plasticit - school of engineering · plasticit y in reinforced concrete bridge piers eric m hines f...

Experimental Spread of Plasticity

in Reinforced Concrete Bridge Piers

Eric M� Hines Frieder Seible

August ��

ABSTRACT

Experimental values pertaining to the spread of plasticity� such as the equivalentplastic hinge length� are presented for a variety of large�scale structural tests� Thegeometry of these large�scale test units includes standard circular bridge piers� struc�tural walls and hollow rectangular bridge piers� The reported experimental valuesare mostly calculated values that are based on assumptions about the test unit be�havior� This report outlines these assumptions and discusses their relevance to theactual inelastic force�displacement behavior of the reinforced concrete bridge piers inquestion�

ii

Contents

� Introduction �

� Experimental Characterization

of Lp �

�� Observed Mechanisms ofFlexure�Shear Deformation � � � � � � � � � � � � � � � � � � � � � � � � �

�� The Argument for Curvature � � � � � � � � � � � � � � � � � � � � � � � �� Relating � to � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Approach to the Experimental Data � � � � � � � � � � � � � � � � � � � �

�� Plastic Curvature Distribution � � � � � � � � � � � � � � � � � � �� Average Curvature Proles � � � � � � � � � � � � � � � � � � � � �� Calculation of ��c and ��s � � � � � � � � � � � � � � � � � � � � � �� Calculation of Lsp and L�

sp � � � � � � � � � � � � � � � � � � � � � �� Explanation of Appendices B � D � � � � � � � � � � � � � � � � � � � � ��

� Conclusions ��

A Test Setups and Properties ��

B Circular Columns ��

C Structural Walls ��

D East Bay Skyway Piers ��

iii

List of Tables

A�� General test unit properties� � � � � � � � � � � � � � � � � � � � � � � � ��A�� Test unit material properties� � � � � � � � � � � � � � � � � � � � � � � ��A�� Test unit yield properties� � � � � � � � � � � � � � � � � � � � � � � � � �

B�� Average experimental plasticity values �� B�� Peak curvature values �� B�� Flexural strain values �� B�� Average experimental plasticity values �� B�� Peak curvature values �� B� Flexural strain values ��

C�� Average experimental plasticity values Test �A �� C�� Peak curvature values Test �A �� C�� Flexural strain values� Test �A �� C�� Average experimental plasticity values� Test �B �� C�� Peak curvature values Test �B� �� C� Flexural strain values� Test �B �� C�� Average experimental plasticity values Test �A �� C�� Peak curvature values Test �A �� C�� Flexural strain values� Test �A �� C�� Average experimental plasticity values Test �B �� C�� Peak curvature values Test �B �� C�� Flexural strain values� Test �B �� C�� Average experimental plasticity values Test �C �� C�� Peak curvature values Test �C �� C�� Flexural strain values� Test �C �� C�� Average experimental plasticity values Test �A �� C�� Peak curvature values Test �A �� C�� Flexural strain values� Test �A �� C�� Average experimental plasticity values Test �B �� C�� Peak curvature values Test �B �� C�� Flexural strain values� Test �B �� C�� Average experimental plasticity values Test �C �� C�� Peak curvature values Test �C �� C�� Flexural strain values� Test �C ��

v

D�� Average experimental plasticity values� SFOBB LPT �� D�� Peak curvature values� SFOBB LPT �� D�� Flexural strain values� SFOBB LPT �� D�� Average experimental plasticity values� SFOBB DPT�L� �� D�� Peak curvature values� SFOBB DPT�L� �� D� Flexural strain values� SFOBB DPT�L� �� D�� Average experimental plasticity values� SFOBB DPT�T� �� D�� Peak curvature values� SFOBB DPT�T� �� D�� Flexural strain values� SFOBB DPT�T� ��

vi

List of Figures

�� Schematic representation of proposed Bay Area bridge piers� � � � � � �

�� UCSD Column �A crack pattern� � � � � � � � � � � � � � � � � � � � � �� UCSD Column �A strain proles� � � � � � � � � � � � � � � � � � � � � �� UCSD Column �A �plane sections�� Test �A �Hines et al� �� curvature at �� Test �A �Hines et al� �� curvature at �� UCSD �A Hines et al� �� Detail of curvature instrumentation� �a�

elevation� �b� section and rotation scheme� � � � � � � � � � � � � � � � �� Average curvature proles Test �A �� Experimental values of L�

sp for Test �A from Hines et al� ��

A�� Well�conned circular column �TU�� test setup east elevation� columnsection and curvature instrumentation layout� ��

A�� Poorly�conned circular column �C�� test setup east elevation� columnsection and curvature instrumentation layout� ��

A�� Test Units �A� �B setup� east elevation �� A�� Test Units �A� �B� �C setup� east elevation �� A�� Cross sections of Test Units �A� �B� �A� �B and �C with reinforcement

�� A� Test Units �A� �B� �A� �B and �C� curvature instrumentation layout�

east elevations �� A�� Test Unit �A setup� east elevation �� A�� Test Unit �B setup� east elevation �� A�� Test Unit �C setup� east elevation� � � � � � � � � � � � � � � � � � � � ��A�� Cross sections of Test Units �A� �B and �C with reinforcement �� A�� Test Units �A� �B and �C� curvature instrumentation layout� west

elevations �� A�� SFOBB Longitudinal Pier Test �SFA�� Test setup� isometric view �� A�� SFOBB Diagonal Pier Test �SFB�� Test setup� isometric view �� PT

rods not shown�� A�� SFOBB Longitudinal Pier Test Unit and Diagonal Pier Test Unit �SFA�

SFB�� cross section with dimensions and reinforcement �� A�� SFOBB Longitudinal Pier Test �SFA�� Curvature instrumentation� west

elevation and section ��

vii

A�� SFOBB Diagonal Pier Test �SFB�� Curvature instrumentation� southelevation and section ��

B�� Average curvature proles �� B�� Curvature proles at �� and �� B�� Curvature proles at �� and �� B�� Curvature proles at �� and �� B�� Average �exural strain proles �� B� Pre�yield �exural strain proles �� B�� Post�yield �exural strain proles �� B�� Average curvature proles �� B�� Curvature proles at �� and �� B�� Curvature proles at �� and �� B�� Curvature proles at �� B�� Average �exural strain proles �� B�� Pre�yield �exural strain proles �� B�� Post�yield �exural strain proles ��

C�� Average curvature proles Test �A �� C�� Curvature proles at �� and �� Test �A �� C�� Curvature proles at �� and �� Test �A �� C�� Curvature proles at �� and �� Test �A �� C�� Average �exural strain proles� Test �A �� C� Pre�yield �exural strain proles� Test �A �� C�� Post�yield �exural strain proles� Test �A �� C�� Average curvature proles Test �B� �� C�� Curvature proles at �� and �� Test �B �� C�� Curvature proles at �� and �� Test �B �� C�� Curvature proles at �� and �� Test �B �� C�� Average �exural strain proles� Test �B �� C�� Pre�yield �exural strain proles� Test �B �� C�� Post�yield �exural strain proles� Test �B �� C�� Average curvature proles Test �A �� C�� Curvature proles at �� and �� Test �A �� C�� Curvature proles at �� and �� Test �A �� C�� Curvature proles at �� and �� Test �A �� C�� Average �exural strain proles� Test �A �� C�� Pre�yield �exural strain proles� Test �A �� C�� Post�yield �exural strain proles� Test �A �� C�� Average curvature proles Test �B �� C�� Curvature proles at �� and �� Test �B �� C�� Curvature proles at �� and �� Test �B �� C�� Curvature proles at �� and �� Test �B �� C�� Average �exural strain proles� Test �B ��

viii

C�� Pre�yield �exural strain proles� Test �B �� C�� Post�yield �exural strain proles� Test �B �� C�� Average curvature proles Test �C �� C�� Curvature proles at �� and �� Test �C �� C�� Curvature proles at �� and �� Test �C �� C�� Curvature proles at �� and �� Test �C �� C�� Average �exural strain proles� Test �C �� C�� Pre�yield �exural strain proles� Test �C �� C�� Post�yield �exural strain proles� Test �C �� C�� Average curvature proles Test �A �� C�� Curvature proles at �� and �� Test �A �� C�� Curvature proles at �� and �� Test �A �� C�� Average �exural strain proles� Test �A �� C�� Pre�yield �exural strain proles� Test �A �� C�� Post�yield �exural strain proles� Test �A �� C�� Average curvature proles Test �B �� C�� Curvature proles at �� and �� Test �B �� C�� Curvature proles at �� and �� Test �B �� C�� Average �exural strain proles� Test �B �� C�� Pre�yield �exural strain proles� Test �B �� C�� Post�yield �exural strain proles� Test �B �� C�� Average curvature proles Test �C �� C�� Curvature proles at �� and �� Test �C �� C�� Curvature proles at �� and �� Test �C �� C�� Average �exural strain proles� Test �C �� C�� Pre�yield �exural strain proles� Test �C �� C�� Post�yield �exural strain proles� Test �C ��

D�� Average curvature proles� SFOBB LPT �� D�� Curvature proles at �� and �� SFOBB LPT �� D�� Curvature proles at �� and �� SFOBB LPT �� D�� Curvature proles at �� and �� SFOBB LPT �� D�� Average �exural strain proles� SFOBB LPT �� D� Pre�yield �exural strain proles� SFOBB LPT �� D�� Post�yield �exural strain proles� SFOBB LPT �� D�� Average curvature proles� SFOBB DPT�L� �� D�� Curvature proles at �� SFOBB DPT�L� �� D�� Curvature proles at �� SFOBB DPT�L� �� D�� Curvature proles at �� SFOBB DPT�L� �� D�� Curvature proles at �� SFOBB DPT�L� �� D�� Average pre�yield �exural strain proles� SFOBB DPT�L� �� D�� Average post�yield �exural strain proles� SFOBB DPT�L� �� D�� Pre�yield �exural strain proles at positive peaks� SFOBB DPT�L� ��D�� Pre�yield �exural strain proles at negative peaks� SFOBB DPT�L� ��

ix

D�� Post�yield �exural strain proles at positive peaks� SFOBB DPT�L��

D�� Post�yield �exural strain proles at negative peaks� SFOBB DPT�L��

D�� Average curvature proles� SFOBB DPT�T� �� D�� Curvature proles at �� SFOBB DPT�T� �� D�� Curvature proles at �� SFOBB DPT�T� �� D�� Curvature proles at �� SFOBB DPT�T� �� D�� Curvature proles at �� SFOBB DPT�T� �� D�� Average pre�yield �exural strain proles� SFOBB DPT�T� �� D�� Average post�yield �exural strain proles� SFOBB DPT�T� �� D�� Pre�yield �exural strain proles at positive peaks� SFOBB DPT�T� ��D�� Pre�yield �exural strain proles at negative peaks� SFOBB DPT�T�

�� D�� Post�yield �exural strain proles at positive peaks� SFOBB DPT�T�

�� D�� Post�yield �exural strain proles at negative peaks� SFOBB DPT�T�

��

x

List of Symbols

ACI American Concrete InstituteASCE American Society of Civil EngineersCaltrans California Department of TransportationDPT Diagonal Pier Test �SFOBB East Bay Skyway�LPT Longitudinal Pier Test �SFOBB East Bay Skyway�SFOBB San Francisco�Oakland Bay BridgeUCSD University of California� San Diego

Ag gross area of a sectionco cover concrete depthc�o depth from concrete surface to extreme steel berC net compressive forcedb bar diameterD member total section depthD� distance between extreme ber steel andconned concrete strainsD� distance between curvature potentiometersE elastic modulusf �c unconned concrete strengthfu ultimate steel stressfy steel yield stressF lateral force applied to a test unitFy ideal yield forceF �y rst yield force

I moment of inertiaL shear span �L � M�V �Lgb gage length at base of columnLp plastic hinge lengthLpc compressive plastic hinge region lengthLpr plastic hinge region lengthLpt tensile plastic hinge region lengthLsp strain penetration lengthL�sp articial strain penetration value

M bending moment

xi

My ideal yield moment� yield momentM �

y rst yield moment �directional components are namedsimilar to ideal yield moment components�

M�V D member aspect ratioP axial load� steel strain hardening exponentP

f �

cAg

axial load ratio

T net tensile forceTy net tensile force at ideal yieldT �y net tensile force at rst yield

� test unit top displacement�c compressive curvature potentiometer displacement�ef elastic �exural displacement�es elastic shear displacement�f �exural displacement�p plastic displacement�pf plastic �exural displacement�ps plastic shear displacement�s shear displacement�t tensile curvature potentiometer displacement�y ideal yield displacement��

y rst yield displacement�c extreme ber conned concrete strain compatible with

linear distribution method�c� extreme ber conned concrete strain compatible with

base curvature method� assuming no strain penetration��c extreme ber conned concrete strain

calculated independently of curvature�s extreme ber steel strain compatible with

linear distribution method�s� extreme ber steel strain compatible with

base curvature method� assuming no strain penetration��s extreme ber steel strain calculated independently

of curvature�sh steel strain at rst hardening� member rotation�p plastic rotation�� displacement ductility�� curvature ductility�l boundary element longitudinal reinforcement ratio�s volumetric connement ratio� curvature� strength reduction factor�b base curvature

xii

�b� curvature calculated from gages at the column base�bsp base curvature assuming strain penetration�y ideal yield curvature��y rst yield curvature

xiii

Chapter �

Introduction

Moment�curvature analyses are widely used in California as the basis for assessing

the non�linear force�displacement response of a reinforced concrete member that is

subjected to inelastic deformation demands under seismic loads �� In the

seismic design of bridges� it is consistent with the principles of capacity design ��

�� to design bridge piers as ductile members that are expected to form plastic

hinges under seismic loads �� The inelastic displacement capacity of a bridge as

a whole is therefore often assumed to depend heavily on the inelastic displacement

capacities of the individual piers supporting the bridge�

Ideally� the inelastic force�displacement response of a bridge pier can be approxi�

mated by applying Bernoulli�s hypothesis that plane sections remain plane and per�

pendicular to one another� Bernoulli�s hypothesis results in the well�known equation

� �M

EI��

where �� the curvature at a given section� depends on the ratio of M � the moment

at that section� to EI� the combined material and geometric �exural sti�ness of

the member at the same section� Accounting for non�linear material behavior in the

moment�curvature analysis and assuming small displacements� the total displacement

of a bridge pier subjected to given loads is approximated according to beam theory�

This is accomplished by integrating the curvatures over the pier height as

� �

Z L

�

��x�xdx ��

where � is the displacement at the top of the pier� L is the length of the pier� ��x� is

the curvature distribution along the height of the pier� and x is a variable representing

distance along the length of the pier measured from its base� This ideal approach�

�

accounting for non�linear material behavior� has been discussed by researchers since

the �� s ��

For decades� however� it has been well�established that overall member behavior

and boundary conditions complicate the inelastic deformation response of reinforced

concrete members by providing additional modes of inelastic deformation ��

�� As a result� Equations �� and �� tend to under�predict the ultimate

displacement capacity of reinforced concrete bridge piers by accounting only for one

of the phenomena associated with the spread of plasticity in reinforced concrete� the

moment gradient� Typically� a short member will have a high moment gradient�

This results in a quick transition between the ultimate and yield moments and� as a

consequence� plasticity spreads very little� A longer member� on the other hand� will

have a lower moment gradient� resulting in a longer transition between ultimate and

yield moments and greater spread of plasticity�

The problem of overall member behavior manifests itself through the shear transfer

mechanism inside a plastic hinge region� The result of this shear transfer which

in�uences the spread of plasticity is called tension shift� The term �tension shift�

refers to the tendency of �exural tensile forces to decrease only minimally over a

certain distance above the base until these forces can be e�ectively transferred to the

compression zone by adequately inclined compression struts� Thus� maximum �exural

tension is observed not only at the base of a pier but also is observed to be �shifted�

a certain distance along the height of the pier� This complicates the relatively simple

relationships between moment� curvature� rotation� and displacement expressed in

Equations �� and �� by falsifying the assumption that plane sections remain plane

and perpendicular to one another under bending demands�

The problem of boundary conditions manifests itself in the form of strain pene�

tration into the footing or bentcap� The term �strain penetration� refers to the fact

that longitudinal bar strains can reach signicant inelastic levels some distance into

the footing� These strains taper to zero over a length required to develop su�cient

bond strength for anchoring the bars under ultimate tensile loads� The accumulation

of such strains inside the footing allows the �exural tension zone at the base of the

pier to lift o� the footing� This results in a nite rotation at the base of the pier�

In summary� three independent phenomena have been observed to in�uence plastic

hinging in reinforced concrete members� These are the moment gradient� tension

�While Equation �� does not explicitly include rotation� it is implied as the integral of curvature

only�

�

shift and strain penetration� The moment gradient and tension shift in�uence the

spread of plasticity in the member and strain penetration in�uences the concentrated

rotations at the member�s boundaries� The moment gradient depends primarily on

the length of a member�s shear span� and the ratio of yield moment to ultimate

moment� Tension shift depends primarily on the height reached by an �adequately

inclined� compression strut and depends therefore primarily on the member�s section

depth and its level of transverse reinforcement� Strain penetration depends on the

ability of the footing to anchor the longitudinal bars in tension and can therefore be

related to the longitudinal bar diameter�

Alternatives to Equations �� and �� for predicting inelastic deformation have

long been proposed in various forms� The form that has had the greatest impact

on seismic design in California has been that of the equivalent plastic hinge length

�� This alternative assumes a given plastic curvature to be lumped

in the center of the equivalent plastic hinge� The length of the equivalent plastic hinge

is the length over which this plastic curvature� if assumed constant� is integrated to

solve for the total plastic rotation� This length is referred to as either the �equivalent

plastic hinge length� or simply the �plastic hinge length��

The word �equivalent� implies that this length has no physical meaning and is

simply a number� calibrated according to experimental results� to produce the correct

plastic rotation and plastic displacement from a given plastic curvature� Mathemati�

cally� it has been denoted by the symbol Lp and has been applied in the equation for

plastic displacement as

�p � �pLp

�L�

Lp

�

��

where it is employed as a multiplier with no direct physical signicance� This docu�

ment refers to Lp simply as the �plastic hinge length��

In addition to Lp� this document refers to Lpr� or the length of the plastic hinge

region� The plastic hinge region length is the length of pier over which plasticity

actually spreads� Inside the plastic hinge region� �exural strains are observed to be

inelastic� Outside the plastic hinge region� �exural strains are observed to be elastic�

The plastic hinge region length refers to a length along the pier only� and therefore

does not account for any penetration of inelastic strains into the footing or bentcap�

It is natural and logical to expect that while Lp is not equivalent to Lpr� a value

that actually has physical signicance� Lp should be proportional to Lpr� If� for a

given plastic curvature� plasticity spreads further along the pier� the resulting plas�

�

tic displacement should be greater than the case where plasticity did not spread as

far� This should be re�ected by a greater value of Lp in Equation �� If� however�

plastic curvature capacity is assumed to vary for di�erent spreads of plasticity� this

expectation need not be satised� This caveat is given in order to emphasize the

fact that Equation �� can be formulated correctly based on inaccurate values of �p

and Lp as long as the combined value �pLp is accurate� The accuracy of Equation

��s individual components has therefore remained primarily academic in interest�

Meanwhile� the consistency and conservatism of Equation �� validated experimen�

tally� have found general acceptance as both necessary and su�cient conditions for

the successful estimation of a bridge pier�s inelastic force�displacement response�

California�s widespread use of the plastic hinge length has centered primarily on

an equation developed to model the behavior of simple circular and rectangular rein�

forced concrete bridge piers� This equation is similar in principle to earlier equations

proposed in the �� s �� and �� s �� but was developed primarily over the past

two decades �� into its present form

Lp � � �L� ��dbfy � �� dbfy �ksi�� L� � ��dbfy � � ��dbfy �MPa�

��

Equation �� can be seen to have a moment gradient component and a strain pene�

tration component� It has no tension shift component� because the data base used to

construct the basic equation implied that the e�ects of tension shift were statistically

insignicant �� As previously stated� this database consisted primarily of solid cir�

cular and rectangular bridge piers� While equations including the combined e�ects of

moment gradient and tension shift on the spread of plasticity on reinforced concrete

members �� and reinforced concrete structural walls in particular �� have

been proposed� they have received little or no attention in the discipline of seismic

bridge design�

The design of three new toll bridges in the San Francisco Bay Area has recently

prompted a reevaluation of Lp in its application to reinforced concrete piers supporting

long span bridges �unsupported span � � ft�� These three bridges are the East

Bay Skyway of the San Francisco�Oakland Bay Bridge� the Second Benicia Martinez

Bridge and the Third Carquinez Strait Bridge� The supporting structures for all

three bridges are designed as hollow rectangular reinforced concrete members with

highly�conned corner elements and are shown schematically in Figure ��

Experimental results from large scale tests based on these bridge piers have brought

to light the importance of all three components of the spread of plasticity� Working

�

Benecia Martinez

Carquinez

East Oakland Bay

East Bay Skyway Pier Detail

highly-confined

corner elements

Toll Bridge Cross Sections

Figure �� Schematic representation of proposed Bay Area bridge piers�

with the data from these tests to derive the experimental plastic hinge length for

each test has also brought to light the di�culties inherent in calculating experimental

values such as the plastic hinge length and the base curvature� In order to develop

an analytical model that accounts for the three phenomena associated with plastic

rotation� it is necessary to develop a method for consistently evaluating experimen�

tal plastic deformations so that a database can be constructed from which to draw

accurate conclusions about actual plastic deformations in bridge piers� This report

proposes an accurate and consistent approach to experimental data� It applies the

approach to twelve diverse large scale reinforced concrete bridge pier tests�

One problem with assessing the plastic hinge length of bridge piers in the past has

been that researchers have compared their theoretical models to experimental plastic

hinge length values acquired by di�erent methods� This report proposes a method for

calculating the experimental plastic hinge length that is conceptually simple and �ex�

ible enough to be applied to a wide variety of member types� reinforcement schemes

and loading conditions� This report intends to guide future researchers who set out to

characterize the experimental behavior of plastic hinge regions� Knowing the prob�

�

lems inherent in calculating these experimental values and the possible solutions�

researchers can instrument their tests� observe their tests and report their results in

a way that is relevant both to the actual behavior of the member and to the practical

task of calculating displacements based on curvatures�

Unfortunately� it is not possible to measure experimental values such as curvature

and the plastic hinge length directly� These values must be calculated from the results

given by several independent instruments� Such calculations require assumptions

about the plastic deformations� Test data must therefore be evaluated carefully in

order to make sure that experimental values are calculated according to accurate

assumptions and that the accuracy of the assumptions can be observed in the test

data�

This document presents test results from the tests reported in �� from

previous tests by other researchers �� for the sake of comparing the spread of plas�

ticity over a range of bridge pier types� While this study is by no means exhaustive�

the data reported herein are thought to be su�cient for highlighting general trends

observed in the spread of plasticity of specic reinforced concrete bridge piers�

The data reported has been calculated from measured test results based on spe�

cic assumptions regarding reinforced concrete member behavior� The assumptions

undergirding the relationship of moment�curvature analysis and plastic hinge length

to force�displacement characterization as well as the assumptions regarding the cal�

culation of instrument readings to arrive at experimental values such as �p and Lp

are outlined in Chapter ��

Chapter �

Experimental Characterization

of Lp

This chapter begins by introducing the problem of assessing experimental curvature

values and emphasizes in particular the problem with base curvature� It explores the

actual inelastic strain behavior of bridge piers inside their plastic hinge regions and

proposes a means for evaluating curvature that is consistent with this actual behavior�

The chapter then explores the relationship between curvatures and displacements�

After outlining a consistent relationship between curvature� �� and displacement� ��

the chapter explains how this relationship and the previously explained method for

evaluating experimental curvatures were employed to produce the graphs and tables

in Appendices B � D�

�� Observed Mechanisms of

Flexure�Shear Deformation

True �exural deformation mechanisms have long been recognized as prohibitively

complex for the creation of a consistent and lasting theory of �exural deformation�

�Flexural deformation� itself is a misnomer� since the complexity is due in part to

the fact that plastic �exural deformations in reinforced concrete are almost always

coupled with the shear behavior of the plastic hinge region�namely the fanning

crack pattern� Therefore� the term ��exure�shear deformation� is a more accurate

description of the phenomenon�

Currently� the most widely used approach for characterizing the force�displacement

relationship in reinforced concrete members assumes that plastic rotation is the inte�

gral of some distribution of plastic curvatures over the so�called plastic hinge region�

�

This plastic rotation is in turn integrated to arrive at a plastic displacement� Re�

searchers who have advanced this approach have recognized the di�culty of including

the e�ects of tension shift and strain penetration� both of which defy the assumption

of plane sections� If strain penetration and tension shift could be described purely

P

F

CF T

Lprc

Lprt

M

Figure �� UCSD Column �A crack pattern� Comparison of idealized crack patternwith the real crack pattern ��

in terms of spreading plasticity� then they could be accounted for with a fair amount

of rigor� The action of tension shift is� however� intimately linked to a concentration

of compression strains at the point of maximum moment�� This complicates the no�

tion of base curvature and increases the di�culty of basing calculations for �exural

deformation on actual strain levels at the column base�

Figure �� compares an idealized crack distribution to the real crack distribution

at �� in Test Unit �A �� Figure �� shows compression struts radiating from

the compression toe at the column base up to full height of the plastic hinge region

in the tension boundary element� This representation of the fanning crack pattern

implies that most of the plastic tensile strains in the tension boundary element are

associated with plastic compressive strains at or near the base of the column� Figure

�For the purposes of this discussion� the point of maximum moment is assumed to be synonymous

with the column base�

�

-0.0

3

-0.0

2

-0.0

1

0.0

0

0.0

1

0.0

2

0.0

3

0.0

4

0.0

5

0.0

6

Strain at positive peak (in./in.)

-24

-12

0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

footi

ng

(in.)

-600

-300

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y = 0.00214 Hines 3A �� = 4 x +1L = 120 in.D = 48 in.

External potentiometer

Internal strain gage

P

F

CF T

Lprc

Lprt

tensile strain at basecalculated by addingstrain penetration tothe base gage length

compressive strain at basecalculated by addingstrain penetration tothe base gage length

Figure �� UCSD Column �A strain proles� Comparison of idealized crack patternwith the experimental strain distributions ��

�� compares the same idealized crack pattern with measured tensile and compres�

sive strain distributions along the column height� This comparison reinforces the

notion that while the tensile strains spread up the column height� the compression

strains remain concentrated near the base� The concentration of compression strains

at the base of the column varies according to column geometry� reinforcement� ma�

terial properties and axial load� Columns in Appendices B�D subjected to higher

shear stresses were likely to exhibit a higher concentration of compression strains at

their base� since the �exure�shear crack angles were generally steeper and forced more

compression struts to radiate directly out of the compression toe at the base of the

column� Taller columns subjected to lower shear stresses exhibited only partial con�

centration of compression strains at their base due to their more shallow �exure�shear

crack angles�

If increased shear force and broader fanning of compression struts out of the

compression toe increases the concentration of compression strains at the column

base� then strain limit states should also depend on the level of shear applied to a

column and the manner in which this shear is transferred within the member� While

�

Lprc

Lprt

P

F

CF T

M

�s

�c

�

A

A

A

A

Figure �� UCSD Column �A �plane sections�� Comparison of idealized crack pat�tern with the �plane sections� assumption ��

this knowledge may be applicable in setting up a spectrum of strain limits that varies

according to applied shear� it is not yet possible to apply it to the prediction of actual

strain limit states�

Three more phenomena a�ect the compression strain demand and capacity at the

base of a column� The rst is the assumption of plane sections� Even if a section cuts

through a column according to an idealized crack pattern� as Section A�A is shown to

do in Figure �� it is di�cult to assess whether or not that section remains plane� The

righthand side of Figure �� shows both a segment of the columny near Section A�A

and an idealized rotation of Section A�A� An arrow inside the column segment point�

ing into the compression toe at the column base shows the expected direction of force

transfer from the compression strut into the compression toe� There is no guarantee

that the resulting compression strains at the column base will correspond in magni�

tude to the compression strains derived from a �plane sections� moment�curvature

analysis for Section A�A� Second� the connement provided to the compression toe at

the column base by the footing is not understood in detail� Third� strain penetration

occurring on both the tension and compression sides of the column obscures �exural

strain compatibility along the base of the column� In this region� strains penetrate

into the boundary and show up at the base of the column as a net rotation�

yThe forces on this segment necessary for equilibrium are not shown for the sake of clarity and

simplicity in the drawing�

�

�� The Argument for Curvature

Although the relationship between longitudinal tensile and compressive strains to

column deformation is rather complicated� the spread of plasticity can be modeled

accurately based on the assumption that plastic curvatures are distributed linearly

over the length of the plastic hinge region� Lpr� Looking at Figure �� one can see

that if strain penetration is accounted for in the calculation of the experimental lon�

gitudinal tensile strains at the column base� the magnitude of tensile strains near the

column base remains relatively constant� Since the compression strains in this same

region increase to very high values near the column base� the curvatures calculated

as

� ��s � �cD�

��

where tension is assumed positive� compression is assumed negative and D� is the

distance between the tension and compression strains in question� will maintain a

relatively constant slope toward the column base� Higher up the plastic hinge region�

the curvature distribution follows the tensile strain distribution and lower in the

plastic hinge region� it follows the compressive strain distribution�

Observation of experimental data has consistently shown that plastic curvatures

have an approximately linear distribution inside the plastic hinge region� This lin�

earity is disturbed only at the column base by the presence of both compressive and

tensile strain penetration into the footing�

Figure �� shows this linear distribution of plastic curvature for both positive and

negative peaks� If strain penetration is calculated as

Lsp � ��dbfy � �� dbfy �ksi�� dbfy � � ��dbfy �MPa�

��

and added to the base gage length such that the base curvature is calculated as

�b ��tb ��cb

D��Lgb � Lsp��

the resulting base curvatures match the base curvature projected as a least squares

line from plastic curvature values higher up in the plastic hinge region� While such

a close match was not observed on many columns� this example demonstrates the

conceptual viability of the assumption that plastic curvatures are distributed linearly�

This assumption� which is based on observed physical behavior� helps to evaluate

��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

Hei

ghtab

ove

foot

ing

(in.

)

-60

-40

-20

0 20 40 60

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

[mm

]

Hines et al. 3A�� = 4

� y

�

�b

� bspb0

Figure �� Test �A �Hines et al� �� curvature at ��

experimental base curvature and spread of plasticity more consistently and e�ectively

than evaluations made from the potentiometers at the base of the column� The linear

distribution method employs all of the available plastic curvature data in the column

in order to create and evaluate an articial base curvature that is consistent with the

assumption that plastic curvatures are distributed linearly�

The assumption that plastic curvatures are distributed linearly provides a second

advantage in that it also estimates the extent of plasticity spread up from the column

base� With experimental values for both the base curvature and length of the plas�

tic hinge region� there is a greater degree of redundancy in determining the correct

experimental plastic behavior of a column�

Assuming that plastic curvature is linearly distributed from the column base up

to a height of Lpr� and assuming that plastic rotation occurs primarily about the

column base� Lp can be evaluated as

Lp �Lpr

�� Lsp ��

This equation also implies that Lsp represents the depth beneath the footing that the

��

base curvature can be assumed uniform in order to account for the total base rotation

due to strain penetration�

Based on this discussion� it is possible to construct a consistent method for evalu�

ating the experimental base curvature� �exural strains and plastic hinge length for all

levels of plastic deformation� Sections �� and �� discusses this method in detail with

the aim of explaining how the graphs and tables in Appendices B � D were created�

�� Relating � to �

Assuming that the total shear span displacement of a reinforced concrete member

is characterized by the addition of independent �exural and shear components such

that

� � �f ��s ��

these components can be broken down further into elastic and plastic components�

giving

�f � �ef ��pf ��

and

�s � �es ��ps ��

where the subscript e denotes the elastic displacement and the subscript p denotes

the plastic displacement� Practically speaking� this report distinguishes only between

the elastic and plastic components of �exural displacement� Shear displacement is

always considered in its totality�

Combining Equations �� and �� gives

��s � �f � �ef ��pf ��

If the elastic component of �exural displacement is assumed to be the �exural dis�

placement at rst yield of the longitudinal reinforcement scaled up by the increase in

moment demand due to strain hardening� then

�ef � ��yf

M

M �y

��

where ��yf is the �exural displacement at rst yield� M is the maximum moment at

the column base� and M �y is the moment at rst yield�

��

Combining Equations �� and �� gives

�pf � ��s ��yf

M

M �y

��

It is practical and accurate to assume that �pf can be expressed as

�pf � �pLpL ��

where �p is the plastic curvature at the column base� Lp is the plastic hinge length�

and L is the column shear span� Assuming that the plastic curvature can be derived

from the total curvature as

�p � �� y

M

M �y

��

where ��y is the rst yield curvature� Equations �� and �� can be combined

to solve for Lp as

Lp ��s ��

yfMM �

y��

yMM �

y

�L

��

Before proceeding further� Equation �� must be justied on the basis of its ability

to model plastic rotation realistically� This equation assumes that the plastic rotation

acts about the column base and therefore di�ers from the more widely accepted

assumption that the plastic rotation acts about the center of the plastic hinge length�

Park and Paulay popularized this prevailing assumption in �� They that

�neglecting shear displacements� �p could be calculated according to the equation

�p � �pLp�L� Lp��

Subsequently� the seismic research on reinforced concrete columns at the University

of Canterbury� Christchurch� New Zealand �� and at the

University of California� San Diego �� used Equation �� for calculating the

plastic displacement�

Other researchers prior to �� such as Corley in �� used a similar approach

to Equation �� for the testing of simply�supported beams� This approach has since�

however� largely given way to Equation �� in the published literature�

Although the di�erences between Equations �� and �� are only slight �generally

on the order of less than � �� there are several reasons to apply the simpler Equation

�� in favor of Equation ��

��

�� Equation �� is simpler than Equation ��

�� Equation �� re�ects the fact that in many types of bridge piers� due to tension

shift� most plastic rotation occurs about the column base� where there is a

concentration of compression�

�� The plastic rotation calculated as �p � �pLp can be lumped simply at the ends of

beam elements� over zero distance� for numerical analysis of structural systems�

�� Equation �� generally provides more physical insight into the problem of cal�

culating �exural displacements than Equation �� The one exception is for

very tall� slender columns with relatively small diameter longitudinal bars whose

plastic behavior is not in�uenced heavily by tension shift and strain penetration�

�� Equation �� includes a renement that is not rigorous�

� When working with experimental results� Equation �� the most transparent

relationship possible between �p and �p� This allows Lp to be recalculated

easily for use with other methods�

Equation �� assumes a correct lever arm for the plastic curvature in the column

if� and only if� it is assumed to be distributed uniformly over the plastic hinge length�

Observations of test data has shown� however� that plastic curvatures are not actually

distributed uniformly� but rather have distribution which is much closer to linear than

it is constant� The correct lever arm for a triangular distribution of plastic curvatures

is �L�Lp�� Furthermore� Equation �� assumes the incorrect lever arm for strain

penetration� Strain penetration results in uplift at the column base and its e�ect

on the plastic displacement is hence best approximated by assuming a concentrated

rotation at the column base multiplied by the entire column shear span� The strain

penetration depth cannot be averaged into the column shear span and then used in

Equation �� without considerable e�ort� In the event that the strain penetration

constitutes more than half of the plastic hinge length� the point of rotation would be

calculated to occur below the base of the column�

In reality� the base of the column is the lowest possible location for the center of

rotation� since all strain penetration into the footing is assumed to act as uplift at the

column base� Furthermore� diagonal �exure�shear cracking contributes signicantly to

the spread of plasticity� and the center of rotation resulting from the tension shift e�ect

occurs at the base of the tension shift zone rather than in the middle� If the tension

��

shift e�ect is separated from the moment gradient e�ect and assumed to occur at the

column base� Equation �� is more rigorous than Equation �� because it assumes

one incorrect center of rotation and two correct centers of rotation� Furthermore�

if tension shift is included� Equation �� does not assume any center of rotation

correctly and needlessly complicates both the calculation of experimental values of

Lp and the calculation of the plastic displacement�

For these reasons� Equation �� is considered the more accurate and useful formu�

lation of the relationship between plastic curvature and plastic �exural displacement�

Therefore� this report calculates Lp experimentally based on Equation �� Finally�

since Lp is a numerical multiplier and not a direct physical quantity� the assumed

center of rotation used in analysis must only be consistent with the center of rotation

used to determine experimental values� This consistency is the most important aspect

of interpreting experimental results� It is possible for particular elements of a method

to have no physical signicance� while they still yield a correct numerical answer� As

stated in Chapter �� this is the reason that it has never been critical for Lp to have

physical signicance�

�� Approach to the Experimental Data

This section outlines the approach used to reduce the data that are presented in Ap�

pendices B � D� The structural wall with boundary elements �Test �A� tested by

Hines et al� �� and presented in Appendix C is used as an example� The appen�

dices were assembled with the aim of providing experimental �plasticity values� over

several levels of displacement ductility� Each test unit featured in the appendices is

presented in a format that includes curvature proles up the height of the test unit

and strain proles created from the same external potentiometers used to calculate

experimental curvatures� Also presented are curvature proles that consist of values

averaged from a positive and negative excursion at the same cycle and level of dis�

placement ductility� and tabulated values for base curvature� plastic hinge length and

other �experimental plasticity values�� The curvature and strain prole plots give

insight into the actual spread of plasticity� concentration of curvature in the com�

pression toe� strain penetration into the footing and uniformity of the experimental

data� The plots of average curvature proles� show the development of plastic cur�

vature with increasing displacement ductility� The experimental moment�curvature

plots are available for comparison with theoretical moment�curvature relationships�

�

The tabulated plasticity values provide numerical data for the interpretation of the

equivalent plastic hinge length� e�ective base curvature and e�ective concrete and

steel strains�

�� Plastic Curvature Distribution

Figure �� shows the curvature proles for a structural wall with highly�conned

boundary elements �Test �A� at the rst cycle of �� The assumed linear

plastic curvature distributions are shown as straight lines� The experimental curva�

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

Hei

ghtab

ove

foot

ing

(in.

)

-60

-40

-20

0 20 40 60

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

[mm

]


� y

�

�b

� bspb0

Figure �� Test �A �Hines et al� �� curvature at ��

ture distributions which are shown as solid square data points connected by straight

lines� The base curvature� �b is assumed to be the curvature where the �best t� line�

taken as the least squares t to all of the plastic curvatures above the base� reaches

the base� In Figure �� b� is the base curvature� assuming no strain penetration�

calculated as

�b� ��tb ��cb

D�Lgb

��

��

where �tb and �cb are the displacements measured by the tension and compression

linear potentiometers at the base of the column� D� is the distance between the two

potentiometers and Lgb is the gage length over which the potentiometers measure

displacement� shown in Figure �� Assuming strain penetration on the tension and

6"[152]

1"[25]

3/4" [19] PVC pipe

as blockout for threadrod

butt weld 6" [152] threadrod

to transverse bar

LCA LCG

6"[152]

L =g b

L + Lg b sp

D�

c’o

c’o

�’c

�cb

�tb�’s

�g b

(a) (b)

Lg

Lg

Figure �� UCSD �A Hines et al� �� Detail of curvature instrumentation� �a�elevation� �b� section and rotation scheme�

compression sides according to the strain penetration component in Equation ��

�bsp is the base curvature calculated as

�bsp ��tb ��cb


�� Average Curvature Pro�les

Curvature proles on several columns tested were not symmetric between the positive

and negative excursions because the cracks were not symmetric between the positive

and negative excursions� It was therefore common that one gage level would record a

large rotation in the positive direction� while the next gage level higher would record

the majority of the same rotation in the negative direction� This left the second gage

level recording proportionally less rotation in the positive direction and the rst gage

level recording proportionally less rotation in the negative direction� For this reason�

the positive and negative curvature proles were averaged for each cycle� The base

curvatures used to determine the plastic hinge length and �exural strains were derived

from the averaged proles� The averaged proles for the circular column in Figure

��

�� are plotted in Figure �� where the base curvatures are the curvatures� �b taken

from the best t of the individual averaged proles�0.

0000

0.00

01

0.00

02

0.00

03

0.00

04

0.00

05

0.00

06

0.00

07

0.00

08

0.00

09

0.00

10

Average curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

Hei

ghtab

ove

foot

ing,

h(i

n.)

0 5 10 15 20 25 30 35

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

[mm

]

� y = 0.0000986 rad/in. Hines et al. Test 3AL = 120 in.D = 48 in.

�� = 1

�� = 2

�� = 3

�� = 4

Figure �� Structural wall with conned boundary elements� Test �A �Hines et al�� Average curvature proles�

�� Calculation of ��

cand �

�

s

If� instead of calculating curvatures by Equation �� strains are calculated directly�

the concentration of compression at the base of a column can be shown� Diagram �b�

in Figure �� shows a possible rotation calculated from the curvature potentiometers

on either side of the column� This rotation is calculated as

� ��t ��c

D�

��

Typically� an average curvature over the gage length is then calculated from the

rotation as

� ��

Lg

��

��

Both this rotation and this curvature assume that plane section remain plane�

Strains calculated directly from the gages give results that di�er from the strains

implied by calculating experimental curvature� This is due primarily to the concen�

tration of compression at the base of the column� Ideally� strains could be calculated

directly from the curvature potentiometers as

��c ��c

Lg

��

and

��s ��t

Lg

��

The problem with this method is that the potentiometers necessarily lie some distance

away from the actual location of interest�namely the extreme conned concrete or

steel ber�

This report has dealt with this problem in part by assuming that displacement

values can be scaled back linearly before they are converted to strains� This is shown

in Figure �� by the values ��c and ��s� This approach� however� reintroduces the

assumption that plane sections remain plane� Therefore� while this approach reduces

the experimental strains to a more realistic level� it does not give the correct strains�

The resulting change in strain level by scaling down the values of �c and �t is most

often very low� The entire process of translating gage displacements to expected

internal column displacements is therefore� mostly a futile exercise� Unfortunately�

this fact does not allow the values ��c and ��s to indicate real strains� These values are

therefore limited as approximate indications of actual �exural column strains�

�� Calculation of Lsp and L�

sp

Appendices B � D report values labeled Lsp and L�sp� These values both represent

strain penetration� but they are calculated by two completely di�erent methods� The

value assumed to be the real experimental strain penetration� Lsp� is calculated di�

rectly from experimental values of Lp and Lpr as

Lsp � Lp �Lpr

��

This value of strain penetration is the real strain penetration required to create an

experimental Lp that is both consistent with the assumption that plastic curvatures

are distributed linearly and Equation �� In this sense� the strain penetration is the

�

leftover value that complements the assumption that plastic curvatures are distributed

linearly� As long as these values tend to remain close to the values calculated according

to Equation �� Equation �� can be considered to be compatible with the assumption

of a linear distribution of plastic curvatures�

If� on the other hand� the base curvature potentiometers are employed to calculate

a value of strain penetration� the values are slightly di�erent� These values have been

called articial strain penetration values� L�sp� They are of little use for constructing

realistic values of Lp� however they provide some insight into the consistency between

the linear plastic curvature distribution assumption and the actual base curvature

readings�

If the calculated base curvature takes into account strain penetration� the re�

lationship between the assumed value of strain penetration and the resulting base

curvature are related hyperbolically� For example� given displacement readings from

a potentiometer on the tension side and the compression side of a column� the base

curvature �b can be calculated as

�b ��tb ��cb


where D� is the distance between the north and south potentiometers and Lg is

the potentiometer gage length� The inverse proportionality between �b and Lsp in

Equation �� implies that at low values of Lsp base curvature is more sensitive to

imperfections in this value than at high values of Lsp� This fact causes the base cur�

vature assuming strain penetration� �bsp to vary greatly with any initial change in

gage length� but then to become less sensitive to increases in this change� This phe�

nomenon is demonstrated in Figure �� where strain penetration values are plotted

for varying levels of curvature at di�erent displacement ductilities� The symbols on

the plot represent the values of articial strain penetration calculated by Equation

�� for the base curvatures determined according to the linear distribution method�

L�sp �

�tb ��cb

D��b� Lgb �

��b��b� �

�Lgb ��

Ultimately� it is more consistent to assume a linear distribution of plastic cur�

vatures� to calculate Lsp from Lp and Lpr� and to list the values of articial strain

penetration than to rely on measured base curvatures and theoretical strain penetra�

tion values for the calculation of a denitive base curvature� In addition to using the

maximum amount of experimental data as the basis for �b as opposed to the possibly

��

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

0.00

12

0.00

14

0.00

16

0.00

18

0.00

20

Curvature (rad/in.)

0

1

2

3

4

5

6

7

8

9

10

11

12

L' sp

0 10 20 30 40 50 60 70

[� rad / mm]

0

25

50

75

100

125

150

175

200

225

250

275

300

[mm

]

��= 2

��= 3

�� = 4

Lsp= 0.15 f ydb= 7.0"

Figure �� Experimental values of L�sp for Test �A from Hines et al� �� assuming a

linear plastic curvature distribution�

inconsistent data of only the base gages� the assumed linear distribution of plastic

curvature makes it possible estimate the base curvature even in the case where the

test unit footing was post�tensioned and it is impossible to measure realistic strain

penetration values�

�� Explanation of Appendices B � D

The following section explains in detail the tables and graphs for Test �A found in

Appendix C�

Table C�� presents the average experimental plasticity values that are believed

to re�ect most accurately the experimental behavior of the test unit� This is the table

that is recommended for reference in any attempt to construct an analytical model

that re�ects the spread of plasticity in Test �A� The fteen columns of Table C��

list the following data according to displacement ductility level and cycle�

�� Level� displacement ductility levels are listed according to their value and cycle�

For instance� the row beginning with �� will contain data that have

been averaged from the positive and negative excursions during the second cycle

of displacement ductility ��

��

�� the average total measured experimental displacement at the point of con�

tra�exure�

�� M � the average experimental moment at the column base calculated based on

the average measured experimental force�

�� f � the average total experimental �exural displacement at the point of con�

tra�exure� calculated as �f � ��s�

�� l�s� points� the number of curvature values used for tting the least squares

lines� Greater spread of plasticity can include more points� The minimum

number of points required is �� A double dash signies that there were less

than � points available at that level of displacement ductility�

� �b� the average base curvature� dened as the projection of the best t line

through the average plastic curvature prole to the base of the column� If

italicized� this base curvature was dened based on a least squares or assumed

value of Lpr� an assumed value of Lsp and the resulting value of Lp� Lpr was

assumed to have a minimum value of Lpr � �Lsp� in order to ensure that

Lp � �Lsp�

�� c� the extreme ber conned concrete compression strain from a moment cur�

vature analysis of the section at a curvature level corresponding to �b�

�� s� the extreme ber steel tensile strain from a moment curvature analysis of

the section at a curvature level corresponding to �b�

�� curvature ductility level dened as �b��y� where �y is the theoretical ideal

yield curvature calculated as �y � ��yMy

M �

y

� In this case� ��y is the theoretical

rst yield curvature� M �y is the theoretical rst yield moment� and My is the

theoretical moment at either �c � � � or �s � � �� whichever corresponds

to the lower theoretical curvature value�

� � �p� plastic base curvature calculated as �p � �b � ��yMM �

y

�

�� p ��pf�� plastic �exural displacement at the point of contra�exure� calcu�

lated as �p � �f � ��yf

MM �

y

� where ��y is the experimental �exural rst yield

displacement at the point of contra�exure�

��

�� Lp� experimental equivalent plastic hinge length calculated as Lp � �p

�pL� If

the value is written in italics� a proper value of �p could not be calculated and

Lp was calculated as Lp � Lpr

�� Lsp� where Lsp was calculated according to

Equation ��

�� Lpr� experimental plastic hinge region length� This value tells the height to

which plasticity spread during the experiment� The value either increases or

stays the same with increasing curvature ductility� It is determined as the

height at which the least squares line intersects the theoretical value of �y� If

this value is given in italics� it was either interpreted directly from a plot of

the average curvature proles� or it was assumed to be the minimum value

Lpr � �Lsp�

�� Lsp� experimental strain penetration length� calculated as Lsp � Lp �Lpr

�� If

this value is given in italics� it was calculated according to Equation ��

�� L�sp� experimental ctitious strain penetration value� This value indicates the

level of strain penetration implied by the least squares base curvature� �b�

through Equation ��

Figure C�� displays the average curvature proles for every level of displacement

ductility listed in Table C�� The base curvatures displayed in Figure C�� are the

base curvatures given in Table C�� These base curvatures were found either by

projecting a least squares line tted to at least three average plastic curvature point

further up the column or by being back�calculated from Lp as calculated from Lpr

and Lsp�

Table C�� lists a variety of values obtained for every force and displacement peak

during the test� This table is provided to complement Figures C�� and C�� as well

as provide values for the experimental force�displacement envelopes in the positive

and negative directions� The twelve columns of Table C�� are explained below�

�� Level� force or displacement ductility peak�

�� the total measured experimental displacement at the point of contra�exure�

�� f � the total experimental �exural displacement at the point of contra�exure�

calculated as �f � ��s�

�� F � the measured experimental lateral force applied to the column�

��

�� M � the experimental moment at the column base calculated based on the mea�

sured experimental force� M � FL�

� �b�� the experimental base curvature assuming a base gage length equivalent to

the physical length of the gage� This value was calculated according to Equation

��

�� bsp� the experimental base curvature assuming a base gage length equivalent

to the physical length of the gage plus a strain penetration length calculated

according to Equation �� This value was calculated according to Equation

��

�� b� the base curvature determined by the projection onto the column base of

a least squares line tted to the plastic curvatures higher up the column� If

the values are given in italics� they were calculated not from the least squares

project� but based on Lp as derived from Lpr and Lsp�

�� points� the number of points used to t a line according to least squares to the

plastic curvature distribution�

� � Lp�� the experimental plastic hinge length calculated� assuming zero shear dis�

placements� according to the equation

� � ��y

M

M �y

�

��b� � ��y

M

M �y

�L

�L�

Lp

�

�

such that

Lp � L

��

s��

��p

L��p

�

�� Lpsp� the experimental plastic hinge length calculated� assuming zero shear

displacements� according to the equation

� � ��y

M

M �y

�

��bsp � ��y

M

M �y

�L

�L�

Lp

�

�

such that

Lp � L

��

s��

��p

L��p

�

��

�� Lp� the experimental plastic hinge length calculated according to the method

used for Table C��

Figures C�� and C�� show the curvature proles at the positive and negative

peaks of the rst full cycle of a given displacement ductility level� Positive curvature

corresponds to a positive peak and negative curvature corresponds to a negative peak�

Values for �b� and �bsp are shown as part of the curvature prole� The least squares

projections are shown with the curvature distributions and their value at the column

base is �b�

Table C�� compares average �exural strain values derived using three di�erent

approaches� The rst two approaches were already demonstrated in Tables C�� and

C�� These two approaches correlate experimental base curvatures to theoretical

extreme ber �exural strain values based that are based on moment�curvature anal�

ysis� The third approach calculates �exural strains at the column base directly based

on the extension or compression of the linear potentiometers used for calculating

experimental curvature�

This third approach yields di�erent strains than either of the other two approaches

since it captures the phenomenon of compression concentration� Due to compression

concentration� the �exural compression strains ��c are consistently higher than the

values �c�� which were calculated according to the curvature yielded by the base gages�

These three strains are compared in order to demonstrate that there are any number

of possibilities for evaluating experimental �exural strains� Therefore� �exural strains

in and of themselves are meaningless� Only when they are intimately bound to a

particular method for deriving them experimentally or for applying them analytically

do they acquire some meaning�

The eleven columns of Table C�� are explained below�

�� Level� force or displacement ductility peak�

�� the average total measured experimental displacement at the point of con�

tra�exure�

�� F � the average measured experimental lateral force applied to the column�

�� b�� the average experimental base curvature assuming a base gage length equiv�

alent to the physical length of the gage� This value was calculated according to

Equation ��

�

�� b� the base curvature determined by the projection onto the column base of

a least squares line tted to the plastic curvatures higher up the column� If

the values are given in italics� they were calculated not from the least squares

project� but based on Lp as derived from Lpr and Lsp�

� �c�� the extreme ber conned concrete compression strain from a moment

curvature analysis of the section at a curvature level corresponding to �b��


vature analysis of the section at a curvature level corresponding to �b�


vature analysis of the section calculated directly from a linear potentiometer as

the average strain along a given gage length�

�� s�� the extreme ber steel tension strain from a moment curvature analysis of

the section at a curvature level corresponding to �b��

� � �s� the extreme ber steel tension strain from a moment curvature analysis of

the section at a curvature level corresponding to �b�

�� s� the extreme ber steel tension strain from a moment curvature analysis of

the section calculated directly from a linear potentiometer as the average strain

along a given gage length�

Figure C�� displays the average �exural strains calculated directly from the linear

potentiometers from which curvatures were calculated� The top graph displays these

�exural strains prior to yield� The bottom graph displays these �exural strains after

yield� Figure C�� displays the same �exural strains prior to yield at both positive

and negative peaks of given lateral force levels� Figure C�� displays the same �exural

strains after yield at both positive and negative peaks of given lateral force levels�

��

��

Chapter �

Conclusions

A detailed examination of the experimental data from twelve diverse large�scale

bridge pier structural tests under fully�reversed� incrementally increasing cyclic load�

ing demonstrates both the complexity of the real inelastic �exural deformations in

such piers and the accurate simplicity with which the piers can be modeled if the

the plastic curvatures are assumed linearly distributed� The phenomena of tension

shift� compression concentration and strain penetration clearly defy the two lineariz�

ing assumptions that plane sections remaining plane and that boundary conditions

do not in�uence member behavior� In spite of their faulty conceptual nature� how�

ever� plastic curvature distributions tend to remain for the most part linear� If the

linearity of the plastic curvature distribution is adopted as the key assumption� then

the concepts of curvature and related �exural strains can be applied to the inelastic

deformations of such bridge piers� When such an assumption is made� it absolutely

critical to evaluate �exural strain limits according to the method used for deriving

the experimental plastic hinge length� Three experimental plastic hinge lengths were

introduced in Chapter �� each implying its own strain limits�

As long as strain limits are coupled with a corresponding plastic hinge length

for a particular bridge pier� it is not necessary for the plastic hinge length to carry

any physical signicance� If a method is to be created that can be generalized to

bridge piers that have not been tested� however� the plastic hinge length must have a

physical meaning� It is proposed that the plastic hinge length be proportional to the

actual spread of plasticity and the actual level of strain penetration in a bridge pier�

By assuming that plastic curvatures are distributed linearly within the plastic hinge

region� the relationship between Lp� Lpr and Lsp can be expressed simply as

Lp �Lpr

�� Lsp ��

��

Tension shift and strain penetration have long been known to a�ect the spread

of plasticity� This report o�ers nothing new in the identication of their e�ects�

The great degree to which compression strains concentrate at the base of the twelve

bridge piers in question is� however� a nding that has been seldom demonstrated

and discussed� Furthermore� the assumption that plastic curvatures can be assumed

uniformly distributed was introduced by Priestley et al� in �� Priestley et

al� introduced this idea in the context of inadequate instrumentation within the

plastic hinge region� The application of the uniform distribution of plastic curvatures

inside the plastic hinge region and the use of lines t by the least squares method for

determining both the base curvature and the length of the plastic hinge region is a

new approach which promises a great degree of consistency�

This linear distribution approach is more labor intensive than previous approaches

which applied only experimental base curvature readings for determining Lp exper�

imentally� It is also� however� much more useful and accurate� For instance� in the

case of the twelve piers presented in Appendices B � D the method of evaluating Lp

in relation to the actual spread of plasticity at each level of displacement ductility

yielded values of Lp that increased with increasing curvature ductility� This conrm

both intuition and the visual appearance of curvature proles that plasticity spreads

further with increasing deformation demands�

Recognizing that it is possible to conceive of Lp as proportional to the actual

spread of plasticity� Lpr� it becomes clear that Lp also tends to grow with increasing

curvature ductility demand� This implies that while it may be possible to predict

ultimate displacements based on a xed value of Lp� predicting accurate displacements

for all levels of curvature ductility requires the ability to model the spread of plasticity

and its e�ect on plastic displacements� Such a requirement is consistent with the

philosophy of performance�based engineering� The closer engineers� understanding of

inelastic deformations comes to the reality� the more accurately� they will be able to

predict the actual behavior of bridge piers for a given performance level�

Experimental curvatures and their corresponding �exural strains are intimately

related to the method by which they were calculated� This report introduced three

viable methods for calculating �exural strains experimentally� The results of these

di�erent methods are directly compared in tables such as Table C�� Existing strain

limits� based primarily on axial load tests of prismatic members �� do not

re�ect the e�ects of compression concentration� strain penetration and inelastic longi�

tudinal bar buckling due to large strain demand reversals� While these existing strain

�

limits have been calibrated to provide generally conservative results with Equations

�� and �� they also do not re�ect actual inelastic deformations in bridge piers�

Unfortunately� realistic estimates of compression strain capacity that account for

compression concentration� connement provided by the footing� and inelastic longi�

tudinal bar buckling are not available� Even if they were available� translating them

into estimates of curvature capacity would remain problematic� For now� it is recom�

mended to accept the experimental strains given in Appendices B � D as appropriate

limits� Specically� it is recommended to limit strains according the the equation

� �� c � �s � � � ��

which tends to re�ect the range of data presented in this report� Naturally� Equation

�� applies only to the linear distribution method� with which it was derived to be

compatible�

Ultimately� an engineer using strain limits and plastic hinge lengths to predict

inelastic deformation capacity should be aware of the need for consistency between

the assumed inelastic curvature and assumed plastic hinge length� It is possible

for neither one to have physical signicance and still produce correct results� The

implications of such an approach� however� are certainly cause for concern�especially

in the case of members that have not be tested comprehensively and proven to t

with an engineer�s assumptions�

The two most important future contributions related to this work are the de�

velopment of a performance�based model for the spread of plasticity and reliable

strain limits that are compatible with such a model� It is conceivable to develop

such a model as a function of curvature ductility� column geometry� reinforcement

and loads� It is also conceivable that such strain limits may not need to re�ect real

ultimate strains in a zone subject to compression concentration� Strain limits must�

however� be consistent with the assumptions undergirding the spread of plasticity�

��

��

Appendix A

Test Setups and Properties

This appendix contains information on the test setups� geometries and reinforcement

schemes and material properties for all of the columns discussed in the report�

��

Table A�� General test unit properties�

Test Reference L D M/VD P Ag �l �s db

(in.) (in.) (kips) (in.2) (in.)

TU1 Hose et al. (1997) 144 24.0 6.00 400 452 0.0265 0.00889 0.875

C3 Chai et al. (1991) 144 24.0 6.00 400 452 0.0253 0.00174 0.750

1A Hines et al. (1999) 194 48.0 4.04 198 396 0.0143 0.0138 0.500

1B Hines et al. (1999) 194 48.0 4.04 198 396 0.0143 0.0138 0.500

2A Hines et al. (1999) 96.0 48.0 2.00 198 396 0.0143 0.0138 0.500

2B Hines et al. (1999) 96.0 48.0 2.00 198 396 0.0143 0.0138 0.500

2C Hines et al. (1999) 96.0 48.0 2.00 171 342 0.0146 0.0138 0.500

3A Hines et al. (2001) 120 48.0 2.50 171 342 0.0429 0.0207 0.750

3B Hines et al. (2001) 75.0 30.0 2.50 135 270 0.0429 0.0207 0.750

3C Hines et al. (2001) 180 72.0 2.50 219 438 0.0429 0.0207 0.750

LPT Hines et al. (2002) 138 54.0 2.56 1370 2740 0.0200 0.0170 0.625

DPT(L) Hines et al. (2002) 138 54.0 2.56 1370 2740 0.0200 0.1700 0.625

DPT(T) Hines et al. (2002) 306 84.0 3.64 1370 2740 0.0200 0.0170 0.625

Test Reference L D M/VD P Ag �l �s db

[mm] [mm] [kN] [m2] [mm]

TU1 Hose et al. (1997) 3658 610 6.00 1780 0.292 0.0265 0.00889 22.2

C3 Chai et al. (1991) 3658 610 6.00 1780 0.292 0.0253 0.00174 19.1

1A Hines et al. (1999) 4928 1219 4.04 881 0.255 0.0143 0.0138 12.7

1B Hines et al. (1999) 4928 1219 4.04 881 0.255 0.0143 0.0138 12.7

2A Hines et al. (1999) 2438 1219 2.00 881 0.255 0.0143 0.0138 12.7

2B Hines et al. (1999) 2438 1219 2.00 881 0.255 0.0143 0.0138 12.7

2C Hines et al. (1999) 2438 1219 2.00 761 0.221 0.0146 0.0138 12.7

3A Hines et al. (2001) 3048 1219 2.50 761 0.221 0.0429 0.0207 19.1

3B Hines et al. (2001) 1905 762 2.50 601 0.174 0.0429 0.0207 19.1

3C Hines et al. (2001) 4572 1829 2.50 975 0.283 0.0429 0.0207 19.1

LPT Hines et al. (2002) 3505 1372 2.56 6097 1.77 0.0200 0.0170 15.9

DPT(L) Hines et al. (2002) 3505 1372 2.56 6097 1.77 0.0200 0.1700 15.9

DPT(T) Hines et al. (2002) 7772 2134 3.64 6097 1.77 0.0200 0.0170 15.9

Imperial

Metric

��

Table A�� Test unit material properties�

Test Reference f'c fy fu �sh �su Esh

(psi) (ksi) (ksi) (ksi)

TU1 Hose et al. (1997) 6010 61.0 103 0.0090 0.085 1500

C3 Chai et al. (1991) 4730 45.7 72.2 0.0100 0.100 1500

1A Hines et al. (1999) 5530 67.0 100 0.0050 0.118 1150

1B Hines et al. (1999) 6210 67.0 100 0.0050 0.118 1150

2A Hines et al. (1999) 5310 66.0 96.0 0.0100 0.100 850

2B Hines et al. (1999) 6020 66.0 96.0 0.0100 0.100 850

2C Hines et al. (1999) 4509 72.0 105 0.0080 0.100 1100

3A Hines et al. (2001) 5930 62.0 90.5 0.0080 0.100 850

3B Hines et al. (2001) 5930 62.0 90.5 0.0080 0.100 850

3C Hines et al. (2001) 5930 62.0 90.5 0.0080 0.100 850

LPT Hines et al. (2002) 6500 60.0 88.0 0.0100 0.100 800

DPT(L) Hines et al. (2002) 7730 60.0 87.0 0.0100 0.100 800

DPT(T) Hines et al. (2002) 7730 60.0 87.0 0.0100 0.100 800

Test Reference f'c fy fu �sh �su Esh

[Mpa] [Mpa] [Mpa] [Gpa]

TU1 Hose et al. (1997) 41.4 421 710 0.0090 0.085 10.3

C3 Chai et al. (1991) 32.6 315 498 0.0100 0.100 10.3

1A Hines et al. (1999) 38.1 462 690 0.0050 0.118 7.93

1B Hines et al. (1999) 42.8 462 690 0.0050 0.118 7.93

2A Hines et al. (1999) 36.6 455 662 0.0100 0.100 5.86

2B Hines et al. (1999) 41.5 455 662 0.0100 0.100 5.86

2C Hines et al. (1999) 31.1 496 724 0.0080 0.100 7.58

3A Hines et al. (2001) 40.9 427 624 0.0080 0.100 5.86

3B Hines et al. (2001) 40.9 427 624 0.0080 0.100 5.86

3C Hines et al. (2001) 40.9 427 624 0.0080 0.100 5.86

LPT Hines et al. (2002) 44.8 414 607 0.0100 0.100 5.52

DPT(L) Hines et al. (2002) 53.3 414 600 0.0100 0.100 5.52

DPT(T) Hines et al. (2002) 53.3 414 600 0.0100 0.100 5.52

Imperial

Metric

��

Table A�� Test unit yield properties�

Test Reference M'y My �'y �y �'y �y �'yf �yf

(kft) (kft) (rad/in.) (rad/in.) (in.) (in.) (in.) (in.)

TU1 Hose et al. (1997) 660 797 0.000179 0.000216 0.934 1.57 0.934 1.57

C3 Chai et al. (1991) 536 628 0.000147 0.000172 0.880 1.08 0.880 1.08

1A Hines et al. (1999) 838 1080 0.0000744 0.0000959 0.854 1.11 0.800 1.05

1B Hines et al. (1999) 853 1100 0.0000734 0.0000947 0.840 1.10 0.800 1.04

2A Hines et al. (1999) 830 1030 0.0000743 0.0000922 0.250 0.350 0.204 0.277

2B Hines et al. (1999) 846 1050 0.0000734 0.0000911 0.232 0.351 0.219 0.312

2C Hines et al. (1999) 809 1040 0.0000797 0.000103 0.329 0.355 0.255 0.290

3A Hines et al. (2001) 1220 1620 0.0000746 0.0000986 0.725 0.947 0.569 0.731

3B Hines et al. (2001) 556 754 0.000130 0.000176 0.421 0.619 0.349 0.502

3C Hines et al. (2001) 2360 3060 0.0000482 0.0000625 0.981 1.32 0.823 1.08

LPT Hines et al. (2002) 5410 7120 0.0000713 0.0000938 0.659 0.866 0.602 0.780

DPT(L) Hines et al. (2002) 5230 7080 0.0000724 0.0000980 0.645 0.945 0.595 0.810

DPT(T) Hines et al. (2002) 10400 13000 0.0000408 0.0000510 1.73 2.17 1.64 2.00

Test Reference M'y My �'y �y �'y �y �'yf �yf

[kNm] [kNm] [�rad/mm] [�rad/mm] [mm] [mm] [mm] [mm]

TU1 Hose et al. (1997) 894 1080 7.05 8.51 23.7 39.9 23.7 39.9

C3 Chai et al. (1991) 726 851 5.79 6.78 22.4 27.4 22.4 27.4

1A Hines et al. (1999) 1136 1463 2.93 3.78 21.7 28.2 20.3 26.7

1B Hines et al. (1999) 1156 1491 2.89 3.73 21.3 27.9 20.3 26.4

2A Hines et al. (1999) 1125 1396 2.93 3.63 6.35 8.89 5.18 7.04

2B Hines et al. (1999) 1146 1423 2.89 3.59 5.89 8.92 5.56 7.92

2C Hines et al. (1999) 1096 1409 3.14 4.06 8.36 9.02 6.48 7.37

3A Hines et al. (2001) 1653 2195 2.94 3.88 18.4 24.1 14.5 18.6

3B Hines et al. (2001) 753 1022 5.12 6.93 10.7 15.7 8.86 12.8

3C Hines et al. (2001) 3198 4146 1.90 2.46 24.9 33.5 20.9 27.4

LPT Hines et al. (2002) 7331 9648 2.81 3.70 16.7 22.0 15.3 19.8

DPT(L) Hines et al. (2002) 7087 9593 2.85 3.86 16.4 24.0 15.1 20.6

DPT(T) Hines et al. (2002) 14092 17615 1.61 2.01 43.9 55.1 41.7 50.8

Metric

Imperial

�

D = 28"�

L = 6"g b

24"[610]

#7 [22]20 tot.

#3 [10]spiral

s = 2 1/4" [57]1/2" [13]

cover(assumed)

144"[3658]

6"[152]

8"[203]

8"[203]

8"[203]

8"[203]

8"[203]

8"[203]

8"[203]

8"[203]

14"[356]

84"[2134]

North

Figure A�� Well�conned circular column �TU�� test setup east elevation� columnsection and curvature instrumentation layout� ��

��

South

144"[3658]

5" [127]

5" [127]

5" [127]

5" [127]

10"[254]

10"[254]

10"[254]

D = 27"

24"[610]

#6 [19]26 tot.

#2 [6]spiral

s = 5" [127]1/2" [13]

cover(assumed)

Lgb = 5"

Figure A�� Poorly�conned circular column �C�� test setup east elevation� columnsection and curvature instrumentation layout� ��

��

1 - Actuator MTS

capacity = 220 kips [979 kN]

stroke = +/- 24 in. [610 mm]

Column Capacity

75 kips [334 kN]

Max Displacement

10 in. [254 mm]

Axial Load Apparatus

Axial Load = 198 kips [881 kN]

P/(f'cAg) = 0.10

Strong Floor

Test Unit:

Structural Wall with

Boundary Elements

17'-0" [5180 mm]

4'-0" [1220 mm]

16

'-1

1/2

"[4

92

0m

m]

Axial Load Jacks

2 x 200 kips [2 x 890 kN]

load cell

Figure A�� Test Units �A� �B setup� east elevation ��

��

1 - Actuator MTS

capacity = 220 kips [979 kN]

stroke = +/- 24in [610 mm]

Column Lateral Load

Capacity

150 kips [668 kN]

Max Displacement

4 in [102 mm]

Axial Load Apparatus:

2 x 200 kip [2 x 890 kN] jacks,

calibrated load cells,

Axial Load = 198 kips [881 kN]

P/(f'cAg) = 0.10

Strong Floor

Test Unit:

Structural Wall with

Boundary Elements

17'-0" [5180 mm]

4'-0" [1220 mm]

A A

8'-

0"

[24

40

mm

]

Figure A�� Test Units �A� �B� �C setup� east elevation ��

�

2C

2A, 2B

1A, 1B

48"[1219]

3"[76]

6"[152]

24"[610]

3"[76]

6"[152]

3"[76]

3"[76]

3"[76]

6"[152]

3"[76]

6"[152]

3"[76]

4"[102]

#4 [13]9 tot.

#4 [13]5" [127] o.c.

1A, 2A: #3 [10]s = 6" [152]1B, 2B: #2 [6]s = 8" [203]

#3 [10] spiral

s = 3" [10]

o.d. = 11" [279]

1/2" [13]

cover

22"[559]

4"[102]

4"[102]

4"[102]

#3 [10]s = 9" [229]

Figure A�� Cross sections of Test Units �A� �B� �A� �B and �C with reinforcement��

��

16

'-1

1/2

"[4

91

5m

m]

2x

1'-

0"

[30

5m

m]

Lin

ear

Po

ten

tio

met

ers

11

/2"

[38

mm

]

(28

tot)

1'-

6"

[45

7m

m]

1'-

0"

[30

5m

m]

4x

[15

2m

m]

6"

8'-

0"

[2438

mm

]

Lin

ear

Pote

nti

om

eter

s

11/2

"[3

8m

m]

(28

tot)

1'-

6"

[457

mm

]

3'-

0"

[914

mm

]

wa

llcu

rva

ture

tota

lcu

rva

ture

3x

1'-

0"

[305

mm

]

4x

6"

[152

mm

]

4'-

0"

[12

19

mm

]

Tes

tU

nit

s1

Aan

d1

BT

est

Un

its

2A

,2

Ban

d2

C

FigureA��TestUnits�A��B��A��Band�C�curvatureinstrumentationlayout�eastelevations��

��

Figure A�� Test Unit �A setup� east elevation ��

��

Figure A�� Test Unit �B setup� east elevation ��

��

Figure A�� Test Unit �C setup� east elevation�

��

3"[76]

6"[152]

3"[76]

3"[76]

6"[152]

3"[76]

3"[76]

6"[152]

3"[76]

4"[102]

4"[102]

#6 [19]12 tot.

#3[10]

spiral, s = 2” [52]

11"[279]o.d.

#3[10]s = 4” [102]

#3[10]5"[127]o.c.

1/2"[13]cover3C

1/2" [25] polystyrene blockout,

bottom only

1/2"

[13]

7/8"

[23]

1/2"

[13]

48"[1219]

72"[1829]

6"[152]

30"[762]

24"[610]

48"[1219]

3B

3A

Recess Detail

(applies to 3A, 3B & 3C)

Figure A�� Cross sections of Test Units �A� �B and �C with reinforcement ��

�

3A

3B

3C

12

0"

[30

48

]

6"

[15

2]

1"

[25

]3

/4"

[19

]P

VC

pip

e

asb

lock

ou

tfo

rth

read

rod

bu

ttw

eld

6"

[15

2]

thre

adro

d

totr

ansv

erse

bar

6"

[15

2]

48

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21

9]

12

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05

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terv

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48

"[1

21

9]

24

"[6

10

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terv

als

LC

A6

LC

A1

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LC

A2

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LC

A3

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LC

A4

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LC

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LC

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LC

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LC

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LC

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6"

[15

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2"

[51

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LC

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6"

[15

2]

48

"[1

21

9]

12

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05

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terv

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LC

A6

LC

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LC

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A3

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LC

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LC

E6

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E4

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LC

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2"

[51

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6"

[15

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21

9]

12

"[3

05

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G3

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G2

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[51

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6"

[15

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(E)

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��

Appendix B

Circular Columns

This appendix contains data from circular columns tested at the University of Cali�

fornia� San Diego�

��

Table B�� Well�conned circular column test �Hose et al� �� Average experimentalplasticity values�

Level � M l.s. points �b �c �s �� p �p Lp Lpr Lsp L'sp

�� (in.) (kft) (#) (rad/in.) (rad/in.) (in.) (in.) (in.) (in.) (in.)

1 x 1 1.56 686 -- 0.00041 -0.0031 0.0057 1.9 0.00022 0.510 16 16 8.0 0.92

1 x 2 1.54 667 -- 0.00041 -0.0031 0.0057 1.8 0.00023 0.519 16 16 8.0 0.86

1 x 3 1.55 664 -- 0.00041 -0.0031 0.0057 1.9 0.00023 0.533 16 16 8.0 0.83

1.5 x 1 2.36 822 4 0.00060 -0.0043 0.0086 2.7 0.00037 1.10 20 24.9 8.0 2.16

1.5 x 2 2.36 778 4 0.00063 -0.0045 0.0091 2.9 0.00042 1.17 19 22.9 8.0 1.91

1.5 x 3 2.39 782 4 0.00063 -0.0045 0.0091 2.9 0.00042 1.19 20 23.5 8.0 2.14

2 x 1 3.14 846 4 0.00076 -0.0052 0.011 3.5 0.00053 1.85 24 32.4 8.0 4.28

2 x 2 3.15 813 4 0.00077 -0.0053 0.011 3.5 0.00055 1.90 24 31.9 8.0 4.16

2 x 3 3.14 824 4 0.00078 -0.0053 0.011 3.5 0.00055 1.88 24 31.1 8.0 4.05

3 x 1 4.71 883 4 0.00111 -0.0075 0.016 5.16 0.000874 3.46 27.5 33.4 10.8 3.92

3 x 2 4.71 855 4 0.00111 -0.0075 0.016 5.16 0.000882 3.50 27.6 33.6 10.8 3.93

3 x 3 4.71 845 4 0.00114 -0.0077 0.017 5.26 0.000908 3.52 26.9 33.5 10.2 3.14

4 x 1 6.29 908 4 0.00163 -0.011 0.024 7.57 0.00139 5.00 25.0 36.0 7.04 2.46

4 x 2 6.30 884 4 0.00161 -0.011 0.023 7.45 0.00137 5.05 25.6 36.8 7.22 2.45

4 x 3 6.28 865 4 0.00156 -0.011 0.023 7.24 0.00133 5.06 26.4 37.4 7.73 2.60

6 x 1 9.45 937 5 0.00218 -0.015 0.031 10.1 0.00193 8.12 29.3 45.1 6.72 3.05

6 x 2 9.42 878 5 0.00225 -0.016 0.032 10.4 0.00201 8.18 28.2 44.8 5.81 2.63

8 x 1 12.6 873 5 0.00306 -0.023 0.043 14.1 0.00282 11.3 27.9 46.3 4.77 2.37

8 x 2 12.6 859 5 0.00290 -0.021 0.041 13.4 0.00267 11.3 29.5 46.6 6.24 3.21


�� [mm] [kNm] (#) [�rad/mm] [�rad/mm] [mm] [mm] [mm] [mm] [mm]

1 x 1 1.56 686 -- 16 -0.0031 0.0057 1.9 8.7 12.9 400 0 200 23

1 x 2 1.54 667 -- 16 -0.0031 0.0057 1.8 8.9 13.2 400 0 200 22

1 x 3 1.55 664 -- 16 -0.0031 0.0057 1.9 9.1 13.5 400 0 200 21

1.5 x 1 60.0 1110 4 24 -0.0043 0.0086 2.7 15 28.0 490 632 200 55

1.5 x 2 60.0 1050 4 25 -0.0045 0.0091 2.9 16 29.8 460 581 200 48

1.5 x 3 60.7 1060 4 25 -0.0045 0.0091 2.9 16 30.2 470 596 200 54

2 x 1 79.8 1150 4 30 -0.0052 0.011 3.5 21 46.9 610 822 200 109

2 x 2 79.9 1100 4 30 -0.0053 0.011 3.5 22 48.3 600 812 200 106

2 x 3 79.8 1120 4 31 -0.0053 0.011 3.5 22 47.8 590 790 200 103

3 x 1 120 1200 4 43.9 -0.0075 0.016 5.06 34.4 88 699 849 274 100

3 x 2 120 1160 4 43.9 -0.0075 0.016 5.06 34.7 89 701 853 274 100

3 x 3 120 1140 4 44.8 -0.0077 0.017 5.17 35.8 89 684 851 258 80

4 x 1 160 1230 4 64.4 -0.011 0.024 7.43 54.7 127 636 914 179 62

4 x 2 160 1200 4 63.4 -0.011 0.023 7.32 54.0 128 650 934 183 62

4 x 3 160 1170 4 61.6 -0.011 0.023 7.11 52.4 128 671 950 196 66

6 x 1 240 1270 5 85.9 -0.015 0.031 9.91 75.9 206 744 1146 171 78

6 x 2 239 1190 5 88.7 -0.016 0.032 10.2 79.3 208 717 1138 148 67

8 x 1 319 1180 5 120 -0.023 0.043 13.9 111 288 710 1177 121 60

8 x 2 319 1160 5 114 -0.021 0.041 13.2 105 288 750 1183 158 82

Imperial Units

Metric Units

��

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Aver

age

curv

ature

(rad

/in.)

012

24

36

48

60

72

84

96

108

120

132

144

Heightabovefooting,h(in.)

0

25

50

75

100

125

[ �ra

d/

mm

]

0300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm]

�y=

0.0

00220

rad/i

n.

Hose

etal

.T

U1

L=

144

in.

D=

24

in.

��

=1

��

=1.5

��

=2

��

=3

��

=4

��

=6

��

=8

FigureB��Well�connedcircularcolumntest�Hoseetal��Averagecurvatureproles�

��

Table B�� Well�conned circular column test �Hose et al� �� Peak curvaturevalues�

� F M �b0 �bsp �b points Lp0 Lpsp Lp

(in.) (kips) (kft) (rad/in.) (rad/in.) (rad/in.) (in.) (in.) (in.)

zero 0.0000 0.00 0.00 0.0000000 0.0000000 -- -- -- -- --

+1/4 F'y -0.0396 -5.33 -64.0 -0.0000150 -0.0000060 -- -- -- -- --

-1/4 F'y 0.0624 7.09 85.1 0.0000130 0.0000060 -- -- -- -- --

+1/2 F'y -0.235 -19.7 -236 -0.0000750 -0.0000320 -- -- -- -- --

-1/2 F'y 0.271 19.5 234 0.0000750 0.0000320 -- -- -- -- --

+3/4 F'y -0.507 -31.7 -380 -0.000154 -0.0000660 -- -- -- -- --

-3/4 F'y 0.558 31.4 376 0.000145 0.0000620 -- -- -- -- --

+ F'y -0.938 -44.7 -537 -0.000307 -0.000131 -- -- -- -- --

- F'y 0.929 41.5 498 0.000267 0.000114 -- -- -- -- --

�� = +1 x 1 1.56 58.2 698 0.000458 0.000196 0.00041 -- 13.9 -- 16

�� = -1 x 1 -1.57 -56.2 -674 -0.000482 -0.000207 -0.00043 -- 13.5 -- 16

�� = +1 x 2 1.55 56.4 677 0.000443 0.000190 0.00042 -- 15.1 -- 16

�� = -1 x 2 -1.54 -54.8 -658 -0.000485 -0.000208 -0.00042 -- 13.1 -- 16

�� = +1 x 3 1.55 55.6 667 0.000448 0.000192 0.00042 -- 15.3 -- 16

�� = -1 x 3 -1.55 -55.1 -661 -0.000489 -0.000209 -0.00042 -- 13.2 -- 16

�� = +1.5 x 1 2.33 68.8 825 0.000767 0.000329 0.00061 -- 14.7 -- 20

�� = -1.5 x 1 -2.35 -68.3 -820 -0.000859 -0.000368 -0.00061 -- 12.7 -- 20

�� = +1.5 x 2 2.36 68.3 819 0.000759 0.000325 0.00062 -- 15.6 -- 20

�� = -1.5 x 2 -2.35 -66.8 -801 -0.000898 -0.000385 -0.00062 -- 12.2 65.3 20

�� = +1.5 x 3 2.35 65.7 788 0.000764 0.000327 0.00062 -- 15.6 -- 20

�� = -1.5 x 3 -2.34 -64.8 -777 -0.000948 -0.000406 -0.00062 -- 11.4 53.8 20

�� = +2 x 1 3.14 70.9 851 0.00113 0.000483 0.00085 -- 15.3 71.5 21

�� = -2 x 1 -3.12 -71.1 -853 -0.00145 -0.000620 -0.00085 -- 10.9 39.0 21

�� = +2 x 2 3.03 66.2 795 0.00118 0.000504 0.00083 -- 14.0 57.1 21

�� = -2 x 2 -3.12 -69.3 -832 -0.00144 -0.000616 -0.00085 -- 11.1 39.4 21

�� = +2 x 3 3.12 68.3 820 0.00118 0.000504 0.00085 -- 14.5 61.7 21

�� = -2 x 3 -3.10 -69.0 -828 -0.00143 -0.000614 -0.00084 -- 11.1 39.2 21

�� = +3 x 1 4.68 73.4 881 0.00174 0.000744 0.00116 4 16.6 59.6 25.7

�� = -3 x 1 -4.67 -73.8 -885 -0.00195 -0.000835 -0.00117 4 14.3 47.5 25.2

�� = +3 x 2 4.71 71.4 857 0.00182 0.000781 0.00115 4 15.8 54.5 26.1

�� = -3 x 2 -4.70 -71.2 -854 -0.00187 -0.000799 -0.00116 4 15.4 52.0 25.8

�� = +3 x 3 4.70 70.1 841 0.00178 0.000761 0.00111 4 16.4 57.0 27.3

�� = -3 x 3 -4.67 -70.7 -848 -0.00169 -0.000723 -0.00119 4 17.2 62.4 24.8

�� = +4 x 1 6.28 76.1 913 0.00247 0.00106 0.00158 4 16.3 52.0 25.7

�� = -4 x 1 -6.28 -75.3 -904 -0.00214 -0.000915 -0.00159 4 19.5 68.2 25.6

�� = +4 x 2 6.32 69.7 837 0.00249 0.00107 0.00159 4 16.5 51.7 25.9

�� = -4 x 2 -6.27 -73.7 -885 -0.00205 -0.000879 -0.00150 4 20.5 73.6 27.4

�� = +4 x 3 6.26 71.8 861 0.00250 0.00107 0.00154 4 16.1 50.5 26.6

�� = -4 x 3 -6.27 -72.5 -870 -0.00198 -0.000849 -0.00147 4 21.4 78.9 28.2

�� = +6 x 1 9.46 77.6 931 0.00375 0.00161 0.00209 5 17.0 50.6 30.7

�� = -6 x 1 -9.41 -78.5 -942 -0.00283 -0.00121 -0.00226 5 23.5 82.3 27.8

�� = +6 x 2 9.42 73.3 879 0.00355 0.00152 0.00216 5 18.2 54.6 29.4

�� = -6 x 2 -9.39 -76.5 -918 -0.00293 -0.00126 -0.00230 5 22.6 76.0 27.2

�� = +8 x 1 12.6 74.3 892 0.00453 0.00194 0.00308 5 19.6 57.9 27.6

�� = -8 x 1 -12.6 -81.0 -972 -0.00418 -0.00179 -0.00286 5 21.3 66.0 29.8

�� = +8 x 2 12.5 71.2 855 0.00426 0.00183 0.00308 5 20.9 63.2 27.5

�� = -8 x 2 -12.6 -76.8 -922 -0.00460 -0.00197 -0.00273 5 19.1 56.5 31.4

Level

�

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

Hei

ghtab

ove

foot

ing

(in.

)

-75

-50

-25

0 25 50 75

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm

]

� y

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm

]

� y-7

5

-50

-25

0 25 50 75

[� rad / mm]

�� = 1

�� = 2

inaccurateleast squaresline

alternativebase curvature



Figure B�� Well�conned circular column test �Hose et al� �� Curvature prolesat �� and at ��

��

-0.0

040

-0.0

030

-0.0

020

-0.0

010

0.00

00

0.00

10

0.00

20

0.00

30

0.00

40

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

Hei

ghtab

ove

foot

ing

(in.

)

-150

-100

-50

0 50 100

150

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm

]

� y

-0.0

040

-0.0

030

-0.0

020

-0.0

010

0.00

00

0.00

10

0.00

20

0.00

30

0.00

40

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm

]

� y-1

50

-100

-50

0 50 100

150

[� rad / mm]

�� = 3

�� = 4


��

-0.0

040

-0.0

030

-0.0

020

-0.0

010

0.00

00

0.00

10

0.00

20

0.00

30

0.00

40

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

Hei

ghtab

ove

foot

ing

(in.

)

-150

-100

-50

0 50 100

150

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm

]

� y

-0.0

040

-0.0

030

-0.0

020

-0.0

010

0.00

00

0.00

10

0.00

20

0.00

30

0.00

40

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm

]

� y-1

50

-100

-50

0 50 100

150

[� rad / mm]

�� = 6

�� = 8


��

Table B�� Well�conned circular column test �Hose et al� �� Flexural strainvalues�

� F �b0 �b �c0 �c �'c �s0 �s �'s

(in.) (kips) (rad/in.) (rad/in.)

+1/4 F'y 0.0510 6.21 0.000014 -- -0.00032 -- -0.00046 -0.00002 -- -0.00016

+1/2 F'y 0.253 19.6 0.000075 -- -0.00089 -- -0.0013 0.00072 -- 0.00026

+3/4 F'y 0.533 31.5 0.000150 -- -0.0015 -- -0.0021 0.0018 -- 0.0011

+ F'y 0.933 43.1 0.000287 -- -0.0023 -- -0.0034 0.0038 -- 0.0027

�� = 1 x 1 1.56 57.2 0.000470 0.00041 -0.0035 -0.0031 -0.0053 0.0066 0.0057 0.0048

�� = 1 x 2 1.54 55.6 0.000464 0.00041 -0.0034 -0.0031 -0.0055 0.0065 0.0057 0.0045

�� = 1 x 3 1.55 55.3 0.000469 0.00041 -0.0035 -0.0031 -0.0056 0.0066 0.0057 0.0045

�� = 1.5 x 1 2.34 68.5 0.000813 0.00060 -0.0055 -0.0043 -0.0085 0.012 0.0086 0.0089

�� = 1.5 x 2 2.35 67.5 0.0008285 0.00063 -0.0056 -0.0045 -0.0093 0.012 0.0091 0.0085

�� = 1.5 x 3 2.34 65.2 0.000856 0.00063 -0.0058 -0.0045 -0.0095 0.013 0.0091 0.0088

�� = 2 x 1 3.13 71.0 0.00129 0.00076 -0.0087 -0.0052 -0.013 0.019 0.011 0.015

�� = 2 x 2 3.07 67.8 0.00131 0.00077 -0.0089 -0.0053 -0.013 0.019 0.011 0.015

�� = 2 x 3 3.11 68.6 0.00130 0.00078 -0.0088 -0.0053 -0.013 0.019 0.011 0.015

�� = 3 x 1 4.68 73.6 0.00184 0.00111 -0.013 -0.0075 -0.020 0.027 0.016 0.020

�� = 3 x 2 4.70 71.3 0.00184 0.00111 -0.013 -0.0075 -0.021 0.027 0.016 0.018

�� = 3 x 3 4.68 70.4 0.00173 0.00114 -0.012 -0.0077 -0.019 0.025 0.017 0.018

�� = 4 x 1 6.28 75.7 0.00230 0.00163 -0.016 -0.011 -0.023 0.033 0.024 0.026

�� = 4 x 2 6.30 71.7 0.00227 0.00161 -0.016 -0.011 -0.024 0.032 0.023 0.025

�� = 4 x 3 6.26 72.1 0.00224 0.00156 -0.016 -0.011 -0.025 0.032 0.023 0.023

�� = 6 x 1 9.43 78.1 0.00329 0.00218 -0.024 -0.015 -0.034 0.046 0.031 0.037

�� = 6 x 2 9.40 74.9 0.00324 0.00225 -0.024 -0.016 -0.039 0.045 0.032 0.030

�� = 8 x 1 12.6 77.7 0.00436 0.00306 -0.032 -0.023 -0.054 0.061 0.043 0.039

�� = 8 x 2 12.5 74.0 0.00443 0.00290 -0.033 -0.021 -0.066 0.061 0.041 0.028

Level

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Average post-yield strains ( �'c & �'s)

0

12

24

36

48

60

72

84

96

108

120

132

144

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm

]

Hose et al. TU1Post-Yield Strains

L = 144 in.D = 24 in.

�� = 1 x 1

�� = 2 x 1

�� = 3 x 1

�� = 4 x 1

�� = 6 x 1

�� = 8 x 1

� y = 0.00210

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

Average pre-yield strains ( �'c & �'s)

0

12

24

36

48

60

72

84

96

108

120

132

144

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm

]

� y = 0.00210 Hose et al. TU1Pre-Yield Strains

L = 144 in.D = 24 in.

1/4 F'y

1/2 F'y

3/4 F'y

F'y

Figure B�� Well�conned circular column test �Hose et al� �� Average �exuralstrain proles�

�

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

Pre-yield strains at negative peak ( �'c & �'s)

0

12

24

36

48

60

72

84

96

108

120

132

144

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm

]

Hose et al. TU1Pre-Yield Strains

L = 144 in.D = 24 in.

1/4 F'y

1/2 F'y

3/4 F'y

F'y

� y = 0.00210

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

Pre-yield strains at positive peak ( �'c & �'s)

0

12

24

36

48

60

72

84

96

108

120

132

144

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm

]

� y = 0.00210 Hose et al. TU1Pre-Yield Strains

L = 144 in.D = 24 in.

1/4 F'y

1/2 F'y

3/4 F'y

F'y

Figure B�� Well�conned circular column test �Hose et al� �� Pre�yield �exuralstrain proles�

�

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Post-yield strains at negative peak (�'c & �'s)

0

12

24

36

48

60

72

84

96

108

120

132

144

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm

]

Hose et al. TU1Post-Yield Strains

L = 144 in.D = 24 in.

�� = 1 x -1

�� = 2 x -1

�� = 3 x -1

�� = 4 x -1

�� = 6 x -1

�� = 8 x -1

� y = 0.00210

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Post-yield strains at positive peak (�'c & �'s)

0

12

24

36

48

60

72

84

96

108

120

132

144

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm

]

� y = 0.00210 Hose et al. TU1Post-Yield Strains

L = 144 in.D = 24 in.

�� = 1 x +1

�� = 2 x +1

�� = 3 x +1

�� = 4 x +1

�� = 6 x +1

�� = 8 x +1

Figure B�� Well�conned circular column test �Hose et al� �� Post�yield �exuralstrain proles�

�

Table B�� Unconned circular column test �Chai et al� �� Average experimentalplasticity values�


�� (in.) (kft) (#) (rad/in.) (rad/in.) (in.) (in.) (in.) (in.) (in.)

1 x 1 1.08 516 -- 0.00031 -0.0026 0.0040 1.8 0.000165 0.238 10 0 10 0.282

1 x 2 1.08 504 -- 0.00031 -0.0026 0.0040 1.8 0.00018 0.254 10 0 10 0.2064

1 x 3 1.08 502 -- 0.00032 -0.0027 0.0041 1.9 0.00018 0.260 10 0 10 0.1898

1.5 x 1 1.63 582 -- 0.00052 -0.0040 0.0070 3.0 0.00036 0.672 13 16 5.1 1.044

1.5 x 2 1.62 581 -- 0.00051 -0.0040 0.0069 3.0 0.00035 0.668 13 16 5.1 1.057

1.5 x 3 1.62 577 -- 0.00052 -0.0040 0.0070 3.0 0.00036 0.675 13 16 5.1 1.061

2 x 1 2.17 630 4 0.00062 -0.0047 0.0084 3.6 0.00045 1.13 18 24.9 5.1 3.07

2 x 2 2.17 617 4 0.00063 -0.0048 0.0086 3.7 0.00046 1.15 17 24.6 5.1 3.18

2 x 3 2.17 610 4 0.00063 -0.0048 0.0086 3.7 0.00047 1.16 17 24.5 5.1 3.20

3 x 1 3.25 649 4 0.000929 -0.0073 0.012 5.40 0.000751 2.19 20.2 24.4 8.03 2.74

3 x 2 3.25 633 4 0.000873 -0.0068 0.012 5.08 0.000700 2.21 22.0 25.6 9.17 3.32

3 x 3 3.25 626 4 0.000867 -0.0068 0.012 5.04 0.000696 2.22 22.2 25.8 9.28 3.36

4 x 1 4.33 654 4 0.00117 -0.0098 0.015 6.78 0.000987 3.26 22.9 29.9 7.98 2.88

4 x 2 4.33 633 4 0.00113 -0.0094 0.015 6.60 0.000961 3.29 23.8 30.6 8.47 2.88

4 x 3 4.33 615 4 0.00116 -0.0097 0.015 6.76 0.000994 3.32 23.2 30.5 7.92 2.49

5 x 1 5.42 605 4 0.00148 -0.013 0.018 8.63 0.00132 4.43 23.3 32.7 6.96 2.41

Level � M l.s. points �b �c �s �� p �p Lp Lpr Lsp Lsp

�� [mm] [kNm] (#) [�rad/mm] [�rad/mm] [mm] [mm] [mm] [mm] [mm]

1 x 1 27.43 699 -- 12 -0.0026 0.0040 1.8 6.5 6.04 254 0 254 7

1 x 2 27.43 683 -- 12 -0.0026 0.0040 1.8 7.0 6.46 254 0 254 5

1 x 3 27.43 680 -- 13 -0.0027 0.0041 1.9 7.1 6.61 254 0 254 5

1.5 x 1 41.35 789 -- 20 -0.0040 0.0070 3.0 14 17.1 333 406 130 27

1.5 x 2 41.20 788 -- 20 -0.0040 0.0069 3.0 14 17.0 333 406 130 27

1.5 x 3 41.21 782 -- 20 -0.0040 0.0070 3.0 14 17.1 333 406 130 27

2 x 1 55.02 853 4 24 -0.0047 0.0084 3.6 18 28.8 445 632 130 78

2 x 2 55.05 836 4 25 -0.0048 0.0086 3.7 18 29.3 442 625 130 81

2 x 3 55.00 827 4 25 -0.0048 0.0086 3.7 18 29.6 440 622 130 81

3 x 1 82.63 880 4 36.6 -0.0073 0.012 5.4 29.6 55.6 514 620 204 69

3 x 2 82.64 858 4 34.4 -0.0068 0.012 5.1 27.6 56.2 558 650 233 84

3 x 3 82.55 848 4 34.2 -0.0068 0.012 5.0 27.4 56.5 563 656 236 85

4 x 1 109.97 886 4 45.9 -0.0098 0.015 6.8 38.9 82.7 582 758 203 73

4 x 2 109.89 858 4 44.7 -0.0094 0.015 6.6 37.9 83.5 603 777 215 73

4 x 3 109.93 834 4 45.8 -0.0097 0.015 6.8 39.2 84.3 589 775 201 63

5 x 1 137.63 820 4 58.5 -0.013 0.018 8.6 52.0 112 592 830 177 61

Imperial Units

Metric Units

�

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Aver

age

curv

ature

(rad

/in.)

012

24

36

48

60

72

84

96

108

120

132

144


0

25

50

75

100

125

[ �ra

d/

mm

]

0300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

[mm]

�y=

0.0

00172

rad/i

n.

Chai

etal

.C

3L

=144

in.

D=

24

in.

��

=1

��

=1.5

��

=2

��

=3

��

=4

��

=5

FigureB��Unconnedcircularcolumntest�Chaietal��Averagecurvatureproles�

�

Table B�� Unconned circular column test �Chai et al� �� Peak curvature values�

� F M �b0 �bsp �b points Lp0 Lpsp Lp

(in.) (kips) (kft) (rad/in.) (rad/in.) (rad/in.) (in.) (in.) (in.)

zero 0 0 0 0 0 -- -- -- -- --

+ 1/5 F'y x 1 0.107 8.61 103 0.0000270 0.0000130 -- -- -- -- --

- 1/5 F'y x 1 -0.068 -7.44 -89.3 -0.0002700 -0.0000130 -- -- -- -- --

+ 1/5 F'y x 2 0.105 8.58 103 0.0000270 0.0000130 -- -- -- -- --

- 1/5 F'y x 2 -0.092 -8.99 -108 -0.0002700 -0.0000130 -- -- -- -- --

+ 3/8 F'y 0.193 15.4 185 0.0000270 0.0000130 -- -- -- -- --

- 3/8 F'y -0.163 -14.4 -173 -0.0002700 -0.0000130 -- -- -- -- --

+ 0.7 F'y x 1 0.432 27.7 333 0.000144 0.0000700 -- -- -- -- --

- 0.7 F'y x 1 -0.385 -26.9 -323 -0.000104 -0.0000510 -- -- -- -- --

+ 0.7 F'y x 2 0.439 27.5 330 0.000154 0.0000760 -- -- -- -- --

- 0.7 F'y x 2 -0.412 -27.0 -324 -0.000114 -0.0000560 -- -- -- -- --

+ 0.7 F'y x 3 0.448 27.5 330 0.000159 0.0000780 -- -- -- -- --

- 0.7 F'y x 3 -0.421 -27.0 -325 -0.000118 -0.0000580 -- -- -- -- --

+ 0.7 F'y x 4 0.452 27.5 330 0.000160 0.0000780 -- -- -- -- --

- 0.7 F'y x 4 -0.424 -27.0 -324 -0.000119 -0.0000580 -- -- -- -- --

+ 0.7 F'y x 5 0.457 27.6 331 0.000163 0.0000800 -- -- -- -- --

- 0.7 F'y x 5 -0.427 -27.0 -324 -0.000120 -0.0000590 -- -- -- -- --

F'y 0.900 40.0 480 0.000293 0.000143 -- -- -- -- --

F'y -0.850 -39.0 -468 -0.000231 -0.000113 -- -- -- -- --

�� = +1 x 1 1.09 42.4 509 0.000356 0.000175 0.00031 -- 8.2 62.6 10

�� = -1 x 1 -1.08 -43.5 -522 -0.000292 -0.000143 -0.00030 -- 11.0 -- 10

�� = +1 x 2 1.08 42.2 506 0.000359 0.000176 0.00031 -- 8.2 59.0 10

�� = -1 x 2 -1.08 -41.9 -503 -0.000296 -0.000145 -0.00032 -- 11.8 -- 10

�� = +1 x 3 1.08 41.8 502 0.000362 0.000178 0.00032 -- 8.2 54.9 10

�� = -1 x 3 -1.08 -41.8 -501 -0.000299 -0.000146 -0.00032 -- 11.7 -- 10

�� = +1.5 x 1 1.62 48.9 587 0.000659 0.000323 0.00051 -- 9.5 31.8 13

�� = -1.5 x 1 -1.63 -50.7 -608 -0.000564 -0.000277 -0.00050 -- 11.4 47.3 13

�� = +1.5 x 2 1.62 48.0 576 0.000679 0.000333 0.00052 -- 9.3 30.0 13

�� = -1.5 x 2 -1.62 -48.9 -587 -0.000564 -0.000277 -0.00051 -- 11.8 47.1 13

�� = +1.5 x 3 1.62 47.7 573 0.000687 0.000337 0.00052 -- 9.2 29.3 13

�� = -1.5 x 3 -1.62 -48.5 -582 -0.000564 -0.000276 -0.00052 -- 12.0 47.7 13

�� = +2 x 1 2.17 51.8 621 0.00107 0.000523 0.00064 -- 9.2 24.7 17

�� = -2 x 1 -2.17 -53.2 -638 -0.000938 -0.000460 -0.00063 -- 10.6 30.5 17

�� = +2 x 2 2.17 50.8 610 0.00112 0.000550 0.00064 -- 8.8 23.0 17

�� = -2 x 2 -2.17 -52.0 -624 -0.000939 -0.000460 -0.00064 -- 10.7 30.7 17

�� = +2 x 3 2.17 50.3 604 0.00114 0.000560 0.00065 -- 8.6 22.5 17

�� = -2 x 3 -2.16 -51.4 -617 -0.000937 -0.000459 -0.00064 -- 10.8 30.9 17

�� = +3 x 1 3.26 53.0 636 0.00155 0.000761 0.000850 4 11.6 29.1 22.8

�� = -3 x 1 -3.25 -55.2 -663 -0.00132 -0.000648 -0.000855 4 13.8 36.9 22.3

�� = +3 x 2 3.26 51.7 620 0.00159 0.000780 0.000767 4 11.4 28.3 26.1

�� = -3 x 2 -3.25 -53.9 -647 -0.00132 -0.000645 -0.000802 4 14.0 37.3 24.3

�� = +3 x 3 3.25 51.0 612 0.00160 0.000785 0.000765 4 11.3 28.0 26.1

�� = -3 x 3 -3.25 -53.3 -640 -0.00130 -0.000638 -0.000793 4 14.3 38.0 24.7

�� = +4 x 1 4.33 53.2 638 0.00203 0.000993 0.00109 4 12.9 31.3 24.9

�� = -4 x 1 -4.33 -55.8 -670 -0.00165 -0.000809 -0.00116 4 16.2 42.0 22.9

�� = +4 x 2 4.33 51.5 618 0.00196 0.000959 0.00108 4 13.5 32.9 25.1

�� = -4 x 2 -4.32 -54.0 -648 -0.00162 -0.000794 -0.00112 4 16.7 43.2 24.1

�� = +4 x 3 4.33 49.6 595 0.00187 0.000919 0.00116 4 14.3 35.0 23.4

�� = -4 x 3 -4.33 -53.0 -635 -0.00161 -0.000789 -0.00110 4 16.9 43.8 24.8

�� = +5 x 1 5.42 48.1 577 0.00238 0.00116 0.00152 4 14.8 35.2 22.9

�� = -5 x 1 -5.42 -52.8 -633 -0.00202 -0.000992 -0.00133 4 17.5 43.8 26.3

Level





-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

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05

0.00

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20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

Hei

ghtab

ove

foot

ing

(in.

)

-75

-50

-25

0 25 50 75

[� rad / mm]

0

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3600

[mm

]

Chai et al. C3�� = 2

� y

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

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Curvature (rad/in.)

0

12

24

36

48

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144H

eigh

tab

ove

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(in.

)

0

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3600

[mm

]

� y Chai et al. C3�� = 1

-75

-50

-25

0 25 50 75

[� rad / mm]

Figure B�� Unconned circular column test �Chai et al� �� Curvature proles at�� and at ��

�

-0.0

040

-0.0

030

-0.0

020

-0.0

010

0.00

00

0.00

10

0.00

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Curvature (rad/in.)

0

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48

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Hei

ghtab

ove

foot

ing

(in.

)

-150

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[� rad / mm]

0

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[mm

]

Chai et al. C3�� = 4

� y

-0.0

040

-0.0

030

-0.0

020

-0.0

010

0.00

00

0.00

10

0.00

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Curvature (rad/in.)

0

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eigh

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ove

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(in.

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[mm

]


-150

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0 50 100

150

[� rad / mm]

Figure B�� Unconned circular column test �Chai et al� �� Curvature proles at�� and at ��

�

-0.0

040

-0.0

030

-0.0

020

-0.0

010

0.00

00

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Curvature (rad/in.)

0

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Hei

ghtab

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(in.

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[mm

]


-150

-100

-50

0 50 100

150

[� rad / mm]

Figure B�� Unconned circular column test �Hose et al� �� Curvature proles at��

�

Table B�� Unconned circular column test �Chai et al� �� Flexural strain values�

� F �b0 �b �c0 �c �'c �s0 �s �'s


1/5 F'y x 1 0.0875 8.03 0.000027 -- -0.00005 -- -0.00082 0.00008 -- -0.00033

1/5 F'y x 2 0.0987 8.79 0.000027 -- -0.00005 -- -0.00089 0.00008 -- -0.00032

3/8 F'y 0.178 14.9 -- -- -- -- -0.00114 -- -- -0.00007

0.7 F'y x 1 0.409 27.3 0.000124 -- -0.0013 -- -0.0018 0.0013 -- 0.00093

0.7 F'y x 2 0.426 27.3 0.000134 -- -0.0014 -- -0.0019 0.0015 -- 0.0010

0.7 F'y x 3 0.434 27.3 0.0001385 -- -0.0014 -- -0.0020 0.0016 -- 0.0011

0.7 F'y x 4 0.438 27.3 0.0001395 -- -0.0014 -- -0.0020 0.0016 -- 0.0011

0.7 F'y x 5 0.442 27.3 0.0001415 -- -0.0014 -- -0.0020 0.0016 -- 0.0011

F'y 0.875 39.5 0.000262 -- -0.0023 -- -0.0030 0.0034 -- 0.0028

�� = 1 x 1 1.08 43.0 0.000324 0.00031 -0.0027 -0.0026 -0.0036 0.0042 0.0040 0.0036

�� = 1 x 2 1.08 42.0 0.0003275 0.00031 -0.0027 -0.0026 -0.0037 0.0042 0.0040 0.0035

�� = 1 x 3 1.08 41.8 0.0003305 0.00032 -0.0027 -0.0027 -0.0037 0.0043 0.0041 0.0036

�� = 1.5 x 1 1.62 49.8 0.0006115 0.00052 -0.0047 -0.0040 -0.0051 0.0083 0.0070 0.0084

�� = 1.5 x 2 1.62 48.4 0.0006215 0.00051 -0.0047 -0.0040 -0.0055 0.0084 0.0069 0.0082

�� = 1.5 x 3 1.62 48.1 0.0006255 0.00052 -0.0048 -0.0040 -0.0057 0.0085 0.0070 0.0081

�� = 2 x 1 2.17 52.5 0.00100 0.00062 -0.0080 -0.0047 -0.0076 0.013 0.0084 0.014

�� = 2 x 2 2.17 51.4 0.00103 0.00063 -0.0083 -0.0048 -0.0080 0.014 0.0086 0.015

�� = 2 x 3 2.17 50.9 0.00104 0.00063 -0.0084 -0.0048 -0.0083 0.014 0.0086 0.015

�� = 3 x 1 3.25 54.1 0.00144 0.000929 -0.013 -0.0073 -0.013 0.018 0.012 0.019

�� = 3 x 2 3.25 52.8 0.00145 0.000873 -0.013 -0.0068 -0.014 0.018 0.012 0.018

�� = 3 x 3 3.25 52.1 0.00145 0.000867 -0.013 -0.0068 -0.015 0.018 0.012 0.017

�� = 4 x 1 4.33 54.5 0.00184 0.00117 -0.017 -0.0098 -0.020 0.022 0.015 0.020

�� = 4 x 2 4.33 52.8 0.00179 0.00113 -0.016 -0.0094 -0.022 0.022 0.015 0.018

�� = 4 x 3 4.33 51.3 0.00174 0.00116 -0.016 -0.0097 -0.022 0.021 0.015 0.016

�� = 5 x 1 5.42 50.4 0.00220 0.00148 -0.022 -0.013 -0.028 0.025 0.018 0.020

Level

�

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Average post-yield strains (�'c & �'s)

0

12

24

36

48

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108

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144

Hei

ghtab

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(in.

)

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300

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3600

[mm

]

Chai et al. C3Post-Yield Strains

L = 144 in.D = 24 in.

�� = 1 x 1

�� = 2 x 1

�� = 3 x 1

�� = 4 x 1

�� = 5 x 1

� y = 0.00158

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

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4

Average pre-yield strains (�'c & �'s)

0

12

24

36

48

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Hei

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(in.

)

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[mm

]

� y = 0.00158

Chai et al. C3Pre-Yield Strains

L = 144 in.D = 24 in.

1/5 F'y x 1

3/8 F'y x 1

0.7 F'y x 1

F'y x 1

Figure B�� Unconned circular column test �Chai et al� �� Average �exural strainproles�

��

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

Pre-yield strains at negative peak ( �'c & �'s)

0

12

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Hei

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(in.

)

0

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[mm

]


L = 144 in.D = 24 in.

F'y x -1

0.7 F'y x -1

3/8 F'y x -1

1/5 F'y x -1

� y = 0.00158

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

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4

Pre-yield strains at positive peak ( �'c & �'s)

0

12

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Hei

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(in.

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[mm

]

� y = 0.00158


L = 144 in.D = 24 in.

F'y x +1

0.7 F'y x +1

3/8 F'y x +1

1/5 F'y x +1

Figure B�� Unconned circular column test �Chai et al� �� Pre�yield �exuralstrain proles�

��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

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0.06


0

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Hei

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[mm

]


L = 144 in.D = 24 in.

�� = 1 x -1

�� = 2 x -1

�� = 3 x -1

�� = 4 x -1

�� = 5 x -1

� y = 0.00158

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

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0

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Hei

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[mm

]

� y = 0.00158


L = 144 in.D = 24 in.

�� = 1 x +1

�� = 2 x +1

�� = 3 x +1

�� = 4 x +1

�� = 5 x +1

Figure B�� Unconned circular column test �Chai et al� �� Post�yield �exuralstrain proles�

��

��

Appendix C

Structural Walls

This appendix contains data from structural walls with highly�conned boundary

elements tested by Hines et al� at UCSD ��

��

Table C�� Structural wall with conned boundary elements� Test �A �Hines et al�� Average experimental plasticity values�

Level � M �f l.s. points �b �c �s �� p �p Lp Lpr Lsp L'sp

�� (in.) (kft) (in.) (#) (rad/in.) (rad/in.) (in.) (in.) (in.) (in.) (in.)

1 x 1 1.11 868 1.05 -- 0.00020 -0.0020 0.0071 2.0 0.000114 0.221 10 10 5.0 0.51

1 x 2 1.11 852 1.04 -- 0.00021 -0.0021 0.0074 2.1 0.000124 0.240 10 10 5.0 0.30

1 x 3 1.11 830 1.04 -- 0.00021 -0.0021 0.0074 2.1 0.000130 0.253 10 10 5.0 0.15

1.5 x 1 1.68 990 1.57 5 0.00026 -0.0024 0.0095 2.6 0.00016 0.624 20 30 5.0 1.50

1.5 x 2 1.68 957 1.57 5 0.00027 -0.0025 0.0099 2.7 0.00017 0.654 20 29 5.0 1.40

1.5 x 3 1.68 947 1.57 5 0.00027 -0.0025 0.0099 2.7 0.00018 0.662 19 29 5.0 1.36

2 x 1 2.23 1050 2.07 5 0.00032 -0.0028 0.012 2.87 0.00022 1.07 25 40.8 5.0 2.50

2 x 2 2.23 995 2.07 5 0.00033 -0.0029 0.012 2.97 0.00023 1.12 25 39.3 5.0 2.27

2 x 3 2.23 995 2.07 5 0.00034 -0.0030 0.013 3.02 0.00024 1.12 24 38.8 5.0 2.30

3 x 1 3.32 1070 3.07 5 0.000521 -0.0042 0.020 5.43 0.000426 2.05 25 41.6 4.06 2.55

3 x 2 3.34 1040 3.08 5 0.000528 -0.0042 0.020 5.51 0.000437 2.09 24.7 41.3 4.05 2.63

3 x 3 3.34 1030 3.08 5 0.000523 -0.0042 0.020 5.46 0.000432 2.10 25.1 41.3 4.40 2.90

4 x 1 4.45 1090 4.09 5 0.000729 -0.0056 0.028 7.60 0.000632 3.05 24.9 43.3 3.29 2.80

4 x 2 4.45 1050 4.08 5 0.000712 -0.0055 0.027 7.43 0.000619 3.08 25.7 43.6 3.93 3.16

4 x 3 4.45 1040 4.08 5 0.000704 -0.0054 0.027 7.34 0.000612 3.08 26.0 43.7 4.20 3.34

6 x 1 6.67 1130 6.11 5 0.00108 -0.0082 0.042 11.2 0.000977 5.03 26.6 46.3 3.44 3.34

6 x 2 6.67 1090 6.09 5 0.00122 -0.0092 0.047 12.7 0.00112 5.05 23.3 44.1 1.21 1.23

6 x 3 6.67 1010 6.08 5 0.00124 -0.0094 0.048 12.9 0.00115 5.12 23.1 43.8 1.20 1.13

8 x 1 8.82 1030 7.99 5 0.00162 -0.012 0.062 16.9 0.00153 7.00 23.6 42.8 2.25 3.13

Level � M �f l.s. points �b �c �s �� p �p Lp Lpr Lsp Lsp

�� [mm] [kNm] [mm] (#) [�rad/mm] [�rad/mm] [mm] [mm] [mm] [mm] [mm]

1 x 1 28.2 1080 26.7 -- 7.9 -0.0020 0.0071 2.0 4.5 5.62 250 254 130 12.9

1 x 2 28.2 1150 26.4 -- 8.3 -0.0021 0.0074 2.1 4.9 6.10 250 254 130 7.6

1 x 3 28.2 1130 26.4 -- 8.4 -0.0021 0.0074 2.1 5.1 6.42 250 254 130 3.7

1.5 x 1 42.6 1340 39.9 5 10.3 -0.0024 0.0095 2.6 6.4 15.8 510 758 130 38.0

1.5 x 2 42.6 1300 39.8 5 10.5 -0.0025 0.0099 2.7 6.8 16.6 500 745 130 35.4

1.5 x 3 42.6 1280 39.8 5 10.7 -0.0025 0.0099 2.7 6.9 16.8 490 734 130 34.5

2 x 1 56.6 1420 52.6 5 12.7 -0.0028 0.012 2.9 8.54 27.2 646 1038 127 63.6

2 x 2 56.6 1350 52.5 5 13.2 -0.0029 0.012 3.0 9.21 28.4 625 997 127 57.5

2 x 3 56.6 1350 52.5 5 13.2 -0.0030 0.013 3.0 9.30 28.4 620 985 127 58.5

3 x 1 84.4 1450 77.9 5 20.5 -0.0042 0.020 5.4 16.8 52.1 632 1057 103 64.8

3 x 2 84.9 1410 78.1 5 20.8 -0.0042 0.020 5.5 17.2 53.0 628 1050 103 66.8

3 x 3 84.9 1400 78.2 5 20.6 -0.0042 0.020 5.5 17.0 53.2 637 1050 112 73.5

4 x 1 113 1480 104 5 28.7 -0.0056 0.028 7.6 24.9 77.4 633 1099 83 71.2

4 x 2 113 1420 104 5 28.1 -0.0055 0.027 7.4 24.4 78.2 653 1106 100 80.2

4 x 3 113 1410 104 5 27.8 -0.0054 0.027 7.3 24.1 78.3 661 1109 107 84.8

6 x 1 170 1530 155 5 42.5 -0.0082 0.042 11.2 38.5 128 676 1177 87 84.9

6 x 2 170 1480 155 5 48.1 -0.0092 0.047 12.7 44.3 128 591 1120 31 31.3

6 x 3 169 1370 154 5 48.7 -0.0094 0.048 12.9 45.2 130 586 1111 31 28.8

8 x 1 224 1370 203 5 63.9 -0.012 0.062 16.9 60.3 178 601 1087 57 79.6

Metric

Imperial

�

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0.0020

Av

erag

ecu

rvat

ure

(rad

/in

.)

012

24

36

48

60

72

84

96

10

8

12

0

13

2

14

4

15

6

16

8

18

0

19

2


0

10

20

30

40

50

60

70

[ �ra

d/

mm

]

030

0

60

0

90

0

12

00

15

00

18

00

21

00

24

00

27

00

30

00

33

00

36

00

39

00

42

00

45

00

48

00

[mm]

�y=

0.0

00

10

1ra

d/i

n.

Hin

eset

al.

1A

L=

19

4in

.D

=4

8in

.

��

=1

��

=1

.5

��

=2

��

=3

��

=4

��

=6

��

=8

FigureC��Structuralwallwithconnedboundaryelements�Test�A�Hinesetal��Average

curvatureproles�

��

Table C�� Structural wall with conned boundary elements� Test �A �Hines et al�� Peak curvature values�

� �f F M �b0 �bsp �b points Lp0 Lpsp Lp

(in.) (in.) (kips) (kft) (rad/in.) (rad/in.) (rad/in.) (in.) (in.) (in.)

+1/4 F'y 0.0784 0.0778 10.6 172 0.0000167 0.0000091 -- -- -- -- --

-1/4 F'y -0.0709 -0.0669 -9.6 -155 -0.0000118 -0.0000065 -- -- -- -- --

+1/2 F'y 0.181 0.173 21.8 352 0.0000497 0.0000272 -- -- -- -- --

-1/2 F'y -0.171 -0.160 -20.8 -335 -0.0000410 -0.0000225 -- -- -- -- --

+3/4 F'y 0.374 0.351 32.9 531 0.0000628 0.0000344 -- -- -- -- --

-3/4 F'y -0.398 -0.368 -31.9 -515 -0.000134 -0.0000736 -- -- -- -- --

+ F'y 0.862 0.810 46.6 751 0.000102 0.0000559 -- -- -- -- --

- F'y -0.889 -0.830 -46.5 -750 -0.000242 -0.000133 -- -- -- -- --

�� = +1 x 1 1.12 1.05 54.1 872 0.000140 0.000077 0.000162 -- 20.2 -- 10

�� = -1 x 1 -1.11 -1.04 -53.4 -860 -0.000297 -0.000163 -0.000162 -- 5.6 14.7 10

�� = +1 x 2 1.12 1.05 53.0 855 0.000145 0.000080 0.000170 -- 19.4 -- 10

�� = -1 x 2 -1.11 -1.04 -51.8 -836 -0.000295 -0.000162 -0.000171 -- 6.2 16.2 10

�� = +1 x 3 1.12 1.06 52.3 844 0.000146 0.000080 0.000175 -- 19.8 -- 10

�� = -1 x 3 -1.11 -1.04 -51.0 -822 -0.000293 -0.000160 -0.000178 -- 6.6 17.1 10

�� = +1.5 x 1 1.68 1.58 62.5 1007 0.000198 0.000108 0.000231 -- 33.9 -- 20

�� = -1.5 x 1 -1.67 -1.56 -60.3 -973 -0.000455 -0.000249 -0.000233 -- 9.8 23.0 20

�� = +1.5 x 2 1.68 1.57 60.1 970 0.000212 0.000116 0.000237 -- 30.7 -- 20

�� = -1.5 x 2 -1.67 -1.56 -58.5 -944 -0.000448 -0.000245 -0.000238 -- 10.4 24.3 20

�� = +1.5 x 3 1.68 1.57 59.4 958 0.000218 0.000120 0.000238 -- 29.3 -- 20

�� = -1.5 x 3 -1.67 -1.56 -58.0 -936 -0.000445 -0.000244 -0.000240 -- 10.6 24.7 20

�� = +2 x 1 2.23 2.08 66.1 1065 0.000307 0.000168 0.000318 -- 30.0 -- 23

�� = -2 x 1 -2.23 -2.06 -64.1 -1034 -0.000607 -0.000333 -0.000318 -- 12.1 26.9 23

�� = +2 x 2 2.23 2.07 61.9 998 0.000328 0.000180 0.000325 -- 28.1 -- 23

�� = -2 x 2 -2.23 -2.06 -61.6 -992 -0.000592 -0.000325 -0.000324 -- 12.8 28.6 23

�� = +2 x 3 2.23 2.08 62.4 1005 0.000334 0.000183 0.000325 -- 27.3 -- 23

�� = -2 x 3 -2.23 -2.06 -61.1 -985 -0.000597 -0.000327 -0.000324 -- 12.8 28.4 23

�� = +3 x 1 3.31 3.06 67.2 1083 0.000606 0.000332 0.000615 5 23.7 56.4 20.1

�� = -3 x 1 -3.34 -3.07 -64.9 -1046 -0.000878 -0.000481 -0.000406 5 15.6 33.0 34.2

�� = +3 x 2 3.34 3.08 65.0 1049 0.000663 0.000363 0.000626 5 21.8 49.7 20.1

�� = -3 x 2 -3.34 -3.07 -63.4 -1022 -0.000856 -0.000469 -0.000407 5 16.1 34.3 34.2

�� = +3 x 3 3.34 3.09 64.7 1043 0.000691 0.000379 0.000574 5 20.7 46.7 22.4

�� = -3 x 3 -3.34 -3.07 -62.9 -1015 -0.000860 -0.000471 -0.000403 5 16.1 34.2 34.6

�� = +4 x 1 4.45 4.10 68.8 1110 0.000968 0.000530 0.000840 5 20.8 44.8 21.1

�� = -4 x 1 -4.45 -4.08 -66.7 -1075 -0.00117 -0.000642 -0.000589 5 16.8 34.8 31.9

�� = +4 x 2 4.45 4.09 65.9 1063 0.00101 0.000556 0.000773 5 19.9 42.3 23.4

�� = -4 x 2 -4.45 -4.07 -64.3 -1037 -0.00116 -0.000636 -0.000604 5 17.1 35.4 31.0

�� = +4 x 3 4.45 4.09 65.8 1060 0.00103 0.000564 0.000776 5 19.6 41.4 23.2

�� = -4 x 3 -4.45 -4.07 -63.6 -1025 -0.00116 -0.000638 -0.000592 5 17.1 35.3 31.8

�� = +6 x 1 6.68 6.13 71.1 1147 0.00159 0.000869 0.00116 5 20.2 41.4 24.5

�� = -6 x 1 -6.67 -6.09 -69.0 -1113 -0.00177 -0.000970 -0.000960 5 17.9 36.1 30.1

�� = +6 x 2 6.68 6.12 68.4 1102 0.00108 0.000591 0.00150 5 31.8 71.0 18.6

�� = -6 x 2 -6.67 -6.07 -66.7 -1075 -0.00186 -0.00102 -0.000902 5 17.0 34.1 32.2

�� = +6 x 3 6.67 6.09 59.8 964 0.000982 0.000538 0.00151 5 36.0 82.2 18.8

�� = -6 x 3 -6.68 -6.07 -65.3 -1053 -0.00196 -0.00107 -0.000875 5 16.2 32.2 33.4

�� = +8 x 1 8.82 8.03 65.3 1052 0.00170 0.000929 0.00205 5 26.8 55.8 18.5

�� = -8 x 1 -8.82 -7.94 -62.9 -1014 -0.00324 -0.00178 -0.00115 5 13.2 25.5 33.8

Level

��





-0.0

010

-0.0

008

-0.0

006

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004

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002

0.00

00

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Curvature (rad/in.)

0

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156

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Hei

ghtab

ove

foot

ing

(in.

)

-30

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-10

0 10 20 30

Curvature [� rad / mm]

0

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[mm

]


� y

-0.0

010

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008

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006

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004

-0.0

002

0.00

00

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Curvature (rad/in.)

0

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eigh

tab

ove

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(in.

)

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[mm

]

� y


-30

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0 10 20 30


Figure C�� Structural wall with conned boundary elements� Test �A �Hines et al�� Curvature proles at �� and at ��

��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

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05

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Curvature (rad/in.)

0

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Hei

ghtab

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(in.

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[mm

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� y

-0.0

020

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015

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010

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005

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00

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Curvature (rad/in.)

0

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192H

eigh

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ove

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[mm

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� y


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�

-0.0

020

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015

-0.0

010

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005

0.00

00

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Curvature (rad/in.)

0

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Hei

ghtab

ove

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(in.

)

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0

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[mm

]


� y

-0.0

020

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015

-0.0

010

-0.0

005

0.00

00

0.00

05

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10

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Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

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192H

eigh

tab

ove

foot

ing

(in.

)

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[mm

]

� y


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0 20 40 60



��

Table C�� Structural wall with conned boundary elements� Test �A �Hines et al�� Flexural strain values�

� F �b0 �b �c0 �c �'c �s0 �s �'s


+1/4 F'y 0.0746 10.1 0.000014 -- -0.00040 -- -0.00078 0.00022 -- -6.7E-05

+1/2 F'y 0.176 21.3 0.000045 -- -0.00079 -- -0.0012 0.0013 -- 0.00084

+3/4 F'y 0.386 32.4 0.000099 -- -0.0013 -- -0.0020 0.0032 -- 0.0028

+ F'y 0.876 46.5 0.000172 -- -0.0019 -- -0.0031 0.0061 -- 0.0051

�� = 1 x 1 1.11 53.7 0.000218 0.000201 -0.0021 -0.0020 -0.0037 0.0078 0.0071 0.0068

�� = 1 x 2 1.11 52.4 0.000220 0.000210 -0.0022 -0.0021 -0.0037 0.0080 0.0074 0.0069

�� = 1 x 3 1.11 51.7 0.000219 0.000214 -0.0022 -0.0021 -0.0037 0.0080 0.0074 0.0068

�� = 1.5 x 1 1.68 61.4 0.000326 0.000261 -0.0029 -0.0024 -0.0052 0.012 0.0095 0.010

�� = 1.5 x 2 1.68 59.3 0.000330 0.000268 -0.0029 -0.0025 -0.0053 0.012 0.0099 0.011

�� = 1.5 x 3 1.68 58.7 0.000332 0.000271 -0.0029 -0.0025 -0.0053 0.012 0.0099 0.011

�� = 2 x 1 2.23 65.1 0.000457 0.000322 -0.0038 -0.0028 -0.0065 0.017 0.012 0.016

�� = 2 x 2 2.23 61.7 0.000460 0.000334 -0.0038 -0.0029 -0.0065 0.017 0.012 0.015

�� = 2 x 3 2.23 61.7 0.000465 0.000336 -0.0038 -0.0030 -0.0066 0.018 0.013 0.016

�� = 3 x 1 3.32 66.0 0.000742 0.000521 -0.0057 -0.0042 -0.0093 0.028 0.020 0.026

�� = 3 x 2 3.34 64.2 0.000760 0.000528 -0.0058 -0.0042 -0.0092 0.029 0.020 0.027

�� = 3 x 3 3.34 63.8 0.000776 0.000523 -0.0059 -0.0042 -0.0092 0.030 0.020 0.028

�� = 4 x 1 4.45 67.7 0.00107 0.000729 -0.0081 -0.0056 -0.012 0.041 0.028 0.040

�� = 4 x 2 4.45 65.1 0.00109 0.000712 -0.0083 -0.0055 -0.012 0.042 0.027 0.040

�� = 4 x 3 4.45 64.7 0.00110 0.000704 -0.0084 -0.0054 -0.012 0.042 0.027 0.041

�� = 6 x 1 6.67 70.1 0.00168 0.00108 -0.013 -0.0082 -0.017 0.065 0.042 0.064

�� = 6 x 2 6.67 67.5 0.00147 0.00122 -0.011 -0.0092 -0.0053 0.057 0.047 0.065

�� = 6 x 3 6.67 62.6 0.00147 0.00124 -0.011 -0.0094 -0.0064 0.057 0.048 0.065

�� = 8 x 1 8.82 64.1 0.00247 0.00162 -0.019 -0.012 -0.017 0.095 0.062 0.10

Level

��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Average post-yield strains ( �'c & �'s)

0

12

24

36

48

60

72

84

96

108

120

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156

168

180

192

Hei

ghtab

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(in.

)

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[mm

]

� y = 0.00231

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

Average pre-yield strains ( �'c & �'s)

0

12

24

36

48

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84

96

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Hei

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[mm

]

� y = 0.00231 Hines et al. 1APre-Yield Strains

L = 194 in.D = 48 in.

F'y

3/4 F'y

1/2 F'y

1/4 F'y

Hines et al. 1APost-Yield Strains

L = 194 in.D = 48 in.

�� = 1 x 1

�� = 2 x 1

�� = 3 x 1

�� = 4 x 1

�� = 6 x 1

�� = 8 x 1

Figure C�� Structural wall with conned boundary elements� Test �A �Hines et al�� Average �exural strain proles�

��

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

Pre-yield strains at negative peak (�'c & �'s)

0

12

24

36

48

60

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84

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Hei

ghtab

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(in.

)

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[mm

]

Hines 1APre-Yield Strains

L = 194 in.D = 48 in.

1/4 F'y

1/2 F'y

3/4 F'y

F'y

� y = 0.00231

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

Pre-yield strains at positive peak (�'c & �'s)

0

12

24

36

48

60

72

84

96

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Hei

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(in.

)

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[mm

]

� y = 0.00231Hines et al. 1A

Pre-Yield StrainsL = 194 in.D = 48 in.

1/4 F'y

1/2 F'y

3/4 F'y

F'y

Figure C�� Structural wall with conned boundary elements� Test �A �Hines et al�� Pre�yield �exural strain proles�

��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

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0.04

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0.06


0

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Hei

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[mm

]

Hines 1APost-Yield Strains

L = 194 in.D = 48 in.

�� = 1 x -1

�� = 2 x -1

�� = 3 x -1

�� = 4 x -1

�� = 6 x -1

�� = 8 x -1

� y = 0.00231

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

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0

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Hei

ghtab

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)

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[mm

]

� y = 0.00231 Hines et al. 1APost-Yield Strains

L = 194 in.D = 48 in.

�� = 1 x +1

�� = 2 x +1

�� = 3 x +1

�� = 4 x +1

�� = 6 x +1

�� = 8 x +1

Figure C�� Structural wall with conned boundary elements� Test �A �Hines et al�� Post�yield �exural strain proles�

��

Table C�� Structural wall with conned boundary elements� Test �B �Hines et al�� Average experimental plasticity values�



1 x 1 1.12 859 1.06 -- 0.000160 -0.0017 0.0057 1.69 0.0000857 0.257 15.5 21 5.0 -0.43

1 x 2 1.12 843 1.06 -- 0.000152 -0.0017 0.0054 1.61 0.0000796 0.271 17.6 25 5.0 -0.09

1 x 3 1.12 838 1.06 -- 0.000151 -0.0017 0.0054 1.59 0.0000787 0.274 18.0 26 5.0 -0.06

1.5 x 1 1.68 985 1.59 5 0.000234 -0.0021 0.0087 2.47 0.000149 0.664 23.0 33.7 6.13 -0.24

1.5 x 2 1.68 943 1.58 5 0.000233 -0.0021 0.0087 2.46 0.000152 0.698 23.8 33.2 7.19 -0.31

1.5 x 3 1.68 942 1.58 5 0.000234 -0.0021 0.0087 2.48 0.000153 0.701 23.6 33.1 7.02 -0.36

2 x 1 2.23 1040 2.09 5 0.000317 -0.0027 0.012 3.35 0.000228 1.12 25.3 40.2 5.23 -0.31

2 x 2 2.23 993 2.08 5 0.000313 -0.0027 0.012 3.31 0.000228 1.15 26.1 39.4 6.41 -0.26

2 x 3 2.23 989 2.08 5 0.000317 -0.0027 0.012 3.35 0.000232 1.15 25.7 38.8 6.30 -0.34

3 x 1 3.34 1060 3.10 5 0.000511 -0.0039 0.020 5.39 0.000419 2.10 25.9 43.2 4.37 0.45

3 x 2 3.34 1020 3.10 5 0.000510 -0.0039 0.019 5.39 0.000422 2.14 26.2 43.0 4.67 0.76

3 x 3 3.34 1010 3.09 5 0.000504 -0.0039 0.019 5.33 0.000418 2.14 26.5 43.1 4.98 1.03

4 x 1 4.45 1080 4.10 5 0.000698 -0.0051 0.027 7.37 0.000605 3.09 26.4 45.3 3.72 1.39

4 x 2 4.45 1040 4.09 5 0.000690 -0.0051 0.027 7.29 0.000601 3.11 26.8 45.6 3.99 1.58

4 x 3 4.45 1030 4.08 5 0.000680 -0.0050 0.026 7.18 0.000591 3.11 27.2 45.6 4.43 1.77

6 x 1 6.67 1110 6.08 5 0.000928 -0.0067 0.036 9.80 0.000832 5.04 31.3 50.8 5.88 3.49

6 x 2 6.67 1060 6.06 5 0.000883 -0.0064 0.034 9.32 0.000791 5.06 33.1 51.6 7.24 4.01

6 x 3 6.67 1050 6.05 5 0.000856 -0.0062 0.033 9.04 0.000766 5.07 34.2 52.1 8.15 4.57

8 x 1 8.89 1050 8.02 5 0.00129 -0.0092 0.050 13.6 0.00120 7.04 30.3 49.3 5.67 3.41



1 x 1 28.3 1160 27.0 -- 6.29 -0.0017 0.0057 1.60 3.38 6.54 394 534 130 -10.9

1 x 2 28.4 1140 27.0 -- 5.99 -0.0017 0.0054 1.52 3.13 6.88 447 639 130 -2.28

1 x 3 28.3 1140 26.9 -- 5.94 -0.0017 0.0054 1.51 3.10 6.95 457 659 130 -1.47

1.5 x 1 42.6 1340 40.3 5 9.22 -0.0021 0.0087 2.34 5.88 16.9 584 857 156 -6.11

1.5 x 2 42.6 1280 40.2 5 9.17 -0.0021 0.0087 2.33 5.97 17.7 605 844 183 -8.00

1.5 x 3 42.6 1280 40.2 5 9.24 -0.0021 0.0087 2.34 6.05 17.8 599 842 178 -9.22

2 x 1 56.6 1410 53.0 5 12.5 -0.0027 0.012 3.17 8.97 28.3 643 1020 133 -7.84

2 x 2 56.6 1350 52.9 5 12.4 -0.0027 0.012 3.13 8.98 29.2 663 1000 163 -6.56

2 x 3 56.6 1340 52.9 5 12.5 -0.0027 0.012 3.17 9.14 29.3 653 986 160 -8.73

3 x 1 84.8 1440 78.8 5 20.1 -0.0039 0.020 5.10 16.5 53.5 659 1096 111 11.5

3 x 2 84.8 1380 78.6 5 20.1 -0.0039 0.019 5.10 16.6 54.3 664 1092 119 19.3

3 x 3 84.8 1370 78.5 5 19.9 -0.0039 0.019 5.04 16.5 54.5 674 1095 127 26.2

4 x 1 113 1460 104 5 27.5 -0.0051 0.027 6.97 23.8 78.5 670 1152 94 35.2

4 x 2 113 1410 104 5 27.2 -0.0051 0.027 6.90 23.7 79.1 680 1158 101 40.0

4 x 3 113 1400 104 5 26.8 -0.0050 0.026 6.79 23.3 79.1 692 1158 113 44.8

6 x 1 170 1510 154 5 36.6 -0.0067 0.036 9.27 32.8 128 795 1291 149 88.5

6 x 2 170 1440 154 5 34.8 -0.0064 0.034 8.82 31.2 129 840 1312 184 102

6 x 3 170 1420 154 5 33.7 -0.0062 0.033 8.55 30.2 129 869 1323 207 116

8 x 1 226 1420 204 5 50.8 -0.0092 0.050 12.9 47.3 179 770 1251 144 86.7

Imperial

Metric

�

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0.0020

Av

erag

ecu

rvat

ure

(rad

/in

.)

012

24

36

48

60

72

84

96

10

8

12

0

13

2

14

4

15

6

16

8

18

0

19

2


0

10

20

30

40

50

60

70

[ �ra

d/

mm

]

030

0

60

0

90

0

12

00

15

00

18

00

21

00

24

00

27

00

30

00

33

00

36

00

39

00

42

00

45

00

48

00

[mm]

�y=

0.0

00

10

0ra

d/i

n.

Hin

es1

BL

=1

93

.5in

.D

=4

8in

.

��

=1

��

=1

.5

��

=2

��

=3

��

=4

��

=6

��

=8

FigureC��Structuralwallwithconnedboundaryelements�Test�B�Hinesetal��Average

curvatureproles�

��

Table C�� Structural wall with conned boundary elements� Test �B �Hines et al�� Peak curvature values�



+1/4 F'y 0.0806 0.0797 10.6 172 0.000010 0.000006 0.000017 -- -- -- --

-1/4 F'y -0.0724 -0.0687 -9.59 -155 -0.000017 -0.000009 0.000000 -- -- -- --

+1/2 F'y 0.187 0.180 21.7 352 0.000030 0.000016 0.000027 -- -- -- --

-1/2 F'y -0.183 -0.172 -21.2 -343 -0.000039 -0.000021 -0.000017 -- -- -- --

+3/4 F'y 0.414 0.394 32.3 522 0.000058 0.000032 0.000091 -- -- -- --

-3/4 F'y -0.391 -0.365 -32.9 -532 -0.000051 -0.000028 -0.000093 -- -- -- --

+ F'y 0.880 0.833 47.2 764 0.000106 0.000058 0.000127 -- -- -- --

- F'y -0.850 -0.769 -47.2 -763 -0.000107 -0.000059 -0.000126 -- -- -- --

�� = +1 x 1 1.12 1.07 53.4 863 0.000151 0.000083 0.000148 5 18.8 -- 18.4

�� = -1 x 1 -1.12 -1.05 -52.9 -855 -0.000141 -0.000077 -0.000165 5 22.2 -- 14.2

�� = +1 x 2 1.12 1.07 52.4 846 0.000152 0.000083 0.000138 5 19.5 -- 21.8

�� = -1 x 2 -1.12 -1.05 -51.9 -840 -0.000148 -0.000081 -0.000154 5 20.8 -- 16.7

�� = +1 x 3 1.12 1.07 51.9 839 0.000150 0.000082 0.000136 5 20.3 -- 22.8

�� = -1 x 3 -1.11 -1.05 -51.8 -837 -0.000148 -0.000081 -0.000154 5 20.8 -- 16.7

�� = +1.5 x 1 1.68 1.60 61.6 996 0.000228 0.000125 0.000209 5 27.2 -- 28.0

�� = -1.5 x 1 -1.67 -1.57 -60.3 -974 -0.000221 -0.000121 -0.000231 5 29.0 -- 23.1

�� = +1.5 x 2 1.67 1.59 58.4 944 0.000222 0.000122 0.000205 5 29.5 -- 29.5

�� = -1.5 x 2 -1.68 -1.57 -58.2 -941 -0.000219 -0.000120 -0.000247 5 30.4 -- 21.5

�� = +1.5 x 3 1.68 1.59 58.6 948 0.000224 0.000123 0.000213 5 29.2 -- 27.6

�� = -1.5 x 3 -1.68 -1.58 -57.9 -936 -0.000216 -0.000119 -0.000248 5 31.3 -- 21.4

�� = +2 x 1 2.23 2.10 65.0 1051 0.000286 0.000157 0.000279 5 34.5 -- 30.5

�� = -2 x 1 -2.23 -2.08 -63.3 -1023 -0.000303 -0.000166 -0.000347 5 31.8 -- 22.2

�� = +2 x 2 2.22 2.09 61.2 990 0.000274 0.000150 0.000255 5 37.8 -- 35.1

�� = -2 x 2 -2.23 -2.08 -61.7 -997 -0.000326 -0.000179 -0.000360 5 28.9 -- 21.5

�� = +2 x 3 2.23 2.09 61.0 986 0.000270 0.000148 0.000241 5 38.9 -- 38.3

�� = -2 x 3 -2.23 -2.08 -61.3 -992 -0.000328 -0.000180 -0.000372 5 28.7 88.1 20.6

�� = +3 x 1 3.34 3.11 65.9 1066 0.000417 0.000229 0.000377 5 40.4 -- 38.2

�� = -3 x 1 -3.34 -3.09 -65.5 -1059 -0.000681 -0.000373 -0.000619 5 21.2 47.9 20.4

�� = +3 x 2 3.34 3.11 63.1 1020 0.000456 0.000250 0.000384 5 36.0 -- 37.3

�� = -3 x 2 -3.34 -3.09 -63.2 -1022 -0.000694 -0.000380 -0.000614 5 21.0 46.8 20.8

�� = +3 x 3 3.34 3.10 62.2 1005 0.000476 0.000261 0.000366 5 34.1 90.5 39.7

�� = -3 x 3 -3.34 -3.08 -62.3 -1008 -0.000705 -0.000386 -0.000631 5 20.7 45.8 20.2

�� = +4 x 1 4.45 4.11 66.8 1080 0.000731 0.000400 0.000533 5 29.6 69.1 36.3

�� = -4 x 1 -4.45 -4.09 -66.3 -1072 -0.000988 -0.000541 -0.000845 5 20.6 44.0 21.1

�� = +4 x 2 4.45 4.10 64.3 1039 0.000778 0.000427 0.000513 5 27.6 62.3 38.1

�� = -4 x 2 -4.45 -4.08 -64.8 -1047 -0.000965 -0.000529 -0.000839 5 21.3 45.5 21.3

�� = +4 x 3 4.45 4.09 63.7 1029 0.000792 0.000434 0.000495 5 27.1 60.8 39.7

�� = -4 x 3 -4.45 -4.07 -64.3 -1040 -0.000968 -0.000530 -0.000831 5 21.3 45.4 21.5

�� = +6 x 1 6.67 6.06 68.7 1111 0.00141 0.000772 0.00074828 5 23.3 48.6 39.6

�� = -6 x 1 -6.68 -6.10 -68.7 -1110 -0.00153 -0.000836 -0.00109 5 21.3 43.8 26.2

�� = +6 x 2 6.67 6.04 65.8 1063 0.00147 0.000804 0.000722 5 22.4 46.2 41.2

�� = -6 x 2 -6.67 -6.08 -65.8 -1064 -0.00147 -0.000806 -0.00103 5 22.3 46.1 27.9

�� = +6 x 3 6.67 6.03 64.5 1042 0.00154 0.000841 0.000711 5 21.3 43.6 41.9

�� = -6 x 3 -6.68 -6.07 -64.9 -1049 -0.00148 -0.000812 -0.000987 5 22.2 45.7 29.2

�� = +8 x 1 8.90 8.06 65.1 1052 0.00230 0.00126 0.00134 5 19.3 38.5 29.2

�� = -8 x 1 -8.89 -7.99 -64.3 -1039 -0.00175 -0.000960 -0.00121 5 26.1 54.1 32.3

Level

��

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

Hei

ghtab

ove

foot

ing

(in.

)

-30

-20

-10

0 10 20 30


0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

[mm

]

Hines et al. 1B�� = 2

� y

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192H

eigh

tab

ove

foot

ing

(in.

)

0

300

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2700

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3300

3600

3900

4200

4500

4800

[mm

]

� y


-30

-20

-10

0 10 20 30


Figure C�� Structural wall with conned boundary elements� Test �B �Hines et al�� Curvature proles at �� and at ��

��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

Hei

ghtab

ove

foot

ing

(in.

)

-60

-40

-20

0 20 40 60


0

300

600

900

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1500

1800

2100

2400

2700

3000

3300

3600

3900

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4800

[mm

]


� y

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

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2100

2400

2700

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3300

3600

3900

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4500

4800

[mm

]

� y


-60

-40

-20

0 20 40 60



�

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

Hei

ghtab

ove

foot

ing

(in.

)

-60

-40

-20

0 20 40 60


0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

[mm

]


� y

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

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2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

[mm

]

� y


-60

-40

-20

0 20 40 60



��

Table C�� Structural wall with conned boundary elements� Test �B �Hines et al�� Flexural strain values�

� F �b0 �b �c0 �c �'c �s0 �s �'s


+1/4 F'y 0.0765 10.1 0.0000128 -- -0.00038 -- -0.00059 0.00020 -- 0.00007

+1/2 F'y 0.185 21.5 0.0000342 -- -0.00066 -- -0.00088 0.00094 -- 0.00075

+3/4 F'y 0.403 32.6 0.0000556 -- -0.00088 -- -0.0014 0.0017 -- 0.0012

+ F'y 0.865 47.2 0.000106 -- -0.0013 -- -0.0029 0.0035 -- 0.0023

�� = 1 x 1 1.12 53.2 0.000148 0.0001597 -0.0016 -0.0017 -0.0034 0.0052 0.0057 0.0036

�� = 1 x 2 1.12 52.2 0.000150 0.0001521 -0.0016 -0.0017 -0.0035 0.0052 0.0054 0.0037

�� = 1 x 3 1.12 51.8 0.000149 0.0001508 -0.0016 -0.0017 -0.0035 0.0052 0.0054 0.0037

�� = 1.5 x 1 1.68 60.9 0.000225 0.000234 -0.0021 -0.0021 -0.0051 0.0083 0.0087 0.0057

�� = 1.5 x 2 1.68 58.3 0.000220 0.0002327 -0.0021 -0.0021 -0.0052 0.0083 0.0087 0.0054

�� = 1.5 x 3 1.68 58.3 0.000220 0.0002345 -0.0021 -0.0021 -0.0052 0.0083 0.0087 0.0053

�� = 2 x 1 2.23 64.1 0.000301 0.000317 -0.0026 -0.0027 -0.0066 0.011 0.012 0.0075

�� = 2 x 2 2.23 61.4 0.000300 0.0003135 -0.0026 -0.0027 -0.0069 0.011 0.012 0.0075

�� = 2 x 3 2.23 61.1 0.000299 0.0003171 -0.0026 -0.0027 -0.0070 0.011 0.012 0.0074

�� = 3 x 1 3.34 65.7 0.000549 0.0005107 -0.0042 -0.0039 -0.011 0.021 0.020 0.015

�� = 3 x 2 3.34 63.2 0.000575 0.0005101 -0.0043 -0.0039 -0.012 0.022 0.019 0.016

�� = 3 x 3 3.34 62.2 0.000591 0.0005043 -0.0045 -0.0039 -0.013 0.023 0.019 0.016

�� = 4 x 1 4.45 66.6 0.000859 0.0006979 -0.0062 -0.0051 -0.017 0.033 0.027 0.025

�� = 4 x 2 4.45 64.5 0.000872 0.0006905 -0.0063 -0.0051 -0.018 0.034 0.027 0.024

�� = 4 x 3 4.45 64.0 0.000880 0.00068 -0.0064 -0.0050 -0.018 0.034 0.026 0.024

�� = 6 x 1 6.67 68.7 0.00147 0.000928 -0.011 -0.0067 -0.027 0.057 0.036 0.043

�� = 6 x 2 6.67 65.8 0.00147 0.000883 -0.011 -0.0064 -0.030 0.057 0.034 0.040

�� = 6 x 3 6.67 64.7 0.00151 0.0008559 -0.011 -0.0062 -0.032 0.059 0.033 0.041

�� = 8 x 1 8.89 64.7 0.00203 0.0012906 -0.015 -0.0092 -0.034 0.079 0.050 0.063

Level

��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Average peak post-yield strains ( �'c & �'s)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

[mm

]

� y = 0.00231

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

Average peak pre-yield strains ( �'c & �'s)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

[mm

]

� y = 0.00231 Hines et al. 1BPre-Yield Strains

L = 194 in.D = 48 in.

F'y

3/4 F'y

1/2 F'y

1/4 F'y

Hines et al. 1BPost-Yield Strains

L = 194 in.D = 48 in.

�� = 1

�� = 2

�� = 3

�� = 4

�� = 6

�� = 8

Figure C�� Structural wall with conned boundary elements� Test �B �Hines et al�� Average �exural strain proles�

��

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

Pre-yield strains at negative peaks (�'c & �'s)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

[mm

]

Hines et al. 1BPre-Yield Strains

L = 194 in.D = 48 in.

-F'y

-3/4 F'y

-1/2 F'y

-1/4 F'y

� y = 0.00231

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

Pre-yield strains at positive peaks (�'c & �'s)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

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3000

3300

3600

3900

4200

4500

4800

[mm

]

� y = 0.00231


L = 194 in.D = 48 in.

+1/4 F'y

+1/2 F'y

+3/4 F'y

+F'y

Figure C�� Structural wall with conned boundary elements� Test �B �Hines et al�� Pre�yield �exural strain proles�

��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Post-yield strains at negative peaks (�'c & �'s)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

[mm

]


L = 194 in.D = 48 in.

�� = 1 x -1

�� = 2 x -1

�� = 3 x -1

�� = 4 x -1

�� = 6 x -1

�� = 8 x -1

� y = 0.00231

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Post-yield strains at positive peaks (�'c & �'s)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

[mm

]

� y = 0.00231 Hines et al. 1BPost-Yield Strains

L = 194 in.D = 48 in.

�� = 1 x +1

�� = 2 x +1

�� = 3 x +1

�� = 4 x +1

�� = 6 x +1

�� = 8 x +1

Figure C�� Structural wall with conned boundary elements� Test �B �Hines et al�� Post�yield �exural strain proles�

��




1 x 1 0.350 796 0.276 -- 0.00017 -0.0019 0.0060 1.8 0.000084 0.0804 10 10 5.0 0.48

1 x 2 0.352 802 0.280 -- 0.00017 -0.0019 0.0060 1.9 0.000086 0.0829 10 10 5.0 0.47

1 x 3 0.350 796 0.277 -- 0.00017 -0.0019 0.0060 1.8 0.000085 0.0819 10 10 5.0 0.49

1.5 x 1 0.527 959 0.414 -- 0.00029 -0.0026 0.0110 3.1 0.00019 0.179 10 10 5.0 0.47

1.5 x 2 0.523 932 0.405 -- 0.00028 -0.0026 0.0100 3.1 0.00018 0.176 10 10 5.0 0.23

1.5 x 3 0.520 878 0.384 -- 0.00027 -0.0025 0.0100 2.9 0.00018 0.168 10 10 5.0 -0.46

2 x 1 0.701 921 0.540 4 0.00029 -0.0026 0.0110 3.1 0.00019 0.313 17 23.5 5.0 3.37

2 x 2 0.696 963 0.537 4 0.00026 -0.0025 0.0096 2.8 0.00016 0.301 19 28.7 5.0 4.21

2 x 3 0.695 954 0.532 4 0.00026 -0.0025 0.0096 2.8 0.00016 0.298 19 28.2 5.0 3.97

3 x 1 1.05 956 0.798 4 0.000338 -0.0029 0.013 3.66 0.0002521 0.563 23.3 31.4 7.56 6.36

3 x 2 1.05 914 0.795 4 0.000375 -0.0032 0.014 4.07 0.0002936 0.571 20.2 30.1 5.21 4.76

3 x 3 1.05 942 0.798 4 0.000379 -0.0032 0.014 4.11 0.0002947 0.566 20.0 29.9 5.08 4.78

4 x 1 1.40 1000 1.06 4 0.000589 -0.0046 0.022 6.39 0.0004995 0.818 17.1 31.2 1.45 2.92

4 x 2 1.40 967 1.06 4 0.000590 -0.0046 0.023 6.40 0.0005037 0.819 16.9 31.4 1.24 2.75

4 x 3 1.40 918 1.05 4 0.000585 -0.0046 0.022 6.35 0.000503 0.828 17.2 31.6 1.38 2.76

6 x 1 2.10 990 1.58 5 0.000810 -0.0061 0.031 8.79 0.0007214 1.33 19.2 40.5 -1.01 2.77

6 x 2 2.10 958 1.56 5 0.000792 -0.0060 0.030 8.59 0.0007064 1.33 19.6 40.8 -0.82 2.95

6 x 3 2.10 935 1.55 5 0.000772 -0.0058 0.030 8.37 0.0006883 1.32 20.0 40.6 -0.27 3.28

8 x 1 2.80 980 2.08 5 0.000982 -0.0074 0.038 10.6 0.0008942 1.84 21.4 43.5 -0.38 3.80

Level � M �f l.s. points �b �c �s �� p �p Lp Lpr Lsp Lsp


1 x 1 8.88 1078 7.01 -- 6.6 -0.0019 0.0060 1.8 3.3 2.04 250 254 130 12.2

1 x 2 8.94 1087 7.11 -- 6.7 -0.0019 0.0060 1.9 3.4 2.11 250 254 130 12.1

1 x 3 8.90 1078 7.05 -- 6.7 -0.0019 0.0060 1.8 3.4 2.08 250 254 130 12.5

1.5 x 1 13.4 1299 10.5 -- 11.3 -0.0026 0.0110 3.1 7.3 4.54 250 254 130 12.0

1.5 x 2 13.3 1263 10.3 -- 11.1 -0.0026 0.0100 3.1 7.2 4.46 250 254 130 5.8

1.5 x 3 13.2 1190 9.76 -- 10.6 -0.0025 0.0100 2.9 6.9 4.28 250 254 130 -11.6

2 x 1 17.8 1248 13.7 4 11 -0.0026 0.0110 3.1 7.7 7.96 425 597 130 85.5

2 x 2 17.7 1305 13.6 4 10 -0.0025 0.0096 2.8 6.4 7.64 491 729 130 107

2 x 3 17.6 1292 13.5 4 10 -0.0025 0.0096 2.8 6.4 7.56 485 717 130 101

3 x 1 26.7 1296 20.3 4 13.3 -0.0029 0.013 3.66 9.93 14.3 591 799 192 161

3 x 2 26.6 1238 20.2 4 14.8 -0.0032 0.014 4.07 11.6 14.5 514 764 132 121

3 x 3 26.7 1276 20.3 4 14.9 -0.0032 0.014 4.11 11.6 14.4 509 759 129 122

4 x 1 35.6 1355 27.0 4 23.2 -0.0046 0.022 6.39 19.7 20.8 433 793 36.7 74.2

4 x 2 35.6 1310 26.8 4 23.3 -0.0046 0.023 6.40 19.8 20.8 430 798 31.6 69.9

4 x 3 35.6 1244 26.8 4 23.1 -0.0046 0.022 6.35 19.8 21.0 436 801 35.0 70.1

6 x 1 53.4 1341 40.0 5 31.9 -0.0061 0.031 8.79 28.4 33.8 489 1029 -25.7 70.3

6 x 2 53.4 1299 39.7 5 31.2 -0.0060 0.030 8.59 27.8 33.7 497 1036 -20.9 74.9

6 x 3 53.4 1267 39.4 5 30.4 -0.0058 0.030 8.37 27.1 33.6 509 1031 -6.9 83.3

8 x 1 71.2 1327 52.7 5 38.7 -0.0074 0.038 10.6 35.2 46.6 543 1105 -9.6 96.6

Imperial

Metric

�

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.0010

Aver

age

curv

ature

(rad

/in.)

012

24

36

48

60

72

84

96


0

5

10

15

20

25

30

35

[ �ra

d/

mm

]

0300

600

900

1200

1500

1800

2100

2400

[mm]

�y=

0.0

000959

rad/i

n.

Hin

eset

al.

2A

L=

96

in.

D=

48

in.

��

=1

��

=1.5

��

=2

��

=3

��

=4

��

=6

��

=8


curvatureproles�

��




+1/4 F'y 0.0232 0.0381 22.1 177 0.0000300 0.0000160 -- -- -- -- --

-1/4 F'y -0.0211 -0.0066 -23.6 -189 0.0000000 0.0000000 -- -- -- -- --

+1/2 F'y 0.0556 0.0651 46.4 371 0.0000470 0.0000260 -- -- -- -- --

-1/2 F'y -0.0482 -0.0268 -46.1 -369 -0.0000180 -0.0000100 -- -- -- -- --

+3/4 F'y 0.126 0.123 70.9 567 0.0000960 0.0000520 -- -- -- -- --

-3/4 F'y -0.119 -0.084 -70.9 -567 -0.0000650 -0.0000360 -- -- -- -- --

+ F'y 0.262 0.231 94.5 756 0.000160 0.0000880 -- -- -- -- --

- F'y -0.238 -0.177 -93.8 -751 -0.000124 -0.0000680 -- -- -- -- --

�� = +1 x 1 0.341 0.293 104 832 0.000199 0.000109 0.00017 -- 7.9 32.9 10

�� = -1 x 1 -0.353 -0.261 -104 -836 -0.000173 -0.0000950 -0.00013 -- 11.4 -- 10

�� = +1 x 2 0.347 0.299 103 823 0.000206 0.000113 0.00017 -- 8.2 31.5 10

�� = -1 x 2 -0.355 -0.263 -102 -813 -0.000171 -0.0000940 -0.00014 -- 12.5 -- 10

�� = +1 x 3 0.346 0.297 102 816 0.000205 0.000112 0.00017 -- 8.3 32.3 10

�� = -1 x 3 -0.350 -0.258 -101 -807 -0.000172 -0.000094 -0.00013 -- 11.9 -- 10

�� = +1.5 x 1 0.523 0.430 121 967 0.000327 0.000179 0.00029 -- 10.6 31.1 10

�� = -1.5 x 1 -0.531 -0.399 -119 -950 -0.000293 -0.000161 -0.00026 -- 13.2 43.4 10

�� = +1.5 x 2 0.516 0.414 118 942 0.000299 0.000164 0.00027 -- 12.0 37.8 10

�� = -1.5 x 2 -0.530 -0.395 -115 -922 -0.000285 -0.000156 -0.00026 -- 14.0 47.6 10

�� = +1.5 x 3 0.514 0.362 116 927 0.000177 0.000097 0.00022 -- 31.0 -- 10

�� = -1.5 x 3 -0.526 -0.388 -114 -909 -0.000276 -0.000151 -0.00025 -- 14.6 51.7 10

�� = +2 x 1 0.694 0.554 127 1020 0.000463 0.000254 0.00028 4 11.5 29.2 17

�� = -2 x 1 -0.702 -0.549 -123 -986 -0.000460 -0.000252 -0.00028 4 12.1 30.7 17

�� = +2 x 2 0.691 0.552 121 967 0.000454 0.000249 0.00028 4 12.1 30.5 17

�� = -2 x 2 -0.700 -0.522 -120 -958 -0.000428 -0.000234 -0.00026 4 13.5 35.5 17

�� = +2 x 3 0.692 0.553 120 957 0.000453 0.000248 0.00029 4 12.2 30.8 16

�� = -2 x 3 -0.698 -0.511 -119 -950 -0.000405 -0.000222 -0.00027 4 14.5 39.4 16

�� = +3 x 1 1.04 0.820 132 1050 0.000697 0.000382 0.000361 4 13.5 31.5 21.9

�� = -3 x 1 -1.05 -0.785 -127 -1020 -0.000714 -0.000391 -0.000251 4 13.3 30.6 34.9

�� = +3 x 2 1.04 0.813 125 1000 0.000686 0.000376 0.000370 4 13.9 32.4 21.1

�� = -3 x 2 -1.05 -0.781 -123 -985 -0.000664 -0.000364 -0.000284 4 14.7 34.6 28.7

�� = +3 x 3 1.05 0.817 124 990 0.000687 0.000376 0.000358 4 14.1 32.8 22.2

�� = -3 x 3 -1.05 -0.780 -122 -974 -0.000676 -0.000371 -0.000289 4 14.5 33.7 28.0

�� = +4 x 1 1.40 1.08 132 1050 0.000903 0.000495 0.000522 4 15.2 34.3 20.0

�� = -4 x 1 -1.40 -1.05 -128 -1030 -0.000858 -0.000470 -0.000487 4 16.2 37.5 21.0

�� = +4 x 2 1.40 1.08 127 1020 0.000897 0.000492 0.000502 4 15.4 34.7 21.0

�� = -4 x 2 -1.40 -1.04 -125 -1000 -0.000835 -0.000457 -0.000491 4 16.9 39.2 20.7

�� = +4 x 3 1.39 1.07 126 1010 0.000884 0.000484 0.000540 4 15.5 35.3 19.1

�� = -4 x 3 -1.39 -1.03 -124 -989 -0.000821 -0.000450 -0.000480 4 17.1 39.8 20.9

�� = +6 x 1 2.09 1.61 136 1090 0.00133 0.000727 0.000801 5 16.3 35.9 19.8

�� = -6 x 1 -2.09 -1.54 -132 -1050 -0.00105 -0.000575 -0.000921 5 21.8 53.1 16.1

�� = +6 x 2 2.09 1.61 130 1040 0.00136 0.000743 0.000815 5 16.0 34.9 19.5

�� = -6 x 2 -2.10 -1.52 -128 -1020 -0.00100 -0.000550 -0.000852 5 23.2 58.5 17.4

�� = +6 x 3 2.08 1.59 128 1020 0.00134 0.000736 0.000795 5 16.1 35.1 19.8

�� = -6 x 3 -2.10 -1.51 -126 -1010 -0.00103 -0.000567 -0.000824 5 22.4 54.9 17.9

�� = +8 x 1 2.80 2.13 135 1080 0.00179 0.000982 0.00101 5 16.7 35.8 21.3

�� = -8 x 1 -2.79 -2.02 -130 -1040 -0.00141 -0.000772 -0.000957 5 22.1 52.1 21.3

Level

��





-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

-30

-20

-10

0 10 20 30

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

[mm

]


� y

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y


-30

-20

-10

0 10 20 30

[� rad / mm]


��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

-60

-40

-20

0 20 40 60

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

[mm

]


� y

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y


-60

-40

-20

0 20 40 60

[� rad / mm]


�

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

-60

-40

-20

0 20 40 60

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

[mm

]


� y

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y


-60

-40

-20

0 20 40 60

[� rad / mm]


� �


� F �b0 �b �c0 �c �'c �s0 �s �'s


+1/4 F'y 0.0225 22.9 0.0000150 -- -0.00043 -- -0.00036 0.00027 -- 0.00033

+1/2 F'y 0.0610 43.2 0.0000326 -- -0.00065 -- -0.00052 0.00084 -- 0.00096

+3/4 F'y 0.139 66.8 0.0000819 -- -0.0012 -- -0.0010 0.0026 -- 0.0027

+ F'y 0.277 88.0 0.000141 -- -0.0017 -- -0.0017 0.0049 -- 0.0049

�� = 1 x 1 0.350 99.5 0.000181 0.000128 0.00 -0.0019 -0.0022 0.0064 0.0060 0.0064

�� = 1 x 2 0.352 100 0.000184 0.000131 0.00 -0.0019 -0.0022 0.0066 0.0060 0.0065

�� = 1 x 3 0.350 99.4 0.000183 0.000129 0.00 -0.0019 -0.0022 0.0066 0.0060 0.0064

�� = 1.5 x 1 0.527 120 0.000310 0.000239 -0.0027 -0.0026 -0.0031 0.012 0.0110 0.011

�� = 1.5 x 2 0.523 117 0.000292 0.000235 -0.0026 -0.0026 -0.0029 0.011 0.0100 0.010

�� = 1.5 x 3 0.520 110 0.000248 0.000224 -0.0024 -0.0025 -0.0024 0.009 0.0100 0.0080

�� = 2 x 1 0.701 115 0.000449 0.000263 -0.0037 -0.0026 -0.0042 0.017 0.0110 0.017

�� = 2 x 2 0.696 120 0.000441 0.000256 -0.0036 -0.0025 -0.0034 0.017 0.0096 0.017

�� = 2 x 3 0.695 119 0.000429 0.000264 -0.0035 -0.0025 -0.0030 0.016 0.0096 0.017

�� = 3 x 1 1.05 120 0.000695 0.000338 -0.0053 -0.0029 -0.0047 0.027 0.013 0.028

�� = 3 x 2 1.05 114 0.000673 0.000375 -0.0051 -0.0032 -0.0043 0.026 0.014 0.027

�� = 3 x 3 1.05 118 0.000681 0.000379 -0.0052 -0.0032 -0.0047 0.026 0.014 0.027

�� = 4 x 1 1.40 125 0.000876 0.000589 -0.0066 -0.0046 -0.0054 0.034 0.022 0.035

�� = 4 x 2 1.40 121 0.000861 0.000590 -0.0065 -0.0046 -0.0054 0.033 0.023 0.034

�� = 4 x 3 1.40 115 0.000854 0.000585 -0.0064 -0.0046 -0.0053 0.033 0.022 0.034

�� = 6 x 1 2.10 124 0.00118 0.000810 -0.0088 -0.0061 -0.0071 0.046 0.031 0.048

�� = 6 x 2 2.10 120 0.00118 0.000792 -0.0088 -0.0060 -0.0082 0.046 0.030 0.046

�� = 6 x 3 2.10 117 0.00119 0.000772 -0.0089 -0.0058 -0.0087 0.046 0.030 0.046

�� = 8 x 1 2.80 122 0.00160 0.000982 -0.012 -0.0074 -0.01206 0.062 0.038 0.062

Level

� �

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y = 0.00228

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y = 0.00228 Hines et al. 2APre-Yield Strains

L = 96 in.D = 48 in.

F'y

3/4 F'y

1/2 F'y

1/4 F'y


L = 96 in.D = 48 in.

�� = 1

�� = 2

�� = 3

�� = 4

�� = 6

�� = 8


� �

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

Hines et al. 2APre-Yield Strains

L = 96 in.D = 48 in.

-F'y

-3/4 F'y

-1/2 F'y

-1/4 F'y

� y = 0.00228

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y = 0.00228


L = 96 in.D = 48 in.

+1/4 F'y

+1/2 F'y

+3/4 F'y

+F'y


� �

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]


L = 96 in.D = 48 in.

�� = 1 x -1

�� = 2 x -1

�� = 3 x -1

�� = 4 x -1

�� = 6 x -1

�� = 8 x -1

� y = 0.00228

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]


L = 96 in.D = 48 in.

�� = 1 x +1

�� = 2 x +1

�� = 3 x +1

�� = 4 x +1

�� = 6 x +1

�� = 8 x +1


� �




1 x 1 0.351 841 0.312 -- 0.00018 -0.0019 0.0065 2.0 0.000098 0.094 10 10 5.0 -0.33

1 x 2 0.351 808 0.313 -- 0.00019 -0.0019 0.0069 2.1 0.00011 0.104 10 10 5.0 -0.27

1 x 3 0.352 804 0.313 -- 0.00019 -0.0019 0.0069 2.1 0.00011 0.105 10 10 5.0 -0.35

1.5 x 1 0.527 913 0.434 -- 0.00030 -0.0026 0.011 3.3 0.00021 0.197 10 10 5.0 -0.77

1.5 x 2 0.528 904 0.436 -- 0.00030 -0.0026 0.011 3.3 0.00021 0.202 10 10 5.0 -0.84

1.5 x 3 0.528 900 0.436 -- 0.00030 -0.0026 0.011 3.3 0.00021 0.203 10 10 5.0 -0.88

2 x 1 0.701 951 0.567 4 0.00029 -0.0025 0.011 3.2 0.00020 0.320 17 24 5.0 1.18

2 x 2 0.703 960 0.567 4 0.00029 -0.0025 0.011 3.2 0.00020 0.319 17 24 5.0 1.45

2 x 3 0.703 952 0.565 4 0.00029 -0.0025 0.011 3.2 0.00020 0.319 17 24 5.0 0.93

3 x 1 1.05 1020 0.840 4 0.00041 -0.0032 0.016 4.5 0.00031 0.575 19 29.0 5.0 1.97

3 x 2 1.05 975 0.835 4 0.00040 -0.0032 0.015 4.4 0.00030 0.583 20 29.9 5.0 2.12

3 x 3 1.05 967 0.836 4 0.00039 -0.0031 0.015 4.3 0.00030 0.585 21 31.3 5.0 2.34

4 x 1 1.40 1020 1.09 6 0.00040 -0.0032 0.015 4.4 0.00030 0.825 29 48.2 5.0 4.20

4 x 2 1.40 980 1.09 6 0.00040 -0.0032 0.015 4.4 0.00030 0.833 29 48.6 5.0 4.37

4 x 3 1.40 965 1.08 6 0.00039 -0.0031 0.015 4.3 0.00029 0.829 30 49.0 5.0 4.40

6 x 1 2.11 1020 1.54 6 0.000510 -0.0039 0.020 5.60 0.000421 1.27 31.4 54.7 4.04 5.57

6 x 2 2.11 983 1.50 6 0.000494 -0.0038 0.019 5.42 0.000408 1.25 31.8 55.5 4.01 5.25

6 x 3 2.11 963 1.47 6 0.000472 -0.0036 0.018 5.18 0.000388 1.22 32.8 56.2 4.70 5.37

8 x 1 2.81 1010 1.85 6 0.000635 -0.0046 0.025 6.97 0.000547 1.59 30.2 57.7 1.40 4.61



1 x 1 8.91 1140 7.92 -- 7.3 -0.0019 0.0065 2.0 3.9 2.39 250 130 130 -8

1 x 2 8.91 1095 7.95 -- 7.5 -0.0019 0.0069 2.1 4.2 2.63 250 130 130 -7

1 x 3 8.95 1089 7.96 -- 7.6 -0.0019 0.0069 2.1 4.3 2.67 250 130 130 -9

1.5 x 1 13.4 1237 11.0 -- 12 -0.0026 0.011 3.3 8.1 5.01 250 130 130 -20

1.5 x 2 13.4 1225 11.1 -- 12 -0.0026 0.011 3.3 8.3 5.12 250 130 130 -21

1.5 x 3 13.4 1220 11.07 -- 12 -0.0026 0.011 3.3 8.3 5.15 250 130 130 -22

2 x 1 17.8 1288 14.4 4 12 -0.0025 0.011 3.2 7.7 8.14 380 610 130 30

2 x 2 17.9 1301 14.4 4 12 -0.0025 0.011 3.2 7.7 8.09 430 610 130 37

2 x 3 17.9 1290 14.4 4 12 -0.0025 0.011 3.2 7.7 8.09 340 610 130 24

3 x 1 26.8 1382 21.3 4 16 -0.0032 0.016 4.5 12.11 14.6 500 736 130 50

3 x 2 26.8 1321 21.2 4 16 -0.0032 0.015 4.4 12.0 14.8 510 759 130 54

3 x 3 26.8 1310 21.2 4 16 -0.0031 0.015 4.3 11.6 14.9 530 796 130 59

4 x 1 35.7 1382 27.7 4 16 -0.0032 0.015 4.4 11.6 21.0 739 1224 130 107

4 x 2 35.7 1328 27.6 4 16 -0.0032 0.015 4.4 11.7 21.2 745 1236 130 111

4 x 3 35.7 1308 27.4 4 15 -0.0031 0.015 4.3 11.5 21.1 749 1245 130 112

6 x 1 53.5 1382 39.0 5 20.1 -0.0039 0.020 5.60 16.6 32.2 797 1390 103 142

6 x 2 53.5 1332 38.1 5 19.5 -0.0038 0.019 5.42 16.1 31.6 807 1410 102 133

6 x 3 53.5 1305 37.4 5 18.6 -0.0036 0.018 5.18 15.3 31.1 834 1429 119 136

8 x 1 71.3 1369 47.0 5 25.0 -0.0046 0.025 6.97 21.6 40.4 768 1465 36 117

Imperial

Metric

�

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.0010

Aver

age

curv

ature

(rad

/in.)

012

24

36

48

60

72

84

96


0

5

10

15

20

25

30

35

[ �ra

d/

mm

]

0300

600

900

1200

1500

1800

2100

2400

[mm]

�y=

0.0

000951

rad/i

n.

Hin

eset

al.

2B

L=

96

in.

D=

48

in.

��

=1

��

=1.5

��

=2

��

=3

��

=4

��

=6

��

=8


curvatureproles�

� �




+1/4 F'y 0.02 0.01 22 177 0.0000310 0.0000170 -- -- -- -- --

-1/4 F'y -0.02 -0.03 -24 -191 -0.0000020 -0.0000010 -- -- -- -- --

+1/2 F'y 0.06 0.04 47 372 0.0000420 0.0000230 -- -- -- -- --

-1/2 F'y -0.05 -0.06 -45 -357 -0.0000280 -0.0000150 -- -- -- -- --

+3/4 F'y 0.11 0.10 69 554 0.0000530 0.0000290 -- -- -- -- --

-3/4 F'y -0.12 -0.13 -69 -549 -0.0000960 -0.0000530 -- -- -- -- --

+ F'y 0.24 0.22 94 749 0.0000750 0.0000410 -- -- -- -- --

- F'y -0.22 -0.22 -92 -734 -0.000153 -0.000084 -- -- -- -- --

�� = +1 x 1 0.35 0.31 107 853 0.000198 0.000108 0.00017 -- 10.5 48.4 10

�� = -1 x 1 -0.35 -0.32 -104 -829 -0.00015 -0.0000820 -0.00018 -- 18.0 -- 10

�� = +1 x 2 0.35 0.31 101 811 0.000223 0.000122 0.00018 -- 9.2 30.9 10

�� = -1 x 2 -0.35 -0.31 -101 -805 -0.000141 -0.0000770 -0.00018 -- 21.4 -- 10

�� = +1 x 3 0.35 0.31 101 807 0.000220 0.000120 0.00018 -- 9.6 33.0 10

�� = -1 x 3 -0.35 -0.31 -100 -802 -0.000141 -0.0000770 -0.00018 -- 21.5 -- 10

�� = +1.5 x 1 0.53 0.46 119 948 0.000292 0.000160 0.00030 -- 14.4 47.8 10

�� = -1.5 x 1 -0.53 -0.42 -120 -959 -0.000239 -0.000131 -0.00026 -- 19.5 -- 10

�� = +1.5 x 2 0.53 0.48 113 902 0.000273 0.000150 0.00033 -- 16.4 58.8 10

�� = -1.5 x 2 -0.53 -0.39 -113 -906 -0.000247 -0.000135 -0.00025 -- 19.2 -- 10

�� = +1.5 x 3 0.53 0.48 112 899 0.000273 0.000150 0.00034 -- 16.5 58.7 10

�� = -1.5 x 3 -0.53 -0.39 -113 -902 -0.000244 -0.000134 -0.00024 -- 19.6 -- 10

�� = +2 x 1 0.70 0.64 125 1000 0.000406 0.000222 0.00037 4 14.9 41.3 14

�� = -2 x 1 -0.70 -0.49 -123 -981 -0.000290 -0.000159 -0.00026 4 25.4 -- 14

�� = +2 x 2 0.70 0.66 121 970 0.000441 0.000242 0.00035 4 13.7 35.3 16

�� = -2 x 2 -0.70 -0.48 -119 -951 -0.000288 -0.000158 -0.00023 4 26.0 -- 16

�� = +2 x 3 0.70 0.66 120 959 0.000392 0.000215 0.00044 4 16.2 45.6 12

�� = -2 x 3 -0.70 -0.47 -118 -945 -0.000284 -0.000156 -0.00028 4 26.6 -- 12

�� = +3 x 1 1.05 0.94 130 1040 0.000612 0.000335 0.00046 4 16.8 41.7 19

�� = -3 x 1 -1.05 -0.74 -126 -1010 -0.000482 -0.000264 -0.00035 4 23.4 76.1 19

�� = +3 x 2 1.05 0.94 123 986 0.000602 0.000330 0.00044 4 17.3 42.7 20

�� = -3 x 2 -1.05 -0.73 -121 -969 -0.000491 -0.000269 -0.00033 4 22.9 69.9 20

�� = +3 x 3 1.05 0.95 121 971 0.000614 0.000337 0.00045 4 17.0 41.3 20

�� = -3 x 3 -1.05 -0.72 -120 -962 -0.000482 -0.000264 -0.00033 4 23.6 74.7 20

�� = +4 x 1 1.40 1.20 129 1030 0.000736 0.000403 0.00042 6 20.2 50.5 29

�� = -4 x 1 -1.40 -0.98 -127 -1010 -0.000625 -0.000342 -0.00035 6 25.2 77.3 29

�� = +4 x 2 1.40 1.21 122 979 0.000727 0.000399 0.00043 6 20.6 51.4 29

�� = -4 x 2 -1.40 -0.97 -123 -981 -0.000642 -0.000352 -0.00034 6 24.3 69.4 29

�� = +4 x 3 1.40 1.21 121 969 0.000716 0.000392 0.00043 6 21.1 53.2 29

�� = -4 x 3 -1.40 -0.95 -120 -961 -0.000641 -0.000351 -0.00033 6 24.4 69.8 29

�� = +6 x 1 2.11 1.72 130 1040 0.00102 0.000556 0.000527 6 23.3 58.6 34.6

�� = -6 x 1 -2.11 -1.35 -127 -1020 -0.000953 -0.000522 -0.000475 6 25.4 68.1 29.3

�� = +6 x 2 2.11 1.73 122 977 0.000959 0.000526 0.000509 6 25.2 66.3 36.3

�� = -6 x 2 -2.11 -1.27 -124 -989 -0.000893 -0.000489 -0.000467 6 27.7 85.7 27.7

�� = +6 x 3 2.10 1.73 122 976 0.000927 0.000508 0.000497 6 26.2 72.2 37.4

�� = -6 x 3 -2.11 -1.21 -120 -962 -0.000859 -0.000471 -0.000439 6 29.2 -- 28.2

�� = +8 x 1 2.81 2.24 127 1020 0.00120 0.000659 0.000717 6 27.5 77.1 32.7

�� = -8 x 1 -2.79 -1.45 -126 -1010 -0.00104 -0.000569 -0.000539 6 33.4 -- 27.5

Level

� �





-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

-30

-20

-10

0 10 20 30

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

[mm

]


� y

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y


-30

-20

-10

0 10 20 30

[� rad / mm]


� �

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

-60

-40

-20

0 20 40 60

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

[mm

]


� y

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y


-60

-40

-20

0 20 40 60

[� rad / mm]


��


� F �b0 �b �c0 �c �'c �s0 �s �'s


+1/4 F'y 0.0249 22.1 0.0000162 -- -0.00043 -- -0.00038 0.00031 -- 0.00038

+1/2 F'y 0.057 43.7 0.0000343 -- -0.00066 -- -0.00080 0.00092 -- 0.00080

+3/4 F'y 0.1196 67.8 0.0000736 -- -0.0011 -- -0.0017 0.0024 -- 0.0017

+ F'y 0.2442 89.0 0.000116 -- -0.0014 -- -0.0026 0.0039 -- 0.0026

�� = 1 x 1 0.351 105 0.000174 0.00610 -0.0018 -0.0019 -0.0026 0.0061 0.0065 0.0054

�� = 1 x 2 0.351 101 0.000182 0.00650 -0.0019 -0.0019 -0.0025 0.0065 0.0069 0.0058

�� = 1 x 3 0.352 101 0.000181 0.00650 -0.0019 -0.0019 -0.0026 0.0065 0.0069 0.0057

�� = 1.5 x 1 0.527 114 0.000261 0.00970 -0.0023 -0.0026 -0.0033 0.0097 0.011 0.0089

�� = 1.5 x 2 0.528 113 0.000260 0.00970 -0.0023 -0.0026 -0.0036 0.0097 0.011 0.0084

�� = 1.5 x 3 0.528 113 0.000259 0.00950 -0.0023 -0.0026 -0.0036 0.0095 0.011 0.0083

�� = 2 x 1 0.701 119 0.000352 0.01300 -0.0029 -0.0025 -0.0031 0.013 0.011 0.013

�� = 2 x 2 0.703 120 0.000365 0.01400 -0.0029 -0.0025 -0.0032 0.014 0.011 0.014

�� = 2 x 3 0.703 119 0.000338 0.01300 -0.0028 -0.0025 -0.0031 0.013 0.011 0.012

�� = 3 x 1 1.05 128 0.000547 0.02100 -0.0041 -0.0032 -0.0039 0.021 0.016 0.021

�� = 3 x 2 1.05 122 0.000547 0.02100 -0.0041 -0.0032 -0.0041 0.021 0.015 0.021

�� = 3 x 3 1.05 121 0.000548 0.02100 -0.0041 -0.0031 -0.0040 0.021 0.015 0.021

�� = 4 x 1 1.40 128 0.000680 0.02600 -0.0049 -0.0032 -0.0047 0.026 0.015 0.027

�� = 4 x 2 1.40 123 0.000685 0.02700 -0.0050 -0.0032 -0.0054 0.027 0.015 0.026

�� = 4 x 3 1.40 121 0.000678 0.02600 -0.0049 -0.0031 -0.0055 0.026 0.015 0.026

�� = 6 x 1 2.11 129 0.000984 0.000510 -0.0070 -0.0039 -0.0074 0.038 0.020 0.038

�� = 6 x 2 2.11 123 0.000926 0.000494 -0.0066 -0.0038 -0.0074 0.036 0.019 0.035

�� = 6 x 3 2.11 120 0.000894 0.000472 -0.0064 -0.0036 -0.0072 0.035 0.018 0.034

�� = 8 x 1 2.81 126 0.00112 0.000635 -0.0079 -0.0046 -0.0075 0.044 0.025 0.044

Level

��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y = 0.00228

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y = 0.00228


L = 96 in.D = 48 in.

�� = 1

�� = 2

�� = 3

�� = 4

�� = 6

�� = 8


L = 96 in.D = 48 in.

1/4 F'y

1/2 F'y

3/4 F'y

F'y


��

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]


L = 96 in.D = 48 in.

� y = 0.00228

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y = 0.00228


L = 96 in.D = 48 in.

Flexural strain profiles are the same forboth the positive and negative directionssince they were created from potentiometerson only one side of the test unit. The potentiometerchannels on the other side experienced electronicmalfunction prior to first yield.


��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]


L = 96 in.D = 48 in.

�� = 1 x -1

�� = 2 x -1

�� = 3 x -1

�� = 4 x -1

�� = 6 x -1

�� = 8 x -1

� y = 0.00228

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]


L = 96 in.D = 48 in.

�� = 1 x +1

�� = 2 x +1

�� = 3 x +1

�� = 4 x +1

�� = 6 x +1

�� = 8 x +1


��

Table C�� Structural wall with conned boundary elements� Test �C �Hines et al�� Average experimental plasticity values�



1 x 1 0.355 727 0.264 -- 0.00011 -0.0015 0.0036 1.0 0.000036 0.0344 10 10 5.0 2.2

1 x 2 0.353 684 0.279 -- 0.00013 -0.0017 0.0044 1.3 0.000066 0.0630 10 10 5.0 0.7

1 x 3 0.354 694 0.260 -- 0.00011 -0.0015 0.0036 1.1 0.000042 0.0407 10 10 5.0 0.8

1.5 x 1 0.531 878 0.402 -- 0.00022 -0.0023 0.0078 2.1 0.00013 0.125 10 10 5.0 -0.1

1.5 x 2 0.531 840 0.399 -- 0.00022 -0.0023 0.0078 2.2 0.00014 0.134 10 10 5.0 -0.2

1.5 x 3 0.531 840 0.402 -- 0.00023 -0.0024 0.0082 2.2 0.00014 0.137 10 10 5.0 -0.2

2 x 1 0.708 957 0.523 -- 0.00021 -0.0022 0.0074 2.0 0.00012 0.221 20 30 5.0 2.3

2 x 2 0.709 934 0.536 -- 0.00022 -0.0023 0.0078 2.1 0.00013 0.241 20 30 5.0 2.4

2 x 3 0.708 918 0.508 -- 0.00020 -0.0021 0.0070 2.0 0.00011 0.218 20 30 5.0 2.5

3 x 1 1.06 991 0.788 5 0.00032 -0.0030 0.012 3.1 0.00022 0.476 23 35.0 5.0 3.2

3 x 2 1.06 966 0.772 5 0.00030 -0.0029 0.011 2.9 0.00021 0.468 23 36.8 5.0 3.5

3 x 3 1.06 961 0.777 5 0.00031 -0.0029 0.011 3.0 0.00021 0.474 23 36.9 5.0 3.6

4 x 1 1.42 1026 1.03 5 0.0003891 -0.0034 0.014 3.78 0.000288 0.711 25.7 40.5 5.44 3.89

4 x 2 1.42 968 1.03 5 0.0003837 -0.0034 0.014 3.73 0.000288 0.730 26.4 41.1 5.83 4.02

4 x 3 1.42 972 1.04 5 0.0003855 -0.0034 0.014 3.74 0.000290 0.734 26.4 41.4 5.70 4.03

6 x 1 2.12 1055 1.59 5 0.0006488 -0.0053 0.025 6.30 0.000545 1.26 24.0 46.3 0.90 3.12

6 x 2 2.12 987 1.57 5 0.0006213 -0.0051 0.024 6.03 0.000524 1.25 24.9 46.4 1.73 3.24

6 x 3 2.12 968 1.54 5 0.0005934 -0.0049 0.022 5.76 0.000498 1.23 25.8 46.6 2.53 3.55

8 x 1 2.83 993 2.06 5 0.0008117 -0.0066 0.031 7.88 0.000714 1.74 25.4 47.6 1.64 4.14



1 x 1 9.03 985 6.69 -- 4.2 -0.0019 0.0065 1.0 1.4 0.87 250 250 130 55

1 x 2 8.96 926 7.07 -- 5.2 -0.0019 0.0069 1.3 2.6 1.60 250 250 130 17

1 x 3 8.99 941 6.59 -- 4.4 -0.0019 0.0069 1.1 1.7 1.03 250 250 130 21

1.5 x 1 13.5 1190 10.2 -- 8.5 -0.0026 0.011 2.1 5.1 3.18 250 250 130 -2

1.5 x 2 13.5 1138 10.1 -- 8.8 -0.0026 0.011 2.2 5.5 3.41 250 250 130 -5

1.5 x 3 13.5 1139 10.2 -- 8.9 -0.0026 0.011 2.2 5.6 3.48 250 250 130 -5

2 x 1 18.0 1296 13.3 -- 8.3 -0.0027 0.012 2.0 4.5 5.62 510 760 130 59

2 x 2 18.0 1266 13.6 -- 8.6 -0.0026 0.011 2.1 4.9 6.12 510 760 130 60

2 x 3 18.0 1243 12.9 -- 8.0 -0.0029 0.013 2.0 4.5 5.54 510 760 130 63

3 x 1 27.0 1343 20.0 5 13 -0.0032 0.016 3.1 8.7 12.1 570 890 130 80

3 x 2 27.0 1309 19.6 5 12 -0.0032 0.015 2.9 8.2 11.9 590 934 130 90

3 x 3 27.0 1302 19.7 5 12 -0.0031 0.015 3.0 8.3 12.0 600 937 130 91

4 x 1 36.0 1390 26.3 5 15.3 -0.0032 0.015 3.78 11.3 18.1 653 1029 138 99

4 x 2 35.9 1311 26.3 5 15.1 -0.0032 0.015 3.73 11.4 18.5 669 1043 148 102

4 x 3 36.0 1318 26.4 5 15.2 -0.0031 0.015 3.74 11.4 18.6 670 1051 145 102

6 x 1 53.9 1430 40.4 5 25.6 -0.0039 0.020 6.30 21.5 31.9 611 1176 23 79

6 x 2 53.9 1337 39.8 5 24.5 -0.0038 0.019 6.03 20.6 31.9 633 1178 44 82

6 x 3 53.9 1312 39.1 5 23.4 -0.0036 0.018 5.76 19.6 31.4 656 1184 64 90

8 x 1 71.9 1345 52.2 5 32.0 -0.0046 0.025 7.88 28.1 44.2 646 1208 42 105

Imperial

Metric

��

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.0010

Aver

age

curv

ature

(rad

/in.)

012

24

36

48

60

72

84

96


0

5

10

15

20

25

30

35

[ �ra

d/

mm

]

0300

600

900

1200

1500

1800

2100

2400

[mm]

�y=

0.0

00103

rad/i

n.

Hin

eset

al.

2C

L=

96

in.

D=

48

in.

��

=1

��

=1.5

��

=2

��

=3

��

=4

��

=6

��

=8

FigureC��Structuralwallwithconnedboundaryelements�Test�C�Hinesetal��Average

curvatureproles�

��

Table C�� Structural wall with conned boundary elements� Test �C �Hines et al�� Peak curvature values�



+1/4 F'y 0.0282 0.0257 20.4 163 0.0000199 0.0000109 -- -- -- -- --

-1/4 F'y -0.0303 -0.0282 -19.8 -159 -0.0000120 -0.0000066 -- -- -- -- --

+1/2 F'y 0.0701 0.0590 47.9 383 0.0000514 0.0000282 -- -- -- -- --

-1/2 F'y -0.0728 -0.0652 -46.3 -370 -0.0000384 -0.0000211 -- -- -- -- --

+3/4 F'y 0.160 0.120 70.6 565 0.000098 0.0000535 -- -- -- -- --

-3/4 F'y -0.167 -0.139 -67.3 -538 -0.0000866 -0.0000474 -- -- -- -- --

+ F'y 0.358 0.270 90.8 726 0.000147 0.0000805 -- -- -- -- --

- F'y -0.342 -0.240 -90.6 -725 -0.000113 -0.0000616 -- -- -- -- --

�� = +1 x 1 0.356 0.260 91.3 731 0.000143 0.0000781 0.00010 -- 9.0 -- 10

�� = -1 x 1 -0.355 -0.267 -90.4 -723 -0.000150 -0.0000822 -0.00011 -- 8.5 -- 10

�� = +1 x 2 0.351 0.305 84.7 678 0.000160 0.0000878 0.00016 -- 8.8 50.9 10

�� = -1 x 2 -0.354 -0.252 -86.2 -689 -0.000135 -0.0000738 -0.00010 -- 12.3 -- 10

�� = +1 x 3 0.352 0.272 84.9 679 0.000129 0.0000705 0.00013 -- 13.7 -- 10

�� = -1 x 3 -0.356 -0.247 -88.6 -709 -0.000124 -0.0000677 -0.00009 -- 14.1 -- 10

�� = +1.5 x 1 0.529 0.439 109 874 0.000226 0.000124 0.00026 -- 13.9 87.4 10

�� = -1.5 x 1 -0.533 -0.365 -110 -883 -0.000201 -0.000110 -0.00018 -- 17.4 -- 10

�� = +1.5 x 2 0.529 0.421 106 846 0.000221 0.000121 0.00024 -- 15.2 -- 10

�� = -1.5 x 2 -0.534 -0.377 -104 -834 -0.000210 -0.000115 -0.00020 -- 17.4 -- 10

�� = +1.5 x 3 0.529 0.434 106 845 0.000231 0.000126 0.00026 -- 14.2 71.3 10

�� = -1.5 x 3 -0.533 -0.370 -104 -836 -0.000206 -0.000113 -0.00019 -- 18.0 -- 10

�� = +2 x 1 0.706 0.543 121 966 0.000302 0.000165 0.00022 -- 17.3 78.3 20

�� = -2 x 1 -0.710 -0.503 -118 -947 -0.000278 -0.000152 -0.00020 -- 20.5 -- 20

�� = +2 x 2 0.707 0.565 118 942 0.000316 0.000173 0.00023 -- 16.5 61.7 20

�� = -2 x 2 -0.712 -0.506 -116 -927 -0.000289 -0.000159 -0.00020 -- 19.6 -- 20

�� = +2 x 3 0.705 0.537 112 898 0.000299 0.000164 0.00022 -- 18.6 81.3 20

�� = -2 x 3 -0.711 -0.478 -117 -937 -0.000278 -0.000152 -0.00019 -- 20.7 -- 20

�� = +3 x 1 1.06 0.821 125 1003 0.000434 0.000238 0.00033 5 23.0 -- 23

�� = -3 x 1 -1.06 -0.755 -122 -979 -0.000535 -0.000293 -0.00030 5 17.4 46.5 23

�� = +3 x 2 1.06 0.803 121 964 0.000441 0.000242 0.00032 5 22.8 85.6 23

�� = -3 x 2 -1.07 -0.741 -121 -968 -0.000523 -0.000287 -0.00029 5 18.1 49.2 23

�� = +3 x 3 1.06 0.828 119 950 0.000463 0.000254 0.00033 5 21.4 67.8 23

�� = -3 x 3 -1.07 -0.725 -121 -972 -0.000512 -0.000281 -0.00029 5 18.6 51.7 23

�� = +4 x 1 1.41 1.09 128 1026 0.000595 0.000326 0.000446 5 24.0 76.9 23.2

�� = -4 x 1 -1.42 -0.978 -128 -1025 -0.000688 -0.000377 -0.000287 5 19.8 51.8 36.7

�� = +4 x 2 1.41 1.08 118 941 0.000588 0.000322 0.000436 5 24.8 79.9 23.8

�� = -4 x 2 -1.42 -0.989 -124 -994 -0.000693 -0.000380 -0.000290 5 19.8 51.1 36.6

�� = +4 x 3 1.41 1.10 120 962 0.000599 0.000328 0.000446 5 24.2 74.8 23.6

�� = -4 x 3 -1.42 -0.981 -123 -982 -0.000690 -0.000378 -0.000285 5 20.0 51.6 37.1

�� = +6 x 1 2.12 1.65 132 1057 0.000927 0.000508 0.000685 5 24.5 66.9 23.6

�� = -6 x 1 -2.12 -1.53 -132 -1054 -0.00104 -0.000573 -0.000570 5 21.1 51.5 26.8

�� = +6 x 2 2.12 1.64 123 983 0.000913 0.000500 0.000660 5 25.3 69.9 24.6

�� = -6 x 2 -2.13 -1.49 -124 -990 -0.00100 -0.000549 -0.000546 5 22.5 56.3 27.3

�� = +6 x 3 2.12 1.62 120 961 0.000900 0.000493 0.000623 5 25.9 73.0 26.0

�� = -6 x 3 -2.13 -1.46 -122 -976 -0.000990 -0.000542 -0.000525 5 22.9 57.7 28.0

�� = +8 x 1 2.83 2.16 126 1009 0.00133 0.000727 0.000834 5 23.4 57.1 26.1

�� = -8 x 1 -2.83 -1.95 -122 -977 -0.00142 -0.000776 -0.000756 5 21.7 50.7 25.9

Level

��





-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

-30

-20

-10

0 10 20 30

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

Hines et al. 2C�� = 2

� y

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y


-30

-20

-10

0 10 20 30

[� rad / mm]

Figure C�� Structural wall with conned boundary elements� Test �C �Hines et al�� Curvature proles at �� and at ��

��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

-60

-40

-20

0 20 40 60

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

[mm

]


� y

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y


-60

-40

-20

0 20 40 60

[� rad / mm]


��

Table C�� Structural wall with conned boundary elements� Test �C �Hines et al�� Flexural strain values�

� F �b0 �b �c0 �c �'c �s0 �s �'s


+1/4 F'y 0.0293 20.1 0.0000160 -- -0.00046 -- -0.00068 0.00026 -- 0.000055

+1/2 F'y 0.0715 47.1 0.0000449 -- -0.0008 -- -0.00117 0.00130 -- 0.000928

+3/4 F'y 0.163 69.0 0.0000921 -- -0.0013 -- -0.00192 0.0029 -- 0.0026

+ F'y 0.350 90.7 0.000130 -- -0.0017 -- -0.00194 0.0044 -- 0.0040

�� = 1 x 1 0.355 90.9 0.000146 0.00011 -0.0018 -0.0015 -0.00288 0.0050 0.0036 0.0043

�� = 1 x 2 0.353 85.5 0.000147 0.00013 -0.0018 -0.0017 -0.00288 0.0050 0.0044 0.0042

�� = 1 x 3 0.354 86.8 0.000126 0.00011 -0.0016 -0.0015 -0.00257 0.0042 0.0036 0.0039

�� = 1.5 x 1 0.531 110 0.000214 0.00022 -0.0023 -0.0023 -0.00379 0.0076 0.0078 0.0066

�� = 1.5 x 2 0.531 105 0.000216 0.00022 -0.0023 -0.0023 -0.00386 0.0076 0.0078 0.0064

�� = 1.5 x 3 0.531 105 0.000218 0.00023 -0.0023 -0.0024 -0.00391 0.0077 0.0082 0.0064

�� = 2 x 1 0.708 120 0.000290 0.00021 -0.0028 -0.0022 -0.00457 0.011 0.0074 0.0094

�� = 2 x 2 0.709 117 0.000303 0.00022 -0.0029 -0.0023 -0.00465 0.011 0.0078 0.0093

�� = 2 x 3 0.708 115 0.000289 0.00020 -0.0027 -0.0021 -0.00468 0.010 0.0070 0.0087

�� = 3 x 1 1.06 124 0.000485 0.00032 -0.0041 -0.0030 -0.00561 0.018 0.012 0.016

�� = 3 x 2 1.06 121 0.000482 0.00030 -0.0041 -0.0029 -0.00592 0.018 0.011 0.017

�� = 3 x 3 1.06 120 0.000488 0.00031 -0.0042 -0.0029 -0.00591 0.018 0.011 0.017

�� = 4 x 1 1.42 128 0.000641 0.000389 -0.0053 -0.0034 -0.00671 0.024 0.014 0.024

�� = 4 x 2 1.42 121 0.000641 0.000384 -0.0053 -0.0034 -0.00666 0.024 0.014 0.024

�� = 4 x 3 1.42 122 0.000645 0.000386 -0.0053 -0.0034 -0.0068 0.024 0.014 0.023

�� = 6 x 1 2.12 132 0.000986 0.000649 -0.008 -0.0053 -0.00923 0.037 0.025 0.036

�� = 6 x 2 2.12 123 0.000957 0.000621 -0.0077 -0.0051 -0.00954 0.036 0.024 0.034

�� = 6 x 3 2.12 121 0.000945 0.000593 -0.0076 -0.0049 -0.010 0.036 0.022 0.034

�� = 8 x 1 2.83 124 0.00137 0.000812 -0.011 -0.0066 -0.016 0.052 0.031 0.048

Level

��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y = 0.00248

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y = 0.00248

Hines et al. 2CPost-Yield Strains

L = 96 in.D = 48 in.

�� = 1

�� = 2

�� = 3

�� = 4

�� = 6

�� = 8

Hines et al. 2CPre-Yield Strains

L = 96 in.D = 48 in.

1/4 F'y

1/2 F'y

3/4 F'y

F'y

Figure C�� Structural wall with conned boundary elements� Test �C �Hines et al�� Average �exural strain proles�

��

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]


L = 96 in.D = 48 in.

-1/4 F'y

-1/2 F'y

-3/4 F'y

-F'y

� y = 0.00248

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y = 0.00248


L = 96 in.D = 48 in.

+1/4 F'y

+1/2 F'y

+3/4 F'y

+F'y

Figure C�� Structural wall with conned boundary elements� Test �C �Hines et al�� Pre�yield �exural strain proles�

��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]


L = 96 in.D = 48 in.

�� = 1 x -1

�� = 2 x -1

�� = 3 x -1

�� = 4 x -1

�� = 6 x -1

�� = 8 x -1

� y = 0.00248

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

[mm

]

� y = 0.00248 Hines et al. 2CPost-Yield Strains

L = 96 in.D = 48 in.

�� = 1 x +1

�� = 2 x +1

�� = 3 x +1

�� = 4 x +1

�� = 6 x +1

�� = 8 x +1

Figure C�� Structural wall with conned boundary elements� Test �C �Hines et al�� Post�yield �exural strain proles�

��




1 x 1 0.947 1411 0.732 -- 0.00013 -0.0019 0.0042 1.3 0.000045 0.0756 14 14 7.0 13

1 x 2 0.947 1371 0.728 -- 0.00014 -0.0019 0.0046 1.4 0.000054 0.0906 14 14 7.0 12

2 x 1 1.90 1609 1.47 3 0.000469 -0.0042 0.017 4.76 0.000371 0.718 16.1 28.7 1.80 6.41

2 x 2 1.90 1568 1.45 3 0.000407 -0.0038 0.015 4.12 0.000311 0.726 19.4 28.9 4.97 8.78

3 x 1 2.85 1702 2.20 4 0.000538 -0.0047 0.020 5.45 0.000434 1.41 27.1 43.5 5.38 10.0

3 x 2 2.85 1659 2.18 4 0.000526 -0.0047 0.020 5.33 0.000425 1.40 27.5 42.8 6.14 10.5

4 x 1 3.80 1750 2.91 5 0.000679 -0.0057 0.026 6.89 0.000573 2.10 30.6 50.4 5.38 10.8

4 x 2 3.80 1715 2.85 5 0.000634 -0.0054 0.024 6.43 0.000529 2.05 32.3 49.4 7.55 12.3



1 x 1 24.0 1912 18.6 -- 5.2 -0.0019 0.0042 1.3 1.8 1.92 356 356 178 319

1 x 2 24.1 1858 18.5 -- 5.4 -0.0019 0.0046 1.4 2.1 2.30 356 356 178 300

2 x 1 48.2 2180 37.2 3 18.5 -0.0042 0.017 4.8 14.6 18.2 410 728 46 163

2 x 2 48.2 2124 36.9 3 16.0 -0.0038 0.015 4.1 12.3 18.4 494 735 126 223

3 x 1 72.3 2306 56.0 4 21.2 -0.0047 0.020 5.5 17.1 35.9 689 1105 137 255

3 x 2 72.3 2248 55.3 4 20.7 -0.0047 0.020 5.3 16.7 35.7 700 1088 156 266

4 x 1 96.4 2371 74.0 5 26.8 -0.0057 0.026 6.9 22.6 53.4 777 1280 137 275

4 x 2 96.4 2324 72.3 5 25.0 -0.0054 0.024 6.4 20.9 52.0 819 1255 192 311

Imperial

Metric

��

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.0010

Av

erag

ecu

rvat

ure

(rad

/in

.)

012

24

36

48

60

72

84

96

10

8

12

0


0

5

10

15

20

25

30

35

[ �ra

d/

mm

]

030

0

60

0

90

0

12

00

15

00

18

00

21

00

24

00

27

00

30

00

[mm]

�y=

0.0

00

09

86

rad

/in

.H

ines

etal

.T

est

3A

L=

12

0in

.D

=4

8in

.

��

=1

��

=2

��

=3

��

=4


curvatureproles�

��




+1/4 F'y 0.0995 0.0995 37.8 378 0.0000390 0.0000150 -- -- -- -- --

-1/4 F'y -0.102 -0.101 -32.9 -329 -0.0000400 -0.0000160 -- -- -- -- --

+1/2 F'y 0.209 0.191 59.7 597 0.0000830 0.0000330 -- -- -- -- --

-1/2 F'y -0.308 -0.268 -60.1 -601 -0.000122 -0.0000490 -- -- -- -- --

+3/4 F'y 0.438 0.357 89.7 897 0.000174 0.0000700 -- -- -- -- --

-3/4 F'y -0.485 -0.414 -90.0 -900 -0.000192 -0.0000770 -- -- -- -- --

+ F'y 0.703 0.565 120 1197 0.000275 0.000110 -- -- -- -- --

- F'y -0.747 -0.580 -120 -1199 -0.000291 -0.000117 -- -- -- -- --

�� = +1 x 1 0.946 0.748 144 1445 0.000411 0.000165 0.00013 -- 2.4 10.2 14

�� = -1 x 1 -0.947 -0.715 -138 -1378 -0.000399 -0.000159 -0.00013 -- 3.5 15.6 14

�� = +1 x 2 0.947 0.747 139 1393 0.000418 0.000167 0.00014 -- 3.1 13.1 14

�� = -1 x 2 -0.947 -0.708 -135 -1349 -0.000398 -0.000159 -0.00013 -- 4.0 17.4 14

�� = +2 x 1 1.90 1.50 165 1651 0.000988 0.000395 0.000366 3 9.0 29.6 23.0

�� = -2 x 1 -1.90 -1.43 -157 -1566 -0.000945 -0.000378 -0.000525 3 9.9 33.2 13.7

�� = +2 x 2 1.90 1.49 158 1575 0.00102 0.000406 0.000367 3 9.1 29.5 23.2

�� = -2 x 2 -1.90 -1.42 -156 -1560 -0.000987 -0.000395 -0.000410 3 9.5 31.1 18.4

�� = +3 x 1 2.85 2.25 173 1727 0.00145 0.000578 0.000531 4 11.9 38.2 28.3

�� = -3 x 1 -2.84 -2.16 -168 -1677 -0.00143 -0.000571 -0.000539 4 12.3 39.4 26.3

�� = +3 x 2 2.85 2.22 167 1669 0.00143 0.000570 0.000501 4 12.3 39.6 30.1

�� = -3 x 2 -2.85 -2.13 -165 -1649 -0.00146 -0.000585 -0.000519 4 12.0 38.3 27.2

�� = +4 x 1 3.79 2.99 176 1762 0.00185 0.000740 0.000643 5 14.0 44.5 33.7

�� = -4 x 1 -3.80 -2.84 -174 -1738 -0.00196 -0.000783 -0.000694 5 13.2 41.1 28.8

�� = +4 x 2 3.79 2.91 170 1701 0.00182 0.000729 0.000597 5 14.4 45.9 35.9

�� = -4 x 2 -3.80 -2.78 -173 -1729 -0.00204 -0.000814 -0.000635 5 12.6 38.9 31.1

Level

��

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

Hei

ghtab

ove

foot

ing

(in.

)

-30

-20

-10

0 10 20 30

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

[mm

]


� y

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

[mm

]

� y


-30

-20

-10

0 10 20 30

[� rad / mm]


��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

Hei

ghtab

ove

foot

ing

(in.

)

-60

-40

-20

0 20 40 60

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

[mm

]


� y

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

[mm

]

� y


-60

-40

-20

0 20 40 60

[� rad / mm]


��


� F �b0 �b �c0 �c �'c �s0 �s �'s


+1/4 F'y 0.101 35.4 0.0000395 -- -0.00077 -- -0.00149 0.0010 -- 0.000566

+1/2 F'y 0.258 59.9 0.000102 -- -0.0016 -- -0.0030 0.0030 -- 0.0023

+3/4 F'y 0.461 89.9 0.000183 -- -0.0022 -- -0.00513 0.0064 -- 0.0044

+ F'y 0.725 120 0.000283 -- -0.0023 -- -0.00747 0.010 -- 0.0073

�� = 1 x 1 0.947 141 0.000405 0.000131 -0.0038 -0.0019 -0.00975 0.015 0.0042 0.011

�� = 1 x 2 0.947 137 0.000408 0.000137 -0.0038 -0.0019 -0.00993 0.015 0.0046 0.011

�� = 2 x 1 1.897 161 0.000971 0.000469 -0.0080 -0.0042 -0.020 0.037 0.017 0.030

�� = 2 x 2 1.897 157 0.00100 0.000407 -0.0083 -0.0038 -0.019 0.038 0.015 0.033

�� = 3 x 1 2.85 170 0.00144 0.000538 -0.012 -0.0047 -0.027 0.055 0.020 0.047

�� = 3 x 2 2.85 166 0.00144 0.000526 -0.012 -0.0047 -0.027 0.055 0.020 0.048

�� = 4 x 1 3.80 175 0.00190 0.000679 -0.016 -0.0057 -0.034 0.072 0.026 0.065

�� = 4 x 2 3.80 171 0.00193 0.000634 -0.016 -0.0054 -0.034 0.073 0.024 0.067

Level

��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

108

120

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

[mm

]

� y = 0.00214

-0.0

08

-0.0

07

-0.0

06

-0.0

05

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8


0

12

24

36

48

60

72

84

96

108

120

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

[mm

]

� y = 0.00214


L = 120 in.D = 48 in.

�� = 1

�� = 2

�� = 3

�� = 4


L = 120 in.D = 48 in.

1/4 F'y

1/2 F'y

3/4 F'y

F'y


��

-0.0

08

-0.0

07

-0.0

06

-0.0

05

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8


0

12

24

36

48

60

72

84

96

108

120

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

[mm

]


L = 120 in.D = 48 in.

-1/4 F'y

-1/2 F'y

-3/4 F'y

-F'y

� y = 0.00214

-0.0

08

-0.0

07

-0.0

06

-0.0

05

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8


0

12

24

36

48

60

72

84

96

108

120

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

[mm

]

� y = 0.00214


L = 120 in.D = 48 in.

+1/4 F'y

+1/2 F'y

+3/4 F'y

+F'y


��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

108

120

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

[mm

]


L = 120 in.D = 48 in.

�� = 1 x -1

�� = 2 x -1

�� = 3 x -1

�� = 4 x -1

� y = 0.00214

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

108

120

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

[mm

]


L = 120 in.D = 48 in.

�� = 1 x +1

�� = 2 x +1

�� = 3 x +1

�� = 4 x +1


��




1 x 1 0.619 670 0.502 -- 0.00023 -0.0022 0.0044 1.3 0.000072 0.0756 14 14 7.0 8

1 x 2 0.618 656 0.500 -- 0.00024 -0.0023 0.0045 1.4 0.000086 0.0906 14 14 7.0 7

2 x 1 1.24 837 0.982 3 0.000582 -0.0043 0.012 3.31 0.000386 0.456 15.7 23.0 4.22 6.81

2 x 2 1.24 789 0.967 3 0.000558 -0.0042 0.012 3.17 0.000374 0.471 16.8 22.9 5.35 7.15

3 x 1 1.86 861 1.46 3 0.000880 -0.0062 0.019 5.00 0.000679 0.919 18.1 28.2 3.97 6.40

3 x 2 1.88 829 1.46 3 0.000838 -0.0059 0.018 4.76 0.000644 0.935 19.4 28.0 5.35 7.08

4 x 1 2.50 877 1.96 3 0.00111 -0.0076 0.024 6.33 0.000910 1.41 20.6 30.8 5.22 7.26

4 x 2 2.48 844 1.92 3 0.00109 -0.0075 0.023 6.18 0.000890 1.39 20.8 30.2 5.69 7.24



1 x 1 15.7 908 12.8 -- 9.0 -0.0022 0.0044 1.3 2.8 1.92 356 356 178 196

1 x 2 15.7 888 12.7 -- 9.4 -0.0023 0.0045 1.4 3.4 2.30 356 356 178 185

2 x 1 31.5 1134 24.9 3 22.9 -0.0043 0.012 3.3 15.2 11.6 400 585 107 173

2 x 2 31.5 1070 24.6 3 22.0 -0.0042 0.012 3.2 14.7 12.0 427 582 136 182

3 x 1 47.2 1167 37.1 3 34.7 -0.0062 0.019 5.0 26.7 23.3 459 715 101 163

3 x 2 47.8 1123 37.0 3 33.0 -0.0059 0.018 4.8 25.4 23.8 492 712 136 180

4 x 1 63.6 1188 49.8 3 43.9 -0.0076 0.024 6.3 35.8 35.8 524 783 133 184

4 x 2 62.9 1143 48.7 3 42.8 -0.0075 0.023 6.2 35.1 35.3 529 768 145 184

Imperial

Metric

��

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0.0020

Aver

age

curv

ature

(rad

/in.)

012

24

36

48

60

72


0

10

20

30

40

50

60

70

[ �ra

d/

mm

]

0300

600

900

1200

1500

1800

[mm]

�y=

0.0

00176

rad/i

n.

Hin

eset

al.T

est

3B

L=

75

in.

D=

30

in.

��

=1

��

=2

��

=3

��

=4

��

=6

(on

ly+

)


curvatureproles�

��




+1/4 F'y 0.0553 0.0534 24.6 153 0.0000550 0.0000250 -- -- -- -- --

-1/4 F'y -0.0827 -0.0790 -27.6 -172 -0.0000770 -0.0000360 -- -- -- -- --

+1/2 F'y 0.150 0.141 45.0 282 0.000133 0.0000620 -- -- -- -- --

-1/2 F'y -0.196 -0.169 -45.1 -282 -0.000161 -0.0000740 -- -- -- -- --

+3/4 F'y 0.256 0.218 62.6 391 0.000223 0.000103 -- -- -- -- --

-3/4 F'y -0.315 -0.276 -63.3 -395 -0.000258 -0.000119 -- -- -- -- --

+ F'y 0.393 0.318 83.6 522 0.000322 0.000149 -- -- -- -- --

- F'y -0.449 -0.381 -84.2 -526 -0.000358 -0.000165 -- -- -- -- --

�� = +1 x 1 0.618 0.503 111 691 0.000528 0.000244 0.00023 -- 3.51 17.2 14

�� = -1 x 1 -0.620 -0.502 -104 -649 -0.000518 -0.000239 -0.00024 -- 4.83 23.2 14

�� = +1 x 2 0.618 0.502 107 671 0.000541 0.000250 0.00023 -- 3.89 17.8 14

�� = -1 x 2 -0.619 -0.498 -102 -640 -0.000519 -0.000240 -0.00024 -- 5.03 23.6 14

�� = +2 x 1 1.24 0.991 137 859 0.00136 0.000628 0.000465 3 7.07 21.3 22.8

�� = -2 x 1 -1.24 -0.973 -130 -815 -0.00113 -0.000519 -0.000655 3 9.49 32.2 13.2

�� = +2 x 2 1.24 0.970 126 788 0.00131 0.000606 0.000460 3 7.99 24.1 23.0

�� = -2 x 2 -1.24 -0.963 -127 -791 -0.00114 -0.000524 -0.000626 3 9.61 32.0 14.1

�� = +3 x 1 1.86 1.46 140 876 0.00187 0.000864 0.000790 3 10.2 30.2 20.8

�� = -3 x 1 -1.86 -1.46 -135 -846 -0.00177 -0.000815 -0.000957 3 11.2 34.1 16.2

�� = +3 x 2 1.86 1.43 133 831 0.00175 0.000809 0.000726 3 11.4 34.6 22.7

�� = -3 x 2 -1.91 -1.48 -132 -826 -0.00184 -0.000850 -0.000897 3 11.2 33.6 18.3

�� = +4 x 1 2.48 1.93 142 888 0.00234 0.00108 0.00103 3 12.3 36.5 22.1

�� = -4 x 1 -2.53 -1.99 -138 -865 -0.00259 -0.00120 -0.00118 3 11.3 32.0 19.8

�� = +4 x 2 2.48 1.81 135 845 0.00221 0.00102 0.00105 3 13.4 40.9 19.9

�� = -4 x 2 -2.48 -2.03 -135 -843 -0.00259 -0.00119 -0.00111 3 11.1 31.1 21.9

Level

��

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

Hei

ghtab

ove

foot

ing

(in.

)

-30

-20

-10

0 10 20 30

[� rad / mm]

0

300

600

900

1200

1500

1800

[mm

]


� y

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

[mm

]

� y


-30

-20

-10

0 10 20 30

[� rad / mm]




��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

Hei

ghtab

ove

foot

ing

(in.

)

-60

-40

-20

0 20 40 60

[� rad / mm]

0

300

600

900

1200

1500

1800

[mm

]


� y

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

[mm

]

� y


-60

-40

-20

0 20 40 60

[� rad / mm]


��


� F �b0 �b �c0 �c �'c �s0 �s �'s


+1/4 F'y 0.0690 26.1 0.0000662 -- -0.00078 -- -0.0011 0.00096 -- 0.000718

+1/2 F'y 0.173 45.1 0.000147 -- -0.0016 -- -0.00228 0.0025 -- 0.0018

+3/4 F'y 0.285 63.0 0.000241 -- -0.0023 -- -0.00341 0.0045 -- 0.0032

+ F'y 0.421 83.9 0.000340 -- -0.0029 -- -0.0046 0.0068 -- 0.0048

�� = 1 x 1 0.619 107 0.000523 0.000229 -0.0040 -0.0022 -0.00656 0.011 0.0044 0.0078

�� = 1 x 2 0.618 105 0.000530 0.000240 -0.0040 -0.0023 -0.00671 0.011 0.0045 0.0079

�� = 2 x 1 1.24 134 0.00124 0.000582 -0.0084 -0.0043 -0.013 0.027 0.012 0.021

�� = 2 x 2 1.24 126 0.00122 0.000558 -0.0083 -0.0042 -0.012 0.026 0.012 0.021

�� = 3 x 1 1.86 138 0.00182 0.000880 -0.012 -0.0062 -0.018 0.039 0.019 0.032

�� = 3 x 2 1.88 133 0.00183 0.000838 -0.012 -0.0059 -0.015 0.039 0.018 0.034

�� = 4 x 1 2.50 140 0.00246 0.00111 -0.017 -0.0076 -0.020 0.053 0.024 0.048

�� = 4 x 2 2.48 135 0.00240 0.00109 -0.016 -0.0075 -0.017 0.051 0.023 0.049

Level

��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

[mm

]

� y = 0.00214

-0.0

08

-0.0

07

-0.0

06

-0.0

05

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8


0

12

24

36

48

60

72H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

[mm

]

� y = 0.00214


L = 75 in.D = 30 in.

�� = 1

�� = 2

�� = 3

�� = 4


L = 75 in.D = 30 in.

1/4 F'y

1/2 F'y

3/4 F'y

F'y


��

-0.0

08

-0.0

07

-0.0

06

-0.0

05

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8


0

12

24

36

48

60

72

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

[mm

]


L = 75 in.D = 30 in.

-1/4 F'y

-1/2 F'y

-3/4 F'y

-F'y

� y = 0.00214

-0.0

08

-0.0

07

-0.0

06

-0.0

05

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8


0

12

24

36

48

60

72H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

[mm

]

� y = 0.00214


L = 75 in.D = 30 in.

+1/4 F'y

+1/2 F'y

+3/4 F'y

+F'y


��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

[mm

]


L = 75 in.D = 30 in.

�� = 1 x -1

�� = 2 x -1

�� = 3 x -1

�� = 4 x -1

� y = 0.00214

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

[mm

]


L = 75 in.D = 30 in.

�� = 1 x +1

�� = 2 x +1

�� = 3 x +1

�� = 4 x +1


��

Table C�� Structural wall with conned boundary elements� Test �C �Hines et al�� Average experimental plasticity values�



1 x 1 1.319 2690 1.078 -- 0.00013 -0.0022 0.0070 1.3 0.000056 0.1404 14 14 7.0 9

1 x 2 1.319 2615 1.058 -- 0.00013 -0.0022 0.0070 1.3 0.000058 0.1473 14 14 7.0 8

2 x 1 2.68 2958 2.19 4 0.000347 -0.0041 0.020 5.55 0.000286 1.16 22.5 43.9 0.51 4.58

2 x 2 2.65 2932 2.14 4 0.000323 -0.0038 0.019 5.17 0.000263 1.12 23.6 44.7 1.25 5.22

3 x 1 3.97 3145 3.26 6 0.000408 -0.0046 0.024 6.52 0.000343 2.17 35.0 66.2 1.96 8.57

3 x 2 3.97 3092 3.23 6 0.000382 -0.0043 0.022 6.11 0.000319 2.15 37.5 67.9 3.55 9.59

4 x 1 5.29 3252 4.27 6 0.000501 -0.0054 0.030 8.02 0.000435 3.14 40.1 73.5 3.31 10.4



1 x 1 33.5 3645 27.4 -- 5.0 -0.0022 0.0070 1.3 2.2 3.57 356 356 178 220

1 x 2 33.5 3543 26.9 -- 5.0 -0.0022 0.0070 1.3 2.3 3.74 356 356 178 205

2 x 1 68.0 4009 55.6 4 13.7 -0.0041 0.020 5.55 11.3 29.4 571 1116 13 116

2 x 2 67.2 3972 54.3 4 12.7 -0.0038 0.019 5.17 10.4 28.4 599 1135 32 133

3 x 1 101 4262 82.9 6 16.1 -0.0046 0.024 6.52 13.5 55.0 890 1680 50 218

3 x 2 101 4190 82.1 6 15.1 -0.0043 0.022 6.11 12.6 54.7 952 1724 90 244

4 x 1 134 4406 108 6 19.8 -0.0054 0.030 8.02 17.1 79.7 1018 1867 84 264

Imperial

Metric

��

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.0010

Av

erag

ecu

rvat

ure

(rad

/in

.)

012

24

36

48

60

72

84

96

10

8

12

0

13

2

14

4

15

6

16

8

18

0


0

5

10

15

20

25

30

35

[ �ra

d/

mm

]

030

0

60

0

90

0

12

00

15

00

18

00

21

00

24

00

27

00

30

00

33

00

36

00

39

00

42

00

45

00

[mm]

�y=

0.0

00

06

25

rad

/in

.H

ines

etal

.T

est

3C

L=

18

0in

.D

=7

2in

.

��

=1

��

=2

��

=3

��

=4

FigureC��Structuralwallwithconnedboundaryelements�Test�C�Hinesetal��Average

curvatureproles�

��

Table C�� Structural wall with conned boundary elements� Test �C �Hines et al�� Peak curvature values�



+1/4 F'y 0.120 0.120 38.2 573 0.0000310 0.0000140 -- -- -- -- --

-1/4 F'y -0.101 -0.094 -27.9 -419 -0.0000320 -0.0000150 -- -- -- -- --

+1/2 F'y 0.303 0.282 65.9 989 0.0000770 0.0000350 -- -- -- -- --

-1/2 F'y -0.307 -0.291 -66.0 -990 -0.000075 -0.0000350 -- -- -- -- --

+3/4 F'y 0.629 0.543 114 1709 0.000141 0.0000650 -- -- -- -- --

-3/4 F'y -0.631 -0.551 -110 -1652 -0.000124 -0.0000570 -- -- -- -- --

+ F'y 1.02 0.902 140 2099 0.000250 0.000115 -- -- -- -- --

- F'y -0.959 -0.772 -152 -2279 -0.000183 -0.0000840 -- -- -- -- --

�� = +1 x 1 1.32 1.13 179 2692 0.000346 0.000160 0.00013 -- 3.88 11.0 14

�� = -1 x 1 -1.32 -1.02 -179 -2687 -0.000275 -0.000127 -0.00009 -- 5.17 16.3 14

�� = +1 x 2 1.32 1.09 175 2624 0.000332 0.000153 0.00012 -- 4.63 13.3 14

�� = -1 x 2 -1.32 -1.02 -174 -2606 -0.000266 -0.000123 -0.00010 -- 6.27 19.9 14

�� = +2 x 1 2.65 2.21 203 3052 0.000648 0.000299 0.000362 4 13.6 36.0 21.3

�� = -2 x 1 -2.71 -2.17 -191 -2865 -0.000571 -0.000263 -0.000328 4 17.3 47.4 24.1

�� = +2 x 2 2.65 2.18 195 2930 0.000673 0.000310 0.000345 4 13.5 35.2 22.5

�� = -2 x 2 -2.65 -2.10 -196 -2933 -0.000536 -0.000247 -0.000295 4 17.5 49.1 25.6

�� = +3 x 1 3.97 3.34 212 3179 0.000911 0.000421 0.000420 6 18.3 47.7 34.9

�� = -3 x 1 -3.97 -3.19 -207 -3111 -0.00108 -0.000496 -0.000390 6 15.4 38.5 35.8

�� = +3 x 2 3.97 3.26 205 3081 0.000915 0.000422 0.000392 6 18.5 48.0 37.0

�� = -3 x 2 -3.97 -3.20 -207 -3104 -0.00107 -0.000494 -0.000367 6 15.4 38.7 38.7

�� = +4 x 1 5.29 4.24 216 3239 0.00127 0.000584 0.000503 6 19.3 48.9 39.6

�� = -4 x 1 -5.29 -4.30 -218 -3264 -0.00147 -0.000680 -0.000489 6 16.3 40.1 41.6

Level

��


-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

Hei

ghtab

ove

foot

ing

(in.

)

-30

-20

-10

0 10 20 30

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

[mm

]


� y

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

[mm

]

� y


-30

-20

-10

0 10 20 30

[� rad / mm]


��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

Hei

ghtab

ove

foot

ing

(in.

)

-60

-40

-20

0 20 40 60

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

[mm

]


� y

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

[mm

]

� y


-60

-40

-20

0 20 40 60

[� rad / mm]


��

Table C�� Structural wall with conned boundary elements� Test �C �Hines et al�� Flexural strain values�

� F �b0 �b �c0 �c �'c �s0 �s �'s


+1/4 F'y 0.111 33.1 0.0000300 -- -0.00085 -- -0.00159 0.0013 -- 0.00060

+1/2 F'y 0.305 66.0 0.0000714 -- -0.0016 -- -0.00313 0.0034 -- 0.0021

+3/4 F'y 0.630 112 0.000133 -- -0.0022 -- -0.00506 0.0072 -- 0.0042

+ F'y 0.989 146 0.000232 -- -0.0031 -- -0.00722 0.013 -- 0.0078

�� = 1 x 1 1.32 179 0.000310 0.000127 -0.0037 -0.0022 -0.00994 0.018 0.0070 0.012

�� = 1 x 2 1.32 174 0.000299 0.000128 -0.0036 -0.0022 -0.010 0.017 0.0070 0.010

�� = 2 x 1 2.68 197 0.000611 0.000347 -0.0063 -0.0041 -0.019 0.036 0.020 0.024

�� = 2 x 2 2.65 195 0.000604 0.000323 -0.0063 -0.0038 -0.018 0.036 0.019 0.024

�� = 3 x 1 3.97 210 0.000990 0.000408 -0.0098 -0.0046 -0.023 0.059 0.024 0.046

�� = 3 x 2 3.97 206 0.000993 0.000382 -0.0098 -0.0043 -0.023 0.060 0.022 0.046

�� = 4 x 1 5.29 217 0.00137 0.000501 -0.013 -0.0054 -0.031 0.082 0.030 0.064

Level

��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

[mm

]

� y = 0.00214

-0.0

08

-0.0

07

-0.0

06

-0.0

05

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8


0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

[mm

]

� y = 0.00214


L = 180 in.D = 72 in.

�� = 1

�� = 2

�� = 3

�� = 4


L = 180 in.D = 72 in.

1/4 F'y

1/2 F'y

3/4 F'y

F'y

Figure C�� Structural wall with conned boundary elements� Test �C �Hines et al�� Average �exural strain proles�

��

-0.0

08

-0.0

07

-0.0

06

-0.0

05

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8


0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

[mm

]


L = 180 in.D = 72 in.

-1/4 F'y

-1/2 F'y

-3/4 F'y

-F'y

� y = 0.00214

-0.0

08

-0.0

07

-0.0

06

-0.0

05

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8


0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

[mm

]

� y = 0.00214


L = 180 in.D = 72 in.

+1/4 F'y

+1/2 F'y

+3/4 F'y

+F'y

Figure C�� Structural wall with conned boundary elements� Test �C �Hines et al�� Pre�yield �exural strain proles�

��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

[mm

]


L = 180 in.D = 72 in.

�� = 1 x -1

�� = 2 x -1

�� = 3 x -1

�� = 4 x -1

� y = 0.00214

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

[mm

]

� y = 0.00214 Hines et al. 3CPost-Yield Strains

L = 180 in.D = 72 in.

�� = 1 x +1

�� = 2 x +1

�� = 3 x +1

�� = 4 x +1

Figure C�� Structural wall with conned boundary elements� Test �C �Hines et al�� Post�yield �exural strain proles�

��

Appendix D

San Francisco�Oakland Bay Bridge

East Span Skyway Piers

This appendix contains data from the East Bay Skyway Piers of the New San Francisco�

Oakland Bay Bridge tested at UCSD �� The pier geometry is a hollow rectangular

reinforced concrete pier with highly�conned corner elements�

��

Table D�� Hollow rectangular pier with conned corner elements� SFOBB LPT�Hines et al� �� Average experimental plasticity values�



1 x 1 0.932 5801 0.833 3 0.00020 -0.0025 0.0067 2.2 0.00013 0.188 11 10 5.6 -0.04

1 x 2 0.925 5838 0.826 3 0.00020 -0.0024 0.0065 2.1 0.00012 0.176 11 10 5.6 0.13

1 x 3 0.927 5755 0.826 3 0.00020 -0.0024 0.0066 2.1 0.00013 0.186 11 10 5.6 0.06

2 x 1 1.85 6787 1.57 3 0.00034 -0.0038 0.0120 3.7 0.00026 0.818 23 35.2 5.6 10.4

2 x 2 1.80 6705 1.60 3 0.00035 -0.0039 0.0120 3.8 0.00027 0.858 23 35.4 5.6 8.39

2 x 3 1.79 6559 1.60 3 0.00036 -0.0039 0.0120 3.8 0.00027 0.865 23 35.1 5.6 8.09

3 x 1 2.69 6914 2.21 3 0.00052 -0.0054 0.018 5.5 0.00043 1.443 25 37.9 5.6 7.47

3 x 2 2.69 6936 2.22 3 0.00051 -0.0053 0.018 5.4 0.00042 1.450 25 39.1 5.6 7.90

3 x 3 2.69 6693 2.22 3 0.00051 -0.0053 0.018 5.5 0.00043 1.474 25 38.9 5.6 7.63

4 x 1 3.71 7290 3.05 4 0.00065 -0.0066 0.023 6.9 0.00056 2.24 29 47.2 5.6 10.3

4 x 2 3.55 6833 2.89 4 0.00062 -0.0063 0.022 6.6 0.00053 2.13 29 47.3 5.6 10.4

4 x 3 3.79 6004 3.07 4 0.00069 -0.0069 0.024 7.3 0.00061 2.40 29 46.1 5.6 8.55

6 x 1 5.45 6883 4.40 4 0.00097 -0.0096 0.034 10 0.00088 3.63 30 48.5 5.6 10.7

6 x 2 5.42 6947 4.35 4 0.00096 -0.0095 0.034 10 0.00087 3.57 30 48.3 5.6 11.4

6 x 3 5.43 6585 4.33 4 0.00096 -0.0095 0.034 10 0.00088 3.60 30 48.2 5.6 9.60



1 x 1 23.7 7860 21.2 3 8.0 -0.0025 0.0067 2.2 5.0 4.77 272 259 142 -1

1 x 2 23.5 7911 21.0 3 7.8 -0.0024 0.0065 2.1 4.8 4.47 267 250 142 3

1 x 3 23.5 7798 21.0 3 7.9 -0.0024 0.0066 2.1 4.9 4.72 273 262 142 2

2 x 1 47.0 9197 40.0 3 14 -0.0038 0.0120 3.7 10 20.8 590 895 142 265

2 x 2 45.7 9085 40.7 3 14 -0.0039 0.0120 3.8 11 21.8 592 900 142 213

2 x 3 45.6 8888 40.5 3 14 -0.0039 0.0120 3.8 11 22.0 588 891 142 205

3 x 1 68.3 9369 56.2 3 20 -0.0054 0.018 5.5 17 36.6 624 964 142 190

3 x 2 68.2 9398 56.4 3 20 -0.0053 0.018 5.4 16 36.8 638 992 142 201

3 x 3 68.4 9070 56.4 3 20 -0.0053 0.018 5.5 17 37.4 636 988 142 194

4 x 1 94.3 9878 77.4 4 26 -0.0066 0.023 6.94 22 56.8 741 1198 142 262

4 x 2 90.3 9259 73.3 4 24 -0.0063 0.022 6.58 21 54.0 743 1202 142 265

4 x 3 96.3 8135 78.0 4 27 -0.0069 0.024 7.32 24 61.0 728 1172 142 217

6 x 1 138 9327 112 4 38 -0.0096 0.034 10 35 92.3 758 1232 142 272

6 x 2 138 9413 110 4 38 -0.0095 0.034 10 34 90.8 755 1226 142 290

6 x 3 138 8922 110 4 38 -0.0095 0.034 10 35 91.4 755 1225 142 244

Imperial

Metric

��

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0.0020

Av

erag

ecu

rvat

ure

(rad

/in

.)

012

24

36

48

60

72

84

96

10

8

12

0

13

2Heightabovefooting,h(in.)

0

10

20

30

40

50

60

70

[ �ra

d/

mm

]

030

0

60

0

90

0

12

00

15

00

18

00

21

00

24

00

27

00

30

00

33

00

[mm]

�y=

0.0

00

09

38

rad

/in

.H

ines

etal

.L

PT

2L

=2

76

in.

D=

54

in.

��

=1

��

=2

��

=3

��

=4

��

=6

FigureD��Hollowrectangularpierwithconnedcornerelements�SFOBBLPT�Hinesetal��

Averagecurvatureproles�

��

Table D�� Hollow rectangular pier with conned corner elements� SFOBB LPT�Hines et al� �� Peak curvature values�



+1/4 F'y 0.0785 0.0508 99.4 1143 0.0000140 0.0000090 -- -- -- -- --

-1/4 F'y -0.0789 -0.1032 -110 -1262 -0.0000100 -0.0000070 -- -- -- -- --

+1/2 F'y 0.192 0.160 231 2653 0.0000380 0.0000260 -- -- -- -- --

-1/2 F'y -0.183 -0.200 -231 -2662 -0.0000330 -0.0000230 -- -- -- -- --

+3/4 F'y 0.415 0.367 345 3966 0.000084 0.0000570 -- -- -- -- --

-3/4 F'y -0.387 -0.372 -349 -4015 -0.0000770 -0.0000530 -- -- -- -- --

+ F'y 0.674 0.605 464 5332 0.000129 0.0000880 -- -- -- -- --

- F'y -0.644 -0.600 -466 -5357 -0.000127 -0.0000870 -- -- -- -- --

�� = +1 x 1 0.865 0.773 519 5964 0.000177 0.000120 0.00015 3 10.6 -- 11

�� = -1 x 1 -0.868 -0.787 -532 -6121 -0.000184 -0.000126 -0.00015 3 8.8 -- 11

�� = +1 x 2 0.921 0.818 504 5799 0.000196 0.000133 0.00019 3 13.7 31.0 11

�� = -1 x 2 -0.902 -0.812 -519 -5967 -0.000199 -0.000135 -0.00018 3 11.0 -- 11

�� = +1 x 3 0.915 0.811 512 5894 0.000195 0.000133 0.00018 3 12.8 -- 11

�� = -1 x 3 -0.931 -0.835 -512 -5888 -0.000207 -0.000141 -0.00020 3 12.5 -- 11

�� = +2 x 1 1.82 1.56 629 7234 0.000627 0.000427 0.00033 3 13.4 22.3 23

�� = -2 x 1 -1.86 -1.59 -622 -7154 -0.000628 -0.000427 -0.00034 3 14.2 -- 23

�� = +2 x 2 1.75 1.64 593 6820 0.000597 0.000406 0.00037 3 13.8 22.9 23

�� = -2 x 2 -1.81 -1.54 -611 -7024 -0.000595 -0.000405 -0.00033 3 14.6 -- 23

�� = +2 x 3 1.76 1.65 573 6589 0.000600 0.000408 0.00038 3 14.3 23.7 23

�� = -2 x 3 -1.82 -1.54 -580 -6673 -0.000596 -0.000406 -0.00034 3 15.2 -- 23

�� = +3 x 1 2.67 2.23 649 7461 0.000806 0.000549 0.00050 3 19.4 -- 25

�� = -3 x 1 -2.63 -2.18 -657 -7553 -0.000849 -0.000578 -0.00049 3 17.6 28.9 25

�� = +3 x 2 2.88 2.38 649 7462 0.000902 0.000614 0.00055 3 19.0 31.1 25

�� = -3 x 2 -2.47 -2.05 -592 -6812 -0.000784 -0.000534 -0.00046 3 18.4 30.1 25

�� = +3 x 3 2.84 2.34 633 7279 0.000886 0.000603 0.00054 3 19.2 31.4 25

�� = -3 x 3 -2.43 -2.01 -584 -6711 -0.000756 -0.000514 -0.00045 3 18.8 31.0 25

�� = +4 x 1 4.15 3.41 679 7809 0.00135 0.000919 0.00074 4 20.0 32.2 29

�� = -4 x 1 -3.26 -2.69 -635 -7299 -0.00105 -0.000717 -0.00056 4 19.3 31.3 29

�� = +4 x 2 4.07 3.33 665 7646 0.00132 0.000900 0.00072 4 20.1 32.3 29

�� = -4 x 2 -2.94 -2.40 -620 -7126 -0.000946 -0.000644 -0.00050 4 18.9 30.6 29

�� = +4 x 3 4.72 3.85 563 6470 0.00141 0.000960 0.00087 4 23.5 37.7 29

�� = -4 x 3 -2.78 -2.27 -580 -6672 -0.000899 -0.000612 -0.00047 4 18.9 30.6 29

�� = +6 x 1 6.65 5.39 726 8353 0.00222 0.00151 0.0012 4 20.9 33.0 30

�� = -6 x 1 -4.21 -3.38 -606 -6965 -0.00143 -0.000970 -0.00072 4 19.7 31.3 30

�� = +6 x 2 6.65 5.35 701 8067 0.00229 0.00156 0.0012 4 20.3 32.0 30

�� = -6 x 2 -4.17 -3.32 -574 -6599 -0.00145 -0.000990 -0.00071 4 19.2 30.4 30

�� = +6 x 3 6.65 5.34 686 7890 0.00238 0.00162 0.0012 4 19.5 30.5 30

�� = -6 x 3 -4.17 -3.30 -565 -6494 -0.00133 -0.000904 -0.00071 4 21.4 34.1 30

Level

��

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

Hei

ghtab

ove

foot

ing

(in.

)

-30

-20

-10

0 10 20 30

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

[mm

]

Hines et al. LPT�� = 2

� y

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

[mm

]

� y


-30

-20

-10

0 10 20 30

[� rad / mm]

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB LPT�Hines et al� �� Curvature proles at �� and at ��

��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

Hei

ghtab

ove

foot

ing

(in.

)

-60

-40

-20

0 20 40 60

[� rad / mm]

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

[mm

]


� y

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

[mm

]

� y


-60

-40

-20

0 20 40 60

[� rad / mm]


��

Table D�� Hollow rectangular pier with conned corner elements� SFOBB LPT�Hines et al� �� Flexural strain values�

� F �b0 �b �c0 �c �'c �s0 �s �'s


+1/4 F'y 0.0787 105 0.0000118 -- -0.00047 -- -0.00050 0.00036 -- 0.00004

+1/2 F'y 0.193 219 0.0000367 -- -0.00072 -- -0.00088 0.00094 -- 0.00077

+3/4 F'y 0.406 329 0.0000801 -- -0.0013 -- -0.0016 0.0024 -- 0.0021

+ F'y 0.704 444 0.000136 -- -0.0018 -- -0.0025 0.0043 -- 0.0034

�� = 1 x 1 0.932 504 0.000203 0.00020 -0.0025 -0.0025 -0.0033 0.0067 0.0067 0.0050

�� = 1 x 2 0.925 508 0.000200 0.00020 -0.0024 -0.0024 -0.0035 0.0066 0.0065 0.0056

�� = 1 x 3 0.927 500 0.000202 0.00020 -0.0024 -0.0024 -0.0036 0.0066 0.0066 0.0057

�� = 2 x 1 1.85 590 0.000645 0.00034 -0.0065 -0.0038 -0.0081 0.023 0.012 0.021

�� = 2 x 2 1.80 583 0.000603 0.00035 -0.0061 -0.0039 -0.0083 0.021 0.012 0.019

�� = 2 x 3 1.79 570 0.000598 0.00036 -0.0061 -0.0039 -0.0086 0.021 0.012 0.019

�� = 3 x 1 2.69 601 0.000838 0.00052 -0.0083 -0.0054 -0.014 0.029 0.018 0.024

�� = 3 x 2 2.69 603 0.000845 0.00051 -0.0084 -0.0053 -0.014 0.030 0.018 0.025

�� = 3 x 3 2.69 582 0.000842 0.00051 -0.0084 -0.0053 -0.013 0.030 0.018 0.024

�� = 4 x 1 3.71 634 0.00121 0.000651 -0.012 -0.0066 -0.019 0.043 0.023 0.037

�� = 4 x 2 3.55 594 0.00115 0.000617 -0.011 -0.0063 -0.018 0.041 0.022 0.034

�� = 4 x 3 3.79 522 0.00118 0.000687 -0.012 -0.0069 -0.018 0.042 0.024 0.035

�� = 6 x 1 5.45 599 0.00184 0.000972 -0.018 -0.0096 -0.026 0.065 0.034 0.058

�� = 6 x 2 5.42 604 0.00188 0.000963 -0.019 -0.0095 -0.026 0.066 0.034 0.060

�� = 6 x 3 5.43 573 0.00173 0.000964 -0.017 -0.0095 -0.023 0.061 0.034 0.063

Level

�

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

108

120

132

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

[mm

]

� y = 0.00207

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4


0

12

24

36

48

60

72

84

96

108

120

132H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

[mm

]

� y = 0.00207


L = 96 in.D = 48 in.

�� = 1

�� = 2

�� = 3

�� = 4

�� = 6

�� = 8

Hines et al. LPTPre-Yield Strains

L = 138 in.D = 54 in.

1/4 F'y

1/2 F'y

3/4 F'y

F'y

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB LPT�Hines et al� �� Average �exural strain proles�

��

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4


0

12

24

36

48

60

72

84

96

108

120

132

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

[mm

]


L = 138 in.D = 54 in.

-1/4 F'y

-1/2 F'y

-3/4 F'y

-F'y

� y = 0.00207

-0.0

04

-0.0

03

-0.0

02

-0.0

01

0.00

0

0.00

1

0.00

2

0.00

3

0.00

4


0

12

24

36

48

60

72

84

96

108

120

132H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

[mm

]

� y = 0.00207


L = 138 in.D = 54 in.

+1/4 F'y

+1/2 F'y

+3/4 F'y

+F'y

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB LPT�Hines et al� �� Pre�yield �exural strain proles�

��

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

108

120

132

Hei

ghtab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

[mm

]

Hines et al. LPTPost-Yield Strains

L = 138 in.D = 54 in.

�� = 1 x -1

�� = 2 x -1

�� = 3 x -1

�� = 3.6 x -1

�� = 4.6 x -1

�� = 6.5 x -1

� y = 0.00207

-0.0

6

-0.0

5

-0.0

4

-0.0

3

-0.0

2

-0.0

1

0.00

0.01

0.02

0.03

0.04

0.05

0.06


0

12

24

36

48

60

72

84

96

108

120

132H

eigh

tab

ove

foot

ing

(in.

)

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

[mm

]

� y = 0.00207 Hines et al. LPTPost-Yield Strains

L = 138 in.D = 54 in.

�� = 1 x +1

�� = 2 x +1

�� = 3 x +1

�� = 4.6 x +1

�� = 7.3 x +1

�� = 9.1 x +1

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB LPT�Hines et al� �� Post�yield �exural strain proles�

��

Table D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Longitudinal Direction� �Hines et al� �� Average experimental plasticity values�



1 x 1 bot 0.943 4870 0.808 -- 0.00023 -0.0028 0.0076 2.3 0.00016 0.248 11 11 5.6 2.26

2 x 1 bot 1.93 5148 1.58 4 0.00029 -0.0034 0.0098 2.9 0.00032 0.990 23 34.1 5.6 15.8

3 x 1 bot 2.89 5646 2.36 4 0.00050 -0.0052 0.018 5.13 0.00042 1.71 29 48.6 4.9 10.2

4 x 1 bot 3.78 5140 3.05 4 0.00062 -0.0063 0.022 6.31 0.00055 2.46 33 55.5 4.8 12.0

6 x 1 bot 5.80 5563 4.66 4 0.00094 -0.0091 0.034 9.54 0.00086 4.02 34 57.8 5.0 12.7

1 x 1 top 0.943 4870 0.808 -- 0.00023 -0.0028 0.0076 2.3 0.00016 0.248 11 11 5.6 -4.01

2 x 1 top 1.93 5148 1.58 3 0.00046 -0.0049 0.016 4.7 0.00039 0.990 18 25.5 5.6 0.2

3 x 1 top 2.89 5646 2.36 3 0.00050 -0.0052 0.018 5.10 0.00042 1.71 29 36.5 11.1 3.9

4 x 1 top 3.78 5140 3.05 3 0.00066 -0.0066 0.023 6.74 0.00059 2.46 30 43.5 8.5 5.5

6 x 1 top 5.80 5563 4.66 3 0.00097 -0.0094 0.035 9.88 0.00089 4.02 33 46.8 9.3 6.6



1 x 1 bot 24.0 6599 20.5 -- 9.0 -0.0028 0.0076 2.3 6.3 6.29 284 284 142 58

2 x 1 bot 48.9 6976 40.2 4 11 -0.0034 0.0098 2.9 12.5 25.2 575 865 142 401

3 x 1 bot 73.5 7650 59.9 4 20 -0.0052 0.018 5.13 16.7 43.4 741 1235 124 259

4 x 1 bot 96.0 6965 77.5 4 24 -0.0063 0.022 6.31 21.6 62.5 827 1409 123 305

6 x 1 bot 147 7537 118 4 37 -0.0091 0.034 9.54 33.8 102 862 1469 128 323

1 x 1 top 24.0 6599 20.5 -- 9 -0.0028 0.0076 2.3 6 6.3 284 284 142 -102

2 x 1 top 48.9 6976 40.2 3 18 -0.0043 0.014 4.7 15 25.2 467 649 142 6

3 x 1 top 73.5 7650 59.9 3 20 -0.0052 0.018 5.10 16.6 43.4 746 928 282 99

4 x 1 top 96.0 6965 77.5 3 26 -0.0066 0.023 6.74 23.2 62.5 769 1104 217 138

6 x 1 top 147 7537 118 3 38 -0.0094 0.035 9.88 35.1 102 830 1188 236 168

Imperial

Metric

��

0.00

00

0.00

01

0.00

02

0.00

03

0.00

04

0.00

05

0.00

06

0.00

07

0.00

08

0.00

09

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

Hei

ghtab

ove

foot

ing,

h(i

n.)

0 5 10 15 20 25 30 35

Curvature [ � rad/mm]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

h/L

� y= 0.000098 rad/in.

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

[mm

]

SFOBB DPTHines et al.

Longitudinal Peaks2L = 276 in.D = 54 in.

�� = 6

�� = 4

�� = 3

�� = 2

�� = 1

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Longitudinal Direction� �Hines et al� �� Average curvature proles�

��

Table D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Longitudinal Direction� �Hines et al� �� Peak curvature values�



+1/4 F'y (bot) 0.0914 0.0904 105 1205 -- -- -- -- -- -- --

-1/4 F'y (bot) -0.0739 -0.0723 -107 -1229 -- -- -- -- -- -- --

+1/2 F'y (bot) 0.204 0.198 210 2417 -- -- -- -- -- -- --

-1/2 F'y (bot) -0.180 -0.174 -213 -2450 -- -- -- -- -- -- --

+3/4 F'y (bot) 0.399 0.378 317 3640 0.000103 0.0000700 -- -- -- -- --

-3/4 F'y (bot) -0.368 -0.346 -319 -3668 -0.0000720 -0.0000490 -- -- -- -- --

+ F'y (bot) 0.666 0.617 422 4852 0.000163 0.000111 -- -- -- -- --

- F'y (bot) -0.624 -0.570 -425 -4888 -0.000127 -0.0000870 -- -- -- -- --

�� = 1 (bot) 0.923 0.789 481 5530 0.000239 0.000163 0.00018 4 11.2 21.9 11

�� = -1 (bot) -0.963 -0.827 -414 -4766 -0.000302 -0.000206 -0.00025 4 12.1 21.0 11

�� = 2 (bot) 1.86 1.51 593 6825 0.000636 0.000433 0.00033 4 14.3 23.8 23

�� = -2 (bot) -2.00 -1.66 -388 -4466 -0.000701 -0.000478 -0.00042 4 17.5 28.0 23

�� = 3 (bot) 2.78 2.25 594 6829 0.000879 0.000600 0.000467 4 19.3 31.4 28.7

�� = -3 (bot) -3.01 -2.47 -300 -3451 -0.000979 -0.000668 -0.000538 4 21.8 34.4 30.7

�� = 4 (bot) 3.72 3.00 586 6742 0.00114 0.000775 0.000589 4 21.7 35.1 32.6

�� = -4 (bot) -3.85 -3.10 -381 -4383 -0.00134 -0.000912 -0.000649 4 20.2 31.8 32.1

�� = 6 (bot) 5.57 4.49 605 6954 0.00171 0.00117 0.000904 4 23.0 36.7 33.1

�� = -6 (bot) -6.04 -4.83 -338 -3886 -0.00214 -0.001460 -0.000966 4 20.9 32.5 34.9

+1/4 F'y (top) 0.0914 0.0904 105 1205 -0.0000130 -0.0000090 -- -- -- -- --

-1/4 F'y (top) -0.0739 -0.0723 -107 -1229 0.0000100 0.0000070 -- -- -- -- --

+1/2 F'y (top) 0.204 0.198 210 2417 -0.0000300 -0.0000210 -- -- -- -- --

-1/2 F'y (top) -0.180 -0.174 -213 -2450 0.0000260 0.0000180 -- -- -- -- --

+3/4 F'y (top) 0.399 0.378 317 3640 -0.000065 -0.0000450 -- -- -- -- --

-3/4 F'y (top) -0.368 -0.346 -319 -3668 0.0000580 0.0000400 -- -- -- -- --

+ F'y (top) 0.666 0.617 422 4852 -0.000116 -0.0000790 -- -- -- -- --

- F'y (top) -0.624 -0.570 -425 -4888 0.000101 0.0000690 -- -- -- -- --

�� = 1 (top) 0.923 0.789 481 5530 -0.000166 -0.000113 0.00018 3 21.1 61.6 11

�� = -1 (top) -0.963 -0.827 -414 -4766 0.000137 0.000093 -0.00025 3 46.0 -- 11

�� = 2 (top) 1.86 1.51 593 6825 -0.000492 -0.000336 0.00033 3 19.9 34.8 23

�� = -2 (top) -2.00 -1.66 -388 -4466 0.000451 0.000308 -0.00042 3 30.2 52.6 23

�� = 3 (top) 2.78 2.25 594 6829 -0.000728 -0.000496 -0.000511 3 24.3 41.2 25.7

�� = -3 (top) -3.01 -2.47 -300 -3451 0.000596 0.000407 0.000489 3 39.9 69.6 34.1

�� = 4 (top) 3.72 3.00 586 6742 -0.00100 -0.000679 -0.000661 3 25.5 42.1 28.5

�� = -4 (top) -3.85 -3.10 -381 -4383 0.000925 0.000630 0.000660 3 31.2 51.8 31.5

�� = 6 (top) 5.57 4.49 605 6954 -0.00162 -0.00111 -0.00101 3 24.6 39.5 29.3

�� = -6 (top) -6.04 -4.83 -338 -3886 0.00139 0.000945 0.000927 3 34.6 57.0 36.5

Level

�

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

Hei

ghtab

ove

foot

ing,

h(i

n.)

-30

-20

-10

0 10 20 30


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

[mm

]

SFOBB (DPT) Hines et al.Longitudinal Peaks

2L = 276 in.D = 54 in.

�� = 2

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Longitudinal Direction� �Hines et al� �� Curvature proles at ��

��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

Hei

ghtab

ove

foot

ing,

h(i

n.)

-60

-40

-20

0 20 40 60


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

[mm

]


2L = 276 in.D = 54 in.

�� = 3


��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

Hei

ghtab

ove

foot

ing,

h(i

n.)

-60

-40

-20

0 20 40 60


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

[mm

]


2L = 276 in.D = 54 in.

�� = 4


��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

Hei

ghtab

ove

foot

ing,

h(i

n.)

-60

-40

-20

0 20 40 60


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

[mm

]


2L = 276 in.D = 54 in.

�� = 6


��

Table D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Longitudinal Direction� �Hines et al� �� Flexural strain values�

� F �b0 �b �c0 �c �'c �s0 �s �'s


+1/4 F'y (bot) 0.0826 106 -- -- -- -- -- -- -- --

+1/2 F'y (bot) 0.192 212 -- -- -- -- -- -- -- --

+3/4 F'y (bot) 0.383 318 0.0000875 -- -0.00138 -- -0.00175 0.0026 -- 0.0023

+ F'y (bot) 0.645 423 0.000145 -- -0.0020 -- -0.00263 0.0045 -- 0.0042

�� = 1 (bot) 0.943 448 0.000271 0.00023 -0.0032 -0.0028 -0.00414 0.0092 0.0076 0.0086

�� = 2 (bot) 1.93 491 0.000669 0.00029 -0.0067 -0.0034 -0.00779 0.024 0.0098 0.024

�� = 3 (bot) 2.89 447 0.000929 0.000502 -0.0090 -0.0052 -0.011 0.033 0.018 0.033

�� = 4 (bot) 3.78 484 0.00124 0.000619 -0.012 -0.0063 -0.014 0.044 0.022 0.044

�� = 6 (bot) 5.80 471 0.00193 0.000935 -0.019 -0.0091 -0.022 0.069 0.034 0.068

+1/4 F'y (top) 0.0826 106 0.0000115 -- -0.00035 -- -0.00055 0.000160 -- --

+1/2 F'y (top) 0.192 212 0.0000280 -- -0.00061 -- -0.00091 0.000640 -- 0.000414

+3/4 F'y (top) 0.383 318 0.0000615 -- -0.0011 -- -0.0015 0.0017 -- 0.001404

+ F'y (top) 0.645 423 0.000109 -- -0.0016 -- -0.00234 0.0033 -- 0.0028

�� = 1 (top) 0.943 448 0.000152 0.00023 -0.0021 -0.0028 -0.0030 0.0048 0.0076 0.0041

�� = 2 (top) 1.93 491 0.000472 0.00046 -0.0050 -0.0049 -0.0062 0.016 0.016 0.016

�� = 3 (top) 2.89 447 0.000662 0.000500 -0.0066 -0.0052 -0.0096 0.023 0.018 0.022

�� = 4 (top) 3.78 484 0.000960 0.000660 -0.0093 -0.0066 -0.015 0.034 0.023 0.030

�� = 6 (top) 5.80 471 0.00150 0.000969 -0.014 -0.0094 -0.028 0.054 0.035 0.043

Level

��

-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 0.004

Average pre-yield strains (�' s and�' c )

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

Hei

ghtab

ove

foot

ing,

h(i

n.)

-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 0.004

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

[mm

]



F'y

3/4F'y

1/2F'y

1/4F'y

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Longitudinal Direction� �Hines et al� �� Average pre�yield �exural strain proles�

��

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Average post-yield strains ( �' s and�' c )

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

Hei

ghtab

ove

foot

ing,

h(i

n.)

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

[mm

]



�� = 1

�� = 2

�� = 3

�� = 4

�� = 6

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Longitudinal Direction� �Hines et al� �� Average post�yield �exural strain proles�

��

-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 0.004

Pre-yield strains at positive peaks ( �' s and�' c )

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276H

eigh

tab

ove

foot

ing,

h(i

n.)

-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 0.004

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

[mm

]



+F'y

+3/4F'y

+1/2F'y

+1/4F'y

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Longitudinal Direction� �Hines et al� �� Pre�yield �exural strain proles at posi�tive peaks�

��

-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 0.004

Pre-yield strains at negative peaks ( �' s and�' c )

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276H

eigh

tab

ove

foot

ing,

h(i

n.)

-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 0.004

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

[mm

]



-F'y

-3/4F'y

-1/2F'y

-1/4F'y

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Longitudinal Direction� �Hines et al� �� Pre�yield �exural strain proles at neg�ative peaks�

��

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Post-yield strains at positive peaks ( �' s and�' c )

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276H

eigh

tab

ove

foot

ing,

h(i

n.)

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

[mm

]



�� = +1

�� = +2

�� = +3

�� = +4

�� = +6

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Longitudinal Direction� �Hines et al� �� Post�yield �exural strain proles atpositive peaks�

��

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Post-yield strains at negative peaks ( �' s and�' c )

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276H

eigh

tab

ove

foot

ing,

h(i

n.)

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

[mm

]



�� = -1

�� = -2

�� = -3

�� = -4

�� = -6

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Longitudinal Direction� �Hines et al� �� Post�yield �exural strain proles atnegative peaks�

��

Table D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Transverse Direction� �Hines et al� �� Average experimental plasticity values�



1 x 1 2.17 10184 2.00 -- 0.00015 -0.0023 0.0091 3.0 0.00011 0.393 11 11 5.6 1.08

2 x 1 4.36 10762 3.97 4 0.000264 -0.0035 0.017 5.18 0.000222 2.27 33.4 47.1 9.86 9.19

3 x 1 6.53 10688 5.95 4 0.000404 -0.0049 0.026 7.92 0.000362 4.26 38.5 61.3 7.83 7.97

4 x 1 8.70 10573 7.92 6 0.000490 -0.0058 0.031 9.60 0.000448 6.25 45.6 77.8 6.66 9.44

6 x 1 13.0 10463 11.8 6 0.000716 -0.0081 0.046 14.0 0.000675 10.1 48.9 85.3 6.28 10.2



1 x 1 55.2 13799 50.8 -- 6.1 -0.0023 0.0091 3.0 4.5 9.98 284 284 142 27

2 x 1 111 14583 101 4 10.4 -0.0035 0.017 5.18 8.75 57.6 848 1196 250 233

3 x 1 166 14482 151 4 15.9 -0.0049 0.026 7.92 14.3 108 978 1558 199 202

4 x 1 221 14327 201 6 19.3 -0.0058 0.031 9.60 17.7 159 1158 1977 169 240

6 x 1 331 14177 299 6 28.2 -0.0081 0.046 14.0 26.6 257 1243 2167 159 258

Metric

��

0.00

00

0.00

01

0.00

02

0.00

03

0.00

04

0.00

05

0.00

06

0.00

07

0.00

08

0.00

09

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

288

300

Hei

ghtab

ove

foot

ing,

h(i

n.)

0 5 10 15 20 25 30 35


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

h/L

� y= 0.000051 rad/in.

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

7200

7500

[mm

]


Transverse PeaksL = 306 in.D = 84 in.

�� = 6

�� = 4

�� = 3

�� = 2

�� = 1

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Transverse Direction� �Hines et al� �� Average curvature proles�

��

Table D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Transverse Direction� �Hines et al� �� Peak curvature values�



+1/4 F'y 0.188 0.187 96.0 2448 0.0000110 0.0000080 -- -- -- -- --

-1/4 F'y -0.224 -0.225 -96.1 -2450 -0.0000090 -0.0000060 -- -- -- -- --

+1/2 F'y 0.507 0.497 192 4897 0.0000310 0.0000210 -- -- -- -- --

-1/2 F'y -0.548 -0.542 -192 -4898 -0.0000300 -0.0000210 -- -- -- -- --

+3/4 F'y 0.952 0.934 288 7345 0.0000610 0.0000420 -- -- -- -- --

-3/4 F'y -0.997 -0.967 -288 -7345 -0.0000620 -0.0000420 -- -- -- -- --

+ F'y 1.77 1.68 384 9784 0.000107 0.0000730 -- -- -- -- --

- F'y -1.69 -1.59 -364 -9280 -0.000102 -0.0000700 -- -- -- -- --

�� = 1 2.17 2.00 379 9674 0.000184 0.000126 0.00018 -- 12.8 21.5 11

�� = -1 -2.18 -2.00 -419 -10693 -0.000153 -0.000104 -0.00013 -- 12.0 21.9 11

�� = 2 4.35 3.98 377 9625 0.000501 0.000341 0.000270 4 20.0 31.2 34.7

�� = -2 -4.36 -3.95 -467 -11899 -0.000432 -0.000295 -0.000259 4 20.9 33.2 32.0

�� = 3 6.54 5.93 364 9289 0.000736 0.000502 0.000394 4 24.3 37.3 40.8

�� = -3 -6.53 -5.97 -474 -12086 -0.000608 -0.000415 -0.000414 4 27.6 43.2 36.3

�� = 4 8.69 7.94 342 8717 0.000962 0.000656 0.000473 6 26.7 40.8 48.9

�� = -4 -8.72 -7.89 -487 -12430 -0.000788 -0.000537 -0.000506 6 31.0 48.3 42.4

�� = 6 13.0 11.8 325 8299 0.00148 0.00101 0.000671 6 27.4 41.7 53.8

�� = -6 -13.1 -11.7 -495 -12627 -0.00116 -0.000793 -0.000762 6 34.1 52.7 44.6

Level

��

-0.0

010

-0.0

008

-0.0

006

-0.0

004

-0.0

002

0.00

00

0.00

02

0.00

04

0.00

06

0.00

08

0.00

10

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

288

300

Hei

ghtab

ove

foot

ing,

h(i

n.)

-30

-20

-10

0 10 20 30


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

7200

7500

[mm

]

SFOBB (DPT) Hines et al.Transverse Peaks

L = 306 in.D = 84 in.

�� = 2

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Transverse Direction� �Hines et al� �� Curvature proles at ��

��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

288

300

Hei

ghtab

ove

foot

ing,

h(i

n.)

-60

-40

-20

0 20 40 60


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

7200

7500

[mm

]


L = 306 in.D = 84 in.

�� = 3


��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

288

300

Hei

ghtab

ove

foot

ing,

h(i

n.)

-60

-40

-20

0 20 40 60


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

7200

7500

[mm

]


L = 306 in.D = 84 in.

�� = 4


��

-0.0

020

-0.0

015

-0.0

010

-0.0

005

0.00

00

0.00

05

0.00

10

0.00

15

0.00

20

Curvature (rad/in.)

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

288

300

Hei

ghtab

ove

foot

ing,

h(i

n.)

-60

-40

-20

0 20 40 60


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

7200

7500

[mm

]


L = 306 in.D = 84 in.

�� = 6


��

Table D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Transverse Direction� �Hines et al� �� Flexural strain values�

� F �b0 �b �c0 �c �'c �s0 �s �'s


+1/4 F'y 0.2061 96 0.000010 -- -0.00043 -- -0.00081 0.00033 -- -0.00003

+1/2 F'y 0.527 192 0.000031 -- -0.00084 -- -0.0014 0.0015 -- 0.0010

+3/4 F'y 0.975 288 0.0000615 -- -0.0013 -- -0.0020 0.0033 -- 0.0027

+ F'y 1.73 374 0.000105 -- -0.0018 -- -0.0030 0.0062 -- 0.0050

�� = 1 2.17 399 0.000169 0.00015 -0.0025 -0.0023 -0.0040 0.010 0.0091 0.0090

�� = 2 4.36 422 0.000467 0.00026 -0.0055 -0.0035 -0.0088 0.030 0.017 0.027

�� = 3 6.53 419 0.000672 0.000404 -0.0077 -0.0049 -0.012 0.043 0.026 0.039

�� = 4 8.70 415 0.00088 0.000490 -0.0099 -0.0058 -0.016 0.057 0.031 0.052

�� = 6 13.0 410 0.00132 0.000716 -0.015 -0.0081 -0.024 0.085 0.046 0.078

Level

��

-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 0.004

Average pre-yield strains ( �' s and�' c )

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

288

300

Hei

ghtab

ove

foot

ing,

h(i

n.)

-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 0.004

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

7200

7500

[mm

]


Longitudinal PeaksL = 306 in.D = 84 in.

F'y

3/4F'y

1/2F'y

1/4F'y

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Transverse Direction� �Hines et al� �� Average pre�yield �exural strain proles�

��

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Average post-yield strains ( �' s and�' c )

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

288

300

Hei

ghtab

ove

foot

ing,

h(i

n.)

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

7200

7500

[mm

]



�� = 1

�� = 2

�� = 3

�� = 4

�� = 6

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Transverse Direction� �Hines et al� �� Average post�yield �exural strain proles�

��

-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 0.004

Pre-yield strains at positive peaks ( �' s and�' c )

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

288

300

Hei

ghtab

ove

foot

ing,

h(i

n.)

-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 0.004

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

7200

7500

[mm

]



+F'y

+3/4F'y

+1/2F'y

+1/4F'y

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Transverse Direction� �Hines et al� �� Pre�yield �exural strain proles at positivepeaks�

��

-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 0.004

Pre-yield strains at negative peaks ( �' s and�' c )

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

288

300

Hei

ghtab

ove

foot

ing,

h(i

n.)

-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 0.004

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

7200

7500

[mm

]



-F'y

-3/4F'y

-1/2F'y

-1/4F'y

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Transverse Direction� �Hines et al� �� Pre�yield �exural strain proles at negativepeaks�

��

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Post-yield strains at positive peaks ( �' s and�' c )

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

288

300

Hei

ghtab

ove

foot

ing,

h(i

n.)

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

7200

7500

[mm

]



�� = +1

�� = +2

�� = +3

�� = +4

�� = +6

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Transverse Direction� �Hines et al� �� Post�yield �exural strain proles at positivepeaks�

��

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Post-yield strains at negative peaks ( �' s and�' c )

0

12

24

36

48

60

72

84

96

108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

288

300

Hei

ghtab

ove

foot

ing,

h(i

n.)

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

h/L

� y

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

3900

4200

4500

4800

5100

5400

5700

6000

6300

6600

6900

7200

7500

[mm

]



�� = -1

�� = -2

�� = -3

�� = -4

�� = -6

Figure D�� Hollow rectangular pier with conned corner elements� SFOBB DPT�Transverse Direction� �Hines et al� �� Post�yield �exural strain proles at nega�tive peaks�

��

��

References

�� B�G� Ang� M�J�N� Priestley� and R� Park� Ductility of Reinforced Concrete Bridge

Piers Under Seismic Loading� Department of Civil Engineering Research Report

�� University of Canterbury� Christchurch� New Zealand� February ��

�� ATC� Improved Seismic Design Criteria for California Bridges� Applied Tech�

nology Council� Redwood City� California� ��

�� A�L�L� Baker� Recent Research in Reinforced Concrete and its Application to

Design� Institution of Civil Engineers� Structural and Building Engineering Di�

vision� �� February ��

�� A�L�L� Baker and A�M�N� Amarakone� Inelastic Hyperstatic Frames Analysis� In

Proceedings of the International Symposium on Flexural Mechanics of Reinforced

Concrete� ASCE�ACI� pages �� Miami� Florida� ��

�� R�H� Brown and J�O� Jirsa� Reinforced Concrete Beams under Reversed Loading�

ACI Journal� �� May ��

�� Caltrans� Seismic Design Criteria� Version �� California Department of Trans�

portation� Sacramento� California� � ��

�� Y�H� Chai� M�J�N� Priestley� and F� Seible� Flexural Retrot of Circular Rein�

forced Concrete Bridge Columns by Steel Jacketing� Structural Systems Research

Project �� University of California� San Diego� ��

�� W�W�L� Chan� The Ultimate Strength and Deformation of Plastic Hinges in

Reinforced Concrete Frameworks� Magazine of Concrete Research� ��

�� November ��

�� W�G� Corley� Rotational Capacity of Reinforced Concrete Beams� ASCE Journal

of the Structural Division� ��ST�� October ��

��

�� B�E� Davey and R� Park� Reinforced Concrete Bridge Piers Under Seismic Load�

ing� Department of Civil Engineering Research Report �� University of Can�

terbury� Christchurch� New Zealand� February ��

�� W�D� Gill� Ductility of Rectangular Reinforced Concrete Columns with Axial

Load� Department of Civil Engineering Research Report �� University of

Canterbury� Christchurch� New Zealand� February ��

�� E�M� Hines� A� Dazio� and F� Seible� Seismic Performance of Hollow Rectangular

Reinforced Concrete Piers with Highly�Conned Corner Elements� Phase III�

Web Crushing Tests� Structural Systems Research Project � � �� University

of California� San Diego� � ��

�� E�M� Hines� A� Dazio� and F� Seible� Structural Testing of the San Francisco�

Oakland Bay Bridge East Span Skyway Piers� Structural Systems Research

Project � � �� University of California� San Diego� � ��

�� E�M� Hines� F� Seible� and M�J�N� Priestley� Seismic Performance of Hollow

Rectangular Reinforced Concrete Piers with Highly�Conned Corner Elements�

Phase I� Flexural Tests� and Phase II� Shear Tests� Structural Systems Research


�� Y�D� Hose� F� Seible� and M�J�N� Preistley� Strategic Relocation of Plastic Hinges

in Bridge Columns� Structural Systems Research Project �� University of

California� San Diego� September ��

�� D�C� Kent� Flexural Members with Con�ned Concrete� PhD thesis� University

of Canterbury� Christchurch� New Zealand� ��

�� J�B� Mander� Seismic Design of Bridge Piers� PhD thesis� University of Canter�

bury� Christchurch� New Zealand� ��

�� J�B� Mander� M�J�N� Priestley� and R� Park� Observed Stress�Strain Behavior of

Conned Concrete� ASCE Journal of Structural Engineering� ��

August ��

�� I�R�M� Munro� R� Park� and M�J�N� Priestley� Seismic Behavior of Reinforced

Concrete Bridge Piers� Department of Civil Engineering Research Report ��

University of Canterbury� Christchurch� New Zealand� February ��

��

�� K�H� Ng� M�J�N� Priestley� and R� Park� Seismic Behavior of Circular Reinforced

Concrete Bridge Piers� Department of Civil Engineering Research Report ��

University of Canterbury� Christchurch� New Zealand� ��

�� R� Park� Theorisation of Structural Behavior with a View to Dening Resistance

and Ultimate Deformability� Bulliten of the New Zealand Society for Earthquake

Engineering� �� June ��

�� R� Park and T� Paulay� Reinforced Concrete Structures� Wiley� New York� ��

�� T� Paulay and M�J�N� Priestley� Seismic Design of Reinforced Concrete and

Masonry Buildings� Wiley� New York� ��

�� R�T� Potangaroa� M�J�N� Priestley� and R� Park� Ductility of Spirally Reinforced

Concrete Columns Under Seismic Loading� Department of Civil Engineering

Research Report �� University of Canterbury� Christchurch� New Zealand�

��

�� M�J�N� Priestley and R� Park� Strength and Ductility of Bridge Substructures�

RRU Bulliten �� Road Research Unit� National Roads Board� Wellington� New

Zealand� ��

�� M�J�N� Priestley and R� Park� Bridge Columns under Seismic Loading� ACI

Structural Journal� �� January�February ��

�� M�J�N� Priestley and F� Seible� Seismic Assessment and Retrot of Bridges�

Structural Systems Research Project �� University of California� San Diego�

July ��

�� M�J�N� Priestley� F� Seible� and G� Benzoni� Seismic Performance of Circu�

lar Columns with Low Longitudinal Steel Ratios� Structural Systems Research


�� M�J�N� Priestley� F� Seible� and Calvi G�M� Seismic Design and Retro�t of

Bridges� Wiley� New York� ��

�� H�A� Sawyer� Design of Concrete Frames for Two Failure States� In Proceedings

of the International Symposium on Flexural Mechanics of Reinforced Concrete�

ASCE�ACI� pages � �� Miami� Florida� ��

��

�� B�D� Scott� R� Park� and M�J�N� Priestley� Stress�Strain Behavior of Concrete

Conned by Overlapping Hoops at Low and High Strain Rates� ACI Journal�

�� January�February ��

�� H� Tanaka and R� Park� E�ect of Lateral Conning Reinforcement on the Ductile

Behavior of Reinforced Concrete Columns� Department of Civil Engineering

Research Report � � �� University of Canterbury� Christchurch� New Zealand�

��

�� S�P� Timoshenko� Theory of Elastic Stability� McGraw�Hill Inc�� New York� ��

�� T�Y�Lin International and Mo�at ! Nichol Engineers� San Francisco�Oakland

Bay Bridge East Span Seismic Safety Project� Design Criteria� Draft� ��

Revision � T�Y�Lin International Mo�at ! Nichol Engineers� San Francisco�

California� ��

�� R� Yamashiro� Moment�Rotation Characteristics of Reinforced Concrete Mem�

bers Subjected to Bending� Shear and Axial Load� PhD thesis� University of

Illinois� Urbana� ��

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