‘ ....w.h..‘...u..«
u.

3.14:...-
)9... a)

Efi

.t..r£:.
. I}

 

 

 

 

 

U. 21.3.31!
z. 04¢:

‘ .9... .. ‘ .5 .
$.11... 52.
a!) it...

.51! u: . !
:zuhufii.
1.9 J

 

I

I.\ > ‘.

1!:
w? 31s.:
. 1.. A. I 3

Lu] o.

. .3!

.9445?
.

:5...

 

LIBRARY

\ ,- Michigan State
.7 3 3"" {‘0 University
' W 2 / f .7 3

This is to certify that the
thesis entitled

Parametric Material Layout Optimization of Natural Fiber
Composite Panels

presented by

Christina Isaac

has been accepted towards fulfillment
of the requirements for the

MS. degree in Mechanical Enfleen‘ng

 

 

 

MSUIsanAMmatiwAwoNEquaIOppommfiylnsumflon

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 c:/ClRC/DateDue.p65-p.15

 

PARAMETRIC MATERIAL LAYOUT OPTIMIZATION OF NATURAL FIBER
COMPOSITE PANELS

By

Christina Isaac

A THESIS

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

MASTER OF SCIENCE
Department of Mechanical Engineering

2005

ABSTRACT

MATERIAL LAYOUT OPTIMIZATION OF NATURAL FIBER COMPOSITE
PANELS

By
Christina Isaac

Natural fiber composites, have recently gained renewed interest due to increased
concern for the long-terrn sustainability of structural materials. Nonetheless, their use for
load-bearing applications has been restricted because of their low mechanical properties.
Yet, recent developments have shown that the performance of biocomposite components
can be overcome by using properly engineered or optimized designs.

This thesis presents a finite, parametric approach to optimize the material
distribution in biocomposite cellular panels. The approach combines optimization and
finite-element software through a parametric problem formulation. Unlike traditional
topology optimization, the presented approach leads to optimized material layouts while
permitting the use of multiple objectives and constraints. The optimization procedure was
validated using benchmark topology problems. Small-scale component testing was
conducted to validate the optimized solutions and evaluate the vacuum assisted resin
transfer molding as a method to create the optimized biocomposite cellular panels.

The parametric optimization results compared favorably to the multi-resolution
topology solutions and lead to designs that are easier to manufacture than those obtained
by the power-law method. As expected, the manufactured optimized designs exhibited
improved performance. Thus, the material layout optimization technique implemented for
this study has proven to be viable for achieving optimal structural layouts to enhance the

performance of biocomposites while simultaneously accounting for manufacturing.

Copyright by
Christina Isaac

2005

To my parents, Sam and Rema Isaac, my sister, Marie, and brothers,
Danny and Jacob for all of their love, support, and guidance
throughout the years

iv

ACKNOWLEDGEMENTS

I am most thankful for my advisor Dr. Rigoberto Burguefio, for his support,
patience, and guidance, throughout this research. His time, dedication, and
encouragement provided me with the knowledge and strength to become successful in
my research as well as my future aspirations. The topic and ideas presented
throughout this thesis were proposed by him.

I would also like to acknowledge the members of my graduate committee, Dr.
Ronald Averill and Dr. Alejandro Diaz for their interest, and insightfirl ideas
throughout my research.

I am also very grateful for the guidance and support offered by Dr. Amar
Mohanty, Dr. Manjusri Mirsa, and Dr. Hiro Miyagawa. They expertise in
biocomposites provided me with assistance to progress in my work. Furthermore, I
would like to thank them for providing me with necessary supplies needed for
manufacturing of my cellular biocomposite samples.

I would like to thank Siavosh Ravanbakhsh for his help with fabrication and
testing of biocomposite cellular samples at the Civil Infrastructure Laboratory. He
was always willing to assist and to provide insight and encouragement when needed.

I am also grateful for the assistance of Robert Jurek for his advice and
expertise with the vacuum assisted resin transfer molding (V ARTM) manufacturing
process and for providing me with VARTM materials when needed.

I am also very appreciative of the staff at Red Cedar Technology for training
me on and providing me with effective optimization software to implement in my

research.

Lastly, I wish to thank all my colleagues for their assistance, and most
importantly their friendship throughout my years of graduate school at MSU. Their

encouragement helped me get through the all difficult times in my research.

vi

3.1

3.2

3.3
3.3.1
3.3.2
3.3.3
3.3.4
3.3.5
3.3.6

3.4

TABLE OF CONTENTS

 

 

 

 

 

 

ACKNOWLEDGEMENTS _ - - - - - v
TABLE OF CONTENTS -- - - - - - -- - - _ _ - - vii
LIST OF TABLES _ -- -- - - - - ix
LIST OF FIGURES _ x
1 INTRODUCTION 1
1.1 Overview ..................................................................................................... 1
1.2 General Research Objective ........................................................................ 2
1.3 Natural Fiber Composites ........................................................................... 3
1.4 Material Structures in Biology .................................................................... 5
1.5 Achieving Strategic Material Layouts ........................................................ 8
1.6 Manufacturing Limitations ......................................................................... 9
1.7 Parametric Layout Optimization ............................................................... 10
1.8 Objective and Scope ................................................................................. 11
2 OPTIMIZATION BACKGROUND _ _ - -- - - - 13
2.1 Overview of Structural Optimization ........................................................ 13
2.1.1 Size and Shape Optimization .................................................................... 15
2.1.2 Topology Optimization ............................................................................. 18
2.1.3 Parametric Material Layout Optimization ................................................ 24
2.2 Optimization through Genetic Algorithms ............................................... 24
2.2.1 Definition and Implementation of Genetic Representation ...................... 27

2.2.2 Definition and Implementation of Genetic Operators and Control
Parameters ................................................................................................. 28
2.2.3 The Objective Function ............................................................................. 34
2.2.4 The Fitness Function ................................................................................. 34

3 DEVELOPMENT AND IMPLEMENTATION OF

PARAMETRIC MATERIAL LAYOUT OPTIMIZATION

 

TECHNIQUE 36
Overview ................................................................................................... 36
Overview of the Optimization Technique ................................................. 36
Validation of the Optimization Procedure ................................................ 40
Problem Definition .................................................................................... 41
Parametric Modeling ................................................................................. 42
Optimization Problem Formulation .......................................................... 44
Implementation of Optimization Process .................................................. 52
Evaluation of Optimal Solutions ............................................................... 53
Validation of Optimal Solutions ............................................................... 61
Optimization of Continuous Panel Systems-Case Study .......................... 67

vii

3.4.1
3.4.2
3.4.3

4.1

4.2

4.3
4.3.1

4.4

4.5

5.1

5.2
5.2.1
5.2.2
5.2.3
5.2.4

5.3

Problem Definition .................................................................................... 68

 

 

 

Parametric Modeling and Optimization Problem Formulation ................ 71
Implementation of Optimization Process and Evaluation of Optimal

Solutions ................................................................................................... 77
EXPERIMENTAL VALIDATION - _ _ - 93
Overview ................................................................................................... 93
Material Systems and Structural Forms .................................................... 93
Automated Manufacturing Process ........................................................... 96
Manufacturing of VARTM Cellular Panels .............................................. 98
F lexural Testing of VARTM Hierarchical Samples ............................... 107
Results and Discussion ........................................................................... 113
CONCLUSIONS AND RECOMMENDATIONS - 125
Conclusions ............................................................................................. 125
Recommendations for Future Work ........................................................ 126
Material Layout Optimization Techniques ............................................. 127
Optimization of Material and Resin Systems ......................................... 128
Design of Optimized Structural Layouts ................................................ 128
Improving the VARTM Technique ........................................................ 129
Applications of Biocomposite Structural Components ........................... 129
REFERENCES--- - _ - -- - - - - - - 131

viii

LIST OF TABLES

Table 1-1. Basic fiber properties (Lackey et al. 2004) ..................................... 4

Table 3-1. Summary of genetic operators and control parameters ............................. 45
Table 3-2. Optimization problem formulation for MBB-beam .................................. 50
Table 3-3. Summary of formulation elements for the MBB-beam ............................. 50
Table 3-4. Optimization problem formulation for the 8-bar truss .............................. 51
Table 3-5. Summary of formulation elements for the 8-bar truss ............................... 51
Table 3-6. Evolution of designs for the MBB-beam problem .................................... 56
Table 3-7. Evolution of designs for the 8-bar truss problem ...................................... 58
Table 3-8. Comparison in results for the MBB-beam ................................................. 63
Table 3-9. Comparison in results for the 8-bar truss .................................................. 64

Table 3-10. Optimization problem formulation for the simplified continuous panel
system ................................................................................................................... 75

Table 3-11. Summary of formulation elements for the simplified continuous panel
system ................................................................................................................... 76

Table 3-12. Evolution of designs for case study with increased discrete variable set 83

Table 3-13. Evolution of designs for case study with decreased variable set to account

for manufacttuing .................................................................................................. 85
Table 3-14. Summary of results for the designs solutions obtained for the case study87
Table 3-15. Comparison in computational results for the initial and optimal designs

for the case study .................................................................................................. 92
Table 4-1. Resin System ............................................................................................. 95
Table 4-2. Flexural test results summary with respect to the central load ................ 116
Table 4-3. Comparison in initial stiffness ................................................................. 124

ix

LIST OF FIGURES

Figure 1-1. Examples of natural fibers ......................................................................... 3
Figure 1-2. Transverse view of wood cellular structure ............................................... 7
Figure 2-1. Main types of structural optimization techniques .................................... 15
Figure 2-2. Example of shape optimization with internal cavities or holes ................ 17

Figure 2-3. Common example used for shape optimization (Taken from [Hafika and

Gurdal 1993]) ........................................................................................................ 1 8
Figure 2-4. Example of t0pology optimization using the power law approach
(Adapted fiom [Sigmund and Tcherniak 2001]) .................................................. 21
Figure 2-5. Result fi'om implementation of power law approach (Taken fi'om
[Sigmund and Tcherniak 2001]) ........................................................................... 21
Figure 2-6. Result from implementation of the multiresolution approach (Adapted
fiom [Chellappa et a1. 2004]) ................................................................................ 23
Figure 2-7. Language of genetic algorithms ............................................................... 26
Figure 2-8. Schematic of GA (Adapted from [Turkkan 2003]) .................................. 26
Figure 2-9. Possible structures used for representation of individual chromosomes
(Adapted from [Daves 1991]). .............................................................................. 28
Figure 2-10. Examples of mutation ............................................................................ 31
Figure 3-1. Optimization flowchart ............................................................................ 38
Figure 3-2. Optimization process ................................................................................ 39
Figure 3-3. Schematic of optimization procedure ....................................................... 40
Figure 3-4. MBB-beam ............................................................................................... 42
Figure 3-5. 8-bar truss ................................................................................................. 42
Figure 3-6. Parametric model for MBB-beam (intial design) ..................................... 43
Figure 3-7. Parametric model for 8-bar truss (initial design) ..................................... 43

1 INTRODUCTION

1.1 Overview

Increased environmental awareness and interest in long-term sustainability of
structural materials have recently challenged the development of environmentally
friendly alternatives to fiber reinforced polymer (FRP) composites (Mohanty et al.
2000). FRP composites provide advantages such as high stiffness and strength to
weight ratios in comparison to conventional construction materials; however, due to
their high initial material costs, restricted use in efficient structural forms, and
environmental impact, their application for structural components has been limited
(Burgueflo et al. 2004).

Natural fiber reinforced polymer composites, or biocomposites, have been
gaining renewed interest as an environmentally fiiendly alternative to synthetic F RP
composites and have become appealing for diverse applications for reasons which
also include low cost and light weight. Despite these advantages, their main
disadvantage is that their mechanical properties, such as strength and stiffiress, are
much lower than those of synthetic F RP composites and other conventional structural
materials, which do not seem to make them useful for load-bearing applications.
However, recent studies have shown that through proper engineering and processing,
biocomposites are capable of competing with E-glass FRP composites (Mohanty et a1.
2000).

The performance of a component depends on both its material and structural
properties. The lower material stiffness biocomposite materials can be thus overcome

using properly engineered structural configurations that place material in specific

locations to achieve the highest structural performance while using minimum material
(Gibson and Ashby 1988). This idea of has been recently gaining much attention due
to renewed interest and knowledge into the way nature uses materials to achieve
complex structural configurations for adequate structural efficiency.

The concepts introduced throughout this section have been integrated into this
research to display that properly engineered biocomposites can serve as
environmentally friendly, cost-effective, and adequate load bearing components. The
remaining sections of this chapter will give a general overview of the research
objectives, followed by the background information which motivated these
objectives. The background incorporates an introduction and discussion of natural
fiber composites, material structures in biology, numerical methods that are capable
of achieving strategic material layouts, and manufacturing limitations. The chapter is
concluded with the specific research objectives and scope of the research work

presented in this thesis.

1.2 General Research Objective

The objective of this research was to explore and implement an optimization
technique to improve the performance of natural fiber reinforced polymer composites
(biocomposites) for load-bearing applications. The optimization procedure was
validated using benchmark data from well-accepted optimization. techniques and

design solutions were validated through small-scale component testing.
The research work involved the following tasks:
0 Development and implementation of the optimization procedure

0 Validation of the optimization procedure

0 Manufacturing of designs solutions
0 Experimental validation and analysis
A general description of the objectives was outlined prior to providing a
background on this study to display the direction of this research at the onset. Specific

objectives will be outline in greater detail in a later section in this chapter.

1.3 Natural Fiber Composites

Natural fibers have recently gained much attention for reasons including low
cost, lightweight, increased enviromnental awareness, interest in long term
sustainability, and for their ability to compete with glass fiber composites (Mohanty
et al. 2000, Burgucfio et al. 2004) Types of common natural fibers include sisal,
hcnequen, flax, hemp, jute, kenaf, cotton, and coir (coconut husk). The quality of the
fiber depends highly on size, maturity, processing method used for fiber extraction,
and origin of the fiber. Depending on the origin, biofibers can be grouped into one of
the following categories: leaf, bast, seed, or fruit. Examples of each include: leaf:
sisal and henequen, bast: flax, hemp, kenaf, and jute, seed: cotton and fruit: coconut

husk (coir), (Mohanty et a1. 2000) (Figure 1-1).

, a: .3. 5mm

. 5%“

‘i-éi; » t6“. .2! Ii 3

 

 

 

Figure 1-1. Examples of natural fibers

Natural fibers have two basic components: cellulose and lignin. These two
constituents work to provide rigidity to the walls of plants stems and also the bonding
in between them. The contents of both cellulose and lignin vary from one natural
fiber to another.

The density of natural fibers is relatively low in comparison to that of synthetic
composites, such as E-glass, thus, it can be shown that the bast fibers (e. g. flax, hemp,
and jute) have specific strengths, elastic modulus, and specific modulus that are
comparable or even superior to E-glass (Mohanty et al. 2000) (Table 1-1). Other
advantages include, ease of separation and enhanced energy recovery. Due to their
acceptable properties and numerous environmental advantages, natural fibers have

attracted renewed interest as a glass fiber substitute in a variety of industries.

Table 1-1. Basic fiber properties (Lackey et al. 2004)

 

 

Property E-Glass Flax Hemp Jute
Density (lbs/m3) 0.092 0.050 0.053 0.052
Tensile Strength (ksi) 348 116-217.5 79.7-130.5 58-116
. 8.70E+03- l.SOE+03 -

E-Modulus (ksr) 1.06E+04 1.2013403 1.00E+04 4.40E+03

. . 1.74E+05 - 2.88E+04 -
Specrfic Modulus (E/Densrty) 1.15E+05 2. 40E +0 5 1.89E+05 8. 4 6E +04
Moisture Absorption (%) -- 7 8 12
Cost (S/Ib) Raw 0.59 0.23-0.68 0.27-0.81 0.16

 

Although natural fibers possess sufficient strength and stiffness, and have
environmental qualities that make them attractive over synthetic composites, they
also have several major drawbacks. Among the most relevant drawbacks are that

natural fibers tend to be highly moisture absorbent, generally need to be processed

under low temperatures to reduce the possibility of fiber degradation, and have
extremely low tolerance to processing damage (Rout et al. 2001). Manufacturing
limitations are also important due to the “springy” nature of the fibers, their cotton-
like texture, and extreme randomness, i.e. varying shapes and sizes. Natural fibers
tend to clump together and should be separated prior to manufacturing to ensure
uniform dispersion throughout the composite. Furthermore, since they are “fluffy"
and possess a cotton-like texture, it is suggested that they remain compressed
throughout manufacturing (Brouwer 2000). By compressing the fibers, the natural
adhesive lignin is released thus forming a slight bond between fibers. This bond
assists in preventing the fibers from shifting during manufacturing and may also help
in maintaining uniform distribution throughout the composite. If the compression of
these fibers is accomplished through automated processing, the mechanical properties
of the fibers could decrease due high temperatures and possible chemical and/or
physical treatments (Hepworth et al. 2000). Therefore, natural fiber composites are
typically manufactured through hand-lay procedures or by vacuum assisted resin

transfer molding (V ARTM) depending on geometric complexity.

1.4 Material Structures in Biology

Nature has displayed that through proper combination of natural constituents in
advanced geometries, adequate structural efficiency can be achieved even when the
raw constituents have rather low individual properties. Engineers are beginning to
take advantage of this knowledge and are attempting to mimic nature with intentions
of developing materials that may be useful for various structural applications. The

study of natural materials has also introduced advanced engineering forms at the

material and structural levels (Srinivasan 1995). For example, materials of living
organisms, such as tendons and bones, which possess unique hierarchical structural
forms that provide them with increased toughness and ductility. The rope was the first
application to mimic these natural materials and has proved to be applicable in
various structural components (Srinivasan 1995). Wood is another example of a
natural composite that exhibits a remarkable combination of strength, stiffness, and
toughness (Srinivasan 1995). It is comprised of parallel columns of hollow cells
joined end-to-end, around which fibers of cellulose are wound in spirals and
embedded to a matrix of lignin, a natural adhesive. The structural formation of wood
(Figure 1-2), together with its material constituents provides this material with the
capability to absorb large amounts of energy. When a wood cell buckles inward, it
fractures the surrounding cells, absorbing energy in the process. A long spiral crack
develops along the wound fibers, dissipating energy over a short length of the wood.
Since the crack runs along and in between the fibers and within the lignin, the fibers
are able to stay intact so that the macroscopic wood material will not fall apart. Even
though the overall wood material is fractured, it is still capable of supporting load

(Srinivasan 1995, Hepworth et al. 2000)

 

Figure 1-2. Transverse view of wood cellular structure
Natural materials do have various disadvantages that limit their use. As stated

previously, natural fibers easily absorb excess moisture, causing swelling and
degradation. They also have limited processing temperatures due to degradation and
flammability. Natural fibers, or biofibers, have variable quality depending on origin
and weather influences. They have irregular lengths, shapes, and sizes, which makes
them difficult to handle and manufacture (V erheus 2002). The major limitations of
natural fibers are their low mechanical properties, such as strength and stiffiress,
which restricts their use to non load-bearing applications. However, nature elegantly
shows that materials with low mechanical properties can be enhanced through hybrid
formations and strategic material organization such as complex hierarchical
configurations. Biofibers are extremely fibrous and therefore can be easily combined
with synthetic fibers, such as carbon, glass, nylon, polyester and/or polyethylene to
form hybrid designs. Furthermore, complex hierarchical configurations seem
appropriate for biocomposites due to the complexity in manufacturing pieces with
aligned reinforcement, which suggests their use as a continuous distributed material
(Burguefio et al. 2005). By combination of advanced structural forms and hybrid

designs a more enhanced structural material with higher performance can be

achieved, while simultaneously minimizing cost and weight (Allinger et al. 1996).
Previous research on biocomposites for structural applications has considered
analytical and experimental evaluations of the mechanical properties of various
natural fiber laminates and cellular structures (Burguefio et a1. 2004, Burgueiio et al.
2005, Quagliata 2003). The research concluded that biocomposites in
cellular/sandwich structures are capable of serving as load bearing components and
are able to compete with conventional structural materials. In general, this work has
shown that by altering the material layout of a structure, strategic arrangements or
hierarchical forms can effectively improve the mechanical properties of

biocomposites thus making them competitive for load-bearing applications.

1.5 Achieving Strategic Material Layouts

In order to achieve strategic material layouts to improve the mechanical
properties of biocomposites, without going through the trial and error of
experimentation and when solutions might be non-intuitive, engineers use structural
optimization algorithms. Optimization algorithms are now feasible due to diverse
computational tools which allow for modeling and analysis of complex shapes and
internal geometries for better understanding of the performance of various structural
layouts (Bendsde and Sigmund 2003).

Structural optimization is employed to improve the performance of a design,
e.g. maximizing stiffness or minimizing weight. An objective function is used to
locate a solution by measuring the “fitness” or efficiency of each design.
Optimization problems are typically formulated with limitations or constraints such as

maximum and minimum stresses, strains, and deflections that control the design

selection (Arora 1989). Various optimization algorithms exist which can be combined
with the use of finite element software to perform structural optimization techniques
that are capable of optimizing a structure’s topology, size, and/or shape (Arora 1989,
Hafika and Gurdal 1993). Topology optimization is typically employed to determine
the material arrangement, or layout, within a structure in a way that will allow it to
serve in its the most efficient manner. Size optimization generally focuses on
achieving an Optimal cross-sectional area, typically centering on obtaining an optimal
shape of the domain or internal geometry (Hafika and Gurdal 1993). It should be
noted that while designs obtained through optimization techniques may be optimal
they are not always feasible, or cost-effective, to manufacture. This can be

particularly important when using natural fibers, as discussed next.

1.6 Manufacturing Limitations

Designs obtained through the use of structural optimization techniques can be
rather intricate and difficult to manufacture using natural fibers; therefore significant
post-processing techniques may be required to simplify the geometry of the optimal
design solutions to ensure manufacturability. Rather than relying on post-processing
measures, an alternate solution to geometric simplification is to implement an
optimization technique which deals with finite and well-defined geometrical changes,
eliminating the geometric intricacies that tend to create difficulties in manufacturing
with natural fibers. Chellappa et al. (2004) proposed an elegant topology optimization
scheme with finite size inclusions using a multiresolution method. This technique is
capable of optimizing the material layout within a given domain using a predefined

set of inclusions, therefore minimizing the need for rigorous post-processing prior to

manufactruing. This technique locates the optimal material layout of a given structure
while maximizing stiffness (Chellappa et al. 2004). However, the disadvantage of this
method is that it is not designed for ease of problem reformulation. Specifically,
altering loading and boundary conditions, incorporating system constraints, i.e.
stresses, strains, and deformations, or employing multiple objective firnctions would
require code restructuring. Thus, in order to account for manufacturing and redefining
system conditions and constraints, it is necessary to develop and implement an
optimization procedure with finite resolution that also minimizes the complexities of
problem reformulation. Such a technique would allow multiple constraints and
objective functions to be incorporated into the design optimization problems with

case.

1.7 Parametric Layout Optimization

Due to the limitation of the multiresolution technique pr0posed by Chellappa
et al. 2004 (see Section 1-6), an alternate optimization procedure was explored to
perform material layout optimization for the present work. A parametric approach to
material layout optimization was investigated and implemented. This parametric
based optimization technique uses existing finite element software and a general
purpose optimization program to optimize the material layout within a structural
domain subject to loading and boundary conditions. This technique allows for
incorporation of multiple design constraints and objective fimctions without

significant code restructuring and simultaneously accounts for manufacturing.

10

1.8 Objective and Scope

The objective of this research was to investigate and implement a material
layout optimization technique to improve the performance of natural fiber reinforced
polymer composites (biocomposites) for load-bearing paneling applications. A
material layout topology with finite geometrically defined features was sought due to
the known complexities in manufacturing biocomposite panels. This optimization
technique was applied in the context of a specific problem, specifically, the material
distribution within the transverse cross-section of a continuous panel system. Such a
panel system was optimized and optimal design solutions were validated through
small-scale component testing. The goal is to develop a design optimization approach
that would allow for the use of biocomposites in applications ranging from civil
structures (such as bridges, decks, and flooring systems) to aerospace structures (such
as fuselages, wing skins, and various other integrated components). As stated
previously, the material layout optimization technique should be powerful enough to
incorporate multiple objective functions and loading conditions for problems relating
to solid mechanics, fluid mechanics, and dynamic/vibration analysis.

The research work involved the following tasks:

0 Development and implementation of the parametric material layout
optimization technique:

0 Validation of the parametric material layout optimization technique

0 Manufacturing of optimal designs

0 Experimental validation and analysis

11

The research presented in this thesis incorporates concepts of topology, shape,
and material layout optimization techniques; fabrication of natural fiber composite
components, and experimental structural testing. All of these concepts are linked
through the motivation to improve the feasibility of natural fiber composites for load-
bearing applications. A parametric material layout optimization process is introduced
and implemented to locate optimal material layouts throughout panel cross-sections
that will enhance structural performance. The designs obtained through the developed
structural optimization approach are used to validate the use of natural fiber
composites for load-bearing applications.

Based on the above research tasks, the chapters of the thesis are organized in
the following order:

0 Chapter 2: Optimization Background

0 Chapter 3: Development and Implementation of Parametric Material
Layout Optimization Technique

0 Chapter 4: Experimental Validation

0 Chapter 5: Conclusions and Recommendations

12

2 OPTIMIZATION BACKGROUND

2.1 Overview of Structural Optimization

Structural optimization has become a chief concern in the desigr of mechanical
systems, civil infrastructure, and aeronautical and aerospace integrated components.
Engineers are no longer satisfied with simple desigr improvements, but are now
striving for structurally efficient design solutions through global optimization of
weight, cost, and/or stiffness. Generally, engineers will attempt to improve desigrs
through trial and error, and an optimal solution is found by intuition. The
disadvantage of this approach is that it is very costly, time-consuming, and may result
in erroneous desigr solutions (Red Cedar Technology 2004). Therefore, numerical
optimization strategies have become very attractive and valuable tools in creating
efficient and adequate desigrs for structural components without encountering the
problems introduced by the trial and error approach.

Typical structural optimization problems involve the search for the minimum or
maximum value of objective functions subjected to a set of constraints and/or
restrictions on the sizes or shapes of structural component members. The constraints
are usually dependent on performance measures such as stresses and deflections
(Arora 1989, Haftka and Gurdal 1993, Turkkan 2003). Structural optimization
becomes even more powerfirl when combined with computational tools such as finite
element software and computer aided design. This combination can assist in creating
cost-effective, lightweight structures, while minimizing desigr time (Hafika and
Gurdal 1993). Various optimization techniques exist that are suitable to use when

optimizing the material distribution of a structural component and can be categorized

13

by search method or goal (i.e. what is being optimized).

Two common search algorithms exist and include gradient search algorithms
and the stochastic search algorithms. The gadient search technique uses information
of the first and possibly second order derivatives of the objective function are used to
determine an optimal search direction towards the optimum. Information from these
derivatives guarantees decreasing values for the objective function in consecutive
iterations. The disadvantage of this method is that it is sensitive to the initial estimates
of the unknown variables if the objective function has more than one optimum. This
means that it is possible for the algorithm to converge to a local optimum instead of
the wanted global optimum (Arora 1989). Stochastic search methods, commonly
implemented by genetic algorithms (Section 2-2), use randomized decisions while
searching for solutions to a given problem. They operate on a population of solutions
to locate a global optimum and are less likely to get “stuck” at local optimums, in
comparison to gadient search methods. A genetic algorithm will continue to search
for optimal solutions until the allotted nrunber of user defined cycles has completed.
Convergence can be detected if the fitness value of the last successive desigrs is
relatively close in value, or if the last desigr found was obtained approximately 10
cycles prior to completion of the last cycle (Arora 1989).

Structural optimization techniques are typically implemented to achieve one of
three goals and are categorized in this manner throughout this study. Researchers
have proposed ideas such size, shape, and/or topology optimization (Haftka and
Gurdal 1993). For example, given a defined domain, (2, boundary and loading

conditions (see Figure 2—1), size optimization generally refers to a change in the

14

cross-sectional area of Q; shape optimization typically refers to change in shape of
the domain, but not the topology; and topology optimization can be implemented to
incorporate size and shape optimization as well as the material layout and/or topology
within (2. Note that topology optimization generally does not affect the size of the

defined domain (Figure 2-1).

 

I Defined domain, (2, and boundary and loading conditions

 

 

 

 

 

 

 

 

 

Figure 2-1. Main types of structural optimization techniques

2.1.1 Size and Shape Optimization

Of the three types of structural optimization techniques introduced above, size
and shape optimization are the most commonly used. Size optimization is typically
concerned with optimization of “sizing variables” such as the thickness or cross-

sectional area of a structure, where modification of the cross-section is performed by

15

altering the size of the individual finite elements. This technique is typically
implemented to homogenize the stress or strain distribution throughout a structure’s
cross-section (Hafika and Gurdal 1993, Allinger 1996, Spath et al. 2002)

Shape optimization problems are typically more difficult to solve than size
optimization problems. Shape optimization generally refers to two types of problems:
(1) optimizing the shape of the boundary of a structural component (either 2 or 3
dimensional) or (2) optimization of the shape of internal cavities or holes.
Implementing shape optimization by altering the boundary or domain of a structure is
performed by changing the configuration of the structure by creating new boundaries
as well as modifying the existing boundaries in the model (Canonaco et a1. 1997).
These boundaries are generally altered by changing the position of existing nodes or
removing nodes in a finite element mesh based on calculated stress levels. The goal is
to homogenize the stress distribution according to the specified stress constraint. As
the shape of the structure is modified, it is necessary to re-mesh the finite element
model, which can lead to element distortion and loss of accuracy in computational
solutions (Haftka and Gurdal 1993). This problem is typically addressed through
manual re-meshing or through implementation of mesh generators (Haftka and
Gurdal 1993).

The optimization of boundaries or holes within a domain can also be defined
as shape optimization. This technique has the ability to produce optimal desigrs with
internal cavities; however, these vacancies cannot be generated without prior
knowledge of their existence. Specifically, this optimization procedure can easily

locate the optimum shape of a cavity once one is assumed, but it cannot determine

l6

how many cavities should exist (Hafika and Gurdal 1993). For example, if shape
optimization is implemented on a structure with four holes the optimal solution will
also have four holes. The positioning and/or size of these holes may differ from those
in the original structure, but this technique is incapable of increasing the number of
holes, removing holes, or merging them without resorting to specialized post-
processing or interpreting algorithms. An example of shape optimization in a

cantilever beam subjected to a concentrated load is shown in Figure 2—2.

Original Design Optimized Design
Shape Optimization
0 O O O O o .

Figure 2-2. Example of shape optimization with internal cavities or holes

 

 

 

 

 

 

A common problem in shape optimization is to optimize the shape of a hole in
a plate subjected to a uniform tension field (Figure 2-3) to reduce stress
concentrations around the hole (Haftka and Gurdal 1993). The optimization
procedure is formulated using the nodes surrounding the hole together with other
desigr parameters. Even though the original shape of the hole may be an adequate
desigr, it may not be the optimal (Hafika and Gurdal 1993). In comparison to the
original desigr (see Figure 2-2), the optimal configuration was desigred to
homogenize the stress distribution around the hole according to the specified stress

constraint.

1?

 

 

 

 

 

Figure 2-3. Common example used for shape optimization (Taken from [Haftka
and Gurdal 1993])

Shape optimization techniques are very robust and have been under improving
development for years; however they have limitations with respect to modifying the
number of existing cavities in a domain to truly optimize the material distribution
within a structure (Hafika and Gurdal 1993). These restrictions led to the
development and implementation of an alternate optimization technique for optimal

material layout desigr: topology optimization.

2.1.2 Topology Optimization

Topology optimization is the third commonly implemented structural
optimization technique and will be discussed in greater detail because it portrays
ideas most closely related to those represented by the parametric material layout
optimization approach (see Chapter 3). The goal of topology optimization is to
efficiently distribute material throughout a defined structural domain subject to
prescribed loading and boundary conditions, such that the stifliress of the structure is
maximized (Hafika and Gurdal 1993, Signond and Tcherniak 2001, Canonaco et al.
1997). Topology optimization uses finite element formulations to generate optimal

desigr concepts. Given a problem definition consisting of a defined domain, loading

18

and boundary conditions, and mass and deflection constraints, the most structurally
efficient material layout can be determined. Initially, the available material is evenly
distributed throughout the domain, and then re-distributed within the desigr space
until the structure has achieved adequate stiffness. The redistribution of material is
achieved by addition, removal, or merging of cavities within the domain, resulting in
an optimal starting point for a refined desigr (Hafika and Gurdal 1993). Diverse
topology optimization techniques have been developed to optimize the material
distribution of various structures. Topology optimization algorithms can be extremely
robust provided a sufficient problem definition exists (Canonaco et al. 1997). In spite
of its power, topology optimization techniques locate optimal solutions (i.e. designs)
that feature extensively detailed geometries, which can be difficult to manufacture. If
this is the case, shape optimization strategies can be explored to “fine-tune” the
design; or a topological algorithm which is capable of preserving efficient and
adequate structural desigrs can be implemented (Canonaco et al. 1997).

The homogenization method is one common method used to implement
topology optimization. This idea was introduced by Bendsoe and Kikuchi, and is
implemented to optimize the material distribution in a perforated structure with
infinite rnicro-scale voids. This is accomplished by discretizing a specified domain
into multiple finite elements where a material density is then prescribed over the
defined domain. The optimal density of each element is determined from the stress
limit and a constraint is placed on the percentage on material used in the space and
affects the type of solution that is generated (Bendsde and Kikuchi 1988).

The “hard-kill’ and “soft-kill” methods are also examples of common

19

approaches to topology optimization. Both of these methods focus on removing
unnecessary material to generate an optimal topological desigr. The hard kill option
eliminates material much faster than the sofi kill method. The hard kill technique
removes elements of material experiencing low stress concentrations by replacing the
element’s elastic modulus with the stress it experiences during each finite element
analysis. This means that regions of high stress become harder, while less loaded
areas become softer. Thus, the initially homogeneous domain becomes non-
homogeneous with varying elastic moduli. This technique however, produces designs
that contain stiff material in the load-bearing zones; therefore desi gr solutions may
possess regions of considerably large stress concentrations (Mattheck 1997). Tire sofi
kill approach weakens or softens the elements by replacing the elastic modulus of
each element with its original modulus plus the relative increment of stress it
experiences. Specifically, Em = En + Ac“, where, n is the number of finite element
analyses and A0,, = on - any. The reference stress, any, is defined as the desired
component stress. The soft kill option does not remove material as the quickly as the
hard kill, but will eventually eliminate the material that is not beneficial to the
structure’s mechanical properties such as strength and stiffness (Mattheck 1997).

The power law technique is another well-accepted approach to topology
optimization. This method is approached by discretizing the design domain into
multiple finite elements where the relative density of the material within each element
is the desigr variable. In this technique, the Young’s modulus of the material is
proportional to the relative density raised to a power. Consider the domain displayed

in Figure 2-4, where the gey region represents the desigr domain with loading and

20

boundary conditions, the black region denotes the solid material, and the white area is
to be a void. Using the power law approach to topology optimization, an optimal
material distribution for the cantilever beam subjected to a concentrated load (Figure
2-4) can be obtained. The optimal design solution is shown in Figure 2-5 (Sigmund

and Tcherniak 2001).

S l'dM t ' 1
Design Domain 01 a erra

 

Void
Figure 2-4. Example of topology optimization using the power law approach
(Adapted from [Sigmund and Tcherniak 2001])

 

Figure 2-5. Result from implementation of power law approach (Taken from
[Sigmund and Tcherniak 2001])

As previously stated, the optimized structure shown in Figure 2-5 is a design
solution produced by the power-law technique, but also resembles a typical design
solution that would have been obtained by implementation of the homogenization and
hard and sofi kill approaches to topology optimization. This design solution (see

Figure 2-5) displays an effective distribution of material, however it is geometrically

21

intricate thus difficult and expensive to manufacture. Therefore, alternate techniques
must be explored which are capable of optimizing the material distribution within a
pro-defined domain, while simultaneously accounting for manufacturing.

In order to account for manufacturing while implementing topology
optimization, an alternate technique was investigated. Chellappa et al. proposed a
topology optimization technique using a multiresolution method with finite-size
features. This approach employs a wavelet-based decomposition (Chellappa et al.
2004) of material distribution followed by a multi-resolution analysis which is then
used to generate a library of stiffness matrices for various elements or coupons. Each
element can be of different size and, in general, can include more than one
perforation. Each coupon is of finite dimension and the effective stiffness matrices of
two coupons of different dimensions are typically different. Structures are built and
optimized using diverse combinations of the coupons within the library. A typical
optimization problem locates optimal patterns of perforations, e.g., to minimize
weight (Chellappa et al. 2004). Figure 2-6 displays atypical desigr solution obtained
for a cantilever structure subjected to a concentrated load using the multiresolution
approach. The optimized structure has finite features therefore reducing geometric

intricacies, thus making it feasible and cost-effective to manufacture.

22

 

Figure 2-6. Result from implementation of the multiresolution approach
(Adapted from [Chellappa et al. 2004])

Given a problem consisting of a defined domain, loading and boundary
conditions, and mass and deflection constraints, the most structurally efficient
material layout can be determined by implementing topology optimization. Despite
the advantages of this technique, most topological algorithms are written to allow for
the use of only one objective function, are generally written to define a single loading
condition, and do not typically incorporate multiple constraints (e. g. stress and strain).
Redefining loading and boundary conditions, integrating constraints, and
implementing multiple objective functions would require modifications to the original
program or redevelopment of the program in its entirety. Due to this limitation, an
alternate optimization procedure was explored to perform material layout
optimization for the present work. Without difficult progam restructuring, the
presented technique permitted the use of multiple objective fimctions and constraints.

This approach is introduced next and presented in detail in Chapter 3.

23

2.1.3 Parametric Material Layout Optimization

The Optimization approach that was implemented for this research was
defined to be a material layout optimization technique because it accomplishes goals
similar to those intended for shape and topology optimization; although it does not
technically fulfill all the criteria associated with either. The optimization technique,
which is discussed in more detail in Chapter 3, was conducted on structures with a
predefined domain and an initial material layout. The technique was not implemented
to optimize the shape of the domain, but the material distribution within the domain
using a parametric modeling approach. Specifically, certain parameters were defined
prior to optinrization and only those parameters were used to control the material
layout within the defined boundaries. A specific number of cavities which essentially
defined material layout within the structure were established in the initial desigr. The
size and positioning of these theses cavities were capable of changing, however
cavities could not be added, removed, or merged. Despite this limitation, the
parametric based optimization approach does allow for incorporation multiple
objective functions and desigr constraints and is capable of producing desigr

solutions that account for manufacturing.

2.2 Optimization through Genetic Algorithms

The optimization procedure proposed in Chapter 3 focuses on the use of genetic
algorithms (GA’s). Genetic algorithms refer to a class of adaptive search procedures
using the principle of “survival of the fittest” to locate optimal solutions where the
fittest members of an initial population are given better chances of reproducing and

transmitting part of their genes to the next generation. (Turkkan 2003, Chou et al.

24

2001). The method is based on stochastic or random search methods to operate on a
population of solutions to locate a global optimum. They are less likely to get “stuck”
at local optimums, in comparison to gadient search methods. Genetic algorithms are
methods for optimization that work well on combinations of discrete and continuous
problem sets. Implementation of a GA requires representation of the solution to a
problem as a chromosome. The GA will then create an initial population of solutions
and apply a genetic operator such as selection, mutation and/or crossover to evolve
into “new” populations of solutions and eventually locate the optimum. Figure 2-7
provides a visual description of the implementation of a genetic operator. Selection,
mutation, and crossover operators are applied to the existing population are a
population to generate a “new” population of chromosomes. This “new” population is
generated by the most “fit” chromosomes from the initially population. A schematic
of the GA procedure is shown in Figure 2-8 and presents all the important steps
required for proper implemented of a genetic algorithm. These steps will be discussed

in greater detail throughout the remainder of Chapter 2.

25

Gene

Chromosome -7////////-7//A-7/AIZ_

 

Selection
Mutation

Crossover

N-tlk\\\‘-

   

9
/.
l a
Z

9' I
32
7,
I

 

Z
-
fl
=

   

Population “New” Population

Figure 2-7. Language of genetic algorithms

 

Initial Population

 

 

 

 

 

 

 

 

 

 

‘——> Fitness Objective Frurction

 

 

 

 

 

Selection

Crossover/Mutation

' Yes

 

 

 

 

 

 

 

 

 

 

 

 

Generation > Max Generation

 

 

 

 

No

 

 

 

Generation = Generation + l

 

 

 

Figure 2-8. Schematic of GA (Adapted from [Turkkan 2003])

26

Three important aspects must be defined prior to implementing 3a GA. These
include: (1) definition and implementation of the genetic representation, (2) definition
and implementation of genetic operators and control parameters, (3) definition of an
objective function, and (4) definition of the fitness function (Wall 2001). The

following sections will discuss the basic aspects of GA’s in more detail.

2. 2.1 Definition and Implementation of Genetic Representation

In order to implement a genetic algorithm an appropriate data structure and
representation must be defined. The problem characteristics and data structure are
usually represented in the format of a chromosome string. The data structure should
include all relevant parameters of the problem and uniformly symbolize all possible
solutions. For example, if the fimction being optimized is of real numbers, real
numbers should be chosen in the chromosome. On the other hand, if imaginary
numbers and integer values are observed in the objective function, the chromosome
should be defined with those characteristics. The chosen representation must be able
to represent all solutions to the problem, but if possible, representation of infeasible
solutions should be eliminated. If the chromosome is capable of representing
infeasible solutions, the objective frmction must be desigred to give them partial
credit, so that they do not reproduce and/or exist in multiple future generations. The
representation should not contain information beyond what is necessary to represent a
solution to the problem since this tends to increase the size of the search space and
impede the performance of the GA (Wall 2001). For examples, if the problem
depends on a sequence of items, an order-based representation can be selected. If this

is the case, the genetic operators must be chosen so that the reliability of the sequence

27

is maintained. Common structures used for representation include a list, an array, or a

tree structure (Figure 2-9) (Wall 2001).

 

O
15>}.

 

 

 

 

 

 

Tree
1C" “31195 Q
0 0 List 0202020202:

 

Figure 2-9. Possible structures used for representation of individual
chromosomes (Adapted from [Daves 1991]).

Some problems include a combination of continuous and discrete variables. In
this case, it may be necessary to create a new structure to store the mix Of information
and the genetic Operators must be defined so that the structure of the solution is not
affected. Specifically, a solution containing integer values and real numbers might
use a crossover Operator that crosses the integer values with one another and the real
numbers with one another, but does not mix the two values. Ultimately, the genetic
Operators must be chosen so that they are appropriate for whichever representation is

selected (Wall 2001).

2.2.2 Definition and Implementation of Genetic Operators and Control
Parameters

Genetic Operators are used in genetic algorithms to generate diversity and to

combine existing solutions with others (Guervos 1997). Three primary genetic

28

Operators can act upon a chromosome and include: selection, mutation, and crossover.
Using these operators, an initial population can be favored, a mutation or crossover
corresponding tO the genetic representation can be defined, and alterations Of the

genetic algorithm can be made as the population evolves (Wall 2001).

2.2.2.1 The Selection Operator

The selection Operator gives preference tO the better, or more “fit,” designs, or
individuals, allowing them to pass their “genes” onto the next generation. Researchers
recommend using an enlarged sampling approach for the selection process, which is
used to reduce the amount Of duplicating chromosomes entering a population during
selection. Two enlarged sampling approaches exist, the (u + A) and the (u, A), where
p. is the number Of parents and A is the number Of Offspring. The “comma” or “plus”
determines the type of selection process used in choosing the new parental generation
(Beyer 2000, Chou et al. 2001, Lagaros 2002).

The “comma” selection method is a process of competing Offspring. It begins
when u parents produce 2. children through mutation. The p parents are discarded,
leaving only the children to compete directly with one another. The children are then
assigred a fitness value based on their quality considering the problem specific
conditions that they are in. The best individuals Of the A children are selected as the
next parental generation. This technique is capable Of diverging due to the fact each
parent can only produce children once in the entire process, resulting in possible
elimination Of the most-fit individuals (Lagaros 2002, Beyer 2000).

The “plus” selection method involves competing parents and offspring. This

process is implemented when u parents produce it children by mutation. Each child is

29

then assigred a fitness value and the best or fittest individuals Of both the parents and
offspring become the next generation’s parents. This selection process has proven to
be more effective due to fact that both parents and children can survive through
multiple generations. The more elite or fit will remain, and the weak are eliminated,
therefore possible solutions will not be discarded throughout the process (Lagaros

2002)

2.2.2.2 The Mutation Operator

The mutation operator plays two important roles in GA’s: (1) it provides and
maintains diversity within a populatiOn to prevent premature convergence and (2) it
can work as a search Operator. The mutation Operator makes alterations to an
individual chromosome rather than combining parts of two or more chromosomes as
is done with the crossover technique. Mutation can alter a single field, multiple fields,
or all fields of an existing chromosome. For example, if a child has a given
chromosome structure consisting of five 2-bit fields, the single, multiple, and all-field

mutation would be as shown in Figure 2-10. (Daves 1991 , Goodman 2002).

30

Offspring, = 10/11/00/01/10

 

fl Mutation
Single-Field Mutation Multiple-Field Mutation All-Field Mutation

l l U

Offspring2= 10/11/01/01/10 Offspring2= 10/11/01/11/10 Oflspfing2= 00/10/01/11/11

Figure 2-10. Examples of mutation

2.2.2.3 The Crossover Operator

Like the mutation Operator, the crossover operator also allows for exploration
Of a new solution space. Three main types Of crossover methods exist in GA’s and
include: one point, two point, and uniform crossover. In one-point crossover, a
random position along a pair Of chromosomes is generated. The bits from a fixed
position on the first chromosome to the end are swapped with the second
chromosome in the same range. In the process, the bits from the second chromosome
are transferred to the corresponding bits in the first chromosome. For example, if

parent 1 and parent 2 have the following chromosome structure (Chou et al. 2001):

Parentl=101010510101
Parent2= 111001 200111

I
Crossover Point

The generated Offspring would be:

Offspring] = 1010105 00111

Offspring; =111001§ 10101

Crossover Point

Two-point crossover is an enhancement Of one-point crossover approach and
is desigred to explore a wider search space. With this method, the chromosomes are
thought of as rings with the first and last gene connected. The rings are cut in two
positions and the resulting portions are swapped. For example, if parent 1 and parent

2 have the following chromosome structure (Chou et al. 2001):

Parent] = 10 510105 00111

Parent, = 11 1001i 10101

\/

Crossover Points

32

The generated offspring would be:

Offspringl=1151010510101

Offspring, = 10E 1001i 00111
V
Crossover Points
Uniform crossover is a process which involves random selection Of bits to
form Offspring. This crossover technique results in only one Offspring, as opposed to
the two produced by the one and two-point crossover methods. Specifically, each bit
of the Offspring is randomly chosen from the corresponding bits of the parents. For
example, if parent 1 and parent 2 have the following chromosome structure (Chou et
al. 2001):
Parentl = 11101010101
Parent; = 10100100111
The generated Offspring could be:

Offspringl = 11100000111

2.2.2.4 Control Parameters

Various control parameters can be used to vary a genetic algorithm. These
parameters include: (1) crossover rate, (2) mutation rate, (3) generation gap, (4)
population size, (5) scaling and (6) stopping criteria. The crossover rate defines the
probability of crossover occurring between two chromosomes. Mutation rate is the
probability that a value in a chromosome will be changed. The generation gap

identifies the proportion Of the population that will be replaced with new Offspring.

33

The scaling parameter controls the fitness function to magrify the differences
between chromosomes. This magrification assists in maintaining competition in the
search space to prevent premature convergence which may be based on a few highly
fit chromosomes. The stopping criteria are directly related to termination Of the
evolution process. This could be based on a number Of cycles, fitness convergence, or

other criteria (Chou et al. 2001).

2.2.3 The Objective Function

The objective function, also known as the goal function, is the function to be
Optimized and is used to determine how “gOOd” or “fit” each individual is. This
process involves the evaluation Of the individuals and determination of a fitness
value. After a fitness value is determined, the selection process is then implemented
again and the best individuals from the present generation are selected for the new
generation. The fitness of each individual is defined by a value, which reflects how
well an individual solves the task at hand. This value, along with generational age is
used to determine the number Of times any individual is replicated (see Section 2.2.4)

(Daves 1991).

2.2.4 The Fitness Function

The “fitness function” quantifies the Optimality of a solution in a genetic
algorithm such that a particular desigr may be ranked against all other designs. This
function is generated on the basis Of the Objective values of the individual in
comparison with all other individuals in the selection pool. The fitness function is

used tO map a chromosome to a fitness value and may be dependent on the Objective

34

function, different constraints and/or stochastic influences. The best fit individuals are
those who will survive in their existing environments and whose descendents will

most likely adapt or thrive in future environments (Daves 1991).

35

3 DEVELOPMENT AND IMPLEMENTATION OF
PARAMETRIC MATERIAL LAYOUT OPTIMIZATION
TECHNIQUE

3.1 Overview

This chapter presents the development and implementation Of a parametric
approach to material layout Optimization. This computational study was completed
through implementation of the following Objectives: (1) development Of the
Optimization procedure, (2) validation Of the proposed Optimization technique, and (3)
application of this technique to a specific case study. These three tasks are introduced

and discussed in detail throughout the remainder of this chapter.

3.2 Overview of the Optimization Technique

The optimization technique presented in this thesis focuses on an idea similar to
that Of topology Optimization, which is tO distribute material efficiently throughout a
defined structural domain subject to loading and boundary conditions. Topology
Optimization techniques make possible the identification of Optimal solutions to
material distribution; however these solutions (i.e. Optimal desigrs) may be rather
extensive with detailed geometries that may create difficulties in manufacturing
biocomposite components due to their random nature and fluffy, cotton-like texture
(Chapter 1). This limitation has been addressed through recent efforts that propose
methods resulting in desigr solutions with simple domain geometries. One example
used to generate simplified geometries is through implementation of the
multiresolution technique (Chellappa et al. 2004) (Chapter 2), which involves the
formulation Of a finite element code that pro-defines the boundaries Of the structure

and only allows the material distribution within the domain to change. The limitation

36

Of the multiresolution technique, however, is that in order to incorporate of multiple
objective functions and constraints, re-forrnulation Of the finite element progam is
necessary. TO avoid this limitation, it is also possible to approach the problem by
using existing finite element progams together with parametric modeling. In this
method, simultaneous implementation Of finite element software and a general
purpose Optimization progam allow the Optimal material distribution within a defined

structural domain for maximum stiffness to be Obtained.

The global Optimization problem studied for this thesis was approached using
parametric modeling and finite element analyses conducted with the commercial
program ABAQUS (Hibbitt et al. 2004) and solved using a genetic algorithm
implemented through the commercial software package, HEEDS (Red Cedar
Technology 2004). HEEDS is a general purpose optimization software package that
automates the search for Optimal solutions within a given design space through the
use of different mathematical Optimization algorithms (gradient-based, genetic
algorithms, simulated annealing, desigr of experiments, etc.). The development Of the
Optimization procedure was initiated by modeling and analyzing an initial desigr in
ABAQUS CAE, ABAQUS pro-processor, by means of a script file that defined
geometric parameters within the domain, i.e. void size, positioning, etc. HEEDS
altered the defined parameters within the existing base desigr by means Of the
ABAQUS CAE script file to create a modified or “new” script file. A finite element
analysis was then conducted with ABAQUS to evaluate the “new” design. The fitness
function Of each desigr was then assessed using HEEDS according to the

optimization problem formulation. The process is iterative and was repeated until

37

design time was exhausted (Figure 3-1). Exhausting the desigr time was the stopping
criteria of the particular GA implemented by HEEDS and was dependent on the
number of cycles defined by the user. Even though this criterion was not based on
fitness convergence, it was still capable Of being detected. For example, if the
comparison in the fitness value of the last few desigrs was within a small percentage
Of one another, convergence can be assumed. Furthermore, convergence can be
assumed if the last desigr saved was Obtained approximately 10 cycles prior to

completion Of the last cycle.

 

 

 

 

 

 

 

 

 

 

 

 

 

. . . . FEM model is
l t 1
6:82;“ Geometry and F112“ generated in HEEDS - I °
——p parameters are ————-> ABAQUS CAB ——> c o o
initially defined in and analyzed in A / I \
ABAQUS CAE AB AQUS O O o o o o
Parametric Modeling FEM Model Genetic Optimization Algorithm
If constraints 1
are not satisfied
._ Design is updated in

 

reevaluated in
ABAQUS

1

Possible optimal
solutions

 

 

 

 

 

 

 

Figure 3-1. Optimization flowchart

38

'- ............................................................................
I

ABAQUS

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. . . . Scri t File FEM Mesh is
Inrtraolddesrgr rs _P_. generated and
creat model is evaluated
New desigr is
5: created
New population is [
--11——-1 used to run the HEEDS '
algorithm Frtrress of the new
design is assessed
New population of ~ .
designs is .._, Fitness of each Randompopulatron of
generated by design is evaluated desrgns rs generated

 

 

 

 

 

 

evaluation Of the
previous generation

 

 

 

 

 

Figure 3-2. Optimization process

The Optimization process starts with HEEDS randomly selecting values for
specified desigr variables within user-defined limitations (i.e. variable bounds). The
desigr variables are defined as the geometric parameters i.e. void size, positioning,
etc. within the initial ABAQUS CAE script file and are altered by HEEDS to control
the re-modeling Of each existing desigr to create a new one. The ABAQUS CAE
generated script file, typically referred tO as the python file, (model.py), which is
modified by HEEDS, is then fed back into ABAQUS for re-meshing and analysis.
This process is repeated with ABAQUS CAB creating multiple re-modeled desigrs
and the fitness (fitness formulation will be discussed in later section) Of each being

evaluated by HEEDS (i.e. analyses were performed). The first desigr is saved to the

39

HEEDS working files as the benchmark desigr and new desigrs are added when a
better desigr is found. HEEDS continues to alter the desigr variables within the
ABAQUS CAE script file to generate new desigrs until the number Of cycles has
completed. The Optimization process described above (Figure, 3-1 and Figure 3-2) is

schematically shown in Figure 3-3.

 

DIG) DIG) D36) Script File

Initial design with defined FEM mesh is generated

geometric parameters (diameter)
Re-create FE Analysis
’/ mesh

D1 D2 D3 System constraints are
° CD (I) <:’ checked and fitness of

Design is design is evaluated

updated

Possible Optimal

solutions

0,0 DIG) 13,0)
”(DUKE ”(D

Figure 3-3. Schematic of optimization procedure

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
  

7
Optimal D1 D2 1),
=> (1) o .

 

 

 

 

 

3.3 Validation Of the Optimization Procedure
The proposed Optimization procedure was validated by comparing its

performance in solving standard topology optimization problems against those

40

Obtained with well-accepted topology Optimization methods. Thus, the Optimal
solutions Obtained through the parametric approach were compared tO “known”
solutions generated by employing actual topology optimization techniques. If the
results were similar, the proposed Optimization method was considered valid for the
overall goal Of Obtaining an optimized material layout. The validation procedure
included six major tasks: (1) problem definition, (2) parametric modeling, (3)
Optimization problem formulation, (4) implementation Of the optimization process,
(5) evaluation Of Optimal solutions, and (6) validation Of optimal solutions. These

tasks are discussed in detail in the following sections.

3. 3.1 Problem Definition

Validation Of the proposed parametric optimization procedure was conducted
through the Optimization of two classical topology problems: The standard problems
were borrowed from literature and include: (1) the Messerschmitt-BOlkow-Blohm
Beam (MBB-beam) (Olhoff et al. 2004) and (2) the 8-bar truss (Rozyany et al. 1992).
These problems have been used extensively by researchers to evaluate topology
Optimization algorithms. The MBB-beam is a simply supported beam with an aspect
ratio of 1/5 (Figure 3-4) and the 8-bar truss is a short cantilevered beam with an
aspect ratio Of 5/8 (Figure 3-5). Both problems are defined as having concentrated

unit loads and with a unit Young’s modulus.

41

 

6 in.

 

 

 

30 in.

Figure 3-4. MBB-beam

 

 

 

‘1?-
AL

12.8 in.

_ Figure 3-5. 8-bar truss

3.3.2 Parametric Modeling

Since the goal was to Optimize the structural or material arrangement within
the pre-defined domain, the parametric model must allow for redistribution of
material. TO accomplish this, an initial desigr with a pre-defined layout was created
for both the MBB-beam and the 8-bar truss (see Figure 3-6 and Figure 3-7) and the
radii of each void were identified as the parameters to control the desigr features or
material layout. The parameters (i.e., the radii of each void) were identified through

ABAQUS CAE by means Of a script file that was then linked tO the Optimizer. The

42

Optimizer uses this ABAQUS script file to read the parameters (i.e radii) and
randomly alter them to generate a modified script file. This modified script file is then
used by ABAQUS to create a pararnetrically remodeled desigr. This parametric
remodeling occurs throughout each of the desigr iterations prior to re-meshing and

FE analysis.

 

A I

 

Figure 3-7. Parametric model for 8-bar truss (initial design)

Changes to the parameter values (radii) throughout the Optimization process

are controlled by the definition of desigr variables within the optimization problem

43

formulation. Details Of the problem formulation are discussed in the following

section.

3.3.3 Optimization Problem Formulation

The Optimization problem formulation includes identifying and defining the
Optimization algorithm, genetic Operators and control parameters, and formulation
elements such as desigr variables, system constraints, Objective function, and the
fitness function for the both the MBB-beam and the 8-bar truss.

The algorithm chosen within HEEDS capabilities for the presented parametric
layout Optimization approach was a general genetic algorithm. Genetic algorithms use
a stochastic or random search method to create a population Of desigrs. Desigr
variables, which define the desigr solution, are typically represented in binary form
as a continuous string resembling a chromosome. Through the implementation Of
crossover and mutation Operators, individuals in the current population evolve into
new populations of individuals and the fitness of each chromosome is evaluated. The
process continues until the total number of user defined cycles is completed (Chapter
2) (Wall 2001).

The genetic Operators chosen for the Optimization procedure were multi-field
mutation (Chapter 2) and one point crossover (Chapter 2). Multi-field mutation
allowed for alteration in multiple fields within each chromosome (Goodman 2002).
The one-point crossover operator selects a common, random crossover point in two
parents and then swaps the corresponding bits to generate Offspring (Daves 1991)

The control parameters defined for the general genetic Optimization technique

include: (1) mutation rate, (2) crossover rate, (3) population size and (4) stopping

criteria. For the presented parametric Optimization technique, the mutation rate was
defined to be 20%, which means that the values in each chromosome would have a
20% probability of changing. Similarly, the crossover rate was defined to be 50% (i.e.
a 50% crossover probability between chromosomes). Lastly, the population size was
set equal to 60. Choosing a larger population size increases the number of evaluations
performed per cycle, enhancing diversity within each desigr and, most importantly,
assisting in preventing premature convergence. The stopping criterion for this genetic
algorithm was dependent on the number Of cycles, which was set to a value Of 75.
Once the 75 cycles was completed the Optimization problem was terminated. Even
though this stopping criterion is not based on fitness convergence, it is capable Of
being detected. For example, if the comparison in the fitness value of the last few
desigrs is relatively close in value, convergence can be assumed. Furthermore,
convergence can also be assumed if the last design saved was Obtained approximately
10 cycles prior to the completion of the last cycle. A summary of the genetic

operators and control parameters is shown in Table 3-1.

Table 3-1. Summary of genetic operators and control parameters

Genetic Operators

Multi-field Mutation
One-Poirrt Crossover

Control Parameters

 

 

Mutation Rate Crossover Rate Population Size Stopping Criterion
20% 50% 60 75 Cycles

 

45

Desigr variables are used tO model the specific parameters that influence the
system performance. Variables can be represented as continuous or discrete sets,
and/or as dependent variables. Optimization for the MBB-beam and 8-bar truss was
performed twice, once using continuous variable sets and once using discrete variable
sets the radii to account for manufacturing. In both attempts at Optimization,
dependent variables were also defined. The desigr variables were used to represent
the radii of each void in the initial desigr layout (Figure 3-6 and Figure 3-7). The
continuous and discrete sets were formed. by defining virtual regions around each
void and restraining them to increase or decrease in diameter within that region. The
surrounding areas encompassing each void were 2 in. X 2 in. for the MBB-beam,
resulting in 45 virtual square elements (Figure 3-8); and 1.6 in. X 1.6 in. for the 8-bar
truss, resulting in 40 elements (Figure 3-9). Each void was numbered so that a
corresponding radius could be defined. For example, the voids in the upper left hand
corners of Figure 3-8 and Figure 3-9 were assigned a corresponding radius of n, and
continued along the rows, completing with the voids in the lower right hand corners

which were identified as having a radius Of r45 and r40, respectively

46

2.00 6.00 10.00 14.00 18.00 22.00 26.00 30.00
0.00 I 4.00 I 8.00 I 12.00 I 16.00 I 20.00 I 24.00 I 28.00 I

 

Figure 3-8. Virtual square elements within the domain of the MBB-beam
(dimensions shown in inches)

0.00 1.60 3.20 4.80 6.40 8.00 9.60 11.20 12.80
8.00——
6.40—
4.80—-
3.20——

1.60 —-

 

0.00 —

Figure 3-9. Virtual square elements within the domain of the 8—bar truss
(dimensions shown in inches)

The continuous and discrete variable sets used in both attempts at
Optimization of the MBB—beam and 8-bar truss problems were defined such that each
void in the base desigr was allowed to increase to a maximum of 90% of its
surrounding virtual region, thus preventing void overlap and exceeding Of the
domain. The continuous variable set was defined such that the radius of each void
could range in value from 0.05 in. to 0.9 in. with increments Of 0.01 in., while the

discrete variable set was defined using a set of 4 values: 0.05 in., 0.25 in. 0.55 in., and

47

0.9 in. Some of the variables were defined as dependent due to the symmetric
response present in both systems. Thus, a symmetry condition was enforced along a
vertical plane for the MBB-beam (see Figure 3-10) and along a horizontal plane for

the 8-bar truss (see Figure 3-11).

 

Verticail Plane
of Symmetry

Figure 3-10. Plane of symmetry enforced on MBB-beam

___ Horizontal Plane of
Symmetry

 

Figure 3-11. Plane of symmetry enforced on 8-bar truss

Constraints represent the lirrritations or specifications that a variable is
subjected to and are used to guide the optimization process since Optimal solutions
are searched only amongst those that satisfy them (Arora 1989). The optimization

problems for the MBB-beam and the 8-bar truss were formulated using a single

48

system constraint. A mass constraint governed the desigr and was defined by
implementing a relative density (Equation 3.1) Of 0.65 for the MBB-beam and 0.70
for the 8-bar truss which were values borrowed from literature (Chellappa et al.
2004). Specifically, the mass Of the optimized MBB structure was constrained to be
no more than 65% of the mass of an identical structure with zero voids. Similarly, the
mass of the optimized 8-bar truss was unable to exceed 70% of the mass of an

identical structure with zero voids.

M

optimizeddesign _ p
— relative (3.1)
M solid

 

The objective fimction is the Optimized target that minimizes or maximizes a
specific aspect of the model (Daves 1991). The objective function for the MBB-beam
and 8-bar truss Optimization problems was to minimize strain energy. This Objective
was chosen since the goal is to Obtain the stiffest structure possible with a given
amount Of material. Clearly, strain energy and mass are indirectly related, since as
material is removed the mass reduces, which increases the strain energy. Table 3-2
through Table 3-5 summarize the formulation of the optimization problems
implemented for the two validation models. These formulations were defined in the

HEEDS Optimizer prior to process initiation.

49

Table 3-2. Optimization problem formulation for MBB-beam

 

 

 

 

 

 

Find: r" , where n = void number = 1-8, 16-23, 31-38 (integers)
That minimizes: Strain Energy

Subject to: Mass constraint

Find: ’n , where n = void number 1-8, 16-23, 31-38 (integers)
That minimizes f(x) = l v or: dv

Subject to: 81' (x) S 0, i = l- n constraints

 

 

Table 3-3. Summary of formulation elements for the MBB-beam

 

Formulation

 

 

 

. 't' .
Elements Vanable Type Defim ron of Forrnulatron Element
r" , where n = void
Discrete/Continuous Void Radii number = 1-8, 16-23, 31..
38 (integers)
Desigr Variables " n . where n = V°id
Dependent Void Radii number = 9-15, 24-30,
39-45 (integers)
Parameter Length 12.8 in.
Parameter Height 8 in.
Constraints Mass M S 117 lbs
Objective Function Mininrize Strain Energ

 

50

Table 3-4. Optimization problem formulation for the 8-bar truss

 

 

 

Find: r n , where n = void number = 1-24 (integers)
That minirrrizes: Strain Energy

Subject to: Mass constraint

Find: r n , where n = void number = 1-24 (integers)
That minimizes f(x) =1 ,, oe dv

Subject to: g ,- (x) S 0, i = 1- n constraints

 

Table 3-5. Summary of formulation elements for the 8-bar truss

 

Formulation

 

 

 

Variable Type Definition of Formulation Element
Elements
Discrete/Continuous Void Radii r" ’ where n = vord
number = 1-24 (integers)
Desigr Variables r n , where n = VOid
. Dependent Void Radii number = 25.40
A (integers)
Parameter Length 12.8 in.
Parameter Height 8 in.
Constraints Mass M 5 71.68 lbs
Objective Function Minimize Strain Energr

 

As stated in Chapter 2, the fitness function is used to determine the optimality
Of a solution in a genetic algorithm and may be dependent on different constraints
and/or stochastic influences. The fitness function defined in the Optimizer, HEEDS,
for a genetic algorithm is dependent on the Objective function and desigr constraints.

The fitness value is Obtained as follows (see Equation 3.2):

51

Fitness _ Objective

Value — Function (Penalty thction "' Constraint Violation)

Specifically, for optimization Of the two validation problems:

E

= —10,000* M —M. (32)
N0 m a 11.21.” gC 0 efli Ci 6 n t [ ( allowable wolated )]

 

where, E is the strain energy and M is the mass.

This function is evaluated by HEEDS at the end Of each desigr iteration and
the fitness level of the desigr is dependent the Obtained value. The most “fit” desigr
(i.e, the desigr with the highest fitness value) when the number of user defined cycles
has exhausted is considered the best desigr. It is only considered Optimal if the fitness
value of the prior desigrs was similar to that of the last desigr Obtained or the last
desigr Obtained was found approximately 10 cycles prior to completion of the last

cycle.

3. 3.4 Implementation of Optimization Process

In order to implement the Optimization procedure, it was necessary to link the
parametric model tO the problem formulation, which was done within the HEEDS
optimizer. The connection was performed by means Of the script file that is generated
after creation Of the initial desigr through ABAQUS pro-processor: ABAQUS CAE.
The task Of the Optimizer was to randomly select values for the design variables, or
controlling parameters, within the defined continuous and discrete variable sets (see
Section 3.3.2) to alter the parameters in the existing ABAQUS script file. The
modified script file was then used by ABAQUS CAE, to generate and re-mesh a new

model. The parametrically remodeled design could then be analyzed by ABAQUS

52

which would retrieve the output values needed to evaluate the fitness in HEEDS.
Through continuous alteration Of the design variables/controlling—parameters a
collection of new desigrs could be created. The Optimizer saves the first desigr as a
benchmark desigr and a new desigr is added when a better one is found. The process
is repeated until the number of user defined cycles has been reached. The last desigr
saved by the Optimizer can be considered Optimal if its fitness value compares
favorably tO the two or three prior desigrs that were saved or if the last desigr saved
was Obtained approximately 10 cycles before the maximum amount Of cycles was

reached.

3.3.5 Evaluation of Optimal Solutions
Using the Optimization problem formulations displayed in Table 3-2 through

Table 3-5 in the previous section, the proposed procedure was implemented using
both the continuous and discrete variable sets to Optimize the material layout Of the
MBB-beam and the 8-bar truss problems. It was found that the difference in results
when using the continuous variable sets and discrete variable sets were minimal; with
Optimization using the discrete variable sets producing desigr solutions for both the
MBB-beam and the 8-bar truss problems that were more comparable tO those
Obtained through the well-established topology Optimization techniques (Section
3.3.6). Therefore, the desigr solutions and evaluation history displayed in this section
only represent those Obtained through Optimization with the discrete variable sets.
The parametric modeling approach to Optimization produced multiple desigrs
for both problems throughout the multiple cycle run time. In the early stages of

optimization various desigrs were produced, but a steady and immediate decrease in

53

strain energy was not apparent (Figure 3-12 and Figure 3-13). However, as the
procedure advanced the desigrs seemed to evolve from one another and progressively
improved (see Table 3-6 and 3-7). The relative densities of each structure began to
approach the maximum allowable value and the strain energy continued to decrease.
The desigr solutions obtained were those with the lowest strain energy and with
relative densities of 0.65 for the MBB-beam and 0.70 for the 8-bar truss (Figure 3-14
and Figure 3-15). These solutions could be considered optimal since they were saved
by HEEDS approximately 10 cycles prior to completing the allowed number Of 75
cycles. Thus, no improved desigrs were found for several cycles before concluding

the Optimized search.

62.0 -

60.0 n
58.0 n J
56.0 -

54.0 r

52.0 -
50““ 50.0 .

Em” 48.0 11.
46.0 1
44.0 -
42.0 «
40.0 -

38.0 n
36.0 f r r r F r r r I a—

0 200 400 600 800 1000 1200 1400 1600 1800 2000
DesignNunirer

 

 

 

 

 

Figure 3-12. Optimization history of MBB-beam problem using a discrete
variable set

54

Shaun

thugy

(Psi)

30115
291)5
2811‘
2711‘
261)-
251)—
241)“
231)“
2211*
211)5
201)-
191)“

 

 

—|__.—.

 

 

181)
0

I I I I I I 7 7 I I

220 440 660 880 1100 1320 1540 1760 1980 2200
Ineshprbhnnber

Figure 3-13. Optimization history of 8-bar truss problem using a discrete

variable set

55

Table 3-6. Evolution of designs for the MBB-beam problem

Design Design Number Mass (lbs) Strain Energy (psi) Relative Density

137.13 3.9ZE+01 0.762

129.74 4.26E+01 0.721

126.38 4.87E+01 0.702

125.00 4.80E+01 0.694

124.87 4.755401

120.83 5.64E+01

120.24 5.631901

111.24 5.97E+01

115.18 5.16E+01

1 16.45 5.06E+01

1 16.64 4.8715401

1 14.95 4.7OE+01

117.27 4.62E+01

117.14 4.39E+01

117.05 4.38E+01

116.83 4.355401

117.69 4.22E+01

117.22 4.24E+01

116.62 41le1

1 16.34 4.04E+01

117.23 3.99E+01

116.10 3.92E+01

 

56

Table 3-3. Evolution of designs for the MBB-beam problem (cont.)

Design Design Number Mass (lbs) Strain Energy (psi) Relative Density

1215 117.11 3.88E+01 0.651

1502 115.87 3.8113+01 0.644

1528 117.00 38115-101 0.650

1560 117.58 3.7313401 0.653

117.62 3.66E+01 0.653

117.48 3.68E+01 0.653

117.14 3.66E+01 0.651

117.42 3.63E+01 0.652

117.36 3.60E+01 0.652

117.04 3.61 E+01 0.650

 

57

Table 3-7. Evolution of designs for the 8-bar truss problem

 

 

Design

 

 

 

 

 

 

 

 

 

 

 

Design Number Mass (lbs) Strain Energy (psi) Relative Density
1 73.90 3.18E+01 0.722
3 71.72 2.52E+01 0.700

94 67.9 29413-101 0.663
96 70.77 2.59E+01 0.691
123 70.76 2.58E+01 0.691
269 69.05 2.50E+Ol 0.674
289 70.18 2.50E+01 0.685
406 69.46 2.41E+01 0.678
457 70.32 2.41E+01 0.687
464 70.81 2.26E+01 0.692

 

 

 

 

58

 

Table 3-4. Evolution of designs for the 8-bar truss problem (cont.)

 

Design

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Design Number Mass (lbs) Strain Energy (psi) Relative Density
648 70.58 2.28E+01 0.689
794 69.81 2.28E+01 0.682
840 70.18 2.27E+01 0.685
924 69.02 2.23E+01 0.674
939 70.59 2.22E+01 0.689
1045 70.96 2.1 1E+01 0.693
1052 70.78 2.10E+01 0.691
1058 69.96 2.10E+01 0.683
1096 69.99 2.03E+01 0.683
1551 70.33 2.02E+01 0.687

 

 

 

 

59

 

Table 3-4. Evolution of designs for the 8-bar truss problem (cont.)

 

 

 

 

 

 

 

 

 

 

 

 

Design Design Number Mass (lbs) Strain Energy (psi) Relative Density

0 O 0

o O. 1708 70.72 1.98E+01 0.691
O 0 0

Q 0 0

0.. 1870 70.36 1.98E+01 0.687
O 0 o

.0 o 0

00. - 1994 70.03 1.97E+01 0.684
O 0 0 0

0 o 0

O. - o 2071 70.50 1.951<:+01 0.688
0 O 0

0 o 0

00 - - 2206 70.33 1.93E+01 0.687
0 o 0

P

 
 

00000000000000.
000° 00. ° 00. '000
A 0/

Figure 3-14. Optimal design generated for MBB-beam using the discrete
variable set

60

 

Figure 3-15. Optimal design generated for the 8-bar truss using the discrete
variable set
3.3.6 Validation of Optimal Solutions

To ensure proper functionality of the proposed Optimization technique, the
optimal solutions to the MBB-beam and 8-bar truss problems (Figure 3-14 and Figure
3-15) were compared to those Obtained with two well-established topology
optimization techniques. Specifically, the optimal solutions generated by employing:
the numerical multiresolution approach and a power-law approach were examined
(see Chapter 2).

The multiresolution approach proposed by Chellappa et al. (2004) also
evaluated solutions to the MBB-beam and 8-bar truss problems. Thus, these results
are used as one set of benchmark solutions to compare the Optimal desigr Obtained by
the proposed approach in this thesis. The optimal solutions Obtained through
implementation of the proposed parametric approach compare favorably to those
achieved by Chellappa et al. (2004) through their multiresolution technique. A visual
comparison of the Optimal solutions for the MBB-beam shows that the material layout
in both solutions is quite similar (Figure 3-16 and Figure 3-17). Upon analyzing the

respective Optimal structures computationally (i.e. performing finite element

61

analysis), it was found that minor variations do exist in the resulting mass, strain
energy, and deflection (or compliance). A summary of results is given in Table 3-8.
The relative density of the MBB-beam Obtained using both approaches were identical.
However, a 5% difference was Observed in the compliance and strain energy.

The optimal material layouts and analytical results for the 8-bar truss were
also compared (Figure 3-18 and Figure 3-19). Again, a visual comparison indicates
that the Obtained solutions have similar material distributions. Similar to the MBB-
bearn comparison, the analytical results for the 8-bar truss varied minimally between
the two Optimization approaches. The comparison of results for the 8-bar truss
problem is given in Table 3-9. Similar to the results Obtained by the MBB-beam, the
relative density of the 8-bar truss obtained using both approaches were identical.

However, an 8% difference was Observed in the compliance and strain energy.

 

Figure 3-16. Optimal solution for MBB-beam using the multiresolution
approach by Chellappa et al. (2004)

62

 
 

00000000000000.
000° 00. ° 00. '000
,A ,0

Figure 3-17. Optimal solution for MBB-beam using the proposed parametric
modeling approach

Table 3-8. Comparison in results for the MBB-beam

 

Parametric Modeling Multiresolution

 

Approach (A) Approach (B) (NB)
Relative Density 0.65 0.65 0.0
Compliance 72.20 69.06 5.0
Strain Energy 36.10 34.53 5.0

 

 

Figure 3-18. Optimal solution for 8-bar truss using the multiresolution approach
by Chellappa et al. (2004)

63

 

Figure 3-19. Optimal solution for the 8-bar truss using the proposed parametric
modeling approach

Table 3-9. Comparison in results for the 8-bar truss

 

Pararrretric Modeling Multiresolution

 

Approach (A) Approach (B) (NB)
Relative Density 0.70 0.70 0.0
Conrpliance 38.55 35.67 8.0
Strain Energy 19.27 17.84 8.0

 

The material distributions Obtained for both the MBB-beam and the 8-bar
truss through the parametric procedure and the multiresolution approach were quite
similar. Material was shifted so as to transfer the load directly to the supports, placing
more material in regions of higher stresses and removing material from regions under
low stresses. The variation in computational solutions between the parametric
modeling approach and the multiresolution approach (Table 3-8 and Table 3-9) were
minor and may be due to the differences in the employed Optimization techniques
and/or the stopping criteria implemented by the two methods. As mentioned
previously, Chellappa et a1 (2004) implemented a multiresolution approach to
topology optimization (Chapter 2), which is a gradient-based technique. The search
method implemented in gradient-based approaches differs fiom that of a genetic

algorithm (see Chapter 2) (Arora 1989). The stopping criteria implemented in the

64

multiresolution approach also differs from that Of the GA. Unlike the multiresolution
approach which uses a convergence based stopping criterion, the stopping criterion of
this genetic algorithm is based on the number Of user defined cycles. In this case,
Optimization continues until the amount Of cycles is completed, however convergence
can still be detected. As stated previously, if the fitness value Of the last desigr saved
is relatively close tO the 2 or 3 desigrs saved prior, convergence can be assumed.
Furthermore, convergence can assumed if the last design saved was found

approximately 10 cycles prior to completion Of the last cycle.

The power-law approach to topology Optimization (Chapter 2) proposed by
Bendstie was also implemented to generate an Optimal topology for the MBB-beam
and the 8-bar truss. The power-law technique is capable of Optimizing topology
problems by distributing an initial amount Of material in the desigr domain such that
the compliance of a structure is minimized, i.e., maximum stiffiress (Signund and
Tcherniak 2001). The power-law method is implemented with mesh generation
routines and finite element analysis algorithms to perform topology Optimization. The
Optimal solutions generated for both validation problems using an educational version
Of a topology progam that uses the power-law approach (Signund and Tcherniak

2001) are displayed in Figure 3-20 and Figure 3-21.

65

     

, ,1 .24.... ....-. “an.

Figure 3-20. Optimal design for MBB-beam problem using a power-law
approach

 

Figure 3-21. Optimal design for 8-bar truss problem using a power-law
approach
When comparing the solutions in Figure 3-20 and Figure 3-21 to those
obtained through parametric and multiresolution techniques (Figure 3-16-Figure 3-
18), it can be noticed that the Optimal material distributions are displayed differently.
The power-law approach to topology Optimization created truss-like Optimum
structures, while the other two techniques achieved a similar structure through finite,
i.e., well-defined, geometric features. As discussed in Chapter 2, unlike the proposed

pararrretric approach to optimization, the power law approach has the advantage of

66

distributing material by addition or removal of cavities within the domain, allowing
for more detailed and complex geometries (Haftka and Gurdal 1993). The
multiresolution approach can also achieve this by means Of coupon libraries with
multiple cavities and void coalescence algorithms. The disadvantage Of the fine
material distribution in the power law solutions, however, becomes apparent when
Optimizing systems with multiple load cases, boundary conditions, constraints, and
objective functions. In order tO incorporate these changes, code modification is
required (Signund and Tcherniak 2001). Furthermore, since solutions may feature
complex geometries, modifications or “smoothing” of the Optimal desigr is usually

necessary to reduce difficulties in manufacturing.

3.4 Optimization of Continuous Panel Systems-Case Study

The proposed parametric Optimization procedure is considered valid based on
the comparisons with established topology Optimization techniques in the previous
section. The parametric approach is now applied in this section within the context of a
specific problem, a continuous panel system. The goal for this case study was tO
Optimize the material/structural layout Of the transverse cross-section of a continuous
panel system for maximum stiffness using the proposed parametric modeling
approach. Specifically, the continuous panel system will be defined as a bridge deck
with multiple, equally spaced supports. The following sections introduce and discuss
the problem definition, parametric modeling and Optimization problem formulation,

and optimization results in detail.

67

3. 4.1 Problem Definition

The case study was performed by Optimizing the material distribution within
the transverse cross-section of a continuous panel system (i.e. bridge deck). The
geometric domain and loading and boundary conditions are discussed in further detail

throughout this section.

The Optimization process was formulated for a bridge deck continuous panel
system subjected to concentrated loads, P, at the midspan of each bay (Figure 3-22).
The supports or girders Of bridge deck are equally spaced at 6 ft apart. The deck is
desigred with a thickness Of 6 in. This system experiences moving loads, and thus
was desigred for the maximum case scenario which is a point load at each span
(Figure 3-23). The loading condition displayed in Figure 3-22 is experienced by an
effective width along the length Of the system. For bridge decks on girders, such
effective width can be determined from code recommendations (Barker and Puckett
1997). It is suggested that the effective width be calculated using the following
equations: (1) for the region experiencing a positive moment, Sw+=26.0+6.60S and (2)
for the region experiencing a negative moment, SW” = 48.0+3.0S; where, in both cases,

S is the spacing between supports in feet.

68

 

 

 

 

Figure 3-22. Loading and boundary conditions of entire system

 

 

 

 

P P P P P P P
f I I I I i I /
l /
A 0 0 0 0 0 0 0 w
/. Al / ‘ / Al _ I I
1‘ S '1‘ S '1‘ S '1‘ S '1‘ S '1‘ S '1‘ S '1

Figure 3-23. Deformation of continuous panel system

A bridge deck is a three-dirnensional complex system and simplifications are
necessary for modeling purposes. Due to the periodic boundary conditions, modeling
Of the entire bridge deck was simplified into a single representative span (see Figure
3-24). Furthermore, the structure and loading conditions are symmetric about that

span, allowing for further simplification as shown in Figure 3-25.

69

 

 

 

 

 

7 1

F s

Figure 3-24. Representative span due to periodic boundary conditions

 

 

 

(r>/2)/sw
e
Z!
/
2‘ a
a. a,
S/ 2

Figure 3-25. Simplified model of continuous panel system due to symmetry

The applied load on the system was defined as that produced by one wheel Of
a design truck. Bridge desigr specifications state that the wheel load of a desigr truck
without impact is 16 kips. This load is transmitted to the deck system over a
longitudinal effective width, S... (Barker and Puckett 1997). In this case, since the load
was applied to the region experiencing the positive moment, the effective width was
determine using, Sw+=26.0+6.60S. The bridge deck continuous panel system has a

spacing of 6 it between girder supports, resulting in an effective width, 5..., of

70

approximately 66 inches. Since the load is applied at the symmetry line, only half of
it is considered. Thus, the load per unit width of longitudinal deck was equal to 8

kips/ 66 in. = 121.2 lbs/in.

3.4.2 Parametric Modeling and Optimization Problem Formulation

Like the validation procedure, the simplified structure displayed in Figure 3-
25 was modeled with an initial hierarchy and the optimization was conducted using a
parametric approach as shown in Figure 3-26. The radii of each void were defined as
the controlling parameters and their size change essentially modified the material
distribution within the domain.

(Pm/3w

 

 

36in.

Figure 3-26. Parametric model (initial design)

The optimization problem formulation implemented for the continuous panel
was more complex than that used for the validation process. Unlike the validation
problem formulations, multiple design variables and constraints were implemented
for optimization.

In order to define the design variables for the continuous panel system, it was
necessary to discretize the initial desigrs into small rectangular virtual elements. As

mentioned previously, these elements were created to assist in defining variable sets

71

for the controlling parameters which were the void radii. The model was discretized
into fifty-three 1.71 in. X 2.0 in. virtual regions (see Figure 3-27). Similar tO the
validation procedure, each void was numbered so that a corresponding radius could
be defined. The voids in the upper left hand comer Of Figure 3-26 was assigred a
corresponding radius of r,, and continued along the rows, completing with the void in
the lower right hand corner which was identified as having a radius of r63.

Discrete and dependent variables were defined in the formulation. The
discrete variable sets were identified such that each void could increase to a
maximum radius Of 0.835 in. to prevent overlapping and/or exceeding the pro-defined
domain. Since the structure is expected tO exhibit a symmetric bending moment
distribution along its length, dependent variables were defined to enforce a line of
symmetry along a “diagonal”. This implies a symmetry line condition about the

middle of the bean subjected tO an 180° rotation.

1.71 5.14 8.57 12.00 15.43 18.86 22.29 25.71 29.14 32.57 36.00
0'0 I 3‘43 I 6i86 I 1o|.29 I 13|.71 I 17'." I ztI.57 I 2I00I zI.43 I 3<I.86 I 31-29 I

2.1-{“3 1 1 11: 1 -1 1 1 33G 3:; 1 1 1~1',
1 1 1 1 1 1 1 14 1 1 1 .
d°1q1°1°1°1°1m o .O 01 ct, zonot,o..o *1.~

1 . I? 411% 53;.

1,1633- 51:;t1m1n1-1-i’514a1. 1 a 1.4 __1.

:. .'OIODOIOIOIIOI.0'IO 0101,01.“ 1:."O;.1'..*.,-
" , 1 1 1 -1-_- .1 . .11.; .u, .111. .. 1;-,.1 ,3 1- '

 

Figure 3-27. Virtual regions surrounding each void in the initial design
(dimensions shown in inches)

The simplified panel system was subjected stress, strain, deflection, and mass
constraints. The deflection constraint was determined by using the standard

conditions and regulations Of bridge deck desigr. Using a code recommendation for a

72

 

bridge deck, deformations were limited to S/800 (Barker and Puckett 1997). In this
case, S is defined as the distance between bridge supports, 72 in., resulting in a
maximum allowable deflection of .09 inches.

As discussed previously, the motivation of this research was to use natural fiber
composites for load-bearing applications (Chapter 1), thus, the stress and strain
constraints were detemrined for randomly oriented, short fiber composites,
specifically, industrial green hemp/unsaturated polyester (UPE) composite systems.
The longitudinal strain constraint was determined from the longitudinal stress
assuming a linear elastic isotropic material. Halpin and Pagano (1996) developed
approaches to determine equivalent isotropic elastic constants for randomly oriented
fiber composites by approximating the composite as a quasi-isotropic laminate
(Gibson 1994, Quagliata 2003). Defining the isotropic modulus of a randomly
oriented fiber composite is accomplished by assuming the composite consists of
various superimposed unidirectional layers, each layer characterized by its fiber
orientation and with layer behavior being fully additive over all layers. Therefore, the
elastic modulus can be modeled by geometrically averaging the properties of the
unidirectional layers over all fiber orientations (Gibson 1994). The “average”
isotropic elastic modulus for randomly oriented, industrial green hemp/UPE
composites was evaluated experimentally through material testing in previous
research by Quagliata (2003) to be 900,000 psi, which is approximately half that of a
randomly distributed chopped-strand-mat E- glass composite.

The maximum allowable longitudinal stress, a“ was also determined using

results fi'om previous research obtained through ultimate tensile strength (UT S)

73

testing for biocomposite systems. Testing of the industrial green hemp/UPE
composite samples withstood, on average, a maximum stress of approximately 3 ksi
(Quagliata 2003). Incorporating a factor of safety of 2, the maximum longitudinal
stress was limited to 1.5 ksi. Furthermore, a Von Mises stress constraint was set to
5770;“, to limit shear-induced failures, obtaining a maximum allowable value of
865.75 psi. Assuming a linear elastic material, an ultimate longitudinal strain of .0033
can be back calculated from the ultimate tensile strength and the modulus obtained
from previous material testing. Incorporating a factor of safety of 2, the allowable
longitudinal strain was defined as 1.65e-4.

Lastly, the mass constraint was identified by defining the system’s relative
density. Specifically, the mass of the optimized design should be 7/10 that of the solid
structure, that is, a pmam of 0.70.

Like the validation formulation for the MBB-beam and the 8-bar truss, the
continuous panel system was optimized using strain energy as the objective function
since the overall goal is to obtain the stiffest possible design. Stiffness is chosen as
the main design objectives since it usually governs the design of fiber reinforced
polymer composites, and consequently any designs to be made from natural fiber
reinforced polymer composites. Therefore, the goal was to minimize the overall
structural strain energy while simultaneously satisfying all design constraints. Again,
due to the indirect relationship between mass and strain energy, mass was the primary
governing constraint. Thus, it was necessary for the optimizer to locate a design that
was light enough to minimize the strain energy, but not too light that constraints were

violated. Table 3-10 and Table 3-11 outline the problem formulation used for

74

optimization of the simplified continuous panel system. This formulation was defined

in the optimizer prior to process initiation.

Table 3-10. Optimization problem formulation for the simplified continuous
panel system

 

Find: r" , where n = void number = 1-32 (integers)
That minimizes: Strain Energy

Mass constraint
Subject to: Stress constraint

Strarn Constramt

 

 

Find: r n , where n = void number 1-32 (integers)

That minimizes f(x) = I v as dv

. gi(x) SO,i=1-n constraints
Subject to:

hi(x) 20, i= l-n constraints

 

75

Table 3-11. Summary of formulation elements for the simplified continuous

panel system

 

Formulation Vanable Definition of F orrnulation Element

Elements Type

Discrete Void Radii

Design Variables . ..
Dependent Vord Radn

Parameter Length
Parameter Height
Maximum Allowable Longitudinal Stress

r n , where n = void
number = l-32 (integers)

r n , where 11 = void
number = 33-63 (integers)
L

H
aSlSOOpsi

Minimum Allowable Longitudinal Stress a 2 -1500 psi
Maximum Allowable Von Mises Stress a 1 865.5 psi
. Minimum Allowable Von Mises Stress a 2 —865.5 psi
mm“ Maximum Principal Stra'm s 5 .00165
Minirmm Principal Strain e 2 -.00165
Allowable Deformation 6 S .09 in.
Mass M :5 151.20 be

 

M’ 'eFImctron' Minimize StrainEm

 

As stated in previously, a fitness function was used to determine the

optimality of the design solutions generated by the genetic algorithm for this case

study. The fitness value was dependent on the objective function and the design

constraints. The fitness firnction defined in the optimizer, HEEDS, for this case study

was as follows:

((M allowable 'M violated ) J

E (0' allowable " 0W )- (3.3)

 

= . -[10,000*
NormalrzingCoeflicient

 

(5.111.111.1111: " 3111.11.11“! )-

\(5a11mbze "’ 611101111111 ) J

 

This function was evaluated by HEEDS at the end of each design iteration and

the fitness level of the design is dependent the obtained value. The last design saved

76

had the highest fitness value and was considered optimal if the fitness value of the
prior designs was similar to the last design obtained or the last design obtained was

found approximately 10 cycles prior to completion of the last cycle.

3. 4.3 Implementation of Optimization Process and Evaluation of Optimal
Solutions

Using the problem formulation given in Table 3-10, the formulation elements
shown in Table 3-11, and the parametric model displayed in Figure 3-26, the
proposed optimization procedure was implemented to optimize the transverse cross-
section of the continuous panel system in Figure 3-25. The material distribution
within the cross-section was optimized twice, each time with a modified discrete
variable set. The first attempt was implemented with a discrete variable set consisting
of 17 values where the void radii ranged from 0.05 to 0.835 inches in increments of
.05. The second optimization trial was performed with a much smaller discrete
variable set, where of only 4 void radii values between 0.05 inches and 0.835 inches
were possible. The decreased variable set size was implemented to reduce
manufacturing difficulties in creating random size voids. The results obtained from
both optimization trials were compared to ‘observe if variation in solutions existed
when accounting for manufacturing constraints.

Throughout the early stages of optimization an immediate or consistent
decrease in strain energy was not observed. As noted in Figure 3-28 and Figure 3—29,
the optimization history exhibited an oscillatory nature due to the stochastic search
method of the GA. and also the fact the fitness value for all designs is not solely

dependent on the objective function, but is also dependent on the constraint

77

violations. However, as the designs evolved (Table 3-12 and Table 3-13) the
reduction in strain energy was more consistent in the newer designs and an optimum
was found. The design solutions obtained satisfied all design constraints (Figure 3-30
through Figure 3-34) while simultaneously achieving adequate strain energy. The last
designs generated and saved by HEEDS for the panel system had the highest fitness
values and were considered optimal since both were obtained approximately 10
cycles before completion of the 75th or last cycle. The designs generated are shown in
Figure 3-35 and Figure 3-36 for the increased and decreased discrete variable sets,
respectively.

Figure 3-30 through Figure 3-34 display the constraint history for the case
study optimized with the increased discrete variable set. The mass constraint history
is shown in Figure 3-30. . This constraint was violated through the majority of the
optimization procedure; however, as the design numbers increased the mass
approached the maximum allowable value of 151.2 lbs. The mass constraint was met
by design 2386 at an approximate cycle number of 60. When examining the plot, it
may look like design 2386 was the last to be generated by HEEDS which makes
satisfying the constraint look like a coincidence; however, since the cycle number
was set to 75, the search for a “better” solution continued until the last cycle had
completed and was not found. Figure 3-31 displays the behavior of the deflection
constraint which remained less than the maximum allowable value of .09 in.
throughout the entire optimization process. The behaviors of the longitudinal and Von
Mises stresses are shown in Figure 3-32 and Figure 3-33. Similar to the structure’s

deflection, large stress concentrations did not seem to be an issue; thus, the stress

78

constraints were not violated. The strains experienced by the designs generated
throughout the optimization process (Figure 3-34) were also much less the maximum

allowable value of .0033.

2L88 -

2L86 -

3L83 -
Shaun

Emergyztgl -

(95‘) 2.78 -
2.76 -
2.73 -

2L7] r

 

 

 

 

2.68 I r I I I j
0 400 800 1200 1600 2000 2400

Design Nunber

Figure 3-28. Optimization history of continuous panel system with increased
discrete variable set

79

2.95 -
2.88 7

Strain 2.80 -
Energy

(W7 2.73 ~ I r I |

2.65 -

 

 

 

 

2.58 -

 

 

2.50 I T fl I I I I I
0 150 300 450 600 750 900 1050 1200

Design Nunirer

Figure 3-29. Optimization history of continuous panel system with decreased
variable set to account for manufacturing

174.0 -

170.0 -

 

166.0 ~

  
 
 
 
 
 

Mass

(lbs) 162.0 7

158.0 7

154.0 -

 

\ Maxim Allowable = 151.2 lbs

 

1 46.0 I If T I I I I
0 400 800 1200 1600 2000 2400

Design Number

Figure 3-30. Mass constraint history evaluation

80

9.508-02

 

 

 

 

 

 

2400

8.75E—02 ~
\ Maxim Deflection = .09 in.
8.00E-02 ~
Deflectbn 7.25E’02 "

(ill)
6.50E-02 .
5.75E-02 -
5.00E-02 ‘

? —"'\.1 —
4255-02 7
3.50E‘02 1? I I I I I l
0 400 800 1200 1600 2000 2400
Design Nunber
Figure 3-31. Deflection constraint history evaluation
1.55E+03
1.451~:+03 « \ Mm ,
1.35E+03 1 ”W
stress= 1500 psi
1.25E+03 7
longitldnal 1.15E+03 .

Stress

(psi) 1.05E+03 «
950sz ~
8.50E+02 1
750sz 4
6.50E+02 1 1 fl . . .

O 400 800 1200 1600 2000
Design Nunirer

Figure 3-32. Longitudinal stress evaluation history

81

 

 

 

 

l.00E+03 7
9.50E+02 7
Maxinmn Von Mises
9.00E+02 7
/ Stress = 865.75 psi
Von Mises
Stress 8.50E+02 7
(psi)
8.00E+02 7
7.50E+02 7
71mE+02 T I I I I 1
0 400 800 1200 1600 2000 2400
Design Number
Figure 3-33. Von Mises stress evaluation history
1.70E-03 I .
1. E- 7
60 03 \M ,
1.50503 ~ longitudinal
1.401303 ~ stram=.0033
1.3013-0 J
Lo 3
Strain 1.20E-03 i
1.10E-03 7
1.00E-03 7
9.00E-04 77
8.00E-04 7
7.00E-04 1 f 1 1 T 1
0 400 800 1200 1600 2000 2400
Design Nunirer

Figure 3-34. Longitudinal strain evaluation history

82

Table 3-12. Evolution of designs for case study with increased discrete variable
set

Design . , Strain Relative
Mass (lbs) Deflection (1n) Energy (psi)

Desrgn N umber

Density
1 148.69 6.87E-02 4.16E+00 0.688
2 163.29 4.59E-02 2.79E+00 0.756

129 160.14 4.64E-02 2.81E+00 0.741

312 159.73 4.64E-02 2.81E+00 0.739

350 158.17 4.6 I 13-02 2.79E+00 0.732

454 158.35 4.59E-02 2.78E+00 0.733

466 157.66 4.60E—02 2.79E+00 0.730

588 157.85 4.59E—02 2.78E+00 0.731

651 155.13 4.68E-02 2.83E+00 0.718

683 158.23 4.535-02 2.74E+00 0.733

691 156.30 4.63E-02 2.81E+00 0.724

833 156.35 4.60E-02 2.79E+00 0.724

919 156.28 4.54E-02 2.75E+00 0.724

952 154.83 4.6 l E-02 2.79E+00 0.717

155.34 4.58E-02 2.78E+00 0.719
155.81 4.49E-02 2.72E+00 0.721
155.58 4.44E-02 2.69E+00 0.720
153.80 4.57E—02 2.77E+00 0.712
153.86 4.57E-02 2.77E+00 0.712
154.36 4.52E—02 2.74E+00 0.715
153.47 4.525—02 2.74E+OO 0.711
153.18 4.52E-02 2.74E+00 0.709
152.96 4.52E—02 2.74E+00

152.84 4.50E-02 2.73E+00

 

83

Table 3-8. Evolution of designs for case study with increased discrete variable set
(cont.)

Design Strain Relative
Energy (psi) Density

Design Mass (lbs) Deflection (in)

Number
1692 152.37 4.53E-02 2.74E+00 0.705
1902 152.60 4.51E-02 2.73E+00 0.706
2008 152.50 4.50E-02 2.73E+00 0.706
2128 151.97 4.53E-02 2.75E+00
2187 152.27 4.51E—02 2.74E+00
2198 151.86 4.52E—02 2.74E+00
2207 152.07 4.51E-02 2.74E+00
2287 152.05 4.51E-02 2.74E+00
2375 151.35 4.51E-02 2.74E+00

23 86 151.76 4.50E-02 2.73E+00

 

84

Table 3-13. Evolution of designs for case study with decreased variable set to
account for manufacturing

Design Number Mass (lbs) Deflection (in) Strain Energy (psi) Relative Density

1 154.43 6.30E-02 3.83E+00 0.715

l64.35 4.90E-02 3.015400 0.761

159.30 4.90E-02 2.98E+00 0.738

164.55 4.70E-02 2.86E+00 0.762

161.75 4.60E-02 2.81E+00 0.749

162.48 4.50E-02 2.73E+00 0.752

I62.12 4.50E-02 2.75E+00 0.751

159.27 4.7OE-02 2.86E+00 0.737

162.13 4.205—02 2.56E+00 0.751

160.57 4.465-02 2.7lE+00 0.743

157.90 4.59E—02 2.78E+00 0.731

154.75 4.38E-02 2.66E+00 0.716

154.21 4.48E-02 2.72E+00 0.714

153.29 4.46E-02 2.71E+00 0.710

151.40 4.505-02 2.74E+00 0.701

152.09 4.46E-02 2.7IE+00 0.704

 

(Pm/sw

 

Figure 3-35. Optimal design generated for the continuous panel system using the
increased discrete variable set

85

(P/2)/Sw

°-00°.0...0.0°0-
4

/
IOCO'OO 0°0'.°. 00'...
°0°OOOOOOOO-OO°-

 

Figure 3-36. Optimal design generated for the continuous panel system with
decreased variable set to account for manufacturing

The optimal solutions for the case study presented for the increased discrete
variable set formulation in Figure 3-35 and the decreased variables set formulation in
Figure 3-36 display similar trends. Material was placed in regions experiencing large
stress concenUations, specifically in areas of high tension and compression as
illustrated in See Figure 3-37. In both cases the material layout is arranged so as to
more efficiently transfer the applied loads to the support conditions. A quantitative
comparison of the two design solutions obtained for the cases study is displayed in
Table 3-14. The computational results were quite similar displaying only a 1%
difference in mass, deflection, and strain energy, a 20% difference in the longitudinal

stress, and an 8% difference in the Von Mises stress.

mwg

 

 

a...

\\\\\\\\\\\\\\\\\\‘Q
i .
\\\\ \\\\\\\\\\\\\\‘

Figure 3-37. Panel Response analysis highlighting

86

Table 3-14. Summary of results for the designs solutions obtained for the case

 

 

study
. Decreased
£22155]: 3:236 Discrete Variable (A/B)
Set (B)

Relative Density 0.70 0.70 0.0
Mass (lbs) 151 152 1.0
Deflection (in.) 4.50E-02 4.46E-02 1.0
Longitudinal Stress (psi) 813 676 20.0
Von Mises Stress (psi) 806 872 8.0
Strain 8.26E-O4 7 .40E-04 12.0

Strain Energy (psi) 2.73 2.70 1.0

 

Afier examining finite element analysis results for both the initial and optimal
designs for the case study, it was apparent the optimized structure exhibited more
homogenous stress and strain fields (Figure 3-38 through Figure 3-43) than the initial
design. However, the optimal configuration exhibited larger deformations (see Figure
3—44 and Figure 3-45) than the initial design. This difference is a direct consequence
of the available material since the initial design has a relative density of 0.98
compared to 0.70 for the optimal, or 40% more material. Nonetheless, the
deformation constraint defined in the optimization problem formulation (Table 3-11)
was still satisfied. Thus the optimal design is a much more efficient and economical
(less material) solution. A comparison in computational results for both the initial and
optimal designs is summarized in Table 3-15.

The computational design solution obtained for this case study, using the
proposed parametric approach to optimization, was examined fimher through small-
scale component testing. Flexure tests were implemented to study the difference in

performance of a base periodic cellular structure and a structure with the optimized

87

material distribution. This experimental portion of this study is introduced and

discussed in fiirther detail in Chapter 4.

 

Figure 3-38. Longitudinal stress field experienced by initial design for the case
study

 

Figure 3-39. Longitudinal stress field experienced by optimal design obtained for
the case study

88

 

Figure 3-40. Von Mises stress field experienced by initial design obtained for the
case study

 

Figure 3-41. Von Mises stress field experienced by optimal design obtained for
the case study

89

 

Figure 3-42. Strain field experienced by initial design for the case study

 

Figure 3-43. Strain field experienced by optimal design obtained for the case
study

90

 

Figure 3-44. Deformation experienced by initial design for the case study

 

Figure 3-45. Deformation experienced by optimal design obtained for the case
study

91

Table 3-15. Comparison in computational results for the initial and optimal

 

 

designs for the case study

Optimal Design (A) Initial Design (B) (A/B)
Relative Density 0.70 0.98 40.0
Mass (lbs) 151.00 213.00 41.0
Deflection (in.) 0.045 0.030 50.0
Max. Longitudinal Stress (psi) 813.00 448.00 81.0
Max.Von Mises Stress (psi) 806.00 1158.00 44.0
Max. Strain 8.26E-04 6.23E-04 33.0
Strain Energy (psi) 2.73 1.84 48.0
Strain Enefl/Mass (psi/lbs) 0.018 0.009 209.0

 

92

4 EXPERIMENTAL VALIDATION

4.1 Overview

The findings from the computational studies were used to assess the feasibility
of natural fiber composites for load-bearing applications through small-scale
component testing. This chapter provides an account of the manufacturing and
experimental testing of an optimal and a non-optimal biocomposite panel system built
using industrial green hemp and unsaturated polyester resin (UPE). The small-scale
cellular samples fabricated this study. (1) a base periodic cellular design for the
continuous panel system and (2) the generated optimal hierarchical cellular solution.
Both samples were fabricated using the vacuum assisted resin transfer molding
(V ARTM) manufacturing method and their performance was evaluated through

flexural testing.

4.2 Material Systems and Structural Forms

This section discusses in detail the material and resin systems, as well as the
structural forms implemented for the experimental study. In order to obtain a higher
performance natural fiber composite, a hybrid material system in multiple advanced
structural configurations were fabricated. The hybrid natural fiber system used for
this experimental study and was comprised of jute and industrial green hemp (Figure
4-1). The jute fabric consisted of Hessian jute mats (IJIRA, Calcutta, India), which
was used as a face sheet. The industrial green hemp fibers (F laxcrafi, Inc., Cresslcill,
NJ) were random chopped fibers with average length of 0.3 to 0.5 inches and an
aspect ratio (L/d) of 100, which were used to make up the core of the sample (Figure

4-2).

93

         

(a) Jute mat (b) Green hemp short fibers

Figure 4-1. Pictures of natural fibers used for fabrication

Top Face Sheet: Jute/UPE

Green Hemp/UPE
Core

 

Bottom Face Sheet: Jute/UPE

Figure 4-2. Arrangement of hybrid natural fiber material system

The resin system employed for fabrication of the biocomposite cellular samples
was orthounsaturated polyester (UPE) resin (Kemlite Co., Inc., Joliet, IL) with a
methyl ethyl ketone peroxide (MEKP, Sigma Aldrich) catalyst, and cobalt
naphthenate (CoNAP, Sigma Aldrich) promoter. The resin system composition is

shown in Table 4-1.

94

Table 4-1. Resin System

 

Constituent Amount by Weight % by Resin Weight

 

UPE (33% Styrene) 1000 g '-
MEKP 1.5 g 0’15
CN 03 g 0.03

 

 

The natural fiber and resin materials discussed in the previous paragraphs were
used to create two test samples, namely a base and an optimal design. The geometry
and material layout of the base and optimal panel system for verification of the design
case study are shown in Figure 4-3. The designs displayed in Figure 4-3 have larger
dimensions than that displayed by the initial and optimal design solutions presented in
Section 3.4.2. This is because the initial and optimal designs for the case study were
fabricated to simulate the entire distance between supports (Figure 3-24) with a
segment of overhang. This was done for testing purposes, and will be addressed in the

following sections.

95

 

 

(a) Base periodic cellular design

 

 

 

 

s//¢=1/” ¢=3n6’

eroOOoo Doe/00000000000 o OOOOOOOOOOOOOOOO
OOOOOOOOOOOO OOOOOOO OOOOOOOOOOOO OOOOOOO 000000
$00 0 0 0000000000000000 0000000000000000 o
\ (l) _ 1/8” Relative Density =O.7O

(b) Optimal design obtained through parametric modeling

Figure 4-3. Test unit dimensions and cross-sectional geometry of cellular
biocomposite panels (16 X 3 X 1 inches)

4.3 Automated Manufacturing Process

This section presents the processing technique implemented for fabrication of
the cellular biocomposite samples. In principle, the processing methods available for
natural fiber composites can be similar to those for glass fiber composites. Several
processes exist such as traditional hand lay-up techniques, autoclaving, compression
molding, sheet-molding compound (SMC), and resin transfer molding. However,
vacuum assisted resin transfer molding (V ARTM) has been shown to be an effective
process for fabrication of both glass fiber and natural fiber composites (Brouwer
2000, Quagliata 2003).

VARTM is a clean closed mold manufacturing technique that can be used for
the fabrication of natural fiber composites. The use of a closed mold has made this

process environmentally safe since it constrains hazardous styrene emission from the

96

resin. VARTM has also proven to decrease cost in comparison to traditional hand lay-
up or autoclaving techniques due to reduced labor and low equipment cost. It is
possible to perform tailored lay-up and achieve high fiber volume contents through
VARTM (Brouwer 2000, Brouwer 2003). The VARTM process involves the
placement of dry fibers into a mold and then enclosing the mold with its other half or
with vacuum bagging film. After an airtight seal is obtained by this enclosure the
resin can then be injected by the use of vacuum pressure. A schematic of a VARTM
process is shown in Figure 4-4.

For laboratory scale samples, a VARTM setup can be implemented on a
movable cart with two shelves. The sample setup can be arranged in the top shelf
while the bottom shelf is used to hold the vacuum pump. A picture of an actual

laboratory VARTM setup is shown in Figure 4-5

Vacuum Bag Fiber
Sealant Tape / Reinforcement
/

 

////////1/////'//////|\_

     
 

Vacuum
Bagging
Materials Pressrn'e Gage

  
 
  

Resin Reservoir Resin Trap

Figure 4-4. Schematic of VARTM manufacturing process

97

 

Figure 4-5. Actual VARTM setup

4.3.] Manufacturing of VARTM Cellular Panels

The automated manufacturing of cellular biocomposite panels was performed
using the VARTM process discussed in the previous section (Figure 4-4). This
technique was studied experimentally to show that VARTM is a viable technique for
manufacturing cellular components with biocomposites. This study included the
manufacturing two cellular samples (Figure 4-3) with identical material systems
(Section 4.2). The aim was to explore the performance of the biocomposite structures
with varying degrees of optimal and non-optimal cellular designs, validate the
optimization results, and evaluate VARTM as a viable manufacturing method for
these designs

The VARTM manufacturing process was divided into four main steps: (1)
preparation of vacuum bagging materials, (2) placement of jute fabric face sheet, dry
green hemp core fibers, and cells, (3) vacuum bagging the setup, and (4) resin

infusion. Each of these steps is discussed in more detail below.

98

Special vacuum bagging materials were used to assist in resin transfer and
easy release of the sample after resin infusion. The bagging materials were positioned
to surround the entire sample. Specifically, all materials lined the bottom of the
cellular plate mold and were cut long enough to wrap the entire sample. The bagging
material included 5 different plies: (1) a non-porous Teflon release ply, (2) a breather
ply, (3) a resin transfer media, (4) a polyester peel ply, and (5) a porous Teflon
release ply. First, the non-porous Teflon release ply was placed at the bottom of the
mold to prevent the sample from sticking to the mold or the vacuum bag after curing
(Figure 4-6a). The breather ply was then placed on top the non-porous Teflon release
ply to absorb excess resin from the sample (Figure 4-6b). On top of the breather ply
was the resin transfer media (Figure 4-6c). This ply was used to assist in resin transfer
for uniform distribution of the resin throughout the sample. A layer of polyester peel
ply was then placed on top of the transfer media, which allowed for excess resin to be
taken out of the sample (Figure 4-6d). Lastly, the porous Teflon release ply was
placed on top of the peel ply (closest to the sample) to allow for excess resin flow and
to prevent the other bagging materials from sticking to the sample after curing (Figure

4-6e). Note that the materials were cut large enough to wrap around the entire sample.

99

 

(b) Breather ply

 

(e) Porous Teflon release ply

Figure 4-6. Picture sequence of vacuum bagging material placement

After all vacuum bagging materials were positioned properly the fibers and
cells were ready for placement. First, a jute fabric layer was placed on top of the non-
porous Teflon release ply and was cut large enough to longitudinally enclose the
entire sample (Figure 4-7a). A layer of green hemp was then uniformly distributed on
top of the jute layer as shown in Figure 4-7b. A row of metal rods was then inserted
through the mold (Figure 4-7c) to create the first layer of voids. Each metal rod was
wrapped with rubber tubing to prevent them fiom sticking to the sample afier curing
and thus facilitate their removal. A second layer of green hemp was then placed in

between and on top of each rod until they were completely covered (Figure 4-7d).

100

This process was repeated for all rods layers until all rows of voids were created (see

Figure 4-7e).

 

(a) First layer of green hemp fibers (b) First layer of rods (cell voids)

‘ 1J1 uni "11,": i" . ("A Hui-.111 \

‘ ’l’l'!;’"1‘1“‘11111‘2'71111‘.‘i11

(0) second layer of hemp fibers and rods (d) Last layer of hemp fibers

 

Figure 4-7. Placement of green hemp and rods (void spaces)

The jute and vacuum bagging materials were then wrapped around the green
hemp cellular core, ultimately producing a mirror image of the materials on both sides

of the sample. (Figure 4-8a and Figure 4-8b).

101

W ”’l'lfli'mil‘liflilWill

”’1' [.I"\l]I/\ llH‘

(a) Jute wrapped around green hemp core (b) Mirroring of vacuum “bagging materials

 

Figure 4-8. Completed sample ready for bagging

Once they dry sample was in place, a VARTM setup was prepared as shown
in Figure 4-4. As previously stated, the setup was implemented on a movable cart
with two shelves, one for the sample and the other for the vacuum (see Figure 4-5).
Nylon bagging film was cut and placed on the top surface of the cart underneath the
mold and was large enough to enclose the entire mold. The resin and vacuum ports
were then prepared. A small square wood block, approximately the same height as the
mold (1 inch), was used to secure the vacuum port. A strip of sealant tape was placed
on top of the block and around the vacuum port and pressed down firmly until stable
(Figure 4-9a). The vacuum port was covered with additional breather ply and resin
transfer media to improve resin transfer and prevent blocking of the vacuum port. The
port was then placed in between the peel ply and resin transfer media on the right
edge of the sample. Pieces of breather cloth and resin transfer media were placed
around the rods on either side of the mold, around the edges of the mold, and around
the wood block to prevent puncturing of the nylon bag (Figure 4-9b). The resin port
was created using a vacuum connector. The nylon bagging film was folded on top of
the mold to identify the resin inlet. A small puncture was placed in the nylon film

toward the left edge of the sample to properly secure a vacuum connector. Once the

102

vacuum connector was properly secured, the bagging film was opened and sealant
tape was placed on the vacuum bagging film surrounding the entire mold and also
around the vacuum port. Once all materials and ports were in place the sealant tape
cover was removed and the nylon bagging film was folded over the entire mold and
tightly fastened to the vacuum port and the portion of the bagging film lying on the
top surface of the cart. To complete the setup, a valve was added between the resin

port and the resin reservoir to control resin flow during resin infusion.

103

 

Breather cloth and resin
transfer media placed
around the rods

  

(b) Placement of the vacuum port, breather cloth, and resin transfer media

Valve for resin port

Vacuum connector attachment

affiliation

l' I J
.. ‘w
'4- ‘a-Jl' "3,. 7

 

Sealant Tape Nylon bagging film

(c) Placement of all ports and sealing of vacuum bag

Figure 4-9. Preparation for resin infusion in the VARTM setup

Lastly, the free ends of the resin and vacuum ports were secured into the resin
reservoir and trap (Figure 4-10) and the vacuum pressure was tested to ensure a

pressure between 25 and 27 in. of Hg.

Resin reservoir Resin trap

   

Figure 4—10. Resin and vacuum port attachment to the reservoir and trap

The resin infusion process into the sample began by first compressing the
sample under the vacuum pressure for approximately 15 minutes (Figure 4-11). While
the dry fibers in the sample were kept under pressure, the polyester resin system was
proportioned (see Table 4-1) and added to the resin reservoir. The resin port was
secured into the reservoir and the valve was opened to initiate infusion. Initially, the
resin moved quickly into the sample. However, after saturation of approximately '/4 of
the sample the flow began to slow down saturating the entire sample after

approximately 50 minutes. The resin infusion process is shown in Figure 4-12.

105

 

    

5:. A“:

(a) minutes

 

 

(c) 50 minutes

Figure 4-12. Resin infusion process

Once the sample was completely impregnated with resin, the valve between
the resin reservoir and the sample was closed so that no additional resin would be
pulled into the system. The system was kept under vacuum for approximately 45
minutes to allow the resin to begin gelling and to minimize the amount of air voids
that could develop in the sample. After gelling, the vacuum was turned off and the

resin and vacuum ports were detached from the mold. The sample was oven cured for

106

6 hours at 100°C. A lower curing temperature than that used in prior studies
(Quagliata 2003) was necessary to avoid melting of the resin transfer media, which is
made of polyethylene. Steel plates were placed on top of the sample during curing to
assist in creating uniformity on the sample surfaces, squeeze out excess resin, and
help in compaction. The manufactured cellular panels are displayed in Figure 4-13.
Figure 4-13a shows the base design with periodic cells and Figure 4-13b shows the
optimized design obtained through the parametric optimization approach featuring the

hierarchical cellular arrangement.

 

(b) Optimal design with hierarchical cellular arrangement

Figure 4-13. Manufactured base and optimal cellular biocomposite panels for
case-study

4.4 Flexural Testing of VARTM Hierarchical Samples

The performance of the cellular panels manufactured for the case study was
evaluated through flexural testing. This section introduces and discusses the testing

procedure and setup, instrumentation, and loading scheme used.

107

F lexural testing was performed in such a way that the tests replicated the
loading conditions experienced by a continuous panel system (i.e., a bridge deck). A
schematic of a continuous panel system on multiple supports is shown in Figure 4-
14a. The continuous panel system is assumed to be loaded in between supports,
representing maximum loading conditions. Taking symmetry into consideration, the
system can be reduced to a two-span panel arrangement as shown in Figure 4-l4b.
From the simplified system, the bending moment diagram demands are obtained as
given in Figure 4-15a. In order to achieve the desired design moments shown in
Figure 4-15b the panels were tested as a simply supported beam with a cantilevered
overhang and the beam was loaded at the cantilever tip and at the middle of the span.
The load levels at midspan and at the cantilever overhang were selected so as to
match the desired design moment diagram. A schematic of the testing setup is shown
in Figure 4-16. The test setup was realized by mounting a stiff support beam on a
universal loading frame. The two loading points and their respective magnitudes were
achieved by means of a loading spreader beam. A picture of the actual test setup is
shown in Figure 4-17.

The test unit was loaded monotonically until failure at a loading rate of 0.01
min/sec. The deflection of the samples was measured at both points of loading using
externally mounted displacement transducers (Figure 4-16 and Figure 4-17). Four
electrical resistance strain gages were mounted the tensile and compressive faces of
the two maximum moment locations on regions of maximum tension and
compression (Figure 4-16). The strain gage readings, external displacements, and

applied load were simultaneously recorded with a digital data acquisition system.

108

P919

1 151 1]
A A A A A

 

L L ' L L
(a) Continuous panel system with multiple supports

it ‘1]
Lee

(b) Reduced two-span panel arrangement

 

 

 

Figure 4-14. Loading and boundary conditions of a simplified continuous panel
system

109

 

 

+.170 PL
+.116 PL

(a) Bending moment diagram of simplified continuous panel system

 

+ .170PL

(b) Bending moment diagram on tested unit

Figure 4-15. Bending moment diagram to be replicated during testing

110

Displacement
Transducer

     

 
    

OVOVOVOVC . g 0

EEE. 0
Displacement
Transducer

  
 

   

.1
6.5" 6.5"

 

Figure 4-17. Picture of continuous beam test setups on universal loading frame

111

While the ultimate load capacity of both panels was similar (4% difference),
their response was very different. The base non-optimal design was very flexible
(Figure 4-1 8a) with large cell distortions while the response of the optimal design was
essentially linear elastic exhibiting relatively small deformations (approximately
200% smaller at failure) (Figure 4—18b). Results obtained from the flexure tests and
comparisons between the base and optimal structures obtained for the case study are

discussed in detail in the following section.

 

l1.
(0) Deformation of optimal design during testing

Figure 4-18. View of test unit response

112

4.5 Results and Discussion

This section provides the experimental results and comparison between the base
and optimal structures obtained for the design case study of this thesis. The load-
deflection responses of the jute/green—hemp/UPE cellular panels are shown in Figure
4—19 and Figure 4-20 for the base and optimized designs, respectively while a
summary of results is given in Table 4-2. The response of the base design was much
more flexible (200% higher) than the optimal panel with a compounded response that
showed stiffening after significant deformations. Conversely, the response of the
optimal panel was essentially linear elastic up to failure with comparatively smaller
deformations (twice as small).

Both the base and optimal cellular structures failed due to tensile stresses at the
peak moment zones. However, the base design had a tensile rupture under the
positive moment region, at a peak load, P, of 0.43 kips. The Optimal design failed in
tension at the negative moment region at a peak load of 0.45 kips. Thus, the failure
capacity of both designs was essentially the same, with the optimal design reaching a
failure load that was only 5% higher than that of the base design.

The effect of the optimized material layout on the performance of the cellular
biocomposite panels was much more evident in their deformation response. Figure 4-
19 shows the central load (P)-displacement response for the base and optimal designs,
while Figure 4-20 gives the edge load (.87P)-displacement response. The reason to
display the load-displacement responses with respect to the load at either the midspan
or at the cantilever tip is to distinguish the respective stiffnesses of the panel regions.
As expected, the optimized design displayed a higher initial stiffness (i.e. the slope of

the linear portion of the load-displacement response) and secant stiffness to maximum

113

load at both midspan and the cantilever tip than the base design (Table 4-2). A
schematich description of approach used to determine the initial and secant stiffnesses
in Figure 4-21. Specifically, with respect to the central load, the optimal panel system
was found to have an initial stiffness approximately 41% larger then the initial design
and a 64% larger secant stiffiiess to maximum load. It should be noted, however, that
the optimized design did have a 16% higher relative density than the base design.
However, it is clearly the optimized material layout has the biggest influence in

allowing this sample to have a significantly higher stiffness.

 

_Initial Design for Case Study
——Optirml Design for Case Study

 

 

 

 

0.50

0.45 ‘ 37 wt.%

0.40 ~

Load 0'35 i
P 0.30 .
mm) 0.25 «

0.20 ‘
0.15 _ .87P P

0.10 - [1 l j

0.05 . Wm ............ L5. """"""" 3,},—

o.oo . , . r . .

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Deflection (in.)

31 wt.%

 

 

 

Figure 4-19. Force-displacement response at midspan

114

 

—— initial Design for Case Study
—— Optirml Design for Oise Study

 

 

 

0.45
0.40 _
0.35 .

Load 0‘30 ‘

37” 0.25 .
(”98)
0.20 .

0.15 ~
0.10 *

 

 

0.05 r

 

 

 

 

0.” T l r T l l
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Deflection (in.)

Figure 4-20. Force versus displacement response at cantilever tip

     

Initial Secant

Load

 

 

Displacement

Figure 4-21. Initial and secant lines used to determine sample stiffnesses

115

 

Table 4-2. Flexural test results summary with respect to the central load

 

 

Initial stiffness Secant stiffness to Initial stiffness Secant stiffness to
Fibers by Strength with respect to Max. Load with . Max. Load with
o . With respect to
Wt. A: (laps) central 6 respect to central 6 edge 5 (kips/in) respect to edge 6
(Rips/in) (Rips/in) (kins/in)
”it?“ 31 0.43 1.70 1.41 2.20 1.55
Des1gn
01ml“! 37 0.45 2.40 2.31 4.15 3.07
Desrgn

 

 

The loadedisplacement behavior plots shown in Figure 4-19 and Figure 4-20
show that the test units has similar capacity, however, their response was markedly
different. The optimal design exhibited essentially a linear response up to failure.
Conversely, the base design showed two distinct nonlinear responses, the second one
showing significant stiffening. This behavior is a result of shear distortion of the
periodic cells in the base design. Shear deformations occur when a structure has few
or no transverse diaphragrns or internal bracing, so that the vertical shear force across
a cell causes the internal bracing and webs to flex independently out-of-plane. This
deformation is similar to that of a Vierendeel truss as shown in Figure 4-22a (I-Iambly
1991). The shear distortion of the Vierendeel truss can be minimized by addition of
internal diagonal bracing as shown in Figure 4-22b, which is the well-known
traditional design of truss structures. This same distortion may also be present in
periodic cellular structures as illustrated in the cantilever beam with a tip load in
Figure 4-23a. Cell distortion is not just resisted by out-of—plane flexure of the top and
bottom slabs and webs but also by in-plane bending and shear of the plate elements.

Cell distortion can be minimized by aligning the material such that the stiffness of the

116

structure is maximized as shown in Figure 4-23b (Hambly 1991). It can be noticed
that the rearranged material essentially follows the direction of inclined diagonal
braces of a traditional truss. Not surprisingly, traditional topology optimization results
(see Section 2.1.2) look like truss structures

P

z [I I P

(a) Shear deformation

 

 

 

 

 

 

P

 

 

 

 

 

 

\

Diagonal truss members

(b) Internal diagonal braces added to minimize shear deformations

Figure 4-22. Shear deformation of Vierendeel truss

117

 

’OOOO

 

 

 

 

 

 

(a) Cell distortion
P
O O 0 °
(b) Proper material alignment

Figure 4-23. Cell distortion in a cantilever beam with tip load

Periodic cells exhibit large distortion thus leading to a softer structure with
slight nonlinear behavior. Through this shear distortion, the circular cells essentially
ovalize in a deformation mode that requires little energy. However, once the cells
“reorient” themselves, the material is aligned and thus more effective, leading to the
stiffening branch of the response. By comparison, the linear response of the optimal
design follows fiom the fact that the optimization procedure has essentially lead to a
material layout that is already for the stiffest performance, i.e. without shear
deformations. It can be further observed that upon “realignment” of the cells in the
base design, the slope (i.e., stiffness) of the load-deformation response is initially
equal (parallel) to that of the optimal design. Unfortunately, the base system later
develops other areas of shear-dominated response leading again to softening
behavior. It can thus be concluded that the response of the tested samples would

correspond to the behavior of the Vierendeel truss, for the base design, and an actual

118

triangulated truss structure for the optimal design (see Figure 4-22). Thus, the
computationally optimized design essentially has arranged the material in a truss-like
fashion, just as what is expected in results from traditional topology optimization
approaches.

The strain behavior was also examined at four locations (see strain gage
locations in Figure 4-16), which was different in comparison to the load-displacement
response. Typical load-strain behavior is shown in Figure 4-24 through Figure 4-25.
In these figures it can be seen that while the load—displacement response for the base
design was highly nonlinear, all of the load-strain histories were essentially linear
elastic. The reason is that the shear-induced deformations in the base design are a
cross-sectional effect away from the critical sections, whose strain distribution
continues to increase essentially in a linear way. The load-strain responses also show
a stiffer response for the optimal design and the strains measured for the base design
at failure were larger than for the optimal design. The only explanation for failure of
the optimal design at a lower strain is material variability or manufacturing
imperfections. This highlights the importance of implementing safety factors in the

strain limit formulation of the design optimization problem.

119

 

—- Initial Design for Case Study
— Optimal Design for Case Study

 

 

 

 

0.50

0 45 _ Strain Gage Location:
' Bottom mder the central load

Load

(Rips)

 

 

 

 

 

 

 

Figure 4-24. Load-strain response at extreme tension fiber on maximum positive
moment section

 

— Initial Design for Case Study
— Optirml Design for Case Study

 

 

 

 

 

 

 

 

 

 

0.45
Strain Gage Location:
0'40 ‘ Top left, above the support
0.35 4
0.30 ~
Load
_mp 0.25 ‘
mp” 0.20 -
0.15 " .87? P
0.10 - so i
l J
0.05 —,"'1///Y ...................... 3%
0.00 r a l I ' I j .
010002000300040005000600070008000
strain (nicro)

Figure 4-25. Load-strain response at extreme tension fiber on negative moment
section

120

 

—- Initial Design for Case Study
— Optirml Design for Case Study

 

 

 

 

0.45

0 40 Strain Gag Location:
' Top, under tin central bad

    

0.35

0.30
Load
P 0.25
(kins)
0.20
0.15
0.10

0.05 x, ..... Q ............................. Q

0.00

 

 

 

 

 

 

   

M I l T l l T

0 500 1000 1500 2000 2500 3000 35004000
strain(nicro)

Figure 4-26. Load-strain response at extreme compression fiber on maximum
positive moment section

 

 

 

 

 

 

 

 

 

 

 

 

—— Initial Design for Case Study
'— Optirml Design for Case Study
0.45
Strain Gage Location:
”-40 ‘ Bottom 1e11, inlet the support
0.35 -
0.30 ~
Load 0.25 5
P
(Rips) 0.20 4
0.15 - .87? P
0.10 4 1
0.05 - " W, 55 ....................... S);
0.” n I l l I T fi
0 500 1000 1500 2000 2500 3000 3500 4000
strain (nicro)

Figure 4-27. Load-strain response at extreme compression fiber on negative
moment section

121

The initial stiffnesses obtained for both the base and optimal designs (see
Table 4-2) were compared to those obtained through finite element analysis (FEA).
The stiffnesses were obtained from FEA by taking the force/displacement at the
location of the concentrated load. The deformations of the base and optimal designs
obtained from the FE analysis are displayed in Figure 4-28 and Figure 4-29 below.
Table 4-3 displays the comparison in computational and experimental initial
stiffnesses. The findings show a 58% difference in the ratio of base to optimized
stiffnesses. This difference could be due to the difference in the elastic modulus
implemented in the FEM model and the actual modulus of the manufactured samples.
Furthermore, unlike the manufactured samples, the FEM model was not created to
include the hybrid jute mat/green-hemp design.

Upon close inspection of Figure 4-28, a slight distortion of the periodic cells
can be noticed. As previously stated, the circular cells essentially ovalized (see Figure
4-23) until they were “reoriented”. This reorientation assisted in aligning the material,

thus creating a stiffer structure.

122

 
 

 
 
    
 

 

 

Figure 4-28. Deformation experienced by base design obtained for the case study

 

Figure 4-29. Deformation experienced by optimal design obtained for the case
study

123

Table 4-3. Comparison in initial stiffness

 

Experimental ratio of Computational ratio
stiffness (kE) of stiffness (kc) kE/kc

 

kbase/koptml 0.71 0.45 58.0

124

5 CONCLUSIONS AND RECOMMENDATIONS

5.1 Conclusions

The results presented from the computational and experimental studies have
reported that through the use of a finite parametric approach to structural optimization
the material distribution within a given domain can be enhanced to improve the load-
bearing capacity of components while simultaneously accounting for manufacturing.
Specifically, the findings in this study have led to the following conclusions regarding
obtaining optimal and “buildable” material distributions to enhance the properties of
biocomposites for load-bearing components.

0 A finite structural optimization technique can be implemented using a
parametric modeling approach that combines robust optimization
algorithms with finite element analysis through file scripting.

o The parametric modeling approach is capable of optimizing the material
distribution of a structural component within a given domain while
incorporating multiple objective fimctions and design constraints.

0 Due to the use of finite geometrically defined features the proposed
optimization technique is also capable of producing design solutions
that account for manufacturing constraints.

0 Homogeneous designs with optimized cellular herarchical
configurations in the form of voids, or cells, are appropriate for

biocomposite components due to the complexity in fabricating them in

125

geometrically detailed components or with aligned reinforcement
designs.

Vacuum assisted resin transfer molding (V ARTM) is a practical
technique for manufacturing biocomposite cellular samples.

Material layout optimization techniques can obtain optimal cellular
design solutions with multiple hierarchies to improve the mechanical
properties and structural performance of biocomposites.

The parametric approach to material layout optimization was chosen
because it allowed for the definition of additional constraints and
multiple objective functions. However, for the case study that was
examined, the constraints did not control the design. Under these
conditions the problem could have also been solved with the
multiresolution approach proposed by Chellappa et al. 2004.
Nonetheless, it should be realized that this might not always be the case
for all problems, or problem formulations, i.e., when design constraints
or multiple objectives may have a stronger influence in the resulting

optimal design.

5.2 Recommendations for Future Work

Recommendations for future work based on the findings of this research are

regarding both the computational and experimental studies. Future work for the

computational portion of this study should focus not only on optimizing the material

distribution within the domain but also the material and resin systems that are

incorporated into the fabrication of biocomposites. Optimizing these systems may

126

also contribute to improving the mechanical properties of biocomposites desired for
structural applications. Future work in the experimental portion of this study involves
the improvement in achieving the optimized structural layouts and progress in
automated manufacturing techniques, such as the VARTM method, for the fabrication

of large-scale cellular, hierarchical components.

5.2.1 Material Layout Optimization Techniques

The proposed parametric modeling approach to optimization is a viable
technique for achieving optimal material layouts; however, in order improve this
procedure to obtain components with enhanced structural performance, further
research and development is necessary. A variety of geometric parameters for
improved void/cell positioning should be explored for the material layout within a
given domain. Instead of only relying on the size of a void, or cell, to alter the
material layouts, void positioning and shape should also be included as a factor.
Incorporating more geometric parameters may assist in producing a variety of unique
design solutions; however, it may also lead to void overlapping and exceeding the
pre-defined domain. To account for this problem, post-processing techniques can be
explored to “correct” or enhance designs as they evolve throughout the optimization
procedure.

Another enhancement to the material layout optimization procedure could be
addressed through modeling the material systems. Hybrid designs could be included
in the FEM models to check for performance enhancements. Furthermore, the

exterior of each void, or cell could be assigned a different material property,

127

simulating a synthetic or natural fiber wall lining which can also have a positive

effect on the overall structure performance.

5. 2.2 Optimization of Material and Resin Systems

Future work is necessary for biocomposite materials to examine the physical
properties of the natural fibers and resin systems for fabrication. Properties such as
flammability, degradation, and robustness of both the fiber and resin systems should
be noted, for they are all important in structural applications. In addition to the study
of the physical properties, cost effective fiber treatment methods should also be
assessed for possible improvement of fiber-matrix adhesion, fiber sizing for resin, and
processing of fibers (V aia and Krishnamoorti 2002). Furthermore, studies should be
implemented regarding the replacement of unsaturated polyester resin systems, with

bio-based systems which are more environmentally fiiendly.

5. 2.3 Design of Optimized Structural Layouts

After achieving a design with an optimal cellular, hierarchical distribution, it
is necessary to prepare this design for fabrication. Future work is needed to examine
the various possibilities of fabricating the obtained optimal cellular hierarchies. For
example, hierarchical cells can be created using non-structural foams wrapped with
synthetic or natural fiber composites or by lining the exterior of each void with a
continuous composite material. These concepts may assist in enhancing the
performance of cellular biocomposite panels and is a design technique that should be

addressed in future work.

128

5.2.4 Improving VART M Manufacturing

Based on the results from this experimental study, VARTM has proven to be a
viable technique for the manufacturing of biocomposite cellular components. Future
work is needed to allow for full-scale manufacturing of these components in order to
ensure their feasibility for structural applications. Fabrication of biocomposites
should be optimized to achieve load-bearing components with high fiber volume
fiactions efficiently and in minimal time such that production of mass quantities is
possible. Manufacturing time could be minimized by using multiple resin and vacuum
ports which will be more feasible to include when manufacturing large—scale
components. Furthermore, in order for VARTM to be considered as a cost-effective
manufacturing process for the proposed optimal biocomposite panels, it will be
necessary to locate reusable and/or integrated materials for implementation of this

technique.

5.3 Applications of Biocomposite Structural Components

Biocomposites do have environmental and health safety advantages over
conventional structural materials and have proven to compete with E-glass FRP
composites. However, further research and development is still required in order to
ensure the use of biocomposites as structural components. With further development,
biocomposites may be able of serving in applications ranging from civil structures,
such as pedestrian bridges, decks, and housing or building flooring systems, to

aerospace structures, such as fuselages, wing skins, and other integrated components.

129

REFERENCES

130

6 REFERENCES

Allaire G., Francfort, G., Bendsoe, M. "Topology design of structures." Nata ASI
Series E, Vol. 227, 239-248.

Allinger, F., Milller, M., Puchinger, and S. (1996). “A New Approach for Sizing,
Shape, and Topology Optimization.” SAE International Congress and Exposition.
Detroit, MI.

Arora, J. (1989), Introduction to (btimpm Desigp, McGraw Hill Intemational
Editions, Singapore.

Barker, R., Puckett, J. (1997). Design of Highway Bridges. Wiley Interscience.

Bendsde, M. and Sigmund, O. (2003), Tomlogy Optimization — Theory, Methods,
and Applications. Springer Verlag, Berlin.

Beyer, G. (2000).University of Dortmund. Department of Computer Science
Homepage [online] Available: lltpz/flsl l-www.cs.i_mi-dortm1£d.de (Accessed
February 2005).

Brouwer, W.D.(2000). “Natural Fibre Composites: Where can Flax Compete with
Glass?” SAMPE Journal, Vol 36, 18-23.

Brouwer, W.D., Van Herpt, E.C..and, Labordus, M. (2003). “Vacuum Injection
Molding for Large Structural Applications.” Composites, Vol 34, 551-558.

Burguefio, R., Quagliata, M.J., Mohanty, A.K., Mehta, G., Drzal, L.T., and Misra, M.
(2004), “Load-Bearing Natural Fiber Composite Cellular Beams and Panels”
Composites Part A, Vol. 35, 645-656.

Burguefio, R., Quagliata, M.J., Mohanty, A.K., Mehta, G., Drzal, L.T., and Misra, M.
(2005), “Hybrid Biofiber-Based Composites for Structural Cellular Plates”
Composites Part A, Vol. 36, 581-593.

Canonaco, M., Loher, U., Esslinger, V., Weisse, B (1997). “Structural Optimization
Methods.” Computer Aided Optimum Design of Structures V, Vol 28, 381-389.

Chellappa, S., Diaz, A.R., and Bendsoe, MP (2004). “Layout Optimization of

Structures with Finite-Size Features using Multiresolution Analysis.” Structural and
Multidisciplinary Optimization, Vol. 26, 77-91.

131

Chou, H., Premkumar, G., and Chu, C. H. (2001). "Genetic Algorithm for
Communication Network Design -An Empirical Study of the Factors that Influence
Performance," IEEE Transactions on Evolutionary Computation, Vol. 5, 236-249.

Daves, L (1991). Handbook of Genetic Algorithms. Van Nostrand Reinhold, New
York.

Gibson R.(l994). Principles of Composite Material Meflmics. McGraw Hill, New
York, NY.

Gibson, L. Ashby, M. (1988). Cellifl_ar Solids: Strucpires afll Properties. 1St Edition,
Pergarnon Press, Oxford, UK.

Goodman, E. (2002). GALOPPS User’s Manual version 3.2.4, East Lansing, MI.

 

Guervos, J. (1997). University of Grenada. Department of Architecture and Computer
Technology [online] Available: ht_tp://kal-el.ugr.es/GAGS/gags-tutorial/node3.html)
(Accessed February 2005).

Haflka, R., Gurdal, Z. (1993). Elements of Structural thimization. Kluwer
Academic Publishers.

Harnbly, E. (1991). Bridge Decl_c Behaviour. Chapman & Hall, London, England.

Hepworth, D., Bruce, D. M., Vincent, J., and Jeronimidis, G. (2000). The
Manufacture and Mechanical Testing of Thermosetting Natural Fiber Composites.
Journal of Materials Science Vol 35, 293-298.

I-Iibbitt, Karlsson, Sorensen (2004) ABAQUS Users Manual, version 6.4, Pawtucket,
RI.

Lackey, E., Vaughan, J., Inarndar, K., Hancock, B. (2004). “Statistical
Characterization of Pultruded Composites with Natural Fiber Reinforcements.”
Composites. Tampa, Florida, October 6-8.

Lagaros, N. and Papadrakakis, K. (2002). “Structural Optimization using
Evolutionary Algorithms.” Computers and Structures, Vol. 80, 571-589.

Mattheck, C. (1997) DESIGN IN NATURE-Leaming From Trees. Springer
Publishing.

Mohanty, A.K., Misra, M., Hinrichsen, G. (2000), “Biofibres, Biodegradable

Polymers, and Biocomposites: An Overview.” Macromolecular Materials and
Engineering, Vol.276/277. 1-24.

132

Olhoff, N., Bendsoe, MP, and Rasmussen, J. (1991). “On CAD-Integrated Structural
Topology and Design Optimization.” Computer Methods in Applied Mechanics and
Engineering, Vol. 89, 259-279.

Poli, R., Langdon, W.B. “Genetic Programming with One-Point Crossover and Point
Mutation.” Proceedings of the Second International Conference on Genetic
Programming, 278-285.

Quagliata, MJ. (2003). “Development and Characterization of Biocomposite Cellular
Beams and Plates for Load-Bearing Components”. Michigan State University, East
Lansing, MI. 1-200.

Red Cedar Technology (2004). HEEDS User’s Manual version 4.1, East Lansing,
MI.

 

Rout, J., Misra, M., Tripathy, S.S., Nayak, SK and Mohanty, A.K. (2001). “The
Influence of Fibre Treatment on the Performance of Coir-Polyester Composites.”
Composites Science and Technology, Vol. 61, 1303-1310.

Rozyany, G.I.N., Zhou, M., and Birker, T. (1992). “Generalized Shape Optimization
without Homogenization.” Structural Optimization, Vol 4, 250-254.

Sigmund, O. (2001). “99 Line Topology Optimization Code written in MATLAB.”
Structural Multidisciplinary Optimization, Vol 21, 120-127.

Sigmund, 0., Tcherniak, D. (2001). “A Web-Based Topology Optimization
Program.” Structural Multidisciplinary Optimization, Vol 22, 179-187.

Spath, D., Neithardt, W., Bangert, C (2002). “Optimized Design with Topology and
Shape Optimization.” Proceedings of the I MECH E Part B Journal of Engineering
Mnaufacture, Vol. 216, 1187-1191.

Srinivasan, A.V. (1995). “Smart Biological Systems as Models for Engineered
Structures.” Material Science and Engineering, Vol. 4, 19-26.

Turkkan, N. (2003). “Discrete Optimization of Structures using a Floating Point
Genetic Algorithm.” Annual Conference of the Canadian Society for Civil
Engineering. Moncton, NB, Canada, June 4-7.

Vaia, R. and Krishnamoorti, R. (2002). “Polymer Nanocomposites: Synthesis,
Characterization, and Modeling.” American Chemical Society Conference,
Washington, DC.

Verheus, A. (2002). Centre of Lightweight Structures TUD— 7N0. Company

Homepage. [online] Available: <ht_tp://www.clc.tno.nl/hom§page.html > (Accessed
April 2005).

133

Wall, M. (2001 ). Mechanical Engineering at Michigan Tech. Mechanical
Engineering Homepage [online] Available: <http://lancit.mit.edu> (Accessed
March 2005).

134

"111111111111111'