PhD Thesis G a Chang

8/9/2019 PhD Thesis G a Chang

1/298

SEISMIC

ENERGY

BASED

DAMAGE

ANALYSIS

OF

BRIDGE

COLUMNS

Gilberto

Axel

Chang

Advisor

Dr

John

B

Mander

A dissertation

submitted

o

the

Faculty

of

the Graduate

School

of

the

State

UniversitY

of

New

York

in

partial fulfillment of the requirementsor the degreeof

Doctor

of

PhilosoPhY

by

July

1993


2/298


3/298

Abstract

This study is concernedwith the computationalmodeling of energy absorption

(fatigue)

capacit),

of

reinforced

concretebridge

columnsby using a cyclic dynamic

Fiber

Element

computational

model.

The results are used

with

a smooth

hysteretic

rule

to

generate

eismic

energydemand.

By comparing

he ratio

of energy demand

o

capacity,

inferences f column

damageability

r

fatigue osistance re

made

The

complete

analysis

methodology

or

bridge columns

s

developedstarting

rom

basic

principles.

The hysteretic

behaviorof steel

einforcement s

dealt

with in

detailed:

stability, degradationand consistency f cyclic behavior s explained. An energy based

universally

applicable

ow

cycle

fatiguemodel for steel

s

proposed.

A hysteretic

model

for confined and

unconfined

concretesubjected

o

both

tension

or compressioncyclic

loading s developed,

which

is

also

capableof simulating

gradual

crack closure.

A

Cyclic

Inelastic Stnrt-Tie

(CIST)

model

is developed,

n

which

the

comprehensiveconcrete

model

proved to

be suitable.

The

CIST

model is

capableof assessing

nelastic

shear

deformations

with high accuracy,

within

the

context

of a

Fiber Element

(FE) program.

A

parabolic fiber

element

with

parabolic

stress unction element

for uniaxial flexure is

developed,as well as a rectangular iber element with a quadratic interpolation function

suitable

or

biaxial

flexure.

A

smooth ule-based

macro

model for

the

simulation

of

the

hystereticbehaviorof

reinforced

concrete

elements

s developed.

The model

s

capableof accurately imulating

cyclic behavior

when compared

with actual experimental data,

through

use of an

automated system

dentification

procedure

which

proved to

be

very effective

in

finding

the model

parameters o

best approximate

member

behavior.

The macro model was

oalibrated o simulate he behaviorof a full sizebridgepier and then implemented nto a

SDOF

non-lineardynamic analysis

rogram

o

generate

nelastic

esponse

pecffa.

In

addition

to

the

usual

ductility-based

nelasticspecffa,severaladditional

energy

spectra are also

generated

which

include: viscous

damping,

hysteretic energy, cyclic

(fatigue)

demand.

These

spectra

are used

as

part

of a

rational methodology n which

the

cyclic

demand on bridge

columns

is compared with

the capacity

predicted

by

Fiber-Elementanalvsis.


4/298

Acknowledgments

My profoundgratitudes expressedo my advisorDr. JohnB. Mander,whosebrilliant

ideasand

helpful

comments

ade

possiblehe

completion

f

this work.

His

competent

guidance asstamped

great

mpressionn

my

academic

hought.

The

financialsupport

of

my home

nstitution

Universidad

ecnol6gica e

Panam6s

deeply

appreciated.

Special

hanks o Ing

H6ctor

Montemayor or

his

profound

commitment

and encouragement,

o Ing.

Luis Mufroz

or his sincere

riendshipand

to

Ing. Jorge

uis Rodriguez

or his spiritual

ellowship.

I am also thankful o LASPAU for its financialsupportand specially o Ms. Sonia

Wallenberg

or her

personal

nterest.

I

wish

to thank

my friends at

the

Civil

Engineering

Department

or their friendship,

specially

oy Lobo

My appreciation

oes o

my

family for

their

patience

nd support,specially

o my

parents-in-law hose

oveand

caro

werealways

elt.

This

researchwas conducted

t

the department

f Civil

Engineeringat

the

State

Universityof New York at Buffalounder he supervisionf Dr. JohnB. Mander. Drs.

Andrei

M., Reinhornand

Ian G.

Bucklealso served n

the

committee, nd

Dr Peter

Gergely

s

outside

eader.

The

NationalCenter

or Earthquake

ngineering

esearch t

the

State

Universityof

New

York at

Buffalo

provided

inancial

upport

or the work contained

n

Sections

and 6,

this

assistance

s

gratefully

cknowledged.

To my beloved

Marfa,

or

your ove.


5/298

Table

of Gontents

1. ntroduction

1.1 Background. . . .

1.2 Integrationf Previous esearch ork

1.3

Seismic valuation ethodologies

1.4

Scope f Presentnvestigation

2.Hysteretic

nd

Damage

odeling

f

Reinforced

teel

Bars

2.1 ntroduction

2.2

Monotonic

tress-Strain

urve

2.2.1

The

Elastic ranch

2.2.2

TheYield

Plateau

2.2.3

Strain

Hardened

ranch

2.3The

Menegotto-Pinto

quation

2.3.1

Computat ion

f

Parameters

,

n

nd

R .. .

2.3.2

Menegotto-Pinto

quation

imiting

ase

2.4

CyclicProperties

f Reinforcing

teel

2.4.1

Envelope

ranches

Rules

and2)

2.4.2Reversal ranchesRules and4)

2.4.3

Returning

ranches

Rules

and

6) .

.

.

2.4.4

First

Transition

ranchesRules

and

8)

2.4.5

Second

ransition

ranches

Rules

and

10)

2.4.6

Strength egradation

2.5

Stress-Strain

odel

Verification

2.6Damage

Modeling

1-1

1-2

1-3

1-5

2-1

2-1

2-1

2-2

2-2

2-2

24

2-7

2-10

2-10

2-12

2-15

2-23

2-24

2-26

2-26

2-36

2.7

Damage

ModelmplementationndVerification . 241

2.8

Stra in ate

Ef fects

. . . . . .

2-ST

2.9

Conclusions

.

2-Sg

3. Modeling

tress-Strain

yclic

Behavior

f Concrete

3.1 ln t roduct ion. . . .

3-1

3.2

Review

f Previous

Work n

Stress-Strain

elations

or

iii

Concrete

3-2


6/298

3.5

3.6

3.7

3 2.1

Monotonic

ompression

tress-Strain

quation

3-2

3.2.2

nitial

Modulus

f

Elasticity

. 3-12

3.2.3 Strain

t

PeakStress

or

Unconfined

oncrete

3'14

3.2.4 Characterist ic

f

heDescending

ranch

f heMonotonic

. . 3-16

Stress-strainurveor Unconfinedoncrete

3.3

Recommended

omplete

tress-Strain

urve

or

Unconfined

. . .

3-17

Concrete

3 .4Con f inemento f

oncre te

. . . . .3 -22

3.4.1Conf inement

odels

. . . . .

3-23

3.4.2Confinement

echanism

..

. . .

3-29

3.4.2.1 onf inementof

i rcu larSect ions

. . . . .3-29

3.4.2.2 onfinement

f Rectangularections

. 3-30

3.4.3Conf inementEf fectonStrength. . . . .3-32

3.4.4Conf inementEf fecton

uct i l i ty

. . . . . . 3-34

3.4.5 Confinement

ffect

n the

Descending

ranch

. . . 3-35

Concre te inTens ion

. . . .3 -35

Dynamic

f fects

nConcrete

ehavior

. . .

. . . . . . 3-37

Modeling

ysteretic

ehavior

. 3-39

3.7.1

Basic

Components

f a Hysteretic

odel

. . 3-39

3.7.2A General pproacho Assessing

egradation

ithin

. . . 3-40

Partial

ooping

n a

Rule-Basedysteretic

odel

3 .7 .2 .1

i rs tPar t ia l

eversa l

. . . . .341

3.7.2.2

Part ia l

eloading

,.

342

3.7.2.3

Partial nloading

roma

Partial eloading

. 345

3.7.3A Smooth

ransition urve

orMathematical

odeling

. 3-46

3.8 Cyclic

Properties

f Confined

nd

Unconfined

oncrete

. .

. . 3-49

3.8.1Compression

nvelopeurve

Rules

and5)

...

3-49

3,8.2Tension nvelope urveRules and6) .. 3-51

3.8.3

Pre-Cracking

nloading

ndReloading urves

.. 3-52

3.8.4

Post-Cracking

nloading

nd

Reloading urves

. 3-58

3.8.5

Pre-Cracking

ransit ion

urves

..

.. 3-59

3.8.6

Post-Cracking

ransi t ionurve

. . . . . . 3€1

3.9

Model erif ication

..

. .

3€4

3.10

DamageAnalys is . .

. . . .3€4

iv


7/298

3.11

Conc lus ions

. . . . .3€6

4. Damage odeling

f

Reinforced

oncrete

olumns

usingFiber-Element

nalysis

4.1 ntroduction

4.2Moment-Curvature

nalysis

or

Uniaxial

ending

4.3Moment-Curvature

nalysis

or

Biaxial

ending

4.4 Force-Displacement

nalysis

. .

4.4.1Elastic

lexural

eformation

4

4.2

Plastic

lexural

eformation

4.4.3

Elastic

hearDeformation

. .

4.4.4 nelastic

hear

Deformation

4.4.4.1

Proposed

yclic

nelastic

trut-Tie

CIST)

Model

or

Shear

Deformations

.

4.4.4.2

Crack nclination

ngle

4-30

4.5 Validation

f Fiber-Element

odel

..

4-92

4.6

Conc lus ions

. . . . . .

4-3S

5. SmoothAsymmetric

egrading

ysteretic

odel

with

Para

meter

dentifi

ation

5.1 ntroduction 5-1

4-1

4-1

4-9

4-16

4-16

4-17

4-18

4-20

4-21

5.2A

Smooth

Curve

o FitTwo

Tangents

5.2.1The

Menegotto-Pinto

quation

5.2.2

Computationf Parameters

,

n

ndR

5.3

Description

f

Smooth

ysteretic

odel

5.3.1

Monotonic

nvelope

urves

5.3.2Reverse

urves

5.3.3Transition

urves

5.3.4

Model

Summary

5.4 Parameter

dentification

5.4.1

Optimization

ethod

5.4.2

Scaling

5.4.3

ConstrainingheParameters

..

5.4.4Init ial

st imate

5.4.5

Order

f Parameter

dentification

5-2

5-2

54

5-7

5-7

5-9

5-12

5-16

5-18

5-19

5-20

5-21

5-22

5-22


8/298

5.5

Verification

f

S

5.6

Conclusions

mooth

Modeland

System

dentification

Method

6.

Assessmentf

Hysteretic

nergy EMAND

6.1 ntroduction

6-1

6.2

Elastic esponse

f a

SDOFSystem 6-1

6.3 nelastic

esponse

f a

SDOFSystem 6-4

6.4

Inelastic

esponse

pectra

6-7

6.4.1

Displacement

ucti l i ty

pectra 6-7

6.4.2

Energy ased pectra

6-8

6.5 mplementation

nd

Results . . . 6-11

6.6An ll lustrative

xample

. 6-14

6.7Conclusions . 6-15

7.

Summary,

onclusionsnd

Recommendations

7.1

Summary

.

7-1

7 2SomeSpecific onclusions

7-2

7.3Recommendationsfor

uture esearch...

74

Appendix

A. References

5-23

5-24

VI


9/298

List

of

Symbols

a = sffess lock depth

A ,

:

atea

of

the

confined

core

concrete

measured

o the

centerline

of

the

perimeter

hoop

Aror,

:

area

of bound

concrete

nder

compression

A

:

effectively

confined

area

or rectangular ection

As

:

gross

section

area

Aq

:

shear

area

Au

:

total area

of

hoop steel

Au

:

areaof

the

shear

einforcement

A,

:

areaof

longitudinal

steel

Arn

:

hoop cross

sectional

rea

A,,

:

total longitudinal

steelarea

Ar,

:

total areaof

transverse

einforcement

parallel to the x

axis

lsr : total areaof transverseeinforcementparallel to the y axis

b

:

breadth

of

the

section

b

:

fatigue strength

exponent

b,

:

concrete ore

dimension

n

the x

direction

c

:

distance

rom where e-

is measured

o the neutral axis

c

:

damping

coefficient

c

:

fatigue ductility

exponent

D

:

total

damageaccumulated

d

:

column

diameter

dt

:

bar diameter

of steel

d

:

concrete

ore dimension

n the

y

direction

D

:

damage

or

one cycle

of a

given

amplitude

Ae

d,

:

diameter

of circular

or

spiral

hoops

v11


10/298

f,ep

:

energy

absorbed

n

a elastic-perfectly

lastic

oop

Ey

:

final slope

E1,

:

hystereticenergy

absorbed

E1

:

absolute

inetic

energy

E

o

:

tangentmodulus

of elasticity

at

the nitial

point for

a softened

urve

e

:

average

strain

E2

:

viscousdamping

energy

8,,

:

tangent

modulus

at

the

returning

point

E,

:

sffain

energy

E,

:

elastic

modulus

of

elasticity

or

steel

,E.""

:

secant

modulus

Et

:

tangent Modulus

Et

:

total

energy

at

a

given time

8,"

:

tangential

Young's

Modulus of concrete

F

"

:

compressive

orce

in concrete

strut

Ft

:

tensile

orces

n concrete

ie

F,

:

forceon steelhoop

ft

:

strain

under

the confining

fluid

pressure

f

:

form factor

f,

:

stress

amplitude

f,

:

concretestress

f ,

:

confined

concrete

strength

f

:

unconfined

concrete

strength

fo

=

damping

force

fro

:

unbalanced

orce

ft

:

final

stress

or

a

softened

curve

f,

:

inertia force

fl

:

confining

sffess

fu

:

confining

sffess

n the x direction

vll l


11/298


12/298

M

:

moment about

he centroid

Mj

:

moment at

first

yielding

M^o

:

maximum

moment

Mr,

:

crackingmoment

m

=

total mass

N

:

effective

or

equivalent

numberof equi-amplitude ycles

Ny

:

number of cycles

o failure

P,

:

appliedaxial

oad

O

:

post

yielding

slope

atio

R

:

radius of curvafure

parameter

R

:

symmetry

parameter

R ,

:

critical value

of

R

Rmo

:

critical

value of

R for

the M-P

equation

R,,

-

force reduction

factor

s

:

center

o

center

hoop spacing

s/

:

clear

ongitudinalspacing

between pirals

n which archingaction of

the

concrete

develops

,So

:

spectralacceleration

S7

:

spectraldisplacement

Su

=

pseudo

velocity

T

:

period

of

the structure

/

:

time

To

:

period that

separates

long and

medium

period

behavior

tr

:

displacementof

the system

espect

o the

ground (deformation)

V

=

shear

orce

V,

:

shearstrength

of concrete

V,

:

shearstrengthofsteel

w

:

specificweight

W,n

:

modulus

of

toughness f

the hoop

steel


13/298

.r

:

system

isplacement

xs

:

ground

displacement

Xp

:

plastic

displacement

t : system elocity

i-s

:

ground

isplacement

i

:

system cceleration

is

=

ground

cceleration

vsp

:

non-dimensional

palling

train

Xv

:

leld

displacement

X,

:

maximum

nelastic isplacement

esponse

y(x)

:

non-dimensionaltress

unction

z(x)

=

non-dimensional

angent

modulus

unction

o

:

fraction

of shear

orce which

is

added

o the

axial

load

y

:

sheardistortion

e

:

strain at any

fiber

eo

:

strain

at

the

centroid

A

:

total deformation

L,

:

elastic

flexure

deformation

Lp

:

plastic

flexure deformation

Lr

:

elasticshear

deformation

[sp

:

inelastic sheardeformation

A€o : strain amplitude

Ay

=

yield

displacement

2eo

:

total

strain

range amplirude

et,

:

strainat

peak

sffess

or

confinedconcrete

et

:

unconfined

concretestrain

at

peak

stress

eca

:

ultimate

compressionstrain

E1

:

strain at

final

point

for a softenedcurve

xi


14/298

Etr^

:

location of

the tensionenvelope ranch

eo

=

strainat

initial

point for softened

urve

Ei^

:

location of

the compression

nvelope

ranch

ep

:

plasticstrainamplitude

Ept

:

Plastic

strain

Er,

:

strainat

the returning

point to the

envelope urve

es

=

steelstrain

tsfr

:

strain

hardening strain

€sro

:

standard eviationof

the

shain

history response

E,u

:

strain at ultimate sffess

Eun

:

unloading strain

from

an envelopecurve

etf

:

fatigue ductility coefficient

ey

:

leld

strain

Eo

=

average ongitudinal strain on

the

concretestruts

ev

:

sfain on

the

transverse oops

Q,

:

magnitude

of

plastic

curvature

0u : ultimate curvafure

Qy

:

leld

curvature

p

:

ductility

LLef

:

effectiveor equivalent

equi-amplitude

ycle ductility

€

:

damping ratio

9

:

volumeffic ratio

of

the ongitudinal

steel

n the

confined

core

p,

=

ratio

of

hoop reinforcement

o

volume of concrete

ore

measured o

outsideof

the hoops

P,n

=

volumetric ratio

of

transverse oops

P'

:

Ar'/sd

py

=

Arrlsb

6tf

:

fatigue

strengthcoefficient

0

:

angleof inclination of cracks

espect o the

ongitudinalaxis

xl l


15/298

0p

@4

Cr) 1

:

plastic otation

:

damped

requency

=

naturalangular

requency

xl l l


16/298

List

of

Figures

Section

1

Fig.

-1Summaryof

esearch

igni f icanceofh isStudy

ntheContextfa . . . . . .1-4

Seismic valuation ethooologY

Section

2

Fig.2-1The

enegot to-Pinto

quat ion

. . . . . . .

2-3

Fig.2-2Different

urves

avingheSame tart ing

ndEnding ropert ies

.... . . . .

2-4

Fig.2-3

Tension

ndCompression

nvelope urves

2-1'l

Fig.2-4

Effect

f heStrain

mplitudef

heReversaln he Equation

. . . .

2-15

Parameters

Fig.2-5Reversal

rom

YieldPlateau

2-16

Fig.2-6Definition

f

heReversal nloading

ranch

2-16

Fig.2-7Effect f heStrain mplitudef Loop n he nitialModulus ndR

parameter

or

Reinforcingars

fy

=

Sgksi)

Loading)

. .

Fig.2-8Effect f

heStrain

mplitudef Loop

n he nitialModulus ndR

parameter

or

Reinforcing ars

fy

=

53 ksi)

Unloading)

Fig.2-9Effect f he

Strain

mplitudef Loop n he

nitialModulus ndR 2-19

parameter

or

HighStrength ars

fy

=

123ksi)

Loading)

Fig.2-10Effect f

heStrain

mplitudef Loop n he nitial

Modulus ndR 2-20

parameterfor

igh

Strength ars

fy

=

123ksi)

Unloading)

Fig.2-11 itting

f M-P

Equationo

a Loading oop f Reinforcing

teelBars

. . . .

2-21

(fY

=

53 ksi)

..

Fig.2-12 itting f M-PEquationo an Unloadingoop f Reinforcingteel 2-21

Bars

fy

=

53 ksi)

Fig.2-13Fitting

f M-PEquation

o a Loading

oop f HighStrength teel 2-22

Bars

fY

=

123

ksi)

Fig.2-14

itting f M-P

Equationo

an Unloadingoop f HighStrength 2-22

Steel

Bars

tY

=

'tZg

ksi)

.

.

Fig.2-15Sequence

f

Partial eversals

2-23

Fig.2-16Flow

of Rulesat

EveryReversal

ndTargetStrain 2-25

Fig.2-17Degradation

f Reinforcing

ndHighStrength teelBars

2-27

Fig. -18Comparison

f

Degradingodelwith

xperimentalesults .. .

2-27

Fig.2-19StressDegradation

imulation nd

Fracture rediction n Steel

2-28

Bars

Fig.2-20

tress-Strain

xperimenty

Kent ndPark

1973),

pecimen

.... . . . .

2-29

Fig.2-21Stress-Strain

xperimenty

Kent ndPark

1973),

pecimen

.... . . . .

2-29

Fig.2-22 tress-strain

xperiment

yKent nd

Park(1973),pecimen

.... . . . .

2-30

Fig.2-23 tress-Strain

xperimentbyKentand

ark(1973),pecimen

5 ... . . . . 2-30


xperiment

y Kent

andPark

1973),

pecimen 1 2-31

Fig.2-25 tress-Strain

xperiment

y Kent

ndPark

1973),

pecimen 7 .

.

..

. . .

2'31

2-17

2-18

XiV


17/298

Fig.

2-26

Stress-Strain

xperimenty Ma,Bertero ndPopov

1976),

2-32

Specimen


xperiment

y Ma,Bertero ndPopov

1976),

2-32

Spec imen

. . . .


xperiment

y Panthaki

1991),

pecimen 2

Fig.2-29Stress-StrainxperimentyPanthaki1991), pecimen 3

Fig.

2-30

Stress-Strain

xperimenty Panthaki

1991),

pecimen ' 6


xperimenty Panthaki

1991),

pecimen 19

Fig.2-32Stress-Strainxperiment

y Panthaki

1991),

pecimen l


xperimenty Panthaki

1991),

pecimen 4

2-33

2-33

2-34

2-34

2-35

2-35

2-36

2-40

2-42

2-43

2-44

2-45

2-46

2-47

2-48

2-49

2-50

2-51

2-52

2-53

2-54

2-55

2-56

2-57

Fig.2-34Stress-Strainxperimenty Panthaki

1991),

pecimen 5 . . .

Fig.2-35Determination

f Equivalenttrain mplitude

Fig.2-36

HighStrength ar,Specimen 18

Panthaki,

991)

Fig.2-37

High

Strength ar,Specimen 10 .

Fig.2-38 ighStrengthar,Specimen13 .

Fig.2-39HighStrength

ar,Specimen 12 .

Fig.

2-40

HighStrength ar,

Specimen

4

Fig.2-4'lHighStrength

ar,Specimen 7

Fig.2-42

High

Strength ar,Specimen 14 .

Fig.2-43

High

Strength ar,Specimen 9

Fig.2-44HighStrength ar,Specimens11,P2andP3

Fig.2-45

Reinforcing

ar,Specimen l

Fig.2-46Reinforcingar,Specimen 9

Fig.2-47Reinforcingar,

Specimen

5

Fig.2-48 einforcingar,

Specimens11,R7and

R10

Fig.2-49HighStrength ar,

Specimen 20,

Low-High tepTest

Fig.2-50HighStrength ar,Specimen

21,

High-Low tepTest

Fig.2-51 ncipient ailure

rediction

Section

3

Fig.3-1Characteristics

f

the

Stress-Strain

elationor Concrete

. . . . 3-2

Fig.3-2Comparisonf Different tress-Strainquationsor Concrete

.. . ... 3-6

Fig. -3Equat ionuggestedyYoung1960) . . . " . , . . . 3-6

Fig. -4Equat ionuggestedySaenz

1964)

. . . . . . . . .

3-7

Fig.

3-5Equationroposed

yPopovics

1973)

... . . . ^

3-7

Fig.3-6Equation uggestedySaenz

1964)

3-10

Fig.3-7Equation uggestedy

Sargin

1968)

3-10

Fig.

3-8Equation roposed

yTsai

1988)

3-11

Fig.3-9Comparisonf Different quationsor heSecantModulus

f 3-20

Concrete

Fig.3-10Comparisonf Different quationsor he Strainat Peak

Stress 3-20


18/298

Fig.3-11

Proposed

heoreticaltress-Strainurvesor Unconfined

3-21

Concrete

Fig.

3-12Theoretical

tress-Strainurves uggestedyColl ins ndMitchell

. . . 3-21

Fig.3-13

Tsai sEquation

arametersor Unconfined

oncrete

3-22

FiE. -14Some

Proposed

tress-Strainurvesor

Confined oncrete

3-28

Fig.3-15ConfinementechanismorCircularndRectangularross 3-31

Sections

Fig.3-16Confined oncrete

trength atio .

Fig.3-17Comparison

f Different odelsor

Triaxial onfinement.

. . .

Fig.3-18Characteristicf

heFalling

ranchorConfined oncrete

Fig.3-19Definitionf

Falling

ranchorConfined oncrete

Fig.3-20Relationshipetween urves

n

a Rule-Basedodel

Fig.3-21 arget oint nd

Reloadingoint

n a Complete eversal

Fig.3-22

Reloading

roma Partial nloading

Fig.3-23Unloadingroma Partial eloading

Fig.3-24 Smooth ransitionurve

Fig.

3-25Tension

ndCompressionnvelope urves

Fig.3-26

Cyclic

Compressionharacteristicsf Concrete

Fig.3-27Complete nloading

ranch

Fig.3-28Complete oading

ranch

Fig.3-29

Loading

ndUnloadingurves fterCracking . . .

Fig.3-30Partial nloading

urves

orTension ndCompression

Fig.3-31Transition urves

Before

racking)


After

Cracking).

Fig.3-33RelationshipmongheModelRules

Fig.3-34UnconfinedyclicCompressionestby

Sinha,Gerstle ndTulin

(1

e64)

Fig.

3-35UnconfinedyclicCompressionestby Karsan

ndJirsa

1969)

Fig.3-36

Unconfined

yclicCompressionestby

Okamoto

1976)

Fig.3-37

Unconfined

yclicCompressionest

by Okamoto

1976)

Fig.3-38

Unconfined

yclicCompressionestby

Tanigawa

1979)

Fig.3-39Cyclic ension estbyYankelevskyndReinhardt1987)

Fig.

3-40Confined oncrete yclic

est

by Mander t al.

1984)

Fig.

3-41Confined oncrete yclic estby Mander t al.

(1984)

Fig.3-42Comparisonf theProposedension ranch

quation ithother

Analytical quations

3-33

3-33

3-38

3-38

3-40

3-43

3-43

3-44

3-48

3-50

3-54

3-55

3-56

3-58

3-60

3-62

3-62

3-63

3-68

3-68

3-69

3-69

3-70

3-70

3-71

3-71

3-72

Chapter

Fig.4-1Definition

f

Global ndLocalCoordinates

. . . . 4-5

Fig.4-2Definitionf Variablesna FiberElement

. . . . 4-7

Fig.4-3

Definitionf

Variablesor Biaxial ending

4-10

XVI


19/298

Fig.4-4Element

ode

Numbering 4-13

Fig.

4-5Flexural

eformation

n a

Column

4-'16

Fig.

4-6Shear

Deformation

na Column 4-22

Fig.4-7

Equivalent

trut-Tie

odelor ShearDeformations. . . .

4-22

Fig.4-8EquilibriumndStrainDeformationn heCyclicneiastic trut-Tie 4-23

Shear

Model

Fig.4-9Definition

f Average ongitudinaltrain n ShearConcrete trut

4-24

Fig.4-10Comparison

f heAnalytical tress-Strainelationshipith

he 4-33

Experimental

ehavior f PlainConcreteromAycardi t al.

(1992)

or

Specimens

and4

Fig.4-11Comparisonf

Propossediber

Element

odelwith

Experimental 4-37

ResultsromAycardi

t al.

1992)

pecimen, P

=

0.10 c Ag . .

Fig.4-12Comparison

f Proposed

iberElement odelwith

Experimental 4-38

Resultsrom

Aycardi t al.

1992)

pecimen,

P

=

0.30 c Ag .

Fig.4-13Comparisonf ProposediberElement nalysis ithExperimental 4-39

andAnalytical

esultsromMander t al.

(1984)

olumn

Fig.4-14

Predictionf LowCycle atigue

racture

f Longitudinalars or 4-40

Co lumn . . . .

Fig.4-15Comparison

f ProposediberElement nalysis ithExperimental

4-41

andAnalytical

esultsromMander t al.

(1984)

olumn

. . . .

Fig.4-16Comparisonf

ProposediberElement

nalysis

ith

Experimental 4-42

andAnalytical esults

romMander t

al.

1984)

olumn

Fig.4-17 nalytical imulationf a

Full

SizeShearCritical ridge ier 4-43

Tested y Mander t al.

1993)

Chapter5

F ig .5 -1 heMenegot to -P in toEquat i on

. . . . . . . . 5 -3

Fig.5-2Computationf Parametersor

he

M-PEquation ., . .

5-4

Fig.5-3Monotonicnvelopeurves .. . . 5-8

Fig.5-4Reverse oading

urve

5-10

Fig.5-5Reverse nloading

urve

5-13


5-14

Fig.

5-7

Logical ranching iagram

5-17

Fig.5-8Comparisonf MacroModelSimulationseneratedhrougha) 5-28

Experimentalata,

c)

FiberElement xperimentimulation

Fig.5-9MacroModelSimulationf a

Full

SizeBridge ierBased nActual 5-29

Experimentalata

Fig.

5-10Simulationf

heCyclic ehaviorf a FullSizeBridge ierBased

5-30

on a FiberElement imulated xperiment

Fig.

5-11Macro

ModelSimulationf a 113 caleColumn ased n 5-31

Experimentalata

Fig.

5-12Macro

ModelSimulationf a 113 caleColumn ased n Fiber 5-32

Model

Simulated

xperiment

XVII


20/298

Fig.5-13Macro

Model

imulationf a Bridge ol low olumn ased n Fiber

5-33

Element

imulated

xPeriment

Chapter

6

Fig. -1

Proposedhree

evel

eismic valuationethodology

.

.. . . . . .

6-2

Fig.

-2Equivalent

ingle-Degree-Of-Freedom

ystem

.

...

^

6-6

Fig.6-3aForce

Correction

actor . ...

. . . 6-6

Fig.6-3bStepBy-Step

ntegration . . . ..

6-6

Fig.6-4Symmetry

arameter 6-10

Fig.6-5

nputGround

Motions sed

or

SpectralAnalysis

6-12

Fig.6-6Energy,

uctility

nd

Low

CycleFatigue emand pectraor El

6-16

Centro

1940)

-S,

with5olo iscous amping atio ndPGA

=

0.348

Fig.6-7Energy, uclility

ndLowCycle

atigue

emand pectraor 6-18

Pacoima

1971) ,

i th %Viscousampingat io ndPGA

=

1.17 . . . . .

Fig.6-8Energy, uctility

ndLowCycleFatigue emand pectraor San 6-20

Salvador1986), i th olo iscous amping atio ndPGA =0.695 ... .

Fig.6-9Energy,

uctilityndLowCycle

Fatigue

emand pectraor Taft 6-22

(1952)

21E,with5%

Viscous amping atio ndPGA

=

0.156

Fig.

6-10Energy,

uctility

ndLowCycleFatigue emand pectraor

6-24

MexicoCity

1985),

ith5%Viscous amping atio ndPGA

=

0.171


ndLowCycle atigue emand pectraor

6-26

Sinusoidalnput ,wi th

olo iscousDampingat ioand GA

=

1.09 . . . . . .


ndLowCycle

atigue

emand pectra

or

El 6-28

Centro

1940)

-S,with

5%

Viscous

amping

atio nd

PGA

=

0.348

.

(Elasto-Perfectly

lastic

Model)

Fig.6-13Energy,

uctilityndLowCycle

atigue

emand pectraor 6-30

Pacoima

1971),

i th

olo iscous amping atio ndPGA

=

1.17

.

(Elasto-Perfectly

lasticModel)


ndLowCycleFatigue

emand

pectraor

San 6-32

Salvador

1986),

ith

5%Viscous amping atio ndPGA

=0.695

.

(Elasto-Perfectly

lastic

Model)


ndLowCycle

atigue

emand pectraorTaft 6-34

(1952)

21E,

with5olo iscous amping atio ndPGA

=

0.156

.

(Elasto-Perfectly

lastic

Model)

Fig.

6-16Energy,

uctility ndLowCycle

atigue

emand pectraor 6-36

Mexico ity 1985), ith5olo iscous amping atio ndPGA = 0.171 .

(Elasto-Perfectly

lasticModel)


ndLowCycle atigue emand pectraor 6-38

Sinusoidal

nput, ith5%Viscous amping atio ndPGA

=

1.0

g

(Elasto-Perfectly

lasticModel)

Section

7

Fig.7-1 ummaryf Research

ignif icancef hisStudyn heContext f a . . . . . . 7-6

Seismic valuation

ethodology

XVII I


21/298

Section

Introduction

1.1 Background

In order to designor analyzehe behaviorof bridge substructurespiles and

columns f

piers)

hatmay

be either

einforced,

r

fully or

partially

prestressed

oncrete,

it is

essential

hat

analyticalmodels

be developed

hat

accurately eflect

the true

non-linear

dynamic cyclic

loading

behavior

of

those

members.

Current

analytical

modeling echniques

f structural lement se eithera macromodeling

approach

e.g.

DRAIN, Kanaan

nd

Powell,

1973;Allahabadi

nd

Powell,

1988)

or micro inite

element

approache.g.ANSYS,Kohnke,1983). t is consideredhata coarsemacroapproachn

which

lumped

plasticity

within elements

s

used o

predict

esponse

ehavior, n

many

instances,s too

crudewhen

ooking

at detailed

ehavior f

joints

and

plastic

hinges.

On

the

other

hand,

ophisticated

inite

elementmodels

may equire

mesh epresentation

hat

is too fine, thus

prohibiting

he

analysis f large

or evenmoderate

ize bridges. It is

considered

hat the most

appropriate

ompromises to

use a combination

f the two.

Fiber

elements

anbe used

or this

purpose.

Fiber

elements

an be incorporated

nto

a

non-linear time-history

structural analysis

computer

program

using two

different

approaches:

irect iber modeling,

r

indirect

iber modeling.

The first

has ecently

een

incorporated

nto the atest ersion

f DRAIN-2DX,

but

s in

a relatively

rude orm

and

still

may require

some urther

refinement,

ut

the

approach

hows

great

promise.

The

second pproachs the

subject f

this

study

or

the

purpose

f use

with

programs

uchas

IDARC

Park

et al., 1987)

or

DRAIN-2DX).

A fiber model

epresentation

an

eapture

details of features uchas the critical concrete nd steel strainsas part of the analysis

t-1


22/298

process hrough

he direct

ntegrationof stress-strainesponse.Most

existing ime-history

computer

programs

ocus on

determiningthe nelasticdemands

aused

y a

given

seismic

excitation.

As

part

of a

fiber

element

analysisof components he

inelastic

capacity

of

members can also be determinedas part of a preprocessing post-processinganalysis.

Further, as

part

of a

post-processing nalysis,

he

damage

sustained

y components

and

subassemblages an

be determined as

the ratio

of demand versus

capacity. This

dissertation

ocuses

on

this

damageability

concept

as

part

of

the

modeling for

bridge

substructures.

1.2 Integrationf Previous esearch ork

Considerable

work has beenundertaken

y

Mander,

Priestley

and Park

(198a)

in

developing

moment curvatureand

force-deformation

models

based

on a

fiber

approach,

directly

integrating

stress-strain

elations or reinforced

concretemembers

Mander

et

al.,

1988a,

1988b).

Dynamic

reversed

cyclic

loading

of

members s

accounted or and

inelastic

buckling of

longitudinal reinforcement, ransverse

oop fracture,

and

concrete

crushingmodesof failure are determined rom energy considerations.Good agreement

has been demonstratedwhen

tested

against a variety

of

physical

model

experimental

results. This

fundamentalwork was

followed

by Zahn

et al.

(1990)

who developed

energy-based

esigncharts or bridge

pier

with

ductile detailing.

The

need for sophisticated

ools to

analyze

sffuctures

subjected

to

earthquake

loadingshas

produced

a

great

deal of research. Much

of

this

research s the

coordinated

effort of

many researchers hat

share a common

purpose,

to

gain

insight into this very

complex

problem.

The

complexity of

the

problem

underlies

both

the randomness

of

earthquakemotions

and

the nonlinearhysteretic

behavior

of structuralcomponents. At

the

end,

the

goal

is to

develop

rational

methods

of design,

hat

will consider

both

the

demand

that the ground

motion will impose

on

the

structure and

the

capacity of

the

sffucture

o

meet

those

equirements.

The

demand on a sffucture can be of

two types:

displacementductility demand and

energy demand. The

former

dictates bearing set width requirements

and secondaryP-A

|

-^/.


23/298

load

effects,

while the

latter leads

to

failure

of the

constituent

materials.

steel

and

concrete, hrough

ow

cycle fatigue.

It

will

subsequently

e

shown

that

the

two

are

also

interrelated.

Much

of the research

ffort had

been

concentrated

n

the

ductility

demand,

although energy demand esearch s gaining popularity among researchers The

capacity

of structural

elementss,

of course,

a fundamental

roblem.

A

computer program

to

simulate

he

cyclic

behavior

of

reinforced

concrete

s

presented

n this

study. Every

major

aspect

of

its

development

s

presented.

Advanced

models

for

concrete

and

steel are

proposed,

with

improvements

over previous

models.

Mathematical

models or

the

description

of

damage

n

steel

elements

re ncorporated.

A

uniaxial

moment-curvature

nd force-deformation

micro model is presented s well as a

biaxial

moment-curvature

iber

element

model.

A general

pulpose

macro

model

with

system

identification

for

uniaxial

moment-curvature

or force

deformation

was

implemented.

Theseprograms

can

be inte$ated

as

part

of

an analysis

methodology

outlined

in

F ig

1 -1 .

1.3

Seismic

valuation

ethodologies

Herein

a three

evel

seismic

valuation

methodology

s proposed.

The

first

is

based

n well-known

oncepts

f ductility

and

uses

imit

analysis

echniques

rom

which

capacity/demand

C/D)

ratios

are calculated

or

structural

trength

nd

ductility.

This

is

called irst-order

pproach

s t

doesnot

concern

tself

with

cyclic

oading

effects

and s

similar

to the

procedures

iven

n

ATC

6-2.

The

second

s new

approach

dvanced

herein,

s

based

on fatigue

or damage

oncepts

nd s

concerned

ith

comparing

energy

absorption

apacities

with

seismic

energy

demands.

This

is

called

a

second-order

approach,

s t is

more

refined

aking

nto

account he

earthquake

uration

and

would

be

used

when the

results

rom

a first-order

analysis

re n

doubt.

A third

and

more

refined

analysis evel

concern

multi-degree

f freedom

system

analysis,

n

which

rationally

implemented

ysteretic

erformance

s

used.

l-3


24/298

Step

r.

Strength

emand,

(d)

Step2" Strength apacity (c) (Limit Analysis)

C ( c t

I

Step.

r*

=

ffi

=

t

lIf

r,,

>

1.5

STOPI

Step

.I Ductility

Demand

(d)

Step

.1

DuctilityCapacity

(c)

per

ATC

6-2

Step . rt = g? [If ru 2 1.5 STOP]

$la1

Step

.2

Rotational

emand

0

p(A,

N(A

=

f

(R

$,

E

Q,

HYst.moe[)

Step

.2

Rotational apacity

o(c),N(c)

M/r)

Step.2 ,r

=ff i

[If r^'21.5 STOP]

Step

7. Generate

emberSpecific

Hysteretic

Models

From

Steps

.2and

5.2)

Step

8.

Perform

imeHistoryAnalysis

(IDARC

or DRAIN-2DX)

Step9.

ExamineCritical

Members erformance.

Use

Fiber-Element

o

predict

detailed

behavior

ased n

memberime-history.

SEISNtrC

EVALUATION

METHODOLOGY

Fig.

1-1

Summary

of

Research ignificance f

this

Study

n the

Contextof a Seismic valuationMethodology.

F a

; v

:''l d

a t r

<

v E

t s €

€ a r

- a

p

.=

al

0

, >,

. = E

- E

b<

13

q,)

L b D

o

EF

6 t t

>F

O A

= a

= ?

2 2 2

F F

b i

: ; o

v c t

a E

;A

Sections & 5

Sections

-

4

Section 5

Section

l-4


25/298

1.4 Scope

of

Present

nvestigation

Firstly.

this

investigation

deals with

the modeling of

the

hysteretic and fracture

characteristics

f

reinforcing

steel.

The low

cycle

fatigue behavior of

steel

s modeled

based

on experimental

data.

The importance of

this modeling is that it allows the

prediction of

the

fatigue

ife of

longitudinal bars

n

the

context of a

reinforced concrete

member

subjected

o

cyclic

loading.

Thus,

his modeling will allow

to predict the failure

of

a

member due

to

low

cycle

fatigue,which

is

predominant

on

well

detailed

beams

and

columns

with low

levels of axial

load. Numerousexamples

are

presented

o

show

the

capacity

of

the model

to simulate

both

the

sffess-strain

yclic behavior

and

the

fatigue

fracture.

Secondly,

his investigation

egards

with the modeling of

the

behavior of both

confined

and unconfined

concretesubjected

o

cyclic

compression nd

tension

Section

2).

This

is the first

time

any

model

have

attempted

o model

cyclic behavior of concrete

in both

tension and compression.

The need

for

such

model is more

obvious

when

considering

shear

deformations

where

the tension

capacity

of reinforced steel

plays

an

important

role,

as

in

the Modified Compression

Field Theory

Collins

and Mitchell.

1992),and

the

Softened

Truss

Model

Hsu,

1993).

Section

4

deals

with

the Fiber Elements

modeling

of

the

moment-curvature

behavior

of a concrete ection

and

with

the

assessmentf deformations.

A cyclic strut-tie

model

is

developed

o

assess hear

deformations.

This

cyclic

sffut-tie

model for

shear

deformation,

which

makes

good

use of

the

comprehensive

onstitutive

models

developed

in

sections

and3, allows

to

simulate

he

behaviorof shear

dominated

members.

Section5 presents smoothmacro model that can be incorporatednto non-linear

dynamic

analysis

programs

o accurately

epresent he hysteretic

behavior

of concrete

elements.

The advantage f

this model,

over

previous

models,

ies

upon

the

capability of

the model

to represent ealistically

the hysteretic

behavior of concrete elements.

A

parameter dentification

procedure s also

presented,

which

is use for

the

automatic

identification of

the model

parameterso representa

provided

hysteretic

behavior.

This

l -5


26/298

automation

makes

the

use

of

a more comprehensive

macro model more

appealing

han

unrealistic

simpler

macro

models.

Section

6

presents he development

of spectralcharts

for both

elasto-perfectly

plastic structuresand typical bridge

pier

sffuctures.

The hysteretic

behavior of bridge

piers

is simulated

by

the

macro

model developed

n

section 5

which was calibrated o

simulate

he

behavior

of an

actual ull

size bridge

pier

testedcyclically.

The model was

also calibrated

o

simulate

he

hysteretic

behavior

of an analytically

produced

srmulated

experiment

using

actual

material and

section

properties

nto

the

fiber

model

column

analysis

program

developed

n

section

4.

The

development

of

reliable inelastic spectra

is

an

important aspect when

assessing

he

energy

and

ductility demands n

ductile structures.

A methodology or the

consideration

f energy-based

nelasticdamage ssessment

s

given

in

this

section.

This

approach

of

simulating stnrctural

behavior by

means

of a

Fiber Elements

model analysis

and

then

calibrate

a

macro model to represent

he

actual behavior

is

presented n this investigation

as

the most rational way

of

simulating

the

behavior of a

reinforcedconcrete trucnlre

without

the

costly

mplementation

more refined

procedure

asFinite Elements.

Finally

some

conclusions

nd recommendations

or

further

research

re

presented

in

the

last section.

This

investigation

has

shed some

ight into the need for some well

designed

experiments

o

look

into

the

behaviorof somespecific

variables.

l -6


27/298

Section

Hystereticnd

DamageModeling

f Steel

Reinforcingars

2.1 lntroduction

The hystereticbehavior

of

the reinforcing

and

prestressing

teel bars

influences

the hysteretic

behavior

of a

sffucturalconcrete

member. Fracture

of a

reinforcing

bar may

also

be defined as

ailure of

the member tself. It is

very

important

o

thus model

both

the

hysteretic

and

the

fatigue

properties

of

the

reinforcing

bars

accurately. Tests

performed

by

Kent and Park

(1973),

Ma

et

al.

(1976)

and Panthaki

1991)

were

used

o calibrate he

stress-strain

model advanced

herein. The degradingcharacteristicof

steels

with

leid

stresses

anging from 50

ksi to 120 ksi were

studied, and

damage relationships

were

incorporated

nto the model. The Menegotto-Pinto quation

1973)

used

by

Mander

et al

(1984)

s

used

herein o represent

he

oadingand unloading

stress-sfrain

elations.

2.2

Monotonic

tress-Strain

urve

Numerous estshave shown

hat the monotonic

stress-strain urve for reinforcing

steel

can be describedby

three

well defined branches. The corresponding elations for

stress

f,)

and

angentmodulus

E,)

after

Mander

et. al.

(1984)

are

given

below:

2.2.1

The

Elastic

Branch

0

(

e,

(

eu

f,

=

E,E,

P T

_

D S

(2-r)

(2-2)

2- l


28/298

wher

f

e:

er=f i

in which,e_u

yield

strain,

r:yield

stress,E,

:

Elastic

Modulus

of Elasticity.

2.2.2 TheYieldPlateau y( r-r r,1

f'

=fy

Et=0

in

which,

€r1

:

sffain hardening

train.

2.2.3

Strain

Hardened

ranch

s 2

r,n

f,=f*+(fy-f,,)

l##l'

r r -,

(

e :--*-\l

*

-f, l( '-;,)

Lt

=

Lsh

tgn[€ ,

€ ,

) l

1,

_1ry

where:

=

E,r13

- J

f r u - f ,

in

which,

,r, i, the

stress

t ultimate

tress nd

,u

:ultimate

(maximum)

tress.

These

relations

an

be

represented

y a single

quation

s

given

n

Eq.

(2-45)

2.3 TheMenegotto-Pinto

quation

The Menegotto-Pinto

1973)

(M-P

hereafter)

s

useful for

describing

curve

connectingwo tangents

ith

a

variable

adius

of curvature

t

the

intersection

oint

of

thosewo

angents,

sshownn Fig.2-1.

The

M-P

equations

expressed

s:

f ,

=f,

+

Eo(e,

e )

1Q+

I -Q

(2-7)

The angent

modulus

t any

point

s

given

by:

(2-3)

(2-4)

(2-s)

{2-6}

.,

.,


29/298

E*,

_

QEO

(2-8)

r*lr,9'-t:|-^

I Jrn-J

with a

secant

modulus connecting

he origin coordinates

eo,

fo)

and

the

coordinatesof

the

point underconsideration

e,,/) defined s:

t ,

=*=

E* .

dtr

{ _ {

/ s l o

h 1

l S € C - a

a

c s

-

L o

in which

€,

:

steel

strain,

f,

:

steel

stress,eo

=

sffain at

initial

point,

/,

:

sffessat

initial

point, E

o

:

tangent

modulusof elasticity

at

initial

point,

Q,

R

and

fs11

rc equation

parameterso control

the

shape

of

the curve.

It

should

be

noted

that as t is

presented,

q.

(2-7)

has he following

properties:

(1)

a slope

Eo at

the starting

coordinate

(eo,

fo),

(2)

it

approaches

he

slope

QEo

as

€5

oo.

For

computational

tractability

R needs

to

be

limited

to

about

25. This

essentially

epresents

bilinear

curve

given

by a single

equation.

To

use

this

equation

t

is necessary

o

develop

an algorithm

to

compute

the

parameters

Q,

*

andR.

A

procedure o

compute

hese

parameters

s

presented

n the next

section.

0.8

f _ f

':

':

0.6

t . - l

C N J

O

0.4

(2-e)

R

F

J

)

I

Fig.2-1

TheMenegotto-Pinto

quation

Eo@,

eo)

2-3


30/298

2 3.1Computation

f

Parameterc

,

f ^

and

R

Let

the

denominator

n

the M-P

equationbe such

hat,

-

,Oa l

I l - €,-e, l l^

A = l l - / - l p n = ' - - - - ' : t

t

L

I

l ,n- l l

)

The

derivative

of

I is therefore:

dA

_A( l -A -R)

de'

t '

- to

Eq.

(2-7)

canbe expressed

n terms

of

,,4

as:

f , =fo Eo(e,-d(e+y)

\ , ' r )

and

then the

derivativeof

f,

respect o

e,

gives

a

tangentmodulus

which is:

(2-10)

E,=#=s (e*+)-' #(+#)

(2-1

)

(2-12)

(2-13)

Fig.2-2 Different

Curves

Having he

SameStarting

and Ending Properties

A A

z-+


31/298

By substituting

q.

(2-11)

nto

(2-13)

and

earranging.

L

=

o

-'-.?

E

'

trR+l

By evaluating his equation&t 0, = Ey, andsolving for Q,

tf

-

o^^u'

(2-t4)

Q=

(2-1s)

(2-r6)

|

-A-8+r)

Solving

or

Q

in Eq.

(2-12),

Er..

A

l

t r

-n

Q =

o -

t -A- ,

Eq.

(2-15)

wasobtained

rom an equation

elated

o

the final

slope

{),

thus

his

equation

guarantees

hatat

the target

point the

slope

condition

s met Elef)

=

E.f

Eq.

(2-16)

was derived

rom

the

ordinate

quation o,

by satisfying

his

equation,

he

ordinate

ondition

s

met

f,(e)

=fJ.

To satisfy

othconditions,

t is necessary

o

equate

both

Eqs.

2-15)

and

2-16).

(2-r7)

I

where :

A- ' .

The

solution

rocedures as ollows:

fr-f

(1)

E. .

=;--

- 'f

-€ o

(2) R*o

=

t#

, the derivation f this expressrons

given

in the next

subsection

.3.2.

It is not

possible

o reach

he

point

(ef,ff)

with

the

slope

E7

with

a

value of

R

<

R6o.

Evaluation

f

the M-P equation

or

the

caseof

R

=

R,,;o s only

possible

y

taking he

imit of

the expression,

o a

value

of

R slightly

greater

hen

Rn;o

has

o

be

used,

n order

o apply

he

expression

s

t is

shown

n Eq.

(2'7)'

(3)

If R, io 0,

it means

hat

the three

points

are

aligned,

hus

take

Q:

I and

f n

=fr.

Thevalueof

R

need

ot o

be

modified.

1 _ n R + r

_

a ( l - a R )

n

Et-

E*#-

+Ei

'

--

'

-

0

'

r -d t -a

2-5


32/298

(4)

If R

<

R-o

then

akeR

:

R*n

+

0.01

(5)

Solve

or the

valueof a

in the ollowing

expression:

E

-

E, , "4

+

E

"a( , -

aR

-

o

'

t - a I - a

To find

the

value

of a

the ollowing

procedure

s

used:

(a)

Define afunctionf(a) as:

f(a)=r.,-t,,#*t,ff

@)

Evaluate

(1-e)

andf

e),

where is

a small

(c)

If

f

(I-e)

*

f

(e)

>

0,

no

solutions found,

so

ofe and epeat tep b) .

(e)

Takeasan nitial

estimate:

R-o

ao=_f

(2-1e)

value

=0.01).

decrease

he

vaiue

(2-18)

(2-20)

(d)

If

f

(l-e)*f

(e)

<

0

then

a solution s found

n

this

interval"

The

quadratically

onverging

ewton-Raphson

rocedure

an

hen

be used

o

find

the

solution.

(f)

If

/(a")"/(l

-e)

<

0

then

replace

ao by

J,r.

untii

the

inequality s false o

ensure

roper

convergence. f this

condition s not met the

algorithm

will find a solutionoutside he meanineful anee.

(g)

With

o" u,lninitial-estimate the following

recursive

expression

shouldbe applied until convergence

s

met. It is important o note that

the

functionf

(a)

hasa singularityat d= l, so the valueof Aa shouldbe the smallerof 0.5(1

-

a") and

0.00.

2f(a) A,a

(2-2r)

i + l

=

a i -

f

(a,

+

A,a)

f

(ai

-

La)

(8)

After

the value

of a has

beendefined

hen,

I

P . ;

^ -

( l

- 4 " ) "

u

-

--v-

(9) The valuesof fo andQ are hencalculated s:

2-6

(2-22)


33/298

J"n

=

fo

+

f@

-

e")

(2-23)

Er..

t r

-u

Q=ft Q-21

2.3.2 Menegotto-Pinto

quation imiting

ase

In step

2

of

the

procedure

outlined above, a

factor

R^o was introduced. The

derivation

of

that factor and

the relation

of

the Menegotto-Pinto

equation

to

a

power

equation

s the

subject

of

this

subsection.

The Menegotto-Pinto quation an be expressed y:

(2-2s)

where:

If the

curve

s to

pass

hrough

xr,yr),

it canbe

rewritten

as:

1

l f

- f

o

-

8 r . .

-

A t l -Q

Eo

x1-xo

-;=

a+;

Q'26)

and

ts

derivativeas:

=n*0-Q)

E o

Y '

A R + l

(

t Y - v

I

,q=lr+

r"---=- lR)F

\

|

, v c h l o l

. /

rf"

e

=

A-t

(2-29)

then

by solving

or

Q

in Eqs.

2-26)

and2-27),he ollowing

expression

s

obtained:

n r

|

-aR* t

-

a ( l

- cR )

n

Er-E*"ft+E"-ft/=o

(2-30

(2-27)

(2-28)

,''

1


34/298

By solving

for

a

in this equation,

he

parameters

"1,

dnd

Q

are

given

by:

/ \

l ch= lo *Eo (x1 -xo ) l

J

, I

l ' ( t

o^)8,

and.

?

-

o^*'

Q =

" o

|

-

AR+l

Eq.

(2-30)

cannot

e

evaluated

s t

is

written

or

Q:1, but

i t

The limit

value of

the fraction

n the

second

erm s:

(2-31)

(2-32)

presents

limit.

(2-33)

(2-34)

(2-3s)

(2-36)

l i rn lo** ' =R+ I

a--+ l- I

-

A

while

the other

imit is:

, .

a ( l

-

a R )

h

lllil

---:-

=.t(

a - + l

l - A

So

the limit for

the

equation

when a

-+

I is:

Ey-

E,""(R+ )

+E,R

=

0

Solving

for R,

the following equation

or

the

critical

value

of

R can be derived:

^"r=ffi

This value, as

can be shown

numerically, epresents

he minimum value that R is

to have, so that a solution o meetthe conditionsof both slope and ordinatevalue at the

ending

point.

What is of interest

now, is

to know what the limit

for the

original equation

would

be.

Both

y"o

and

Q

tend to

infinity

as a

tends o one.

Eq.

(2-25)

can

be expressed

in

terms

of

a as:

=

lo

+ Eo(x xo)fm+

Q(l

-

m)l

where,

2-8

(2-37)


35/298

[ , , I

x-xol^t -o^ ' l i

L'-

|

xrx"

,^

I

(2-38)

(2-3e)

(2-40)

(2-4r)

(2-42.)

(2-43)

(2-44)

When + l,

l im

m=

a- )1-

The imi t of

QQ

-

m),

s

a

complicated xpression:

Er

)yort-m)=jjl.#

E,

J 7

r

I t K

L o

l _ T - r , l "

=

- t - l

R+l

l x1 -xo l

l -

[ ' .

So,

Eq.

2-37)

anbe expressed

s:

The inal form of

the

imitingcase f

the

Menegotto-Pinto

quation

s:

=

o

+

E

o(x

x

)

+

A(x

-

x

")lx

-

x

olR

with,

I

x-xo

l ^

-o^ lF

ry4"1 "- )

and,

, -E - f -E* "

I \

-

F

Ln"

-

Lo

, - E n " - E o

" - ,

t R

l x t -

xol

Eq.

Q-aD

is

dealt with

in more detail n

section3.6.3.

It is worth noting here hat

this

equation

epresentshe most

"relaxed"

of all

the

curves

given

by

the M-P

equation,

but at

the

same

ime, the M-P equationcannotbe evaluated

or this

case,as it is

a

limit

expression.

2-9


36/298

2 4 Cyclic

Properties

f

Reinforcing

teel

In

this

section

a

universally applicable

cyclic

stress-stralnmociel

s

advanced

or

ordinary

reinforcing

and

high strength

presffessing

ars. The model is

composed

of

ten

different rules, five for the tensionside and five for the compression rde. Each of the

rules

s

described eparately

n the following

sections.

2.4.1Envelope

ranches

Rules

and2)

The

envelopebranches re definedby the monotonic

sffess-strain

elation

which is relocated and scaled

to

simulate strength

degradation. The

shape

of

the

monotonic branch is kept

intact,

except hat

at

the

points

of

reversal

a scale factor

is

calculated. This combinedmodel ensures

egradationwithin local

cyclic,

a

phenomenon

not been modeled

before.

The model was

calibratedusing

experimental esults

given

by

Panthaki

(1991).

The

sffess-sffain

elation

for

the

tension

envelope

curve

can be

expressed s a single expression y:

Rule I

(Tension

Envelope Branch)

r=

;ff *

.=P rr;r;tl,|ffi l-]

L'-(.

.

.]

l

Q-asa)

-T

'

(2-4sb)

(2-45c)

(2-4sd)

where: t r r = t r - € j ,

o.=t:,ffi

in

which €t^

=

location

of

the tension

nvelope ranch. Eq.

(2-a5)

s

shown

plotted

n

Fig. 2-3. Also

shown

n this figure

s

the

compression

nvelope ranch

defined n

an

analogous

orm

as

ollows:

2- t0


37/298

Rule

2

(Compressionnvelope

ranch)

^

=

;ffr 1-

s'9

i'

'

-r'-'l'

|

fi

l'

L'-(.t J ]

(2-46a\

(2-46b)

(2-46d)

(2-46e)

p--

t

p-

t r -

u I

-

E,

sign(e,,

ess)

+

I

sign(e,, e;)E,n

Fig. 2-3 Tensionand Compression

nvelope

Curves

f;

-f'

f;

-f;

['.[?) ]

where:

€ = tt

- t l t

n-

-

F-

g;

-

etl,

y

-

'h

-f;

_f;

in which Ei^

:

locationof

the compression nvelope ranch.

2-1r


38/298

2.4.2 Reversal

ranches

Rules

and

4)

When a

reversal

akes

place

on an envelopebranch

a

reversal

cun'e connects

his

point

of

reversalwith a

target

point

on

the

opposite

envelope.

The curve

that

connects

these wo points will be referred o as a reversalbranch. In general, eversalbranchesare

completely

defined

by

the

extremum

points:

maximum excursion into

the tenslon

envelope

branch

€*o

,

and

maximum excursion nto

the

compressionenvelopebranch

€*n,

(Fig.

2-6). If a

reversal

akes

place

rom within the

yield

plateau

on

the tension

envelope

curve at

a coordinate

e;,

l;),

with

f;

=fi,

then

e'" .,

is

definedas:

€max

=

e;

-e]o^

The

target

strain

on

the compressive

nvelope urve

s

calculated s:

(2-47)

(2-48)

(2-4e)

(2-s0)

(2-s1)

(2-s2)

(2-s4)

(2-ss)

(2-s6)

where,

and

with,

While the targetslope s givenby:

with:

E n = E i ^ * t . l n

t-io

=

e;

+pr(e-rh-er)

f"-

e;^=ei-E

t 'no

-€J

n - -

' '

-

eIo-ei

L r -

-

- u

t

(

'

l )

**P'lr;-

E,

Ei^

=

€i

k;,,

+

euQ

-

k;e")

f+

eI=eI^+e n-?

L S

et=eX^*rmax

I*

E,

and

the target

sftess

f

the

leld

stress n

the

compressive nvelopebranch

(Fig.2-5).

In

the

case when

the reversal

akes

place from the

strain

hardened

curve

of

the tension

envelopebranch,

hen Eqs.

(2-49)

through

2-52)

aremodified as follows.

The

straine6o

is taken

as

he actual

maximum excursion

within

the

compressive nvelopebranchbut,

l r -" l t

l r ; l

(2-s3)

The

shifted

riginabscissa

or

thecompressionnvelope ranch

s

calculated s:

2-12


39/298

in

which

k; , is a

factor

o locate

he

compression

nvelope

ranch

between

he

ej and

ei

as shown

n

Fie. 2-5. and

was

ound

o

be:

/ \

le-* i

Freu

=

exp

I

--;

1

[

5000

il- )

Finally

he

arget

tress

ndslope

[

and

E;

are

calculated sing

Eq.

Q-aQ.

for the

oading

reversal ranch,

he

shifted

ension

riginstrain

s

grven

by:

ei^

=

e;(l

-

kl,,)

+

e

b

kle

r-

t . .

E i = E i ^ * E - r n -

E r :

f

/ m ' ^

E i = E i ^ f € - i o -

, ,

( , , \

I le-i ' | |

Nreu=expl-------

t I

[

5000

il-

)

with:

where:

(2-s7)

Similarly,

(2-s8)

(2-se)

(2-60)

(2-6r)

(2-62)

envelope

ranchs

the initial Young's

(unloading)

can be

Then

the target

strain

on

the

tension

envelope

ranch

s

given

by:

eL

=et^*t.r*

In a

similarway,

he

arget

tress

,i

andslope

EI on

the ension

calculated

sing

Eq.

Q-

5) .

Experiments

performed

by

Panthaki

(1991)

have

shown

that

modulus

at

the

point

of reversal

rom

the tension

envelope

branch

expressed

s:

E

=

(1

-3

Leo)E,

(2-63)

While,

or a

reversalrom

he compressionnvelope ranch

loading),

he nitial Young's

modulus anbesivenby:

EI

=

(l

-A€,)E,

(2-64)

The

M-P

parameter was also

ound

expressed

s:

R-

=rc(

\

to

be

f i \ t

- l

E,)

a functionof

the

vield

sness.

hat

can be

(1-

l0Ae, )

2-13

(2-6s)


40/298

for

the unloadins

branch.

and

(2-661

where

Leo

:

sffain

amplitude

for

the

cycle

and

E,

:

initiai Youn-e's

modulus

for the

reversalbranch,as

shown

n Fig.

2-4. Analytical

calibrationof

these

variables

are

shown

in

Figs.2-7

to 2-10 rom experiments

y

Panthaki

1991),

and

Figs.2-l l to 2-i4

show

someof

the

actualexperimental

oops hat

rvere

used

o fit the M-P equation.

The

unloading

and

unloading

branchare define

as:

Rule

3

(Unloading

Reversal

Branch)

Eol=Etrm*tln*

fot

=f*

Eot

=

Ei

En

=Elo

r _ f -

J

b3

-J

a

En=EL

The

nitial

slope

E; and

he

Menegotto-Pintoquation

arameter

-

are unctions

f the

strainamplitude

Aeo

of

the

oop,Eqs.

2-63)

and(2-65),

hich s defined s:

R.

=

20(*)

,

,

-

2oAe,

c . - _ E a 3

Aeo

=:--

L

Rule 4

(Loading

Reversal

ranch)

€ a + = E i n * t ^o

fo+

=fr

o

Eoq

=

El

E M = E L

fiq

=f i

Etc

=

EL

Leo=lryl

whereEI andR+ are calculated

sing

Eqs.

(2-64)

and

2-66),

respectively, y

havrng.

(2-67)

(2-68)

(2-6e)

2- t4

d2-10)


41/298

2Leo-------2

Fig.2-4 Effectof the StrainAmplitude of the Reversal

on

the Equation

Parameters

2.4.3

Returning

ranches

Rules

and6)

When

partial

unloading

n

the reversal nloading

ranch

rule

3)

takes

place,

he

reloading

ranch

will be

called

oading eturning

ranch

rule

5).

An

analogous

ranch

will exist

when a

reversal

akes

place

on

the

loadingreversalbranch

(rule

4),

and

unloading

s done

hrough

he

unloading

eturning ranch

rule

6), asshown

n Fig. 2-15.

At the occurrence

f

a

reversal

on rule 3, rule

5 will start

and

the target strain €as

s

calculated s:

with,

€ns=EI^*€ro+Lele

Ae "=€a3ta5 L

t .2E,

f.:

0


42/298

f A

,,a'

.o

f

-

,

G,n'Jt

\ t ;

a j ' a

(

E;.

f;

)

te;,fr-

)

Fig.

2-5

Reversal rom Yield Plateau

€max

---------)

Fig.

2-6 Definitionof

the ReversalUnloading

Branch

€s

T

I

f

/

max

f^o

I

I

J

./.- o


43/298

+

Eo

E;

1 2

1

N R

0.6

0.4

v . z

0

1 A

1 . 6

1 . 4

1 . 2

I

N R

Fig.2-7

Effect

of

the

Strain

Amplitude

of

Loop on

the

Initial

Modulus

and

R Parameter

or

Reinforcing

Bars

(fu:

53

ksi)

(Loading)

R

N A

0 . 4

v . 1

0

2-17


44/298


45/298

+

Eo

E"

1 . 2

1

0.8

0.4

u.z

0

? E

4

0 . 5

t . c

4 . 5

5

e"

(%)

(a)

R

2.s

0 .5

n

z

I

z.?

. 5

A

4.5

5

e"

(%)

o)

Fig.

2-9 Effect

of

the Strain

Amplitudeof

Loop

on

the Initial

Modulus

and

R Parameter

or High

Strength

Bars

({

=

123ksi)

(Loading)

2-t9


46/298

t . z

0 . 8

f

L O

L a

( )

4 4 . 5

€a

3 . 5

. 5

6

(a)

2 . 5

l 4

R

? q

2 . 5

. 5

(b)

4 . 5 5

eo

( )

Fig.

2-10 Effect of

the Strain

Amplitudeof

Loop

on

t-he nitial Modulus

and R

Parameter

or High Strength

Bars

(1,

:123

ksi)

(Unloading)

2-24


47/298

Fig.2-f

l Fitting

of

M-P Equation

to

a

Loading

Loop of

Reinforcing

Steel

Bars

(

{,:53

ksi)

8 o i

6 0*

l

l

2 0*

Experiment

Curve itting

-0 .015

-0.005

0.01

0.005

0.0 1

Fig.2-12

Fitting

of M-P

Equation

o

an Unloading

Loop of

Reinforcing teel

Bars

{:

53

ksi)

2-21


48/298

1 5 0

Experiment

Curve

itting

-0.02

-n nn6 fr

Y

- h t t

+

- 1 0 0 -

- 1 5 0

n n l t r

u .uz

Experiment

Curve

itting

/

/:

/,

Fig. 2-13

Fitting of

M-P Equation

o

a

LoadingLoop

of

High StrengthSteel

Bars

{:

123ksi)

1 4 n

. -

l

100

-l-

I

-0.005

q

-50

-U .UZ

- 0 .015 -0.01

0.005

/

n n l - / n n l (

0.02

Fig. 2-14

Fitting of

M-P Equation o

an Unloading

oop of

High StrengthSteel

Bars

({

=

123

ksi)

. , , , . ,

L L L


49/298

In a similar

way, a

partial

oading

rom the oading

reversalbranch

rule

4), which defines

ruie

6.

is calculated

s:

Eua=Ei^ t t ' , ,

+Le,e

(2-74)

(2-75a)

(2-7sb)

with,

Le r

= €a4 - Ea6

*

{-

0

>

Ae,. :+

3 L s

2.4.4 First

Transition

ranches

Rules

and

8)

The curve

followed

after

a reversal

from an envelopebranch

curve has been

named

eversalbranch,

he one

followed by a reversal

rom

a

reversalbranch

s

called he

returning

branch

The curve

then followed after

a reversal

from

a

returning branch is

called

the

irst

transition

branch

and a

reversal rom

this

will

lead

to

a second

ransition

branch.

These ive

typesof curvesare

llustrated

n Fig. 2-15.

It

shouldbe

noted

hat the

reversaland

the returning branches

orm

a

closed

oop and

the

first and second

ransition

branches

ycle

nside

his loop.

{

1 , 2

3 ,4

5 ,6

7 , 8

9 ,10

Envelope ranches

Reversal ranches

Returning

ranches

First ransitionranches

Second

ransitionranches

Fig.

2-15

Sequence

f

PartialReversals

2-23


50/298

The targetsffain

of rule 5

eas s

given

n Eq.

(2-71),

his

equation s

different rom

the

starting

sffain of

rule 3

€.ot,but

if rule

5

would have reached

he

end

the

a

reversal

from

this

point

would

have

been

he

starting

point

for rule

3 agaln.

It means hat

in the

case of

a

reversal from rule 5

(incomplete

oading),

a

redefined

rule 3 needs to

be

calculated.

The

starting

strain

for this redefined

ule ought

to

be between

he previous

startingsffain and

the

targetstrainof rule 5.

By

using a

linear

proportion,

c *

_ . . - E a | - E a 5

r . - t b 5 - € o 7

cai

:

Yb5

ebs

-

%

-

eol€;;4;

(2-76)

It can be

noted that if the reversal

happens

when rule 5

has

ust

started Eat

=

Ea5,

hen

from

Eq.

(2-76)

Elt

=Eoz,

what

means hat

an

insinuation

of reversal

occurredat rule

3,

so

the

path

followed shouldbe on

the

unchanged

ule

3.

While if the reversal

occurredat

the end of rule 5 when Eai=Ebs, hat means t is a

PhD Thesis G a Chang

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