CE 500 Final Reportx

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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

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


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