3-D FINITE ELEMENT MODELING OF REINFORCED CONCRETE BEAM-COLUMN
CONNECTIONS – DEVELOPMENT AND COMPARISON TO NCHRP PROJECT 12-74
A Project
Presented to the faculty of the Department of Civil Engineering
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF SCIENCE
in
Civil Engineering
by
Harpreet Singh Hansra
FALL 2012
© 2012
Harpreet Singh Hansra
ALL RIGHTS RESERVED
ii
3-D FINITE ELEMENT MODELING OF REINFORCED CONCRETE BEAM-COLUMN
CONNECTIONS – DEVELOPMENT AND COMPARISON TO NCHRP PROJECT 12-74
A Project
by
Harpreet Singh Hansra
Approved by:
__________________________________, Committee Chair
Eric E. Matsumoto, Ph.D., P.E.
__________________________________, Second Reader
Mark Schultz, M.S., P.E., S.E.
____________________________
Date
iii
Student: Harpreet Singh Hansra
I certify that this student has met the requirements for format contained in the University format
manual, and that this project is suitable for shelving in the Library and credit is to be awarded for
the project.
________________________________, Department Chair ____________________
Kevan Shafizadeh, Ph.D., P.E., PTOE
Date
Department of Civil Engineering
iv
Abstract
of
3-D FINITE ELEMENT MODELING OF REINFORCED CONCRETE BEAM-COLUMN
CONNECTIONS – DEVELOPMENT AND COMPARISON TO NCHRP PROJECT 12-74
by
Harpreet Singh Hansra
This report, 3-D Finite Element Modeling of Reinforced Concrete Beam-Column
Connections – Development and Comparison to NCHRP 12-74, investigates the use of
finite element modeling (FEM) to predict the structural response of the cast-in-place
(CIP) reinforced concrete bent cap-column test specimen reported in NCHRP Report 681
– Development of a Precast Bent Cap System for Seismic Regions.
Analysis was performed using LS-DYNA as the finite element processor. The
Karagozian & Case Damaged Concrete model, material MAT_072, was used as the
constitutive model for all concrete elements and material MAT_003, a plastic kinematic
model, was used as the constitutive model for the reinforcing steel. Strain-hardening
effects of steel were neglected for this analysis. Boundary conditions on the FE model
were identical to the vertical and horizontal restraints used on the CIP specimen during
testing. The FE model only considered a monotonic push loading sequence, whereas the
CIP specimen was subjected to reverse cyclic loading. To account for the difference in
v
loading, the FE model results were compared to the hysteretic envelope from the CIP
specimen.
The lateral load-lateral displacement response of the FE model (Model 1) compared
reasonably well to the actual and theoretically predicted response of the CIP specimen.
For lateral displacements less than that corresponding to a displacement ductility of 4.1,
the FE model showed a larger stiffness than the actual CIP response. The model stiffness
degraded as a greater number of concrete elements in the column plastic hinging region
accumulated damage. The degradation and lateral load-displacement response matched
the predicted response within 5% for a displacement ductility larger than 2.0; however,
the model degradation was not as severe as that observed for the CIP specimen.
Concrete damage in the FE model correlated reasonably well with observed
cracking and spalling of the CIP specimen. Significant damage was observed in the
column of the FE model, near the joint, reflecting flexural cracking. Initial yielding of
column longitudinal bars in the FE model occurred at a displacement ductility 26% larger
than the CIP specimen. Based on contours of concrete damage and principal stress
vectors, the primary shear crack formed diagonally through the joint of the FE model at a
lateral load 6% higher than that of the CIP specimen. Joint rotation for the FE model was
significantly less than that of the CIP specimen, approximately half of the specimen
values.
Conclusions include: 1) finite element modeling using appropriate constitutive
models and element formulation can accurately capture the nonlinear behavior of
vi
reinforced concrete beam-column connections, including flexural cracking, joint shear
cracking, steel reinforcement yielding and overall stress distribution; 2) element size for
concrete and steel reinforcement significantly impacts the overall response and accuracy
of results and therefore must be carefully selected for convergence; 3) the Karagozian &
Case damaged concrete model, material MAT_072, can accurately capture the cracking
of concrete using limited inputs (f ’c and aggregate size).
Recommendations include: 1) additional analysis should be performed to
appropriately incorporate a strain hardening model for the reinforcing steel; 2) strain
distribution of the steel reinforcement in the joint (longitudinal reinforcement, joint
hoops, and joint stirrups) should be further investigated as well as the hoop strain
distribution in the column plastic hinge region; 3) a concrete constitutive model capable
of reverse cyclic loading should be investigated; 4) a bar slip model for bond between the
concrete and reinforcing steel should be investigated.
_______________________, Committee Chair
Eric E. Matsumoto, Ph.D., P.E.
_______________________
Date
vii
TABLE OF CONTENTS
Page
List of Tables ............................................................................................................................ x
List of Figures ........................................................................................................................ xii
Chapter
1. INTRODUCTION ............................................................................................................... 1
1.1 Background ............................................................................................................ 1
1.2 Project Objective.................................................................................................... 3
1.3 Significance of Project Results .............................................................................. 3
1.4 Literature Review .................................................................................................. 4
1.5 Project Approach ................................................................................................... 7
1.6 Scope of Report ..................................................................................................... 8
2. DEVELOPMENT OF FINITE ELEMENT MODEL ...................................................... 11
2.1 CIP Specimen Information ................................................................................... 11
2.2 Finite Element Model Geometry ......................................................................... 12
2.3 Element Formulation ........................................................................................... 13
2.4 Constitutive Models ............................................................................................. 14
2.5 Mesh Development .............................................................................................. 16
2.6 Boundary Conditions ........................................................................................... 18
2.7 Loading Application ............................................................................................ 18
3. FINITE ELEMENT ANALYSIS RESULTS ................................................................... 34
3.1 Lateral Load-Lateral Displacement Response ..................................................... 34
3.2 Concrete Damage Parameter................................................................................ 36
3.3 Principal Stress Vectors ....................................................................................... 39
viii
3.4 Joint Shear Stress ................................................................................................. 39
3.5 Joint Rotation ........................................................................................................ 40
3.6 Longitudinal and Principal Stress Distribution ..................................................... 41
3.7 Response of Reinforcing Steel .............................................................................. 43
4. COMPARISON OF RESULTS TO SPECIMEN DATA ................................................. 63
4.1 Lateral Load-Lateral Displacement Response ...................................................... 63
4.2 Joint Shear Stress ................................................................................................. 66
4.3 Joint Rotation ....................................................................................................... 67
4.4 Principal Stress .................................................................................................... 67
4.5 Stages of Concrete Cracking ................................................................................ 68
4.6 Reinforcing Steel Stain Profiles ........................................................................... 69
5. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS ..................................... 84
5.1 Summary .............................................................................................................. 84
5.2 Conclusion ........................................................................................................... 86
5.3 Recommendations ................................................................................................ 87
References ............................................................................................................................... 89
ix
LIST OF TABLES
Page
Table 2.1
A) Summary of CIP Concrete Properties B) Comparison of
GD and CIP properties .................................................................................... 19
Table 2.2
Summary of Reinforcement for Bent Cap, Joint, and Column ....................... 20
Table 2.3
Loading Sequence for FEM ............................................................................ 21
Table 3.1
Summary of Stresses in Concrete Elements ................................................... 45
Table 3.2
Summary of Stresses in Reinforcing Steel...................................................... 45
Table 4.1
Comparison of Lateral Load-Lateral Displacement Response between
Model 1, CIP Specimen, GD Specimen, and Specimen Predictions .............. 71
Table 4.2
Comparison of Lateral Load-Lateral Displacement Response between
Model 2, CIP Specimen, GD Specimen, and Specimen Predictions .............. 71
Table 4.3
Comparison of Average Joint Shear Stress between Model 1, CIP
Specimen, and the GD Specimen.................................................................... 71
Table 4.4
Comparison of Joint Rotation between Model 1, the CIP Specimen,
and the GD Specimen ..................................................................................... 72
Table 4.5
Comparison of Specimen Cracking – Including 38 kip Axial Load and
Self-Weight ..................................................................................................... 72
Table 4.6
Strain Profile Table – Column Longitudinal Rebar (LC8),
CIP Specimen ................................................................................................ 73
Table 4.7
Strain Profile Table – Stirrups in Bent Cap (Mid-height) and
Joint (Top), CIP Specimen.............................................................................. 74
x
Table 4.8
Strain Profile Table – Stirrups in Bent Cap (Mid-height) and Joint
(Bottom), CIP Specimen ................................................................................. 75
Table 4.9
Strain Profile Table – Hoops in Column and Joint (East),
CIP Specimen ................................................................................................. 75
xi
LIST OF FIGURES
Page
Figure 1.1
Prototype Structure from NCHRP Project 12-74.............................................. 9
Figure 1.2
Portion of Prototype Used for Testing and Modeling ..................................... 10
Figure 2.1
Elevation of CIP Specimen Dimensions ......................................................... 22
Figure 2.2
Cross Sections of CIP Specimen Showing Rebar ........................................... 23
Figure 2.3
Column with Instrumentation in Place Prior to Casting of Bent Cap ............. 24
Figure 2.4
Bent Cap Formwork Placed Over Column ..................................................... 25
Figure 2.5
Test Set-up ...................................................................................................... 26
Figure 2.6
Individual Bent Cap and Column Components .............................................. 27
Figure 2.7
Dimensions of FE Model ................................................................................ 27
Figure 2.8
A) Rebar configuration in FE Model, B) CIP Specimen Bent Cap Rebar
Cage ................................................................................................................ 28
Figure 2.9
Half of Bent Cap Cut-away (with Stirrups Removed) to Show
Longitudinal Rebar Connecting the Bent Cap and Column ........................... 29
Figure 2.10
Mesh of Solid Elements .................................................................................. 29
Figure 2.11
3-D Isometric View of Meshed Structure ....................................................... 30
Figure 2.12
Close-up of Meshed Beam Elements in Joint Region..................................... 31
Figure 2.13
Strength Model for Concrete (Malvar et al., 1997): (a) Failure
Surfaces in Concrete Model; (b) Concrete Constitutive Model ................. ….31
Figure 2.14
Comparison of #3 Hoop Steel Element Length in Model 1and Model 2 ....... 32
Figure 2.15
Vertical and Horizontal Boundary Conditions ............................................... 32
Figure 2.16
Boundary Conditions for Axial Load and Lateral Displacement .................... 33
xii
Figure 3.1
Model 1 Lateral Load-Lateral Displacement Curve ....................................... 46
Figure 3.2
Model 2 Lateral Load-Lateral Displacement Curve ....................................... 46
Figure 3.3
Comparison of Lateral Load-Lateral Displacement Response for
Models 1 and 2................................................................................................ 47
Figure 3.4
Relationship between Damage and Strength (compression plotted as
positive stress to show relationship for representative element in column
plastic hinging region) .................................................................................... 47
Figure 3.5
Relationship between Tensile Strength and Damage (representative
element in column plastic hinging region)...................................................... 48
Figure 3.6
Concrete Damage Parameter Contours at 0.045 in (μ0.1) of Lateral
Displacement (Initial Flexural Cracking of Column) ..................................... 49
Figure 3.7
Concrete Damage Parameter Contours Showing Initial Flexural
Cracking in Bent Cap at 0.165 in (μ0.35) of Lateral Displacement
(Bottom View) ............................................................................................... 49
Figure 3.8
Development of Cracking in Joint Region at 0.42 (μ0.9) in of Lateral
Displacement .................................................................................................. 50
Figure 3.9
Plot of Damage Parameter Contours at 4 in (μ8.6) of Lateral
Displacement .................................................................................................. 50
Figure 3.10
Principal Stress Vectors at 0.27 in (μ0.58) of Lateral Displacement .............. 51
Figure 3.11
Close-up of Principal Stress vectors at 0.27 in (μ0.58) of Lateral
Displacement .................................................................................................. 52
Figure 3.12
Principal Stress Vectors Overlaid on Damage Parameter Contours at
1.0 in (μ2.15) of Lateral Displacement ........................................................... 52
xiii
Figure 3.13
Maximum Shear Stress at 0.35 in (μ0.75) of Lateral Displacement,
prior to Development of Cracking in Joint ..................................................... 53
Figure 3.14
Close-up of Maximum Shear Stress in Joint Region at 0.35 in (μ0.75)
of Lateral Displacement, prior to Development of Cracking in Joint ............ 53
Figure 3.15
Maximum Shear Stress Distribution at 4 in (μ8.6) of Lateral
Displacement .................................................................................................. 54
Figure 3.16
Close-up of Maximum Shear Stress in Joint Region at 4 in (μ8.6) of
Lateral Displacement ...................................................................................... 54
Figure 3.17
Average Joint Shear Stress vs. Column Lateral Displacement ....................... 55
Figure 3.18
Joint Rotation vs. Column Lateral Displacement ........................................... 55
Figure 3.19
Joint Rotation at 4 in (μ8.6) of Lateral Displacement (scaling
factor of 50) .................................................................................................... 56
Figure 3.20
Longitudinal (X-Stress) Distribution under Gravity and 38 kip Axial
Load on Column ............................................................................................. 56
Figure 3.21
Longitudinal Stress Distribution at 4 in (μ8.6) of Lateral Displacement ........ 57
Figure 3.22
Minimum Principal Stress Contours under Gravity and Column Axial
Load ................................................................................................................ 57
Figure 3.23
Minimum Principal Stress Contours at 4 in (μ8.6) of Lateral
Displacement .................................................................................................. 58
Figure 3.24
Maximum Principal Stress Contours under Gravity and Column Axial
Load ................................................................................................................ 58
Figure 3.25
Maximum Principal Stress Contours at 4 in (μ8.6) of Lateral
Displacement .................................................................................................. 59
xiv
Figure 3.26
Stress in Column and Bent Cap Longitudinal Rebar under Gravity and
Axial Column Load ........................................................................................ 59
Figure 3.27
Stress in Column and Bent Cap Longitudinal Rebar at 4 in (μ8.6) of
Lateral Displacement ...................................................................................... 60
Figure 3.28
Plot of Stress versus Strain for #5 Column Longitudinal on the South
Side of the Bent Cap and Column Joint .......................................................... 60
Figure 3.29
Plot of Compression Stress vs. Compression Strain for #5 Column
Longitudinal on the North Side of Column .................................................... 61
Figure 3.30
Plot of Tension Stress vs. Tension Strain for #5 Column Longitudinal
on the South Side of Column for an Element Located at Mid-depth of
Bent Cap ......................................................................................................... 61
Figure 3.31
Stress in Stirrups and Hoops at 4 in (μ8.6) of Lateral Displacement .............. 62
Figure 3.32
Plot of Tensile Stress vs. Tensile Strain for #3 Stirrup in Column near
the Bent Cap ................................................................................................... 62
Figure 4.1
Model 1 and CIP specimen Lateral Load-Lateral Displacement Response
Comparison ..................................................................................................... 76
Figure 4.2
Model 2 and CIP Specimen Lateral Load-Lateral Displacement
Comparison ..................................................................................................... 76
Figure 4.3
Model 1 and GD Specimen Lateral Load-Lateral Displacement
Comparison ..................................................................................................... 77
Figure 4.4
Model 2 and GD Specimen Lateral Load-Lateral Displacement
Comparison ..................................................................................................... 77
xv
Figure 4.5
Comparison of Average Joint Shear Stresses – Model 1 and CIP and
GD Specimens ................................................................................................ 78
Figure 4.6
Comparison of Joint Rotation - Model 1 and CIP and GD Specimens........... 78
Figure 4.7
Maximum Principal Joint Stresses from Model 1and CIP and GD
Specimens ....................................................................................................... 79
Figure 4.8
Minimum Principal Joint Stress - Model 1 and CIP and GD Specimens ....... 79
Figure 4.9
Comparison of Concrete Cracking at 48 kips of Lateral Load –
Model 1 and CIP Specimen ............................................................................ 80
Figure 4.10
Comparison of Concrete Cracking at 1.4 in (μ3) of Lateral
Displacement – Model 1 and CIP Specimen .................................................. 80
Figure 4.11
Comparison of Concrete Cracking at 2.8 in (μ6) of Lateral
Displacement – Model 1 and CIP Specimen .................................................. 80
Figure 4.12
Comparison of Concrete Cracking at 3.2 in (μ6.9) of Lateral
Displacement – Model 1 and CIP Specimen .................................................. 81
Figure 4.13
Section of Column Shows Location of LC8 Rebar ........................................ 81
Figure 4.14
Location of Longitudinal Rebar Strain Gauges on the CIP Specimen ........... 82
Figure 4.15
Location of Stirrup Strain Gauges on the CIP Specimen................................ 83
xvi
1
Chapter 1
INTRODUCTION
1.1
Background
This report, 3-D Finite Element Modeling (FEM) of Reinforced Concrete Beam-
column Connections – Development and Comparison to National Cooperative Highway
Research Program (NCHRP) Project 12-74, investigates the use of finite element
modeling to predict the structural response of the cast-in-place and grouted duct test
specimen from NCHRP Report 681 – Development of a Precast Bent Cap System for
Seismic Regions.
Bridges throughout the United States are in need of immediate repair or
replacement because of being deemed structurally deficient or obsolete. Accelerated
Bridge Construction techniques are sought as a viable option to quickly replace or
rehabilitate structures while minimizing effects on traffic flow. Significant research has
been conducted to develop constructible details with reliable performance but use of
these details has been limited in seismic regions (Matsumoto, 2009).
The prototype structure studied in NCHRP Project 12-74 is a two-span nonintegral three-column precast bridge bent. Figure 1.1 shows the prototype structure. All
test specimens were a 42% scale version of the center column and center bent cap region
of the prototype structure as shown in Figure 1.2. The cast-in-place (CIP) test specimen,
specifically the joint, will be modeled and analyzed in this report.
FEM is widely used throughout civil engineering as a method for analyzing
complex systems, especially in structural engineering. A finite element model is a
2
mathematical representation of a physical problem and the results of an analysis depend
on the type of analysis, element type, element aspect ratio, mesh density, load application
and rate, boundary conditions, and material models used in the analyses (Mills-Bria,
2006). In structural engineering applications, the two common finite element analysis
(FEA) methods are implicit and explicit analysis. Both of these methods can solve many
different types of nonlinear, static, or dynamic problems. The implicit procedure relies on
a stiffness matrix and a known set of forces that result in a set of linear equations that can
be solved for displacements. The explicit procedure relies on kinematic relationships to
solve for accelerations. Force being applied to a structure causes movement and elements
in the model strain at certain rates and resist loads (Mills-Bria, 2006).
LS-DYNA is an explicit finite element analysis program capable of performing
highly nonlinear and dynamic analysis, but this report only considers a quasi-static
loading case. This program includes several constitutive models for concrete developed
by different researchers for a variety of applications (Predictive Engineering, 2011). One
of the most important aspects of creating a reliable model is selecting an appropriate
constitutive model. Constitutive models for steel have become reliable and relatively
simple to apply to an analysis. Nonlinear constitutive modeling of concrete, however, has
proven to be more difficult and still has limitations on its applicability. It is always up to
the engineer to verify the accuracy and understand the limitations of the constitutive
models being utilized.
An accurate prediction of the behavior of reinforced concrete structures requires
that the concrete constitutive model simulate known behavior at smaller specimen sizes
3
up to full-scale tests (Wu et al, 2012). Consequently, having a test specimen to compare
FEA results against provides an increased level of confidence in the results. Having a
well-calibrated model allows engineers to confidently analyze variations of the test
specimen and using a FE (finite element) model carry to out alternate loading scenarios.
1.2
Project Objective
The overall objective of this project is to create a finite element model that can
accurately replicate the structural response of the CIP and GD bent cap systems from
NCHRP 12-74. The first goal is to replicate the lateral load-lateral displacement response
because it is a strong indicator of overall structural response. Since testing on the
specimen was conducted using cyclic loading, the results of the FE model are compared
to the lateral load-displacement hysteretic envelope in the positive displacement region
under “push” loading. Other structural parameters examined in this report are deflected
shape, cracking of concrete at various stages, shear stress in the joint, magnitude and
orientation of principal stresses in the joint, overall flexural stress distribution, and stress
distribution throughout steel reinforcement.
1.3
Significance of Project Results
FEA results are only meaningful if the model correlates well with test data and
accurately predicts structural behavior. Ultimately, the goal is to have a well-calibrated
and reliable model that can allow for further study on effects of geometric and material
modifications. More directly, the FEA results will be used to develop a strut-and-tie
model of the structure. The orientation of principal stresses and the flow of forces allows
4
engineers to determine proper magnitude and location of compression struts, tension ties
and nodes.
1.4
Literature Review
Several analyses have been published on the nonlinear finite element analysis of
reinforced concrete using LS-DYNA. LS-DYNA includes at least eight different
constitutive models that can be used to model concrete. This analysis uses the Karagozian
and Case Concrete Damage Model Release 3 (K&C model) (LSTC, 2007). Much of the
literature available for this model is based on blast loading applications, but it has proven
to be an effective model for simulating static and quasistatic loading scenarios as well.
The K&C model was selected because numerous studies, referenced throughout this
report, have shown excellent response under various confining pressures, damaged
conditions, and load capacity.
Release 3 of the K&C Concrete Damage Model brings several improvements over
previous versions. Magallanes et al. presented the formulation, improvement, and results
for the K&C model. The first improvement is the automatic input capability for
generating the model parameters. This addition makes the model much easier to use,
especially when limited information is available on the concrete properties. The second
improvement relates to methods embedded into the model to reduce dependencies on the
mesh due to strain-softening. Lastly, guidance is provided on properly modeling strain
rate effects and discusses the effects of the strain rate parameter.
The U.S. Department of the Interior Bureau of Reclamation published the
handbook, “State-of-Practice for the Nonlinear Analysis of Concrete Dams at the Bureau
5
of Reclamation.” The focus of the handbook is on analysis of concrete dams; however,
some of the general information it provides can be applied to many other concrete
structures. Information specifically reviewed for this report was on contacts between
different surfaces, damping, material properties, and boundary conditions.
Malvar and Simons (1996) discusses the development of a Lagrangian finite
element code with explicit time integration for analyzing structures. It explains the
formulation for a damaged concrete constitutive model and the basis of its formation.
Tensile cutoff, volumetric damage, damage accumulation, strain rate effects, and pressure
cutoff are presented along with their formulas.
Wu et al (2007) presented their results of comparing various LS-DYNA
constitutive concrete models in a presentation paper titled “Performance of LS-DYNA
Concrete Constitutive Models.” This report studied the K&C model, Winfrith Concrete
model, and the Continuous Surface Cap model. The models are compared against each
other under an unconfined uniaxial compression test, tension tests, triaxial compression
tests, and response to a blast load. Concluding remarks from this report show that the
K&C model is capable of capturing key concrete behaviors including post-peak
softening, shear dilation, confinement effect, and strain rate effect. The K&C model is
also suitable for quasi-static loads, even though primarily developed for blast and impact
loading.
Sandia National Laboratories also presented their results of comparing four
concrete damage constitutive models in their report titled “Survey of Four Damage
Models for Concrete”. They compared results from the K&C concrete model, the Riedel-
6
Hiermaier-Thoma model, the Brannon-Fossum model, and the Continuous Surface Cap
model. The parameters against which these models were compared include strength
surfaces, strain-rate dependence, damage accumulation, plastic update, and shear and
bulk moduli.
Schwer and Malvar (2005) compared the results of the K&C model against the
well-characterized unconfined compression strength concrete from the Geotechnical and
Structures Laboratory of the US Army Engineering Research & Development Center.
Areas of study included compaction, compressive shear strength, and extension and
tension of concrete. Data from this study shows that the K&C model is capable of
capturing the complex behavior of concrete, especially when only a minimal amount of
information is known (i.e. compressive strength) about the concrete. Many of the internal
parameters have been calibrated to extensive test data and relationships based on
compressive strength were developed.
Sritharan et al. (2000) performed similar nonlinear finite element analysis of
beam-column connections in order to incorporate effects of bar slip. It was found that bar
slip appeared to have a significant effect on the stress and strain contours as well as
cracking in the joint region. Sritharan et al. (2000) modeled the nonlinearity of cracked
concrete using a smeared-concrete constitutive model in ABAQUS with ANAMAT.
Confinement of concrete was accounted for by forcing the concrete model to follow the
Mander model stress- strain curve for confined concrete. The K&C model differs from
this because it is capable of self-generating the effects of confinement. This paper also
7
provides stress distributions and corresponding strut and ties of the model. These results
were studied as a verification of results.
The end goal for the results of this project is to develop a strut-and-tie model of
the beam column connection. Sritharan (2005) presents a strut-and-tie model based
approach to designing concrete bridge joints. Sritharan (2005) discusses the locations and
magnitudes of struts, ties, and nodes.
1.5
Project Approach
This project began with a review of the NCHRP Project 12-74 CIP test set-up,
testing procedure, and analysis of relevant test data. This information was an excellent
starting point for developing an FEA scheme with the appropriate geometry, boundary
conditions, and material properties for the test. LS-DYNA was selected as the most
suitable finite element analysis software after extensive review of available constitutive
models, nonlinear modeling capabilities, and case histories of the software. LS-DYNA
was advantageous over other FEA programs such as ABAQUS because of its ability to
handle highly nonlinear analysis using an explicit solving scheme. The nonlinear
constitutive models available in LS-DYNA have been extensively researched and tested,
thus providing greater confidence in results. In total, seven different models were
developed and analyzed in order to select the most appropriate modeling approach. Each
of the seven models varied with one or more of the following: mesh density, boundary
conditions, constitutive models, and element type. Of the seven models, only the two that
provided the most comparable data will be discussed in this report. The results have been
post-processed and presented in the following chapters.
8
1.6
Scope of Report
This report will serve as a basis for future FEA on other NCHRP 12-74 precast
bent cap models and strut-and-tie models. This report includes extensive comparisons of
FEA results to NCHRP 12-74 results, with a primary focus on the CIP specimen and with
limited comparisons to the GD specimen. This report includes the following chapters:
1.0
Introduction
2.0
Development of Finite Element Model
3.0
Finite Element Analysis Results
4.0
Comparison of Results to Specimen Data
5.0
Summary, Conclusions, and Recommendations
9
Figure 1.1: Prototype Structure from NCHRP Project 12-74 (Matsumoto, 2009)
10
Figure 1.2: Portion of Prototype Used for Testing and Modeling (Matsumoto, 2009)
11
Chapter 2
DEVELOPMENT OF FINITE ELEMENT MODEL
This chapter begins with a description of the CIP specimen being modeled and
then transitions to the geometry, element formulation, constitutive models, boundary
conditions, and the mesh formulation.
2.1
CIP Specimen Information
The CIP specimen was made up of two components, a bent cap and a column (see
Figure 2.1). The bent cap used a 25 in x 25 in cross section and a length of 12 feet. The
steel reinforcing in the bent cap consists of 12-#5’s (0.65%) at top and bottom for flexural
reinforcement and #3’s at 6 in for shear reinforcement. Dimensions and reinforcement
placement are detailed in Figures 2.1 and 2.2. The concrete mix had a compressive
strength of 4,553 psi. The column has a circular cross section with a 20 in diameter. It
includes 16-#5’s (1.58%) for longitudinal reinforcement and #3 hoops at 2 in. The
concrete in the column had a compressive strength of 6,178 psi. Concrete properties are
summarized in Table 2.1. The joint region reinforcement consists of 4-leg #3 stirrups at 5
in (with two sets of #3 cross ties through depth) adjacent to each side of the joint. Hoop
reinforcement through the column consisted of #3s at 5 in. Average yield strength
measured for the #5 rebar was 61.3 ksi and 68.2 ksi for the #3 rebar. Reinforcement
quantity and strengths are summarized in Table 2.2 (Matsumoto, 2009).
The CIP specimen was fabricated as two separate components; the column was
cast first, as shown in Figure 2.3, and then the bent cap was cast over the column as
12
shown in Figure 2.4. The fabrication and assembly process was intended to replicate the
field process as much as possible in order to predict any constructability issues.
After concrete had cured to an adequate strength, the structure was inverted for
testing. The bent cap now formed the bottom of the structure and the column formed the
top of the structure. The bent cap was simply supported, with a pin support at the north
end and a vertical “roller” support at the south end, as shown in Figure 2.5. A force
controlled 38 kip axial load was applied to the top of the column followed by a force
controlled and displacement control sequence of lateral load or displacement applied to
the center of the column stub (Matsumoto, 2009).
2.2
Finite Element Model Geometry
Similar to the CIP specimen, the FE model consists of two individual
components, a bent cap and a column (Figure 2.6), with the steel reinforcing. The FE
model is based on the design drawings of the CIP specimen, not the as-built dimensions,
as obtained from Figures 2.1 and 2.2. The differences between the design and as-built
dimensions are minor and were therefore neglected. Minor changes were made to rebar
lengths in order to simplify modeling. For example, the test specimen had clear cover on
the ends of the longitudinal rebar in both the column and bent cap of approximately 1
inch. The FE model assumes the rebar extends to the end of column, leaving no clear
cover. Since the ends of the longitudinal bars were not in any of the critical regions of the
model, this change had no direct effect on the overall results. Figure 2.7 shows the
dimensions of the FE model and Figure 2.8 shows the geometry of the rebar compared to
the CIP rebar cage.
13
The concrete elements in the bent cap are not bonded or merged in any way to the
concrete elements in the column. The two components are connected by the longitudinal
rebar extending through the column and all the way through the height of the bent cap.
The connection is assumed to be a cold joint. There is a contact surface with a coefficient
of friction of 0.6 (AASHTO requirement for CIP structures) defined between the two
components. Figure 2.9 shows a partial cutaway of the concrete elements in the bent cap
to show the longitudinal rebar connecting the two parts.
2.3
Element Formulation
The two concrete parts of the FE model use eight-node hexahedron elements. The
constant stress solid element formulation was used with varying mesh sizes ranging from
1 in to 1.5 in. The smaller elements were used in the critical joint region and larger
elements outside of the joint. Element size was varied in order to reduce element quantity
in non-critical regions, which decreased computation time. Figure 2.10 shows the relative
mesh size for the solid elements of the entire structure. Figure 2.11 shows a 3-D view of
the meshed structure. The steel reinforcement is modeled using circular Hughes-Liu
beam elements. Figure 2.12 shows a close-up of the meshed beam elements in the joint
region. The FE model consists of 55,860 solid concrete elements and 4,976 beam
elements, for 60,836 total elements.
The concrete and steel elements require a good coupling mechanism in order to
achieve interaction between the two parts. This analysis uses the
CONSTRAINED_LAGRANGE_IN_SOLID formulation to achieve a proper interaction
relationship. Nodes from the rebar beam elements couple with the surrounding concrete
14
element nodes and therefore strain between the two elements is coupled. Consequently,
this technique implies that the concrete and rebar are fully bonded for the entire length of
the rebar and there is no occurrence of bar slip. There is no development length for the
rebar, which would not happen in a real specimen. Although the lack of development
length is not expected to have any significant impact on the results, the stress distribution
in the rebar can show whether or not there is considerable stress near the ends of the rebar
and whether or not refinement is needed to rebar modeling. If stress was present on the
ends of the rebar in the development length region, the coupling technique would need to
be modified. If there is no stress in that region, then it can be assumed that the bar was
not strained in that region. The Lagrange Constraint command is advantageous over other
coupling techniques because it does not require rebar nodes to coincide with the concrete
nodes (Bermejo et al, 2011). This allows for more flexibility with placing rebar in the
model and makes the task of modeling rebar much quicker.
2.4
Constitutive Models
The solid concrete elements are modeled using LS-DYNA material type
MAT_072R3, the MAT_CONCRETE_DAMAGE_REL3 model. Karagozian & Case
(Glendale, CA) developed this model for blast loading and quasi-static loading
applications using lightweight and normal weight concrete. Release 3 of the K&C
concrete model is a three invariant model, uses three shear failure surfaces, and includes
damage and strain-rate effects and is based on Material Type 16, which is the PseudoTENSOR Model (LSTC, 2007).
15
This constitutive model is able to generate input parameters based solely on the
unconfined compressive strength of the concrete. It relies on the concrete strength to
obtain other parameters by using relationships that correlate compressive strength to
tensile strength and bulk modulus. The deviatoric strength is calculated using simple
functions to characterize three independent failure surfaces that define the yield strength,
maximum strength, and the residual strength of the material (Magallanes et al, 2010).
Hardening of the material is captured by interpolating the plasticity surface
between yield and maximum surfaces based on the value of an internal damage
parameter. Softening follows a similar procedure but interpolation is performed between
the maximum and residual surfaces. Figure 2.13 shows the strength model of the concrete
with the three failure surfaces. This model also includes a tension softening parameter
that is scaled using simple relationships for concrete. The tension softening parameter
controls the strain softening and also forms the basis for determining the fracture
energies. This model provides the option to manually enter the fracture energy or allow it
to internally generate the parameter based the unconfined compressive strength and
maximum aggregate size. Parameters for strain rate effects are also self generated by the
model in order to capture inertial effects on the concrete (Magallanes et al, 2010).
For this analysis, the aggregate size and unconfined compressive strength were
defined in the model, leaving all other parameters to be self-generated. Concrete for the
bent cap used a compressive strength of 4,553 psi and the concrete for the column used a
compressive strength of 6,178 psi. Both the column and bent cap usd ¾ - inch aggregate.
16
Limitations of this model, relevant to this analysis, are its performance under
cyclic loading. Elements accumulate damage and are unable to sustain stress after the
maximum damage parameter of 2.0 has been reached. The damage accumulates in this
model based on strain rate, strain, and volumetric strain parameters (Markovich et al,
2011). If an element becomes fully damaged from tensile stress, its ability to carry a
compressive load is greatly diminished. This presents an issue under cyclic loading,
where an element may be subjected to many compression and tension cycles.
The reinforcing steel is modeled using MAT_003, known as
MAT_PLASTIC_KINEMATIC. This model is well suited for isotropic and elastic
behavior. It has the option of including strain-hardening effects using a linear relationship
for plastic behavior (LSTC, 2007). This analysis does not consider strain-hardening
effects of rebar because of issues with obtaining a steady response of the overall FEM.
The stress-strain curve follows a bilinear relationship that has a zero slope after yield.
Further analysis is needed to refine the concrete constitutive model definition and
meshing of the structure in order to include strain-hardening effects to provide more
reliable results.
2.5
Mesh Development
Finite element analysis is reliant on the size and quality of the mesh. Typically, a
greater number of elements results in increased accuracy of analysis and more a refined
distribution of stresses. The drawback to increasing the number of elements is that
computation time tends to increase significantly.
17
For this project, analyses were performed with increasing mesh density (i.e.,
reduced element size) until results were acceptable. Several analyses were discarded
because of unsteady response of lateral load as a function of displacement. Through
iterations of mesh size, the response of the structure became increasingly steady. The
increased mesh density of the concrete, especially in the joint region allowed for
concentrated damage through the joint with distinct diagonal cracking. Analyses with a
course mesh were not detailed enough to isolate specific cracking, but rather just the
overall region where the cracking occurs. Increasing the mesh density in the column
resulted in a more accurate distribution of vertical stress and vertical reactions.
After several analyses with varying mesh density, the length of beam elements
used to model steel reinforcement appeared to have a greater impact on the stability of
the results than the concrete element size. Overall results such as concrete damage
patterns and stress distributions were all much more realistic after a suitable mesh was
settled upon.
This analysis will discuss two of the models that were developed, Model 1 and
Model 2. Model 1 will be the primary focus of discussion but Model 2 will be used to
compare the lateral load-displacement response. The only difference between Model 1
and Model 2 is the length of the beam elements used to model the reinforcing hoops in
the column. Model 1 uses the same length elements for every hoop in the column and the
joint region. Model 2 has a relatively coarse mesh for the hoops in the column but the
hoops in the joint region have the same element length as in Model 1. Except for the
lengths of the hoop elements just described, Model 1 and Model 2 were identical. Figure
18
2.14 compares a column hoop from Model 1 and Model 2. The course hoop from Model
2 has an element length of 3.44 in and the finer meshed hoops from Model 1 have an
element length of 1.58 in.
2.6
Boundary Conditions
Boundary conditions selected in the FE model are consistent with conditions from
the actual specimen test set-up. Figure 2.5 is a photograph of the test set-up and shows
vertical and horizontal restraints at “N” (north) end and a vertical restraint at “S” (south)
end. An axial load and lateral displacement were applied to the top of the column. Figure
2.15 shows boundary conditions applied to the FE model to replicate the vertical and
horizontal restraints applied to the test specimen. Figure 2.16 shows the nodes used to
apply axial load and lateral displacement on the column.
2.7
Loading Application
In order to minimize dynamic effects in the analysis and simulate the quasi-static
test that was actually conducted, loading was applied in a sequence over a ten-second
time interval. Gravity was applied first by linearly ramping from 0 to 100% over a two
second interval. Second, the 38-kip axial load was applied to the column stub over a twosecond interval. Finally, 4.5 inches of lateral displacement was applied over a 6-second
interval. Table 2.3 summarizes the loading sequence.
19
Table 2.1: A) Summary of CIP Concrete Properties B) Comparison of GD and CIP
properties (Matsumoto, 2009)
Parameter
Design
Actual
Slump
5½'' +/- 2½''
< 3 in, cap and column
Unit Weight
143.9 pcf
N/A
Cap: 4553 psi (137 days)
Column: 6178 psi (194 days)
Cap: 361 psi (138 days)
Column: 452 psi (195 days)
Compressive Strength 4000 psi (28 day)
Tensile Strength
(Split Cylinder)
N/A
Table 2.1.A
Parameter
GD
CIP
f'c
Steel Rebar Strength
Cap and Column:
4557 psi
Yield
Tensile
Cap: 4553 psi
Column: 6178 psi
Yield
Tensile
#3 (Bent cap stirrups;
Column hoops)
64.1
99
68.2
95.5
#5 (Bent cap longitudinal;
Column longitudinal)
64.5
95.2
64.5
90
Grout Compressive
Strength (Bedding layer
and ducts)
8026 psi (6421 psi,
equivalent
cylinder strength)
Table 2.1.B
N/A
20
Table 2.2: Summary of Reinforcement for Bent Cap, Joint, and Column (Matsumoto,
2009)
CIP
Prototype
Design
Similitude [or
Design]
Requirement
Test
Specimen
Specimen to
Similitude
[or Design] Ratio
#5 (0.63)
16
64.5
0.0158
222
48.0
1.06
1.14
—
1.14
0.82
0.80
#3 (0.38)
2.0
68.2
0.0125
1.20
[1.11]
—
0.90
#5 (0.63)
12
64.5
0.0065
454
50.2
1.06
—
—
1.27
0.94
1.14
#3 (0.38)
6.0
68.2
1.20
[0.71]
—
Column
Longitudinal Reinforcement
Bar Size (diameter, in)
No. of bars
fy
ksi
ρ
Mn*
K∙ft
Mn/Dc3
K∙ft/ft3
#11 (1.41)
16
66.0
0.0138
3760
58.7
Bar Size (diameter, in)
Spacing
in
fy
ksi
ρ
#6 (0.75)
3.0
66.0
0.0139
0.59
14
—
0.0138
272
58.7
Transverse Reinforcement
0.31
[1.8]
—
0.0139
Bent Cap
Longitudinal Reinforcement
Bar Size (diameter, in)
No. of bars
fy
ksi
ρ
Mn*
K∙ft
Mn/bh2
K∙ft/ft3
#11 (1.41)
12
66.0
0.0051
6680
44.2
Bar Size (diameter, in)
Spacing
in
fy
ksi
#6 (0.75)
12.0
66.0
0.59
—
—
0.0051
483
44.2
Transverse Reinforcement
0.31
[8.4]
—
Joint
Inside Joint
Transverse Reinforcement (ρs)
Bar Size (diameter, in)
Spacing
in
ρs/ρmin
fy
ksi
Side Face Reinforcement (Assf)
No. of bars - Bar Size
As/Acap
in2/ in2
fy
ksi
Construction Stirrups
No. of bars - Bar Size
Area
in2
fy
ksi
#6 (0.75)
3.0
5.56†
66.0
—
—
—
—
#3 (0.38)
5.0
1.22
68.2
—
—
—
—
8 - #6
0.19
60.0
—
[0.10]
—
4 - #3
0.12
68.2
—
[1.20]
—
2 - #6
0.88
66.0
—
—
—
2 - #3
0.22
68.2
—
—
—
Adjacent to Joint
Vertical Stirrups (Asjv)
No. of bars - Bar Size
Spacing
in
Asjv/Ast
in2/ in2
fy
ksi
Horizontal Ties (Asjh)
No. of bars - Bar Size
Spacing
in
Asjh/Ast
in2/ in2
fy
ksi
5 - #6
6.0
0.35
66.0
—
—
[0.20]
—
3 - #3
5.0
0.27
68.2
—
—
[1.33]
—
4 - #6
12.0
0.35
66.0
—
—
[0.10]
—
2 - #3
8.0
0.13
68.2
—
—
[1.33]
—
21
Time
0 to 2 seconds
2 to 4 seconds
4 to 10 seconds
Table 2.3: Loading Sequence for FEM
Loading
Gravity applied over a linear ramp. Gravity is ramped
to 100% over 2 seconds and stays constant for
remainder of analysis.
38-kip axial load applied as a ramp load over this
interval and remains constant for remainder of
analysis.
4.5 inches of displacement applied to the column stub
over this interval.
22
Figure 2.1: Elevation of CIP Specimen Dimensions (Matsumoto, 2009)
23
Figure 2.2: Cross Sections of CIP Specimen Showing Rebar (Matsumoto, 2009)
24
Figure 2.3: Column with Instrumentation in Place Prior to Casting of Bent Cap
(Matsumoto, 2009)
25
Figure 2.4: Bent Cap Formwork Placed Over Column (Matsumoto, 2009)
26
Figure 2.5: Test Set-up (Matsumoto, 2009)
27
Joint
Region
S Support
N Support
Bent Cap
Column
Column
Stub
Figure 2.6: Individual Bent Cap and Column Components
144 in
25 in
3 in diameter
20 in
55 in
Figure 2.7: Dimensions of FE Model
28
#3 stirrups
#3 hoops at 5 in
16-#5
#3 hoops at 2 in
18 in O.D.
A
B
Figure 2.8: A) Rebar configuration in FE Model, B) CIP Specimen Bent Cap Rebar Cage
(Matsumoto, 2009)
29
Bent Cap
Column
Figure 2.9: Half of Bent Cap Cut-away (with Stirrups Removed) to Show Longitudinal
Rebar Connecting the Bent Cap and Column
Elevation View
Figure 2.10: Mesh of Solid Elements
End View
30
Figure 2.11: 3-D Isometric View of Meshed Structure
31
Joint Region
Column
Figure 2.12: Close-up of Meshed Beam Elements in Joint Region
Figure 2.13: Strength Model for Concrete (Malvar et al., 1997): (a) Failure Surfaces in
Concrete Model; (b) Concrete Constitutive Model
32
1.58 in
Model 1
3.44 in
Model 2
Figure 2.14: Comparison of #3 Hoop Steel Element Length in Model 1 and Model 2
Vertical and horizontal
restraints
Vertical restraint
N
S
Figure 2.15: Vertical and Horizontal Boundary Conditions
Axial load
33
Lateral displacement
N
S
Figure 2.16: Boundary Conditions for Axial Load and Lateral Displacement
34
Chapter 3
FINITE ELEMENT ANALYSIS RESULTS
This chapter presents results from the FE analysis of the CIP specimen models of
NCHRP 12-74 described in Chapter 2. The lateral load-displacement response is
presented first. The lateral load-displacement relationship is the most important area of
study for this analysis because it is the best indicator of overall structural response.
The second result presented is the concrete damage parameter that is internal to the K&C
concrete constitutive model. This parameter is an indicator of cracking in the structure at
various stages. This is followed by an analysis of the orientation of principal stresses,
joint rotation, joint shear stress, overall (flexural) stress distribution, and stress
distribution in the reinforcing steel. Each of the parameters described are analyzed at
relevant key points of interest, which include one or more of the following:
a. Initial cracking of bent cap and column
b. Initial cracking of joint region
b. Initial yielding of longitudinal rebar in column
d. Maximum lateral displacement of column
3.1
Lateral Load-Lateral Displacement Response
The lateral load-lateral displacement response for both FE models was developed
from the FEA by acquiring the lateral displacement at the center of the column stub and
the lateral reaction corresponding to a specific displacement (shown in Figure 3.1).
Displacement ductility, μ, was calculated by normalizing displacement by the
displacement at effective yield. Effective yield was calculated as described in
35
Matsumoto (2009). Both curves start with elastic behavior which transitions to plastic
response between 0.2 and 0.5 in of lateral displacement. Model 1 reaches a peak lateral
load of 65.5 kips at a lateral displacement of 4 in (μ8.6). Model 2 reaches a peak lateral
load of 57 kips at a lateral displacement of 2.63 in (μ5.7).
Based on the lateral load-displacement response, Model 1 (shown in Figure 3.1)
appears to have a steadier response relative to Model 2 (shown in Figure 3.2) through
four inches of lateral displacement. The lateral load-displacement response for Model 2
has two sudden dips in force of approximately 8.6 kips at 0.18 in (μ0.34) and 0.465 in
(μ1) of lateral displacement. These dips correspond with times when large areas of
concrete accumulate damage over a lateral displacement interval of less than 0.015 in
(μ0.03). Elements exceed a damage parameter value of 1.0 and LS-DYNA interpolates
strength between the maximum strength surface and the residual strength surface. Since
the damage parameter goes from zero to two over a very small displacement, the
transition from the maximum strength surface to the residual surface is abrupt. A drastic
change in the active surface, from maximum to residual, causes a drop in the force at the
displacement where damage has occurred. Since the residual strength surface is weaker
and less stiff than the other two strength surfaces (as shown in Figure 2.13), the transition
to the residual strength translates to a less stiff response in the lateral load-displacement
following the dip. The “softening” of the response can also be observed on the lateral
load displacement curve for smaller dips that occur at various displacements. Model 1 has
a series of relatively smaller dips on the lateral load-displacement response between
displacements of 0.40 in (μ0.86) and 0.735 in (μ1.6).
36
These dips occur for the same reason as described for Model 2, except the
magnitude of the drop in load is smaller. Figure 3.3 shows the lateral load-displacement
curves for both Model 1 and Model 2 for comparison. Relatively large areas of concrete
elements accumulating significant damage simultaneously were not observed as in Model
2. Damage to the elements is distributed out over a longer interval of lateral
displacement. The smoother response of Model 1 is attributed to the smaller element size
of the hoop steel reinforcing in the column. The increased number of elements in the
hoop steel provides a more accurate confinement of the elements, which limits the
simultaneous damage accumulation. Since the hoop steel element length of Model 2 is
longer than Model 1, when an element reaches the yield stress and begins plastic strain, a
greater length of element yields, which results in less confinement. With the finer rebar
mesh of Model 1, the yielding can be isolated more accurately to a smaller length of
rebar. The smaller length elements therefore produce smaller strains, consequently
resulting in higher confinement. Analysis from this point on is carried out on Model 1,
unless otherwise specified, as it is the more reliable and accurate model.
3.2
Concrete Damage Parameter
The K&C concrete constitutive model determines the strength of the concrete
based on an internal damage parameter that ranges from 0.0 to 2.0. The active strength
surface is based on an interpolation between two of the three strength surfaces in the
model. If the damage parameter is less than 1.0, it is interpolating strength between the
yield surface and the maximum strength surface. When the parameter is equal to 1.0, it
has reached the maximum strength surface. When the damage parameter is greater than
37
1.0, strength is being interpolated between the maximum strength surface and the residual
strength surface. In this analysis, most concrete elements do not reach the fully damaged
state (i.e., a damage parameter equal to 2.0). The reinforcing steel provides confinement,
which increases the failure strength of the concrete and ultimately does not allow
elements to become fully damaged in compression. In this analysis, elements that
accumulated damage would typically reach a maximum damage value that approached
2.0 but was always slightly less (approximately 1.97). For elements under compression,
the damage parameter was close to 2.0 and the damaged element strength is based was
essentially based on the residual strength curve.
Figure 3.4 shows a plot of the vertical compressive stress (Z-stress) and the
damage accumulated for an element in the column. This figure shows the short transition
period of the model interpolating between the various surfaces. After reaching the
maximum stress value, the stress drops rapidly towards the residual strength curve. This
corresponds to the damage parameter going from 1.0 and approaching 2.0 over a short
amount of time. The model does not have a residual strength curve for elements that have
failed in tension, so after reaching the maximum tension strength the stress in the element
rapidly degrades to zero. Figure 3.5 is a plot of the vertical tensile stress (Z-Stress) and
the damage parameter. It shows that as the damage parameter approaches 2.0, the tensile
strength goes to zero.
Contours of this damage parameter are plotted in Figures 3.6 through 3.9. The
damage contours range between blue and red; blue representing nearly no damage
accumulation and red representing damage values approaching 2.0. Figure 3.6 shows the
38
plot of the damage parameter at 0.045 inches (μ0.01) of lateral displacement, which
corresponds to approximately 20 kips of lateral load. The damage parameter for the
concrete elements on the tension side the column near the bent cap and column
connection approaches 2.0, which appears to be the formation of initial cracking due to
flexure. Figure 3.7 shows the damage parameter approaching 2.0 on the bottom south
side of the bent cap at 0.165 (μ0.35) in of displacement, which corresponds to
approximately 36 kips of lateral load. Figure 3.8 shows the initial development of a crack
through the joint at 0.42 (μ0.9) in of lateral displacement, which corresponds to a lateral
load of 51 kips. Lastly, Figure 3.9 shows the damage accumulation at 4.0 in (μ8.6) of
lateral displacement, which corresponds to the maximum lateral load of 65 kips. At 4.0 in
(μ8.6) of lateral displacement, there is significant damage accumulation in the column.
Most of the elements on the compression side of the column near the joint have
transitioned toward the residual strength curve at this stage because the damage
parameter approaches 2.0.
No attempt was made at making a correlation between the damage parameter and
concrete spalling on the column because of the relatively large size of concrete elements
in the column. Larger elements made it difficult to isolate spalling. A cross section of the
center of the bent cap shows that interior joint damage was minimal and remained in the
top 17 in of the bent cap in a 5 in diameter area surrounding the column longitudinal
rebar connecting to the joint.
39
3.3
Principal Stress Vectors
The principal stress vectors are studied to determine where the maximum stresses
are forming and to locate areas of cracking. For this analysis, the focus was on the
principal stresses in the joint region of the bent cap. These stresses are used to predict
cracking through the joint and are compared against cracking predicted by the K&C
model using the damage parameter. Figure 3.10 shows the maximum (tensile) principal
stress vectors at 0.27 inches (μ0.58) of displacement, which just before the development
of cracking in that area. The plot shows principal stress vectors forming at a diagonal
through the joint. Figure 3.11 zooms in on the joint to highlight the vectors through the
joint. The principal vectors show that tension causes the primary diagonal crack through
the joint. Figure 3.12 shows the principal stress vectors overlaid with the contours of the
concrete damage parameter. The concrete becomes damaged along the same diagonal
formed by the maximum principal vectors.
3.4
Joint Shear Stress
Shear stress in the joint region of the bent varies across a large range with small
areas of very high stress concentration. Typical shear stress values range between 100 psi
(1.48√𝑓𝑐′ ) to 500 psi (7.41√𝑓𝑐′ ). Figures 3.13 and 3.14 show the contour plots of the
maximum shear stress at 0.345 in (μ0.74) of lateral displacement, which is prior to
cracking in the joint. A small zone of shear stress concentration, approximately 2x2 in, is
seen where the beam contacts the compression side of the column. Shear stress values in
this region reach a maximum value of 1,041 psi (15√𝑓𝑐′ ) at 4 in (μ8.6) of lateral
displacement. Figures 3.15 and 3.16 show the maximum shear stress values at 4 in (μ8.6)
40
of lateral displacement. The damage parameter of the concrete elements in the shear
stress concentration zone approaches 2.0 at initial joint cracking. Since the element
strength is calculated based on the residual strength surface, the elements reach high
stress values. Maximum stress values are summarized in Table 3.1.
The average shear stress of the joint is estimated by averaging the shear stress
values from twelve elements through the center of the joint. The average values from the
center were comparable with other sections of the joint. The averaging was limited to
twelve elements because of the time-consuming process of post-processing data from
several elements. Selecting every other element through the center of the joint resulted in
twelve elements. Figure 3.17 shows the average shear stress plotted against the lateral
displacement of the column. This plot exhibits a similar trend as the lateral loaddisplacement plot and relatively small stresses.
3.5
Joint Rotation
Joint rotation in the FE model was measured using a similar method as from
Matsumoto (2009). The instrumentation set up from Matsumoto (2009) is shown in
Figure 2.5. Joint rotation on the FE model was calculated using nodal displacement data
at the nodes corresponding to locations of the cap rotation instrumentation on the CIP
specimen. The following equation from Matsumoto (2009) was used to calculate the joint
rotation angle:
𝜃𝑗𝑟 =
𝛿𝑗𝑟,𝑛 − 𝛿𝑗𝑟,𝑠
𝐷𝑐
where,
𝛿𝑗𝑟,𝑛 = 𝐽𝑜𝑖𝑛𝑡 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡, 𝑛𝑜𝑟𝑡ℎ
41
𝛿𝑗𝑟,𝑠 = 𝐽𝑜𝑖𝑛𝑡 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡, 𝑠𝑜𝑢𝑡ℎ
𝐷𝑐 = 𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑟 𝑑𝑒𝑝𝑡ℎ 𝑜𝑓 𝑐𝑜𝑙𝑢𝑚𝑛 𝑖𝑛 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑙𝑜𝑎𝑑𝑖𝑛𝑔
Figure 3.18 shows a plot of the joint rotation against the lateral displacement. The
maximum joint rotation was calculated to be 1.0x10-3 radians at a lateral column
displacement of 4 in (μ8.6). The exaggerated deflected shape of the bent cap is shown in
Figure 3.19 with a scaling factor of fifty applied at 4 in of lateral displacement. The
column is removed from Figure 3.19 for clarity.
3.6
Longitudinal and Principal Stress Distribution
The two primary stresses discussed in this section are the longitudinal stresses
acting in the horizontal direction, the X-stresses, and the principal stresses. The Xstresses in the bent cap are a result of flexure and axial loading caused by the lateral
displacement of the column and the 38 kip axial load on the column. Figure 3.20 shows
the X-stress contours acting on the bent cap after gravity and axial loading on the column
stub is applied (lateral displacement equals zero at this point). This stress distribution
reflects a simply supported beam: tension on the bottom face and compression on the top
face. By observation, the neutral axis runs through mid-height of the bent cap since the
concrete is uncracked and the reinforcing is symmetric. Figure 3.21 shows the stress
distribution of the entire structure at 4 in (μ8.6) of lateral displacement of the column.
Results are only presented at 4 in (μ8.6) of lateral displacement because following lateral
loading, the distribution does not change significantly, but only the magnitude does. As
lateral displacement is incrementally applied to the column stub, the stress distribution
begins to change and forms a diagonal compression strut through the joint. Distinct
42
tensile zones form near the joint and the lateral restraint (as shown in Figure 3.21), which
suggest possible locations of tension-ties. Stress distribution in the column behaves as
expected, with compressive stress acting on the side opposite of where the push load is
applied. Tension stresses develop on the same side as the lateral load application due in
part to the horizontal restraint at the pin.
The moment generated in the column by lateral loading of the column stub is
transferred to the joint region of the bent cap via tension and compression of longitudinal
column reinforcement and compression of the column concrete against the bent cap. The
moment that develops in the joint causes a compression zone on the top face of the right
(S) side of the bent cap and a tension zone on the left (N) side of the bent cap, as shown
in Figure 3.21. The maximum tension stress on the top face of the bent reached
approximately 280 psi (4√𝑓𝑐′ ). The maximum compressive stress on the top face of the
bent reached approximately 700 psi (0.15𝑓𝑐′ ). These values do not include stress
concentrations near the bent-column contact, where stress levels reached higher values.
The concrete elements in the stress concentration area are reaching higher values based
on their residual strength surface.
Figure 3.22 shows the minimum principal stress contours after gravity and the
axial load on the column have been applied. This figure shows how compressive stresses
are transferred from the column to the bent cap, and then transferred to the vertical
restraints. Figure 3.23 is a plot of the principal compressive stress contours at 4 in of
lateral displacement. This plot clearly identifies the location and direction of a
43
compression strut through the joint. Another compression strut can be seen going from
the top left (N) side of the joint to the left (N) restraint.
Figure 3.24 shows the maximum principal stress contours after gravity and axial
loading on the column have been applied. Tensile stresses are observed on the bottom
half of the bent cap and the stress can be observed being transferred to the vertical
restraints. Figure 3.25 shows the maximum principal stress vectors at 4 in of lateral
displacement. Tension stress is observed on the bottom of the right side of the bent cap
beneath where the compressive stresses act. Figure 3.25 also shows significant tension in
the joint region, which contributed to the development of a shear crack. Maximum
principal stress vectors overlaid on the damage parameter contours verifies this result.
3.7
Response of Reinforcing Steel
The stress distribution in the longitudinal rebar under gravity and the axially
loaded column is shown in Figure 3.26. Figure 3.27 shows the stress in the longitudinal
bars at 4 in of lateral displacement. The longitudinal #5 rebar in the column yielded in the
plastic hinging region adjacent to the column connection, and indicates strain penetration
into the joint. Near the joint, column longitudinal rebar yielded at 60.6 ksi and remained
constant at that value for the remaining analysis because strain-hardening effects were
neglected. Rebar in the column yielded due to both compression and tension, although
the zone of tension yielding area is much larger. Longitudinal rebar on the bottom right
(N) side of the bent cap adjacent to the joint reached a maximum stress of 27.4 ksi
(45.2% of yield).
44
Figure 3.28 is a stress-strain plot of the #5 column longitudinal rebar on the north
side of the column. Figure 3.29 is a stress-strain plot of the #5 column longitudinal rebar
on the south side of the column, in the compression region. Figure 3.30 is a stress-strain
plot for an element of #5 column longitudinal rebar 12.5 in into the bent cap on the north
side of the joint. Figure 3.30 shows a stress strain plot for an element of rebar that did not
reach yield.
Hoop reinforcing along the length of the column yielded at 68.4 ksi due to
damage and subsequent dilation of concrete elements. The hoops yielded in tension on
the north side of the column. Stress and strain values at key locations on the hoops is
discussed further in Chapter 4.
Stirrups in the bent cap did not reach the yielding stress. Stirrups in the joint
region reached a stress of 15% of yield, but outside of this region, they had negligible
stress values. Figure 3.31 shows the distribution of stress throughout the column hoops
and the stirrups in the bent cap. Figure 3.32 shows the tensile stress-strain plot of a stirrup
on the compression side of the column that yielded.
45
Table 3.1: Summary of Stresses in Concrete Elements
Stress
Loading State
Component
Gravity + 38 kip
axial on column
4 in of lateral
displacement
(μ8.6)
Maximum
Principal (psi)
Minimum
Principal (psi)
Maximum
Shear (psi)
Joint
Region
335.13
469
21
Column
-
245
105
Joint
Region
434
3472
1566
Column
-
6944
2558
Table 3.2: Summary of Stresses in Reinforcing Steel
Loading
State
Component
Gravity +
38 kip axial
on column
Bent Cap
4 in of lateral
displacement
(μ8.6)
Bent Cap
Column
Column
Bar
Size
Yield
Stress
(ksi)
#5
#3
#5
#3
#5
#3
#5
#3
60.6
68.4
60.6
68.4
60.6
68.4
60.6
68.4
Tension
Max.
% of
Stress
Yield
(ksi)
2.3
3.8
2.0
3.0
27.2
44.9
26.6
38.9
60.6
100
68.4
100
Compression
Max.
% of
Stress
Yield
(ksi)
2.55
4.2
1.78
2.6
3.96
6.5
9.67
16.0
4.67
6.8
60.6
100
31.5
46.1
46
70
Lateral Force (kips)
60
Longitudinal rebar in
column yields at 0.45 in
50
Peak lateral load
of 65.5 kips
40
30
Concrete damage on compression
side of column near joint
20
10
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Lateral Displacement (in)
Figure 3.1: Model 1 Lateral Load-Lateral Displacement Curve
60
Lateral Force (kips)
50
Peak lateral load
of 57 kips
40
Concrete damage on compression
side of column near joint
30
20
10
0
0
0.5
1
1.5
2
2.5
3
3.5
Lateral Displacement (in)
Figure 3.2: Model 2 Lateral Load-Lateral Displacement Curve
4
47
70
Lateral Force (kips)
60
50
40
30
Dip in response results
in lower stiffness
20
Model 1
10
Model 2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Lateral Displacement (in)
Figure 3.3: Comparison of Lateral Load-Lateral Displacement Response for
Models 1 and 2
2
Z-Stress
Damage Paramter
Damage
approaches 2.0
2500
1.5
Damage of 1.0
corresponds to
maximum stress
(2,520 psi)
2000
1500
1
1000
0.5
500
Damage Parameter
Vertical Compressive Stress (psi)
3000
Residual Strength
0
0
0
2
4
6
8
10
Analysis Time (seconds)
Figure 3.4: Relationship between Damage and Strength (compression plotted as positive
stress to show relationship for representative element in column plastic hinging region)
48
2
Analysis Time (seconds)
4
6
8
500
400
Z-stress (psi)
300
200
100
2
Maximum
tensile stress
(468 psi)
Damage
parameter
approaches 2.0
Compressive
stress due to
gravity and axial
load
-200
1.5
1
0
-100
10
Damage Parameter
0
0.5
Z-stress
Damage Parameter
0
Figure 3.5: Relationship between Tensile Strength and Damage (representative element
in column plastic hinging region)
49
Figure 3.6: Concrete Damage Parameter Contours at 0.045 in (μ0.1) of Lateral
Displacement (Initial Flexural Cracking of Column)
Direction of lateral
displacement
Figure 3.7: Concrete Damage Parameter Contours Showing Initial Flexural Cracking in
Bent Cap at 0.165 in (μ0.35) of Lateral Displacement (Bottom View)
50
Introduction of
joint shear crack
development
Figure 3.8: Development of Cracking in Joint Region at 0.42 (μ0.9) in of Lateral
Displacement
Significant damage
accumulation in column
Shear crack through
joint region
Damage due flexure
in bent cap
Figure 3.9: Plot of Damage Parameter Contours at 4 in (μ8.6) of Lateral Displacement
51
Principal stress vectors
indicating diagonal crack
through joint
Figure 3.10: Principal Stress Vectors at 0.27 in (μ0.58) of Lateral Displacement
52
Close-up of
principal vectors
through the joint
Figure 3.11: Close-up of Principal Stress vectors at 0.27 in (μ0.58) of Lateral
Displacement
Figure 3.12: Principal Stress Vectors Overlaid on Damage Parameter Contours at 1.0 in
(μ2.15) of Lateral Displacement
53
Units in psi
Figure 3.13: Maximum Shear Stress at 0.35 in (μ0.75) of Lateral Displacement, prior to
Development of Cracking in Joint
Units in psi
Figure 3.14: Close-up of Maximum Shear Stress in Joint Region at 0.35 in (μ0.75) of
Lateral Displacement, prior to Development of Cracking in Joint
54
Concentration of
shear stress
Units in psi
Figure 3.15: Maximum Shear Stress Distribution at 4 in (μ8.6) of Lateral Displacement
Units in psi
Figure 3.16: Close-up of Maximum Shear Stress in Joint Region at 4 in (μ8.6) of Lateral
Displacement
55
Average Joint Shear Stress (psi)
350
300
250
Peak average shear
stress (299 psi)
200
150
Concrete elements in stress
concentration zone transition to
residual strength after approaching
a damage parameter of 2.0
100
50
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Lateral Displacement (in)
Figure 3.17: Average Joint Shear Stress vs. Column Lateral Displacement
0.0012
Joint Rotation (rad)
0.001
0.0008
0.0006
0.0004
0.0002
0
0
0.5
1
1.5
2
2.5
3
Lateral Displacement (in)
Figure 3.18: Joint Rotation vs. Column Lateral Displacement
3.5
4
56
Figure 3.19: Joint Rotation at 4 in (μ8.6) of Lateral Displacement (scaling factor of 50)
Units in psf
Figure 3.20: Longitudinal (X-Stress) Distribution under Gravity and 38 kip Axial Load
on Column
57
Compressive stress caused
by lateral displacement of
column
Tensile stress across entire
section
Units in psf
Figure 3.21: Longitudinal Stress Distribution at 4 in (μ8.6) of Lateral Displacement
Transfer of compressive
stresses to vertical restraints
Units in psf
Figure 3.22: Minimum Principal Stress Contours under Gravity and Column Axial Load
58
Compression strut
through joint
Units in psf
Figure 3.23: Minimum Principal Stress Contours at 4 in (μ8.6) of Lateral Displacement
Units in psf
Figure 3.24: Maximum Principal Stress Contours under Gravity and Column Axial Load
59
Units in psf
Figure 3.25: Maximum Principal Stress Contours at 4 in (μ8.6) of Lateral Displacement
Bent cap
longitudinal bars
Peak compressive
stress of 3,333 psi
(54% of yield)
Units in psi
Peak tensile stress of
2,308 psi (38% of yield)
Figure 3.26: Stress in Column and Bent Cap Longitudinal Rebar under Gravity and Axial
Column Load
60
Rebar reached
yield stress
Units in psi
Tensile stress dissipated
through joint
Figure 3.27: Stress in Column and Bent Cap Longitudinal Rebar at 4 in (μ8.6) of Lateral
Displacement
70000
Tensile Stress (psi)
60000
50000
Dip in stress possibly
corresponding to
damaged concrete
40000
30000
20000
10000
0
0
0.002
0.004
0.006
0.008
0.01
Tensile Strain
Figure 3.28: Plot of Stress versus Strain for #5 Column Longitudinal on the South Side of
the Bent Cap and Column Joint
61
0
0.002
0.004
0.006
0.008
0.01
0
Compression Stress (psi)
-10000
-20000
-30000
-40000
-50000
-60000
-70000
Compression Strain
Figure 3.29: Plot of Compression Stress vs. Compression Strain for #5 Column
Longitudinal on the North Side of Column
45000
40000
Tensile Stress
35000
30000
25000
20000
15000
10000
5000
0
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
Tensile Strain
Figure 3.30: Plot of Tension Stress vs. Tension Strain for #5 Column Longitudinal on the
South Side of Column for an Element Located at Mid-depth of Bent Cap
62
Hoops reached
yield stress of
68,480 psi
Units in psi
Figure 3.31: Stress in Stirrups and Hoops at 4 in (μ8.6) of Lateral Displacement
70000
Tensile Stress (psi)
60000
50000
40000
30000
20000
10000
0
0
0.001
0.002
0.003
0.004
0.005
Tensile Strain
Figure 3.32: Plot of Tensile Stress vs. Tensile Strain for #3 Stirrup in Column near the
Bent Cap
63
Chapter 4
COMPARISON OF RESULTS TO SPECIMEN DATA
This chapter compares results from the FEA with the results from the CIP and GD
specimen. The lateral load-lateral displacement response of the FE models was compared
against the response from of the CIP specimen, GD specimen, and the predicted
responses of both specimens. Comparisons including Model 2 were limited to the lateral
load-displacement response and all other comparisons were made using data from Model
1. Joint shear stress from Model 1 was compared with the CIP and GD specimen using a
Table of values and a plot of the stress vs. lateral displacement of the column. Similar
comparisons were made for joint rotation, cracking of concrete, the strain profiles of
reinforcing steel.
4.1
Lateral Load-Lateral Displacement Response
The lateral load-displacement response of both FE models compared reasonably
well to the response of the CIP and GD specimens and specimen predictions. Model 2 is
only considered in this section for comparison of lateral load-displacement response.
Figure 4.1 shows the lateral load-displacement response of Model 1 plotted with
the hysteretic envelope of the CIP test specimen and the predicted lateral loaddisplacement response for the CIP specimen. Model 1 closely matches the predicted CIP
response through 4 in (μ8.6) of lateral displacement, except from 0.4 in to 1.0 in. It does
not compare as closely against the hysteretic push envelope of the CIP specimen, as it
predicts higher loads at most displacements greater than 0.5 in (μ1.07). The initial
stiffness (disp. < 0.1 in) of Model 1 is, on average, twice as stiff as the predicted response
64
and 20% larger than the hysteretic envelope of the CIP specimen. Significant change in
the stiffness of Model 1 begins when concrete elements begin accumulating damage on
the south side of the column near the plastic hinging region at 0.36 in (μ0.77) of lateral
displacement, which corresponds to 54 kips of lateral load. A similar change in stiffness
is observed in the predicted response and the hysteretic envelope at approximately the
same lateral displacement but with much different values of force. The predicted
response shows a change in stiffness at approximately 58 kips and the hysteretic envelope
shows stiffness change at approximately 48.2 kips. At displacements greater than 2 in
(μ4.17), the force values for the CIP hysteretic envelope begin to decrease due to column
cover spalling and eventual column bars buckling in the plastic hinging region. This was
not replicated in either Model 1 or the predicted response. The lateral load-displacement
response of Model 1 and the predicted envelope therefore continued to increase through 4
in (μ8.6) of lateral displacement. For the CIP predicted values, the increase in load is
caused by the strain hardening of the rebar. The increase in load in the FE model may
have been related to confinement effects.
Figure 4.2 shows the load-displacement curve of Model 2 plotted with the
hysteretic envelope of the CIP test specimen and the CIP predicted values. The lateral
load-displacement response for Model 2 compares well to the hysteretic envelope
through a 2.0 in (μ4.17) displacement of the test specimens. Model 2 has higher stiffness
values for displacement less than 0.179 in (μ0.38), which is where the first dip in the
lateral load-displacement response of Model 2 appears. After the dip, damaged concrete
decreases the stiffness and appears to match the stiffness more accurately for the
65
hysteretic envelope. The second dip in the lateral load-displacement response again
changes the stiffness even more significantly and this allows Model 2 to continue
matching the response of the CIP test specimen. As previously explained, Model 2 does
not capture the decrease in stiffness after 2.0 inches (μ4.17) of displacement, but it
increases in capacity to approximately 56 kips at 4 in (μ8.6) of lateral displacement.
The predicted lateral load-displacement response of the GD specimen is
essentially the same as the predicted response from the CIP specimen. A similar
comparison is made between Model 1 and the predicted response as was done for the CIP
specimen. The lateral load-displacement response of the GD specimen is plotted with
Model 1 on Figure 4.3.
The lateral load-displacement response of Model 2 is compared against the GD
specimen in Figure 4.4. The hysteretic envelope of the GD specimen compares
reasonably well with Model 1 for lateral displacement less than 1 in and with Model 2
while lateral displacement is less than 2.2 in (μ4.73). Model 1 and Model 2 are, on
average, within 5% of the hysteretic envelope while lateral displacement is less than 0.4
in. After 0.4 in of lateral displacement, Model 1 continues to be in close agreement with
the hysteretic envelope through 1 in (μ2.15) of lateral displacement because it has a
higher load capacity than Model 2. For a lateral displacement less than 1 in, Model 1 is
within 5% of the hysteretic envelope. Both models exhibited the same trends as the CIP
comparison. Tables 4.1 and 4.2 summarize and compare lateral load-displacement
response at lateral displacements of 0.25, 0.5, 1.0, 2.0, and 3.0 in.
66
4.2
Joint Shear Stress
Figure 4.5 shows a plot of the average shear stress in the joint of the CIP
specimen, GD specimen, and Model 1. Overall response of Model 1 in predicting average
shear stress in the joint is in agreement with the CIP specimen and the GD specimen.
Table 4.3 provides a summarized comparison of joint shear stress values for Model 1, the
CIP specimen, and the GD specimen. Model 1 has higher shear stresses compared to the
CIP specimen for a lateral displacement of less than 0.3 in (μ0.65). As concrete elements
begin to accumulate damage, the shear stress values begin to decrease. Average shear
stresses in Model 1 are less than those in the CIP specimen for displacements greater than
0.3 in (μ0.65). Shear stresses from Model 1 are within 17% of the CIP specimen for
displacements greater than 0.5 in (μ1.1). The difference eventually begins to drop below
10% as lateral displacement increases.
Average shear stresses in Model 1, in general, agree more closely with the shear
stress data of the GD specimen. For lateral displacement less than 2.6 in (μ5.6), Model 1
generally has lower shear stress values that compare more closely with the GD data than
with the CIP data. Model 1 has higher shear stresses when the displacement is larger than
0.6 in (μ1.3). Shear stresses in the GD specimen begin to decrease relatively quickly after
2.25 in (μ4.8) of lateral displacement compared to the CIP specimen and Model 1. This
decrease is due to the loss of load corresponding to spalling and buckling of column
longitudinal bars in the plastic hinging region.
The shear stresses in Model 1 remain relatively constant after 1.6 in (μ3.4) of
lateral displacement. The lower shear stress is partially attributed to the non-strain
67
hardening steel model used for the rebar. After yielding, the steel does not pick up any
additional load, which does not put additional tensile force on the joint. Increases in stress
beyond this point are caused by the confinement effects as mentioned earlier.
4.3
Joint Rotation
Figure 4.6 compares the joint rotation for increasing displacement from Model 1,
the CIP specimen hysteretic envelope, and the GD hysteretic envelope. The joint rotation
from Model 1 is significantly less than both CIP and GD specimens, indicating a much
stiffer response in the joint. Table 4.4 summarizes and compares joint rotation data from
the Model and both specimen at various displacements, revealing that Model 1 reached
on average 65% of the CIP rotation and 45% of the GD rotation.
4.4
Principal Stress
Figure 4.7 shows a plot of the average maximum principal joint stress with
increasing displacement in Model 1, the CIP specimen, and the GD specimen. Maximum
principal stresses in Model 1 do not compare well with those of the CIP or GD
specimens. For a displacement greater than 0.2 in (μ0.43), principal stresses in Model 1
are up to 40% less than those in the CIP and GD specimens. For displacements greater
than 2.0 in (μ4.3), stresses in both specimens decrease because of spalling and buckling
of longitudinal rebar in the plastic hinge region of the column, but Model 1 stresses
remain relatively constant.
However, Figure 4.8 shows that the minimum principal joint stresses for Model 1
compare reasonably well with CIP and GD specimen data. For lateral displacement
between 0.2 in (μ0.43) and 2.5 in (μ5.4), Model 1 principal stresses are within 15% of
68
those from the CIP and GD specimens. For displacements larger than 2.5 in (μ5.4),
Model 1 stresses continue increasing whereas those from the CIP and GD specimens
decrease. Principal stresses in Model continued to increase because of confinement
effects and decreased in the CIP and GD specimens because of a spalling and buckling of
longitudinal column reinforcement in the plastic hinge region.
4.5
Stages of Concrete Cracking
Table 4.5 summarizes and compares Model 1 to the CIP specimen for flexural
cracking of the bent cap, diagonal shear cracking in the joint, and flexural cracking of the
column. Concrete elements south of the joint on the bottom face of the bent cap in model
show cracking, based on the damage parameter contours, at 0.15 inches (μ0.32) of lateral
displacement, which corresponded to a lateral load of 35 kips. Similar cracking was
observed in the CIP specimen at 13 kip of lateral load, which is 63% less than Model 1.
Based on the damage parameter contours, a diagonal crack in the joint of Model 1
forms at a lateral displacement of 0.345 inches (μ0.75), which corresponds to a lateral
force of 51 kips. Shear cracking in the CIP specimen joint was observed at 48 kips of
lateral force, which is 6% less than Model 1. Figures 4.9 through 4.12 show the damage
contours from Model 1 and pictures of the CIP specimen at the same displacement, to
compare cracking. Figure 4.9 compares cracking of Model 1 to the CIP specimen at 48
kips of lateral load, which is prior to the development of the primary diagonal joint crack
in Model 1. Figure 4.10 compares cracking of Model 1 to that of the CIP specimen at 1.4
inches (μ3) of lateral displacement. At this displacement, the diagonal shear crack
through the joint is visible on Model 1 and the CIP specimen. Cracking of Model 1 and
69
the CIP specimen at 2.8 in (μ6) of lateral displacement is shown in Figure 4.11. At this
lateral displacement, the diagonal shear crack in Model 1 has formed through the length
of the joint and significant damage is observed in the column. At 3.2 in (μ6.88) of lateral
displacement, Figure 4.12 shows significant damaged concrete in Model 1 and the CIP
specimen, and it appears to correlate well with one another. Much of the damage, in both
Model 1 and the CIP specimen, was limited to the column plastic hinge region, with the
exception of the predicted diagonal joint shear crack. It should be emphasized that
cracking on the test specimen is due to reverse cyclic loading, whereas as damage in
Model 1 is due to monotonic (push) load.
4.6
Reinforcing Steel Strain Profiles
Table 4.6 compares the steel strain profile in the column longitudinal
reinforcement located in the plastic hinge region and through the joint of Model 1 and the
CIP specimen. A cross section of the CIP column showing rebar ID LC8, which is the bar
analyzed in Table 4.6, is shown in Figure 4.13. Because this comparison considers the
push load case, LC8 is the extreme tensile rebar. Figure 4.14 shows the three locations of
the strain gauges: 6 in into the column, 1 in into the joint, and 6 in into the joint.
CIP specimen strain gauge locations for bent cap stirrups and column hoops are
shown in Figure 4.15. Table 4.7 compares the strain profile of the stirrups in the bent cap
at six locations: mid-height of two stirrups adjacent to the south side of the joint, upper
side of two stirrups in the joint, and mid-height of two stirrups adjacent to the north side
of the joint. Strain values from Model 1 and the CIP specimen were well below yield
strain and in many cases so small that a meaningful comparison was difficult. In some
70
cases, the Model 1 strain values were negative of the CIP specimen. Table 4.8 was
formulated similarly to Table 4.7 with the exception of using the bottom of joint stirrups
instead of the top.
Table 4.9 compares the strain profile on the east side of five hoops: three located
in the bent cap and two in the column (Figure 4.15). Model 1 exhibited an unusual strain
pattern in column hoops that was not in agreement with the CIP specimen. Maximum
strain in reinforcing hoops was expected on the transverse face of the stirrups, as seen on
the CIP specimen. However, in Model 1, the maximum strain occurred on the south side
of the column. Because of the unusual strain distribution in Model 1, strain was not
expected to be comparable with the CIP specimen. CIP hoops achieved large strains
beyond yield, hoops strains the same location on Model 1 were significantly less, and
reversed in sign in some cases.
71
Table 4.1: Comparison of Lateral Load-Lateral Displacement Response between Model
1, CIP Specimen, GD Specimen, and Specimen Predictions
Lateral
Disp.
(in)
Disp.
Ductility,
μ
0.25
0.5
1
2
3
0.54
1.08
2.15
4.30
6.45
Lateral Force (kips)
Model 1 CIP
45
52.1
58.5
63
63.9
GD
Ratio
Specimen Model 1/
Predicted
CIP
39
40
49
51
55.8 56.4
55.9 55.5
52
-
48
58.5
59
62
64.4
Model 1/
GD
Model 1/
Predicted
1.13
1.02
1.04
1.14
-
0.94
0.89
0.99
1.02
0.99
1.15
1.06
1.05
1.13
1.23
Table 4.2: Comparison of Lateral Load-Lateral Displacement Response between Model
2, CIP Specimen, GD Specimen, and Specimen Predictions
Lateral Force (kips)
Lateral
Disp.
(in)
Disp.
Ductility
(μ)
Model
2
0.25
0.5
1
2
3
0.54
1.08
2.15
4.30
6.45
41.2
46.1
54.3
56.7
65.8
CIP
GD
39
40
49
51
55.8 56.4
55.9 55.5
52
-
Ratio
Predicted
Model
2/ CIP
Model
2/ GD
Model 2/
Predicted
48
58.5
59
62
64.4
1.06
0.94
0.97
1.01
1.27
1.03
0.90
0.96
1.02
-
0.86
0.79
0.92
0.91
1.02
Table 4.3: Comparison of Average Joint Shear Stresses between Model 1, CIP Specimen,
and the GD Specimen
Average Joint Shear
Stress (psi)
Ratio
Lateral
Displacement
(in)
Disp.
Ductility
(μ)
Model 1
CIP
GD
Model
1/ CIP
Model
1/ GD
0.25
0.5
1
2
3
0.54
1.08
2.15
4.30
6.45
176
236
276
285
284
140
264
325
322
303
197
274
309
303
233
1.26
0.89
0.85
0.89
0.94
0.89
0.86
0.89
0.94
1.22
72
Table 4.4: Comparison of Joint Rotation between Model 1, the CIP Specimen, and the
GD Specimen
Lateral
Displacement (in)
Disp.
Ductility
(μ)
Model 1
CIP
GD
0.25
0.5
1
2
3
0.54
1.08
2.15
4.30
6.45
0.000401
0.000635
0.000778
0.000908
0.000958
0.00045
0.00093
0.00129
0.00162
0.00183
0.00091
0.00143
0.00186
0.00221
0.00183
Rotation (rad)
Ratio
Model 1/ Model 1/
CIP
GD
0.89
0.68
0.61
0.56
0.52
0.44
0.44
0.42
0.41
0.53
Table 4.5: Comparison of Specimen Cracking – Including 38 kip Axial Load and SelfWeight
Lateral Load (kips)
Stage
Bent Cap - Flexural
Column - Flexural
Joint Shear
Ratio
Predicted
CIP Test
Model 1
Model 1/
Predicted
Model
1/ CIP
8.9
14
36.5
13
20
48
35
25
50
3.93
1.79
1.37
2.69
1.25
1.04
73
Table 4.6: Strain Profile Table – Column Longitudinal Rebar (LC8), CIP Specimen
Location
C1
B1
B2
Bar Strain (με)
Lateral
Load/Ductility
Disp. (in)
13 kips
20 kips
30 kips
48 kips
μ1
μ1.5
13 kips
20 kips
30 kips
48 kips
μ1
μ1.5
13 kips
20 kips
30 kips
48 kips
μ1
μ1.5
0.034
0.072
0.154
0.397
0.406
0.533
0.034
0.072
0.154
0.397
0.406
0.533
0.034
0.072
0.154
0.397
0.406
0.533
Ratio
Model 1
CIP
Model 1/
CIP
91.5
577
907
1620
1590
1890
415
580
1220
1850
1850
2002
80.4
124
517
1550
1570
1700
389
999
1875
3249
1757
1711
1391
2228
2457
3019
5589
5595
97
553
1080
1387
813
1192
0.235
0.578
0.484
0.499
0.905
1.105
0.298
0.260
0.497
0.613
0.331
0.358
0.829
0.224
0.479
1.118
1.931
1.426
74
Table 4.7: Strain Profile Table – Stirrups in Bent Cap (Mid-height) and Joint (Top), CIP
Specimen
Lateral
Location Load/Displacement
Ductility
SS5E
SS1E-T
SN1E-T
SN3E
SN5E
Bar Strain (με)
Disp. (in)
Ratio
Model 1
CIP
Model 1/
CIP
13 kips
20 kips
30 kips
48 kips
μ1
0.034
0.072
0.154
0.397
0.406
1.24
2.23
0.154
-6
-6.56
-15
-14
-26
-91
-55
-0.083
-0.159
-0.006
0.066
0.119
μ1.5
13 kips
20 kips
30 kips
48 kips
μ1
0.533
0.034
0.072
0.154
0.397
0.406
1.72
9.24
19.5
11.5
77.5
99.9
-85
63
175
273
690
504
-0.020
0.147
0.111
0.042
0.112
0.198
μ1.5
13 kips
20 kips
30 kips
48 kips
μ1
0.533
0.034
0.072
0.154
0.397
0.406
336
-12.4
-9.78
29.2
-29.8
-36.3
630
-1
35
57
94
104
0.533
12.400
-0.279
0.512
-0.317
-0.349
μ1.5
13 kips
20 kips
30 kips
48 kips
μ1
0.533
0.034
0.072
0.154
0.397
0.406
-41.8
-14.7
-19.2
-1.95
-75.8
-71.5
151
-13
-151
-189
-307
-131
-0.277
1.131
0.127
0.010
0.247
0.546
μ1.5
13 kips
20 kips
30 kips
48 kips
μ1
0.533
0.034
0.072
0.154
0.397
0.406
-93.5
-7.36
-10.5
-23.7
-139
-137
-147
-40
-48
-34
84
40
0.636
0.184
0.219
0.697
-1.655
-3.425
μ1.5
0.533
-143
96
-1.490
75
Table 4.8: Strain Profile Table – Stirrups in Bent Cap (Mid-height) and Joint (Bottom),
CIP Specimen
Lateral
Location Load/Displacement
Ductility
SS5E
SS3E
SS1E-B
SN1E-B
SN3E
SN5E
μ2
μ2
μ2
μ2
μ2
μ2
Bar Strain (με)
Disp. (in)
0.684
0.684
0.684
0.684
0.684
0.684
Ratio
Model 1
CIP
Model 1/
CIP
3.78
121
511
-47.64
-89.4
-145
-81
-10
-205
3307
-180
112
-0.047
-12.100
-2.493
-0.014
0.497
-1.295
Table 4.9: Strain Profile Table – Hoops in Column and Joint (East), CIP Specimen
Lateral
Location Load/Displacement
Ductility
HC4-E
HC1-E
HB3-E
HB4-E
HB5-E
30 kips
μ2
30 kips
μ2
30 kips
μ2
30 kips
μ2
30 kips
μ2
Bar Strain (με)
Disp. (in)
0.154
0.684
0.154
0.684
0.154
0.684
0.154
0.684
0.154
0.684
Ratio
Model 1
CIP
Model 1/
CIP
50.1
498
-58.3
4.3
5.4
-3.5
37.3
9.2
67.6
159
619
1465
295
2234
11
37
42
1288
161
542
0.081
0.340
-0.198
0.002
0.493
-0.095
0.888
0.007
0.420
0.293
76
70
Lateral Force (kips)
60
50
Model 1 continues to
follow predicted response
through 4 in
40
Significant change in stiffness
occurs at similar displacement
but different forces
30
20
Model 1
CIP Test Specimen
Predicted Response
10
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Lateral Displacement (in)
Figure 4.1: Model 1 and CIP specimen Lateral Load-Lateral Displacement Response
Comparison
70
Lateral Force (kips)
60
50
40
CIP load decreases due to
spalling and buckling of
rebar in plastic hinge
Significant change in
stiffness of Model 2 and
continues to follow CIP
specimen
30
20
Slight decrease in
stiffness of Model 2
following concrete
damage
10
Model 2
CIP Test Specimen
Predicted
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Lateral Displacement (in)
Figure 4.2: Model 2 and CIP Specimen Lateral Load-Lateral Displacement Comparison
77
70
Lateral Force (kips)
60
50
40
Model 1 is, on average,
within 5% of GD
Specimen through 1 in of
displacement
30
20
Model 1
Predicted Response
GD Specimen
10
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Lateral Displacement (in)
Figure 4.3: Model 1 and GD Specimen Lateral Load-Lateral Displacement Comparison
70
Lateral Force (kips)
60
50
40
Smaller force than Model 1
but is still within 10% of
GD Specimen
30
20
Model 2
Predicted Response
GD Specimen
10
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Lateral Displacement (in)
Figure 4.4: Model 2 and GD Specimen Lateral Load-Lateral Displacement Comparison
78
350
Average Joint Stress (psi)
300
250
200
150
100
Model 1
CIP Specimen
GD Specimen
Larger initial shear stress
in Model 1
50
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Lateral Displacement (in)
Figure 4.5: Comparison of Average Joint Shear Stresses – Model 1 and CIP and GD
Specimens
0.0024
Joint Rotation (rad)
0.002
0.0016
0.0012
0.0008
Model 1
CIP Specimen
GD Specimen
Concrete damage in column
causes a jump in displacement
but minor joint rotation
0.0004
0
0
0.5
1
1.5
2
2.5
3
3.5
Lateral Displacement (in)
Figure 4.6: Comparison of Joint Rotation - Model 1 and CIP and GD Specimens
4
79
400
Maximum Principal Stress (psi)
350
300
250
200
150
100
Model 1
CIP Specimen
GD Specimen
Initial stress from gravity
and axial load in column
50
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Lateral Displacement (in)
Figure 4.7: Maximum Principal Joint Stresses from Model 1and CIP and GD Specimens
400
Stresses continue to increase
due to confinement effects
Minimum Principal Stress (psi)
350
300
250
200
150
100
Initial stress from gravity
and axial load in column
Model 1
CIP Specimen
GD Specimen
50
0
0
0.5
1
1.5
2
2.5
3
3.5
Lateral Disaplacement (in)
Figure 4.8: Minimum Principal Joint Stress - Model 1 and CIP and GD Specimens
4
80
Figure 4.9: Comparison of Concrete Cracking at 48 kips of Lateral Load – Model 1 and
CIP Specimen
Figure 4.10: Comparison of Concrete Cracking at 1.4 in (μ3) of Lateral Displacement –
Model 1 and CIP Specimen
Figure 4.11: Comparison of Concrete Cracking at 2.8 in (μ6) of Lateral Displacement –
Model 1 and CIP Specimen
81
Figure 4.12: Comparison of Concrete Cracking at 3.2 in (μ6.9) of Lateral Displacement –
Model 1 and CIP Specimen
Figure 4.13: Section of Column Shows Location of LC8 Rebar (Matsumoto, 2009)
82
Figure 4.14: Location of Longitudinal Rebar Strain Gauges on the CIP Specimen
(Matsumoto, 2009)
83
Figure 4.15: Location of Stirrup Strain Gauges on the CIP Specimen (Matsumoto, 2009)
84
Chapter 5
SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS
5.1
Summary
This report, 3-D Finite Element Modeling of Reinforced Concrete Beam-Column
Connections – Development and Comparison to NCHRP 12-74, investigates the use of
finite element modeling (FEM) to predict the structural response of the cast-in-place
(CIP) reinforced concrete bent cap-column test specimen reported in NCHRP Report 681
– Development of a Precast Bent Cap System for Seismic Regions.
The response of the FE model was compared to the response of the CIP bent capcolumn test specimen using several different parameters, including the following:

Column Lateral load-lateral displacement response

Joint shear stress

Joint rotation

Magnitude and orientation of principal stresses

Stages of concrete cracking

Yielding pattern of steel reinforcement in bent and column

Overall distribution of flexural and compressive stresses
The FE model was based on the same dimensions as the 42% scale inverted tee CIP
test specimen, as provided in drawings of NCHRP Report 681, and includes two solid
components, a bent cap and a column, and associated steel reinforcement. The primary
FE model, Model 1, consists of 55,860 solid concrete elements and 4,976 beam elements,
for 60,836 total elements. The solid concrete elements are eight node hexahedron
85
elements with constant stress solid element formulation and the steel reinforcement
elements are circular beam elements. The concrete and rebar is coupled using the
constrained Lagrange card. The Karagozian and Case damaged concrete model
(MAT_072), was used as the constitutive model for the concrete. Rebar was modeled
with a plastic kinematic constitutive model, without strain hardening.
The analysis included monotonic push loading on the FE model (due to various
limitations of the constitutive models) instead of reverse cyclic loading. Therefore FEA
results were compared to the hysteretic envelope from the cyclic test results of the test
specimen. To minimize inertial effects, loading on the FE model was applied over a 10
second interval: gravity and a 38 kip axial load were applied first, followed lateral
displacement applied to the column stub.
The lateral load-lateral displacement response of the FE model (Model 1) compared
reasonably well to the actual lateral load-displacement response of the CIP specimen and
the GD specimen. For lateral displacements less than those corresponding to a
displacement ductility of 4.1, the FE model showed a larger stiffness relative to the
specimen response. Stiffness of Model 1 degraded as an increasing number of concrete
elements accumulated damage in the plastic hinging region, but this degradation was not
a severe as that of the CIP specimen. For displacement ductility greater than 2, the lateral
load-displacement response matched the predicted response within 5%.
Average joint shear stresses in the FE model and the CIP specimen compared
reasonably well with one another. For lateral displacements less than a displacement
ductility of 1.3, the FE model closely matched the average joint shear stress response of
86
the CIP specimen. For lateral displacements larger than a displacement ductility of 1.3,
the FE model, on average, had 10% less average joint shear stress. Joint rotation in the
FE model was on average 60% of the CIP specimen, for lateral displacement larger than a
displacement ductility of 1.1.
Concrete damage parameter contours of the FE model correlated reasonably well with
cracking and spalling in the CIP specimen. The FE model developed damage in the bent
cap due to flexure at 35 kips of lateral (includes 38 kip axial load on column and self
weight) load, whereas the CIP specimen developed cracks at 13 kips of lateral load,
which is 63% less than the FE model. Diagonal joint shear damage developed in the FE
model at 51 kips, compared to 48 kips in the CIP specimen, which is 6% less than the FE
model. Concrete damage contours at a lateral displacement ductility of 8.6 compared
reasonably well with cracking observed in the CIP specimen. The FE model had minimal
damage in the joint region, with the exception of a predicted diagonal shear crack, and
significant damage in the column near the plastic hinging region. The damage reflects the
design mode of failure, which was plastic hinging of the column.
Principal stress vectors in the FE model, prior to joint shear damage, show the
orientation and magnitude of tensile stress development along the diagonal. The
orientation of maximum principal stress vectors is in agreement with flexural damage
observed from the damage parameter contours.
5.2
Conclusion
Based on the results of this analysis, explicit nonlinear finite element analysis is
suitable for predicting the response of reinforced concrete beam-column connections.
87
Particularly, LS-DYNA combined with proper constitutive models has proven to provide
accurate and reliable data when compared against CIP test specimen data. The following
conclusions were made:
a. Finite element modeling, using appropriate constitutive models and element
formulation, can accurately capture the nonlinear behavior of reinforced concrete
beam-column connections. A well-developed model can reasonably approximate
the lateral load-lateral displacement response, flexural cracking, joint shear
cracking, steel reinforcement yielding, and overall stress distribution, as compared
to the CIP specimen.
b. Element size for concrete and steel reinforcement significantly impacts the overall
response and accuracy. Ideally, several iterations with decreasing element size
should be performed in order to more accurately determine the effect of mesh
size. For this analysis, decreasing the element size resulted in a more steady
response of stress distribution, displacement, and more accurate concrete damage
contours.
c. The Karagozian & Case damaged concrete model (MAT_072) can accurately
capture the cracking of concrete using a limited number of inputs (f ’c and
aggregate size) into the model. Flexural and shear cracking patterns correlated
closely with CIP test specimen data when using this constitutive model.
5.3
Recommendations
Recommendations for further analysis include the following:
88
a. Incorporate the use of a strain hardening in the constitutive model for steel
reinforcement in order to replicate the CIP test specimen more closely. This
would also provide further analysis between strain hardening and confinement
effects of concrete.
b. Strain distribution of steel reinforcement in the joint (longitudinal, joint hoops,
and joint stirrups) and plastic hinging region should be further investigated to
address issues with unrealistic distributions observed in some cases.
c. Incorporate a concrete constitutive model capable of accurately modeling reverse
cyclic loading. Incorporating a damaged model capable of coupling the effects of
tension and compression damage would allow the FE model to be analyzed more
closely to specimen testing.
d. Calibration of constitutive model parameters should be performed when
attempting to match specimen data. This would require a more detailed and
rigorous analysis but would result in close match between actual specimen data
and FEA results.
e. Incorporate the effects of bar slip by utilizing a bar slip model for the bond
between concrete and steel reinforcement.
f. Further investigation of the effects of beam element lengths of hoop steel as it
affects the overall response of the structure.
89
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