3

D FINITE ELEMENT MODELING OF REINFORCED CONCRETE BEAM

COLUMN
CONNECTIONS
–
DEVELOPMENT AND COMPARISON TO NCHRP PROJE
CT 12

74
A
Project
Presented to the faculty of the
D
epartment of
Civil Engineering
California State University, Sacramento
Submitted in
partial
satisfaction of
the
r
equirements for the degree
of
MASTER OF
SCIENCE
in
Civil Engineering
b
y
Harpreet Singh Hansra
FAL
L
2012
ii
©
2012
Harpreet Singh Hansra
ALL RIGHTS RESERVED
iii
3

D FINITE ELEMENT MODELING OF REINFORCED CONCRETE BEAM

COLUMN
CONNECTIONS
–
DEVELOPMENT AND COMPARISON TO NCHRP PROJE
CT 12

74
A
Project
by
Harpreet Singh Hansra
Approved by:
__________________________________,
Committee Chair
Eric E. Matsumoto, Ph.D., P.E.
__________________________________,
Second Reader
Mark S
c
hultz, M.S., P.E., S.E.
____________________________
Date
iv
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 sui
t
able
for shelving in the Library and c
redit is to be awarded for
the project
.
______
__________
______________
__
,
Department Chair
___
_______________
__
Kevan Shafizadeh, Ph.D., P.E., PTOE
Date
Department of
Civil Engineering
v
Abstract
of
3

D FINITE ELEMENT MODELING OF REINFORCED CONCRETE BEAM

COLUMN
CONNECTIONS
–
DEVELOPMENT AND COMPARISON TO NCHRP PROJE
CT 12

74
by
Harpreet Singh Hansra
This report, 3

D Finite Element Modeling of Reinforced Concrete Beam

C
olumn
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 68
1
–
Development of a Precast Bent Cap System for Seismic Regions.
A
nalysis was performed
using
LS

DYNA
as
the
finite element processor.
The
K
aragozian
&
C
ase
Damaged Concrete model
, material MAT_072,
was used as the
constitutive
model for all concrete el
ements
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 consider
ed
a monotonic
push
loading sequence, whereas the
CIP specimen
was
subjected to reverse
cyclic loading.
To account for
the difference in
vi
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
theoretical
ly
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
degrad
ed
as a greater number of concrete elements
in the column plastic hinging
region
accumulat
ed
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 lo
ngitudinal 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
later
al 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
vii
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
significant
ly
impact
s
the overall response
and accuracy
of results
and therefore must be carefully selected for convergence
;
3
) the K
aragozian
&
C
ase
damaged concrete model
, material MAT_07
2,
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) stra
in
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 mode
l 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
viii
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
ix
3.4
Joint Shear Stress
................................
................................
................................
.
39
3.5
Join
t 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, CONCLUSION
S
, AND RECOMMENDATIONS
................................
.....
84
5.1
Summary
................................
................................
................................
..............
84
5.2
Conclusion
................................
................................
................................
...........
86
5.3
Recommendations
................................
................................
................................
87
References
................................
................................
................................
...............................
89
x
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 betwee
n 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
xi
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
xii
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 Show
ing 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 Eleme
nts
................................
................................
..................
29
Figure 2.11
3

D
I
sometric
View of Mesh
ed
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 Axi
al Load and Lateral Displacement
....................
33
xiii
Figure 3.1
Model 1 Lat
eral 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 St
rength (compression plotted as
positive stress to show relationship for re
presentative 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 Late
ral 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 Latera
l
Displacement
................................
................................
................................
..
52
Figure 3.12
Principal Stress Vectors Overlaid on Damage Parameter Contours at
1.0 in (μ2.15) of Lateral
Displacement
................................
...........................
52
xiv
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 Displacem
ent
................................
................................
......................
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
Longitu
dinal (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
F
igure 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
xv
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 Longit
udinal 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. Tens
ion 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 Strai
n 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 Lat
eral Load

Lateral Displacement
Comparison
................................
................................
................................
.....
77
xvi
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
Comparis
on 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
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 th
e 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 t
o 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 non

integral three

column precast bridge bent. F
igure 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 p
rototype structure as shown in F
igure 1.2. The
cast

in

place (CIP) test sp
ecimen,
specifically the joint,
will be
m
odeled
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
resu
lts 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 tw
o 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 cert
ain 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 mo
dels 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 veri
fy 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 o
f 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 respon
se 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 princip
al
stress
es
in the joint, overall
flexural
stress distribution, and stress
distribution throughout steel reinforc
ement
.
1.3
Significance of Project Results
FE
A
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
modification
s. More directly, the FE
A
results will be used to develop a strut

and

tie
model of the structure.
T
he orientation of princip
al
stresses and the flow of forces allows
4
engineers to determine
proper
magnitude and
location
of
compression struts
,
tension ties
a
nd 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 anal
ysis 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 quasist
atic
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. L
astly, guidance is provi
ded 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
hand
book, “
State

of

Practice for the Nonlinear Analysis
of Concrete Dams at the Bureau
5
of Reclamation.”
The focus
of the
hand
book
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 rep
ort was on contacts between
different surfaces, damping, material properties, and boundary conditions.
Malvar and Simons
(1996)
discuss
es
the development of a Lagrangian finite
element code with explicit time integration for analyzing structures
.
It expl
ains
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)
prese
nted 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 mo
del.
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 sui
t
able
for quasi

static loads, even though primarily developed for blast and impact
loading.
Sandia National Laboratorie
s 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

Fossu
m 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. comp
ressive 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 o
f
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 nonlinea
rity 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

st
r
ain curve for confined concrete.
T
he
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 resu
lts 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
,
ti
es
, 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
sui
t
able
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 extensi
vely 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 follow
ing 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 model
s.
This report includes extensive comparisons of
FEA results to NCHRP 12

74 results, with a primary focu
s 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, C
onclusions, and
R
ecommendations
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 bot
tom
for flexural
reinforcement and #3’s at 6 in for shear reinforcement.
Dimensions and reinforcement
placement are
detailed
in
Figure
s 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 2
0 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 proper
ties 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 b
ent 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, t
he 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
Finit
e Element
Model
Geometry
Similar to the CIP specimen, the FE
model consists
of
two individual
components, a bent cap and a column
(
F
igure 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
Figure
s 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, t
he 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 c
ritical 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 join
t
.
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 conne
cting 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 u
sed 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.
1
0
shows
the relative
mesh size for the solid element
s of the entire structure.
Figure 2.
1
1
shows a 3

D view of
the
meshed structure
.
The steel reinforcement is modeled
using circular Hughes

Liu
beam elements.
Figure 2.1
2
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.
T
his
analysis
uses the
CONSTRAINED_LAGRANGE_IN_S
OLID 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 con
crete 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
leng
t
h
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 qui
cker.
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
applicatio
ns 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 Pseudo

TENSOR Mod
el
(
LS
T
C
, 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 compres
sive 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 pe
rformed 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. Th
e tens
ion 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
use
d
a compressive strength of 4,553 psi and the
concrete for the column
use
d
a
compressive strength of 6,178 psi. Both the column
and bent
cap
us
d
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 (Mark
ovich
et al
,
2011). If an element becomes fully damaged from tensile stress,
its ability to carry a
compressive load is greatly diminished
. This presen
ts an issue under cyclic
loading
,
where an element may be subjected to many compression and tension cycles.
The reinforc
ing
steel is modeled using MAT_003, known as
MAT_PLASTIC_KINEMATIC.
This model
is well suited for isotropic and elastic
behavior.
It
h
as 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 struc
ture 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 stress
es
. 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
accep
t
able
. 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
c
oncentrated 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
. Increas
ing 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 distribution
s
were
all
much more realistic
after a sui
t
able
mesh was
settled upon
.
This analysis will discuss two of the models that were developed, Mode
l 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 ho
ops 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. Excep
t 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 tes
t 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 mode
l 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
ap
plied
to the column stub
over a two

second 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
Compressive Strength
4000 psi (28 day)
Cap: 4553 psi (137 days)
Column: 6178 psi (194 days)
Tensile Strength
(Split Cylinder)
N/A
Cap: 361 psi (138 days)
Column: 452 psi (195 days)
Table 2.1.A
Parameter
GD
CIP
f'c
Cap and Column:
4557 psi
Cap: 4553 psi
Column: 6178 psi
Steel Rebar Strength
Yield
Tensile
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)
N/A
Table 2.1.B
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
Column
Longitudinal Reinforcement
Bar Size (diameter, in)
#11 (1.41)
0.59
#5 (0.63)
1.06
No. of bars
16
14
16
1.14
f
y
ksi
66.0
—
64.5
—
ρ
0.0138
0.0138
0.0158
1.14
M
n
*
K∙ft
3760
272
222
0.82
M
n
/D
c
3
K∙ft/ft
3
58.7
58.7
48.0
0.80
Transverse Reinforcement
Bar Size (diameter, in)
#6 (0.75)
0.31
#3 (0.38)
1.20
Spacing
in
3.0
[1.8]
2.0
[1.11]
f
y
ksi
66.0
—
68.2
—
ρ
0.0139
0.0139
0.0125
0.90
Bent Cap
Longitudinal Reinforcement
Bar Size (diameter, in)
#11 (1.41)
0.59
#5 (0.63)
1.06
No. of bars
12
—
12
—
f
y
ksi
66.0
—
64.5
—
ρ
0.0051
0.0051
0.0065
1.27
M
n
*
K∙ft
6680
483
454
0.94
M
n
/bh
2
K∙ft/ft
3
44.2
44.2
50.2
1.14
Transverse Reinforcement
Bar Size (diameter, in)
#6 (0.75)
0.31
#3 (0.38)
1.20
Spacing
in
12.0
[8.4]
6.0
[0.71]
f
y
ksi
66.0
—
68.2
—
Joint
Inside Joint
Transverse Reinforcement (ρ
s
)
Bar Size (diameter, in)
#6 (0.75)
—
#3 (0.38)
—
Spacing
in
3.0
—
5.0
—
ρ
s
/ρ
min
5.56
†
—
1.22
—
f
y
ksi
66.0
—
68.2
—
Side Face Reinforcement (A
s
sf
)
No. of bars

Bar Size
8

#6
—
4

#3
—
A
s
/A
cap
in
2
/ in
2
0.19
[0.10]
0.12
[1.20]
f
y
ksi
60.0
—
68.2
—
Construction Stirrups
No. of bars

Bar Size
2

#6
—
2

#3
—
Area
in
2
0.88
—
0.22
—
f
y
ksi
66.0
—
68.2
—
Adjacent to Joint
Vertical Stirrups (A
s
jv
)
No. of bars

Bar Size
5

#6
—
3

#3
—
Spacing
in
6.0
—
5.0
—
A
s
jv
/A
st
in
2
/ in
2
0.35
[0.20]
0.27
[1.33]
f
y
ksi
66.0
—
68.2
—
Horizontal Ties (A
s
jh
)
No. of bars

Bar Size
4

#6
—
2

#3
—
Spacing
in
12.0
—
8.0
—
A
s
jh
/A
st
in
2
/ in
2
0.35
[0.10]
0.13
[1.33]
f
y
ksi
66.0
—
68.2
—
21
Table 2.
3
: Loading Sequence
for FEM
Time
Loading
0 to 2 seconds
Gravity applied over a linear ramp. Gravity is ramped
to 100% over 2 seconds and stays constant for
remainder of analysis.
2 to 4 seconds
38

kip axial load applied as a ramp load over this
interval and remains constant for remainde
r of
analysis.
4 to 10 seconds
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
Figure 2.
6
: Individual Bent Cap and Column Components
Figure 2.
7:
Dimensions of FE Model
144 in
20 in
25 in
55 in
3 in diameter
Joint
Region
Column
Column
Stub
S Support
N Support
Bent Cap
28
Figure 2.
8
: A) Rebar configuration in FE Model, B) CIP Specimen Bent Cap Rebar Cage
(Matsumoto, 2009)
18 in O.D.
#3 hoops at 2
in
#3 hoops at
5 in
#3
stirrups
16

#5
A
B
29
Figure 2.
9
: Half of Bent Cap Cut

away
(with
Stirrups Removed
)
to Show Longitudinal
Rebar Connecting the Bent Cap and Column
Figure 2.
1
0
:
Mesh of Solid Elements
Bent Cap
Column
Elevation View
End View
30
Figure 2.
1
1
: 3

D
Isometric
View of Mesh
ed
Structure
31
Figure 2.1
2
: Close

up of Meshed Beam Elements in Joint Region
Figure 2.
1
3
: Strength Model for Concrete (Malvar et al., 1997): (a) Failure Surfaces in
Concrete Model; (b) Concrete Constitutive Model
Column
Joint
Region
32
Figure 2.1
4
: Comparison of
#3
Hoop
Steel
Element Length in Model 1 and Model 2
Figure 2.
1
5
: Vertical and Horizontal Boundary Conditions
Vertical and horizontal
restraints
Vertical
restraint
Axial
load
Model 1
Model 2
1.58
in
3.44
in
N
S
33
Figure 2.
1
6
: Boundary Conditions for Axial Load and Lateral Displacement
Lateral displacement
N
S
34
Chapter 3
FINITE ELEMENT ANALYSIS RESULTS
This chapter presents results
from the
FE
analysis of the
CIP specimen
model
s
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 respo
nse.
The second
result
presented is the
concrete damage parameter that is internal to the
K&C
concrete constitutive model
. This parameter is a
n
indicator of cracking in the structure at
various stages. This
is
followed by an analysis of the
orientation of
princip
al
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 o
f
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 lo
ad

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
(μ
8.6
)
. Model 2 reaches a peak lateral
load of 57 kips at a lateral displacement of 2.63 in
(μ5.7)
.
Based on the lateral l
oad

displacement
response
, Model 1 (shown in
F
igure 3.
1
)
appears to have a steadier response relative to Model 2 (shown in
F
igure 3.
2
) through
four inches of lateral displacement. The
lateral load

displacement response
for Model 2
has two
sudden
dips
in fo
rce
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
la
rge 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 stren
gth 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 i
n 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 sam
e 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
eleme
nt
length of Model 2
is
longer than Model 1, when an element reaches the yield stress and be
gins
plastic strain, a
greater length of element yiel
ds
,
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, conseque
ntly
resulting in higher confinement
. Analysis from this point on is carried out on Model 1,
unless otherwise specified, as it is the more relia
ble 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 stren
gth 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 reinforci
ng 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
wou
ld
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
re
sidual 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
f
igure
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 a
fter 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. I
t shows that as the damage parameter approaches 2.0, the
tensile
strength goes to zero.
Contours of this damage parameter are plotted in
F
igure
s
3.
6
through 3
.
9
. The
damage contours range between blue and red; blue representing
nearly no
damage
accumulat
ion 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 ele
ments 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 ca
p 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 la
teral
load of 51 ki
ps. Lastly, F
igure
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 dam
age accumulation in the column.
Most of the elements
on the compression side of the column near the joint
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