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This is to certify that the

thesis entitled

DEVELOPMENT OF A MODIFIED FIBER-BASED BEAM-
COLUMN ELEMENT FOR DRAINZDX

presented by

Antonio Cordero—Domenech

has been accepted towards fulfillment
of the requirements for

MS Civil and Environmental

degree in W

 

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Major professor

Date 07/214);

 

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LIBRARY
Michigan State
University
PLACE - - "our record.
1. . - a" , .il :-
MAY .Jti mutt} _ nth ea: 9;: c . +rested.
DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 c:/ClRC/DateDue.p65—p.15

 

 

DEVELOPMENT OF A MODIFIED FIBER-BASED BEAM-
COLUMN ELEMENT FOR DRAINZDX

By

Antonio Cordero-Domenech

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Civil and Environmental Engineering

2002

ABSTRACT

DEVELOPMENT OF A MODIFIED F IBER-BASED BEAM-
COLUMN ELEMENT FOR DRAINZDX

By

Antonio Cordero-Domenech

This work presents the development and validation of a modified fiber-based
beam-column element for the nonlinear structural analysis program DrainZDX @ynamic
Response Analysis of melastic _2_-l_)_imensional Structures, an e_X_tended version). A
major drawback of the existing fiber-based element is that it cannot model the effects of
local buckling and/or biaxial stresses in the steel elements of a member cross-section. A
modified S-type steel stress-strain curve that account for the effects of local buckling
and/or biaxial stresses on the monotonic and cyclic behavior was developed.

For implementation in DrainZDX, the modified S-type steel cyclic stress-strain
curve was discretized into linear segment branches. The changes in stiffness associated
with the fiber response moving from one branch to the other under cyclic loading were
identified using unique event numbers. The event calculation and response state
determination algorithm for the fiber were developed and implemented in Drain2DX
through subroutines SMEVIS and SMSDlS.

A major focus of this report is to demonstrate the capability of the modified fiber—
based element to model the static (monotonic and cyclic) behavior of composite concrete-
steel elements, such as concrete filled steel tube (CFT) members. The results indicate
that the modified fiber-based element is well suited for modeling the behavior of CPT

beam-columns.

ACKNOWLEDGMENTS

I would like to thank God for everything that He has given to me and for
providing me with strength and wisdom to achieve this goal. Additionally I am grateful
to my advisors, professor Arnit H. Varma and professor Ronald S. Harichandran for all
their help, support, advice and encouragement throughout my graduate studies at
Michigan State University. I am also indebted to them and many others from my
graduate and undergraduate years. Thanks also to my college and school friends for their
support. Finally I would never have got this far, in life or academia, without the support

of my close family -— Sonia and Bolivar — thanks for everything.

iii

TABLE OF CONTENTS

LIST OF TABLES vi
LIST OF FIGURES vii
CHAPTER 1. INTRODUCTION 1
1.1 GENERAL 1
1.2 FIBER-BASED BEAM-COLUMN ELEMENT IN DRAINZDX 1
1.3 DRAWBACKS OF EXISTING FIBER-BASED ELEMENT 7
1.4 PROJECT OBJECTIVES AND SCOPE 8
1.5 SUMMARY 9

CHAPTER 2. DEVELOPMENT AND IMPLEMENTATION OF

THE MODIFIED FIBER-BASED ELEMENT 10
2.1 MODIFIED S-TYPE STEEL STRESS-STRAIN CURVE 10
2.2 IMPLEMENTATION OF MODIFIED S-TYPE STEEL
STRESS-STRAIN CURVE 14
2.2.1 BRANCH NUMBERS 15
2.2.2 EVENTS NUMBERS 16
2.2.3 EVENT CALCULATION AND STATE DETERMINATION
ALGORITHM 19
2.2.3.1 COMPRESSIVE STRAIN INCREMENT 20
2.2.3.2 TENSILE STRAIN INCREMENT 33

2.3 SUMMARY 41

iv

CHAPTER 3. VERIFICATION OF MODIFIED FIBER-BASED ELEMENT

3.1

3.2

3.3

3.4

3.5

3.6

GENERAL

FIBER ANALYSIS OF SIMPLE CROSS-SECTION

FIBER ANALYSIS OF A STEEL COLUMN

FIBER ANALYSIS OF CFT BEAM-COLUMNS

3.4.1 MONOTONIC ANALYSIS OF CFT BEAM-COLUMNS

3.4.2 CYCLIC ANALYSIS OF CFT BEAM-COLUMNS

SUMMARY

RECOMMENDATIONS FOR USING THE MODIFIED
FIBER-BASED ELEMENT

CHAPTER 4. SUMMARY AND CONCLUSIONS

4.1

4.2

APPENDIX A

SUMMARY

CONCLUSIONS

APPENDIX A-l

APPENDIX A-2

APPENDIX A—3

APPENDIX A-4

APPENDIX A-S

REFERENCES

MODIFIED SUBROUTINES IN DRAIN2DX

STEEL MATERIAL FIBER
PROPERTIES INFORMATION FILE

STEEL MATERIAL FIBER STATE
INFORMATION FILE

STEEL MATERIAL DATA INPUT

STEEL MATERIAL EVENT
DETERMINATION

STEEL MATERIAL STATE
DETERMINATION

43

43

43

47

53

54

58

7O

71

72

72

73

75

76

77

78

8O

85

93

Table 2.1

Table 2.2

Table 3.1

Table 3.2

Table 3.3

LIST OF TABLES

Description of branch numbers
Description of events
Fiber stress-strain curve for steel bar

Stress-strain points defining the stress-strain curves for
steel and concrete fibers

Yield displacements of CFT beam-columns

vi

16

17

55

59

LIST OF FIGURES

CHAPTER 1. INTRODUCTION

Figure 1.1
Figure 1.2
Figure 1.3

Figure 1.4

CHAPTER 2.

Figure 2.1
Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Fiber-based beam-column element
Monotonic stress-strain behavior for fibers
Cyclic stress-strain behaviors for fibers

Cyclic stress-strain behavior for steel fibers of CFT columns

DEVELOPMENT AND IMPLEMENTATION OF
THE MODIFIED FIBER-BASED ELEMENT

Monotonic behavior of modified S-type stress-strain curve
Cyclic behavior of modified S-type stress—strain curve

Discretization of the modified S-type cyclic stress-strain
curve into branches for fibers

Events for the modified S-type cyclic stress-strain curve

Behavior of branch number 0 for compressive
strain increments

Behavior of branch number 1 for compressive strain
increments

Flowchart for the behavior of branch no. 1 for
compressive strain increments

Behavior of branch 2 for compressive strain increments
Behavior on branch 3 for compressive strain increments

Flowchart of behavior on branch 3 for compressive
strain increments

Behavior on branch -1 for compressive strain increments

vii

ll

13

15

18

21

22

23

24

25

26

27

Figure 2.12

Figure 2.13

Figure 2.14

Figure 2.15

Figure 2.16

Figure 2.17

Figure 2.18
Figure 2.19
Figure 2.20
Figure 2.21

Figure 2.22

Figure 2.23

Figure 2.24

Behavior on branch 4 for compressive strain increments

Flowchart of the behavior on branch 4 for compressive
strain increments

Behavior on branch 5 for compressive strain increments

Behavior on branch 6 for compressive strain increments
and strain reversal

Flowchart of the behavior on branch 6 for
compressive strain increments

Behavior of branches 7 and 8 for compressive
strain increments

Behavior on branch number 0 for tensile strain increments
Behavior on branch -1 for tensile strain increments

Behavior of branches 1, 2, and 8 for tensile strain increments
Behavior on branch 3 for tensile strain increments

Loading path diagram for branch 4 for tensile
strain increments

Flowchart of behavior on branch 3 for tensile
strain increments

Reloading from shooting branches

CHAPTER 3. VERIFICATION OF MODIFIED FIBER-BASED ELEMENT

Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4

Figure 3.5

Schematic of fiber element for steel bar
Fiber discretization of steel bar cross-section
Axial load - tensile monotonic analysis
Axial load — compressive monotonic analysis

Axial load - cyclic analysis

viii

28

29

3O

31

32

33

34

35

36

38

39

4o

41

45

45

46

46

47

Figure 3.6
Figure 3.7
Figure 3.8
Figure 3.9
Figure 3.10
Figure 3.11
Figure 3.12
Figure 3.13
Figure 3.14

Figure 3.15

Figure 3.16
Figure 3.17
Figure 3.18
Figure 3.19
Figure 3.20
Figure 3.21
Figure 3.22
Figure 3.23

Figure 3.24

Pushover analysis load schematic

W14x82 cross-section discretization

Lateral load — monotonic analysis

Moment-curvature analysis

Cyclic loading schematic

Lateral load — lateral displacement response for steel column
Moment-curvature response at failure segment for steel column
Schematic for monotonic analysis of CFT beam-columns

Fiber discretization of the slice (cross-section)

Moment-curvature response of the failure segment for
CFT beam-columns

Schematic for CFT beam-columns

Fiber analysis of CFI‘ beam-column specimen CBC 48-80-20
Fiber analysis of CFT beam-column specimen CBC 48-80-10
Fiber analysis of CFT beam-column specimen CBC 48-46-20
Fiber analysis of CFT beam-column specimen CBC 48-46-10
Fiber analysis of CFT beam-column specimen CBC 32-80-20
Fiber analysis of CFT beam-column specimen CBC 32-80-10
Fiber analysis of CFT beam-column specimen CBC 32-46-20

Fiber analysis of CFT beam-column specimen CBC 32-46-10

ix

48

48

49

50

51

52

52

54

54

57

58

61

62

63

65

66

67

68

CHAPTER 1. INTRODUCTION

1.1 GENERAL

This report presents the development and validation of a modified fiber-based
beam-column finite element for the nonlinear structural analysis program Drain2DX
(Dynamic Response _A_nalysis of _I_N_elastic g-Qimensional Structures, an eXtended
version, Prakash et a1. 1993). Chapter 1 identifies the major drawback of the existing
fiber-based element in Drain2DX. Chapter 2 presents the development and
implementation of a modified fiber-based element that addresses this drawback. The
validation of the modified fiber-based element for static and cyclic loading is presented in
Chapter 3. Finally, Chapter 4 provides recommendations for using the modified fiber-
based element effectively. Appendix A provides a listing of all the files and subroutines

that were modified in the source code of Drain2DX.

1.2 FIBER-BASED BEAM-COLUMN ELEMENT IN DRAIN2DX

Drain2DX is a general-purpose computer program for conducting nonlinear static
and dynamic analysis of two-dimensional (2D) structures. This nonlinear structural
analysis program uses an event-to-event based solution algorithm (Prakash et a1. 1993).
Drain2DX provides several beam, beam-column, and connection/joint element libraries.

One of these is the fiber-based beam-column element (Element Type 15).

The fiber-based beam-column element is a distributed plasticity type finite
element with a flexibility-based formulation. The formulation of the fiber-based element
is presented in detail in Kurama (1997). The assumptions involved in the formulation of
the fiber-based beam-column element are as follows:

(1) Plane sections remain plane and perpendicular to the neutral axis before

and after bending.

(2) Relative motion (slip) does not occur between the materials in the cross-

section.

(3) Inelastic shear deformations are negligible.

(4) The materials in the cross-section are subjected to uniaxial stress-strain

states.

0'

1

fl
—~ —-———-€

Fiber
Stress-strain
I I. '.

Figure 1.1 Fiber-based beam-column element

 

 

A schematic of the fiber-based model is presented in Figure 1.1. As shown in

Figure 1.1, the fiber-based element (defined by the end nodes I and J) can be used to

model the deformable length of a prismatic member. The member length is divided into
segments, where the user controls the number and length of each segment. At the mid-
length of each segment is a slice, which models the member cross-section using fibers.
Each fiber has an associated area, distance from the cross-section centroid, and material
stress-strain curve, all of which are specified by the user. The fiber stress-strain states are

integrated over the cross-section to obtain the slice force—deformation response, i.e., the
section axial force — axial strain — moment — curvature (P — 8 — M — (1)) response. Force

equilibrium is enforced at the slice locations. The slice force-deformation responses are
assumed to remain constant over the length of the corresponding segments and are
integrated along the length of the member to obtain the element force-displacement
response.

The force-displacement response of the fiber-based beam-column element
depends on:

(1) The number and location of segments along the length of the member;

(2) The number and distribution of fibers used to model the member cross-

section; and

(3) The fiber stress-strain relationships.
All of these can be specified by the user for each individual element or model.

Currently, Drain2DX includes two types of stress-strain models that can be used
for the cross-section fibers. These are:

(l) The C-type concrete stress-strain relationship, and

(2) The S-type steel stress-strain relationship.

The monotonic stress-strain relationship, for the C and S-type fibers are shown in Figures
1.2 (a) and (b), respectively. Since Drain2DX uses an event-to-event solution strategy,
both stress-strain relationships must be multi-linear, i.e., consisting of linear segments,

where each linear segment is defined by a pair of stress-strain points as shown in Figure

 

 

 

 

 

1.2.
Stress A
Com resslon
Limitations p
0 Up to 5 points in Compression 02° .
0 Up to 2 points in Tension Ute - - - I
0 In compression. slopes oi ' 1
consecutive segments must keep 1 .
decreasing an L ~ f ----- Jr - ‘- —
: I
I i i
l l l
82T 817 l l '
I T. I i >
I i 81c 82c 83c Strain
I |
l l
l i
l |
Tension : y
. i
| i
l l
I ‘ ‘ 0'17
1
. a
T _ A _ - T _ _ T O2T ( )
Stress A (b)
03 * _ — " ’ ————————— ;d;:::=*f“"——’
02 ————————— -- — —,- ~' ,
/./ l i
./' i |
/
/ I 1
0‘1 — — — f i I
/| l | Limitations
/l j j 0 Up to 5 points
’ i l I 0 Slopes of consecutive segmer
1 j i must keep decreasing
. a I . Symmetric in tension and
/ t : : compression
| I l
I l I
1 I
/ 1 l I
J 1 $
81 £2 8: Strain

Figure 1.2 Monotonic stress-strain behavior for fibers:
(8) C-type; and (b) S-type

CUYVCSI

Drain2DX also imposes the following limitations on the C and S-type stress-strain

The C—type stress-strain curve can be defined with a maximum number of five
stress-strain points in compression and a maximum number of two stress—strain
points in tension.

The stress is assumed to remain constant for strains numerically greater than the
last input points in both compression and tension.

The slopes of consecutive straight-line segments must be decreasing.

The S-type stress-strain curve is assumed to be symmetric in tension and
compression.

The S-type stress—strain curve can be defined with a maximum number of 5 stress-
strain points.

The S-type stress-strain curve cannot undergo strain softening, i.e., the slope of
any linear segment cannot be less than zero.

The cyclic behavior of the C-type and S-type stress-strain curves is controlled by

pre-assigned hysteresis rules and the monotonic stress-strain curves (Figure 1.2 (a) and

(b)), which act as the envelopes for the cyclic stress-strain curves. The cyclic behavior of

the C-type concrete stress-strain curve is shown in Figure 1.3 (a). It can account for

tension crack opening and closing behavior and stiffness degradation under cyclic

loading. The stiffness degradation is controlled by an unloading factor ([3). As shown in

Figure

causes

1.3 (a), a value of B = 0.1 causes almost elastic unloading, and a value of B = 0.9

severe stiffness degradation. Figure 1.3 (b) shows the cyclic behavior of S-type

steel stress-strain curve, which undergoes kinematic hardening behavior under cyclic

 

 

loading.
Stress 4
Compression
om.x
a = 0.5
' 1.
B = 0.9 _ I o_m
B = 0.5 A ' B - 0.1; 2 ’
Strain
Tension crack
opening/closing
behavior

 

 

 

 

(a)
on.
Stress i (b)
Kinematic Pr/ / :
Hardening\ /;/ /
Tension
/ K
‘ /
/ / / ,/ _
f / I / ,/ '
Compression / / / /’ Strain
/ i/ .I/ /
.I/ / I
/ / ,/ .
/ / j/é/ \‘\ Kinematic
. . -3./,1”, Hardening

 

Figure 1.3

Cyclic stress-strain behavior for fibers:

(a) C-type; (b) S-type

1.3 DRAWBACK OF THE EXISTING FIBER-BASED ELEMENT

The existing fiber-based beam-column element in Drain2DX is very useful for
modeling steel, reinforced concrete, and composite steel-concrete members. This beam-
column element has been used by various researchers for investigating the force-
deforrnation behavior of member cross-sections, the force-displacement behavior of
structural members, and also for modeling beams and beam-columns while investigating
the static or dynamic behavior of complete 2D structural frames. For example, El-Sheikh
et al. (1997), Kurama et al. (1997), Ricles et al. (2001), and Shen and Kurama (2002).

More recently, Varma et al. (2001) tried to use the existing fiber-based element
for modeling composite concrete filled steel tube (CFT) beam-columns. These
researchers had developed three-dimensional (3D) finite element model based on
effective uniaxial stress-strain curves for the steel and concrete fibers of the CFT cross-
section. As a result, the effective stress-strain curves for the steel fibers implicitly
accounted for the effects of local buckling and biaxial stresses in the steel tube. Figure
1.4 shows a typical steel cyclic stress-strain curve that Varma et al. (2001) wanted to use
with the fiber-based beam-column element in Drain2DX. However, neither the C nor the
S-type stress-strain curve could be used to model this stress-strain behavior (compare
Figures 1.3 and 1.4).

Thus, Varma et al. (2001) identified a major drawback of the existing fiber-based
beam-column element in Drain2DX. The existing S-type stress-strain curve cannot
model the effects of local buckling or biaxial stresses in the steel plate elements of a

member cross-section for the following reasons:

(1) It is assumed to be symmetric in tension and compression.

(2) It cannot account for strain softening in compression due to local buckling or
biaxial stress effects.

(3) Its pre-assigned hysteresis rule (Figure 1.3 (b)) cannot model the desired

effects of local buckling or biaxial stresses on the hysteretic behavior of plate

elements (Figure 1.4).

 

 

 

 

 

 

 

o A
COMPRESSION
Envelope
Approximate etiect oi , ”
local buckling and ”
biaxial stresses on ' 9,- / I El tic
hysteretic behavior. , 09/ / ”i _J I as .
€99. Es . Unloading
shoo‘lngeww .1
-/ flawe ,L Es
1' I > e
f ‘7’ Loading with
- . elastic slo
TENSION ’ 58/ p°
, Elastic
J Unloading ‘ I.
/ 5' JEs _ fl , _HJ..._I Intersection oi lines with
, ” " slope E- and Es»
' Esll
i— — _ " ' _ _— " . . '
I _ Envelope of cyclic 5—5 — — — Hysteretlc behaVIor

 

Figure 1.4 Cyclic stress-strain behavior for steel fibers of
CFT columns (V arma et al. 2001)

1.4 PROJECT OBJECTIVES AND SCOPE

The objective of this research project is to develop, implement, and verify a
modified S-type stress-strain model for the fiber-based beam-column element in
Drain2DX (Prakash et al. 1993). The modified S-type stress-strain curve will be based
on the recommendations of Varma et al. (2001), and it will be able to model the effects of

local buckling and biaxial stresses under static or cyclic loading.

1.5 SUMMARY

The fiber-based beam-column element is a simple yet versatile tool used for
modeling the nonlinear inelastic behavior of structural members. A fiber-based beam-
column element has been previously implemented in the nonlinear structural analysis
program Drain2DX (Prakash et al. 1993, Kurama 1997). It has been successfully used by
several researchers to investigate the force-deformation behavior of cross-sections,
members, and frames. A major drawback of the existing fiber-based element in
Drain2DX is that it cannot model the effects of local buckling and/or biaxial stresses on
the force—deformation responses of steel or steel-concrete composite members. This
research directly addresses this drawback and focuses on the development,

implementation, and verification of a modified fiber-based element in Drain2DX.

CHAPTER 2. DEVELOPMENT AND IMPLEMENTATION
OF THE MODIFIED F IBER-BASED ELEMENT

The development and implementation of the modified fiber-based beam-colurrm
element in Drain2DX is presented in this chapter. No changes were done to the overall
formulation of the fiber-based element in Drain2DX. The C-type concrete stress-strain
curve was retained as it is and only the S-type steel stress-strain curve was modified.
Section 2.1 presents the modified S-type steel stress-strain curve. Section 2.2 presents
the implementation of the modified S-type stress-strain curve along with the event

calculation and state determination algorithms.

2.1 MODIFIED S-TYPE STEEL STRESS-STRAIN CURVE

A modified S-type steel stress-strain curve was developed for the fiber-based
beam-column element in Drain2DX. The modified S-type stress-strain curve is partially
based on the recommendations of Varma et al. (2001). It is more general in application
and has the following features:

(1) It can model asymmetric behavior in tension and compression;

(2) It can model strain hardening or softening behavior in compression;

(3) It can model kinematic strain hardening behavior under cyclic loading; and

(4) It can also model the effects of local buckling and biaxial stresses on the
hysteretic behavior in a flexible manner.

Figure 2.1 shows the monotonic behavior of the modified S-type steel stress-strain

curve. In Figure 2.1, Gyc, ch, Cue, Gyt, and O'sh are the yield stress in compression, the

10

buckling stress in compression, the ultimate compressive stress, the yield stress in tension
and the ultimate stress in tension, respectively. Eini, Bind, Em and Esh are the initial
stiffness, the inelastic stiffness (strain-hardening in compression), the softening stiffness

in compression and the stain-hardening stiffness in tension, respectively. Finally, Eye,

Ebc, Sue, 8,” and 85., are the strains corresponding to dye, one, due, Gyt, and 05h,

 

 

respectively.
Stress, o A
TENSION
03H ————————— ‘ F
l
l l
I I
l l
EINI I i
Soc Sec Eve 1 l
i l . l l >
I l ' Strain s
COMPRESSION l 8" 8‘" '
l l l EINI
l l I
“—&M—+—-j —.om
l i
EM l EINEL - — "4 Ovc
‘‘‘‘‘‘‘‘‘ '“ O'ec

 

Figure 2.1 Monotonic behavior of modified S-type stress-strain curve

The proposed stress-strain curve has the following limitations:

(1) Three stress-strain points have to be specified in compression;

(2) Two stress-strain points have to be specified in tension;

(3) The tangent stiffness of consecutive linear segments must decrease.

The monotonic stress-strain curve shown in Figure 2.1 serves as the envelope for

the stress-strain behavior under cyclic loading. The hysteresis rules under cyclic loading

11

depend on whether the fiber has undergone strain softening (due to buckling) in
compression during its cyclic response history. Figure 2.2 (a) shows the cyclic stress-
strain behavior and the hysteresis rules for the modified S—type stress—strain curve when it
has not undergone compression strain softening (local buckling) during its cyclic
response history.

As shown in Figure 2.2 (a), kinematic hardening occurs under cyclic loading as
long as the fiber does not undergo compression strain softening during its cyclic response
history. Figure 2.2 (b) shows the cyclic stress-strain behavior and the hysteresis rules for
the modified S-type stress-strain curve after compression strain softening (due to local
buckling and/or biaxial stresses) has occurred during the cyclic response history. As

shown in Figure 2.2 (b), the hysteretic behavior is controlled by the unloading parameter

0t. 0t equal to zero corresponds to a case of no stiffness degradation, i.e., biaxial stress

effects dominate according to Varma et al. (2001), and 0t equal to one corresponds to a
case of significant stiffness degradation, i.e., local buckling effects dominate. The user
can specify any value of 0c between zero and one depending on the application and

model. Thus, the modified S-type steel stress—strain curve also accounts for the effects of

local buckling and biaxial stresses on the hysteretic behavior.

12

 

 

 

 

 

 

and/or biaxial stresses on
hysteretic behavior
Figure 2.2 Cyclic behavior of modified S-type stress-strain curve: (a) before and,
(b) after compression strain softening

13

2.2 IMPLEMENTATION OF MODIFIED S-TYPE STEEL STRESS-
STRAIN CURVE

As mentioned previously, Drain2DX (Prakash et a1. 1993) is a general-purpose
nonlinear structural analysis computer program for static and dynamic analysis of two-
dimensional (2D) structures. Static nonlinear analyses are performed based on an event-
to-event solution strategy, where each event corresponds to a significant change in
stiffness.

Therefore, for implementation into Drain2DX, the linear segments of the picewise
linear modified S-type cyclic stress-strain curve were identified using unique branch
numbers. The changes in stiffnesses associated with the response (stress-strain) states
moving from one branch to the other under cyclic loading were identified using event
numbers.

The modified S-type steel stress-strain behavior was implemented in Drain2DX
using two major subroutines SMEV15 and SMSDIS, which are the event factor
calculation and the state determination subroutines, respectively. Since Drain2DX uses a
displacement control analysis, several events can occur within a strain increment. The
SMEV15 subroutine monitors the magnitude of the strain increment and determines the
occurrence of events using the current state (stress, strain and tangent stiffness) and the
stress-strain curve input by the user.

The SMSDlS subroutine determines the state of the fiber following the applied
strain increment. If an event occurs (according to the SMEV15 subroutine), the strain

increment is scaled down to the value causing the event and the state (stress, strain, and

14

tangent stiffness) of the fiber is updated. The remaining portion of the strain increment is
applied incrementally using the recalculated state, and so on.

Section 2.2.1 presents the discretization of the modified S-type stress-strain
behavior into branch numbers. Section 2.2.2 identifies the events that can occur during
cyclic loading. Section 2.2.3 presents relevant portions of the algorithm that has been
used to implement the modified S-type stress-strain behavior in the SMEV15 and

SMSDlS subroutines. The source code for the subroutines is provided in Appendix A.

2.2.1 Branch Numbers
The modified S-type cyclic stress-strain behavior was discretized into ten unique

branches, which are identified in Figure 2.3 and described in Table 2.1.

 

 

 

 

 

 

0'
i J (a) G (b)
-1 ’/ - 1’» 'J”//".}
'r/f/ '1 / x /-//-1 /
! / / 4 F //4
4 /0 »’ TBGCN / /0 / 133m
./ / / / /
/ ’ / / /
T. 3;; l/ll A;— — 8 173 ,/| i // 8
8 ,I/ // I {/4 8 // 0 //5
/ /° 1 l ,
2 // I? 2 ”6— 4.
1 1 I
W W01

Figure 2.3 Discretization of the modified S-type cyclic stress-strain curve into
branches for fibers: (a) before; and (b) after compression strain softening

15

 

Table 2.1 Description of branch numbers

 

 

Branch Number Description
0 Elastic branch
1 Compression strain-hardening branch
-1 Tension strain-hardening branch
2 Compression softening branch
3 Unloading/reloading from/to compression branch
4 Unloading/reloading from/to tension branch
5 First shooting branch
6 Second shooting branch
7 Ultimate tensile branch
8 Ultimate compressive branch

 

 

 

 

2.2.2 Events Numbers

The changes in stiffnesses associated with the stress-strain response states moving
from one branch to the other under cyclic loading are identified using event numbers.
These event numbers are shown in Figure 2.4, where the arrows indicate the direction of
loading and the event numbers are circled. The branch numbers are also indicated in

Figure 2.4. A brief description of each event is given in Table 2.2 below.

16

 

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Q

 

(a) Before compression strain softening

 

 

 

Afr—’7—
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(b) After compression strain softening

Figure 2.4 Events for the modified S-type cyclic stress-strain curve (a) before;
and (b) after compression strain softening

18

2.2.3 Event Calculation and State Determination Algorithm

The algorithm for determining the response state (stress 0‘, strain 8, tangent

stiffness Emod, and branch number) of the fiber subjected to strain increments (d8) is
presented in this section. The response state of the fiber will depend on its initial
(current) state, cyclic loading history, magnitude and direction of de, and the occurrence
of events.

If an event occurs, the strain increment d8 is scaled down using a calculated

event factor (FACT) to the value (FACT 'd8), and the event, response state and branch
number of the fiber after the event are determined. The remaining portion of the strain
increment (1-FACT)°d8 is applied incrementally using the calculated response state as
the initial state, and so on. If no events occur, then the event factor (FACT) is equal to
1.0 and the complete d8 can be applied without change in stiffness and the response state

can be calculated using the initial state (stress, strain, tangent stiffness) alone.

The algorithm for determining the response state (stress, strain, tangent stiffness,
and branch number) of the fiber subjected to compressive and tensile strain increments is
presented in the following two sub-sections. The state determination algorithm is
presented with reference to the initial state (stress, strain, tangent stiffness, and branch
number) of the fiber and its partial cyclic response history (required for branch number 2,
3, 4, 5, and 6). The partial cyclic response history terms that are monitored and updated
include:

o Lastt — last tensile branch before strain reversal

o Lastc — last compression branch before strain reversal

l9

0 (1'c — compressive stress when the last strain reversal occurred (always on lastc)
0 GT — tensile stress when the last strain reversal occurred (always on lastt)
0 8min — strain corresponding to (3'c

0 am... — strain corresponding to UT

The algorithm is illustrated for each initial branch case using flowcharts and

loading path diagrams.

2.2.3.1 Compressive strain increments
The compressive strain increment has negative sign and the behavior on the

various initial branches is as follows:

0 Initial branch = Branch number 0
The fiber will behave elastically with tangent stiffness equal to Em; until

compression yielding (event no. 1) occurs. As shown in Figure 2.5, if the strain
increment d8 causes event no. 1, then the stresses and strains are updated, and the

branch number and tangent stiffness are set equal to 1 and Einel, respectively.

20

 

d0: Eini * d8

 

 

 

W0) N0

do

-0'

 

YES

 

 

 

fact = (-cyc-o)/do 0: 0mg
o: 43,. a: 8+ fact*de
a: 8+ fact*de d8= 0
de: de“ (l-fact)
I
Brnum = 1
Emod = Eincl

 

 

 

 

 

 

 

 

 

 

(a) Flowchart

___—_.____>8

 

 

(b) Loading path diagram

Figure 2.5 Behavior of branch number 0 for compressive strain increments

0 Initial branch = Branch number 1
The fiber will have a tangent stiffness equal to EN on this compression

strain-hardening branch until strain softening (event no. 2) occurs. As shown 111

Figures 2.6 and 2.7, if the strain increment d8 causes event no. 2, then the stresses

21

and strains are updated, and the branch number and tangent stiffness are set equal
to 2 and Eim, respectively. The occurrence of event no. 2 depends on whether the
fiber has undergone tension yielding during its cyclic response history, which is

established using the cyclic response variable LAST'I‘ as shown in Figure 2.7.

o (a)

 

 

, i - 1 I fi fi,*. 4’8
I/
/
2\\% 1' I‘m
{2 IO‘ec
o (b)
I - /~-
A. _fi IA¥ - /, 8
\ / (Mi (2 ‘
_ BBC—I

Figure 2.6 Behavior of branch number 1 for compressive strain increments: (a)
starting in compression; and (b) starting in tension

22

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0 Initial branch = Branch number 2

The fiber will have a tangent stiffness equal to Em. on this compression

strain-softening branch until the ultimate compressive stress (Cue) is reached

(event no. 3). As shown in Figure 2.8, if the strain increment d8 causes event no.

3, then the stresses and strains are updated and the branch number and tangent

stiffness are set equal to 8 and zero, respectively.

 

do: Eim * de

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fact = (-q,c-o)/do 0’: mdo !/
o: *ch a: 8+ fact*d€ fl _- y _ 8
e = e + fact‘de '8 = 0 /i
as: de* (l-fact) <3 /I
1 _ -_ __ ._ _ _ /_
8 \ GUC
Brnum = 8 x/
Emod = 0 —————— l CBC
(a) Flowchart (b) Loading path diagram

Figure 2.8 Behavior of branch 2 for compressive strain increments

0 Initial branch = Branch number 3

The fiber will have a tangent stiffness equal to Bin: on this

unloading/reloading branch until the compressive stress (0c) is reached (event no.

24

9). 0'c is the compressive stress where the fiber initially went into strain reversal
(branch number 3). As shown in Figure 2.9, 0'c can be on branch numbers 1, 2, 5,

6, or 8. If 0'C is on branch 1 and d8 causes event 9, then the stress-state is updated
and the branch number and the tangent stiffness are set equal to l and Ema,

respectively. As shown in Figure 2.10, corresponding behavior is enforced for 0'c

located on branches 2, 8, 5, or 6. In Figure 2.10, the terms 800 and Eab correspond

to behavior on branch 5 and are explained later.

 

o
//”‘/—/ F
. I,” /
/ i / / 4
3/ 3/ 3/ I/ /
I/ r/ I/ ' / / 6c

 

Figure 2.9 Behavior on branch 3 for compressive strain increment

25

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26

0 Initial branch = Branch number —1

The fiber will have a tangent stiffness equal to Esh on this strain-hardening

branch until a compressive strain increment d8 greater than the tolerance limit is
applied (event no. 7). The tolerance check for de is given by equation (1).
o'tol S Eini ' d8 (1)

where: Owl = specified overshoot tolerance stress.

As shown in Figure 2.11, if event no. 7 occurs, then the branch number

and tangent stiffness are set equal to 4 and Eini, respectively.

 

0'
O'UTI ___________ 7
-1 /-’f
CYT ” 7" //
/
/
l /.
|/O /
#_ # ——:"—— ______> 8

 

Figure 2.11 Behavior on branch -1 for compressive strain increments

0 Initial branch = mnch number 4
The behavior of the fiber on this unloading/reloading branch from tension
will depend on whether compression strain softening has occurred during its
cyclic loading history. If compression strain softening has not occurred, then the

response will remain on branch number 4 until compression yielding (event no. 1)

27

occurs. If compression strain softening has occurred during its cyclic loading
history, then the response will remain on branch number 4 until the zero stress
state (event no. 8) is reached.

As shown in Figure 2.12 (a) and 2.13, if event no. 1 occurs, the branch
number and tangent stiffness are equal to 1 and Band, respectively. As shown in
Figure 2.12 (b) and 2.13, if the event no. 8 occurs, the branch number and the

tangent stiffness are equal to 5 and Bab, respectively.

7
“ ” ‘ 7W7: GUT” ' ‘ ‘ “ 7" / ‘

 

 

! I / /
m5 , z”#___.:-_E£L_, _. @.¢____,
7 f 8 I 5//8°° e
// AEab

 

 

Figure 2.12 Behavior on branch 4 for compressive strain increment if: (a) before
compression strain softening; and (b) after compression softening

28

 

 

 

dU= Eini * d8

 

   

     
   
 

  
   
   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Lastc=l or 0 Lastc = 2 or 5
& Lastt=-l or NO or 6 or 8 &
7 Lastt = -l or 7
NO 800 = 8 - dEini
800-8 NO
> de
fact=(q-oyt-oyc-o)ldo 0.: credo YES
o‘q’OyVOyC 8: 8+ fact*de
e:e+fact*de d8: 0 f
act: 8 -e /de _
de=(lofact)*de (_°° ) 0' “do
8'800 a: 8+ fact‘de
de =(1-fact)*de as: o
o: 0.0
Set:
Brnum =1
Emod = Einel
L35“ = 4 Brnum = 5
El=Oc/(8min‘800)

 

 

 

E,b=a‘El+(l-a)“Eini
EmodzEab

 

 

 

Figure 2.13 Flowchart of the behavior on branch 4 for compressive strain
increments

0 Initial branch = Branch number 5 (first shooting branch)

The fiber will have a tangent stiffness equal to Bab on this shooting branch

 

until the compressive stress (O'C) at the last strain reversal is reached. Eab is given
by equation (2) and it is a function of 0t, Him and E, where 0t is specified by the
user and E1 is calculated as shown in Figure 2.14 (a). For (X equal to zero and

one, Bab will be equal to Em, and E1, respectively. For values of O < (X < 1, E <

29

Bab < Ema. As shown in Figure 2.14, if (1'c is reached due to de, the branch number

and tangent stiffness will be set equal to 6 and zero, respectively.

where:

 

 

E1=Gc/(8min'€ao)
Eab=a*El+(l'a)*Eini
d0: Bab * d8

 

 

Eab = 0L ° E1 + (1'00 ° Eini

(2)

= initial material stiffness

= shooting stiffness from (Eco. 0) to (8min, 0'0)

= shooting stiffness

 

 

 

fact = (Cc-0W0
0': 0'c

e = 8 + fact * d8

d8 = (l-fact) * d8

 

 

 

 

 

Brnum = 6
Emod = 0

 

 

 

(a) Flowchart

 

 

 

o: o+do

8 = 8 + fact*d8

de=0

 

 

 

16/

 

(8.0)
r/ /
4
/ /
/ /
,1 800: e- o/E::::
/ V1 “27/ 8
//7/ I/Ol- 0 Shooting
// ab/ 5 Behavuor
E‘ “‘ ‘JO'c
Loading path diagram

Figure 2.14 Behavior on branch 5 for compressive strain increments

Initial braanh = flinch number 6 (second shooting branch)

The fiber will have a tangent stiffness equal to zero on this second

shooting branch until the compressive strain at last strain reversal (8min) is reached
(event no. 9). As shown in Figures 2.15 (a) and (b), 88 is the length of branch no.

6 and 0'c can be on branch number 2 or 8. If 6c is on branch 2 and d8 causes

event no. 9, then the branch number and the tangent stiffness are set equal to 2
and En“, respectively. Otherwise, as shown in Figure 2.15 (b), the branch number

will be set to 8 and the tangent stiffness will remain equal to zero.

 

 

ll/ 5
___.__ , / / /_—; Eab i?— ”‘F—éc
/il ‘76 ' {0° 3

0’0?

0- 0'
7 II 7
/ / ”T “/7—7h— / f
’ / 4 / /
/ / /,
// ’éO’O - / ——’
/ , '7' ’ 8 / 8

 

 

 

(a) I (m
.— 55 4‘ ——— 88 ——’I

Figure 2.15 Behavior on branch 6 for compressive strain increments and strain

reversal from: (a) branch 2; and (b) branch 8

31

 

88=£mIn-eoo-Oc/Eab

NO /\ NO
abs(em.x - £::::::)/1 8min " 5 >

 

 

 

 

 

fact=(8m:n-8)/de o=o+do
8=e+fact*de e=e+fact*de
de=(1-fact)*de de=0

N0

 

 

 

 

 

 

 

 

 

 

 

0:0“; 6=Gc NO

Brnum = 8 Brnum = 2
EM = Emod = Elnt

oc #5 one

 

 

 

 

YES

 

 

 

 

 

o = one 0' = oc
Brnum = 8 Brnum = 2
Emod = 0 Emod = Eint

 

 

 

 

 

 

Figure 2.16 Flowchart of the behavior on branch 6 for compression strain
increments

0 Initialmnch = Branch number 7

The fiber will have a tangent stiffness equal to zero on this ultimate tensile
branch until a compressive strain increment d8 greater than the tolerance limit is

applied (event no. 7). The tolerance check was presented in equation (1). As
shown in Figure 2.17, if event no. 7 occurs, the branch number and the tangent

stiffness are set equal to 4 and Ham. respectively.

0 Initial branch = Branch number 8
The fiber will have a tangent stiffness equal to zero on this ultimate

compressive branch. No events are associated on this branch while subjected to

32

additional compressive strain increments since there is no change in stiffness (see

Figure 2.17).

 

o
I a.
- i
4/
.-/
I/
[J E ini
“W fir“? ‘fl/‘—+e
/F /
/
‘— — //

 

Figure 2.17 Behavior of branches 7 and 8 for compression strain increments

2.2.3.2 Tensile strain increments
On the other hand, the tensile strain increment has positive sign and the behavior
on the various initial branches is as follows:
0 Initial branch = Branch number 0
The fiber will behave elastically with tangent stiffness equal to Em until

tension yielding (event no. 4) occurs. As shown in Figure 2.18, if the strain
increment d8 causes event no. 4, then the stresses and strains are updated, and the

branch number and tangent stiffness are set equal to -1 and Esh, respectively.

33

 

 

 

 

 

 

 

 

 

 

 

 

.0”. a NO 0
<do I
YES
fact = (-oy,-o)/do O= O+d6
a: 'Oy: e = e + fact*de --
6: 8+ fact*de d5: 0
de = de * (l-fact)

 

 

l

 

 

 

 

 

 

Brnum = -1
Emod = Esh
(a) Flowchart (b) Loading path diagram

Figure 2.18 Behavior of branch number 0 for tensile strain increments

0 Initial branch = Branch number -1
The fiber will have a tangent stiffness equal to Esh on this tension strain-

hardening branch until tension ultimate (event no. 5) occurs. As shown in Figure
2.19, if the strain increment d8 causes event no. 5, then the stresses and strains are

updated, and the branch number and tangent stiffness are set equal to 7 and zero,

respectively.

34

 

dO=E;h*d8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fiact=o -0’ {do 0‘5”“

(it ) e=a+fact*ds
8=8+fiact*d8 I 8Il=-l
de=de*(l-fiact)

|

Bmum=7

Emod=0

Lastt=-l

 

 

 

(a) Flowchart

 

0'
II 7
Om” — —- ran——
O'YT— .1 (5
_L _‘—’8

 

W\,

(b) Loading path diagram

Figure 2.19 Behavior on branch -1 for tensile strain increments

35

0 Initial [Lanch = Bramh number 1

The fiber will have a tangent stiffness equal to Em: on this compression

strain-hardening branch until a tensile strain increment d8 greater than the

tolerance limit is applied (event no. 6). The tolerance check for de is given by

equation (1). As shown in Figure 2.20, if event no. 6 occurs, then the branch

number and tangent stiffness are set equal to 3 and Ema, respectively.

/
a] 3/ 3/ .
Einifl Ems/J / /

 

 

 

Figure 2.20 Behavior of branches 1, 2, and 8 for tensile strain increments

0 Initial branch = Branch number 2

The fiber will have a tangent stiffness equal to Eim on this compression

strain-softening branch until a tensile strain increment d8 greater than the

tolerance limit is applied (event no. 6). The tolerance check for de is given by

equation (1). As shown in Figure 2.20, if event no. 6 occurs, then the branch

number and tangent stiffness are set equal to 3 and Em, respectively.

36

Initial branch = Branch num_btir8

The fiber will have a tangent stiffness equal to zero on this ultimate

compressive branch until a tensile strain increment d8 greater than the tolerance

limit is applied (event no. 6). The tolerance check for de is given by equation (1).

As shown in Figure 2.20, if event no. 6 occurs, then the branch number and

tangent stiffness are set equal to 3 and Bing, respectively.

Initial Branch = Brggch number 3

The fiber will have a tangent stiffness equal to Egg on this
unloading/reloading from compression branch until tension yielding (event no. 4)
occurs. As shown in Figure 2.21, the occurrence of tension yielding will depend
on whether compression strain softening has occurred. If compression strain
softening has not occurred (lastc = l), the fiber will follow kinematic hardening
rules with respect to its current stress. If compression strain softening has

occurred (lastc ¢ 1), the fiber will follow kinematic hardening with respect to the
stress at compression strain softening (O'bc).

As shown in Figure 2.21, if event no. 4 occurs, the branch number and

tangent stiffness are equal to -1 and Esh, respectively.

37

 

do: Em . d8

 

 

 

 
    

Lastc = 2 M8
orS or6

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o: o+do
, e = 8+ fact’de
fact - (0’ +0y +0y -o)/ do d8 = 0 O: Tdod
' c t c = 1"
8 = 8 + fact*de O: -cx.,c+t3y,+oyc
d8 = (18 ‘ (l-fact) d8 = 2+‘f22tit‘fdt30
E = 8 - ac
I
Brnum = -l I
Emod = Est Brnum : -I
Emod = Bill
(a) Flowchart
0'
A—
-1 /.__
I"4\
: 22 W
I / / / / 4
3 3'I . / /
I AElnl / L/ //3//:, _ +
OYT+Ovc / / If / //7 8
I l / / / l/ 5
oc/ \ / / [J
{ 8 \\\ OC// 1 6 CC
__ __ ;.,.//1

(b) Loading path diagram

Figure 2.21 Behavior on branch 3 for tensile strain increments

38

0 Initial branch = Branch number 4
The fiber will have a tangent stiffness equal to Em: on this

unloading/reloading tensile branch until the stress at last tensile strain reversal

(GT) is reached (event no. 0). As shown in Figure 2.22, (ST can be on branch

numbers -1 or 7. If (IT is on branch —1 and d8 causes event 0, then the stress-
state is updated and the branch number and the tangent stiffness are set equal to -l
and Esh, respectively. The same behavior is enforced for CT located on branch 7.

In this case, the branch number and the tangent stiffness will be set to 7 and zero,

respectively.

 

 

Figure 2.22 Loading path diagram for branch 4 for tensile strain increments

39

 

do: E... ' dc

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fact=(q-o)ldo a: (mic fact=(o.-o)/do a: o+da
0:0. e=e+fact‘d8 0:0. e=e+fact"d8
e = e + fact‘de d8 = 0 e = e + fact‘de de- 0
de = dt: ‘ (l-fact) d8 = as ' (l-fact) -
gmum-EJ Brnum = 7
mod " 8h Emod = 0

 

 

 

 

 

 

Figure 2.23 Flowchart of behavior on branch 4 for tensile strain increments

0 Initial branch = Branch number 5 (first shooting branch)

The fiber will have a tangent stiffness equal to Eab on this shooting branch

until a tensile strain increment d8 greater than the tolerance limit is applied (event

no. 6). The tolerance check for de is given by equation (1). As shown in Figure

2.24, if event no. 6 occurs, then the branch number and tangent stiffness are set

equal to 3 and Bing, respectively.

0 Initial brflch = Branch number 6 (second shooting branch)

 

The fiber will have a tangent stiffness equal to zero on this second

shooting branch until a tensile strain increment d8 greater than the tolerance limit

is applied (event no. 6). The tolerance check for d8 is given by equation (1). As

40

2.3

shown in Figure 2.24, if event no. 6 occurs, then the branch number and tangent

stiffness are set equal to 3 and Eini, respectively.

 

—.,,,//a_- —> g

a = 0 Shooting
5 Behavior

 

Figure 2.24 Reloading from shooting branches

Initial branch = Bglnch 7

The fiber will have a tangent stiffness equal to zero on this ultimate tensile
branch. No events occur on this branch under additional tensile strain increments

since there is no change in stiffness (see Figure 2.24).

SUMMARY

A modified S-type steel stress-strain curve was developed for the fiber-based

beam-column element in Drain2DX. This stress-strain curve can model various aspects

of steel behavior such as compression and tension strain hardening, compression strain

softening due to local buckling or biaxial stresses, kinematic hardening under cyclic

loadings, and the effects of local buckling and biaxial stresses on the hysteretic behavior.

41

This is a significant improvement over the S-type steel stress-strain curve previously
available in Drain2DX.

Since Drain2DX uses an event-to-event based solution algorithm, the cyclic
stress-strain curve was discretized into ten unique branch numbers and ten event numbers
corresponding to changes in stiffness due to compressive and tensile strain increments.
The response state depends on the initial (current) state, partial cyclic loading history,
strain magnitude and direction, and the occurrence of events. The event calculation and
state determination algorithm were developed accordingly and implemented in the
corresponding SMEV15 and SMSDlS event calculation and state determination

subroutines in Drain2DX.

42

CHAPTER 3. VERIFICATION OF MODIFIED
FIBER-BASED ELEMENT

3.1 GENERAL

The implementation of the modified S-type steel stress-strain curve in Drain2DX
was verified using three analytical models with increasing levels of complexity. Section
3.2 presents the results of monotonic and cyclic analysis of a two-fiber steel cross-section
subjected to axial loading. This simple analysis was conducted to specifically verify the
implemented modified S-type stress-strain curve. Section 3.3 presents the results of
monotonic and cyclic analysis of a steel column subjected to lateral loading. Section 3.4
presents the results of monotonic and cyclic analysis of high strength CFT beam-columns
tested by Varma et al. (2001). The results of the fiber analysis are also compared with
the experimental results of Varma et al. (2001). Section 3.5 presents a summary of
chapter 3. Section 3.6 presents recommendations for using the modified fiber-based

element in Drain2DX.

3.2 FIBER ANALYSIS OF SIMPLE CROSS-SECTION

In order to verify the implementation of the modified S-type steel stress-strain
curve in Drain2DX, a simple analytical model of a steel bar was developed and analyzed
for monotonic and cyclic axial loading. Figure 3.1 shows a schematic of the analytical

model. As shown in Figure 3.1, the fiber-based element had a length of 1.0 in. The

43

bottom node (1) of the element was fixed and the top node (2) was free. The element had
only one segment, i.e., only one slice located at mid-length. Figure 3.2 shows the
discretization of the cross-section (Area = 1.0 in2) into two fibers. The fiber stress-strain
curve is shown in Figure 3.3 and the numerical values are provided in Table 3.1.

As shown in Figure 3.1, the analytical model was analyzed for three loading
conditions: (1) monotonic tension, (2) monotonic compression; and (3) cyclic loading.

The external load (P) is equal to the stress in the section fiber (area = 1.0 in2) and the
displacement (A) of node 2 (length = 1.0 in) is equal to the strain in the fibers. Thus the

load-displacement from the fiber analysis is also the stress-strain response of the cross-
section fibers. Figures 3.3 and 3.4 show the results from the fiber analysis for the
monotonic tension and compression loadings, respectively. As shown in Figure 3.3 and
3.4, the results from the fiber analysis follow the monotonic stress-strain curve input to
the program perfectly, thus validating the implementation of the modified S-type steel
stress-strain curve in Drain2DX.

Table 3.1 Fiber stress-strain curve for steel bar

 

 

 

 

 

 

Stress Strain

Jksi) (in/in)

90 0.003

Compression 100 0.005
70 0.008

T . 90 0.003

ensnon
100 0.010

 

 

 

 

 

 

 

 

. .9
1.0m ——suce g ——- Slice 8 ‘“ 3“”
. l-
l
1 1 1
'————~~I’L"———“-#_-—Y'H_‘——mn—ELELIL '2'.—
TI

LI V IL ”WM *‘7 :.11I;II"?ITL [II 2‘_
II: I: m_ II i’L-m III: III- M II.—

Figure 3.1 Schematic of fiber element for steel bar

 

Y“_—“_— "_' ———_— ______ ‘_ f j—
g; Fibq Centroid ”2 "i
0.25 "
‘F__ , if i ___i- 1 u
. -o.25 " |
Fiber Centroid ‘ l

 

 

 

L—u—J

Figure 3.2 Fiber discretization of steel bar cross-section

45

 

 

  
    

 
 

 

 

 

 

 

 

 

 

 

 

120
80
A 40
7)
5 0
m
8
3:. -40
to q + Monotonic Analysis-Tension
~80 “ —— Fiber stress-strain curve
-120
0.009 -0.006 -0.003 0 0.003 0.006 0.009 0.012 0.015
Strain (in/In)
Figure 3.3 Axial load - tensile monotonic analysis
120
90
60
3‘ 30
E 0 . . .
5 -30
+Monotonic Analysis -
'60 Compression
— E
_90 . Steel nvelope
-120
-0.012 0.009 0.006 0.003 0 0.000 0.006 0.009 0.012 0.015

Strain (in/In)

Figure 3.4 Axial load - compressive monotonic analysis

The analytical model was subjected to cyclic loading history so that all branches

and events identified in Section 2.2 and 2.3 would occur. The cyclic loading history

included cycles at 0.75 8,, 2.0 8,, and 4.0 8,. The fiber analyses were conducted for

46

values of the unloading parameter 0L equal to 0.0, 0.5 and 1.0. The results from the fiber

analysis for cyclic loading are shown in Figure 3.5.

 

 

 

 

 

 

 

 

 

 

120.0
90.0
60.0 .
c 30.0 .
.2":
a 0.0 I I I , I I
9 1
65 -3o.0 ,- 71
-60.0
. I
-90.0 \‘I' ]
-120.0 »
-0.015 -0.012 -0.009 -0.006 -0.003 0.000 0.003 0.006 0.009 0.012 0.015
Strain (In/In)
Alpha = 0.0 Steel Envelope
Alpha = 0.5 ------ Alpha = 1.0

 

Figure 3.5 Axial load - cyclic analysis

As shown in Figure 3.5, the results from the fiber analysis follows the

implemented modified S-type steel stress-strain behavior (Figure 2.3 and 2.4) closely.

3.3 FIBER ANALYSIS OF A STEEL COLUMN

In order to verify the modified fiber-based element in Drain2DX, a simple
analytical model of a 10 feet long W14x82 cantilever steel beam-column was developed
and analyzed for monotonic and cyclic lateral loading. Figure 3.6 shows a schematic of

the analytical model. As shown in Figure 3.6, the fiber-based element was divided into

47

five equal segments. The element was fixed at one end (node 1) and free at the top (node
2) end. Figure 3.7 shows the discretization of the cross-section into six fibers. The fiber
stress-strain curve was the same used for the steel bar analysis in the previous section. It

is shown in Figure 3.3 and listed in Table 3.1.

 

 

 

 

 

7..
....-. i—wé .— rV - - - »
F ' . ' ' I
- -_ “TM—#7 FLA- -__..I j
2.0 I E
ii. I ;
2.0' i f
33,- a I
2'0. I; [I 12:12 14},
2.0' I I
2.0' I '
~I If g-
L—— .og- ——4
Figure 3.6 Pushover analysis Figure 3.7 W14x82 cross
load schematic -section discretization

The analytical model was analyzed for two loading conditions: (1) constant axial
load (P) and monotonically increasing lateral loading (V), and (2) constant axial load (P)

and cyclically varying lateral load (V). The constant axial load was equal to 50 kips,

which is 11.1% of the axial load capacity (¢an) (AISC 1999). Figure 3.8 shows the

lateral load-displacement response from the monotonic fiber analysis. Under this
analysis, the column reaches it lateral load capacity of 120 kips at a lateral load
displacement of 3.3 in. The column lateral load resistance decreases due to the column
strain softening in compression (local buckling). With increasing lateral deformations,

the lateral load reaches a plateau of 96 kips. The same model was analyzed using the

48

same geometric properties and stress-strain curve but using the original fiber-based

element1 with C—type fibers in Drain2DX. As shown in Figure 3.8, the results were

identical, thus validating the results using the modified fiber-based element.

 

 

 

 

 

 

150-
A120 .............
m I
.9 I
=5 """"""" :"1
'u 90" I I
8 : .
_l | l
E 50' l I
9. i I
I
3 30- E I —ModiiedDra'n
E E ——-OrigindDrdn
o I 1 up“ I I I I t
0 1 2 3 4 5 6 7 8 9

Lateral Top Displacement (in)

Figure 3.8 Lateral load - monotonic analysis

10

Figure 3.9 shows the moment-curvature responses for the five segments of the

column. As shown in Figure 3.9, inelastic deformations occur in the segment at the base

of the column, where the plastic hinge forms, while the remaining segments unload

elastically.

 

' Since the stress-strain curve used for this analysis was asymmetric, to compare the results with the
unmodified version of Drain2DX, the material was analyzed as concrete (only in the unmodified

version).

49

 

 

    
 

 

 

 

1400
1200«
at”!
g 6003 Sagas v
g 400 ’ ., :2: ——SegO1 - $3902
in Segue a $9903 —x—SegO4
200 i} s°°°‘ ®___ ,. $9905
0 I r .
0 0.0005 0.001 0.0015 0.002
Curvature(radlln)

Figure 3.9 Moment-curvature analysis

The cyclic analysis was performed using the model shown in Figure 3.10. The
same geometry and material properties provided previously were used to perform the
analysis. The lateral load required, to yield the most extreme fiber in the cross-section,
was calculated by assuming a linear elastic behavior on the element. The following

equation was used:

V*L*c
0':

I (3)

 

= Yield stress, ksi
Lateral load, kips
= Length of the member, in

Distance from centroid to the extreme fiber
in the cross-section, in

where,

t-‘<Q
II

0
II

I = Moment of inertia, in4
A lateral load of 93.0 kips will cause initial yielding. Only 4.0 % of the calculated axial

load capacity was applied in compression. Since Drain2DX works under displacement

50

control, only lateral displacements need to be input. The lateral displacement at which

the fiber yields was calculated based on Equation (4).

 

V * L3
A’ = 3 * E * 1 (4)
where, Ay = displacement when yielding takes places, in
E = Young’s modulus, ksi

The cyclic loading history consisted of cycles at lateral displacements equal to Ay,

2A,, 3A,, 5Ay and 7A,, where Ay was equal to 2.0 inches.

 

Figure 3.10 Cyclic loading schematic

The results of the analysis are shown in Figures 3.11 and 3.12. The influence of

the unloading factor (0t) is included in both figures. The response is almost identical for

OL = 0 and 0.5. Meanwhile, for CL = 1.0, which represents the most extreme shooting

behavior, the response deviates from the other cases. The effects of stiffness degradation

due to local buckling on the behavior of the steel beam-column is apparent in Figures

51

3.11 and 3.12. Ct =1 has a tendency to make the element modeled in Drain2DX unstable.

A value of a < 0.9 provides a more stable response.

 

150

100

 

Lateral Load (Rips)
O

 

-150 . ------- Alpha=0.5 —Npha=0.0
— - - — - Alpha=1.0
-15 -10 -5 0 5 10 15

Laterd Deplacement (In)
Figure 3.11 Lateral load - lateral displacement response for steel column

 

 

 

 

20000

 

15%

§ 3
\‘a
I

o
I

 

Moment (It-In)

g

“.--
\‘
‘\

'3‘

45“» ------- Alpha=0.5 —Npha=0.0
—- - - - - Npha=1.0

 

 

-20000
0113 -0.(I)4 002 0 01112 0.(D4 0113
Curvature (radlln)
Figure 3.12 Moment-curvature response at failure segment for steel column

 

52

3.4 FIBER ANALYSIS OF CFT BEAM-COLUMNS

Fiber-based models were deve10ped for the high strength square CFl‘ beam-
column specimens tested by Varma et al. (2001). The analyses were conducted to
validate the modified fiber-based element for modeling CFT beam-columns, while
accounting for local buckling and biaxial stress effects.

The CFI‘ specimens tested by Varma et al. (2001) were square tubes with b/t
ratios of 32 or 48, made from conventional (A500 Gr. 46) or high strength (A500 Gr. 80)
steel and filled with high strength (16.0 ksi) concrete. Eight monotonic CFT beam-
column specimens were tested under constant axial load (20% or 40% of axial load
capacity) and monotonic flexural loading. Eight cyclic beam-column specimens were
tested under constant axial load (10% and 20% of axial load capacity) and cyclic flexural
loading. Varma et al. (2001) also developed fiber-based models for the monotonic and
cyclic beam-column specimens using Drain2DX. The steel and concrete fiber stress-
strain curves were derived from three-dimensional (3D) nonlinear finite element analysis
of the CFTs. However, they could not use the previously existing fiber-based beam-
column element in Drain2DX iirfly to conduct the fiber analyses. They were forced to
develop alternate analytical models using Drain2DX to analyze the specimens.

The modified fiber-based beam-column element developed in this research could
be used dir_ectly to analyze the high strength CFT specimens. Section 3.4.1 presents the
fiber analysis of the monotonic CFI‘ beam-column specimens and section 3.4.2 presents

the fiber analysis of the cyclic CFT beam-column specimens of Varma et al. (2001).

53

3.4.1 Monotonic analysis of CFT beam-columns

The monotonic CFT beam-column specimens were tested under constant axial
load (20% or 40% P0) and monotonic flexural loading (Varma et al. 2001). The
specimen test lengths (60 in.) were subjected to uniform primary bending moment. Each
specimen failed with the formation of a 12 in. long inelastic failure segment close to the
mid-length, while the remaining portion of the column unloaded elastically. The

experimental results were reported in the form of the fundamental moment-curvature (M-

0) response of the 12 in. failure segment.

(in I

 

12"

‘ I l._6.-_-
I. “12.. I

L“. «3‘

 

 

Figure 3.13 Schematic for monotonic analysis of CFI‘ beam-columns

b-Gt b

 

 

 

_- .—
T‘ W

3

LCD Shem-p Finn-QM Corn-r th® Steel w» ram ® Come run]

Figure 3.14 Fiber discretization of the slice (cross-section)

54

Figure 3.13 shows the fiber-based model for the monotonic beam-column
specimens. Figure 3.14 shows the fiber discretization of the slice (cross-section). Table
3.2 summarizes the 3D finite element model (FEM)-based steel and concrete fiber stress-
strain curve recommended by Vanna et al. (2001) for conducting the fiber analyses. The
fiber-based models were analyzed for the same loading condition as the test specimens,

i.e., constant axial load and monotonically increasing flexural loading. The moment-
curvature (M-(b) responses of the slices are shown in Figures 3.15 (a) to (h) for the eight

monotonic beam-column specimens. In Figures 3.15 (a) to (h), the nomenclature used to
identify the specimens (e.g., BC 32-80-20) consists of the specimen type (BC stands for
monotonic beam-columns), the nominal b/t ratio, the nominal yield stress in ksi, and the
nominal axial load level (P/Po). The experimental moment-curvature responses of the
corresponding specimens are also shown in Figures 3.15 (a) to (h). Additionally, the
moment-curvature response from the analyses conducted by Varma et al. (2001) using
the same stress-strain curves shown in Table 3.2 and mathematically identical analytical
models in Drain2DX are also shown in Figures 3.15 (a) to (h).

Figures 3.15 (a) to (h) clearly indicate that the results from the fiber analyses
using the modified fiber-based element follow closely the analytical results of Varma et
a1. (2001), thus validating the implementation of the modified fiber-based element.
Additionally, both analytical models predict the experimental results with reasonable
accuracy, which is due to the 3D FEM-based fiber stress-strain curves recommended by

Varrna et al. (2001).

55

Table 3.2 Stress-strain points defining the stress strain curves
for steel and concrete fibers (Varma et al. (2001))

 

 

 

 

CFT , Steel compression fibers Concrete
S , Steel tensnon .
pecImen fibers compressnon
I Flange & web Corner fibers
stress strain stress strain stress strain stress strain
0 0 0 0 0 0 0 0

87.0 0.003 83.1 0.0029 85.6 0.003 16.0 0.0027
b/t = 32

Gr. = 80 97.7 0.124 83.1 0.0033 85.6 0.0032 16.3 0.0033

55.0 0.0076 55.0 0.0076 1 1.5 0.01

 

95.8 0.0033 80.2 0.0028 86.3 0.003 16.0 0.0027
b/t = 48
Gr. = 80 107.0 0.12 80.2 0.0034 86.3 0.0036 16.5 0.0032

54.0 0.0076 54.0 0.006 9.4 0.01

 

39.0 0.0134 39.0 0.0013 39.0 0.0013 16.0 0.0027
b/t = 32

Gr. = 46 40.5 0.0525 40.5 0.0031 40.5 0.0031 16.0 0.003

62.0 0.2277 21.0 0.0057 21.0 0.0057 6.3 0.01

 

 

68.3 0.0024 61.0 0.0021 66.6 0.0023 16.0 0.0027
b]! = 48 77.5 0.148 61.0 0.0028 66.6 0.0029 16.5 0.0032
Gr. = 46
54.1 0.0039 39.0 0.0051 6.4 0.01

35.0 0.0063

 

 

 

 

 

 

 

 

 

56

 

      

 

 

 

 

  
 

 

 

 

 

 

 

 

 

7000
0000
3‘ 5000
a.
4000
3000
w -+Analysis- modified DrainZDx I
--Experiment
1W _Analysia - Vanna et al.
0 eH- — ._.e L L L _ __.
0 0000: 00000 W2 0.0010 QM
Gavan-MM)
(a) BC 48-80-40
m .. ._ _..
9000- ‘ ‘
“"2 («a
i m
m .
—O—Analysis. modified Drain2DX
no I
_. Experiment
um . _Analysia - Vanna et al.
OI - - . M . 7
o 0000: am m2 0mm 00m
mouth)
(c) BC 48-46-22
10000 I
000m
2 0000
E 00° —o——Analysiac modified DrainZDX
Experiment
m Analysis - Vanna et al.
0 00004 0.” N312 0.0010 0.0:!
am (radii)
(e) BC 32-80-40

 

     
   
      

- Analyaia— modified Dm’aZDX
Experiment
_Analysia - Vanna at al.

 

 

f v ye ,_

0 am am 00012 00010
W (rash)

(g) BC 32-46-40

 

 

Moment (II-In)

313383535

-—o—Analysis- modified minzox
- ‘--—— Experiment

_Analysis - Vanna et al.

 

 

I

 

00000 00012
W (radii)

BC 48-80-20

0

00004 QM

(b)

 

 

N

__._Analyaiae modified Drain2DX
_Analyaia - Vmaet al.

a x x .

WM)

Experiment

3
a

 

 

O
L
I

 

f

00000 0W2
m (M)

BC 48—46-20

0.11116 0.1!!

(d)

 

 

 

 

 

 

0 0.11134 00000 00012 0W6

 

0002
mold”
(f) BC 32-80-20
:"ifl 'W*‘“" «A»;

 

 

I Analysia- modified DrainZDX
*
Experiment

-‘| un-

 

 

 

‘m 1 _Analysis - Vanna et al.
0 l -. . . . .
0 am 00000 00312 0.0010 0.!!!
m (m)
(h) BC 32-46-20

Figure 3.15 Moment-curvature response of the failure segment
for CFT beam-columns

57

3.4.2 Cyclic Analysis of CFT beam-columns

The test-length (58.5 in.) of the cyclic beam-column specimens was fixed at the
bottom and subjected to a constant axial load (10% or 20% of P0) and cyclic lateral
loading at the top. Thus, the specimen test-length was subjected to a linearly varying
bending moment. The cyclic loading history is presented in detail by Varma et a1.
(2001). Each specimen failed with the formation of a 12 in. long inelastic failure
segment at the base of the column. The experimental results were reported in the form of
the lateral load-displacement response of the beam-columns and the moment-curvature

response of the 12 in. long inelastic failure segment at the base.

i.

I
115'

 

 

 

 

$5" 12"

 

 

 

 

1

I I
“FEW”

Figure 3.16 Schematic for CFT beam-columns

 

 

Figure 3.16 shows the fiber-based model for the cyclic beam-column specimens.

As shown in Figure 3.16 the fiber-based element was divided into five segments. The

second-order P-A effects were not included in the analysis because they were not

58

presented in the experiments. The fiber-based models were analyzed for the same
loading conditions as the test specimens, i.e., constant axial load and cyclically varying
lateral loading. The lateral loading history was similar to the experimental loading

history. Lateral loading cycles were conducted under displacement control in the

analyses at displacement levels of 0.25 Ay, 0.5 Ay, 0.75 A3,, 1.0 Ay, 1.5 Ay, 2.0 Ay, 3.0

Ay, 5.0 Ay, and 7.0 Ay, where Ay is the displacement at which the steel yields. Two
cycles were conducted at each displacement level. Table 3.3 shows the yield
displacement (Ay) used for each specimen. These values were also reported by Varma et

al. (2001) based on the experimental results.

Table 3.3 Yield displacements of CFT beam-columns

 

 

 

 

 

 

Specimen Yield Dis(1i)lll:;I;ment, Ay
CBC 32-80-20 0.57
CBC 32-80-10 0.60
CBC 48-80-20 0.46
CBC 48-80-10 0.55
CBC 32-46-20 0.39
CBC 32-46-10 0.32
CBC 48-46-20 0,41
CBC 4846-10 0.42

 

 

 

 

The lateral load-displacement responses and the moment-curvature responses of

the failure segments from the fiber analyses are shown in Figures 3.17 to 3.24 for the

59

eight cyclic beam—column specimen. In Figures 3.17 to 3.24, the specimen nomenclature
consists of the specimen type (CBC stands for cyclic beam column). The experimental
lateral load-displacement and moment-curvature responses of the corresponding
specimens are also shown in Figures 3.17 to 3.24. Additionally, Varma et al. (2001)
conducted fiber analyses of the failure segment 911}: using a complex and indirect but
mathematically accurate model in Drain2DX. The cyclic moment-curvature responses
from the fiber analyses of Varma et al. (2001) were also shown in Figures 3.17 to 3.24.
However, Varma et a1. (2001) could not obtain cyclic lateral load-displacement responses

using their model.

60

Lateral Load (kins)

Moment (k-in)

 

 

 

 

 

 

 

 

 

 

 

 

 

150
1
(a)
100
50 ,1
0 *——*———FM ail" .
-50 /,,,,
/ —— Ex riment
_II' 3‘35
ll ----- a — .
-100 . . _ _ _ _ (X ___ 0
(Y. = I n
-150 - - — -
-4 -3 -2 -1 0 1 2 3 4
Lateral Displacement (mch)
8000
030 48-80-20 (b)
6000
4000
2000
0 —————+— ____.
-2000
4000
, . ’ .1 _ Experiment
'6000 ‘J — Analysis-Varma et al.
. Modified Drain2DX
-0000
-0.006 -0.004 -0.002 0 0.002 0.004 0.006

Curvature (rad/in)

Figure 3.17 Fiber analysis of CFI‘ beam-column specimen CBC 48-80-20: (a)
lateral load-displacement; and (b) moment-curvature of the failure segment

61

150

 

(a)

100

 

UI
C

 

 

 

Lateral Load (kins)
O

a‘!\
I
2‘
\
\\
\\\

 

 

 

, L 1,5,7, _._. Experiment
..._........I “(Mal/7”] ----- 0t: 0. 5
-100 «,55/ _ . _ a = 0
-— or = 1.0
-150
-5 4 -3 -2 -1 0 I 2 3 4 5

Lateral Disnlacement (inch)

 

CBC 48-80-10 (b)

 

r‘“ "T.—

Moment (k-in)

 

 

~~ Experiment
_ Analysis-Vanna et al.
Modified Drain2DX

 

 

-3000
-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008
Curvature (rad/in)

Figure 3.18 Fiber analysis of CFT beam-column specimen CBC 48-80-10: (a)
lateral load-displacement; and (b) moment-curvature of the failure segment

 

62

Lateral Load (kins)

Moment (k-in)

 

 

 

 

 

 

 

 

 

 

  
 

 

 

 
  
 
 
   
  
 
  

   
   

    
 
 
 

 

  

150
(a)

100

so ............. £51! _

.34"
0 - I r .:Ji// / I
-50 .‘-.4.....
Experiment
100 q ..... or = 0.5
' _ . . .. on = 0
_ or = 1.0
-150
4 -3 -2 -1 0 I 2 3 4
Lateral Displacement (web)
6000
cac 43-46-20 , (b)
{I 1.-
@4151
’. ‘l—I :—
2000 'r/m'/' I.
o /l/
W 47 , I!”
2000 [/I// 44; 44I 4,,1/1
- .I‘I ,1; " Ill/l-(l/V/
'4‘ ,4. Experiment
4000 4'4 —‘ Analysis-Varma et al.
- - - Modified Drain2DX

-6000

 

 

-0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004 0.005

Curvature (radl'm)

Figure 3.19 ' Fiber analysis of CFT beam-column specimen CBC 48-46-20: (a)
lateral load-displacement; and (b) moment-curvature of the failure segment

63

é

 

 

 

 

 

 

   

 

 

 

 

 

 

    

 

 

(a)
75 l."
g 50 i If”; .‘
E 25
E 0 . .
M"
3
3 -25
_50 Experiment
----- on = 0.5
-75 " ' (1 = 0
—— (Y. = l n
-100
-5 -4 -3 -2 -l 0 l 2 3 4 5
Lateral Displacement (inch)
6000
CBC 4846-10 (b)
4000
A 2000
$
é
*5 0 ———
‘a’
e
2 -2000
Experiment
-4000 — Analysis-Vanna et al.
. Modified Drain2DX
-6000
-0.006 -0.004 -0.002 0 0.002 0.004 0.006

Curvature (radl'm)

Figure 3.20 Fiber analysis of CFI‘ beam-column specimen CBC 48-46-10: (a)
lateral load-displacement; and (b) moment-curvature of the failure segment

Lateral Load (kins)

Moment (k-in)

-100

-150

-200

 

§

fl
Ln
0

§

M
O

(a)

 

O

 

6n
:3

 

 

 

 

 

Lateral Displacement (inch)

10000

 

CBC 32-80-20

 

-8000

 

 

- — —~ Experiment
_ Analysis-Vanna et al.

(b)

   

Modified Drain2DX

 

 

-1 0000
-0.008 -0.006 -0.004 -0.002 0 0.002

Curvature (radl'm)

0.004 0.006

0.008

 

Figure 3.21 Fiber analysis of CFI‘ beam-column specimen CBC 32-80-20: (a)
lateral load-displacement; and (b) moment-curvature of the failure segment

65

 

 

 

 

 

 

 

 

200
(a)
150 .
100 If
33 , 4 I4
"'5 50 4’4 4
3 //
l
3 ° ‘ /_4 1 '
a .1
'4 -so , .;'I
'100 - Experiment
0:: 0.5
-150 . a: 0
a: 1.0
-200
-5 -4 -3 -2 -1 0 1 2 3 4 5
Lateral Displacement (inch)
10000
CBC 32-80-10 (b)
8000
6000
4000
E 2000
f2 0 +~— ”—
8 .
g -2000 .
2 -4000 I
I
"6000 i. - — Experiment
— Analysis-Vanna et al.
'3000 _ _ Modified Drain2DX
-10000 ,
-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008

 

 

 

  
 
 

 

 

 

Curvature (radl'm)

Figure 3.22 Fiber analysis of CFT beam-column specimen CBC 32-80-10: (a)
lateral load-displacement; and (b) moment-curvature of the failure segment

66

 

 

 

 

  
  

 

 

 

 

 

 

  

 

   

 

 

 

 

150 (a)
100
A
m
0° 4,._.—--——~ ._...-«-— “1
V g H ,4
E 50 - {'3 LIP/fl V ’ '4 44’; « 44
E 0 . ,
3 _,./
~50
Experiment
..... on = 0.5
-100 ~ a = 0
_ on = 1.0
-150
-5 4 -3 -2 -1 0 1 2 3 4 5
Lateral Displacement (inch)
8000
CBC 3246-20 (b)
6000 «
4000 - _fi ’
_____ 17
A 2000
cg ‘ _- 4 4‘
i I
’5 " “I . /l ’2’
*5 0 51:! 4; r —*’——#
v '4 .4/ , , .. I
E / I 1“ ,/, / (/
° -2000 , ‘4,— 3 .4 . ..
2 5;” .. . "
‘ " .. -._Experiment
4000 _Analysis-Varma et al.
_Modified Drain2DX
~6000

 

 

 

-0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004 0.005
Curvature (rad/in)

Figure 3.23 Fiber analysis of CFT beam-column specimen CBC 3246-20: (a)
lateral load-displacement; and (b) moment-curvature of the failure segment

67

150

 

 

(a)
100 ‘ M: f- 1 “My?
4 h '9 ‘44; ______ a
.1 Zola! ‘0' - z" ’ A”. “a:
, 4 ‘ '.' 1:23.: -. _— .— ' g
50 J. . I (I /

 
  

 

,/

I.

[I ./
if; 41"] i
/-‘ /’.‘ 4/
All, A/l 944/4";
JI'?'.".0.1.' ‘jlf/é'f‘gl/
I 5"-

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.4
’r

Lateral Load (kins)

 

 

 

 

 

 

 
 

 

 

 

-50
: Experiment
" * a=0.5
~100 "‘ _ _ - _ ' a = O
__ 0: = 1.0
-150
-4 -3 -2 -1 0 l 2 3 4
Lateral Displacement (inch)
6000
CBC 3246-10 (b)
4000
"E:
2000
4
'35 0
H
a
“a’
° -2000
E
Experiment
4000 —— Analysis-Vanna et al.
- - - Modified Drain2DX
-6000
-0.006 -0.004 -0.002 0 0.002 0.004 0.006

Curvature (rad/in)

Figure 3.24 Fiber analysis of CFT beam-column specimen CBC 3246-10: (a)
lateral load-displacement; and (b) moment-curvature of the failure segment

68

Figures 3.17 to 3.24 indicates that:

Cyclic analysis using the modified fiber-based element could be completed for the
full loading history for six specimens.

The cyclic loading history could not be completed for specimens CBC 48-80-20
and CBC 32-46-20 due to convergence problems.

The cyclic analyses using the modified fiber-based element were most stable

(converged and completed the loading history) for the case of 0!. equal to zero. It

was least stable for the case of 0!. equal to one.

The results from the analyses using the modified fiber-based beam—column
element compared well with the experimental results. For specimens CBC 48-80-
10, CBC 48-46-20, CBC 48-46-10, CBC 32—80-20, CBC 32-46-20 0!. equal to
zero provided the best comparison. For specimen CBC 48-80—20, CBC 32-80-10
and CBC 32-46-10 for 0!. equal to one provided the best comparison.

The moment-curvature response from the analyses using the modified fiber-based

element compared favorably with the analytical results from Varma et al. (2001).

They had used the steel fiber cyclic stress-strain behavior shown in Figure 1.4,
which corresponds closely to the case with on equal to one for the modified fiber-

based element.

It is important to note that the analytical model of Varma et al. (2001) was
different than the model shown in Figure 3.16. As a result, they used a slightly
different cyclic loading history, which accounts for the discrepancies between the

analytical moment-curvature responses shown in Figures 3.17 to 3.24.

69

3.5 SUMMARY

The implementation of the modified S-type steel stress-strain curve in Drain2DX
was verified using three analytical models with increasing levels of complexity; (1)
monotonic and cyclic axial load analyses of a two-fiber steel cross-section, (2) monotonic
and cyclic analysis of a steel column subjected to lateral loading, and (3) monotonic and
cyclic analysis of high strength CFT beam-columns tested by Varma et al. (2001).
Additionally, the results from the CFT fiber analysis were compared with the
experimental and analytical results of Varma et al. (2001).

The results from the fiber analyses of the axially loaded steel bar validated the
implementation of the S-type stress-strain curve in Drain2DX. The results from the fiber
analyses of the W14x82 steel beam-column subjected to constant axial load and
monotonic or cyclic lateral loading validated the implementation of the modified fiber-
based element in Drain2DX. The results from the fiber analyses of the CFT beam-
column specimens subjected to constant axial load and monotonic or cyclic flexural
loading validated the modified fiber-based element for modeling members with local
buckling and biaxial stress effects dominating the fundamental force-deformation

response.

70

3.6 RECOMMENDATIONS FOR USING THE MODIFIED FIBER-
BASED ELEMENT

Some recommendations for using the modified fiber-based element effectively are
as follows:
1. Use a reasonable stress-strain curve. The S-type stress-strain curve can be

asymmetric in tension and compression. However, the yield ratios (ratios
of ultimate stress to yield stress, (Tu/0“,) must not be too high. Usually,

this will not be a problem for steel materials.
2. The negative slope of the descending compression strain softening branch

must not be too large (steep). This can lead to convergence problems.

3. Choose the value of 0!. for each application/analytical model carefully. 0L
equal to zero corresponds to elastic-plastic behavior and (X equal to one

corresponds to shooting behavior as shown in Figure 2.2 (b). 0 equal to

zero was found to be more stable for the CFT beam-columns and for large
strain applications. However, 0L equal to one may be more appropriate for

situations were local buckling dominates.

71

CHAPTER 4. SUMMARY AND CONCLUSIONS

4.1 SUMMARY

The existing fiber-based beam-column element in Drain2DX has a major
drawback. It cannot be used to model the effects of local buckling and/or biaxial stresses
in the steel elements of the member cross-section. As a result, it cannot account for their
influence on the force-deformation behaviors of members. Therefore, a modified fiber-
based beam-column element with a modified S-type steel stress-strain curve was
developed in Drain2DX. The modified S-type steel stress-strain curve can be asymmetric
in tension and compression and can model tension and compression strain hardening,
compression strain softening, kinematic hardening under cyclic loading, and the effects
of local buckling and biaxial stresses on the hysteretic behavior.

Drain2DX uses an event-to-event based solution algorithm. Therefore, the multi—
linear S-type stress-strain curve was discretized into ten unique branches. The change in
stiffness associated with the fiber response moving from one branch to the other was
identified using unique event numbers. The response state of the fiber depends on its
current state, cyclic response history, magnitude and direction of the strain increment,
and the occurrence of events. The event calculation and state determination algorithms
were developed and implemented in SMEV15 and SMSDIS subroutines in Drain2DX.

The implementation of the modified fiber-based beam-column element in
Drain2DX was validated using three analytical models with increasing levels of

complexity. These are: (l) monotonic and cyclic axial load analyses of a simple two

72

fiber steel cross-section, (2) monotonic and cyclic lateral load analyses of a steel column,
and (3) monotonic and cyclic analyses of high strength CFT beam-columns tested by
Varma et al. (2001). The results from analyses of the axially loaded steel bar fiber
analyses validated the implementation of the S-type stress-strain curve in Drain2DX.
The results from the fiber analyses of the W14x82 steel beam-column subjected to
constant axial load and monotonic or cyclic lateral loading validated the implementation
of the modified fiber-based element in Drain2DX.

The modified fiber-based element was also used to model and analyze the
monotonic and cyclic high strength square CFT beam-column specimens tested by
Varma et al. (2001). These researchers had reported the specimen geometric and material
properties, the relevant experimental results, and the 3D FEM-based steel and concrete
fiber stress-strain curves to be used with the fiber-based models. The results from the
monotonic and cyclic fiber analyses compared well with the experimental and analytical
results reported by Varma et al. (2001). Some convergence problems were encountered

during the cyclic fiber analyses.

4.2 CONCLUSIONS

1. The modified fiber-based beam-column element developed in this research
can be used to model the effects of local buckling and/or biaxial stresses on

the behavior of the steel and steel-concrete composite members.

73

The modified fiber-based element can model the effects of local buckling

and/or biaxial stresses on the hysteretic behavior through the use of parameter

(X.

0 06 equal to zero corresponds to elastic-plastic behavior upon strain
reversal, which is appropriate for the case where biaxial stresses dominate.

0 0L equal to one corresponds to the shooting behavior upon strain reversal,
which is appropriate for the case where local buckling dominates.

o A value of 0!. between zero and one models both effects.

Cyclic analyses using the modified fiber-based element are most stable for the

case of on equal to zero. They are least stable for the case of on equal to one

combined with large strains.

Some convergence problems may occur while using this element if:
0 The yield ratios (ratio of ultimate stress to yield stress, O'u/O'y) for the S-

type stress-strain curve are too high.
0 The negative slope of the descending compression strain softening branch

is too large (steep).
o For the case of 0!. equal to one and large strains, because the shooting

slope (stiffness) may be too small.

74

APPENDIX A
MODIFIED SUBROUTINES IN DRAIN2DX

75

APPENDIX A-l
STEEL MATERIAL FIBER PROPERTIES

C fittttitfiiitfiiitttittiitttttttiittiifiitifitttfitttitfitfitthitttttiiittiiit
c INFPR15.H

c 30 FIBER FLEXIBILITY BEAM COLUMN ELEMENT

c PROPERTY COMMON BLOCKS

c GEPROP = properties for a geometry type

c FSPROP = properties for a fiber section type

c ESPROP = properties for an elastic section type

c CMPROP = properties for a concrete material type

c SMPROP = properties for a steel material type

c CHPROP = properties for a fiber hinge type

c BMPROP = properties for a pullout fiber type

c GMPROP : properties for a gap fiber type

c ittttttfitiitfiihi0....iDI...fittiittiiltittfifiitiitttifitiiiiittttiiiifitt.
C .fiiflfififlti...IiiiitfitittitlfittttiIififiifiitttfiitfiOtfittitfittiifittitiitiiii
c SMPROP

C ---------------------------------------- STEEL MATERIAL FIBER PROPERTIES

c--Series model.

c—-Set up when steel properties are read.

c--Reca11 whenever a steel fiber is processed.

c--Data does not change.

c--OVERALL DATA (lprsm=5)
real'4 smot ! overshoot tolerance
real'4 smel ! fully yielded modulus (very small)
real'd smrf ! reloading factor
integer'Z lppksm ! length of fiber property packet
integer'z dure ! Dummy
integer‘A kppksm ! pointer to first fiber property packet
integer‘4 ltotsm ! total state data length for this material

ctttttiitNOT USEDtttitiiittt
C--SUBFIBER PACKET (lppksm=3)

c real'4 smsy ! subfiber yield stress

c real'4 3mm ! subfiber elastic modulus

c real'd smey ! subfiber yield strain

c real’4 smeu ! subfiber ultimate strain

c real'4 syrem ! total elastic modulus of all remaining subfibers
ctititfittitflfittttfitttttttttifl

c——FIBER PACKET (1ppksm=10)
real‘4 smcom ! stress-strain matrix - Comp.
real'4 smten ! stress-strain matrix - Tension
C--NOT CURRENTLY SAVED
integer‘Z iduml ! dummy
integer*2 nsmcom ! no. of points on stress-strain curve (Comp.)
integer*2 nsmten ! no. of points on stress-strain curve (tension)

common IsmprIS/ smcom(2,3). smten(2,2) ! current fiber props.
l .smot, smel, smrf, lppksm, kppksm ,ltotsm

2 .nsmcom, nsmten ! overall fiber properties

3 .iduml, dure ! input stress-strain curve
real'4 prsm(6) ! array to transfer basic data
equivalence (prsm,smot)

real'4 ppksm(10) ! array to transfer subfiber packet

equivalence (ppksm,smcom)
C iti...Cifitiitiififififitiiitttttiitiff.tttififi*fiiiiiiififiitfiitfittiiititt...‘

76

APPENDIX A-2
STEEL MATERIAL FIBER STATE

C 't'ttiifiiiitflttifiitiitO...Q00......tiififififitififififiiiiiflifitfit...0.....00

C INFST15.H
C 30 FIBER FLEXIBILITY BEAM COLUMN ELEMENT
c STATE COMMON BLOCKS

FSSTAT a state for a fiber section

ESSTAT = state for an elastic section
CMSTAT = state for a concrete material fiber
SMSTAT = state for a steel material fiber
CHSTAT = state for a fiber hinge

BMSTAT = state for a pullout fiber

GMSTAT = state for a gap fiber

NVSTAT = envelope values

OOQiiiififitflii...fiiii.ififititittifiififitittiiiiifiti'itiItiflfiiiiiiiifi.if...

000000000

c tittiifififiiOIOOOtiifitiifiififiit...fittifi.0itIiii.0"....ifififiiifitttfiiiitfifit

C SMSTAT

c --------------------------------------------- STEEL MATERIAL FIBER STATE
c——Initialized when element is created.

c——Recall whenever a steel fiber is processed.

c--Data changes in each state determination.

c--OVERALL STATE (lstsm=2)
integer'4 kspksm ! pointer to fiber state packet
integer'Z lspksm ! length of fiber state packet
integer'z iismdm ! dummy

c—-FIBER PACKET (lspksm=12)

c real'4 sms ! subfiber stress

c integer'4 iysm ! fiber branch number
c ! O = elastic

c l 1 = yielded positive
c !-1 = yielded negative

current stress

current strain

maximum strain

minimum strain

branch number

last tension branch

last compression branch
compression stress at unloading
tension stress at reloading
current modulus

dummy

rea1'4 ssig
rea1'4 setot
rea1'4 semax
real'4 semin
integer'4 sbrnum
integer'4 slastt
integer'd slastc
rea1'4 ssc
rea1‘4 sst
real'4 smod
integer'A piti
real’4 yab

c--COMMON BLOCK
common /smst15/ ssig,setot.semax,semin,sbrnum,slastt,slastc

1 ,ssc,sst,smod,piti.yab ! current fiber state
2 ,kspksm,lspksm.iismdm ! overall material state
real'4 stsm(2) ! array to transfer overall state

equivalence (stsm,kspksm)

real‘d spksm(12) ! array to transfer fiber packet

equivalence (spksm.ssrg)
C iiiiiitttCIQQI'ifiiititfiiiiiiititt'iiiitififiittiiififiiiittiiiifittiiti...Q

77

APPENDIX A-3
Steel material data input

tt'.itit...ititfiiittiwfiiifitfifiififitfifitittfiitO.QOOOOOOOOOOOOOOGOIOOOQOQit

c
SUBROUTINE SMINlS(ntyp,smot,sme1,nsmcom.nsmten,smcom.smten,MXSMC
l ,MXSMT,kdata,smrf)
C ---------------------------------------------------------------------
c DRAIN FIBER ELEMENTS
c Steel material data input. Parallel model.
c _____________________________________________________________________
c DOUBLE PRECISION / LARGE
include 'double.h'
C ______________________________________________________________________
c CALLED FROM : inellS
c FILE I/O : read data from inp.
c write data to iou.
C ——————————————————————————————————————————————————————————————————————
c ARGUMENTS
c INPUT:
integer'd ntyp ! type no.
c OUTPUT:
rea1'4 smot ! overshoot tolerance
real‘4 smel ! modulus of elastic subfiber (very small)
integer'2 nsmcom ! no. of points on compression stress-strain curve
integer'Z nsmten ! no. of points on tension stress-strain curve
rea1'4 smcom(2,MXSMC) ! Compression stress-strain points
real‘d smten(2,MXSMT) ! Tension stress—strain points
real'4 smrf ! steel material reloading factor
C MODIFY:
integer'd kdata ! error counter. add 1 for each error
c ——————————————————————————————————————————————————————————————————————
c LABELLED COMMONS
include 'cline.h'
include 'ptop.h'
common/tapes/inp,iou
c ----------------------------------------------------------------------
c LOCAL VARIABLES
real'4 small ' small number
real'4 sig ! stress
real‘4 eps ! strain
rea1'4 ymd ! modulus
real‘4 yyp ' previous modulus
real'4 yye ' initial elastic modulus
real'4 ss ' stress
real'd ee ' strain
real'd yy ! modulus
C —————————————————————————————————————————————————————————————————————
c ------------------------------------------------------------- CONSTANTS
small=1.e-6
C --------------------------------------------------------------- READING
if(ntyp.eq.1) write(iou,10)
10 format (/' Steel Material Types'//
1 ' Type’,2x,'Comprn‘,2x,' Tensn'.
2 2x,‘ Unloading',2x,' Overshoot',
3 2x,‘ Stress',2x,' Strain',2x,'Stress',
4 2x.’ Tangent'I
S ' No.',2x.'Points',2x,'Points',
6 2x,‘ Factor ',2x,' Tolerance',
7 2x,’ Value'.2x.' Value',2x,' Type',
8 2x,‘ Modulus’)
C ------------------------------------------- READ AND ECHO CONTROL VALUES
call getlin
read(xx1ine.'(2i5,2f10.0)') nsmcom.nsmten,smrf.smot
write(iou.'(i8.2i8,1pe12.3,1pe12.3)') ntyp.nsmcom,nsmten,smrf.smot
if(nsmcom.gt.MXSMC .or. nsmten.gt.MXSMT) then
write(iou,*) ' "*ERROR - too many points'
kdata=kdata+1
end if
if(smot.lt.small) then
write(iou.') ' "‘ERROR - overshoot tol too small'
kdata=kdata+1
end if
smel=small
c --------------------------------------- READ AND ECHO COMPRESSION VALUES
ss=0
ee=0
yyp=l.e30
do 21 i=l.nsmcom
c--read

call getlin

78

read(xxline,'(2f10.0)') sig,eps
if(sig.1e.0. .or. eps.le.0.) then
write(iou,*) ' "*ERROR - values must be > 0'
kdata=kdata+l
sig=abs(sig)+l.e-1O
eps=abs(eps)+l.e-20

end if
if(eps.le.ee+sma11) then
write(iou,') ' "*ERROR - strains must increase'

ee=eps—small
kdata=kdata+l
end if
c--tangent modulus
smcom(1.i)=sig
smcom(2,i)=eps
yy=(smcom(1,i)-ss)/(smcom(2.i)-ee)
c-—e1astic modulus
if(i.eq.1) yye=yy
c--echo
write(iou,2000) sig,eps,' comprn',yy
2000 format(50x,1p2e12.4,a8,lpe12.4)
if(yy.ge.yyp) then
write(iou,*) ' n*ERROR - modulus must decrease'
kdata=kdata+1
end if
YYD=YY
if(yy.1t.0. .and. abs(yy).ge.0.5*yye) then
write(iou,*) ' "'WARNING - large negative modulus'
end if
c--save
ss=sig
ee=eps
21 continue
c ------------------------------------- READ AND ECHO TENSION VALUES
ss=0.
ee=0.
yy=l.e30
do 22 i=1,nsmten
c-—read
call getlin
read(xxline,'(2f10.0)') sig.eps
if(sig.le.0. .or. eps.le.0.) then
write(iou.*) ' **'WARNING - values must be > 0'
sig=abs(sig)+1.e—10
eps=abs(eps)+1.e-20
end if
smten(1,i)=sig
smten(2,i)=eps
if(sig.le.ss+small) then
write(iou.') '*"ERROR — stresses must increase’
kdata=kdata+l
end if
if(eps.le.ee+small) then
write(iou.') "*‘ERROR - strains must increase'
kdata=kdata+1
ee=ee~sma11
end if
c--tangent modulus
ymd=(smten(l.i)-ss)/(smten(2,i)-ee)
c--echo
write(iou,2000) sig,eps,' tension',ymd
if(ymd.gt.yy—sma11) then
write(iou.') '*"ERROR — modulus must decrease'
kdata=kdata+l
end if
ss=sig
ee=eps
yy=ymd
c--save
22 continue
C ——————————————————————————————————————————————————————————————————————

END

C tifiitfifittifiiifitti...D...ttfiiitiiiiitittfiiitittfittiitfifiitfiififittfiitittit

79

C

c --declaration of variables
syc=smcom(1.1)
sbc=smcom(l.2)

APPENDIX A-4
Steel material event determination

ititiififiitiititiiifltI'ttitfiiiiti.it.fitt‘fififiiiiiififi.fiftiwiiiifiifiiiiiifii

SUBROUTINE SMEV15(smcom, smten, deps, fact, ifact, stol, sig,
1 etot. emax. emin, brnum, lastc, lastt. ssc,
2 sst, smrf, piti)

MODIFIED BY THE MSU CFT RESEACH TEAM FOR STEEL MODEL. 02/20/02

DRAIN-30X FIBER ELEMENTS
DOUBLE PRECISION / LARGE
include 'double.h'

CALLED FROM slevlS
INPUT:

real‘d smcom(2.3) ! stress-strain compressive matrix

real'4 smten(2,2) ! stress-strain tensile matrix

real‘8 deps ! strain increment

real'8 stol ! stress overshoot tolerance

real’4 sig ! current stress

real‘4 etot ! current stress

rea1'4 emin ! minimum strain

real‘4 emax ! maximum strain

real‘d epsu ! strain at zero stress

real'4 ds ! stress increment

integer'd brnum ! branch number

'-1 = strain hardening branch
' 0 = elastic branch
' 1 = inelastic branch
' 2 = intermediate branch
! 3 = unloading/reloading branch - Tension
' 4 = unloading/reloading branch - Compression
' S = shooting branch
' 6 = stress unloading platteu branch
I 7 = yield platteu branch
! 8 = buckled branch
integer'4 lastc ! last branch number (compression)
integer‘A lastt ! last branch number (tension)
MODIFY
rea1'8 fact ! event factor
integer'4 ifact ! event type
' 1 = compression yielding
' 2 = compression buckling
' 3 = compression post-buckling
' 4 = tension yielding
' 5 = tension ultimate
' 6 = unloading from compression envelope
' 7 = unloading from tension envelope
' 8 = shooting from tension to compression
' 9 = loading onto compression envelope
' 0 = loading onto tension envelope
LOCAL VARIABLES

real'4 ssc E stress at unloading

real'4 sst ! stress at reloading

real‘4 ezsig ! strain at zero stress

real'4 syc ! yield stress (compression)

real'4 sbc ! buckling stress (compression)

real'd suc ! ultimate compressive stress

rea1*4 syt ! yield stress (tension)

rea1'4 sut ! ultimate tensile stress

real'4 eyc ! strain at yielding in compression

real'd delsteel ! difference in strains between suc and sbc

real‘d yini ! fiber initial modulus

real'4 yinel ! fiber inelastic modulus

real‘4 yint ! fiber intermediate modulus

rea1'4 ysh ! fiber strain-hardening modulus

real'4 smrf ! reloading factor

real'd sl !

rea1'4 yab ! shooting modulus

real*4 ee ! difference in strain at branch #6

real'd star ! target stress

save yab

80

suc=smcom(l,3)

syt=smten(l,l)

sut=smten(l,2)

epsu=smcom(2,3)

eyc=smcom(2,l)

yini=sycleyc

delsteel=smcom(2,2)-eyc
yinel=(sbc~syc)/delsteel
yint=(suc-sbc)/(epsu-smcom(2.2))
ysh=(sut-syt)/(smten(2,2)-smten(2.1))

C ---------------------------------------- START EVENT FACTOR CALCULATION
C --------------------------------- START NEGATIVE DISPLACEMENT INCREMENT
if(deps.lt.0.) then
C --------------------------------------------------- START ELASTIC FIBER
if(brnum.eq.0) then
c --Branch 0- Fifure 2.5

ds=yini'deps
if(-syc—stol—sig.ge.ds) then
fact=(-syc-stol-sig)/ds

ifact=1
end if
C -------------------------------------------------- END ELASTIC FIBER
C —————————————————————————————————————————————— START INELASTIC FIBER
else if(brnum.eq.l) then
c --Branch 1- Figure 2.6

if(lastt.eq.4) then
if(emax-(syc+syt)lyini-delsteel-etot-stol/yine1.ge.de)then
fact=(emax-(syc+syt)/yini-delsteel—etot—stol/yinel)/de
ifact=2
end if
else
if(sbc.ne.syc) then
ds=yinel'deps
if(-sbc-stol-sig.ge.ds) then
fact=(-sbc-stol-sig)/ds

ifact=2
end if
else
if(emax-(syc+syt)lyini-delsteel-etot
1 -stol/yinel.ge.de) then
fact=(emax-(syc+syt)/yini-delsteel
2 -etot-stol/yinel)/de
ifact=2
end if
end if
end if
piti=316

C ---------------------------------

else if(brnum.eq.2) then
c --Branch 2- Figure 2.8
ds=yint'deps
if(-suc+stol-sig.le.ds) then
fact=(-suc+stol-sig)/ds
ifact=3
end if
c _________________________________

else if(brnum.eq.3) then
c --Branch 3— Figure 2.9
ds=yini'deps
if(lastc.ne.5 .and. 1astc.ne.6) then
if(ssc-sig-stol.ge.ds) then
fact=(ssc-sig—stol)/(ds)
ifact=9
end if
else
if(lastc.eq.5) then
star=(sig+yini*(ezsig-etot))/(l—yini/yab)
else
star=ssc
end if
if (sig.ge.0) then
if(star-sig—stol.ge.ds) then
fact=(star-sig-stol)/ds
ifact=9
end if
else

81

c -------

if(star-sig-stol.ge.ds) then
fact=(star-sig-stol)/ds
ifact=9
end if
end if
end if

-—---------------——--————-

else if(brnum.eq.-1) then

c —-Branch —1 -

C -------

ds=yini*deps

if(-stol.ge.ds) then
fact=-stol/ds
ifact=7

end if

else if(brnum.eq.4) then

c ——Branch 4—

C _______

ds=yini'deps

Figure 2.11

Figure 2.12

if(lastc.eq.l .and.(1astt.eq.-1 .or.lastt.eq.7)) then

if(sst-syt—syc-stol-sig.ge.ds) then
fact:(sst-syt—syc-stol-sig)Ids

ifact=1
end if

else if((lastc.eq.2 .and. (lastt.eq.-1

.or. lastt.eq.7))

.or. (lastc.eq.8 .and. (lastt.eq.-l .or. 1astt.eq.7))) then

ezsig=etot~siglyini

if(ezsig-etot.ge.deps) then
fact=(ezsig-etot)/deps
ifact=8

end if

else if((lastt.eq.-l .or. lastt.eq.7).and.piti.ne.316) then

if(sst-syt-syc-stol—sig.ge.ds) then
fact:(sst-syt—syc-stol-sig)/ds

ifact=1
end if
end if

else if(brnum.eq.5) then

C -~Branch 5-

C ———————

sl=ssc/(emin-ezsig)

yab=smrf'sl+(1-smrf)'yini

ds=yab'deps

if(ssc-sig-stol.ge.ds) then
fact=(ssc-sig—stoll/ds
ifact=8

end if

else if(brnum.eq.6) then

c --Branch 6-

c _______

ee=emin-ezsig-ssc/yab
sl=ssc/(emin—ezsig)
yab=smrf'sl+(l-smrf)‘yini
if(emin-etotestollyab.ge.deps) then
fact:(emin-etot-stol/yab)Ideps
else if(ee.eq.0) then
fact=0
end if
ifact=9

——--———-------———-—_—_----

else if(brnum.eq.7) then

c --Branch 7-

C _______

ds=yini‘deps

if(-stol.ge.ds) then
facts-stol/ds
ifact=7

end if

else if(brnum.eq.8) then

c -—Branch 8—

Figure 2.14

Figure 2.15

Figure 2.17

Figure 2.17

c __ no event
C ------------------------------------------------ END INELASTIC FIBER
end if
C ----------------------------------- END NEGATIVE DISPLACEMENT INCREMENT
end if

82

C --------------------------------- START POSITIVE DISPLACEMENT INCREMENT
if (deps.gt.0.) then
C --------------------------------------------------- START ELASTIC FIBER
if(brnum.eq.0) then
c -- Branch 0- Figure 2.18
ds=yini'deps
if(syt+stol—sig.le.ds) then
fact=(syt+stol-sig)/ds

ifact=4
end if
C -------------------------------------------------- END ELASTIC FIBER
C ---------------------------------------------- START INELASTIC FIBER
else if(brnum.eq.1) then
c --Branch 1— Figure 2.20

ds=yini'deps
if(stol.le.ds) then
fact=stol/ds
ifact=6
end if
c —————————————————————————————————

else if(brnum.eq.2) then
c -—Branch 2- Figure 2.20
ds=yini'deps
if(stol.le.ds) then
fact=stol/ds
ifact=6
end if
c .................................

else if(brnum.eq.3) then
c --Branch 3— Figure 2.21
ds=yini*deps
if(lastc.eq.l) then
if(ssc+syt+syc+stol-sig.le.ds) then
fact:(ssc+syt+syc+stol—sig)/(ds)

ifact=4
end if
else if((lastc.eq.2 .or. lastc.eq.8).or.(lastc.eq.5
1 .or. 1astc.eq.6)) then

if(—sbc+syt+syc+stol—sig.le.ds) then
fact=(-sbc+syt+syc+stol-sig)/(ds)
ifact=4
end if
end if
C —————————————————————————————————

else if(brnum.eq.-l) then
c --Branch -1 - Figure 2.19
ds=ysh'deps
if(sut—sig+stol.le.ds) then
fact=(sut—sig+stol)/ds
ifact=5
end if
c —————————————————————————————————

else if(brnum.eq.4) then
c --Branch 4- Figure 2.22
ds=yini*deps
if(lastt.eq.-1) then
if(sst+stol-sig.le.ds) then
fact=(sst+stol—sig)/ds
ifact=0
end if
else if(lastt.eq.7) then
if(sut-sig+stol.le.ds) then
fact=(sut~sig+stol)/ds
ifact=0
end if
end if
c _________________________________

else if(brnum.eq.5) then
c —-Branch 5- Figure 2.24
ds=yini'deps
if(stol.le.ds) then
fact=stol/ds
ifact=6
end if

83

c ---------------------------------

else if(brnum.eq.6) then

c ——Branch 6- Figure 2.24
ds=yini'deps
if(stol.le.ds) then
fact=stol/ds
ifact=6
end if
C .................................

else if(brnum.eq.7) then

C --Branch 7- Figure 2.24
c _l no event
C .................................

else if(brnum.eq.8) then

c —-Branch 8— Figure 2.20
ds=yini*deps
if(stol.le.ds) then
fact=stol/ds
ifact=6
end if
C ------------------------------------------------ END INELASTIC FIBER
end if
C ----------------------------------- END POSITIVE DISPLACEMENT INCREMENT
end if
C -----------------------------------------------------------------------
RETURN
END

c tit....fiiiitttififiiififiiifiiifiitiiiiiittiiitiiitifififiifiifittfiiitifiiti00....

84

C

C

C LOCAL VARIABLES

real'd
real'd
real'd
real'4
real'4
rea1'4
real'd
real'4
real'4
real'd
rea1'4
real'4
real'4
real'4
real'4
rea1'4
real'4
real'4
rea1‘4
rea1'4
real'd
real‘4

ssc
sst
ezsig
syc
sbc
suc
syt
sut
eyc .
delstee
yini
yinel
yint
ysh
smrf

sl

yab

de

ee

star
piti
smod

-o-o-nco-o-o-no-v-

o-c—o..—n-c-o._o-..o--H

save yab

APPENDIX A-S
Steel material State determination

i...flit...tttttiittttttiifiiiiiiitiiiiittthit.fittiitfiiiiitittiitttttiit
SUBROUTINE SMSD15(smcom, smten, deps, fact, ifact, stol, sig,
1 etot, emax, emin, brnum, lastc, lastt, ssc,
2 sst, smod, smrf, piti)
MODIFIED BY THE MSU CFT RESEACH TEAM FOR STEEL MODEL, 02/20/02
DRAIN-3DX FIBER ELEMENTS
DOUBLE PRECISION I LARGE
include 'double.h'
CALLED FROM slevlS
INPUT:
real‘d smcom(2,3) ! stress-strain compressive matrix
real'4 smten(2,2) ! stress—strain tensile matrix
real'B deps I strain increment
real'4 sig ! current stress
real'4 etot ! current strain
real'4 smod ! current modulus
rea1'4 emin ! minimum strain (Compression)
real’4 emax ! maximum strain (Tension)
real‘4 epsu ! strain at zero stress
real'4 ds ! stress increment
integer'4 brnum ! branch number
!-1 = strain hardening branch
! O = elastic branch
! l = inelastic branch
! 2 = intermediate branch
! 3 = unloading/reloading branch - Tension
! 4 = unloading/reloading branch - Compression
! S = shooting branch
! 6 = stress unloading platteu branch
! 7 = yield platteu branch
! 8 = buckled branch
integer‘d 1astc ! last branch number (compression)
integer'4 lastt ! last branch number (tension)
MODIFY
real'8 fact 1 event factor

stress at unloading

stress at reloading

strain at zero stress

yield stress (compression)

buckling stress (compression)

ultimate compressive stress

yield stress (tension)

ultimate tensile stress

strain at yielding in compression
! difference in strains between suc and sbc

fiber initial modulus

fiber inelastic modulus

fiber intermediate modulus

fiber strain-hardening modulus

reloading factor

shooting modulus

strain increment

difference in strains at branch #6
target stress

Dummy

current moduli

_——---—_~w———_———u~--———————-_-————-——-—-_-——~o—-——_——-———-__.._-.-—-——--n--

c ——declaration of variables
syc=smcom(l,1)
sbc=smcom(1,2)
suc=smcom(l,3)
syt=smten(l,1)
sut=smten(1,2)
epsu=smcom(2,3)
eyc=smcom(2.1)
yini=sycleyc

85

delsteelzsmcom(2,2)-eyc
yinel=(sbc-syC)/delstee1
yint=(suC-sbc)/(epsu-smcom(2,2))
ysh=(sut~syt)/(smten(2.2)-smten(2,1))

C ---------------------------------------- START EVENT FACTOR CALCULATION
C --------------------------------- START NEGATIVE DISPLACEMENT INCREMENT
fact=1
de=deps
if(de.lt.0.) then
C --------------------------------------------------- START ELASTIC FIBER
C -- Branch 0- Figure 2.5

if(brnum.eq.0) then
ds=yini'de
if(-syc—sig.ge.ds) then
fact=(-syc-sig)/ds
sig=-syc
etot=etot+fact‘de
de=(1-fact)'de

brnum=1
smod=yinel
else

sig=sig+ds
etot=etot+fact*de

de=0
smod=yini
end if
c -------------------------------------------------- END ELASTIC FIBER
C ---------------------------------------------- START INELASTIC FIBER
else if(brnum.eq.l) then
C --Branch 1- Figure 2.6

if(lastt.eq.4) then
if(emax-(syc+syt)lyini—delsteel-etot.ge.de) then
fact=(emax-(syc+syt)/yini-delsteel-etot)/de
sig=sst-syc-syt-delsteel*yinel
etot=etot+fact'de
de=(1-fact)'de
brnum=2
smod=yint
else
sig=sig+de'yine1
etot=etot+fact*de
emin=etot
de=0
smod=yinel
end if
else
if(sbc.ne.syc) then
ds=yine1'de
if(-sbc-sig.ge.ds) then
fact=(-sbc—sig)/(ds)
sig=-sbc
etot=etot+fact'de
de=(1-fact)‘de
brnum=2
smod=yint
else
sig=sig+ds
etot=etot+fact‘de
emin=etot
de=0
smod=yinel
end if
else
if(emax-(syc+syt)lyini—delsteel-etot.ge.de) then
fact=(emax-(syc+syt)/yini-delsteel-etot)/de
sig=-sbc
etot=etot+fact'de
de=(l—fact)'de

brnum=2
smod=yint
else

sig=sig+de*yini
etot=etot+fact'de
emin=etot
de=0
smod=yinel
end if
end if

86

end if
piti=3l6
c _________________________________

else if(brnum.eq.2) then
C --Branch 2- Figure 2.8
ds=yint'de
if(-suC-sig.1e.ds) then
fact=(-suc-sig)/(ds)
sig=~suc
etot=etot+fact*de
de=(1-fact)'de
brnum=8
smod=0
else
sig=sig+ds
etot=etot+fact‘de
emin=etot
de=0
smod=yint
end if
c _________________________________

else if(brnum.eq.3) then
C —-Branch 3- Figure 2.9
ds=yini'de
if(lastc.ne.5 .and. 1astc.ne.6) then
if(ssc—sig.ge.ds) then
fact=(ssc-sig)/(ds)
sig=ssc
etot=etot+fact'de
de=(1-fact)*de
if(ssc.eq.-suc) then
brnum=8
smod=0
else if(lastc.eq.1) then
brnum=1
smod=yinel
else if(lastc.eq.2) then
brnum=2
smod=yint
end if
else
sig=sig+ds
etot=etot+fact'de
de=0
smod=yini
end if
else
if(lastc.eq.5) then
star=(sig+yini'(ezsig—etot))/(1-yini/yab)
else
star=ssc
end if
if (sig.ge.0) then
if(star-sig.ge.ds) then
fact=(star-sig)/ds
sig=star
etot=etot+fact'de
de=(1-fact)'de
if(lastc.eq.5) then
brnum=5
smod=yab
else
brnum=6
smod=0
end if
else
sig=sig+ds
etot=etot+fact'de
de=0
smod=yini
end if
else
if(star-sig.ge.ds) then
fact=(star-sig)/ds
sig=star
etot=etot+fact*de
de=(1-fact)‘de
if(lastc.eq.5) then
brnum=5

87

smod=yab
else
brnum=6
smod=0
end if
else
sig=sig+ds
etot=etot+fact'de
de=0
smod=yini
end if
end if
end if

c _________________________________

else if(brnum.eq.-1) then

c —-Branch -1 - Figure 2.11
C etot=etot+fact*de

sst=sig

emax=etot

brnum=4

smod=yini

lastt=-1
C _________________________________

else if(brnum.eq.4) then
C -—BranCh 4— Figure 2.12
ds=yini*de
if(lastc.eq.1 .and. (lastt.eq.-1 .or. 1astt.eq.7)) then
if(sst-syt-syC-sig.ge.ds) then
fact=(sst-syt—syc-sig)/ds
sig=sst-syt—syc
etot=etot+fact*de
de=(l-fact)‘de
brnum=1
smod=yine1
lastt=4
else
sig=sig+ds
etot=etot+fact*de
de=0
smod=yini
end if
else if((lastc.eq.2 .and.(lastt.eq.-l .or. lastt.eq.7)) .or.
1 (lastc.eq.8 .and. (lastt.eq.-l .or. lastt.eq.7))) then
ezsig=etot-sig/yini

if(ezsig-etot.ge.de) then
fact=(ezsig-etot)/de
etot=ezsig
de=(l-fact)*de
sig=0
brnum=5
smod=yab
else
sig=sig+ds
etot=etot+fact*de
de=0
smod=yini
end if
else if((lastt.eq.—1 .or. 1astt.eq.7).and. piti.ne.316) then
if(sst-syt-syC-sig.ge.ds) then
fact=(sst-syt-syC-sig)Ids
sig=sst-syt-syc
etot=etot+fact*de
de=(1-fact)*de
brnum=1
lastt=4
smod=yinel
else
sig=sig+ds
etot=etot+fact*de
de=0
smod=yini
end if
end if
C _________________________________
else if(brnum.eq.5) then
C --Branch 5- Figure 2.14
sl=ssc/(emin—ezsig)

88

yab=smrf'sl+(l—smrf)*yini
ds=yab'de
if(ssc-sig.ge.ds) then
fact=(ssc~sig)/ds
sig=ssc
etotzetot+fact*de
de=(l-fact)*de
brnum=6
smod=0
else
sig=sig+ds
etot=etot+fact‘de

de=0

smod=yab

lastc=5
end if

c —————————————————————————————————

else if(brnum.eq.6) then
c —-Branch 6- Figure 2.15
sl=ssc/(emin—ezsig)
yab=smrf*sl+(1-smrf)'yini
ee=emin-ezsig—ssc/yab
if(emin-etot.ge.de) then
fact=(emin-etot)/de
etot=etot+£act‘de
de=(1-fact)*de
if(ssc.ne.suc) then
sig=ssc
brnum=2
smod=yint
else
sig=suc
brnum=8
smod=0
end if
else if(ee.eq.0) then
fact=0
etot=etot+fact*de
de=(l-fact)'de
if(ssc.ne.suc) then

sig=ssc
brnum=2
smod=yint
else
sig=suc
brnum=8
smod=0
end if
else
etot=etot+fact*de
de=0
smod=0
lastc=6
end if
c —————————————————————————————————
else if(brnum.eq.7) then
C --Branch 7- Figure 2.17
emax=etot
sst=sig
brnum=4
lastt=7
smod=yini
c —————————————————————————————————
else if(brnum.eq.8) then
C -—Branch 8- Figure 2.17
C -- no event
C sig=-suc
etot=etot+fact*de
lastc=8
C emin=etot
de=0
smod=0
C ------------------------------------------------ END INELASTIC FIBER
end if
C ----------------------------------- END NEGATIVE DISPLACEMENT INCREMENT
end if
C --------------------------------- START POSITIVE DISPLACEMENT INCREMENT

if (de.gt.0) then

89

C
if(brnum.eq.0) then
C --BranCh 0-
ds=yini*de
if(syt-sig.le.ds) then
fact=(syt-sig)/ds
sig=syt
etot=etot+fact*de
de=(1-fact)'de

brnum=-l
smod=ysh
else

sig=sig+ds
etot=etot+fact‘de
smod=yini
de=0

end if

C

C ...............................
else if(brnum.eq.1) then

C -—Branch 1-

etot=etot+fact'de

emin=etot

ssc=sig

brnum=3

smod=yini

lastc=l

C ————————————————————————————————

else if(brnum.eq.2) then
c --Branch 2-

etot=etot+fact'de
emin=etot

ssc=sig

brnum=3

smod=yini

lastc=2

c ————————————————————————————————

else if(brnum.eq.3) then
C --Branch 3-
ds=yini'de
if(lastc.eq.1) then

___________________ START ELASTIC FIBER

Figure 2.18

................... END ELASTIC FIBER

--------------- START INELASTIC FIBER

Figure 2.20

Figure 2.20

Figure 2.21

if(ssc+syt+syc-sig.le.ds) then
fact=(ssc+syt+syc-sig)/(ds)
sig=ssc+syt+syc
etot=etot+fact*de

de=(l—fact)'de

brnum=-l

smod=ysh
else

sig=sig+ds

etot=etot+fact'de

de=0
smod=yini
end if
else if((lastc.eq.2

.or.

lastc.eq.8) .or. (lastc.eq.5

.or. lastc.eq.6)) then

if(-sbc+syt+syc-sig.le.ds) then

fact=(-sbc+syt+syc—sig)/ds
sig=~sbc+syt+syc
etot=etot+fact'de

de=(l-fact)‘de

brnum=-1
smod=ysh
else

sig=sig+ds

etot=etot+fact*de

de=0
smod=yini
end if
end if

C ________________________________

else if(brnum.eq.-1)
c --Branch -1 -
ds=ysh*de

then

Figure 2.19

if(sut—sig.le.ds) then
fact=(sut-sig)/ds

sig=sut

90

etot=etot+fact‘de
de=(1-fact)'de
brnum=7
smod=0

else
sig=sig+ds
etot=etot+fact'de
emax=etot
smod=ysh
de=0

end if

lastt=-l

C _________________________________

else if(brnum.eq.4) then
c --Branch 4- Figure 2.22
ds=yini*de
if(lastt.eq.-1) then
if(sst-sig.le.ds) then
fact=(sst-sig)/ds
sig=sst
etot=etot+fact'de
de=(1-fact)'de
brnum=-1
smod=ysh
else
sig=sig+ds
etot=etot+fact'de
de=0
smod=yini
end if
else if(lastt.eq.7) then
if(sut-sig.le.ds) then
fact=(sut-sig)/ds
sig=sut
etot=etot+fact'de
de=(l-fact)'de
brnum=7
smod=0
else
sig=sig+ds
etot=etot+fact*de

de=0
smod=yini
end if
end if
c ---------------------------------
else if(brnum.eq.5) then
C --Branch 5- Figure 2.24
etot=etot+fact*de
brnum=3
smod=yini
lastc=5
c _________________________________
else if(brnum.eq.6) then
C --Branch 6- Figure 2.24
etot=etot+fact'de
brnum=3
smod=yini
lastc=6
c —————————————————————————————————
else if(brnum.eq.7) then
c -—Branch 7- Figure 2.24
C -- no event
etot=etot+fact'de
de=0
smod=0
C _________________________________
else if(brnum.eq.8) then
C -—Branch 8. Figure 2.20
emin=etot
ssc=sig
brnum=3
smod=yini
lastc=8
C ------------------------------------------------ END INELASTIC FIBER

91

end if

C ----------------------------------- END POSITIVE DISPLACEMENT INCREMENT
end if

c ......................................................................
RETURN
END

c t...it.itfitiiti.Otifitfiffififiiiititiititttfitifitit..iitfififiifiifiiiitiiittttt

92

REFERENCES

American Institute of Steel Construction (1999), Load and Resistance Factor Design
Specification for Structural Steel Buildings, Second Edition, Chicago, IL.

El-Sheikh, M., Sause, R., Pessiki, S.P., Lu, L.-W., and Kurama, Y. (1997), “Seismic
Analysis, Behavior, and Design of Unbonded Post-Tensioned Precast Concrete
Frames,” Earthquake Engineering Research Report No. EQ-97-02, Department of
Civil and Environmental Engineering, Lehigh University, Bethlehem,
Pennsylvania.

Kurama, Y. (1997), PhD. Dissertation, “Seismic analysis, behavior, and design of
unbonded post-tensioned precast concrete walls,” Lehigh University.

Kurama, Y., Pessiki, S.P., Sause, R., Lu, L.-W, and El-Sheikh, M. (1996), “Analytical
Modeling and Lateral Load Behavior of Unbonded Post-Tensioned Precast
Concrete Walls,” Earthquake Engineering Research Report No. EQ-96-02,
Department of Civil and Environmental Engineering, Lehigh University,
Bethlehem, PA, 1996.

Prakash, V., Powell, 6., and Campbell, S. (1993), “DRAIN-2DX Base Program
Description and User Guide - Version 1.10,” Report No. UCB/SEMM-93/17 and
93/ 18, Structural Engineering Mechanics and Materials, Department of Civil
Engineering, University of California, Berkeley, CA.

Ricles, J., Sause, R., Garlock, M., and Zhao, C. (2001), “Post-tensioned Seismic-
Resistant Connections for Steel Frames,” Journal of Structural Engineering, Vol.
127, No.2, February 2001, pp. 113-121.

Shen, Q. and Kurama, Y. (2002), "Nonlinear Behavior of Unbonded Post-Tensioned
Hybrid Coupled Wall Subassemblages," Journal of Structural Engineering,
ASCE, Vol. 128, No. 10 (in print).

Varma, A.H., Ricles, J.M., and Sause, R. (2001), “Seismic Behavior, Analysis, and
Design of High Strength Square Concrete Filled Steel Tube (CFI‘) Columns,”
ATLSS Report No. 2001-02, ATLSS Engineering Research Center, Dept. of Civil
and Env. Eng., Lehigh University, Bethlehem, PA.

93

mm