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This is to certify that the
dissertation entitled

Shape and Laminate Optimization of Fiber-Reinforced-
Polymer Structures

presented by

Jun Wu

has been accepted towards fulfillment
of the requirements for the

Department of Civil and

Ph'D' degree In Environmental Engineering

 

 

 
    

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6/01 cz/CIRC/DateDuepGS-pts

Shape and Laminate Optimization of Fiber-Reinforced-Polymer
Structures

By

Jun Wu

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

2004

ABSTRACT

Shape and Laminate Optimization of Fiber-Reinforced-Polymer
Structures

By

Jun Wu

Fiber Reinforced polymer (FRP) composites, or advanced composite materials, which
have shown outstanding mechanical characteristics, such as high strength-to-weight and.
high stiffness-to—weight, and high chemical and environmental endurance compared to
conventional materials, have been adapted from aerospace and defense industries to civil
infrastructure applications. The nature of FRP composites makes them strong and stiff in
planes along fiber orientations but rather weak through their thickness. Thus, they are
most efficient when used under global in-plane stress demands. Due to their high material
costs, a way to employ their high material stiffness and strength is to use them in
structural forms in which the structural efficiency is maximized by shaping the structural
geometry to achieve mainly in-plane behavior under the applied loads. Therefore, an
approach is needed to develop the structural forms where the in-plane properties of FRP
composites can be used efficiently.

An integrated approach is developed for the shape and laminate optimization of FRP
structures. The integrated approach is implemented in a two-level uncoupled procedure:
shape and laminate—property optimization followed by laminate design optimization,
where structural shape and laminate design are optimized simultaneously with the aim of

maximizing structural stiffness.

Examples of optimizing laminated FRP shell structures are provided to validate the
procedure. The performance of the integrated approach is evaluated by comparing a
shape-and-material optimized FRP shell with two shape-only optimized FRP shells. The
numerical results show that the proposed integrated approach is reliable and provides a
useful tool for designing optimized laminated FRP structures.

The integrated approach is also used to aid the rational implementation of FRP
composites in civil infrastructure by developing innovative bridge design concepts.
Innovative FRP composite membrane—based bridge systems were explored through
analytical studies of the integrated approach. Two types of bridge systems are developed,
FRP membrane beam bridges and FRP membrane suspension bridges. Both bridge
systems consist of a shape-and-laminate optimized FRP membrane/shell carrying the in—
plane tensile and shear forces together with a conventional reinforced concrete deck
providing the live load transfer. The analytical studies of optimizing the FRP membrane
bridges provide the initial development of innovative systems that use FRP composites in
their inherent behavioral characteristics for new high-performance structures.

Results from this work demonstrate that FRP composites can be used with higher
efficiency in new structural systems as long as their advantageous properties of
directional strength, lightweight, and tailored properties are properly considered in the
design process. The work further discusses the feasible implementation of the optimum
design for composite membrane-based bridges in practical engineering construction. The
work also provides insight to further development and applications of the integrated

Optimization approach.

 

 

ACKNOWLEDGMENTS

I would like to thank my advisor, Professor Rigoberto Burguefio for his excellent
academic guidance, continuous encouragement, and patient support throughout this work.
It has been a very enlightening and fruitful experience to work with Dr. Burguefio. My
sincere thanks are also extended to the other members of my Ph.D. committee, Professor
Ronald S. Harichandran, Professor Charles R. MacCluer, and Professor Amit H. Varma. I
have my deep gratitude for Mr. Graig Gunn’s selfless help and tremendous assistance
during my studies at Michigan State University. I am also grateful for all the help granted
to me throughout the course of my work by the faculty and staff members, colleagues and
many friends.

I would like to acknowledge the financial support by Michigan State University
through the Department of Civil and Environmental Engineering and an Intramural
Research Grant Program award from the Office of the Vice President for Research and
Graduate Studies.

Grateful thanks should also go to my parents. Their love, encouragement and
confidence in my abilities are the backbone of my persistence in the endeavor. Most of all
I would like to thank my wife, Yue. Her endless love is the source of my dedication

through the completion of this research.

iv

TABLE OF CONTENTS

 

 

 

 

 

LIST OF FIGURES _ -- - viii
LIST OF TABLES -- - -- xii
Shape and Laminate Optimization of Fiber-Reinforced-Polymer Structures ............ 1
1 Introduction - 1
1.1 General ................................................................................................................ 1
1.2 FRP composites for civil infrastructure .............................................................. 1
1.2.1 Introduction to FRP composites .................................................................. 2
1.2.2 Mechanical properties of FRP laminates .................................................... 2
1.2.3 FRP composites in civil structures .............................................................. 4

1.3 Outstanding issues and motivation ..................................................................... 6
1.4 Shape and material optimized FRP structures .................................................... 7
1.4.1 Shape resistant structures and optimal shape finding ................................. 8
1.4.2 FRP laminates and optimal laminate design ............................................... 9
1.4.3 Shape and laminate optimization of FRP structures ................................. 10

1.5 Objective and scope of the dissertation ............................................................ 11
References ..................................................................................................................... 13

2 Laminated Fiber Reinforced Polymer (F RP) Composites - 14
2.1 Constitutive relations for FRP laminates .......................................................... 15
2.2 Stacking sequence of laminates ........................................................................ 22
2.3 Constrained relations of lamination parameters ............................................... 24
2.3.1 Relation between in—plane lamination parameters .................................... 24
2.3.2 Relation between out-of—plane lamination parameters ............................. 27
2.3.3 Relation between in-plane and out-of-plane lamination parameters ........ 29

2.4 Summary ........................................................................................................... 34
References ..................................................................................................................... 35

3 Integrated Optimization Process for Laminated FRP Structures ..................... 36
3.1 Form finding and structural shape optimization ............................................... 37
3.1.1 Form finding of tension membrane structures .......................................... 37
3.1.2 Form finding of shell structures ................................................................ 41
3.1.3 Structural shape optimization ................................................................... 45

3.2 Material optimization of laminated FRP composites ........................................ 52
3.3 Integrated shape and FRP laminate optimization ............................................. 55
References ..................................................................................................................... 58

4 Shape and Laminate-Property Optimization of FRP Structures 60
4.1 Formulation of the shape and laminate-property optimization problem .......... 60
4.1.1 Design variables ........................................................................................ 60

4.1.2 Objective function ..................................................................................... 61

 

 

 

4.1.3 Constraints ................................................................................................ 63
4.2 Algorithm of shape and laminate—property optimization .................................. 64
4.2.1 Linearization of objective and constrain functions ................................... 64
4.2.2 Definition and determination of a searching direction ............................. 65
4.2.3 Definition of the constrained steepest descent function ........................... 67
4.2.4 Determination of the step size ................................................................... 68
4.3 Implementation of shape and laminate—property optimization ......................... 69
References ..................................................................................................................... 71
5 FRP Laminates Design Optimization 72
5.1 Formulation of laminate design optimization ................................................... 72
5.1.1 Design variables ........................................................................................ 72
5.1.2 Objective function ..................................................................................... 74
5.1.3 Constraints ................................................................................................ 74
5.2 Algorithm for laminate design optimization ..................................................... 74
5.2.1 Design representation by genetic coding .................................................. 75
5.2.2 Definition of a population size .................................................................. 76
5.2.3 Definition of fitness values and a selection scheme ................................. 76
5.2.4 Implementation of genetic operators ........................................................ 77
5.3 Implementation of laminate design optimization ............................................. 78
References ..................................................................................................................... 80
6 Shape and Laminate Optim'nation of FRP Shells 81
6.1 Shape and laminate optimized FRP shell .......................................................... 81
6.2 Shape-only optimized laminated FRP shells .................................................... 86
6.3 Comparison of three optimal shells .................................................................. 89
6.4 Buckling analyses of optimized laminated FRP shells ..................................... 95
6.5 Summary ........................................................................................................... 97
References ..................................................................................................................... 98
7 FRP Composite Membrane-Based Bridge Systems- -- -- 99
7.1 FRP composite membrane beam (CMB) bridges ........................................... 100
7.1.1 Bridge system description ....................................................................... 100
7.1.2 Integrated optimization of CMB bridges ................................................ 103
7.1.3 System characteristics of optimal CMB bridges ..................................... 115
7.1.4 Post optimality analyses .......................................................................... 121
7.2 FRP composite membrane suspension (CMS) bridges ................................... 127
7.2.1 Bridge system description ....................................................................... 128
7.2.2 Integrated optimization of CMS bridges ................................................. 130
7.2.3 System characteristics of Optimal CMS bridges ..................................... 138
7.2.4 Post optimality analyses .......................................................................... 142
7.3 Discussion of FRP membrane-based bridge systems ..................................... 150
7.3.1 Comparison of FRP membrane-based bridge systems ........................... 150
7.3.2 Discussion of the objective function ....................................................... 154
7.3.3 Selection of the design variables ............................................................. 158

vi

 

7.4 Summary ......................................................................................................... 159

 

References ................................................................................................................... 16 l
8 Conclusions and Future Research Needs ..... 162
8.1 Conclusions ..................................................................................................... 162
8.2 Future research needs ...................................................................................... 163
8.2.1 About the integrated approach of shape and laminate optimization ....... 163
8.2.2 About applications of the integrated approach ....................................... 165
8.2.3 About the engineering and construction of optimal FRP composite
structures ................................................................................................ 166

vii

 

LIST OF FIGURES

Figure 1—1 FRP elastic modulus as a function of fiber orientation .................................... 4
Figure 1—2 Applications of FRP composites in civil infrastructures .................................. 5
Figure 1—3 Shape resistant structures [Otto, 1969] ............................................................. 8
Figure 2—1 Laminated FRP composites ............................................................................ 18
Figure 2—2 Feasible domain of in—plane lamination parameters ....................................... 27
Figure 2—3 Feasible domain of out-plane lamination parameters ..................................... 28
Figure 2—4 Boundary determination of VID for a given {V1A, V3A} .................................. 31
Figure 2—5 Boundary determination of V30 for a given {V1A, V3A, V10} .......................... 32

Figure 2—6 Feasible domain of combining in-plane and out-plane lamination parameters
................................................................................................................................... 33

Figure 3—1 Shape resistant structures ............................................................................... 36

Figure 3—2 Soap film analogy method and tension membrane structures and [Hildebrandt

and Tromba, 1983] .................................................................................................... 38
Figure 3—3 Hanging method and shell structures and [Isler, 1991, 1994] ........................ 43
Figure 3—4 Structural shape optimization [Ram, 1992] ................................................. 45
Figure 3—5 Structural shape optimization process ............................................................ 47
Figure 3—6 Two-level approach of shape and laminate optimization ............................... 56
Figure 4—1 Shape and laminate-property optimization ..................................................... 69
Figure 5—1 Typical symmetric and balanced laminate ..................................................... 73
Figure 5-2 Design representation by genetic coding ........................................................ 76
Figure 5—3 Crossover operator .......................................................................................... 77
Figure 5—4 Mutation operator ........................................................................................... 78
Figure 5—5 Laminate design optimization ........................................................................ 79

viii

Figure 6—1 Geometry, loading, and control points for shell structures ............................ 82

Figure 6—2 History of objective function - FRP Shell 1 .................................................. 84
Figure 6—3 Histories of partial geometric design variables — FRP Shell 1 ....................... 84
Figure 64 Histories of laminate—property design variables — FRP Shell 1 ..................... 85
Figure 6—5 Second-level optimization process — FRP Shell 1 .......................................... 86
Figure 6—6 History of objective function —- FRP Shell 2: [45°/—45°/—45°/45°]65 .............. 87
Figure 6—7 Histories of geometric design variables — FRP Shell 2: [450/-450/-450/450]63
................................................................................................................................... 88
Figure 6—8 History of objective function — FRP Shell 3: [0°/90°/90°/0°] ........................ 88

Figure 6—9 Histories of geometric design variables — FRP Shell 3: [0°/90°/90°/0°] ........ 89

Figure 6—10 Global and local coordinate systems ............................................................ 91
Figure 6—11 Structural responses of initial and optimal shells ......................................... 92
Figure 6—12 Membrane force distribution of optimal shells ............................................ 94
Figure 6—13 Bending moment distribution of Optimal shells ........................................... 94
Figure 7—1 Composite membrane beam (CMB) bridge ................................................. 100
Figure 7—2 Computational model of the CMB bridge .................................................... 101
Figure 7—3 Geometric key points for CMB bridge shape optimization .......................... 104
Figure 7—4 History of objective function — CMB Model A ............................................ 106
Figure 7—5 History of geometric design variables — CMB Model A .............................. 107
Figure 7—6 History of laminate-property design variables — CMB Model A ................. 107
Figure 7—7 History of objective function — CMB Model B ............................................ 108
Figure 7—8 History of geometric design variables — CMB Model B .............................. 108
Figure 7—9 History of laminate-property design variables — CMB Model B ................. 109
Figure 7—10 History of objective function — CMB Model C .......................................... 109

Figure 7—11 History of geometric design variables — CMB Model C ............................ 110

Figure 7—12 History of laminate-property design variables — CMB Model C ............... 110
Figure 7-13 Laminate design optimization for 8—layer laminates .................................. 113
Figure 7—14 Laminate design optimization for 48-layer laminates ................................ 114
Figure 7—15 Section force (SFl) distribution ................................................................. 118
Figure 7—16 Section force (SF2) distribution ................................................................. 118
Figure 7—17 Shear section force (SF3) distribution ........................................................ 119
Figure 7—18 Load case for maximum torsion in the strength limit state ........................ 126
Figure 7—19 Load case for maximum bending in the strength limit state ...................... 126
Figure 7—20 Concept for Composite Membrane Suspension (CMS) Bridges ................ 128
Figure 7—21 Geometry of the CMS bridge ..................................................................... 129
Figure 7—22 Loading and geometric key points for the CMS bridge ............................. 131
Figure 7—23 Histories of geometric design variables — CMS Model A .......................... 133
Figure 7—24 Histories of laminate-property design variables — CMS Model A ............. 133
Figure 7—25 History of objective function — CMS Model A .......................................... 134
Figure 7—26 Histories of geometric design variables — CMS Model B .......................... 134
Figure 7—27 Histories of laminate-property design variables — CMS Model B ............. 135
Figure 7—28 History of objective function — CMS Model B .......................................... 135
Figure 7—29 Second-level optimization process of Model A ......................................... 137
Figure 7—30 Second-level optimization process of Model B .......................................... 138
Figure 7—31 Coordinate systems ..................................................................................... 139
Figure 7—32 Loading pattern of spaced line loads .......................................................... 145
Figure 7—33 Section force (SFl) distribution in CMS optimal designs .......................... 149

Figure 7—34 Section force (SF2) distribution in CMS optimal designs .......................... 149
Figure 7—35 Structure material distribution of relative stress index ............................... 153

Figure 7—36 Performance of different objective functions for optimal CMB bridge ..... 157

xi

LIST OF TABLES

Table 2—1 A*, B*, and D* matrices in terms of lamina invariants and lamination

parameters ................................................................................................................. 21
Table 6—1 Optimal laminate designs of FRP shells .......................................................... 86
Table 6—2 Summary of strain energies and displacements of optimal shells ................... 89
Table 6—3 Summary of structural responses of initial and optimal shells ........................ 91
Table 6—4 Buckling analyses of Shell 1 ............................................................................ 96
Table 6—5 Buckling analyses of Shell 2 ............................................................................ 96
Table 6—6 Buckling analyses of Shell 3 ............................................................................ 97
Table 7—1 Load case specification for CMB bridge optimization .................................. 101
Table 7—2 Dimensional constraints for geometric design variables ............................... 105
Table 7—3 Initial designs of CMB Bridges ..................................................................... 106
Table 7—4 Optimum results of shape and material-property optimization ..................... 111
Table 7—5 Optimal laminates from laminate design optimization process ..................... 115
Table 7—6 Section forces for the optimal CMB bridge ................................................... 117
Table 7—7 Section moments for the optimal CMB bridge .............................................. 117
Table 7—8 Buckling eigenvalues and eigenvectors of the optimal CMB bridge ............ 120
Table 7—9 Structural response of optimal CMB bridge with different optimal laminates

................................................................................................................................. 122
Table 7—10 Geometric variations of optimal CMB bridges ............................................ 123

Table 7—11 Laminate stresses of the optimized CMB bridge under different limit states
................................................................................................................................. 125

Table 7—12 Stress indices of the optimized CMB bridge under different limit states.... 125
Table 7—13 Laminate stresses of the optimized CMB due to different extreme effects. 127

Table 7—14 Stress indices of the optimized CMB due to different extreme effects ....... 127

xii

Table 7—15 Load cases specification for CMS bridges .................................................. 130

Table 7—16 Coordinates constraints for geometric design variables .............................. 132
Table 7—17 Initial designs of CMS Bridges .................................................................... 132
Table 7-18 Optimal results of the first-level optimization for the CMS bridges ........... 136
Table 7—19 Section forces for optimal CMS bridges ...................................................... 140
Table 7—20 Section moments for optimal CMS bridges ................................................. 140
Table 7—21 Buckling eigenvalues and eigenshapes of the optimal CMS bridges .......... 142

Table 7—22 Structural response of optimized CMS bridges for different limit states 144
Table 7—23 Stress indices of optimized CMS bridges for different limit states ............. 144

Table 7—24 Structural responses of the optimized CMS bridges subject to spaced line
loads under the service limits state ......................................................................... 146

Table 7—25 Stress indices of the optimized CMS bridges subject to spaced line loads
under the service limits state ................................................................................... 146

Table 7—26 Section forces of optimal CMS bridges subject to loading pattern changes 147

Table 7—27 Section moments of optimal CMS bridges subject to loading pattern changes

................................................................................................................................. 147
Table 7—28 Stress indices distribution of three bridges .................................................. 152
Table 7—29 Statistics of relative stress index distribution .............................................. 153

Table 7—30 Optimal CMB bridge designs with different membrane thickness constraints

................................................................................................................................. 155
Table 7—31 Structural response with different membrane thickness .............................. 156
Table 7—32 Evaluations of alternative objective functions ............................................. 157

xiii

1 Introduction

1.1 General

Fiber reinforced polymer (FRP) composites have been adopted from aerospace and
defense industries to civil infrastructure due to their unique properties, such as high
strength-to-weight and stiffness-to-weight ratios, and high chemical and environmental
endurance. However, to date, the use of FRP composites for civil infrastructure is mainly
for structural rehabilitation and strengthening. The use of FRP composites to fabricate
primary structural components has been limited, primarily due to high material costs and
designs based on shapes suitable to conventional materials. The layered and fiber
dominated structure of FRP composites makes them most efficient when used under in—
plane stress demands. The shape resistance concept that structures carry loads by shaping
structural geometries to achieve in-plane or membrane resultants is introduced in this
chapter, followed by a brief discussion of shape finding for shape resistant structures.
Laminate Optimum designs are then discussed to improve in—plane behavior of shape
resistant structures by tailoring the material properties of FRP composites property. The
rationale to find material-adapted structural shapes and tailor shape-adapted material
properties for efficient use of FRP composites in new construction is thus recognized.
The aim of this research is then presented so as to develop an approach that accomplishes

these integrated design tasks.

1.2 FRP composites for civil infrastructure
This section provides an overview of the mechanical characteristics of laminated FRP

composites and a brief review of current applications in civil infrastructure.

 

1.2.1 Introduction to FRP composites

Fiber reinforced polymer (FRP) composites are materials in which a reinforcement
constituent of fibers is surrounded by a continuous and uniform matrix constituent of
polymer. The fibers are generally strong and stiff to provide a primary load-carrying
capacity. The polymer matrix holds the fibers together, protects the fibers from damage,
and allows loads to be distributed among individual fibers.

FRP composites are typically used in laminates, which consist of thin layers of fiber-
reinforced material fully bonded together. Each individual layer, or lamina, contains an
arrangement of fibers embedded within a thin layer of polymer matrix material.
Depending on the arrangement of the fiber constituent, an individual layer of a laminated
composite may be constructed by a number of forms. A typical form is to align
unidirectional continuous fibers in a given direction. As it will be discussed in Section
1.2, this arrangement provides a unique feature of the material properties of FRP
laminates, which can be altered by varying the percentage of layers in the laminate with
different orientations. Therefore, FRP laminates composed of unidirectional continuous-
fiber composite layers are commonly used in designs of high-performance structures and

are thus primarily considered in this research.

1.2.2 Mechanical properties of FRP laminates

The properties of FRP composite layers, such as stiffness and strength, strongly
depend on the directional nature of the fibers. The fibers used in FRP composites usually
have much higher stiffness than the polymer matrix. Consequently, unidirectional FRP

composites have different stiffness properties along the fiber direction and perpendicular

to the fiber direction. The stiffness of FRP composites in the fiber direction is governed
by the fibers and the stiffness perpendicular to the fiber direction is mostly dominated by
the polymer matrix. Such materials, in which material properties have two mutually
perpendicular planes of symmetry, are referred to as orthotropic.

The stiffness of Orthotropic FRP laminates can be fully described by four elastic
stiffness properties, which are referred to as the engineering constants, i.e., two Young’s
moduli, E1 and E2, along the fiber and transverse to the fiber directions respectively, and
the shear modulus 612 and Poisson ratio v12 in the plane of the layer. Typical stiffness
properties of various unidirectional FRP composites are given in Table 1.1.

Table 1.1 Properties of various fiber-reinforced cmosite layers

1 E2 G12
(GPa) (GPa) (GPa)
300/5208 Graphite/Epoxy 181 10.3 7.17 0.28
AS4/3501 Graphite/Epoxy 138 8.96 7.10 0.30
B(4)/ 5 505 Boron/Epoxy 204 18.5 5 .59 0.23

Kevlar49/Ep Aramid/Epoxy 76 5.50 2.30 0.34
Scotchply 1003 Glass/Epoxy 38.6 8.27 4.14 0.26

 

Material Constituents v12

 

 

It needs to be pointed out that FRP layers are highly dependent on the fiber
orientation as shown in Figure 1—1, which demonstrates how the elastic moduli are
strongly influenced by deviation of the fiber orientation. In addition, FRP laminates,
which are constructed of unidirectional-fiber layers stacked at different orientations,
inherit the properties of the individual layers and thus lead to high in-plane but rather low
through-thickness properties. However, material properties of an entire laminate in
different directions along the plane of fibers can be adjusted to maximize the utility of the
directional nature of the material properties by varying the number, orientation, and

stacking sequence of the layers.

 

 

 

 

 

 

 

 

 

 

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0....»4..:....i...ji....4.....i....L.'..i,'.jo
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Figure 1—1 FRP elastic modulus as a function of fiber orientation

1.2.3 FRP composites in civil structures

Modulus (GPa)

Traditionally, FRP composites have been used extensively in aerospace and consumer

sporting goods where their high stiffness and strength-to—weight characteristics were first

exploited. These properties, together with their high chemical and environmental

endurance and non-magnetic properties compared to conventional materials, have

increased interests in the use of FRP composites for civil infrastructure applications

[Bakis et al., 2002].

Particularly in the rehabilitation of existing structural systems, advanced composites

have shown significant promise in recent laboratory and field applications. The seismic

retrofitting of bridge columns (Figure l—2a) with carbon fiber wraps or pre-formed

jackets has been demonstrated to be technically just as effective, and in some

circumstances more economical, as conventional steel jacketing. The benefits of light
weight and high strength also make FRP composites attractive for the flexural and shear
strengthening of existing concrete structures (Figure 1—2b) [Teng et al., 2002; Hollaway

and Leeming, 1999].

 

     

c)Rehabrlrtaion
Figure 1—2 Applications of FRP composites in civil infrastructures

Structural rehabilitation with FRP composites includes not only repairing and
strengthening of aging infrastructure but also replacement of substandard structural
components such as new bridge decks [Black, 2000] (Figure 1—2c). The advantages of
employing FRPs in new bridge decks include its corrosion resistance, their reduced
weight, and ease of installation. Decks made from composite materials can be customized
to dimensions of traditional decks and can avoid modifications or replacement to the
existing substructure.

Another application of using FRPs in structural components is reinforcing bars

(rebars) fabricated from either glass-fiber or carbon-fiber reinforced plastic composites

[Chaallal and Benmokrane, 1996]. Due to corrosion of steel reinforcement in concrete
structures, deterioration of concrete results in costly maintenance, repairs and shortening
of the structure’s service life. FRP rebars have thus shown to be a viable alternative to
steel reinforcement and prestressing tendons in areas where the use of steel can lead to a
limited life span due to the effects of corrosion.

Finally, the use of all—FRP composite designs for major structural components in new
construction has also become of great interest. Two design and construction systems for
short- and medium-span bridges consisting of tubular FRP members in combination with
conventional materials have been developed and applied to two vehicular bridges in

California [Burguefio, 1999].

1.3 Outstanding issues and motivation

In spite of the many applications of FRP composites in civil infrastructure cited above,
their use is predominantly in structural rehabilitation and strengthening [Bakis et al.,
2002, Karbhari and Seible, 2000]. Construction of new structures composed in their
majority by FRP composite materials is still limited to a few highly subsidized
demonstration projects. Most of the projects are unique and wide acceptance by the civil
engineering community and commercialization has not yet occurred. The limited use is
partially due to a high cost of FRP composites compared to the conventional materials
such as concrete and steel. Another reason is using FRP composites for primary structural
members in shapes copied from those used for conventional materials (i.e., linear shapes
such as I-section) does not take advantage of the predominant in—plane stiffness and

strength of laminated FRP composites [Burguefio, 1999; Keller, 2002].

In order to make composite structures competitive with metallic counterparts, the
overall cost of producing FRP composites needs to be reduced. Advances in
manufacturing techniques, such as pultrusion, resin transfer molding, filament winding
and the automated or semi-automated manufacturing of large components, has
significantly reduced costs. A more efficient way to use FRP composites is in conjunction
with conventional structural materials rather than for individual component replacement
or complete FRP composite designs [Burguer'io, 1999]. This requires new structural
concepts and systems that combine the dominant characteristics of concrete in
compression and steel in inelastic deformation capacity with the superior mechanical
characteristics of directional strength and stiffness in the direction of the composite
fibers. Therefore, efficient use of laminated FRP composites requires developing
structural concepts and systems that employ them under in-plane stress demands

[Burguefio and Wu, 2002].

1.4 Shape and material optimized FRP structures

The development of FRP structures working under in-plane structural demands
requires combining shape finding methods and material-property tailoring. The concepts
of shape resistant structures and material tailoring are thus introduced next, followed by a
brief discussion of the integrated optimization approach for laminated FRP structures

presented in this dissertation.

1.4.1 Shape resistant structures and optimal shape finding

A genre of structural systems that can be used to improve the efficiency of laminated
FRP composites is called shape-resistant structures (Figure 1—1), which carry design
loads in large spans mainly through in-plane or membrane resultants acquired by shaping
the material according to the applied loads. Membrane structures and thin-shells are two
types of shape resistant structures in which the material is efficiently used under in-plane

stress demands developed by the structural shapes.

 

Arch Bridge

Shell Structure

Figure 1-3 Shape resistant structures [Otto, 1969]

Traditional design methods using trial-and-error are not applicable for the design
shape-resistant structures. The methods typically used for the design of shape-resistant

structures are called form-finding or shape optimization. These methods, which range

from experimental to diverse numerical approaches, have been successfully developed
for determining the optimal shapes and form finding of shell/membrane structures.
However, they can not be readily applied to the optimal shape design of laminated FRP
composite structures since they apply primarily to structures made from isotropic
materials, maintaining the same material properties during the form finding process.
Designs for FRP structures obtained through these methods are thus often far from

optimal because other competitive material properties cannot be explored.

1.4.2 FRP laminates and optimal laminate design

As previously mentioned in Section 1.2.2, the material properties of FRP laminates
can be altered by varying the number, fiber orientations, and the stacking sequence of
layers in a laminate. This advantage of tailoring the material properties provides the
possibility of improving the structural performance of a shape resistant structure by
adjusting the properties of FRP laminates to strengthen the in—plane behavior of the shape
resistant design. Evolutionary design optimization methods have been successfully
developed and applied for optimal laminate designs requiring discrete changes of fiber
orientations and the stacking sequence. However, these methods cannot be directly
incorporated with shape optimization procedures, which depend on continuous changes
of the structural geometry. Material optimum designs are thus isolated from the shape-
finding process of structural resistant designs. Therefore, development of efficient
designs for FRP structures in infrastructure requires a general approach accomplishing

simultaneous shape and material optimization.

1.4.3 Shape and laminate optimization of FRP structures

From the previous discussion it is understood that finding an efficient structural
design Of a laminated FRP structure that meets all the requirements for a specific
application should be achieved not only by shaping the geometric configuration of the
structure, but also by tailoring the material properties, It is thus considerably more
complex to find efficient designs for laminated FRP structures than for those made of
isotropic materials.

Several researchers have investigated the optimum structural design of FRP structural
components. For example, Aref [2001] investigated an approach using genetic algorithms
to minimize the weight of structural components by simultaneously changing the cross-
sectional shape and ply orientations of the FRP laminates. In another effort, Qiao et al.
[1998] developed a global approximation method to optimize material architecture by
volume fractions Of cross-plies and the cross-sectional area of laminated FRP beams. In
spite of these and other developments, the methods developed thus far for the
optimization Of FRP laminate structures are only applied to what can be termed as sizing
optimization of standard linear shapes with limited laminate optimization of thickness
and/or orientation of limited number of plies.

Although research on the optimum design of FRP laminates and components has been
under continued investigation, it is much less developed than shape optimization
algorithms for conventional structures. The optimum design of an FRP laminate structure
involves both the shape optimization Of the structure and the material Optimizations of the
laminate. Thus, a general approach is required to simultaneously solve for structural

shape and material properties thus improve the performance of laminated FRP structures.

10

1.5 Objective and scope of the dissertation

The main Objective Of the research reported in this dissertation was to develop an
implement a general analytical approach for the development and optimal design c
material-adopted shapes for laminated FRP composites in civil structures. The propose
analytical approach was evaluated and validated by Optimizing FRP shell structures an
applied to develop innovative bridge systems using FRP membrane/shell elements.

The dissertation outlines the developed analytical procedure for simultaneous]
finding the Optimal shape and optimal laminate design of FRP structures. The integrate
approach serves as an analytical tool to aid in the design of laminated FRP composit
bridge systems and provides insight to the concept of efficiently using FRP laminates b
taking advantage Of their directional strength and material tailorability.

Chapter 1 provides with an introduction to FRP composites, particularly laminate
FRP composites and their mechanical prOperties. Current applications of FRP composite
in civil structures are briefly reviewed. The research motivation is discussed, followed b
the research scope.

Chapter 2 presents the fundamental relations that govern the linear elastic propertie
of FRP laminates. The chapter then discusses the computation Of elastic properties 21
functions of variables that can be changed during the design process, followed by th
formulation Of constraints between these variables.

Chapter 3 reviews the existing techniques and their limits for form finding c
membrane structures and shape Optimization of shell structures. The concept of structurz

shape Optimization is then introduced with a brief discussion of requirements for materiz

11

optimization of FRP laminates. Finally, an uncoupled integrated approach is proposed for
shape and material optimization.

Chapter 4 and Chapter 5, respectively, describe the optimization problem
formulation, the chosen algorithm and its corresponding implementation for each
Optimization level in the integrated approach.

Chapter 6 focuses on the investigation of the performance and stability of the
integrated approach by investigating the optimization Of FRP shells. The proposed
approach is evaluated by studying and comparing the structural behavior between a
shape-and-material Optimized shell and two shape-only Optimized shells.

Chapter 7 proposes two types of FRP composite membrane-based bridge systems that
effectively employ laminated FRP composites. The developed integrated approach is
implemented and evaluated through analytical studies on these two bridges.

Chapter 8 summarizes and concludes the current research efforts for the integrated

shape and laminate Optimization and provides recommendations for future research.

12

References

Aref, A.J. (2001). A genetic algorithm-based approach for design optimization of fibe
reinforced polymer structural components. Mechanics and Materials Summer Conferencl
2001 sponsored by ASME, ASCE, SES, San Diego, CA.

 

Bakis, C.E., Bank, L.C., Brown, V.L., Cosenza, E., Davalos, Lesko, IF, A., Machid
J .J ., Rizkalla, SH. and Triantafillou, TC. (2002). Fiber-Reinforced Polymer Composite

for Construction — Sate-of—the—Art Review, Journal of Composites for Construction: 6(2:
73-87.

Black, S., (2000). A survey Of composite bridges, Composites Technology: 6(2), 14-18.

Burguefio, R. (1999). System characteristics and design of modular fiber reinforce:
polymer (FRP) short- and medium-span bridges, Doctoral Dissertation, Univ. 0
California, San Diego, California.

Burguefio, R. and Wu, J. (2002, September). Development of an FRP Membrane Bridg
System. IABSE Symposium: Towards a Better Built Environment — Innovatior
Sustainability, Information Technology, Melbourne, Australia.

 

Chaallal, O. and Benmokrane, B. (1996). Fiber-Reinforced Plastic Rebars for Concret
Applications. Composites: Part B 27B, 245-252.

 

Hollaway, LC. and Leeming, M.B., (1999). Strengthening of reinforced concret.
structures: using externally-bonded FRP composites in structural and civil engineering

Boca Raton, FL: CRC Press.

Otto, F, (1969). Tensile structures: design, structure, and calculation of buildings 0
cables, nets, and membranes. Cambridge, Mass: MIT. Press.

Karbhari, V.M. and Seible, F. (2000). Fiber Reinforced Composites — Advance
Materials for the Renewal Of Civil Infrastructure. Applied Composite Materials: 7, 95
124.

Keller, T. (2002). Overview of Fiber-Reinforced Polymers in Bridge Constructior
Structural Engineering International: Journal Of the International Association for Bridg
and Structural Engineering: 2, 66-70.

 

Qiao, P., Davalos, J .F. and Barbero, E.J. (1998). Design optimization of fiber-reinforce
plastic composite shapes, Journal of Composite Materials: 32(2) 177-196.

Teng, J .G., Chen, J F Smith, ST. and Lam, L., (2002). FRP strengthened RC structure:
New York: Wiley.

13

2 Laminated Fiber Reinforced Polymer (FRP) Composites

Laminated FRP composites are a class Of composite materials where the layers of
unidirectional fiber—reinforced composites are staked at different orientations. By varying
volume fractions and the fiber orientations Of layers in a laminate, material properties of
the entire laminate can be adjusted to meet the requirements of a design. FRP laminates
have a wide range of application in structural design, especially for high—performance
structures that have stringent requirements of specific (property divided by weight)
stiffness and strength. In this research, FRP laminates are the only type Of FRP composite
considered in the composite structures.

Applicable structural designs require meeting certain response quantities such as
displacements, stresses, buckling loads and natural frequencies. These response quantities
depend on the constitutive behavior of materials, which are determined by the elastic
properties such as Young’s modulus, shear modulus, and Poisson’s ratio. For laminated
FRP composites, the constitutive behavior is determined not only by the these elastic
prOperties but also by the parameters Of fiber orientations and stacking sequence that can
be altered in order to improve structural performance.

The Objective of this chapter is to discuss how the sectional stifi‘ness calculations of
FRP laminates depend on the parameters that can be changed during a design process.
The chapter begins by presenting the stress-strain relation that governs linear elastic
responses of an orthotropic lamina. The section stiffness of FRP laminates composed of
orthotropic laminas oriented at angles is then formulated based on the constitutive
relations and the lamination stacking sequence. In addition, the formulation of the section

stiffness is given in mathematical terms of lamina invariants and lamination parameters.

14

 

This alternative formulation of the section stiffness requires establishing the relationships
between lamination parameters when altering fiber orientations in a stacking sequence.
Thus, the feasible domain of out-plane lamination parameters is explored with respect to

the in-plane lamination parameters at the end of this Chapter.

2.1 Constitutive relations for F RP laminates
The stress-strain relation for a three—dimensional anisotropic linear material, also
known as Hooke’s law, is expressed in the following tensor form:
023': ijmnéinn (2-1)
Because of symmetry, 03:0},- and £;,,,,=a.m, there are only 21 independent material

constants in the C matrix, in which Cijmn= 11”,": gm. SO, Eq. (2.1) can be rewritten in the

matrix form:

r N r N r i
0.11 C11 C12 C13 C14 C15 C16 811
0.22 C22 C23 C24 C25 C26 822
03 C33 C3 C15 C36 8 3

i 3>=< 4 ‘ <3 f. (2.2)
0' 23 C44 C45 C46 823
031 C55 C56 831

(0'12 , isym C66 J .812 i

 

 

 

 

 

 

In the case of a three—dimensional orthotropic material, such as a unidirectional fiber-
reinforced lamina, there are two perpendicular planes Of symmetry that define two
principal axes of material properties. These principal axes correspond to the direction of
the fibers and the directions perpendicular to the fibers, denoted by subscripts 1, 2 and 3,
respectively. The stress-strain relation in the principal material directions of an

orthotrOpic lamina is given by Eq. (2.3)

15

ro-lli C11 C12 C13 0 0 0 \ r811‘
022 C22 C23 0 0 0 822
0' C 0 0 0 e
1 33 i=4 33 l1 ‘3 y. (2.3)
023 C44 0 0 £33
031 C55 0 £31
1012, tsym C66, i512,

 

 

 

 

 

 

Due to the thinness of typical lamina, all layers of the FRP laminates are assumed to
behave in a plane stress state in the 1-2 principal material plane so that 0'33 2 0, 023 = 0

and 0,3 = 0. The strain—stress relation is then simplified to Eq. (2.4)

011 Q1] Q12 0 £11
022 2 Q12 Q22 0 822 , (2-4)
012 0 O Q66 812

where the Qij are the reduced material stiffness coefficients which are given in terms of

four independent engineering material constants in principal material directions as:

 

E E
Q11 _ 1 v Q22 _ 2 ,
1_V12V21 1 V17V21
v E V E
Q12 _ 12 2 _ 21 l , (2.5)
1_V12V21 1—V12V21
Q66 : G12 '

Since an orthotropic FRP layer can be generally oriented at a certain angle with
respect to a structural coordinate system of x-y-z, the stress—strain relations (Eq. (2.4)) in
the material coordinate system must be transformed to the structural coordinate system.

The transformation of stresses and strains can be accomplished by

0.11 an 81] xx
0'22 = T 0'yy and 6‘22 = T 8», , (2.6)
0'12 ny 812 8”

16

where T is a transformation matrix. Assuming that the 1-2-3 axis originally coincides
with the x—y-z axis and considering a rotation 49 about the 3 (or z) axis, the

transformation matrix is given by:

cos2 (9 sin2 (9 2cost93inr9
T = sin26’ c0326 -—2cosr9$int9 . (2.7)

. . 7 . ’3
——cos€s1n6l cost9srn6 cos“ 6 —sm“ 6’

With Eq. (2.6) and Eq. (2.7) substituted into Eq. (2.4), the strain-stress relation in the

structural coordinate system is transformed to:

0.1x 8)“ Q1 1 Q12 Q16 8.0

-1 — —" '—
0), =T QT e), = Q, Q,, Q,, a), . (2.8)
axy 813’ Q1 6 Q26 Q66 8.x):

Q}!- are called the transformed reduced stiffness coefficients, which can be expressed

in a simpler form by introducing the lamina invariants [Tsai and Hahn, 1980] as:

Q11 =U,+U,cos26+U,cos46 (2.9a)
5, = U, — U, c0346 (2%)
5,, = Ul — U, cos 2.9 + U3 cos 4e (2.9c)
5, = éU, sin 26 + U, sin 46 (2.9d)
5,, = éU,sm26 — U,Sin46 (2%)
5,, = US — U, cos46’ (2.91)

where the Us are the lamina invariants defined as [Tsai and Hahn, 1980]:

1
U1 = g(3Qll + 35,, + 25,, + 4Q,,) (2.10a)

l7

1
U2 25(Q11 _Q22) (Z-IOb)

1
U3 = §(Qr1 + Q22 — 2Q12 _4Q66) (Z-IOC)
U4 :%(Q11 +Q22 +6Q12 _4Q66) (210(1)
1
Us : §(Q11 + Q22 _ 2Q12 +4Q66) (2-106)

When n orthotropic layers oriented at different angles are stacked together, a set of
additional assumptions are needed in order to derive the equations that govern the
constitutive behavior of the laminate. Each layer is assumed to be perfectly bonded to the
adjacent layers so that the laminated layers deform in unison without experiencing any
discontinuity in displacements. Furthermore, Classical Kirchhoff plate theory is assumed
for the bending behavior of the laminate so that the strains in the out-of—plane direction Z
are neglected and the w displacement in direction z is constant through the thickness Of
the laminate. These two assumptions, along with the assumption of the layers being in a
state of plane stress, define what is known as classical lamination theory (CLT), which

will be applied herein to formulate the constitutive behavior of the laminates.

 

 

 

 

 

 

 

 

 

 

 

 

 

Z ll hk/:Zk+l'Zk
‘1— \ 1/
\_ l/
he, © 9L A ———,j
\ ® OK i/ \ ZkiIZL L+l
) /7’<II IT 1 i =x
Z
mi 3 i“ Z1
i to 92 i i l
/ G) 91 \

 

Figure 2—1 Laminated FRP composites

18

According to CLT, the strains in a plate under a state of stress can be defined by the

addition of the constant mid—plane strain {8”} and the linear variation due to the bending

curvatures { K} as:

0
XY ELY KXX
_ 0
)’y — 8y)» + Z Ky), ' (2.11)
8,, 8:, KB,

The stresses in the km ply expressed by the material reduced stiffness of that particular

ply are:
(k) — — — (k) 0
an Q1 1 Q12 Q16 8n Kn
_ — — — (k)
0'”, — Q12 Q22 Q26 8:, + 2: Km . (2.12)
0.x); Q16 Q26 Q66 83y Kay

The stress resultants and moment resultants for unit width of the cross section are

Obtained by integrating the stresses of all plies through the laminate thickness:

(k)

NX an n 0.“
Ny = ill/22 0“,, dz=§ I z:1 0'), dz, (2.13a)
ny 0x), — X)
Mx 0's. (k) 0'1:
My 41/: 0,, zdz=;I: a, zdz. (2.13b)
M x, 03,, ' 0,,

Substituting the stress-strain relations of Eq. (2.12) into Eq. (2.13), the following

section stiffness of the laminate is Obtained:

19

Nx A11 A12 A16 Bll BIZ Blfi 8:)
Ny A22 A26 BIZ 82?. 326 6:,
N ,3. : A66 B 16 BIG 866 8:; (2.14)
M , Du D12 D,, K,
M y D 22 D 26 K,
M A), sym D 66 Kn

where, [MM is the extensional (in—plane) stiffness matrix; [B]3X3 is the bending—
extension coupling stiffness matrix; and [D]3X3 is the flexural (out—plane) stiffness matrix.

The components of the A-B-D matrix can be expressed in terms of the reduced stiffness

Q1“) ’5 of each layer by

A, =Z5j")(z —z,_ 1) (2.15a)
1 _ .

3, =5 ,1“.(zf_z;_1) (2.151»
1 1 ,
=§H Q,“ (zi —z,_,) (2.15c)

If all layers are of the same material, the components Of the stiffness matrix can be

further simplified by substituting (2.9) into (2.15) Obtaining:

A, =h~A,‘.(U,v) (2.16a)
h2 II:

B, =—2—-B,(U,V) (2.16b)
I13 .

D, =E-D,(U,v) (2.16c)

where the VS are termed lamination parameters. The use of lamination parameters was
first proposed by Tsai and Hahn [1980]. However, it should be noted that the notation

used in Eq. (2.16) uses a non—dimensional formulation of the lamination parameters,

20

which deviates from the classical definition from Tsai and Hahn [1980] by the multipliers
h, h2/2, and h3/12, respectively. The non—dimensional formulation has been found to be
more convenient for use in Optimization problems [Miki, 1982; 1985], and is defined by

the following expressions.

V0{A.B.D) : {1, 09 1}» (2.17a)
" , 1 2 . , 12 _

VllA.B,D} : ZCOS 26m —(Zk _ Zk—l )a 7(Zf — Zk—l )> —3(Zf — Z134 )} ’ (2-17b)
k=l h h h
" . , 1 2 7 7 l2

V21A.B.Dl : Zsm 29(k){_ (Z1 _ Zk—l )2 Tizi _ Z1.2—1) ‘Tizi _ Zi—1)}’ (2-170)
k=1 h h h
" , 1 2 2 2 12 ,

V31A,B.Dl : ZCOS 46m{— (Zk _ Zk—l )’ 7(Zk — Zk—l )7 —3(ZI: ’— Z2-1)}’ (217(1)
M h h h
" . 1 2 2 2 12 ,

V4lA.B,D} : Zsrn 46(k){_(zk _ Zk—l ) __2(Zk _ ZH ) _3(Zf — 224)}. (2.176)
k:1 h ll 11

The A; , B; ,D; coefficients in terms of the lamina invariants Us and the lamination

parameters V’s are summarized in Table 2—1.

Table 2—1 A*, B*, and D... matrices in terms of lamina invariants and lamination

 

 

parameters
V0{A,B,D} VllA,B.D) V2{A,B,D} V3]A,B.D} V4{A,B,D)
{AiirBierfii U1 U2 0 U1 0
{A;..B;..D;.} U1 -U2 0 u. 0
{Aferfzerzi U4 0 0 U1 0
{A;,,B;,,D;,} U5 0 0 U1 0
{£6,812.01} 0 0 U2 0 2U3
{ester} 0 0 U2 0 -211,

 

The lamina invariants US are independent Of the ply orientation and are determined
only by the property of a single layer with respect tO the material coordinate system. The

lamination parameters V’s are related to the fiber orientations and stacking sequence of

21

the layers with respect to the structural coordinate system. Thus, as long as the properties
Of the FRP lamina are known, the section stiffness of an FRP laminate can be determined

by lamination parameters representing the fiber orientations and the stacking sequence.

2.2 Stacking sequence of laminates

In the previous section, the section stiffness matrix describing the elastic responses of
a laminate was presented. According to the section stiffness matrix, it should be noted
that coupling effects may exist. These coupling effects are easily introduced by random
stacking sequences. However, these coupling effects can hamper the effective use of FRP
laminates, and their source and effect should be clearly understood.

The [B] matrix represents the coupling effects between in-plane and out-Of-plane
deformations. Then, according to Eq. (2.17), the [B] matrix vanishes when the stacking
sequence Of a laminate is restricted to be symmetric with respect to the laminate mid-
plane. According to Eq. (2.14), for symmetric laminates, the in-plane responses and out-
of—plane responses can be solved independently. The in-plane resultants are only related
to the mid—plane strains and the moment resultants are only related to the section
curvatures.

The stiffness terms A16 and A26 in the in-plane stiffness matrix creates a shear-and-
extension coupling effect. This coupling effect will result in in-plane normal stress
resultants by shear deformations. The shear-and-extension coupling can be eliminated by
balancing layers of Off-axis fiber orientations. Balanced laminates require that for each

layer with a negative orientation angle 6there is a layer with a positive orientation angle

6. Although the layers in a balanced pair do not need to be placed adjacent to each other,

22

the distance of two balanced layers will determine the value Of the bending—twisting
coupling terms of D16 and D26.

The bending-twisting coupling terms D16 and 026 exist for all laminates that have off-
axis fiber orientations. The out-Of-plane bending—twisting coupling effect can not be
eliminated even when balanced laminates are used. However, balanced laminates in
which layers Of positive and negative Off—axis angles are placed adjacent to one another
will reduce the effect of out-Of-plane bending-twisting coupling. Furthermore, for a
certain laminate thickness, the bending—twisting terms, 016 and D26, vanish with an
increasing number Of the layers. This behavior will be proved in the following section
dealing with constrained relations between lamination parameters.

Designs with consideration of coupling effects are generally avoided in the use Of
FRP laminates, unless the coupling behavior is sought for pseudo-active structures. Thus,
only balanced and symmetric laminates, which have the least-pronounced coupling
characteristics, were investigated in the current research. Coupling of the in-plane and
out-Of-plane responses, denoted by the [B] matrix, vanishes for symmetric laminates.
According to the Eq. (2. 17), the in-plane shear-extension coupling defined by A16 and A26
vanishes when the laminates are balanced. The out-of—plane bending-twist coupling is
considered zero on condition that the number of layers is large enough. Based on the
above-mentioned assumptions of the staking sequences, the section stiffness Of laminates

can be simplified to:

23

 

 

 

 

 

 

N, All A,, 0 0 0 o a:
N, A,, A,, 0 0 0 0 5;?
N“, i 0 O A,, 0 0 O 8:;
, . = . i (2.18)
Mx 0 O 0 D11 D12 0 KX
M, 0 0 0 1),, 1),, 0 K,
[M 1,, 1 0 0 0 0 0 1),, - ,K,

2.3 Constrained relations of lamination parameters
Use Of the lamination parameters to define the section stiffness of laminates requires
the definition of an allowable domain for each lamination parameter so that they meet the

coupled constraints in relation with other lamination parameters. The constrained
relationship between the lamination parameters V1,, V3,, V10, V20, V3,, and V4,, and

their feasible domain are established in the following sections.

2.3.1 Relation between in-plane lamination parameters
As discussed previously, in—plane shear-extension coupling effects vanish for
balanced and symmetric laminates. Then, according to Eq. (2.18), the in-plane section

stiffness is determined by:

A11 A12 0
A: A,, A,, 0 (2.19)
0 0 A,,

Recalling Table 2.1and Eq. (16), the components Of the stiffness matrix A can be

determined in terms of lamination parameters and lamina invariants by:

24

 

 

 

 

i A1 1 Ah U1 V1 A V3A 1
A *, v v
, 22 =h<AE-1=h U1 1A M U, , (2.20)
A12 A12 U4 0 TVs/1 U
* 3
(A66 iA66, _U5 0 _V3A_

 

According to Eq. (2.20), when the properties Of the lamina are known (i.e., U1, U2,
and U5), the in—plane stiffness is fully determined by the axial lamination parameters

VIA and V3, , which are defined as follows:

VIA = 2v, cos 219, , (2.2la)
r=l

v,, = 2v, cos4o, , (2.11b)

r=l

where n is the number of different i6 groups, and v, is the volume fraction of the layer

with i6,- orientation angle

 

2z 2z _1
: r _ r , 2.22
' h h ( )
and satisfies:
2v, =1. (2.23)
r=l

where z, in Eq. (2.22) is defined with respect to Figure 2—1.
From Eq. (2.21) and Eq. (2.23), it can be noted that the lamination parameters VIA and

V3,, , respectively, have the boundary:

—ISVM,V,ASI (2.24)
Obviously, the allowable value Of lamination parameters of V1,, and V3,, should

satisfy the inequality equation (2.24). Furthermore, the possible in-plane elastic

25

properties are constrained by an allowable combination of V1, and V3,. Using the

trigonometric relation, c0346 = 2cos2 219+1, the allowable region Of VIA and V3,, should

satisfy:

V3,, — (2V1: — 1): :12, cos 46 —— [2[: v, cos 26?]~ - I]
r=l

r=l

(2.25)

= 2iv,[cos 26, —:v, cos 28,]~ .

r=l r=l
The term V3,, — (2V,: —1) is non-negative, and achieves zero only on the condition of

n = 1, that is, when the laminate has only one layer. Therefore, the relation stated by Eq.

(2.25) can be expressed as:

v 2 2v“,2 —1. (2.26)

3A

Consequently, the allowable combination domain of in-plane lamination parameters
specified by Eq. (2.24) and Eq. (2.26) can be depicted in a diagram (Figure 2—2). The
diagram has been effectively used to design laminates with prescribed in—plane stiffness

properties in a general graphical procedure introduced by Miki [1982].

26

 

 

 

-1.0

 

Figure 2—2 Feasible domain of in—plane lamination parameters

2.3.2 Relation between out-of—plane lamination parameters

For symmetric and balanced symmetric laminates, the flexural lamination parameters

V”, and V30 are defined as:

V”, = Zs, cos 26, , (2.27a)
r=l
V,,, = 2s, cos4e,, (2.27b)

r=l

where n is the number of different i6 groups, and s, is defined as:

3 3
8. =6?) {git—J . (2.28)

As before, both lamination parameters must satisfy the boundary defined by

 

—1 S V10, V30 S 1. In addition, in a procedure similar to that presented previously, it can

27

be shown that the allowable region (Figure 2—3) of combining V”, and V3,, satisfies

[Miki, 1985]:

1 v...

 

 

 

Figure 2—3 Feasible domain Of out-plane lamination parameters

As previously discussed, the bending-twisting coupling terms 016 and D26, which are

caused by any Off-axis fiber orientation layer, exist even for balanced and symmetric

laminates. The bending-twisting coupling terms D16 and D26 are determined by the V20

and V4,, lamination parameters, where:

V,,, = Zs, sin 26, , (2.30a)
r=l

V4,, = 2s, sin 46, . (2.30b)
r=l

28

Considering symmetric and balanced laminates in which each layer has constant

thickness and the balanced layers are placed adjacent with each other,

expressed by:

 

3 3
Sr : 2Zr __ 2 Zr + <'r—l + 2Zr-l
h h h

 

 

=lilB-2t’22.”l+l";‘i
=Z:7[4r3 +4(r-1)3 —(2r—1)3]

6r—3
4n3 '

 

Then, V2,) and V4,, satisfy

 

 

”0:216:29 3. n26 <2:67*n—3 3:

r=l n

V4D:Z6r— 33in 46f <z6r: 3—

,_1 4n 4n3

 

 

:4n'

3, can be

(2.31)

(2.32a)

(2.32b)

According to Eq. (2.32), as the number of layers increases, the lamination parameters

of V2,, and V40 that define the bending—twisting coupling tend to vanish.

2.3.3 Relation between in-plane and out-Of-plane lamination parameters

In optimization practices that use lamination parameters to design FRP laminates, it

has been observed that the in-plane lamination parameters (V1,, V3,) and out-of-plane

lamination parameters (VH3, V30) are not independent of each other when altering fiber

orientations and the stacking sequence of a laminate. This implies that determination of

laminate properties by in-plane and out-of—plane lamination parameters is constrained by

an allowable combination domain.

29

Considering symmetric and balanced laminates, in which each layer has a constant
thickness and the balanced layers are placed adjacent to each other, the in-plane and out-

of—plane lamination parameters can be expressed as:

V1.1 = 212005219, % , (2.33a)
V3,, 2 zoos 46, % , (2.33b)
V”, zgcos2flriL’13hL—l, (2.33c)
V3,, = :cos 419, 1%. (2.33d)

,z, n n“

The allowable combination domain is defined as a feasible region of (V10, V30) with
respect to a given in—plane feasible design (VI A , V3,, ). The mathematical equivalence can
be alternatively implemented to find out the feasible region bound of (V1,, , V30) in terms

of functions of (V1,, , V3,) throughout whole feasible domain previously defined for the
in-plane lamination parameters (Eq. (2.26)) and Figure 2—2.

The region boundaries of (V10, V30) with respect to (VIA , V3,) can be implemented
by a two-step procedure. The first step is to define the bounds of V10 with respect to a
given feasible design (V1,, , V3,, ). This step can be formulated by the Optimization

problem stated in Eq. (2.34). Figure 2—4 shows an example of determining the feasible

boundary of V10 for a given point of {V1A, V3A} = {-0.4, -O.4};

30

Objective:

" 1 3 2 — 3 +1
Max/Min V”, — V, = Zcos 26, —[—:—,—C— — 1] (2.34a)
,2, n n“
Constraints given by:
" 1
V“, = Zoos 26, — (2.3413)
r=l n
n 1
v,, = Zeos46, —, (2.34c)

ll

r=l

where the 6,,’s are chosen as design variables, and t9, 6 [— 90°,90° J.

 

 

 

  

A V30 (V3A)
l
1.0
Min (V1D'V1A) .
/ +0.5
' ‘ V113 (V111)
A! I I l . , . I 1 T #7
-1.0 i 1.0
(Vw V3A) '

  

 

 

Max (Vro‘vrA)
Figure 2—4 Boundary determination of V10 for a given {Vm V3A}

The second step is then to define the bounds of V3,, for a given value within the

domain of V”, found in the first step with respect to the given design of (V1, , V3,, ). The

procedure can be accomplished by solving another Optimization problem as stated in Eq.

31

(2.35) below. Figure 2—5 shows an example of determining the feasible boundary of V30

for a given point of {Vim V3,, V10} 2 {-0.4, 04, V10} for VIDE [max VID, min V30].

Objective:
" 1 2 — +1
Max/Min V30 — V3, 2 Zoos 46, _[3r__3r___ - I] (2.35a)
,:1 n n”
Constraints given by:
V10 : V 6 [Vlein ’VlD.maxJ (2-35b)
" 1
VI, = Zcos26, — (2.350)
r=l n
V3, 2 2008 46’, 1, (2.35d)
n

r=l

where the @’s are chosen as design variables, and 6,, E J— 90°,90° J.

11 V30 (V311)

 

 

1.0

Max (V3D-V3A)

I-1io I W -o.5

1

1 1 L l 1 L 1 LL

V10 (V111)

I l

1.0

 

 

 

 

 

y...

 

Min (V3D-V3A)

 

 

_ -1.0

Figure 2—5 Boundary determination of V3,; for a given {V1A, V3.1, V19}

 

32

Due to the highly non-linear objective functions, numerical methods are best suited

for use in the optimization process. Given a pair of (V1,, , V3.1 ), a set of 61 was searched to
achieve the maximum/minimum objective. The optimization search is performed for the
different feasible sets of (V1,, VM) throughout the feasible design domain. Thus, the
bound for any given point (VID, V30) can be constructed by interpolating between
discrete points within the (V1,, , V3, ) space.

The numerical solution was implemented with the optimization toolbox provided in

Matlab [MathWorks, 2000]. Figure 2—6 illustrates the allowable regions of out-of-plane

lamination parameters (V1,), V30) for three in-plane lamination parameter sets {VIA ,

V,, }: {-04, —0.4}, {0,0} and {0.7, 0.8}.

f v30 (V3A)
,‘

 

 

1.0

 

V10 (V111)

 

Figure 2—6 Feasible domain of combining in-plane and out-plane lamination parameters

33

As shown in Figure 2—6, the feasible domain for a pair of out-plane lamination
parameters can not achieve the feasible region illustrated in Figure 2—3 in combination
with a given pair of in-plane lamination parameters. Moreover, its feasible region will

depend on the given pair of in-plane lamination parameters.

2.4 Summary

In this chapter, the section stifi‘hess matrix Of FRP laminates based on classical
lamination theory were derived in terms of lamina invariants and lamination parameters.
The elastic properties and coupling effects were discussed, followed by the assumptions
about the stacking sequences of laminates considered in this work. The section stiffness
matrix of the laminates that will be used throughout the research was derived based on
these assumptions. Finally, constrained relations of lamination parameters were

established for laminate designs.

34

References

MathWorks, Inc. (2001). Matlab - Using Matlab Version 6.

 

Miki, M. (1982). Material design of composite laminates with required in—plane elastic
properties, Progress in Science and Engineering of Composites Vol. 2, 1725-1731.
Hayashi, T., Kawata, K., and Umekawa, S. (eds) ICCM-IV, Tokyo.

 

Miki, M. (1985). Design of laminated fibrous composite plates with required flexural
thickness. Recent advances in composites in the United States and Japan, ASTM 864.
Vinson, J .R. and Tava, M. (eds) American Society for testing and materials, Philadelphia:
387-400.

Tsai, SW. and Hahn, HT. (1980). Introduction to composite materials. Lancaster:
Technomic.

 

35

3 Integrated Optimization Process for Laminated FRP Structures

The nature of laminated FRP composites makes them strong and stiff in-plane, along
the fiber orientation, but rather weak through their thickness. Thus, they are most
efficient when used under global in—plane stress demands. Therefore, the efficient use of

laminated FRP composites in civil infrastructure requires finding structural forms that

can maximize the in-plane carrying capacity of laminated FRP composites.

    

(a) Membrane structure if (b) Shellc
Figure 3—1 Shape resistant structures

Tension membrane structures (Figure 3—1 (a)) and thin-shells (Figure 3—1 (b)) are two
structural form types for lightweight structures that carry loads in large spans mainly
through in-plane, or membrane resultants. These two types of lightweight structures
belong to the genre of shape resistant structures in which structural strength is obtained
primarily by shaping the material to achieve membrane behavior under the applied loads.
The shape-resistant nature of in—plane structural response makes structures highly
efficient since only a minimal amount of material is used. Therefore, the form finding of
lightweight structures is to determine the layout, or shape, in which the material is
optimally used so that the structure is subjected primarily to membrane forces rather than

bending.

36

This chapter presents an overview of the capabilities and limitations of different
approaches to the Optimization of laminated FRP structures. An integrated approach
accomplishing the combined tasks of shape and FRP laminate optimization is
conceptually presented. The algorithms for each level of optimization are identified,

followed by a discussion of the proposed integrated approach.

3.1 Form finding and structural shape optimization

Different experimental methods and computational methods have been developed for
the form finding of lightweight structures. Methods for form finding of tension
membrane structures and compression thin-shell structures, which depend on different
types of membrane action, are introduced in this chapter. The computational methods
derived from the experimental methods are also presented. A general computational

method for form finding of lightweight structures is then discussed.

3.1.1 Form finding of tension membrane structures

Tension membrane structures, which may consist of prestressed cable nets or fabric
membranes, are very attractive alternatives to span large distances. They are very light,
elegant, and structurally efficient. The material is optimally used since membranes are
subjected to constant surface stresses. Therefore, the form finding of a membrane
structure implies finding a shape that results in a constant surface stress distribution under
prestressing and externally applied loads.

A membrane that exhibits a constant surface stress is described as a stable minimal

surface. Naturally, stable minimal surfaces are achieved by soap films, which have zero

37

mean curvature at every point on the surface and maintain a minimum surface area when
in stable equilibrium. The soap film analogy (Figure 3—2) [Hildebrandt and Tromba,
1983] is an experimental method that mimics the characteristics of soap films to
determine an optimal shape for a tension membrane structure, which is to act in a similar

state to that of a soap film.

 

Figure 3—2 Soap film analogy method and tension membrane structures and
[Hildebrandt and Tromba, 1983]

The inherent basis of the soap film analogy method for form finding is to find an
Optimal membrane shape due to a uniform stress distributed on a deformed structure.
Mathematically, this procedure leads to minimal surfaces sustained in a stable
equilibrium state. The mathematical principle of the soap film analogy was used to
develop several numerical methods, such as the force density method [Schek, 1974;
Linkwitz, 1999], the dynamic relaxation method [Barnes, 1994; Lewis and Lewis, 1996],
and the updated reference strategy method [Bletzinger, 1997; Bonet and Mahaney, 2001].

The force density method [Schek, 1974; Linkwitz, 1999] is primarily used for cable
structures. A cable net structure is discretized into branches. The force-length ratios, or
force densities, Of the branches are defined as structural degrees of freedom. Each node,
or the intersection of the branches, is subject to external loads and internal forces from

the connecting branches. The optimal shape is the deformed shape of the cable structure

38

that achieves an equilibrium state with the prescribed cable forces. Equilibrium of the
structure is obtained by shaping the structure to achieve force balance at every node. The
problem can be solved by a set of linear equations in terms of nodal coordinates and
force-densities at the branches. However, based on special discretization and linearization
techniques, such as modified Newton-Raphson iterations [Haug and Powell, 1972;
Suzuki and Hangai 1991], the force density method is mainly intended for the form
finding of cable structures, for which equilibrium can be enforced by linear equilibrium
equations of force-density parameters.

The dynamic relaxation method [Barnes, 1994; Lewis and Lewis, 1996] finds optimal
shapes Of tension membranes by the principle of minimum potential energy of surface
tension, which governs the formation of soap films. This method solves a static problem
by non-linear equilibrium equations simulating a pseudo—dynamic process in time.
Initially, the membrane surface is discretized by finite elements. The mass of the
structure is equivalently distributed to the nodes of the elements. Initially, the nodal
velocities and displacements of the structure are set to zero. The residual forces, which
are calculated as a difference of external and internal loads, excite an oscillation of the
structure. The nodes of the structure start to vibrate with increasing velocities and
accumulating displacements. The kinetic energy of the structure is monitored during this
process. At the peak of the kinetic energy, the oscillation is interrupted. The nodal
velocities are reset to zero and the residual forces are recomputed for the current
deflected structure. The iteration process is restarted from the current deformed geometry
and continued until the residual forces are reduced to a minimum. The dynamic

relaxation method is particularly suitable for structural analyses involving large

39

displacements. However, this technique has no reliable means of controlling the in-plane
distortion of finite element meshes [Bonet and Mahaney, 2001]. Therefore, the method
can produce unreliable results or, in some cases, result in unstableperformance
procedures.

The updated reference strategy (URS) method [Bletzinger, 1997; Bonet and
Mahaney, 2001] is another form-finding approach to solve minimal surfaces. This
method is consistently derived from continuum mechanics of elastic bodies with respect
to large deflections and small strains. The fundamental concept of the method is to
supplement the stiffness matrix in terms of Cauchy stress tensors in the actual deformed
configuration with an additional stiffness matrix. The additional stiffness matrix is
derived in terms of the second Piola—Kirchhoff stress tensor in a reference configuration.
According to the URS, the reference configuration is chosen as the previous deformed
shape of the last iteration. The additional stiffness matrix is known as a geometrical
stiffness matrix, which measures the amount of shear distortion of the deformed
structures between two continuous steps. The reference configuration is iteratively
adapted and the optimization process is terminated when the differences of shear
distortion in subsequent reference configurations are within an acceptable tolerance. The
structural shape leads to the minimum surface area of a membrane. This method can be
applied to cable net structures and membrane structures. If applied to one-dimensional
elastic bodies such as cable structures, it will reduce to the force density method.
However, update of the reference configuration requires nodal movements normal to the
structural surface. This enforcement reduces the nodal degrees of freedom to a 1D

direction, which is in conflict with the generation of a free-form surface in 3D space.

40

Also, the additional stiffness matrix will be singular for the special cases in which nodal
displacements are tangential to the surface. This main deficiency exists for all the
procedures previously presented for minimal surface. From a mathematical point of view,
this deficiency is inherited for all numerical solutions of minimal surfaces. Furthermore
the stress filed of a deformed structure is prescribed and not related to structural
deformations.

Other computational approaches that involve non-finite element analyses have been
developed for the form finding of tension membrane structures [Brew and Lewis, 2003].
However, all of these methods are limited to special applications due to their inherent
simulation of the soap film analogy. The soap film analogy, which can physically only be
implemented in isotropic films, leads to uniform membrane stresses by minimizing the
surface area of the membrane. The computational methods inherent the limitations of the
physical approach that they simulate. These computational methods solve an Optimal
shape by achieving either a minimum area of a stable surface or uniform membrane

stresses, which can be only implemented in tension membrane structures.

3.1.2 Form finding of shell structures

Shell structures are another type of shape resistant structures that use materials
efficiently. Their excellent structural performance for high load resistance over large
spans results from their double-arching behavior that allows them to carry loads by
membrane resultants rather than bending. Hanging chains are the simplest structures to
carry external loadings by tangent tension resultants obtained through catenary shapes.

Several experimental methods have been developed for the form finding of shell

41

structures on the basis of the mechanical principle of bending-free hanging chains, or
catenaries.

The hanging method (Figure 3—3) is an experimental approach that directly employs
the mechanical principle of hanging chains. The models obtained by the hanging method
are free of bending and only subject to tension. Therefore, the inverse of the hanging
shapes can be considered as the optimal shapes for masonry arches and vaults in pure
compression. Applying this principle to three-dimensional structures, Heinz Isler [1991,
1994] successfully implemented optimal shapes of shell structures defined by physical
hanging experiments. In the experimental manipulation, shell structures are simulated by
similar-dimension membrane materials such as textile cloths that are not able to resist
bending and shear. The membranes are hanged according to the boundary conditions of
the real structure. The Optimal shape of a shell is Obtained by taking the inverse shape of
the deflected structure subject to a variety of forces, preferably self—weight. In addition to
the hanging method, other physical handling techniques are also used in form-finding
experimental methods for shell structures, such as the pneumatic method and the flow

method [Isler, 1991].

42

 

(c) ’ “(11)
Figure 3—3 Hanging method and shell structures and [Isler, 1991, 1994]

The inherent nature of the hanging method generates an optimal shape that is the
deflected shape of a structure made of “membrane materials” or materials with negligible
bending stiffness. This process can be numerically simulated by finite element analyses
based on geometrical nonlinear theory that allows finite deformations. The structure is
discretized into a finite element model using membrane elements. A geometric nonlinear
analysis starts with a plane structural geometry. Two approaches are used to achieve
optimal shapes. The first approach is to fasten the membrane on the final supports and to
load the structure until it reaches its equilibrium state. The second one is to define a
membrane with oversize dimensions and to allow the boundary points to slip into their
final positions by large rigid body motions. In the numerical simulation of hanging
models, in-plane shear stiffness needs to be a fraction of the modulus of elasticity in order

to avoid a singular structural stiffness matrix and obtain a fold-free deformed shape.

43

Additionally, the critical stresses and deflections measured on controlling points are used
to control the deformation states in the form finding process.

A variety of concrete shell structures designed by the hanging method or other
physical experiments have shown good structural performance and also have good
aesthetical qualities. However, shape Optimum designs by the hanging method are limited
to specific applications. Membrane materials used in the experiments are usually
isotropic and thus do not relate even to the actual materials used for actually constructing
the shells. In addition, because the optimal shapes achieved by hanging methods result
from mechanical deformations, design factors that alter the optimal results can not be
considered simultaneously in the Optimization processes. For example, the anisotropic
property of laminated FRP composites can be tailored by fiber orientations and stacking
sequences. However, these factors, in which material alterations may change the optimal
shape from a positive to a negative curved surface, can not be taken into account by
hanging methods. Stability effects such as buckling also can not be considered by
hanging models since the hanging model is only subjected to tension [Isler, 1991]. A
sophisticated analysis or a test on a stiff and slender real model is needed to clarify
instability issues. The cutting pattern for initial flat membranes, which are critical to
avoid developing wrinkles and folds in the deformed shapes, has to be determined by trial
and error in a variety of possible solutions. Therefore, a more general approach, which
can take into account more design freedom and constraints that are not implemented in
physical experiments and their corresponding numerical simulation has become of
growing research interest for the form finding of lightweight structures (Figure 3—4)

[Ramm et al., 1991; 1993].

44

 

 

 

Initial Shape
Figure 3—4 Structural shape optimization [Ramm, 1992]

3.1.3 Structural shape optimization

The experimental form—finding methods previously presented rely on the selection of
experimental materials and are limited to specific structural forms. The numerical
methods used inherit the limitations of the physical approaches that they simulate. A
better approach to form finding is structural shape optimization, which solves the form
finding problem by defining an optimization problem through an objective function that
measures the “goodness” or “efficiency” of a structural shape design. The Objective
function is defined by design variables, which are parameters that change during the
design process. Design variables can be continuous or discrete depending on whether
they can take values from a continuum or are limited to a set of discrete values. The
process of shape optimization is then to improve the goodness Of the design.
Optimization processes are generally performed with some limits that constrain the
choice of a design. Such limits are called constraints and usually include performance
criteria. Structural shape Optimization is thus a general and versatile technique for

structural form finding.

45

Shape optimization problems are typically defined in standard mathematical terms as:
Objective: Min f(x)
Subject to: gi(x) _<0 i = 1 to m

hj(x)=0 j=1top

where x is the design variables vector, XL and X U, are the lower and upper bound vectors
of the design variables, f(x) is the objective function, and g,~(x) and [1,-(x) are the equality
and inequality constraints, respectively.

As far as structural shapes is concerned, the design variables are obviously chosen as
the coordinates of points that govern the geometry of a structure. The selection of the
Objective function depends on the application. For example, the objective for the shape
optimization of shell structures consists in the form finding of membrane structures such
as to obtain a minimal area of stable surfaces while maximizing the structural stiffness.
The constraints are used to implement design limits such as maximum stresses and
displacements for the achievement of an optimum objective.

Structural shape optimization combines highly specialized techniques from different
disciplines including computer aided geometrical design (CAGD) [Bohm et al., 1984],
structural finite element analyses [Cook et al., 2001], structural sensitivity analyses
[Arora and Haug, 1979; Haftka and Adelman, 1989], and mathematical algorithms

[Arora, 1989].

46

The classical approach employed in structural shape optimization is based on the
following steps:

1. A finite element model is generated by computer aided geometric design;

2. Structural responses, such as displacements and stresses, are evaluated through
structural analyses;

3. Sensitivities are computed to formulate the Optimization problem in mathematical
terms;

4. A mathematical algorithm is chosen to solve the optimization problem and a new
design of the structure is generated until an optimum is achieved.

The above—mentioned procedure is structural depicted by an optimization procedure

of a shell structure in Figure 3—5.

 

 

     
 

Initialize Generate a finite
the design element model .
—> —-—>

 

 

 

 

 

Geometric key points

 

 

Computer-Aided Geometric Design

L

Update the design by
mathematical algorithms

 

Geometric key points

 

 

 

Sensitivity Analyses
Figure 3—5 Structural shape optimization process

47

3.1.3.1 Computer aided geometric design

In a shape Optimization problem, the design variables are geometric parameters
defining a structural shape. The generality Of a structural shape requires as many
geometric design variables as possible. However, considering the efficiency of
computational approaches, the number of design variables should be restricted as far as
possible. CAGD techniques [Bohm et al., 1984] can assure the generality of a structural
shape defined by fewer design variables. The coordinates of the key nodes governing the
structural shape are thus typically chosen as the design variables. The shape of an entire
structure can then be generated by appropriate shape functions in terms of the coordinates
of the key nodes. A finite element mesh is then defined discretize the resulting shape for

conducting the finite element analyses.

3.1.3.2 Finite element analysis

Finite element analyses used in structural optimization provide the evaluation of
structural responses required by the objective and constraint functions, and also by the
design sensitivity analyses. The finite element method provides a numerical procedure for
analyzing continua and structures that are too complicated to be solved satisfactorily by
analytical methods. In most of practical optimization problems, structures are modeled by

the finite element method to calculate their response to applied loads.

3.1.3.3 Design sensitivity analysis
The design sensitivity analysis (DSA) is a fundamental requirement for structural
shape optimization. Design sensitivity analysis supplies gradient information on objective

and constraint funcions with respect to the design variables for formulating the

48

Optimization problem. Considering that finite element analyses are used, two major
techniques, numerical and analytical, can be used to calculate structural sensitivity
derivatives for finite element modeled structures [Adelman and Haftka, 1986].

Numerical techniques using finite difference methods are straightforward methods in
which derivatives are calculated by a finite difference approximation. The accuracy of the
derivatives obtained by this technique is depended on the selection of the perturbation
step size for the finite difference. Analytical techniques, such as direct differentiation
methods and adjoint variable methods, are reliable methods for most kinds of
applications at the cost of a higher programming and computational effort. They can be
more efficient than numerical techniques if only parts of the structure are affected by
shape variants. However, especially for the sensitivity analyses in general shape
Optimization, analytical techniques will not be able to calculate structural sensitivity
derivatives when structural responses implicitly depend on the design variables. In
general, the techniques employed for design sensitivity analyses rely on the algorithm to

be chosen to solve the optimization problem.

3.1.3.4 Mathematical algorithm

Given a constrained optimization problem, design updates and the convergence to an
optimum design rely on a mathematical algorithm. Most of mathematical algorithms are
based on the general iteration:

x(k+1) = x(k) + akdu)

(k+1)

where x00 is the design variables vector Of the kth iteration, x is the design estimate of

the (k+1)th iteration, on, is a step size, and da‘) is a search direction.

49

Depending on techniques used to update the designs, mathematical algorithms for
solving constrained Optimization problems are broadly classified as transformation
methods and direct methods [Belegundu and Arora, 1985]. Transformation methods were
developed to solve a constrained optimization problem by transforming it into an
unconstrained problem whose solution converges to a solution of the original problem.
The basic concept of transformation methods is to construct a transformed function by
adding a penalty for constraint violations to the objective function. Transformation
methods include the penalty (exterior) and barrier (interior) function methods as well as
multiplier (augmented Lagragian) methods.

Due to the limitation inherited from the transformed functions, direct methods, also
known as primal methods, were developed to directly solve the original constrained
Optimization problem. Many methods such as the method of feasible directions, the
gradient projection method and the constrained steepest descent method have been
developed and successfully used for engineering design problems. Compared to other
methods, the constrained steepest descent (CSD) method is robust and effective for
solving nonlinear constrained optimization problems. Direct methods are generally based
on the following four basic steps:

1. Linearize the objective and constraint functions about the current desing estimate;

2. Define a subproblem formulated by the linearized objective and constraint

functions to determine a searching direction;

9°

Solve the subproblem to give a search direction in the design space;

9“

Calculate a step size to minimize a descent fimction in the search direction.

50

The objective and constraint functions are linearized by the design sensitivity
analyses previously discussed. The searching direction is determined by a subproblem
based on the linearized Objective and constraint functions of the current design. In
numerical practices, the subproblems using nonlinear programming have a better
algorithm performance and optimum convergence rate than those using linear
programming.

The descent function plays a very important role in CSD methods for constrained
problems. The basic concept is to compute a step size along the search direction such that
the descent function is reduced. Therefore, the descent function is used to monitor the
progress of the algorithm towards an optimum point. For constrained problems, the
descent function is generally constructed by adding a penalty for constraint violations to
the current value of the objective function. Several descent functions have been proposed
and applied successfully in practical optimization designs [Han, 1971; Powell, 1978;
Schittkowski, 1981].

Based on a descent function, a step size is determined by searching for a minimum Of
the descent function along the desirable direction in the design space. The step size
determination is also known as a one-dimensional search, or line search, problem.
Several numerical techniques have been employed for the step size calculation, such as
equal interval search, golden section search and polynomial interpolation. However, such
exact line search techniques can be inefficient for constrained optimization methods.
Therefore, an inexact line search is preferred to determine a step size for practical

implementation of constrained problems.

51

All of these classical mathematical algorithms make use of derivatives of the
objective function and constraint functions with respect to the design variables to
construct an approximate model of the initial optimization problem. These algorithms
rely on the gradients derived by linearizing the original functions to perform the
optimization processes. Such gradient-based algorithms can not be generally applied on
the optimization of FRP laminates, which typically involve discrete design variables in
the Objective function and constraints. The rationale behind this limitation brings about
the concept and technique presented here for material optimization of laminated FRP

composites as discussed in the next section.

3.2 Material optimization of laminated FRP composites

Laminated FRP composites consist of thin layers of fiber-reinforced material fully
bonded together. Each individual layer, or lamina, contains an arrangement of
unidirectional fibers embedded within a thin layer of polymer matrix material. The
material properties of the entire laminate can be adjusted to meet the requirements of a
design by varying the percentage of layers in the laminate, their individual orientations
and the stacking sequence of the layers. Therefore, finding an efficient structure design
can be achieved by tailoring the material prOperties. FRP laminates are thus the most
commonly used type of composite material in the design Of hi gh-performance structures.

From the above statements, and as presented in Chapter 3, it follows that the design
space for the material optimization of laminated FRP composites is typically a set of
discrete parameters that define the material elastic properties. Such optimization

problems in which design variables are restricted to take only discrete/integer value are

52

referred to as integer programming (IP) problems. Classical gradient-based algorithms
for structural optimizations requiring the gradient information of objective and constraint
functions can not generally be employed to solve IP optimization problems. In recent
years, there has been considerable interest in exploring new optimization methods that do
not rely on derivatives of the Objective function and constraints to solve this type of
problems. For example, the branch-and—bound algorithm [Lawler and wood, 1966;
Tomlin, 1970] was advanced to solve linear integer programming problems. Due to the
robustness and reliability to achieve global optimums in nonlinear integer programming
problems, genetic algorithms [Holland, 1975] and simulated annealing methods
[Kirkpatrick et al., 1983] have emerged as strong contenders against classical gradient-
based algorithms. These methods belong to a genetic category of stochastic search
techniques. Such evolution-based algorithms rely on the selection of fittest “individuals”
in a “population” to obtain an optimum design by implementing the operations based on
the principles of natural genetics. Unlike classical gradient-based algorithms that move
from one point to another in the design space, such algorithms work with a population of
designs. By keeping the solutions that have the potential of being the optimum in the
population, such algorithms generally result in a global or near-global Optimum rather
than converging to a local optimum.

The evolution-based algorithms, such as genetic algorithms have been successfully
applied to the material design of FRP composites [Giirdal et al., 2000; Liu el al., 2000].
However, most of the research on FRP optimum design for use in high-performance
structures has focused only on material optimization, that is, altering material properties

by arranging the stacking sequence and fiber orientations. Such optimization is isolated

53

from the structural shape, which also influences the mechanical behavior of a structure.
This approach may thus lead to designs that make limited use of the dominant in—plane
strength and stiffness of FRP laminates. On the other hand, the structural shape
optimization procedures discussed in Section 1.4.1 were primarily focused on Optimum
designs of structures made from isotropic materials. Optimum designs obtained by
traditional shape optimization methods maintain the same material properties during the
form finding process. These designs are often far from optimal because other competitive
material properties cannot be explored. Therefore, efficient structural designs can be best
obtained through shape efficient structures constructed with tailored laminated FRP
composites.

Finding an efficient structure design that meets all the requirements for a specific
application can then be achieved not only by shaping the geometric configuration of the
structure but also by tailoring the material properties. It follows that not only continuous
parameters (defining the shape Of the structure) but also discrete parameters of lay-up and
angles (defining the elastic properties of the material) should be taken into account in an
optimization procedure. For the optimal design of FRP composite structures, the design
space is thus a mixed set of discrete and continuous design variables.

Gradient-based algorithms, which depend upon sensitivity analyses requiring
continuity and derivative existence, are thus not applicable. Although genetic algorithms
have been used with some success in problems where a mixed set of integer, discrete, and
continuous variables are presented, they generally require more iteration steps to arrive at
an optimum design. The random search in genetic algorithms requires more finite

element analyses for evaluation of the objective and constraint functions, which decreases

54

computational efficiency and may result in a slower convergence compared to gradient-
based algorithms. Therefore, an efficient optimization process needs to be explored for

the combined shape and laminate optimization Of FRP composite structures.

3.3 Integrated shape and FRP laminate optimization

From the above discussions, shape optimization problems and FRP laminate
optimization problems are generally highly nonlinear. However, gradient-based and
genetic-based algorithms, which are robust and reliable to solve shape optimization
problems and laminate Optimization problems respectively, can not be directly applied to
solve optimum designs of shaped-optimized laminated FRP structures. Therefore, an
integrated approach that utilizes the computational advantages offered by both algorithms
is proposed to accomplish the combined task of shape and material optimization. Due to
difficulties to solve the entire problem through a single optimization procedure, the
optimization task is thus to be implemented in a two-level uncoupled approach (Figure 3—
6):

Level 1: Shape and laminate-property optimization: to achieve an optimal shape
with optimal stiffness properties;

Level 2: Laminate design optimization: to design an FRP laminate that has the

same stiffness properties as those obtained in the first step.

55

<_§@
Level-1 3

Shape and Laminate-Property
Optimization
(Gradient based algorithm)

Level-2 fl

Laminate Design
Optimization
(Genetic algorithm)

3

End

 

 

 

 

Figure 3—6 Two-level approach of shape and laminate Optimization

The alternative formulation of the laminated composite material stiffness derived in
terms of lamina invariants and lamination parameters (see Section 2.1) allows
implementation of the proposed optimization scheme. First, the lamination parameters
(Vim, 0)) that determine the sectional stiffness of laminates are combined with geometric
parameters that define the shape of the structure. The optimum set of continuous design
variables (sectional stiffness and geometry) is found so as to maximize the stiffness of the
structure. As a second and final step, a set of stacking sequences of variable fiber
orientations is then searched to achieve the desired lamination parameters, as obtained
from the first step.

The two-level optimization approach combines shape optimization of the structural
geometry with material optimization for the FRP laminate design. In addition to the
computational efficiency provided by gradient-based algorithms, diverse laminate
designs can be achieved by implementing genetic-based algorithms in the proposed

integrated approach. The proposed approach takes advantage of decoupling the two

56

different Optimization processes and corresponding optimization Objectives with respect
to their own set of design variables.

According to the two levels employed in the integrated approach, each optimization
level involves an independent problem formulation by identifying and defining design
variables, objective functions and constraints, and requires the implementation of the
corresponding algorithm. The following two chapters describe in detail the procedure for

each optimization level.

57

References

Adelman, HM. and Haftka, RT. (1986). Sensitivity analysis of discrete structural
systems. AIAA Journal: 24(5), 823-832.

Arora, J .S. (1989). Introduction to Optimum Design. New York: McGraw-Hill.

Arora, J.S. and Haug, E.J. (1979). Methods of design sensitivity analysis in structural
optimization. AIAA Journal: 17, 970-974.

Barnes, M. (1994). Form and stress engineering of tension structures. Structural
Engjreering Review: 6, 175-202.

Bletzinger, K.U. (1997) Form finding of tensile structures by the updated reference
strategy. IASS International Colloquium -— Towards the new millennium, University of
Nottingham

Bohm, W., Farin, G. and Kahmann, J. (1984). A survey of curve and surface methods in
CAGD. Computer Aided Geometric Design: 1(1), 1-60.

Bonet, J. and Mahaney, J. (2001). Form finding of membrane structures by the updated
reference method with minimum mesh distortion. International Journal of Solids and
Structures: 38, 5469-5480.

Booker, L. (1987). Improving search in genetic algorithms. In Genetic alggrithms and
simulated annealing, Davis, L. (ed.). Los Altos, CA: Morgan Kaufmann Inc., 61-73.

 

Brew, J .S. and Lewis, W.J. (2003). Computational form finding of tension membrane
structures — Non-finite element approaches (part 1-3). International Journal for numerical
methods in engineering: 56(3), 651-698.

Cook, R.D., Malkus, D.S. and Plesha, ME. (2001). Concepts and applications of finite
element analysis. NewYorK: Wiley.

 

Giirdal, Z., Haftka, RT. and Hajela, P. (1999). Design and optimization of laminated
composite materials. New York: Wiley.

 

Haftka, RT. and Adelman, HM. (1989). Recent developments in structural sensitivity
analysis. Structural Optimization: 1, 137-151.

Han, SP. (1979). A globally convergent method for nonlinear programming. Journal of
ggtimization theory and applications: 22, 297-309.

Haug, E. and Powell, OH. (1972). Finite element analysis of nonlinear membrane
structures, Proc 1971 IASS Pacific Symposium Part II on Tension and Space Structures,
Tokyo and Kyoto: 165-175.

58

Hildebrandt, S. and Tromba, A. (1983). Mathematics and optimal form. New York:
Scientific American Library.

 

lsler, H. (1991). Generating shell shapes by physical experiments, IASS Bulletin: 34, 53-
63.

 

Isler, H. (1994). Concrete shells derived from experimental shapes. Structural
Engineering international: 3, 142-147.

Lewis, W.J. and Lewis TS. (1996). Application of formian and dynamic relaxation to the
form finding of minimal surfaces, Journal of the International Association for the Shell
and Spatial Structures: 37(122) 165-186.

Linkwitz, K. (1999). Formfinding by the “direct approach” and pertinent strategies for
the conceptual design of prestressed and hanging structures. International Journal of
Space Structures: 14 (2), 73-87.

 

Liu, B., Haftka, R.T., Akgiin, MA. and Todoroki, A. (2000). Permutation genetic
algorithm for stacking sequence design of composite laminates. Computer Methods in
Applied Mechanics and Engineering: 186, 357-372.

 

Powell, M.J.D. (1978). Algorithms for nonlinear functions that use Lagrange functions.
Mathematical Programming: 14, 224-248

Ramm, E., (1992). Shape finding methods of shells. IASS Bulletin: 33, 89-99.

Ramm, E., Bletzinger, K. U., and Kimrnich, S., (1991). Strategies in shape optimization
of free form shells. Nonlinear comgrtational mechanics — state of the art, P. Wriggers, W.
Wagner (eds): 163-192.

Ramm, E., Bletzinger, K.U., and Reitinger, R., (1993). Shape optimization of shell
structures. IASS Bulletin: 34, 103-121.

Schek, H]. (1974). The force density method for form finding and computation of
general networks. Computer Methods in Applied Mechanics and Engineering: 3, 115-
134.

Suzuki, T. and Hangai, Y. (1991). Shape analysis of minimal surface by the finite
element method, Int. IASS Symposium on Spatial Structures at the Turn of the
Millenium, Copenhagen: 103-110.

59

4 Shape and Laminate-Property Optimization of FRP Structures

This optimization level seeks the most efficient structural geometry and laminate
sectional stiffness subject to constraints such as structural performance, geometrical
dimensions, and laminate failure criteria. This chapter firstly presents the formulation of
optimization problem by identifying the design variables and the objective function used
for this level of optimization. The algorithm used. to solve the optimization problem is

given first, followed by its implementation.

4.1 Formulation of the shape and laminate-property optimization problem

Solving optimization problems involves transcribing a verbal description of an
optimization problem into a well-defined mathematical statement. Therefore, this
optimization level starts by identifying design variables, an objective function and the

corresponding constraints, which are presented in the following three sections.

4.1.1 Design variables

The design variables are independent parameters chosen to describe the design of a
system in mathematical terms. Proper formulation of an optimization problem always
starts by identifying the design variables for the system. In this optimization level, the
design variables are the continuous geometric parameters defining the structural shape
and the continuous lamination parameters defining the section stiffness properties of the
FRP laminate.

The generality of an optimal shape requires as many geometric design variables as

possible. However, in order to maximize the efficiency of the computational algorithm,

60

only key dimensional points that define the geometric shape of a structure are chosen as
the shape (geometric) design variables. The entire structure is then constructed by CAGD
techniques by interpolating the key points.

The laminate coupling effects, which are introduced by random stacking sequences of
angle plies (see Section 2.2), are generally avoided for effective use of engineering
materials. Therefore, symmetric and balanced laminates, which have the least-
pronounced coupling characteristics, are considered in this research. The in-plane and
out-of—plane coupling defined by the [B] matrix, and the in-plane shear-extension
coupling, defined by the A16 and A26 terms, vanish due to the use of symmetric and
balanced laminates. In order to reduce the effects Of out-of-plane bend-twist coupling,
defined by the D16 and D26 terms, 48-layer symmetric and balanced laminates are used, in
which the lamination parameters V20 and V40 are considered to be equal to zero. Thus,
only the lamination parameters VIA, V3A, V10 and V30 (see Section 2.2) [Giirdal et al.,
1999], which fully define the in-plane and out-of-plane section stiffness properties, are

chosen as the material-property design variables.

4.1.2 Objective function

The objective of the shape optimization procedure to determine the layout of a shape
resistant structure is to maximize structural stiffness as measured by the structural
stiffness matrix. However, the two-order tensor of the structural stiffness matrix can not
be used to evaluate the objective function, which requires a scalar value. Therefore, the
structural strain energy, a scalar that quantifies the structural stiffness, is calculated

instead in the objective function.

61

 

Considering a finite-element-discretized structure without initial strains and stresses,

the total strain energy, U, can be stated as
1 l
U = Ejlef [Elieldv =§{D}T [K 1D} (4.1)
V

where [E] is the material stiffness, {D} is the global degree-of-freedom (d.o.f.) vector of
the entire structure, and [K] is the structural stiffness matrix.

The global d.o.f. vector {D} can be solved by the equilibrium equation as:

{D}=[Kl“{R}. (4.2)
where {R} is the consistent global nodal load vector.

By substituting Eq. (4.2) into (4.1), the strain energy can be evaluated as:

u = $11164 {at min-11111}

=§1rt11<1111<111<111n (43>

=11er [Kl“{R}.

2

According to Eq. (4.3), for a given statically equivalent load {R}, the total strain
energy U decreases as the structural stiffness [K] increases. Furthermore, the strain
energy contribution by different strain types can be determined by recalling Kirchhoff’s
theory, which is assumed in classical lamination theory. Thus, the strain in a plate under a
state of plane stress is defined by the addition of the mid-plane strain {60 } and the linear
variation due to bending curvatures {K} as:

{e} = {5" }+ z{K}. (4.4)

Substituting Eq. (4.4) into Eq. (4.1) one obtains:

62

U:

l (1601+ 2181 19111808 zinildv
j({s~ i [QKSO }+ 2z{8" }T [gym any (grain.

(4.5)

It is thus shown in Eq. (4.5) that, within a unit volume, the strain energy contributed
by the bending curvature and axial-bending coupling are quadratically and linearly
proportional to the section thickness, respectively. Therefore, minimization of the
structural strain energy implies minimizing bending and axial-bending effects, thus

maximizing in-plane response and consequently increasing the system stiffness.

4.1.3 Constraints

As previously stated, the objective of the shape and laminate-property optimization is
to maximize structural stiffness. However, maximizing the stiffness without restricting
material use is meaningless since stiffness can be improved if the amount of material is
increased. Thus, the mass of FRP composite used is restricted. Instead of using the
equality constraint that the volume of FRP laminates remains the constant during the
Optimization process, an inequality constraint is chosen to limit the volume of FRP
laminates within a certain level. This can be implemented by constraining the thickness
of the laminate and the structural geometry.

Among the several function constraints that need to be satisfied for a laminated FRP
structure, service and failure constraints are the two most important design criteria for
engineering applications. These design criteria can be investigated by the structural
responses of maximum structural deflection and maximum section stresses as evaluated

by finite element analyses.

63

4.2 Algorithm of shape and laminate-property optimization

In the shape and laminate—property Optimization of laminated FRP structures, the
objective function and constraints are nonlinear functions with respect to the design
variables. The constrained steepest descent (CSD) method is a simple, yet effective,
gradient-based algorithm used for solving this type of nonlinear problems [Arora, 1989].

The CSD method is based on four basic steps as presented in the following.

4.2.1 Linearization of Objective and constrain functions

For shape and material-property optimization, the objective and constraint functions
are implicit functions of the design variables. The solution strategy employed by the CSD
method involves solving an approximate problem obtained by linearizing the original
objective and constraint functions. The derivatives of objective and constraints functions
used to construct the approximate problem are derived by Taylor series expansions. The
approximate optimization problem using the first-order Taylor series expansion is

formulated as:

Objective:
n
f :ZCldl’ or f :CTd (4'63)
i=1
Subject to:
n
Za,d,sb,, j=ltom; or ATdsb (4.6b)
i=1
n
2,1,6,- =ej, j=1to p; or NTd=e (4.6c)
1'21

64

where e,- is the negative of the j"’ equality constraint function value at the current design

x , e, = —h,(x(k’); bj is the negative of the j"' inequality constraint function value at the

current design it“), b, =—gj(xa"); Cj is the derivative Of the objective function with

respect to the i"’ design variable, c]. = af(xa‘))/ 8x1; n, is the derivative of the j”' equality

constraint with respect to the i"' design variable, n, 2 8h]. (x‘k’ )/ axl; a, is the derivative of

the j”' inequality constraint with respect to the i"' design variable, a, 2 8g 1. (x‘k) )/ 8x, .

The derivatives obtained by a linear Taylor series expansion can be numerically
evaluated by finite difference approaches. In this optimization level, the forward
difference technique is used to evaluate the numerical differentiation of a multi-variable

function fix) with respect to variables x,- , which is defined as:

8f

_ = limit f(x1,...,x, +Axi""’xit)—f(x1"“’xi""’xn)
ax; ax,—>O Axl-

 

(4.7)

where Ax,- is a small perturbation in the variables xi.

The accuracy of function gradients depend on the selection of the perturbation Ax.
Value selection of the perturbation is presented in detail by Gill, Murray and Wright
[1981]. In general, a perturbation of 1% of the current value, which works fairly well for

most of optimum designs, was chosen for this optimization problem.

4.2.2 Definition and determination of a searching direction
In order to solve the constrained optimization problem, the CSD method requires a
desirable searching direction towards the optimum design. A searching direction is

determined by solving an subproblem. Due to its high efficiency and quadratic

65

convergence rate, a subproblem using a quadratic programming (QP) formulation is
chosen to determine the searching direction.

The QP subproblem employs a Hessian matrix constructed by a quadratic objective
function and linear constrains, which are defined in the following:

Minimize:

f=ch+%-dTHd

Subject to:

NTd = e

ATd s b (4.8)
where H is the Hessian matrix, which represents the second-order derivatives of the
Objective function with respect to the design variables.

For large-scale optimization problems it is difficult and inefficient to calculate the
second-order derivatives matrix. However, due to the usefulness of incorporating the
Hessian matrix into the optimization algorithm, the Quasi-Newton method was developed
to approximate the Hessian matrix by making use of information obtained from previous
iterations. In the Quasi-Newton method, an approximate Hessian matrix is updated by
using design changes and the gradient vector of the previous iteration. Several updating
procedures have been developed [Gill et al., 1981]. The modified BFGS (Broyden-
Fletcher-Goldfarb-Shanno) method [Powell, 1978] is implemented to numerically

approximate the Hessian matrix in this optimization level.

66

4.2.3 Definition of the constrained steepest descent function

For shape and laminate-property optimization the objective is to minimize the strain
energy of a structure. Achievement of this optimum requires a reduction of the objective
function value in each iteration. A descent function is used to ensure the reduction in
optimization processes towards the minimum. For constrained Optimization problems, the
descent function needs to take into account violations of the constraints by adding a
penalty for constraint violations to the current value of the objective function.

Several descent functions [Han, 1977; Powell, 1978] have been proposed for
constrained optimization problems. Pshenichny’s descent function [Pshenichny and
Danilin, 1978, Arora, 1989, 1997] is chosen as the descent function of the CSD method
in the shape and laminate-property optimization due to its wide use in engineering
optimization problems.

Pshenichny’s descent function (I) at point xfl‘) is defined as

<D(x(k))= f(x("))+RV(x(")), or o, = f, +RV, (4.9)
where fk is the k"' value of objective function, R is a penalty parameter initially specified
by the user, V1, is the k” value of the maximum among all the constraint violations and
zero.

The penalty parameter R is a positive number and may be changed during the iterative
process. It must be greater than or equal to the sum of all the Lagrange multipliers of the

QP subproblem at the point xlk).

67

 

P
R22
i=1

"1
vfk)| + Zuf") (4.10)
i=1

) h

where v,“ and ufk’ are the values of Lagrange multiplier at the k”' iteration for the i’

equality and inequality constraint, respectively.
The maximum constraint violation V is nonnegative and is determined at the point x“)

of the kth iteration by

V = max{O,Jh,J,|h,|,...,|hp|;g,,g,,...,gmJ (4.11)

4.2.4 Determination of the step size

Given a descent function, a step size will be searched to minimize the descent
function along with the given direction in the design space. The step size determination is
often called a one-dimensional search (or line search) problem. Several zero-order
methods [Cooper and Steinberg, 1970] are available for one-dimensional search
problems, such as the Equal Interval Search and the Gloden Section Search. Other
methods using polynomial interpolation can be an efficient technique to solve one-
dimensional searches. However, sometimes it is difficult to approximate a function of the
step size at the current point x“) by, e. g., a second-order polynomial interpolation. Thus,
in the numerical implementations of this optimization level, the step size determination is
solved by an inexact line search using a bisection procedure as described in the
following.

At the km iteration, an acceptable step size at is determined as t,- with the smallest

integer j to satisfy the descent function:

68

CI) SCI), —t,,8, (4.12)

k+l,j

(k+lJ)

where (I) is the descent function evaluated at the trial design point x that is

k+l,j

x("+"j):x(k’+tjd(k), tj is the j” trial step determined by using the bisection

t, = (1/2)’, j = 0,1,2,..., and [it is the constant determined by ,6, = yJ|d(k)lJ2,}/E [0,1],

4.3 Implementation of shape and laminate-property optimization

The scheme of shape and laminate-property optimization (Figure 4—1) was
implemented in a fully automated custom program written in Matlab [MathWorks, Inc.,
2000]. An initial reference structure, generated by a custom program for computer-aided-
geometric-design (CAGD), is chosen as the starting design for the Optimization

procedure. The objective strain energy function and constraint values are evaluated by the

 

 

Start Initialize Design Structural Analysis1
- Initial geometric shape - Evaluation of strain energy
- Initial lamination parameters - Evaluation of stresses

 

- Evaluation of displacement
- Evaluation of buckling(if needed

Geometric Model Generation . _ _ . ]
( Computer aided modeling) l sensmVltIi’ AnalySIS

 

 

 

 

 

 

 

 

 

 

Mathematical Programming
Analytical Model Generation - Formulate optimum design
- Automatic mesh generation - Determine searching direction2
- Sectional stiffness generator - Determine step size

 

 

 

 

 
  

 

 

 

N
———[Update DesignJ< O

Converged?
Yes

   

1.ABAQUS
2. MatLab

To Level-2
Figure 4—1 Shape and laminate-property Optimization

69

finite element software ABAQUS [HKS, 2001]. A numerical searching strategy
implemented with Matlab functions determines a search direction and a step size towards
an optimal solution. The new geometry and section stiffness properties of the structure
are updated and provided as input for the next ABAQUS analysis. The iterative process is
continued until no further modifications are required and the optimality conditions are

satisfied.

70

References

Arora, J. S. (1997). Guide to structural optimization. ASCE Manuals and Reports on
Engineering Practice No. 90, Reston. VA: American Society of Civil Engineers.

Belegundu, AD. and Arora, J .S., (1985). A study of mathematical programming methods
for structural optimization. International Journal for Numerical Methods in Engineering:
21, 1583-1599.

Cooper, L. and Steinberg, D. (1970). Introduction to methods of optimization.
Philadelphia: W. B. Saunders.

Gill, P. E., Murray, W. and Wright, M. H. (1981). Practical Optimization. NY: Academic
Press.

Lawler, EL. and Wood, DE. (1966). Branch-and-bound methods — A Survey.
Operations research: 14, 699-719.

MathWorks, Inc. (2001). Matlab — Using Matlab Version 6.

Miki, M. (1983). A graphical method for designing fibrous laminated composites with
required in-plane stiffness. Trans. JSCM: 9, 2, 51-55.

Tomlin, J.A. (1070). Branch-and—bound methods for integer and non-convex
programming. In integer and nonlinear programmingLAbadie, J. (ed), 437-450. New
York: Elsevier Publishing CO.

Tsai, S. W. and Pagano, N. J. (1968). Invariant properties of composite material.
Composite Matreials Workshop: 233-253.

71

5 FRP Laminates Design Optimization

This optimization level is to design laminates based on the optimal section stiffness,
defined by the optimal lamination parameters, which was Obtained in the first
optimization level. This chapter starts with the formulation of the laminate optimum
design problem by identifying the design variables and objective function. The algorithm
chosen to solve the optimum design is then presented, followed by a description of its

implementation.

5.1 Formulation of laminate design optimization

Similar to the shape—and-material optimization, the process of formulating this
optimization level, as described in Section 4.1, involves identifying the design variables,
the Objective function and the corresponding constraints, which are presented in the

following three sections.

5.1.1 Design variables

The goal of the second optimization level is to find stacking sequences of several
orthotropic layers with certain fiber orientations that will lead to the lamination
parameters obtained in the first optimization level. The fiber orientations for each of a
predefined number of plies, which determine the lamination parameters for the section
stiffness properties of an FRP laminate, are defined as discrete design variables. Because
balanced laminates are used (see Figure 5—1), negative and positive fiber orientations will
occur in the laminate designs. Therefore, the definition domain of fiber orientations is

from -90° to 90°.

72

 

 

 

 

 

 

 

Balanced Lamina
[01 / -0,]

 

 

Balanced Lamina
[02/ —02]

 

X
Balanced. and Symmetric Laminate Material coordinate system: axis-1,2,3
[01/ ‘91 / 02 / 432]S Structural coordinate system: axis-x,y,z

Figure 5—1 Typical symmetric and balanced laminate

As shown in Section 2.3.2, the lamination parameters of V20 and V40, which
determine the out-plane bend-twist coupling terms of D16 and D26, will vanish when the
numberof plies increases. However, a large number of plies will make the computation
inefficient. According to Eq. (2.31), a 48-layer balanced and syrmnetric laminate has
lamination parameters V20 and V4,) smaller than 0.0625, which leads to the bend twist
terms of D16 and D26 (see Table 2—1) much smaller than D11, D12, D22, and D26 terms. Its
out-plane bend-twist coupling effect can thus be neglected compared to the other out-
plane stiffness terms. Therefore, 48-layer balanced and symmetric laminates are
employed in this research and their optimum designs are to be determined in this

Optimization level.

73

5.1.2 Objective function

The objective function is to minimize the discrepancy between the lamination
parameters (i.e. VIA, V3A, etc.) evaluated with the design variables from this (2nd)
optimization level and the optimal lamination parameters obtained from the first

optimization level. The objective function can be expressed as

Minimize: A = \/2(V,. —17,. )2 .

 

where V, are the lamination parameters evaluated by the design variables of fiber

orientations, and V are the lamination parameters of the optimal laminates derived in

l

the first optimization level.

5.1.3 Constraints
The laminate design optimization problem as formulated here is an unconstrained
problem. Therefore, no constraints are required in this laminate design optimization

problem.

5.2 Algorithm for laminate design optimization

For the laminate design Optimization problem, the objective function is not
continuous and derivable with respect to the discrete design variables representing the
fiber orientations. Gradient-based algorithms, which depend upon the requirements of
continuity and derivative existence, are thus not applicable. Conversely, genetic

algorithms, which do not rely on derivatives, are suitable to solve this type of problem.

74

A genetic algorithm is a guided random search technique that works on a population
of designs. Each individual in the population represents a design obtained by genetic
coding techniques. A fitness value is selected for each individual in order to determine its
probability of being chosen as a future “parent design” to create a “children design.”
Therefore, the application of a genetic algorithm to an Optimization problem is based on

the steps [Gen and Cheng, 2000] presented in the following sections.

5.2.1 Design representation by genetic coding

In order to reduce the bending-twisting coupling terms D16 and D26, balanced
laminates pairs of positive (+6) and negative (-6) off-axis layers adjacent to each other
are assumed for this research. Therefore, a pair of balanced off-axis layers can be
represented by the fiber orientation of the top layer in the pair. Considering that the
laminates are also symmetric, the 48—layers balanced and symmetric laminates designed
in this optimization level can thus be interpreted by 12 integer design variables, which
represent 12 different fiber orientations of it9with a certain stacking sequence.

The optimum achieved by genetic algorithms depends on genetic Operators that
mimic the principle of natural genetics. The application of genetic operators requires
representing the possible combinations Of design variables in terms of bit strings that
simulate genetic chromosomes. Binary numbers, which are predominantly used for the
purpose of computer implementation [Back, et. al., 1997], are therefore employed as the
technique Of genetic coding to represent 12 design variables of different fiber orientations
as bit strings. The process of genetic coding for the balanced and symmetric laminate

designs, with the reduction mentioned above, is illustrated in Figure 5—2.

75

1:6,°/.--/:45°/---/-1-6,°/~-/-_- 61,0],
Coding rule
[010/---/450/---/0k°/~-/012°]
Genetic coding

l (-7... (4505.. (9.. (... (9120

Genetic chromosome

1
101101 0

Binary string
Figure 5—2 Design representation by genetic coding

 

 

 

 

5.2.2 Definition of a population size

A population in genetic algorithms includes all the designs in a generation. The size
of the population is determined by the initial generation and is fixed for all. future
generations. Currently there are no general rules or solutions for determining the optimal
population size for a specific problem. Typically, a population size is chosen at least as
large as the chromosome string length in terms of binary bits. The current laminate
design problem has 12 integer design variables with values ranging between —90 and 90.
Each design variable requires at least 8 bits for binary representation. Therefore, a

population size of 100 is chosen for the total 96-bit length of a genetic chromosome.

5.2.3 Definition of fitness values and a selection scheme

The forward movement of genetic algorithms depends on a selection process by
giving higher chances for better individual designs to pass their characteristics to future
generations. Thus, the selection process requires a value to evaluate the fitness of an
individual design in a generation. For unconstrained optimization problems, the value of

the objective function can be used as the fitness value of a design.

76

Once the fitness values of the designs in a generation are determined, the designs with
higher fitness values will be selected to create child designs by a selection scheme.
Several selection schemes [Goldberg 1989; Davis, 1991] have been developed for
genetic procedure, such as roulette wheel, tournament selection and ranking selection.
Due to its small spread about the desired distribution and its freedom from bias,
stochastic universal sampling [Baker, 1987] is used in this work as the selection scheme

in the laminate design optimization.

5.2.4 Implementation of genetic operators

Crossover (Figure 5—3) is one of two major genetic operators used in genetic
algorithms. The task of the crossover operator [Goldberg, 1989; Davis, 1991] is to
explore new designs alternatives by combining the existing family traits. The crossover
operation generates children designs by exchanging genetic segments of the selected
parent designs. The crossover operation is typically to be executed with a probability
number. A random number, uniformly distributed between 0 and 1, will determine the
implementation of the crossover operator. The crossover operation is performed when the

random number is less than the probability of occurrence.

 

 

 

 

 

 

 

 

 

 

 

Chromosome 1 Chromosome 1
(91 i D (m (91 i O (91 i 191le Crossover [91 i” ("7 (91 ((91 L (9120
. _ X . ,2 x _ _ --. .. _ -
(9(1) ((910 1.9.1:: 19120 (*0.- (- l (6. ( (91190
Chromosome 2 Chromosome 2

Figure 5—3 Crossover operator

In addition to the crossover, mutation (Figure 5—4) is needed to prevent the premature

of the Optimization procedure to achieve an optimum design by exploring new traits with

77

 

the genes that are not represented in the initial population. This function is implemented
by the mutation operator [Goldberg 1989; Davis, 1991], which introduces occasional
random alteration of a genetic bit string. Based on the small rates of occurrence of this
phenomenon in biological systems and on numerical experiments, mutation is generally

performed with a small probability of occurrence.

 

Mutation A A

(... 1Q 101101 mjojlo ...... O _, (m 1}) 101101 mm“) ..... J)

' v v _ v
Binary chromosome Binary chromosome

Figure 5—4 Mutation operator

 

 

5.3 Implementation of laminate design optimization

The laminate design optimization by a genetic algorithm begins with the random
generation of a population of design alternatives. It is processed by means of genetic
operators (see Figure 5—5), crossover and mutation create a new population. Among
several types of crossover operators, two-point crossover is considered to be superior
[Booker, 1987] and is thus employed in this work for the laminate design optimization.
The new generation, which combines the most desirable characteristics of the Old
population with less discrepancy to the optimal lamination parameters, will replace the
old population. The process is repeated until a stopping criterion is satisfied. The
stopping criterion is implemented in the form of a maximum number of generations
without improvement in the best design. The scheme of the genetic algorithm described
above was implemented in the program GALOPPS by Goodman [1996]. As part of this
work, custom programs were specifically developed to be used together with GALOPPS.

The additional programs allow implementation of the genetic representation of design

78

variables and evaluation of the objective function to accomplish the laminate design
optimization.

From Level-l

V
Initialize Laminates
- Fiber orientations
- Stacking sequence

11

[ Generate Population J I Implement Genetic Operator

3 - Selection crossover
J Evaluate Objective Function J - Mutation

 

Represent Design Variables
By Genetic Chromosome

 

 

 

 

 

 

 

 

 

 

 

Figure 5—5 Laminate design optimization

79

References

Baker, J.E. (1987). Reducing bias and inefficiency in the selection algorithm, E
Proceedings of the Second International Conference on Genetic Algorithms Lawrence
Erlbaum Associates Hillsdale: 14-21.

Back, T., Hammel, U., and Schwefel, HP. (1997). Evolutionary computation: Comments
on the history and current state. IEEE Transactions on evolutionary Computation: 1, 1.

Davis, L. (1991). Handbook of genetic algorithms. New York: Van Nostrand Reinhold.

Gen, M. and Cheng, R., (2000). Genetic algorithms and engineering optimization. New
York: Wiley.

Goldberg, D. (1989). Genetic Algorithms in Search. Optimization, and Machine
Learning. Massachusetts: Addison-Wesley, Reading.

Goodman, ED. (1996). An Introduction to GALOPPS — the Genetic Algorithm
Optimized for Portability and Parallelism System, Release 3.2. Technical Report 96-07-
01, Intelligent Systems Laboratory and Case Center for Computer-Aided Engineeringand
Manufacturing. East Lansing, MI: Michigan State University.

 

8O

6 Shape and Laminate Optimization of FRP Shells

As discussed in Section 3.1.2, shells are shape-resistant structures in which strength
and stiffness is Obtained by shaping the material to achieve primarily in-plane or
membrane compression resultants due to the applied loads. The efficiency of FRP
structures can then be enhanced by their use in this type of geometry. Thus, the shape and
laminate optimization of FRP shell structures is presented here to evaluate the proposed
integrated approach for shape and material optimization. To demonstrate the capability
and effectiveness of the integrated optimization approach, the results and performance of
a shape-and-material optimized shell are compared to two other shape-only optimized
shells. Shape-only optimized shells, Optimized only with the shape optimization

technique used in the integrated approach utilize laminates with fiber orientations of i45°

and 0°/90. The optimum designs of three FRP shells are compared to similar optimal
studies obtained for isotropic materials [Ramm et al., 2000, 1992; Isler, 1991, 1994]. The
optimum results for all three FRP shells are also compared with respect to the Objective
of strain energy and other types of structural performance in terms of displacements and

section resultants.

6.1 Shape and laminate optimized FRP shell

The shell selected for studies of integrated optimization, referred later as Shell 1, is of
arbitrary dimensions with has a projected area of 4.88 m by 4.88 m and a thickness of
25.4 mm. The shell is taken to be simply supported at the comers and is subjected to a
uniform gravity load of 6.89 kN/m2 (Figure 6-1). The laminates used in the shell studies

are based on atypical medium-performance carbon/epoxy material system for use in civil

81

structures. Thus the lamina orthotropic properties are taken as: E 1 1 = 155.025 GPa, E22 =
10.335 GPa, 012 = 6.89 GPa, G13 2 6.89 GPa, 023 = 0.689 GPa, and V12 2 0.3.

P = 6.9 kN/m2

 

key points

 

Figure 6—1 Geometry, loading, and control points for shell structures

Taking advantage of symmetry, the geometric shape of the shell can be controlled by
14 design variables that define the vertical coordinates of key points as shown in Figure
6—1. The key points are distributed non-uniformly over 1/8 of the shell area in order to
sample geometric details about the free edges of the shell. The shape of the entire shell is
then constructed by spline interpolation of the key points. Four-node doubly curved
general-purpose shell elements, which use reduced integration with hourglass control
[HKS, 2001], are used to discretize the entire shell into a 64x64 finite element mesh.

The first-level Optimization histories of the objective, partial geometric design
variables and FRP laminate-property design variables are presented in Figure 6—2, Figure

6—3 and Figure 6—4 respectively. The optimization starts with a flat shell with a section

82

stiffness corresponding to a [900/00/00/900] laminate, which lamination parameters are
{V121, V321, V10, V3D}={0, 1, —0.75, 1}. In the course of the Optimization process, the strain
energy of the shell, starting at 112.98 kN—m converges stably to 0.0496 kN-m after 66
iterations with the optimal shape shown and the Optimal material property defined by
lamination parameters {V1A, V32], V10, V3D}={-0.12654, -0.08372, -0.00498, -0.79898}.
Stiffness variation is small (Figure 6—2) during the last steps with minor changes in shape
and material prOperties (Figure 6—3 and Figure 6—4). Figure 6—2 also shows a collection
of the shell shapes as they evolved during the optimization process. It can be seen that the
central point of the shell rises in order to reduce bending moments while a negative
curvature appears at the edges of the shell. This featuring shape is clearly illustrated in
the elevation view of the shell (see Figure 6—3). These features of the optimal shape have
also been observed by other researchers on the shape Optimization of concrete shells (see
Figure 3—3 (b) and Figure 3—4) [Ramm et al., 2000, 1992; lsler, 1991, 1994; Ram and

Mehlhom, 1991].

83

Strain energy (N-m)

F’
a:

1111111111111141111

Dimension (m)

 

106,

 

 

 

l | | l ' l l i l l 1 | l
1O1IIYITIIIIT1j—[TIIIIIIII rrrrrrr IIIITT"-I[ITITIIIIIIII[ITIIIIYIYTIIIIYI

 

0 5 10 15 20 25 30 35 4O 45 50 55 60 65 70

Iteration no.
Figure 6—2 History of objective function — FRP Shell 1

 

L4

‘L2

9
O)
l
j

.0
h
11111111
I

53

N
__L_J——-|—J_‘LL
J.

 

 

 

0 10 20 30 4o 50 60 7o
Iteration no.

Figure 6—3 Histories of partial geometric design variables — FRP Shell 1

84

 

 

_+_ _._Y;_1 +_ ___V19_+Vao ‘-

1:3:1' __

IA

9’
0|
1 I 1

 

O
1 1 4

 

Lamination parameter
5
in

 

 

 

 

-1 . . r . ' I . 1 1 =4

I I T Y 1 | T I I .‘ I 7 l V I l T l 1 I V I l I 1

0 10 20 3O 40 50 60 I I I [70
Iteration no.

Figure 6—4 Histories of laminate-property design variables — FRP Shell 1
After the optimal laminate properties represented by lamination parameters were
achieved, the optimal design of the laminate by fiber orientations and stacking sequence
was carried out to achieve the optimal properties. The genetic algorithm optimization
process included 10,000 generations and each of them had a population of 50 individuals.
The second-level optimization history is presented in Figure 6—5. Genetic-based
optimization algorithms can lead to multiple designs that satisfy the Optimization criteria.
Thus, multiple optimal designs of laminates were found with the optimal laminate
properties derived in the first optimization. The optimal laminate designs, with

corresponding lamination parameters within an error of 1% compared to the optimal

lamination parameters obtained in the first-level optimization, are listed in Table 6—1. It

85

should be further noted that all optimal laminates have several common laminas with the

same fiber orientations arranged in the same stacking sequence.

 

 

 

 

 

 

 

 

 

 

35% _
e2
30%
25%
>1
:2: 20% 1 Optimal Lamination Parameters
0. VM = -0.127, V3,, 2 -0.084
2 v”, = -0005, v,,, = -0799
0 15% n
m _
E :1 Optimal Laminate Design
10% [:46°/i43°/i45°/1~45°/:48°/:39°/
51 i66°/i2°/:t88°/0°2/ i89°li90°2]S
,
°o :L
5/ : Error=0.157%
o%‘....i.e..:!—--F-%I...............
0 1 000 2000 3000 4000 5000 6000

Iteration no.
Figure 6—5 Second-level optimization process — FRP Shell 1

Table 6—1 Optimal laminate designs of FRP shells

 

 

Laminate designs Error
[: 46/: 43/: 46/: 45/: 39/: 47/: 66/ : 2/ : 88/02/902 /: 87], 0.610%
[: 46 /: 43/: 46/: 45 /: 46/: 39/: 66/ : 2/ : 88/02/ : 89/: 87] S 0.445%
[: 46/: 43/: 46/: 45/: 46/: 39/: 66/ : 2/ : 88/02/902/112 87] 5 0.427%
[: 46/: 43/: 46/: 45/: 46/: 39/: 66/ : 2/ : 88/02/ : 89/9021s 0.300%
[: 46/: 43/: 46/: 45/: 46/: 39/: 66/ : 2/ : 88/02/902/902JS 0.287%
[: 46/: 43/: 45 /: 45/: 48/: 39/: 66/ : 2/ : 88/02/ : 89/902], Q 157%

 

6.2 Shape-only optimized laminated FRP shells
The optimum result above (Shell 1) is compared to the shape optimization of two
laminated FRP shells (Shell 2 and Shell 3) with constant material properties. These two

shells have the same geometry and loading of the shell analyzed in the previous example.

86

The geometric design variables used here were the same as before, see Figure 6—1.
Guided by the experiments of hanging models [Ramm et al., 2000], two laminate designs
were chosen. Shell 2 used a [450/-450/-450/450J65 laminate with the lamination
parameters {V]A, V3A, V10, V30}={0, -1, 0, —1} while Shell 3 used a [00/900/90°/0°]
laminate with the lamination parameters {VI/b V321, V11), V3D}={0, l, 0.75, 1}. Therefore,
the laminate-property design variables of the lamination parameters determining the
stiffness matrix of each shell are constant throughout the optimization procedure. The
Optimization process histories for Shell 2 and Shell 3 are presented in Figure 6—6 and
Figure 6—7, and Figure 6—8 and Figure 6—9, respectively. It should be noted that the
obtained optimal shapes for the two shells using different laminate designs are different

(see Table 6—2), although the initial shapes were the same.

 

 

 

 

 

105;
1043*
A
ei
Z ,
V
>.
9
0103“
c
0
.E
m
h
H
U)
1023-
1014:111.
0 5 10 15 20 25 30 35 40

Iteration no.
Figure 6—6 History of objective function — FRP Shell 2: [45°/-45°/-45°/45°]63

87

 

Dimension (m)
.o s:
0') 00

P
&
IJAII

 

 

 

5 1o 15 20 25 30 35
Iteration no.

 

40

Figure 6—7 Histories of geometric design variables — FRP Shell 2: [45°/-45°/-45°/45°los

 

105,

1043

103?

Strain energy (N-m)

_|
O
M
i 1
I

 

 

 

101 1 I 1 1 1 1 1 1 1
fifTrlyffiililllfl’IIII

 

 

I'lll|l'll[ltll[ 111111111 [1

5 10 15 20 25 30 35 40 45
Iteration no.
Figure 6—8 History of objective function — FRP Shell 3: [0°/90°/90°/0°]

88

50

 

(m)

P
Q
ll_11111

 

Dimgnsion
a»

C
h
1
T
3

N
3
u
3
b
3
or

 

p
N
1111
l
\
\
/

 

 

0- 4m
0 5 10 15 20 25 30 35 4o 45 50
Iteration no.

Figure 6—9 Histories of geometric design variables — FRP Shell 3: [00/900/900/00]

6.3 Comparison of three optimal shells

The structural response of all three Optimal shells regarding strain energies,
displacements and section resultants is summarized in Table 6—2 and Table 6—3,
respectively.

Table 6—2 Summary Of strain energies and displacements of optimal shells

Strain Energy Max Deflection
(N-m) (mm)

Shell 2 M 72.76 1.23
Shell 3 r...‘ 92.98 1.59

89

Optimal Shape

According to Table 6—2, the optimal shell obtained through the shape and laminate
Optimization (Shell 1) has the maximum structural stiffness, that is, the smallest vertical
displacement and the smallest strain energy among all three shells. Shell 2, constructed

by the laminate of [45°/-45°/-45°/45°]6s, was found to be better than Shell 3, constructed

by the laminate of [0°/90°/90°/0°] with respect to the structural stiffness.

The inherent nature of shape Optimization for membrane structures and thin shells is
to minimize bending stresses thus ensuring in-plane only response. Achievement of this
Objective was evaluated by comparing the section resultants between initial and optimal
structures for all three shells. As shown in Table 6—3, the bending moments are reduced
significantly after the optimization process. Meanwhile, membrane compressions are
developed in all three Optimal shells compared to the zero in-plane response of the initial
structures. Contours of the force (SF1) and moment (SMl) section resultants (see Figure
6—10 for element axis orientation) are also compared for the initial and Optimal structures
in Figure 6—11. Figure 6—11 makes evident that bending behavior is minimized while the
in-plane behavior becomes dominant. Additionally the results in Figure 6—11 show that

the section resultants seek to be uniformly distributed throughout the entire structure.

90

Local coordinate systems

 

Global coordinate system

Figure 6—10 Global and local coordinate systems

Table 6—3 Summary of structural responses of initial and optimal shells

 

 

 

 

 

 

Shell 1 Shell 2 Shell 3

Initial Optimum Initial Optimum Initial Optimum

SFl Max 20 235.195 :0 192.815 :0 412.599

(kN/m) Min :0 -744.289 =0 -755147 =0 -820.294
SF2 Max 2:0 241.325 :0 192.990 zO 415.226

(kN/m) Min 2:0 -945.860 :0 -953.566 20 -1012.41
SMl Max 0287 0.329 108.288 0.395 123.407 0.355

(kN-m/m) Min -24.957 -0.368 -24.932 -0.283 -64.117 -0.171
SM2 Max -0.158 0.298 108.288 0.387 114.982 0.195

(kNvm/m) Min -54.077 -0.478 -24.932 0358 -41.885 -0.067

 

SF 1 , SF2: Direct membrane force per unit width in local l-, 2-axis of elements
SMl, SM2: Bending moment force per unit width about local 2-, 1-axis of elements

91

Membrane Force (SFl) Bending Moment (SMl)
Initial Optimal Initial Optimal

23x10'12kN 235.0 kN 220.0 kN-m/m 20.09 kN-m/m

 

s—3><10'12 kN $0.09 kN-m/m

    

$20.0 kN-m/m .

Shell 1

Shell 2

 

 

Shell 3

 

i
i
i
i
i
i
i
i
l
i
i
l
i

SMl: Bending moment force per unit width about local 2-axis of elements

Figure 6—11 Structural responses of initial and optimal shells

Images in this thesis/dissertation are presented in color.

92

It should be noted that stress leveling within the shell is not an optimization criterion
to achieve maximum structural stiffness. For example, the shape-and-material optimal
shell (Shell 1) attained the minimum strain energy and had the lowest membrane force
and highest bending moment specific values (i.e., value at a given location) than the other
two shape-only optimal shells. However, this does not conflict with the objective to
minimize the strain energy of the entire structure by reducing bending moments. The in-
plane structural performance can be best evaluated by comparing the stress-state in
individual elements, which are summarized in Table 6—3. Thus, as shown in Figure 6—12,
90% of Shell 1 is subject to membrane forces ranging between —45 and 0 kN/m,
compared to 79% and 84% for Shell 2 and Shell 3, respectively. The efficiency of Shell 1
is more obviously shown in Figure 6—13 regarding the bending moment SMl. Thus,
although Shell 3 has the smallest range of bending moments among all three optimal
shells, only 32% of Shell 3 is subject to bending moments of small magnitude, i.e.,
between —30 and 30 kN-m/m, compared to 74% and 47% for Shell 1 and Shell2,
respectively.

According to the analyses described above, the stiffness of bending structures is in
fact improved by achieving the objective of minimizing the structure’s strain energy. In
addition, the structural performance of laminated FRP shells can be further enhanced by
the integration of shape and material optimization. Therefore, the shape and material
objective function of minimizing strain energy leads to structural geometries and material

properties that are dominated by in-plane response with minimum bending demands.

93

 

2500 22 60%
. [SSW :

JESheII .

flamefl

2000 5:

.A

500 -- \

Ele_r_nent Count
0
O
O
.21

 

 

30 45 60 Max

Min -60 -45 -30 -15 0 15
Force Range (kN/m)
Figure 6—12 Membrane force distribution of optimal shells

 

45%

IShell1

ll
EShe ‘ 40%
@Shell

i 35%

 

5 30%
4 25%
5 20%
, 15%
10%

f 5%

 

 

 

 

. ~ ,, 10%
Min -150 -120 -90 —60 -30 0 30 60 90 120 150 Max
Moment Range (N-m/m)

Figure 6—13 Bending moment distribution Of Optimal shells

94

 

Structure Fraction

Structure Fraction

6.4 Buckling analyses of optimized laminated FRP shells

As it is well known, shape-resistant structures carry their design loads primarily by
axial or membrane action, rather than by bending action. When subjected to compression
forces as in the case of shells, these stiff thin structures are sensitive to buckling.
Considering that a structural stability criterion was not implemented in the presented
integrated Optimization approach, the structural buckling performance was evaluated and
compared among the Optimized shells.

Eigenvalue buckling analyses are generally used to provide useful estimates of the
critical buckling loads and collapse mode shapes of stiff structures. To evaluate the
buckling response of the optimized shells, the unloaded optimal shells are chosen as the
base state and buckling loads are calculated relative to the base state of the structure. The
optimization load-generating load case of 6.9 kN/m2 applied in the gravity direction is
used gain to introduce the linear perturbation. The subspace iteration extraction method is
used to explore the eigenvalues and eigenvectors of the first five buckling modes. The
results for Shell 1, Shell 2, and Shell 3 are summarized in Table 6—4, Table 6—5, and
Table 6—6, respectively.

As judged by the values for the first buckling eigenvalue given in Table 6—4 through
Table 6—6, structural buckling capability is improved by the shape and FRP laminate

optimization along with maximizing structural stiffness. The general-optimized shell

(Shell 1) has the highest buckling capacity (4.: = 8.05 ), compared to Shell 2 (4,2 -- 4.29)

and Shell 3 (if = 3.49 ). On the other hand, although the buckling mode shapes vary

significantly in character, it can be seen that the first three eigenvalues are closely spaced.

The series of closely spaced eigenvalues indicates that the optimized shells are

95

imperfection sensitive. In addition, the smaller spacing of eigenvalues for Shell 1 further
implies Shell 1 is more buckling-sensitive compared to Shell 2 and Shell3. Therefore, it
can be concluded that a stiffer structure will increase buckling resistance but suffer from

higher buckling sensitiveness.

Table 6—4 Buckling analyses of Shell 1
. Buckling shapes
Mode Eigenvalue Side view Axis 1-3 Side view Axis 2-3

Base

State _ fir ti

3 8.88 W m
4 13.38 m n
5 13.49 n m

 

Table 6—5 Buckling analyses of Shell 2
Buckling shapes

MOde Elgenvalue Side view Axis 13 Side view Axis 2—3

 

Base

_ r 3 ‘
State L1 L2

1 4.29
2 4.37
3 4.82
4 6.70
5 9.46

 

 

96

Table 6—6 Buckling analyses of Shell 3

Buckling Shapes (eigenvectors)

MOde Eigenvalue Side view Axis 13 Side view Axis 2-3

     

 

 

if}: 1L 1:2
1 3.49 h
2 3.53 n
3 3.64 r fl
4 4.63 n
5 11.55 m

 

6.5 Summary

In this chapter, the Optimization procedure of the integrated approach was evaluated
and validated by optimizing FRP shells. It was illustrated in the comparison of a general-
Optirnized Shell and shape—only optimized FRP Shells that the structural performance of a
Shape resistant structure can be improved by using FRP laminates with Optimized
material design. Stability evaluation of the optimized shells showed that buckling
capacity should be an essential constraint of the integrated approach for shape resistant

structures subject to compression.

97

References

Hibbitt, Karlsson & Sorensen, Inc. (2001). ABAQUS standard user’s manual Version
_6_.2.

lsler, H. (1991). Generating Shell shapes by physical experiments, IASS Bulletin: 34, 53-
63.

lsler, H. (1994). Concrete shells derived from experimental Shapes. Structural
Eggineeringintemational: 3, 142-147.

Ramm, E., (1992). Shape finding methods of Shells. IASS Bulletin: 33, 89-99.

Ramm, E., Kemmler, R. and Schwarz, S., (2000). Form finding and optimization of shell
structures, In: Proc IASS-ICAM 2000. Fourth International Colloquium on Computation
of Shell and Spatial Structures: Chania—Crete, Greece.

Ramm, E. and Mehlhom G., (1991). On Shape finding methods and ultimate load analysis
of reinforced concrete Shells. Engineering Structures: 13, 178-198.

98

7 FRP Composite Membrane-Based Bridge Systems

Laminated FRP composites have been widely applied in strengthening and
rehabilitating aging bridge systems and proved to be efficient. However, the application
of laminated FRP composites for an entire structure is still limited. Although technical
advances in manufacturing FRP composites have significantly reduced the manufacturing
cost, their high material cost and the lack of proven design philosophies compared to
conventional materials still confines the use of laminated FRP composites in primary
structural components of bridge systems. However, the use of FRP composites in
conjunction with conventional structural materials has Shown that technical efficiency
can be achieved within competitive economical constraints [Burguefio, 1999; Seible et al.,
1999]. Furthermore, it was proved that the use of the high in-plane strength and Stiffness
of FRP composites can be maximized by material—adapted concepts of employing Shape
resistant structures of membrane/shells. Therefore, FRP composites membrane-based
bridge systems are proposed as analytical studies for the application of laminated FRP
composites in bridge systems.

The use of Optimized laminated FRP composites in bridge systems is conceptualized
in two bridge designs that can effectively employ the in-plane stiffness of FRP laminates.
Both bridge types consist of a laminated FRP membrane together with a conventional
reinforced concrete deck, in which the laminated FRP membrane provides the in-plane
strength and the concrete deck provides the live load transfer and resists the majority Of
the compression forces. The developed optimization approach will be implemented and

evaluated through analytical studies on these two types of FRP membrane-based bridges.

99

7.1 FRP composite membrane beam (CMB) bridges

Based on the concept of girder—slab bridges, an FRP Composite Membrane Beam
(CMB) bridge (Figure 7—1) is considered as a case study for the proposed optimization
approach. Because of the high Stiffness provided by double-curvature surfaces, the
system uses a hyperbolic paraboloid laminated membrane working compositely with a
conventional reinforced concrete deck as schematically shown in Figure 7—1. The
proposed system behavior is such that the FRP laminate is to carry the in-plane tensile
and Shear forces while the concrete deck carries the compressive force. The developed
Shape and FRP laminate Optimization approach was thus used to obtain the optimal Shape

and laminate design of the membrane element for the CMB bridge system.

   
 
 

Abutment

Membrane

Abutment
Figure 7—1 Composite membrane beam (CMB) bridge

7.1.1 Bridge system description

The proposed CMB Bridge for optimization studies is a 10.06 m long simple-span
bridge girder with a 2.44 m wide compression flange (Figure 7—2). The flange is a 203.2
mm thick concrete deck with a density of 2402.8 kg/m3. In order to reduce difficulties of
finite element modeling, the abutments were not modeled in the finite element analyses.
Rather, the abutment stiffness was taken into account in the computational model by

constraining the plane section motion that it imposes on the rotations and displacements

100

at both ends of the bridge (Figure 7—2). The laminates used in the CMB bridge systems
are assumed to be from medium-performance carbon/epoxy FRP lamina, of which the
lamina orthotropic properties are taken as: E / 1 = 88.253 GPa, E22 = 48.884 GPa, 012 =

4.557 GPa, 013 = 4.557 GPa, G23 = 0.456 GPa, and v12 = 0.513.

     

Truck Load

Figure 7—2 Computational model of the CMB bridge

Table 7—1 Load case specification for CMB bridge ommization

 

 

. . . . Design Criteria
lelt State Load Case Specrficatlon Stress Deflection
Optimization l.0><DL + l.0><LL 0.0501. -

Service 1.0xDL + l.0><(LL + 1.33xTL) 0.250'u L/800
Strength 1.25xDL + l.75x(LL + 1.33xTL) 0.506u -

According to the AASHTO Bridge Design Specification [AASHTO, 1998], the
simple-span CMB bridge is to be designed to load combinations that include dead load

(DL), live lane load (LL) and concentrated loads from a three-axle truck (TL). The design

101

lane load consists of a uniformly distributed pressure Of 3.06 kPa over each lane while the
design truck load consists of three concentrated loads spaced at 4.27 m (Figure 7—2). The
load case used in the integrated Optimization procedure takes into account the dead and
uniform live load with unit, or service load factors (Table 7—1). In addition to the
optimization load case, the service and strength load cases (Table 7—1), in which the
loads are combined with different load factors also need to be considered [AASHTO,
1998]. The influence of these two load cases on the Optimal CMB bridge design was
studied through post optimality analyses.

Two design criteria, namely stress and deflection are used for the CMB bridge
designs. The stress criterion chooses a percentage, or stress index, of ultimate material
failure for each limit state. The prediction of the ultimate material failure varies with
different failure stress criteria, which define different failure surfaces surrounding the
origin in three-dimensional space {01], 0'22, 0'12}. The stress index is used to measure the
proximity to the failure surface. The material failure determinations by different stress
criteria are all depended on the stress limits of the individual lamina measured along the
material directions. Following knowledge gained through experiments on carbon/epoxy
FRP laminates, 1% and 0.5% of the Young’s modulus of the corresponding direction
respectively are commonly used as the tensile and compressive stress limits, while 0.5%
of the shear modulus was chosen as the Shear limit. Therefore, the tension and
compression stress limits along the l-axis (fiber-longitudinal direction) and 2-axis (fiber-
transverse direction) were chosen to be: X2 = 0.883 GPa, XC = -O.442 GPa and Y2 = 0.489
GPa, YC = -0.244 GPa, respectively, and the shear stress limit in the 1-2 plane was taken

as S = 22.9 MPa.

102

In addition, the maximum structural deflection of U800 [AASHTO, 1998] was
chosen as another design criterion for the service limit state. Although the AASHTO
[1998] bridge design Specifications recommend the deflection limit Of U800 to apply for

live load only, this limit is used conservatively in this work together with the system dead

load.

7.1.2 Integrated Optimization of CMB bridges

Based on the above-mentioned geometry, properties, and loading of the CMB bridge
system, the integrated Optimization approach was applied to design CMB bridges by
sequentially employing the shape and laminate-property optimization and the laminate

design optimization procedures.

7.1.2.1 Shape and material-property optimization

The shape design variables for the CMB bridge are the width and depth of the FRP
membrane at the middle (W2, [12) and ends (w), h) of bridge component (Figure 7—3).
These variables are used to define the hyperbolic paraboloid shape of the membrane
through a CAGD algorithm. The laminate design variables are the lamination parameters
{V,A, V3A, V10, V30} controlling the in-plane tension stiffness and out-plane bending

stiffness.

103

   
 
     

Depth (h2), . * /
Thickness (t)

. ..» ” Width (w2)
Wldth (w l) ,,/‘
//

/

 

Figure 7—3 Geometric key points for CMB bridge shape optimization

It is well known that structural stiffness is improved when the amount of material
increases. Therefore, the mass of materials needs to be constrained during shape
optimization when the Objective is maximizing structural stiffness. An inequality
constraint, instead of using an equality constraint, is used to limit the mass of materials
used in the structure. Considering that the geometry of the concrete deck remains
constant during the optimization process, the material constraint is thus implemented by
setting a lower and upper bound for the geometric design variables that are controlling
the geometric dimensions and the thickness of laminated FRP composite (Table 7—2).
Maximum stresses evaluated at the top and bottom surface of the laminate and the
maximum deflection of the entire structure are two additional constraints taken into

account in the structural design criteria for the integrated optimization procedure.

104

Table 7—2 Dimensional constraints for geometric design variables
W1 W2 ’1] [12 1
Bound (mm) (mm) (mm) (mm) (mm)
Lower 609 609 609 305 6.35
Upper 2438 2438 1219 1219 19.05

 

 

 

The structural design achieved by an optimization process is normally obtained for a
single load case. Due to the random nature of the truck loading position the optimization
load case for the CMB bridge was taken as the combination of the dead load and live lane
load with unit (service) factors (Table 7—1). This assumption was found to be
conservative in the post optimality analyses (Section 7.2.4) as truck loading governs the
demands in Short-Span bridges.

In the finite element analyses, the strains on the FRP laminates and concrete are
expected to remain in the linear elastic range. Furthermore, the approximation of
formulating the structure stiffness matrix in the reference (original) configuration is
expected to have an error of 10'3 order compared to unity because of the small rotations
and displacements under the loads. Thus, material linear elasticity and geometric linearity
were used in the structural finite element analyses.

The stability of the proposed optimization approach was studied by searching for the
Optimal CMB bridge design from three different starting or initial designs. The initial
designs considered are listed in Table 7—3. The optimization histories of the first-level
shape and laminate-property optimization for the CMB bridges are shown in Figure 7-4

through Figure 7—12.

105

Table 7—3 Initial designs of CMB Bridges

 

Initial design Model A Model B Model C

 

{WI’WZ’hlyw2,t} {1625.6, 1016,

{1625.6, 1016, {1219.2, 1219.2,
(mm) 609.6, 609.6, 19.05} 609.6, 609.6, 19.05} 812.8, 508, 6.35}
{V1A,V3A,V,D, V30} {0,1,0.75,1} {O,-1,0,-1} {0,1,0.75,1}
Laminate Design [0°/90°]s [:45°]g [0°/90°]3

 

 

 

 

 

 

O
_l_.

10 15 20
Iteration no.

Figure 7—4 History of Objective function — CMB Model A

106

25

 

 

 

 

 

 

 

 

2500
4
i
i
1
2000 r
E
@1500 ~—
I:
.2
(D
51000
.5.
D
500 ~-
- [+w, +w2 +11, +11, +t(x20)J
0.1..li...14...lr.li .
O 5 10 15 20 25
Iteration no.
Figure 7—5 History of geometric design variables — CMB Model A
1.5

1__

parameters
s:
01

O
1 1 1 1 l
l

Lamination
5':
0|

 

J+V1A +V3A +V10 +VQ‘J

.1,5 . = . . , . . . . , . . , . , . . . . . 4
0 5 10 15 20 25
Iteration no.
Figure 7—6 History of laminate-property design variables — CMB Model A

 

 

 

 

107

(

Dimension

 

 

 

 

 

 

 

mm)

 

 

 

 

 

 

 

 

800 _
700 f—
AGOO :—
E _
' 1
E500 :—
> i
m .1
L-
0) 400 --
c
LIJ ,4
.E 300 —— / /.
a . we”
a: ‘ JV] .2}/
(D 200 3- K/ 12”” w/‘r‘ fl,
: / 201h
100 17 3 ¢ ¢ c A. e e e e e g g
I
o . : . . . . . . . . . . . =1
0 5 10 15 20
Iteration no.
Figure 7—7 History of objective function - CMB Model B
2500
2000 ~-
1500 ~—
500 r
_ P—w, +w2 +|11 +112 +t(x20) ]
0 l I I I i l I I r i ' ' ' I I l I I T I
0 5 10 15 20

Iteration no.
Figure 7—8 History of geometric design variables — CMB Model B

108

 

1.5

0.5 :

Lamination parameters
O
'o.

 

NV ' ‘

 

 

J+V1A +V3A +V10 +V30J

 

I I T I j

5 10
Iteration no.

20

Figure 7—9 History of laminate-property design variables — CMB Model B

 

01 05
O O
O O
1p11111_LJI

Strain energy (N-m)
8 8
O O

 

 

0

 

5 10
Iteration no.

109

Figure 7—10 History of objective function — CMB Model C

 

 

2500

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2000--
A
E
£1500“
:
.2
in
5
1000 *5
.E
D
500 ‘5
q / {Va—w, +W2 +111 —o—h2 +t(x20)J
o . . . . l . . . . : . . . . :
0 5 10 15
Iteration no.
Figure 7-11 History of geometric design variables — CMB Model C
1.5
1- vkegfigkvkvgrefike
. _ IIII - . . . . .2
0 _
905. Y r
E . V
10
h
8
c 0‘
.2
up
2
---0.5~
E
10
.1
-1—
4 J—o—VM —I—v3A +V10 +V£J
0 5 10 15

Iteration no.
Figure 7-12 History of laminate-property design variables — CMB Model C

110

 

AS summarized in Table 7—4, the shapes for all three CMB bridges (see

Table 7—3) converged to a Similar geometry with negligible differences. The
optimization processes starting from three different initial attempts also resulted in the
same in-plane lamination parameters (see Table 7—4). However, the out-plane lamination
parameters were different.

Table 7—4 Optimum results of shape and material-property optimization

 

 

Optimal design Model A Model B Model C
Stra‘“ Energy 83.85 83.85 83.85
(N-m)
{W1 W2 h) W2 1} { 1359.0, 2184.4, { 1365.2, 2184.4, { 1352.6, 2222.4,
(mm) ’ 984.3, 1219.2, 1022.4, 1219.2, 997.0, 1219.2,
19.05} 19.05} 19.05}

{V1A, V321, V11), V30} {081,031,421} {0.81,0.3l,-0.8,0.7} {O.81,0.31,0.75,1}

 

In fact, it was noted that the lamination parameters governing out-plane bending
stiffness of membranes remain unchanged beyond certain iteration steps. This behavior
does not significantly disturb the solution since the strain energy minimization process
leads to a system dominated by in-plane stress resultants, i.e., bending resultants become
negligible. Therefore, the out-plane lamination parameters, which determine the flexural
stiffness, have a smaller influence on the optimization process when the Structure

becomes dominated by in-plane behavior.

7.1.2.2 Laminate design optimization

Assuming that only the in—plane lamination parameters govern the sectional stiffness
of the FRP laminate, the optimal lay-up of the laminate can be solved by an alternate
method presented by G‘Lirdal et al. [1999]. This method allows finding the optimal lay-up

for a 4-layer laminate with only two distinct orientation angles for a given volume

111

fraction. By this method, the lay-up corresponding to the optimal section stiffness, which
is defined by the optimal lamination parameters of {VI/1, V3A }={0.81, 0.31} is found to be
[l7°/19°]g. This result is later compared to the solution found by the developed laminate
design optimization.

Symmetric and balanced laminates with certain number of distinct fiber orientations,
which are defined by the design variables, were determined by the laminate design
optimization process. In order to be able to compare the results with the previous optimal
laminate design, optimal 8-layer symmetric and balanced laminates that have two distinct
orientation angles were searched by the laminate design optimization algorithm.
Furthermore, as it was discussed in Section 2.3, the bend-twist (0,6 and 026 terms)
coupling effect is minimized by increasing the number of plies in a symmetric and
balanced laminate. Therefore, optimal 48-layer symmetric and balanced laminates (an
arbitrarily selected large number of plies), which have twelve distinct fiber orientations,
were also investigated.

In the second-level optimization, an optimal laminate is achieved by minimizing the
difference of the lamination parameters of the searched laminates from the Optimal
lamination parameters derived from the first-level optimization. Therefore, the optimum
criterion of the laminate design optimization is an allowable value of discrepancy. For the
current research, a maximum error of 1% was considered acceptable for the optimum

laminate design.

112

 

1 2%

10% 3— //

on
o\°
1

 

Optimal Lamination Parameters

VM = 0.81, v3, = 0.31

 

 

Discrepency
CD
.\°

Optimal Laminate Desigr

 

 

4% 4
[:17°/¢19°]S
2% Error = 0.22%

 

 

 

 

_.
O l 1 l l l
o IIIIIIIII | IIIIIIIIIIIIIIIIIII r IIIIIIIII irrrrrrrrtrrrrrrrrrr

Iteration no.

Figure 7—13 Laminate design optimization for 8-layer laminates

The optimal 8-layer laminate was found to be composed of [i17°/i'19°]g with an error
of 0.22% with respect to the optimal in-plane section properties from the first-level
optimization. The resulting laminate design is thus the same as that previously obtained
neglecting out-of—plane behavior through Gurdal et al.’s [1999] method. The optimization
history of the objective function (lamination parameter error minimization) is shown in
Figure 7—13. The optimal 48-layer symmetric and balanced laminate was found to be
[il7°/i21°/i'19°/i25°/i21°/i5°/ir22°/i12°/i25°/il4°/ i17°li12°]3 with an error of 0.73%.

Its optimization history is shown in Figure 7—14.

113

 

 

 

 

16%~r
14% y
>‘12% {—
g -’ . . .
8,10% _’ OJgtzmal Lamination. Parameters
e _; VIA=0.81, V3A=0.3l
U 8°/o :’
g . Optimal Laminate Design

6% 3- [:17°/:2 1°/:19°/-1_-25°/:21°/:S°/
- :220/112°/~_~25°/¢14°/ i17°/:12°]S

 

 

 

 

 

 

 

4% {~
Error = 0.73%
2% 7 _
: _
00/0 1 1 1 1 i 1 1 1 1 1L 1 1 1 1 i 1 1 1 V 11 T I i I i r 1 1 1 1L 1 r l r
0 500 1000 1500 2000 2500 3000 3500

Iteration no.

Figure 7—14 Laminate design optimization for 48-layer laminates

The lamination parameters and the section stiffness of the three optimal laminate are
compared and summarized in Table 7—5. As shown in the table, all three optimal
laminates have the same in-plane section properties and in-plane lamination parameters
while the major terms of the out—of-plane section stiffness {D11, D12, D22, D66} and the
out-plane lamination parameters {V113, V30} are only slightly different. However, the
bend-twist coupling terms (D16 and D26) of the out-plane stiffness are different for all
three laminates. Thus, as expected, balanced and symmetric laminates have less bend-
twist coupling effects than symmetric-only laminates. It also can be noted that the bend-
twist coupling terms can be further reduced by increasing the distinct groups of balanced

fiber orientations.

114

 

Table 7—5 Optimal laminates from laminate design optimization process

 

Lamination
O timal laminate parameters A matrix D matrix
p {VI/r, V34, V10. (103 kN) (kN-m)

V20, V30, V40}
{0,809,0308, ’362 6.01 0 1 [43.6 6.67 10.8)

 

 

 

4—layer [17°/l9°]s 0824,0566, 20.7 0 25.0 -4.41
0358,0933} L 6.98, 79!)
{0,809,0308, "36.2 5.96 0 “ ”43.6 6.67 4.07)
8-layer [i17°/i19°]3 0824,0215, 20.7 0 25.0 —l.58
0358,0352} _ 6.98, l 7.91 l

48-layer i21°/i5°/i22°/i12°/ 0.800, -0.037, 20.8 24.4 0'23
i25°li14°lil7°li12°1s 0.231.-0.056} 893-

 

 

[il7°/i21°/i19°/i-25°/ {0.803, 0.310, |736.1 5.96 0] ”42.1 7.80 —0.68“

 

7.1.3 System characteristics of optimal CMB bridges

The system characteristics of the optimized CMB bridge systems were investigated
by studying their structural behavior for the optimal load case and their buckling
sensitivity. Due to the slight geometric variations existing in the optimum designs, the
CMB bridge chosen for the system evaluation studies has a shape with average
dimensions of the obtained optimal bridges (see Table 7—4). The optimal 8-1ayer laminate
design is used for FRP membrane in system characteristic studies. The variations to the
Optimum design by using an average geometry and the 8-layer laminate with non-zero

bend-twist coupling will be studied in post Optimality analyses.

7.1.3.1 Structural behavior of optimal CMB bridges
As shown in Figure 7—4, Figure 7—7 and Figure 7—10, the optimal geometry Of the
FRP membrane is that of a hyperbolic paraboloid. A hyperbolic paraboloid is obtained by

translating a parabola along the longitudinal direction of another parabola. This shape

115

obviously follows as a direct consequence of the moment diagram of the loaded CMB
bridge to assure in-plane behavior of the FRP membrane.

The results illustrated in Table 7—6 and Table 7—7 clearly show that the FRP
membrane of the Optimized CMB bridge is dominated by an in-plane stress-state by
comparing the in-plane and out-of—plane section resultants. In addition, the FRP
membrane also provides a shear force resistance, which reaches maximum values at both
ends of the bridge structure. However, the bar chart in Figure 7—15 shows that 91% of the
FRP membrane is subjected to longitudinal tension ranging from 0 kN/m to 133.9 kN/m,
while only 9% of the membrane is subjected to longitudinal compressions with values les
than 25 kN/m. Figure 7—16 reveals that the transverse section stress resultants of the FRP
membrane due to the Poisson effect are mainly compression forces with values close to
zero. The shear membrane force distribution shown in Figure 7—17 further indicates that
the high level of shear forces is a local behavior and most of the FRP membrane is
subjected to a low level shear forces, i.e., values close to zero. Therefore, the FRP
membrane is primarily subjected to longitudinal tensile stresses rather than transverse
stresses.

According to Table 7—6 and Table 7—7, although the concrete deck is subjected to
higher out-of—plane section resultants than the FRP membrane, the concrete deck is still
dominated by in—plane membrane forces. Furthermore, the deck is mainly subjected to
longitudinal compression (Figure 7—15), which introduces a small level of transverse
tension due to the Poisson effects (Figure 7—16). Therefore, the optimized FRP CMB
bridge behavior is such that the FRP membrane carries the in-plane longitudinal tensile

and shear forces while the concrete deck carries the compressive forces.

116

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Force Range (kN/m)
Figure 7—15 Section force (SF 1) distribution

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Figure 7—17 Shear section force (SF3) distribution

 

 

7.1.3.2 Stability evaluation

As previously discussed, shape optimization of a shape resistant structure will
improve their structural stiffness. Such optimized structures become stiffer and carry their
design loads primarily by axial or membrane action rather than bending. This leads to a
response that usually involves very small deformations prior to buckling. Therefore,
shape optimized structures are stability sensitive. The stability of the optimized CMB
bridge was investigated by eigenvalue buckling analyses, which are generally used to
estimate the critical buckling loads and potential collapse shapes of stiff structures.

Considering that the material properties of the optimal 8-layer laminate were only
slightly different to the optimal 48-layer laminate, the laminate design of [il7°/i19°]s

was employed to reduce the computational cost of the buckling analyses. The buckling

119

analysis of the optimal CMB bridge considered a perturbation uniform load of 6.9 kN/m2
on the concrete deck acting in the gravity direction. The method of subspace iteration
eigenvalue extraction [HKS, 2000] was used to predict the first five eigenvalues and the
corresponding eigenmodels for the optimal CMB bridge as summarized in Table 7—8.

Table 7—8 Buckling eigenvalues and eigenvectors of the optimal CMB bridge

 

 

Mode Eigenvalue Eigenmodel
1 -76.909
2 —76.909
3 -77.137
4 -77.137
5 —79.923

 

 

As shown in Table 7—8, negative eigenvalues were obtained from the buckling
analyses. Such negative eigenvalues indicate that the optimized CMB bridge would
buckle if the perturbation load were applied Opposite to the gravity direction. However,
the first five buckling modes show that the structure has closely spaced eigenvalues,
which indicates that the structure is imperfection sensitive. Therefore, further
consideration to buckling stability is required for CMB bridges if the applied loading can

act in a direction opposite to that of gravity.

120

7.1.4 Post optimality analyses

Structural designs by Optimization methods are obtained with optimal parameters
achieved regarding certain optimization conditions. For example, the optimum design
defined by the optimal geometric and material parameters for the CMB bridge is
achieved with respect to an Optimization load case. However, the CMB bridge might not
be constructed exactly to the Optimal geometry and the optimal laminate specified by the
optimal design variables may not be easily achieved in practice. Furthermore, multiple
load cases that must be resisted with corresponding performance limit states are typically
considered for the design of bridge structures. The performance of an optimized structure
could thus be compromised due to changes in structural geometry, loading and material
properties. Therefore, the response of the optimum solution to the CMB bridge system
due to geometric imperfections, loading alterations and material property changes (bend-
twist coupling effects) was investigated. The CMB bridge model used in the post
optimality analyses was constructed by averaging dimensions of the three optimized

CMB bridges previously obtained (see Table 7—4).

7.1.4.] Influence of bending twisting coupling effect of laminate properties

The influence of material coupling effects to the optimal CMB model was
investigated by analyzing CMB bridge systems with different optimal laminates for the
FRP membrane that varied only in their coupling stiffness terms (i.e., 016 and D26). The
previously obtained optimal 8-layer and 48-layer laminates, which varied in the out-plane
material properties (see Table 7—5), are thus used in this analysis. The structural
responses of the two CMB models constructed by these two balanced-and-symmetric

laminates are compared in Table 7—9.

121

Table 7—9 Structural response of optimal CMB bridge with different Optimal laminates

CMB In-plane section Out-of—plane section Max Strain
membrane resultants (kN/m) resultants (kN-m/m) Deflection Energy

model SFl SF2 SF3 SMl SM2 3M3 (mm) WW

8- Max 133.87 1.20 52.33 0.15 0.06 0.02
layer Min -1051 -13.79 —52.31 -O.13 -008 -002
48- Max 133.88 1.29 52.31 0.15 0.06 0.02
layer Min -1047 -13.76 -52.31 —013 -007 -002

SFl, SF2: Direct membrane force per unit width in local l-, 2-axis of elements.
SF3: Shear membrane force per unit width in local 1-2 plane.

SMl, SM2: Bending moment force per unit width about local 2-, l-axis of elements.
SM3: Twisting moment force per unit width in local 1-2 plane.

 

 

1.692 84.51

 

1.692 84.51

 

According to Table 7—9, both of the material-modified CMB bridges have the same
strain energy while maximum deflection and the in-plane section resultants in the
membrane vary slightly. This can be explained by the in-plane dominated behavior of the
optimized CMB bridge. The optimized CMB bridge is primarily subjected to in-plane
longitudinal tension rather than out-of—plane section resultants. Furthermore, the small
magnitudes of the bend-twist coupling terms (D 16 and D26) compared to the other major
bending stiffness terms (D11 and D22) also partially reduce the bending—twisting coupling
effect. Therefore, material property changes in bend-twist coupling can be neglected for

the structural behavior of the optimal CMB bridge.

7.1.4.2 Influence of geometric variations

Table 7—10 summarizes the geometric variations of the CMB bridge using average
shape dimensions with respect to the three optimal CMB bridges in terms of geometric
design variables. It can be noted that the maximum variation of 1.94% occurs for h 1 (the
height at the end of the bridge) by using average dimensions. Consequently, the strain

energy of the CMB bridge with average dimensions has an 0.7% increase (Table 7—4 and

122

Table 7—9). Although the CMB bridge combining several slightly-different optimal
designs by averaging shape geometries can be still considered an optimum design in the
current application, it can be anticipated that adequate performance of optimum designs
can be compromised by geometric imperfections.

Table 7—10 Geometric variations of optimal CMB bridges

 

 

W1 W2 I11 I22 I
(mm) (turn) (turn) (Hun) (turn)
Model A 1359.0 2184.4 984.3 1219.2 19.05
Model B 1365.2 2184.4 1022.4 1219.2 19.05
Model C 1352.6 2222.4 997.0 1219.2 19.05
Average 1358.9 2197.1 1001.2 1219.2 19.1
Variation 0.46% 1.00% 1.94% 0.00% 0.00%

 

Variation = (Standard deviation of a dimension) / (Average dimension)

7.1.4.3 Influence of loading alterations

Structural designs require a structure that meets design criteria (i.e. limit states) with
respect to different load cases. However the CMB bridge was optimized with respect to a
single load case. Other load cases, i.e., the service load case and the strength load case,
are required to comply with structural design requirements for different limit states
[ASSHTO, 1998]. Therefore, the performance of the optimized CMB bridge due to these
two load cases was studied with respect to the corresponding design criteria.

Considering that most unidirectional laminated composites behave in a brittle
manner, local failures generally can lead to complete fracture and total loss of load-
carrying capacity. Therefore, the first-ply failure theory, which assumes failure of a
composite laminate to take place when failure initiates in the most critical layer of the
laminate, is used to evaluate the material strength of laminates. Several stress criteria
have been developed to predict the material failure of FRP lamina under different limit

states [Jones, 1999]. In addition to the maximum stress criterion implemented in the

123

integrated optimization procedure, other three commonly used stress criteria were
employed to calculate the ultimate strength of laminates in the study of load alternations.
The criteria used were the Tsai-Hill [Tsai, 1968], Tsai-Wu [Wu, 1974] and Azzi-Tsai-
Hill [Azzi and Tsai, 1965] models. These models are more accurate in comparison to the
maximum stress criteria since they consider polynorrrialcombination of the stress state
rather independent relations [J ones, 1999].

The structural responses and the stress indices of the optimized CMB bridge with
respect to the different limit states are summarized in Table 7—11 and Table 7—12
respectively. As expected, the strength limit state has the maximum influence on the
optimized CMB bridge as it results in maximum in-plane stress demands and a maximum
deflection. With regards the deflection criterion, the maximum deflection obtained in the
service limit state reached 5.237mm, which is 41.65% deflection limit (U800) of 12.573
mm. It should be pointed out that the maximum stress index reached 5.39% of the
material failure, that is 21.54% of the allowed service stress limit (Table 7—1). Therefore,
the deflection criterion governed the design of the Optimum CMB bridge under service
loading. Furthermore, it can be noted that the stress indices are below the design limits
proposed for each limit state (Table 7—1).

Although the optimized C1VIB bridge satisfied the design criteria specified in Table 7—
1 for all limit states, the maximum stress criterion has the lowest stress index, which
underestimates the stress level of laminates compared to the other failure stress criteria.
Therefore, a more accurate material strength criteria should be applied in the stress

constraint in future optimization studies.

124

Table 7—11 Laminate stresses of the optimized CMB bridge under different limit states

 

 

 

 

Load case (N/Siirlmz) (N/Erzrinz) (NS/i312) Max $23M
Optimization 11:44: 2:: if; -8233 4.692
3...... lit: 32:23 431?}. $32 -5537
Streng‘“ ii: 322:? i213: -3133 -8-649

 

S11, S22: Direct stress in local 1-, 2-axis of elements.
S12: Shear stress in local 1-2 plane.

Table 7—12 Stress indices of the optimized CMB bridge under different limit states

 

 

Load Case MStrs (%) TsaiH (%) TsaiW (%) AzziT (%)
Optimization 1.73 2.17 2.03 2.05
Service 4.31 5.39 5.02 5.07
Strength 7.01 8.76 8.17 8.24

 

In addition to the loading alteration by load cases corresponding different limit states,
the structural response of the optimized CMB bridge due to the extreme loading effects
introduced by the loading pattern of moving trucks in the strength limit state was also
explored. Two extreme truck loading positions were considered. One loading pattern was
introduced to create maximum torsion by placing the truck loads at the edge of the
optimized CMB bridge (Figure 7—18). A second loading position sought to develop the
maximum mid-span moment by placing the two axel loads (142.4 kN) in the middle of
the bridge (Figure 7—19). The structural responses and stress indices for the Optimized
CMB bridge due to these two critical loading patterns are summarized in Table 7—13 and

Table 7—14.

125

Truck Load

Live Load

Live Load

 

Figure 7—19 Load case for maximum bending in the strength limit state

As shown in Table 7—13 and Table 7—14, the maximum structural response is
developed due the load case of maximum torsion. The maximum deflection reached
11.82 mm and the maximum stress demand achieved about 21% of the material failure

stress state (Table 7—14), which is 2 times greater than the maximum stress level of the

126

optimized CMB bridge under strength loads. Also, the transverse stresses (S22) are
obviously increased due to the twisting behavior introduced by the truck loads (Table 7—
13). Although the optimized CMB bridge is still within the stress limit for the strength
limit state (see Table 7—1), the structural responses under the critical load patterns are
clearly different than those obtained for the optimum load case. Therefore, structural
Optimum designs with respect to a typical load pattern should be designed with strength
reserves to allow possible increases of structural response due to non-optimal loading
alterations while in service.

Table 7—13 Laminate stresses of the optimized CMB due to different extreme effects

 

 

 

S S S Max deflection
Load case (N/rrlrinz) (N/rrzrinz) (N/rrlrinz) (mm)
Max 33.96 12.94 3.03
Strength Min -26.71 -l6.52 -319 '8'65
Maximum Max 48.74 23.54 4.07 -1 1.82
Torsion Min -38.36 -50.43 -4.94
Maximum Max 44.83 18.07 4.13 _9 50
Bending Min -37.27 -22.73 -4.22 '

 

S“, S22: Direct stress in local 1-, 2-axis of elements.
312: Shear stress in local 1-2 plane.

Table 7—14 Stress indices of the optimized CMB due to different extreme effects

 

 

Load Case MStrs (%) TsaiH (%) TsaiW (%) Azzit (%)
Strength 7.01 8.76 8.17 8.24
Maximum Torsion 20.63 21.14 21.44 21.14
Maximum Bending 9.30 11.94 10.76 10.83

 

7.2 FRP composite membrane suspension (CMS) bridges
Inspired by conventional cable suspension bridges, an FRP composite membrane bi-
suspended (CMS) bridge (see Figure 7—20) was conceptualized as a second case study for

the developed optimization approach. In CMS bridge systems, a deck system (either a

127

conventional concrete slab or an FRP panel) is assumed to be placed on top of the FRP
membrane such that it transfers the applied live loads through internal diaphragms or
“bulkheads.” Thus, the in—plane stiffness of the FRP membrane is employed in two—way
tension under uniform loading from the deck system. A conceptual depiction of a three—

span CMS bridge is shown in Figure 7—20.

Membrane Deck

Diaphragm

 

‘.
Figure 7—20 Concept for Composite Membrane Suspension (CMS) Bridges

7.2.1 Bridge system description

The CMS bridge proposed for Optimization studies is a 60.96 m long three-span
bridge with a 10.36 m wide concrete deck to accommodate two 3.66 m traffic lanes and
essential shoulders (Figure 7—21). The concrete deck is 304.8 mm thick with a density of
2400 kg/m3 for normal concrete. The loads acting on the bridge include the self-weight of
the concrete deck, and the live lane and truck vehicular loads. The live lane load, which
acts over the 3.05 m wide lane, is assumed to be distributed over the entire surface of the
deck, resulting in a pressure of 3.06 kN/mz. The total truck loads introduced by two
trucks for two lanes are also considered to be ideally distributed over the FRP membrane
through diaphragms as a pressure of 1.014 kN/mz. The laminates used for this bridge

system are based on typical medium—performance carbon/epoxy material system for use

128

in civil structures. Thus the lamina orthotropic properties are taken as: E“ = 155.025

GPa, E22 1' 10.335 GPa, 012 = 6.89 GPa, 013 = 6.89 GPa, (323 = 0.689 GPa, and v,2 = 0.3.

 

 

 

 

 

 

 

 

~ L = 60.96 m
(a) Side view

 

 

N N
r\ l\
0 El 0
..l N N 18.
—m. ooooooo i—aofi ———————————————— + ————————————————————
2+ 15 at r

 

 

 

.11
1 5.54 m

 

C.

N

A

B
|
L_

 

L = 60.96 m

 

ll

 

(b) Plan view (c) Cross view

Figure 7—21 Geometry of the CMS bridge

The CMS bridge is subjected to dead loads (DL), two lanes of uniformly distributed
lane loads (LL) and two three-axe] truck loads (TL) [AASHTO, 1998]. Two load cases
using different combinations of the applied loads, which are specified in Table 7—15,
were selected to analyze the CMS bridges for service and strength limit states.

As previously used for CMB bridges, the tension and compression stress limits based
on the material properties for l-axis (fiber-longitudinal direction) and 2-axis (fiber-

transverse direction) were chosen as X1 = 1.55 GPa, XC = -O.775 GPa and YT = 0.103

129

GPa, YC = —0.052 GPa respectively, and the shear stress limit in 1-2 plane is taken as S
234.45 MPa. The ultimate material failure 0;, is then determined by combining the stress
limits measured on the material directions according to a stress criterion.

Table 7—15 Load cases specification for CMS bridges
Design Criteria

Load Cases Specification Stress
Strength

0250'u L/800 Service

 

Limit
Deflection State

 

Service load l.0><DL + l.0><(LL + 1.33xTL)
case

Strength load 1.25xDL + 1.75X(LL + 1.33xTL)
case

 

0.500Ll — Strength

 

7.2.2 Integrated optimization of CMS bridges

Based on the above-mentioned geometric, material, and loading definition for the
bridge systems, the integrated optimization approach was applied to design FRP CMS
bridges starting with the shape and laminate-property optimization and then followed by

the laminate design Optimization.

7.2.2.1 Shape and FRP laminate-property optimization

Taking advantage of structural symmetry, the hyperbolic profile of the CMS
membrane can be controlled by four key points as shown in Figure 7-22. The shape
design variables were therefore the six degrees of freedom of the four geometric key
points. The membrane system was completely defined by spline interpolation of the key
points. The positions of the pier supports (Figure 7—21 (a)) are constant throughout the
optimization process. The rational of this assumption will be discussed in Section 7.3.3.
The laminate design variables are the lamination parameters {VI/r, V3A, V10, V312}

controlling the in-plane tension stiffness and out-plane bending stiffness.

130

Geometric key points

 

Figure 7—22 Loading and geometric key points for the CMS bridge

As applied in the integrated optimization of CMB bridges, an inequality constraint
was employed to limit the used mass of the membrane FRP laminate. Therefore, a
material constraint was implemented by setting a lower and upper bound for the
coordinates of the shape key points (Table 7—16). Maximum stresses monitored at the top
and bottom surface of the FRP laminate and maximum deflections of the complete
system constituted two additional constraints required to enforce the structural design
criteria in the integrated optimization procedure. Instead of only using dead load and live
lane in the optimization load case, the load case for the service limit state (Table 7—15),
which includes truck loads, was chosen as the optimization load case for the CMS

bridges.

131

Table 7—16 Coordinates constraints for geometric design variables

 

 

Coordinate P1 (m) P2 (m) P3 (m) P4 (m)
bound x y z x y z x y z x y 2
Lower -6.1 —6.1 -10.4 -6.1 ~104 -6.1
Upper 30.48 0.0 0.0 0.0 0.0 0.0 30.48 -7.8 0.0 0.0 -7.8 0.0

 

Two typical boundary conditions for suspension bridges were considered in the CMS
bridge analyses (Table 7—17) to evaluate the effect of boundary conditions on the
optimized shape. Model A was provided with simple supports at the positions of the piers

while Model B featured extra roller supports at both ends of the bridge.

Table 7—17 Initial designs of CMS Bridges
Initial design Model A Model B

 

 

Boundary conditions

{ 21,22, Z3, 24, y3, y4} (m) {0, 0,0,0, -10.4, -104}
Laminate design [O°/90°]s
{V1A,V3A,VrD,V3D} {01.0-75.1}

 

The integrated optimization approach was implemented as previously described and
the performance and stability of the obtained results was assessed by studying a single
initial design (geometry and laminate layup) for each CMS bridge model (Table 7—17).
Both integrated optimizations were initiated with a plane FRP membrane constructed
with a cross-ply laminate, i.e., a [0°/90°]s. The first-level optimization history of the two

CMS bridges is shown in Figure 7—25 and Figure 7—28, respectively.

132

 

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A_4 __
E .
v
C
.2 7‘
a) -6 -— ffffffi . . .
c _
Q) 4
g -
D

 

 

 

 

 

 

Iteration no.
Figure 7—23 Histories of geometric design variables — CMS Model A

 

 

 

 

    

 

 

 

 

 

1 .1 1|
T
U) .
h 7'- 1
G) 0.5
4-0 .
a)
E \ I
(U l \/\,
h .
m - , l ,
O 7 4 i
t: .
N
Is
g - ......
_l'0.5 “ .....
I+V1A +V3A +V10 +Vao l
-1 rrrrrrrrr i rrrrrrrrr i 111111111 i 111111111 i 11111111 ri11111T111i 111111111
0 10 20 30 40 50 60 70

Iteration no.
Figure 7—24 Histories of laminate-property design variables — CMS Model A

133

107 ;

106 -
’15
2 ii.- j, , _ ‘ . Optimal Shape
‘x’ 1 05 $49195:gé‘méfigwfifwwmnmkk 1:} 1;" , ;«:07 2.9:“ ..
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a)
5
c 10" —-
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H
(I)

103

102 1 . . . . .

0 10 20 30 40 50 60 70
Iteration no.
Figure 7—25 History of objective function — CMS Model A

0

.2 ;
’E‘ -4
c
.9
w -6
c
0.)
.§ :
D -8 .

-10
. 1:”21 +Z+Zo;z“:* +Y
.12 . . .. . ,; .Wr 1 #44
0 10 20 30 40 50 60 70

 

 

 

  

 

 

 

 

 

 

 

 

 

Iteration no.
Figure 7—26 Histories of geometric design variables — CMS Model B

134

 

parameters

Lamination

I
.°
01

Strain energy (kN-m)

 

P
a:

O
. i .

 

 

‘ v1 A VJA v1 D VJD ‘

 

0 10 20 30 40 50 60 70
Iteration no.
Figure 7—27 Histories of laminate-property design variables — CMS Model B

 

107;

105.

Optimal Shape

   
  

Plan View

—I
O
0!

 

IsoView

..x
O
p.

I l

Elevation View

A

O
u
i

 

 

 

1o2

 

Iteration no.
Figure 7—28 History of objective function — CMS Model B

135

The optimal results obtained from the first-level optimization are summarized in
Table 7—18. It can be seen that although both models have a similar geometric shape, the
optimized objective of Model B has a strain energy value 20% lower than that of Model
A. The improvement in minimizing the strain energy came from changes in the laminate
properties due to the different boundary conditions. In Chapter 6 it was demonstrated that
structural optimization can be further improved by material optimization. Optimum
laminates were thus also determined for the CMS bridges using the second-level
optimization so as to achieve the obtained optimal lamination parameters.

Table 7—18 Optimal results of the first-level optimization for the CMS bridges

 

 

Optimum design Model A Model B
Strain energy (kN-m) 183.26 149.93
{2 z z z } (m) {-6.05, -6.10, —0.43, -0.37, {—6.10, -6.10, -O.22, -0.30,
1’ 2’ 3’ 4’ Y3’ W -8.20, —7.99} -779, -7.85}

{V1A,V3A,V1D,V3D} {0.138, 0.033, —0.317, —0.533} {0.155, 0.106, —0.309, -0.494}

It should be noted in Figure 7—28 that the objective function experienced an abrupt
decrease after a relative smooth development in the iteration history of CMS bridge
optimization. The occurrence is thought to be inherent from the gradient properties of the

objective function with respect to the design variables for this optimization case.

7.2.2.2 Laminate design optimization

In the second level optimization, the genetic optimization procedure was employed to
design optimal laminates with a maximum error of 1% with respect to the optimal
lamination parameters obtained from the first-level optimization.

The genetic iteration histories for the laminate designs of the two CMS bridge models

are shown in Figure 7—29 and Figure 7—30, respectively. The optimal laminate for Model

136

 

A was found to be [i51°/i57°/i60°/i63°/i500/i670/i620/i21°/0°g]3 with a minimum error
of 0.807% with respect to the Optimal lamination parameters. The optimal laminate for

Model B was [i53°/i51°Ii64°2/i53°/i62°/i64°/i4°/0°8]5 with a minimum error of

 

 

 

 

 

 

 

 

 

0.643%.
35% _
30%
25% ‘
5‘ : . . .
c 20% -. Optimal Lamination Parameters
Q) -
t - 35:: (2.-23;.
h A, :- _ , :- _
8 15% __ ID 30
E Optimal Laminate Design
10% j [:51°/1-57°/i60°/i63°/i50°
: /i>67°/i62°/fl1°/0°8]S
% 3~
5 : Error=0.807%
0%4Tr:r'rrruriur-riwrrviwrrwirrrui

o 1000 2000 3000 4000 5000 6000
Iteration no.

Figure 7—29 Second-level optimization process of Model A

137

 

 

 

 

 

 

 

 

 

 

40% _
35% f-
30% f—
325% - , , .
c Optimal Lamination Parameters
d)
B VID : -Oo309, V3D : '0.494
(D
D 15% " Optimal Laminate Design
100/ [fly/:5 1°li64°2/-_t5 3°
° /:62°/:64°/1-4°/0°8]S
5% i' Error = 0.643%
0%- fl % mfimfi rrrrrrrrr i vvvvvvvvv r 111111111 . ..........
0 1 000 2000 3000 4000 5000 6000

Iteration no.

Figure 7—30 Second-level optimization process of Model B

7.2.3 System characteristics of optimal CMS bridges

The system characteristics of the optimized CMS bridges was investigated by
studying the structural response of the bridge systems under the optimal load case and the
stability of the optimized CMS FRP membrane. The influence of boundary conditions on

the optimum designs was evaluated by comparing the results from both models.

7.2.3.1 Structural behavior of optimal CMS bridges

The section forces and moment resultants for the CMS bridges are given in Table 7—
19 and Table 7—20, respectively. From these tables it is clearly seen that the FRP
membrane of both CMS bridges is dominated by in-plane stress demands. In addition, a

further review of the section forces along the local 1- and 2-direction (Figure 7—31)

138

shows that the FRP membrane is subjected to two-way tension, as the relative magnitude
of both of these stress fields is similar. Therefore, the behavior of FRP CMS bridges is
such that the FRP membrane acts as a membrane structure carrying distributed loads

primarily through bi-directional tension.

 

Figure 7—31 Coordinate systems

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140

7.2.3.2 Stability

From above-discussed results, the optimized CMS bridges are shape resistance
structures in which high structural stiffness is primarily gained by in-plane membrane
actions rather than flexural actions. As previously addressed, such structures are buckling
sensitive and usually require only small deformations prior to experiencing buckling.
Therefore, the stability of the optimized CMS bridges was investigated through buckling
sensitivity analyses.

Eigenvalue buckling analyses determined with ABAQUS [HKS, 2001] were used to
predict critical buckling loads and potential collapse modes of CMS bridges. The loading
of the service limit state was employed as the perturbation load required for the buckling
analyses. The method of subspace iteration extraction was chosen to obtain the first four
eigenvalues and corresponding eigenshapes, which are summarized in Table 7—21.

According to the results shown in Table 7—21, negative eigenvalues are reported for
both CMS bridge models. Such eigenvalues imply that buckling of the optimal CMS
bridges would happen only if the perturbation load were applied opposite to the gravity
direction. Further comparison of the two models shows how the different boundary
conditions will influence the buckling modes of the structures. Model A has obvious
buckling deformations at the free ends of the bridge while buckling of Model B will
primarily happen at the positions where the columns support the FRP membrane.
Considering that Model B is stiffer due to its lower strain energy, it is understood that
Model B is more sensitive to buckling and, therefore, has higher and more closely spaced

eigenvalues than Model A.

141

Table 7—21 Buckling eigenvalues and eigenshapes of the optimal CMS bridges

 

 

      

 

 

Mode Model A Model B
Eigenvalue Eigenshape Eigenvalue Eigenshape

Base
State ‘ '

1 —0.037 —0.2526

2 -O.114 -0.3151

3 -0.120 —0.3152

4 —0.125 -0.3153

 

 

7.2.4 Post optimality analyses

Optimal structural designs are achieved by satisfying certain optimization paramters
and constraints such as loading and boundary conditions. Change of such conditions can
alter the optimum designs whose response may violate the constraints originally satisfied.
Therefore, it is necessary to evaluate the influence of such changes on the behavior of
optimum designs through post optimality analyses. The following analyses focus on the
influence of loading and boundary—condition changes on the performance of the

previously-obtained optimal CMS bridges.

142

7.2.4.1 Influences ofloading alternations

Since the load case for the service limit state was used to achieve the optimal CMS
bridges, the constraints of maximum deflection and maximum stresses were verified for
the strength limit state load case on the optimal CMS bridges.

A first—ply failure analysis was applied to predict the material ultimate strength of the
laminated FRP membrane. In addition to the independent maximum stress criterion
implemented in the integrated optimization procedure, the coupled polynomial stress
failure theories by Tsai-Hill [Tsai, 1968], Tsai-Wu [Wu, 1974] and Azzi-Tsai-Hill [Azzi
and Tsai, 1965] were also used in a first-ply-failure criterion. These models are readily
available in ABAQUS to estimate the stress failure of FRP laminates.

The structural responses and the stress indices of the optimized CMS bridges with
respect to the different limit states, see Table 7—15, are summarized in Table 7—22 and
Table 7—23, respectively. As shown in the tables, although the maximum stresses and
maximum deflection are increased from the service limit state to the strength limit state,
the stress indices for both models of the optimized CMS bridges indicate that the load
case associated to the service limit state is still the critical load case for the bridge system.
It should be noted that the stress index evaluated by the maximum stress criterion
employed in the optimization process predicted that the stress level in the laminates was
within the allowable limits and the material strength. However, as seen in Table 7—23, the
bi—axial stress failure criteria indicate that the stresses on the FRP laminate exceed the
service stress limits. Therefore, multiple stress failure criteria should be considered as

stress constraints during optimization processes.

143

Table 7—22 Structural response of optimized CMS bridges for different limit states

 

SH $22 812 Max (16116011011
MOdel Load case (N/mmz) (N/mmz) (N/mmz) (mm)
. Max 525.08 39.22 20.96
A semce Min -133.91 -15.08 -11.41 40030

Max 715.88 53.48 28.57
Strength Min -182.57 20.55 -1555 54559

Service Max 613.59 46.30 19.57 139.95

Min -187.88 ~19.06 —15.52

Max 836.54 63.12 26.68
Strength Min -256.15 -2599 —21.17 19””

Sn, S22: Direct stress in local l-, 2-axis of elements.
S12: Shear stress in local 1-2 plane.
SP1, SP2: Minimum, maximum principal stresses

 

 

Table 7—23 Stress indices of optimized CMS bridges for different limit states
Model Load Case MStrs (%) TsaiH (%) TsaiW (%) AzziT (%)

 

Service 23.47 32.38 29.23 30.73
Strength 32.01 44.14 39.84 41.88
Service 20.78 27.20 25.30 25.83
Strength 33.86 45.70 52.90 45.70

 

MStrs: Maximum stress theory failure measure
TsaiH: Tsai-Hill theory failure measure.
TsaiW: Tsai-Wu theory failure measure.
AzziT: Azzi-Tsai-Hill theory failure measure.

In the optimization procedure, the design loads applied on the FRP membrane were

ideally considered to be uniformly distributed over the area that the concrete deck

projects onto the FRP membrane. However, in practice, it is conceptualized that the

concrete deck weight and live loads will be transferred to the FRP membrane through

diaphragms or “bulkheads.” Therefore, spaced line loads (Figure 7—32), which simulate

the pattern of the loads transferred from the diaphragms, were applied on the FRP

membrane as a non-optimal loading pattern. The structural response of the optimal CMS

bridges subject to spaced line loads at the service limit state will be used to verify the

design criteria and is compared to the structural behavior of the optimal CMS bridge

subjected to the distributed loading.

144

Spaced line loads
Membrane

  
   
  

Diaphragm

Figure 7—32 Loading pattern of spaced line loads

The structural response of the optimal CMS bridges subject to the spaced diaphragm
line loading pattern is summarized in Table 7—24, Table 7—25, Table 7—26 and Table 7—
27. By comparing the section forces and moments, it is shown that the response of both
models still remains governed by in-plane stresses. However, it is easily seen in the
figures of Table 7—26 and Table 7—27 that the distribution of the section forces and
moments in the FRP membrane has obvious changes along the loading pattern, while the
magnitude variation of the section forces and moments varied only modestly. A
comparison of Table 7—22 and Table 7—24 shows that compressive forces noticeably
increased in the FRP membrane of Model A after the loading pattern was altered. This
lead to the stress index of the FRP membrane of Model A (see Table 7-23 and Table 7—
25) to be almost twice than the stress demand limits chosen for the service limit state.
However, the stress level and stress indices of Model B had smaller variations compared
to Model A due to changes in the loading pattern.

It can be further noted for Model A that, although its section moments remained small
compared to the in-plane forces, the magnitudes of the section moments increased

considerably for the non-optimal loading pattern as compared to values obtained under

145

the uniform load distribution (Table 7—26 and Table 7—27). Accordingly, the structural
strain energy under the non-optimal load pattern is more than two times that of the
Optimal value previously obtained (Table 7—25). The increase in strain energy is caused
by the increase in the section moments throughout the FRP membrane, which can be
clearly seen in the figures and values given in Table 7—27. Consequently, optimal Model
A, obtained by a uniform loading pattern, might not be considered as an optimal shape
for the loading pattern of spaced line loads. On the other hand, the strain energy of Model

B increased only moderately over the optimal strain energy value.

Table 7—24 Structural responses of the optimized CMS bridges
subject to spaced line loads under the service limits state

 

Stress
Model S1 1 S22 S 12 SP1 SP2
(N/mmz) (N/mmz) (N/mmz) (N/mmz) (N/mmz)
A Max 439.21 25.99 27.65 17.43 439.29
Min -303.67 -2002 -26.61 —304.75 -18.82
B Max 506.56 36.96 19.74 36.71 506.74
Min —135.39 -1340 —10.02 —135.75 -1153

 

811. 822: Direct stress in local 1-, 2-axis of elements.
Sn: Shear stress in local 1-2 plane.
SP], SP2: Minimum, maximum principal stresses

Table 7—25 Stress indices of the optimized CMS bridges
subject to spaced line loads under the service limits state

 

 

Stress Index Max Strain

M d 1 '
0 e MStrs (%) TsaiH(%) TsaiW (%) AzziT(%) (1623:?“ fit?)
A 40.10 58.83 68.73 58.83 427.74 414.20
B 23.76 26.65 28.65 25.95 139.65 193.20

 

MStrs: Maximum stress theory failure measure
TsaiH: Tsai-Hill theory failure measure.
TsaiW: Tsai-Wu theory failure measure.

Azzit: Azzi-Tsai-Hill theory failure measure.

146

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147

The summarized results point out that the performance of an optimum design can be
significantly altered by changes in loading patterns. However, Optimal CMS bridge
Model B was found to be less sensitive to loading pattern changes than Model A (their
difference arising from their boundary conditions). Thus, the optimal CNHB bridge of

Model B would behave better under changes in loading patterns during service.

7.2.4.2 Influence of boundary conditions

As previously discussed, the optimal CMS bridge designs show different performance
detriments due to changes in loading patterns. The change in performance due to non-
optimal loading patterns is largely originated from the difference in boundary conditions.
Therefore, it is necessary to investigate the influence of boundary conditions on the
optimum design of CMS bridges. This can be done by comparing the in-plane behavior
of the two CMS bridge models under the service limit state.

The membrane forces of Model A and Model B are compared in Figure 7—33 and
Figure 7—34. According to Figure 7—34, both models are primarily subjected to in-plane
tensile forces and have an almost equal force distribution in the transverse direction of the
bridge. However, although both models are subjected to tension and compression in
longitudinal direction, their in—plane structural behavior, demonstrated by the distribution
of transverse section forces, is different. Figure 7—33 shows that about 85% of Model B is
subjected to membrane forces with 300 kN/m or less while 15% is subjected to
membrane forces higher than 300 kN/m. In contrast, about 30% of Model A has
membrane forces over 300 kN/m. This is the reason that Model A has a higher optimal
strain energy than Model B. The variation in section force distribution is caused by the

difference in boundaryconditions between the two models. These results, combined the

148

 

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issues discussed in the system characteristics and the load variation studies, lead to the
conclusion that Model B has a more effective and stable in-plane behavior than Model A.
This improved in-plane behavior is achieved by the addition of supporting boundaries at
both ends of Model B (Table 7—17), which strengthens the dual hanging concept of the

CMS bridge system.

7 .3 Discussion of FRP membrane-based bridge systems

In this section, the conceptualized FRP composite membrane-based bridge systems
(CMB, CMS—A, and CMS-B) are evaluated by comparing their structural behavior in
terms of material use efficiency. In addition, the selection of an objective function that
could further improve the integrated shape and laminate optimization approach is also

discussed.

7.3.1 Comparison of FRP membrane-based bridge systems

As shown through the studies of system characteristics, the performance of both types
of composite membrane-based bridges is improved by their in-plane behavior as tension
structures after the integrated optimization. However, efficient performance under in-
plane behavior is gained through different schemes for both composite membrane-based
bridges.

The system characteristic studies of the optimal CMB bridges showed that the
structural stiffness used by CMB bridges to carry external loadings is obtained by
developing a force couple between the longitudinal tension forces in the FRP membrane

and the longitudinal compression in the concrete deck. The in-plane section resultants of

150

 

the FRP membrane and the concrete deck in the transverse direction are due primarily to
the Poisson effect and are two orders of magnitude less than the longitudinal in-plane
resultants. Therefore, CMB bridges achieve mainly unidirectional in-plane behavior.
However, the system characteristic studies for the optimal CMS bridges indicated that the
FRP membrane in optimal CMS bridges are subject to in-plane longitudinal and
transverse tensions. The structural stiffness of CMS bridges is thus acquired by achieving
the bi-directional membrane behavior of membrane and shell structures.

The two types of composite membrane-based bridges presented in this work differ not
only in their structural behavior, as previously discussed, but also in their efficiency in
material use. As it is well known, efficient use of materials requires achieving not only
in—plane or membrane resultants but also a uniform stress distribution throughout the
structure. The distribution of stress demands in a generally anisotropic structure can be
represented by strength safety factors of the elements in a finite element model of the
structure. The strength safety factor of an element is evaluated by the stress index of that
element with respect to a specific failure criterion.

However, due to the different structural geometry, loading and material volume used
for CMB and CMS bridge systems, their structural response is different even for the same
limit state. Therefore, a concept of relative stress indices is introduced to evaluate and
compare the efficient use of the FRP membrane. Thus, the stress indices of individual
elements are normalized by the maximum stress index for the complete structure (clearly
applied to each structure separately). Table 7—28 summarizes the distribution of
normalizing stress indices defined by the Tsai-Wu failure theory for the FRP membranes

of all membrane-based bridges under the service limit state load demands. The structural

151

fraction distributions, defined as the number of elements within a given relative index
range divided by the total number of elements in the structure, for all three bridges with
respect to the relative stress indices are compared in Figure 7—35. The average and the

standard deviations of the three curves are given in Table 7—29.

Table 7—28 Stress indices distribution of three bridges

CMB CMS - Model A CMS — Model B
Normalized Fraction Normalized Fraction Normalized Fraction

 

Stress # EL Stress # EL Stress # EL
Index 317;; Index :75}: Index ‘21:;
0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
6.67% 0.00% 6.25% 26.73% 6.25% 22.16%
13.33% 0.33% 12.50% 28.52% 12.50% 30.73%
20.00% 1.31% 18.75% 20.41% 18.75% 22.47%
26.67% 2.47% 25.00% 10.70% 25.00% 13.69%
33.33% 6.78% 31.25% 5.83% 31.25% 4.27%
40.00% 19.75% 37.50% 2.27% 37.50% 2.97%
46.67% 24.08% 43.75% 1.70% 43.75% 1.77%
53.33% 19.31% 50.00% 1.94% 50.00% 1.14%
60.00% 14.14% 56.25% 1.22% 56.25% 0.31%
66.67% 3.81% 62.50% 0.19% 62.50% 0.09%
73.33% 3.33% 68.75% 0.06% 68.75% 0.03%
80.00% 2.42% 75.00% 0.13% 75.00% 0.06%
86.67% 1.67% 81.25% 0.00% 81.25% 0.00%
93.33% 0.56% 87.50% 0.06% 87.50% 0.06%
100.00% 0.06% 93.75% 0.00% 93.75% 0.19%
0.00% 100.00% 0.25% 100.00% 0.06%

 

152

 

 

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Normalized Strength Index
Figure 7—35 Structure material distribution of relative stress index

Table 7—29 Statistics of relative stress index distribution

 

 

Relative Stress Index CMB CMS —- Model A CMS - Model B
Average 50.07% 14.51% 17.77%
Standard deviation 13.1 1% 10.43% 11.06%

 

Evaluation of the data presented in Figure 7—35 and Table 7—29 indicates that a
higher average of relative stress index implies a higher average stress level for that
structure. Thus, CMS bridges have a higher strength reserve in terms of FRP laminate
capacity than CMB bridges of the same structural size and material use when the
structures are subjected to an equal loading. On the other hand, a smaller standard
deviation of the stress index distribution indicates a large portion of the FRP membrane is
located within a smaller range of the average stress index. From this point of view, CMS

bridges have a better structural performance of uniform stress distribution than CMB

153

bridges. Therefore, it can be concluded that CMS bridges utilize the FRP material more

efficiently than CMB bridges.

7.3.2 Discussion Of the objective function

As proved in Section 4.1.2, structural stiffness is maximized by achieving in-plane
structural response, which is implemented by minimizing the structural strain energy in
the integrated shape and laminate optimization approach. On the other hand, structural
stiffness can be also improved by adding material. Therefore, structural geometric
dimensions and the thickness Of the FRP membrane were constrained tO limit the use Of
laminated FRP composites. However, it is difficult to distinguish the individual
contribution from structural shape and material quantity towards improved structural
stiffness by minirm'zing the structural strain energy. For example, strain energy can be
further decreased with increased FRP materials even though the structural response be
already in-plane dominant. This limitation Of the objective function could result in an
over-designed optimum as shown in the post optimality analyses for CMB bridges (Table
7—12 and Table 7—14). The results from Section 7 .1.4 showed that the stress indices Of
the optimal CMB bridges were far below the allowable stress levels for all limit states.
An over-bound constraint on the FRP membrane thickness is the most possible reason.
Therefore, an integrated Optimization employing a narrow-bound constraint on the
membrane thickness was investigated for the Optimum design Of CMB bridge.

The effort Of a narrow bound constraint on the FRP membrane thickness was
implemented by evaluating the optimal design for the CMB bridge with the upper bound

on thickness reduced from 19.05 mm to 9.525 mm. The newly Optimized CMB bridge

154

(Model D) is summarized in Table 7—30. It can be noted that the FRP membrane for
Model D achieved essentially the same Optimal geometric shape and the same Optimal
laminate properties compared tO the previously Obtained Optimum designs (see

Table 7—3). Also, it is noted that the FRP membrane thickness still reached the upper
bound of the thickness constraint. Due tO the reduced thickness (50% reduction) the
structural strain energy was almost doubled. However, this increase Of the strain energy
does not indicate that the in-plane structural behavior was impaired since the FRP
membrane still achieved a close Optimal geometric shape. In fact, the increase Of the
strain energy is caused by the increase Of in-plane stress resultants due to the thickness
reduction. Therefore, strain energy minimization with geometric constraints can not avoid
over-designed Optimal structures although the structural stiffness is properly improved by

achieving membrane response.

Table 7—30 Optimal CMB bridge designs with different membrane thickness constraints

 

 

Optimal design Model A Model B Model C Model D
3m“ Energy 83.85 83.85 83.85 161.90
mm
{w W k W t} {1359.0, {1365.2, {1352.6, {1327.3,
1’ 2' 1’ 2’ 2184.4,9843, 2184.4, 1022.4, 2222.4, 997.0, 2174.2, 990.6,
(mm) 121921905} 121921905} 121921905} 121929525}
{081,031, {081,031, {081,031, {081,031,

{VIA,V3A,V10,V3D} _1’ 1} -03, 07} 0.75, 1} 0.75, 1}

 

According to the Optimization experience Obtained above, Optimal CMB bridges with
an average Optimal geometry were investigated to study the structural response with
respect to the FRP membrane thickness (Table 7—31). It can be Observed that, while the

structure remained under in-plane demands, reductions in membrane thickness increased

155

the structural strain energy while the maximum deflection and stress index approached

the prescribed design criteria values, which is also indicative Of a more efficient design.

Table 7—31 Structural response with different membrane thickness
Thickness Strain Energy Deflection Stress Index

 

 

T (m) SE (kN-mm) D (m) SI (%)
19.050 84.51 1.692 34.56
15.875 100.33 1.966 41.16
12.700 127.33 2.464 51.13
9.525 165.07 3.226 66.15
6.350 235.12 4.765 95.17

 

Several alternative Objective functions that could seek to maximize structural stiffness
and minimize material use were investigated:

0 Objective function f1: Minimize (Strain Energy)><(Thickness)

0 Objective function f2: Minimize (Strain Energy) / (Stress Index)

0 Objective function f3: Minimize (Strain Energy)><(Thickness) / (Stress Index)

The performance Of these Objective functions evaluated with Optimal CMB bridge
designs is summarized in Table 7—32 and graphically represented in Figure 7—36. It can
be seen that the proposed Objective function (f3 = SE-T/SI) linearly decreases with
reduction Of the membrane thickness. Conversely, as shown in Table 7-31, the decrease
in the membrane thickness leads to stresses and deflections that are closer to the design
criteria. Therefore, the Objective Of maximizing structural stiffness while minimizing the
material can be achieved by implementing a function that incorporates minimization Of
the structural strain energy and the membrane thickness as well as maximization of stress

levels.

156

Table 7—32 Evaluations Of alternative Objective functions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f1=SE~T f2=SElT f3=SE-T/SI
(kN-mmz) (kN-mm) (kN-mmz)
1610.0 244.5 4658.5
1592.7 243.8 3869.6
1617.1 249.0 3162.8
1572.3 249.5 2376.9
1493.0 247.1 1568.8
5000
- +1, = SE-T (kN-mmz)
‘ +12: SEISI (kN-mm)
: +f3=SE-T/SI(kN-mm2)
4000 “r
c
.2 j
2 3000 ~~
3
LL
0
.2
8 2000 ~—
3 .
o a
1000 a
: I 4 f I J
0 r 1 . . r . 4
5 10 15 20

Membrane Thickness (mm)
Figure 7—36 Performance Of different Objective functions for Optimal CMB bridge
In addition, the Optimum designs for FRP composite membrane-based bridges are
currently Obtained under an Optimization load case taking into account a single loading
effect and a single loading pattern. However, as evaluated in the post optimality analyses
of both bridges, the structural behavior Of the Optimum designs can be significantly under
non-Optimization loading effects and patterns. Therefore, multiple loading effects and

loading patterns should be considered simultaneously in a limit state in order tO reduce

157

significant variations in the response Of an Optimum design. This can be implemented by
a general Optimization load case in which different loading effects and patterns are
combined with different weight factors. For example, the extreme loading effects Of
maximum torsion, maximum mid-span bending and the normal load effect might be
considered with a uniform weight factor Of 1/3 in the Optimization load case for the
Optimum design Of CMB bridges. Similarly, the Optimization load case for CMS bridges
might combine the loading patterns Of uniform and spaced line loads with a weight factor
Of 1/2.

Instead Of using a general Optimization load case, multiple loading effects and
patterns can be taken into account by using a multi-Objective Optimization approach.
Multi-Objective Optimization consists Of defining different Objective functions assigned
with weight factors with respect to their respective loading effects and patterns. The
Optimum design that compromises among the different loading effects and patterns is
Obtained by Optimizing the multi—Objective functions with respect to their own set Of

constraints.

7.3.3 Selection Of the design variables

It should be noted that the structural performance Of CMS bridges (Section 7.2) will
vary with the position Of the pier supports. Therefore, the pier position should be taken
into account as one of the geometric design variables in Optimizing CMS bridges.
However, preliminary Optimization efforts revealed that the Objective function of
minimizing the strain energy varied discretely with respect to changes Of the pier

positions. This resulted in premature Optimization solutions at local Optimum points.

158

Thus, the pier positions Of CMS bridges were currently fixed at constant locations
throughout the Optimization process.

Moreover, it should be noted that, for all Of the Optimization examples presented in
this dissertation, the geometric key points that determine the structural shape were
selected tO achieve a higher computational efficiency. A structural shape is then
generated by interpolating the geometric coordinates Of the key points. Therefore, the
structural shapes that can be Obtained are limited by the number Of geometric key points
and their degrees Of freedom. For example, the number Of geometric key points selected
for shape finding Of FRP membranes in the bridge systems can only determine the
surfaces high up to a 2nd-Order curvature by interpolation techniques. The accuracy Of
Optimal shapes Obtained by structural shape Optimization for free form finding can be
improved by increasing the number Of the geometric key points. However, free form
finding is achieved at the cost of computational efforts due to the increase Of design
variables. In addition, general free-forms, although efficient, may not be practically
feasible. The determination Of the number Of design variables therefore requires
considering the computational efforts, the accuracy Of the Optimal shape sought, and the
ease Of implementation. Thus, in spite Of the power provided by these mathematical
Optimization tools, it is still the design’s judgment that dictates the successful design Of

an efficient structural form.

7.4 Summary

In this chapter, the integrated shape and laminate Optimization approach was applied

to the design and Optimization Of two innovative and newly proposed FRP composite

159

 

membrane-based bridge systems. The characteristic studies for both bridge systems show
that the high in-plane stiffness and strength characteristics Of FRP laminates are utilized
to maximize the performance and efficiency of the bridge systems due to the
effectiveness provided by shape resistant membrane structures. Furthermore, based on the
Observation Of over-designed Optimal CMB bridges, several alternative Objective
functions, in addition to the minimization Of structural strain energy, were investigated to
achieve in-plane structural response with minimum material use. The Objective function
Of minimizing the strain energy and the membrane thickness while maximizing the
membrane stress level seems tO have the best performance for CMB bridges and is
thought tO be a viable Objective function for maximizing structural stiffness with
minimum material use. The use Of Objective functions that take intO account multiple
loading effects and patterns were discussed, and the importance behind the selection of
geometric design variables was noted to place the current work and its results into

context.

160

References

AASHTO—American Association Of State Highway and Transportation Officials (1998).
AASHTO LRFD bridge design specifications. Washington, DC: The Association.

Azzi, V.D. and Tsai, SW. (1965). Anisotropic strength Of composites. Experimental
mechanics: 5, 283-288.

 

Hibbitt, Karlsson & Sorensen, Inc. (2001). ABAQUS standard user’s manual Version
6;.

 

Jones, RM. (1999). Mechanics of composite materials. Philadelphia, PA: Taylor &
Francis.

Seible, F., Karbhari, V.M., and BurguefiO,R. (1999). Kings Stormwater and 1-5/Gilman
Bridges. Structural Engineering International: Journal of the International Association for
Bridge and Structural Engineering: 9, 4, 250-253.

 

Tsai, SW. (1968). Strength theories of filamentary structures. In Fundamental Aspects Of
Fibeer Reinforced Plastic Composites Schwartz, RT. and Schwartz H.S. (eds), 3-11.
New York: Wiley Interscience.

 

Wu, EM. (1974). Strength and Fracture Of composites. In Composite Materials,
Broutman, LI. and Krock, R.H. (eds) Vol. 5, 191-247. New York: Academic Press.

161

8 Conclusions and Future Research Needs

8.1 Conclusions

The presented integrated Optimization approach Offers a procedure for simultaneously
finding the Optimal shape and Optimal laminate design Of FRP structures. The integrated
approach was investigated by analytical studies on the development Of FRP shell
structures that combine FRP laminates and shape resistant structures for improved FRP
structural designs. The approach provides an efficient tOOl to determine maximum
stiffness designs by Optimizing not only the geometry Of the membrane or shell but also
the material properties and the composite laminate design.

The integrated Optimization approach can also serve as an analytical tOOl to aid the
rational implementation Of laminated FRP composites in civil infrastructure by
developing innovative design concepts. The developed concepts were that Of membrane-
based FRP composite bridge systems. The bridge systems consist Of a shape-and-material
Optimized laminated FRP membranes that carry the in-plane tensile and shear forces
while using a conventional reinforced concrete slab or FRP deck system to provide the
live load transfer. The dual-level Optimization approach can thus support the analytical
studies required for initial development Of innovative systems that use FRP composites in
their inherent behavioral characteristics for new high-performance structures.

During the efforts Of integrated shape and material Optimization, it was Observed that
an over-bounded constraint Of material use can lead to an over-designed Optimum
achieved for maximizing structural stiffness. Several feasible Objective functions using
combination Of the structural strain energy, the membrane thickness, and stresses were

investigated. The Objective function Of minimizing the strain energy and the membrane

162

thickness while maximizing the membrane stress level seems tO have the best
performance and is thought tO be a viable Objective function for maximizing structural
stiffness with minimum material use.

The work has provided insight tO the concept that FRP laminates can be used with
higher efficiency in new structural systems as long as their advantageous properties Of
directional strength, light weight, and tailored properties are properly considered in the

design process.

8.2 Future research needs
8.2.1 About the integrated approach Of shape and laminate Optimization

As it is well known, shape-resistant structures, such as thin shells subject to
compression forces, are sensitive to buckling. While not considered in the presented
work, structural buckling performance can be easily included in the proposed integrated
Optimization procedure. This can be done by two distinct approaches. One is to consider a
determined buckling capacity as a constraint in the shape and laminate Optimization
procedure, thus achieving the Optimization Objective Of maximum stiffness while
satisfying the required buckling resistance. Another approach is to search for the
maximum buckling capacity along with maximum stiffness in a multi-Objective
Optimization process, where the different design Objectives can be weighted to maximize
structural performance. Therefore, the presented integrated approach can further evolve
into a multi-Objective decoupled approach for both shape and laminate optimization Of

FRP lightweight structures by adding appropriate Objectives and constraint functions.

163

A gradient algorithm is used in the first step Of the integrated approach to achieve the
Objective Of maximizing structural stiffness. Due to the expectation Of unaffordable
computation demands by the large-scale optimizations, the numerical technique Of finite
difference methods was employed to supply the required gradients information tO
advance the gradient algorithm. The main disadvantage Of calculating gradients by finite
difference methods is the uncertainty in the choice Of a perturbation step size. A
perturbation that is either tOO large or tOO small will result in inaccurate gradient
evaluations and misguide the search direction Of the algorithm. The techniques from
analytical methods have a better performance, in terms Of robustness and accuracy, tO
derive gradient information than finite difference methods. However, the differentiation
Of Objective and constraint functions in analytical methods requires the structural stiffness
matrix, which will change completely even for small shape changes. This makes gradient
derivation computationally costly. Therefore, future reSearch is needed to provide the
proposed integrated approach with an efficient and accurate technique for sensitivity
analyses.

In the current approach, the shape and laminate Optimization processes were
decoupled into a two-step procedure, in which laminate designs in the second-step
Optimization are based on the lamination parameters from the first step. This requires that
feasible larrrination parameters be Obtained during the first-step optimization. Thus, the
constrained relationship between the lamination parameters is very important and has to
be solved for the first-step Optimization. At present, only the constraints between the
essential lamination parameters, {V,A, V3A, V10, V30}, required by the current approach

were numerically established (with certain assumptions) and applied in the Optimization

164

process. However, this might not be enough for general applications. A competitive
approach that can avoid the formulation Of explicit constraints and implicitly satisfy the
feasibility of laminate designs is to choose fiber orientations as material design variables.
For this alternative approach, the integrated Optimization has to be implemented with
genetic algorithms in which geometric and material design variables are both treated as
discrete variables. Although the volume of computations needed for sensitivity analyses
can be avoided by using genetic algorithms, the computational effort will probably not be
reduced because Of the random search approach Of genetic algorithms. The formulation
Of this alternative approach and its computational efficiency can maximize the broad

implementation Of the developed method.

8.2.2 About applications Of the integrated approach

Although not explored in the current research effort, the integrated approach can be
applied for the Optimum design Of different lightweight structures seeking to maximize
structural stiffness, such as domes and membrane roof systems. Furthermore, with
appropriate formulation of Objective and constraint functions, the integrated optimization
approach can be employed for many diverse applications. For example, a potential
application is vibration tuning Of structures. Structures subject to dynamic forces or
excitation at their supports might need to be designed to avoid resonance or other
negative vibration related problems. By relating the Objective function to the fundamental
vibration frequency, the integrated approach can yield Optimum designs Of structures that

can avoid vibration problems.

165

The concept employed in the current approach to consider continuous and discrete
design variables separately in a two decoupled procedure can also be applied to other
Optimization applications, for example, for smart structures made using piezoelectric
materials. Piezoelectric materials are the materials that can convert electric energy into
mechanical energy and vice versa. They produce an electrical response when
mechanically stressed (sensors) and high motion can be Obtained when an electric field is
applied to them (actuators). Composite structures integrated with piezoelectric sensors
and actuators Offer potential benefits in a wide range of engineering applications such as
vibration suppression and shape control. The development and efficiency Of this type of
structures rely on Optimizing the location of the actuators and sensors that control the
behavior of adaptive structures. This type of optimization applications involves
continuous design variables of structural geometric shapes and discrete design variables
that define the position of actuators and sensors. Therefore, the current integrated

approach can be directly applied to investigate this type Of Optimization problems.

8.2.3 About the engineering and construction Of Optimal FRP composite structures

It needs to be noted that the fiber orientations Of the Optimal laminates achieved in the
previous Optimization examples are based on the local structural coordinates of the finite
elements describing the FRP membrane, which vary with the normal of the membrane
surface. Therefore, the construction Of an FRP membrane with a laminate design in
which the layout is specified based on local coordinates requires further study.

In addition, the construction of bridges like those presented in this work requires

providing an effective membrane/slab connection to assure composite behavior between

166

the FRP membrane and the concrete slab, or the FRP panel, Of the bridge system.
Particularly for CMB bridges, shear force resistance is required at the membrane/slab
interface under flexural actions. A study on possible forms Of shear connectors is thus
needed to allow implementation for the developed CMB bridge systems. Finally, post
Optimality studies on the response Of Optimal CMS bridges indicated that the distribution
of in-plane stresses depends on the form and spacing Of diaphragms, which affect the
efficiency of load transfer to the FRP membrane. A future study is thus required to

investigate geometries and Optimal placement Of diaphragms for CMS bridges.

167

 

 

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