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Modeling of fractured steel girders strengthened with CFRP sheets

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Title:
Modeling of fractured steel girders strengthened with CFRP sheets
Creator:
Bodenstab, William Eric Jr.
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Master's ( Master of science)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
Department of Civil Engineering, CU Denver
Degree Disciplines:
Civil engineering
Committee Chair:
Rens, Kevin L.
Committee Members:
Kim, Yail J.
Li, Cheng

Notes

Abstract:
Computer modeli ng of bridge structures might seem deceptively easy to the novice, untrained in the field of structural engineering. Howe ver, to those who have studied the discipline, it is clear that modeling is not as simple as build ing blocks. The mathematically deri ved relationships are at the backbone of a well developed three dimensional, finite element analysis computer program. It is aim of this paper to provide the educated structural engineer a background in computer modeling, while investigating a real life i ssue, fatigue. Perhaps one of the reasons that steel has fallen out of favor in Colorado for use in girders of bridges is presence of fatigue in many of the primary and secondary elements within a bri dge structure. Reinforced, pre stressed or cast in plac e concrete has become a dominant material in bridge construction, partly be cause it is much more forgiving building material, and is generally not vulnerable to fatigue. Yet many structures require the us e of steel for economic reasons or due to design co nstraints. While there are few cases of fracture for steel fatigue details greater than category C, it is still imperative to model damaged structures using computer programs and accurately determine stresses. This thesis investigates cases of cracked and CFRP-strengthened girders of bridges modeled in a three-dimensional finite-element analysis program. It includes tabulated and graphed results, along with sectional analysis of the technical data. This thesis also includes calculations of flexural capacity, modal frequencies and LRFR (rating factors, Inventory and Operating.)

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University of Colorado Denver
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Auraria Library
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Copyright William Eric Bodenstab, Jr. Permission granted to University of Colorado Denver to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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Full Text
MODELING OF FRACTURED STEEL GIRDERS STRENTHENED WITH CFRP
SHEETS
by
WILLIAM ERIC BODENSTAB, JR. B.A., Williams College, 2000
A thesis submitted to Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering Program
2016


This thesis for the Master of Science Degree by William Eric Bodenstab, Jr. has been approved for the Civil Engineering Program by
Kevin L. Rens, Chair Yail J. Kim, Advisor Cheng Li
Date: December 17th, 2016


Bodenstab, William Eric, Jr. (M.S., Civil Engineering)
Modeling of Fractured Steel Girders Strengthened with CFRP Sheets Thesis directed by Professor Yail J. Kim
ABSTRACT
Computer modeling of bridge structures might seem deceptively easy to the novice, untrained in the field of structural engineering. However, to those who have studied the discipline, it is clear that modeling is not as simple as building blocks. The mathematically derived relationships are at the backbone of a well-developed three-dimensional, finite-element analysis computer program. It is aim of this paper to provide the educated structural engineer a background in computer modeling, while investigating a real-life issue, fatigue.
Perhaps one of the reasons that steel has fallen out of favor in Colorado for use in girders of bridges is presence of fatigue in many of the primary and secondary elements within a bridge structure. Reinforced, prestressed or cast-in-place concrete has become a dominant material in bridge construction, partly because it is much more forgiving building material, and is generally not vulnerable to fatigue. Yet many structures require the use of steel for economic reasons or due to design constraints. While there are few cases of fracture for steel fatigue details greater than category C, it is still imperative to model damaged structures using computer programs and accurately determine stresses.
This thesis investigates cases of cracked and CFRP-strengthened girders of bridges modeled in a three-dimensional finite-element analysis program. It includes tabulated and graphed results, along with sectional analysis of the technical data. This
m


thesis also includes calculations of flexural capacity, modal frequencies and LRFR (rating factors, Inventory and Operating.)
The form and content of this abstract are approved. I recommend its publication.
Approved: Yail J. Kim


ACKNOWLEDGEMENTS
I am very grateful to faculty of the Civil Engineering Department at the University of Colorado, Denver, including Y. Jimmy Kim, Cheng Li, Kevin Rens, N. Y. Chang, Fred Rutz and Rui Liu. I would not have become proficient enough in civil and structural engineering to undertake this thesis without their instruction, knowledge and experience. I am grateful for Roxanne Pizano’s assistance with the requirements and with meeting the deadlines for this degree.
My family and friends were also instrumental to my completion of this thesis. My wife, son, parents, close friends and employer made big sacrifices of their time and resources during my time in the program.
I am blessed to live in this Great Country, allowing its citizens the freedom to pursue their dreams, providing us with unlimited opportunity.
Eric Bodenstab November 2016 Denver, Colorado
v


TABLE OF CONTENTS
CHAPTER
I. Introduction...............................................................1
1.1 Modeling with EE A. Software.......................................1
1.2 The Typical Section..................................................2
1.3 Silver Bridge Collapse...............................................3
1.4 Definition of Fatigue................................................5
1.5 Redundancy and Load Shedding.........................................9
1.6 Other Types of Fatigue..............................................10
1.7 Literature Review: Modeling in Finite Element Programs.............12
1.8 Literature Review: Sources of Error in 3-D Modeling................13
1.9 Literature Review: Various Modeling Approaches and Their Accuracy .... 14
1.10 Carbon Fiber Reinforced Polymer and Its Use......................16
II. Model Development and Validation........................................23
2.1 Beam Comparison: SAP-2000 to Theory................................28
2.2 Full-Scale Model Validation: Nebraska Bridge......................29
2.3 Full-Scale Model Validaiton: Flint, Michigan Bridge................31
2.4 Developing a Fatigue-Prone Model...................................33
III. Analysis and Results....................................................44
3.1 The Control Bridge.................................................44
vi


3.2 Cracked Sections
45
3.3 Fatigue life of damaged sections...........................47
3.4 CFRP Strengthening.........................................48
3.5 Fatigue Life Improvement...................................50
3.6 Flexural Capacities of Sections............................50
3.7 Load and Resistance Factor Rating..........................53
3.8 LRFRFatigue................................................54
3.9 Load Factor Rating.........................................54
3.10 Modal Responses...........................................55
IV. Conclusions.....................................................75
REFERENCES..........................................................77
APPENDIX A..........................................................80
APPENDIX B..........................................................84
APPENDIX C..........................................................93
vii


LIST OF TABLES
TABLE
1.1 AASHTO Detail category constants and thresholds...........................17
2.1 Material properties used in study models.................................35
2.2 Summary of beam validation................................................35
2.3 Full-scale model validation summary, Nebraska............................36
2.4 Summary of Flint, MI bridge model validation..............................37
3.1 Control bridge dimensions .................................................56
3.2 Control girder dimensions.................................................56
3.3 Damaged girder dimensions (without CFRP).................................57
3.4 Maximum stress as a function of crack % (of girder height),
fatigue truck loading......................................................57
3.5 Fatigue life of a category B’ detail for damaged girders.................58
3.6 CFRP dimesions (for damaged girders)......................................58
3.7 Girder sectional properties: transformed area (steel, mm3)................59
3.8 Girder sectional properties: moment of inertia (10"3 m3)................59
3.9 LRFR rating factors.......................................................60
3.10 Modal frequencies for the unstrengthened bridge models...................60
3.11 Modal frequencies of the first five modes of a 10% cracked girder with
strengthening cases.......................................................61
viii


LIST OF FIGURES
FIGURE
1.1 Typical section of a steel I-girder...................................17
1.2 Architectural elevation of the Silver Bridge showing Warren trusses and
the point of failure...................................................17
1.3 Eyebar detail (category E for fatigue) ...............................18
1.4 Photo of Silver Bridge collapse.......................................18
1.5 Diagram of a truck axle (point) loads on a simply-supported bridge....18
1.6 AASHTO fatigue category curves....................................19
1.7 AASHTO HS-20 truck load ...............................................19
1.8 S-N curve for a category B’ fatigue detail ............................20
1.9 Photo of the girder crack of the 1-79 Bridge at Neville Island ........20
1.10(a) Detail of out-of-plane distortion .................................21
1.10 (b) Photo of web gap region with crack on transverse weld ............21
1.11 (a), (b) & (c) Three different modeling techniques ...................21
1.12 Use of springs in modeling friction in unrestrained joint DOF’s .....22
2.1 Soffit, extruded view of 5% crack modeled in steel girder
from SAP 2000 .........................................................38
2.2 Soffit view of CFRP-strengthened bottom flange .......................38
2.3 Deflection of a W21xl32 in SAP-2000 ..................................39
2.4 Standard view of the Nebraska bridge model in SAP 2000 ................39
2.5 Extruded view of the Nebraska bridge model in SAP 2000 ................39
2.6 Nebraska bridge and girder dimensions (mm) ............................40
2.7 Aerial view of loaded Nebraska bridge model in SAP 2000 ...............40
IX


2.8 Nebraska bridge 70-ft span (21.3m) with loading weights and locations...41
2.9 Bridge rail dimensions in (mm), modeled as structural element...........42
2.10 Flint, MI bridge section and girder dimensions (mm).....................42
2.11 SAP 2000 aerial view of Flint, MI bridge with loading pattern...........43
2.12 Loading configuration for the Flint, MI bridge validation, axle weights
shown between wheels....................................................43
3.1 The control bridge.......................................................62
3.2 Control bridge and girder dimensions (damage location indicated for
variable cases)..........................................................62
3.3 Control bridge: stress in lower flange of exterior girder ..............63
3.4 (a) SAP 2000 aerial view of the control bridge with loading pattern.....63
3.4 (b) Loading configuration producing most adverse effects in the control
bridge, dimensions in mm and kN (not to scale)........................64
3.5 Assigned dimensions in millimeters for damaged girder sections..........64
3.6 Maximum stress of 5% cracked girder, exterior...........................65
3.7 Maximum stress as a function of crack percentage, fatigue loading.......65
3.8 Category B’ detail fatigue life vs. crack percentage....................66
3.9 Assigned dimensions in mm for CFRP-strengthened girders.................66
3.10 Neutral axis depth, from top of slab...................................67
3.11 Strain of CFRP at crushing of concrete.................................67
3.12 Stress range vs. crack percentage, with four cases of CFRP- strengthening
(387 mm2, 1161 mm2, 1806 mm2 and 3613 mm2)..............................68
3.13 Fatigue life vs. crack percentage, control with 387 mm2 and 1161 mm2 cases of CFRP-strengthening. 1806 mm2 and 3613 mm2 cases have
infinite fatigue life...................................................68
3.14 Calculation of Mn capacity of girders, strengthened and control.........69
x


3.15 Inventory rating factor (LRFR Strength).................................69
3.16 Operating rating factor (LRFR Strength).................................70
3.17 Fatigue rating factor (LRFR)............................................70
3.18 Load Rating Factor (Inventory)...........................................71
3.19 Load Rating Factor (Operating)...........................................71
3.20 Mode 1: T =0.3115 seconds, f =3.210 Hz..................................72
3.21 Mode 2: T = 0.2684 seconds, f =3.725 Hz.................................72
3.22 Mode 3: T = 0.18130 seconds, f = 5.516 Hz...............................73
3.23 Mode 4: T= 0.11791 seconds, f =8.481 Hz.................................73
3.24 Mode 5: T = 0.08535, f=11.716 Hz........................................74
xi


COMMON ABBREVIATIONS
3-D Three-dimensional
AASHTO American Association of State Highway and Transportation Officials
ADTT Average Daily Truck Traffic
AISC American Institute of Steel Construction
BIRM Bridge Inspector's Reference Manual
CFRP Carbon Fiber-Reinforced Polymer
CoV Coefficient of Variation
DOF Degree(s) of Freedom
F.E.A. Finite Element Analysis
F.E.M. Finite Element Modeling
LFR Load Factor Rating
LRFR Load and Resistance and Factor Rating
PNA Plastic Neutral Axis
Xll


CHAPTER I
INTRODUCTION
1.1 Modeling with EE.A. Software
It is surprising to discover that there is actually limited academic research within the field of structural engineering on the problems encountered in finite-element modeling (F.E.M.), even with the array of computer programs that have been available for the last 30 years. An aim of this study is to develop a helpful guide for study in structural engineering, while using three-dimensional F.E.M. computer programs.
One might ask if there are user manuals for these programs, which have become a necessary part of engineering practice. There are indeed, but the function of the program’s user manual is equivalent to a cookbook. They show the procedures of gaining familiarity with the software, and the steps needed to produce a model with basic outputs of reactions and forces. However, software developers do not generally address in the manuals how the programs work. Structural engineering research publications are also limited in this respect. Therefore, developing a guide for future studies in this area will be helpful for all those who follow.
Fatigue in steel bridges will be frequently referenced and investigated throughout the course of this thesis and an attempt will be made to merge these copious studies with the Finite-Element Analysis (F.E.A.) software available to the modern engineer. Beginning with the very basics of both structural engineering and the procedures of F.E.A. software, and evolving into some of the esoteric problems of steel fatigue and the intricacies of the supporting software, the reader will develop understanding of the core principles of F.E.A. and fatigue. The remainder of Chapter I will focus more so the
1


history of fatigue studies in structural engineering, while Chapter II: Model Development and Validation will introduce computer modeling that is integral to this and many studies. Chapter II also is integral to the overall study in that the computer models that are used as the backbone of the investigations are validated against previous work of past researchers. Chapter III: Analysis and Results presents the bulk of the F.E. A. studies and models cracked and CFRP-strengthened, composite steel-girder bridges. Chapter IV draws conclusions from the studies. Short Appendices follow to allow the student of structural engineering to see the models’ inputs and some of the calculations in the process.
1.2 The Typical Section
In structural engineering, the basis of all design is the type of section one uses to support the loads that are externally applied to the structure and also to support the selfweight of the structure. If a section is too small or narrow and experiences too much stress, then nine times out of ten the designing engineer “sizes-up” the section. An ancient example of such a practice can be seen in the monument at Stonehenge. The girth of both the posts and lintels far exceed the necessary required resistance of the column axial loads, buckling considerations and flexural strengths needed to stand its own weight, lateral forces such as wind or earthquakes, and so on.
The focus in this paper is steel, a far stronger material than stone, and especially so in tension. A couple types of sections will be discussed; however, one key type of section is the basis of consideration for the investigation in this paper. The so-called “I-girder” takes its name from the shape of the capital letter. A girder is essentially a beam,
2


but will often support other beams that are perpendicular in plan. I-girders are typically made of steel for bridge spans over 20 feet (a so-called “bulb-tee,” having an I-shaped profile, is used for precast concrete girders.) There have been different specified steels for this common type of girder over the past hundred years, and concurrently the American Society for Testing and Materials (now ASTM International) has kept pace with the development of steel standards. In the 1910’s it began with A7 steel, progressing through A36 Steel. Presently, A992 Grade 50 Steel is the most common. Regardless of the type of steel used, the I-girder has kept its characteristic geometry. Along with the given span length of a beam or girder, the section profile is the other key parameter in determining how a beam will function or behave. An I -girder section (shown in Figure 1.1) is comprised of two key elements: a set of two flanges joined perpendicularly with a “web.” These elements can either be “rolled” or fabricated together in a steel mill, or as in the case of a plate-girder bridge, welded together.
1.3 Silver Bridge Collapse
To briefly review some of the history of the problem of fatigue, one might go to the year 1967 when the Silver Bridge collapsed. Steel bridges had been in place for 80 years by then, and had of course spanned some of the most scenic waterways. (See Figure 1.2 for an architectural elevation of the Silver Bridge.) The Bridge’s history dates from 1927, when construction began to provide a passing over a key point along the Ohio River. Unlike many large over-water crossing bridges of today, it was erected in a little over a year. It was known also as the Point Pleasant Bridge, but also called the “Silver Bridge” due to its aluminum paint. (Lichtenstein 1993.) Various design alternatives were
3


considered, but chain link, Warren-type trusses ultimately comprised the main span suspension structure.
The eyebar element (51cm thick) was the key connection element, and its detail is shown in Figure 1.3, as well as it is visible in the wreckage of Figure 1.4. As Lichtenstein points out, “The design and construction of the chain was such that it brought together two very dangerous elements - extremely high tensile stresses and corrosion on the inside of the eyebar.” (Lichtenstein 1993.) During the holiday season of that year there was an increase in loading due to heavy traffic, and furthermore, the bridge had undergone 40 years of use and corrosion. Loads were significantly higher than in the days of the 1200-lbs. Model T. Truck loads, which are the principle governing loads for fatigue, had also increased substantially. Despite the fact that the engineers specified 75 ksi (517 MPa) steel, it could not support the loads of the heavy traffic in its fatigue-weakened state. A small (A-inch) defect at the point noted progressed rapidly or instantaneously into a full crack, and the entire structure fell suddenly into the Ohio River, taking 46 lives along with it.
It was later determined that the eyebars (only one fractured in this case) were the breaking point in the structure. Since the 1970’s through today, these elements have been classified in the fatigue category of most concern to the bridge engineer, which are categories E and E’. These categories have largely been avoided since their introduction. Certainly, after the tragic collapse, fatigue had caught the full attention of researchers and designers alike. The National Bridge Inspection Standards was established not too long after the Silver Bridge failure, in order to prevent future accidents of this kind.
It may be hard to imagine how a seemingly sufficient-sized section of a super-
4


structure could suddenly split, sending it sinking into the river. Considering that all engineering takes place between the stressed and over-stressed condition, the margin for error can be razor-thin when dealing with sensitive elements such as the eyebar. Capacity of the structure then is paramount in order to provide the necessary resistance to the forces and stresses put on a structure. We know now that the eyebar element can only resist 4.5 ksi (31 MPa) of net section stress, as indicated in Table 1.1; 75,000 pounds per square inch of tension had been assumed for this structural member, due to the specified high-strength steel. Note that there is no such thing as “high-fatigue resistance steel.” In addition, a small defect can further reduce resistance of member section and be subject to greater stress concentrations. As stated, this was also the case with the Silver Bridge.
In summary of the problems that led to the eventual collapse of the Silver Bridge, engineering historian Henry Petroski notes that the “design inadvertently made inspection all but impossible and failure all but inevitable.” (Petroski 2012) Both insights are true, but a third must be added: without an understanding of fatigue, it is impossible to know that a section and its material will fail at stress lower than its yielding value. Moreover, without a clear understanding material fatigue, the exact mechanism of this particular collapse would remain a mystery.
1.4 Definition of Fatigue
At this point, it is necessary to define fatigue in more detail. The Bridge
Inspector’s Reference Manual gives the definition of fatigue as:
“Fatigue is the tendency of a member to fail at a stress level below its yield stress when subject to cyclical loading. Fatigue is the primary cause of failure in fracture critical members. Describing the process by which a member fails when subjected to fatigue is called failure mechanics.” (Federal Highway
5


Administration 2012)
Fatigue was the principle reason for failure of the Silver Bridge; the eyebar was a tension element subjected to many cycles of loading. Fatigue was also a crucial reason for the I-35W Mississippi River Bridge collapse in 2007. (Federal Highway Administration 2012) Fatigue is categorized in detail categories A through E’, with the latter being the most sensitive. A-Categories have a threshold for infinite-life of 24 ksi (165 MPa), and E’ only 2.6 ksi (17.9 MPa). Table 1.1 at the end of Chapter I shows the threshold and the finite-life constants that describe the behavior of the member details of a steel-girder
A
bridge. The constant A is from the equation, N = 3 , where N is the number of cycles,
(AF)n
A is the constant listed in Table 1.1, and (AF)n is the nominal fatigue resistance.
Fatigue is induced in a material by cycles of tension, and is not directly affected by compression. Tri-axial tension, that is pulling force in three distinct directions, is by far the most difficult condition for a material to endure, as it relates to fatigue of metals. (Dexter and Fisher 1999) Small regions of this type of tension can exist in both bridges and buildings. The reason highway bridges experience more fatigue problems is due to the weight and regularity of trucks passing by on the deck of the bridge, as sketched in Figures 1.5,1.7. (In this study, wheel loads are modeled as “point-loads”, as opposed to distributed loads). Generally, the problems of fatigue with bridges do not happen after just a few cycles, but after tens of thousands or millions of trucks passing above the supporting girder-beams.
In many cases, the passing of 100,000,000 heavy trucks will not induce fatigue damage, nor cause cracking, especially in the well-detailed bridge built after 1975. The structures that can withstand tens of hundreds of millions of cycles of loading are said to
6


possess infinite life details. Indeed, Fisher notes “there have been very few if any failures which have been attributed to details which have a fatigue strength greater than category C ” (Dexter and Fisher 1999) The stresses in the bridge will not be high enough to cause fatigue problems, a fact that will be observed in the subsequent research. Theoretically, the bridge might be able to sustain loading of the fatigue trucks for as long as the bridge is in service; no amount of truck traffic over time will typically result in fatigue damage.
Examining the graphs in Figure 1.6, the stress range is induced by the standard HL-93 fatigue truck. Figure 1.7 only includes the values for a category B’ detail, which will be the type of steel detail that will be the focus of analysis in subsequent chapters. Because the observed stresses in the bridge models that will be presented in Chapter III indicate that the bridge would be susceptible to fatigue category B’ details, it is necessary to consider the impact of further damage. Particularly, this damage is likely in the region of fatigue and present with the loss of section due to cracking. The remaining life of the material, section and system are of natural concern to all charged with the possible strengthening of such a section, and this will be the focal point of analysis later on in this paper.
The S-N curves are used to predict the remaining fatigue life, the number of cycles that a section can take in an applied stress range. For the situations modeled in Chapter III: Analysis and Results, the B’ category became of primary interest. Such a detail is described in the AISC Manual (14th Ed.), pg. 16.1-200 Table A-3.1 §3.2: “Base metal and weld metal in members without attachments built up of plates or shapes, connected by continuous longitudinal complete-joint-penetration groove welds with backing bars not removed, or by continuous partial-joint-penetration groove weld.”
7


(American Institute of Steel Construction 2011) Such welds would be present in a built-up member such as the steel plate girders of the validation and control bridges, as will be presented in Chapters II and III. The B’ category detail threshold for infinite life is 12.0 ksi or 82.7 MPa and is indicated by the flat line of Figure 1.8.
All regions below any of the curves are probabilistically safe from fatigue; note that the lower categories dip farther down on the chart and have less fatigue resistance. The number of cycles grows exponentially across the x-axis. In spite of loading cycles in the tens or hundreds of millions, it still may not always be fatigue that could cause failure. Also important to note, cars or pick-up trucks are not included in fatigue cycles since they do not generate enough stress in bridges. Their loads are not considered in the fatigue limit-state and are not a part of the later analysis.
In addition, fatigue limit stresses are not a function of the base metal yielding strength, but instead they are only limited by the geometry of the detail. In addition, while the notion of stress is central to the calculations, it is more appropriately conceived that strain is the governing phenomenon. (Fisher, Kulak and Smith 1998) Microinconsistencies perpendicular to the load path will further result in growing inconsistencies or crack tip propagating. As was pointed out, it was not the lack of strong steel that contributed to the Silver Bridge’s collapse, but rather the reduction of capacity due to damage of fatigue sensitive elements, existing imperfections and the development of corrosion at key points in the structure. The lack of structural redundancy was also a key factor in the collapse of the Silver Bridge. Redundancy is discussed in the following section.
8


1.5 Redundancy and Load Shedding
In most cases in structural engineering, a building or bridge might have additional reserve capacities, or redundancy, to exceed the maximum loading effects. An additional problem with the Silver Bridge was that the eyebar chain-link system specified was not redundant; only a pair was used for each link joint. Furthermore, a single eyebar is not sufficient to hold the system in place; it would be similar to the situation of a missing link on one side of a bike chain, at which point the entire chain would likely come apart.
Many similar bridge structures were built in the same time period of the 1930’s, but would have four or six eyebars mounted in parallel. Even with one of these eyebars cracking in failure, sufficient warning and reserve capacity were available in these structures to hold and allow time to remedy.
Even when redundancy might not theoretically exist, a non-redundant structure can avoid collapse if the applied loads are being redirected to other intact members of the structure. Again, this phenomenon exists in sharp contrast to the Silver Bridge Failure of 1967. With “load-shedding,” the concrete deck (in most cases used instead of steel) and the other girder(s), could assume the applied loads in the area of the failure discontinuity.
One notable case of load shedding that involved a significant fatigue crack occurred in Pittsburgh in 1979. (Dexter and Fisher 1999) The crack is shown in Figure 1.9. The 1-79 Bridge at Neville Island, an essentially non-redundant two-girder bridge, held in spite of a full-flange crack that developed and propagated up nearly the entirety of the web of the girder. It is interesting to consider this case also because the bridge was relatively young when it developed this debilitating crack. Only being three years old, this bridge had a manufacturing defect and an unadvisable weld. The stress-
9


concentration in the affected section caused the defect to develop into a quickly propagating crack after relatively few loading cycles.
A towboat passing underneath just happened to notice the crack, and soon after, the state DOT shut down traffic over the bridge for two months in order to complete repairs. The structure, as Dexter and Fisher note, was subject to “displacement control”, and the stiff surrounding members, including cross-frames, took on the applied loads after the crack had formed. The authors say that not only did the structure avoid collapse, but also the load shedding slowed down crack propagation. (Dexter and Fisher 1999)
The question may arise to the engineer if this type of damage could be included in a bridge model. With a simple span and a reduced section at the point of damage discontinuity, it is relatively easy to model bridges in a variety of available Finite Element Analysis programs. This type of full-flange crack, perpendicular to span length of the girder, will be investigated in the Section 3.2.
1.6 Other Types of Fatigue
One might ask whether fatigue problems fall outside the detailed categorization provided by AASHTO, AISC and other agencies. Indeed, this is the case. Distortion-induced fatigue affects specific details of a bridge that do not experience the stresses that normally induce fatigue. Instead, the fatigue is caused by the distortion or warping of a particular element. The phenomena has been studied extensively and represents a major subtopic of fatigue within steel structures. The problem has been examined by many previous researchers: “It has been concluded that distortions with the magnitudes even on the order of only 0.5mm (0.02in) may induce high cyclic stress-ranges up to 276 MPa (40
10


ksi) in small welded gaps.” (Lui, Frangopol and Kwon 2010) The authors go on to say that although these problems are common their effects are manageable, insofar as the cracks occur in planes parallel to the primary longitudinal stresses in a bridge girder, and that they can be detected early and the bridge retrofitted appropriately.
An example of distortion-induced fatigue was thoroughly covered in recent research at Kansas University. The Tuttle Creek Bridge, built in 1962 in northeastern Kansas, was retrofitted in 1986 and 2005 to mitigate the deleterious loading effects due to the cracking that had developed. A concurrent study was published in part by the Kansas Department of Transportation. (Anderson, et al. 2007) The study notes that cracking had developed shortly after the bridge was built in the web gap region. The cause was distortion in the web of the girder, which was being pulled out of its design plane by the lateral tension forces in the cross-bracing. While the longitudinal weld of the flange to the web is normally a category-B fatigue design element, the distortion and the presence of tension in the transverse direction made the element highly susceptible to the fatigue induced by the passing truck live loads. Normally this category-B element will not succumb to fatigue at any cycle of loading with stresses less than 16.0 ksi (110 MPa).
The transverse fillet weld is also highly susceptible, if not more so, to distortion-induced fatigue cracking shown in Figures 1.10 (a) & (b).
These types of fatigue cannot be ignored in design or during in-service inspection. Cracks such as these can propagate into adjacent elements and cause failure in the entire structure; hence, the need for retrofitting. Despite the importance of both retrofitting in situations like these, and restoration of distortion-induced fatigued elements, this paper will not focus on this type fatigue, but instead will center on load-cycle induced fatigue.
11


1.7 Literature Review: Modeling in Finite Element Programs
The previous work of other bridge modelers, including Nowak and Sotelino, other academic studies and field tests, along with guidance from CSI in their manuals for SAP 2000, is essential in developing F.E.M. Sotelino and Chung indicated in their 2005 study that “the design bending moment for steel girders can be determined more accurately using Finite-Element Analysis of a bridge superstructure rather than using lateral load distribution factor[s] specified in AASHTO.” (Sotelino and Chung 2006) The goal of any 3-D model is to determine the bending forces, stresses and moments in the key elements of the bridge structure, particularly the lower flange of the girders, the supports, and the connection details. While it was not discussed in the introduction, AASHTO provides a shortcut to conservatively estimating these forces through simplified formulas. Yet since the 3-D models make use of the finite-element theory to form the basis for (giant) matrix equilibrium, they should provide the most accurate results.
The simplest 3-D F.E.M. is the one discussed above with the rigid links joining the nodes of the centroids of the concrete slab and the steel girders. Again, the concrete slab is modeled as shell elements, and the steel girders as beam elements. In some other past modeling techniques, the I-girder itself was broken into three shell elements. Other past techniques used three-dimensional solid elements. (Brockenbrough 1986) (Mabsout, et al. 1997) Figures 1.10 (a), (b) & (c) show some of the proposed alternatives to bridge girder modeling in the past.
Chung and Sotelino conclude that despite the simplicity, the model represented in Figure 1.10 (c), which they call the “eccentric beam model,” provides as accurate results as the other more complicated modeling techniques and does not require fine mesh
12


density. They compared the model to a full-scale test done at the University of Nebraska (which will also be used to validate the modeling technique here in this paper.) They found comparable results for deflection observed, and error within 6%. Also in comparison, the other models used three times the number of degrees of freedom. They conclude that the simplest model is the best model.
1.8 Literature Review: Sources of Error in 3-D Modeling
While the results of the proposed modeling scheme are sufficient to proceed with fatigue modeling of bridges, it is still necessary to consider at this point how error is introduced in this or any 3-D F.E.M. One source of error that it is important to consider is the end restraints modeling. With the SAP 2000 program, the “assignment of joint restraints” is completed in order to provide a foundation of sorts for the model of the superstructure, limiting the 3-D degrees of freedom of the individual nodes established. There are six DOFs available for any node in a model; they are movement or translation in the direction of any of the three principal axes and the three possibilities of rotation around any of the three principal axes.
For field conditions, no joint can be perfectly restrained or exhibit complete freedom from restraint (or friction) in any of the six degrees of freedom. For instance, with a simply-supported bridge, translational restraint along the longitudinal horizontal axis in reality is subject to friction at the bearing pad. This friction was not modeled in the validations in the Chapter II, however good suggestions for accounting for it have been proposed in previous literature. (Kim and Nowak 2001) By placing a spring at the shown locations in Figure 1.12, a 3-D F.E.M. can account for friction.
13


CSI notes that shear deformation is included in the deflection in the SAP 2000
analysis. (Computers & Structures, Inc. 2013) As a result, observed deflection in the 3-D F.E.M.s are slightly more than the theoretical equations yield. In addition, the shear deformation may also cloak the increased moment and stress in the damaged sections in some methods of composite modeling. Shear deformation error is also noted in APPENDIX A. The next section will ways to model composite bridges within SAP 2000.
1.9 Literature Review: Various Modeling Approaches and Their Accuracy
There are more ways to model a composite steel girder bridge in SAP 2000 that have not been discussed up to this point. This next section details further approaches that ultimately produce equivalent results as the “eccentric beam” model that had been detailed in method Section 1.8. The tables and figures in APPENDIX A show eight ways of creating a model in SAP 2000 that capture, to varying degrees, the composite nature of a well-designed steel-girder bridge.
Of the eight ways to model a steel girder section, there are four ways noted here that capture the composite nature of a (shear-studded) steel girder with a concrete deck. “Model 8” is the “eccentric beam” model from Sotelino and Chung, and as mentioned that was the primary model developed and used in this analysis. Generally, the principle that governs the bridge mechanics of all modeling techniques suggested is identical to “eccentric beam” model. The four methods listed here accomplish the change in the moment of inertia of a composite section:
1. Offset the shell joints (away from the centroid of the steel girder) - Model 2.
2. Use the body constraint tool to hold the deck and girder centroid so that the joints
14


are restrained in all six degrees of freedom - Model 3.
3. Offset the girder centroid from the plane of the deck centroid - Model 4. (This type of modeling technique will be investigated in the coming section.)
4. Use a rigid link to join the centroids of the deck and girder - Model 8.
The results are identical for these four methods of increasing the net section moment of inertia. Without offset, the eccentric beam does not govern the sectional properties of the section, and essentially it becomes non-composite. Figure A1 depicts the extruded view modeling in SAP 2000 showing an example of Model 1 from Table Al, with its noncomposite properties and the characteristic single-centroid plane.
While Model 1 is not a useful structure, nor does it bare verisimilitude toward a real bridge or bridge model, it is a helpful launching point and can help give the engineer an approximate estimate of flexural properties. It is still necessary to compare results from computer model trials of the various models. The eight models shown in Table Al were subjected to the following loading conditions in a software study. A simply-supported span of 20 meters was given the following section with a composite moment of inertia of 0.15433 meters4.
A 20-meter span with the section shown in Figure A2 was subjected to a point load at midspan of 100 kN. From structural analysis theory, the beam should have
p-i3
deflection of------= 3.2725 mm. This is indicated in row one of the summary of
48-e-i J
calculations in Table A2. From the table of results, the four methods that most closely model composite behavior have identical results of 3.2624 mm of deflection at midspan. This result should alleviate the concerns of the bridge engineer over what modeling technique to choose. Since there is no difference, as would be suspected due to creation
15


of four model with the same constraints and the same net section moments of inertia, the researcher can proceed. Furthermore, the results of all these four models are within 0.3% of the theoretical beam.
1.10 Carbon Fiber Reinforced Polymer and Its Use
Carbon fiber reinforced polymer (or CFRP, henceforth) is a practical, widely available material from several manufacturers domestically and abroad. It has been used in place of rebar in concrete, due to its ability to resist corrosion in a way that steel is not capable. Generally, CFRP is not conceived of a strengthening material for applications other than concrete applications. The reasons are widely attributed to the bonding failure of CFRP and its bonding failure to a steel substrate. Some studies indicate a bond failure beginning around 0.006 strain for CFRP epoxied to steel. (Kim and Brunell 2011) In more detail: “All of the [tested] beams showed the maximum CFRP-strains ranging from 0.0061 to 0.0076 that were 36.5% to 45.5% of the rupture strain (Sfrpu = 0.0167).”
Because the performance results of CFRP have been promising, one of the goals of this study in this paper is to examine its F.E.M.
16


Table 1.1
AASHTO detail category constants and thresholds (AASHTO 2012)
Detail Constant Fatigue Fatigue
Category A (1011) Threshold Threshold
(ksi)3 (ksi) (MPa)
A 250.0 24.0 165
B 120.0 16.0 110
B’ 61.0 12.0 82.7
C 44.0 10.0 69.0
C’ 44.0 12.0 82.7
D 22.0 7.0 48.3
E 11.0 4.5 31.0
E’ 3.9 2.6 17.9
[
Web
Flanges
Figure 1.1
Typical section of a steel I-girder
Figure 1.2
Architectural elevation of the Silver Bridge showing Warren trusses and the point of failure (Lichtenstein 1993.)
17


2.4 Baie metal at the net section E
of eyebar heads or pin plates
(Note: for base metal in the
shank of eyebar* or through the
gross section of pin plates, see
Condition l.l or 1.2. as
applicable).
4.5
In the net section originating at the side of the hole
Figure 1.3
Eyebar detail (category E for fatigue)
Figure 1.4
Photo of Silver Bridge collapse (Lonaker 2006)
Figure 1.5
Diagram of a truck axle (point) loads on a simply-supported bridge (Federal Highway Administration 2012)
18


AASHTO Curves
Figure 1.6
AASHTO fatigue category curves cited in (Dexter and Fisher 1999)
8,000 lbs 32,000 lbs 32,000 lbs
CLEARANCE AND LOAD LANE WIDTH
lO'-O"
Figure 1.7
AASHTO HS-20 truck load (Federal Highway Administration 2012)
19


Stress
Number of cycles
Figure 1.8
S-N Curve for a Category B’ Fatigue Detail
Figure 1.9
Photo of the girder crack of the 1-79 Bridge at Neville Island (Purdue University 2015)
20


Girder Top Flange
Figure 1.10
(a) Detail of out-of-plane distortion (b) Photo of web gap region with crack on transverse weld (Anderson, et al. 2007)
(a) (b)
(c)
Figure 1.11 (a), (b) & (c)
Three different modeling techniques (Sotelino and Chung 2006)
21


Figure 1.12
Use of springs in modeling friction in unrestrained joint DOF’s (Kim and Nowak 2001)
22


CHAPTER II
MODEL DEVELOPMENT AND VALIDATION
A structural analysis program had to be chosen for analysis using the principles of
F. E.M. Due to its ready availability, its success in many other applications, and its continued development over thirty years, SAP 2000 was selected for the analysis. (Computers & Structures, Inc. 2013) Some have described the interface as “gooey” (or
G. U.I. - graphical user interface), meaning that there was a limited amount of control over the design parameters; however, for the purposes of this study there was sufficient allowance for varying and tailoring the key elements for further analysis of the desired system. Developing the model in SAP 2000 required using many, if not all of the predetermined features imbedded in the program.
Only simply-supported bridges were modeled in this study. This is the case for both the validations and the test modeling of the control, cracked and CFRP restored bridges. Restraints of the end supports are briefly discussed below. None of the bridges modeled included skew, though this is easily modeled in SAP 2000.
The girders of all the models used for validation and for fatigue, cracking and CFRP-strengthening studies, are represented in the program by beam-elements. The beam-element is drawn between two nodes, nodes that also define two of the four points of the shell-slab elements discussed below. The section properties can be inputted easily in SAP 2000 (See APPENDIX C.) Note that the material properties are also assigned alongside the geometric parameters. Material properties of steel, concrete and CFRP used in the modeling research are shown in Table 2.1. Again despite the limitations of the graphical interface, the ease of input makes for quick modeling, and so the next important element that is represented in SAP 2000 is the concrete deck. Four nodes each
23


with 6 degrees of freedom allowing for out-of-plane movement define the shell element: three translational, three rotational. In SAP 2000, concrete decks are modeled through the shell-elements and that method is chosen here.
The beam and the shell elements are joined together compositely in the models, meaning they interact and enhance each other’s strength and flexural resistance (the primary structural engineering property examined in this paper) as opposed to the deck not contributing anything to the resistance of the beam elements mentioned above. In order to accomplish the composite nature, there are several methods mentioned in the in APPENDIX A. The various bridge cases, both damaged and CFRP-strengthened, are modeled with an eccentric beam technique (see Model #4 on page 80). That is all the nodes of the model exist on the same xy-plane and the insertion point of the girders, represented by beam-elements noted above, is the top center of the girder(s). The center of the top flange is aligned with the soffit of the deck. The stiffness of the girder section is transformed (increased) with the offset from the base xy-plane of the model. Alternatively, the nodes of the xy-planes can be drawn parallel and offset, connected by rigid links as was discussed previously. The inputs and step-by-step guide shows this procedure in APPENDIX C.
Node meshing of the deck is of concern to the modeler, and finer meshing equaling less than 1% (~ 1 ft or 0.29 m) of the overall length of the span is chosen for the models. This refines the responses between the nodes observed in the results, producing appropriate stress and loading effects in the primary flexural members, the girders. Reducing the girder lengths of the beam-elements does not have the same kind of effects that fine meshing of the shell elements of the deck has.
24


A 3-D quality of the elements is visible in the extruded view. While this is not necessary for the calculations that the program makes, it is helpful for the user to develop an accurate model to analyze. For instance, one can quickly check in this view mode if the soffit of the deck is at or slightly above the top plane of the top flange of the girders. For the preliminary models and ultimately for test models, overhangs were modeled in all cases. During the course of the model verification, one system included the presence of both overhangs and barriers; this will be shown later in this section.
The ends of the simply-supported girders are restrained with pins and rollers. The pins allow freedom of rotation, but not translation. The roller allows both rotation in all three directions and translation in the long direction of the bridge span. As was mentioned in the previous section, springs at unrestrained degrees of freedom could be used at ends of the girders to model friction in the system; this was not done in the course of modeling in this study.
Without getting into the results of the modeling, the targeted values of the model were maximum deflection and moment. From the moment observed in the beam-elements, stress or strain could be back-calculated from a composite properties
Me
spreadsheet drafted in Excel, and then using the simple relationship, O = —. The
extreme fiber of section at the base point of the steel flange would experience the highest stress and strain in tension due to the flexural behavior of the simply supported bridge. The highest values, generally near the midspan, under the highest concentration of loading (transverse location), are sought from the modeling to investigate the effect of fatigue loads.
The loading effect was modeled through linear-static point loads applied to the
25


nodes.1 The loads are static as opposed to dynamic; analysis is linear (elastic behavior of materials) as opposed to non-linear, inelastic behavior of materials. The nodes of course are connectivity locations for the drawing of beam and shell elements. For some of the validations, the loading location were predetermined from the study. For other validations from field tests (that were compared to F.E.M. analysis), the longitudinal placement of the loads along the span was varied to produce the greatest loading effect. For this research, this maximum loading effect of the fatigue truck was determined and used for further structural analysis.
For simplicity, cracking of the girder was modeled in one dimension, which is a percentage of overall height. Several ways of logically reducing the section properties were considered, including corrosion of the steel plates, however the simplest simulation of damage was a crack. While real cracks can have all sorts of unusual geometries and propagation paths, for the purposes of modeling in SAP 2000, it made clear sense to keep the parameters simple. One anticipated consequence of this is that cracking from the undamaged state up to three percent (2.77% to be exact) constitutes the greatest reduction of capacity of the girder, breaking through the bottom (tension) flange. When the crack had broken through the flange, the crack was essentially modeled with a beam-element T-section, as opposed to I-section, having length of one inch (25.4mm). This length represents less than one-tenth of one percent of the overall length of the span. With the
1 SAP 2000 allows for the use of a moving load that can be applied to beam-elements of the structure model, either statically or dynamically with varying speed. The moving loads could be defined with truck loads separated by fixed or variable trailing axles. This was done on a 92-ft beam in conjunction with a simple trial-and-error method. It is important to note that in SAP 2000 moving load can only be applied to beam elements, whereas the modeling technique used required placement at shell-only node (due to the fixed width of the axle not necessarily corresponding to the girder spacing). Thus, moving loads were only used to get an indication of where along the length of the span the axle loads should start in order to get the maximum loading effect.
26


eccentric beam model, the sharp spike in stress can be observed at the short beam-element where the section is reduced to a T-beam. Discontinuity of the bottom flange in a cracked model is shown in Figure 2.1.
The increase of stress at the point of damage can be easily determined. When pinned from the top flange of the girder (see APPENDIX C), and without interference from shear deformation, the sections will show the characteristic increases in stress.
Stress range of the bridge section will be then used to compare with the fatigue threshold of the assumed category detail; here in this study the category is B’ with a threshold stress of 12.0 ksi (82.7 MPa).
Later in the analysis CFRP (Carbon Fiber-Reinforced Polymer) was used in the model as strengthening. See Figure 2.2 for an extruded view of the CFRP modeling and see Table 2.1 for material properties of concrete, steel and CFRP used in this study. Materials such as 4,000 psi (27.58 MPa) concrete and ASTM A992 Steel are preloaded and easy to use within SAP 2000 modeling. However, CFRP is not an available material to model with in the program. Therefore the material is created from another material (steel, concrete, it’s not important), and the material properties such as the modulus of elasticity, rupture and yield stresses, Poisson’s ratio, etc., are all manipulated to match the properties of CFRP (given by a manufacturer.) Also critical to the modeling is that the CFRP is attached to the eccentric beam model (of the composite concrete slab and steel girder assembly) by means of rigid links. It is not possible in SAP 2000 to have a compound beam elements between two nodes; the program recognizes one section type or another, not both.
27


2.1 Beam Comparison: SAP-2000 to Theory
If the F.E.M. software cannot predict the bending of a simple beam, let alone a bridge model, it might be best to seek another program or method. For this reason, a comparison to beam theory is sought. Consider the basic problem:
A simply-supported beam of 40 feet (12.2 meters) is subjected to a 100-kip (444.8 kN) point-load at midspan. The type of rolled steel section chosen was a W21xl32, which is a 21-inch (0.533 m) deep beam with a self-weight of 132 pounds per liner foot (1.93 kN/m). The AISC gives the moment of inertia of this section of 3200 inch4 (American Institute of Steel Construction 2011), through the strong axis (where the beam is loaded with the flanges normal to the load direction.) Beam theory gives the maximum deflection [in inches] from the following formula:
PI3 100 kips (40 x 12m)3
Am„T=------=-----------------------— = 2.47 in
max 48El 48 (29,000/rst)(3200m4)
This value was compared with the basic finite-element model computed in SAP-2000. One simple beam with a pre-loaded W21xl32 section was drawn to length of 40 feet (12.2 m), with pinned and roller joint restraints. It was loaded with 100 kips (444.8 kN) at midspan. After computation, the program gave the maximum deflection to be 2.54 inches (6.45 centimeters), bringing the calculation to within 3% of the theoretical value. The user output of the beam deflection from SAP 2000 is shown in Figure 2.3. The producers of SAP 2000 note that with regard to this discrepancy “that SAP 2000 calculations produce slightly greater values because shear deformation is considered in deflection.” (CSi Knowledge Base 2013) Because the error was so small, it is reasonable to proceed with the analysis in this paper. This is summarized in Table 2.2.
28


2.2 Full-Scale Model Validation: Nebraska Bridge
With a beam section investigated and confirmed, it is necessary to compare the F.E.M. with a full-scale test and compare deflections due to live loading, as above. A full-scale laboratory test bridge was constructed at the University of Nebraska in 1995 to examine the loading effects on a composite steel girder bridge. (Kathol, Azizinamini and Luedke 1995) The study was of sufficient quality to attract future researchers; (Sotelino and Chung 2006) used the Nebraska results and compared them with their own F.E.M. Sotelino determined the most accurate modeling of composite steel girder bridges.
Several of these developments were previously discussed in Chapter I, and APPENDIX A summarizes the modeling comparisons. Therefore, the Nebraska bridge was useful not only for the modeling validation, but also for the model development previously discussed.
The Nebraska bridge is used as a reference point for validation and verifying the accuracy of the SAP 2000 bridge modeling. Specifically, the experimental and modeling values of deflection were compared. There are a few unique features of both the laboratory Nebraska bridge and the SAP 2000 model developed here. The Nebraska Bridge had rails and overhangs in the full-scale test and these were modeled as concrete beam-elements with the appropriate section dimensions. (See Figure 2.9.) In the model, they were attached at the centroids by rigid links, normal to the principle bending axis. Overhangs were accomplished by widening the shell element areas of the deck. Figure 2.4 shows the Nebraska bridge with the rails and rigid links, as well as the overhangs, modeled.
The Nebraska test-bridge was also loaded differently than a standard AASHTO
29


truck, fatigue or otherwise. The goal of the Nebraska testing was to determine the ultimate loading capacity; therefore, two “trucks” or sets of loads were placed on the bridge. Furthermore, the standard axle loads were scaled up two and a half times (2.5), which were well beyond the standard load factors. This scaling made the responses of the test bridge more salient. There were two 10-kip (44.48 kN) loads at each front axle tire, and four 40-kip (177.9 kN) loads at each trailer tire. Figures 2.7 and 2.8 show the loading locations for the Nebraska bridge.
(Kathol, Azizinamini and Luedke 1995) use four feet as the distance between side-by-side vehicles; which is standard. Of greater effect is the longitudinal spacing in the direction of the span. Between the first and second axles on the Nebraska test there was a two-foot distance reduction. Also the variable length between the second and third axles (14 to 30 feet; 4.27 to 9.14 m) on the loading truck(s) was given the advantageous placement of 15 feet (4.57 m). When the point loads are placed closer to the midspan of the bridge, higher bending moments and lower flange stresses occur. The Nebraska test took advantage of this basic static equilibrium fact, in order to measure the ultimate capacity of the bridge. The non-standard reduction of two feet is reasonable and helps to ensure a sufficiently large moment for testing is imparted on the supporting girders.
The seventy-foot span that was modeled in SAP 2000 also had to have accurate section modeling for the girders. The Nebraska test built three 54-inch (1.37 m) deep girders, with 3/s-inch (9.525 mm) web, flanges 9 by 3/4 inches (228.6 by 19.05 mm) at the top and 14 by VA inches (355.6 by 31.75 mm) at the bottom. At 17 'A feet (5.33 m) from the bridge-end supports, where less moment resistance is necessary, the bottom flanges were spliced to a reduced thickness of % inch (19.05 mm). This upsizing for increased
30


moment capacity of the steel girder at the middle region is shown in the two sections depicted in Figure 2.6. Table 2.3 shows all of the geometric parameters of the Nebraska test that are used in the verification for modeling.
To mimic the wheel loading of the two trucks on the Nebraska test-bridge (12) twelve l3/s inch post-tensioning (DYWIDAG or Dyckerhoff & Widmann AG) rods were used. The test bridge live-loading deflection was almost exactly 3/4 of an inch (0.749 inches = 19.025 mm). (Kathol, Azizinamini and Luedke 1995) In comparison, when the SAP 2000 model developed here was executed, 0.797 inches (20.23 mm) was computed for the midspan deflection due to all the 10 and 40 kip (44.5 and 177.9 kN) loads indicated. The results of the verification are within 6% agreement and concur with the previous research at Nebraska. Furthermore, Sotelino and Chung’s results for maximum deflection from their modeling also are close to 20 mm and agree with both the Nebraskan testing and the results in this thesis. (Sotelino and Chung 2006) Centerline deflection at midspan is a convenient way to establish the flexural characteristics of either a real in-service bridge or a 3-D F.E.M. With this agreement established here, the modeling approach taken in this research shows validity.
2.3 Full-Scale Model Validation: Flint, Michigan Bridge
The University of Michigan, on behalf of MDOT, conducted field tests on an in-service steel girder bridges. (Nowak and Eom 2001) The study took place across the state of Michigan, but the bridge that is the focus here is located in Flint, MI over Interstate 75 along Stanley Road. The span is 38.4 meters and has seven steel girders (S=2.2m), with an out-to-out width of 14.75 meters. The slab is 20 cm thick, and is
31


assumed to be 4000 psi (27.58 MPa) concrete. Barriers are not modeled, but overhang of the deck is included in the composite section. There is no skew, and the bridge is modeled as simply-supported. Girder geometry is shown in Figure 2.10 (note that these girders are also upsized for increased moment capacity between the 7.3 meter and 31.1 meter marks of the span) Other properties are shown in Table 2.4. Girder geometry is obtained from Sotelino’s study, which used Nowak’s field test for her own validations. (Sotelino, et al. 2004)
The results of Nowak’s field test are recorded in microstrain, with a maximum of 150 microstrain. The results are valuable to validation procedures because the truck was larger than a standard AASHTO vehicle. Nowak notes “In Michigan, the maximum midspan moment for medium span bridges is caused by 11-axle trucks, with gross vehicle weight up to 730 kN depending on axle configuration.” (Nowak and Eom 2001) GVW or “Gross Vehicle Weight” while being a convenient shorthand for truckers and DOT officials, it is not completely useful for structural engineers as the location of a load is consequential, as is the magnitude. The specific locations of loading are noted in Figure 2.12, and correspond to the location inputted into the models. From the loading pattern indicated in Figures 2.11 & 2.12, one can easily foresee that the greatest loading effect will occur in the center girder. Indeed, that is where Nowak records the highest value of 150 microstrain.
Sotelino also used this study to verify one of her models in F.E.A. In her modeling, the micro-strain also is 5 to 10% higher than the observed values in the field. (Sotelino, et al. 2004) The discrepancy in this study is 8.7%, and therefore compares well with the past tests and modeling of this particular bridge. With three validations of
32


modeling completed, the modeling technique and development described at the beginning of the Chapter II shows sufficient accuracy.
2.4 Developing a Fatigue-Prone Model
When the stress range observed in a bridge model is greater than 12 ksi (82.7 MPa), B’ category details will have finite fatigue life. The loads due to a fatigue truck (8 kips, 32 kips, 32 kips or 35.5 kN, 142.3 kN, 142.3 kN) remain constant, but the loading effects will vary depending on the bridge geometry. Dead loads do not contribute to the stress range of member, as only live or fatigue loading can only do so by definition. Since that was the case, the ways to increase flexural stress in the bridge were to modify the geometry of the structure.
Perhaps the most obvious way to increase the loading effect is to lengthen the span. The number of girders can be reduced, or the space between them can be increased. The section of the girders themselves can be reduced; particularly the lower flange can be reduced in both width and thickness. Increasing the depth of the girder will also increase the moment observed in the bridge model in SAP 2000, if all other variables are held constant. When developing the model for the control bridge, these strategies were employed to have a fatigue-prone bridge model.
As has been noted (Dexter and Fisher 1999), after the mid-1970s, D, E and E’ categories are largely avoided in design, if not eliminated. Depending on the category detail, between two and twenty million cycles, infinite life of that detail is assumed. At two million cycles on a Category-A element, fatigue resistance is the highest of all details with stress ranges of 24 ksi (165 MPa). This stress range was not observed in the course
33


of modeling bridges, so lower categories had to be considered. Category-B’ was promising, since a stress range of 12 ksi (82.7 MPa), as noted, had to be demonstrated in the model. Again, due to their presumed removal from most bridges in the past 40 years, details D and E were not considered.
From some of the early models completed on SAP 2000, stress in the lower flange from a fatigue-truck-1 oaded bridge was typically not observed above 10 ksi (69 MPa). This will be shown in Chapter III. In fact, in order to produce stress much above 8 or 9 ksi (55.2 or 62.1 MPa) in composite section, the span needs to be beyond 100 feet (30.5 m), and the sections undersized (lower flange thickness = 3/4 inch =19 mm). It is so much so that there is significant deflection not only due to the dead load, which can be modified and compensated for in design, but more critically with the live load which can produce a large deflection of 4+ inches (100+ mm). This is unsuitable for driving, with a deflection to span ratio of 1/300. The AASHTO optimal limit is 1/800. While structurally, the bridge will not necessarily fail and has well beyond the sufficient strength not to collapse, deflection of this kind is not psychologically reassuring for the (truck) driver and other present vehicles experiencing the unnecessary dip.
34


Table 2.1
Material properties used in study models
E F(c) or Fy Fu Poisson
Concrete 3605 ksi 4000 psi n/a 0.2
24,900 MPa 27.6 MPa n/a
Steel (A992 29,000 ksi 50 ksi 65 ksi 0.3
Grade 50) 200 GPa 345 MPa 448 MPa
CFRP2 23,900 ksi n/a 449 ksi 0.25
165 GPa n/a 3.10 GPa
Table 2.2
Summary of beam validation
Parameter Dimension/ Value
Beam Length 12.192 m
Beam Self-Weight 1.93 kN/m
Moment of Inertia 0.00133 m4
Esteel 200 GPa
Depth 0.533 m
Load Subjected to 444.8 kN
Load Location 6.096 m (mid-span)
Node Insertion Bottom of Flange
Supports Pin/roller
Results:
Beam Theory Deflection 6.274 cm
SAP 2000 Model Deflection 6.452 cm
Discrepancy 2.8%
2 (Sika Corporation 2011)
35


Table 2.3
Full-scale model validation summary, Nebraska
Parameter Dimension/Value
Span Length 21.336 m
Bridge Deck Width 7.925 m
Number of Girders 3
Girder Spacing 3.048 m
Deck Thickness 190.5 mm
Concrete f 'c (deck) 41.37 MPa
Concrete f 'c (rails) 44.82 MPa
Overhang(s) 0.914 m
Concrete Bridge Rails (b h) 357 x 406 mm
Girder Depth (total) 1.372 m
Esteel 200 GPa
Web Thickness 9.525 mm
Top Flange 19.05 x 228.6 mm
Bottom flange (middle 50% of span) 31.75 x 355.6 mm
Bottom flange (end quarters of span) 19.05 x 355.6 mm
Load Locations See Figure 2.8
Results:
Max Observed Deflection 19.025 mm
SAP 2000 Model Deflection 20.23 mm
Discrepancy: 6.3%
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Table 2.4 Summary of Flint, MI bridge model validation
Parameter Dimension/Value
Span Length 38.4 m
Bridge Deck Width 14.75 m
Number of Girders 7
Skew None
Girder Spacing 2.2 m
Deck Thickness 200 mm
Concrete f 'c (deck) 27.58 MPa
Esteel 200 GPa
Overhang(s) 0.75 m
Overall Girder Depth (center 23.8 meters of span) 1.34m
Overall Girder Depth 1.28 m
(7.3 meters from ends of span)
Web Thickness (all sections) 12.7 mm
Top flange (center 23.8 meters of span) 48 x 457 mm
Bottom flange (center 23.8 meters of span) 70 x 457 mm
Top flange (7.3 meters from ends of span) 22 x 457 mm
Bottom flange (7.3 meters from ends of span) 35 x 457 mm
Load Locations See Figure 2.12
Nowak’s observed strain 150 ps
Results:
SAP 2000 Max Moment 1305 kN-m
Section Stress 32.61 MPa
Section Strain 163 ps
Discrepancy: 8.7%
37


Figure 2.1
Soffit, extruded view of 5% crack modeled in steel girder from SAP 2000
Figure 2.2
Soffit view of CFRP-strengthened bottom flange
38


Deflection of a W21xl32 in SAP-2000
Figure 2.4
Standard view of the Nebraska bridge model in SAP 2000
39


229
1
19 f
229
â–¡ â–¡


G3 G2 G1 914 3048 3048 i914
- n r t n 7925 r -
19
i
T
_L
T
190
9.5-
1372
19
OUTER 1/4 SPANS
Figure 2.6
Nebraska bridge and girder dimensions (mm)
Figure 2.7
SAP 2000 aerial view of Nebraska bridge with loading pattern
40


21336
7925
â–  177.9 kN = 40 kips
Figure 2.8
Nebraska bridge 70-ft span (21.3m) with loading weights and locations
355.6

406.4
330.2
190.5 1
Figure 2.9
Nebraska rail dimensions in (mm), modeled as structural element
41


203

m
G7 G6 G5 G4 C 3 G2 G1
747 2210 2210 2210 2210 2210 2210 747
* *r *l r t" r t *r n r *
—*
14754
OUTER 7.3 m
CENTER 23.8 m
Figure 2.10 Flint, MI bridge section and girder dimensions (mm)
42
H h


Figure 2.11
SAP 2000 aerial view of Flint, MI bridge with loading pattern
38405
14754
Figure 2.12 Loading configuration (mm) for the Michigan bridge validation, axle weights (kN) shown between wheels
43


CHAPTER III
ANALYSIS AND RESULTS
3.1 The Control Bridge
The control bridge consisted of an approximately 28-meter (92’-0”) steel-girder, simply-supported span. The extruded view of the 3-D model is shown in Figure 3.1. A 190 mm (IV2 inch) deck was modeled compositely with the four girders, using the frame insertion offset method discussed in the previous model development section. As the modeling techniques were also verified in Chapter II and the results prove to be most consistent with a composite-deck model, the best practice was to proceed accordingly. The deck consisted of 27.6 MPa (4,000 psi), normal-weight concrete; the steel was A992 Grade 50 (344.7 MPa). A full set of input global geometric and sectional properties is listed in Tables 3.1, 3.7 and 3.8, as well as in more detail in APPENDIX B.
The girders of the control bridge are built-up members, not standard AISC sections. They would include the welds (and base metal of the A992 steel) that are indicated in AISC’s fatigue design guides, especially including the B’ detail defined earlier. No further problematic details are assumed in the control girder, such as cover plates and transverse welds. Therefore, it is safe to assume that the fatigue threshold of 82.7 MPa will govern the member. The dimensions of the uncracked Control Girder are shown in Figure 3.2.
In order to investigate the phenomena of fatigue within the 3-D model, the bridge is subjected to an HL-93 fatigue truck (axle loads of 8, 32 and 32 kips, or 35.6, 142.3 and 142.3 kN). Full inputs and geometries are given in APPENDIX B. The maximum loading effect from the fatigue truck was determined. Since the axle width did not match
44


the girder spacing, it was necessary to apply loads to nodes that only had shell (area) elements assigned. The maximum loading effect did not occur at midspan for the control bridge, as is visible in Figure 3.3. In terms of transverse placement of the loads, it is assumed that a 0.61-meter offset would prevail from the exterior edge of the area (deck); note that the overhang is 0.91 meters. (See Figure 3.4.) This is a reasonable assumption for fatigue loading since these types of loads would have to occur repeatedly to qualify as a fatigue load. Loading at the very edge of slab structure is unlikely or infrequent. Furthermore, it was not the objective of the modeling to design the overhang (or barriers), but rather to investigate the girder stresses and their susceptibility to fatigue and fracture.
In order to calculate stresses at the extreme fibers of the sections concerned, classic formulation for bending is assumed. Therefore stress at that extreme fiber is
Me
simply from the equation, a = —. The sectional properties, given by the moment of
inertia, I, are varied from the control case as various cracking and fatigue states are investigated. For instance, the control girder (depth = 45 inches = 1143 mm) possesses a moment of inertia of 15900 in4 (6.62 x 10"3 m4). However, when composite section properties are calculated, the net section moment of inertia increases to 41690 in4 (0.01735 m4). The maximum resulting stress for this composite condition is 3.49 ksi (24.1 MPa).
3.2 Cracked Sections
Of course, a stress of ~25 MPa is too low to induce fatigue in all but the lowest category of fatigue-resistant details. For this reason, damage has been assumed in the steel girder, at a point of maximum loading effect. Several ways of logically reducing the
45


section properties were considered, including corrosion of the steel plates, however the simplest simulation of damage was a crack. This was further simplified by considering the crack as a percentage of height, giving this parameter only one dimension. While real cracks can have all sorts of unusual geometries and propagation paths, for the purposes of modeling in SAP 2000, it made clear sense to keep the parameters simple. Due to the geometry of the control bridge, a crack assumed as such in the exterior girder would produce the highest stresses, where the loading effects were determined to be the most extreme. While modeling the crack in one of the interior girders would yield worthwhile results, in this study the emphasis is on the exterior girder.
While the typical, undamaged girder, supported by pin and roller, consists of 92 ^ 4 +1 = 24 nodes, the damage was chosen to occur very near midspan at 48.0 feet (550 in = 14.63 m). The section is modeled with an inch-long (1.0 in = 25.4 mm) member, either an I-girder or a tee, depending on the level of damage. It is straightforward to set the insertion point at the top center of the member, rather than the confusion of the changing height of the centroids of the various sections (See APPENDIX C.)
Generally, the highest stresses and moments on the bridge occur beneath the first 32-kip (142.3 kN) axle, or more accurately the 16-kip (71.2 kN) wheel load directly above. When the load is applied to the model with the damage simulated as above, the highest stresses and moments are found to be in the damaged section. Table 3.2 and Figure 3.7 show the expected results of higher stresses as the crack propagates. It should be noted that the stress of the material in a section where a crack tip is propagating will theoretically have infinite stress. It is assumed that the point of integration for the stress calculation is just off the boundary condition.
46


3.3 Fatigue life of damaged sections
A big jump also occurs as expected when the crack fully penetrates the lower flange and begins to propagate up the web. These stresses put the damaged section in the range of a detail vulnerable to fatigue at the relatively-high resistive Category B’ (=82.7 MPa =12.0 ksi).
The number of loading cycles is given by N, and fatigue life is calculated with
A
N=(AF)3 (Barker and Puckett 2013). The other variables are as follows: A is the detail
category constant shown previously in Table 1.1 and AF„ is the nominal fatigue resistance (here it is the fatigue threshold of 82.7 MPa for the B’ Detail.) The stresses are calculated in cracked sections in Table 3.5, the fatigue life of the section as the crack increases is compiled in Figure 3.8.
While the flange is still intact (with less than a 2.77% crack), the section has infinite life. The stress range is below the threshold of 82.7 MPa at values of 17.1 MPa, 27.8 MPa, and 54.1 MPa, respectively for 0%, 1% and 2% cracking. However when the crack propagates to 3%, the effectiveness of the section is severely compromised with respect to fatigue resistance, if not ultimate strength (evaluated later in Section 3.6) Here, the fatigue life loses close to 70,000 cycles of resistance for each percentage increase in the crack height. With the total number of cycles being well under a million cycles, even a road with a low ADTT (Average Daily Truck Traffic) will have but a few years to persist against fatigue in the cracked condition.
For each percentage in increase in the affected section, a model was created or adjusted from the control case, and executed to produce the resulting live moment. Each
Me
of moments was then used to back-calculate stress using the simple equation: a = —.
47


(Contributions in the y-direction and z-directions are minimal.) Stress could then be used to relate the global geometry of the bridge to the fatigue life, resulting in the graph above. Furthermore, the information from this graph can be used to convert cycles of life, into years of life, using independent information, like the ADTT. Here at this stage of analysis, the observed live-load moment increase and simultaneously the sectional properties are reduced, resulting in the higher stress and lower fatigue lives.
In order for the situation to be remedied, external reinforcement or other restorative processes must be conducted. Stress will continue to degrade the material condition and cause the crack to grow, as suggested in this modeling. The next section will demonstrate that it is possible to not only to restore actual bridges, but their respective models.
3.4 CFRP Strengthening
The development of a crack in a steel structure is not the end of its potential serviceability, but strengthening methods must be employed at this point. In the study, carbon-fiber reinforced polymer (CFRP) was materially modeled and used in the cracked bridge models. The CFRP is commercially available (Sika Corporation 2011) in sheets or strips that have characteristic dimensions, 1.2 mm by 100 mm, although to some degree these can be manipulated in the manufacturing process, in order to best suit the intended use. The dimensions of the CFRP sections that are used in modeling reinforcement, to be applied to the underside of the cracked girder flange, will vary according to the number of plies attached to bottom.
In this study, four levels of strengthening were used to incrementally demonstrate
48


the improvement of overstressed conditions. The basic level of CFRP was selected as a six-inch wide strip that was 1/10 inch (2.54 mm) thickness. The total area of the CFRP section is then 0.60 in2 or 387 mm2. The next case was created by doubling the width of the CRFP strip and increasing its thickness by 50% at the same time (1161 mm2). A third case of strengthening consisted of a width that matched the width of the girder’s lower flange (14 in = 355.6mm), and doubled the thickness of the base level of CFRP to 2/10 inch (5.08 mm). The cross section area of the third level of strengthening was 1806 mm2. The most strengthened case also had the same width as the bottom flange and the thickness was doubled to 4/10 inch (10.16 mm.) Thereby, the cross-sectional area of the fourth case was twice that of the third case, equaling 3613 mm2. According to the manufacturer, the standard thickness of CFRP strips is 1.2 mm, so several plies of the material would have to be applied in actuality.
The four levels of CFRP-strengthening demonstrate incrementally lower stress ranges from the damaged condition. Adding the CFRP sheet presents another element to be included in the composite moment of inertia calculation. In the model, CFRP is applied to the soffit of the girder across the length of approximately 90% of the span that is 84 feet (25.6 m) long, centered at midspan. In actuality, the CFRP would require end anchoring as a failsafe against potential debonding. Debonding could happen through natural decay or strain exceeding debonding strain. Figure 3.12 shows the reduction of stress due to the four cases of CFRP strengthening.
In comparison to the damaged case, the additions of CFRP, take the stresses in the extreme fibers of the cracked steel section into stress ranges around or below the infinite life point. The case of 1161 mm2 provided reinforcement puts the model very close to
49


being under the critical value of 82.7 MPa (12 ksi) for Category B’ infinite life. In fact, as the crack fully penetrates the lower flange the stress value slips past 83 MPa. Thus, the next level of CFRP of 1806 mm2 studied will allow the steel member to stay below the stress range of infinite life.
3.5 Fatigue Life Improvement
Similar to the calculations above to determine the fatigue life for the cracked section, the fatigue life of the CFRP-strengthened sections was determined. The fatigue lives of the four cases of the strengthening are shown in Figure 3.13 against the damaged case. When the lower flange has not fully cracked, the section (of category B’) remains with infinite fatigue life; the graphed lines extend up to infinity when the line is traced to the left. However, for cracking greater than 2.77%, the first two levels of strengthening (387 mm2 and 1161 mm2) relieve and extend fatigue life. The higher two cases of strengthening (1806 mm2 and 3613 mm2) extend fatigue life for a Category B’ to infinity for all levels of cracking examined in modeling. The higher cases of strengthening do not appear on the graph.
3.6 Flexural Capacities of Sections
To determine flexural capacities of sections, calculations are done through spreadsheets, not SAP 2000 calculations. SAP 2000 does not compute moments of inertia compound sections like CFRP-strengthened members, let alone for composite bridge sections. It does however compute moments of inertia for individual defined frames and stresses in beam elements.
50


The process for calculating moment of capacity of any of the composite sections begins with the determination where the plastic neutral axis (PNA) lies. The test of whether the PNA lies within the depth range of the slab is determined by the inequality:
Pslab ~ (j3compression flange "F Pweb "F Ptension flange F PcFRP) ^ 0 If the answer is a positive number, then indeed the PNA will lie in the slab. Pcfrp is the plastic force in the CFRP, which is limited by the crushing of the concrete (this will be explained in more detail below.) Henceforth, Ps is the plastic force in the slab; Pc is the plastic force in the compression flange; Pw is the plastic force in the web and Pt is the plastic force in the tension flange. Where ts represents the thickness of the slab.
Next, y or PNA is calculated:
(U) [Pc + Pw + Pt + Pcfrp] y =--------------P-------------
FS
Using y, the nominal moment capacities of (CFRP-strengthened) sections is calculated with the following formulas:
(y 2)(PS)
Mn - Mp - ——---------1- [Pcdc + Pwdw + Ptdt + PcFRpdcFRp]
Depth to the particular element is given by dx.
If Ps — (Pc + Pw + Pt + Pcfrp) < 0, then the PNA lies below the slab. If the sum of the plastic forces of the slab and the top flange exceeds the sum of the plastic forces of the web, the bottom flange and the CFRP, the PNA will lie in the depth range of the top (compression) flange:
(Ps + Pc) ~ (Pw + Pt + Pcfrp) > 0
51


If the PNA lies in the top (compression) flange, the formulas for Mn and y are
slightly modified:
y =
(!)
^c\ [Pw "f Pt ~f PcFRP Ps] 2) Pc
+ 1
Here y is the distance below the slab (and below the 31.75 mm non-structural haunch). The procedure for the calculation of Mn continues in this manner, however none of the cases examined in this study had a PNA below the top (compression) flange in the web region.
The procedure above for calculation of Mn is used not just for CFRP-strengthened cases; it is also used for the control and cracked (unstrengthened) cases. The sections without CFRP are easier to calculate, because of the elimination of one term (two terms if the section’s lower flange is completely cracked). Unlike steel as structural material where a yielding range is known and well-defined, the rupture of the CFRP is unknown and sudden. Because of this, the crushing of concrete (0.003) in an equilibrium section limits the flexural capacity of that section, and the full strength of the CFRP cannot be assumed. To determine the strain of CFRP at crushing of concrete, an iterative technique was employed to simultaneously determine the PNA and strain of the CFRP. Generally, the stress in the CFRP for these sections was less than half of the rupture stress. The strain of the CFRP at the crushing of concrete is shown in Figure 3.11. These values and their corresponding stresses were used for the calculation of the flexural capacities of CFRP-strengthened sections.
52


Mn improves incrementally with each level of strengthening, as shown in Figure 3.14. The most significant losses of capacity occur, as expected, as the lower flange cracks completely. The most-strengthened case of 3613 mm2 CFRP increases the moment capacity of a damaged section to exceed the control case as is seen from the diagram.
3.7 Load and Resistance Factor Rating
The LRFR - Strength I rating system has two different variants: Inventory and Operating. The LRFR (rating) is calculated using the Mn capacity determined in the previous section and the flexural loading effects, in addition to various reduction factors:
Capacity-1.25 (Dead Loading)
Rating Factor =-------------------------------------
y (Live Loading+Dynamic Loading)
where Capacity = (pc(ps(pRMn.
(AASHTO 2011)
The live load factor (yLL) is 1.75 for inventory and 1.35 for operating. Dynamic loading
(IM) is taken as 33% of the live (truck) loading, as provided by Table 3.6.2.1-1 from (AASHTO 2012). The other resistance, condition and system factors are shown in Table 3.9. The calculations of the rating factors for all modeling conditions are shown in Figures 3.15 & 3.16.
While LRFR (ratings) in theory need only be greater than 1.0, in practice, they typically are greater than 3.0. Here the LRFR (rating) for the control bridge is around 5.0, but it should also be noted that this bridge was not loaded for ultimate capacity, but loaded instead for fatigue resistance.
53


3.8 LRFR Fatigue
Developing a Fatigue Rating (Figure 3.17) for the bridge models consists of a ratio of the applied live load stress to the fatigue threshold (AF)TH of the desired category.
For the category B’ detail, the fatigue threshold is 82.7 MPa. The dynamic loading (IM) is lowered to 15% of the live loading (truck); the live load factor (yLL) is 0.75 for fatigue.
(AASHTO 2012) The dead load moments or stress is not a variable in this calculation, as it is in the Strength I - LRFR Rating. The fatigue-rating factor for LRFR is given by the following equation:
RFfatigue—
(_AF)th
A/lL+ZM
where AfLL+IM is the live loading plus the dynamic loading.
3.9 Load Factor Rating
Another method for evaluating the operating and inventory conditions of a bridge is the LFR, Load Factor Rating. (Results shown in Figures 3.18 & 3.19.) Inventory designation covers the bridge for an indefinite period of service (set it and forget it), while the operating condition evaluates the maximum permissible load. (AASHTO 2011). The equation is as follows:
C—AiD
T>P’ — ______±__
LFR A2L{1+1)
As with the more current LRFR system, C represents capacity or Mn, but does not have any reduction factor associated with it. D represents the dead loading effect (MDL) and
does have a factor of A i that is 1.3 for both operating and inventory conditions. L
54


represents live loading effect (MLL), where d 2 is 1.3 for operating and 2.17 for inventory.
The / or the impact factor, which accounts for speed, vibration and so forth, is defined through another equation:
3.10 Modal Responses
One important characteristic of a structure or model to observe is the modal responses. For this study, the first five fundamental modal frequencies were recorded from SAP 2000 for the control, cracked, and cracked and restored conditions. There is not much variation between the crack and control cases, and similarly little variation from the damaged and the CFRP-strengthened cases. The data in Tables 3.10 & 3.11 shows the narrow range of the results, with little change in fundamental and modal frequencies as crack length propagates. In addition, the modal shapes are depicted in Figures 3.20-24 to show the dynamic behavior of the model. The only control cases are presented, as the modal frequencies vary only very slightly.
where L is the length of the span in feet (92 ft).
55


Table 3.1
Control bridge dimensions
Parameter Number Dimension Note
Span Length Single 28.04 m Simply-supported
Girders 4
Girder Spacing 2.44 m
Supports 4 per end For four girders
Bridge Width 9.14 m
Design Lanes 2 3.05 m
Overhangs 2 0.91 m
Barriers None Not considered
Deck Thickness 0.19 m
Haunch Depth 3.2 mm Non-structural
Table 3.2
Control girder dimensions
Parameter Dimension
Overall Height 1.14m
Web Height 1.09 m
Web Thickness 7.94 mm
Top Flange Width 305 mm
Top Flange Thickness 25.4 mm
Bottom Flange Width 356 mm
Bottom Flange Thickness 31.75 mm
56


Table 3.3
Damaged girder dimensions in mm (without CFRP)
Parameter Dimension
Bottom Flange Width (Constant) 305 mm
Bottom Flange Thickness:
Damage 1% 20.3 mm
Damage 2% 8.9 mm
Web Height:
Damage 3% 1083 mm
Damage 4% 1072 mm
Damage 5% 1060 mm
Damage 6% 1049 mm
Damage 7% 1038 mm
Damage 8% 1026 mm
Damage 9% 1015 mm
Damage 10% 1003 mm
Table 3.4
Stress range as a function of crack percentage (of girder height), fatigue truck loading
% ksi MPa
0 3.49 24.1
1 1 4.96 34.2
Flange 2 8.45 58.3
Web 3 18.42 126.5
1 4 19.05 131.3
5 19.70 135.8
6 20.38 140.5
7 21.10 145.5
8 21.84 150.6
9 22.61 155.9
10 23.42 161.5
57


Table 3.5
Damaged girders: fatigue life (category B’ detail)
% N
0 CO
1 CO
2 CO
3 1,050,000
4 983,000
5 923,000
6 864,000
7 809,000
8 756,000
9 707,000
10 663,000
Table 3.6
CFRP dimesions (for damaged girders)
Parameter Dimension Note
CFRP Width:
Level 1 152 mm Modular sizes
Level 2 305 mm
Level 3 356 mm Full-flange width
Level 4 356 mm Full-flange width
CFRP Thickness:
Level 1 2.54 mm Modular thicknesses
Level 2 3.81 mm
Level 3 5.08 mm
Level 4 10.16 mm
CFRP Area:
Level 1 387 mm2 Modular sizes (standard)
Level 2 1161 mm2
Level 3 1806 mm2
Level 4 3613 mm2
58


Table 3.7
Girder sectional properties: transformed area (mm3)
Strengthening Level
Damage % None 1 2 3 4
0 78,458 n/a n/a n/a n/a
1 74,393 74,800 75,616 76,296 78,199
2 70,329 70,735 71,552 72,232 74,135
3 67,149 67,555 68,372 69,052 70,955
4 67,058 67,464 68,281 68,961 70,864
5 66,966 67,374 68,189 68,869 70,772
6 66,875 67,284 68,098 68,778 70,681
7 66,784 67,193 68,007 68,687 70,590
8 66,693 67,097 67,916 68,596 70,499
9 66,602 67,009 67,825 68,505 70,408
10 66,512 66,917 67,735 68,415 70,318
Table 3.8
Girder sectional properties: moment of inertia (10"3 m3)
Strengthening Level
Damage % None 1 2 3 4
0 17.4 n/a n/a n/a n/a
1 13.1 13.8 14.7 15.5 17.5
2 8.5 11.1 12.0 12.8 16.7
3 4.6 5.1 6.2 7.1 9.5
4 4.5 5.0 6.1 7.0 9.4
5 4.3 4.9 6.0 6.9 9.3
6 4.2 4.8 5.9 6.8 9.2
7 4.1 4.7 5.8 6.7 9.1
8 4.0 4.6 5.7 6.6 9.0
9 3.9 4.5 5.6 6.5 8.9
10 3.8 4.4 5.5 6.4 8.8
59


Table 3.9
LRFR rating factors
Strength I
Resistance factor:
CPr 1.0 for flexure
Condition factor:
(pc
(pc
System factor: (ps
Load factors:
Y
Y
Y
DL
LL
LL
1.0 for good condition, control case 0.95 for fair condition, cracked cases
0.9 for a 3 or 4 girder steel bridge, simply-supported 1.25
1.75 (Inventory)
1.35 (Operating)
Table 3.10
Modal frequencies for the unstrengthened bridge models
Crack Frequency (Hz)
0% 3.2105
1% 3.2102
2% 3.2099
3% 3.2096
4% 3.2093
5% 3.2090
6% 3.2087
7% 3.2084
8% 3.2081
9% 3.2078
10% 3.2075
60


Table 3.11
Modal frequencies of the first five modes
of a 10% cracked girder, with strengthening cases
Mode No.
Bridge Case: 1 2 3 4 5
Unstrengthened 3.207 3.722 5.513 8.480 11.716
387 sq mm CFRP 3.216 3.732 5.517 8.480 11.736
1161 sq mm CFRP 3.230 3.756 5.524 8.498 11.769
1806 sq mm CFRP 3.241 3.776 5.530 8.517 11.791
3613 sq mm CFRP 3.254 3.791 5.536 8.532 11.818
61


Figure 3.1
The control bridge
Figure 3.2
Control bridge and girder dimensions (damage location indicated for variable cases)
62


30
25 -
Figure 3.3
Control bridge: stress in lower flange of exterior girder
Figure 3.4 (a)
SAP 2000 aerial view of the control bridge with loading pattern
63


28042
8534
4267
9144
6096
9144
Loading configuration producing most adverse effects in the control bridge,
dimensions in mm and kN (not to scale)
25 , 305 J 25
* r *1 _L
T' l T
1109 (3%)
1097 (4%)
1086 (5%)
8 -*• 1074 (6%)
1063 (7%)
1052 (8%)
1040 (9%)
1029 (10%) 20
9

Immmmmmmmm T
305
(2%)
356
Figure 3.5
Assigned dimensions in millimeters for damaged girder sections.
(Damage occurs at 14.63-meter location at exterior girder)
(Tension flange is absent, indicated with dotted lines, for crack percentage > 2.77 %.)
64


Figure. 3.6
Maximum stress of 5% cracked girder, exterior
Figure 3.7
Maximum stress as a function of crack percentage, fatigue loading
65


oo |
in
O
k'a
u
• i-H
bo
• i-H
ts
Ph
1.300.000 -|
1.200.000 -1,100,000 -1,000,000 -
900.000 -
800.000 -
700.000 -
600.000 -
500,000 -
0 2 4 6
Crack %
—I
10
Figure 3.8
Category B’ detail fatigue life vs. crack percentage
305
JL|
T
305
h
CFRP
(varies)
_L T
CFRP
(varies)
1143
CFRP
(varies)
Figure 3.9
Assigned dimensions in mm for CFRP-strengthened girders. (Yellow for CFRP; tension flange absent for crack percentage > 2.77%.)
66


< =â–  Control
— . . CFRP 3613 sqmm CFRP 1806 sq mm
— — — CFRP 1161 sqmm
-----CFRP 387 sq mm
-----Unstrenthened
Figure 3.10
Neutral axis depth, from top of slab
-----CFRP 387 sq mm
-----CFRP 1161 sqmm
-----CFRP 1806 sq mm
— . . CFRP 3613 sqmm
Figure 3.11
CFRP strain at concrete crushing
67


Fatigue life (cycles)
Unstrengthened
------CFRP 387 sq mm
------CRFP 1161 sq mm
------CFRP 1806 sq mm
-----CFRP 3613 sq mm
Figure 3.12
Stress range vs. crack %, with 4 cases of CFRP strengthening (387 mm2, 1161 mm2, 1806
mm2 and 3613 mm2)
7.000. 000 -|
6.000. 000 -
5.000. 000 -
4.000. 000 -
3.000. 000 -
2.000. 000 -1,000,000 -
0 -
0
oo |
Unstrengthened CFRP 387 sq mm
----CFRP 1161 sqmm
n-----1------1------1-----1
2 4 6 8 10
Crack %
Figure 3.13
Fatigue life vs. crack percentage, control with 387 mm2 and 1161 mm2 cases of CFRP-strengthening. 1806 mm2 and 3613 mm2 cases have infinite fatigue life.
68


12000 n
a
0 -I-----,-----1------1------i------1
0 2 4 6 8 10
Crack %
-----CFRP3613 sq mm
------CFRP 1806 sqmm
------CFRP 1161 sqmm
------CFRP 387 sq mm
- Unstrengthened
Figure 3.14
Calculation of Mn (capacity) of girders, strengthened and control.
■ =» Control
— . • 3613 sqmm
— — — 1186 sqmm
— — 1806 sq mm
- 387 sq mm
- Unstrengthened
0% 2% 4% 6% 8% 10%
Crack
Figure 3.15
Inventory rating factor (LRFR Strength)
69


0% 2% 4% 6% 8% 10%
Crack
â–  Control
— ■ • 3613 sq mm
------ 1806 sq mm
------1186 sq mm
------387 sq mm
------Unstrengthened
Figure 3.16
Operating rating factor (LRFR Strength)
S
3
btj
03
Ph
Figure 3.17
Fatigue rating factor (LRFR)
70


Crack
Figure 3.18
Load Rating Factor (Inventory)
Control
--- • • 3613 sq mm
--- — 1806 sq mm
— — —1186 sq mm
-------387 sq mm
Unstrengthened
0% 2% 4% 6% 8% 10%
Crack
Figure 3.19
Load Rating Factor (Operating)
71


Figure 3.20
Mode 1: T =0.3115 seconds, f =3.210 Hz
Figure 3.21
Mode 2: T = 0.2684 seconds, f =3.725 Hz


Figure 3.22
Mode 3: T = 0.18130 seconds, f = 5.516 Hz
Figure 3.23
Mode 4: T= 0.11791 seconds, f =8.481 Hz
73




CHAPTER IV
CONCLUSIONS
The addition of CFRP as a strengthening material adds capacity to a structure and its members. The process of quantifying such an increase is not as straightforward. The results of this F.E.A. study, using the speed and ease of the modeling programs available, show that not only can a cracked section be modeled, but also its can be strengthened within the model. A research objective of investigating and understanding the effects of fatigue for a bridge and its model was achieved in the literature review, modeling studies and development, and test trials.
Modeling discovery was critical at the outset to the research. There is a steep learning curve for learning the software of SAP 2000, perhaps even greater with other less user-friendly structural engineering or finite-element analysis software. Upon arrival at proficiency, the results of modeling cannot be accepted at face value; they must be checked and verified against past accepted research and accurate results. Models were successfully compared against results from researchers such as Nowak, Sotelino, et. al.
In addition, agreement with beam theory was found. By achieving consistency of approach, the models used in the studies were validated. The degree of accuracy was +2.8% (beam theory), +6.3% (Nebraska), +8.7% (Michigan); furthermore due to shear deformation and lack of system friction noted in Chapter I, this very slight overshooting was predicted.
After validation, a control bridge was established with a consistent modeling technique. The control bridge allowed for various manipulations such as the modeling of cracking and strengthening. As expected, cracking reduces capacity and strengthening augments it. Stress in the damaged sections are significantly increased. In the highest
75


case of strengthening of 3613 mm3 full recovery of moment capacity is achieved. At the same time, the neutral axis of the composite section is lowered, stiffening the system.
The improvement due to CFRP restore the bridge to its original LRFR and LFR ratings. Fatigue ratings do not reach the same degree of recovery, but fatigue life attains infinity for the two higher strengthening cases (1806 and 3613 mm3.) Only these two higher cases of applied CFRP take the rating factor above 1.0 (when the lower flange is fully cracked). The rating dips slightly below 1.0 as the crack propagates up the web for the 1806 mm2 of strengthening.
In the future, researchers could consider developing (phi) OR-factors for CFRP in (re)strengthening design of damaged or fatigue-prone structures. One obstacle exists: sufficient studies of the coefficient of variation (CoV) of cracked steel do not exist. In order to develop accurate phi-factors, the CoV of cracked steels needs to show an increase as crack length increases. General categories of damage would not be sufficient. After a CoV for cracked steel is established, Monte Carlo simulations can be completed to develop a phi factor for CFRP-strengthening. Other possibilities for further research could include varying the locations of damage or the type of crack.
In summation, fatigue studies and finite-element modeling whether through SAP 2000 or another F.E.M. program can yield valuable results. The bridge researcher or restoration engineer best understand not only the procedures of inputting data into the programs, but also be knowledgeable in the theory behind the software. This thesis has been a step in the direction of understanding structural modeling, and much more is needed to produce qualified structural engineers to tackle the most difficult problems that lie ahead.
76


REFERENCES
AASHTO. 2012. LRFD Bridge Design Specifications. Washington, DC: American Association of State Highway and Transportation Officials.
AASHTO. 2011. Manual for Bridge Evaluation. Washington, DC: American Association of State Highway and Transportation Officials.
Akesson, Bjorn. 2010. Fatigue Life of Riveted Steel Bridges.
American Institute of Steel Construction. 2011. Steel Construction Manual. 14th Edition. Chicago: AISC.
Anderson, B., S. Rolfe, C. Matamoros, C. Bennett, and S. Bonetti. 2007. Post-Retrofit Analysis of the Tuttle Creek Bridge Br. No. 16-81-2.24. SM Report No. 88, Lawrence: The University of Kansas Center for Research, Inc.
Barker, Richard M., and Jay A. Puckett. 2013. Design of highway bridges: An LRFD approach. 3rd Edition. Hoboken: John Wiley & Sons.
Brockenbrough, R. L. 1986. "Distribution Factors for Curved I-Girder Bridges." Journal of Structural Engineering (ASCE) 110 (10): 2200-15.
Computers & Structures, Inc. 2013. "SAP2000 vl6.2, Integrated Software for Structural Analysis and Design." Walnut Creek.
CSi Knowledge Base. 2013. Composite Section. Edited by Ondrej Kalny. Computers and Structures, Inc. Accessed March 2015.
https://wiki.csiamerica.com/display/tutori als/Composite+section.
1999. "Fatigue and Fracture." Chap. 53 in Handbook of Bridge Engineering, by Robert J. Dexter and John W. Fisher, edited by Wai-Fah Chen and Lian Duan. CRC Press.
Eom, Junsik, and S. Andrzej Nowak. 2001. "Live Load Distribution for Steel Girder Bridges." Journal of Bridge Engineering (ASCE) 489-497.
Federal Highway Administration. 2012. Bridge Inspector's Reference Manual. Vol. 1,
chap. 5, "Bridge Mechanics" and 6, "Bridge Materials", 1234-2468. Washighton, D.C.: U.S. Department of Transportation.
Fisher, John W., Geoffrey L. Kulak, and Ian F. C. Smith. 1998. A Fatigue Primer for Structural Engineers. National Steel Bridge Alliance.
Jaramilla, Becky, and Sharon Huo. 2005. Looking to Load And Resistance Factor Rating. Vers. Vol. 69 No.l. FHwA. July/August. Accessed March 2015. https://www.fhwa.dot.gov/publications/publicroads/05jul/09.cfm.
77


Kathol, Steve, Atorod Azizinamini, and Jim Luedke. 1995. Final Report: Strength Capacity of Steel Girder Bridges. Lincoln: Nebraska Department of Roads.
Kim , Yail J., and Garrett Brunell. 2011. "Interaction between CFRP-repair and initial
damage of wide-flange steel beams subjected to three-point bending." Composite Structures (93): 1986-1996.
Kim, Sangjin, and Andrzej S. Nowak. 2001. "Load Distribution and Impact Factors for I-Girder Bridges." Journal of Bridge Engineering (ASCE) (Nov/Dec): 489-97.
Lichtenstein, Abba G. 1993. "The Silver Bridge Collapse Recounted." ASCE: Journal of Constructed Facilities 7: 249-261.
Lonaker, Timothy. 2006. Silver Bridge Collapse. Accessed March 2015.
http://www.freewebs.com/silverbridgeaccident/thebridgecollapse.htm.
Lui, Ming, Dan M. Frangopol, and Kihyon Kwon. 2010. "Fatigue Reliability Assessment of Retrofitted Steel Bridges Integrating Monitored Data." Structural Safety 77-89.
Mabsout, Mounir E., Kassim M. Tarhini, Gerald R. Frederick, and Charbel Tayar. 1997. "Finite-Element Analysis of Steel Girder Highway Bridges." Journal of Bridge Engineering {August): 83-87.
Nowak, Andrezj S., and Junsik Eom. 2001. Verification of girder distribution factors for steel girder bridges. Final Report, Department of Civil and Environmental Engineering, University of Michigan, Lansing: Michigan Department of Transportation.
Petroski, Henry. 2012. To Forgive Design: Understanding Failure. Cambridge, MA: Harvard University Press.
Purdue University. 2015. Steel Bridge Fatigue: Knowledge Base. Local Technical
Assistance Program. Accessed March 2015. http://rebar.ecn.purdue.edu/fatigue.
Sika Corporation. 2011. "Sika CarboDur, Edition 5.4." pds-cpd-SikaCarboDur-us.pdf http://usa.sika.com/en/home-page-features/product-finder/ifram eanddropdown/carb odur. html.
Sotelino, Elisa D., Judy Liu, Wonseok Chung, and Kitjapat Phuvoravan. 2004. Simplified load distribution factor for use in LRFD design. Final Report, School of Civil Engineering, Purdue University, Indianapolis: Indiana Department of Transportation.
Sotelino, Elisa, and Wonseok Chung. 2006. "Three-Dimensional Finite Element
Modeling of Composite Girder Bridges." Engineering Structures (Elsevier) (28): 63-71.
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Timoshenko, Stephen, and J. N. Goodier. 1970. Theory of Elasticity. 3rd Edition. McGraw-Hill.
79


APPENDIX A
EIGHT WAYS TO MODEL IN SAP 2000
(Computers & Structures, Inc. 2013)
Table Al:
Actual Section Modeling in SAP2000 Legend
Model 1 - fictitious noncomposite
(frames and shells are drawn at the elevation of
girder centroid sharing the same joints)
Model 2 - composite
(frames and shells drawn the elevation of girder centroid sharing the same joints; shell joint offsest are used to place the deck above the girder)
Model 3 - composite
(frames and shells are drawn at the elevations of their respective centroids and connected using body constraints; separate body constraint is used for each pair of connected joints)
Model 4 - composite
(frames and shells are drawn the elevation of deck centroid sharing the same joints; frame joint offsets and top center insertion points are used to place the deck above the girder)

Shell Joint Offset
t
] Slab
Shear Studs (Symbol for Composite Behavior)
Girder
0 Slab Location
as Drawn in SAP2000
• Girder Location
as Drawn in SAP2000
Symbol for Links, Constraints or Joint Offsets
Model 5 - noncomposite
(frames and shells are drawn at the elevations of their respective centroids and connected using equal constraint in Z direction; separate equal constraint is used for each pair of connected joints)

Equal Z Constraint
Model 6 - noncomposite
(frames and shells are drawn at the elevations of their respective centroids and connected using links that are fixed in vertical direction and free for all other degrees of freedom)
z
Link for Noncomposite Behavior
Model 7 - partially composite (frames and shells are drawn at the elevations of their respective centroids and connected using links that are fixed in vertical direction, have stiffness in girder longitudinal direction and are free for all other directions)
Link for Partially Composite Behavior
Model 8 - composite
(frames and shells are drawn at the elevations of their respective centroids and connected using fixed links)

Fixed Link for Composite Behavior
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Figure A1
Model 1, a Fictitious Rendering in SAP 2000 (Nebraska Bridge)
Section of a Composite Beam Model
81


Table A2:
Beam Designation Behavior Midspan Deflection [mm] Comments
Theoretical Beam composite 3.2725 Theoretical deflection is based p-i3 on the formulation. 48-E-I Please note that SAP 2000 calculations produce slightly greater values because shear deformation is considered in deflection.
Beam 1 (top beam) Non- composite 7.1752 The deck-slab center line coincides with the section neutral axis. Therefore, the deck-slab contribution to section flexural stiffness will be negligible. Further, because there is no composite action, midspan deflection should be close to that of a naked girder.
Beam 2 composite 3.2624 In this model, slab shell objects are drawn at the girder center of gravity (COG), and then offset vertically, above the girder, to model composite action. The shells are offset such that the slab soffit is located above the girder top flange.
82


Table A2 (continued):
Beam Designation Behavior Midspan Deflection [mm] Comments
Beam 3 composite 3.2624 In this model, the girder and the slab are drawn at their respective center-lines. The corresponding girder and slab joints are then connected through body constraints.
Beam 4 composite 3.2624 In this model, composite action is modeled using frame insertion points.
Beam 5 Non- composite 7.1752 Equal constraints are used to model non-composite behavior.
Beam 6 Non- composite 7.1752 Links are used to model noncomposite behavior.
Beam 7 partially composite 3.5036 Links are used to model partially composite behavior.
Beam 8 (bottom beam) composite 3.2624 Links are used to model composite behavior.
83


APPENDIX B
SAP 2000 INPUTS FOR CONTROL BRIDGE
(Computers & Structures, Inc. 2013)
1. Model geometry
This section provides model geometry information, including items such as joint coordinates, joint restraints, and element connectivity.
Figure 1: Finite element model (Computers & Structures, Inc. 2013)
1.1. Joint coordinates
Table 1: Joint Coordinates
Joint Coord. Sys Coord. Type Global X feet Global Y feet Global Z feet
1 GLOBAL Cartesian 0.00 0.00 4.167
2 GLOBAL Cartesian 4.00 0.00 4.167
3 GLOBAL Cartesian 4.00 6.00 4.167
4 GLOBAL Cartesian 0.00 6.00 4.167
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Joint
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
Table 1: Joint Coordinates
Coord. Sys Coord. Type Global X feet Global Y feet Global Z feet
GLOBAL Cartesian 8.00 0.00 4.167
GLOBAL Cartesian 8.00 6.00 4.167
GLOBAL Cartesian 12.00 0.00 4.167
GLOBAL Cartesian 12.00 6.00 4.167
GLOBAL Cartesian 16.00 0.0 4.167
GLOBAL Cartesian 16.00 6.00 4.167
GLOBAL Cartesian 20.00 0.0 4.167
GLOBAL Cartesian 20.00 6.00 4.167
GLOBAL Cartesian 24.00 0.0 4.167
GLOBAL Cartesian 24.00 6.00 4.167
GLOBAL Cartesian 28.00 0.0 4.167
GLOBAL Cartesian 28.00 6.00 4.167
GLOBAL Cartesian 32.00 0.0 4.167
GLOBAL Cartesian 32.00 6.00 4.167
GLOBAL Cartesian 36.00 0.0 4.167
GLOBAL Cartesian 36.00 6.00 4.167
GLOBAL Cartesian 40.00 0.0 4.167
GLOBAL Cartesian 40.00 6.00 4.167
GLOBAL Cartesian 44.00 0.0 4.167
GLOBAL Cartesian 44.00 6.00 4.167
GLOBAL Cartesian 48.00 0.0 4.167
GLOBAL Cartesian 48.00 6.00 4.167
GLOBAL Cartesian 52.00 0.0 4.167
GLOBAL Cartesian 52.00 6.00 4.167
GLOBAL Cartesian 56.00 0.0 4.167
GLOBAL Cartesian 56.00 6.00 4.167
GLOBAL Cartesian 60.00 0.0 4.167
GLOBAL Cartesian 60.00 6.00 4.167
GLOBAL Cartesian 64.00 0.0 4.167
GLOBAL Cartesian 64.00 6.00 4.167
GLOBAL Cartesian 68.00 0.0 4.167
GLOBAL Cartesian 68.00 6.00 4.167
GLOBAL Cartesian 72.00 0.0 4.167
GLOBAL Cartesian 72.00 6.00 4.167
GLOBAL Cartesian 76.00 0.0 4.167
GLOBAL Cartesian 76.00 6.00 4.167
GLOBAL Cartesian 80.00 0.0 4.167
85


Table 1: Joint Coordinates
Joint Coord. Sys Coord. Type Global X feet Global Y feet Global Z feet
42 GLOBAL Cartesian 80.00 6.00 4.167
43 GLOBAL Cartesian 84.00 0.0 4.167
44 GLOBAL Cartesian 84.00 6.00 4.167
45 GLOBAL Cartesian 88.00 0.0 4.167
46 GLOBAL Cartesian 88.00 6.00 4.167
47 GLOBAL Cartesian 92.00 0.0 4.167
48 GLOBAL Cartesian 92.00 6.00 4.167
240 GLOBAL Cartesian 92.00 27.00 4.167
241 GLOBAL Cartesian 0.0 -3.00 4.167
242 GLOBAL Cartesian 4.00 -3.00 4.167
243 GLOBAL Cartesian 8.00 -3.00 4.167
263 GLOBAL Cartesian 88.00 -3.00 4.167
264 GLOBAL Cartesian 92.00 -3.00 4.167
1.2. Joint restraints Table 2: Joint Restraint Assignments
Joint U1 U2 U3 R1 R2 R3
1 Yes Yes Yes No No No
47 No No Yes No No No
50 Yes Yes Yes No No No
72 No No Yes No No No
98 Yes Yes Yes No No No
120 No No Yes No No No
146 Yes Yes Yes No No No
168 No No Yes No No No
1.3. Element connectivity
86


Table 3: Connectivity - Frame
Frame Joint I Joint J Length feet
1 1 2 4.00
2 2 5 4.00
4.00
92 167 168 4.00
93 170 25 2.083
94 169 170 0.083
Table 4: Frame Section Assignments
Frame Analysis Section Design Section Material Property
1 45" girder 45" girder Default
2 45" girder 45" girder Default
94 45" girder 45" girder Default
Table 5: Connectivity - Area
Area Jointl Joint2 Joint3 Joint4
1 1 2 3 4
2 2 5 6 3
229 262 263 45 43
230 263 264 47 45
Table 6: Area Section Assignments
Area Section Material Property
1 deck Default
2 deck Default
230 deck Default
2. Material properties
This section provides material property information for materials used in the model.
87


Table 7: Material Properties 02 - Basic Mechanical Properties
Material Unit Unit Mass El G12 U12 A1
Weight Kip/in3 Kip-s2/in4 Kip/in2 Kip/in2 1/F
4000 psi 8.680E-05 2.248E-07 3604 1502 0.200 5.500E-06
A992 Fy=50 2.835E-04 7.344E-07 29000 11153 0.300 6.500E-06
Table 8: Material Properties 03a - Steel Data
Material Fy Kip/in2 Fu Kip/in2 Final Slope
A992 Fy=50 50.00 65.00 -0.100
Table 9: Material Properties 03b - Concrete Data
Material Fc Final
Slope
Kip/in2
4000Psi 4.000 -0.100
3. Section properties
This section provides section property information for objects used in the model.
3.1. Frames
Table 11: Frame Section Properties 01 - General, Part 1 of 4
Section Material Shape t3 t2 tf t\v t2b tfb
Name
inch inch inch inch inch inch
45" girder A992 Fy=50 I/Wide Flange 45.00 12.00 1.00 0.3125 14.00 1.250
88


Full Text

PAGE 1

MODELING OF FRACTURED STEEL GIRDERS STRENTHENED WITH CFRP SHEETS by WILLIAM ERIC BODENSTAB, JR . B.A., Williams College, 2000 A thesis submitted to Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the d egree of Master of Science Civil Engineering Program 201 6

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ii This thesis for the Master of Science Degree by William Eric Bodenstab, Jr. has been approved for the Civil Engineering Program by Kevin L. Rens, Chair Y ail J. Kim, Advisor Cheng Li Date: December 17 th , 2016

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iii Bodenstab, William Eric, Jr. (M.S., Civil Engineering) Modeling of Fractured Steel Girders Strengthened with CFRP Sheets Thesis directed by Professor Yail J . Kim ABSTRACT Computer modeli ng of bridge structures might seem deceptively easy to the novice, untrained in the field of structural engineering. Howe ver, to those who have studied the discipline, it is clear that modeling is not as simple as build ing blocks. The mathematically deri ved relationships are at the backbone of a well developed three dimensional, finite element analysis computer program. It is aim of this paper to provide the educated structural engineer a background in computer modeling, while investigating a real life i ssue, fatigue. Perhaps one of the reasons that steel has fallen out of favor in Colorado for use in girders of bridges is presence of fatigue in many of the primary and secondary elements within a bri dge structure. Reinforced, pre stressed or cast in plac e concrete has become a dominant material in bridge construction, partly be cause it is much more forgiving building material, and is generally not vulnerable to fatigue. Yet many structures require the us e of steel for economic reasons or due to design co nstraints. While there are few cases of fracture for steel fatigue details greater than category C, it is still imperative to model damaged structures using computer programs and accurately determine stresses. This thesis investigates cases of cracked and CFRP strengthened girders of bridges modeled in a three dimensional finite element analysis program. It includes tabulated and graphed results, along with sectional analysis of the technic al data. This

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iv thesis also includ es calculation s of flexural capac ity, modal fr equencies and LRFR (r ating factors , Inventory and Operating . ) The form and content of this abstract are approved. I recommend its publication. Approved: Y ail J . Kim

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v ACKNOWLEDGEMENTS I am very grateful to faculty of the Civil Engineer ing Department at the University of Colorado, Denver, including Y. Jimmy Kim , Cheng Li , Kevin Rens , N.Y. Chang , Fred Rutz and Rui Liu. I would not have become proficient enough in civil and structural engineering to undertake this thesis without their ins truction, knowledge and experience. I am grateful for Roxanne Pizan the requirements and w i t h meeting the deadlines for this degree. My family and friends were also instrumental to my completion of this thesis. My wife, son, parents, close friends and employer made big sacrifices of their time and resources during my time in the program . I am blessed to live in this G reat C ountry , allow ing its citizens the freedom to pursue their dreams, providing us with unlimited opportunity. Eric Bodenst ab November 2016 Denver, Colorado

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vi TABLE OF CONTENTS CHAPTER I . Introduction ................................ ................................ ................................ .............. 1 1 .1 Modeling with F.E.A. Software ................................ ................................ .... 1 1.2 The Typical Section ................................ ................................ ...................... 2 1.3 Silver Bridge Collapse ................................ ................................ .................. 3 1.4 Definition of Fatigue ................................ ................................ ..................... 5 1.5 Redundancy and Load Shedding ................................ ................................ .. 9 1.6 Other Types of Fatigue ................................ ................................ ................ 10 1.7 Literature Review: Modeling in Finite Element Programs ......................... 12 1.8 Literature Review: Sources of Error in 3 D Modeling ............................... 13 1.9 Li terature Review: Various Modeling Approaches and Their Accuracy .... 14 1.10 Carbon Fiber Reinforced Polymer and Its U se ................................ ......... 16 I I . Model Dev elopment and Validation ................................ ................................ ...... 23 2 .1 Beam Comparison: SAP 2000 to Theory ................................ ................... 28 2.2 Full Scale Model Validation: Nebraska Bridge ................................ .......... 29 2.3 Full Scale Model Validaiton: Flint, Michigan Bridge ................................ 31 2.4 Developing a Fatigue Prone Model ................................ ............................ 33 I I I . Analysis and Results ................................ ................................ ............................ 44 3 .1 The Control Bridge ................................ ................................ ..................... 44

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vii 3.2 Cracked Sections ................................ ................................ ......................... 45 3.3 Fatigue life of damaged sections ................................ ................................ . 47 3.4 CFRP Strengthening ................................ ................................ ................... 48 3.5 Fatigue Life Improv ement ................................ ................................ .......... 50 3.6 Flexural Capacities of Sections ................................ ................................ ... 50 3.7 Load and Resistance Factor Rating ................................ ............................. 53 3.8 LRFR Fatigue ................................ ................................ .............................. 54 3.9 Load Factor Rating ................................ ................................ ..................... 54 3.10 Modal Responses ................................ ................................ ...................... 55 I V . Conclusions ................................ ................................ ................................ .......... 75 R E F E R E N C E S ................................ ................................ ................................ ........... 77 APPENDIX A ................................ ................................ ................................ ............. 80 APPENDIX B ................................ ................................ ................................ ............. 84 APPENDIX C ................................ ................................ ................................ ............. 93

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viii LIST OF TABLES TABLE 1.1 AASHTO Detail c ategory c onstants and t hresholds ................................ ......... 17 2. 1 Material properties used i n study models ................................ .......................... 35 2. 2 Summary of beam validation ................................ ................................ ............ 35 2. 3 Full scale model validation summary, Nebraska ................................ ............... 36 2. 4 Summary of Flint, MI bridge model validation ................................ ................ 37 3.1 Control b ridge dimensions ................................ ................................ ............... 56 3.2 Control g irder dimensions ................................ ................................ ................. 56 3.3 Damaged girder dimensi ons (without CFRP) ................................ ................... 57 3.4 Maximum s tress as a f unction of c rack % (of g irder h eight), fatigue t ruck l oading ................................ ................................ ......................... 57 3.5 Fatigue l ife of a c ................................ ... 58 3.6 CFRP dimesions (for damaged girders) ................................ ............................ 58 3.7 Girder sectional properties: transformed area (steel, mm 3 ) .............................. 59 3.8 Girder sectional properties: moment of inertia (10 3 m 3 ) ................................ .. 59 3.9 LRFR rating factors ................................ ................................ .......................... 60 3.10 Modal f requencies for the unstrengthened bridge models .............................. 60 3.11 Modal f requencies of the first five m odes of a 10% cracke d girder with strengthening cases ................................ ................................ .......................... 61

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ix LIST OF FIGURES FIGURE 1. 1 Typical section of a steel I g irder ................................ ................................ ...... 17 1. 2 Architectural elevation of the Silver Bridge s howing Warren t russes and the point of failure ................................ ................................ ............................. 17 1.3 Eyebar detail ( c ategory E for fatigue) ................................ .............................. 18 1. 4 Photo of Silver Bridge c ollapse ................................ ................................ ......... 18 1. 5 Diagram of a truck axle (point) loads on a simply supported bridge ............... 18 1.6 AASHTO f atigue c ategory c urves ................................ ................................ ..... 19 1. 7 AASHTO HS 20 tr uck load ................................ ................................ ............. 19 1. 8 S N c urve for a c f atigue d etail ................................ ....................... 20 1. 9 Photo of th e girder crack of the I 79 Bridge at Neville Island ......................... 20 1. 10 (a) Detail of out of plane distortion ................................ ............................... 21 1. 10 (b) Photo of web gap region with crack on transverse weld .......................... 21 1. 11 (a), (b) & (c) Three different modeling techniques ................................ ........ 21 1. 12 Use of springs in ................... 22 2.1 Soffit, extruded view of 5% crack modeled in steel girder from SAP 2000 ................................ ................................ ................................ . 38 2. 2 Soffit view of CFRP strengthened bottom flange ................................ ............ 38 2. 3 Deflection of a W21x132 in SAP 2000 ................................ ............................ 39 2. 4 Standard view of t he Neb raska b r idge model in SAP 2000 ............................. 39 2.5 Extruded view of the Neb raska b ridge model in SAP 2000 ............................. 39 2.6 Neb raska b ridge and girder dimensions (mm) ................................ ................. 40 2. 7 Aerial view of loaded Nebraska b ridge model in SAP 2000 ............................ 40

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x 2. 8 Nebraska b ridge 70 ft s pan (2 1.3m) with loading weights and locations ......... 41 2. 9 Bridge r ail dimensions in (mm), modeled as structural element ...................... 42 2. 10 Flint, MI b ridge s ection and g irder d imensions (mm) ................................ ..... 42 2.1 1 SAP 2000 aerial view of Flint, MI b ridge with loading pattern ..................... 43 2. 1 2 L oading con fig uration for the Flint, MI b ridge validation, axle weights shown between wheels ................................ ................................ .................... 43 3.1 The c ontrol b ridge ................................ ................................ ............................. 62 3. 2 Control b ridge and g irder d imensions ( d amage location indicated for variable cases) ................................ ................................ ................................ ... 62 3. 3 Control b ridge: s tress in lower fl ange of exterior girder ................................ .. 63 3.4 (a) SAP 2000 aerial view of the c ontrol b ridge with loading pattern ................ 63 3. 4 (b) Loading configuration producing most adverse effects in the c ontrol b ridge , d imensions in mm and kN ( n ot to scale) ................................ .......... 64 3. 5 Assigned dimensions in millimeters for d amaged g irder sections .................... 64 3. 6 Maximum stress of 5% cracked girder, exterior ................................ ............... 65 3. 7 Maximum s tress as a f unction of c rack percentage , fatigue l oading ................. 65 3. 8 d etail f atigue l ife vs. c rack percentage ................................ ......... 66 3. 9 Assigned d imens ions in mm for CFRP strengthened girders ........................... 66 3.10 Neutral a xis d epth , from top of slab ................................ ................................ 67 3.11 Strain of CFRP at crushing of concrete ................................ ........................... 67 3. 12 Stress r ange vs. crack percentage , with four cases of CFRP strengthening (387 mm 2 , 1161 mm 2 , 1806 mm 2 and 3613 mm 2 ) ................................ .......... 68 3. 13 Fatigue life vs. crack percentage, control with 387 mm 2 and 1161 mm 2 cases of CFRP strengthening. 1806 mm 2 and 3613 mm 2 cases have infinite fatigue life. ................................ ................................ .......................... 68 3. 14 Calculation of M n capacity of girders , strengthened and control .................... 69

PAGE 11

xi 3. 15 Invent ory r ating f actor (LRFR Strength) ................................ ......................... 69 3. 16 Operating r ating f actor (LRFR Strength) ................................ ........................ 70 3. 17 Fatigue ra ting f actor (LRFR) ................................ ................................ ........... 70 3. 18 Load Rating Factor (Inventory) ................................ ................................ ....... 71 3. 19 Load Rating Factor (Operating) ................................ ................................ ...... 71 3. 20 Mode 1: T =0.3115 seconds, f =3.210 Hz ................................ ....................... 72 3. 21 Mode 2: T = 0.2684 seconds, f =3.725 Hz ................................ ...................... 72 3. 22 Mode 3: T = 0.18130 seconds, f = 5.516 Hz ................................ ................... 73 3. 23 Mode 4: T= 0.11791 seconds, f =8.481 Hz ................................ ..................... 73 3. 24 Mode 5: T = 0.08535, f =11.716 Hz ................................ ................................ 74

PAGE 12

xii COMMON ABBREVIATIONS 3 D Three dimension al AASHTO American Asso ciation of State Highway and Transportation Officials ADTT Average Daily Truck Traffic AISC American Institute of Steel Construction BIRM Bridge Inspector's Reference Manual CFRP Carbon Fiber Reinforced Polymer CoV Coefficient of Variation DOF Degree(s ) of Freedom F.E.A. Finite Element Analysis F.E.M. Finite Element Modeling LFR Load Factor Rating LRFR Load and Resistance and Factor Rating PNA Plastic Neutral Axis

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1 CHAPTER I I NTRODUCTION 1.1 Modeling with F.E.A. Software It is surprising to disco ver that there is actually limited academic research within the field of structural engineering on the problems encountered in finite element modeling (F.E.M.) , even with the array of computer programs that have been available for the last 30 years. An ai m of this study is to develop a helpful guide for stud y in structural engineering , while using three dimensional F.E.M. computer programs. One might ask if there are user manuals for these programs, which have become a necessary part of engineering prac tice. There are indeed , but the function of the program s user manual is equivalent to a cookbook. They sh ow the procedures of gaining familiarity with the software , and the steps needed to produce a model with basic output s of reactions and forces. Howe ver, s oftware developers do not generally address in the manuals how the programs work . S tructural engineering research public ations are also limited in this respect . Therefore, developing a guide for future studies in this area will be helpful for all t hose who follow. Fatigue in steel bridges will be frequently referenced and investigated throughout the course of this thesis and an attempt will be made to merge these copious studies with the Finite Element Analysis (F.E.A . ) software available to the mo dern engineer. Beginning with the very basics of both structural engineering and the procedures of F.E.A. software, and evolving into some of the esoteric problems of steel fatigue and the intricacies of the supporting software, the reader will develop un derstanding of the core principles of F.E.A. and fatigue . The remainder of Chapter I will focus more so the

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2 history of fatigue studies in structural engineering, while Chapter II : Model Development and Validation will introduce computer modeling that is i ntegral to this and many studies. Chapter II also is integral to the overall study in that the computer models that are used as the backbone of the investigations are validated against previous work of past researchers. Chapter III : Analysis and Results presents the bulk of the F.E.A. studies and models cracked and CFRP strengthened, composite steel girder bridges. Chapter IV draws conclusions from the studies. Short Appendices follow to allow the student of uts and some of the calculations in the process. 1.2 T he T ypical S ection In structural engineering, the basis of all design is the type of section one uses to support the loads that are externally applied to the structure and also to support the self wei ght of the structure. If a section is too small or narrow and experiences too much stress, then nine times out of ten the designing engineer sizes up the section. An ancient example of such a practice can be seen in the monument at Stonehenge. The gir th of both the post s and lintels far exceed the necessary required resistance of the column axial loads, buckling considerations and flexural strength s needed to stand its own weight, lateral forces such as wind or earthquakes, and so on . The focus in t his paper is steel, a far stronger material than stone, and especially so in tension . A couple types of sections will be discussed; however, one key type of section is the basis of consideration for the investigation in this paper. The so takes its name from the shape of the capital letter. A girder is essentially a beam,

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3 but will often support other beams that are perpendicular in plan. I girders are typically made of steel for bridge spans over 20 feet (a so n I shaped profile, is used for precast concrete girders . ) There have been different specified steel s for this common type of girder over the past hundred years, and concurrently the American Society for Testing and Materials (now ASTM International) has k ept pace through A36 Steel . Presently , A992 Grade 50 Steel is the most common. Regardless of the type of steel used, the I girder has kept its characteristic geom etry. Along with the given span length of a beam or girder, the section profile is the other key parameter in determining how a beam will function or behave. An I girder section (shown in Figure 1.1 ) is comprised of two key elements: a set of two flange s joined perpendicularly with a in the case of a plate girder bridge, welded together. 1.3 Silver Bridge Collapse To briefly review some of the history of the problem of fatigue, one might go to the year 1967 when the Silver Bridge collapsed . Steel bridges had been in place for 80 years by then, and had of course spanned some of the most scenic waterways. ( See F igure 1.2 for an architectural elevation of the S ilver B ridge. ) dates from 1927 , when construction began to provide a passing over a key point along the Ohio River . Unlike many large over water crossing bridges of today, it was erected in a little over a year. It was known also as the Point Pleasant Bridge, but also called ilver B (Lichtenstein 1993.) Various design alternatives were

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4 considered, but chain link, Warren type trusses ultimately comprised the main spa n suspension structure. The eyebar element (51 c m thick) was the key connection element , and its detail is shown in F igure 1 .3 , as well as it is visible in the wreckage of F igure 1. 4 . As d esign and construction of the chain w as such that it brought together two very dangerous elements extremely high tensile stresses and (Lichtenstein 1993.) D uring the holiday season of that year there was an increase in lo ading due to heavy traffic, and furthermore, the bridge had undergone 40 years of use and corrosion. Loads were significantly higher than in the days of the 1200 lbs. Model T. T ruck loads, which are the principle governing loads for fatigue, had also inc reased substantially. Despite the fact that the engineers specified 75 ksi (517 MPa) steel , i t could not support the loads of the heavy traffic in its fatigue weakened state. A small (¼ inch) defect at the point noted progressed rapidly or instantaneousl y in to a full crack, and the entire structure fell suddenly into the Ohio River, taking 46 lives along with it. It was later determined that t he eyebar s (only one fractured in this case) were the breaking point i n the structure. Since the 19 through t oday, these elements have been classified in the fatigue category of most concern to the bridge engineer, which are categories E and E . These categories hav e l argely been avoided since their introduction . Certainly, after the tragic collapse, fatigue ha d caught the full attention of researchers and designers alike. The National Bridge Inspection Standards was established not too long after the Silver Bridge failure, in order to prevent future accidents of this kind. It may be hard to imagine how a seem ingly sufficient sized section of a super -

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5 structure could suddenly split, sending it sinking into the river . C onsidering that all engineering takes place between the stressed and over stressed condition , the margin for error can be razor thin when dealing with sensitive elements such as the eyebar. Capacity of the structure then is paramount in order to provide the necessary resistance to the forces and stresses put on a structure. We know now that the eyebar element can only resist 4.5 ksi (31 MPa) of n et section stress, as indicated in T able 1. 1 ; 75,000 pounds per square inch of tension had been assumed for this structural member , due to the specified high strength steel . In addition , a small defect can further reduce resistance of member section and be subject to greater stress concentrations. As stated, this was also the case with the Silver Bridge. In summary of the problems that led to the eventual collapse of the Silver Bridge, all but impossible and failure all but inevi t able (Petroski 2012) Both insights are true, but a third must be added: without an under standing of fatigue , it is impossible to know that a section and its material will fail at stress lower than its yielding value. Moreover, without a clear understanding material fatigue, the exact mechanism of this particular collapse would remain a myste ry. 1.4 Definition of Fatigue At this point , it is necessary to define fatigue in more detail . The Brid ge : Fatigue is the tendency of a member to fail at a stress level below its yield str ess when subject to cyclical loading. Fatigue is the primary cause of failure in fracture critical members. Describing the process by which a member fails when (Federal Highway

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6 Administration 2012) Fatigue was the principle reason for failure of the Silver Bridge ; the eyebar was a tension element subjected to many cycles of loading . Fatigue was also a crucial reason for the I 35W Mississippi River Bridge c ollapse in 2007. (Federal Highway Administration 2012) Fatigue is categorized in detail categ t ter being the most sensitive. A Categories have a threshold for infinite life of 24 ksi (165 MPa), and (17.9 MPa). T able 1. 1 at the end of C hapter I shows the threshold and the finite life constants that describe the behavior of the member details of a steel girder bridge. The constant A is from the equation, N , where N is the number o f cycles, A is the constant listed in Table 1.1 , and ( ) n is the nominal fatigue resistance. Fatigue is induced in a material by cycles of tension , and is not directly a ffected by compression . Tri axial tension, that is pulling force in three distinct dir ections, is by far the most difficult condition for a material to endure , as it relates to fatigue of metals . (Dexter and Fisher 1999) Small regions of this type of tension can exist in both bridges and buildings. The reason highway bridges experience more fatigue problems is due to the weight and regularity of trucks passing by on the deck of the bridge , as sketched in F igures 1. 5, 1.7 . ( , as opposed to distributed loads ). Generally, the problems of fatigue with bridges do not happen after just a few cycle s , but after tens of thousands or millions of trucks passing ab ove the supporting girder beams. In many cases, the passing of 100,000,000 heavy trucks wil l not induce fatigue damage , nor cause cracking, especially in the well detailed bridge built after 1975. The structures that can withstand tens of hundreds of millions of cycles of loading are said to

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7 possess infinite life details. I ndeed, Fisher notes there have been very few if any failures which have been attributed to details which have a fatigue strength greater than c ategory (Dexter and Fisher 1999) The stresses in the bridge will not be high enough to cause fatigue problem s , a fact that will be observed in the subsequent resea r ch . Theoretically, the bridge might be able to sustain loading of the fatigue trucks for as long as the bridge is in service; no amount of truck traffic over time will typically result in fatigue dam age. Examining the graph s in F igure 1. 6 , t he stress range is induced by the standard H L 93 fatigue truck. F igure 1. 7 etail, which will be the type of steel detail that will be the focus of analysis in subsequent chapters. Because the observed stresses in the bridge models that will be presented in Chapter III indicate that the bridge would be suscepti it is necessary to consider the impact of further damage . P articularly , this damage is likely in the region of fatigue and present with the loss of section due to cracking . The remaining life of the material, section and system are of natural concern to all charged with the possible strengthening of such a section, and this will be the focal point of analysis later on in this paper. The S N curves are used to predict the remaining fatigue life, the number of cycles that a section can take in an applied stress range. For the situation s modeled in Chapter III : Analysis and Result s c ategory became of primary interest. Such a detail is described in the AISC Manual (14 th Ed.), pg. 16.1 200 Table A 3.1 metal and weld metal in members without attachments built up of plates or shapes, connected by continuous longitu dinal complete joint penetration groove welds with backing bars not removed, or by continuous partial joint

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8 (American Institute of Steel Construction 2011) Such welds would be present in a built up me mber such as the steel plate girders of the v alidation and c ontrol b ridges , as will be presented in Chapters II and III c ategory d etail threshold for infinite life is 12.0 ksi or 82.7 MPa and is indicated by the flat line of F igure 1 .8 . All regi ons below any of the curves are probabilistically safe from fatigue; note that the lower categories dip farther down on the chart and have less fatigue resistance. The number of cycles grows exponentially across the x axis . In spite of loading cycles in the tens or hundreds of millions, it still may not always be fatigue that c ould cause failure. Also important to note, c ars or pick up trucks are not included in fatigue cycles since they do not generate enough stress in bridges . Their loads are not cons idered in the fatigue limit state and are not a part of the later analysis. In addition , fatigue limit stresses are not a function of the base metal yielding strength, but instead they are only limited by the geometry of the detail. In addition, w hile the notion of stress is central to the calculations, it is more appropriately conceived that strain is the governing phenomenon. (Fisher, Kulak and Smith 1998) Micro i n consistencies perpendicular to the load path will further res ult in growing inconsistencies or crack tip propagating. As was pointed out, it was not the lack of strong due to damage of fatigue sensitive elements , existing i mperfections and the development of corrosion at key points in the structure . The lack of structural redundancy was also a key factor in the collapse of the Silver Bridge . Redundancy is discussed in the following section .

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9 1.5 Redundancy and Load Sheddin g In most cases in structural engineering, a building or bridge might have additional reserve capacitie s, or redundancy, to exceed the maximum loading effects . An additional problem with the Silver Bridge was that the eyebar chain link system specified wa s not redundant ; only a pair was used for each link joint . Furthermore, a single eyebar is not suffici ent to hold the system in place; it would be similar to the situation of a missing link on one side of a bike chain, at which point the entire chain woul d likely come apart . Many similar bridge structures were built in the same time period but would have four or six eyebars mounted in parallel. Even with one of these eyebars cracking in failure, sufficient warning and reserve capacity were available in these structures to hold and allow time to remedy. Even when redundancy might not theoretically exist, a non redundant structure can avoid collapse if the applied loads are being redirected to other intact members of the structure. Again , th is phenomenon exists in sharp contrast to the Silver Bridge Failure of the other girder (s) , could assume the applied loads in the area of the failure discontinuity. On e no t able case of load shedding that involved a significant fatigue crack occurred in Pittsburgh in 1979. (Dexter and Fisher 1999) T he crack is shown in F igure 1. 9 . The I 79 Bridge at Neville Island, a n essentially non redund ant two girder bridge , held in spite of a full flange crack that developed and propagated up nearly the entirety of the web of the girder. It is interesting to consider this case also because the bridge was relatively young when it developed this debilita ting crack. Only being three years old, this bridge had a manufacturing defect and an unadvisable weld . The stress -

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10 concentration in the affected section caused the defect to develop into a quickly propagating crack after relatively few loading cycles. A towboat passing underneath just happened to notice the crack, and soon after , the state DOT shut down traffic over the bridge for two months in order to complete repairs . an d the stiff surrounding members , including cross frames, took on the applied loads after the crack had formed. The authors say that not only did the structure avoid collapse, but also the load shedding slow ed down crack propagation . (Dexter and Fisher 1999) The question may arise to the engineer if this type of damage could be included in a bridge model. With a simple span and a reduced section at the point of damage discontinuity, it is relatively easy to model bridges in a v ariety of available Finite Element Analysis programs. This type of full flange crack, perpendicular to span length of the girder, will be investigated in the Section 3.2 . 1.6 Other Types of Fatigue One might ask whether fatigue problems fall outside th e detailed categorization provided by AASHTO , AISC and oth er a gencies. Indeed, this is the case. Distortion induced fatigue affects specific details of a bridge that do not experience the stresses that normally induce fatigue. Instead , the fatigue is ca used by the distortion or wa rping of a particular element. The phenomena has been studied extensively and represents a major subtopic of fatigue within steel structure s . The problem has been examined by many previous researchers hat distortions with the magnitudes even on the order of only 0.5mm (0.02in) may induce high cyclic stress ranges up to 276 MPa (40

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11 (Lui, Frangopol and Kwon 2010) The authors go on to say that altho ugh these problems are common their effects are manageable, insofar as the cracks occur in planes parallel to the primary longitudinal stresses in a bridge girder, and that they can be detected early and the bridge retrofitted appropriately. An example of distortion induced fatigue was thoroughly covered in recent research at Kansas University. The Tuttle Creek Bridge , built in 1962 in northeastern Kansas , w as retrofitted in 1986 and 2005 to mitigate the deleterious loading effects due to the cracking that had developed. A concurrent study was published in part by the Kansas Department of Transportation. (Anderson, et al. 2007) The study note s that cracking had developed shortly after the bridge was built in the web gap region . The cause was distortion in the web of the girder , which was being pulled out of its design plane by the lateral tension forces in the cross bracing. While the longitudinal weld of the flange to the web is normally a c ategory B f atigue design element, the distortion and the presence of tension in the transverse direction made the element highly susceptible to the fatigue induced by the passing truck live loads. Normally this c ategory B element will not succumb to fatigue at any cycle of loading with st resses less than 16.0 ksi ( 110 MPa). The transverse fillet weld is also highly susceptible , if not more so, to distortion induce d fatigue cracking shown in F igures 1. 10 (a) & (b) . These types of fatigue cannot be ignored in design or during in service ins pection. Cracks such as these can propagate into adjacent elements and cause failure in the entire structure; hence, the need for retrofitting. Despite the importance of both retrofitting in situations like these, and restoration of distortion induced fa tigued elements, this paper will not focus on this type fatigue, but instead will center on load cycle induced fatigue.

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12 1.7 Literature Review: Modeling in Finite Element Programs T he previous work of other br idge modelers, including Nowak and Sotelino, o ther academic studies and field tests , along with guidance from CSI in their manual s for SAP 2000, is essential in developing F.E.M. Sotelino and Chung indicated in their 2005 study accurately using Finite Element Analysis of a bridge superstructure rather than using lateral load (Sotelino and Chung 2006) The goal of any 3 D model is to determine the bending for ces, stresses and moments in the key elements of the bridge structure, particularly the lower flange of the girders, the supports, and the connection details. While it was not discussed in the introd uction, AASHTO provides a short cut to conservatively est imating these forces through simplified formulas. Yet since the 3 D models make use of the finite element theory to form the basis for (giant) matrix equilibrium, they should provide the most accurate results. The simplest 3 D F.E.M. is the one discussed above with the rigid links joining the nodes of the centroids of the concrete slab and the steel girders. Again , the concrete slab is modeled as shell elements, and the steel girders as beam elements. In some other past modeling techniques, the I girder itself was broken into three shell elements. Other past techniques used three dimensional solid elements. (Brockenbrough 1986) (Mabsout, et al. 1997) F igures 1. 10 (a), (b) & (c) show some o f the proposed alternatives to bridge girder modeling in the past . Chung and Sotelino conclude that despite the simplicity, the model represented in F igure 1. 10 (c) as the other more complicated modeling techniques and does not req uire fine mesh

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13 density. They compared the model to a full scale test done at the University of Nebraska (which will also be used to validate the modeling technique here in this paper.) They found comparabl e results for deflection observed, and error within 6%. Also in comparison, the other models used three times t he number of degrees of freedom. They conclude that the s implest model is the best model . 1.8 Literature Review: Sources of Error in 3 D Mode ling While the results of the proposed modeling scheme are sufficient to proceed with fatigue modeling of bridges, it is still necessary to consider at this point how error is introduced in this or any 3 D F.E.M. One source of error that it is important to consider is the end restraints modeling. With the SAP 2000 superstructure, limiting the 3 D degrees of freedom of the individual n odes established. There are six DOFs available for any node in a model; they are movement or translation in the direction of any of the three principal axes and the three possibilities of rotation around any of the three principal axes. For field conditi ons, no joint can be perfectly restrained or exhibit complete freedom from restraint (or friction) in any of the six degrees of freedom. For instance, with a simply supported bridge, translational restraint along the longitudinal horizontal axis in realit y is subject to friction at the bearing pad. This friction was not modeled in the validations in the C hapter II , however good suggestions for accounting for it have been proposed in previous literature. (Kim and Nowak 2001) B y placing a spring at the shown locations in Figure 1. 12 , a 3 D F.E.M. can account for friction.

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14 CSI notes that shear deformation is included in the deflection in the SAP 2000 analysis. (Computers & Structures, Inc. 2013) As a result, observed deflection in the 3 D F.E.M . s are slightly more than the theoretical equations yield. In addition, the shear deformation may also cloak the increased moment and stress in the damaged sections in some methods of composite modeling. Shear deformation error is also noted in APPENDIX A . The next section will ways to model composite bridges within SAP 2000 . 1.9 Literature Review: Various Modeling Approaches and Their Accuracy There are more ways to model a composite steel girder brid ge in SAP 2000 that have not been discussed up to this point. This next section details further approaches that detailed in method Section 1.8 . The t ables and figures in AP PENDIX A show eight ways of creating a model in SAP 2000 that capture, to varying degrees, the composite nature of a well designed steel girder bridge. Of the eight ways to model a steel girder section, there are four ways noted here that capture the comp osite nature of a (shear studded) steel girder with a concrete deck. and Chung , and as mentioned that was the primary model devel oped and used in this analysis. Generally , the principle that governs t he bridge mechanics of all modeling techniques suggested is identical to The four methods listed here accomplish t he change in the moment of inertia of a composite section: 1. Offset the shell joints (away from the centroid of the ste el girder) Model 2. 2. Use the body constraint tool to hold the deck and girder centroid so that the joints

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15 are restrained in all six degrees of freedom Model 3. 3. Offset the girder centroid from the plane of the deck centroid Model 4. (This type of mode ling technique will be investigated in the coming section.) 4. Use a rigid link to join the centroids of the deck and girder Model 8. The results are identic al for these four methods of increasing the net section moment of inertia. Without offset, the ecc entric beam does not govern the sectional properties of the section , and essentially it becomes non composite. F igure A1 depicts the extruded view modeling in SAP 2000 show ing an example of Model 1 from T able A 1 , with its non composite properties and the chara cteristic single centroid plane. While Model 1 is not a useful structure , nor does it bare verisimilitude toward a real bridge or bridge model, it is a helpful launching point and can help give the engineer an approximate estimate of flexural properti es. It is still necessary to compare results from computer mode l trials of the various models. The eight models shown in T able A1 were subjected to the following loading conditions in a software study. A simply supported span of 20 meters was given the following section with a composite moment of inertia of 0.15433 meters 4 . A 20 meter span with the section shown in F igure A2 was subjected to a point load at midspan of 100 kN. From structural analysis theory, the beam should have deflection of 3.2725 mm. This is indicated in row one of the summary of calculations in T able A2 . From the t able of results, the four methods that most closely model composite behavior have identical results of 3.2624 mm of deflection at midspan. This res ult should alleviate the concerns of the bridge engineer over what modeling technique to choose. Since there is no difference, as would be suspected due to creation

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16 of four model with the same constraints and the same net section moments of inertia, the r esearcher can proceed. Furthermore, the results of all these four models are within 0.3% of the theoretical beam. 1.10 Carbon Fiber Reinforced Polymer and Its U se Carbon f iber r einforced p olymer (or CFRP , henceforth) is a practical, widely available m aterial from several manufacturers domestically and abroad. It has been used in place of rebar in concrete, due to its ability to resist corrosion in a way that steel is not capable. Generally, CFRP is not conceived of a strengthening material for applic ations other than concrete applications. The reasons are widely attributed to the bonding failure of CFRP and its bonding failure to a steel substrate. Some stu dies indicate a bond failure beginning around 0.006 strain for CFRP epoxied to steel. (Kim and Brunell 2011) In strains ranging from 0.0061 to 0.0076 that were 36.5% to 45.5% of the rupture strain ( frpu Because the performance results of CFRP have been promising, one of the goals of this study in this paper is to examine it s F.E.M.

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17 Table 1.1 AASHTO d etail c ategory c onstants and t hresholds (AASHTO 2012) Detail Category Constant A (10 11 ) (ksi) 3 Fatigue Threshol d (ksi) Fatigue Threshold (MPa) A 250.0 24.0 165 B 120.0 16.0 110 61.0 12.0 82.7 C 44.0 10.0 69.0 44.0 12.0 82.7 D 22.0 7.0 48.3 E 11.0 4.5 31.0 3.9 2.6 17.9 Figure 1. 1 Typical section of a steel I g irder Figure 1. 2 Architectural elevation of the Silver Bridge s howing Warren t russes and the point of failure (Lichtenstein 1993.)

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18 Figure 1.3 Eyebar detail ( c ategory E for fatigue ) Figure 1.4 Photo of Silver Bridge c ollapse (Lonaker 2006) Figure 1. 5 Diagram of a truck axle (point) loads on a simply supported bridge (Federal Highway Administration 2012)

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19 Figure 1. 6 AASHTO f atigue c ategory c urves c ited in (Dexte r and Fisher 1999) Figure 1. 7 AASHTO HS 20 t ruck load (Federal Highway Administration 2012)

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20 Figure 1. 8 S Figure 1. 9 Photo of the girder crack of the I 79 Bridge at Neville Island (Purdue University 2015) 10 100 1000 1.E+05 1.E+06 1.E+07 1.E+08 Stress Range (MPa) Number of cycles

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21 (a) (b) Figure 1. 10 (a) Detail of out of plane distortion (b ) Photo of web gap region with crack on transverse weld (Anderson, et al. 20 07) (a) (b) (c) Figure 1. 11 (a), (b) & (c) Three different modeling techniques (Sotelino and Chung 2006)

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22 Figure 1. 12 (Kim and Nowak 2001)

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23 CHAPTER II MODEL DEVELOPMENT AND VALIDATION A structural analysis program had to be chosen for analysis using the principles of F.E.M. Due to its ready availability, its success in many other applications, and its continued development over thirty years, SAP 2000 was selected for the analysis. (Computers & Structures, Inc. 2013) G.U.I. graphical user interface), meaning that there w as a limited amount of control over the design parameters; however, for the purposes of this study there was sufficient allowance for varying and tailoring the key elements for further analysis of the desired system. Developing the model in SAP 2000 requi red using many, if not all of the predetermined features imbedded in the program. Only simply supported bridges were modeled in this study. This is the case for both the validations and the test modeling of the control, cracked and CFRP restored bridges. Restraints of the end supports are briefly discussed below. None of the bridges modeled included skew, though this is easily modeled in SAP 2000. The girders of all the models used for validation and for fatigue, cracking and CFRP strengthening studie s, are represented in the program by beam elements. The beam element is drawn between two nodes, nodes that also define two of the four points of the shell slab elements discussed below. The section properties can be inputted easily in SAP 2000 (See APPE NDIX C . ) Note that the material properties are also assigned alongs ide the geometric parameters. Material pro p erties of s teel, concrete and CFRP used in the modeling research are shown in Table 2. 1 . Again despite the limitations of the graphical interfa ce, the ease of input makes f or quick modeling, and so the next important element that is represented in SAP 2000 is the concrete deck. Four nodes each

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24 with 6 degrees of freedom allowing for out of plane movement define the shell element : three translatio nal, three rotational . In SAP 2000 , concrete deck s are mod eled through the shell elements and that method is chosen here. The beam and the shell elements are joined together compositely in the models , ngth and flexural resistance (the primary structural engineering property examined in this paper) as opposed to the deck not contributing anything to the resistance of the beam elements mentioned above. In order to accomplish the composite nature, there a re several methods mentioned in the in APPENDIX A . The various bridge cases , both damaged and CFRP strengthened, are modeled with an eccentric beam technique (see Model # 4 on page 80 ) . That is all the nodes of the model exist on the same xy plane and the insertion point of the girders, represented by beam elements noted above, is the top center of the girder(s). The center of the top flange is aligned with the soffit of the deck. The stiffness of the girder section is transformed (increased) with the of fset from the base xy plane of the model. Alternatively, the nodes of the xy planes can be drawn parallel and offset, connected by rigid links as was discussed previously. The inputs and step by step guide shows this procedure in APPENDIX C. Node meshing of the deck is of concern to the modeler, and finer meshing equaling less than 1% (~ 1 ft or 0.29 m) o f the overall length of the span is chosen for the models. This refines the responses between the nodes observed in the result s , produc ing appropriate s tress and loading effects in the primary flexural members, the girders. Reducing the girder lengths of the beam elements does not have the same kind of effects that fine meshing of the shell elements of the deck has.

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25 A 3 D quality of the elements is visi ble in the extruded view . While this is not necessary for the calculations that the program makes, it is helpful for the user to develop an accurate model to analyze. For instance, one can quickly check in this view mode if the soffit of the deck is at o r slightly above the top plane of the top flange of the girders. For the preli minary model s and ultimately for test models, overhangs were modeled in all cases . During the course of the model verification , one system included the presence of both overhan gs and barriers; this will be shown later in this section. The ends of the simpl y supported girders are restrained with pins and rollers. The pins allow freedom of rotation, but not translation. The roller allows both rotation in all three directions and translation in the long direction of the bridge span. As was mentioned in the previous section, springs at unrestrained degrees of freedom could be used at ends of the girders to model friction in the system; this was not done in the course of modelin g in this study. Without getting into the results of the modeling, the targeted values of the model were maximum deflection and moment . From the moment observed in the beam elements , stress or strain could be back calculated from a composite properties sp readsheet drafted in Excel, and then using the simple relationship, . The extreme fiber of section at the base point of the steel flange would experience the highest stress and strain in tension due to the flexural behavior of the simply supported bridge . The highest values, generally near the midspan, under the highest concentration of loading (transverse location), are sought from the modeling to investigate the effect of fatigue loads. The loading effect was modeled through linear static point loads applied to the

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26 nodes. 1 The loads are static as opposed t o dynamic; analysis is linear (elastic behavior of materials) as opposed to non linear, inelastic behavior of materials. The nodes of course are connectivity locations for the drawing of beam and shell elements. For some of the validations, the loading l ocation were predetermined from the study. For other validations from field tests (that were compared to F.E.M. analysis), the longitudinal placement of the loads along the span was varied to produce the greatest loading effect. For this research, t h is m aximum loading effect of the fatigue truck was determined and used for further structural analysis. For simplicity, c racking of the gir der was modeled in one dimension, which is a percentage of overall height. Several ways of logically reducing the sect ion properties were considered, including corrosion of the steel plates, however the simplest simulation of damage was a crack. While real cracks can have all sorts of unusual geometries and propagation paths, for the purposes of modeling in SAP 2000 , it made clear sense to keep the parameters simple. One anticipated consequence of this is that cracking from the undamaged state up to three percent (2.77% to be exact) constitutes the greatest reduction of capacity of the girder , break ing thro ugh the bottom (tension) flange . When the crack had broken through the flange, the crack was essential ly modeled with a beam element T section, as opposed to I section, having length of one inch (25.4mm). This length represents less than one tenth of one percent of th e overall length of the span. With the 1 SAP 2000 allows for the use of a moving load that can be appli ed to beam elements of the structure model, either statically or dynamically with varying speed. The moving loads could be defined with truck loads separated by fixed or variable trailing axles. This was done on a 92 ft beam in conjunction with a simple t rial and error method. It is important to note that in SAP 2000 moving load can only be applied to beam elements, whereas the modeling technique used required placement at shell only node (due to the fixed width of the axle not necessarily corresponding t o the girder spacing). Thus, moving loads were only used to get an indication of where along the length of the span the axle loads should start in order to get the maximum loading effect.

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27 eccentric beam model, the sharp spike in stress can be observed at the short beam element where the section is reduced to a T beam . Discontinuity of the bottom flange in a cracked model is shown in F igure 2. 1. T he i ncrease of stress at the point of damage can be easily determined. When pinned from the top flange of the girder (see APPENDIX C ) , and without interference from shear deformation, the sections will show the characteristic increases in stress . Stress rang e of the bridge section will be then used to compare with the fatigue threshold stress of 12.0 ksi (82.7 MPa). Later in the analysis CFRP (Carbon Fiber Reinforced Polyme r) was used in the model as strengthening. See F igure 2. 2 for an extruded view of the CFRP modeling and see Table 2. 1 for material properties of concrete, steel and CFRP used in this study . Materials such as 4,000 psi (27.58 MPa) concrete and ASTM A992 S teel are preloaded and easy to use within SAP 2000 modeling. However , CFRP is not an available material to model with in the program. Therefore the material is created from another material es such as the modulus of properties of CF RP (given by a manufacturer . ) Also cr itical to the modeling is that the CFRP is attached to the eccentric beam model (of the composite concrete slab and steel girder assembly) by means of rigid links. It is not possible in SAP 2000 to have a compound beam elements between two nodes; the program recognizes one section type or another, not both.

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28 2.1 Beam Comparison: SAP 2000 to Theory If the F.E.M . software cannot predict the bending of a simple beam, let alone a bridge model, it might be best to seek another program or method. For this reason, a comparison to beam theory is sought. Consider the basic problem: A simpl y supported beam of 40 feet (12.2 m eters ) is subjected to a 100 kip (444.8 kN) point load at midspan. The type of rolled steel section chosen was a W21x132, which is a 21 inch (0.533 m) deep beam with a self weight of 132 pounds per liner foot (1.93 kN/m) . The AISC give s the moment of inertia of this section of 3200 inch 4 (American Institute of Steel Construction 2011) , through the strong axis (where the beam is loaded with the flanges normal to the load direction.) Beam theo ry gives the maximum deflection [in inches] from the following formula: This value was compared with the basic finite element model computed in SAP 2000. One simple beam with a pre loaded W21x132 section was drawn to length of 40 feet (12.2 m) , with pinned and roller joint restraints. It was loaded with 100 kips (444.8 kN) at midspan. After computation, the program gave the maximum deflection to be 2.54 inches (6.45 centimeters), bringing t he calculation to within 3 % of the t heoretical value. The user output of the beam deflection from SAP 2000 is shown in F igure 2. 3 . The producers of SAP SAP 2000 calculations produce slightly greater values because shear deformation is co nsidered in (CSi Knowledge Base 2013) Because the error was so small, it is reasonable to proceed with the analysis in this paper. This is summarized in T able 2. 2 .

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29 2.2 Full Scale Model Validation : Nebraska Bridge With a beam section investigated and confirmed, it is necessary to compare the F.E.M. with a full scale test and compare deflections due to live loading, as above. A full scale lab oratory test bridge was constructed at the University of Nebraska in 1995 to examine the loading effects on a composite steel girder bridge. (Kathol, Azizinamini and Luedke 1995) The study was of sufficient quality to attract future researchers; (Sotelino and Chung 2006) used the Nebraska results and compared them with their own F.E.M. Sotelino dete rmine d the most accurate modeling of c omposite steel girder bridges. Several of these developments were previously discussed in C hapter I, and APPENDIX A summarizes the mode ling compari sons. Therefore, the Nebraska b ridge was useful not only for the modeling validation, but also for the model development previously discussed. The Nebraska b ridge is used as a reference point for validation and verifying the accuracy of the SA P 2000 bridge modeling. S pecifically , the experimental and modeling values of deflection were compared. There are a few unique feature s of both the laboratory Ne braska b ridge and the SAP 2000 model developed here . The Nebraska Bridge had rails and overh an gs in the full scale test and t h e se were modeled as concrete beam elements with the appropriate section dimensions. (See Figure 2. 9 . ) In the model, they were attached at the centroids by rigid links, normal to the principle bending axis. O verhangs wer e accomplishe d by widening the shell element area s of the deck. Figure 2.4 shows the Nebraska b ridge with the rails and rigid links, as well as the overhang s , modeled . The Nebraska test bridge was also loaded differently than a standard AASHTO

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30 truck, fa tigue or otherwise. The goal of the Nebraska testing was to determine the ultimate loading capacity; therefore , two s or sets of loads w ere placed on the brid g e. Furthermore, the standard axle loads were scaled up two and a half times (2.5), which were well be yond the standard load factors. This scaling made the responses of the test bridge more salient. There were two 10 kip (44.48 kN) loads at each front axle tire, and four 40 kip (177.9 kN) loads at each trailer tire. Figures 2.7 and 2.8 show the loading locati ons for the Nebraska b ridge . (Kathol, Azizinamini and Luedke 1995) use f our feet as the distance between side by side vehicles ; which is standard . Of greater effect is the longitudinal spacing in the directi on of the span. B etween the first and second axles on th e Nebraska test there was a two foot distance reduction. Also the variable length between th e second and third axles (14 to 30 feet ; 4.27 to 9.14 m ) on the loading truck (s) was given the advantageou s placement of 15 feet (4.57 m) . When the point loads are placed closer to the midspan of the bridge, higher bending moments and lower flange stresses occur. The Nebraska test took advantage of this basic static equilibrium fact, in order to measure the ultimate capacity of the bridge. The non standard reduction of two feet is reasonable and help s to ensure a sufficiently large moment for testing is imparted on the supporting girders. The seventy foot span that was modeled in SAP 2000 also had to have a ccurate section modeling for the girders. The Nebraska t est built three 54 inch (1.37 m) deep girders , with i nch (9.525 mm) web, flanges 9 by ¾ inches (228.6 by 19.05 mm) at the top and 14 by 1¼ inches (355.6 by 31.75 mm) at the bottom . A t 17 ½ feet ( 5.33 m) from the bridge end supports , where less moment resistance is necessary, the bottom flanges were spliced to a reduced thickness of ¾ inch (19.05 mm). This upsizing for increase d

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31 moment capacity of the steel girder at the middle region is shown in the two sections depicted in Figure 2.6. Table 2. 3 shows a ll of the geometric parameter s of the Nebraska test that are used in the verification for modeling . To mimic the wheel loading of the two trucks on the Nebraska test b ridge (12) twelve 1 i nch post tensioning (DYWIDAG or Dyckerhoff & Widmann AG ) ro ds were used . The test bridge live loading deflection was almost exactly ¾ of an inch (0.749 inches = 19 .025 mm). (Kathol, Azizinamini and Luedke 1995) In comparis on, when the SAP 2000 model developed here was ex e cuted , 0.797 inches (20.23 mm) was c omputed for the midspan deflection due to all the 10 and 40 kip (44.5 and 177.9 kN) loads indicated . Th e results of the verification are within 6% agreement and concur w ith the previous research at Nebraska. Furthermore, Sotelino results for maximum deflection from their modeling also are close to 20 mm and agree with both the Nebraskan testing and the results in this thesis . (Sotel ino and Chung 2006) Centerline deflection at midspan is a convenient way to establish the flexural characteristics of either a real in service bridge or a 3 D F.E.M . With th is agreement established here, th e modeling approach taken in this research sho ws validity. 2.3 Full Scale Model Validation: Flint, Michigan B ridge The University of Michigan, on behalf of MDOT, conducted field test s on an in service steel girder bridges . (Nowak and Eom 2001) The study took place acr oss the state of Michigan, but the bridge that is the focus here is located in Flint, MI over Interstate 75 along Stanley Road. The span is 38.4 meters and has seven steel girders (S=2.2m), with an out to out width of 14.7 5 meters. The slab is 20 c m thic k, and is

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32 assumed to be 4000 psi (27.58 MPa) concrete. Barriers are not modeled, but overhang of the deck is included in the composite section. There is no skew, and the bridge is modeled as simply supported. Girder geometry is shown in Figure 2. 10 (not e that these girders are also upsized for increased moment capacity between the 7.3 meter and 31.1 meter marks of the span) O ther properties are shown in Table 2. 4 . Girder geometry is , own validations. (Sotelino, et al. 2004) The test are recorded in microstrain, with a maximum of 150 microstrain. The results are valuable to validation procedures because the truck was larger than a standard AASHTO vehicle. Nowak note s In Michigan, the maximum midspan moment for medium span bridges is caused by 11 axle trucks, with gross vehicle weight up to 730 kN depending on axle con figur ation (Nowak and Eom 2001) GVW officials, it is not completely useful for structural engineers as the location of a load is consequential, as is the magnitude. The specific locations of loading are noted in F igure 2. 1 2 , and correspond to the location inputted into the models. From the l oading pattern indicated in Fig ures 2.11 & 2.12 , one can easily foresee that the greatest loading effect will occur in the center girder. Indeed, that is where Nowa k records the highest value of 150 microstrain. Sotelino also used this study to verify one of her models in F.E.A. In her modeling, the micro strain also is 5 to 10% higher than the observed values in the field. (Sotelino, et al. 2004) The discrepancy in this study is 8.7%, and therefore compares well with the past tests and modeling of this particular bridge. With three validations of

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33 modeling completed, the modeling technique and development described at the beginning of the Chapter II shows sufficient accuracy. 2.4 Developing a Fatigue Prone Model When the stress range observed in a bridge model is greater than 12 ksi ( 82.7 MPa) ategory details will have finite fatigue life. The loads due to a fatigue truck (8 k ips, 32 kips, 32 kips or 35.5 kN, 142.3 kN, 142.3 kN ) remain constant, but the loading effects will vary depending on the bridge geometry. Dead loads do not contribute to the stress range of member, as o nly live or fatigue loading can only do so by defini tion. Since that was the case, the ways to increase flexural stress in the bridge were to modify the geometry of the str ucture. Perhaps the most obvious way to increase the loading effect is to lengthen t he span . The number of girders can be reduced, or the space between them can be increased. The section of the girders themselves can be reduced; particularly the lower flange can be reduced in both wid th and thickness. Increasing the depth of the girder will also increase the moment observed in the brid ge model in SAP 2000 , if all other variables are held constant. When developing the model for the control bridge, these strategies were employed to have a fatigue prone bridge model . As ha s been noted (Dexter and Fisher 1999) , after the mid 1970s, D categories are largely avoided in design , if not eliminated. Depending on the category detail, b etween two and twenty million cycles, infinite life of that detail is assumed . At two million cycles on a Category A elemen t, fatigue resistance is the highest of all details with stress ranges of 24 ksi (165 MPa). This stress range was not observed in the course

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34 of modeling bridges , so lower categories had to be considered. Category was promising, since a stress range of 1 2 ksi ( 82.7 MPa) , as noted, had to be demonstrated in the model. Again, due to their presumed removal from most bridges in the pa st 40 years, details D and E w ere not considered. F rom some of the early model s completed on SAP 2000 , stress in the lowe r flange from a fatigue truck loaded bridg e was typically not observed above 10 ksi (69 MPa) . This will be shown in C hapter III . In fact, in order to produce stress much above 8 or 9 ksi (55.2 or 62.1 MPa) in composite secti on, the span needs to be beyon d 100 feet (30.5 m) , and the sections undersi zed (lower flange thickness = ¾ inch = 19 mm ). It is so much so that there is significant deflection not only due to the dead load, which can be modified and compensated for in design, but more critically with the live load which can produce a large deflection of 4 + inches ( 100+ mm) . This is unsui t able for driving, wit h a deflection to span ratio of 1/300. Th e AASHTO optimal limit is 1/800. While structurally, the bridge will not necessarily fail and has well beyond the sufficient strength not to collapse, deflection of this kind is not psychologically reassuring for the (truck) driver and other present vehicles experiencing t he unnecessary dip.

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35 Table 2. 1 Material properties used in study models E F` ( c) or F y F u Poisson Concrete 3605 ksi 4000 psi n/a 0.2 24,900 MPa 27.6 MPa n/a Steel (A992 Grade 50) 29,000 ksi 50 ksi 65 ksi 0.3 200 GPa 34 5 MPa 448 MPa CFRP 2 23,900 ksi n/a 449 ksi 0.25 165 GPa n/a 3.1 0 GPa Table 2. 2 Summary of be am validation Parameter Dimension/ Value Beam Length 12.192 m Beam Self Weight 1.93 kN/m Moment of Inertia 0.00133 m 4 E steel 200 GPa Depth 0.533 m Load Subjected to 444.8 kN Load Location 6.096 m (mid span) Node Insertion Bottom of Flange Supports Pin/roller Results: Beam Theory Deflection 6.274 cm SAP 2000 Model Deflection 6.452 cm Discrepancy 2.8% 2 (Sika Corporation 2011)

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36 Table 2. 3 Full scale model validation summary, Nebraska Parameter Dimension/Value Span Length 21.336 m Bridge Deck Width 7.925 m Numbe r of Girders 3 Girder Spacing 3.048 m Deck Thickness 190.5 mm Concrete f ' c (deck) 41.37 MPa Concrete f ' c (rails) 44.82 MPa Overhang(s) 0.914 m Concrete Bridge Rails (b·h) 357 x 406 mm Girder Depth (total) 1.372 m E steel 200 GPa Web Thickness 9. 525 mm Top Flange 19.05 x 228.6 mm Bottom flange (middle 50% of span) 31.75 x 355.6 mm Bottom flange (end quarters of span) 19.05 x 355.6 mm Load Locations See Figure 2. 8 Results: Max Observed Deflection 19.025 mm SAP 2000 Model Deflection 20.23 mm Discrepancy: 6.3%

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37 Table 2. 4 Summary of Flint, MI bridge model validation Parameter Dimension/Value Span Length 38.4 m Bridge Deck Width 14.75 m Number of Girders 7 Skew None Girder Spacing 2.2 m Deck Thick ness 200 mm Concrete f ' c (deck) 27.58 MPa E steel 200 GPa Overhang(s) 0.75 m Overall Girder Depth (center 23.8 meters of span) 1.34 m Overall Girder Depth (7.3 meters from ends of span) 1.28 m Web Thickness (all sectio ns) 12.7 mm Top flange (center 23.8 meters of span) 48 x 457 mm Bottom flange (center 23.8 meters of span) 70 x 457 mm Top flange (7.3 meters from ends of span) 22 x 457 mm Bo ttom flange (7.3 meters from ends of span) 35 x 457 mm Load Locations See Figur e 2.1 2 1 50 Results: SAP 2000 Max Moment 1305 kN m Section Stress 32.61 MPa Section Strain 163 Discrepancy: 8.7%

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38 Figure 2 . 1 Soffit , extruded view of 5% crack modeled in steel girder from SAP 2000 Figure 2. 2 Soffit view of CFRP strengthened bott om flange

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39 Figure 2. 3 Deflection of a W21x132 in SAP 2000 Figure 2.4 Standard view of the Nebraska b ridge model in SAP 2000 Figure 2.5 Extruded view of the Nebraska b ridge model

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40 Figure 2. 6 Neb raska b ridge and girder dimensions (mm) Fig ure 2.7 SA P 2000 aerial view of Nebraska b ridge with loading pattern

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41 Figure 2. 8 Nebraska b ridge 70 ft s pan (21.3m) with loading weights and locations Figure 2. 9 Nebraska r ail dimensions in (mm), modeled as structural element

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42 Figure 2. 10 Flin t, MI b rid ge section and girder d imensions (mm )

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43 Figure 2.1 1 SAP 2000 aerial view of Flint, MI b ridge with loading pattern Figure 2. 1 2 Loading con fig uration (mm) for the Michigan bridge validation , axle weights (kN) shown between wheels

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44 CHAPTER III ANALYSIS AND RESULTS 3.1 The Control Bridge The control bridge consisted of an approximately 28 meter steel girder, simply supported span. The extruded view of the 3 D model is shown in F igure 3.1 . A 190 mm (7½ inch) deck was model ed compositely with the four girders, using the frame insertion offset method discussed in the previous model development section . As the modeling techniques were also verified in C hapter II and the results prove to be most consiste nt with a composite deck model, the b est practice was to proceed accordingly. The deck consisted of 27.6 MPa (4,000 psi) , normal weight concrete; the steel was A992 Grade 50 (344.7 MPa). A full set of input global geometric and sectional pro perties is listed in T able s 3.1 , 3.7 and 3.8 , as well as in more detail in A PPENDIX B . The girders of the control b ridge are built up members, not standard AISC sections. They would include the welds (and base metal of the A992 steel ) that are earlier. No further problema tic details are assumed in the control g irder, such as cover plates and transverse welds. Therefore , it is safe to assume that the fatigue threshold of 82.7 MPa will govern the member. The dimensions of the uncracked Control Girder are shown in F igure 3. 2 . In order to investigate the phenomena of fatigue within the 3 D model, the bridge is subjected to an HL 93 fatigue truck (axle loads of 8, 32 and 32 kips, or 35.6, 142.3 and 142.3 kN). Full inputs and geometries are given in A PPENDIX B . T he maximum loading effect from the fatigue truck was determined. Since the axle width did not match

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45 the girder spacing, it was necessary to apply loads to nodes that only had shell (area) elements assigned. The maximum loading effect did not occur at midspan for the control bridge , as is visible in F igure 3. 3 . In terms of transverse placement of the loads, it is assumed that a 0.6 1 meter offset would prevail from the exterior edge of the area (deck); note that t he overhang is 0.91 meters . (See Figure 3. 4 .) This is a reasonable assumption for fatigue loading since these types of loads would have to occur repeated ly to qualify as a fatigue load. Loading at the very edge of slab structure is unlikely or infrequen t. Furthermore, it was not the objective of the modeling to design the overhang (or barriers), but rather to investigate the girder stresses and their susceptibility to fatigue and fracture. In order to calculate stresses at the extreme fibers of the se ctions concerned, classic formulation for bending is assumed. Therefore stress at that extreme fiber is simply from the equation, . The sectional properties, given by the moment of inertia, I, are varied from the control case as various cracking and fatigue states are investigated. For instance, the control girder (depth = 45 in ches = 1143 mm) possesses a moment of iner tia of 15900 in 4 (6.62 x 10 3 m 4 ). However, when composite section properties are calculated, the net section moment of inertia increases to 41690 in 4 (0.01735 m 4 ). The maximum resulting stress for this composite condition is 3.49 ksi ( 24.1 MPa). 3.2 Cracked Sections Of course, a stress of ~25 MPa is too low to induce fatigue in all but the lowest category of fatigue resistant details. For this reason, damage has been assumed in the steel girder, at a point of maximum loading effect . Several ways of logically reducing the

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46 section properties were considered, including corrosion of the steel plates, however the simplest simulation of damage was a crack. This was further simplif ied by considering the crack as a percentage of height, giving this paramete r only one dimension. While real cracks can have all sorts of unusual geometries and propagation paths, for the purposes of modeling in SAP 2000 , it made clear sense to keep the parameters simple. D ue to the geometry of the control bridge, a crack assume d as such in the exterior girder would produce the highest stresses , where the loading effects were determined to be the most extreme . While m odeling the crack in one of the interior girder s would yield wo rthwhile results, in this study the emphasis is on the exterior girder. While the typical, undamaged girder, supported by pin and roller, consists of 92 ÷ 4 +1 = 24 nodes, the damage was chosen to occur very near midspan at 48.0 feet (550 in = 14.63 m). The section is modeled with an inch long (1.0 in = 25.4 mm) member, either an I girder or a tee, depending on the level of damage . It is straightforward to set the insertion point at the top center of the member, rather than the confusion of the changing height of the centroids of the various sections (See APPENDIX C . ) Generally, the highest stresses and moments on the br idge occur beneath the first 32 kip (142.3 kN) axle, or more accurately the 16 kip (71.2 kN) wheel load directly above. When the load is applied to the model with the damage simulat ed as above, the highest stresses and moments are found to be in the damaged section. T able 3.2 and F igure 3. 7 show the expected results of higher stresses as the crack propagates. It should be noted that the stress of the material in a section where a c rack tip is propagating will theoretically have infinite stress. It is assumed that the point of integration for the stress calculation is just off the boundary condition.

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47 3.3 Fatigue life of damaged sections A big jump also occurs as expected when the c rack fu lly penetrates the lower flange and begins to propagate up the web. These stresses put the damaged section in the range of a detail vulnerable to fatigue at the relatively MPa =12.0 ksi). The number of l oading cycl es is given by N, and f atigue life is calculated with N= (Barker and Puckett 2013) . The other variables are as follows: A is the d etail c ategory constant shown previously in Table 1.1 and n is the nominal fatigue resistance ( here it is the fatigue threshold of 82.7 . ) T he stresses are calculated in cracked section s in T able 3. 5 , the fatigue life of the section as the crack increases is compiled in Figure 3.8 . While the flange is still intact (with less than a 2.77 % crack) , the section has infin ite life. The stress range is below the threshold of 82.7 MP a at values of 17.1 MPa, 27.8 MPa, and 54.1 MPa, respectively for 0%, 1% and 2% cracking . However when the crack propagates to 3%, the effectiveness of the section is severely compromised with r espect to fatigue resistance, if not ultimate strength (evaluated later in Section 3.6 ) Here, the fatigue life loses close to 70,000 cycles of resistance for each percentage increase in the crack height. With the total number of cycles being well under a million cycles , even a road with a low ADTT (Average Daily Truck Traffic) will have but a few years to persist against fatigue in the cracked condition . For each percentage in increase in the affected section, a model was created or adjusted from the cont rol case, and executed to produce the resulting live moment. Each of moments was then used to back calculate stress using the simple equation: .

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48 (Contributions in the y direction and z directions are minimal.) Stress could then be used to r elate the global geometry of the bridge to the fatigue life, resulting in the graph above. Furthermore, the information from this graph can be used to convert cycles of life, into years of life, using independent information, like the ADTT. Here at this stage of analysis, the observed live load moment increase and simultaneously the sectional properties are reduce d, resulting in the higher stress and lower fatigue lives. In order for the situation to be remedied, external reinforcement or other restorat ive processes must be conducted. Stress will continue to degrade the material condition and cause the crack to grow, as suggested in this modeling. The next section will demonstrate that it is possible to not only to restore actual bridges, but their res pective models. 3.4 C FRP Strengthening The development of a crack in a steel structure is not the end of its potential serviceability, but strengthening methods must be employed at this point . In the study , c arbon f iber r einforced p olymer (CFRP) was mat erially modeled and used in the cracked bridge models. The CFRP is commercially available (Sika Corporation 2011) in sheets or strips that have characteristic dimensions , 1.2 mm by 100 mm , although to some degree these can be manipulated in the manufacturing process, in order to best suit the intended use. The dimensions of the CFRP sections that are used in modeling reinforcement , to be applied to the underside of the cracked girder flange, will vary according to the number o f plies attached to bottom. In this study, four levels of strengthening were used to incrementally demonstrate

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49 the improvement of overstressed conditions. The basic level of CFRP was selected as a six inch wide strip that was 1/10 inch (2.54 mm) thickn ess. The total area of the CFRP section is then 0.60 in 2 or 387 mm 2 . The next case was created by doubling the width of the CRFP strip and increasing its thickness by 50% at the same time (1161 mm 2 ). A third case of strengthening consist ed of a width lower flange (14 in = 355.6mm), and doubl ed the thickness of the base level of CFRP to 2/10 inch ( 5 .08 mm ) . The cross section area of the third level of strengthening was 1806 mm 2 . The most strengthened case also h ad the same width as the bottom flange and the thickness was doubled to 4/10 inch ( 10.16 mm. ) Thereby, the cross sectional area of the fourth case was twice that of the third case, equaling 3613 mm 2 . According to the manufacturer, t h e standard thickness of CFRP strips is 1. 2 mm , so several plies of the material would have to be applied in actuality . The four levels of CFRP strengthening demonstrate incrementally lower stress ranges from the damaged condition . A dding the CFRP sheet presents another el ement to be included in the composite moment of inertia calculation. In the model, CFRP is applied to the soffit of the girder across the length of approximately 90% of the span that is 84 feet (25.6 m) long, centered at midspan. In actuality, the CFRP w ould require end anchoring as a failsafe against potential debonding. Debonding could happen through natural decay or strain exceeding debonding strain. F igure 3 . 12 shows the reduction of stress due to the four cases of CFRP strengthening . In comparison to the damaged case, the additions of CFRP, take the stresses in the extreme fibers of the cracked steel section into stress ranges around or below the infinite life point. The case of 1161 mm 2 provided reinforcement puts the model very close to

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50 being un i nfinite l ife. In fact, as the crack fully penetrates the lower flange the stress value slips past 83 MPa. Thus, the next level of CFRP of 1806 mm 2 studied will allow the steel member to stay be low the stress range of infinite life. 3.5 Fatigue Life Improvement Similar to the calculations above to determine the fatigue lif e for the cracked section, the fatigue l ife of the CFRP strengthened se ctions was determined. The fatigue lives of the four cases of the strengthening are shown in Figure 3.13 against the damaged case. When the lower flange has not fully crack ed , the section with infinite fatigue l ife; the graphed lines extend up to infinity when the line is traced to the left. However, for cracking greater than 2.77%, the first two levels of strengthening (387 mm 2 and 1161 mm 2 ) relieve and extend fatigue life . T he higher two cases of strengthening (1806 mm 2 and 3613 mm 2 ) nity for all levels of cracking examined in modeling . The higher cases of strengthening do not appear on the graph. 3.6 Flexural Cap acities of Sections To determine flexural capacit ies of sections, calculations are done through spreadsheets, not SAP 200 0 calculations. SAP 2000 does not compute moments of inertia compound sections like CFRP strengthened members, let alone for composite bridge sections. It does however compute moments of inertia for individual defined frames and stresses in beam elements .

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51 The process for calculating moment of capacity of any of the composite sections begins with the determination where the plastic neutral axis (PNA) lies. The test of whether the PNA lies within the depth range of the slab is determined by the inequalit y : If the answer is a positive number, then indeed the PNA will lie in the slab. P CFRP is the plastic force in the CFRP, which is limited by the crushing of the concrete (this will be explained in more detail below.) Henceforth, P s is the plastic force in the slab ; P c is the plastic force in the compression flange ; P w is the plastic force in the web and P t is the plastic force in the tension flange . Where t s represents the thickness of the slab. Next, or PNA is calculated: , the n ominal moment capacities of ( CFRP strengthened ) sections is calculated with the following formulas : Depth to the particular element is given by d x . I f , then the PNA lies below the slab . I f the sum of the plastic forces of the slab and the top flange exceeds the sum of the plastic forces of the web, the bottom flange and the CFRP , the PNA will lie in the depth range of the top (compression) flange :

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52 If the PNA lies in the top (compression) flan ge, t he formulas for M n slightly modified: below the 31.75 mm non structural haunch). Th e procedure for the calculation of M n continues in this manner, however none of the cases examined in this study had a PNA below the top (compression) flange in the web region. The procedure above for calculation of M n is used not just for CFRP strengthened cases ; it is also used for the control and cracked (unstrengthened) cases. The sections without CFRP are easier to calculate, because of the elimination of one t erm (two terms if r flange is completely cracked). Unlike steel as structural material where a yielding range is known and well defined, the rupture of the CFRP is unknown and sudden. Because of this , the crushing of concrete (0.003) in an equilibrium section limit s the flexural capacity of that section, and the full strength of the CFRP cannot be assumed . To determine the strain of CFRP at crushing of concrete , an iterative technique was employed to simultaneously determine the PNA and strain of the CFRP. Generally, the stress in the CFRP for these sections was less than half of the rupture stress. The strain of the CFRP at the crushing of concrete is shown in Figure 3.11. These values and their corresponding stresses were used for t he calculation of the flexural capacities of CFRP strengthened sections.

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53 M n improve s incrementally with each level of strengthening , as show n in F igure 3. 14 . The most significant losses of capacity occur, as expected, as th e lower flange cracks completely . The most strengthened case of 3613 mm 2 CFRP increase s the moment capacity of a damaged section to exceed the control case as is seen from the diagram. 3.7 Lo ad and Resistance Factor Rating The LRFR Strength I rating system has two different varian ts: Inventory and Operating . The LRFR (rating) is calculated using the M n capacity determined in the previous section and the flexural loading effects, in addition to various reduction factors: Rating Factor , where Capacity . (AASHTO 2011) The live load factor ( LL ) is 1.75 for i nventory and 1.35 for o perating. Dynamic l oading (IM) is taken as 33% of the live (truck) loading , as provided by Table 3.6.2.1 1 from (AASHTO 2012) . The other resistance, condition and system factor s are shown in Table 3. 9 . The calculations of the rating factors for all mod eling conditions are shown in F igures 3. 15 & 3. 16 . While LRFR (ratings) in theory need only be greater than 1.0, in practice, they typically are greater than 3.0. Here the LRFR (rating) for the control bridge is around 5.0, but it should also be noted that this bridge was not loaded for ultimate capacity, but loaded instead for fatigue resistance.

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54 3.8 LRFR Fatigue Developing a Fatigue Rating ( F igure 3.17 ) for the bridge models consist s of a ratio of the applied live load stress to the fatigue thres ho ld F ) T H of the desired category. The d ynamic l oading (IM) is lowered to 15% of the live loading (truck) ; the live load factor ( LL ) is 0.75 for fatigue . (AASHTO 2012) The dead load moments or stress is not a variable in this calculation, as it is in the Strength I LRFR Rating. The fatigue rating factor for LRFR is given by the following equation: RF fatigue where is the live loading plus the dynamic loading. 3.9 L oad Factor Rating Another method for evaluating the operating and inventory conditions of a bridge is the LFR, Load Factor Rating. (Results shown in Figure s 3.18 & 3.19 .) Inventory designation cove rs the bridge for an indefinite period of service (set it and forget it) , while the operating condition evaluates the maximum permissible load. (AASHTO 2011) . The equation is as follows: RF LFR A s with the more current LRFR system, C represents capacity or M n , but does not have any reduction factor associated with it. D represents the dead loading effect (M DL ) and does have a factor of A 1 that is 1.3 for both operating and inventory conditions. L

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55 represents live loading effect (M LL ), where A 2 is 1.3 for operating and 2.17 for inventory. The I or the impact factor, which accounts for speed, vibration and so forth, is defined through another equation: where L is the length of the span in feet (92 ft). 3.10 M odal Responses One important characteristic of a structure or model to observe is the modal response s . For this study, the first five fundamental modal frequencies wer e reco rded from SAP 2000 for the control, cracked, and cracked and restored conditions. There is not much variation between the crack and control cases, and similarly little variation from the damaged and the CFRP strengthened cases. The data in T ables 3. 10 & 3. 11 show s the narrow range of the results , with little change in fundamental and modal frequencies as crack length propagates . In addition , the modal shapes are depicted in Figures 3.20 24 to show the dynamic behavior of the model. The only control case s are presented , as the modal frequencies vary only very slightly .

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56 Table 3.1 Control b ridge dimensions Parameter Number Dimension Note Span Length Single 28.04 m Simply supported Girders 4 Girder Spacing 2.44 m Supports 4 per end For four girders Bridge Width 9.14 m Design Lanes 2 3.05 m Overhangs 2 0.91 m Barriers None Not considered Deck Thickness 0.19 m Haunch Depth 3.2 mm Non s tructural Table 3.2 Control g irder dimensions Parameter Dimension Overall Height 1.14 m Web Height 1.09 m Web Thickness 7.94 mm Top Flange Width 305 mm Top Flange Thickness 25.4 mm Bottom Flange Width 356 mm Bottom Flange Thickness 31.75 mm

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57 Table 3.3 Damaged girder dimensions in mm (without CFRP) Parameter Dimension Bottom Fla nge Width ( C onst ant ) 305 mm Bottom Flange Thickness: Damage 1% 20.3 mm Damage 2% 8.9 mm Web Height: Damage 3% 1083 mm Damage 4% 1072 mm Damage 5% 1060 mm Damage 6% 1049 mm Damage 7% 1038 mm Damage 8% 1026 mm Damage 9% 1015 mm Damage 10 % 1003 mm Table 3.4 Stress range as a f unction of c rack percentage (of g irder h eight), fatigue truck l oading % ksi MPa 0 3.49 24.1 1 4.96 34.2 Flange 2 8.45 58.3 Web 3 18.42 126.5 4 19.05 131.3 5 19.70 135.8 6 20.38 140.5 7 21.1 0 145.5 8 21.84 150.6 9 22.61 155.9 10 23.42 161.5

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58 Table 3. 5 Damaged girders: f atigue l ife ( c % N 0 1 2 3 1,050,000 4 983,000 5 923,000 6 864,000 7 809,000 8 756,000 9 707,000 10 663,000 Table 3.6 CFRP dim esions (for damaged girders) Parameter Dimension Note CFRP Width: Level 1 152 mm Modular sizes Level 2 305 mm Level 3 356 mm Full flange width Level 4 356 mm Full flange width CFRP Thickness: Level 1 2.54 mm Modular thicknesses Level 2 3.81 mm Level 3 5.08 mm Level 4 10.16 mm CFRP Area: Level 1 387 mm 2 Level 2 1161 mm 2 Modular sizes Level 3 1806 mm 2 (standard) Level 4 3613 mm 2

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59 Table 3.7 Girder sectional properties: transformed area (mm 3 ) Strengthening Level Damage % None 1 2 3 4 0 78,458 n/a n/a n/a n/a 1 74,393 74,800 75,616 76,296 78,199 2 70,329 70,735 71,552 72,232 74,135 3 67,149 67,555 68,372 69,052 70,955 4 67,058 67,464 68,281 68,961 70,864 5 66,966 67,374 68,189 68,869 70,772 6 66,875 67,284 6 8,098 68,778 70,681 7 66,784 67,193 68,007 68,687 70,590 8 66,693 67,097 67,916 68,596 70,499 9 66,602 67,009 67,825 68,505 70,408 10 66,512 66,917 67,735 68,415 70,318 Table 3. 8 Girder sectional properties: m oment of i nertia (10 3 m 3 ) Strengthe ning Level Damage % None 1 2 3 4 0 17.4 n/a n/a n/a n/a 1 13.1 13.8 14.7 15.5 17.5 2 8.5 11.1 12.0 12.8 16.7 3 4.6 5.1 6.2 7.1 9.5 4 4.5 5.0 6.1 7.0 9.4 5 4.3 4.9 6.0 6.9 9.3 6 4.2 4.8 5.9 6.8 9.2 7 4.1 4.7 5.8 6.7 9.1 8 4.0 4.6 5.7 6.6 9.0 9 3. 9 4.5 5.6 6.5 8.9 10 3.8 4.4 5.5 6.4 8.8

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60 Table 3.9 LRFR rating factors Strength I Resistance f actor : R 1.0 for flexure Condition f actor : c 1.0 for good condition, control case c 0.95 for fair condition, crack ed case s System f actor : s 0.9 for a 3 or 4 girder steel bridge, simply supported Load f actors: DL 1.25 LL 1. 75 (Inventory) LL 1.35 (Operating) Table 3. 10 Modal f requencies for the unstrengthened bridge models Crack Frequency (Hz) 0% 3.2105 1% 3.2102 2% 3.2099 3% 3.2096 4% 3.2093 5% 3.2090 6% 3.2087 7% 3.2084 8% 3.2081 9% 3.2078 10% 3.2075

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6 1 Table 3. 11 Modal f requencies of the f irst f ive m odes of a 10% cracked girder, with strengthening cases Mode No. Bridge Case: 1 2 3 4 5 Unstrengthen e d 3.207 3.722 5.513 8.480 11.716 387 sq mm CFRP 3.216 3.732 5.517 8.480 11.736 1161 sq mm CFRP 3.230 3.756 5.524 8.498 11.769 1806 sq mm CFRP 3.241 3.776 5.530 8.517 11.791 3613 sq mm CFRP 3.254 3.791 5.536 8.532 11.818

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62 Figure 3.1 The c ontrol b ridge Figure 3. 2 Control b ridge and g irder d imensions (d amage location indicated for var iable cases)

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63 Figure 3. 3 Control b ridge: s tress in lower flange of exterior girder Figure 3.4 (a) SAP 2000 aerial view of the control b ridge with loading pattern 0 5 10 15 20 25 30 0 4 8 12 16 20 24 28 Stress (MPa) Span Length (m)

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64 Figure 3. 4 (b) Loading configuration producing most adverse effects in the control b ridge , d imensions in mm and kN (n ot to scale) Figure 3 . 5 Assigned dimensions in millimeters for d amaged g irder sections. (D amage occurs at 14.63 me ter location at exterior girder ) ( Tension flange is absent, indicated with dotted lines, for crack perc entage > 2 .77 %. )

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65 Figure . 3 . 6 M aximum stress of 5% cracked girder, exterior Figure 3. 7 Maximum s tress as a f unction of c rack percentage , fatigue loading 0 20 40 60 80 100 120 140 160 0 4 8 12 16 20 24 28 Stress (MPa) Span Length (m) 0 20 40 60 80 100 120 140 160 180 0 2 4 6 8 10 Stress (MPa) Crack %

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66 Figure 3. 8 d etail f atigue l ife vs. c rack percentage Figure 3 . 9 Assigned d imensions in mm for CFRP strengthened girders . ( Yellow for CFRP; t ension f lange absent for crack percentage > 2 .77 % . ) 500,000 600,000 700,000 800,000 900,000 1,000,000 1,100,000 1,200,000 1,300,000 0 2 4 6 8 10 Fatigue Life (Cycles) Crack %

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67 Figure 3. 10 Neutral axis depth, from top of slab Figure 3. 1 1 CFRP strain at concrete crushing 350 400 450 500 550 0 2 4 6 8 10 Neutral axis depth (mm) Crack % Control CFRP 3613 sq mm CFRP 1806 sq mm CFRP 1161 sq mm CFRP 387 sq mm Unstrenthened 0.0040 0.0045 0.0050 0.0055 0.0060 0.0065 0.0070 0.0075 0.0080 0 2 4 6 8 10 Strain (mm/mm) Crack % CFRP 387 sq mm CFRP 1161 sq mm CFRP 1806 sq mm CFRP 3613 sq mm

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68 Figure 3.12 Stres s r ange vs. crack % , with 4 cases of CFRP strengthening (387 mm 2 , 1161 mm 2 , 1806 mm 2 and 3613 mm 2 ) Figure 3. 13 Fatigue life vs. crack percentage , c ontrol with 387 mm 2 and 1161 mm 2 cases of CFRP strengthening. 1806 mm 2 and 3613 mm 2 cases have infinite fatigue life. 0 20 40 60 80 100 120 140 160 0 2 4 6 8 10 Stress Range (MPa) Crack % Unstrengthened CFRP 387 sq mm CRFP 1161 sq mm CFRP 1806 sq mm CFRP 3613 sq mm

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69 Figure 3. 14 Calculation of M n (capacity) of girders, strengthened and control . Figure 3. 15 Inventory r ating f actor (LRFR Strength) 0 2000 4000 6000 8000 10000 12000 0 2 4 6 8 10 M n Capacity (kN m) Crack % Control CFRP 3613 sq mm CFRP 1806 sq mm CFRP 1161 sq mm CFRP 387 sq mm Unstrengthened 0 1 2 3 4 5 6 7 8 0% 2% 4% 6% 8% 10% Inventory RF Crack Control 3613 sq mm 1186 sq mm 1806 sq mm 387 sq mm Unstrengthened

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70 Figure 3. 16 Operating r ating f actor (LRFR Strength) Figure 3. 17 Fatigue r ating f actor (LRFR) 0 1 2 3 4 5 6 7 8 9 10 0% 2% 4% 6% 8% 10% Operating RF Crack Control 3613 sq mm 1806 sq mm 1186 sq mm 387 sq mm Unstrengthened 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0 2 4 6 8 10 Fatigue RF Crack % 3613 sq mm 1806 sq mm 1161 sq mm 387 sq mm Unstrengthened

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71 Figure 3. 18 Load Rating Factor (Inventory) Figure 3. 19 Load Rating Factor (Operating) 0 1 2 3 4 5 6 7 0% 2% 4% 6% 8% 10% Inventory RF Crack Control 3613 sq mm 1806 sq mm 1186 sq mm 387 sq mm Unstrengthened 0 2 4 6 8 10 12 0% 2% 4% 6% 8% 10% Operating RF Crack Control 3613 sq mm 1806 sq mm 1186 sq mm 387 sq mm Unstrengthened

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72 Figure 3.20 Mode 1: T =0.3115 seconds, f =3.210 Hz Figure 3. 21 Mode 2: T = 0.2684 seconds, f =3.725 Hz

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73 Figure 3. 22 Mode 3: T = 0.18130 seconds, f = 5.516 Hz Figure 3. 23 Mode 4: T= 0.11791 seconds, f =8.481 Hz

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74 Figure 3. 24 Mode 5: T = 0.08535, f =11.716 Hz

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75 CHAPTER IV CONCLUSIONS The addition of CFRP as a strengthening material adds capacity to a structure and its members. The process of quantifying such an increase is not as straightforward. The results of this F.E.A. stu dy, using the speed and ease of the modeling programs available, show that not only can a cracked section be modeled, but also its can be strengthened within the model . A research objective of investigating and understanding the effects of fatigue for a b ridge and its model was achieved in the literature review, modeling studies and development, and test trials. Modeling discovery was critical at the outset to the research. There is a steep learning curve for learning the software of SAP 2000, perhaps ev en greater with other less user friendly structural engineering or finite element analysis softwa re . Upon arrival at proficiency, the results of modeling cannot be accepted at face value; they must be checked and verified against past accepted research an d accurate results. M odel s were successfully compared against results from researchers such as Nowak, Sotelino, et. al. In addition, agreement with beam theory was found. By achieving consistency of approach, the models used in the studies were validat ed. The degree of accuracy was +2.8% (beam theory), +6.3% (Nebraska), +8.7% (Michigan); furthermore due to shear deformation and lack of system friction noted in Chapter I , this very slight overshooting was predicted . After validation, a control bridge was established with a consistent modeling technique. The control bridge allowed for various manipulations such as the modeling of crack ing and strengthening. As expected, crack ing reduces capacity and strengthening augments it. Stress in the damaged s ection s are significantly increased . In the highest

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76 case of strengthening of 3613 mm 3 full recovery of moment capacity is achieved. At the same time, the neutral axis of the composite section is lowered, stiffening the system. The improvement due to CFR P restore the bridge to its original LRFR and LFR ratings . Fatigue ratings do not reach the same degree of recovery, but fatigue life attains infinit y for the two higher strengthening cases (1806 and 3613 mm 3 . ) Only the se two higher cases of applied CFRP take the ratin g factor above 1.0 ( when the lower flange is fully cracked ) . The rating dips slightly below 1.0 as the crack propagates up the web for the 1806 mm 2 of strengthening . In the future, researchers could consider R factors for CFRP in ( re ) strengthening design of damaged or fatigue prone structures. One obstacle exists: sufficient studies of the coefficient of variation (CoV) of cracked steel do not exist. In order to develop accurate phi factor s, the CoV of cracked steels needs to show an increase as crack length increases. General categories of damage would not be sufficient. After a CoV for cracked steel is established, Monte Carlo simulations can be completed to develop a phi factor for CFR P strengthening. Other possibilities for further research could include varying the locations of dam age or the type of crack. In summation, fatigue studies and finite element modeling whether through SAP 2000 or another F.E.M. program can yield valuabl e results. The bridge researcher or restoration engineer best understand not only the procedures of inputting data into the programs, but also be knowledgeable in the theory behind the software. This thesis has been a step in the direction of understandi ng structural modeling, and much more is needed to produce qualified structural engineers to tackle the most difficult problems that lie ahead.

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77 REFERENCES AASHTO. 2012. LRFD Bridge Design Specifications. Washington, DC: American Associati on of State Highway and Transportation Officials. AASHTO. 2011. Manual for Bridge Evaluation. Washington, DC: American Association of State Highway and Transportation Officials. Akesson, Bjorn. 2010. Fatigue Life of Riveted Steel Bridges. American Inst itute of Steel Construction. 2011. Steel Construction Manual. 14th Edition. Chicago: AISC. Anderson, B., S. Rolfe, C. Matamoros, C. Bennett, and S. Bonetti. 2007. Post Retrofit Analysis of the Tuttle Creek Bridge Br. No. 16 81 2.24. SM Report No. 88, Lawr ence: The University of Kansas Center for Research, Inc. Barker, Richard M., and Jay A. Puckett. 2013. Design of highway bridges: An LRFD approach. 3rd Edition. Hoboken: John Wiley & Sons. Brockenbrough, R. L. 1986. "Distribution Factors for Curved I Gir der Bridges." Journal of Structural Engineering (ASCE) 110 (10): 2200 15. Computers & Structures, Inc. 2013. "SAP2000 v16.2, Integrated Software for Structural Analysis and Design." Walnut Creek. CSi Knowledge Base. 2013. Composite Section. Edited by Ond rej Kalny. Computers and Structures, Inc. Accessed March 2015. https://wiki.csiamerica.com/display/tutorials/Composite+section. 1999. "Fatigue and Fracture." Chap. 53 in Handbook of Bridge Engineering , by Robert J. Dexter and John W. Fisher, edited by Wai Fah Chen and Lian Duan. CRC Press. Eom, Junsik, and S. Andrzej Nowak. 2001. "Live Load Distribution for Steel Girder Bridges." Journal of Bridge Engineering (ASCE) 489 497. Federal Highway Administration. 2012. Bridge Inspector's Reference Manual. Vol. 1, chap. 5, "Bridge Mechanics" and 6, "Bridge Materials", 1234 2468. Washighton, D.C.: U.S. Department of Transportation. Fisher, John W., Geoffrey L. Kulak, and Ian F. C. Smith. 1998. A Fatigue Primer for Structural Engineers. National Steel Bridge Allia nce. Jaramilla, Becky, and Sharon Huo. 2005. Looking to Load And Resistance Factor Rating. Vers. Vol. 69 No.1. FHwA. July/August. Accessed March 2015. https://www.fhwa.dot.gov/publications/publicroads/05jul/09.cfm.

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78 Kathol, Steve, Atorod Azizinamini, and J im Luedke. 1995. Final Report: Strength Capacity of Steel Girder Bridges. Lincoln: Nebraska Department of Roads. Kim , Yail J., and Garrett Brunell. 2011. "Interaction between CFRP repair and initial damage of wide flange steel beams subjected to three po int bending." Composite Structures (93): 1986 1996. Kim, Sangjin, and Andrzej S. Nowak. 2001. "Load Distribution and Impact Factors for I Girder Bridges." Journal of Bridge Engineering (ASCE) (Nov/Dec): 489 97. Lichtenstein, Abba G. 1993. "The Silver Bri dge Collapse Recounted." ASCE: Journal of Constructed Facilities 7: 249 261. Lonaker, Timothy. 2006. Silver Bridge Collapse. Accessed March 2015. http://www.freewebs.com/silverbridgeaccident/thebridgecollapse.htm. Lui, Ming, Dan M. Frangopol, and Kihyon Kwon. 2010. "Fatigue Reliability Assessment of Retrofitted Steel Bridges Integrating Monitored Data." Structural Safety 77 89. Mabsout, Mounir E., Kassim M. Tarhini, Gerald R. Frederick, and Charbel Tayar. 1997. "Finite Element Analysis of Steel Girder Hi ghway Bridges." Journal of Bridge Engineering (August): 83 87. Nowak, Andrezj S., and Junsik Eom. 2001. Verification of girder distribution factors for steel girder bridges. Final Report, Department of Civil and Environmental Engineering, University of Mi chigan, Lansing: Michigan Department of Transportation. Petroski, Henry. 2012. To Forgive Design: Understanding Failure. Cambridge, MA: Harvard University Press. Purdue University. 2015. Steel Bridge Fatigue: Knowledge Base. Local Technical Assistance Pr ogram. Accessed March 2015. http://rebar.ecn.purdue.edu/fatigue. Sika Corporation. 2011. "Sika CarboDur, Edition 5.4." pds cpd SikaCarboDur us.pdf. http://usa.sika.com/en/home page features/product finder/iframe_and_dropdown/carbodur.html. Sotelino, Elis a D., Judy Liu, Wonseok Chung, and Kitjapat Phuvoravan. 2004. Simplified load distribution factor for use in LRFD design. Final Report, School of Civil Engineering, Purdue University, Indianapolis: Indiana Department of Transportation. Sotelino, Elisa, an d Wonseok Chung. 2006. "Three Dimensional Finite Element Modeling of Composite Girder Bridges." Engineering Structures (Elsevier) (28): 63 71.

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79 Timoshenko, Stephen, and J. N. Goodier. 1970. Theory of Elasticity. 3rd Edition. McGraw Hill.

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80 APPENDIX A EIGHT WAYS TO MODEL IN SAP 2000 (Computers & Structures, Inc. 2013) Table A1:

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81 Figure A1 Model 1, a Fictitious Rendering in SAP 2000 (Nebraska Bridge) Figure A 2 Section of a Composite Beam Model

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82 Table A2: Beam Designation Behavior Midspan Deflection [mm] Comments Theoretical Beam composite 3.2725 Theoretical deflection is based on th formulation. Please note that SAP 2000 calculations produce slightly greater values because shear defor mation is considered in deflection. Beam 1 (top beam) Non composite 7.1752 The deck slab center line coincides with the section neutral axis. Therefore, the deck slab contribution to section flexural stiffness will be negligible. Further, because there i s no composite action, midspan deflection should be close to that of a naked girder. Beam 2 composite 3.2624 In this model, slab shell objects are drawn at the girder center of gravity (COG), and then offset vertically, above the girder, to model composit e action. The shells are offset such that the slab soffit is located above the girder top flange.

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83 Table A2 (continued): Beam Designation Behavior Midspan Deflection [mm] Comments Beam 3 composite 3.2624 In this model, the girder and the slab are dr awn at their respective center lines. The corresponding girder and slab joints are then connected through body constraints. Beam 4 composite 3.2624 In this model, composite action is modeled using frame insertion points. Beam 5 Non composite 7.1752 Equal constraints are used to model non composite behavior. Beam 6 Non composite 7.1752 Links are used to model non composite behavior. Beam 7 partially composite 3.5036 Links are used to model partially composite behavior. Beam 8 (bottom beam) composite 3. 2624 Links are used to model composite behavior.

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84 APPENDIX B SAP 2000 INPUTS FOR CONTROL BRIDGE (Computers & Structures, Inc. 2013) 1. Model geometry This section provides model geometry information, including items such a s joint coordinates, joint restraints, and element connectivity. Figure 1: Finite element model (Computers & Structures, Inc. 2013) 1.1. Joint coordinates Table 1: Joint Coordinates Joint Coord. Sys Coord. Type Global X Global Y Global Z feet feet feet 1 GLOBAL Cartesian 0.00 0.00 4.167 2 GLOBAL Cartesian 4.00 0.00 4.167 3 GLOBAL Cartesian 4.00 6.00 4.167 4 GLOBAL Cartesian 0.00 6.00 4.167

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85 Table 1: Joint Coordinates Joint Coord. Sys Coord. Type Global X Global Y Global Z feet feet feet 5 GLOBAL Cartesian 8.00 0.00 4.167 6 GLOBAL Cartesian 8.00 6.00 4.167 7 GLOBAL Cartesian 12.00 0.00 4.167 8 GLOBAL Cartesian 12.00 6.00 4.167 9 GLOBAL Cartesian 16.00 0.0 4.167 10 GLOBAL Cartesian 16.00 6.00 4.167 11 GLOBAL Cartesian 20.00 0.0 4.167 12 GLOBAL Cartesian 20.00 6.00 4.167 13 GLOBAL Cartesian 24.00 0.0 4.16 7 14 GLOBAL Cartesian 24.00 6.00 4.167 15 GLOBAL Cartesian 28.00 0.0 4.167 16 GLOBAL Cartesian 28.00 6.00 4.167 17 GLOBAL Cartesian 32.00 0.0 4.167 18 GLOBAL Cartesian 32.00 6.00 4.167 19 GLOBAL Cartesian 36.00 0.0 4.167 20 GLOBAL Cartesian 36.00 6. 00 4.167 21 GLOBAL Cartesian 40.00 0.0 4.167 22 GLOBAL Cartesian 40.00 6.00 4.167 23 GLOBAL Cartesian 44.00 0.0 4.167 24 GLOBAL Cartesian 44.00 6.00 4.167 25 GLOBAL Cartesian 48.00 0.0 4.167 26 GLOBAL Cartesian 48.00 6.00 4.167 27 GLOBAL Cartesian 5 2.00 0.0 4.167 28 GLOBAL Cartesian 52.00 6.00 4.167 29 GLOBAL Cartesian 56.00 0.0 4.167 30 GLOBAL Cartesian 56.00 6.00 4.167 31 GLOBAL Cartesian 60.00 0.0 4.167 32 GLOBAL Cartesian 60.00 6.00 4.167 33 GLOBAL Cartesian 64.00 0.0 4.167 34 GLOBAL Carte sian 64.00 6.00 4.167 35 GLOBAL Cartesian 68.00 0.0 4.167 36 GLOBAL Cartesian 68.00 6.00 4.167 37 GLOBAL Cartesian 72.00 0.0 4.167 38 GLOBAL Cartesian 72.00 6.00 4.167 39 GLOBAL Cartesian 76.00 0.0 4.167 40 GLOBAL Cartesian 76.00 6.00 4.167 41 GLOBA L Cartesian 80.00 0.0 4.167

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86 Table 1: Joint Coordinates Joint Coord. Sys Coord. Type Global X Global Y Global Z feet feet feet 42 GLOBAL Cartesian 80.00 6.00 4.167 43 GLOBAL Cartesian 84.00 0.0 4.167 44 GLOBAL Cartesian 84.00 6.00 4.167 45 GLOBAL Cartesian 88.00 0.0 4.167 46 GLOBAL Cartesian 88.00 6.00 4.167 47 GLOBAL Cartesian 92.00 0.0 4.167 48 GLOBAL Cartesian 92.00 6.00 4.167 240 GLOBAL Cartesian 92.00 27.00 4.167 241 GLOBAL Cartesian 0.0 3.00 4.167 242 GLOBAL Cartesian 4.00 3.00 4.167 243 GLOBAL Cartesian 8.00 3.00 4.167 263 GLOBAL Cartesian 88.00 3.00 4.16 7 264 GLOBAL Cartesian 92.00 3.00 4.167 1.2. Joint restraints Table 2: Joint Restraint Assignments Joint U1 U2 U3 R1 R2 R3 1 Yes Yes Yes No No No 47 No No Yes No No No 50 Yes Yes Yes No No No 72 No No Yes No No No 98 Yes Yes Yes No No No 120 No No Yes No No No 146 Yes Yes Yes No No No 168 No No Yes No No No 1.3. Element connectivity

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87 Table 3: Connectivity Frame Frame Joint I Joint J Length feet 1 1 2 4.00 2 2 5 4.00 4.00 92 167 168 4.00 93 170 25 2.083 94 169 170 0.083 Tab le 4: Frame Section Assignments Frame Analysis Section Design Section Material Property 1 45" girder 45" girder Default 2 45" girder 45" girder Default 94 45" girder 45" girder Default Table 5: Connectivity Area Area Joint1 Joint2 Joint3 J oint4 1 1 2 3 4 2 2 5 6 3 229 262 263 45 43 230 263 264 47 45 Table 6: Area Section Assignments Area Section Material Property 1 deck Default 2 deck Default 230 deck Default 2. Material properties This section provides materi al property information for materials used in the model.

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88 Table 7: Material Properties 02 Basic Mechanical Properties Material Unit Weight Unit Mass E1 G12 U12 A1 Kip/in 3 Kip s 2 /in 4 Kip/in 2 Kip/in 2 1/F 4000 psi 8.680E 05 2.248E 07 3604 1502 0.200 5.5 00E 06 A992 F y =50 2.835E 04 7.344E 07 29000 11153 0.300 6.500E 06 Table 8: Material Properties 03a Steel Data Material F y F u Final Slope Kip/in 2 Kip/in 2 A992 F y =50 50.00 65.00 0.100 Table 9: Material Properties 03b Concrete Data Material Fc Final Slope Kip/in 2 4000Psi 4.000 0.100 3. Section properties This section provides section property information for objects used in the model. 3.1. Frames Table 11: Frame Section Properties 01 General, Part 1 of 4 Section Name Material Shap e t3 t2 t f t w t2 b t fb inch inch inch inch inch inch 45" girder A992 F y =50 I/Wide Flange 45.00 12.00 1.00 0.3125 14.00 1.250

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89 Table 11: Frame Section Properties 01 General, Part 2 of 4 Section Name I23 Area Tors Const. I33 I22 AS2 AS3 in 4 in 2 in 4 in 4 in 4 in 2 in 2 45" girder 0.00 42.86 12.82 15900 429.9 14.06 24.58 Table 11: Frame Section Properties 01 General, Part 3 of 4 Section Name S33 S22 Z33 Z22 R33 R22 in 3 in 3 in 3 in 3 in in 45" girder 630.12 61.42 766.08 98.29 19.26 3.18 Table 11 : Frame Section Properties 01 General, Part 4 of 4 NONE 3.2. Tendons NONE 3.3. Areas Table 1 2 : Area Section Properties, Part 1 of 3 Section Materia l Area Type Type Drill DOF Thick. Bend Thick. F11 Mod in in deck 4000 psi Shell Shell Thin Yes 7.50 7.50 1.00 Table 12 : Area Section Properties, Part 2 of 3 Section F22 Mod F12 Mod M11 Mod M22 Mod M12 Mod V13 Mod V2 3Mod M Mod deck 1.00 1.00 0.20 1.00 1.00 1.00 1.00 1.00

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90 Table 1 2 : Area Section Properties, Part 3 of 3 Section W Mo d deck 1.00 4. Load patterns This section provides loading information as applied to the model. 4.1. Definitions Table 14: Load Pattern Definitions Load Pat Design Type Self Wt. Mult. Auto Load DEAD DEAD 1.00 live truck LIVE 0.00 5. Lo ad cases This section provides load case information. 5.1. Definitions Table 15: Load Case Definitions Case Type Initial Cond Des Act Opt. Design Act. DEAD Lin. Static Zero Prog. Det. Non Composite MODAL Lin. Modal Zero Prog. Det. Other live truck Li n. Static Zero Prog. Det. Short Term Composite 5.2. Static case load assignments Table 16: Case Static 1 Load Assignments Case Load Type Load Name Load SF DEAD Load pattern DEAD 1.00 live truck Load pattern live truck 1.00

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91 5.3. Response spectr um case load assignments Table 17: Function Response Spectrum User Name Period Sec. Accel. Func Damp UNIFRS 0.0 1.00 0.050 UNIFRS 1.00 1.00 6. Load combinations This section provides load combination information. Table 18: Combination Defini tions Combo Name Combo Type Case Name Scale Factor DSTL1 Linear Add DEAD 1.40 DSTL2 Linear Add DEAD 1.20 DSTL2 live truck 1.60 DSTL3 Linear Add DEAD 1.00 DSTL4 Linear Add DEAD 1.00 DSTL4 live truck 1.00 7. Design preferences This section provid es the design preferences for each type of design, which typically include material reduction factors, framing type, stress ratio limit, deflection limits, and other code specific items. 7.1. Steel design Table 19: Preferences Steel Design AISC LRFD 99, Part 1 of 2 Frame Type Pat LLF Ratio Limit Seis . Cat Plug Weld Phi B Phi C Phi TY Phi TF OMF 0.7500 0 0.9500 0 D Yes 0.9000 0 0.8500 0 0.9000 0 0.7500 0

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92 Table 19: Preferences Steel Design AISC LRFD99, Part 2 of 2 Phi V Phi VT Phi CA DL Rat SDL And LL Rat LL Rat Total Rat Net Rat 0.9 00 0.75 0 0.900 120.00 120.00 360.00 240.00 240.00 8. Design overwrites This section provides the design overwrites for each type of design, which are assigned to individual members of the structure. 8.1. Steel design Table 22: Overwrites Steel Design AISC LRFD99, NONE

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93 APPENDIX C PROCEDURE FOR SAP 2000 MODELING DEFINE GRID SYSTEM DATA, establish x, y, z grid for model basis:

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94 DEFINE MATERIALS, (note 4,000Psi concrete and A992Fy50 steel are pre loaded): MATERIAL PROPERTY DATA, for 4,000 psi concrete:

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95 MATERIAL PROPERTY DATA, A992 Grade 50 steel:

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96 DEFINE SECTION PROPERTIES, FRAME SECTIONS: STEEL SECTION (I/WIDE FLANGE SECTION) , enter girder dimension as follows for the co ntrol girder (inches shown) :

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97 DEFINE SECTION, AREA SECTIONS, select shell type : SHELL SECTION DATA , enter deck thickness as (7.5 inches) for both membrane and bending:

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98 STIFFNESS MODIFIERS of Shell Section Data, PROPERTY /STIFFNESS MODIFICATI ON FACTORS, leave all factors at 1.0: DEFINE LOAD PATTERNS, live load (fatigue truck): Draw frames and shell area elements to match dimensions of control bridge (or dimenstions of desired model). Make use of Replicate feature to reduce time draw ing identical elements. Nodes for frames and shells will all be assigned on the same xy plane, when modeling compositely. (Frame offset method, described below.)

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99 ASSIGN JOINT RESTRAINTS, pinned condition, free to rotate: ASSIGN JOINT RESTRAINTS, roller condition, movement in x, y direction allowed:

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100 Select all (girder) frames and ASSIGN FRAME INSERTION POINT. Choose Cardinal 5 for Local directi unchecked: This will place the top flange of the composite section under the soffit of the deck. The condition of a non structural haunch of 5 3.75 =1.25 inches will persist.

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101 Assign loads to test locations (this can be done at any point after frames and shells are drawn.) ASSIGN JOINT LOADS, live truck ( fatigue loads ) th e trailing axles: ANALYZE, RUN ANALYSIS, SET LOAD CASES TO RUN , run all cases:

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102 Deformed Shape induced by dead load is the default graphic outputs. Use toggle arrows at the bottom right of screen to view other loading cases and patterns output. Or use DEFORMED SHAPE :

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103 For other critical output select icon, MEMBER FORCE/STRESS DIAGRAM FOR 3, Axial Force, Stress M ax, etc.): Use output in further analysis.

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104 For cracked sections and CFRP reinforced sections add the following additional material and sectional properties. ASSIGN SECTION PROPERTIES, TEE SECTION for cracked girder with lower flange fully penetr ated:

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105 ASSIGN MATERIAL PROPERTY DATA, Note: CFRP is not a default material and must be assign properties from another base mat erial.

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106 ASSIGN SECTION PROPERTIES for CFRP sheets: