mesh independent analysis of 3d shell-like laminate...
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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
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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.
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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
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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
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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
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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
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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
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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
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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.
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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,
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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.
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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
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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.
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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)
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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
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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 .
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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
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( , , ) ( , )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
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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)
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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
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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
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( , , ) ( )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)
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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.
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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
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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)
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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.
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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.