z z z z z Q

D Qz z z z z z

(4-5)

Also equation (4-5) can be written explicitly as

*1 *1 *11 0 2 1 3 2

11 12 3311 12 33

2 2 2 2 2 2 *2 *2 *2

11 12 33 1 0 2 1 3 2 11 12 33

*3 *3 *311 12 33

11 12 333 3 3 3 3 3

1 0 2 1 3 2

1 1 1( ) ( ) ( )

2 2 2

1 1 1( ) ( ) ( )

3 3 3

z z z z z z Q Q QA A A

B B B z z z z z z Q Q Q

D D D Q Q Qz z z z z z

(4-6)

By solving the equations above, the *

ijQ ( 1,2,3i ; 1,2,3j ) for the new 3-ply

laminate can be obtained.

Out-of-plane shear stresses also have to be taken into consideration. The shear

force resultants can be determined by the equation (3-58)

46

4 44 45 4

5 45 55 5

A AK

A A

(4-7)

Where 𝐾 is a constant and 𝐴44, 𝐴45 and 𝐴55 can be determined by

1 0

1

, 4,5k

k

n zk

ij ijz

k

A Q dz i j

(4-8)

1

3*

1

, 4,5k

k

zk

ij ijz

k

A Q dz i j

(4-9)

As it has been discussed, the in-plane stresses are much larger than the out-of-

plane stresses in most case for plane and shell-like structures. Also, because the matrix

of the composite laminate will dominate the laminate properties in the thickness

direction, the modulus 𝐺31 and 𝐺23 are relatively small and won’t vary much from plane

to plane in most cases. So, it is good enough to assume that the out-of-plane shear

properties are the same for all 3 plies for the new 3-ply laminate. Under this assumption,

equation (4-9) will simplified as

* * *

1 0 2 1 3 2( ) ( ) ( )ij ij ij ijA Q z z Q z z Q z z (4-10)

And it can be further simplified as

*

ij ijA Q h (4-11)

Out-of-plane properties *

ijQ ( 4,5i ; 4,5j ) of the new 3-ply laminate can be

obtained as

* /ij ijQ A h (4-12)

Where ℎ is the total thickness of the laminate.

The modulus of the thickness direction of the new 3-ply laminate is assumed to

be constant as the out-of-plane shear stiffness and it can be approximated as the

47

harmonic average of the Young’s Modulus of the thickness direction of each ply of the

original multi-ply laminate. Reasons for this assumption is similar to the reasons for

setting the out-of-plane shear stiffness as constants for the new 3-ply laminate, that is,

the stress in the thickness is negligible and modulus of thickness direction is small and

will not vary much from ply to ply. So, it can be computed as

*3

0 0 *1 133 33 33

k kn

k kk k

h h h

E E E

(4-13)

By solving the equations (4-13) the modulus of thickness direction of the new 3-

ply laminate can be determined.

As out-of-plane shear properties, the composite matrix material will dominate the

properties of the laminate in the thickness direction, the out-of-plane stresses are

negligible and the modulus 𝐸33 are relatively small and won’t change much from plane

to plane in most cases. So, it is good enough to assume that the properties normal to

the plane are same for all 3 ply for the new 3-ply and all the plies in the original multi-ply

laminate. Under this assumption equation (4-13) can be simplified as (4-14) in most

cases

0 *

33 33E E (4-14)

To sum up, for a typical laminate:

Solving equation (4-6), the in-plane properties of the new 3-ply laminate *( 1,2,3; 1,2,3)ijQ i j can be determined

Solving equation (4-12), the out-of-plane properties of the new 3-ply laminate *( 4,5; 4,5)ijQ i j can be determined

Finally, solving equation(4-13), the Young’s Modulus of the thickness direction can be determined

48

All in all, all the material properties of the 3 plies of the new 3-ply laminate can be

obtained.

Local and Global Stiffness Matrix

The material properties discussed above are all with respect to the coordinate

system attached to the surface passing through an element such that its z-axis is

normal to the surface. In a real problem, lots of element will be involved, the element

coordinate system attached to the surface may be translated or rotated with respect to

the global coordinate system. Since the variables like displacement, force are defined

with respect to the global coordinate system, it is important to set up the transformation

matrix between the global and the local or element coordinate system.

Figure 4-1. Coordinate Systems

There are three coordinate systems involved and the relationship between each

of them are shown in Figure 4-1.

1-2-3 coordinate system: material coordinate system;

x-y-z coordinate system: element coordinate system;

X-Y-Z coordinate system: global coordinate system;

𝑧(3)

𝑋

𝑍

𝑌

𝑥

𝑦

1

2

49

As equation in (4-15) and (4-16) if the stress-strain relationship are set up in a x-

y-z element coordinate system and an X-Y-Z global coordinate systems as

{ } { }xyz xyzQ (4-15)

{ } { }XYZ XYZQ

(4-16)

To transform a stress tensor from X-Y-Z coordinate system (global coordinate

system in most cases) to a given x-y-z coordinate system (element coordinate system in

most cases), equation (4-17) can be used.

[ ] [ ][ ] [ ]T

xyz XYZ (4-17)

Where the stress tensor are defined as

[ ]

XX XY XZ

XYZ XY YY YZ

XZ YZ ZZ

(4-18)

[ ]

xx xy xz

xyz xy yy yz

xz yz zz

(4-19)

The transformation matrix are defined as

1 1 1

2 2 2

3 3 3

[ ]

l m n

l m n

l m n

(4-20)

In the equations above 𝑙1 is the cosine of the x-axis with respect to X-axis, 𝑚1 is

the cosine of the x-axis with respect to Y-axis and 𝑛1 is the cosine of the x-axis with

respect to Y-axis. The rest of variables are defined similarly. 𝑙2 𝑚2 and 𝑛2 are the

cosines of the y-axis with respect to X, Y and Z axis, respectively, and 𝑙3 𝑚3 and 𝑛3 are

the cosines of the z-axis with respect to X, Y and Z axis, respectively.

50

The strain tensor is transformed in the same manner.

[ ] [ ][ ] [ ]T

xyz XYZ (4-21)

Where strain tensor are defined as

[ ]

XX XY XZ

XYZ XY YY YZ

XZ YZ ZZ

(4-22)

[ ]

xx xy xz

xyz xy yy yz

xz yz zz

(4-23)

The stresses are always put in to a vector form instead of a matrix form in finite

element analysis. The stress vector can be written as

{ } [ ]T

XYZ XX YY ZZ YZ ZX XY (4-24)

{ } [ ]T

xyz xx yy zz yz zx xy (4-25)

So the translation matrix should be formed as

{ } [ ]{ }xyz XYZT (4-26)

By doing the matrix multiplication in equation (4-17) and rearranging the terms to

satisfy the form of equation (4-26), the transformation matrix for stress vector can be

obtained as

2 2 2

1 1 1 1 1 1 1 1 1

2 2 2

2 2 2 2 2 2 2 2 2

2 2 2

3 3 3 3 3 2 2 3 3

2 3 2 3 2 3 2 3 3 2 2 3 3 2 2 3 3 2

1 3 1 3 1 3 1 3 3 1 1 3 3 1 1 3 3 1

1 2 1 2 1 2 1 2 2 1 1 2 2 1 1 2 2 1

2 2 2

2 2 2

2 2 2[ ]

l m n m n l n l m

l m n m n l n l m

l m n m n l n l mT

l l m m n n m n m n l n l n l m l m

l l m m n n m n m n l n l n l m l m

l l m m n n m n m n l n l n l m l m

(4-27)

51

Similarly, strains are also used in a vector (or column matrix) form instead of the

square matrix form for finite element method most often as

{ } [ ]T

XYZ X Y Z YZ ZX XY (4-28)

{ } [ ]T

xyz x y z yz zx xy (4-29)

And the transformation matrix are defined as

{ } [ ]{ }xyz XYZT (4-30)

By doing the matrix multiplication, it can be shown that transformation matrix 𝑇

for the strain tensor is same as the transformation matrix used for stress tensor defined

as equation (4-27).

It is also very important to derive the equation that is used to transform the

stresses and strain from local x-y-z coordinate system back to global X-Y-Z coordinate

system, which means

1{ } [ ] { }XYZ xyzT (4-31)

From equation (4-17), it can be obtained that

1[ ] [ ] [ ] [ ] T

XYZ xyz (4-32)

By using the definition of transformation matrix [] in equation (4-20), it can be

concluded that the inverse and the transverse of it are equal, that is

1[ ] [ ]T (4-33)

So equation (4-32) can be simplified as

[ ] [ ] [ ] [ ]T

XYZ xyz (4-34)

52

By doing the matrix multiplication in equation (4-34) and rearranging the terms to

satisfied the form of equation(4-31), the inverse of the transformation matrix for stress

tensor can be obtained as

2 2 2

1 2 3 2 3 1 3 1 2

2 2 2

1 2 3 2 3 1 3 1 2

2 2 2

1 1 2 3 2 3 1 3 1 2

1 1 2 2 3 3 2 3 3 2 1 3 3 1 1 2 2 1

1 1 2 2 3 3 2 3 3 2 1 3 3 1 1 2 2 1

1 1 2 2 3 3 2 3 3 2 1 3 3 1 1 2 2 1

2 2 2

2 2 2

2 2 2[ ]

l l l l l l l l l

m m m m n m m m m

n n n n n n n n nT

m n m n m n m n m n m n m n m n m n

l n l n l n l n l n l n l n l n l n

l m l m l m l m l m l m l m l m l m

(4-35)

The same process can be used for the strain tensor to get inverse of

transformation matrix that used for strain tensor in the equation (4-36)

1{ } [ ] { }XYZ xyzT (4-36)

It can be shown that transformation matrix is the same as the transformation

matrix for the stress vector defined as equation (4-35).

Engineering strain instead of true strain are used more frequently in FEM. The

normal strains are the same in both cases but the shear strains are different and the

relationship between engineering shear strain ij and true strain

ij can be written as:

1

( )2

ij ij i j (4-37)

If the strain are written in the vector form, the relationship can be shown as

'{ } [ ]{ }xyz xyzR (4-38)

Where

'{ } { }T

xyz x y z yz zx xy (4-39)

53

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0[ ]

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

R

(4-40)

Strain stress relationship in X-Y-Z coordinate system can be written as

{ } { }XYZ XYZQ

(4-41)

Where the transformed stiffness matrix can be computed as

1 1[ ] [ ][ ][ ]Q T Q R T R (4-42)

This can be derived as

1

1

1 '

1 '

1 1

{ } [ ] { }

[ ] { }

[ ] [ ]{ }

[ ] [ ][ ]{ }

[ ] [ ][ ][ ] { }

XYZ xyz

xyz

xyz

XYZ

XYZ

T

T Q

T Q R

T Q R T

T Q R T R

(4-43)

The transformed 𝑄 matrix can be put in the equation(4-44) to calculate the virtual

strain energy in the system according to the global x-y-z coordinate system as

0

{ } { }XYZ

T

XYZ

V

U Q dv (4-44)

Another approach to calculate the virtual strain energy in the system, can be

started with directly deriving the transformation matrix for stress vector and strain vector

respectively, shown as

{ } [ ]{ }xyz XYZT (4-45)

54

' '{ } [ ]{ }xyz XYZT (4-46)

Where

2 2 2

1 1 1 1 1 1 1 1 1

2 2 2

2 2 2 2 2 2 2 2 2

2 2 2

3 3 3 3 3 2 2 3 3

2 3 2 3 2 3 2 3 3 2 2 3 3 2 2 3 3 2

1 3 1 3 1 3 1 3 3 1 1 3 3 1 1 3 3 1

1 2 1 2 1 2 1 2 2 1 1 2 2 1 1 2 2 1

2 2 2

2 2 2

2 2 2[ ]

l m n m n l n l m

l m n m n l n l m

l m n m n l n l mT

l l m m n n m n m n l n l n l m l m

l l m m n n m n m n l n l n l m l m

l l m m n n m n m n l n l n l m l m

(4-47)

2 2 2

1 1 1 1 1 1 1 1 1

2 2 2

2 2 2 2 2 2 2 2 2

2 2 2

3 3 3 3 3 2 2 3 3

2 3 2 3 2 3 2 3 3 2 2 3 3 2 2 3 3 2

1 3 1 3 1 3 1 3 3 1 1 3 3 1 1 3 3 1

1 2 1 2 1 2 1 2 2 1 1 2 2 1 1 2 2 1

[ ]2 2 2

2 2 2

2 2 2

l m n m n l n l m

l m n m n l n l m

l m n m n l n l mT

l l m m n n m n m n l n l n l m l m

l l m m n n m n m n l n l n l m l m

l l m m n n m n m n l n l n l m l m

(4-48)

So the virtual strain in the x-y-z coordinate system can be obtained as

' '{ } { } [ ]T T T

xyz XYZ T (4-49)

And the virtual strain energy can be computed as

0 0

' ' ' '{ } { } { } { }T T

xyz xyz XYZ XYZ

V V

U Q dv Q dv (4-50)

Where

[ ] [ ]TQ T Q T

(4-51)

The same transformed stiffness matrix can be obtained from both approach.

In the IBFEM, the element stiffness matrix can be calculated as in equation

(2-21)

1

12

e

T

e

hK B CB d d

(4-52)

55

Instead of having a constant 𝐶 matrix inside one element as for isotropic

materials, the properties of composite laminate are typically orthotropic and the material

property matrix 𝐶 will be different from layer to layer. For n-ply composite, the stiffness

matrix for one element should be rewritten as

1

1

i

e i

znT i

e

i z

K B Q Bd d

(4-53)

Where the 𝑧𝑖+1 and 𝑧𝑖 denote the upper and bottom location in thickness direction

of the ith layer. In ABD-equivalent material model of laminate, the number of ply are 3

instead of 𝑛, so equation (4-53) will become

13

*

1

i

e i

z

T i

e

i z

K B Q Bd d

(4-54)

As the [𝐵] matrix is constant in the thickness direction and the material

properties are assumed to be constant through each layer for the new 3-ply laminate,

the integration of the thickness direction can be separate in to 3 parts and equation

(4-54) will become

1/3 1/3 1

*1 *2 *3

1 1/3 1/3

( )2

e e e

T T T

e

hK B Q Bd d B Q Bd d B Q Bd d

(4-55)

As can be seen, the integration in the thickness direction will reduce to 3 instead

of n for a multi-ply laminate if the new 3-ply laminate is used. By employing the ABD-

equivalent material model for the laminate, the computation will become much faster

especially for a laminate containing a large number of layers.

56

CHAPTER 5 RESULTS AND DISCUSSION

Overview

It is very important to implement the laminate ABD-equivalent model in to

programs and test it with examples to prove its validly. By comparing the answers with

the analytical solution, if available, or with answers from commercialized FEM software,

the laminate ABD-equivalent material model can be verified.

The commercialized software that has been used is Solidworks. SolidWorks is

a solid modeling Computer-Aided Design (CAD) and Computer-Aided

Engineering (CAE) software program that been widely used in engineering. SolidWorks

has been marketed by the Dassault Systemes since 1997. It is a very popular software

that can be used for solid modeling and drawing generation. It also contains

functionality for FEA which allows users to perform structural analysis. .

Implicit Boundary Finite Element Method (IBFEM) can directly use the geometry

created in CAD software for analysis without generating a mesh to approximate the

geometry so that the accuracy of the geometry can be guaranteed. It allows loads and

boundary conditions to be applied directly on the solid and provides a platform for

implementing the ABD-equivalent material model of composite laminate for 3D analysis.

Several examples have been analyzed both in Solidworks and IBFEM, answers

are been compared with each other and analytical solutions, if available, and discussed

in this chapter.

Example of Square Plate

The first example is a square plate clamped on all four edges, and it is subject to

a uniform pressure normal to the plate as shown in Figure 5-1.

57

Figure 5-1. Geometry and Load of Clamped Square Plate

A plate under these boundary conditions shows pure bending, with the maximum

displacement at the center. The plate is 10 inch by 10 inch and the uniform pressure

applied is 0.01 psi. Various thickness (1 inch, 0.1 inch and 0.01 inch) are used and the

ratio r is defined as

/r h a (5-1)

The ratio is defined to indicate whether the plate is thick or not. A typical thin

plate should have a ratio smaller than 0.1.

The material properties are used are:

6 6 6

1 2 3

12 13 23

5 5 5

12 13 23

25 10 , 1 10 , 1 10

0.25, 0, 0

5 10 , 5 10 , 2 10

E psi E psi E psi

G psi G psi G psi

(5-2)

Two type of laminate, angled ply (−45 / 45)𝑛 and crossed ply (0/ 90)𝑛 are used

in this example and the maximum displacement are calculated. The mesh and the

displacement of 10 layer angled ply when 𝑟=1/100 are plotted (Figure 5-2). In addition,

the results are listed in Table 5-1, Table 5-2 and Table 5-3 (The SW, ANSYS and SnS

𝑦

𝑥

𝑦

𝑎

𝑎

ℎ

𝑝

𝑧

𝑎

58

results are reported by Kumar and Shapiro [12], and the analytical solution is report by

Reddy [27]).

Figure 5-2. Clamped Square Plate in IBFEM. A) Geometry and Mesh; B) Displacement

of 10 layer angled ply (𝑟=1/10)

Table 5-1. Maximum Displacement (×10-7 inch) 𝑟=1/10

Reddy IBFEM SW ANSYS SnS Element Number 225 1k 10k 1k 3k

2 angled 3.891 7.846 6.984

2 crossed 3.814 7.848 6.515

10 angled 4.286 4.36 5.094 4.057 4.152

10 crossed 3.981 4.058 4.748 3.855 3.762

Table 5-2. Maximum Displacement (×10-4 inch) 𝑟=1/100

Reddy IBFEM SW ANSYS SnS Element Number 225 1k 10k 1k 3k

2 angled 3.891 4.122 4.110

2 crossed 3.814 3.987 3.978

10 angled 1.621 1.62 1.629 1.597 1.611

10 crossed 1.55 1.55 1.543 1.532 1.543

B A

59

Table 5-3. Maximum Displacement (×10-1 inch) 𝑟=1/1000

Reddy IBFEM SW ANSYS SnS Element Number 225 1k 10k 1k 3k

2 angled 3.891 4.061 4.072

2 crossed 3.814 3.954 3.951

10 angled 1.58 1.578 1.581 1.163 1.684

10 crossed 1.525 1.552 1.51 1.145 1.661

As is shows in the tables, the IBFEM shows good agreement with SolidWorks

using a lower density mesh because cubic B-spline elements are for modeling shells in

IBFEM. The result also shows good agreement with ANSYS when the ratio equal to 100

and 1000 but not the 10. That is because of the element type used in ANSYS is SHELL

181, which is designed for moderately-thick shell, and it is not suitable for the case 𝑟=10

(thick structure). In addition, the analytical solution reported by Reddy are calculated

based on the Kirchhoff thin shell theory which is also only good for thin shell, not for

thick shell. So it is not suitable to analyze the structure with the ratio 𝑟 equal to 0.1. The

answers from IBFEM are better than the answers given by the SnS (Scan and Solve)

with much less mesh density because the results for SnS were obtained using quadratic

elements.

The converge study are also performed in IBFEM, to verify the convergence and

converge rate of this method. The relative error (in log scale) and the maximum

displacement of different mesh density are plotted and the relative error is computed

based on the strain energy of the entire structure.

60

Figure 5-3. Converge Plot of Strain Energy

Figure 5-4. Converge Plot of Maximum Displacement

IBFEM converges very fast with respect to size of the elements because the

elements are cubic and a good answer can be achieved with a mesh density which is

not too high. However the computational time for the analysis is higher than the typical

shell elements in commercial software because of the size of the element stiffness

matrix and cost of computing and assembling it.

Example of Pressured Cylinder

The second example is a thin shell-like cylinder subjected to internal pressure

and fixed along the edges at both ends. The geometry of the cylinder is shown as

61

Figure 5-5 and its radius is 20 inch, height is 20 inch, thickness is 1 inch and the internal

pressure is 2.04 ksi.

Figure 5-5. Geometry and Load of Pressured Cylinder

The material properties are:

6 6 6

1 2 3

12 13 23

5 5 5

12 13 23

7.5 10 , 2 10 , 2 10

0.25, 0, 0

12.5 10 , 6.25 10 , 6.25 10

E psi E psi E psi

G psi G psi G psi

(5-3)

Using symmetry one-eighth of the structure is modeled and meshed in IBFEM

as shown in Figure 5-6. The maximum radial displacement are listed in the Table 5-4

(The SW, ANSYS and SnS results are reported by Kumar and Shapiro [12], and the

Reddy solution is report by Reddy [27])

𝑥

𝑦

𝑥

𝑎

𝑧

𝑝 𝑝

ℎ

𝑅

62

Figure 5-6. Pressured Cylinder in IBFEM A) Geometry and Mesh; B) Displacement of 10 angled ply

Table 5-4. Maximum Radius Displacement of Cylinder Subjected to Internal Pressure

(10-1 inch)

Reddy IBFEM SW ANSYS SnS Element Number 190 1.2k 15k 1k 3k

[0]T 3.754 3.763 3.752

2 crossed 1.870 1.763 1.848 1.706 1.820 1.773

2 angled 2.287 2.350 2.204 2.356 2.291

10 crossed 1.759 1.830 1.719 1.814 1.776

10 angled 2.271 2.340 2.21 2.334 2.282

It can be seen that answers obtained by IBFEM show good agreement with SW

and analytical solutions with lower mesh density.

Example of Scordelis-Lo Roof

Scordelis-Lo roof is a thin shell-like structure subjected to gravity loads and

supported at both sides by diaphragms while the side edges are free as shown in the

Figure 5-7. The geometry of the structure are the radius equal to 300 inch, length is 600

A B

63

inch and the pressure equal to 0.625 psi. The two edges are constrained such that they

cannot move in the z and x axis direction and cannot rotate about the y axis.

Figure 5-7. Geometry and Load of Scordelis-Lo Roof

The material properties are:

6 6 6

1 2 3

12 13 23

5 5 5

12 13 23

25 10 , 1 10 , 1 10

0.25, 0, 0

5 10 , 5 10 , 2 10

E psi E psi E psi

G psi G psi G psi

(5-4)

Various thickness (3 inch, 6 inch and 15 inch) were tested. The ratio between the

radius and shell thickness is defined as

/r h R (5-5)

Figure 5-8 shows the model and the mesh used for analysis using IBFEM. The

maximum displacement of Scordelis-Lo Roof with different thicknesses has been

computed and the results are listed in Table 5-5, Table 5-6 and Table 5-7. (The SW,

𝑧z 𝑥

𝑦

𝑥

𝑎 80°

ℎ

𝑅

𝑝

64

ANSYS and SnS results are reported by Kumar and Shapiro [12], and the Reddy

solution is report by Reddy [27])

Figure 5-8. Scordelis-Lo Roof in IBFEM. A) Geometry and Mesh of Scordelis-Lo Roof;

B) Displacement of 10 Crossed Ply (𝑟=100)

Table 5-5. Maximum Displacement of Scordelis-Lo Roof (inch) 𝑟=100

Reddy IBFEM SW ANSYS SnS

Element Number 16 1160 1.2K 10K 1K 3K

10 crossed 1.415 1.473 1.564 1.434 1.542 1.593

2 crossed 2.339 2.396 2.46 2.407 2.307 2.415

10 angled 1.818 1.82 1.955 1.836 1.821 1.912

2 angled 3.597 3.482 3.866 3.871 3.411 3.743

Table 5-6. Maximum Displacement of Scordelis-Lo Roof (10-1 inch) 𝑟=50

Reddy IBFEM SW ANSYS SnS

Element Number 16 1160 1.2K 10K 1K 3K

10 crossed 2.94 3.233 3.27 2.979 3.335 3.412

2 crossed 5.082 5.463 5.659 5.291 5.48 5.81

10 angled 4.096 4.16 4.089 4.082 3.796 3.94

2 angled 6.76 6.557 7.17 7.652 6.675 7.157

B A

65

Table 5-7. Maximum Displacement of Scordelis-Lo Roof (10-2 inch) 𝑟=20

Reddy IBFEM SW ANSYS SnS

Element Number 16 1160 1.2K 10K 1K 3K

10 crossed 5.234 5.969 5.37 5.246 5.361 5.398

2 crossed 7.292 8.304 7.56 7.449 7.877 8.067

10 angled 10.04 10.27 9.594 9.727 7.856 8.009

2 angled 12.05 13.30 8.959 13.97 1.061 11.27

The IBFEM shows good agreement with both the analytical solutions and the

numerical results given by the commercial software SW and ANSYS. But the mesh

density used in IBFEM are much less that the mesh used in SW, ANSYS and SnS.

Again, the element type in IBFEM is 3D cubic B-spline in this example, the answers

match better with analytical solutions based on thin shell theory when the structure is

thinner.

Example of Doubly-Curved Shell

Doubly-Curved Shell is a thin shell-like structure subjected to internal pressure

loads and simple supported at all of the four edges as shown in the Figure 5-9. The

geometry of the structure is defined by the two radius of two curves 𝑅1 and 𝑅2, and the

length of the two edges 𝑎 and 𝑏. In this case the radii are assumed to be the same

(𝑅1 = 𝑅2) equal to 100 inch and 𝑎 = 𝑏. The pressure are applied on all of the structure

and acting outward. The value of it is set to be 10 psi in this example.

66

Figure 5-9. Geometry and Load of Doubly-Curved Shell

The material properties are:

6 6 6

1 2 3

12 13 23

5 5 5

12 13 23

25 10 , 1 10 , 1 10

0.25, 0, 0

5 10 , 5 10 , 2 10

E psi E psi E psi

G psi G psi G psi

(5-6)

Various /R a ratio and /a h ratio are tested and the center deflation values are

non-dimensionalized as in the equation below

3 4 3

2 0[ / ( )]10w wE h q a (5-7)

The structure as modeled and analyzed in IBFEM is shown in Figure 5-10. The

maximum displacement will occur at the center of the structure. The results have been

non-dimensionalized and listed in the Table 5-8 and Table 5-9. (The Reddy solution is

reported by Reddy [27])

𝑏 𝑎

𝑅1

𝑅2

𝑦

𝑧

𝑥

𝑝

67

Figure 5-10. Doubly-Curved Shell in IBFEM A) Geometry and Mesh Doubly-Curved Shell; B) Displacement of [0/90]S Ply (𝑟/𝑎 = 2)

Table 5-8. Maximum Displacement (Non-dimensionalized) [0/90]T

𝑅/𝑎 Reddy SW IBFEM

2 0.2855 0.2816 0.2952

3 0.6441 0.6393 0.6564

4 1.1412 1.1360 1.1564

5 1.7535 1.7487 1.7725

10 5.5428 5.544 5.583

1030 16.98 17.06 17.10

Table 5-9. Maximum Displacement (Non-dimensionalized) [0/90]S

𝑅/𝑎 Reddy SW IBFEM

2 0.2844 0.2796 0.2938

3 0.6246 0.6201 0.6366

4 1.0559 1.0516 1.07

5 1.5358 1.5315 1.5525

10 3.7208 3.72 3.74

1030 6.8331 6.846 6.85

IBFEM shows good agreement with both the analytical solutions and the

numerical results given by the SolidWorks with much less mesh density.

B A

68

CHAPTER 6 CONCLUSION

Summary

In this thesis, ABD-equivalent material model for laminate, for which the original

multi-ply laminate is replaced by a new 3-ply laminate with the same laminate stiffness

matrices, has been implemented in the Implicit Boundary Finite Element Method

(IBFEM). Initially, the concepts and equations that required for the Implicit Boundary

Finite Element Method are discussed followed by the discussion of the properties of

composite laminate, Classical Lamination Plate Theory (CLPT) and Shear Deformable

Plate Theory (SDPT). In addition, the formation of ABD-equivalent material model for

the laminate has been discussed in detail along with the description of the

transformations required. In addition, the model has been tested with examples in

IBFEM and the answers are compared with analytical solutions, if available, as well as

answers from other FEA software.

The main advantage of ABD-equivalent material model is that it can reduce time

taken by the numerical integration through thickness direction by reducing the original

multi-ply laminate to a 3-ply laminate without losing the ability to catch the macro-

behavior of the laminate structure. The examples tested using ABD-equivalent material

model for laminate shows it’s validity. An accurate solution can be obtained using in

IBFEM with a lower mesh density than SolidWorks, ANSYS and SnS. The

computational time taken by the integration through thickness direction is reduced

significantly. As in any approximation, the ABD-equivalent material model for laminate

may sacrifice some accuracy, but the advantages of this mothed are significant with no

observed loss of accuracy.

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Future Work

In this thesis, ABD-equivalent material model for laminate has been implemented

in the Implicit Boundary Finite Element Method (IBFEM) and it shows it’s validly,

however, more work is needed to improve this model.

First of all, the results presented here is using 3D cubic B-spline elements that

use a 3D stress-strain formulation which requires computing the equivalent 3-ply model.

It would be advantageous to develop a 3D shell element that is based on the Kirchoff or

Midlin shell theory so that the effective ABD matrix can be directly used.

Secondly, the ABD-equivalent material model for laminate can be further

improved to endow the capability of modeling thick laminate, honey cone laminate, etc.

Furthermore, the ABD-equivalent material model for laminate is not capable of

catch the micro-behavior of each layer. The method can be endowed with the ability to

analyze the stress and strain distribution within each lamina.

70

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BIOGRAPHICAL SKETCH

Li Liang was grew up in Guangzhou, Guangdong province, China. He did his

high school in No.4 Guangzhou High School located in Guangzhou and graduated in

2010. He got his bachelor’s degree of engineering mechanics and finance at 2014 after

4 years of studying beginning in 2010 in Southwest Jiaotong University, Chengdu,

China. Meanwhile, he gained some research experiences by working on two research

projects: “Study of Micro-Mechanism of Metallic Glass and Metallic Glass Matrix

Composites” and the “Study of Cyclic Deformation Behavior of Polycarbonate Polymer

in Hydrothermal Environment”. Also he did internship in Guangdong Hydropower

Planning & Design Institute, Guangdong, China, from July to August 2013. After that, he

studied in the University of Florida, Gainesville, Florida, USA, for 2 years and received a

master’s degree in mechanical engineering in May of 2016. His areas of specialization

includes Finite Element Method, Composites Material and Computational Method.

During his master’s degree studies, he did research on implementing Composite

Laminate for 3D Shell-Like Structure in Implicit Boundary Finite Element Method and

worked as a Teaching Assistant for the graduate course on Finite Element Analysis and

Application in the Spring and Fall semesters of 2015 and the undergraduate class on

Finite Element Analysis and Application in Spring 2016.