finite element analysis to simulate reinforced concrete
TRANSCRIPT
FINITE ELEMENT ANALYSIS TO SIMULATE
REINFORCED CONCRETE CORROSION
IN BEAMS AND BRIDGE DECKS
by
Diane Wurst
A thesis submitted to the Faculty of the University of Delaware in partial
fulfillment of the requirements for the degree of Master of Civil Engineering
Fall 2013
© 2013 Diane Wurst
All Rights Reserved
FINITE ELEMENT ANALYSIS TO SIMULATE
REINFORCED CONCRETE CORROSION
IN BEAMS AND BRIDGE DECKS
by
Diane Wurst
Approved: __________________________________________________________
Jennifer E. Righman McConnell, Ph.D.
Professor in charge of thesis on behalf of the Advisory Committee
Approved: __________________________________________________________
Harry W. Shenton III, Ph.D.
Chair of the Department of Civil and Environmental Engineering
Approved: __________________________________________________________
Babatunde A. Ogunnaike, Ph.D.
Dean of the College of Engineering
Approved: __________________________________________________________
James G. Richards, Ph.D.
Vice Provost for Graduate and Professional Education
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ACKNOWLEDGMENTS
I would like to thank my advisor, Dr. Jennifer Righman McConnell. Without
her, I would not have had the opportunity to perform research and write a thesis. She
also provided invaluable guidance and support throughout the entire process. In
addition, I would like to thank my family and friends. Without their support, I
wouldn’t have had the motivation to apply and attend graduate school.
I would like to thank all of the other structures graduate students at the
University of Delaware for making graduate school a fun and exciting experience, and
for providing a listening ear when my research wasn’t going the way I had hoped.
Finally, I would like to thank the University of Delaware University
Transportation Center (UD-UTC) for providing financial support for my project.
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TABLE OF CONTENTS
LIST OF TABLES ....................................................................................................... vii
LIST OF FIGURES ........................................................................................................ x ABSTRACT ................................................................................................................ xiii
Chapter
1 INTRODUCTION .............................................................................................. 1
1.1 Motivation ................................................................................................. 1 1.2 Objectives and Scope ................................................................................ 2
1.3 Thesis Outline ............................................................................................ 3
2 LITERATURE REVIEW ................................................................................... 5
2.1 Previous Experimental Results .................................................................. 5
2.1.1 Destructive Bridge Testing ............................................................ 5 2.1.2 Corroded Reinforced Concrete ...................................................... 6
2.2 Reinforced Concrete and Corrosion Modeling ........................................ 11
3 MODELING APPROACH .............................................................................. 15
3.1 Concrete ................................................................................................... 15
3.1.1 Elastic Behavior ........................................................................... 16
3.1.2 Non-Linear Behavior ................................................................... 16
3.2 Rebar ........................................................................................................ 23
3.2.1 2-Dimensional Rebar ................................................................... 23
3.2.2 3-Dimensional Rebar ................................................................... 24
3.3 Boundary Conditions ............................................................................... 29 3.4 Analysis Method ...................................................................................... 30
4 MODEL CALIBRATION ................................................................................ 33
4.1 Beam Geometry and Material Properties ................................................ 34
4.1.1 Concrete ....................................................................................... 34 4.1.2 Rebar ............................................................................................ 36
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4.2 Calibration Metrics .................................................................................. 39 4.3 Strength and Deflection Calculations ...................................................... 42
4.4 2-Dimensional Concrete Model .............................................................. 43
4.4.1 Uncorroded Base Model Input Values ........................................ 44 4.4.2 Mesh Sensitivity Analysis and Uncorroded Base Model Input ... 48 4.4.3 Corroded Model ........................................................................... 54
4.5 3-Dimensional Concrete Model .............................................................. 72
4.5.1 2-Dimensional Rebar ................................................................... 73 4.5.2 3-Dimensional Rebar ................................................................... 79
4.6 Conclusions ............................................................................................. 89
5 BRIDGE MODELS .......................................................................................... 92
5.1 7R ............................................................................................................ 92
5.1.1 Bridge Information ...................................................................... 92 5.1.2 Results ......................................................................................... 94
5.2 SR 1 over US 13 .................................................................................... 109
5.2.1 Bridge Information .................................................................... 109
5.2.2 Modeling Results ....................................................................... 111
5.3 SR 299 over SR 1 .................................................................................. 115
5.3.1 Bridge Information .................................................................... 116 5.3.2 Modeling Results ....................................................................... 118
5.4 Conclusions ........................................................................................... 119
6 CONCLUSIONS ............................................................................................ 122
6.1 Summary ................................................................................................ 122
6.2 Results ................................................................................................... 125 6.3 Future Work ........................................................................................... 129
REFERENCES ........................................................................................................... 132
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Appendices
A SAMPLE STRENGTH CALCULATIONS ................................................... 136
B COMPLETE RESULTS FOR 2-D BEAM CALIBRATION
VARIATIONS ................................................................................................ 143 C COMPLETE RESULTS FOR 3-D BEAM CALIBRATION
VARIATIONS ................................................................................................ 166 D PERMISSION LETTERS .............................................................................. 183
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LIST OF TABLES
Table 2.1. Comparison of strength of DRBA Bridge 7R (Ross 2007). ...................... 6
Table 4.1. Yield and ultimate strengths of SD345 and SD295 rebar found in
literature compared to the minimum specified yield strengths. .............. 37
Table 4.2. True stress and true plastic strain input values for SD295 and SD345
rebar. ........................................................................................................ 39
Table 4.3. Input values for post-cracking commands for CDP model. .................... 47
Table 4.4. Strength results of mesh sensitivity analysis with percent differences
based on the calculated theoretical strength values. ................................ 50
Table 4.5. Deflection results of mesh sensitivity analysis with percent differences
based on the calculated deflection values for the uncracked and cracked
sections. ................................................................................................... 50
Table 4.6. Input and results of final uncorroded 2-D beam model. .......................... 54
Table 4.7. Selected results for initial calibration of 2-D corroded models with values
expressed as a percentage of the base model. .......................................... 57
Table 4.8. Input and results of calibration of Ec, f’c, and f’t for corroded 2-D beam
model. ...................................................................................................... 59
Table 4.9. Input and results of 2-D beam model after calibrating input
parameters. ............................................................................................... 67
Table 4.10. 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......................................... 69
Table 4.11. Strength results of rebar mesh sensitivity analysis for 3D concrete beam
with 2D rebar including percent differences based on the calculated
theoretical strength values. ...................................................................... 75
Table 4.12. Strength results of concrete mesh sensitivity analysis for 3D concrete
beam with 2D rebar including percent differences based on the calculated
theoretical strength values. ...................................................................... 76
Table 4.13. Base model input for 3-D concrete elements with 2-D rebar elements for
brittle cracking technique. ....................................................................... 77
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Table 4.14. Selected results of optimization of corrosion for 3-D beam model with 2-
D rebar. .................................................................................................... 78
Table 4.15. Results of varying friction input for 3-D concrete beam models with 3-D
rebar elements. ......................................................................................... 83
Table 4.16. Comparison of variations in friction and pressure-overclosure of 3-D
concrete beam models with 3-D rebar using brittle cracking. ................. 88
Table 4.17. Comparison of strength values utilizing different analysis techniques of
3-D concrete beams with 3-D rebar using brittle cracking. ..................... 89
Table 5.1. Original capacity results for Bridge 7R. .................................................. 96
Table 5.2. Results of bridge 7R when varying tension stiffening values. .............. 102
Table 5.3. Peak stress values in cross-frames as maximum loading in different 7R
bridge models. ....................................................................................... 105
Table 5.4. Distribution factors of bridge 7R previously determined in research
(McConnell et al. in review). ................................................................. 106
Table 5.5. DF values for finite element models created of Bridge 7R. .................. 107
Table 5.6. Initial results from analyzing bridge US13 including non-linear concrete
and rebar commands. ............................................................................. 112
Table 5.7. Results of Bridge US13 when varying tension stiffening values. ......... 115
Table 5.8. Initial results from analyzing Bridge US299. ........................................ 118
Table 5.9. Results of Bridge US299 when varying tension stiffening values. ....... 119
Table B.1. Results of mesh sensitivity analysis for 2-D beam models. .................. 143
Table B.2. Complete input and results of all concrete damaged plasticity models
tested during 2-D beam calibration. ...................................................... 144
Table C.1. Results of mesh sensitivity analysis for 3-D beam with 2-D rebar models
utilizing the brittle cracking technique………………………………………...... 166
Table C.2. Results of mesh sensitivity analysis for 3-D beam with 2-D rebar models
utilizing the CDP and SC techniques……………………………………………. 167
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Table C.3. Results for 3-D beam with 2-D rebar models utilizing the brittle cracking
technique…………………………………………………………………….……………. 169
Table C.4. Results for 3-D beam with 3-D rebar models utilizing the CDP
technique………………………………………..…………………………………………. 170
Table C.5. Results for 3-D beam with 3-D rebar models utilizing the SC
technique………………………………………..…………………………………………. 178
Table C.6. Results for the 3-D beam with 3-D rebar models utilizing the brittle
cracking technique…………………………..…………………………………………. 181
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LIST OF FIGURES
Figure 2.1. Strength losses reported in literature (McConnell et al. 2012). ................. 9
Figure 2.2. Stiffness losses reported in literature (McConnell et al. 2012). ................ 9
Figure 3.1. Tension stiffening model ABAQUS employs in the smeared crack
technique (Simulia 2011). ....................................................................... 18
Figure 3.2. Response of concrete to uniaxial loading in (a) tension and (b)
compression for CDP model (Simulia 2011). ......................................... 20
Figure 3.3. Exponential decay friction model (Simulia 2011). .................................. 25
Figure 3.4. Default “hard” pressure-overclosure relationship (Simulia, 2011). ........ 27
Figure 3.5. “Softened” exponential pressure-overclosure relationship
(Simulia 2011). ........................................................................................ 28
Figure 4.1. Geometry of finite element calibration models, elevation view (top) and
cross-section view (bottom) (dimensions in inches). .............................. 35
Figure 4.2. Example of determination of (a) the last step before failure and (b) the
first step of failure in 3-D models utilizing ABAQUS/Explicit. ............. 41
Figure 4.3. Uncorroded 2-D base model stress contours (units of psi). ..................... 53
Figure 4.4. Load and deflection comparison between final uncorroded model and
theoretical calculations for the uncracked and cracked sections. ............ 53
Figure 4.5. Comparison of strength between results from calibration models and
theoretical values caused by varying Ec. ................................................. 60
Figure 4.6. Comparison of deflection between results from calibration models and
theoretical values of the uncracked section caused by varying Ec. ......... 60
Figure 4.7. 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. . 62
Figure 4.8. 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..... 62
Figure 4.9. 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..... 63
xi
Figure 4.10. 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. ....................................................................................................... 64
Figure 4.11. 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. ....................................................................................................... 64
Figure 4.12. 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. ....................................................................................................... 65
Figure 4.13. Stress-strain response of element in 2-D beam models reported as percent
decrease in As. ......................................................................................... 66
Figure 4.14. Deflection response of elements in 2-D beam models reported as percent
decrease in As. ......................................................................................... 66
Figure 4.15. Stress contours of (a) final uncorroded and (b) final corroded 2-D beam
models (units of in psi). ........................................................................... 71
Figure 4.16. Load and deflection comparison between uncorroded and corroded finite
element model and theoretical calculations for the uncracked section. .. 72
Figure 4.17. Comparison of compression concrete output to input values for 3-D
beam with 3-D rebar. ............................................................................... 84
Figure 4.18. Comparison of tensile concrete output to input values for 3-D beam with
3-D rebar. ................................................................................................. 85
Figure 4.19. Time comparison for 3-D concrete beam models with 3-D rebar using
brittle cracking. ........................................................................................ 87
Figure 5.1. Map of location of Bridge 7R (Ross 2007). ............................................ 94
Figure 5.2. Finite element model of bridge 7R viewed from (a) the top, (b) the
bottom, and (c) isoparametrically. ........................................................... 95
Figure 5.3. 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. ................................................................... 97
Figure 5.4. Representative example of location of deck element used for stress
distribution analysis in (a) the elastic, (b) uncorroded and (c) corroded
deck. ......................................................................................................... 99
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Figure 5.5. Tension stiffening values for uncorroded model. .................................. 101
Figure 5.6. Stress-strain response of representative deck element of Bridge 7R at a
stress concentration location using the original tension stiffening input.
............................................................................................................... 103
Figure 5.7. Stress-strain response of element in the deck of Bridge 7R at a stress
concentration location using a tension stiffening value of 0.035. ......... 104
Figure 5.8. Comparison of nonlinear response of concrete for bridge 7R for the (a)
uncorroded and (b) corroded deck. ........................................................ 108
Figure 5.9. Satellite View of SR 1 over US 13 (Ambrose 2012). ............................ 110
Figure 5.10. Finite element model of bridge US13 viewed from (a) the top, (b) the
bottom, and (c) in cross-section. ............................................................ 111
Figure 5.11. 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. ................................................................................. 113
Figure 5.12. Compressive stress-strain response of elements in deck of US13 causing
inability to converge. ............................................................................. 114
Figure 5.13. Tensile stress-strain response of elements in deck of US13 causing
inability to converge. ............................................................................. 114
Figure 5.14. Satellite view of SR 299 over SR 1 (Ambrose 2012). .......................... 117
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ABSTRACT
Current analysis techniques do not acknowledge the existence of load
redistribution between girders, or system capacity, of bridges due to a lack of
understanding on the redistribution mechanisms. This lack of understanding is the
primary motivation for this research. Specifically, the deck as a load redistribution
mechanism is analyzed. It is thought that including the system capacity of bridges
would help to prioritize repairs and allocate the limited funding available for
infrastructure. For this reason, this research aims to aid in quantifying the system
capacity effects of bridges due to corrosion of reinforcement in the deck.
Previously, a literature review was executed to determine the effects of
corrosion in reinforced concrete. It was determined through this review of testing that
the change in performance due to corrosion is best estimated as a strength decrease of
50% and an ultimate deflection increase of 82%, mimicking 25 years of corrosion.
These expected performance metrics were used to create finite element models of
uncorroded and corroded reinforced concrete beams. Different concrete material
modeling techniques available within the commercial software ABAQUS were
assessed; these include brittle cracking, smeared crack, and concrete damaged
plasticity techniques. This was done for both 2-dimensional and 3-dimensional
concrete elements, as well as for both 2-dimensional and 3-dimensional rebar elements
within the 3-dimensional concrete elements. In the end, it was determined that using
the concrete damaged plasticity approach with 2-dimensional beam and rebar elements
produced the most accurate results and was also the easiest approach to implement in
existing full-scale bridge models; this approach will also reduce computational effort
in any future full-scale bridge models.
xiv
After the modeling technique was determined, the input was calibrated to
determine the optimal approach to model uncorroded and corroded reinforced
concrete. It was determined which input values to use for the uncorroded concrete,
and how to alter these values to simulate corrosion. Through this optimization
process, it was determined that a 40% decrease in the modulus of elasticity of
concrete, 40% decrease in tensile strength of concrete, 64% decrease in compressive
strength of concrete, 20% decrease in area of compressive steel, and 61.5% decrease
in area of tensile steel resulted in the optimum simulation of reinforced concrete
corrosion.
This uncorroded and corroded input was applied to 3 different full-scale bridge
models which were previously created; these models were created and calibrated
based on actual bridges located in Delaware that had been previously field tested, all
having steel girders. It was found that this modeling approach created convergence
difficulties in some of the bridge models when attempting to load the structures to
their ultimate capacities and subsequently only the results of one of the bridges, for
which convergence was obtained up through a peak loading, was analyzed in depth.
This bridge is referred to as Bridge 7R and served as an exit ramp for Interstate 295
North, just south of the Delaware Memorial Bridge. Initially, convergence with
corroded models was not reached. However, after changing the input parameters
governing tension stiffening, convergence was achieved.
The maximum loading and the distribution factors of the bridge models were
analyzed. It can be seen in these results that the corrosion in the deck caused a more
uniform stress distribution in the deck, and consequently the girders, than with an
uncorroded deck. Contrary to the expected response, the corroded model resulted in
xv
DF values between those of the elastic and those of the uncorroded models; however,
in all models the DF approached the theoretical inelastic values. The results of these
models also indicated that the corroded models reached higher strengths than their
uncorroded counterparts. It was thought that the cause of the differences in both the
strength and DF values was due to greater load sharing in the corroded model when
compared to the corresponding uncorroded model.
1
Chapter 1
INTRODUCTION
1.1 Motivation
Current analysis techniques for bridges, as specified by the American
Association of State Highway and Transportation Officials (AASHTO) (2013),
specify that each individual component of a bridge be designed separately to be
capable of carrying the maximum possible loading which may be applied. This
includes utilizing a line girder analysis. However, in reality, bridges act as a system,
redistributing the load when members begin yielding. This produces a higher strength
than through the analysis of individual components. This true strength can then be
used to help better prioritize bridge repair and replacement.
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.
It was previously shown by McCarthy (2012) that including the load path
redundancy of bridges in the rating process can create a significant savings when
prioritizing bridge repairs. This analysis reviewed 14 steel girder-concrete deck
2
bridges located in Delaware and determined that, when including load path
redundancy in the load carrying capacity, a savings between $2 and $4.7 million from
the estimated $23.4 million could be achieved. This $23.4 million is the approximated
cost of repairing the selected structures based on their current conditions.
One of the current impediments to implementing system capacity analysis of
bridges is the lack of information regarding transverse load distribution mechanisms
and how, in the case of transverse distribution through concrete decks (which tend to
suffer the most accelerated degradation of condition of all structural components due
to factors such as direct traffic loads and applications of deicing agents), these
mechanisms may vary with age and condition. For this reason, the deck is the load
distribution mechanism to be investigated and, due to corrosion of rebar being one of
the most prevalent causes of deterioration facing bridge decks, the influences of this
corrosion on the system capacity of steel girder bridges is the focus of this study.
Previously, a literature review was performed to quantify the effects of corrosion on
the behavior of reinforced concrete specimens and these results were synthesized and
extrapolated to a deck design life of 25 years (McConnell et al. 2012). This research
focuses on these behavior targets and uses them to calibrate a finite element model of
reinforced concrete members, through varying selected input parameters. These
calibrated values of input parameters are then applied to bridge models to estimate the
change in system capacity.
1.2 Objectives and Scope
The two primary objectives of this project were to calibrate a finite element
modeling technique to model reinforced concrete which has experienced corrosion due
to deicing agents and then apply this technique to full-scale bridge models in order to
3
determine how corrosion of reinforcing bars in the deck effects the system capacity of
bridges. Specifically, a reinforced concrete beam was modeled to determine the
material input values required to emulate corrosion using ABAQUS. After
performing a literature review to determine the different modeling approaches for
modeling reinforced concrete within ABAQUS, models were created using three
different approaches. These include the smeared crack, concrete damaged plasticity,
and brittle cracking techniques. The previous strength and deflection results for a 25
year old deck were utilized as the response goals of the corroded concrete. The input
parameters, as well as the different modeling approaches and commands, were then
calibrated to reflect these corrosion goals.
Once the input was calibrated, it was applied to three previously created and
calibrated full-scale bridge models; these models were based off of bridges which are
located in Delaware and previously field tested (Michaud 2011 and Ambrose 2012).
All 3 of these bridges are steel girder bridges with composite reinforced concrete
decks due to steel girder bridges being a common bridge configuration for which the
advantages of system analysis has been previously demonstrated (Michaud 2011).
One of the three bridges is a simple span bridge and the other two are 2-span
continuous structures. These models were loaded in attempts to estimate their ultimate
capacity. The ultimate capacity with the uncorroded input and the corroded input
were compared and this change of strength, caused by the corrosion in the deck, was
analyzed to determine the effects on load redistribution ability.
1.3 Thesis Outline
In the pages that follow is an investigation of how corrosion of reinforced
concrete bridge decks due to deicing agents effects the system capacity of bridges.
4
The primary research tasks were the development of the finite element models as well
as the calibration of these models to demonstrate corrosion, followed by the
application of this calibration to full-scale bridge models. The material has been
divided into the following chapters.
Chapter 2 presents the background of the project, including a literature review of
how to model reinforced concrete beams and how corrosion effects the material
properties of reinforced concrete.
Chapter 3 describes the relevant modeling approaches available within ABAQUS
and an associated discussion of how these approaches and corresponding
commands were utilized.
Chapter 4 presents the results of the reinforced concrete beam models which were
created and the calibration process applied.
Chapter 5 discusses the application of the calibrated reinforced concrete input to
the full-scale bridge models and the results of this analysis.
Chapter 6 presents the conclusion of the research; this chapter also provides
recommendations for future work for better calibrating beam models and
processing bridge result data.
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Chapter 2
LITERATURE REVIEW
2.1 Previous Experimental Results
A literature review was performed focusing on the proof of concept underlying
the motivation for this work and experimental testing performed involving reinforced
concrete that had undergone corrosion. This was done to prove how system capacity
is significantly larger than the strength calculated using standard analysis techniques
and to determine how the strength and deflection of reinforced concrete changes as
reinforced concrete corrodes. Many different tests have been performed to cause
accelerated corrosion; however, many of these studies used an electrical current to
induce corrosion. This method has been questioned by other researchers and therefore
all tests found using this method were disregarded (Li 2000 and Melchers et al. 2006).
Below, the proof of concept and corroded reinforced concrete studies which were
utilized are discussed.
2.1.1 Destructive Bridge Testing
Previous research was performed which destructively tested a full-scale bridge
to determine the ultimate strength (Chajes et al. 2010). For this testing, hydraulic
jacks were placed on a bridge to mimic the presence of an HS-20 truck. However, the
hydraulic jacks and associated load resistance mechanism did not have sufficient
capacity to cause the bridge to fail, although previous predictions using current
evaluation methods (AASHTO 2012) indicated that this should not be the case. A
finite element model was then created of the same bridge. This model was loaded
until failure and the results were compared to the predictions using current evaluation
methods. The results of this comparison can be seen in Table 2.1. Note that the
6
strength reported for the loading system is the maximum force which could be applied
to the bridge, not the force causing failure. It can be seen in these results that the code
predictions are very conservative compared to the actual results and the finite element
predictions.
Table 2.1. Comparison of strength of DRBA Bridge 7R (Ross 2007).
Quantification Method
Strength
Number of Equivalent
HS-20 Trucks Load (MN)
Code-Predicted, with load factors 5 1.6
Code-Predicted, no load factors 12 3.8
Loading System 17 5.4
FEA, Girder Yielding 19 6.1
FEA, System Failure 30 9.6
These results reinforce the concepts that load redistribution in bridges
increases the system capacity and that using finite element analysis can help predict
the true strength. The code-predicted capacity of 12 trucks is based on the same
criteria, load to induce yielding in a bottom flange, as the FEA girder yielding strength
of 19 trucks. This demonstrates conservatism in the existing methods for distributing
load to individual girders. However, these system capacities are based on linear-
elastic concrete deck properties. The approach proposed in this work is to extend this
analysis to include the inelastic response of concrete post-cracking and determine how
this change affects the system capacity of bridges.
2.1.2 Corroded Reinforced Concrete
Li and Zheng (2005), Oyado et al. (2011), and, less significantly, Gu et al.
(2010) were three different studies relied upon in the present work. In these studies,
7
reinforced concrete beams were constructed and accelerated corrosions techniques
were utilized, with the exception of Gu et al. (2010) which analyzed an existing
structure which has undergone natural corrosion. These studies and their relevant
results are described below.
2.1.2.1 Experimental Set-Up
Each of the tests utilized a different test set-up. The first test, Li and Zheng
(2005), used an accelerated salt-spraying technique and then converted the accelerated
time to real time. A marine environment was simulated by alternating wetting and
drying cycles of saltwater spray in a chamber constructed specifically for the testing.
That is, the beams were sprayed for 2 hours then let sit for an hour before being
sprayed for another 2 hours. The relative humidity of the chamber was also
controlled. The test accelerated corrosion by spraying sodium chloride directly on the
cracks and intensifying the drying phases of the wetting and drying cycles. This
accelerated test was calibrated using identical specimens under the natural
environment and the accelerated time was converted to real time. The accelerated test
took place over a seven month period, equivalent to approximately 9 years in real
time. During this testing, both strength and deflection data were recorded initially, at
3 months, 5 months, and 7 months.
In the second test, Oyado et al. (2011), the beams were placed outdoors, in an
urban area away from the coast, for 3 months; after this time, the beams were sprayed
with a saline solution 3 times a day for 17 months in attempts to accelerate corrosion.
After this time, some of the beams were strength tested and the remaining beams were
left outdoors for a total of 12 years, after which time they were also strength tested.
8
The first set of testing, performed at 20 months, provided only strength data. The
second set of testing, performed at 12 years, provided strength and deflection data.
The final test, Gu et al. (2010), only included stiffness data. However, this
report was vague regarding the condition of the concrete which was tested, stating it
was removed from a building which had “gone through decades of natural corrosion”.
For this reason, less emphasis was placed on the results of this testing and it was used
more for comparison than specific quantitative data.
2.1.2.2 Results of Testing and Concrete Performance Goals
The strength losses reported by Li and Zheng (2005) and Oyado et al. (2011)
are summarized in Figure 2.1. In this figure, strength loss is quantified by the
decrease in strength between the corroded and uncorroded specimens normalized by
the strength of the original, uncorroded, beam. The stiffness losses reported by the
previously mentioned literature are summarized in Figure 2.2. In this figure, change
in stiffness is quantified by the increase in ultimate deflection between corroded and
uncorroded specimens normalized by the ultimate deflection of the uncorroded
specimen.
9
Figure 2.1. Strength losses reported in literature (McConnell et al. 2012).
Figure 2.2. Stiffness losses reported in literature (McConnell et al. 2012).
When reviewing the strength loss values, a trend can be observed in that the
data from Li and Zheng (2005) which suggests a linear relationship between chloride
exposure time and strength loss. Comparing this data to Oyado et al. (2011) at 20
10
months, it can be observed that greater strength losses are reported. This is possibly
due to the fact that, for months 3 to 20, the Oyado et al. (2011) specimens were
sprayed with a chloride solution 3 times per day. It can be hypothesized that this
accelerated the corrosion of these specimens over this time period. If the linear trend
using the Li and Zheng (2005) data is extrapolated, it intersects with the 12 year
Oyado et al. (2011) data.
Using this relationship, it can be concluded that the strength loss data from Li
and Zheng (2005) and Oyado et al. (2011) are in general agreement with one another.
For this reason, a linear trend using the data from Li and Zheng (2005) and the 12 year
data from Oyado et al. (2011) was fit. The 20 month data from Oyado et al. (2011)
was not included for the previously stated reasons involving the unexpectedly high
strength loss values. This resulted in the following relationship, shown in Equation
2.1.
𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ 𝑙𝑜𝑠𝑠(%) = 2.0002 ∗ (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠) [2.1]
This relationship was extrapolated for a 25 year design life, resulting in a prediction of
50% strength loss. With a lack of better data, this is used as the assumed strength loss
chosen to represent a corroded deck condition in the following analyses.
When reviewing the stiffness data, only the results from Li and Zheng (2005)
includes data from multiple time periods. This data indicates a bi-linear relationship
between ultimate deflection and time where change in deflection increases rather
rapidly for the first 4 years of exposure and then becomes more gradual. Extrapolating
the second linear portion of the trend line results in the Li and Zheng (2005) data
intersecting with the Oyado et al. (2011) data; for this reason, the second portion of the
11
bi-linear curve is fitted with a linear trend line and used to extrapolate the increase in
deflection that would be expected after 25 years of chloride exposure. This trend line
resulted in the following relationship, shown in Equation 2.2.
𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒(%) = 2.0347 ∗ (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑦𝑒𝑎𝑟) + 31.559 [2.2]
This equation indicated an 82% increase in ultimate deflection after 25 years. With a
lack of better data, this is used as the assumed deflection increase chosen to represent
a corroded deck condition in the following analyses.
It should also be noted that information regarding changes in mass, due to
spalling of concrete or corroding of reinforcing rebar, was also evaluated. However, it
was decided that these changes in mass were encompassed by the observed changes in
strength and deflection; therefore, it can be considered a cause of observed changes
versus an effect in the following analyses.
2.2 Reinforced Concrete and Corrosion Modeling
In order to determine the optimal way to model reinforced concrete, a literature
review on available modeling techniques was performed. When modeling rebar and
reinforced concrete, different methods were investigated. The main difference
between these modeling approaches is whether the rebar is modeled as 2-dimensional
(2-D) or 3-dimensional (3-D) elements (Simulia 2011). The 2-D rebar can be
embedded into the concrete, creating a perfect bond between the concrete and rebar.
However, the 3-D rebar allows for more properties discussed in prior research to be
incorporated; these properties included debonding of concrete and rebar which causes
the friction between the concrete and rebar to vary, considered by both Val et al.
12
(2009) and Amleh and Ghosh (2006); increased contact pressure between the rebar
and the surrounding concrete due to the creation of rust during the corrosion process
(Amleh and Ghosh 2006); and associated mass loss of rebar (Amleh and Ghosh 2006).
For the purposes of this research, papers which used ABAQUS were isolated.
These were reviewed and emphasis was placed on these techniques as they would be
easiest to replicate due to models in this file format already existing. Val et al. (2009)
describes a technique to map cracking of corroded reinforced concrete beams. This
approach used 2-D elements for both concrete and rebar, using Rankine criterion to
define crack initiation. Friction between the reinforcing bars and the surrounding
concrete is modeled using Coulomb’s friction model with a constant coefficient of
friction. The corroded rebar was modeled using a penalty contact algorithm for the
interaction between the rebar and concrete, which defines the master and slave
surfaces and how far the slave surface can penetrate into the master surface.
Deformability of corrosion products is neglected as a simplification; however, the
cross-sectional area of the rebar is increased to reflect the creation of corrosion
products. In this model, only corroded reinforced concrete is analyzed.
Amleh and Ghosh (2006) describes a pull-out test which was performed on
corroded concrete then modeled. This approach describes using 3-D elements for both
concrete and rebar, defining the concrete as the master surface and the rebar as the
slave surface. It describes using a pressure-overclosure relationship between the
contact surfaces to include the creation of corrosion by-products. In this study,
friction was defined between the concrete and rebar surfaces as decaying
exponentially from the static value to the kinetic value as the rebar begins to slip
within the concrete. This decay was then related to the mass loss of rebar caused by
13
corrosion. A more comprehensive description of these modeling techniques and how
they apply to ABAQUS are provided in Chapter 3.
In addition to articles which specifically used ABAQUS, other approaches to
modeling reinforced concrete and corrosion were reviewed to determine if they could
be applied to ABAQUS models. Fang et al. (2006) modeled the rebar as 3-D elements
with rust modeled as a granular material between the rebar and concrete elements. A
similar approach to modeling corrosion was used by Dekoster et al. (2003), modeling
rust as a component between the steel and concrete with mechanical properties almost
identical to that of water. Kallias and Rafiq (2010) modeled the rebar as truss
elements, representing the bond between the rebar and concrete using 2-D interface
elements. To model the corrosion, stress was increased in certain locations to mimic
pitting. Chen and Mahadevan (2007) also considered that corrosion by-product takes
up more volume than the structurally-sound amount of rebar lost to create the
corrosion products and therefore creates pressure. Coronelli and Gambarova (2004)
modeled the concrete as 2-D elements, with the rebar modeled as truss elements.
Bond-link elements were used to couple the concrete and rebar elements to one
another. To model corrosion, the cross-sectional area of the rebar was decreased and
the ultimate strains of the steel were decreased. Biondini and Vergani (2012)
specifically focused their model on the difference between uniform corrosion, pitting,
and a combination of the two. Three-dimensional elements were used for both the
concrete and rebar, and equations were created and input into the model to change the
cross-sectional area and shape of the rebar to reflect the effects of the corrosion. The
cross-sectional area was uniformly decreased due to the creation of corrosion by-
product and/or decreased locally due to pitting by altering the cross-sectional shape.
14
All of the previously described models utilized a smeared crack approach to modeling
the cracking initiation and propagation in concrete.
An alternative approach to modeling reinforced concrete is called the
multifiber approach (Richard et al. 2011 and Adelaide et al. 2012). This approach is
primarily used to decrease the global computational costs. The concrete and steel are
represented by fibers and the equilibrium equations provide the nodal displacements
and rotations. However, this technique does not provide accurate local results in terms
of cracking pattern, crack opening, crack spacing etc. and was therefore considered not
able to accurately model small scale beams as required in this research.
15
Chapter 3
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 3.1) and
rebar that are used as well as the different approaches to modeling corrosion of the
rebar (Section 3.2), 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 3.3) 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 3.4).
3.1 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-
linear behavior of the material governs, including the post-cracking response. The
elastic commands are described in Section 3.1.1 and the non-linear behavior is
described in Section 3.1.2.
16
3.1.1 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 3.1 and 3.2, where Ec, f’c, and f’t are expressed in psi.
𝑓′𝑡
= 7.5√𝑓′𝑐 [3.1]
𝐸𝑐 = 57,000√𝑓′𝑐 [3.2]
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.
3.1.2 Non-Linear Behavior
As stated previously, three different modeling techniques for modeling non-
linear behavior of concrete are available within ABAQUS: the smeared cracking
model (SC), the concrete damaged plasticity model (CDP), and the brittle cracking
model. These are described in the following Sections 3.1.2.1 to 3.1.2.3, respectively.
17
3.1.2.1 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
tension stiffening relationship can be seen in Figure 3.1. ABAQUS also takes into
consideration the change in shear modulus of the concrete, which affects the shear
18
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 3.4.
Figure 3.1. Tension stiffening model ABAQUS employs in the smeared crack
technique (Simulia 2011).
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
strain minus the direct strain at cracking. The final command used is the “failure
19
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 4.4.1.3.
3.1.2.2 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 3.2 (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
undamaged elasticity matrix. A similar approach is used to model the compressive
20
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 3.2 (b).
Figure 3.2. Response of concrete to uniaxial loading in (a) tension and (b)
compression for CDP model (Simulia 2011).
21
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 3.4.
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 post-
cracking 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 post-
cracking 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
22
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 4.4.1.2.
3.1.2.3 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 3.4.
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
23
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 4.5.1.1.
3.2 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 3.2.1 and 3.2.2.
3.2.1 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
24
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 cross-
sectional area and strength; no changes can be made to affect the bond between the
rebar and concrete.
3.2.2 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
(γ̇eq) and a decay coefficient (dc), as shown in Figure 3.3 (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 3.3.
25
𝜇 = 𝜇𝑘 + (𝜇𝑠 − 𝜇𝑘)𝑒−𝑑𝑐�̇�𝑒𝑞 [3.3]
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 4.5.2.2.
Figure 3.3. Exponential decay friction model (Simulia 2011).
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 γ̇eq is calculated at each loading increment automatically by ABAQUS,
then applied using Equation 3.3. In addition to the “friction command,” the “surface
interaction” and “contact pair” commands are required. The “surface interaction”
26
command creates a surface interaction property definition; this command defines a
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 3.4. 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.
27
Figure 3.4. Default “hard” pressure-overclosure relationship (Simulia, 2011).
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
exponential law (Amleh and Ghosh 2006), which is shown in Figure 3.5. 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,
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 4.5.2.2.
28
Figure 3.5. “Softened” exponential pressure-overclosure relationship (Simulia 2011).
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 3.4, where p0 is expressed in MPa and C is expressed
in mm.
𝑝0 = 0.128𝐶 + 1.5 [3.4]
29
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 3.5, which is then multiplied by p0 to
determine a new pressure at zero clearance for corroded rebar (Amleh and Ghosh
2006).
𝐿 = [(−0.00024𝑓′𝑐
− 0.0028)𝐶 + 4.3]𝑀 [3.5]
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 3.6 and 3.7,
respectively (Amleh and Ghosh 2006), where M is expressed as a percentage.
𝜇𝑠 = 𝑒−0.035𝑀 [3.6]
𝑑𝑐 = 0.0261𝑀 + 0.45 [3.7]
3.3 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
30
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
and the elements attached to those nodes displayed no deformation before the model
failed.
3.4 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 time-
dependent material effects such as creep, swelling, and viscoelasticity; however, it
31
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
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,
32
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
input values which can be more readily estimated based on common concrete
properties.
33
Chapter 4
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 3.1 and 3.2. 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
Section 2.1.2.2. The geometry and material properties which were modeled are
described in Section 4.1, with the concrete modeling described in Section 4.1.1 and the
rebar modeling described in Section 4.1.2. Section 4.2 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 4.3.
The 2-D beam model is described in Section 4.4. Within this section, the
uncorroded base model input to simulate an uncorroded reinforced concrete beam is
described in Section 4.4.1, the mesh sensitivity analysis and resulting final uncorroded
model input are described in Section 4.4.2, and the calibration and determination of
the input for the corroded model are described in Section 4.4.3.
In addition to the 2-D beam model, a 3-D beam model was created. This
model is described in Section 4.5. Initially, this modeling was performed with 2-D
rebar elements, as described in Section 4.5.1; the model was then refined by using 3-D
34
rebar elements, as described in Section 4.5.2. The conclusions drawn from these
different modeling techniques can be found in Section 4.6.
4.1 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 4.1.1 and 4.1.2,
respectively.
4.1.1 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 type conducted by
McConnell et al. (2012), which is described in Section 2.1.2. The geometry of this
beam can be seen in Figure 4.1. 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.
35
Figure 4.1. 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 3.2 and 3.1, 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
36
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).
4.1.2 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 4.1.1, 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 4.1, 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 4.1. 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
37
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 4.1. 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)
SD345 53,809 8% 82,671 55,694 11% 78,465
SD295 54,679 28% 77,885 48,588 14% 91,519
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 4.1, where σy is the yield stress value and εy is the
corresponding yield strain value.
𝐸𝑠 =𝜎𝑦
𝜀𝑦 [4.1]
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
38
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 4.2, where σ1 is in psi.
𝜎1 = 𝜎𝑦 + 145,000(0.011 − 𝜀𝑦) [4.2]
The strain corresponding to the ultimate stress was calculated using Equation 4.3,
where εu is the strain corresponding to the ultimate stress and σu is the ultimate stress
in psi.
𝜀𝑢 = 0.011 +𝜎𝑢−𝜎1
720,000 [4.3]
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 4.4,
where σ2 is in psi.
𝜎2 = 𝜎𝑢 + 145,000(0.3 − 𝜀𝑢) [4.4]
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 4.5, where σtrue is the true stress value, σeng is the engineering stress value,
and εeng is the corresponding engineering strain.
𝜎𝑡𝑟𝑢𝑒 = 𝜎𝑒𝑛𝑔(1 + 𝜀𝑒𝑛𝑔) [4.5]
39
Lastly, the plastic logarithmic strain is calculated by using Equation 4.6, where εlnplastic
is the logarithmic plastic strain.
𝜀𝑙𝑛𝑝𝑙𝑎𝑠𝑡𝑖𝑐 = ln(1 + 𝜀𝑒𝑛𝑔) −
𝜎𝑡𝑟𝑢𝑒
𝐸𝑠 [4.6]
The plastic input values for SD295 and SD345 rebar resulting from these equations
can be found in Table 4.2.
Table 4.2. True stress and true plastic strain input values for SD295 and SD345 rebar.
SD295 SD345
σtrue (psi) εlnplastic σtrue (psi) εln
plastic
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
4.2 Calibration Metrics
In order to directly compare the different modeling techniques discussed in
Section 3.1, 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
40
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 3.4; 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
41
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 4.2, where Figure 4.2 (a) shows the last
step before failure and Figure 4.2 (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.
Figure 4.2. 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
(a) (b)
42
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.
4.3 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 seen in
Appendix A.
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
4.1. This corresponds to an LPF of 0.914 based on a total applied load of 5901.4 lbs
43
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 4.4.3.
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.
4.4 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,
44
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 4.4.1. 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 4.4.2.
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 4.4.3.
4.4.1 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 4.4.1.1 through 4.4.1.3.
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4.4.1.1 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 4.1), 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 4.1. 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.
4.4.1.2 Concrete Damaged Plasticity
The elastic input values previously described in Section 3.1.1 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”,
46
and “concrete compression damage” commands, as described in Section 3.1.2.2. 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 4.3. It should be
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.
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Table 4.3. 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
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.
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4.4.1.3 Smeared Crack
As with the CDP model, the elastic commands and corresponding input values
used are as described in Section 3.1.1. A description of each command and how it
related to the SC model is detailed in Section 3.1.2.1. The first command for post-
cracking 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
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.
4.4.2 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
49
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 4.4.1 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 4.4 and Table 4.5, 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
calculations described in Section 4.3, 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).
50
Table 4.4. Strength results of mesh sensitivity analysis with percent differences based
on the calculated theoretical strength values.
Mesh Size
(mm)
Concrete
Model
Highest Load
(LPF)
% Difference from
Expected Load
100 CDP1 1.75 92%
SC1 4.72 417%
50 CDP 2.55 179%
SC1 3.02 231%
25 CDP1 0.89 -3%
SC 4.10 349%
20 CDP 1.10 20%
SC1 2.98 226%
10 CDP 1.08 18%
SC1 2.39 162%
5 CDP 3.69 304%
SC1 10.2 1017% 1Model terminated at peak load, indicating inability to converge.
Table 4.5. 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
Model
Highest
Deflection (in)
% Difference,
Uncracked
% Difference,
Cracked
100 CDP1 0.0533 196% -57%
SC1 0.3831 2022% 208%
50 CDP 0.2820 1462% 127%
SC1 1.4130 7728% 1035%
25 CDP1 0.0233 29% -81%
SC 0.2109 1068% 69%
20 CDP 0.0435 141% -65%
SC1 0.1361 654% 9%
10 CDP 0.0425 135% -66%
SC1 0.0993 450% -20%
5 CDP 0.0024 -87% -98%
SC1 0.0069 -62% -94% 1Model terminated at peak load, indicating inability to converge.
51
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 4.4. Thus, the 20% increase in strength
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 4.4.1.2 were used as
the uncorroded material input and the results, outlined in Tables 4.4 and 4.5, were
used as a baseline for comparison to determine if a 50% decrease in strength and 82%
increase in deflection, for reasons described in Section 2.1.2.2, were achieved during
the calibration process. The stress contours of this uncorroded model can be seen in
52
Figure 4.3, 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 4.3.
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 4.3. Thus, this
deflection was plotted versus load in Figure 4.4, 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
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 4.5, 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 4.4 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 4.6.
53
Figure 4.3. Uncorroded 2-D base model stress contours (units of psi).
Figure 4.4. Load and deflection comparison between final uncorroded model and
theoretical calculations for the uncracked and cracked sections.
0
0.2
0.4
0.6
0.8
1
1.2
0 0.1 0.2 0.3 0.4 0.5
Load
(L
PF
)
Deflection (in)
FEA Uncracked Cracked
54
Table 4.6. Input and results of final uncorroded 2-D beam model.
Inp
ut
Dilation Angle 38 Es (ksi) 29000
ϵ 1 f'y (psi) 47000
σb0/σc0 1.12 fy (psi) 55000
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) 29000
Res
ult
s Strength (LPF) 1.1
Deflection (in) 0.0435
4.4.3 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
55
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 4.4.3.1. 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 4.4.3.2, the optimization of A’s and As is described in Section 4.4.3.4,
followed by the calibration of the dilation angle, described in Section 4.4.3.4. During
the optimization process, the results of the models were compared to the strength and
deflection results expected through the theoretical calculations described in Section
4.3, where the basis for this comparison is also described.
4.4.3.1 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 Appendix B. For brevity, only the models which affected the
strength and deflection in the targeted manner (i.e., helped towards achieving the 50%
56
decreased strength and 82% increase in ultimate deflection goals) are included; these
can be found in Table 4.7. 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 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.
57
Table 4.7. 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
Inp
ut
Dilation Angle 38 100% 100% 105% 100% 100%
ϵ 1 100% 100% 100% 100% 100%
σb0/σc0 1.12 100% 100% 100% 100% 100%
Kc 2/3 100% 100% 100% 100% 100%
A's (in2) 0.0438 100% 75% 100% 100% 100%
As (in2) 0.20563 75% 75% 100% 100% 100%
E's (ksi) 29000 100% 100% 100% 100% 100%
Es (ksi) 29000 100% 100% 100% 100% 90%
f'y (psi) 47000 100% 100% 100% 100% 100%
fy (psi) 55000 100% 100% 100% 100% 100%
f'c (psi) 3147.3 100% 100% 100% 100% 100%
f't (psi) 420.8 100% 100% 100% 100% 100%
Ec (psi) 3197746 100% 100% 100% 75% 100%
Res
ult
s
Strength (LPF) 1.10 1.06 1.01 1.10 1.06 1.09
Deflection (in) 0.0435 0.0451 0.0430 0.0452 0.0501 0.0441
Change in
Strength N/A -3.64% -8.18% 0.00% -3.64% -0.91%
Change in
Deflection N/A 3.84% -1.05% 4.03% 15.14% 1.40%
It can be seen by the results in Table 4.7 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
most subsequent models will include a decrease in this value. Due to f’c, f’t, and Ec
being related through Equations 3.1 and 3.2, f’c and f’t were modified in conjunction
with the change in Ec for subsequent models according to this relationship.
58
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 4.4.3.2, 4.4.3.3, and 4.4.3.4, respectively.
4.4.3.2 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 4.8. 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 4.5. 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
seen in Figure 4.6. 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.
59
Table 4.8. 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
Inp
ut
Dilation Angle 38 100% 100% 100% 100% 100%
ϵ 1 100% 100% 100% 100% 100%
σb0/σc0 1.12 100% 100% 100% 100% 100%
Kc 2/3 100% 100% 100% 100% 100%
A's (in2) 0.0438 100% 100% 100% 100% 100%
As (in2) 0.20563 100% 100% 100% 100% 100%
E's (ksi) 29000 100% 100% 100% 100% 100%
Es (ksi) 29000 100% 100% 100% 100% 100%
f'y (psi) 47000 100% 100% 100% 100% 100%
fy (psi) 55000 100% 100% 100% 100% 100%
f'c (psi) 3147.3 90% 81% 56% 49% 36%
f't (psi) 420.8 95% 90% 75% 70% 60%
Ec (psi) 3197746 95% 90% 75% 70% 60%
Res
ult
s
Strength (LPF) 1.10 1.06 1.02 0.852 0.798 0.689
Deflection (in) 0.0435 0.0435 0.0435 0.0412 0.0406 0.0394
Change in
Strength N/A -3.64% -7.27% -22.5% -27.5% -37.4%
Change in
Deflection N/A 0.04% 0.03% -5.32% -6.71% -9.26%
60
Figure 4.5. Comparison of strength between results from calibration models and
theoretical values caused by varying Ec.
Figure 4.6. Comparison of deflection between results from calibration models and
theoretical values of the uncracked section caused by varying Ec.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 5 10 15 20 25 30 35 40
Str
ength
(L
PF
)
% Decrease in Ec
Model Theoretical
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 5 10 15 20 25 30 35 40
Def
lect
ion (
in)
% Decrease in Ec
Model Theoretical Uncracked
61
Using a decrease in Ec of 40%, as seen in Model 10 of Table 4.8, 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
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.
4.4.3.3 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 4.7, Figure 4.8,
and Figure 4.9 corresponding to an A’s of 100%, 90%, and 80% of the uncorroded
model value, respectively. The remaining input variables are consistent with those
used in Model 10, as shown in Table 4.8. Again, the black line represents a 50%
strength decrease from the uncorroded base model.
62
Figure 4.7. 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.
Figure 4.8. 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.
0.2
0.3
0.4
0.5
0.6
0.7
0.8
25 35 45 55 65 75
Str
ength
(L
PF
)
% Decrease As
Model Theoretical
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
45 50 55 60 65 70
Str
ength
(L
PF
)
% Decrease As
Model Theoretical
63
Figure 4.9. 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.
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 4.10, Figure 4.11, and Figure 4.12 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.
0.25
0.35
0.45
0.55
0.65
0.75
48 53 58 63
Str
ength
(L
PF
)
% Decrease As
Model Theoretical
64
Figure 4.10. 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.
Figure 4.11. 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.
0.023
0.033
0.043
0.053
0.063
0.073
0.083
40 45 50 55 60 65 70
Def
lect
ion (
in)
% Decrease As
Model Theoretical
0.023
0.033
0.043
0.053
0.063
0.073
0.083
50 55 60 65
Def
lect
ion (
in)
% Decrease As
Model Theoretical
65
Figure 4.12. 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.
As can be seen in the results of Figures 4.7 through 4.12, 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
4.13, 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 4.14 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 4.14 that the erratic
0.023
0.033
0.043
0.053
0.063
0.073
0.083
50 55 60 65
Def
lect
ion (
in)
% Decrease As
Model Theoretical
66
deflection results are caused by the erratic strengths rather than both parameters
varying unpredictably.
Figure 4.13. Stress-strain response of element in 2-D beam models reported as percent
decrease in As.
Figure 4.14. Deflection response of elements in 2-D beam models reported as percent
decrease in As.
0
100
200
300
400
500
600
700
800
0 10 20 30 40 50
Str
ess
(psi
)
Strain
50%
60%
62%
62.25%
62.50%
62.75%
63%
65%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.01 0.02 0.03 0.04 0.05 0.06
Load
(L
PF
)
Deflection (in)
50%
60%
62%
62.25%
62.50%
62.75%
63%
65%
67
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 4.9.
Table 4.9. Input and results of 2-D beam model after calibrating input parameters.
Inp
ut
Dilation Angle 38
A's (in2) 0.03504
As (in2) 0.07711
f'c (psi) 1133
f't (psi) 252.4
Ec (psi) 1,918,648
Res
ult
s Strength (LPF) 0.603
Deflection (in) 0.0485
Change in Strength -45.2%
Change in Deflection 11.6%
4.4.3.4 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 4.4.3.1 indicated that changing the dilation angle increased the deflection.
Since this is the goal which was most difficult to achieve, it was thought that
68
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 4.10. 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
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.
69
Table 4.10. 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.
Inp
ut
Dilation
Angle 38 40 41 42 43 44 45
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) 36% 36% 36% 36% 36% 36% 36%
f't (psi) 60% 60% 60% 60% 60% 60% 60%
Ec (psi) 60% 60% 60% 60% 60% 60% 60%
Res
ult
s
Strength
(LPF) 0.603 0.624 0.624 0.624 0.625 0.482 0.484
Deflection
(in) 0.0485 0.0481 0.0480 0.0480 0.0480 0.0328 0.0330
Change in
Strength -45.2% -43.3% -43.3% -43.3% -43.2% -56.2% -56.0%
Change in
Deflection 11.6% 10.6% 10.5% 10.4% 10.4% -24.7% -24.1%
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
4.9. 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
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 4.15 (b) and (a),
70
respectively. To help determine the validity of the model, this data is compared to the
deflection results calculated for the uncracked and cracked sections using the same
loading applied in the model, as shown in Figure 4.16, 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 4.10, 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 4.16
(which plots the loading and unloading responses of the specimens, although the
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.
71
Figure 4.15. Stress contours of (a) final uncorroded and (b) final corroded 2-D beam
models (units of in psi).
(a)
(b)
72
Figure 4.16. Load and deflection comparison between uncorroded and corroded finite
element model and theoretical calculations for the uncracked section.
4.5 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 4.5.1 and 4.5.2, respectively.
0
0.2
0.4
0.6
0.8
1
1.2
0 0.1 0.2 0.3 0.4 0.5
Load
(L
PF
)
Deflection (in)
Uncorroded FEA Uncorroded Uncracked Uncorroded Cracked
Corroded FEA Corroded Uncracked Corroded Cracked
73
4.5.1 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 3.2.1. 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 4.5.1.1 through 4.5.1.3, where Section 4.5.1.1 describes input values
associated with the use of the brittle cracking technique, Section 4.5.1.2 details the
results of the mesh sensitivity analysis which was performed, and Section 4.5.1.3
outlines the attempted calibration of the corroded model input.
4.5.1.1 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 4.1 were used as
the uncorroded base model for the brittle cracking technique. The nonlinear input
commands were previously described in Section 3.1.2.3. 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 3.1 and 3.2 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
74
direct cracking strain using the same logic as previously explained in Section 4.1.1.
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.
4.5.1.2 Mesh Sensitivity Analysis
A mesh sensitivity analysis was performed for 2-D rebar elements. This brittle
cracking technique using the input described in Section 4.5.1.1 was utilized, along
with the CDP and SC techniques, described in Sections 4.4.1.2 and 4.4.1.3,
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 4.11 shows representative results of the rebar sensitivity in this analysis
and Table 4.12 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
75
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
results were also closest to the theoretical results while also providing the largest mesh
size to reduce computational effort.
Table 4.11. 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
5
CDP1 0.334 -63.4%
SC1 0.287 -68.6%
Brittle Cracking 0.450 -50.7%
10
CDP1 0.341 -62.7%
SC1 0.287 -68.6%
Brittle Cracking 0.450 -50.7%
20
CDP1 0.337 -63.1%
SC 0.230 -74.8%
Brittle Cracking 0.450 -50.7%
50
CDP1 0.339 -62.9%
SC1 0.234 -74.4%
Brittle Cracking 0.400 -56.2%
100
CDP 0.333 -63.5%
SC 0.234 -74.4%
Brittle Cracking 0.350 -61.7%
1Model terminated at peak load, indicating inability to converge.
76
Table 4.12. 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
5
CDP1 0.321 -64.8%
SC1 0.321 -64.8%
Brittle Cracking 0.400 -56.2%
10
CDP1 0.333 -63.5%
SC 0.234 -74.4%
Brittle Cracking 0.350 -61.7%
20
CDP 0.354 -61.2%
SC1 0.395 -56.7%
Brittle Cracking 0.600 -34.3%
25
CDP1 0.323 -64.6%
SC1 0.434 -52.5%
Brittle Cracking 0.450 -50.7%
50
CDP1 0.650 -28.8%
SC1 1.56 70.8%
Brittle Cracking 1.50 64.3%
100
CDP1 1.25 36.9%
SC1 1.67 82.9%
Brittle Cracking 2.25 146.4% 1Model terminated at peak load, indicating inability to converge.
A summary of this input and the results of the base model can be seen in Table
4.13. It should be noted that the expected strength calculated in Section 4.3 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 4.4.3;
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.
77
Table 4.13. Base model input for 3-D concrete elements with 2-D rebar elements for
brittle cracking technique.
Inp
ut
f't (psi) 420.8
Direct Cracking Strain 0.0027
Direct Cracking
Failure Strain 0.0027
Crack Opening Strain 0.0027
A’s (in2) 0.0438
As (in2) 0.20563 R
esu
lts Strength (LPF) 0.850
Deflection (in) 0.616
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
4.5.2.2.2.
4.5.1.3 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 4.14. It should be noted that in these models the
78
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
corroded model. The results from all of the calibration models can be found in
Appendix C.
Table 4.14. Selected results of optimization of corrosion for 3-D beam model with 2-
D rebar.
Model # Base 1 2 3 4
Inp
ut
f't (psi) 420.8 50% 25% 50% 100%
Direct Cracking
Strain 0.0027 50% 200% 50% 100%
Direct Cracking
Failure Strain 0.0027 50% 200% 50% 100%
Crack Opening
Strain 0.0027 50% 200% 200% 100%
A's (in2) 0.04235 100% 100% 100% 50%
As (in2) 0.20574 100% 100% 100% 50%
Res
ult
s
Strength (LPF) 0.425 0.375 0.425 0.4 0.225
Deflection (in) 0.616 0.34674 0.5629 0.41765 0.11086
Change in Strength N/A -11.8% 0.0% -5.9% -47.1%
Change in
Deflection N/A -43.7% -8.6% -32.2% -82.0%
79
It can be seen from Table 4.10 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 4.5.2.
4.5.2 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 4.5.2.1 and 4.5.2.2.
4.5.2.1 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
80
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
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.
4.5.2.2 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
3.1.2.3. 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 pressure-
overclosure 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 3.4. The
81
“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
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.
4.5.2.2.1 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 4.4.1.3 and 4.4.1.2, 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
82
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
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.
83
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 4.15; all of the models
which were tested, including both input and results, can be found in Appendix C.
These results show that the strengths of these models was less than 1/3 of the
calculated strength LPF of 0.914.
Table 4.15. Results of varying friction input for 3-D concrete beam models with 3-D
rebar elements.
µ 0.5 1.0 1.5 None
Concrete
Model CDP SC CDP SC CDP SC CDP SC
Strength
(LPF) 0.139 0.276 0.103 0.350 0.104 0.328 0.084 0.247
Deflection
(in) 0.0097 0.0175 0.0071 0.0223 0.0071 0.0207 0.0068 0.0184
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 4.17 and Figure 4.18, respectively, where the stress values reported are
84
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.
Figure 4.17. Comparison of compression concrete output to input values for 3-D
beam with 3-D rebar.
0
500
1000
1500
2000
2500
3000
3500
0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120 0.0140
Str
ess
(psi
)
Strain
Friction 0.5 Friction 1.5 Friction 1.0 Input
85
Figure 4.18. Comparison of tensile concrete output to input values for 3-D beam with
3-D rebar.
4.5.2.2.2 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
“MPC” commands were required. However, a few of the commands defining the
contact surface between the concrete and rebar, which were described in Section
4.5.2.2, 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 3.2.2, 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
0
50
100
150
200
250
300
350
400
450
0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012
Str
ess
(psi
)
Strain
Friction 0.5 Friction 1.5 Friction 1.0 Input
86
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 4.19, 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.
87
Figure 4.19. 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 4.16 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 4.19 as the 1 second model. These changes did not have a positive effect on
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.05 0.1 0.15
Load
(L
PF
)
Deflection (in)
1 Second 10 Seconds 60 Seconds
300 Seconds Uncracked Cracked
88
the results. Specifically, removing the pressure-overclosure relationship significantly
decreased the resulting strength.
Table 4.16. Comparison of variations in friction and pressure-overclosure of 3-D
concrete beam models with 3-D rebar using brittle cracking.
Time
(sec)
Pressure-
Overclosure
Used
Friction
Used
Strength
(LPF)
1
Exp Exp 0.750
N/A Exp 0.250
Exp 1.0 0.650
N/A 1.0 0.300
10
Exp Exp 0.460
N/A Exp 0.290
Exp 1.0 0.570
N/A 1.0 0.290
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 4.2. 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 4.17. It should be
noted that this new technique of determining the load at failure produced significantly
89
lower strength values than the previous technique. Considering that the initial strength
values were lower than the calculated values, these results were not ideal.
Table 4.17. Comparison of strength values utilizing different analysis techniques of 3-
D concrete beams with 3-D rebar using brittle cracking.
Time
(sec)
Pressure-
Overclosure
Used
Friction
Used
Original
Strength
(LPF)
Maximum
Stress
Strength
(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.
4.6 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
90
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
11.6% increase in deflection. These were the input values to be applied to the full-
scale 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
91
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
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.
92
Chapter 5
BRIDGE MODELS
The techniques and input which were calibrated in Section 4.4 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 5.1, 5.1.2.2,
and 5.3, respectively.
5.1 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 4.4, and was then used for
analysis. A description of the bridge geometry and location can be found in Section
5.1.1 and the results of the modeling are discussed in Section 5.1.2.
5.1.1 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.
93
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 5.1 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”.
94
Figure 5.1. Map of location of Bridge 7R (Ross 2007).
5.1.2 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 5.1.2.1 and the DF values are
discussed in Section 5.1.2.2.
5.1.2.1 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 5.2. The
95
uncorroded (Tables 4.3 and 4.6) and corroded (Table 4.9) non-linear input which was
calibrated, as described in Sections 4.4.2 and 4.4.3, 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 non-
linear concrete and plastic rebar commands.
Figure 5.2. 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 5.1. 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.
(a) (b) (c)
96
Table 5.1. Original capacity results for Bridge 7R.
Model Maximum Load
(kips)
Equivalent Number of
Trucks
Elastic 2178.4 30.3
Uncorroded1 1402.4 19.5
Corroded1 1084.5 15.1 1Model 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 5.3, showing (a) the elastic model with no non-linear concrete
and rebar commands, (b) the uncorroded non-linear model, and (c) the corroded non-
linear 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.
97
Figure 5.3. 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.
(a)
(b)
(c)
98
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 non-
linear 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 5.4 (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.
99
Figure 5.4. Representative example of location of deck element used for stress
distribution analysis in (a) the elastic, (b) uncorroded and (c) corroded
deck.
(a)
(b)
(c)
100
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
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 5.7, 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).
101
Figure 5.5. Tension stiffening values for uncorroded model.
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
input. It was also previously shown in Section 4.4.3.1 that including or removing
these commands does not affect the strength results.
0
100
200
300
400
500
600
700
800
0 0.02 0.04 0.06 0.08 0.1
Dir
ect
Str
ess
Aft
er C
rack
ing (
psi
)
Direct Cracking Strain
Original 0.01 0.035 0.1
102
These tension stiffening values were applied to the bridge models and the results
of these models can be seen in Table 5.2, 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.
Table 5.2. Results of bridge 7R when varying tension stiffening values.
Tension
Stiffening Model
Maximum
Loads (kips)
Equivalent
Number of Trucks
N/A Elastic 2178.4 30.3
Original Uncorroded1 1402.4 19.5
Corroded1 1084.5 15.1
0.1 Uncorroded 1795.0 24.9
Corroded 1813.7 25.2
0.035 Uncorroded 1795.0 24.9
Corroded 2047.5 28.4
0.01 Uncorroded 1888.5 26.2
Corroded 2047.5 28.4
1Model 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 stress-
strain response using the original tension stiffening and a value of 0.035 can be seen in
Figures 5.6 and 5.7, respectively, where the elastic input stress-strain response is also
103
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
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.
Figure 5.6. Stress-strain response of representative deck element of Bridge 7R at a
stress concentration location using the original tension stiffening input.
0
1000
2000
3000
4000
5000
6000
7000
0 20 40 60 80 100 120
Str
ess
(psi
)
Strain (x105)
Elastic Uncorroded Corroded
104
Figure 5.7. Stress-strain response of element in the deck of Bridge 7R at a stress
concentration location using a tension stiffening value of 0.035.
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 5.3. 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.
0
1000
2000
3000
4000
5000
6000
7000
0 50 100 150 200 250
Str
ess
(psi
)
Strain (x105)
Elastic Uncorroded Corroded
105
Table 5.3. Peak stress values in cross-frames as maximum loading in different 7R
bridge models.
Tension Stiffening Model Stress (psi)
N/A Elastic 62889
Original Uncorroded1 36000
Corroded1 34551
0.1 Uncorroded 43127
Corroded 73095
0.035 Uncorroded 53599
Corroded 88669
0.01 Uncorroded 59906
Corroded 88778
1Model terminated at peak load, indicating inability to converge.
5.1.2.2 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 bridge-
specific 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 in Section 2.1.1. The girders are numbered G1 through G4; this
numbering can be associated with Figure 5.2 (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 5.4.
106
Table 5.4. Distribution factors of bridge 7R previously determined in research
(McConnell et al. in review).
Distribution Factor
Analysis Method G1 G2 G3 G4
AASHTO (Elastic loading) 0.480 0.371 0.371 0.480
In-Service Field Test (Elastic loading) 0.334 0.368 0.330 0.289
Theoretical Inelastic Distribution Factor 0.250 0.250 0.250 0.250
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
following Table 5.5. 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.
107
Table 5.5. DF values for finite element models created of Bridge 7R.
Model Distribution Factor
Tension Stiffening Type G1 G2 G3 G4
N/A Elastic 0.257 0.267 0.252 0.224
Original Uncorroded1 0.265 0.321 0.224 0.190
Corroded1 0.260 0.314 0.223 0.203
0.01 Uncorroded 0.268 0.270 0.256 0.205
Corroded 0.260 0.262 0.252 0.226
0.035 Uncorroded 0.274 0.277 0.251 0.198
Corroded 0.260 0.262 0.252 0.226
0.1 Uncorroded 0.287 0.215 0.278 0.221
Corroded 0.272 0.274 0.253 0.201
1Model terminated at peak load, indicating inability to converge.
It should be noted that the DF results shown in Table 5.5 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
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 5.8 (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
108
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.
Figure 5.8. Comparison of nonlinear response of concrete for bridge 7R for the (a)
uncorroded and (b) corroded 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.
(a)
(b)
109
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.
5.2 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
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 5.2.1 and the results are described in Section 5.2.2.
5.2.1 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
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 5.9 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,
110
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 5.9. 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 cross-
frames consist of two 3 1/2” x 3 1/2" x 3/8” steel angles that comprise the inclined
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).
111
5.2.2 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 5.10. Again, the
uncorroded (Tables 4.3 and 4.6) and corroded (Table 4.9) non-linear input which was
calibrated, as described in Sections 4.4.2 and 4.4.3, 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 non-
linear concrete and plastic rebar commands. The results of this initial analysis can be
seen in Table 5.6. It can be seen in these results that the models which include non-
linear 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.
Figure 5.10. Finite element model of bridge US13 viewed from (a) the top, (b) the
bottom, and (c) in cross-section.
(a)
(b)
(c)
112
Table 5.6. Initial results from analyzing bridge US13 including non-linear concrete
and rebar commands.
Model Maximum Load (kips) Equivalent Number of
Trucks
Elastic1 3461.5 48.1
Uncorroded1 41.2 0.6
Corroded1 30.0 0.4 1Model 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 5.11, showing (a) the elastic model with no non-linear concrete
and rebar commands, (b) the uncorroded non-linear model, and (c) the corroded non-
linear 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
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 5.12 and 5.13 for compression and tension, respectively. It can be noted in the
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.
113
Figure 5.11. 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.
(a)
(b)
(c)
114
Figure 5.12. Compressive stress-strain response of elements in deck of US13 causing
inability to converge.
Figure 5.13. Tensile stress-strain response of elements in deck of US13 causing
inability to converge.
0
1000
2000
3000
4000
5000
6000
0 200 400 600 800 1000 1200 1400
Str
ess
(psi
)
Strain (x105)
Element 1 Element 2 Input
0
100
200
300
400
500
600
700
0 20 40 60 80 100 120
Str
ess
(psi
)
Strain (x105)
Element 1 Element 2 Input
115
As with bridge 7R, and as described in Section 5.1.2.1, 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 5.7. 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 5.7 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 5.7. Results of Bridge US13 when varying tension stiffening values.
Tension
Stiffening Model
Maximum
Loads (kips)
Equivalent
Number of Trucks
0.1 Uncorroded1 2049.2 28.5
Corroded1 924.9 12.8
0.035 Uncorroded1 1340.3 18.6
Corroded1 947.1 13.2
0.01 Uncorroded1 1301.5 18.1
Corroded1 786.5 10.9 1Model terminated at peak load, indicating inability to converge.
5.3 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
116
manner as performed for the models discussed above. A description of the bridge
geometry and location is located in Section 5.3.1 and the results are described in
Section 5.3.2.
5.3.1 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 5.14. 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).
117
Figure 5.14. Satellite view of SR 299 over SR 1 (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
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).
118
5.3.2 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 4.3 and 4.6) and corroded (Table 4.9) non-linear input which was
calibrated, as described in Sections 4.4.2 and 4.4.3, 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 non-
linear concrete and plastic rebar commands. The results of this initial analysis can be
seen in Table 5.8. 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 5.8. Initial results from analyzing Bridge US299.
Model Maximum Load (kips)
Equivalent Number of
Trucks
Elastic1 1157.5 16.1
Uncorroded1 40.9 0.6
Corroded1 26.4 0.4 1Model 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
119
damage coefficient commands were removed. The results of these models can be seen
in Table 5.9. 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
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 5.9. Results of Bridge US299 when varying tension stiffening values.
Tension
Stiffening Model
Maximum
Loads (kips)
Equivalent
Number of Trucks
0.1 Uncorroded1 620.3 8.6
Corroded1 692.3 9.6
0.035 Uncorroded1 620.3 8.6
Corroded1 664.6 9.2
0.01 Uncorroded1 969.2 13.5
Corroded2 603.7 8.4 1Model terminated at peak load, indicating inability to converge. 2Model terminated at maximum number of analysis increments specified.
5.4 Conclusions
It can be seen in Sections 5.1 through 5.3 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
120
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
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
121
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
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.
122
Chapter 6
CONCLUSIONS
6.1 Summary
The objectives of this thesis were to create and calibrate a reinforced concrete
model that considers the effects of corrosion due to deicing agents using finite element
analysis, then apply this modeling method to full-scale bridge models to determine
how deck corrosion effects the system capacity of bridges. The system capacity of
bridges is a topic not well quantified; however, the system capacity can significantly
affect the strength of a bridge and cause it to be higher than that of the design strength.
Previously, a literature review was performed to determine how corrosion
effects the behavior of reinforced concrete members (McCarthy 2012), which was
described in Section 2.1.2. The results of this review suggested that reasonable
expectations for corroded reinforced concrete, after 25 years of deterioration, were a
50% decrease in maximum strength and an 82% increase in ultimate deflection
(McConnell et al. 2012). These material properties were used as baseline goals for
calibrating a reinforced concrete beam finite element model.
Different modeling techniques for modeling reinforced concrete were also
researched. Specifically, this search was narrowed down to include modeling
techniques using the commercial finite element software ABAQUS. These modeling
techniques included the brittle cracking, smeared crack (SC), and concrete damaged
plasticity (CDP) approaches (Simulia 2011). The input commands and values for
these modeling techniques were investigated to determine the values which needed to
be utilized. These commands and associated input for the concrete and rebar are
described in Sections 3.1 and 3.2, respectively.
123
Once the modeling approaches were determined, finite element models of
reinforced concrete beams were created based on actual beams that were corroded and
tested in literature (Oyado et al. 2010). Two-dimensional and 3-D beam models were
created. Initially, 2-D beams were utilized, as this modeling approach is simpler and
directly applicable to previously created bridge models. This modeling technique is
described in Section 4.4. However, it was thought that using 3-D beams would be a
more accurate representation of the physical changes caused by corrosion in reinforced
concrete members; the 3-D modeling allowed for encompassing more properties,
including the bond between the concrete and rebar and the pressure caused by the
creation of corrosion by-products. This modeling technique is described in Section
4.5. For the 3-D beam models, all 3 of the approaches were analyzed. However, in
the 2-D beam models, the material points never reach the failure point for the brittle
cracking technique due to the elements carrying both compressive and tensile forces
and therefore the brittle cracking technique was not analyzed.
In addition to these modeling techniques and material properties, hand
calculations were performed to determine the theoretical strength and deflection
values associated with the experimental specimen and the corresponding values as the
material and geometric properties were varied in the calibration models. A description
of these calculations is presented in Section 4.3. The results of the models were
compared to these theoretical values to assess the accuracy of the modeling
approaches and input values.
The 2-D beam models were calibrated to simulate the strength and deflection
effects of corrosion to the extent possible. This calibration process is described in
Section 4.4.3. This was done by varying the different input parameters. These
124
parameters included the values used for the modulus of elasticity of concrete, the
tensile strength of concrete, the compressive strength of concrete, the area of
reinforcing bars, the yield strength of the reinforcing bars, and the dilation angle,
along with including or disregarding the use of the optional damage coefficients in the
CDP technique. A mesh sensitivity analysis was performed to determine the
appropriate element sizes and establish a strength and deflection baseline. The results
of the literature review (i.e., 50% decrease in maximum strength with 82% increase in
ultimate deflection) were then targets for the strength and deflection results of this
corroded model. Once the input parameters were calibrated, these input values were
then applied as the corroded concrete input for future bridge models.
The 3-D beam models which were created used two different types of rebar: 2-
D rebar and 3-D rebar elements. Using 3-D rebar elements allowed for the inclusion
of the effects of the varying bond between the concrete and rebar as the rebar
corrodes. As with the 2-D beams, the results of the literature review were the criteria
applied to the uncorroded base model to attempt to determine the new material
properties for a corroded model. This process for the 2-D and 3-D rebar is described
in Sections 4.5.1 and 4.5.2, respectively.
Once the input values for the uncorroded and corroded reinforced concrete
models were determined, the input was applied to full-scale bridge models. These
models were previously created and calibrated based on existing bridges within
Delaware and prior field testing performed on the bridges (Ross 2007; Michaud 2011;
McConnell et al. in review; Ambrose 2012; Radovic, personal communication 2013).
The concrete decks of these models were already created using 2-D elements allowing
for direct application of the concrete material property input which was calibrated for
125
2-D uncorroded and corroded reinforced concrete. The application of the uncorroded
and corroded deck input are described in Chapter 5.
6.2 Results
A 2-D reinforced concrete beam model was created. A mesh sensitivity
analysis determined that the optimal mesh size to be 0.7874 in (20 mm) squares. The
mesh sensitivity analysis also indicated that the SC model results were erratic and not
analogous to the expected value along with an inability to converge. The input values
for the uncorroded model were determined; this model used a dilation angle of 38,
flow potential eccentricity (ϵ) of 1, ratio of initial equibiaxial compressive yield stress
to initial uniaxial compressive stress (σb0/σc0) of 1.12, ratio of the second stress
invariant on the tensile meridian to that of the compressive meridian at initial yield
(Kc) of 2/3, area of compressive steel (A’s) of 0.0438 in2, area of tensile steel (As) of
0.20563 in2, modulus of elasticity of compressive steel (E’s) and modulus of elasticity
of tensile steel (Es) of 29,000 ksi, yield strength of compressive steel (f’y) of 47,000
psi, yield strength of tensile steel (fy) of 55,000 psi, compressive strength of concrete
(f’c) of 3147.3 psi, tensile strength of concrete (f’t) of 420.8 psi, and modulus of
elasticity of concrete (Ec) of 3,197,746 psi. This model resulted in a load
proportionality factor (LPF) of 1.10 (6,491 lbs) and a corresponding deflection at mid-
span of 0.0435 inches at the same level of load.
Through the literature review, it was determined that the corroded model
should experience a 50% decrease in strength and an 82% increase in deflection. This
is synonymous with a targeted LPF of 0.55 and a targeted deflection of 0.0792 in.
Through a parametric calibration analysis, the corroded model input best achieving
these targets was determined; the corroded model used an Ec of 1,918,648 psi, f’t of
126
252.4 psi, f’c of 1,133 psi, As of 0.07711 in2, and an A’s of 0.03504 in2. These values
correspond to a: 40% decrease in Ec, 40% decrease in f’t, 64% decrease in f’c, 20%
decrease in A’s, and 61.5% decrease in As. All the remaining values were unchanged
from the uncorroded model. These Ec, f’t, and f’c were determined by decreasing the
modulus of elasticity and calculating the tensile and compressive strengths based on
Equations 3.1 and 3.2, respectively. This model resulted in a 45.2% decrease in
strength and 11.6% increase in deflection at the maximum loading when compared to
the 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 model with 2-D rebar elements was analyzed next. A mesh
sensitivity analysis was performed using the brittle cracking, CPD, and SC concrete
models; it was determined to use 0.7874 in (20 mm) cubic elements with 3.937 in (100
mm) long rod elements for the rebar. Base input values for the uncorroded model
were determined using the brittle cracking technique; these values were an f’c of
3,147.3 psi, f’t of 420.8 psi, Ec of 3,197,746 psi, direct cracking strain of 0.0027, direct
cracking failure strain of 0.0027, crack opening strain of 0.0027, A’s of 0.0438 in2, and
127
an As of 0.20563 in2. These resulted in a LPF of 0.850 (5,016 lbs) and a deflection at
mid-span of 0.616 in. This leads to a targeted LPF of 0.425 LPF (2,508 lbs) and a
deflection goal of 1.12 in for the corroded model.
A parametric study was performed. The results of this study indicated that
utilizing 3-D beam elements with 2-D rebar elements was not conducive to accurately
modeling corrosion. Although the models did display a decrease in strength, the
deflection also decreased proportionately to the strength; this is the opposite of the
desired change in deflection. For this reason, 3-D concrete with 3-D rebar models
were analyzed in attempts to more accurately model corrosion.
The final reinforced concrete model which was analyzed was the 3-D concrete
beam elements using 3-D rebar elements. A mesh sensitivity analysis was not
performed, as the cross-sectional area of the rebar limited the beam element sizes.
These elements measured 0.3937 in x 0.5249 in x 0.3937 in (10 mm x 13.33 mm x 10
mm); all of these dimensions of the elements are smaller than the previously
determined optimal mesh sizes. Initially, these models were analyzed using the SC
and CDP techniques. However, when using these techniques, the model failed to
converge and terminated early; the results of these models were less than 1/3 of the
calculated expected strength using the theoretical equations as well as compared to the
previously created models. A technique which allowed these models converge was
not determined; for this reason, this modeling technique was not used.
A time analysis was performed using the brittle cracking technique. As
opposed to the SC and CDP modeling techniques, the brittle cracking approach
utilizes ABAQUS/Explicit rather than ABAQUS/Standard. For this reason, the time
over which the loading is applied is included in the modeling and an analysis was
128
performed to determine the appropriate length of time to apply the loading. These
results concluded with both 1 second and 10 being utilized for subsequent analyses.
As with the models using SC and CDP, the brittle cracking models also produced
strengths significantly lower than the calculated theoretical strength and those
produced using the 2-D beam models, ranging from 27% to 82% of the theoretical
strength. For this reason, the corroded input was not calibrated and this modeling
approach was not applied to the full-scale bridge models.
Once the input for the uncorroded and corroded beams was determined, it was
applied to 3 different full-scale bridge models previously created based on actual
bridges located in Delaware. Due to time constraints and convergence difficulties,
only one of these models was thoroughly analyzed; this bridge is referred to as 7R and
served as an exit ramp for Interstate 295 North, just south of the Delaware Memorial
Bridge before being decommissioned in 2010. Initially, the input which was
calibrated using the 2-D beam models was applied. However, this input prevented the
models from converging on a final result. For this reason, the tension stiffening input
associated with the “concrete tension stiffening” command was modified and strain
values at which the stress decreased to zero of 0.01, 0.035, and 0.1 were analyzed.
Modifying these values allowed all of the 7R models to converge, while the remaining
bridges still experienced an inability to converge.
The maximum loading of the 7R bridge models was determined. In addition to
the strength results, the distribution factors (DF) 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 experienced during the field test. It can
also be seen that, in general, the corrosion in the deck caused a more uniform
129
distribution between the girders than with an uncorroded deck. The results of these
models also indicated that the corroded models reached higher strengths than their
uncorroded counterparts. It was thought that the cause of the differences in both the
strength and DF values was due to greater load sharing in the uncorroded model when
compared to the corresponding corroded model.
The DF values between different models were compared; 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 the girders closest to the location of the applied
loading, followed by the uncorroded and then the corroded. However, it was seen that
this is not the case. The results of the corroded model produced DF values between
those of the elastic and those of the uncorroded models due to the corroded deck
distributing the load more uniformly throughout the deck as the peak stress for any
individual element was limited to a smaller value. 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; the percent values reported here
are for the 0.035 tension stiffening model but similar results are obtained for models
with other tension stiffening values.
6.3 Future Work
A modeling technique utilizing 3-D elements that was able to converge was
not able to be concluded; the 3-D modeling technique offers a more direct approach to
modeling the physical effects of corrosion within concrete, as opposed to 2-D models
130
which require the general strength and deflection effects of corrosion to be modeled
indirectly. Therefore, future work should be done to determine the appropriate means
of modeling reinforced concrete using 3-D elements. The results of the uncorroded
beam models utilizing the brittle cracking technique did not meet the strength
expectations and the CDP and SC techniques were unable to converge. More effort
should be put into researching modeling techniques and input parameters for 3-D
beam modeling to potentially determine a more accurate corroded reinforced concrete
beam model.
When applying the 2-D reinforced concrete input to the full-scale bridge
models, the continuous bridge models did not converge. A more thorough analysis
into the cause which prevented convergence should be performed. The results from
additional bridge models would help to determine a general relationship between deck
corrosion and the system capacity of bridges. It is suggested that, when analyzing
additional bridges, simple span bridges (to initially minimize potential convergence
problems) be used as a starting point to determine if the results of 7R are consistent
amongst simple span steel I-girder bridges. In addition, future work is required to
determine the correlation of the deck condition based on current rating systems to the
material input parameters used in the model. This would help create a system to better
prioritize the repair of bridges categorized as structurally deficient or functionally
obsolete based on their actual system capacity.
In addition, the results of Bridge 7R should be further investigated.
Specifically, the results indicated that the input was not performing in an anticipated
way and more effort should be placed in determining why the concrete elements were
not unloading after reaching the defined maximum stress. It is thought that this may
131
be due to discrepancies between the direct input values and the evolution of the failure
surface as the multi-axial stresses develop in the model and further evaluation of this
hypothesis should be considered. Alternatively, additional analysis of the stress
distribution throughout the bridge at different loadings would help to better understand
the system behavior with different deck conditions. Evaluating the regions of the deck
in tension and compression over a girder line is also suggested to determine if the
changes from tension to compression observed in the present results are realistic.
Lastly, the results presented here were calibrated based on relatively small,
simple beam and cylinder tests with limited physical specimens. Additional research
should be performed to more accurately understand how corrosion effects the material
properties of and load transfer mechanisms within reinforced concrete decks and
members along with the quantification of the variables affecting these responses in
terms consistent with FEA input, as well as the global response generally expected for
these members. Specifically, more large-scale tests based on realistic exposure
conditions would provide valuable insight. Furthermore, calibrating FEA models to
existing corroded cylinder data could provide additional insights.
132
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136
Appendix A
SAMPLE STRENGTH CALCULATIONS
Variable Definitions:
T = tensile force
C = total compressive force
Cc = compressive force in concrete
Cs = compressive force in steel
As = area of tensile steel
A’s = area of compressive steel
f’c = compressive strength of concrete
f’t = tensile strength of concrete
fy = ultimate strength of tensile rebar
f’y = ultimate strength of compressive rebar
b = length of base of concrete cross-section
β1 = 0.85
c = distance from top of beam to neutral axis
f’s = force in compressive rebar
Ec = modulus of elasticity of concrete
Es = modulus of elasticity of tensile steel
E’s = modulus of elasticity of compressive steel
εs = strain in tensile steel
ε’s = strain in compressive steel
εcu = strain in concrete at cracking
d = distance from top of beam to tensile rebar
d’ = distance from top of beam to compressive rebar
h = total height of beam
I = moment of inertia of reinforced concrete beam
Ic = cracked moment of inertia of reinforced concrete beam
Ast = transformed area of tensile steel
A’st = transformed area of compressive steel
n = modular ratio of tensile steel
n’ = modular ratio of compressive steel
P = force applied at one point on beam during loading
L = total length of beam
x = distance from support to P
137
Generic Equations
Neutral Axis:
𝑇 = 𝐶 = 𝐶𝑐 + 𝐶𝑠
𝐴𝑠𝑓𝑦 = (0.85)𝑓′𝑐 ∗ 𝑏 ∗ 𝛽1 ∗ 𝑐 + 𝐴′𝑠(𝑓′𝑠 − 0.85 ∗ 𝑓′𝑐)
𝑓′𝑠 = 𝐸′𝑠𝜀′𝑠
𝜀′𝑠 = (𝑐 − 𝑑′
𝑐) 𝜀𝑐𝑢
𝐴𝑠𝑓𝑦 = 0.85 ∗ 𝑓′𝑐 ∗ 3.937 ∗ 0.85 ∗ 𝑐 + 𝐴′𝑠
∗ {[(𝑐 − 0.7874
𝑐) ∗ 0.003] ∗ 𝐸′𝑠 − 0.85 ∗ 𝑓′𝑐}
Solve for c:
𝑐 = 0.175779 ∗
√
𝐴𝑠2𝑓𝑦
2 − 0.006𝐴𝑠𝐴′𝑠(𝐸′
𝑠 − 283.333𝑓′𝑐)𝑓𝑦
+0.000009𝐴′𝑠(𝐴′
𝑠(𝐸′𝑠2 − 566.667𝐸′
𝑠𝑓′𝑐
+ 80277.8𝑓′𝑐2)
+2986.33𝐸′𝑠𝑓′𝑐)
𝐴𝑠𝑓𝑦 − 0.003𝐴′𝑠(𝐸′
𝑠 − 283.333𝑓′𝑐)
𝑓′𝑐
Moment of Inertia:
𝑛 =𝐸𝑠
𝐸𝑐
𝑛′ =𝐸′𝑠
𝐸𝑐
𝐴𝑠𝑡 = 𝑛 ∗ 𝐴𝑠
𝐴′𝑠𝑡 = 𝑛′ ∗ 𝐴′𝑠
𝐼 =𝑏𝑐3
12+ 𝑏𝑐 (
𝑐
2)
2
+𝑏(ℎ − 𝑐)3
12+ 𝑏(ℎ − 𝑐) (
ℎ − 𝑐 − 𝑑′
2)
2
+ 𝐴𝑠𝑡(ℎ − 𝑐 − 𝑑′)2
+ 𝐴′𝑠𝑡(𝑐 − 𝑑′)2
138
𝐼𝑐 =𝑏𝑐3
12+ 𝑏𝑐 (
𝑐
2)
2
+ 𝐴𝑠𝑡(ℎ − 𝑐 − 𝑑′)2 + 𝐴′𝑠𝑡(𝑐 − 𝑑′)2
Moment Capacity:
𝐶𝑐 = 0.85 ∗ 𝑓′𝑐 ∗ 𝑏 ∗ 𝛽1 ∗ 𝑐
𝐶𝑠 = 𝐴′𝑠(𝑓′𝑠 − 0.85𝑓′𝑐)
𝑀𝑛 = 𝐶𝑐 (𝑑 −𝛽1𝑐
2) + 𝐶𝑠(𝑑 − 𝑑′)
𝑀𝑛 = 𝑃𝑥
Deflection:
∆𝑚𝑎𝑥=𝑃𝑥
24𝐸𝑐𝐼(3𝐿2 − 4𝑥2)
Check Rebar Yielded:
𝜀𝑦 =𝑓𝑦
𝐸𝑠
𝜀𝑠 = (𝑑 − 𝑐
𝑐) 𝜀𝑐𝑢
𝜀𝑠 ≥ 𝜀𝑦
139
Sample Calculation of Base Model
Given Values:
As = 0.20563in2
A’s = 0.0438in2
fy = 55,000psi
f’y = 47,000psi
b = 3.937in
β1 = 0.85
Es = E’s = 29000000psi
d’ = 0.7874in
d = 7.0866in
εcu = 0.003
f’c = 3147.2psi
Ec = 3197746psi
h = 7.874in
x = 27.5586in
𝐴𝑠𝑓𝑦 = (0.85)𝑓′𝑐 ∗ 𝑏 ∗ 𝛽1 ∗ 𝑐 + 𝐴′𝑠(𝑓′𝑠 − 0.85 ∗ 𝑓′𝑐)
𝑓′𝑠 = 𝐸′𝑠𝜀′𝑠
𝜀′𝑠 = (𝑐 − 𝑑′
𝑐) 𝜀𝑐𝑢
𝜀′𝑠 = (
𝑐 − 0.7874𝑖𝑛
𝑐) ∗ 0.003
𝑓′𝑠 = 29,000,000𝑝𝑠𝑖 ∗ (𝑐 − 0.7874𝑖𝑛
𝑐) ∗ 0.003
0.20563𝑖𝑛2 ∗ 55,000𝑝𝑠𝑖= 0.85 ∗ 3147.3𝑝𝑠𝑖 ∗ 3.937𝑖𝑛 ∗ 0.85 ∗ 𝑐 + 0.0438𝑖𝑛2
∗ [29,000,000𝑝𝑠𝑖 (𝑐 − 0.7874𝑖𝑛
𝑐) ∗ 0.003 − 0.85 ∗ 3147.3𝑝𝑠𝑖]
11,309.7 = 8942.44𝑐 +3693.43(𝑐 − 0.81238)
𝑐
𝑐 = 1.14377𝑖𝑛
𝑛 = 𝑛′ =𝐸𝑠
𝐸𝑐=
29,000,000𝑝𝑠𝑖
3197746𝑝𝑠𝑖
𝑛 = 𝑛′ = 9.06889
𝐴𝑠𝑡 = 𝐴𝑠 ∗ 𝑛 = 0.20563𝑖𝑛2 ∗ 9.06889
𝐴𝑠𝑡 = 1.86484𝑖𝑛2
𝐴′𝑠𝑡 = 𝐴′𝑠 ∗ 𝑛′ = 0.0438𝑖𝑛2 ∗ 9.06889
𝐴′𝑠𝑡 = 0.397217𝑖𝑛2
141
𝐼 =𝑏𝑐3
12+ 𝑏𝑐 (
𝑐
2)
2
+𝑏(ℎ − 𝑐)3
12+ 𝑏(ℎ − 𝑐) (
ℎ − 𝑐 − 𝑑′
2)
2
+ 𝐴𝑠𝑡(ℎ − 𝑐 − 𝑑′)2
+ 𝐴′𝑠𝑡(𝑐 − 𝑑′)2
𝐼 =(3.937𝑖𝑛) ∗ (1.14377𝑖𝑛)3
12+ (3.937𝑖𝑛) ∗ (1.14377𝑖𝑛) ∗ (
1.14377𝑖𝑛
2)
2
+(3.937𝑖𝑛) ∗ (7.874𝑖𝑛 − 1.14377𝑖𝑛)3
12+ (3.937𝑖𝑛)
∗ (7.874𝑖𝑛 − 1.14377𝑖𝑛) ∗ (7.874𝑖𝑛 − 1.14377𝑖𝑛 − 0.7874𝑖𝑛
2)
2
+ 1.86484𝑖𝑛2 ∗ (7.874𝑖𝑛 − 1.14377𝑖𝑛 − 0.7874𝑖𝑛)2 + 0.397217𝑖𝑛2
∗ (1.14377𝑖𝑛 − 0.7874𝑖𝑛)2
𝐼 = 401.851𝑖𝑛4
𝐼𝑐 =𝑏𝑐3
12+ 𝑏𝑐 (
𝑐
2)
2
+ 𝐴𝑠𝑡(ℎ − 𝑐 − 𝑑′)2 + 𝐴′𝑠𝑡(𝑐 − 𝑑′)2
𝐼𝑐 =(3.937𝑖𝑛) ∗ (1.14377𝑖𝑛)3
12+ (3.937𝑖𝑛) ∗ (1.14377𝑖𝑛) ∗ (
1.14377𝐼𝑁
2)
2
+ 1.86484𝑖𝑛2 ∗ (7.874𝑖𝑛 − 1.14377𝑖𝑛 − 0.7874)2 + 0.397217𝑖𝑛2
∗ (1.14377𝑖𝑛 − 0.7874𝑖𝑛)2
𝐼𝑐 = 67.875𝑖𝑛4
𝐶𝑐 = 0.85 ∗ 𝑓′𝑐
∗ 𝑏 ∗ 𝛽1 ∗ 𝑐 = 0.85 ∗ 3147.3𝑝𝑠𝑖 ∗ 3.937𝑖𝑛 ∗ 0.85 ∗ 1.14377𝑖𝑛
𝐶𝑐 = 10239.5𝑙𝑏
𝜀′𝑠 = (1.14377𝑖𝑛 − 0.7874𝑖𝑛
1.14377𝑖𝑛) ∗ 0.003
𝜀′𝑠 = 0.000935
𝑓′𝑠
= 29,000,000𝑝𝑠𝑖 ∗ 0.000935
𝑓′𝑠 = 27,107𝑝𝑠𝑖
𝐶𝑠 = 𝐴′𝑠 ∗ (𝑓′
𝑠− 0.85 ∗ 𝑓′
𝑐) = 0.0438𝑖𝑛2 ∗ (27,107𝑝𝑠𝑖 − 0.85 ∗ 3147.3𝑝𝑠𝑖)
𝐶𝑠 = 1070.1𝑙𝑏
𝑀𝑛 = 𝐶𝑐 ∗ (𝑑 −𝛽1 ∗ 𝑐
2) + 𝐶𝑠 ∗ (𝑑 − 𝑑′)
142
= 10239.5𝑙𝑏 ∗ (7.0866𝑖𝑛 −0.85 ∗ 1.14377𝑖𝑛
2) + 1070.1𝑙𝑏
∗ (7.0866𝑖𝑛 − 0.7874𝑖𝑛)
𝑀𝑛 = 74,327𝑙𝑏 − 𝑖𝑛
𝑃 =𝑀𝑛
𝑥=
74,327𝑙𝑏 − 𝑖𝑛
27.5586𝑖𝑛
𝑃 = 2,697.1𝑙𝑏
∆𝑢𝑛𝑐𝑟𝑎𝑘𝑐𝑒𝑑=𝑃𝑥
24 ∗ 𝐸𝑐 ∗ 𝐼∗ (3 ∗ 𝐿2 − 4 ∗ 𝑥2)
=2697.1𝑙𝑏 ∗ 27.5586𝑖𝑛
24 ∗ 3197749𝑝𝑠𝑖 ∗ 401.41𝑖𝑛4∗ (3 ∗ (82.677𝑖𝑛)2 − 4 ∗ (27.5586𝑖𝑛)2)
∆𝑢𝑛𝑐𝑟𝑎𝑐𝑘𝑒𝑑= 0.0421𝑖𝑛
∆𝑐𝑟𝑎𝑐𝑘𝑒𝑑=𝑃𝑥
24 ∗ 𝐸𝑐 ∗ 𝐼𝑐∗ (3 ∗ 𝐿2 − 4 ∗ 𝑥2)
=2697.1𝑙𝑏 ∗ 27.5586𝑖𝑛
24 ∗ 3197749𝑝𝑠𝑖 ∗ 67.871𝑖𝑛4∗ (3 ∗ (82.677𝑖𝑛)2 − 4 ∗ (27.5586𝑖𝑛)2)
∆𝑐𝑟𝑎𝑐𝑘𝑒𝑑= 0.2493𝑖𝑛
𝜀𝑦 =𝑓𝑦
𝐸𝑠=
55,000𝑝𝑠𝑖
29,000,000𝑝𝑠𝑖
𝜀𝑦 = 0.001897
𝜀𝑠 = (𝑑 − 𝑐
𝑐) 𝜀𝑐𝑢 = (
7.0866𝑖𝑛 − 1.14377𝑖𝑛
1.14377𝑖𝑛) ∗ 0.003
𝜀𝑠 = 0.015587
𝜀𝑠 ≥ 𝜀𝑦
0.015587 ≥ 0.001897 √
143
Appendix B
COMPLETE RESULTS FOR 2-D BEAM CALIBRATION VARIATIONS
Table B.1. Results of mesh sensitivity analysis for 2-D beam models. In
pu
t
Tension Coefficient Y N N/A Y N N/A
Compression
Coefficient Y N N/A Y N N/A
Concrete Model CDP CDP SC CDP CDP SC
Mesh Size (mm) 100 100 100 50 50 50
Res
ult
s Highest Load (LPF) 1.75 1.74 4.72 2.55 2.47 3.02
Highest Deflection (in) 0.053 0.053 0.383 0.282 0.283 1.413
Converged N N N Y Y N
Table B.1. cont’d
Inp
ut
Tension Coefficient Y N N/A Y N N/A
Compression
Coefficient Y N N/A Y N N/A
Concrete Model CDP CDP SC CDP CDP SC
Mesh Size (mm) 25 25 25 20 20 20
Res
ult
s Highest Load (LPF) 0.890 0.931 4.10 1.10 1.12 2.98
Highest Deflection (in) 0.023 0.022 0.211 0.043 0.041 0.136
Converged N Y Y Y Y N
Table B.1. cont’d
Inp
ut
Tension Coefficient Y N N/A Y N N/A
Compression
Coefficient Y N N/A Y N N/A
Concrete Model CDP CDP SC CDP CDP SC
Mesh Size (mm) 10 10 10 5 5 5
Res
ult
s Highest Load (LPF) 1.08 1.08 2.39 3.69 3.65 10.20
Highest Deflection (in) 0.042 0.042 0.099 0.002 0.002 0.007
Converged Y N N Y N N
144
Table B.2. Complete input and results of all concrete damaged plasticity models
tested during 2-D beam calibration.
Inp
ut
Dilation Angle 38 38 38 38 36 40
Flow Potential
Eccentricity 1 1 0.1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.16 1.12 1.12
Tension
Coefficient Y N Y Y Y Y
Compression
Coefficient Y N Y Y Y Y
A's (in2) 0.0438 0.0438 0.0438 0.0438 0.0438 0.0438
As (in2) 0.2056 0.2056 0.2056 0.2056 0.2056 0.2056
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 29000 29000 29000 29000 29000 29000
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Ec (psi) 3197746 3197746 3197746 3197746 3197746 3197746
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 1.10 1.10 1.11 1.10 1.10 1.10
Deflection (in) 0.043 0.041 0.044 0.043 0.043 0.045
Converged N Y Y N Y Y
145
Table B.2. cont’d
Inp
ut
Dilation Angle 38 38 38 38 38 38
Flow Potential
Eccentricity 1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.0438 0.0438 0.0438 0.0438 0.0438 0.0438
As (in2) 0.1542 0.1028 0.2056 0.2056 0.2056 0.2056
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 29000 29000 29000 29000 29000 29000
f'y (psi) 47000 47000 47000 47000 42300 37600
fy (psi) 55000 55000 49500 44000 55000 55000
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Ec (psi) 3197746 3197746 3197746 3197746 3197746 3197746
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 1.06 0.767 1.10 1.10 1.10 1.10
Deflection (in) 0.045 0.032 0.043 0.043 0.043 0.043
Converged N Y Y N N N
146
Table B.2. cont’d
Inp
ut
Dilation Angle 38 38 38 38 38 38
Flow Potential
Eccentricity 1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.0438 0.0438 0.03285 0.0438 0.0438 0.0438
As (in2) 0.2056 0.2056 0.1542 0.1542 0.1028 0.1542
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 29000 29000 29000 29000 29000 29000
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 2832.57 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 378.72 420.8 378.72 378.72 315.6
Ec (psi) 3197746 3197746 3197746 3197746 3197746 3197746
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 1.10 1.02 1.01 0.976 0.705 0.856
Deflection (in) 0.043 0.041 0.043 0.041 0.029 0.034
Converged Y N Y Y Y Y
147
Table B.2. cont’d
Inp
ut
Dilation Angle 38 38 38 38 38 38
Flow Potential
Eccentricity 1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.0438 0.0438 0.0438 0.0438 0.0438 0.0438
As (in2) 0.2056 0.2056 0.2056 0.2056 0.2056 0.2056
E's (ksi) 29000 29000 29000 26100 29000 26100
Es (ksi) 29000 29000 29000 29000 26100 26100
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Ec (psi) 3037859 2877971 2398310 3197746 3197746 3197746
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 1.11 1.11 1.06 1.08 1.09 1.07
Deflection (in) 0.045 0.047 0.050 0.042 0.044 0.043
Converged Y N Y Y N N
148
Table B.2. cont’d
Inp
ut
Dilation Angle 38 38 38 38 38 38
Flow Potential
Eccentricity 1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.0438 0.0438 0.0438 0.0438 0.0438 0.0438
As (in2) 0.2056 0.2056 0.2056 0.2056 0.2056 0.2056
E's (ksi) 29000 29000 29000 31900 29000 31900
Es (ksi) 29000 29000 29000 29000 31900 31900
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Ec (psi) 3357633 3517521 3997183 3197746 3197746 3197746
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 1.1 1.09 1.08 1.12 1.12 1.13
Deflection (in) 0.042 0.040 0.036 0.044 0.043 0.043
Converged N Y Y Y N N
149
Table B.2. cont’d
Inp
ut
Dilation Angle 38 38 38 38 38 38
Flow Potential
Eccentricity 1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.0438 0.0438 0.0438 0.0438 0.0438 0.0438
As (in2) 0.1542 0.1028 0.2056 0.2056 0.1542 0.1542
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 29000 29000 29000 29000 29000 29000
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 378.72 315.6 378.72 315.6
Ec (psi) 2398310 2398310 2398310 2398310 2398310 2398310
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.929 0.803 0.985 0.976 1.00 0.908
Deflection (in) 0.046 0.042 0.046 0.046 0.053 0.046
Converged Y Y N N Y N
150
Table B.2. cont’d
Inp
ut
Dilation Angle 38 38 38 38 38 38
Flow Potential
Eccentricity 1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.0438 0.0438 0.0438 0.0438 0.0438 0.0438
As (in2) 0.1028 0.1028 0.1542 0.1542 0.1542 0.1542
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 29000 29000 26100 26100 29000 29000
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 378.72 315.6 378.72 315.6 378.72 378.72
Ec (psi) 2398310 2398310 2398310 2398310 2238422 1918648
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.734 0.523 0.932 0.843 0.994 1.11
Deflection (in) 0.039 0.026 0.050 0.042 0.055 0.069
Converged Y N Y Y Y N
151
Table B.2. cont’d
Inp
ut
Dilation Angle 38 38 38 38 38 38
Flow Potential
Eccentricity 1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.0438 0.0438 0.0438 0.0438 0.0438 0.0438
As (in2) 0.1542 0.1542 0.1542 0.1028 0.2056 0.2056
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 26100 29000 29000 29000 29000 29000
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 3147.3 3147.3 3147.3 3147.3 2840.4 2549.3
f't (psi) 315.6 315.6 315.6 315.6 399.718 378.68
Ec (psi) 2398310 2238422 1918648 1918648 3037859 2877971
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.843 0.853 0.567 0.509 1.06 1.02
Deflection (in) 0.042 0.044 0.031 0.030 0.043 0.043
Converged Y Y Y N Y N
152
Table B.2. cont’d
Inp
ut
Dilation Angle 38 38 38 38 38 38
Flow Potential
Eccentricity 1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.0438 0.0438 0.0438 0.0438 0.0438 0.0438
As (in2) 0.2056 0.1542 0.1542 0.1028 0.1028 0.1542
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 29000 29000 29000 29000 29000 26100
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 1770.4 1542.2 1133.0 1542.2 1133.0 1133.0
f't (psi) 315.6 294.5 252.5 294.5 252.5 252.5
Ec (psi) 2398310 2238422 1918648 2238422 1918648 1918648
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.852 0.825 0.719 0.772 0.694 0.726
Deflection (in) 0.041 0.046 0.045 0.049 0.049 0.047
Converged N Y N Y Y Y
153
Table B.2. cont’d
Inp
ut
Dilation Angle 38 38 38 38 38 38
Flow Potential
Eccentricity 1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.0438 0.0438 0.0438 0.0438 0.0438 0.0438
As (in2) 0.1028 0.1542 0.1542 0.1028 0.1028 0.1542
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 26100 29000 29000 29000 29000 26100
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 44000 44000 44000 44000 44000
f'c (psi) 1133.0 1542.2 1133.0 1542.2 1133.0 1133.0
f't (psi) 252.5 294.5 252.5 294.5 252.5 252.5
Ec (psi) 1918648 2238422 1918648 2238422 1918648 1918648
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.680 0.825 0.719 0.772 0.681 0.726
Deflection (in) 0.050 0.046 0.045 0.049 0.049 0.047
Converged Y Y N Y Y Y
154
Table B.2. cont’d
Inp
ut
Dilation Angle 38 38 38 38 38 38
Flow Potential
Eccentricity 1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.0438 0.0438 0.0438 0.0438 0.03942 0.03504
As (in2) 0.1028 0.0823 0.0617 0.0514 0.1028 0.1028
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 26100 26100 26100 26100 26100 26100
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 44000 55000 55000 55000 55000 55000
f'c (psi) 1133.0 1133.0 1133.0 1133.0 1133.0 1133.0
f't (psi) 252.5 252.5 252.5 252.5 252.5 252.5
Ec (psi) 1918648 1918648 1918648 1918648 1918648 1918648
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.680 0.654 0.439 0.420 0.662 0.644
Deflection (in) 0.050 0.051 0.031 0.030 0.049 0.047
Converged Y Y N Y Y N
155
Table B.2. cont’d
Inp
ut
Dilation Angle 38 38 38 38 38 38
Flow Potential
Eccentricity 1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.03504 0.03942 0.0438 0.03942 0.03504 0.0438
As (in2) 0.0823 0.0823 0.0720 0.0720 0.0720 0.0771
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 26100 26100 26100 26100 26100 26100
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 1133.0 1133.0 1133.0 1133.0 1133.0 1133.0
f't (psi) 252.5 252.5 252.5 252.5 252.5 252.5
Ec (psi) 1918648 1918648 1918648 1918648 1918648 1918648
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.619 0.637 0.457 0.457 0.453 0.467
Deflection (in) 0.048 0.050 0.031 0.032 0.032 0.032
Converged N N N Y Y Y
156
Table B.2. cont’d
Inp
ut
Dilation
Angle 38 38 38 38 38 38
Flow
Potential
Eccentricity
1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.03942 0.03504 0.0438 0.03942 0.03504 0.0438
As (in2) 0.0771 0.0771 0.0781 0.0781 0.0781 0.0802
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 26100 26100 26100 26100 26100 26100
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 1133.0 1133.0 1133.0 1133.0 1133.0 1133.0
f't (psi) 252.5 252.5 252.5 252.5 252.5 252.5
Ec (psi) 1918648 1918648 1918648 1918648 1918648 1918648
Concrete
Model CDP CDP CDP CDP CDP CDP
Res
ult
s
Max Load
(LPF) 0.465 0.603 0.649 0.631 0.466 1.37
Deflection
(in) 0.032 0.049 0.051 0.050 0.032 0.128
Converged Y Y Y Y Y N
157
Table B.2. cont’d
Inp
ut
Dilation
Angle 38 38 38 38 38 38
Flow
Potential
Eccentricity
1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.03942 0.03504 0.0438 0.03942 0.03504 0.0438
As (in2) 0.0802 0.0802 0.0761 0.0761 0.0761 0.0766
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 26100 26100 26100 26100 26100 26100
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 1133.0 1133.0 1133.0 1133.0 1133.0 1133.0
f't (psi) 252.5 252.5 252.5 252.5 252.5 252.5
Ec (psi) 1918648 1918648 1918648 1918648 1918648 1918648
Concrete
Model CDP CDP CDP CDP CDP CDP
Res
ult
s
Max Load
(LPF) 0.634 0.617 0.463 0.462 0.611 0.562
Deflection
(in) 0.050 0.048 0.031 0.032 0.049 0.052
Converged N N N Y N N
158
Table B.2. cont’d
Inp
ut
Dilation
Angle 38 38 38 38 38 38
Flow
Potential
Eccentricity
1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.03942 0.03504 0.0438 0.03942 0.03504 0.0438
As (in2) 0.0766 0.0766 0.0776 0.0776 0.0776 0.0779
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 26100 26100 26100 26100 26100 26100
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 1133.0 1133.0 1133.0 1133.0 1133.0 1133.0
f't (psi) 252.5 252.5 252.5 252.5 252.5 252.5
Ec (psi) 1918648 1918648 1918648 1918648 1918648 1918648
Concrete
Model CDP CDP CDP CDP CDP CDP
Res
ult
s
Max Load
(LPF) 0.464 0.462 0.572 0.465 0.613 0.581
Deflection
(in) 0.032 0.032 0.043 0.032 0.049 0.044
Converged Y Y Y Y Y Y
159
Table B.2. cont’d
Inp
ut
Dilation
Angle 38 38 38 38 38 38
Flow
Potential
Eccentricity
1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.03942 0.03504 0.0438 0.03942 0.03504 0.0438
As (in2) 0.0779 0.0779 0.0777 0.0777 0.0777 0.0775
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 26100 26100 26100 26100 26100 26100
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 1133.0 1133.0 1133.0 1133.0 1133.0 1133.0
f't (psi) 252.5 252.5 252.5 252.5 252.5 252.5
Ec (psi) 1918648 1918648 1918648 1918648 1918648 1918648
Concrete
Model CDP CDP CDP CDP CDP CDP
Res
ult
s
Max Load
(LPF) 0.659 0.707 0.567 0.466 0.613 0.647
Deflection
(in) 0.053 0.057 0.043 0.032 0.049 0.051
Converged Y Y Y Y Y Y
160
Table B.2. cont’d
Inp
ut
Dilation
Angle 38 38 38 38 38 38
Flow
Potential
Eccentricity
1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.03942 0.03504 0.0438 0.03942 0.03504 0.0438
As (in2) 0.0775 0.0775 0.0773 0.0773 0.0773 0.0769
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 26100 26100 26100 26100 26100 26100
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 1133.0 1133.0 1133.0 1133.0 1133.0 1133.0
f't (psi) 252.5 252.5 252.5 252.5 252.5 252.5
Ec (psi) 1918648 1918648 1918648 1918648 1918648 1918648
Concrete
Model CDP CDP CDP CDP CDP CDP
Res
ult
s
Max Load
(LPF) 0.464 0.576 0.468 0.683 0.464 0.466
Deflection
(in) 0.032 0.045 0.032 0.054 0.032 0.032
Converged Y Y Y Y Y Y
161
Table B.2. cont’d
Inp
ut
Dilation
Angle 38 38 38 38 38 38
Flow
Potential
Eccentricity
1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.03942 0.03504 0.0438 0.03942 0.03504 0.0438
As (in2) 0.0769 0.0769 0.0767 0.0767 0.0767 0.0765
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 26100 26100 26100 26100 26100 26100
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 1133.0 1133.0 1133.0 1133.0 1133.0 1133.0
f't (psi) 252.5 252.5 252.5 252.5 252.5 252.5
Ec (psi) 1918648 1918648 1918648 1918648 1918648 1918648
Concrete
Model CDP CDP CDP CDP CDP CDP
Res
ult
s
Max Load
(LPF) 0.463 0.463 0.567 0.465 0.463 0.562
Deflection
(in) 0.032 0.032 0.043 0.032 0.032 0.043
Converged Y Y Y N Y Y
162
Table B.2. cont’d
Inp
ut
Dilation
Angle 38 38 38 38 38 40
Flow
Potential
Eccentricity
1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.03942 0.03504 0.0438 0.03942 0.03504 0.0438
As (in2) 0.0765 0.0765 0.0763 0.0763 0.0763 0.1542
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 26100 26100 26100 26100 26100 29000
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 1133.0 1133.0 1133.0 1133.0 1133.0 3147.3
f't (psi) 252.5 252.5 252.5 252.5 252.5 378.7
Ec (psi) 1918648 1918648 1918648 1918648 1918648 1918648
Concrete
Model CDP CDP CDP CDP CDP CDP
Res
ult
s
Max Load
(LPF) 0.462 0.461 0.466 0.463 0.461 0.919
Deflection
(in) 0.032 0.032 0.032 0.032 0.032 0.054
Converged Y Y N Y Y Y
163
Table B.2. cont’d
Inp
ut
Dilation
Angle 40 45 40 40 40 40
Flow
Potential
Eccentricity
1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.0438 0.0438 0.0438 0.0438 0.0438 0.0438
As (in2) 0.1542 0.1542 0.1542 0.1542 0.1542 0.2056
E's (ksi) 29000 29000 29000 30000 29000 29000
Es (ksi) 29000 29000 30000 30000 29000 29000
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 3147.3 3147.3 3147.3 3147.3 1133.0 1133.0
f't (psi) 378.7 378.7 378.7 378.7 252.5 252.5
Ec (psi) 2238422 2238422 2238422 2238422 1918648 1918648
Concrete
Model CDP CDP CDP CDP CDP CDP
Res
ult
s
Max Load
(LPF) 1.00 1.01 1.03 1.03 0.717 0.694
Deflection
(in) 0.056 0.056 0.057 0.057 0.045 0.040
Converged Y Y Y Y N Y
164
Table B.2. cont’d
Inp
ut
Dilation
Angle 40 40 40 40 45 45
Flow
Potential
Eccentricity
1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.03942 0.0438 0.03504 0.0438 0.0438 0.03504
As (in2) 0.0761 0.1542 0.0771 0.0766 0.0766 0.0771
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 29000 29000 29000 29000 29000 29000
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 1133.0 1542.9 1133.0 1133.0 1133.0 1133.0
f't (psi) 252.5 294.5 252.5 252.5 252.5 252.5
Ec (psi) 1918648 2238422 1918648 1918648 1918648 1918648
Concrete
Model CDP CDP CDP CDP CDP CDP
Res
ult
s
Max Load
(LPF) 0.640 0.825 0.624 0.658 0.485 0.484
Deflection
(in) 0.049 0.046 0.048 0.051 0.033 0.033
Converged Y Y Y Y Y N
165
Table B.2. cont’d
Inp
ut
Dilation
Angle 41 42 43 44 38 38
Flow
Potential
Eccentricity
1 1 1 1 1 1
σb0/σc0 1.12 1.12 1.12 1.12 1.12 1.12
Tension
Coefficient Y Y Y Y Y Y
Compression
Coefficient Y Y Y Y Y Y
A's (in2) 0.03504 0.03504 0.03504 0.03504 0.0438 0.0438
As (in2) 0.0771 0.0771 0.0771 0.0771 0.2056 0.2056
E's (ksi) 29000 29000 29000 29000 29000 29000
Es (ksi) 29000 29000 29000 29000 29000 29000
f'y (psi) 47000 47000 47000 47000 47000 47000
fy (psi) 55000 55000 55000 55000 55000 55000
f'c (psi) 1133.0 1133.0 1133.0 1133.0 1542.2 1133.0
f't (psi) 252.5 252.5 252.5 252.5 294.5 252.5
Ec (psi) 1918648 1918648 1918648 1918648 2238422 1918648
Concrete
Model CDP CDP CDP CDP CDP CDP
Res
ult
s
Max Load
(LPF) 0.624 0.624 0.625 0.482 0.798 0.689
Deflection
(in) 0.048 0.048 0.048 0.033 0.041 0.039
Converged N Y Y Y Y Y
166
Appendix C
COMPLETE RESULTS FOR 3-D BEAM CALIBRATION VARIATIONS
Table C.1. Results of mesh sensitivity analysis for 3-D beam with 2-D rebar models
utilizing the brittle cracking technique.
Inp
ut
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Direct
Cracking
Strain
0.0027 0.0027 0.0027 0.0027 0.0027 0.0027
Direct
Cracking
Failure Strain
0.0027 0.0027 0.0027 0.0027 0.0027 0.0027
Crack Opening
Strain 0.0027 0.0027 0.0027 0.0027 0.0027 0.0027
Load Rate
(lb/s) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
Rebar Size
(mm) 10 100 5 10 20 50
Mesh Size
(mm) 5 5 10 10 10 10
Res
ult
s
Last Step
Before Failure 0.40 0.25 0.45 0.45 0.45 0.40
Deflection (in) 0.0384 0.0181 0.0282 0.0279 0.0289 0.0266
167
Table C.1. cont’d
Inp
ut
f't (psi) 420.8 420.8 420.8 420.8 420.8
Direct Cracking
Strain 0.0027 0.0027 0.0027 0.0027 0.0027
Direct Cracking
Failure Strain 0.0027 0.0027 0.0027 0.0027 0.0027
Crack Opening
Strain 0.0027 0.0027 0.0027 0.0027 0.0027
Load Rate (lb/s) 5901.39 5901.39 5901.39 5901.39 5901.39
Rebar Size
(mm) 100 100 100 100 100
Mesh Size (mm) 10 20 25 50 100
Res
ult
s Last Step Before
Failure 0.35 0.60 0.45 1.50 2.25
Deflection (in) 0.0209 0.0520 0.2028 0.1876 0.1230
Table C.2. Results of mesh sensitivity analysis for 3-D beam with 2-D rebar models
utilizing the CDP and SC techniques.
Inp
ut
Tension
Stiffening Strain N/A 0.0027 N/A 0.0027 N/A 0.0027
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 N/A 420.8 N/A 420.8 N/A
Model Type CDP SC CDP SC CDP SC
Rebar Size (mm) 5 5 10 10 100 100
Mesh Size (mm) 5 5 5 5 5 5
Res
ult
s
Max Load (LPF) 0.321 0.171 0.321 0.321 0.29 0.165
Converged N N N N N N
168
Table C.2. cont’d
Inp
ut
Tension Stiffening
Strain N/A 0.0027 N/A 0.0027 N/A 0.0027
f'c (psi) 3147.3 3147 3147.3 3147 3147.3 3147
f't (psi) 420.8 N/A 420.8 N/A 420.8 N/A
Model Type CDP SC CDP SC CDP SC
Rebar Size (mm) 5 5 10 10 20 20
Mesh Size (mm) 10 10 10 10 10 10
Res
ult
s
Max Load (LPF) 0.334 0.287 0.341 0.287 0.337 0.23
Converged N N N N N Y
Table C.2. cont’d
Inp
ut
Tension Stiffening
Strain N/A 0.0027 N/A 0.0027 N/A 0.0027
f'c (psi) 3147.3 3147 3147.3 3147 3147.3 3147
f't (psi) 420.8 N/A 420.8 N/A 420.8 N/A
Model Type CDP SC CDP SC CDP SC
Rebar Size (mm) 50 50 100 100 100 100
Mesh Size (mm) 10 10 10 10 20 20
Res
ult
s
Max Load (LPF) 0.339 0.234 0.146 0.234 0.354 0.395
Converged N N Y Y Y N
Table C.2. cont’d
Inp
ut
Tension Stiffening
Strain N/A 0.0027 N/A 0.0027 N/A 0.0027
f'c (psi) 3147.3 3147 3147.3 3147 3147.3 3147
f't (psi) 420.8 N/A 420.8 N/A 420.8 N/A
Model Type CDP SC CDP SC CDP SC
Rebar Size (mm) 100 100 100 100 100 100
Mesh Size (mm) 25 25 50 50 100 100
Res
ult
s
Max Load (LPF) 0.323 0.434 0.65 1.56 1.25 1.67
Converged N N N N N N
169
Table C.3. Results for 3-D beam with 2-D rebar models utilizing the brittle cracking
techniques.
Inp
ut
Load Rate (lb/s) 11240 11240 11240 11240 11240
As (in2) 0.20563 0.20563 0.20563 0.20563 0.20563
A's (in2) 0.0438 0.0438 0.0438 0.0438 0.0438
f't (psi) 210.4 210.4 210.4 105.2 105.2
Direct Cracking
Strain 0.0027 0.002025 0.00135 0.0027 0.0054
Direct Cracking
Failure Strain 0.0027 0.002025 0.00135 0.0027 0.0054
Crack Opening
Strain 0.0027 0.002025 0.00135 0.0027 0.0054
Mesh Size (mm) 25 25 25 25 25
Res
ult
s 1st Removed Element 0.150 0.125 0.125 0.100 0.150
Max Load (LPF) 0.400 0.400 0.375 0.400 0.425
Deflection (in) 0.342 0.404 0.347 0.387 0.563
Table C.3. cont’d
Inp
ut
Load Rate (lb/s) 11240 11240 11240 11240 11240
As (in2) 0.20563 0.20563 0.20563 0.20563 0.20563
A's (in2) 0.0438 0.0438 0.0438 0.0438 0.0438
f't (psi) 52.6 105.2 105.2 105.2 210.4
Direct Cracking
Strain 0.0054 0.0054 0.0054 0.0054 0.00135
Direct Cracking
Failure Strain 0.0054 0.0054 0.108 0.0108 0.0054
Crack Opening Strain 0.0054 0.0108 0.0054 0.0108 0.00135
Mesh Size (mm) 25 25 25 25 25
Res
ult
s 1st Removed Element 0.125 0.225 0.150 0.225 0.125
Max Load (LPF) 0.400 0.425 0.425 0.450 0.400
Deflection (in) 0.345 0.476 0.507 0.546 0.418
170
Table C.3. cont’d
Inp
ut
Load Rate (lb/s) 11240 11802.78 11240 11240
As (in2) 0.10287 0.20563 0.20563 0.20563
A's (in2) 0.02117 0.0438 0.0438 0.0438
f't (psi) 420.8 420.8 420.8 420.8
Direct Cracking
Strain 0.0027 0.0027 0.0027 0.00119
Direct Cracking
Failure Strain 0.0027 0.0027 0.0027 0.00119
Crack Opening Strain 0.0027 0.0027 0.0027 0.00119
Mesh Size (mm) 25 10 25 25
Res
ult
s 1st Removed Element 0.225 N/A 0.25 0.2
Max Load (LPF) 0.225 0.45 0.4 0.2
Deflection (in) 0.111 0.143 0.366 0.035
Table C.4. Results for 3-D beam with 3-D rebar models utilizing the CDP techniques.
Inp
ut
Tension Damage Y Y Y N N N
Compression Damage Y Y Y N N N
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Hard Hard Hard Hard Hard Hard
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient 0.5 1.5 N/A 1.0 0.5 1.5
MPC Slider
Nodes/Cross-Section 2 2 2 2 2 2
MPC Beam
Nodes/Rebar 1 1 1 1 1 1
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.135 0.145 0.238 0.319 0.142 0.145
Deflection (in) 0.010 0.010 0.021 0.025 0.011 0.010
Converged N Y N N N N
171
Table C.4. cont’d
Inp
ut
Tension Damage N Y Y Y Y N
Compression Damage N Y Y Y Y N
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Hard Hard Hard Hard Hard Hard
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient N/A 1.0 0.5 1.5 N/A 1.0
MPC Slider
Nodes/Cross-Section 2 2 2 2 2 2
MPC Beam
Nodes/Rebar 1 N/A N/A N/A N/A N/A
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.258 0.142 0.135 0.145 0.122 0.319
Deflection (in) 0.023 0.010 0.010 0.010 0.010 0.025
Converged Y Y N Y N N
Table C.4. cont’d
Inp
ut
Tension Damage N N N Y Y Y
Compression Damage N N N Y Y Y
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Hard Hard Hard Penalty Exp Hard
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient 0.5 1.5 N/A 1.0 Exp 1.0
MPC Slider
Nodes/Cross-Section 2 2 2 2 2 1
MPC Beam
Nodes/Rebar N/A N/A N/A N/A N/A N/A
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.142 0.305 0.124 0.134 0.103 0.140
Deflection (in) 0.011 0.023 0.010 0.010 0.007 0.010
Converged N Y N N Y N
172
Table C.4. cont’d
Inp
ut
Tension Damage Y Y Y Y Y N
Compression Damage Y Y Y Y Y N
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Penalty Hard Hard Hard Hard Hard
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient 1.0 0.5 1.0 1.5 N/A 0.5
MPC Slider
Nodes/Cross-Section 1 4 4 4 4 4
MPC Beam
Nodes/Rebar N/A 1 1 1 1 1
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.136 0.111 0.115 0.116 0.118 0.135
Deflection (in) 0.010 0.008 0.007 0.007 0.009 0.010
Converged Y Y Y Y Y Y
Table C.4. cont’d
Inp
ut
Tension Damage N N N Y Y Y
Compression Damage N N N Y Y Y
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Hard Hard Hard Penalty Penalty Penalty
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient 1.0 1.5 N/A 0.5 1.0 1.5
MPC Slider
Nodes/Cross-Section 4 4 4 2 2 2
MPC Beam
Nodes/Rebar 1 1 1 N/A N/A N/A
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.139 0.141 0.239 0.131 0.134 0.149
Deflection (in) 0.009 0.009 0.020 0.010 0.010 0.011
Converged Y Y Y N N N
173
Table C.4. cont’d
Inp
ut
Tension Damage Y N N N N Y
Compression Damage Y N N N N Y
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Penalty Penalty Penalty Penalty Penalty Exp
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient N/A 0.5 1.0 1.5 N/A 0.5
MPC Slider
Nodes/Cross-Section 2 2 2 2 2 2
MPC Beam
Nodes/Rebar N/A N/A N/A N/A N/A N/A
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.125 0.251 0.144 0.111 0.133 0.139
Deflection (in) 0.010 0.021 0.011 0.008 0.011 0.010
Converged N Y Y Y Y Y
Table C.4. cont’d
Inp
ut
Tension Damage Y Y Y N N N
Compression Damage Y Y Y N N N
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Exp Exp Exp Exp Exp Exp
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient 1.0 1.5 N/A 0.5 1.0 1.5
MPC Slider
Nodes/Cross-Section 2 2 2 2 2 2
MPC Beam
Nodes/Rebar N/A N/A N/A N/A N/A N/A
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.103 0.104 0.084 0.119 0.121 0.117
Deflection (in) 0.007 0.007 0.007 0.008 0.008 0.008
Converged Y Y N N N N
174
Table C.4. cont’d
Inp
ut
Tension Damage N Yes Yes Yes Yes Yes
Compression Damage N Yes Yes Yes Yes Yes
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Exp Exp Penalty Hard Hard Hard
No. Surfaces in
Contact 2 1 2 2 2 2
Friction Coefficient N/A Exp 1.0 1.0 0.5 1.5
MPC Slider
Nodes/Cross-Section 2 2 1 1 2 2
MPC Beam
Nodes/Rebar N/A N/A N/A N/A N/A N/A
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.088 0.103 0.136 0.140 0.135 0.145
Deflection (in) 0.007 0.007 0.010 0.010 0.010 0.010
Converged N N Y N N Y
Table C.4. cont’d
Inp
ut
Tension Damage Yes Yes No No No No
Compression Damage Yes Yes No No No No
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Hard Hard Hard Hard Hard Hard
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient 0.5 1.5 0.5 1.5 N/A 1.0
MPC Slider
Nodes/Cross-Section 2 2 2 2 2 2
MPC Beam
Nodes/Rebar 1 1 1 1 1 1
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.135 0.145 0.142 0.145 0.124 0.319
Deflection (in) 0.010 0.010 0.011 0.010 0.010 0.025
Converged N Y N N Y N
175
Table C.4. cont’d
Inp
ut
Tension Damage Yes No No No No Yes
Compression Damage Yes No No No No Yes
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Hard Hard Hard Hard Hard Hard
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient N/A 0.5 1.5 N/A 1.0 N/A
MPC Slider
Nodes/Cross-Section 2 2 2 2 2 2
MPC Beam
Nodes/Rebar 1 N/A N/A N/A N/A N/A
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.238 0.142 0.150 0.124 0.319 0.122
Deflection (in) 0.021 0.011 0.011 0.010 0.025 0.010
Converged N N Y N N N
Table C.4. cont’d
Inp
ut
Tension Damage Yes Yes Yes Yes Yes Yes
Compression Damage Yes Yes Yes Yes Yes Yes
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Penalty Exp Hard Hard Hard Hard
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient 1.0 1.0 1.0 0.5 1.0 1.5
MPC Slider
Nodes/Cross-Section 2 2 2 4 4 4
MPC Beam
Nodes/Rebar N/A N/A N/A 1 1 1
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.134 0.103 0.142 0.110 0.086 0.085
Deflection (in) 0.010 0.007 0.010 0.008 0.007 0.007
Converged N Y Y Y Y Y
176
Table C.4. cont’d
Inp
ut
Tension Damage Yes No No No No Yes
Compression Damage Yes No No No No Yes
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Hard Hard Hard Hard Hard N/A
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient N/A 0.5 1.5 N/A 1.0 1.0
MPC Slider
Nodes/Cross-Section 4 4 4 4 4 2
MPC Beam
Nodes/Rebar 1 1 1 1 1 N/A
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.119 0.135 0.141 0.093 0.139 0.142
Deflection (in) 0.009 0.010 0.009 0.020 0.009 0.010
Converged Y Y Y Y Y Y
Table C.4. cont’d
Inp
ut
Tension Damage Yes Yes Yes Yes Yes Yes
Compression Damage Yes Yes Yes Yes Yes Yes
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Exp Exp Exp Exp Exp Exp
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient Exp 0.5 1.0 1.5 Exp 0.5
MPC Slider
Nodes/Cross-Section N/A 2 2 2 2 2
MPC Beam
Nodes/Rebar N/A N/A N/A N/A N/A N/A
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.062 0.139 0.125 0.104 0.103 0.139
Deflection (in) 0.005 0.010 0.010 0.007 0.007 0.010
Converged N Y Y Y Y Y
177
Table C.4. cont’d
Inp
ut
Tension Damage Yes Yes Yes No No No
Compression Damage Yes Yes Yes No No No
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Exp Exp Exp Exp Exp Exp
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient 1.0 1.5 N/A 0.5 1.0 1.5
MPC Slider
Nodes/Cross-Section 2 2 2 2 2 2
MPC Beam
Nodes/Rebar N/A N/A N/A N/A N/A N/A
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.103 0.104 0.084 0.119 0.121 0.117
Deflection (in) 0.007 0.007 0.007 0.008 0.008 0.008
Converged Y Y N N N N
Table C.4. cont’d
Inp
ut
Tension Damage No Yes Yes Yes Yes Yes
Compression Damage No Yes Yes Yes Yes Yes
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Pressure-Overclosure Exp Exp Penalty Penalty Penalty Penalty
No. Surfaces in
Contact 2 2 2 2 2 2
Friction Coefficient N/A Exp 0.5 1.0 1.5 N/A
MPC Slider
Nodes/Cross-Section 2 2 2 2 2 2
MPC Beam
Nodes/Rebar N/A N/A N/A N/A N/A N/A
Concrete Model CDP CDP CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.088 0.103 0.131 0.134 0.149 0.125
Deflection (in) 0.007 0.007 0.010 0.010 0.011 0.010
Converged N Y N N N N
178
Table C.4. cont’d
Inp
ut
Tension Damage No No No No
Compression Damage No No No No
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4
f'c (psi) 3147.3 3147.3 3147.3 3147.3
f't (psi) 420.8 420.8 420.8 420.8
Pressure-Overclosure Penalty Penalty Penalty Penalty
No. Surfaces in Contact 2 2 2 2
Friction Coefficient 0.5 1.0 1.5 N/A
MPC Slider
Nodes/Cross-Section 2 2 2 2
MPC Beam
Nodes/Rebar N/A N/A N/A N/A
Concrete Model CDP CDP CDP CDP
Res
ult
s Max Load (LPF) 0.140 0.144 0.111 0.133
Deflection (in) 0.011 0.011 0.008 0.011
Converged Y Y Y Y
Table C.5. Results for 3-D beam with 3-D rebar models utilizing the SC technique.
Inp
ut
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
Friction Coefficient 0.5 1.0 1.5 N/A 0.5 1.0
Pressure-Overclosure Hard Hard Hard Hard Hard Hard
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
MPC Slider
Nodes/Cross-Section 4 4 4 4 2 2
MPC Beam
Nodes/Rebar 1 1 1 1 1 1
Res
ult
s Max Load (LPF) 0.376 0.391 0.393 0.286 0.340 0.355
Deflection (in) 0.025 0.026 0.025 0.021 0.024 0.024
Converged N Y N N N Y
179
Table C.5. cont’d
Inp
ut
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147.3 3147.3
Friction Coefficient 1.5 N/A 0.5 1.0 1.5 N/A
Pressure-
Overclosure Hard Hard Penalty Penalty Penalty Penalty
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.4 5901.4
MPC Slider
Nodes/Cross-
Section
2 2 2 2 2 2
MPC Beam
Nodes/Rebar 1 1 1 1 1 1
Res
ult
s Max Load (LPF) 0.325 0.273 0.273 0.277 0.339 0.296
Deflection (in) 0.021 0.020 0.019 0.019 0.024 0.023
Converged N N N N N N
Table C.5. cont’d
Inp
ut
f'c (psi) 3147.3 3147.3 3147.3 3147.3 3147 3147
Friction Coefficient 0.5 1.0 1.5 N/A 0.5 1.0
Pressure-
Overclosure Exp Exp Exp Exp Hard Hard
Applied Load (lb) 5901.4 5901.4 5901.4 5901.4 5901.39 5901.39
MPC Slider
Nodes/Cross-
Section
2 2 2 2 2 2
MPC Beam
Nodes/Rebar 1 1 1 1 1 1
Res
ult
s Max Load (LPF) 0.276 0.350 0.328 0.247 0.340 0.355
Deflection (in) 0.018 0.022 0.021 0.018 0.024 0.024
Converged N N N N N Y
180
Table C.5. cont’d
Inp
ut
f'c (psi) 3147 3147 3147 3147 3147 3147
Friction
Coefficient 1.5 N/A 0.5 1.0 1.5 N/A
Pressure-
Overclosure Hard Hard Hard Hard Hard Hard
Applied Load (lb) 5901.39 5901.39 5901.39 5901.39 5901.39 5901.39
MPC Slider
Nodes/Cross-
Section
2 2 4 4 4 4
MPC Beam
Nodes/Rebar 1 1 1 1 1 1
Res
ult
s Max Load (LPF) 0.325 0.273 0.376 0.391 0.393 0.286
Deflection (in) 0.021 0.020 0.025 0.026 0.025 0.021
Converged N N N Y N N
Table C.5. cont’d
Inp
ut
f'c (psi) 3147 3147 3147 3147 3147
Friction Coefficient 0.5 1.0 1.5 N/A 0.5
Pressure-
Overclosure Exp Exp Exp Exp Penalty
Applied Load (lb) 5901.39 5901.39 5901.39 5901.39 5901.39
MPC Slider
Nodes/Cross-Section 2 2 2 2 2
MPC Beam
Nodes/Rebar 1 1 1 1 1
Res
ult
s Max Load (LPF) 0.276 0.350 0.328 0.247 0.273
Deflection (in) 0.018 0.022 0.021 0.018 0.019
Converged N N N N N
181
Table C.5. cont’d
Inp
ut
f'c (psi) 3147 3147 3147
Friction Coefficient 1.0 1.5 N/A
Pressure-Overclosure Penalty Penalty Penalty
Applied Load (lb) 5901.39 5901.39 5901.39
MPC Slider
Nodes/Cross-Section 2 2 2
MPC Beam
Nodes/Rebar 1 1 1
Res
ult
s Max Load (LPF) 0.277 0.339 0.296
Deflection (in) 0.019 0.024 0.023
Converged N N N
Table C.6. Results for 3-D beam with 3-D rebar models utilizing the brittle cracking
technique.
Inp
ut
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Direct Cracking
Strain 0.0027 0.0027 0.0027 0.0027 0.0027 0.0027
Crack Opening
Strain 0.0027 0.0027 0.0027 0.0027 0.0027 0.0027
Direct Cracking
Failure Strain 0.0027 0.0027 0.0027 0.0027 0.0027 0.0027
Load Rate (lb/s) 5901.4 5901.4 5901.4 5901.4 5901.4 4426.0
No. Surfaces in
Contact 2 2 2 2 2 2
Mechanical
Constraint Penalty Penalty Penalty Penalty Penalty Penalty
Pressure-
Overclosure Exp Exp N/A N/A Exp Exp
Friction Exp 1 Exp 1 Exp Exp
Res
ult
s
Max Load (LPF) 0.75 0.65 0.25 0.25 0.75 0.45
Deflection (in) 0.106 0.053 0.019 0.019 0.106 0.035
182
Table C.6. cont’d
Inp
ut
f't (psi) 420.8 420.8 420.8 420.8 420.8 420.8
Direct Cracking
Strain 0.0027 0.0027 0.0027 0.0027 0.0027 0.0027
Crack Opening
Strain 0.0027 0.0027 0.0027 0.027 0.0027 0.0027
Direct Cracking
Failure Strain 0.0027 0.0027 0.0027 0.0027 0.0027 0.0027
Load Rate (lb/s) 11802.8 5901.4 590.1 590.1 98.4 98.4
No. Surfaces in
Contact 2 2 2 2 2 2
Mechanical
Constraint Penalty Penalty Penalty Penalty Penalty Penalty
Pressure-
Overclosure Exp Exp Exp Exp Exp Exp
Friction Exp Exp Exp Exp Exp Exp
Res
ult
s
Max Load (LPF) 0.60 0.45 0.46 0.44 0.27 0.48
Deflection (in) 0.071 0.031 0.037 0.030 0.018 0.032
Table C.6. cont’d
Inp
ut
f't (psi) 420.8 420.8 420.8 420.8 420.8
Direct Cracking
Strain 0.0027 0.0027 0.0027 0.0027 0.0027
Crack Opening
Strain 0.0027 0.0027 0.0027 0.0027 0.0027
Direct Cracking
Failure Strain 0.0027 0.0027 0.0027 0.0027 0.0027
Load Rate (lb/s) 19.7 19.7 590.1 590.1 590.1
No. Surfaces in
Contact 2 2 2 2 2
Mechanical
Constraint Penalty Penalty Penalty Penalty Penalty
Pressure-
Overclosure Exp Exp Exp N/A N/A
Friction Exp Exp 1 Exp 1
Res
ult
s Max Load (LPF) 0.07 0.07 0.57 0.29 0.29
Deflection (in) 0.006 0.005 0.043 0.020 0.020
183
Appendix D
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 I-
girder Bridges".
Sincerely,
Kelly L. Ambrose
(302)-528-2263
1405 Riverside Ave.
Baltimore, MD 21230