Investigation of Load-Path Redundancy in
Aging Steel Bridges
By Jennifer McConnell, Diane Wurst, Gillian McCarthy, and
Matthew Sparacino
A report submitted to the University of Delaware University
Transportation Center (UD-UTC)
December, 2013
DISCLAIMER:
The contents of this report reflect the views of the authors, who
are responsible for the facts and the accuracy of the information
presented herein. This document is disseminated under the
sponsorship of the Department of Transportation University
Transportation Centers Program, in the interest of information
exchange. The U.S. Government assumes no liability for the
contents or use thereof.
Investigation of Load-Path Redundancy in Aging Steel Bridges
UDUTC FINAL REPORT
PI:
Jennifer McConnell
Research Assistants:
Diane Wurst,
Gillian McCarthy, and
Matthew Sparacino
TABLE OF CONTENTS
EXECUTIVE SUMMARY .......................................................................................... vii 1
INTRODUCTION .............................................................................................. 1 1.1 Motivation ................................................................................................. 1 1.2 Objectives and Scope of Work .................................................................. 2 1.3 Report Organization .................................................................................. 3 2
BACKGROUND ................................................................................................ 5 2.1 Federal Funding and the Highway Bridge Program .................................. 5 2.1.1 The Rating Process behind the National Bridge Inventory ........... 5 2.1.2 The Sufficiency Rating Equation .................................................. 6 2.1.3 The Importance of Load Carrying Capacity in the Rating
Process ........................................................................................... 6 2.1.3.1 Structural Adequacy and Safety (S1) ............................. 7 2.1.3.2 “Inventory Rating” ......................................................... 8 2.1.3.3 Observations Related to Sufficiency Rating................... 8 2.2 Load Path Redundancy and Load Carrying Capacity ............................... 9 2.2.1 What is Load Path Redundancy? ................................................... 9 2.2.2 How Does this Affect the Load Carrying Capacity? ................... 10 2.3 Load Carrying Capacity is a System Effect ............................................ 11 2.4 Scope for Improvement in the Bridge Rating and Funding Process ....... 11 3
EFFECTS OF AGING ON REINFORCED CONCRETE BRIDGE DECKS 13 3.1 Deterioration of Reinforced Concrete ..................................................... 13 3.1.1 Chloride Ion Ingress and the Corrosion of Reinforcing Steel ..... 14 3.1.2 Effects of Chloride Induced Corrosion ........................................ 14 ii
3.1.3 Deterioration Effects on Concrete Mechanical Properties .......... 15 3.2 Review of Studies on Concrete Deterioration due to Chloride ion
Ingress...................................................................................................... 16 3.2.1 Study 1 – “Propagation of Reinforcement Corrosion in
Concrete and its Effects on Structural Deterioration” (Li &
Zheng, 2005)................................................................................ 17 3.2.1.1 3.2.1.2 3.2.1.3 3.2.1.4 Concept of Time Transformation ................................. 18 Rebar Mass Loss ........................................................... 21 Strength Loss ................................................................ 22 Stiffness Loss................................................................ 23 3.2.2 Study 2 – “Bending Performance of Reinforced Concrete
Member Deteriorated by Corrosion” (Oyado, Kanakubo, Sato,
& Yamamoto, 2011) .................................................................... 25 3.3.1 Effects on Strength ...................................................................... 36 4
MODELING APPROACH .............................................................................. 42 4.1 Concrete ................................................................................................... 42 4.1.1 Elastic Behavior........................................................................... 43 4.1.2 Non-Linear Behavior ................................................................... 43 4.1.2.1 Smeared Crack Model .................................................. 44 4.1.2.2 Concrete Damaged Plasticity Model ............................ 46 4.1.2.3 Brittle Cracking Model ................................................. 49 4.2 Rebar........................................................................................................ 50 4.2.1 2-Dimensional Rebar ................................................................... 50 4.2.2 3-Dimensional Rebar ................................................................... 51 4.3 Boundary Conditions ............................................................................... 56 4.4 Analysis Method ...................................................................................... 57 5
MODEL CALIBRATION ................................................................................ 60 5.1 Beam Geometry and Material Properties ................................................ 61 5.1.1 Concrete ....................................................................................... 61 5.1.2 Rebar............................................................................................ 63 iii
5.2 Calibration Metrics .................................................................................. 66 5.3 Strength and Deflection Calculations ...................................................... 69 5.4 2-Dimensional Concrete Model .............................................................. 70 5.4.1 Uncorroded Base Model Input Values ........................................ 71 5.4.1.1 Rebar............................................................................. 72 5.4.1.2 Concrete Damaged Plasticity ....................................... 72 5.4.1.3 Smeared Crack.............................................................. 74 5.4.2 Mesh Sensitivity Analysis and Uncorroded Base Model Input... 75 5.4.3 Corroded Model........................................................................... 81 5.4.3.1 5.4.3.2 5.4.3.3 5.4.3.4 Individual Parameter Variation .................................... 82 Calibration of Ec, f’c, and f’t ......................................... 84 Calibration of A’s and As .............................................. 87 Calibration of Dilation Angle ....................................... 93 5.5 3-Dimensional Concrete Model .............................................................. 97 5.5.1 2-Dimensional Rebar ................................................................... 97 5.5.1.1 Brittle Cracking Base Model Input............................... 98 5.5.1.2 Mesh Sensitivity Analysis ............................................ 99 5.5.1.3 Corroded Model.......................................................... 102 5.5.2 3-Dimensional Rebar ................................................................. 104 5.5.2.1 Mesh Sensitivity Analysis .......................................... 104 5.5.2.2 Uncorroded Base Models Using Brittle Cracking ...... 105 5.5.2.2.1 Smeared Crack and Concrete Damaged
Plasticity Models ...................................... 106 5.5.2.2.2 Brittle Cracking Model ............................. 108 5.6 Conclusions ........................................................................................... 113 6
BRIDGE MODELS ........................................................................................ 116 6.1 7R .......................................................................................................... 116 6.1.1 Bridge Information .................................................................... 116 6.1.2 Results ....................................................................................... 118 6.1.2.1 Finite Element Modeling ............................................ 118 iv
6.1.2.2 Distribution Factors .................................................... 128 6.2 SR 1 over US 13 .................................................................................... 130 6.2.1 Bridge Information .................................................................... 131 6.2.2 Modeling Results ....................................................................... 133 6.3 SR 299 over SR 1 .................................................................................. 139 6.3.1 Bridge Information .................................................................... 139 6.3.2 Modeling Results ....................................................................... 140 6.4 Conclusions ........................................................................................... 142 7
RELATIONSHIPS TO BRIDGE RATING PRACTICE............................... 144 7.1 FEA-Determined System-Level Strength.............................................. 145 7.2 FEA-Determined Member-Level Distribution Factors ......................... 147 7.2.1 Procedure ................................................................................... 147 7.2.2 Summary of Results .................................................................. 150 7.3 System-Level Rating Calculations ........................................................ 153 7.4 Member-Level Distribution Factors ...................................................... 156 7.4.1 Inelastic Member-Level Distribution Factors ........................... 156 7.4.2 Elastic Member-Level Distribution Factors Considering
Aging ......................................................................................... 157 7.4.2.1 7.4.2.2 7.4.2.3 7.4.2.4 7.4.2.5 Background................................................................. 157 Sensitivity Study ......................................................... 159 Results ........................................................................ 162 Discussion................................................................... 163 Conclusions ................................................................ 167 7.5 Relationship to Bridge Inspection Reports ............................................ 168 7.5.1 7.5.2 7.5.3 7.5.4 Background................................................................................ 168 Bridge Inspection Reports Data Set .......................................... 172 Preliminary Study, Delaware Bridges ....................................... 172 Primary Study ............................................................................ 178 7.5.4.1 Procedure .................................................................... 178 7.5.4.2 Results and Discussion ............................................... 179 v
7.5.5 Conclusions ............................................................................... 186 8
PARALLEL AND FUTURE WORK ............................................................ 190 8.1 Parallel Considerations .......................................................................... 190 8.1.1 Cost-Effectiveness Analysis ...................................................... 190 8.1.2 Life365 Model for Concrete Deterioration................................ 192 8.2 Future Work........................................................................................... 166
REFERENCES ............................................................................................... 170 A
B
SUPPLEMENTAL DATA FOR DISTRIBUTION FACTOR
CALCULATIONS ........................................................................................ 176 PERMISSION LETTERS .............................................................................. 183 vi
EXECUTIVE SUMMARY
National statistics on numbers of structurally-deficient bridges and the
associated limited financial resources for infrastructure renewal require ever more
thoughtful approaches to bridge management. One significant opportunity for better
prioritizing the limited funding available for this purpose is through better
quantification of the existing strength of aging structures. Specifically, one promising
means for achieving this goal is leverage the load-path redundancy inherent to steel
girder bridges to calculate a so-called system capacity of the structure, in contrast to
the present approach of calculating a strength that is governed by the weakest or mostheavily loaded member. When this more realistic approach is utilized, bridges have
significant reserve capacity above what is predicted from modern analysis and design
methods predicated on the analysis of individual members.
The present research was motivated by this promising concept of system
capacity and the identification of information needed to transition this concept to
practice. These needs include both knowledge needed with respect to structural
mechanics (which was the subject of Phase 1 of this project) as well as practical needs
to synthesize the research result into a format that is synergistic with existing
evaluation methods used by bridge engineers (which was the subject of Phase 2 of this
work). The structural mechanics that must be assessed and quantified relates to the
fact that in system level analysis the transverse load distribution mechanisms become
more influential than in current methods, so further understanding of this phenomena
are needed. Furthermore, the reliability of these transverse load distribution
mechanisms, particularly those achieved through reinforced concrete bridge decks, in
aging structures with various states of deterioration was unknown. For the results of
these efforts to be most beneficial, they must be translated into a rating approach
consistent with American Association of State Highway and Transportation Officials
(AASHTO) Load and Resistance Factor Rating (LRFR) procedures for highway
bridges. In addition, a means to incorporate the actual condition of the deck as
quantified by the inspection report specific to each structure would further improve the
applicability of the research.
Considering these needs while limiting the research to a practical scope for this
project, the key objectives of this work were:
vii
1. To assess and quantify the influence of corrosion caused by deicing-agents on
the ability of reinforced-concrete bridge decks to serve as a transverse load
distribution and redistribution mechanism in steel girder bridges;
2. To formulate a rating procedure to quantify the system capacity of bridges that
includes the effects of reinforced-concrete deck deterioration; and
3. To explore the correlations deck condition ratings and level of deterioration
and how this may affect predicted system capacities.
This was accomplished through several steps, which included the following
primary tasks and key conclusions, which are organized as follows.

Chapters 1 and 2 introduce this work. Specifically, Chapter 1 describes the
motivation, objectives, and scope of the research as well as a summary of the
organization of this report. Chapter 2 provides background on the basic
premises leading to system capacity, the bridge rating process, and financial
aspects of this process.

The first research task was a literature review (Chapter 3), performed to
quantify the effects of aging on reinforced concrete bridge decks. This review
focused on chloride–induced corrosion as this is one of the most ubiquitous
causes of reinforced concrete deck deterioration throughout a large portion of
the United States.
The review then identified rebar mass loss, concrete strength degradation, and
changes in concrete ductility as the primary effects of chloride-induced
deterioration which would affect the structural performance of reinforced
concrete members. Experimental test results were reviewed to quantify these
changes versus time. These trends were then extrapolated to estimate changes
after 25 years of regular exposure to chlorides, which led to estimates that the
strength (as a result of both changes to concrete strength and rebar mass loss)
would decrease by 50% and ultimate deflection would increase by 80%
relative to the original good-condition material and geometric properties over
this time period. These performance characteristics would become the targets
for calibrating a finite element model (FEM) simulating chloride-induced
reinforced concrete corrosion.
viii

Prior to conducting this modeling, a review of the FEM approaches available
for modeling: the non-linear concrete constitutive response, two-dimensional
(2-D) versus three-dimensional (3-D) rebar elements, and various contact and
bond properties between the concrete and rebar elements was necessary in
order to understand the strengths and weaknesses in the accuracies and
capabilities of the various combinations of input which are available (Chapter
4).
This review was specific to the capabilities of the commercial finite element
(FE) software Abaqus due to the ultimate desire to implement these methods
into existing bridge models in order to expedite the work and maintain a
reasonable scope for the present work. The prior models were created with
good-condition decks in a format specific to the Abaqus platform.

With the targeted responses determined and the available modeling techniques
identified, a calibrated FEA approach for modeling the effects of chlorideinduced corrosion in reinforced-concrete members was sought (Chapter 5).
This was attempted using both 2-D and 3-D models of reinforced-concrete
beams tested before and after chloride-induced corrosion in the literature.
This effort was successful for the 2-D approach but not the 3-D approach. This
was considered a suitable outcome because the existing bridge models that
were to be used in later tasks contained 2-D elements representing the
reinforced-concrete decks and to use 3-D elements in these models would
significantly increase both the modeling effort and the computational demands
(in the latter case, possibly to the extent that the effort would be impossible
with modern resources).
For the 2-D approach, the calibrated model utilized a concrete damaged
plasticity modeling approach with embedded rebar elements. This is a
limitation of the 2-D approach in that the degradation of the bond between the
concrete and rebar as deterioration occurs cannot be directly simulated; rather a
perfect bond between the materials is assumed. Thus, this and other effects of
deterioration were simulated by varying the rebar areas as well as
simultaneously varying the concrete tensile strength, compressive strength, and
modulus of elasticity in a manner consistent with existing empirical
ix
relationships that describe the inter-dependent relationships among these latter
three properties. As a result a 45% decrease in strength was achieved, which
was deemed to be acceptably close to the 50% decrease which was targeted.
The assessment of the change in ultimate deflection is less straight-forward as
a clear definition of ultimate deflection does not exist. However, when
comparing the deflections between the uncorroded and corroded models at the
maximum loading achieved in the uncorroded model a 120% increase in
deflection was obtained; when comparing the deflections corresponding to the
maximum loading obtained in each model, the corresponding difference is
12%. Thus, it was deemed that because these results bound the target of an
80% increase in deflection, the deflection of the models was also acceptably
calibrated. Furthermore, it should be noted that many preliminary models
produced a decrease in deflection in the corroded model (counter to the goal of
increased deflection), so simply achieving the goal of increased ductility, while
also achieving the quantitative targets to a reasonable extent, was not a trivial
matter.

After the calibration of the reinforced-concrete beam models, these inputs were
applied to existing FEM models of three bridges previously field-tested and
calibrated in prior work (Chapter 6). One of these was a simple-span bridge
and the other two were continuous-span bridges. Successful results were
obtained for the simple-span bridge, but the indeterminate structure of the twospan bridges prevented convergence of these models to a realistic result.
The results of the simple-span bridge showed that, contrary to expectations, not
only did deck deterioration not have a detrimental effect on the system
capacity of the structure, but the deck deterioration led to greater load-sharing
between girders, which ultimately led to the corroded deck producing a higher
system strength than the uncorroded model. These findings are quantified
through FEM-predicted system strengths of 25 and 28 HS-20 trucks for the
uncorroded and corroded decks, respectively. It appears that this phenomenon
manifests because the load distributes more uniformly throughout the deck in
the corroded model because the peak stress for any individual element are
limited to a smaller value.
To put these strengths in context, these system-level strengths should be
considered relative to: (1) a previously-determined system strength of 30 HS-
x
20 trucks for the same bridge modeled with a linear-elastic reinforced-concrete
deck of infinite strength, and (2) a predicted capacity of 15 HS-20 vehicles
based on current AASHTO methods. Thus, it is demonstrated that systemlevel analysis of bridges can dramatically increase the strength of this steel
girder bridge, with the estimates in this work suggesting the increase can be
between 67 and 87% for realistic assumptions of the reinforced-concrete deck
characteristics. Consequently, it is concluded that corrosion caused by
deicing-agents does not significantly diminish the ability of reinforcedconcrete bridge decks to serve as a transverse load distribution and
redistribution mechanism in steel girder bridges for this representative
structure (satisfying the goals of Objective 1).
The distribution factors resulting from the uncorroded and corroded models
were also calculated and compared in order to synthesize the results into this
format that is well-understood by bridge engineers. These calculations also
supported that there is greater load sharing between the girders in the corroded
model, where the maximum distribution factor among the two most-heavily
loaded girders is 0.262 in the corroded model compared to a corresponding
value of 0.277 in the uncorroded model. However, for all three models (elastic,
uncorroded, and corroded deck models), the distribution factor results for the
two-most heavily loaded girders (which are the two girders whose loading is
synonymous with valid load positions for calculating distribution factors) are
within 11% of the theoretical inelastic values and are a 25 to 46% reduction
relative to the current AASHTO (2013) elastic distribution factors, showing
both the potential for adapting these results into a format convenient for use in
routine bridge analysis and the dramatic improvement in economy that could
be achieved by more-accurately calculating inelastic distribution factors in
contrast to the present methods based on elastic distribution factors.

The final task of this project was to explore ways in which the results of the
research above could be adopted in bridge rating practices. Thus Chapter 7
first discusses how FEM output can be synthesized into quantification of
strengths and distribution factors, which are quantities bridge engineers are
familiar with. Then, two alternative options for adopting the research results
into this format to serve as a component of generalized rating procedures are
discussed (satisfying the goals of Objective 2). The first of these is a systemlevel rating calculation based where the bridge is rated based on a summation
xi
of girder strengths. The second is to maintain the present line-girder method of
analysis, but to revise the distribution factors used in this approach to consider
inelastic load redistribution.
It was originally envisioned that the summation of the strength of all girders in
the cross-section may produce a simple upper-bound of system capacity, but
that reductions to this capacity would be necessary for compromised deck
conditions or other situations. However, the weakest of the three models
evaluated herein (the uncorroded model, see Table 6.2) achieved this strength
and the other two models exceeded this strength, with the corroded model
exceeding the simple theoretical prediction by 14%. By looking at the
distribution of stress in the FEA, it is clear that this is largely due to
conservatism AASHTO estimates for individual girder capacities. Thus, it is
suggested that the summation of girder strengths is a reasonable estimate of a
system capacity, although future research is needed to investigate potential
reductions to this capacity (which could be expressed in specifications as
additional limit states) that are needed to account for the possibility of
transverse load redistribution mechanisms being insufficient for redistributing
forces between girders in order for the idealized system capacity to be
achieved.
This chapter concludes with a discussion between the relationship to bridge
inspection reports and deck condition based on reviewing element-level
condition states and environmental conditions of a sample of bridges, and a
methodology for incorporating this information in revised distribution factor
calculations. As a result, a relationship that allows the DF for a bridge to be
estimated by knowing its NBI deck rating and the environmental factor from
the element-level inspection data now recorded by many agencies (in addition
to the geometry of the bridge and other variables necessary to the DF equation
of choice) was established. While the data used to derive this equation was
limited, a valid process has been determined that could be refined with a larger
data set to broaden the applicability of the resulting estimates. Furthermore,
while these results were limited to elastic distribution factors as this is
presently the only format in which distribution factors are quantified for a
broad range of parameters, the process that has been developed could be
equally applied to inelastic distribution factors as these equations become
available. Thus an efficient methodology to estimate distribution consider the
xii
current condition state of the bridge has been created (satisfying the goals of
Objective 3).

During the course of this research, two additional parallel routes were
explored. The first of these aimed to further evaluate the potential financial
impacts of this research and is discussed in Section 8.1.1. As a result of this
effort, significant economic savings through implementing system-level
analysis on steel girder bridges with both good and poor condition reinforcedconcrete decks are estimated. Specifically, it is estimated that if system-level
analysis is implemented for good condition decks only, at least 10% of the
estimated budget can be saved through deferred spending. However, if the
methodology is also extended to poor condition decks, the number of bridges
in Delaware to which this economic savings can be applied nearly triples.
Additional capabilities for modeling reinforced-concrete deck deterioration
through a deterioration model known as Life365 are reviewed in Section 8.1.2.
The potential usefulness in this program lies in its ability to plot the infiltration
of chloride as a function of time. However, because this data could not be
directly related to the numerical change in material properties of the concrete it
was of limited value to the present approaches used in this research.
Section 8.2 concludes this report through discussing future research tasks and
themes inspired by this research.
In addition to the above technical contributions resulting from this research,
this work has led to additional research products including 2 MS theses (Wurst 2013
and McCarthy 2012, the latter of which was completed under the co-advisement of
Prof. Sue McNeil), a technical report authored by an undergraduate researcher
(Sparacino 2013), and one conference paper (McConnell et al 2012) to date. The
student authors are credited with much of the text in the following chapters, which is
reproduced in the above reports.
Preparation of future publications on this work is also planned. Furthermore,
this research has led to the creation of several analytical models, which continue to be
evaluated to reveal greater understanding of system behavior of steel bridges. Lastly,
xiii
the research has revealed other promising avenues for future research which are likely
to be pursued in future work.
xiv
Chapter 1
INTRODUCTION
1.1
Motivation
National statistics on numbers of structurally-deficient bridges and the
associated limited financial resources for infrastructure renewal require ever more
thoughtful approaches to bridge management. According to the American Society of
Civil Engineers (ASCE) report card (2009), approximately 27% of bridges in the
United States are either structurally deficient or functionally obsolete. ASCE
estimates that, over 50 years, $650 billion would be required to maintain the current
level of bridges; that is, leave approximately 27% of bridges structurally deficient or
functionally obsolete. To eliminate all of these deficiencies, $850 billion over 50
years is estimated to be required. However, in 2004 only $10.5 billion was spent on
bridge improvements. This indicates a need for a prioritization method to properly
allocate the limited funding.
One significant opportunity for better prioritizing the limited funding available
for this purpose is through better quantification of the existing strength of aging
structures. Specifically, one promising means for achieving this goal is leverage the
load-path redundancy inherent to steel girder bridges to calculate a so-called system
capacity of the structure, in contrast to the present approach of calculating a strength
that is governed by the weakest or most-heavily loaded member. When this more
realistic approach is utilized, bridges have significant reserve capacity above what is
predicted from modern analysis and design methods predicated on the analysis of
individual members.
1
The present research was motivated by this promising concept of system capacity and
the identification of information needed to transition this concept to practice. These needs
include both knowledge needed with respect to structural mechanics (which was the subject of
Phase 1 of this project) as well as practical needs to synthesize the research result into a format
that is synergistic with existing evaluation methods used by bridge engineers (which was the
subject of Phase 2 of this work). The structural mechanics that must be assessed and quantified
relates to the fact that in system level analysis the transverse load distribution mechanisms
become more influential than in current methods, so further understanding of this phenomena
are needed. Furthermore, the reliability of these transverse load distribution mechanisms,
particularly those achieved through reinforced concrete bridge decks, in aging structures with
various states of deterioration was unknown. For the results of these efforts to be most
beneficial, they must be translated into a rating approach consistent with American Association
of State Highway and Transportation Officials (AASHTO) Load and Resistance Factor Rating
(LRFR) procedures for highway bridges. In addition, a means to incorporate the actual
condition of the deck as quantified by the inspection report specific to each structure would
further improve the applicability of the research.
1.2
Objectives and Scope of Work
Considering these needs while limiting the research to a practical scope for this project,
the key objectives of this work were:
1.
To assess and quantify the influence of corrosion caused by deicing-agents on
the ability of reinforced-concrete bridge decks to serve as a transverse load
distribution and redistribution mechanism in steel girder bridges;
2.
To formulate a rating procedure to quantify the system capacity of bridges that
includes the effects of reinforced-concrete deck deterioration; and
2
3.
To explore the correlations deck condition ratings and level of deterioration and
how this may affect predicted system capacities.
The primary components of the scope of work began with reviewing the literature to
quantify the effect of deicing agents on the structural performance of reinforced concrete
members. A finite element methodology was then formulated and calibrated to simulate this
effect, first in analytical reproductions of the experimental specimens in the literature and then
through FEA models of three existing steel I-girder bridges that had been calibrated to field
data in prior work. Theoretical calculations supported by the data generated from these models
and data from bridge inspection reports was used to accomplish the second and third
objectives.
1.3
Report Organization
The organization of this report is as follows.

Chapters 1 and 2 introduce this work. Specifically, Chapter 1 describes the
motivation, objectives, and scope of the research as well as a summary of the
organization of this report. Chapter 2 provides background on the basic
premises leading to system capacity, the bridge rating process, and financial
aspects of this process..

Chapter 3 discusses the literature review performed to quantify the effects of
aging on reinforced concrete bridge decks.

Chapter 4 presents a review of the FEM approaches available for performing the
modeling necessary to complete the research objectives. This includes
consideration of the non-linear concrete constitutive response, two-dimensional
(2-D) versus three-dimensional (3-D) rebar elements, and various contact and
bond properties between the concrete and rebar elements.
3

Chapter 5 discusses the calibration of the FEA approach for modeling the effects
of chloride-induced corrosion in reinforced-concrete members.

In Chapter 6 the methodologies and results of applying the calibrated FEA
approach for reinforced concrete members to the concrete decks in FEA models
of composite steel I-girder bridges are discussed.

In Chapter 7 the relationships between the analytical results and bridge rating
practice are discussed in terms of live load distribution factors and system-level
capacities.

Chapter 8 concludes this report by summarizing the results of parallel efforts
that were conducted as part of this research which revealed other findings worth
documenting and by discussing future research tasks and themes that are shortand long-term logical extensions of this research.
4
Chapter 2
BACKGROUND
2.1
Federal Funding and the Highway Bridge Program
The Highway Bridge Program (HBP) is the source of federal funding for repairs,
rehabilitation and replacements of national infrastructure. It is through this program that each
state receives funding for the management of its bridges (U.S. Department of Transportation,
2012).
When bridges within a state are rated, the data is entered in the National Bridge
Inventory database (NBI). The ratings as they appear in the NBI allow the federal government
to prioritize allocation of funding through the HBP to each state. Under current practice, to be
eligible for funding, a bridge must have a ‘Sufficiency Rating’ of 80% or less (Xanthakos,
1996). This sufficiency rating essentially serves as a quantitative indication of the degree of
deficiency of a bridge, with a lower rating indicating a higher priority for remedial works.
While the limitations of the sufficiency rating have been recognized, the sufficiency rating is
the quantitative measure of bridge health used by the federal government (Roberts & Shepard,
2007).
Building on this description of the performance measure and decision variable for the
allocation of funding (i.e. the sufficiency rating), the rating process behind the determination of
that measure is briefly outlined. It can then be better understood where the shortcomings of this
process lie and how they might affect management decisions.
2.1.1
The Rating Process behind the National Bridge Inventory
Determination of a bridge’s sufficiency rating begins with a visual inspection. The data
gathered from the inspection is entered into a ‘Structure Inventory and Appraisal’ itemized
form (see McCarthy 2012 for sample of this form). The guidelines for data entry of the
5
inspection process are found in the Recording and Coding Guide for the Structure Inventory
and Appraisal of the Nation’s Bridge’s, published by the U.S. Department of Transportation
(Federal Highway Administration, 1995). This process provides information for input into the
sufficiency rating equation.
2.1.2
The Sufficiency Rating Equation
The sufficiency rating equation is built on four factors: Structural Safety; Serviceability
and Functional Obsolescence; Essentiality for Public Use; and an allowance for Special
Reductions to the overall rating if items such as detour length or traffic safety features, like
guard rails for example, are applicable; however it is the former three factors which are at the
heart of the overall bridge sufficiency rating. The sufficiency rating equation is set out below
(Equation 2-1), where S1, S2, S3 and S4 represent the four factors respectively.
S.R. = S1 + S2 +S3 + S4
[2-1]
Each of the four factors is calculated separately upon completion of the structural
appraisal form before being combined to produce a single value for the overall bridge
sufficiency rating. The maximum obtainable sufficiency rating is 100%, which would represent
a bridge which is entirely sufficient but this is not commonplace.
The weighted influence carried by each of the four factors is not however equal. The fourth
factor, Special Reductions, may or may not be included in the equation at all depending on the
circumstances. This inequity between the four factors is now explained in more detail along
with the importance it places in turn on load carrying capacity and decision making in the
bridge rating process.
2.1.3
The Importance of Load Carrying Capacity in the Rating Process
The maximum percentage attributable to each of the three primary factors in the
sufficiency equation is depicted in Figure 2-1 below. It is apparent from Figure 2-1 that S1, or
6
Structural Adequacy and Safety, is attributed a 55% weighting in the sufficiency rating
equation. Therefore, Structural Adequacy and Safety has the greatest influence on determining
the overall sufficiency rating.
Figure 2-1. Sufficiency Rating Factors.
2.1.3.1
Structural Adequacy and Safety (S1)
The sample structural appraisal form and the worked example of the calculation of the
sufficiency rating for one bridge provided by the FHWA (see McCarthy 2012 for additional
details) document the process for computing this component. The Structural Adequacy and
Safety of the bridge is calculated in two parts.
The first, “A” is the combined value of the condition ratings for the super structure,
sub-structure and culverts, which are items 59, 60 and 62, respectively. These items are defined
in Recording and Coding Guide for the Structure Inventory and Appraisal of the Nation’s
Bridges.
7
The second, part “B” is a value based on the load carrying capacity and the “Inventory
Rating” of the bridge. The load carrying capacity of the bridge, can be determined by “analysis
and, in some cases, load testing”, which might involve “the Load Factor Method, Working
Stress Design Method and Load and Resistance Factor Method”. The reader is referred to
Chapter 4 of the DelDOT Bridge Design Manual for a more detailed description (Delaware
Department of Transportation, 2009). The ‘Inventory Rating’ is item 66 as defined in
Recording and Coding Guide for the Structure Inventory and Appraisal of the Nation’s
Bridges.
The values of both “A” and “B” carry equal weight and when they are combined and
subtracted from the upper bound of 55%, the S1 factor is obtained. This factor represents the
Structural Adequacy and Safety of the bridge.
2.1.3.2
“Inventory Rating”
In the example in McCarthy (2012), it can be seen under part “1. B” that the “Load
Carrying Capacity” is related to the “Inventory Rating” on the structural appraisal form. The
“Inventory Rating” is referred to in the Recording and Coding Guide for the Structure
Inventory and Appraisal of the Nation’s Bridge’s under items 65 and 66. Item 65 being an
indication of the type of load rating method used in assessing the bridge and item 66 indicating
the load rating itself in metric tons. The load rating is in fact the “capacity rating, referred to as
the inventory rating” (Federal Highway Administration, 1995). In other words the guide says
that the “inventory rating” is synonymous with the load carrying capacity of the bridge.
2.1.3.3
Observations Related to Sufficiency Rating
The inventory rating influences the determination of the first factor, S1, the Structural
Adequacy and Safety of the bridge. This first factor, in turn is weighted the most influential in
the calculation of the bridge’s overall sufficiency rating, having a weighting of 55% in the
8
sufficiency rating equation, Equation 2-1. Comparatively, the second and third factors, S2 and
S3, respectively, are weighted less, at 30% and 15% each, respectively, in the sufficiency rating
equation.
Therefore, it could be said that the load carrying capacity of the bridge holds the
“highest weight in sufficiency rating formula” (Akgul & Frangopol, 2004). It is also “a crucial
measure for bridge management and decision making”, as is the argument promoted by Akgul
and Frangopol (2004). How the load carrying capacity of the bridge relates to load path
redundancy is now outlined.
2.2
Load Path Redundancy and Load Carrying Capacity
The action of load path redundancy can be of great benefit to the load carrying capacity
of the bridge system. Before discussing how the current rating process neglects this action, a
basic description of load path redundancy is given in the section immediately following,
followed by a required description of the relationship between load path redundancy and load
carrying capacity in Section 2.2.2.
2.2.1
What is Load Path Redundancy?
The girder is the primary load path in a composite girder and deck bridge system. With
that understanding, load path redundancy then can be explained by the following definition
given by the Missouri Department of Transportation:
With respect to bridge structures redundancy means that should a member or element
fail, the load previously carried by the failed member will be redistributed to other members or
elements which have capacity to temporarily carry additional load and collapse of the structure
may be avoided (Missouri Department of Transportation, 2000).
The term ‘load path’ then refers to the component/element/member of the bridge
structure through which the load redistribution or ‘shedding’ can take place. For the ‘load path’
9
to be redundant, that configuration, or “the number of supporting elements” (Missouri
Department of Transportation, 2000) must be greater than one. Or as Xanthakos (1996) puts it
“at least one alternative load path exists and prevents collapse”. In other words, it is multigirder bridges specifically which are referred to when discussing load path redundancy. The
action of load path redundancy is illustrated in its most basic sense in Figure 2-2 below.
Figure 2-2. Load Path Redundancy.
The principle is that if girder “A” were to experience some type of deterioration, overloading or failure causing it to be unable to carry its load, that load would be redistributed to
girders “B” and “C” adjacent to it. That redistribution of load can occur by travelling through
the concrete deck as indicated by the arrows.
2.2.2
How Does this Affect the Load Carrying Capacity?
The action of load path redundancy allows the bridge to maintain its load carrying
capacity. Rather than losing its capacity due to the reduced performance, over-loading or
failure of one (or more) girders, the bridge maintains that capacity due to the uptake of the shed
load by the other bridge elements. This load sharing is a system effect, which is now explained.
10
2.3
Load Carrying Capacity is a System Effect
Current bridge design practice can be described as conservative. That is, girders are
designed on an individual component level for a worst case scenario. This design approach
means that it is assumed each girder carries a constant proportion of the load throughout its
lifetime.
It should be clear from the above description of the action of load path redundancy that
what happens in reality is significantly different from what is conservatively designed for. That
is, the redistribution of load between girders means that the proportion of load carried by an
individual girder may not be constant. In fact, a girder has the reserve capacity to take an
additional proportion of its neighbor’s load. This tells us that the bridge elements (girders) act
as a system.
From Figure 2-2 it is clear that there is an additional element belonging to the success
of this system performance, which is the deck. Figure 2-2 indicates by way of the arrows that
the deck is a primary means through which the load of a girder unable to carry the weight it is
subjected to alone is redistributed laterally to the other bridge elements. The deck therefore, is a
key component in the realization of load path redundancy and therefore in achieving the full
potential of the bridge’s load carrying capacity, i.e., through a system process.
It should be noted that lateral bracing members are a second additional means for
redistributing load to adjacent load paths, but this work will focus solely on load redistribution
via the deck. Load redistribution through cross-frames of steel girders is the subject of other
work (Ambrose, 2012).
2.4
Scope for Improvement in the Bridge Rating and Funding Process
It should now be clear how due to load path redundancy, the load carrying capacity of a
bridge can be described a system behavior rather than being attributable solely to individual
11
girder capacity. Furthermore, system capacity can afford greater load carrying capacity than
that determined on an individual girder basis.
Referring again to the NBI sample structural appraisal form and sufficiency rating
example calculation, item 66 on the form is ‘Inventory Rating’. Inventory Rating is used in the
appraisal synonymously with load capacity rating or load carrying capacity. It is here under
item 66, assigning the inventory rating, that an acknowledgement of load path redundancy and
the system rather than component load carrying capacity of the bridge could be made. In so
doing, the end product (sufficiency rating) of the bridge could reflect the action of load path
redundancy.
As the rating process is currently completed, assessment is still reflective of a
component level capacity only. This evaluation approach “yields a conservative measure of
actual load carrying capacity” (Wang, O'Malley, Ellingwood, & Zureick, 2011). Based upon
this rationale, if the NBI rating process acknowledged load path redundancy as part of the
sufficiency rating, bridges currently rating “Poor” in sufficiency might have improved ratings.
This would reflect the additional system capacity that is achievable.
Of course this yields the need for improved appraisal procedures that can capture the
system behavior and condition of the bridge. It is sensible to suggest that the most appropriate
place where this may be achievable is via evaluation of items 58, 59 and 66 when completing
the structure inventory and appraisal. These items visually rate the deck and super-structure and
give the inventory rating (load carrying capacity).
12
Chapter 3
EFFECTS OF AGING ON REINFORCED CONCRETE BRIDGE DECKS
As explained in Section 2.3, the success of load path redundancy, which causes
enhanced bridge system load carrying capacity, may be heavily dependent on the ability of
the deck to transfer load between girders. This successful transfer by the deck is in turn
dependent upon its health. This preliminary investigation into the effects of aging on load path
redundancy therefore focuses on understanding the deterioration of the concrete deck with
aging and quantifying that process in terms of its mechanical properties.
3.1
Deterioration of Reinforced Concrete
Deterioration of concrete can be defined as a reduction in the materials durability,
performance and/or life span. Durability is defined herein as the ability to resist
deterioration. Performance is defined here as functional operation in service, for example load
carrying capacity, bending deflection and other related characteristics. While life span is
defined as the functional life time relative to the design life.
Concrete can suffer from an array of both physical and chemical processes which
cause it to deteriorate. Concrete deterioration can occur due to factors originating within the
concrete itself in the form of flawed design and deleterious composition, for example, a
water/cement ratio that is too high causing a weak and permeable concrete, or from
external factors in the form of poor construction and environmental assailants, for example,
poor compaction forming porous joints and chloride ingress.
It is widely believed that chloride induced corrosion of reinforcing steel is one of the
most detrimental problems associated with the deterioration of concrete today, significantly
13
impacting the material’s durability and consequently reducing its service life (Huang, Bao, &
Yao, 2005), (Vorechovska & Podrouzek, 2009).
Zimmermann et al. (2000) go one step further and call this deterioration mechanism not
one of but “the main cause of damage and early failure of reinforced concrete
structures”. The chloride induced corrosion process and its effects are now discussed.
3.1.1
Chloride Ion Ingress and the Corrosion of Reinforcing Steel
In terms of the reinforcing steel, hydroxide ions (OH+), which are constituents of the
cement paste, provide alkalinity to pore water, allowing the formation of a layer of ferric
oxide (Fe2O3) to act as a protective cushion to the reinforcing steel (herein referred to as
rebar). This cushion offers protection against chloride ions whose presence can induce
corrosion of the rebar. In the case of concrete bridge decks, de-icing salt agents, (herein
referred to as deicers), can be a typical sources of chloride ions which ingress into the concrete
to cause deterioration.
Chloride ions (Cal-) which ingress into the concrete and are in the combined
presence of moisture, oxygen and OH+ ions cause a chemical reaction dissolving
the rebar by using up the alkaline OH+ ions and reducing pore water alkalinity, thereby
breaking the protective cushion around the steel rebar. When the rebar is fully protected by
the hydroxide cushion it is referred to as being in a “passive” state, but when that cushion
is broken, the rebar is said to be de-passivated (Rendell, Jauberthie, & Grantham, 2002).
3.1.2
Effects of Chloride Induced Corrosion
Rebar corrosion can present itself as surface spalling or transverse cracking however it
can also be present without giving any outward signs (Weyers & Cady, 1987). Spalling and
cracking of the concrete occurs due to expansion of the rebar. The latter causes
14
compressive pore stress development within the concrete micro-structure due to volumetric
expansions from the formation of rust which is of greater volume than its parent material,
steel. This stress development results in fissuring, cracking, spalling and weakening of the
concrete (Hobbs, 2001).
Meanwhile inside the pore structure there is a loss of rebar cross-section and/or a
reduced bond between the rebar and concrete (Vorechovska & Podrouzek, 2009).
Furthermore, due to rust formation, the most insidious effect of this deterioration is the loss
of rebar cross section and hence ultimate loss of strength of the concrete section.
The effects of spalling, cracking and mass loss of rebar due to chloride induced
corrosion cause more far-reaching and serious effects on the mechanical properties of the
concrete deck. The effects on the mechanical properties and their level of severity are now
discussed.
3.1.3
Deterioration Effects on Concrete Mechanical Properties
The primary mechanical properties of concrete affected by chloride ion ingress and
associated rebar corrosion are mass loss of rebar, strength and stiffness. Both stiffness and
strength are influenced by the degree of porosity/permeability of the concrete. Concrete
afflicted by the deleterious effects of Cl- ion ingress and rebar corrosion will have an
increased porosity and therefore reduced strength and stiffness (Basheer, Kropp, & Cleland,
2001).
How these mechanical properties are affected are now analyzed further and quantified
in the sections following by considering the results of three studies into the deterioration of
reinforced concrete beams due to rebar corrosion induced by chloride ion ingress.
15
3.2
Review of Studies on Concrete Deterioration due to Chloride ion Ingress
There is a minimal amount of literature on the effects of chloride induced corrosion on
the mechanical properties of reinforced concrete (in contrast to plain concrete which is not of
concern within the scope of this research). In particular, a comprehensive and uniform
quantification of those effects is lacking.
This research focuses on three studies which investigate reinforced concrete
deterioration due to chloride induced re-bar corrosion. These studies appear to be the
extent of good data which is available on reinforced concrete deterioration due to chlorideinduced rebar corrosion, and which reflect a natural corrosion methodology. These studies
provide data from which conclusions can be made on the effects of this deterioration on the
mechanical properties of reinforced concrete beams, which may be applied to bridge decks.
These studies were chosen because the techniques used in their experiments closely
represent a natural environment in which salt (chloride ions) might enter concrete and cause
deterioration, for example from percolating snow melt which was melted using deicers. As an
aside, another common method used in experimentation to induce chloride corrosion is
submersion of reinforced concrete specimens in a chloride-laden water bath with
simultaneous application of an electric current to the re-bar. This method however has been
cited as an unrealistic representation of the natural corrosion process because the electric
current causes uniform corrosion along the re-bar (Li & Zheng, 2005). Additionally, some
studies are known to have used corrosion inducing chemicals which were not chloride based
and therefore neither representative of the natural corrosion process. Furthermore, the
time to achieve an advanced degree of corrosion using the electric current method is
significantly less than that required using more natural methods and for this reason it is
difficult to translate the time it may take naturally for the same degree of corrosion which
would be reflective of the effects on mechanical properties seen in the laboratory using this
method. Some examples of these studies are Huang et al. (2005) and Wang et al. (2006).
16
Furthermore, the three studies which are of focus here, namely, Gu et al. (2010), Li
and Zheng (2005) and Oyado et al. (2011), all use specimen sizes which are structurally
significant and therefore the results which are obtained from destructive load testing of
specimens, which allows conclusions to be made on the results presented in the studies on
which can be applied to what can be expected of reinforced concrete in the field. The
details of each study are summarized in the following sections.
3.2.1
Study 1 – “Propagation of Reinforcement Corrosion in Concrete and its Effects on
Structural Deterioration” (Li & Zheng, 2005)
In this study reinforced concrete beams of structurally significant size were corroded in
a laboratory setting within an environmental chamber. To induce corrosion, a method reflective
of natural exposure was utilized: beams were subjected to salt water spraying with alternate
wetting and drying cycles.
The corrosion process was accelerated by two factors. Firstly, the drying periods of the
cycles was intensified so as to speed up the absorption of fluid from the beam surface.
Intensification of the drying periods was achieved by carrying out the experiment in a special
environmental chamber where the climate could be adjusted as required, for example
temperature and relative humidity. Secondly, while the entirety of the beams was under general
salt spray, an additional direct spraying of the sodium chloride (NaCL) or “salt solution” onto
surface cracks was utilized to accelerate the corrosion process. These cracks were produced by
lead weights representing service load conditions and were kept constant on the cantilever ends
of beams until such time as the beams were removed from the environmental chamber for
destructive load testing.
Residual flexural strength was determined by destructive load testing under 4- pointbending and was measured as the ultimate failure loads of corroded beams in comparison to
un-corroded replicate beams. Residual flexural stiffness was measured by the deflection at the
17
cantilever end of corroded beams in comparison to that of un- corroded beams. Three sets of
results are presented in the paper, representing the three time periods over which beams were
corroded, 3, 5 and 7 months respectively. It was at the end of each of these periods that
destructive testing was performed. Mass loss was determined by a different method which is
discussed under the following section.
An explanation of how real time corresponds to accelerated laboratory time in this
study is now outlined below before results of the study are discussed.
3.2.1.1
Concept of Time Transformation
As these laboratory tests involved accelerated corrosion of rebar in beams, a time
transformation concept is outlined by Li and Zheng (2005) to allow results to be interpreted for
‘natural’ exposure to salt (chloride ion ingress). The time transformation is achieved by use of
an ‘acceleration factor’. This factor was determined by the authors using long term calibration
tests on identical beams under natural exposure or un- accelerated conditions. Details of the test
can be found in Li (2000).
The source explicitly and simply states the issue concerning utilization of accelerated
time for testing to determine results in natural/real time: “The essential problem is to determine
the equivalent time period of one cycle of wetting and drying under the accelerated conditions
to the real time period of a wetting and drying cycle under natural conditions”. The parameter
chosen as the basis on which the acceleration factor could be determined was moisture content
of the concrete, measured by weighing of specimens. It was found that: “on the average, one
natural cycle of wetting and drying takes approximately 47 days as measured by weight of
specimen”.
Under accelerated testing meanwhile a cycle of wetting and drying lasted 3 days. From
this, the following observations and conclusion can be made:

1 natural cycle of wetting and drying = 47 days
18

1 accelerated cycle of wetting and drying = 3 days

47 ‘natural’ days of wetting and drying represents 1 laboratory cycle
If it is assumed 30 days to a month, then finally the natural time that is represented by one
month of accelerated laboratory corrosion can be determined:

1 month in laboratory = 30 days/3-day-cycle = 10 accelerated cycles per 1
month in lab.

10*47 = 470
Therefore there are 470 natural days per 1 month of laboratory testing.
Therefore in utilizing data from this study the approach was to multiply each 1 month
of laboratory testing by 470 to obtain the time in natural days. The number of natural days was
then converted to natural months and from that to natural years. For example, for the 3 month
Series the following three steps can be used to convert this accelerated laboratory time period
to natural time:
1.
3*470 = 1410 natural days
2.
1410/30 = 47 natural months
3.
47/12= 3.79 natural years
The acceleration factor of 47 was used to convert both strength and stiffness data to
natural time from accelerated testing time. The situation varied slightly for “mass loss” of rebar
data however.
Data for mass loss data is considered herein to be represented by the ‘corrosion current
density’ (icorr) readings. The rationale behind this is discussed in detail in Section 3.2.1.2
below. These readings are presented in Fig. 3 of the study by Li and Zheng (2005) and shown
below as Figure 3-1. The study uses these corrosion current density (icorr) readings to obtain
strength loss measurements by a non-destructive method. When the authors measured strength
loss by destructive load testing and compared the results with the corrosion current density
19
method, they found the latter to underestimate the strength loss, as is observable between the
data shown in Figure 3-2 below. To account for this underestimation, an acceleration factor of
51.7 was used in the study to transform accelerated time to natural time
for corrosion current density data (or “mass loss” as it is considered here), as opposed to 47
used for strength and stiffness. As destructive testing shows the true strength (mass) losses at
time ‘X’ and the corrosion current density method underestimates those losses at the same time
‘X’, this implies that it requires a longer time if measuring losses by corrosion current density,
to obtain the same losses as seen with measuring by destructive load testing. Consequently, the
acceleration factor for corrosion current data (mass loss data) is greater than that for strength
and stiffness data for transforming accelerated time to natural time (51.7 as opposed to 47,
respectively). Mass loss of rebar is now discussed in more detail.
Figure 3-1.
Basis for Time Transformation for Li and Zheng Mass Loss Data (Li & Zheng,
2005).
20
3.2.1.2
Rebar Mass Loss
Rebar mass loss was not measured by gravimetric means. However, the study
states that from corrosion current density, (icorr), readings “the metal loss of rebar can be
determined using Faraday’s Law” and that “this metal loss is then translated to the
reductions of cross-sectional area of rebar” (Li & Zheng, 2005). The study indicates that
corrosion current density readings have units of “µA/cm2”, and states that is “uses 1
µA/cm2 of corrosion current density to equal to 11.6 µm/year of metal loss of rebar in the
radial direction” (Li & Zheng, 2005). Fig. 15 of the study, referred to herein as Figure 3-2,
showing strength deterioration represented by corrosion current density readings, was
interpreted in this work as the change/reduction in corrosion current density
readings and therefore as being representative of mass loss. The mass loss results
discussed in the following sections of this report were extrapolated from Figure 3-2.
Figure 3-2. Corrosion Current Density Reading Data used to Derive Strength & Mass Loss
Values (Li & Zheng, 2005)
21
In this figure, corrosion current density readings for a period of years in real/natural
time are plotted against a deterioration factor. So for example, in the study a deterioration
factor of 0.98 at 4 years represents a mass loss of rebar of 2% at 4 years. The maximum
mass loss observed in the study is approximately 5.5% after 8.8 years (natural time). Rebar
mass loss is a primary cause of strength loss of the concrete beams, which is now deliberated
in the following section.
3.2.1.3
Strength Loss
As outlined in Section 3.2.1, strength loss was determined in the study by way of
destructive load testing under 4-point-bending of beams which were corroded for 3, 5 and 7
months. Beams were loaded to ultimate failure load and losses were determined by
comparison of results obtained with the ultimate failure load of healthy, un-corroded
replicate specimens and represented by a deterioration factor.
Similar to the approach for mass loss results discussed in the previous section, the
strength loss results discussed in the following sections of this report were extrapolated from
Figure 3-2 where strength loss readings over real/natural time in years is plotted against the
deterioration factor. So for example, in the study a deterioration factor of 0.9 at 4 years
represents a strength loss of 10% after 4 years. The maximum strength loss observed in the
study is 21.25% after 8.8 years (natural time), where all estimates are approximate. Finally,
the method of determining stiffness loss in the study by Li and Zheng is explained in the
next section.
22
3.2.1.4
Stiffness Loss
Beams were loaded under 4-point-bending and deflection at the cantilever end of the
beams was measured to represent stiffness whereby the amount of deflection is correlated
to the amount of stiffness lost. Load versus deflection data for un-corroded beams and
beams corroded for 3 and 7 months (note no data is supplied for 5 month readings by the
authors) is given in Fig. 6 of the study, herein referred to as Figure 3-3.
Figure 3-3. Stiffness Loss Basis (Li & Zheng, 2005)
Stiffness losses are expressed by the authors as changes in deflection between
healthy beams and corroded beams which are loaded incrementally to maximum failure
load. The changes are represented by a deterioration factor. Stiffness losses are explicitly
23
given in Fig. 9 of the study, referred to herein as Figure 3-4, and are represented by the
changes in the deterioration factor plotted over accelerated time in months.
For this research, the losses in stiffness due to corrosion recorded by the study were
interpreted from the data presented in Figure 3-4 by converting accelerated time to real/natural
time using the acceleration factor of 47. So in the study for example, a deterioration factor of
0.587 at 4 months of accelerated time indicates a stiffness loss of 41.3% after 5 years (natural
time), all estimates being approximate. The maximum stiffness loss observed in the study is
about 50% after 8.8 years (natural time). The second study which is reviewed for the
preliminary investigation into the effect of aging on load path redundancy is now described in
detail. As an aside, the strength data seen in Figure 3-4 below is the same as the strength data
shown for destructive load testing in Figure 3-2 above.
Figure 3-4.
Data used to Derive Stiffness Loss Values from Li and Zheng (Li & Zheng,
2005)
24
3.2.2
Study 2 – “Bending Performance of Reinforced Concrete Member Deteriorated by
Corrosion” (Oyado, Kanakubo, Sato, & Yamamoto, 2011)
Reinforced concrete beams of structurally significant size were corroded using the
‘natural’ salt water spraying technique and the greatly accelerated ‘electrical’ technique
described above in Section 3.2. As outlined in that section, the latter method is treated with
skepticism as to the accuracy and reliability of its results because the corrosion achieved does
not realistically represent what occurs in the field i.e. uniform rebar corrosion does not happen
in reality. For this reason, this research neglects the information from this study on those beams
which were corroded using the ‘electrical’ method, namely ‘Series C’ as they are referred to in
the study. Thus, the information analyzed in this report belongs to those beams of ‘Series S’
and ‘Series M’ which were corroded using the ‘natural’ method which is referred to as ‘EX’ in
the study.
In the study, Series S and Series M beams were exposed for 3 months outdoors in an
“urban environment away from the coast” (Oyado, Kanakubo, Sato, & Yamamoto, 2011).
After which time NaCl solution was sprayed onto cracks in the beams 3 times daily for 17
months to expedite corrosion initiation. Series S beams were removed from the outdoor
location at this point, having been exposed for 20 months in total, while Series M continued to
be exposed for a total of 12 years. The mass loss of re-bar observed in the study is discussed
next.
3.2.2.1
Rebar Mass Loss
Rebar mass loss was measured by gravimetric means whereby “the mass loss of every
specimen was measured by weighing rust removed bars” (Oyado, Kanakubo, Sato,
& Yamamoto, 2011). What this means is that corroded, rusted bars were removed of their
rust until just the parent material, steel, was once again visible and weighed. The corroded
weight of the bars was compared against the weight of a replicate un-corroded bar.
25
The mass loss results analyzed here were extrapolated from data found in Table 4 of
the study, referred to herein as Figure 3-5, which shows time plotted against the ‘C’ ratio for
each beam. The ‘C’ ratio represents the ratio of mass lost to the original mass of
the re-bar. The ratios were simply multiplied by a factor of 100 to obtain the percentage (%)
mass loss of re-bar for each specimen. The ‘C’ ratios used for analysis were the “Average”
values given in the study.
So for example, from Figure 3-5, for specimen M1, which was corroded for 12 years
and has a ‘C’ ratio of 0.14, a mass loss of rebar of 14% is indicated and is also the maximum
mass loss observed in this study. The strength losses observed in the study are discussed in the
following section.
3.2.2.2
Strength Loss
Strength loss was determined by way of destructive load testing under 4-point-
bending after corrosion for 20 months and 12 years for the Series S and Series M beams,
Figure 3-5. Data used to Derive Mass & Strength Loss Values from Oyado et al.
(Oyado, Kanakubo, Sato, & Yamamoto, 2011)
26
respectively. Beams were loaded to ultimate failure and losses were determined by comparison
with the ultimate failure load of an un-corroded replicate specimen (S-0N).
The strength loss results discussed here were based on the data in Figure 3-5 which
shows a ratio Puc/Pun for each beam type. This is the ratio of the ultimate strength of a corroded
beam specimen to the ultimate strength of the un-corroded beam specimen. It should be noted
that the beam ‘Name’ allows indication of the length of corrosion (deterioration) time. For
example, beam ‘Name’, “M1”, indicates a beam from the series that has been corroded for
12 years, while beam ‘Name’, “SD-1N”, indicates a beam from the series that has been
corroded for 20 months. To obtain the percentage strength loss, the ratio was subtracted from
1.00 and the result multiplied by and factor of 100. So for example, from Figure 3-5, a Puc/Pun
ratio of 0.9 for beam specimen SD-1N indicates a strength loss of 10% after 20 months of
corrosion. The maximum strength loss observed in this paper was 28% after 12 years. A
similar discussion of stiffness losses is now made.
3.2.2.3
Stiffness Loss
Beams were loaded under 4-point-bending and deflection at the mid-point of the beams
was measured to represent stiffness. Load versus deflection data for Series M was scaled from
Figure 6b of the study, reproduced herein as Figure 3-6. No data is given in the study for
Series S beams, however as Series M were those beams corroded for the longest period of
time the disadvantage of this omission is not so great.
Stiffness losses are expressed as changes in deflection under load between the actual
deflections recorded for corroded beams and the expected deflection of a replicate un-corroded
beam. The latter was calculated based on assumptions related to material characteristics,
information supplied on specimen geometry and the loading case. This information was
27
Figure 3-6
Data used to Derive Stiffness Loss Data from Oyado et al. (Oyado,
Kanakubo, Sato, & Yamamoto, 2011)
input to the following formula which, it should be noted, gives elastic deflection, the
consequence of which being that deflections at the ultimate load will be underestimated:
Fa/24EI (3L2 – 4a2)
[3-1]
The variables in this equation are based both on the loading case seen in Figure 3-6 and
the geometry of beam, where: the breadth of the beam, “b”, is 100 mm; the depth of beam,
“d”, is 200 mm; the moment of inertia, “I”, is found by using the equation “bd3/12” and
is 66670000 mm4; based on typical reinforced concrete characteristics, Young’s Modulus,
“E”, is estimated at 26,000 MPa or 26 kN/mm2 for the beam (MATBASE, 2009); the length
of the beam, “L”, is 1800mm; and finally, the point loads, “P/2”, are represented in Equation
3-1 as “F”, and the distance between them is represented in the Equation 3-1 by “a”, and is
700mm.
28
Based on the data obtained using Equation 3-1 the recorded ultimate load and
ultimate deflection for corroded beams was compared with the calculated deflection of an uncorroded beam under the same load. In this way the difference between both results can
represent the percentage loss in stiffness of the beams due to corrosion. The load and deflection
data which was scaled from Figure 3-6 to calculate the expected ultimate deflection of M
Series beam specimens M1, M2 and M3, is shown below in Table 3-1. The estimated ultimate
deflection of un-corroded Series M beams which was calculated using Equation 3-1 is shown
along with the observed ultimate deflection in Table 3-2. The percentage of stiffness loss is
also indicated there. Maximum losses are estimated to be 73.9% at 12 years, which is for beam
M2. However when compared to the losses seen in beams M1 and M3, this value is treated
as an outlier as will be discussed further in Section 3.3.3.
The details of the third and final study reviewed are now discussed in the following
section.
Table 3-1.
M1
LoadF (kN)
10
20
30
36.5
36.25
Data used to Calculate Expected Ultimate Deflection of Series M beams
M1
Defl.(mm)
1.31
2.61
3.92
4.77
4.73
M2 LoadF (kN)
10
20
30
37
40
M2
Defl.(mm)
1.31
2.61
3.92
4.83
5.22
29
M3 LoadF (kN)
10
20
30
37
42
M3
Defl.
1.31
2.61
3.92
4.83
5.48
Table 3-2.
Estimated Stiffness Losses for Oyado et al. (2011) Data
M1
Expected Ult. Defl.
(mm) un-corroded
beams
Observed Ult. Defl.
(mm) corroded
beams
Ratio of
Expected/Observed
Deflection
% Stiffness Loss
3.2.3
M2
M3
4.77
5.22
5.48
8
20
10
0.596
40.38
0.261
73.90
0.548
45.20
Study 3 – “Flexural Behavior of Corroded Reinforced Concrete Beams” (Gu,
Zhang, Shang, & Wang, 2010)
This study focuses on the use of the ‘electrical’ method for accelerated corrosion of 3
groups of beams as described in Section 3.2. As discussed, results from studies of this nature
are disregarded in this report because of the degree of skepticism in the field over their
accuracy. The authors of this study recognize those discrepancies and state, “It was found that
the accelerated corrosion process, as used in most of the investigations, has quite different
effects from the natural corrosion process” (Gu, Zhang, Shang, & Wang, 2010) .
However, this study refers to ‘Group D’ beam samples taken from “an existing building
which has gone through decades of natural corrosion” which would be representative of the
‘natural’ corrosion process. Unfortunately additional details on the historical particulars of
the corrosion process of these beams and the time frame of exposure are not supplied in
this study. Certain data is available from this study on rebar mass loss, stiffness losses and
strength measurements for those beams. However, it was decided to treat this data as
available for informational purposes only rather than as a basis for any conclusions. Perhaps,
the most unsatisfying aspect related to this set of beam data is the inability to quantify time
30
of exposure any better than with a description of “decades”. Nonetheless, the best available
data on rebar mass loss available in the paper is now discussed followed by strength and
stiffness loss data.
3.2.3.1
Rebar Mass Loss
Mass loss of rebar was determined by the gravimetric method whereby bars were
cleaned and weighed. Given that the beams from Group D are from buildings which had
suffered decades of corrosion, it is assumed in this report that the rebar diameters for those
beams, seen in Figure 3-7 below, were estimated by Gu et al. to be 12mm or that some
healthier, more in-tact rebar was present in the beams to allow confirmation of the original
rebar size. Similarly, it is assumed here that Gu et al. utilized the average weight of an uncorroded 12mm rebar to obtain the mass lost by corrosion, by comparing that un-corroded
Figure 3-7 Data used to Derive Mass Loss Values from Gu et al. (Gu, Zhang, Shang, &
Wang, 2010)
31
weight to the measured weight of the corroded rebars. The average mass loss ratios provided in
Table 1 of the study, herein referred to as Figure 3-7, were multiplied by a factor of 100 to
obtain the percentage mass loss. The maximum mass loss calculated for Group D beams
exposed to “decades of natural corrosion” is estimated at 9.2%. The strength data provided by
the study is now considered. As an aside, it is also interesting to note that beam specimens D3
and D4 (Figure 3-7), with the greatest mass loss, 9.2% and 7.5%, respectively, are also noted
as having spalling, but the remaining specimen, D2, with a mass loss of 3.4%, does not exhibit
this characteristic.
3.2.3.2
Strength
Beams were loaded under 3-point-bending to ultimate failure load. Results are
provided in Table 2 of Gu et al.’s study, herein referred to as Figure 3-8. The original
ultimate strength of the “decades old” beams remains unknown. It is assumed that beam D1
is an un-corroded replicate beam and that from the load versus deflection data for beam D1 the
original ultimate strength could be extrapolated. That data is shown in Figure 3d of Gu et al.,
herein referred to as Figure 3-9. This approach would return a strength gain rather than loss for
beams corroded over decades which may at first appear illogical, but it should be remembered
that these beams were from a building, not a bridge, so they were not subjected to deicing salts,
so that larger strength losses would not be expected. However, the fact that these beams are
from a building and not a bridge, provides yet another reason why the data in this study is treated
as available for informational purposes only. Finally, available data for stiffness losses from
this study are discussed in the section immediately following.
32
Figure 3-8. Data used to Derive Strength Loss Values from Gu et al. (Gu, Zhang, Shang,
& Wang, 2010)
Figure 3-9
Data used to Derive Stiffness Loss Values from Gu et al. (Gu, Zhang,
Shang, & Wang, 2010)
33
3.2.3.3
Stiffness Loss
Beams were loaded under 3-point-bending and deflection at the mid-point of the beams
was measured. Figure 3-9 shows the load versus deflection data for Group D beams and
appears to indicate, by the presence of beam D1 that a replicate un-corroded beam was tested
to obtain stiffness losses, but there is also the possibility that the curve for D1 is a theoretical
approximation, due to the bilinear nature of this curve. Those losses are regarded here as
the change in deflection under load between the un-corroded and corroded beams at the
proportional limit. The “remaining stiffness of corroded RC beams” is given explicitly in
Figure 6 of the study, referred to herein as Figure 3-10. From this figure the stiffness loss
for Group D beams was scaled off. For example Figure 3-10 shows beam D1 to have a
remaining stiffness of 100%, indicating a stiffness loss of 0%. The maximum stiffness losses
seen in this study are approximately 15% after “decades”.
Figure 3-10. Data used to Derive Stiffness Loss Values from Gu et al (Gu, Zhang, Shang,
& Wang, 2010)
34
Having discussed the methods and procedures of the three studies and some of the
significant results observed for rebar mass loss, strength and stiffness losses, a number of
general observations and conclusions on the data are outlined following.
3.3
Observations and Preliminary Conclusions on the Effects of Aging on Load Path
Redundancy
As outlined at the beginning of this chapter, the focus of this preliminary investigation into
the effects of aging on load path redundancy is to quantify the deterioration of the mechanical
properties of the concrete deck with aging. This quantification should better inform
assumptions made on how this deterioration might affect the system load carrying capacity
of the bridge and the successful action of load path redundancy.
The effected mechanical properties are strength and stiffness, and mass loss of rebar
is one mechanism causing the changes in these mechanical properties which is relatively
easily identified. Observations and conclusions in relation to both properties made from the
three studies reviewed are discussed below but firstly a number of noteworthy points on the
approach to data treatment are made.
As discussed previously the data presented by Gu et al. (2000) has been disregarded
for the basis of conclusions relevant to this report. Therefore, observations and conclusions
herein are based upon synthesized data from the results in the papers by Li and Zheng (2010)
and Oyado et al. (2011). The latter studies both use what is considered to be the more
natural and realistic method (salt-spraying) to induce corrosion. Furthermore, the paper by
Li and Zheng (2010) provides the information and means necessary to transform accelerated
corrosion time in the laboratory to ‘natural’ time. Finally, on the point of time
transformation, the details of the procedure laid out within the study by Oyado et al. (2011)
indicate that the corrosion process was more “natural” than accelerated. This is interpreted
35
to mean that results in that study do not require a time conversion. While each of these studies
has some uncertainty regarding the representative “natural time”, results from these two studies
correlate well comparatively as will be discussed in sections following.
The observation times for the experiments carried out by Li and Zheng (2010) and
Oyado et al. (2011) are estimated at 8.8 years and 12 years, respectively. It is acknowledged
that this is a short time frame for analysis from which to draw conclusions on what can be
expected in the relatively longer scope (approximately 30-50 years)associated with bridge
deck life. None the less, it is assumed that both studies likely have sound scientific reasoning
for their choice in the observed testing time.
With the above preface made, observations and conclusions on the results of the studies
which can be applied to concrete deck deterioration with aging due to salt exposure and its
effects on load path redundancy are now discussed, beginning firstly with strength.
3.3.1
Effects on Strength
Strength losses are regarded here as the reduction in the ultimate flexural load
carrying capacity of beams. Results from Li and Zheng (2010) and Oyado et al. (2011)
indicate losses in the range of 21-28% after a period of 8-12 years approximately. A
summary of results can be observed below in Figure 3-11. Based on that data an “upper
bound” of strength loss for the time period studied may be approximated as 30%. This report
is interested in predicting losses over a longer period of time reflective of typical bridge life.
The trends in the data are analyzed in order to make that prediction.
As McConnell et al. (2012) note, the data from the study by Li and Zheng (2010) show a
largely linear trend whose associated equation with an R2 value of 0.987 is:
% strength loss = 2.2989 * number of years
36
[3-2]
In contrast, McConnell et al. (2012) show that the linear trend average results at each
time period, (including zero loss at time zero as a data point), from the study by Oyado et
al. (2011) returns an R2 value of 0.5053. This indicates a poor correlation in the Oyado et al.
(2011) data when using a linear trend fit. McConnell et al. (2012) conclude this to be due to
differences in testing conditions between 20 months and 12 years. That is, after 3 months
exposure, beams were sprayed with salt solution three times daily for 17 months at which
point Series S beams were removed from exposure for testing and Series M beams remained
under natural exposure for 12 years. This difference in exposure conditions between Series S
and Series M beams would explain the increased rate of strength loss seen in earlier on in
years 0 and 1.67.
McConnell et al. (2012) point to the short period of thrice daily salt spraying to explain
the difference between maximum strength losses seen in both studies at the 1.67 year mark.
Figure 3-11. Strength Loss Conclusions (McConnell, Mc Carthy, & Wurst, 2012)
37
The paper then indicates that if the Li and Zheng (2010) data is extrapolated to the 12year mark
using Equation 3-2 above, there is agreement with the upper range of losses observed by
Oyado et al. (2011). McConnell et al. (2012) conclude that both studies are in general
agreement with one another as regards strength loss results.
McConnell et al. (2012) predict expected strength losses over a range of time that agrees
with that which this report is concerned with. They do so by making the following conclusion:
strength losses can be represented linearly if uniform test conditions had existed across
both studies. The paper uses the linear curve fit for the Li and Zheng (2010) and Oyado
et al. (2011) data to make a prediction for a 25 year horizon.
However, the paper states that the Oyado et al. (2011) data for 1.67 years represents
specimens sprayed with salt three thrice daily for 17 months. For this reason McConnell et
al. (2012) exclude the Oyado et al. results at 1.67 years from the final data set used to make
long-term strength predictions.
The final data set which McConnell et al. (2012) use to make strength loss predictions
is shown as Figure 3-12. The associated linear equation which predicts a strength loss of
50% after 25 years is:
% strength loss = 2.000 * number of years
McConnell et al. (2012) acknowledge this to be “an appropriately conservative estimate
based on the inherent conservatism of the data”.
38
[3-3]
Figure 3-12. Final Strength Loss Predictions over 25 years (McConnell, Mc Carthy, & Wurst,
2012)
3.3.2
Rebar Mass Loss
Rebar mass losses are regarded here as the percentage original rebar mass lost to corrosion.
Losses can be estimated in the range of approximately 9-14% after 8-12 years of chloride
induced corrosion from the results seen by Li and Zheng (2010) and Oyado et al. (2011). This
is treated as an average value however as rebar mass loss is not uniform along rebars, with
areas of pitting being commonplace.
Rebar mass loss has been frequently mentioned along with strength in the literature reviewed
and it has been concluded that its effects are linked more strongly with a cause of strength
losses rather than stiffness losses which are discussed now.
3.3.3
Stiffness Loss
Stiffness losses are regarded here as the change in deflection under load between
corroded and un-corroded replicate beams. It is noteworthy that, conversely, when stiffness is
expressed as a change in elastic stiffness there are no significant changes observed in either
of the studies.
39
In making a loss prediction equation for the longer 25-year range, McConnell et al.
(2012) consider only data supplied by Li and Zheng (2010). The Oyado et al (2011) data is
omitted in making this prediction equation because, while mass loss and strength loss data
from that study are plentiful, data for stiffness losses is only supplied for Series M beams (12
year exposure). This means a trend in results was not readily retrievable.
Data for Li and Zheng (2010) (Figure 3-13), shows a bi-linear relationship between
ultimate deflection and time (McConnell, Mc Carthy, & Wurst, 2012). McConnell et al.
(2012) extrapolate the second linear portion of the trend results seen in Figure 3-13 over a
long term, 25-year, horizon to predict expected stiffness losses. Figure 3-13 shows that the
trend results overlap with the Oyado et al. (2011) data. The associated equation for stiffness
loss over time is:
(% loss) = 2.0347*number of years + 31.559
[3-4]
Results from Li and Zheng (2010) and Oyado et al. (2011) show losses can be
expected in the range of 40-50% after an estimated 8-12 years of exposure to chloride ion
ingress. This result ignores the outlier present in the Oyado et al. (2011) results (Figure 313) where over a 70% increase in deflection is recorded at 12 years. The extrapolated results
by McConnell et al. (2012) shown in Figure 3-13, indicate an 82% increase in ultimate
deflection after a 25-year period. In the absence of better data, in particular that related to
elastic stiffness, this result is treated as the assumed degree of stiffness loss by which decks
in poor condition can be represented in subsequent future analysis and research.
40
Figure 3-13. Stiffness Loss Conclusions (McConnell, Mc Carthy, & Wurst, 2012)
41
Chapter 4
MODELING APPROACH
In this chapter, the different ABAQUS commands and techniques which were
utilized in creating a finite element model of reinforced concrete are discussed. This
includes both the mechanics behind each command and the variables which are input
into ABAQUS to quantify the underlying behavior; in addition, the different modeling
techniques available within ABAQUS that were used within this research for the
purposes of modeling corrosion of reinforced concrete due to chlorides are discussed.
Specifically, this includes the different material models of concrete (Section 4.1) and
rebar that are used as well as the different approaches to modeling corrosion of the
rebar (Section 4.2), which differs depending on whether the rebar is modeled as 2-D or
3-D elements. Exact values used in these commands are discussed in Chapter 4. The
boundary conditions (Section 4.3) used in the models are also discussed and this
chapter concludes with a discussion of the different analysis methods available within
ABAQUS as well as the pros and cons of these alternative methods with respect to the
present research objectives (Section 4.4).
4.1
Concrete
ABAQUS offers three different approaches for modeling the non-linear
behavior of concrete; in each of these models, the elastic portion of the material
response is consistent. ABAQUS uses the elastic definition to determine the material
response until the material reaches the defined cracking stress; at this point, the non-
42
linear behavior of the material governs, including the post-cracking response. The
elastic commands are described in Section 4.1.1 and the non-linear behavior is
described in Section 4.1.2.
4.1.1
Elastic Behavior
As stated previously, all of the concrete models utilize the same linear-elastic
behavior. For this behavior, the modulus of elasticity is defined for concrete (Ec), as
well as Poisson’s ratio (ν). A standard relationship between Ec, compressive concrete
strength (f’c), and tensile concrete strength (f’t) was assumed and are expressed in
Equations 4.1 and 4.2, where Ec, f’c, and f’t are expressed in psi.
7.5
57,000
′
[4.1]
′
[4.2]
These material properties are defined using the “elastic” command within
ABAQUS. For the purposes of these analyses, it was assumed that the material was
isotropic, and this parameter was included in the “elastic” command. In addition to
the “elastic” command, the density was also defined for the concrete. This value was
included using the “density” command within ABAQUS. The exact values which
were used for these commands can be found in Section 4.1.1. These elastic commands
do not directly take into consideration f’c or f’t.
4.1.2
Non-Linear Behavior
As stated previously, three different modeling techniques for modeling non-
linear behavior of concrete are available within ABAQUS: the smeared cracking
43
model (SC), the concrete damaged plasticity model (CDP), and the brittle cracking
model. These are described in the following Sections 4.1.2.1 to 4.1.2.3, respectively.
4.1.2.1
Smeared Crack Model
One modeling approach for post-cracking behavior of concrete which was
explored was the SC model. In this model, ABAQUS employs a smeared cracking
technique; rather than tracking individual cracks, the smeared cracking technique
performs constitutive calculations independently at each integration point and the
presence of cracks enters into these calculations through the stress and material
stiffness associated with the integration point (Simulia 2011). These stress and
material stiffness values are defined by the user in the commands associated with the
SC technique. The SC model is intended as a model of concrete behavior for
relatively monotonic loadings under low confining pressures, with cracking assumed
to be the most important aspect of the behavior.
The SC approach utilizes a Rankine criterion to detect crack initiation; a crack
forms in the direction normal to the maximum principle tensile stress when this stress
reaches a failure surface that is a linear relationship between the equivalent pressure
stress and the Mises equivalent deviatoric stress (Simulia 2011). Once it forms at a
point, the crack orientation is stored for subsequent calculations. A new crack at the
same point can form only in a direction orthogonal to the direction of an existing
crack. These cracks may open and close as the integration point goes into tension and
compression, but remain for all subsequent calculations.
Once a crack has formed, the load transfers across cracks through the rebar,
modeled using tension stiffening; tension stiffening defines the stress-strain response
after cracking and unloads to zero stress at a level of strain defined by the user. This
44
Figure 4.1. Tension stiffening model ABAQUS employs in the smeared crack
technique (Simulia 2011).
tension stiffening relationship can be seen in Figure 4.1. ABAQUS also takes into
consideration the change in shear modulus of the concrete, which affects the shear
behavior post-cracking. One of the most defining features of the SC model is that it is
only applicable when utilizing ABAQUS/Standard, as described further in Section 4.4.
In order to utilize the SC model, certain ABAQUS commands are required.
The first command, called the “concrete” command, defines the stress-strain behavior
of plain concrete in uniaxial compression outside the linear elastic range. The next
command is the “tension stiffening” command. This command defines the fraction of
remaining stress to stress at cracking as a function of the absolute value of the direct
45
strain minus the direct strain at cracking. The final command used is the “failure
ratios” command. This defines the shape of the failure surface for a concrete model
by defining the ratio of the ultimate biaxial compressive stress to the uniaxial
compressive ultimate stress, the absolute value of the ratio of uniaxial tensile stress at
failure to the uniaxial compressive stress at failure, the ratio of the magnitude of a
principal component of plastic strain at ultimate stress in biaxial compression to the
plastic strain at ultimate stress in uniaxial compression, and the ratio of the tensile
principal stress value at cracking in plane stress to the tensile cracking stress under
uniaxial tension. The exact values which were input into the model are described in
Section 5.4.1.3.
4.1.2.2
Concrete Damaged Plasticity Model
Another concrete model investigated was the CDP model. This model is a
continuum, plasticity-based damage model for concrete, assuming that the two main
failure mechanisms are tensile cracking and compressive crushing (Simulia 2011).
Under uniaxial tension, the stress-strain response follows a linear elastic relationship
until the value of the failure stress (σt0), is reached; σt0 is calculated by ABAQUS
when the cracking strain is achieved. This relationship can be seen in Figure 4.2 (a).
The failure stress corresponds to the onset of micro-cracking in the concrete material,
beyond which the formation of micro-cracks is represented macroscopically with a
softening stress-strain response. This softening induces strain localization in the
concrete and under uniaxial loading is linear below the value of initial yield (σc0)
(Simulia 2011). Under multiaxial loading, the stress-strain relations are given by a
scalar damage elasticity equation which utilizes a scalar stiffness degradation variable,
calculated by ABAQUS, that is generalized to the multiaxial stress case to modify the
46
undamaged elasticity matrix. A similar approach is used to model the compressive
behavior, defining the stress-strain behavior of plain concrete in uniaxial compression
outside the elastic range and using a generalized scalar stiffness degradation variable
to model multiaxial behavior, as can be seen in Figure 4.2 (b).
Figure 4.2. Response of concrete to uniaxial loading in (a) tension and (b)
compression for CDP model (Simulia 2011).
47
Tension stiffening is again used to model the stress-strain response between the
concrete and rebar after cracking; that is, tension stiffening defines how the load is
transferred to the rebar from the concrete as it cracks. In addition, damage can be
specified. These variables are treated as non-decreasing material point quantities and
correspond to reductions in stiffness. Should this input not be included, the model
behaves as a plasticity model. There are separate variables to define the tension and
compression damage coefficients. In addition to these other values, flow potential,
yield surface, and viscosity parameters can be defined. As with the smeared crack
model, the CDP model is applicable only when using ABAQUS/Standard, as is
described further in Section 4.4.
To properly define the CDP model using ABAQUS, many different commands
need to be utilized. The first of these is the “concrete damaged plasticity” command.
This command defines the dilation angle, flow potential eccentricity (ϵ), ratio of initial
equibiaxial compressive yield stress to initial uniaxial compressive yield stress
(σb0/σc0), and the ratio of the second stress invariant on the tensile meridian to that of
the compressive meridian at initial yield for any given value of the pressure invariant
such that the maximum principal stress is negative (Kc) (Simulia 2011).The next
command is the “concrete tension stiffening” command. This is used to define postcracking properties for concrete that is in tension by using a multi-linear relationship
to define the remaining direct stress after cracking and the associated direct cracking
strain. The “concrete compression hardening” command defines the equivalent postcracking properties for concrete in compression by using a multi-linear relationship
and defining the stress after yielding in compression and associated crushing strain.
The “concrete tension damage” and “concrete compression damage” commands define
48
post-cracking damage properties for concrete in tension and compression,
respectively, by defining a damage variable and associated crushing strain to include
stiffness degradation as the cracking proliferates. The exact values which were used
for this input are discussed in Section 5.4.1.2.
4.1.2.3
Brittle Cracking Model
The final concrete constitutive modeling approach to be discussed is the brittle
cracking model. As with the previous SC model, the concrete is modeled using a
smeared crack model; rather than tracking individual cracks, constitutive calculations
are performed independently at each integration point and the presence of cracks
enters into these calculations through the stress and material stiffness associated with
the integration point. This is in contrast to the CDP model where the formation of
micro-cracks is represented macroscopically with a softening stress-strain response.
As with the SC model, the brittle cracking model considers tension stiffening
and shear retention. However, unique to the brittle cracking model, ABAQUS has the
capability of defining brittle failure of a material. That is, when any of the local direct
cracking strain components at a material point reach the input value for failure strain,
the material point fails and all the stress components are set to zero. If all the material
points fail within an element, the element is removed from subsequent calculations.
The primary difference between the brittle cracking model and the SC model is that,
rather than using ABAQUS/Standard, the brittle cracking model is only applicable
when ABAQUS/Explicit is utilized, as is explained further Section 5.4.
To implement the brittle cracking model in ABAQUS, the first command that
is specified is the “brittle cracking” command. This command defines the stress
which causes a crack to form, as well as the constitutive relationship after the element
49
fails, which is referred to as tension stiffening. The next command is the “brittle
shear” command. A retention factor is employed to specify the post-cracking shear
behavior by entering the ratio of shear strength to original shear strength as a function
of the crack opening strain. The final command controlling the post-cracking behavior
of concrete used is the “brittle failure” command. This is the command which
specifies the strain at which the material points should fail and the element is removed
from subsequent calculations. The exact input values which are used for each of these
commands is discussed in Section 5.5.1.1.
4.2
Rebar
In order to determine the most accurate way to model corroded reinforced
concrete using ABAQUS, previous research was reviewed. Specifically, research
using ABAQUS was isolated and evaluated. Two different techniques were selected;
one which models rebar as 2-D elements and embeds them within the concrete, and
another which models the rebar as 3-D elements, defining how the surfaces of the
rebar and concrete elements interact with one another. These approaches are detailed
in the following Sections 4.2.1 and 4.2.2.
4.2.1
2-Dimensional Rebar
For the 2-D rebar model, the rebar are modeled as 2-D rod elements. These
elements and nodes are defined separately from those defining the concrete portion of
the model. The elements are then embedded in the concrete, constraining the response
of the rebar’s nodal translational degrees of freedom to that of the concrete, using the
“embedded element” command and defining the host and embedded element sets.
This approach is specifically designed by ABAQUS to model rebar in reinforced
50
concrete, although it can be used for other purposes. This approach is the most simple
and straightforward, but provides few opportunities for modeling the effects of
corrosion. The only properties which can be changed for the rebar are the crosssectional area and strength; no changes can be made to affect the bond between the
rebar and concrete.
4.2.2
3-Dimensional Rebar
When modeling the rebar as 3-D elements, the concrete and rebar elements are
modeled with separate nodes and elements from one another. The literature then
suggests a surface interaction between the concrete and rebar surfaces be defined (Val
et al. 2009). Once the surface interaction has been defined, many different properties
can be included. Most notable of these properties are friction and the pressure applied
by the creation of corrosion by-products around the rebar.
ABAQUS has many different options when defining the friction between two
surfaces. The most simple of these is to define a constant friction coefficient (µ). This
coefficient is defined independent of the slip rate, although a dependency can be
manually defined. Another approach is to define an exponential decay curve, where
the friction value starts at the static friction value (μs) when the slip rate is zero, then
exponentially decays to the kinetic friction value (μk) based on increasing slip rate
(γ ) and a decay coefficient (dc), as shown in Figure 4.3 (Amleh and Ghosh 2006).
In this approach, the slip rate is calculated by ABAQUS at each loading increment and
is applicable to both ABAQUS/Standard and ABAQUS/Explicit. It is suggested that
this approach is more appropriate when modeling corrosion of rebar as the effects of
corrosion can be more directly incorporated. The exponential decay function is
defined by Equation 4.3.
51
Figure 4.3. Exponential decay friction model (Simulia 2011).
[4.3]
The decay coefficient is defined by the user. Amleh and Ghosh (2006) used previous
test results and compared them to their model to determine the most accurate dc value,
which was also used for this analysis. This value, along with µs and µk, are detailed in
Section 5.5.2.2.
In order to define the friction, the “friction” command is utilized. When
defining a constant friction coefficient, no other information is required besides the
value of µ. However, the parameter “exponential decay” must be included to use the
exponential decay curve. After including this parameter, μs, μk, and dc are defined by
the user and γ
is calculated at each loading increment automatically by ABAQUS,
then applied using Equation 4.3. In addition to the “friction command,” the “surface
interaction” and “contact pair” commands are required. The “surface interaction”
command creates a surface interaction property definition; this command defines a
52
label that will be used to reference the surface interaction property in the “contact
pair” command and also allows for the inclusion of an interfacial layer between the
contact surfaces. The “contact pair” command defines pairs of node sets or surfaces
that may contact or interact with each other during the analysis. This is where the
surfaces of the concrete and rebar which are in contact are directly defined.
To model the pressure caused by the creation of rust, literature suggests the use
of a pressure-overclosure relationship (Amleh and Ghosh 2006). The most basic of
this approach is to use a “hard” contact relationship where the surfaces transmit no
contact pressure unless nodes of the slave surface contact the master surface. In the
case of reinforced concrete, the slave surface is the rebar and the master surface is the
concrete. This approach does not allow penetration at each constraint location and
there is no limit to the magnitude of contact pressure that can be transmitted when the
surfaces are in contact. A graph of this relationship can be seen in Figure 4.4. A
contact clearance at which the contact pressure is zero (c0) can be defined, allowing
for space between surfaces to be defined before contact is made. In addition, a linear
penalty stiffness value can be defined; typically this value is calculated by ABAQUS
and assumed to be 10 times a representative underlying element stiffness and defines
the relationship between contact pressure and overclosure.
In contrast to this “hard” contact relationship, a “softened” contact relationship
is available. This is used to model a soft, thin layer on one or both surfaces and can be
better numerically because it can be easier to resolve the contact conditions (Simulia
2011). In order to define a “softened” contact relationship, a type has to be chosen.
The available types include using a linear law, a tabular piecewise-linear law, or an
exponential law. The optimal version to model corrosion was found to be the
53
Figure 4.4. Default “hard” pressure-overclosure relationship (Simulia, 2011).
exponential law (Amleh and Ghosh 2006), which is shown in Figure 4.5. This
relationship takes into consideration the increase in pressure as the surfaces get closer,
and allows for the pressure to become zero should the surfaces no longer be in contact.
This exponential relationship is based on c0, the pressure at zero clearance (p0), and,
Figure 4.5. “Softened” exponential pressure-overclosure relationship (Simulia 2011).
54
when employing ABAQUS/Explicit, the maximum stiffness value (kmax). The kmax
value is a required parameter when utilizing ABAQUS/Explicit and is not available in
ABAQUS/Standard; limiting this value can be useful for penalty contact to mitigate
the effect that large stiffnesses have on reducing the stable time increment. Exact
values which were used and input into the model are discussed in Section 5.5.2.2.
In order to use a pressure-overclosure relationship, the “surface behavior”
command must be utilized. The default for this command is to apply the “hard”
relationship. In this approach, c0 is defined, along with a linear penalty stiffness. A
parameter can be added to utilize the exponential pressure-overclosure relationship
where c0 and p0 are defined. When employing ABAQUS/Explicit, a kmax value is also
defined.
One of the benefits of using the exponential pressure-overclosure relationship
over the default “hard” relationship is the ability to include corrosion effects. Using
the results from pull out tests, the pressure and friction were related to the concrete
cover thickness (C) and the mass loss as a percentage (M) caused by corrosion in prior
work by Amleh and Ghosh (2006). The pressure at zero clearance for uncorroded
concrete is defined by Equation 4.4, where p0 is expressed in MPa and C is expressed
in mm.
0.128
1.5
[4.4]
This p0 changes as rebar corrodes and M increases. The percentage loss of contact
pressure (L) is related to M and f’c by Equation 4.5, which is then multiplied by p0 to
determine a new pressure at zero clearance for corroded rebar (Amleh and Ghosh
2006).
55
0.00024
0.0028
4.3
[4.5]
In this empirical equation, L and M are expressed as a percentage, f’c is in MPa, and C
is in mm.
In addition to the pressure changing, μs and dc also change due to mass loss.
These changes can be calculated using the following Equations 4.6 and 4.7,
respectively (Amleh and Ghosh 2006), where M is expressed as a percentage.
.
0.0261
4.3
[4.6]
0.45
[4.7]
Boundary Conditions
The original beam test set-up utilized by Oyado et al. (2010) loaded the beam
under simple support conditions. It was found during initial modeling that using one
row of nodes as a pin and another row of nodes as a roller caused high bearing forces
and deformations at the supports. For this reason, each of the supports were modeled
across multiple rows of nodes. The number of rows of nodes was directly related to
the size of the mesh; the supports were modeled over 1.57 in (40 mm). This was
chosen based on the 0.7874 in (20 mm) mesh size using 3 rows of nodes. This model
was the one used to analyze the support condition, and the size of the support was kept
consistent between mesh sizes. All of the supports nodes were modeled as rollers,
only limiting vertical displacement, with the exception of one node on the pinned end.
This node, located in the center of the defined support, was fixed in all directions with
the exception of rotation about the length of the beam. It was found that this provided
adequate restraint while still accurately modeling the fixity of the support. The
bearing forces were no longer considered high when the nodes defined for the support
56
and the elements attached to those nodes displayed no deformation before the model
failed.
4.4
Analysis Method
ABAQUS offers two different techniques for performing analyses:
ABAQUS/Explicit and ABAQUS/Standard. ABAQUS/Explicit is an explicit
dynamic analysis. It is more computationally efficient for large models with relatively
short dynamic response times and allows for the definition of general contact
conditions (Simulia 2011). This approach uses a consistent, large-deformation theory
where models can undergo large rotations and large deformations. It can also use a
geometrically linear deformation theory where strains and rotations are assumed to be
small. It allows for either automatic or fixed time incrementation to be used and can
be used to perform quasi-static analyses with complicated contact conditions. To
implement this type of analysis, the “dynamic” command is used, specifying the
optional parameter of “explicit.”
As opposed to the explicit dynamic analysis of ABAQUS/Explicit,
ABAQUS/Standard is a static stress analysis. This is used when inertia effects can be
neglected and can be linear or nonlinear (Simulia 2011). This analysis ignores timedependent material effects such as creep, swelling, and viscoelasticity; however, it
takes rate-dependent plasticity and hysteretic behavior for hyperelastic materials into
account. To implement this type of analysis, the “static” command is used.
With each of these different analysis techniques come different loading
definitions. For ABAQUS/Explicit, a loading amplitude is defined using the
“amplitude” command. With this command, the time and load proportion are defined
by the user and can be applied as a ramp or sustained. With the loading defined, the
57
analysis then performs the applicable calculations to determine stress and
displacements.
For ABAQUS/Standard, the “riks” command was utilized. This method is
generally used to predict unstable, geometrically nonlinear collapse of a structure and
can include nonlinear materials and boundary conditions (Simulia 2011). This method
uses the load magnitude as an additional unknown and solves simultaneously for loads
and displacements. This approach provides solutions regardless of whether the
response is stable or unstable and is only applicable to ABAQUS/Standard.
There are different pros and cons associated with these different analysis
techniques. ABAQUS/Standard allows for calculating a static loading corresponding
to the equilibrium condition of the deformed structure and clearly indicates failure by
reaching a peak loading then decreasing; this gives a direct quantitative definition of
failure. Conversely, ABAQUS/Explicit utilizes a dynamic loading approach that may
over-estimate realistic loads, requiring judgment to assess failure; this approach also
includes a time component, adding another parameter needing to be analyzed and
calibrated. These loading differences cause the two methods to require differing levels
of judgment to determine when failure has occurred, with ABAQUS/Standard being
more straightforward. The biggest con with utilizing ABAQUS/Standard is that,
ideally, the concrete input should be taken from actual concrete testing results; that is,
the “concrete compression hardening” and “concrete tension stiffening” commands
require multiple input values which should be based on actual testing results that are
not widely documented. However, applying the results of concrete testing similar to
the problem of interest, the approach which is used in this research, can also result in
calibrated input. This is in contrast to ABAQUS/Explicit modeling techniques with
58
input values which can be more readily estimated based on common concrete
properties.
59
Chapter 5
MODEL CALIBRATION
The ABAQUS variables and commands described in Chapter 3 were utilized to
model corrosion within a reinforced concrete beam. An uncorroded base model was
created using the original material property input and modeling commands as
described in Sections 4.1 and 4.2. Once the input and modeling approach for the final
uncorroded model were determined, these input variables were altered in attempts to
simulate corroded reinforced concrete and obtain the targeted amounts of strength
decrease and deflection increase associated with corrosion, as previously explained in
Chapter 3. The geometry and material properties which were modeled are described
in Section 5.1, with the concrete modeling described in Section 5.1.1 and the rebar
modeling described in Section 5.1.2. Section 5.2 defines how the results are evaluated
and standardized in order to easily compare different modeling techniques. Hand
calculations were performed to compare to the results of the finite element model; this
was done for both strength and deflection. These calculations are described in Section
5.3.
The 2-D beam model is described in Section 5.4. Within this section, the
uncorroded base model input to simulate an uncorroded reinforced concrete beam is
described in Section 5.4.1, the mesh sensitivity analysis and resulting final uncorroded
model input are described in Section 5.4.2, and the calibration and determination of
the input for the corroded model are described in Section 5.4.3.
In addition to the 2-D beam model, a 3-D beam model was created. This
model is described in Section 5.5. Initially, this modeling was performed with 2-D
rebar elements, as described in Section 5.5.1; the model was then refined by using 3-D
60
rebar elements, as described in Section 5.5.2. The conclusions drawn from these
different modeling techniques can be found in Section 5.6.
5.1
Beam Geometry and Material Properties
Certain input was constant throughout all different types of beam modeling.
This includes the strength and modulus of elasticity of the concrete as well as the
elastic and plastic parameters of the rebar. This concrete and rebar input is referred to
as the uncorroded base model and is described in Sections 5.1.1 and 5.1.2,
respectively.
5.1.1
Concrete
For the purposes of this study, a beam design which was previously corroded
and tested (Oyado et al. 2010) was used to calibrate modeling techniques and inputs.
This beam was chosen based on the review of experiments of this discussed in Chapter
3. The geometry of this beam can be seen in Figure 5.1. For ease of creating the
model, the hooks that were located on the ends of the tension and compression
reinforcement as well as the stirrups in the physical specimen were ignored. The rebar
which were used for compression, tension, and stirrups had varying sizes and
strengths; the #2 bars used for compression reinforcement and stirrups were SD295
and the #4 bars used for tensile reinforcement were SD345. These material
designations correspond to a minimum yield strength of 42,786 psi (295 MPa) and
50,038 psi (345 MPa), respectively.
61
Figure 5.1. Geometry of finite element calibration models, elevation view (top) and
cross-section view (bottom) (dimensions in inches).
The material property inputs selected for the uncorroded base model (using the
material properties of the tested specimen and the input values suggested in literature)
are based on those from Oyado et al.’s (2010) uncorroded specimen S-0N, their
specimen which carried the highest load. The tested material properties of this
specimen were reported as a compressive strength of 3,147.3 psi (Oyado et al. 2010),
resulting in a modulus of elasticity of 3,197,746 psi and a tensile strength of 420.8 psi,
when calculated using Equations 4.2 and 4.1, respectively. A standard value of
Poisson’s ratio of 0.2 was also assumed. A failure crack strain value of 0.0027 was
assumed; this is the amount of additional strain which can be carried by the concrete
after an initial crack forms and before complete failure. This value was based on the
difference between the ultimate strain and the strain at first cracking; an ultimate strain
(the strain at which the concrete fails completely and can no longer carry load) of
62
0.003 was utilized and it was assumed that the first crack occurs at a strain of 10% of
the ultimate strain (i.e., 0.0003).
5.1.2
Rebar
Standard elastic steel material property inputs were specified for the rebar,
which included the following assumptions: the modulus of elasticity of the rebar (Es)
was assumed to be 29,000 ksi, the Poisson’s ratio was assumed to be 0.3, and the
density was assumed to be 0.000734 lb/in3 (1.27 lb/ft3). The input into ABAQUS is
based on inputting all values in consistent units, where pounds and inches were used in
these models.
In addition to this value, the plastic properties of the rebar were also included.
For these, the yield and ultimate strengths of the SD295 and SD345 rebar, used by
Oyado et al. (2010) and serving as the calibration specimen, were researched. As
stated in Section 5.1.1, the stirrups and compressive reinforcement are comprised of
#2 bars which were SD295 and the tensile reinforcement are comprised of #4 bars
which were SD345. The minimum yield strengths of SD295 and SD345 are 42,786
psi (295 MPa) and 50,038 psi (345 MPa), respectively. However, two different
research papers were found which included actual yield and ultimate strength values
of both rebar types: Takahashi (2008) and Shirai et al. (2002). These values can be
found in Table 5.1, where Takahashi (2008) provided 1 set of values for the rebar
strengths and Shirai et al. (2002) provided 2 sets of values for the rebar strength,
which were averaged before being included in Table 5.1. In this table, the percent
difference is compared to the minimum specified strengths of 295MPa and 345MPa
for SD295 and SD345, respectively. These values are consistent with a trend which
exists for rebar grades typically used in the US, where a factor of 1.1 is applied in
63
some situations to approximate the actual rebar strength relative to the minimum
specified strength (Morales n.d.). Thus, this factor was multiplied by the minimum
specified yield strength to obtain the yield stress in the models. For the ultimate
strength, the individual values provided by Takahashi (2008) and Shirai et al. (2002)
were averaged and this average was used.
Table 5.1. Yield and ultimate strengths of SD345 and SD295 rebar found in literature
compared to the minimum specified yield strengths.
Takahashi (2008)
Shirai et al. (2002)
Fy (psi) % Difference Fu (psi) Fy (psi) % Difference Fu (psi)
8%
82,671
55,694
11%
78,465
SD345 53,809
28%
77,885
48,588
14%
91,519
SD295 54,679
For both yield and ultimate strengths, these engineering stresses were
converted to true stresses and plastic logarithmic strains for input into ABAQUS. This
was done by first calculating the engineering strain from the engineering yield stress
using Es by utilizing Equation 5.1, where σy is the yield stress value and εy is the
corresponding yield strain value.
[5.1]
In addition to the yield and ultimate stresses, two more values are calculated to
make a more complete plastic response. Together, these data points describe a linear
elastic regime, followed by a yield plateau, followed by strain hardening, followed by
a second plateau after strain hardening terminates. This rationale, as well as the
specific stiffnesses and strain values associated with this multi-linear response, are
64
based on the steel material modeling discussed in Barth et al. (2005). The end of the
yield plateau corresponds to a strain of 0.011. The calculation of this strain, σ1, was
done by using Equation 5.2, where σ1 is in psi.
145,000 0.011
[5.2]
The strain corresponding to the ultimate stress was calculated using Equation 5.3,
where εu is the strain corresponding to the ultimate stress and σu is the ultimate stress
in psi.
0.011
,
[5.3]
The final stress which was utilized was a value larger than the ultimate stress and
corresponding to a strain of 0.3. This value, σ2, was calculated using Equation 5.4,
where σ2 is in psi.
145,000 0.3
[5.4]
Once the 4 pairs of stress-strain input were determined, the true stress was then
calculated by multiplying the engineering stress by the engineering strain using
Equation 5.5, where σtrue is the true stress value, σeng is the engineering stress value,
and εeng is the corresponding engineering strain.
1
65
[5.5]
Lastly, the plastic logarithmic strain is calculated by using Equation 5.6, where εlnplastic
is the logarithmic plastic strain.
ln 1
[5.6]
The plastic input values for SD295 and SD345 rebar resulting from these equations
can be found in Table 5.2.
Table 5.2. True stress and true plastic strain input values for SD295 and SD345 rebar.
SD295
SD345
plastic
σtrue (psi)
εln
σtrue (psi)
εlnplastic
47,076
0.0000
55,104
0.0000
48,892
0.0093
56,940
0.0090
85,563
0.0519
84,668
0.0414
109,893
0.2586
110,102
0.2586
5.2
Calibration Metrics
In order to directly compare the different modeling techniques discussed in
Section 4.1, the strength and deflection values from various models were compared.
The deflection results were obtained directly; a node in the center of the bottom of the
beam was chosen, as this location should provide the highest deflection result, and the
deflection at this node at the maximum loading is the value reported.
The load proportionality factor (LPF) was used to easily compare the results
obtained utilizing ABAQUS/Standard and ABAQUS/Explicit. This value is the
proportion of the loading at a given step time relative to the total load specified in the
input file of the model. In ABAQUS/Standard, this value is printed directly to the
66
output file, as described below. In ABAQUS/Explicit, this value is hand calculated
based on the step time and the total loading applied, as specified in the input file. This
is done by calculating a loading rate and multiplying this by the step time. The
loading rate is calculated based on the linear input defined in the “amplitude”
command, using the total load input divided by the time over which the load is
applied.
Different approaches were used to determine the maximum loading depending
on the analysis technique used. ABAQUS/Standard was straightforward; an output
file is created while an analysis is running which includes the LPF applied during each
step time. The output file containing the LPF values was analyzed to find the highest
LPF, which denotes the maximum loading of the model. In some versions of the
models, the LPF would reach a peak value before the beam would begin to unload. In
these cases, the first peak value was considered the maximum loading. Ideally, the
model would reach a peak LPF value, followed by a decrease in load. This would
indicate that the model did not experience any problems reaching convergence prior to
achieving its maximum capacity.
When utilizing ABAQUS/Explicit, an LPF is not directly printed. The loading
is defined using the “amplitude” command, as described in Section 4.4; the load can
then be calculated directly based on the step. The step is indicated in the results files
created by ABAQUS during analysis. In contrast to ABAQUS/Standard,
ABAQUS/Explicit does not reach a maximum LPF followed by a decrease, but rather
increases until reaching the maximum load input using the “amplitude” command or
the model terminates due to excessive distortion, which may be well past the point of
67
realistic behavior. To determine the maximum realistic load, the visual output (.odb)
file was analyzed as follows.
Two different methods were used to determine the loading at which the model
is classified as having failed. Initially, visual inspection was used to determine the
step time at the point where the beam displayed an abrupt and obvious non-linear
change in deflection and/or element distortion from one step to the next. An example
of this for the 3-D beam can be seen in Figure 5.2, where Figure 5.2 (a) shows the last
step before failure and Figure 5.2 (b) shows the step where failure occurs. Every
version of this model had similarly clear indications of the failure; although the
excessive deformation was not always located in the center, the distortion of elements
was always evident.
(b)
(a)
Figure 5.2. Example of determination of (a) the last step before failure and (b) the first
step of failure in 3-D models utilizing ABAQUS/Explicit.
To determine whether this visual method for determining the failure of the
beam was too qualitative, a second method of assessing maximum load was
formulated. In this method, a row of elements along the bottom of the beam, halfway
between the center of the beam and the loading, was used. It was thought that this
location would provide a more accurate indication of if the beam was globally
68
unloading as the elements are not directly under the load nor experiencing the
maximum shear and moment, such that the results are not sensitive to localized
unloading as individual elements fail; in addition, the center of the beam was the most
common location of deformation in the model. The maximum principal stress at the
integration point of each of these elements was determined using the output file, and
this value for each element was added together for each step. The step where the
highest stresses were seen was then considered to be the failure step.
5.3
Strength and Deflection Calculations
The study that the finite element beam calibration model was based on
reported the tested strength of an uncorroded specimen by indicating the total load
which was applied at failure (Oyado et al. 2010). However, this value of 11,802.5 lbs
(52.5 kN) was thought to be high; therefore, strength calculations based on traditional
reinforced concrete design (Wight and MacGregor 2009) were computed. In order to
easily account for changes in material properties which were performed to model
corrosion, a spreadsheet was created, based on geometry and material properties of the
beam, to calculate the expected strength and deflection for all models. With these
strength and deflection values, the accuracy of each beam modeling technique could
be estimated by comparing the model results to these theoretical values. The
equations and an example of the calculations put into these spreadsheets can be found
in McCarthy (2012).
Using these calculations, the expected strength was calculated for both the
uncorroded and corroded models. For the uncorroded model, the strength was
calculated to be 5,392 lbs based on the loading configuration shown above in Figure
5.1. This corresponds to an LPF of 0.914 based on a total applied load of 5901.4 lbs
69
in the model, which is half of the experimental load. The deflection calculations were
performed based on both the uncracked and cracked sections, as is done in typical
design calculations for reinforced concrete members. These deflections were
calculated, based on the loading which is associated with the calculated strength, to be
0.0181 in for the uncracked section and 0.1245 in for the cracked section. For the
corroded model, these values varied depending on the material properties which were
defined. The exact values as the input parameters are varied are reported in the
discussion of the calibration process, and can be found in Section 5.4.3.
These theoretical values are later used for comparison with the results of the
models to estimate accuracy. It was thought that these values, based on similar
theoretical equations that ABAQUS utilizes when analyzing models, were a more
accurate representation of the model accuracy than the experimental values; there is a
large amount of deviation in concrete and rebar properties that are possible and for
which specific values are not reported for the experimental specimen. These
deviations could account for the higher failure load observed experimentally when
comparing to the theoretical strength of the concrete beam. The experimental strength
was more than twice what the theoretical calculations indicated. While some
discrepancy between the theoretical and experimental values are expected, as the
theoretical equations are relatively conservative when predicting the response of a
reinforced concrete beam, this large deviation suggests that the experimental specimen
displayed unusual characteristics which were not attempted to be simulated.
5.4
2-Dimensional Concrete Model
In attempts to keep computational effort and time to a minimum, 2-D concrete
beams were analyzed. In using this approach for modeling reinforced concrete,
70
previously created bridge models could be utilized with minimal modeling efforts,
which involved only incorporating post-cracking behavior into the decks. These
existing models used 2-D linear-elastic concrete elements for the deck and the “rebar”
command for the rebar. By modeling the reinforced concrete beam the same way, the
only changes to be made were those related to the material input variables. The
definition of the material property input values used in the uncorroded base model,
including the rebar properties applied to all models, the concrete damaged plasticity
input, and the smeared crack input applied when evaluating different modeling
techniques, are described in Section 5.4.1. A mesh sensitivity analysis was performed,
and the results of this, along with the modeling technique determined to most
accurately model uncorroded reinforced concrete, are described in Section 5.4.2.
Once the uncorroded modeling technique and input were determined, a calibration
process was performed to determine the optimal approach for modeling corrosion to
achieve the target changes in deflection and strength. This calibration process and
final corroded model input are described in Section 5.4.3.
5.4.1
Uncorroded Base Model Input Values
To compare how the different modeling approaches affect the accuracy of the
results, both the concrete damaged plasticity (CDP) and smeared crack (SC)
approaches were analyzed. These are both approaches which utilize
ABAQUS/Standard and therefore do not require interpretation for determining
maximum realistic loadings as well as directly simulating static loadings such that the
rate of load application is not applicable and does not need to be included. The
material properties which were used in the input commands for the uncorroded base
model are defined below in Sections 5.4.1.1 through 5.4.1.3.
71
5.4.1.1
Rebar
In order to accurately create the beam models, the values for the inputs
described in Chapter 3 needed to be determined. The “rebar” command which was
utilized involves defining the cover of the rebar and an equal rebar spacing. Due to
there being only 2 rows of rebar present, which are not constantly spaced between
each other and the edge (see Figure 5.1), it was assumed for this command that the
cover distance was equal to a horizontal rebar spacing of 1.2992 in (33 mm); that is,
rather than defining a 0.7874 in (20 mm) cover to the outer edge of the beam with a
2.3622 in (80 mm) spacing between the bars, a constant spacing of 1.2992 in (33 mm)
between the outer edges and between the bars was defined. It was thought that this
slight spacing change should have no significant effect on the strength results of the
beam, as theoretically the horizontal position of the rebar is not influential. The
vertical spacing was consistent with the spacing utilized in the experimental beam, as
described in Section 5.1. The area of the rebar was also used in the “rebar” command,
defining values of 0.0438 in2 (6 mm2) and 0.20563 in2 (13 mm2) for #2 (SD295) and
#4 (SD345) bars, respectively. These are the actual cross-sectional areas of the #2 and
#4 bars.
5.4.1.2
Concrete Damaged Plasticity
The elastic input values previously described in Section 4.1.1 were
used in combination with CDP inputs as one means to model the concrete constitutive
response. These elastic properties included Ec and ν values of 3,197,746 psi and 0.2,
respectively. In addition to these elastic values, the post-cracking values of concrete
needed to be determined for use in the “concrete damaged plasticity”, “concrete
tension stiffening”, “concrete compression hardening”, “concrete tension damage”,
72
and “concrete compression damage” commands, as described in Section 4.1.2.2. A
study was found which used experimental tests to calibrate the input for these CDP
commands based on actual reinforced concrete beams (Jankowiak and Lodygowski
2005). This study provided exact tabulated input for each command. Jankowiak and
Lodygowski’s input (2005) for the “concrete tension stiffening” and “concrete
compression hardening” commands was scaled based on the strength of the concrete
and used directly. This scaling involved taking the proportion of each stress value in
the input relative to the maximum stress value reported in the input for the
corresponding command and multiplying this by the experimental strength, f’c. The
“concrete tension damage” and “concrete compression damage” command input
values were used directly. These input values can be seen in Table 5.3. It should be
Table 5.3. Input values for post-cracking commands for CDP model.
Compression Hardening
Compression Damage
σ (psi)
ε
Coefficient
ε
944
0.0
0.0
0.0
1271
0.0000747
0.0
0.0000747
1881
0.0000988
0.0
0.0000988
2537
0.000154
0.0
0.000154
3147
0.000762
0.0
0.000762
2532
0.00256
0.195
0.00256
1274
0.00568
0.596
0.00568
331
0.0117
0.895
0.01173
Tension Stiffening
Tension Damage
σ (psi)
ε
Coefficient
ε
269
0.0
0.0
0.0
421
0.0000333
0.0
0.0000333
277
0.000160
0.406
0.000160
128
0.000280
0.696
0.000280
34
0.000685
0.920
0.000685
8
0.00109
0.980
0.00109
73
noted that, for the “concrete tension damage” and “concrete compression damage”
commands, multiple rows of 0.0 coefficients are required as the strain for these
commands must be the same as the corresponding tension stiffening and compression
hardening commands, respectively, even if no coefficient value is used. These
coefficients define post-cracking damage properties for concrete in tension and
compression by defining a damage variable and associated crushing strain to include
stiffness degradation as the cracking proliferates.
The final command which is used for the CDP model is the “concrete damaged
plasticity” command. The dilation angle which was used was 38, as determined by
Jankowiak and Lodygowski (2005). The flow potential eccentricity (ϵ) and ratio of
initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress
(σb0/σc0) (Simulia 2011) were also determined by Jankowiak and Lodygowski (2005)
to be 1.0 and 1.12, respectively; these values were also used directly. The final input
values required is Kc, and a default value of 2/3 was used.
5.4.1.3
Smeared Crack
As with the CDP model, the elastic commands and corresponding input values
used are as described in Section 4.1.1. A description of each command and how it
related to the SC model is detailed in Section 4.1.2.1. The first command for postcracking behavior for the SC model is the “concrete” command. The strength of 3,147
psi, which is the experimental strength as determined by Oyado et al. (2010), was
used, along with an absolute value of plastic strain of 0.0. This point defines the yield
stress of the concrete. A second data point of 5,500 psi stress and 0.0015 absolute
plastic strain was used. This point defines the maximum stress of the concrete before
crushing and the associated plastic strain. The “tension stiffening” command specified
74
the relationship between the fraction of remaining stress to stress at initial cracking
and the cracking strain, which is the absolute value of direct strain minus direct strain
at initial cracking. Two data points were used to define this relationship. The first
data point was 1 and 0, where the remaining stress is all of the stress and cracking has
not yet initiated, thus the cracking strain is zero. The second data point was 0 and
0.0027, which specifies the strain at which the stress has been entirely relieved; this
strain value assumes a compressive strain limit of 0.003 and a strain at cracking of
0.0003. For the “failure ratios” command, all of the ABAQUS default values were
used in the absence of any reason for altering these defaults.
5.4.2
Mesh Sensitivity Analysis and Uncorroded Base Model Input
The creation of the 2-D geometry was relatively straightforward for the
concrete elements; the beam was broken up into equally sized square elements. The
exact size of these elements was varied to determine the most computationally
efficient and accurate size. A mesh sensitivity analysis was performed and the mesh
sizes which were tested were 3.937 in (100 mm), 1.9685 in (50mm), 0.7874 in (20
mm), 0.3937 in (10 mm), and 0.19685 in (5 mm) squares. For each of these mesh
sizes, the CDP and SC approaches were utilized. For these initial models, the
uncorroded base model values previously described in Section 5.4.1 were used.
Ideally, as the mesh sizes decrease, the results should asymptote to a constant value.
The load which was applied was 5901.4 lbs, half of the experimental load at failure as
reported by Oyado et al. (2010). The strength and deflection results of this sensitivity
analysis can be seen in Table 5.4 and Table 5.5, respectively. The percent difference
values are based on a strength LPF of 0.914 and deflection values of 0.0181 in for the
uncracked section and 0.1245 in for the cracked section, as determined from the hand
75
Table 5.4. Strength results of mesh sensitivity analysis with percent differences based
on the calculated theoretical strength values.
Mesh Size
Concrete Highest Load
% Difference from
(mm)
Model
(LPF)
Expected Load
1.75
92%
CDP1
100
1
SC
4.72
417%
CDP
2.55
179%
50
1
SC
3.02
231%
CDP1
0.89
-3%
25
SC
4.10
349%
CDP
1.10
20%
20
SC1
2.98
226%
CDP
1.08
18%
10
SC1
2.39
162%
CDP
3.69
304%
5
1
SC
10.2
1017%
1
Model terminated at peak load, indicating inability to converge.
Table 5.5. Deflection results of mesh sensitivity analysis with percent differences
based on the calculated deflection values for the uncracked and cracked
sections.
Mesh Size
(mm)
Concrete
Highest
% Difference,
% Difference,
Model
Deflection (in)
Uncracked
Cracked
1
CDP
0.0533
196%
-57%
100
SC1
0.3831
2022%
208%
CDP
0.2820
1462%
127%
50
SC1
1.4130
7728%
1035%
CDP1
0.0233
29%
-81%
25
SC
0.2109
1068%
69%
CDP
0.0435
141%
-65%
20
SC1
0.1361
654%
9%
CDP
0.0425
135%
-66%
10
SC1
0.0993
450%
-20%
CDP
0.0024
-87%
-98%
5
1
SC
0.0069
-62%
-94%
1
Model terminated at peak load, indicating inability to converge.
76
calculations described in Section 5.3, where positive changes indicate results larger
than the calculated values and negative changes indicate lesser values. Since these
models utilize ABAQUS/Standard, the loading would ideally reach a maximum peak
before decreasing. Should this not occur, convergence issues are the cause. These
results indicate that, with the exception of the 5 mm mesh, the values begin to become
relatively constant as the mesh sizes decrease. It was thought that the reasoning for
the 5 mm mesh results being significantly higher than the previous models with larger
meshes is that, when tracking cracks, the elements are smaller and therefore have a
smaller area effected by each crack, allowing the elements to reach higher strengths.
It should also be noted that, although the models did not converge, the results of the 10
mm, 20 mm, and 50 mm models utilizing the SC approach were relatively close to the
experimental load of an LPF of 1.83 (11,802 lbs).
When only taking into consideration the models which converged, it was seen
that only 1 SC model and 4 CDP models converged. Of these CDP models, 2 had
relatively consistent results; 1.1 and 1.08. Due to the lack of information regarding
converged SC models, and the inconsistency of the available results, this technique
was not applied to the final uncorroded model input. Ultimately, the CDP approach
was chosen with a mesh size of 20 mm rather than 10 mm (of the two models that
converged and gave consistent values) to save on computational time.
After performing the mesh sensitivity analysis, the results of the analysis were
compared to the calculated strength and deflection values to determine accuracy. It
should be noted that the models include post-cracking behavior which is not included
in the corresponding strength and deflection equations. This is likely the explanation
for the increased strength values seen in Table 5.4. Thus, the 20% increase in strength
77
associated with the 20 mm CDP model was considered to be within reason. As for the
deflection, these values were consistently between the cracked and uncracked
expected values based on the theoretically expected strengths, with the exception of
the 50 mm and 5 mm mesh size. Although the values are not the same as the expected
values, it was thought that ABAQUS can take into consideration the post-cracking
behavior more accurately than the theoretical equations and therefore the 20 mm mesh
size was used as the final uncorroded model for comparison with the calibration of the
corroded model. That is, the values described above in Section 5.4.1.2 were used as
the uncorroded material input and the results, outlined in Tables 5.4 and 5.5, were
used as a baseline for comparison to determine if a 50% decrease in strength and 82%
increase in deflection, for reasons described in Chapter 3, were achieved during the
calibration process. The stress contours of this uncorroded model can be seen in
Figure 5.3, where darker colors indicate a higher stress and lighter colors indicate a
lower stress and the stress which is reported is the Mises stress at the integration point
of each element; subsequent stress data in 2-D elements is also reported at integration
points, which means that peak tensile and compressive stresses at the extreme fiber of
the elements is not considered. Similarly, the relative tensile and compressive stress
on the top and bottom cannot be discerned from one another visually in Figure 5.3.
The maximum and minimum principal stress values can be used to determine the
compressive and tensile stresses at the integration points, respectively. Furthermore,
the deformation of the beam cannot be visually seen in Figure 5.3. Thus, this
deflection was plotted versus load in Figure 5.4, where the FEA data includes the
loading response and the unloading response, which nearly traces the loading
response. To determine the validity of the model, this data is compared to the
78
deflection results calculated for the uncracked and cracked sections using the same
loading applied in the model, where the theoretical deflection results are limited to the
elastic regime, which is assumed to occur up to the predicted strength of the beams.
These results indicate that the finite element model follows the expected deflection
results relatively well. It should be noted, however, that these results were calculated
using the actual loading applied in the finite element model and therefore the resulting
deflections are not the same as those compared to in Table 5.5, where the load was
based on the loading associated with the theoretically expected strength and thus
indicated the model deflection results were between the uncracked and cracked
sections deflections. It can be seen in Figure 5.4 that this is not the case. A summary
of the input used in the final uncorroded model for future comparison during the
calibration process of determining the corroded model input can be found in Table 5.6.
Figure 5.3. Uncorroded 2-D base model stress contours (units of psi).
79
1.2
Load (LPF)
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
Deflection (in)
FEA
Uncracked
0.4
0.5
Cracked
Figure 5.4. Load and deflection comparison between final uncorroded model and
theoretical calculations for the uncracked and cracked sections.
Results
Input
Table 5.6. Input and results of final uncorroded 2-D beam model.
38
29000
Dilation Angle
Es (ksi)
ϵ 1
47000
f'y (psi)
1.12
55000
σb0/σc0
fy (psi)
Kc
2/3
f'c (psi)
3147.3
A's (in2)
0.0438
f't (psi)
420.8
As (in2)
0.20563
Ec (psi)
3197746
E's (ksi)
Strength (LPF)
29000
1.1
Deflection (in)
0.0435
80
5.4.3
Corroded Model
While calibrating the optimal method to simulate the effects of corrosion in the
2-D models, different input values and the use of numerous commands were varied.
These included the inclusion of the “concrete compression damage” and “concrete
tension damage” command, as well as variation of the dilation angle, flow potential
eccentricity (ϵ), σb0/σc0, Kc, compression rebar area (A’s), compression rebar strength
(f’y), compression rebar modulus of elasticity (E’s), tensile rebar area (As), tensile
rebar strength (fy), tensile rebar modulus of elasticity (Es), f’c, f’t, and Ec. The
inclusion of the “concrete compression damage” and “concrete tension damage”
commands, the dilation angle, ϵ, σb0/σc0, and Kc were varied as their effects on the
model were not well understood and the sensitivity of the models to these values
needed to be determined. Changing the A’s and As was used to reflect the decrease in
area which occurs during corrosion and varying f’y, fy, E’s, and Es was used as an
indirect means to produce the desired changes in strength and deflection. It was
thought that decreasing f’c, f’t, and Ec would be a valid approach for simulating the
decrease in strength associated with corrosion and the theoretical change in concrete
stiffness; although the theoretical change in stiffness occurs in the non-linear range,
modeling the change in this region was an indirect approach.
Initially, the parameters were varied individually to assess the effects of each
variable on strength and deflection of the beam. A discussion of this calibration can
be seen in Section 5.4.3.1. After this was performed, the model was then optimized
using different variables individually. The optimization of Ec, f’c, and f’t is described
in Section 5.4.3.2, the optimization of A’s and As is described in Section 5.4.3.4,
followed by the calibration of the dilation angle, described in Section 5.4.3.4. During
the optimization process, the results of the models were compared to the strength and
81
deflection results expected through the theoretical calculations described in Section
5.3, where the basis for this comparison is also described.
5.4.3.1
Individual Parameter Variation
Initially, the variations of parameters were performed with each variable
independently. One of the first variables tested was the removal of the commands
“concrete compression damage” and “concrete tension damage.” The values used in
these commands were taken directly from the literature (Jankowiak and Lodygowski
2005); consequently, their importance to the model was not known. It was found that
removing these commands did not affect the results of the uncorroded base model and
therefore subsequent models include these commands.
Next, the remaining variables were changed independently. A complete list of
all models tested, both the input in the model and the resulting strength and deflection
values, can be found in Wurst (2013). For brevity, only the models which affected
the strength and deflection in the targeted manner (i.e., helped towards achieving the
50% decreased strength and 82% increase in ultimate deflection goals) are included;
these can be found in Table 5.7. Thus, the implicit goal in this work is that the
generalized effects of corrosion are intended to be represented rather than calibrating
the model to the specific performance of a given specimen given the ambiguities
regarding the exact properties of the relevant experimental data available in the
literature. For the sake of comparison, the base model results are also included. The
input and output values of the different corroded models are expressed as a percentage
of the base model values. For the deflection results, any value with increased
deflection from the base model is indicated in green and any value with decreased
deflection is indicated in red. For strength, any value with decreased strength from the
82
Results
Input
Table 5.7. Selected results for initial calibration of 2-D corroded models with values
expressed as a percentage of the base model.
Model #
Base
1
2
3
4
5
38
100%
100%
105%
100%
100%
Dilation Angle
ϵ 1
100%
100%
100%
100%
100%
1.12
100%
100%
100%
100%
100%
σb0/σc0
2/3
100%
100%
100%
100%
100%
Kc
0.0438
100%
75%
100%
100%
100%
A's (in2)
As (in2)
0.20563
75%
75%
100%
100%
100%
E's (ksi)
Es (ksi)
f'y (psi)
fy (psi)
f'c (psi)
f't (psi)
Ec (psi)
Strength (LPF)
Deflection (in)
Change in
Strength
29000
29000
47000
55000
3147.3
420.8
3197746
1.10
0.0435
100%
100%
100%
100%
100%
100%
100%
1.06
0.0451
100%
100%
100%
100%
100%
100%
100%
1.01
0.0430
100%
100%
100%
100%
100%
100%
100%
1.10
0.0452
100%
100%
100%
100%
100%
100%
75%
1.06
0.0501
100%
90%
100%
100%
100%
100%
100%
1.09
0.0441
N/A
-3.64%
-8.18%
0.00%
-3.64%
-0.91%
N/A
3.84%
-1.05%
4.03%
15.14%
1.40%
Change in
Deflection
base model is indicated in green, and any model with increased or unchanged strength
is indicated in red. This is done to illustrate which input variables help achieve the
goals for the corroded model.
It can be seen by the results in Table 5.7 that varying As, A’s, Ec, the dilation
angle, and Es have positive effects on the results. However, changing Es was
considered to be unrealistic and was therefore not considered in the final model. This
was not decided until the final calibration steps were being performed and therefore
83
most subsequent models will include a decrease in this value. Due to f’c, f’t, and Ec
being related through Equations 4.1 and 4.2, f’c and f’t were modified in conjunction
with the change in Ec for subsequent models according to this relationship.
Once the variables which positively affected the results were identified, they
were calibrated. The order of this calibration was based on how significantly they
affected the strength and deflection results of the model; variables which changed
these values more drastically were calibrated first. Thus, the values of Ec, f’c, and f’t
were varied to begin with, as their effect was the greatest and their values related. The
values of A’s and As were varied next, followed by the dilation angle. These
calibrations are described in Sections 5.4.3.2, 5.4.3.3, and 5.4.3.4, respectively.
5.4.3.2
Calibration of Ec, f’c, and f’t
The results of the calibration of Ec, f’c, and f’t can be seen in Table 5.8. In this
calibration process, Ec was the value which was changed and the respective f’c and f’t
values were calculated based on this. The input and output values of the different
corroded models are expressed as a percentage of the base model values. For the
deflection results, any value with increased deflection from the base model is indicated
in green and any value with decreased deflection is indicated in red. For strength, any
value with decreased strength from the base model is indicated in green, and any
model with increased or unchanged strength is indicated in red. This is done to
illustrate which input variables help achieve the goals for the corroded model. A
strength comparison between the model results and the theoretical results can be seen
in Figure 5.5. In this graph, the black line represents a 50% decrease in strength from
the base model, an LPF value of 0.55, which is the targeted strength value. A
deflection comparison between the model results and the theoretical results of can be
84
Results
Input
Table 5.8. Input and results of calibration of Ec, f’c, and f’t for corroded 2-D beam
model.
Model #
Base
6
7
8
9
10
38
100%
100%
100%
100%
100%
Dilation Angle
1
100%
100%
100%
100%
100%
ϵ
1.12
100%
100%
100%
100%
100%
σb0/σc0
2/3
100%
100%
100%
100%
100%
Kc
0.0438
100%
100%
100%
100%
100%
A's (in2)
As (in2)
0.20563
100%
100%
100%
100%
100%
E's (ksi)
Es (ksi)
f'y (psi)
fy (psi)
29000
29000
47000
55000
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
f'c (psi)
3147.3
90%
81%
56%
49%
36%
f't (psi)
Ec (psi)
420.8
3197746
95%
95%
90%
90%
75%
75%
70%
70%
60%
60%
Strength (LPF)
1.10
1.06
1.02
0.852
0.798
0.689
Deflection (in)
Change in
Strength
0.0435
0.0435
0.0435
0.0412
0.0406
0.0394
N/A
-3.64%
-7.27%
-22.5%
-27.5%
-37.4%
N/A
0.04%
0.03%
-5.32%
-6.71%
-9.26%
Change in
Deflection
seen in Figure 5.6. In this graph, the black line represents the targeted 82% increase in
ultimate deflection from the base model, represented by a value of 0.0791 in.
Using a decrease in Ec of 40%, as seen in Model 10 of Table 5.8, was chosen
as the calibrated model because this produced the largest decrease in strength.
Although this produces a decrease in deflection, it was anticipated to counteract this
change by varying the remaining parameters. A decrease larger than 40% wasn’t
85
1.20
Strength (LPF)
1.00
0.80
0.60
0.40
0.20
0.00
0
5
10
15
20
25
% Decrease in Ec
Model
30
35
40
Theoretical
Figure 5.5. Comparison of strength between results from calibration models and
theoretical values caused by varying Ec.
0.09
0.08
Deflection (in)
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
0
5
10
Model
15
20
25
% Decrease in Ec
30
35
Theoretical Uncracked
Figure 5.6. Comparison of deflection between results from calibration models and
theoretical values of the uncracked section caused by varying Ec.
86
40
utilized as it was thought that this would be unrealistic as it caused too large a change
in f’c, as a decrease in Ec of 40% causes an associated decrease in f’c of 67%. This
40% decrease in Ec and the associated decreases of 67% and 40% in f’c and f’t,
respectively, were maintained for all subsequent calibration models.
5.4.3.3
Calibration of A’s and As
The next input parameters which were varied were A’s and As; these
parameters were varied concurrently to one another. For ease of comparison, the
strength results of the variation of As and A’s are presented in Figure 5.7, Figure 5.8,
and Figure 5.9 corresponding to an A’s of 100%, 90%, and 80% of the uncorroded
model value, respectively. The remaining input variables are consistent with those
0.8
Strength (LPF)
0.7
0.6
0.5
0.4
0.3
0.2
25
35
45
55
65
75
% Decrease As
Model
Theoretical
Figure 5.7. Strength comparison of results from models and theoretical values while
varying As and using a constant A’s of 100% of the base model value.
87
0.7
0.65
Strength (LPF)
0.6
0.55
0.5
0.45
0.4
0.35
0.3
45
50
55
60
% Decrease As
65
70
Model
Theoretical
Figure 5.8. Strength comparison of results from models and theoretical values while
varying As and using a constant A’s of 90% of the base model value.
0.75
Strength (LPF)
0.65
0.55
0.45
0.35
0.25
48
53
58
63
% Decrease As
Model
Theoretical
Figure 5.9. Strength comparison of results from models and theoretical values while
varying As and using a constant A’s of 80% of the base model value.
88
used in Model 10, as shown in Table 5.8. Again, the black line represents a 50%
strength decrease from the uncorroded base model.
The model results were also evaluated in terms of the deflection values for
each model. For ease of comparison, the deflection results of the variation of As and
A’s are presented in Figure 5.10, Figure 5.11, and Figure 5.12 corresponding to an A’s
of 100%, 90%, and 80% of the base model value, respectively. The black line
represents an 82% increase in deflection from the base model. As with the strength
values, there is a convergence problem at approximately 63% decrease in As. It can
also be seen that, with all the variations of As and A’s, the deflection values were not
reaching the goal 82% increase.
0.083
Deflection (in)
0.073
0.063
0.053
0.043
0.033
0.023
40
45
50
55
60
65
70
% Decrease As
Model
Theoretical
Figure 5.10. Deflection comparison of results from models and theoretical values
while varying As and using a constant A’s of 100% of the base model
value.
89
0.083
Deflection (in)
0.073
0.063
0.053
0.043
0.033
0.023
50
55
60
65
% Decrease As
Model
Theoretical
Figure 5.11. Deflection comparison of results from models and theoretical values
while varying As and using a constant A’s of 90% of the base model
value.
0.083
Deflection (in)
0.073
0.063
0.053
0.043
0.033
0.023
50
55
60
% Decrease As
Model
Theoretical
Figure 5.12. Deflection comparison of results from models and theoretical values
while varying As and using a constant A’s of 80% of the base model
value.
90
65
As can be seen in the results of Figures 5.7 through 5.12, there is a
convergence problem with the resulting values at approximately 63% decrease in As.
This convergence issue is centered on the 50% strength loss goal and the cause of this
is unknown. The stress-strain responses of the different models were analyzed in
hopes of determining this cause. This stress-strain response can be seen in Figure
5.13, where the A’s value is constant at 80% of the original value, the percentages
reported are the percent decrease of As from the original, and the stresses are the
values reported at the integration point. In addition, Figure 5.14 shows the deflection
response compared to the loading, where A’s is again constant at 80% of the original
value and the percentages reported are the percent decrease of As from the original
value. These results indicate that the stress-strain and deflection responses of all
models were consistent. Furthermore, it is observed from Figure 5.14 that the erratic
800
700
50%
Stress (psi)
600
60%
500
62%
400
62.25%
300
62.50%
200
62.75%
100
63%
0
0
10
20
30
40
50
65%
Strain
Figure 5.13. Stress-strain response of element in 2-D beam models reported as percent
decrease in As.
91
0.7
Load (LPF)
0.6
50%
0.5
60%
0.4
62%
0.3
62.25%
0.2
62.50%
62.75%
0.1
63%
0
0
0.01
0.02
0.03
0.04
Deflection (in)
0.05
0.06
65%
Figure 5.14. Deflection response of elements in 2-D beam models reported as percent
decrease in As.
deflection results are caused by the erratic strengths rather than both parameters
varying unpredictably.
When choosing which model input to proceed with, it was decided that a
model before the convergence problems would be best; prior to the erratic spikes in
values, the model results followed a relatively linear trend. This trend is expected, as
can be seen in the theoretical results. By choosing a value prior to the convergence
problems, it was hoped that the convergence issues would be avoided when applying
the input to the full-scale bridge tests. This limited the results to not reducing As by
more than 63%. When considering the strength and deflection results of the different
models, it was found that using an A’s decrease of 20% in combination with an As
decrease of 62.5% produced the strength and deflection values closest to the goal
values. The exact input and results of this model can be seen in Table 5.9.
92
Results
Input
Table 5.9. Input and results of 2-D beam model after calibrating input parameters.
38
Dilation Angle
2
0.03504
A's (in )
2
0.07711
As (in )
5.4.3.4
1133
f'c (psi)
252.4
f't (psi)
1,918,648
Ec (psi)
0.603
Strength (LPF)
0.0485
Deflection (in)
-45.2%
Change in Strength
11.6%
Change in Deflection
Calibration of Dilation Angle
The final value which was optimized was that of the dilation angle. The exact
effects of varying this value were unknown; however, the initial variations done in
Section 5.4.3.1 indicated that changing the dilation angle increased the deflection.
Since this is the goal which was most difficult to achieve, it was thought that
calibrating this value may allow the model to achieve the strength and deflection
goals. The dilation angle was slowly increased to determine the optimum value. The
results of this variation can be seen in Table 5.10. The input and output values of the
different corroded models are expressed as a percentage of the base model values. For
the deflection results, any value with increased deflection from the base model is
indicated in green and any value with decreased deflection is indicated in red. For
strength, any value with decreased strength from the base model is indicated in green,
and any model with increased or unchanged strength is indicated in red. This is done
to illustrate which input variables help achieve the goals for the corroded model. It
can be seen that there is little variation in the strength and deflection results between
models until a dilation angle of 44 or above is used. The largest strength in
93
Results
Input
Table 5.10. Results of optimization of the dilation angle after optimizing Ec, f’c, f’t,
As, and A’s for the corroded 2-D beam model.
Dilation
38
40
41
42
43
44
45
Angle
A's (in2)
80%
80%
80%
80%
80%
80%
80%
As (in2)
37.5%
37.5%
37.5%
37.5%
37.5%
37.5%
37.5%
f'c (psi)
f't (psi)
36%
36%
36%
36%
36%
36%
36%
60%
60%
60%
60%
60%
60%
60%
Ec (psi)
60%
60%
60%
60%
60%
60%
60%
0.603
0.624
0.624
0.624
0.625
0.482
0.484
0.0485
0.0481
0.0480
0.0480
0.0480
0.0328
0.0330
Strength
(LPF)
Deflection
(in)
Change in
Strength
Change in
Deflection
-45.2% -43.3% -43.3% -43.3% -43.2% -56.2% -56.0%
11.6%
10.6%
10.5%
10.4%
10.4%
-24.7% -24.1%
conjunction with deflection change were seen when using a dilation angle of 38, the
uncorroded model value; therefore, a dilation angle of 38 was chosen as the optimal
value for the corroded model.
The exact final input which were used for the corroded model were unchanged
by this final calibration and remain the same as the values previously shown in Table
5.9. Although the final strength and deflection values weren’t exactly the goal values
of 50% decrease in strength and 82% increase in ultimate deflection, it was decided
that the input in this final version was physically possible and well-achieved the
targeted strength with a 45.2% decrease, while also being accompanied by an increase
in deflection; many models tended to fail at a lower deflection in proportion to the
lower strength. It was determined that it would be near impossible to create input for a
94
model which would produce both the desired strength and deflection changes with the
2-D modeling technique used here without utilizing input values which were outside
physical possibilities. The stress contours of this model, along with the contours of
the uncorroded model for comparison, can be seen in Figure 5.15 (b) and (a),
respectively. To help determine the validity of the model, this data is compared to the
(a)
(b)
Figure 5.15. Stress contours of (a) final uncorroded and (b) final corroded 2-D beam
models (units of in psi).
95
deflection results calculated for the uncracked and cracked sections using the same
loading applied in the model, as shown in Figure 5.16, where the FEA data includes
the loading response and the unloading response. These results were also compared to
those of the uncorroded model. It should be noted that the theoretical deflection
results are limited to the elastic regime, which is assumed to occur up to the predicted
strength of the beams; in addition, these results were calculated using the actual
loading applied in the finite element model and therefore the resulting deflections are
not the same as those compared to in Table 5.10, where the load was based on the
theoretically expected strength and thus indicated the model deflection results were
between the uncracked and cracked sections deflections. It can be seen in Figure 5.16
(which plots the loading and unloading responses of the specimens, although the
1.2
Load (LPF)
1
0.8
0.6
0.4
0.2
0
0
Uncorroded FEA
Corroded FEA
0.1
0.2
0.3
Deflection (in)
Uncorroded Uncracked
Corroded Uncracked
0.4
0.5
Uncorroded Cracked
Corroded Cracked
Figure 5.16. Load and deflection comparison between uncorroded and corroded finite
element model and theoretical calculations for the uncracked section.
96
unloading responses trace the loading response) that this is not the case. The results of
the corroded comparison to theoretical indicate that the finite element model follows
the expected deflection results relatively well when compared to the uncracked
section, diverging more as the load increases. The results of the uncorroded
comparison to the corroded follow the expected trend, with the corroded model
reaching higher deflections for the same loading values.
5.5
3-Dimensional Concrete Model
In order to model corrosion with more input variables, 3-D concrete models
were analyzed. Initially, these models were created using 2-D rebar elements.
However, previous research offers techniques for using 3-D rebar elements (Val et al.
2009; Amleh and Ghosh 2006; Fang et al. 2006) to model corrosion which can more
completely simulate all of the effects of corrosion, whereas there are limited modeling
approaches for 2-D rebar elements (Coronelli and Gambarova 2004; Dekoster et al.
2003; Kallias and Rafiq 2010). The 2-D and 3-D rebar element modeling are
described in Sections 5.5.1 and 5.5.2, respectively.
5.5.1
2-Dimensional Rebar
Initially, 2-D elements were used to define the rebar. These elements were
then embedded within the concrete using the “embedded element” command, as
described in Section 4.2.1. The compressive and tensile rebar, as well as the stirrups,
were modeled as 2-D rod elements. This process is described in the following
Sections 5.5.1.1 through 5.5.1.3, where Section 5.5.1.1 describes input values
associated with the use of the brittle cracking technique, Section 5.5.1.2 details the
97
results of the mesh sensitivity analysis which was performed, and Section 5.5.1.3
outlines the attempted calibration of the corroded model input.
5.5.1.1
Brittle Cracking Base Model Input
Initially, the brittle cracking technique was used to model the non-linear
response of the concrete. This technique was chosen as it allowed for visually
tracking cracks within the output file as elements fail and are removed from the mesh.
However, if applied to 2-D models, the elements would not reach the required failure
criteria to be removed, as all of the elements are experiencing both tension and
compression (at opposing section points) and therefore cannot reach the defined
failure criteria at all material points. For this reason, this technique was not applied to
the previous 2-D models.
The elastic and geometric input values described in Section 5.1 were used as
the uncorroded base model for the brittle cracking technique. The nonlinear input
commands were previously described in Section 4.1.2.3. For the “brittle cracking”
command, the values were defined through tabular input. An f’t value of 420.8 psi
was used, based on applying Equations 4.1 and 4.2 to the elastic Ec value, with a value
of 0.0 direct cracking strain; a value of 0.0 stress was then used with a value of 0.0027
direct cracking strain using the same logic as previously explained in Section 5.1.1.
The “brittle shear” command was also defined through tabular input; a shear retention
factor of 1 was used with a crack opening strain of 0.0, and a shear retention factor of
0.0 used with a crack opening strain of 0.0027. The “brittle failure” direct cracking
failure strain was defined as 0.0027.
98
5.5.1.2
Mesh Sensitivity Analysis
A mesh sensitivity analysis was performed for 2-D rebar elements. This brittle
cracking technique using the input described in Section 5.5.1.1 was utilized, along
with the CDP and SC techniques, described in Sections 5.4.1.2 and 5.4.1.3,
respectively. Both the concrete and rebar element sizes were varied; the mesh sizes
for both the concrete and rebar elements ranged from 3.937 in (100 mm) to 0.19685 in
(5 mm) cubes and rods, respectively. Every permutation of the concrete and rebar
sizes was analyzed.
Table 5.11 shows representative results of the rebar sensitivity in this analysis
and Table 5.12 shows representative results of the concrete element sensitivity done in
this analysis. In these tables, the percent difference which is reported is compared to
the theoretical beam strength LPF of 0.914, with negative values being below this
strength and positive values being above. It should be noted that, for the brittle
cracking technique, results were only available in LPF increments of 0.05. It was
found that the 2-D rebar element size was negligible, both in results and in
computational time; the CDP and SC results remained relatively constant and the
brittle cracking varied between 0.35 and 0.45. Due to the limited availability of LPF
increments, it was thought that this difference was minimal. In addition, the CDP and
SC models failed to converge for all rebar sizes with the exception of 100 mm. For
these reasons 3.937 in (100 mm) rods for the rebar were chosen. The results of the
concrete elements began to converge to a constant value with the brittle cracking
approach; however, the CDP and SC models failed to converge on almost all of the
mesh sizes. For this reason, more emphasis was placed on the results of the brittle
cracking models. It was found that a 0.7874 in (20 mm) concrete mesh provided
consistent results and limited computational efforts; these (0.7874 in, 20 mm) mesh
99
Table 5.11. Strength results of rebar mesh sensitivity analysis for 3D concrete beam
with 2D rebar including percent differences based on the calculated
theoretical strength values.
Rebar Size (mm)
Model Type
Highest Load (LPF) % Difference
CDP1
0.334
-63.4%
1
5
SC
0.287
-68.6%
Brittle Cracking
0.450
-50.7%
0.341
-62.7%
CDP1
10
SC1
0.287
-68.6%
Brittle Cracking
0.450
-50.7%
1
CDP
0.337
-63.1%
20
SC
0.230
-74.8%
Brittle Cracking
0.450
-50.7%
CDP1
0.339
-62.9%
1
50
SC
0.234
-74.4%
Brittle Cracking
0.400
-56.2%
CDP
0.333
-63.5%
100
SC
0.234
-74.4%
Brittle Cracking
0.350
-61.7%
1
Model terminated at peak load, indicating inability to converge.
results were also closest to the theoretical results while also providing the largest mesh
size to reduce computational effort.
A summary of this input and the results of the base model can be seen in
Table 5.13. It should be noted that the expected strength calculated in Section
5.3 was an LPF of 0.914; the base model was only 7% below this value. However, the
expected deflection is 0.2593 in. for the cracked section; the base model is 138%
higher than this value. Although this deflection value is higher than the expected, this
higher deflection is similar to that seen in the 2-D beam models described in Section
5.4.3; therefore, these input values were still used for the base model. These results
are the ones which are used for comparison of the corroded model to determine 50%
decrease in strength and 82% increase in deflection.
100
Table 5.12. Strength results of concrete mesh sensitivity analysis for 3D concrete
beam with 2D rebar including percent differences based on the calculated
theoretical strength values.
Concrete Size (mm)
Model Type
Highest Load (LPF) % Difference
CDP1
0.321
-64.8%
1
5
SC
0.321
-64.8%
Brittle Cracking
0.400
-56.2%
0.333
-63.5%
CDP1
10
SC
0.234
-74.4%
Brittle Cracking
0.350
-61.7%
CDP
0.354
-61.2%
20
SC1
0.395
-56.7%
Brittle Cracking
0.600
-34.3%
CDP1
0.323
-64.6%
1
25
SC
0.434
-52.5%
Brittle Cracking
0.450
-50.7%
CDP1
0.650
-28.8%
1
50
SC
1.56
70.8%
Brittle Cracking
1.50
64.3%
1
CDP
1.25
36.9%
100
SC1
1.67
82.9%
Brittle Cracking
2.25
146.4%
1
Model terminated at peak load, indicating inability to converge.
Table 5.13. Base model input for 3-D concrete elements with 2-D rebar elements for
brittle cracking technique.
420.8
f't (psi)
Results
Input
Direct Cracking Strain
Direct Cracking
Failure Strain
Crack Opening Strain
0.0027
0.0027
0.0027
A’s (in2)
0.0438
As (in2)
0.20563
Strength (LPF)
0.850
Deflection (in)
0.616
101
A factor not taken into consideration during this initial modeling using 2-D
rebar is the time associated with the loading command using the brittle cracking
technique. During the time this modeling was performed, it was not considered that
this would significantly affect the results. For this reason, a time of 1 second was used
for all of the models involving 2-D rebar. However, time was taken into consideration
when using 3-D rebar, and this results of this analysis are discussed in Section
5.5.2.2.2.
5.5.1.3
Corroded Model
Due to the brittle cracking technique having few input variables, modeling
corrosion was relatively straight forward. The only variables which were changed
were f’t, strain at cracking, failure strain, shear retention strain, A’s, and As. As with
the 2-D model, these values were varied independently, the effects were analyzed, and
the variations were combined to attempt to produce the targeted results. For brevity,
select results can be found in Table 5.14. It should be noted that in these models the
maximum strength is determined by visually analyzing the result files. These models
represent the results which most closely achieved the 50% strength decrease goal. For
the sake of comparison, the base model results are also included. The input and output
values of the different corroded models are expressed as a percentage of the base
model values. For the deflection results, any value with increased deflection from the
base model is indicated in green and any value with decreased deflection is indicated
in red. For strength, any value with decreased strength from the base model is
indicated in green, and any model with increased or unchanged strength is indicated in
red. This is done to illustrate which input variables help achieve the goals for the
102
Results
Input
Table 5.14. Selected results of optimization of corrosion for 3-D beam model with 2D rebar.
Model #
Base
1
2
3
4
420.8
50%
25%
50%
100%
f't (psi)
Direct Cracking
0.0027
50%
200%
50%
100%
Strain
Direct Cracking
0.0027
50%
200%
50%
100%
Failure Strain
Crack Opening
0.0027
50%
200%
200%
100%
Strain
0.04235
100%
100%
100%
50%
A's (in2)
As (in2)
Strength (LPF)
Deflection (in)
Change in Strength
Change in
Deflection
0.20574
0.425
0.616
N/A
100%
0.375
0.34674
-11.8%
100%
0.425
0.5629
0.0%
100%
0.4
0.41765
-5.9%
50%
0.225
0.11086
-47.1%
N/A
-43.7%
-8.6%
-32.2%
-82.0%
corroded model. The results from all of the calibration models can be found in Wurst
(2013).
It can be seen from Table 5.10 that this particular modeling technique was not
producing the desired deflection increases, but rather was producing decreases in
deflection. Although close strengths were reached, as close as within 5% of a 50%
strength decrease from the base model, it was thought that using a different modeling
approach would produce the desired deflection changes. For this reason, this
modeling technique was not applied to full-scale bridge models and 3-D rebar
elements were analyzed, as described in Section 5.5.2.
103
5.5.2
3-Dimensional Rebar
After determining the use of 2-D rebar elements within 3-D concrete elements
was not producing the desired results, the model was refined using 3-D elements for
rebar. This allowed for consideration of different corrosion factors, including the
effects of the creation of corrosion by-products; these by-products effect the contact
properties between the rebar and the concrete. The creation and optimization of this
model is described in Sections 5.5.2.1 and 5.5.2.2.
5.5.2.1
Mesh Sensitivity Analysis
In the mesh of 3-D concrete with 3-D rebar elements, the cross-sectional area
of the rebar limited the available sizes of the mesh. The initial mesh size used
measured 0.3937 in x 0.5249 in x 0.3937 in (10 mm x 13.33 mm x 10 mm). This size
was chosen as it provided a concrete cross-section height decrease of 0.03%, no
difference in concrete cross-section width, and a rebar area decrease of 0.5%, which
were deemed negligible, while providing a uniform mesh size throughout the model
for both the rebar and concrete elements to simplify the modeling effort. Using this
mesh size placed the centroid of the tensile rebar at approximately the same location
as if the rebar within the specimen created by Oyado et al. (2010). The vertical
distance to the centroid was almost exactly the same as the physical specimen, located
at 0.78735 in (19.995 mm) rather than 0.7874 in (20 mm) from the outer edge. The
horizontal location of the centroid was 0.59055 in (15 mm) from the edge, compared
to the 0.7874 in (20 mm) in the physical specimen. These differences were both
considered negligible, especially when considering that the strength and deflection
characteristics of the beam are theoretically not influenced by the horizontal position
of the rebar. Also, when comparing to the 2-D beam models, whose horizontal rebar
104
spacing was standardized due to input limitations, these 3-D rebar locations were more
accurate when compared to the original specimen. Considering this mesh size was
smaller than those determined through previous sensitivity analyses, and due to
computational limitations when creating smaller mesh sizes, no other mesh sizes were
analyzed.
5.5.2.2
Uncorroded Base Models Using Brittle Cracking
The commands and base values for the different input described by Amleh and
Ghosh (2006) and Val et al. (2009) were used initially. These commands are “surface
interaction”, “surface behavior”, “friction”, and “contact pair”, as described in Section
4.1.2.3. For the “surface interaction” command, a pad thickness of 0 was chosen; this
quantity represents the thickness of an interfacial layer between the contact surfaces.
For the “friction” command, exponential decay was specified, µs was set to 1, µk was
set to 0.4, and dc was set to 0.45, all of which were suggested by Amleh and Ghost
(2006). When utilizing the “surface behavior” command, exponential pressureoverclosure was specified, with c0 set to 0.01 inches because the surfaces should
constantly be in contact, and p0 set to 589 psi, as calculated by Equation 4.4. The
“contact pair” command specified the rebar as the slave surface and the concrete as the
master surface, as suggested by Amleh and Ghosh (2006). Initially, all 4 surfaces of
the rebar in contact with the concrete were used in the “contact pair” command;
however, errors were found involving the surface normals of these surfaces. It was
found that only the top and bottom surfaces of the rebar and concrete had surface
normals pointing in the correct directions and were therefore the only surfaces used; it
was unclear in the ABAQUS documentation how to redefine the surface normal
direction (Simulia 2011). It was thought that only using the top and bottom surfaces
105
would accurately model the rebar as the only loading applied to the model was in the
vertical direction, and presence of the top and bottom contact surface between the
rebar and concrete should induce frictional forces which would limit any horizontal
movement during analysis. During analysis, the horizontal displacement of the rebar
was determined to validate this theory. It was found that the horizontal displacement
was less than 0.004 inches, which is less than 10% of the vertical displacement of 0.05
inches, and thus considered to be negligible.
5.5.2.2.1
Smeared Crack and Concrete Damaged Plasticity Models
The uncorroded base model input values for the reinforced beam using the SC
and CDP techniques were described in Sections 5.4.1.3 and 5.4.1.2, respectively.
These were applied to the 3-D beam with 3-D rebar model. It was found that, when
using ABAQUS/Standard, errors regarding excessive distortion of elements during the
first loading increment resulted in the model terminating. When analyzing the
effected elements, it was found that they were all rebar elements. In order to
accommodate this problem, commands were added which linked the rebar nodes to
concrete nodes. This was done using the ABAQUS command “MPC”, which stands
for multi-point constraint. Initially, the MPC type slider was used. This command
keeps a node on a straight line defined by two other nodes, but allows the possibility
of moving along the line and allows the line to change length (Simulia 2011). The line
was defined using the concrete nodes on the elements which were in contact with the
rebar and the rebar node was confined to move along this line. In the model, the
concrete and rebar elements did not share nodes; however, the nodes were located at
coincident locations. The concrete nodes on either side of a given rebar node were the
2 used to define the line and the rebar node was limited to moving along this. This
106
was done for varying numbers of nodes per cross-section of rebar, ranging from 1
corner of the rebar to all 4. Eventually, it was determined that doing this to 2 opposite
corners provided sufficient restraint. One node on each rebar was simultaneously
linked to the concrete using the MPC type beam. This MPC type provides a rigid
beam between two nodes to constrain the displacement and rotation at the first node to
the displacement and rotation at the second node, corresponding to the presence of a
rigid beam between the two nodes (Simulia 2011). It was thought that doing this
would help prevent rotation of the rebar within the concrete. Again, the rebar node
was linked to the concrete node and constrained to move with the concrete. It was
thought that adding this extra constraint would prevent the rebar from rotating within
the beam. It was found, after running models, that the presence of this MPC beam
made no difference to the results, whereas using the MPC slider provided the required
constraint.
When running this first version of the model, warning messages were produced
stating that the plasticity/creep/connector friction algorithm did not converge. These
models also resulted in strength LPFs that were only 11% of the calculated strength.
Since no commands referencing plasticity or creep were included in the model, it was
determined that this warning was due to the presence of the “friction” command. For
this reason, in contrast to the previous approach of modeling the friction
exponentially, a constant friction value (µ) was also used in attempt to help with
convergence. The µ values which were used were 0.5, 1.0, and 1.5, as a typical value
of 1.0 for the interface between concrete and rebar was suggested (Amleh and Ghosh
2006). Select results from these models can be seen in Table 5.15; all of the models
which were tested, including both input and results, can be found in Wurst (2013).
107
Table 5.15. Results of varying friction input for 3-D concrete beam models with 3-D
rebar elements.
0.5
1.0
1.5
None
µ
Concrete
CDP
SC
CDP
SC
CDP
SC
CDP
SC
Model
Strength
0.139 0.276 0.103 0.350 0.104 0.328 0.084 0.247
(LPF)
Deflection
0.0097 0.0175 0.0071 0.0223 0.0071 0.0207 0.0068 0.0184
(in)
These results show that the strengths of these models was less than 1/3 of the
calculated strength LPF of 0.914.
To help determine the cause of the low strengths of the beams, the stress-strain
data for the beam was analyzed. The element located at the center of the bottom of the
beam was chosen, as this location is expected to have the largest tensile stress values.
The mirror element on the top of the beam was also analyzed to compare compression
and tension data. The results of the stress-strain data was then compared to the input
data. The compression concrete comparison and tensile concrete comparison can be
seen in Figure 5.17 and Figure 5.18, respectively, where the stress values reported are
for the integration points of the elements. It should be noted that none of the models
reach the nonlinear input values. It was determined that using this approach would
not produce the desired strength results while still inputting material properties which
were realistic.
5.5.2.2.2
Brittle Cracking Model
In contrast to the ABAQUS/Standard models, errors regarding excessive
distortion were not an issue when applying the brittle cracking technique; therefore, no
108
3500
Stress (psi)
3000
2500
2000
1500
1000
500
0
0.0000
0.0020
Friction 0.5
0.0040
0.0060 0.0080
Strain
Friction 1.5
0.0100
Friction 1.0
0.0120
0.0140
Input
Stress (psi)
Figure 5.17. Comparison of compression concrete output to input values for 3-D
beam with 3-D rebar.
450
400
350
300
250
200
150
100
50
0
0.0000
0.0002
Friction 0.5
0.0004
0.0006
Strain
Friction 1.5
0.0008
Friction 1.0
0.0010
0.0012
Input
Figure 5.18. Comparison of tensile concrete output to input values for 3-D beam with
3-D rebar.
109
“MPC” commands were required. However, a few of the commands defining the
contact surface between the concrete and rebar, which were described in Section
5.5.2.2, have more input values when used in conjunction with ABAQUS/Explicit.
The first of these commands is the “surface behavior” command. For this, in addition
to defining c0 and p0, as described in Section 4.2.2, a kmax value is also defined. The
default value in ABAQUS is infinity, and this value was used (Simulia 2011). In
addition to the “surface behavior” command, the “contact pair” command also used
additional parameters. In this case, a penalty contact algorithm was used by including
the parameter for mechanical constraint and setting it to penalty, as suggested by Val
et al. (2009). By employing the penalty contact algorithm, ABAQUS/Explicit uses
pure master-slave weighting, which reduces computational time. There are no extra
input values which are associated with including this parameter.
The first factor which was analyzed for the brittle cracking model was a time
comparison for the loading. It was thought that the slower the load is applied, the
more static the analysis results would be and therefore more consistent with the
physical testing on which the models and targeted performances are based. For this
reason, 1 second, 10 seconds, 60 seconds, and 300 seconds were analyzed. The results
of this analysis are shown in Figure 5.19, shown as deflection versus loading, and
including the theoretical results for the uncracked and cracked sections. It can be seen
in these results that 60 seconds and 300 seconds failed prematurely; the cause for this
is unknown. In addition, all of the results follow the theoretical uncracked section
results relatively well, until becoming nonlinear. For these reasons, both 1 second and
10 seconds were used during subsequent analyses.
110
Load (LPF)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1 Second
300 Seconds
0.05
Deflection (in)
10 Seconds
Uncracked
0.1
0.15
60 Seconds
Cracked
Figure 5.19. Time comparison for 3-D concrete beam models with 3-D rebar using
brittle cracking.
As with the CDP and SC models, warning messages were produced regarding
the plasticity/creep/connector friction algorithm not converging. Again, friction was
input as a constant value to attempt to reduce computational complexity; however, in
this case it was held constant at 1.0 as this was suggested as a typical value for the
interface between the concrete and rebar (Amleh and Ghosh 2006). It was also
considered that the pressure-overclosure may be the cause of the warning message and
variations of the model were analyzed that did not include this parameter. The results
of these variations can be seen in Table 5.16 where Exp indicates an exponential
pressure-overclosure or friction relationship. These results indicate that the strength
values are again lower than those expected through calculations; the results varied
from being 18-73% below the theoretical strength LPF of 0.914. The model which
produced the highest strength results of an LPF of 0.750 is the same model indicated
in Figure 5.19 as the 1 second model. These changes did not have a positive effect on
111
Table 5.16. Comparison of variations in friction and pressure-overclosure of 3-D
concrete beam models with 3-D rebar using brittle cracking.
PressureTime
Friction Strength
Overclosure
Used
(LPF)
(sec)
Used
1
10
Exp
N/A
Exp
N/A
Exp
N/A
Exp
N/A
Exp
Exp
1.0
1.0
Exp
Exp
1.0
1.0
0.750
0.250
0.650
0.300
0.460
0.290
0.570
0.290
the results. Specifically, removing the pressure-overclosure relationship significantly
decreased the resulting strength.
In attempts to more quantitatively determine the failure load, a different
approach was used to determine the step at which the beam failed. It was hoped that
by quantitatively determining the failure load, the results may be higher than initially
determined. A row of elements along the bottom of the beam, halfway between the
center of the beam and the loading, was used, as described in Section 5.2. The
maximum principal stress at each of these elements was determined using the output
file, and this value for each element was added together for each step. The step where
the highest stresses were seen was then considered to be the failure step. Due to the
time required to perform this more detailed analysis, this was only done for 3 models.
The models this was done for and the results can be seen in Table 5.17. It should be
noted that this new technique of determining the load at failure produced significantly
lower strength values than the previous technique. Considering that the initial strength
values were lower than the calculated values, these results were not ideal.
112
Table 5.17. Comparison of strength values utilizing different analysis techniques of 3D concrete beams with 3-D rebar using brittle cracking.
Maximum
PressureOriginal
Stress
Friction
Time
Strength
Overclosure
Strength
(sec)
Used
(LPF)
Used
(LPF)
1
Exp
Exp
0.750
0.350
10
Exp
Exp
0.460
0.330
60
Exp
Exp
0.441
0.364
After attempting many different modeling variations, it was determined that
modeling the beams using 3-D concrete elements would need further investigation and
any remaining analyses were outside the scope of this project. More complicated
modeling techniques could be analyzed, including utilizing user subroutines or other
approaches to modeling the bond between the concrete and rebar. Additional
comparison between similar models available in the literature and these models may
also reveal further insights.
5.6
Conclusions
A 2-D beam was calibrated to determine the input parameters required to
model an uncorroded and corroded concrete beam. This was done by first determining
the input parameters required by ABAQUS, then varying these parameters in attempts
to achieve a 50% decrease in strength and 82% increase in deflection. The results of
this calibration process indicated use of a model with a 20% decrease in the
compressive rebar area, 62.5% decrease in the tensile rebar area, 62% decrease in the
compressive strength of the concrete, and a 40% decrease in both the tensile strength
of the concrete and the modulus of elasticity of the concrete from the uncorroded base
model. The model utilizing this input produced a 45.2% decrease in strength and
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11.6% increase in deflection. These were the input values to be applied to the fullscale bridge models.
In addition to 2-D beam models, 3-D beam models were created. This was
done in hopes of creating a model which could take more reinforced concrete
parameters into consideration and ultimately allow for more accurate modeling of
corrosion. However, suitable input compatible with the techniques available in
ABAQUS were not able to be identified to reproduce the desired reinforced concrete
beam behavior. Difficulty was found when creating the base model, and calibrating
for corrosion was never completed. When employing the smeared crack and concrete
damaged plasticity approaches, the model terminated prematurely, with the concrete
elements not reaching a nonlinear response. When utilizing the brittle cracking
approach, a base model was created using 2-D rebar elements, but could not be
calibrated to simulate corrosion because, although the strength decrease close to 50%
was achieved, there was a corresponding deflection decrease rather than increase.
When using 3-D rebar, the strength results of the uncorroded model did not reach the
expected strength based on the experimental values or the theoretical values. The
cause of this was unknown and therefore this modeling approach was not utilized.
The 2-D beam models were straightforward to both create and calibrate. This
is because there were fewer variables to affect the results. The 2-D beam input is also
simple to apply to full-scale bridge model as these models were previously created
using 2-D deck elements. However, the material effects of corrosion within the 2-D
beams was not modeled as directly as possible; only the gross effects of corrosion in
terms of estimated strength and deflection changes are able to be considered rather
than more refined changes in the mechanics of the interaction between the concrete
114
and rebar. The final corroded model resulted in a 45.2% decrease in strength and
11.6% increase in deflection at the maximum loading when compared to the final
uncorroded model at its maximum loading. Thus, the strength goals were deemed to
be satisfied to a reasonable extent. Although the deflection increase was less than the
goal value, this difference in deflection was based on the maximum loading for each
model, while in the experimental results the corresponding change in deflection was
evaluated at the ultimate condition, which would theoretically produce a greater
discrepancy. As a simple means to evaluate this hypothesis, deflections were
compared at a common loading, using the maximum loading for the corroded beam;
the percent difference then becomes 123%, as the uncorroded beam had a deflection of
0.0213 in. and the corroded beam a deflection of 0.0477 in. Thus, the deflection
targets were also deemed to be generally satisfied.
The 3-D beam models could include more of the effects of corrosion, including
the bond deterioration at the concrete and rebar interface and the pressure caused by
the creation of corrosion by-products. However, this greater modeling complexity
resulted in the optimal method of modeling reinforced concrete using 3-D elements
not being determined. Nonetheless, it is thought that his approach has the potential to
produce more accurate results.
115
Chapter 6
BRIDGE MODELS
The techniques and input which were calibrated in Section 5.4 were applied to
full-scale bridge models. These bridge models, previously created and calibrated to
field test results in complementary research (Michaud 2011; McConnell et al. in
review; Radovic, personal communication), assume linear-elastic concrete deck
properties throughout the loading range. In the present work, the concrete material
properties were the only variations made in order to more accurately account for
potential degraded concrete deck conditions. Three bridges were utilized: Bridge 7R,
SR 1 over US 13, and SR 299 over SR 1. These are described in Sections 6.1, 6.1.2.2,
and 6.3, respectively.
6.1
7R
The first bridge which was analyzed was Bridge 7R. A 3-D model of this
bridge using ABAQUS was previously created (Ross 2007) and calibrated (Michaud
2011; McConnell et al. in review). This model was modified to include non-linear
behavior of the deck concrete, as described in Section 5.4, and was then used for
analysis. A description of the bridge geometry and location can be found in Section
6.1.1 and the results of the modeling are discussed in Section 6.1.2.
6.1.1
Bridge Information
Bridge 7R is a three span steel girder bridge with a composite concrete deck;
each of the three spans are simply supported. The main span, which was the subject
span, is 105’-3 9/16” in length with 2 approach spans of similar length. The bridge
was designed in 1961 and built soon after and consists of a four girder cross-section.
116
The plate girders have 60” x 3/8” web plates and the exterior girders have a 20” x 1”
top flange and a 20” x 1 1/4” bottom flange. The interior girders have an 18” x 7/8”
top flange, 20” x 1” bottom flange, and cover plates of 1 1/2" which were attached to
the bottom flanges of the interior girders from 32’-9” on either side of centerline; the
exterior girders use 1 7/8” plates from 32’-3” on each side of centerline (Ross 2007).
The reinforced concrete deck has a total thickness of 8”. Shear connectors are
located on the girders in order to create a composite section and transverse and
longitudinal stiffeners are also present. The bridge girders and abutments are on a 63°
skew from tangent to the supports. Cross frames are located at intermediate locations
along the length of the girders. The “K” shaped cross frames are composed of 4” x 3
1/2” x 3/8” double angles. Additional built-up I-shapes serve as end diaphragms to
connect the girders together at each end of the structure. Concrete parapets with steel
handrails and sidewalks are located along both sides of the bridge. Two concrete piers
provide vertical support (Ross 2007).
Bridge 7R served as an exit ramp for Interstate 295 North through Delaware,
just south of the Delaware Memorial Bridge. Figure 6.1 shows the location of the
bridge. When the bridge was in service, the traffic exiting the interstate continued
onto Route 13 south. The bridge carried one lane of traffic with no existing shoulders;
however, 18” concrete parapets and 30” sidewalks were found on either side of the
travel lane. Baylor Boulevard passed under the bridge and received a minimal amount
of traffic daily. The vertical clearance below the bridge during its service life was 14’6”.
117
Figure 6.1. Map of location of Bridge 7R (Ross 2007).
6.1.2
Results
The results of the finite element models were analyzed. Initially, this was done
by reviewing the results files to determine the maximum loading and the general
qualitative location of any stress concentrations within the deck. Then, in order to
express the results in terms meaningful for bridge design and rating applications,
distribution factors (DF) were calculated from the girder stress results. The strength
and stress results of the model are discussed in Section 6.1.2.1 and the DF values are
discussed in Section 6.1.2.2.
6.1.2.1
Finite Element Modeling
In the previously created model, the reinforced concrete deck already utilized
2-D elements and the “rebar” command. This model can be seen in Figure 6.2. The
118
uncorroded (Tables 5.3 and 5.6) and corroded (Table 5.9) non-linear input which was
calibrated, as described in Sections 5.4.2 and 5.4.3, was then applied directly to the
bridge deck. As a means of comparison, the bridge model was analyzed in three
conditions: without any non-linear concrete or plastic rebar commands, with
uncorroded non-linear concrete and plastic rebar commands, and with corroded nonlinear concrete and plastic rebar commands.
(a)
(b)
(c)
Figure 6.2. Finite element model of bridge 7R viewed from (a) the top, (b) the bottom,
and (c) isoparametrically.
The results of this initial analysis can be seen in Table 6.1. The general trend is
as expected; the elastic model with no non-linear concrete and rebar commands has
the highest strength, while the uncorroded non-linear model has the next highest
strength and the corroded non-linear model has the lowest strength. However, it
should be noted that the elastic model is the only model which achieved a peak load
before decreasing, whereas both the uncorroded and corroded models terminated at the
peak load due to inability to converge.
119
Table 6.1. Original capacity results for Bridge 7R.
Maximum Load
Equivalent Number of
Model
(kips)
Trucks
Elastic
2178.4
30.3
1
Uncorroded
1402.4
19.5
1
Corroded
1084.5
15.1
1
Model terminated at peak load, indicating inability to converge.
In hopes of determining the cause of the convergence problems, the results
files were analyzed visually; the 3 model versions were then compared. These results
can be seen in Figure 6.3, showing (a) the elastic model with no non-linear concrete
and rebar commands, (b) the uncorroded non-linear model, and (c) the corroded nonlinear model. In this figure, the lighter colors represent tension and the darker colors
represent compression. It can be seen that the deck changes from tension to
compression over one of the girders in all 3 models. The exact cause of this is
unknown, but it is thought that this may contribute to the inability to converge which
was experienced.
In addition, the stress-strain response of a deck element was analyzed for each
of the bridge models and compared to determine if the bridge was reaching the nonlinear stress range. The deck element which was chosen was located in a stress
concentration which was common between all the different models. The location of
this element in the elastic, uncorroded, and corroded deck is shown in Figure 6.4 (a),
(b), and (c), respectively. In this figure, the red square indicates the element which
was analyzed. The results of this stress-strain response analysis, along with those of
subsequent models, are described in detail later in the section.
In hopes of achieving convergence, the input for the CDP approach being
utilized was further researched. It was found that the model is sensitive to the tension
120
(a)
(b)
(c)
Figure 6.3. Results of bridge 7R at maximum loading for the (a) elastic version with
no non-linear concrete or rebar commands, (b) the uncorroded version,
and (c) the corroded version.
121
(a)
(b)
(c)
Figure 6.4. Representative example of location of deck element used for stress
distribution analysis in (a) the elastic, (b) uncorroded and (c) corroded
deck.
122
stiffening value; a small value for tension stiffening gives early numerical problems
and therefore could be causing the inability to converge (Baskar et al. 2002). It is also
suggested that the value for tension stiffening be calibrated separately for each
individual problem (Baskar et al. 2002). For this reason, different values of tension
stiffening were used. The tension stiffening value models the load transfer across
cracks through the rebar by defining the stress-strain response after cracking.
Originally, the tension stiffening input was defined as a curved line. However,
Baskar et al. (2002) suggested defining only 2 points; the maximum strength of the
concrete and an associated direct cracking strain of 0, followed by 0 stress and the
tension stiffening value. Baskar et al. (2002) also suggests using a value of 0.1. The
values of tension stiffening which were analyzed for the present bridge models were
0.1, 0.01, and 0.035, as seen in Figure 6.5, along with the original tension stiffening
input. It was thought that a value larger than 0.1 would be too unrealistic; the tension
stiffening value is typically around 10 times the failure tensile strain of the plain
concrete, which equates to 0.027 for the input assumed herein (Basker et al. 2002).
In addition to modifying the tension stiffening value, the tension damage
coefficient commands were removed. As stated previously, the strain values defined
in the “concrete compression hardening” and “concrete tension stiffening” commands
must correspond to the strain input in the “concrete compression damage” and
“concrete tension damage” commands, respectively. The damage coefficients
previously used were taken from literature where they were calibrated to the original
strain values in the tension stiffening input. The relationship is not well enough
understood to appropriately modify these commands for the new tension stiffening
123
Direct Stress After Cracking (psi)
800
700
600
500
400
300
200
100
0
0
0.02
Original
0.04
0.06
Direct Cracking Strain
0.01
0.035
0.08
0.1
0.1
Figure 6.5. Tension stiffening values for uncorroded model.
input. It was also previously shown in Section 5.4.3.1 that including or removing
these commands does not affect the strength results.
These tension stiffening values were applied to the bridge models and the results
of these models can be seen in Table 6.2, along with the previous results utilizing the
original tension stiffening input for comparison. Although the results of these models
produced higher strengths than with the previous tension stiffening values, all of the
corroded models reached a higher strength than their uncorroded equivalents. It
should be also noted that, for each of these models with modified tension stiffening
values, the model reached a peak load before decreasing. This indicates that
convergence is no longer an issue. Lastly, the results between the different tension
stiffening values are relatively close to one another, also indicating convergence.
124
Table 6.2. Results of bridge 7R when varying tension stiffening values.
Maximum
Equivalent
Tension
Model
Loads (kips)
Number of Trucks
Stiffening
N/A
Elastic
2178.4
30.3
1
1402.4
19.5
Uncorroded
Original
1
Corroded
1084.5
15.1
0.1
0.035
0.01
1
Uncorroded
Corroded
Uncorroded
Corroded
Uncorroded
Corroded
1795.0
1813.7
1795.0
2047.5
1888.5
2047.5
24.9
25.2
24.9
28.4
26.2
28.4
Model terminated at peak load, indicating inability to converge.
As stated previously, the stress-strain response of a deck element located in a
stress concentration was analyzed. This was done for the elastic model, as well as the
original tension stiffening values and a tension stiffening value of 0.035. The stressstrain response using the original tension stiffening and a value of 0.035 can be seen in
Figures 6.6 and 6.7, respectively, where the elastic input stress-strain response is also
shown for reference. In these graphs, the principal stress and strain values are
presented. It should be noted that all of these values are tensile stresses. These results
indicate that the stresses within the element are consistent between the uncorroded and
elastic models, and the effects of corrosion can be seen in the offset of the stress
response of the corroded model. However, contrary to the understanding of the
nonlinear input commands, the elements exhibit tensile stresses up to 7 times higher
than the input maximum tensile stress. It was expected that the element stress would
reach the maximum specified value (756.7 psi for the uncorroded and 302.7 psi for the
corroded), then the nonlinear response would enforce a decrease in stress proportional
125
7000
Stress (psi)
6000
5000
4000
3000
2000
1000
0
0
20
Elastic
40
60
80
Strain (x105)
Uncorroded
100
120
Corroded
Figure 6.6. Stress-strain response of representative deck element of Bridge 7R at a
stress concentration location using the original tension stiffening input.
7000
6000
Stress (psi)
5000
4000
3000
2000
1000
0
0
50
Elastic
100
150
Strain (x105)
Uncorroded
200
250
Corroded
Figure 6.7. Stress-strain response of element in the deck of Bridge 7R at a stress
concentration location using a tension stiffening value of 0.035.
126
to the calculated strain. The cause of this discrepancy between the input and the
results is unknown. However, it was determined that the stresses in the deck were not
the cause of the model to reach its maximum load and begin decreasing.
After this stress-strain response analysis was performed, the cause of the
bridge to begin unloading was investigated. The cause of this unloading appears to be
yielding of the web elements to which the cross-frame members are connected. These
cross-frame elements are attached to a node on the web of the girders, which connect
to four surrounding web elements. The cross-frames have no plastic commands
associated with the elements; however, as the cross-frames transfer loading to the
surrounding web elements, they begin to yield and lose the ability to redistribute load,
simulating a realistic response. For this reason, the forces within the cross-frames at
the maximum loading of each model was investigated. These forces can be seen in
Table 6.3. In each of these cases, with the exception of the models with the original
tension stiffening values which did not converge, the force in the cross-frames well
exceeds the yield stress.
Table 6.3. Peak stress values in cross-frames as maximum loading in different 7R
bridge models.
Tension Stiffening
Model
Stress (psi)
N/A
Elastic
62889
1
36000
Uncorroded
Original
Corroded1
34551
Uncorroded
43127
0.1
Corroded
73095
Uncorroded
53599
0.035
Corroded
88669
Uncorroded
59906
0.01
Corroded
88778
1
Model terminated at peak load, indicating inability to converge.
127
6.1.2.2
Distribution Factors
Another factor which was analyzed was the live load distribution factors.
These factors take into consideration how the live load is distributed, or the system
effect of bridges. Many different equations exist which estimate the DF using bridgespecific geometry and material properties, including, but not limited to: span length,
modulus of elasticity, and girder spacing. Previously, the DF for Bridge 7R was
calculated using the AASHTO equations for one lane traffic then applying the skew
reduction factors (McConnell et al. in review). These calculations were also
performed using the field test data recorded when the bridge was loaded until failure,
as described by McConnell et al. (in review). The girders are numbered G1 through
G4; this numbering can be associated with Figure 6.2 (b), where the girders are
numbered 1 through 4 from left to right. The results of the previously calculated DF
values can be seen in Table 6.4.
Table 6.4. Distribution factors of bridge 7R previously determined in research
(McConnell et al. in review).
Distribution Factor
G1
G2
G3
G4
Analysis Method
0.480
0.371
0.371
0.480
AASHTO (Elastic loading)
0.334
0.368
0.330
0.289
In-Service Field Test (Elastic loading)
0.250
0.250
0.250
0.250
Theoretical Inelastic Distribution Factor
For comparison, the DFs of the finite element models were calculated. This was
done by averaging the Mises stress in the elements along the bottom flange at the
centerline of each girder at the time of maximum loading. It was previously shown
that this technique and location are valid approaches for determining DF (Sparacino,
draft internal report, 2013). The results of these calculations can be seen in the
128
following Table 6.5. These results indicate that, as the tension stiffening value is
increased, the DF values approach those of the theoretical inelastic values. The
models with lower tension stiffening values have DF values closer to those which
were experienced during the field test. It can also be seen that, in general, the
corrosion in the deck caused a more uniform distribution between the girders than with
an uncorroded deck, which is the opposite of the predicted results. This indicates that
a more thorough analysis should be performed regarding the best technique for
calculating the DF values from the model.
Table 6.5. DF values for finite element models created of Bridge 7R.
Distribution Factor
Model
Tension Stiffening
Type
G1
G2
G3
Elastic
0.257
0.267
0.252
N/A
1
Uncorroded
0.265
0.321
0.224
Original
1
Corroded
0.260
0.314
0.223
Uncorroded
0.268
0.270
0.256
0.01
Corroded
0.260
0.262
0.252
Uncorroded
0.274
0.277
0.251
0.035
Corroded
0.260
0.262
0.252
Uncorroded
0.287
0.215
0.278
0.1
Corroded
0.272
0.274
0.253
1
G4
0.224
0.190
0.203
0.205
0.226
0.198
0.226
0.221
0.201
Model terminated at peak load, indicating inability to converge.
It should be noted that the DF results shown in Table 6.5 do not exactly follow
the expected trend. It was thought that, if corrosion decreases the load that can be
distributed through the deck, the elastic model should have the lowest DF values for
G1 and G2, followed by the uncorroded and then the corroded. However, it can be
seen that this is not the case. The results of the corroded model resulted in DF values
between those of the elastic and those of the uncorroded models. The cause of this
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was analyzed by visually reviewing the .odb model result files. Here, it was seen that
the deck of the corroded model has more widespread inelastic response at the same
loading as the corresponding uncorroded model; the uncorroded and corroded
responses are shown in Figure 6.8 (a) and (b), respectively. In this figure, the dark
grey indicates any elements with a stress higher than the input f’t value. This
nonlinear response provided greater load sharing between the girders than expected in
the corroded condition, and caused the DF values to be lower than the uncorroded
values. This greater load sharing is also evidenced when comparing the amount of
cross-frame yielding between the models; the uncorroded model shows more yielding
than the elastic, and the corroded model displays more yielding than the uncorroded.
In addition, when analyzing the yielding in the girder visually, the uncorroded girder
exhibited slightly more yielding than the uncorroded, which was almost exactly the
same as the elastic model at a comparable loading.
Thus, from this combined evaluation, it is concluded that it is logical for the
corroded deck to result in a lower distribution factor than the uncorroded model.
However, additional evaluation of the exact cause of termination and corresponding
stress distribution in each model is needed to provide a complete understanding of the
relative differences in DF values observed between the elastic, uncorroded, and
corroded models. Specifically, a plot of DF for each girder versus load throughout the
loading range may be a beneficial first step to provide insight on this topic.
6.2
SR 1 over US 13
The next bridge which was analyzed was SR 1 over US 13, subsequently
referred to as US13. A 3-D model of this bridge using ABAQUS was created and
calibrated by Ambrose (2012) and Radovic (personal communication 2013). This
130
(a)
(b)
Figure 6.8. Comparison of nonlinear response of concrete for bridge 7R for the (a)
uncorroded and (b) corroded deck.
model was modified to include non-linear behavior of the deck concrete and was used
for analysis in the same manner as Bridge 7R. A description of the bridge geometry
and location is located in Section 6.2.1 and the results are described in Section 6.2.2.
6.2.1
Bridge Information
SR 1 over US 13 is a 65 degree skew steel I-girder bridge on Delaware State
Route 1. Twin spans carry the north- and south-bound lanes. The field-tested bridge
used for model validation by Radovic (personal communication 2013) carries the
131
southbound lanes of State Route 1 over US 13 approximately 5 miles south of the
Chesapeake and Delaware Canal in Delaware, immediately south of Road 423 and just
north of Boyd’s Corner, Delaware. Figure 6.9 indicates the location of this bridge. It
consists of two continuous spans of equal (165’) lengths. There are five girders spaced
9’-6” on center with exterior girders spaced 2’-10” and 3’-10” away from the outer
edge of the bridge parapets on the west and east sides, respectively; therefore, the total
width of the bridge is 44’-8”, carrying two 12’ lanes, a 12’ shoulder on the west side,
and a 6’ shoulder on the east side, while also having parapets 1’-4” in width on each
side of the bridge (Ambrose 2012).
Figure 6.9. Satellite View of SR 1 over US 13 (Ambrose 2012).
X-type cross-frames are used to laterally brace girders of the bridge and are
spaced 20’ on center with the exception of the first cross-frame from the end and the
first cross-frame from the support which are spaced at 22’-6” on center. The crossframes consist of two 3 1/2” x 3 1/2" x 3/8” steel angles that comprise the inclined
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members of the cross-frame and a 4” x 4” x 1/2” steel angle serves as the bottom
chord. The two inclined members are bolted at their intersection by a 1/2” x 6” x 1’-1”
fill plate. All of the angles are bolted to the girders with Type 1, 7/8” diameter A325
high strength mechanically galvanized friction bolts via a 1/2” x 10” connection plate
fillet welded along the full height of the web. All structural steel is AASHTO M270
Grade 50 (specified minimum yield strength of 50,000 psi) painted with a urethane
paint. The steel girder is composite with the bridge deck (Ambrose 2012).
6.2.2
Modeling Results
As with bridge 7R, the deck was already created using 2-D elements and
utilizing the “rebar” command. This model can be seen in Figure 6.10. Again, the
uncorroded (Tables 5.3 and 5.6) and corroded (Table 5.9) non-linear input which was
calibrated, as described in Sections 5.4.2 and 5.4.3, was then applied directly to the
bridge deck. As a means of comparison, the bridge model was analyzed in three
conditions: without any non-linear concrete or plastic rebar commands, with
uncorroded non-linear concrete and plastic rebar commands, and with corroded nonlinear concrete and plastic rebar commands. The results of this initial analysis can be
seen in
Table 6.6. It can be seen in these results that the models which include nonlinear concrete and plastic rebar commands terminated prematurely. The results files
were analyzed visually in hopes of determining the cause of this early termination.
The results files indicated that inability to converge was again the cause of
termination.
133
In hopes of determining the cause of the convergence problems, the results
files were analyzed visually; the 3 model versions were then compared. These results
can be seen in Figure 6.11, showing (a) the elastic model with no non-linear concrete
(a)
(b)
(c)
Figure 6.10. Finite element model of bridge US13 viewed from (a) the top, (b) the
bottom, and (c) in cross-section.
Table 6.6. Initial results from analyzing bridge US13 including non-linear concrete
and rebar commands.
Equivalent Number of
Model
Maximum Load (kips)
Trucks
1
Elastic
3461.5
48.1
1
Uncorroded
41.2
0.6
1
Corroded
30.0
0.4
1
Model terminated at peak load, indicating inability to converge.
and rebar commands, (b) the uncorroded non-linear model, and (c) the corroded nonlinear model. In this figure, the lighter colors represent tension and the darker colors
represent compression. Due to the maximum load not reaching a high enough value,
very little information can be obtained from these results. However, when more
134
closely analyzing these models, two different elements were isolated as causing
convergence problems. The stress-strain results for these elements can be seen in
Figures 6.12 and 6.13 for compression and tension, respectively. It can be noted in the
135
(a)
(b)
(c)
Figure 6.11. Results of bridge US13 for (a) the elastic version at s similar loading as
the maximum for the uncorroded/corroded model, (b) the uncorroded
version at the maximum loading, and (c) the corroded version at the
maximum loading.
136
6000
Stress (psi)
5000
4000
3000
2000
1000
0
0
200
400
Element 1
600
800
Strain (x105)
1000
Element 2
1200
1400
Input
Figure 6.12. Compressive stress-strain response of elements in deck of US13 causing
inability to converge.
700
Stress (psi)
600
500
400
300
200
100
0
0
20
40
60
Strain (x105)
Element 1
Element 2
80
100
Input
Figure 6.13. Tensile stress-strain response of elements in deck of US13 causing
inability to converge.
137
120
tensile results that the stress-strain response does not follow the response which was
input. The cause of this is unknown, but it is thought that this difference may
contribute to the cause of termination.
As with bridge 7R, and as described in Section 6.1.2.1, the values of tension
stiffening were modified. The values which were analyzed were 0.1, 0.01, and 0.035.
Again, the tension damage coefficient commands were removed. The results of these
models can be seen in Table 6.7. Although the results of these models produced
higher strengths than with the previous tension stiffening values, as with original
tension stiffening input, these models terminated at the peak load; this again indicated
convergence being the cause of termination. The values in Table 6.7 are the
maximum loading applied which was at the terminating step of the model. For this
reason, these results were not considered to be an accurate representation of the
response of a corroded concrete deck, nor can relative strengths of the differing
models be reliably predicted from these results.
Table 6.7. Results of Bridge US13 when varying tension stiffening values.
Tension
Maximum
Equivalent
Stiffening
Model
Loads (kips)
Number of Trucks
1
Uncorroded
2049.2
28.5
0.1
1
Corroded
924.9
12.8
1
Uncorroded
1340.3
18.6
0.035
1
Corroded
947.1
13.2
1
Uncorroded
1301.5
18.1
0.01
1
Corroded
786.5
10.9
1
Model terminated at peak load, indicating inability to converge.
138
6.3
SR 299 over SR 1
The final bridge which was analyzed was SR 299 over SR 1, subsequently
referred to as SR299. A 3-D model of this bridge using ABAQUS was created and
validated by Ambrose (2012) and Radovic (personal communication 2013). This
model was modified to include non-linear behavior of the concrete deck in the same
manner as performed for the models discussed above. A description of the bridge
geometry and location is located in Section 6.3.1 and the results are described in
Section 6.3.2.
6.3.1
Bridge Information
SR 299 over SR 1 is a 32º skew (measured from tangent to the supports) steel
I-girder bridge on Delaware State Route 299 over Delaware State Route 1. It is located
in the Middletown-Odessa area of Delaware, approximately 9 miles south of the
Chesapeake and Delaware Canal in Delaware. The location of Bridge SR299 is shown
in Figure 6.14. It consists of two continuous spans, of 128’ and 134’. There are eleven
girders in the cross-section, spaced 9’-1” with exterior girders spaced 2’-11” away
from the outer edge of the bridge parapets; therefore, the total width of the bridge is
95’-11”, carrying four 12’ lanes of traffic, two 12’ outside shoulders, a 22’ median and
turning lane which varies position along the length of the bridge, and two 1’-4”
parapets (Ambrose 2012).
K-type cross-frames are used to laterally brace the girders of the bridge and are
spaced 18’-3” on center on the west span and 19’-6” on the east span, with the
exception of the first cross-frame from each support, where the spacing varies. The
typical cross-frames consist of two 3 1/2” x 3 1/2” x 3/8” steel angles that comprise
139
Figure 6.14. Satellite view of SR 299 over SR 1 (Ambrose 2012).
the inclined members of the cross-frame and one 4” x 4” x 1/2” steel angle that serves
as the bottom chord. The two steel angles of the inclined members are welded with a
5/16” fillet weld on both sides to a 1/2” gusset plate, which is also connected by a
5/16” fillet weld on both sides to the midspan of the bottom chord. All fillet welds are
at least 4” in length. All of the angles are connected with 5/16” fillet welds to 1/2”
gusset plates that are connected to the 1/2” x 7” connection plate fillet welded to the
girders along the full height of the web. All structural steel is AASHTO M270 Grade
50 (minimum specified yield strength of 50,000 psi) and is painted (Ambrose 2012).
6.3.2
Modeling Results
As with bridges 7R and US13, the deck was already created using 2-D
elements and utilizing the “rebar” command. This bridge is so large and the elements
used to create it so small, a picture of the model is not included as no details can be
discerned at the size required to fit within these margins. As with 7R and US13, the
uncorroded (Tables 5.3 and 5.6) and corroded (Table 5.9) non-linear input which was
140
calibrated, as described in Sections 5.4.2 and 5.4.3, was then applied directly to the
bridge deck. As a means of comparison, the bridge model was analyzed in three
conditions: without any non-linear concrete or plastic rebar commands, with
uncorroded non-linear concrete and plastic rebar commands, and with corroded nonlinear concrete and plastic rebar commands. The results of this initial analysis can be
seen in Table 6.8. As with Bridge US13, these models all terminate prematurely.
Again, the results files indicated that a convergence problem was the cause of
termination. In hopes of determining the cause of the convergence problems, the
results files were analyzed visually and the 3 model versions were compared.
However, as with US13, due to the maximum load not reaching a high enough value,
very little information can be obtained from these results.
Table 6.8. Initial results from analyzing Bridge US299.
Equivalent Number of
Model
Maximum Load (kips)
Trucks
1
Elastic
1157.5
16.1
1
Uncorroded
40.9
0.6
1
Corroded
26.4
0.4
1
Model terminated at peak load, indicating inability to converge.
As with bridges 7R and US13, the values of tension stiffening were modified.
The values which were analyzed were 0.1, 0.01, and 0.035. Again, the tension
damage coefficient commands were removed. The results of these models can be seen
in Table 6.9. Although the results of these models produced higher strengths than
with the previous tension stiffening values, all of the models again terminated at the
maximum load rather than decreasing and failed to converge. The only exception was
the corroded model with a tension stiffening value of 0.01, which reached the
141
maximum number of increments. Due to time constraints, this model was not able to
be run with a larger number of increments to determine the maximum load or if
convergence was again a problem. As a result of this inability to converge, the
relative accuracy of the results could not be analyzed.
Table 6.9. Results of Bridge US299 when varying tension stiffening values.
Maximum
Equivalent
Tension
Model
Stiffening
Loads (kips)
Number of Trucks
1
Uncorroded
620.3
8.6
0.1
1
Corroded
692.3
9.6
1
620.3
8.6
Uncorroded
0.035
1
Corroded
664.6
9.2
1
Uncorroded
969.2
13.5
0.01
2
Corroded
603.7
8.4
1
Model terminated at peak load, indicating inability to converge.
2
Model terminated at maximum number of analysis increments specified.
6.4
Conclusions
It can be seen in Sections 6.1 through 6.3 that the modeling of an uncorroded
and corroded deck on a full-scale finite element bridge model still needs to be refined.
Although the input using the smaller 2-D beam was calibrated, directly applying this
to the full-scale bridges did not work as desired. Many of the bridges experienced
problems with convergence, terminating prematurely. This caused the models to be
analyzed using different tension stiffening values in order to aid in convergence. In
Bridge 7R, these new tension stiffening values enabled convergence; however, the
SR299 and US13 models did not converge using the updated tension stiffening values.
Due to this lack of convergence, and because more information about the bridge was
known, more emphasis was placed on the analysis of 7R. In addition, both SR299 and
142
US13 were multiple spans and statically indeterminate; it was thought that this likely
contributed to the convergence problems which were experienced.
After modifying the tension stiffening input, the models of 7R converged,
reaching a maximum loading before decreasing. The maximum loading of these
models was determined. In addition to the strength results, the distribution factors of
the bridge models were analyzed. It was seen in these results that the models with
lower tension stiffening values have distribution factors closer to those which were
expected based on theoretical inelastic bridge behavior. It can also be seen that, in
general, the corrosion in the deck caused a more uniform distribution between the
girders than with an uncorroded deck. Consequently, the results also indicated that the
corroded models reached higher strengths than their uncorroded counterparts. It was
thought that the cause of both the strength and DF value discrepancies was due to
greater load sharing in the corroded model when compared to the corresponding
uncorroded model.
Due to time constraints, the proper modeling technique for bridges US13 and
SR299 was not able to be determined. It is hypothesized that one of the reasons for
this is that both bridges are continuous and therefore are statically indeterminate; due
to this, they contain a large region of concrete in tension over the center piers. More
effort should be placed on determining the appropriate modeling technique to allow
for convergence in these situations. The lack of results for these models prevented the
validation of the relationship between deck corrosion and the system effects to the
bridge. Although the results of 7R were able to be analyzed thoroughly, more models
are needed for comparison to validate whether the trends observed in these results are
indicative of a general phenomenon. In addition, it is suggested that more work be
143
performed to determine a relationship between the current rating systems used for
inspecting bridge decks and how these ratings may correlate to the corrosion variables
input into these analyses. Determining this information on a general basis would
allow the relationship between deck corrosion and system capacity to be employed
when evaluating structurally deficient bridges, which could ultimately serve as a
means for prioritizing bridges for repairs and rehabilitation.
144
Chapter 7
RELATIONSHIPS TO BRIDGE RATING PRACTICE
This chapter discusses how the research above can be related to the typical
procedures and quantities used in typical bridge rating practice. First, Sections 7.1 and
7.2 discuss how the FEA results can be synthesized to determine quantities needed for
or could that be beneficial to this process. Specifically, Section 7.1 discusses how the
FEA is used to quantify the system-level strength. This is not a quantity currently
used in the rating process, but this research confirms that use of this quantity has the
potential to greatly improve the accuracy and economy of the rating process. Then,
Section 7.2 discusses how the FEA is used to determine member-level distribution
factors synonymous with the current rating process in order to compare and contrast to
the standard calculations. Sections 7.3 and 7.4 discuss these same quantities (systemlevel strength and member-level distribution factors, respectively) from the
perspective of how the rating process could be revised to include these quantities. The
chapter concludes with a discussion of how the information contained in standard
bridge inspection reports could be used as part of the proposed rating process in
Section 7.5.
7.1
FEA-Determined System-Level Strength
The final models used in this research were analyzed using the Modified Riks
analysis algorithm available in Abaqus Standard. This method can be considered as a
combination of a displacement control procedure, the arc-length method, and Newton
iteration techniques. This method is ideally suited to the present work due to its
capabilities of capturing the loading response up to a peak loading as well as trace the
145
unloading portion of the non-linear equilibrium path. Thus the system-level strength
is directly interpreted as the peak loading that was achieved in the FEA result, prior to
any unloading. This loading is directly output by the analysis through reporting the
load proportionality factor (LPF) at each displacement in the analysis. These
quantities are then multiplied by the loading specified in the input file to determine the
total loading. Ratios between this loading and the load of an HS-20 truck are then
used to quantify the loading in terms of design / rating vehicles.
This approach has the clear advantage of conveniently representing the exact
quantity of interest, system-level capacity. However, it should be noted that these
strengths are taken directly from the model without any safety factors (e.g., resistance
factors) applied. Future work is necessary in order to inform appropriate values for
system-level resistance factors.
The reported system-level strength is associated with a specific limit state, or
cause of failure. This is determined by first using visual inspection of changing stress
contours as the structure reaches its peak loading and then unloads to determine the
regions of the model that display the most dramatic changes and then numerically
analyzing the stress results relative to the input and load redistribution mechanisms to
determine which elements are unloading and why. For example, in the results of the
Bridge 7R model with an elastic deck, the limit state is excessive yielding at the crossframe connection locations, which was revealed because it was observed that these
elements show localized yielding and by realizing that with these elements yielded, the
cross-frame load redistribution mechanisms that had previously existed were lost.
146
7.2
7.2.1
FEA-Determined Member-Level Distribution Factors
Procedure
The DF from the FEA models is calculated based on the peak stresses in each
girder in the given model. These will always occur in one of the two flanges, and
visualization of the stress contours enables the longitudinal location where this occurs
to be generally identified, which is near midspan of each girder. To facilitate this
visualization, different visualizations were created for each girder where the contours
in each of these were manually adjusted to have non-uniform increments such that
relatively narrow increments were used for values approaching the peak stress in the
respective girder. Even with this effort, because there are multiple elements in the
cross-section of each flange and because the girders are subject to lateral bending as
well as vertical bending, it is not always completely obvious which flange crosssection contains the highest average stresses. Thus, queries of multiple cross-sections
of each girder were used and the stresses in each element of the flange cross-section
averaged to determine the flange cross-section with the highest peak stresses. Because
the DF is simply a quantity that approximates what percentage of the total live load is
transferred to each girder, computing the DF for each girder consisted of summing up
the average stress in each girder, and dividing each average stress by that total average
stress of all girders in the structure. Alternatively, averaging the bottom flange
stresses at midspan of each girder and dividing by this sum was investigated as a more
efficient means for calculating distribution factors. For the subject bridge for which
this was evaluated (Bridge 7R) it was found that the discrepancy in DF when using
this process was 2% or less (Sparacino 2013). Given the dramatic time savings
147
afforded by this process, this process simpler process was used in the results presented
herein.
The type of stress output should also be considered. AASHTO methods are
implicitly based on flexural bending of the girders, which would analytically be
expressed by the stress component in the longitudinal direction of the girders.
However, due to the much more refined nature of FEA, the complete stress state in the
models is readily available. Thus, the stresses used to quantify DF in this work are
based on the Mises stresses (an Abaqus quantity based on the von Mises yield
criterion), which are a better indication of whether yielding has or has not occurred in
the steel elements of the model due to all stress components being represented in this
quantity. Preliminary comparisons with calculations based only on flexural bending
stresses indicate that there is not a significant difference in the values resulting from
these different stress outputs.
DF used in practice are based on approximating a worst-case loading condition
for each girder and approximating the percentage of load carried by the respective
girder for that loading condition. This generally involves varying the transverse
position of the loading such that the design or rating vehicle is as close to the girder of
interest as physically possible. However, due to the equal girder spacings used in the
bridges considered in this work, it was possible in some cases to minimize the number
of transverse loading positions considered. For example, for Bridge 7R the loading
was positioned to cause a worst-case loading for Girders 1 and 2. Then due to
symmetry about the vertical axis of the bridge cross-section, it was assumed that the
DF of Girder 2 would equal the DF for Girder 3 and the DF of Girder 1 would equal
the DF of Girder 4 with an acceptable level of accuracy.
148
At low levels of load, the analytical DF should be consistent with DF values
used in practice. As the load increases into the inelastic regime, the DF values should
decrease as greater load-sharing between the girders begins. As an extreme example,
if the stress was entirely uniform between the girders, then each DF would be
equivalent to a value of 1 divided by the number of girders, and this quantity
represents an ideal inelastic distribution factor where sufficient ductility and transverse
load distribution mechanisms exist in order to fully exhaust the girder system-level
reserve capacity. Thus, the DF’s were calculated at several different intervals in the
loading sequence: (1) a load equivalent to one AASHTO HS-20 design / rating vehicle
(which is another means to quantify DF using the same philosophy used in practice)
and (2) at the ultimate loading (as discussed in Section 7.1) achieved in the model,
which can be compared with the ideal inelastic distribution factors to assess the extent
to which ideal transverse load redistribution mechanisms exist.
In preliminary work, additional load levels were also investigated. This was
used to remedy the fact that the undamaged model did not reach its system capacity
during the analysis. Instead, the software terminated the analysis at the first instance it
detected that a single concrete element had entered a non-linear stress state. Because
the DF values that were going to be compared between these two models had to occur
at a comparable load interval, the DF was recalculated for the damaged model at an
interval determined to be the load step at which it experienced its first concrete
element non-linearity. The idea was to compare these DF values as they both occurred
at a comparable point in the loading. However, because of the weakening
modifications made to the inputs in the damaged model, it reached the point of
concrete non-linearity much earlier in the loading than the undamaged model did.
149
This resulted in an undamaged DF which was higher than the damaged DF, which was
an inconclusive result. The last attempt to find comparable values was to calculate the
DF at the same exact load step for both models. The damaged model reached concrete
non-linearity at increment 36 in the loading, and because this increment also occurred
before concrete non-linearity in the undamaged model, this comparison revealed more
logical results.
In all cases, there were three unique models for which the DF was calculated;
an elastic model, an undamaged model, and a damaged model. The elastic model
represented a linear material model for concrete, where only the elastic modulus is
input and the material is assumed to have infinite strength (although the actual
concrete stresses present in these models are on par with typical concrete strengths).
The undamaged and damaged models were designed to exhibit realistic non-linear
behavior. The key difference between the latter two is that the damaged model
reduced strength, meant to simulate the effects of 25 years of aging as discussed in
Chapters 3 and 5.
7.2.2
Summary of Results
Distribution factor results for Bridge 7R, the model for which complete
information was obtained, were previously discussed in Section 6.1.2.2. These results
are discussed in the specific context of their relationship to bridge rating practice
below. The relevant data is first summarized in Table 7.1, where the first three data
lines of the table represent alternative means of calculating DF using the consistent
philosophy of elastic level DF and the last four lines of the table represent inelastic DF
calculated for various differing assumptions.
150
Specifically, the first data line is the DF calculated using the AASHTO
equations for one lane of traffic (which is most appropriate for this single-lane
structure and specifically using AASHTO Tables 4.6.2.2.2b-1 and 4.6.2.2.2d-1 for
interior and exterior girders, respectively) and then applying the skew reduction
factors from AASHTO Table 4.6.2.2.2e-1. The resulting values for interior girders
compare relatively well to those observed during the in-service field test and the FEA
simulation of the same (performed by Ross 2007; note a distribution cannot be
calculated from the decommissioned test because of the lack of data for G1).
However, the field and FEA results show much greater load-sharing between girders
for the exterior girders, where the AASHTO prediction (governed by the lever rule)
over-predicts the girder stress by 40% for G1; given McConnell et al’s (2013) prior
observations on the performance of G4, less emphasis is placed on these results here.
The inelastic results should theoretically be, and are, much less than the DF
based on elastic behavior. Note only the G1 and G2 results are shown here for the
reasons previously discussed in Section 7.2.1. Comparing the ultimate loading results
for the three differing models of concrete behavior, it was expected that the elastic
model would come the closest to achieving the theoretical ideal values of inelastic
distribution factors, followed by the uncorroded model and then the corroded model.
In contrast, it was found that all three of these models came within 11% of the
theoretical ideal (for the results shown in Table 7.1, which are taken from the results
shown in Table 6.5 for the models with the governing tension stiffening value of 0.035
for the reasons explained previously; alternative models did not yield dramatically
different results). Furthermore, the model with the largest discrepancy was the
151
Table 7.1. Summary of Bridge 7R Distribution Factor Calculations
G1
Analysis Method
AASHTO
(Elastic loading)
In-Service Field Test
(Elastic loading)
FEA Simulation of In-Service Field Test
(Elastic Loading)
Elastic Concrete Model
(Ultimate loading)
Uncorroded Concrete Model
(Ultimate Loading)
Corroded Concrete Model
(Ultimate Loading)
Theoretical Inelastic Distribution Factor
Distribution Factor
G2
G3
G4
0.480
0.371
0.371
0.480
0.334
0.368
0.330
0.289
0.353
0.377
0.361
0.405
0.259
0.260
---
---
0.274
0.277
---
---
0.260
0.262
---
---
0.250
0.250
0.250
0.250
uncorroded model in contrast to the expectation that the corroded model would fill this
role.
The cause of this discrepancy between the anticipated and actual performance
of the model containing the corroded concrete material model was analyzed by
visualizing the stress results from the models. Here, it was seen that the deck of the
corroded model had a more widespread inelastic response at the same loading as the
corresponding uncorroded model (see Fig. 6.8). This nonlinear response provided
greater load sharing between the girders with the corroded concrete deck than in the
model with the uncorroded concrete deck, and caused the DF values to be lower than
the uncorroded values. This greater load sharing is also evidenced when comparing
the amount of cross-frame yielding between the models; the uncorroded model shows
more yielding than the elastic, and the corroded model displays more yielding than the
uncorroded. In addition, when analyzing the yielding in the girder visually, the
152
uncorroded girder exhibited slightly more yielding than the uncorroded, which was
almost exactly the same as the elastic model at a comparable loading. Thus, in
general, the corrosion in the deck caused a more uniform distribution between the
girders than with an uncorroded deck.
Thus, from this combined evaluation, it is concluded that it is logical for the
corroded deck to result in a lower distribution factor than the uncorroded model.
Additional evaluation of the exact cause of termination and corresponding stress
distribution in each model could also reveal additional understanding of the relative
differences in DF values observed between the elastic, uncorroded, and corroded
models. Specifically, a plot of DF for each girder versus load throughout the loading
range provide further insight on this topic and also begin to consider these results in a
format where factors of safety could be considered as it is not practical to base the
results on the absolute ultimate loading condition.
7.3
System-Level Rating Calculations
Two alternative methodologies for including system-level behavior into the
rating (or design) process are discussed in this section and in Section 7.4.1. The first
of these is the more significant departure from current practice, but affords greater
opportunities for improved accuracy through more comprehensively addressing
system-level behavior through quantifying system-level capacities. This is the subject
of the present section. Section 7.4.1 will discuss maintaining the current practice of
analyzing individual members, but better accounting for system-level behavior in this
process through adopting inelastic distribution factors in contrast to the present elastic
distribution factors.
153
Assuming adequate transverse load redistribution mechanisms and ductility
exist, the system capacity of a highway bridge can be quantified as the summation of
the girder strengths comprising the structure. This relationship is expressed below by
Equation 7-1.
n
M n  System 
 M nGirder i
i 1
[7-1]
The strength observed in field and FEA results discussed in Chapter 6 can be
compared to the AASHTO predictions for girder strength and to the system strength
estimated by Eq. 7-1, using AASHTO girder strengths for the quantities on the right
side of this equation. First, a difference exists between the AASHTO predictions for
girder strength and the observed results for girder strength. Per AASHTO (2013a)
specifications, the moment capacity of these slender-webbed girders is My, which
corresponds to an applied load of 15 HS-20 vehicles in the loading conditions
considered herein when using the AASHTO distribution factor from Table 7.1.
However, yielding was not observed in any of the girders in the decommissioned field
test, which applied an equivalent of 17 HS-20 vehicles. The FEA results do not
indicate flexural yielding at any point in the girders until a load of 19 HS-20 vehicles
and do not indicate the full yielding across a full-cross section of the bottom flange
(consistent with the stress condition corresponding to My) until a load of 20 HS-20
vehicles. Thus, even if the strength of the bridge is rather modestly limited to the
yielding of the cross-section of one flange, the actual strength of the subject bridge
would be 30% greater than the AASHTO prediction.
Equation 7-1, the summation of the strength of all girders in the cross-section,
results in a (unfactored) capacity of 25 HS-20 vehicles for the subject bridge. It was
originally envisioned that this calculation may produce a simple upper-bound of
154
system capacity, but the weakest of the three models (the uncorroded model, see Table
6.2) achieved this strength and the other two models exceeded this strength, with the
corroded model exceeding the prediction from Eq. 7-1 by 14%. By looking at the
distribution of stress in the FEA, it is clear that in addition to the discrepancies in
individual girder capacity discussed above this is due to the fact that the girders
display significant strength above My, with two of the four girders having final stress
distributions more analogous to the full plastic moment (Mp).
If the above calculations on theoretical upper-bound system strength are
repeated based on girder strengths equal to Mp instead of My, the system strength is 32
HS-20 vehicles based on the loading configuration that was considered. The fact that
the observed strengths of 25 to 30 HS-20 vehicles is between these two theoretical
estimates based on Mp and My is logical since at the ultimate system capacity of the
model all of the girders have a stress state beyond My, but not all girders have
achieved yielding throughout a full cross-section (corresponding to Mp) due to the lack
of ability to transfer forces through the cross-frames that eventually manifested at this
high level of loading (as previously discussed in Chapter 6). Thus, it is suggested that
the summation of girder strengths is a reasonable estimate of a system capacity,
although future research is needed to investigate potential reductions to this capacity
(which could be expressed in specifications as additional limit states) that are needed
to account for the possibility of transverse load redistribution mechanisms being
insufficient for redistributing forces between girders in order for the idealized system
capacity to be achieved.
155
In considering the transition from the expression of system strength given by
Eq. 7-1 to a format complementary to AASHTO LRFR procedures, the current LRFR
rating equation can be expressed in conceptual terms by Equation 7-2.
RF 
Capacity  Dead Load Demand
Live Load Demand
[7-2]
Thus, the rating factor (RF) conceptually represents the multiples of the design live
loading that can be safely applied, where the target value is a quantity greater than 1.
The connection between Eqs. 7-1 and 7-2 lies in the live load demand aspect of Eq. 72. This could be calculated for the structure or using revised DF to calculate the RF
for individual members, which is discussed in greater detail in Section 7.4.2).
7.4
Member-Level Distribution Factors
In addition to the FEA-based determination of member level distribution factors
which were discussed in Section 7.2, an investigation into revising the existing elastic
distribution factors using theoretical methods to account for deicing agent-induced
deterioration of reinforced concrete bridge decks was also considered. The findings of
both of these approaches are summarized herein.
7.4.1
Inelastic Member-Level Distribution Factors
Once quantified, inelastic member-level distribution factors could be readily
adopted into the existing rating process. These factors could simply replace the
present DF values. To fully quantify inelastic DF for a broad range of practical
scenarios (which include the effects of reinforced concrete deterioration as well as
numerous other factors) requires additional effort, which is planned to be completed in
the course of the PI’s pending research funded by the FHWA.
156
7.4.2
Elastic Member-Level Distribution Factors Considering Aging
To better understand the general effects of deterioration of reinforced concrete
bridge decks, irrespective of system behavior and in a format grounded in basic
theoretical assumptions in contrast to FEA, a study was performed to evaluate how the
changes in strength and ductility of reinforced concrete that were quantified in Chapter
3 would affect DF using the current elastic philosophy for determining DF. Because
deterioration causes reduced strength of the bridge deck, it would be expected that
higher DF would result due to deterioration. However, there are also changes in
ductility, which make the ultimate effect of deterioration on the DF values uncertain
using intuition alone (similar to the results of the FEA results discussed in Section
7.2). As a result, a theoretical approach described below was used to investigate this
issue and perhaps provide a conceptual basis for the FEA trends discussed above.
7.4.2.1
Background
From reviewing publications such as “Distribution of Wheel Loads on Highway
Bridges”(Nutt et al, 1988), “An Approach to Evaluating the Influences of Aging on
the System Capacity of Steel I-Girder Bridges”(McConnell et al, 2011), and the
current AASHTO design codes (AASHTO, 2005), it was discerned that it was
possible to develop an equation that could demonstrate how distribution factors
changed with time, which could incorporate the effects of exposure to de-icing agents.
The method for calculating a DF is not universally agreed upon, thus the existence of a
multitude of empirical and semi-empirical equations. With the idea of accounting for
the worst case scenario, a DF equation that could vary the greatest with time was
desired as a starting point. Thus it was first necessary to compile a list of possible
equations, from the publications listed above and from the Ontario Highway Bridge
157
Design Code (OHBDC). Table 7.2 summarizes the options as well as the variables
contained in each.
The last equation contained in this chart seemed like a viable option, but because
D was calculated by hand using a chart and not an explicit equation, it could not be
used to quickly iterate values of DF as a function of time and was excluded from
subsequent calculations. The labels were assigned based on the study or publication
they were presented in, and are strictly for convenience purposes.
The next step was to perform a sensitivity study on the variables listed
separately above, to identify which variables vary the greatest with time, and therefore
which equations vary the greatest.
Table 7.2.
Possible DF Equations for Analysis (see Appendix A for variable
definitions).
Possible Equations Variables Source Label 0.075+((S/9.5)^(0.6))*((S/L)^(0.2))* S, L, ts, EB, AASHTO AASHTO ((Kg/(12*L*(ts^3)))^(0.1)) ED, IB, A, eg Design Code 0.1+((S/3)^(0.6))*((S/L)^(0.2))* S, L, ts, EB, Summary of Summary (((EB/ED)*(I+A*(eg)^2)/(L*(ts^3)))^(0.1) ED, IB, A, eg “Distribution of Wheel Loads on Highway Bridges” S/(4.6+(0.04*L)/(EB*IB/(L*ED*ID))) S, L, EB, ED, “Distribution of Newmark
IB, ID Wheel Loads on Highway Bridges” S/((0.1538+S/150)*(L/SQRT((IB*EB/ S, L, ts, EB, “Distribution of Illinois (L*(ED*(S^3)/(12*(1‐
ED, IB, v Wheel Loads v^2)))))))+4.26+(S/30)) on Highway Bridges” S/D, where D=f(Dx, Dy) S, L, ts, b, OHBDC OHBDC and Di=Ei*ts^3/(12*(1‐vi*vj)) ED, vx, vy 158
7.4.2.2
Sensitivity Study
From “An Approach to Evaluating the Influences of Aging on the System
Capacity of Steel I-Girder Bridges” I obtained two equations which quantified the
effect of time on the strength and deflection of a bridge deck, which are reproduced
below along with the domains listed next to these equations are necessary to bound the
percent loss (or increase) between the physical limits of 0 and 100%. Any percentage
values outside of this range were eliminated by this domain in an attempt to avoid any
algebraic anomalies that might occur in subsequent calculations.
Strength Loss (%) = 2.0002*number of years, for 0 < t < 50 years
[7-3]
Deflection Increase (%) = 2.0347*number of years + 31.559,
for 0 < t < 33 years
[7-4]
Having equations that varied with time, elastic beam theory and other solid
mechanics concepts were used to develop equations that integrate the above effects of
time on strength and deflection into the material properties of the bridge deck that
were thought to significantly vary with a change in strength and/or deflection. These
were the elastic modulus, area moment of inertia, and Poisson’s ratio. The derivations
for the equations containing these three properties as a function of time are contained
in hand calculations which can be found in Appendix A.
Having these equations, the other properties needed to compute DFs were taken
from Bridge 7R as a relevant example. Using the geometry from this bridge, the
values of each material property were computed in 1 year increments, and plotted each
against time, which are shown in Fig. 7.1. Each graph is plotted on its full domain (050 for equations based off of strength loss, and 0-34 for equations based on deflection
increase as discussed above) but for consistency all percent change calculations were
conducted at 33 years, a time value inside of both domains.
159
Elastic modulus (psi)
4000000
3000000
2000000
1000000
0
Area moment of Inertia (in^4)
0
5 10 15 20 25 30 35 40 45
Time (Years)
4200
4000
3800
3600
3400
3200
Poisson's Ratio
0
4
8
12 16 20
Time (Years)
24
28
0.2
0.198
0.196
0.194
0.192
0 3 6 9 12 15 18 21 24 27 30
Time (years)
Figure 7.15. Change in material properties with respect to bridge age.
Summarizing the percent changes observed in Fig 7.1, it can be seen in Table 7.3
that the elastic modulus varies the most with time, followed by the area moment of
inertia. This is likely because the equation for elastic modulus (as presented in the
hand calculations) relies directly on strength (which varies directly with time), in the
presence of no other variables, while the equation for moment of inertia contains many
other parameters. The presence of these other parameters not present in the
160
Table 7.3.
Percent change in properties after 33 years
Elastic Modulus
Moment of inertia
Poisson’s Ratio
38.36%
9.65%
1.49%
elastic modulus equation likely lessens the effect that time has on the output of that
particular equation. Poisson’s ratio shows very small amounts of variability, and as
confirmed by the studies contained in “Development of Design Criteria for Simply
Supported Skew Slab-and-Girder Bridges”, it is appropriate to fix this parameter at a
value of 0.20 for concrete.
Relating the sensitivity of these variables back to the DF equations, Table 7.4
makes note of the variables affected by time and orders the equations from most to
least affected by the aging process. Containing the two variables that were most
affected by time, the Newmark DF equation was determined to be the most sensitive.
Table 7.4.
Ranking of DF equations theoretically affected by age based on
sensitivity of affected variables.
Equation
Affected
Name
Variables
Newmark
ED, ID
Illinois
ED, v
Summary
ED
AASHTO
ED
161
7.4.2.3
Results
With the Newmark equation selected for further evaluation, the material property
equations that were based on time from the sensitivity study were then substituted into
the Newmark DF equation. This resulted in a DF equation that varied with time, the
initial goal of this effort. There were other values such as the geometry of the bridge
and initial material properties that were needed to evaluate this equation, so values
from seven different Delaware bridges were used to fill this need (where for the
reader’s reference the bridge labeled as “Michaud” in the figures below corresponds
with the bridge referred to as Bridge 7R elsewhere in this report). With all of the
necessary values now available, Newmark’s DF was plotted versus time for these
seven bridges in Fig. 7.2. This allows general trends as well as sensitivity to other
bridge parameters to be evaluated.
1.4
1.3
1.2
DF
1.1
Michaud
1
SR299/SR1
Mall Rd.
0.9
McDonough
SR1/SR273
0.8
SR1/US13
SR1/US14_B
0.7
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 32
time (years)
Figure 7.16. Modified Newmark DF equation vs. time for selected bridges
162
From Fig. 7.2 it is shown that the Newmark equation was fairly dependent on
the geometric inputs of the specific bridge. This is also true for all other DF
equations, which are plotted in Figs. 7.3 through Fig. 7.5.
7.4.2.4
Discussion
It is noted that the DF values reported in the figures above are quite high, most
above 1. These are very conservative values. This is attributed to the fact that the
majority of these equations are quite dated. More recent equations such as AASHTO
have been further optimized with more information on bridge behavior, and produce
values that are still conservative but much closer to realistic values. This
consideration was the basis for expanding the analysis beyond simply the equation
which was most sensitive to the influences of time (the modified Newmark equation)
as originally planned. This would generate values to represent the worst-case-scenario
0.9
Michaud
SR299/SR1
0.85
Mall Rd.
McDonough
0.8
SR1/SR273
DF
SR1/US13
SR1/US13_B
0.75
0.7
0.65
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48
Time (Years)
Figure 7.3.
Modified AASHTO DF equation vs. time for selected bridges
163
1.7
1.65
1.6
1.55
DF
1.5
Michaud
SR299/SR1
Mall Rd.
McDonough
SR1/SR273
SR1/US13
SR1/US13_B
1.45
1.4
1.35
1.3
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48
Time (Years)
Figure 7.4.
Modified summary DF equation vs. time for selected bridges
1.7
Michaud
1.6
SR299/SR1
Mall Rd.
1.5
McDonough
DF
SR1/SR273
1.4
SR1/US13
SR1/US14_B
1.3
1.2
1.1
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48
Time (Years)
Figure 7.5.
Modified Illinois DF equation vs. time for selected bridges
164
as indicated by the sensitivity study, as well as more theoretically accurate values
which would be more acceptable for modern design considerations.
With the Newmark DF equation depending on both the elastic modulus and
moment of inertia of the concrete deck (which both change with time based on the
calculations in Appendix A) it turns out that the modified Newmark DF equation only
depends on percent increase in maximum deflection over time, and not percent
decrease in strength. This is due to the inverse relationship between E and I in the
deflection equation (as seen in the hand calculations); with E residing in the
denominator of the equation for I, multiplying E and I eliminates E from the resulting
product. Because of this, the modified Newmark trend lines adopt only the qualities
of the increase in deflection curve, such as the bi-linear relationship with time and the
0-33 year domain. The sensitivity study of the DF equations showed that this
particular equation would change the most with time, and from comparing the function
values on 0-33 of this graph with that of the others in Table 7.5, it can be shown again
that this equation is indeed most sensitive to time. Because of the turbulent nature of
many of these equations at times approaching 50 years, and because of the domain
documented in the report that generated the time equations, it was decided to place
greater emphasis on the data from the domain of 0 to 25 years.
It is noted that the results of the AASHTO and Summary equations are nearly
identical, which is a result of almost identical equations. While some of the
coefficients and exponents are slightly different, the structure of both equations (as
shown in Table 7.2) is identical and both rely only on elastic modulus with time. The
equation spikes as time approaches 50 years because at that value of t the term inside
165
Table 7.5.
Percent Increase in DF from 0 to 25 Years
Bridge
Equation
AASHTO
Summary
Illinois
Newmark
Michaud
3.14%
3.26%
5.25%
24.27%
SR299/SR1
3.14%
3.26%
6.80%
34.76%
Mall Rd.
3.16%
3.28%
7.91%
39.65%
McDonough
3.15%
3.27%
7.20%
33.12%
SR1/SR273
3.16%
3.28%
7.55%
40.39%
SR1/US13
3.17%
3.28%
6.88%
30.68%
SR1/US13_B
3.15%
3.27%
7.18%
31.81%
the radical approaches 0. The domain for these graphs ends at 50 years because
anything greater will result in imaginary numbers.
The Illinois equation is a function of the elastic modulus and Poisson’s ratio of
the concrete deck. From the sensitivity study (and other publications referenced in the
Illinois study) it was shown that the change in Poisson’s ratio with time is negligible,
and with typical values for reinforced concrete ranging from 0.15 to 0.20, Poisson’s
ratio was conservatively selected to be a constant of 0.20. Because the only timedependent parameter in the modified Illinois DF equation is then elastic modulus,
these trend lines assume shapes similar to that of the AASHTO curves. However,
there are two distinct differences, the first of which is the overall increased slope of
this curve. This is because this equation contains an additional square root that the
AASHTO equation does not, making the Illinois equation vary with time to a degree
166
that is (approximately) twice as large. The second distinct difference is that the
individual curves all seem to have different slopes, where the curves in the AASHTO
equation are all vertical translations of one another. This is likely because the term in
the above equation that changes with time is being multiplied by many other input
parameters. Since the term that is varying with time determines the slope, it is logical
that the wide range of input parameters being multiplied by this term will result in
curves with unique slopes. The AASHTO equation has few input parameters being
multiplied by the time term, and most of these few do not change greatly from case to
case, resulting in individual curves that are nearly vertical translations of each other.
7.4.2.5
Conclusions
By analyzing the percent changes in Table 7.5, one can see at a quick glance
the objective effect of time (which is representing aging, weathering, and de-icing
agent exposure) on the DF of a series of bridges through various equations. The first
fact to point out is that the data above is in complete agreement with the sensitivity
study previously conducted, in that the Newmark equation experiences the greatest
changes over time, followed by Illinois, Summary, and AASHTO. The Newmark
equation shows particular variability as explained above because of its dependence on
elastic modulus and moment of inertia, while the more recent AASHTO equations
depend only on elastic modulus and not moment of inertia of the deck, which makes
the latter much less susceptible to changes over time and in initial geometry, as
represented by the lack of fluctuation in percent change between bridges.
As described above, the motive behind this study was to aid in the research on
system level analysis of highway bridge decks. A distribution factor is one of the
167
main parameters that assesses a bridge’s ability to transfer load, and therefore be
analyzed as a system and not a collection of individual elements.
If the value of a bridge deck’s DF becomes too large, that is an indicator that the
bridge does not effectively transfer load between its girders, and therefore behaves less
as a system. With this in mind, the graphs presented in this study provide an efficient
method for determining the extent to which a bridge can be analyzed as a system,
based on the increase of its DF due to the aging process. By knowing the age,
geometry, and material properties of a particular bridge, this data could be used as a
tool during the inspection process to evaluate a bridges’ current DF, and therefore its
ability to transfer load and the effectiveness of system level analysis. However, due to
the numerical issues that arise as the above equations approach their bounds,
additional research would be needed to provide practical quantities for these values.
7.5
7.5.1
Relationship to Bridge Inspection Reports
Background
Another goal of this project was to study the data contained in the inspection
reports of typical highway bridges, in search of a link, in addition to distribution
factors, between data readily-available to bridge owners and the current research. At
least once every two years, it is required for every bridge on public roads to be
inspected, resulting in a comprehensive report of the bridge’s condition. Among
photographs, quantitative ratings, and written descriptions, each report contains one or
more values for the condition of the reinforced concrete deck. These were the values
of interest.
168
These reports utilize two different systems for rating bridge elements. The first
and older of the two is the NBI condition rating system. Such a rating is part of the
NBI that is specified to be recorded for all bridges in the federal inventory. This rating
assigns an integer value between 0 and 9 based on the perceived condition of the deck
at the time of inspection. Table 7.6 outlines the qualitative definitions of each value.
This system attempts to attribute typical inspection examples to each number, but the
subjective descriptors in place make it a system that inherently lacks consistency.
The condition state, or MSPE system, is a newer system that attempts to improve upon
the shortcomings of the NBI system. The first of these modifications is the method of
quantifying the percentage of each element of the bridge in each of various condition
states on a span-by-span basis, to obtain more local values and better document the
range of performance observed within a structure. The condition ratings are a 1 to 5
integer scale. The description of these values, reproduced in Table 7.7 are noticeably
less subjective then the NBI values. While adjectives by their design cannot be
completely objective, this system attempts to remove personal judgment from the
process by attributing “Yes or No” descriptors to the values of the condition states.
The semantics are far more definitive, and while having almost half of the numerical
values as the NBI system available seems to eliminate the element of precision in
rating values, it does help in accomplishing nationwide consistency. For example, it is
much easier for two inspectors to agree on whether protective systems are in place or
not then it would be to agree on the difference between “advanced” and “major”
deterioration.
The last noteworthy feature of the MSPE system is the inclusion of an
Environmental (Env) Factor to describe the environment in which the structure is
169
Table 7.6.
NBI Deck Rating Definitions
9 EXCELLENT CONDITION
8 VERY GOOD CONDITION - no problems noted.
7 GOOD CONDITION - some minor problems.
6 SATISFACTORY CONDITION - structural elements show some minor
deterioration.
5 FAIR CONDITION - all primary structural elements are sound but may
have minor section loss, cracking, spalling or scour.
4 POOR CONDITION - advanced section loss, deterioration, spalling or
scour.
3 SERIOUS CONDITION - loss of section, deterioration, spalling or scour
have seriously affected primary structural components. Local failures are possible.
Fatigue cracks in steel or shear cracks in concrete may be present.
2 CRITICAL CONDITION - advanced deterioration of primary structural
elements. Fatigue cracks in steel or shear cracks in concrete may be present or scour
may have removed substructure support. Unless closely monitored it may be
necessary to close the bridge until corrective action is taken.
1 "IMMINENT" FAILURE CONDITION - major deterioration or section
loss present in critical structural components or obvious vertical or horizontal
movement affecting structure stability. Bridge is closed to traffic but corrective
action may put back in light service.
0 FAILED CONDITION - out of service - beyond corrective action.
170
Table 7.7.
MSPE Condition State Definitions
1. Protected – Protective systems sound and functioning to prevent
deterioration
2. Exposed – Protective systems partially or completely failed
3. Attacked – Element experiencing active attack, but not yet damaged
4. Damaged – Element has lost material such that serviceability is suspect
5. Failed – Element no longer serves intended function.
located. These factors are defined in Table 7.8 and provide interesting opportunities
for correlating condition with environment, which can be considered when evaluating
expected performance due to aging and deterioration. This is further considered in
the sections that follow. It is also noted that the descriptors assigned to these values
do not represent a linear increase in environmental influence from 1 to 4. This fact is
noted and accounted for in the calculations, as will be described below.
Table 10.8
MSPE Environmental Factor Definitions
1. Benign – No environmental conditions affecting deterioration
2. Low – Environmental conditions create no adverse impacts, or are
mitigated by past non-maintenance actions or highly effective protective systems
3. Moderate – Typical level of environmental influence on deterioration
4. Severe – Environmental factors contribute to rapid deterioration.
Protective systems are not in place or are ineffective.
171
7.5.2
Bridge Inspection Reports Data Set
In order to investigate the opportunities for applying the data contained in
bridge inspection reports to the potential quantification methods for system-level
strength that were discussed in Section 7.3 and 7.4, data from a series of inspection
reports was collected. Data that was pulled from these reports included year of
construction or reconstruction, NBI deck rating, the Env factor, and the MSPE deck
condition state. This process was conducted for a collection of 10 bridges from the
Wilmington area, and eventually expanded to other sets of reports from three other
states referred to as East Coast (EC) State, Mountain West (MW) State, and Midwest
(M) State. This variety of reports allowed for the study of local bridges that exist in a
familiar climate, in addition other national bridges that exist in more extreme climates
in terms of snowfall, and therefore increased exposure to de-icing agents.
7.5.3
Preliminary Study, Delaware Bridges
The first analysis of this data was to study the relationship between the rating
systems, time, and Env factor. This study was mainly intended to view, with a small
set of data points from Delaware bridges, what the relationship between these three
parameters would resemble. Both rating systems (NBI and MSPE) were analyzed
separately and condition ratings as a function of age of the structure tabulated. The
Env factors were then reviewed to determine whether this appeared to correlate to any
inconsistencies in the rating factor versus time relationships. This study was also
intended to locate any outliers in the data that could be removed before conducting my
primary study. If the ultimate goal of the primary study is to determine accurate
relationships between the content of these inspection reports and distribution factors,
172
then eliminating outliers is an important preemptive measure to ensure the accuracy of
this relationship.
A table that included columns for age (calculated using the year of
construction), NBI rating, MSPE condition state, and Env factor was then created, a
subset of which is shown in Table 7.9. It is noted that although the MSPE system
allows for different percentages of the deck to be assigned different condition states, in
all cases reviewed in this work, all (or 100%) of each bridge deck was in the same
condition state. The data was then sorted by age, from youngest to oldest. Having the
Env factor adjacent to the deck rating was a useful situation, which allowed for
immediate comparison to assess if the environment was the cause of any data
abnormalities in the trend of the deck ratings versus age.
It was observed that the NBI data generally followed a logical trend of
decreasing deck condition with increasing age, as shown in Fig. 7.6. This graph
represents the NBI deck condition ratings from the Delaware data set as well as their
environmental factor. Each pair of red and blue points represents data from a given
Table 7.9.
Initial Bridge Inspection Data (Limited Rows Included).
Year
Deck Condition,
Condition
NBI Rating
State
2
8
1
1
5
8
1
1
21
7
1
3
173
Env
9
8
NBI Deck Condition
7
6
5
4
3
2
NBI
1
Env
0
0
Figure 7.6.
10
20
30
40
50
60
Time since (re)construction (Years)
70
80
NBI Deck Condition Rating vs. Time (with Environmental Factor)
bridge, 10 in total. Assuming each bridge was exposed to weathering and de-icing
agents since its construction, plotting it against time should show a gradual decrease in
rating which can be observed above. The trend of this line should resemble a staircase
if all bridges have aged uniformly, and the only time where this doesn’t occur (around
32 years) coincides with an environmental rating of 4, the most severe condition. This
suggests the logical conclusion that the severe weather conditions implied by the
highest environmental rating may have had a significant enough effect to prematurely
lower the bridge’s deck rating relative to the other bridges in this data set.
It is also worth noting that irrespective of age, the deck rating and the
environmental factor seem to oppose each other in their trends, which is to say that as
the weather conditions worsen, so does the deck rating. While it would not make
sense to infer that the weather conditions are getting worse with time (since these two
174
quantities are independent of each other), it is logical that the increasing trend in
environmental factor is contributing to the decreasing trend in deck rating, in addition
to age. This furthers the non-linearity of the NBI deck rating vs. time relationship.
The MSPE data however, was much less straight forward, as shown in Fig. 7.7.
There seemed to be a loose connection between the condition state and the Env factor,
but it did not trend smoothly with time. According to this rating system the lower
numbers represent better conditions, hence all elements should theoretically begin
with a rating of 1 and increase with time, rather than decreasing as in the NBI system.
By the same logic presented in the Fig. 7.6, it would be expected of the blue points to
create a staircase (with fewer steps in this case) if all bridges deteriorated at the same
rate and were rated on objective standards. However in this case, there are a
significant amount of data points that do not follow this trend. But, it seems that
5
MSPE
Env
MSPE Condition State
4
3
y = 0.0047x + 1.2405
2
1
0
0
Figure 7.7.
10
20
30
40
50
60
Time since (re)construction (Years)
70
80
MSPE Condition State Rating vs. Time (with Environmental Factor)
175
extreme environmental factors again coincide with the atypical condition state values.
For instance, once again at 32 years it seems the environmental factor of 4 is causing a
bridge deck rating of 2 prematurely. Because of this, and other data points that seem
to be outliers such as the pair at 71 years, the linear trend line is not nearly as steep as
would be expected. It is worth mentioning that even when the deck rating does not
behave expectedly with time, once again there is a noticeable relationship between
deck rating and Env factor. The blue series does not form a nice staircase trend, but
Env factor does seem in general to increase in concert with the deck condition state.
While this relationship is consistent with previous logic, other reasons for the
inconsistencies with the ideal staircase trend were also explored. For example, first
comments associated with condition state from the inspection report, which attempted
to justify the rating that was assigned, were utilized in an attempt to account for the
turbulence in the data. The usefulness of these comments will be touched upon in the
discussion that follows. In light of this information, the data was then modified to
create an idealized data set, which will be justified in the discussion that follows.
Figure 7.8 contains the same data as Figure 7.7, but with alterations to adjust 4 of the
10 data points that did not follow the staircase trend. Ideally, the MSPE data should
form an ascending staircase due to the values being restricted to integers, and the idea
that age will always decrease the quality of a bridge deck. The following
modifications were made only to observe what an ideal MSPE versus time and Env
factor may look like for Delaware bridges.
At 32 and 34 years, the deck condition is recorded as 2, which is not consistent
with the trend. By noticing that the environmental factor was 4 for those bridges, it
176
5
MSPE Condition State
4
y = 0.0204x + 2.1078
3
y = 0.0203x + 0.713
2
MSPE
1
Env
0
0
10
20
30
40
50
60
70
80
Time since (re)construction (Years)
Figure 7.8.
Adjusted MSPE Condition State Rating vs. Time (with Environmental
Factor)
was assumed that this was the cause of the lower condition state, and both of these
condition states were changed to values of 1, making the data representative of Env
states 1 through 3. At times 36 and 71, the deck condition state is given as 1 for both,
but the NBI ratings were 6 and 5 respectively. In all other cases, 6 and 5 related to 2’s
on the MSPE scale. These inconsistencies are difficult to account for; one possibility
emerges from the MSPE report, where it documents “Random Superficial Deck
Cracking”. If the cracking is considered superficial by MSPE standards and
structurally damaging by NBI standards, that could account for the disagreement. In
the case of the oldest bridge, it could be that age alone merited a lower score on the
NBI scale, where the more objective MSPE scale did not find any significant
inadequacies despite its age. Because of these inconsistencies, and the environmental
rating of 3 for both cases, these two MSPE ratings were shifted to 2. With these
177
corrective measures, a perfect staircase trend is created. Interestingly, the slopes of
the two trend lines in Fig. 7.8 are identical. While the precision of their similarity is
strictly coincidental, and the value of the trend line for the Env factor is
inconsequential (as stated before, relating Env factor to age is meaningless), it does
show a correlation between the environment a bridge is exposed to and how the deck
degrades over time. For instance, if the Env factors had instead created a downward
slope with increasing time, perhaps this would have contributed to a decrease in slope
of the MSPE trend line.
7.5.4
7.5.4.1
Primary Study
Procedure
After gaining a basic understanding of what should be expected from relating
deck rating to time, the next step and ultimate purpose of the primary study was to
investigate the relationship between deck rating and distribution factor. Since both
rating and distribution factor can be plotted against time, then a simple substitution
can be conducted to remove time and relate the other two variables directly. However,
as discussed in Section 7.4.2, it would be particularly useful a numerical method for
assessing the impact of the environment on the distribution factor could be developed.
The presence of the Env factor in these inspection reports provided the unique
opportunity to do just that, while simultaneously studying the deck rating relationship.
So in order to develop this relationship, the first step is to relate time to deck rating
and Env factor.
To accomplish this, the deck rating was plotted versus time with separate data
series based on the various Env factors, which is shown in Fig. 7.9. After adding
178
9
NBI Deck Rating
8
1 and 2
y = ‐0.0487x + 8.2173
7
3
y = ‐0.04x + 7.7849
6
4
y = ‐0.0403x + 7.8818
5
4
0
20
40
60
80
Time (Years)
Figure 7.9.
NBI Deck Condition Rating vs. Time (by Env)
separate trend lines to each data set, this resulted in a plot that showed how deck rating
changed with time, in each exclusive environmental condition. From these separate
trend lines, a master “transformation” equation (discussed in the following section)
was created that takes time and (indirectly) Env factor as inputs, and calculates deck
rating.
7.5.4.2
Results and Discussion
The placements of the trend lines in Fig. 7.9 are not completely intuitive, such
as the trend line for Env 4 giving higher NBI condition ratings than the corresponding
trend line for Env 3. Such abnormalities are likely a result of the small sample size
used to generate this approximation, particularly the small number of bridges with Env
values of 4. Thus, the sample size and geographic region was expanded by reviewing
data from three additional agencies, introduced in Section 7.5.2. However, the EC
bridge reports did not contain Env factor, and while all of the bridges in the MW
179
reports were recorded as having Env factors of 4, many of these had identical ages,
which caused creating a trend line as a function of time to be problematic. In contrast,
the M inspection reports generated valuable data as they contained many instances of
extreme weather conditions (as represented by Env ratings of 4) in addition to
exhibiting a wide variety of ages.
It is important to note that the M inspection reports from specific bridges were
selectively included. Including all of the data would technically create desirable trends
as well, but because of existing limitations (there can never be a NBI deck value that
corresponds to a time greater than 50 because the DF equations have a limited domain
of 0-50) they would not be visible on the graphs above. Therefore only certain bridges
from the M data set were selected to modify the existing Delaware data set, in order to
create a trend line that reached an NBI rating of 6 before the 50 year mark by
excluding M bridges that retained a high deck rating with increasing age. This allowed
the resulting master equation to be defined on a domain from an NBI rating of 9 to at
least 6. Without these precautions, the equation would only generate DF’s for a deck
rating of 9, 8, or 7, which would be a significant hindrance.
Further considerations when generating a condition versus age prediction
equation were that, based on the descriptions of the Env values from the MSPE guide,
it seemed that Env 1 and 2 were best combined, as they both essentially mean that
weathering is not a factor. Env 3 represents an average amount of weathering and 4
represents severe corrosion from the elements. Because of this, Env of 3 was used as
the base condition, and factors factors were included to increase or decrease the deck
rating values for Env values other than 3. Using the trend lines from the above graph,
the following equation results.
180
NBIpredicted = 7.7849 - 0.04t + e1(0.4324-0.0087t)+e2(0.969-0.0003t) [7-5]
Where:
NBIpredicted = estimated deck rating
t = time since deck construction in years
e1 = 1 for Env of 1 or 2, 0 for all else
e2 = 1 for Env of 4, 0 for all else
In general, Equation 7-5 operates by first taking the Env input, using it to transform
the master equation into the linear trend line equation associated with that particular
Env value from the graph, and then uses the time input and that specific trend line to
approximate the deck rating.
Equation 7-5 basically assumes an Env factor for 3, but can be transformed
into the equations for Env of 1 and 2 or 4. Furthermore, this equation can be inverted
to solve for t, then substituted this expression for t in the modified AASHTO DF
equation that was discussed in Section 7.4.2, creating a DF equation that varied with
deck rating and Env factor. This result assuming all bridges have an Env of 3 is
plotted in Fig. 7.10. Fig. 7.11 shows similar data if half of the bridges are assumed to
have an Env of 3 and the Env for the remaining bridges is changed to 2. As expected,
lowering the Env factor improves the condition of the deck and load-sharing capability
between girders, which decreases the distribution factor for the bridges with an Env=2.
Fig. 7.12 further evaluates this relationship and shows the effect of
environmental factor on the DF curves for three bridges, where each bridge contains
three curves, 1 for each Env factor (1 and 2 share the same curve). As would be
expected, the distribution factor increases with environmental factor, as both are
181
0.85
0.8
Michaud
SR299/SR1
DF
0.75
0.7
Mall Rd.
McDonough
SR1/SR273
0.65
SR1/US13
SR1/US13_B
0.6
10
9
8
7
6
5
NBI Deck Rating
Figure 7.10. Modified AASHTO DF prediction equation vs. NBI deck condition
rating (Env=3).
0.85
0.8
Michaud
SR299/SR1
DF
0.75
0.7
Mall Rd.
McDonough
SR1/SR273
0.65
0.6
10
9
8
7
6
SR1/US13
SR1/US13_B
5
NBI Deck Rating
Figure 7.11. Modified AASHTO DF prediction equation vs. NBI deck condition
rating (Env= 2 or 3).
182
10
9
8
7
6
5
Michaud‐4
Michaud‐3
Michaud‐1
DF
0.85
0.83
0.81
0.79
0.77
0.75
0.73
0.71
0.69
0.67
0.65
SR299/SR1‐4
SR299/SR1‐3
SR299/SR1‐1
SR1/US13‐4
SR1/US13‐3
SR1/US13‐1
NBI Deck Rating
Figure 7.12. AASHTO DF Equation vs. NBI Deck Rating (All Env's Represented).
inversely proportional to deck rating. Also, it can be observed that the only domain
where there is a significant difference in distribution factor is in between 6 and 7 deck
rating. These ratings correspond to a bridge age of 23 and 48 years respectively, and
in addition, this particular DF equation contains modulus of elasticity as its only timesensitive variable. Because modulus varies with time in a manner that approaches
infinity at an age of 50, it is logical that significant differences in DF based on Env
factor would be observed on the domain with the largest values of time.
To objectify the effect of Env factor on DF, Table 7.10 below displays the
percent change in DF with increasing Env factor between deck ratings of 9 and 6,
which roughly corresponds to an age domain of 0-45 years. The range of 9-6 in deck
ratings is generally equivalent to a range of 0-50 years in age. This table shows that
the difference in individual bridge geometry does not have that large of an effect on
the increase in DF over time. On the contrary, the Env factor has a significant effect
on DF. Considering an Env factor of 3 to be the average once again, then removing
183
Table 7.10.
Percent Change in DF from NBI 9-6
Bridge
Env 1-2
Env 3
Env 4
Michaud
7.95%
13.22%
16.65%
SR299
7.94%
13.20%
16.63%
SR1
8.01%
13.32%
16.78%
the effect of environmental hazards (dropping down to a 1 or 2) shows a 40% decrease
(13.2% versus 8% change) in the effect on DF, while increasing the effect of the
environment to a severe level (increasing to a 4) shows a 25% increase (16.6% versus
13.2% change) in the effect on DF.
Additionally, it was instructive to analyze how the deck rating would vary with
time in each state individually. To help focus on the effects of the individual states the
environmental factors were not considered. Fig. 7.13 demonstrates this data. Though
all states have differing Env factors and domains (due to variability in climate and age
of the bridges), they all ultimately resemble the same decreasing trend. The slopes of
each individual graph are not exactly the same, varying between declining one rating
in 10 years and one rating in 20 years. At worst these slopes have a difference of 50%
which is not overly dissimilar considering the great variability in age and environment.
A point to be made from this data in Fig. 7.13 is the mentality that the NBI
system creates. The subjective descriptors assigned to each numerical value are easily
misinterpreted, especially in light of the age of each structure. For instance, one of the
bridges from the Delaware data pool was 72 years old, and received a deck rating of 5.
However, on the more objective MSPE system, it received a 1 (the highest condition
rating). It seems from examples such as these and from the overall downward trends
184
9
8
NBI Deck Rating
7
6
5
4
DE
3
MD
EC 2
CO
MW 1
M IA
0
0
10
20
30
40
50
Time (Years)
60
70
80
Figure 7.13. NBI Deck Rating vs. Time (States Plotted Exclusively)
that the NBI system creates a mentality where the deck rating will decrease by simply
observing the age of the structure. While the structural integrity is thoroughly
investigated and considered in assigning a rating, a system that is less subjective such
as the MSPE might have an easier time preventing the age of the structure from overly
affecting the rating.
Fig. 7.14 represents a nationwide data pool that depicts both NBI and MSPE
varying with time. While both systems have similar trends (though opposite in
direction because higher values represents increased quality in NBI and decreased
quality in MSPE) the NBI system can be observed to vary more heavily with time. To
take a more objective look at this data, both the NBI and MSPE systems were
normalized to a 10 point scale in order to make their slopes comparable. This involved
a basic calculation of multiplying all of the NBI rating values by (10/9) and the MSPE
values by (10/5). In the process the trend of the MSPE was also reversed so that the
185
9
8
Inspection Rating
7
y = ‐0.0522x + 8.2842
6
5
4
y = 0.0186x + 0.9428
3
2
NBI
1
MSPE
0
0
10
20
30
40
50
60
70
80
Time (Years)
Figure 7.14. NBI and MSPE Deck Ratings vs. Time
trends follow the same philosophical and mathematical direction (by simply replacing
5 with 1, 4 with 2, etc. in order for it to match the NBI trend). Re-plotting the
normalized data creates Fig. 7.15, where the slope of the NBI deck rating trend line
exhibits a 56% greater ((0.058-0.0372)/0.0372=0.56) slope than that of the MSPE
ratings. In general, this shows how a more subjective system such as the NBI will have
a greater variability with time regardless of the structural condition of the bridge deck,
even for the same population of bridges.
7.5.5
Conclusions
As a result of exploring the ways in which information found in bridge
inspection reports can aid in the study of the effect of aging on DF, a relationship that
allows the DF for a bridge to be estimated by knowing its NBI deck rating and the
MSPE Env factor (in addition to the geometry of the bridge and other variables
186
MSPE/NBI Deck Rating (Normalized)
12
10
8
y = ‐0.0372x + 10.114
6
y = ‐0.058x + 9.2046
4
MSPE
2
NBI
0
0
10
20
30
40
Time (Years)
50
60
70
80
Figure 7.15. Normalized NBI and MSPE Deck Ratings vs. Time
necessary to the DF equation of choice) was established. While the data used to
derive this equation was limited, a valid process has been determined that could be
refined with a larger data set to broaden the applicability of the resulting estimates.
Furthermore, while these results were limited to elastic distribution factors as this is
presently the only format in which distribution factors are quantified for a broad range
of parameters, the process that has been developed could be equally applied to
inelastic distribution factors as these equations become available. Thus it is shown
that there is the potential for an efficient way to estimate DF that consider the current
condition state of the bridge.
The most precise way to utilize this information would be to select a DF
equation (a conservative sensitive equation such as Newmark, or a less conservative
and more theoretically grounded equation such as AASHTO) and the necessary data
about the bridge of interest, and input the geometric values and initial material
187
properties into the tables used to create Figures 7.10 through 7.12. This would
effectively re-plot Figure 7.10 for the bridge and equation being analyzed. Knowing
the deck rating and Env factor, one could then use this updated Figure 7.10 to
determine the current DF. This process is quite arduous and would require knowledge
of the files that were used to create these graphs.
A simpler and much more feasible method would be to utilize Table 7.10
included in the discussion. As noted in the discussion, the specific dimensions of a
particular bridge do not seem to greatly affect the DF compared to the deck rating and
Env factor, despite that the three bridges represented in Table 7.10 are significantly
different from one another, within the realm of all three being steel I-girder bridges.
Therefore, for any given bridge, one could simply use the NBI and Env factor values
in combination with Table 7.10 to approximate a percent increase from the theoretical
DF of the bridge based on its as-built condition.
The current limitations to this method would include its limited domain.
Because of the limitations of the charts from the original DF study (the limited domain
from Equation 7-3), an NBI deck rating of 5 or lower cannot be evaluated by the
equations in this study. However, of the almost 50 bridges considered in this study,
only 2 exhibited a deck condition rating of 5 or lower. From a practical standpoint,
this may not represent a huge limitation at the present time because bridges with a
deck condition rating lower than 5 would be unlikely candidates for system-level
evaluation until significant additional research is performed on this bridge type. In
addition to this limitation, the current percent change table in this study only applies
for bridges with a NBI deck rating of 6. If this method proves effective and useful,
188
then it would be relatively easy to reproduce these charts for deck condition rating
values of 7 and / or 8.
189
Chapter 8
PARALLEL AND FUTURE WORK
8.1
Parallel Considerations
This section summarizes the results of two complementary studies performed
in collaboration with the results discussed previously in this report. The first is a
preliminary cost-effectiveness analysis to determine the potential economic impacts of
implementing the system-level analyses investigated in the present work. It is noted
here that the term cost-effectiveness actually refers solely to the economic benefits, as
it is presently envisioned that there are no costs to implementation of this method
beyond the relatively modest research costs of developing the method. The second
parallel study evaluates the potential usefulness to the subject research of a
computational model for predicting the service life of reinforced concrete subjected to
deicing agents.
8.1.1
Cost-Effectiveness Analysis
A study to estimate the potential cost-effectiveness of implementing load-path
redundancy in the bride rating process was completed by McCarthy (2012). In this
analysis the federal funding for a set of 14 steel girder-concrete deck bridges in
Delaware is estimated from data provided in the NBI for that year at $23.4 million.
The set of bridges that were selected represent bridges whose structural condition
appears to easily permit load path redundancy and the additional load carrying
capacity it affords.
Cost-effectiveness scenarios assuming improvements in bridge rating to that
set of bridges show various degrees of budgetary savings and optimized asset
management. Scenario 1 shows that deferral of the annual sum of estimated funds
190
required for that set of bridges by 5 and 10 years permits savings of between $2
million and $3.6 million, respectively, with all values being approximate.
Scenario 2 uses a somewhat loose assumption on the percentage by which
spending can be reduced in so far as that percentage is directly correlated to the
percentage increase in load carrying capacity when referring to the system load
capacity for a prototype bridge field tested in prior work (Michaud, 2011), which is
based on the bridge referred to as 7R and modeled with elastic material properties for
the bridge deck in Chapter 6. This scenario indicates a potential for savings of $4.7
million, or 20% of the estimated budget.
McCarthy (2012) also identified an additional 25 bridges in Delaware whose
deck as well as super-structure shows deterioration. These are bridges whose
capacities could be more accurately predicted using a system-level rating process.
Thus, the significance of the evidence regarding adequate capability for load
redistribution even in bridges with deteriorated concrete deck properties that was
shown in Chapter 6 is that it nearly triples the number of bridges for which systemlevel analysis may be suitable. Incorporation of these additional 25 bridges as
simulated in Scenario 3 indicates deferral of spending can save an additional $15-$29
million. Scenario 4 indicates the ability to reduce spending on these bridges can
potentially save an additional $39 million. All of these cost values are approximate.
Notably, these savings are in addition to those seen under Scenarios 1 and 2.
Therefore, it is concluded that the proposed philosophical changes to current bridge
rating practices, whereby load path redundancy is acknowledged, are cost effective.
Other conclusions form this work were: immediate savings are permitted by deferral
of spending requirements; the dollar amounts saved could potentially be invested to
191
earn interest over time; where monetary savings or deferrals are not made,
prioritization of remedial works can be made by targeting spending which would have
been made on the simulation bridges towards bridges in more critical need of
rehabilitation allowing possibly even the removal of a ‘posted’ status from a number
of bridges and saving time and resources which would have otherwise been spent
elsewhere.
8.1.2
Life365 Model for Concrete Deterioration
Life-365 is a free online software download for predicting the service life and
life cycle cost of reinforced concrete. Given this purpose it was thought that the
service life prediction model featured in this program may be of relevance to the
present research. Thus, this software was download and its tutorial reviewed to gain
an understanding of how the program worked, when it was appropriate to use default
settings, and when unique inputs were imperative. A list was developed of these
inputs and the resulting outputs, which is shown in Table 8.1.
The function of the software is the ability to input multiple options for a
concrete bridge deck design and then view the life cycle of competing designs in
respect to cost and infiltration of chloride from road de-icing agents. The potential
usefulness in this program lies in its ability to plot the infiltration of chloride as a
function of time, for our goal is to evaluate the loss in strength of the concrete over
time, which is a direct result of the infiltration of de-icing agents. However, this data
is only useful to our research if we can relate this infiltration to the numerical change
in material properties of the concrete, for it is only through the material properties that
we can calculate the distribution factors. After thoroughly exploring all that the
program could output, it was concluded that none of it could be related to the change
192
Table 8.1. List of Life-365 necessary inputs (by tab) and useful outputs.
Inputs
Default
Concrete Individual
Tab
Settings
Project
Exposure
Mixtures
Costs
Rebar Cost
Amount of
Mixture
Location
and
Chlorides
Qualities
Quantity
Geometry
Inhibitor
Analysis
Rebar
of
Cost and
Period
Qualities
Parameters
Reinforced
Quantity
Duration of
Concrete
Cost
Chloride
Inhibitor
Deck
Barrier
Parameters
Exposure
Qualities
Cost and
Alternative
Barrier
Quantity
Options
Qualities
Output
Life-Cycle Cost, $/sq. ft for all Alternatives
Component Cost, $/sq. ft for all Alternatives
Service Life
Initiation/Propagation of Chlorides per Year
Cross Section Concentration
Diffusivity vs. Time
in material properties. While there were many different graphical representations of
the degrading conditions, none of them seemed to contain data on the change in the
material properties, or if they did, they were not accessible to the user. It does also not
appear that this direct link between chloride infiltration and mechanical properties has
been documented in the literature. For these reasons, it was discerned that Life-365 is
not presently useful to the goals of this work.
8.2
Future Work
This research has revealed that system-level analysis and rating procedures
have applicability to steel girder bridges with both good and poor-condition concrete
decks. Thus, due to the large number of bridges of this type and the continuously
166
strained budgets for infrastructure renewal, this rating philosophy holds significant
promise for more accurately quantifying bridge capacities and optimizing bridge
management processes such that funding can be directed to the most critical structures.
As a result, continued research on this topic is believed to be highly valuable.
One of the most immediately achievable research needs is to extend the
number of bridges and variables considered to a larger scope than what was able to be
accomplished in the present work. This is necessary in order to reveal general trends
in performance as a function of deck condition and other characteristics which cannot
presently be discerned through the small analytical sample for which data exists.
This work will be necessary in order to confidently validate that the observations made
in this research are generally applicable to the existing population of bridges.
In addition, in order to further validate the analytical results, experimental
results more closely simulating the conditions which are being modeled would be
highly valuable. These results would be difficult to obtain due to the large-scale
nature of the structure and the complexity of accurately simulating aging, but should
nonetheless be pursued. The optimum solution to this difficulty would be destructive
testing of decommissioned structures, which should be pursued whenever possible.
There is also room to improve upon the analytical methods in order to obtain
more refined solutions and solutions to a larger scope of structures. Specifically, 3-D
concrete modeling technique allow a more direct approach to modeling the physical
effects of corrosion within concrete and results from these models would be of great
interest. In addition, numerical difficulties have also prevented successful analysis of
continuous-span bridges and this is another limitation that can hopefully be overcome
167
in future work, perhaps through attempting to reproduce physical results of concrete
cylinder testing and progressing to specimens with increasing complexity.
In addition, there are several uncertainties regarding the relationships between
deicing agent exposure (as well as other causes of deterioration) and quantified
changes in mechanical and material behavior. This research has been based on the
best current estimates of this relationship that are available, but there is significant
opportunities to better understand these relationships. Such understanding would not
only benefit the subject research but also quite likely lead to improved design or
maintenance practices or material developments that would lead to minimizing these
deleterious effects of corrosion or extending the time over which deterioration reaches
a critical point.
Lastly, to further promote the development of system-level rating methods it
would be beneficial to quantitatively evaluate the correlation between deck conditions
(i.e., the NBI deck condition rating and deck element-level data recorded by many
states) that are indicated in existing bridge inspection reports to the material input
parameters used in the model. This could be assessed through selected core sampling
from existing structures, particularly immediately preceding redecking of bridges that
had been in service for various amounts of time. With this information quantified,
system capacities could be estimated as a function of these ratings, greatly improving
the usefulness of the proposed methodology from the perspective of a bridge rating
engineer. Such research is also likely to produce insights that could reduce the
subjectivity of the current inspection processes.
Ultimately, through the above work, a process is envisioned where deck ratings
from inspection reports would be input into equations or simple models to predict a
168
system-level capacity or inelastic distribution factor, which would in turn be used to
revise the estimated structural capacity during the bridge rating process. Through such
a system, the research completed in the present effort indicates that improved
prioritization of the repair of bridges categorized as structurally deficient and / or
significant economic savings would result.
169
REFERENCES
Adelaide, L., Richard, B., Ragueneua, F., and Cremona, C. (2012). “A simplified
numerical approach of global behavior of RC beams degraded by corrosion.”
European J. of Envr. And Civ. Eng., 16(3-4), 414-439.
Akgul, F., & Frangopol, D. M. (2004). Time-dependent interation between load rating
and reliability of deteriorating bridges. Engineering Structures, 1751-1765.
Ambrose, K. (2012). “Field Measurements and Corresponding FEA of Cross-Frame
Forces in Skewed Steel I-Girder Bridges.” M.C.E. thesis, Univ. of Delaware,
Newark, DE.
American Association of State Highway and Transportation Officials (AASHTO).
(2013). “AASHTO LRFD Bridge Design Specifications.” Washington DC, 6th
Edition.
American Society of Civil Engineers (ASCE). (2009). “Bridges.” Report Card for
America’s Infrastructure, <http://www.infrastructurereportcard.org/factsheet/bridges> (Nov. 22, 2012).
ASCE/SEI-AASHTO Ad-Hoc Group on Bridge Inspection, R. R. (2009). White Paper
on Bridge Inspection and Rating. Journal of Bridge Engineering, 1-5.
Amleh, L., and Ghosh, A. (2006). “Modeling the effect of corrosion on bong strength
at the steel-concrete interface with finite-element analysis.” Can. J. Civ. Eng.,
33, 673-682.
Basheer, L., Kropp, J., & Cleland, D. (2001). Assessment of the durability of concrete
from its permeation properties: a review. Construction and Building Materials,
93-101.
Barth, K. E., White, D. W., Righman, J. E., and Yang, L. (2005). “Evaluation of web
compactness limits for singly and doubly symmetric steel I-girders.” J. Constr.
Steel Res., 61(10), 1411-1434.
Baskar, K., Shanmugam, N. E., and Thevendran, V. (2002). “Finite-Element Analysis
of Steel-Concrete Composite Plate Girder.” J. of Struct. Eng., 128(9), 11581168.
170
Bechtel, A., McConnell, J., and Chajes, M. (2011). “Ultimate Capacity Destructive
Testing and Finite-Element Analysis of Steel I-Girder Bridges.” J. Bridge
Eng., 16, 197-206.
Biondini, F. and Vergani, M. (2012). “Damage modeling and nonlinear analysis of
concrete bridges under corrosion.” Proc., Int. Conf. on Bridge Maintenance,
Safety, and Management, Taylor and Francis, Florence, KY.
Bolukbasi, M., Mohammadi, J., & Arditi, D. (2004, February). Estimating the Future
Condition of Highway Bridge Components Using National Bridge Inventory
Data . Practice Periodical on Structural Design and Construction ASCE, pp.
16-25.
Chajes, M., McConnell, J., Shenton, H., Michaud, K., Ross, J., and Russo, C. (2010).
“Full-Scale Destructive Bridge Test Allows Prediction of Ultimate Capacity”,
J. of the Transportation Research Board, 2200, 117-124.
Chen, D., and Mahadevan, S. (2007). “Chloride-induced reinforcement corrosion and
concrete cracking simulation.” Cement and Conc. Composites, 30, 227-238.
Coronelli, D. and Gambarova, P. (2004). “Structural Assessment of Corroded
Reinforced Concrete Beams: Modeling Guidelines.” J. of Struct. Eng., 130(8),
1214-1224.
Dekoster, M., Buyle-Bodin, F., Maurel, O., and Delmas, Y. (2003). “Modelling of the
flexural behavior of RC beams subjected to localized and uniform corrosion.”
Eng. Struct., 25, 1333-1341.
Delaware Department of Transportation. (2009, April). DelDot Bridge Design Manual
Chapter 4 Bridge Load Rating. Retrieved June 19, 2012, from
www.deldot.gov:
http://www.deldot.gov/information/pubs_forms/manuals/bridge_design/pdf/bd
m-04-bridge-rating.pdf?051309
Fang, C., Lundgren, K., Plos, M., and Gylltoft, K. (2006). “Bond behavior of corroded
reinforcing steel bars in concrete.” Cement and Conc. Research, 36, 19311938.
Federal Highway Administration. (1995). Coding Guide for the Structure Inventory
and Appraisal of the Nation's Bridge's. Washington D.C.: U.S. Department of
Transportation.
171
Federal Highway Administration, Department of Transportation. (2010). 2010 NBI
ASCII Files, Delaware. Retrieved January 7, 2012, from U.D. Department of
Transportation Federal Highway Administration:
http://www.fhwa.dot.gov/bridge/nbi/disclaim.cfm?nbiYear=2010&nbiState=D
E10
Gu, X. L., Zhang, W. P., Shang, D. F., and Wang, X. G. (2010). “Flexural Behavior of
Corroded Reinforced Concrete Beams.” Earth and Space 2010: Engineering,
Science, Construction, and Operations in Challenging Environments, ASCE,
Reston, VA, 3545-3552.
Hobbs, D. (2001). Concrete Deterioration: causes, diagnosis and minimizing risk.
International Materials and Reviews, 117-144.
Huang, P., Bao, Y., & Yao, Y. (2005). Influence of HCL corrosion on the mechanical
properties of concrete. Cement and Concrete Research, 584-589.
Jankowiak, T. and Lodygowski, T. (2005). “Identification of Parameters of Concrete
Damage Plasticity Constitutive Model.” Foundations of Civ. and Env. Eng., 6,
53-69.
Kallias, A. N. and Rafiq, M. I. (2010). “Finite element investigation of the structural
response of corroded RC beams.” Eng. Struct., 32, 2984-2994.
Li, C.Q. (2000). “Corrosion Initiation of Reinforcing Steel in Concrete under Natural
Salt Spray and Service Loading – Results and Analysis”. ACI Materials
Journal, 97(6), 690-697.
Li, C. Q. and Zheng, J. J. (2005). “Propagation of Reinforcement Corrosion in
Concrete and its Effects on Structural Steel.” Mag. of Conc. Research, 57(5),
261-271.
MATBASE. (2009). Data Table for Miscellaneous Materials: Concrete: SteelReinforced Concrete. Retrieved June 19, 2012, from www.matbase.com:
http://www.matbase.com/material/other/concrete/steel-reinforcedconcrete/properties
McCarthy, G. (2012). “Improved Bridge Rating Procedures Integrating Load Path
Redundancy: A Cost-Effective Simulation.” M.C.E. thesis, Univ. of Delaware,
Newark, DE.
172
McConnell, J., Chajes, M. and Michaud, K. (in review). “Destructive Testing of a
Decommissioned Skewed Steel I-Girder Bridge: Analysis of System Effects.”
J. of Struct. Eng.
McConnell, J, McCarthy, G, and Wurst, D. (2012). “An Approach to Evaluating the
Influences of Aging on the System Capacity of Steel I-Girder Bridges.” Proc.,
Int. Conf. on Bridge Maintenance, Safety, and Management, Taylor and
Francis, Florence, KY.
Melchers, R. E., Li, C. Q., Lawanwisut, W. (2006). "Modelling deterioration of
structural behavior of reinforced concrete beams under saline environment
corrosion." Mag. of Concrete Research, 58(9), 575-587.
Michaud, K. M. (2011). “Evaluating Reserve Bridge Capacity Through Destructive
Testing of a Decommissioned Bridge.” M.C.E. thesis, Univ. of Delaware,
Newark, DE.
Missouri Department of Transportation. (2000, February). Business with MODOT Bridge Inspection Rating Manual. Retrieved March 11, 2012, from MODOT:
http://www.modot.mo.gov/business/manuals/bridgeinspectrating.htm
Morales, E. M. (n.d.). “Significance of the Ratio of Tensile Strength to Yield Stress
(TS/YS) of Reinforcing Bars.” <
http://www.pgatech.com.ph/documents/Significance%20of%20the%20Ratio%
20of%20Tensile%20Strength%20to%20Yield%20Stress.PDF> (November 15,
2013).
Oyado, M., Kanakubo, T., Sato, T., and Yamamoto, Y. (2010). “Bending performance
of reinforced concrete member deteriorated by corrosion.” Struct. and
Infrastructure Eng., 7(1-2), 121-130.
Rendell, F., Jauberthie, R., & Grantham, M. (2002). Deteriorated Concrete: Inspection
and physiochemical analysis. Reston: ASCE Press.
Richard, B., Ragueneau, F., Adelaïde, L., and Cremona, C. (2011). “A multi-fiber
approach for modeling corroded reinforced concretes structures.” European J.
of Mech., 30, 950-961.
Roberts, J., & Shepard, R. (2007). Bridge Management for the 21st Century.
Transportation Research Record: Journal of the Transportation Research
Board, 197-203.
173
Ross, J. H. (2007). “Evaluating Ultimate Bridge Capacity Through Destructive
Testing of Decommissioned Bridges.” M.C.E. thesis, Univ. of Delaware,
Newark, DE.
Shi, X., Fay, L., Peterson, M. M., & Yang, Z. (2010). Freeze-thaw damage and
chemical change of a Portland cement concrete in the presence of diluted deicers. Materials and Structures, 933-946.
Shirai, N., Watanabe, K., Oh-oka, T., Hakuto, S., and Fujita, T. (2002). “Finite
element analysis of shear wall specimens made of ductile fiber reinforced
cementitious composites subjected to lateral loading.” Proc., Finite Elements
in Civ. Eng. Applications, Taylor & Francis, Tokyo, Japan, 219-227.
Simulia. (2011). ABAQUS/CAE User Manual v. 6.11. ABAQUS, Providence, RI.
Takahashi, Y. (2008). “Flexural Strengthening of RC Beams with CFRP Sheets and
U-Jackets.” Proc., Int. Conference on FRP Composites in Civ. Eng., Zurich,
Switzerland.
The White House. (1992, October 29). Circular No. A-94, Appendix A. Retrieved
February 29, 2012, from The White House Office of Management and Budget:
http://www.whitehouse.gov/omb/circulars_a094#5
The White House. (2011, December). Circular A-94, Appendix C. Retrieved February
29, 2012, from The White House, Office of Management and Budget:
http://www.whitehouse.gov/omb/circulars_a094/a94_appx-c
U.S. Department of Transportation. (2012). SAFETEA-LU: Fact Sheets on Highway
Provisions. Retrieved June 15, 2012, from FHWA.DOT.GOV:
http://www.fhwa.dot.gov/safetealu/factsheets/bridge.htm
U.S. Department of Transportation, Federal Highway Administration. (2011, August).
Bridge Preservation Guide Maintaining State of Good Repair Using Cost
Effective Investment Strategies. Retrieved March 2012, from U.S. Department
of Transportation, Federal Highway Administration:
http://www.fhwa.dot.gov/bridge/preservation/guide/index.cfm
U.S. Department of Transportation, Federal Highway Administration. (2011,
October). Office of Highway Policy Information, National Highway
Construction Cost Index. Retrieved January 5, 2012, from U.S. Department of
Transportation, Federal Highway Administration:
http://www.fhwa.dot.gov/policyinformation/nhcci/pt1.cfm
174
Val, D. V., Chernin, L, and Stewart, M. G. (2009). “Experimental and Numerical
Investigation of Corrosion-Induced Cover Cracking in Reinforced Concrete
Structures.” J. of Struct. Eng., 135, 376-385.
Vorechovska, D., & Podrouzek, J. (2009). Modeling of chloride concentration effect
on reinforcement corrosion. Computer-Aided Civil and Infrastructure
Engineering, 446-458.
Wang, K., Nelson, D., & Nixon, W. (2006). Damaging effects of deicing chemicals on
concrete materials. Cement & Concrete Composites, 173-188.
Wang, N., O'Malley, C., Ellingwood, B. R., & Zureick, A.-H. (2011). Bridge Rating
Using System Reliability Assessment. 1: Assessment and Verification by Load
Testing. Journal of Bridge Engineering, 854-862.
Wight, J. K., and MacGregor, J. G. (2009). "Reinforced Concrete Mechanics And
Design." Pearson Education, Inc., Upper Saddle River, NJ, 158-163.
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Appendix A
Table A.1
List of variable definitions used in Section 7.4.2
Variable
S
Definition
Girder Spacing
L
ts
Span Length
Concrete Deck Thickness
b
EB
Concrete Deck Width
Elastic Modulus of the Girders
ED
Elastic Modulus of the Deck
IB
Area Moment of Inertia of the Girders
ID
Area Moment of Inertia of the Deck
vx
Poisson's Ratio of the Deck in the X-Direction
vy
Poisson's Ratio of the Deck in the Y-Direction
The following pages contain scanned images of the hand calculations used in
Section 7.4.2.
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Appendix B
PERMISSION LETTERS
To whom it may concern:
I give Diane Wurst permission to use Figures 2.1 and 2.9 from my thesis, "Field
Measurements and Corresponding FEA of Cross-frame Forces in Skewed Steel Igirder Bridges".
Sincerely,
Kelly L. Ambrose
(302)-528-2263
1405 Riverside Ave.
Baltimore, MD 21230
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