Earthquake Engineering and Structural Dynamics

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2011; 40:315337Published online 30 July 2010 in Wiley Online Library (wileyonlinelibrary.com/journal/eqe)DOI: 10.1002/eqe.1030

Hysteretic shearflexure interaction model of reinforced concretecolumns for seismic response assessment of bridges

Shi-Yu Xu and Jian Zhang,,

Department of Civil & Environmental Engineering, University of California, Los Angeles, CA, U.S.A.

SUMMARY

This paper presents the methodology, model description, and calibration as well as the application of acoupled hysteretic model to account for nonlinear shearflexure interactive behavior of RC columns underearthquakes, a critical consideration for seismic demand evaluation of bridges. The proposed hystereticmodel consists of a flexure and a shear spring coupled at element level, whose nonlinear behavior aregoverned by the primary curves and a set of loading/unloading rules to capture the pinching, stiffnesssoftening, and strength deterioration of columns due to combined effects of axial load, shear force, andbending moment. The shearflexure interaction (SFI) is considered both at section level when theoreticallygenerating the primary curves and at element level through global and local equilibrium. The model isimplemented in a displacement-based finite element framework and calibrated against a large number ofcolumn specimens from static cyclic tests to dynamic shake table tests. The numerical predictions by theproposed model show very good agreement with experimental data for both flexure- and shear-dominatedcolumns. The application of the proposed model for seismic assessment of bridges has been successfullydemonstrated for a realistic prototype bridge. The factors affecting the SFI and its significance on bridgesystem response are also discussed. Copyright 2010 John Wiley & Sons, Ltd.

Received 25 July 2009; Revised 6 May 2010; Accepted 13 May 2010

KEY WORDS: shearflexure interaction; hysteretic rule; primary curve; seismic response; column; bridge

1. INTRODUCTION

Reinforced-concrete columns are in general the most critical components of highway bridges.They play very important roles in overall structural performance of bridges and their failures oftenresult in bridge collapse or expensive repair cost. A significant number of bridges in current bridgeinventory, built and constructed before the introduction of modern seismic codes in the 1970s, arevulnerable to damage and collapse during major earthquakes, as evidenced by the observed severedamage and collapse of several bridges in the recent earthquakes [13]. One major deficiency ofthose older columns is the insufficient transverse reinforcement which will result in degradationof shear and axial load capacity as well as impacting the nonlinear flexural behavior of columnswhen subject to cyclic lateral loads [4].

The bridge columns are normally under the complex load combinations of bending, shear, axialload, and torsion due to the multi-directional earthquake motions and constraints of structuralor geometric configurations (e.g. short columns, uneven spans, skewed or curved bridges, etc.).

Correspondence to: Jian Zhang, Department of Civil & Environmental Engineering, University of California, LosAngeles, CA, U.S.A.

E-mail: [email protected] Student Researcher.Assistant Professor.

Copyright 2010 John Wiley & Sons, Ltd.

316 S.-Y. XU AND J. ZHANG

Under the combined loadings, the columns inevitably experience considerable nonlinear inelasticbehavior, involving yielding, inelastic deformation, strength, stiffness degradation, etc. Therefore,the bridge responses need to be evaluated more realistically with the consideration of the materialdamage (including the strength deterioration and stiffness degrading due to increasing loadingcycles, as well as pinching behavior resulted from the crack opening and closing during loadingreversals) and the axialshearflexure interaction in columns since neglecting the combined effectswill likely result in overestimation of lateral load capacity and underestimation of lateral deforma-tion demand.

This paper presents the methodology, model description, and calibration as well as the appli-cation of a coupled hysteretic model to account for nonlinear shearflexure interactive behaviorof RC columns under constant axial load when subjected to horizontal earthquake excitations.The proposed hysteretic model consists of a flexure and a shear spring coupled at element level,whose nonlinear behaviors are governed by their respective primary curves and a set of improvedloading/unloading rules to capture the pinching, stiffness softening, and strength deterioration ofcolumns due to combined effects of axial load, shear force, and bending moment. The shearflexureinteraction (SFI) is considered both at section level when theoretically generating the primarycurves and at element level through local and global equilibrium. The model is implemented asan user element (UEL) in a displacement-based finite element program, ABAQUS [5], and cali-brated against a large number of column specimens from static cyclic tests to dynamic shake tabletests showing very good agreement with experimental data for both flexure- and shearflexure-dominated columns. The proposed model is computationally effective and reliable in conductingrealistic seismic assessment of bridges. The significance of considering material damage and SFIis demonstrated for a realistic prototype bridge.

2. HYSTERETIC SFI MODEL DESCRIPTION

2.1. The need for nonlinear SFI model

Ozcebe and Saatcioglu [6] reported that shear displacement can be significant even if a RC memberis not governed by shear failure. They also indicated that RC members with higher shear strengththan flexural strength do not guarantee an elastic behavior in shear deformation. Based on theirobservations, RC members controlled by flexural behavior (as is the case in most of the current RCdesign codes) may still have significant shear displacement which goes into the inelastic stage andthus should not be left ignored. ElMandooh and Ghobarah [7] further pointed out that the variationof axial force in RC columns will cause significant change in the lateral hysteretic momentcurvature relationship and consequently the overall structural behavior. These observations dictatethe importance of including nonlinear SFI in the analyses of RC columns with the considerationof varying axial load effects.

The numerical models for RC columns in the past have focused primarily on the inelastic flexuralbehavior and usually decoupled with axial, shear, and torsion behavior. The shear and torsionbehavior are often modeled with a linear decoupled spring in many analysis software packages. Forexample, the typical concentrated plastic hinge model for flexural behavior uses predefined sectionalresponse independent of columns aspect ratio (which reflects the level of SFI in the column), theapplied axial load (which changes the lateral response), and the loading history. On the other hand,the classical flexural fiber section model [8] as implemented in OpenSees program [9], althoughcouples the axial and flexural behavior at material level, does not include the shear deformation andshear capacity degrading thus incapable of accounting for the accumulated damage resulted fromthe propagating flexural and shear cracks in columns under cyclic loads. Both of these approachesappear to be inadequate for modeling the columns dominated in shear or shearflexure behaviorsince the stiffness degradation, pinching behavior, and strength deterioration as a result of SFI arenot captured [10]. Inelastic shear behavior has been included with the nonlinear flexural behaviorin several previous studies ([1114] among others). Improvements have been made to include theSFI effects for fiber-based element [1518] and Macro-element model [19]. They represent a few

Copyright 2010 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2011; 40:315337DOI: 10.1002/eqe

HYSTERETIC SHEARFLEXURE INTERACTION MODEL 317

advanced models to account for shear effects where different conceptual backgrounds (bi-axialfiber constitutive model, link shear and flexure through kinematic assumptions, etc.) and solutionstrategies (displacement or flexibility based) as well as varying implementation complexity andcalibration requirements are involved. A detailed state-of-art review on fiber elements is offeredby Ceresa et al. [20]. Nevertheless, this paper is focused on the concentrated plastic hinge-typemodel that can be used in displacement-based finite element programs. Although empirical andapproximate, this type of model is relatively easy to be implemented and computationally efficientfor the purpose of conducting seismic assessment of bridges compared to the flexibility-basedspread plastic hinge models and the fiber models.

Similar to the well-known interactive axialflexure capacities expressed in terms of PM inter-action curve, interactive relationship also exists between the shear and flexure capacities of RCcolumns. In a cantilever column, although the axial and shear forces along the element might be thesame, the induced bending moment at each section is different. Therefore, the derived nonlinearmoment-to-curvature (M) and nonlinear shear force-to-shear strain (V ) curves are indeeddifferent at each section due to the varying combinations of axial, shear, and moment loads. Theeffect of M/V ratio on the shear capacity of the RC sections has been experimentally reportedearlier by Ozcebe [21]. In general, larger M/V ratios will result in larger moment capacity butsmaller shear capacity for a given cross section under constant axial load. As a result, the bottomsection of cantilever beam (largest moment) becomes the critical section, whose average bendingmoment-to-shear force ratio (M/V ) and associated M and V curves will govern the responseof column. This is the main reason why columns with same cross section but various aspect ratios(H/D) display quite different behaviors and failure modes. Based on this understanding of thecolumn behavior and to mend for the deficiencies in the current models, a computationally effec-tive analytical approach based on concentrated plastic hinge concept is developed to include thenonlinear SFI of columns. As pointed out by Martinelli [16], even if shear effects actually spreadthroughout the element, the SFI is more pronounced in limited zones, for example, the fixed-endregion in a cantilever.

2.2. The methodology for considering SFI of columns

In this study, the proposed scheme couples the axial force, shear force, and the bending moment atthe section level, similar to the advanced fiber section formulation, pairs with modified hystereticrules, and produces much improved results. The basic idea is to incorporate the interaction betweenaxial load, shear force, and bending moment into the primary curves as the overall responses resultedfrom the combined stress and strain fields due to various load combinations, and subsequently toenforce the interaction by the requirements of local and global equilibrium at any given time.

The total primary curve of a column is equivalent to its monotonic pushover curve by consideringthe combined effects of axial, shear, and moment loads, and it defines the envelope of hysteresisloops of the column. The total primary curves are best derived empirically from column testprograms, or as an alternative, theoretically by applying the modified compression field theory(MCFT) [22], which takes into account the compatibility condition, equilibrium condition, andnonlinear stressstrain relationship when subject to combined axial, shear, and bending loads.After separating the shear displacement from the flexural deformation, the total primary curve isbroken into a flexural and a shear primary curve. The decoupled primary curves then serve as theboundaries for the nonlinear flexural and shear springs (described below) in the proposed scheme.

Figure 1 describes the general procedure of the proposed SFI scheme where MCFT is used toobtain the primary curves for flexural and shear springs in the coupled hysteretic SFI model. Giventhe geometry of the target RC section, the reinforcement configuration, the material properties,and the applied external loads, MCFT can yield the moment-to-curvature (M) and the shearforce-to-shear strain (V ) relationships of the section subject to the combined loading conditions.The software Response-2000 [23], which has incorporated the MCFT theory, is used in this studyto generate these curves, and one of the column specimens (see Table I below) used for modelcalibration in this paper is presented here as an example. Figure 2(a) shows the average momentto shear force ratio (M/V ratio) in each section along the column height of the PEER-93 column



M

h

dy

V

N

yi

V

MCFT

V

V

M

M

+

+

s

V

f

M

M

S-UEL

F-UEL

Integrate curvature and shear strain to get the tip displacement of the cantilever column.

= { i dy yi + i dy } Flexural deformation Shear deformation = f + s = h + s

F-UEL

S-UEL

SSI spring

FNDN

DECK

S-UEL

F-UEL

Rigid Column

Break a RC column at its inflection point into two cantilever columns.

Input the V- s curve to shear spring (S-UEL) & M- curve to flexure spring (F-UEL).

Section profileLoading conditions

Figure 1. Implementation of the SFI scheme.

specimen [24]; Figures 2(d, e) display the corresponding V and M curves in each section.It can be observed that with M/V ratio increasing toward the column base, the shear capacitiesof the sections decrease whereas the moment capacities increase until reaching the pure bendinglimit, agreeing with the previous experimental observation [21]. The flexural deformation (f) andshear deformation (s) can then be obtained by integrating the curvature and shear strain in eachsection along the column length, and results are shown in Figure 2(b). Subsequently, one canobtain the bending moment-to-rotation angle (M) and shear force-to-shear displacement (V s)relationships, and at the same time track the shear-to-total displacement ratio until it reaches theultimate strength (see Figure 2(c)). The M and V s relationships can be regarded as the primarycurves for the flexure and shear springs, respectively. If the inflection point of a RC column isknown (or simply assumed to be at the mid-height of the column), the column can be broken atits inflection point into two cantilever columns and simulated by a rigid bar and a combination offlexure springs (F-UELs) and shear springs (S-UELs), as demonstrated in Figure 1.

The MCFT theory, however, has the following limitations. First of all, it is a force-based approachwhich will stop once the peak strength of the section is reached. Second, although in generalMCFT is considered as capable of accounting for the SFI at section level due to combined loads, itmay not be sufficient for extremely short columns with aspect ratio equal to 2.5 or smaller since ittends to underestimate the ultimate capacity and overestimate the stiffness. Moreover, in essenceit is a sectional analysis tool and thus it does not include the bar slip deformation. To estimatethe yielding platform and softening branch of the primary curves or the response of extremelyshort columns, empirical equations [25] for flexural and shear displacement (or experimental dataif available) can be used as alternatives. Contribution of the bar slip deformation can be added tothe flexural primary curve by applying any of existing prediction models [26]; or it can be singledout by adding a pullout sub-element [27] in series with the flexure spring.

To accurately predict column responses under earthquake loadings, robust hysteretic models areneeded to simulate the concrete damage, including strength deterioration, stiffness degrading, andpinching behavior. It is often difficult to simulate these damages using fiber section models sinceit involves advanced plasticity theory which requires multiple yielding surfaces, appropriate flowrules, return mapping algorithm, and damage mechanics for the concrete material. Therefore, thestudy builds upon the pioneering shear hysteretic model by Ozcebe and Saatcioglu [6] and well-known flexure hysteretic model by Takeda et al. [28] to develop two improved hysteretic models forflexure and shear responses, respectively. To minimize the complexities of computational coding



Tabl

eI.

Geo

met

ry,

rein

forc

emen

t,m

ater

ial

prop

ertie

s,an

dap

plie

dlo

adof

exam

ined

colu

mns

.

Col

umn

Col

umn

Num

ber

Lon

gitu

d.T

rans

vers

eL

ongi

tud.

Tra

nsve

rse

Axi

alA

xial

Col

umn

size

heig

htof

stee

lst

eel

stee

lre

info

rce.

rein

forc

e.fy

fc

load

load

inde

x(m

m)

(mm

)re

bars

diam

eter

(mm

)di

amet

er(m

m)

ratio

(%)

ratio

(%)

(MPa

)(M

Pa)

(kN

)ra

tio(%

)

TP-

021

400

circ

.13

5012

166

1.89

0.26

374

30.0

185

5.0

TP-

031

4004

0013

5020

136

1.58

0.79

374

22.9

470

12.8

TP-

032

4004

0013

5020

136

1.58

0.79

374

23.0

170

4.6

PEE

R-5

315

20ci

rc.

9140

2543

15.9

1.99

0.63

475

35.8

4450

7.0

PEE

R-5

415

20ci

rc.

4570

2543

19.1

1.99

1.49

475

34.3

4450

7.0

PEE

R-6

225

02

5025

08

9.5

8.9

1.01

4.27

375

26.5

322

19.4

PEE

R-9

330

5ci

rc.

1372

219.

54

2.00

0.93

448

29.0

200

9.4

PEE

R-1

0561

0ci

rc.

914.

524

12.7

6.4

1.04

0.17

462

30.0

503

6.0

PEE

R-1

2169

6ci

rc.

1828

.828

19.0

56.

42.

730.

8944

134

.591

1.84

9.0

PEE

R-1

2269

6ci

rc.

4876

.828

19.0

56.

42.

730.

8944

134

.591

1.84

9.0

PEE

R-1

5840

6.4

circ

.18

54.2

1212

.74.

51.

170.

5345

8.5

36.5

00

PEE

R-1

6040

6.4

circ

.10

47.7

514

12.7

4.5

1.37

0.10

458.

534

.70

0PE

ER

-163

609.

6ci

rc.

1219

.220

15.8

754.

91.

360.

1345

429

.818

.80.

0U

NR

-9F1

406.

4ci

rc.

1828

.820

12.7

6.35

1.95

1.00

448

37.4

355.

8610

.0U

NR

-9S1

406.

4ci

rc.

1219

.216

19.0

56.

353.

500.

9244

837

.035

5.86

10.0

Bri

dge

#412

19.2

circ

.60

96.0

(ful

lh.

)34

35.8

1(#

11)

15.9

(#5)

2.93

0.63

414

27.6

3825

.511

.87



0 5 10 150

10

20

30

40

50

60

70

Total Displacement (mm)

She

ar (

kN)

total displ.

shear displ.

flexural displ.

0 5 10 150.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

Total Displ. (mm)

She

ar-t

o-To

tal D

ispl

. Rat

io

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4 M/ V =0.076(m)1

M/ V =0.229(m)2

M/ V =0.381(m)3

M/ V =0.534(m)4

M/ V =0.686(m)5

M/ V =0.838(m)6

M/ V =0.991(m)7

M/ V =1.143(m)8

M/ V =1.296(m)9

M/ V ratio

Col

umn

Hei

ght (

m)

0 2 4 6 8 10 120

50

100

150

200

250

300

Shear Strain (mm/m)

She

ar (

kN)

V-1

V-2

V-3

V-4

V-5

V-6

V-7

V-8

V-9

0 5 10 15 20 25 300

30

60

90

Curvature (rad/km)

Mom

ent (

kN-m

)

M-1

M-2

M-3

M-4

M-5

M-6

M-7

M-8

M-9

(a) (b) (c)

(d) (e)

Figure 2. The primary curves for column specimen PEER-93 (P/P0=9.44%) derived by MCFT:M/V ratio (a), shear forceshear strain (d), and momentcurvature (e) relationships for eachsection along the column height; shear and flexural contribution to total primary curve (b) and

shear-to-total displacement ratio (c).

and implementation for the flexure and shear plastic hinge subroutines, it is preferred to composethe subroutines for both springs using the same numerical model framework, with the least amountof modification made to each spring to account for its specific unloading/reloading characteristics.For this reason, the Ozcebe and Saatcioglus shear model is selected as the basis, combined withthe Takeda model and some improvements, to make up two new hysteretic models for the flexuraland shear responses as outlined later.

2.3. Description and model defects of hysteretic shear model by Ozcebe and Saatcioglu [6]

Ozcebe and Saatcioglus hysteretic shear model was originally established by statistic regres-sion of experimental data, and it is revised and re-calibrated in this study to allow for possiblelarger ductility levels, to improve numerical stability, and to expand its application to the flexuralresponses. Ozcebe and Saatcioglus shear model consists of three major parts: primary curve,unloading branches, and reloading branches. The model is illustrated in Figure 3 and summarizedby the following rules:

1. Initial loading and uncracked unloading/reloading follow the primary curve.2. For pre-yield cracked unloading from points above the cracking load (e.g. AB; ST ),

the unloading stiffness is given by Equation (1), and from points below cracking load (e.g.CD; LM) equal to k1. The reference stiffness k1 is the slope connecting origin to thecrack point, as shown in Figure 5(b).

k=k1 crycr (k1k2) (1)



B

E

FA

C

D

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

Shear Displacement

Shea

r Fo

rce

Vcr

Vy

maximum peak (m,Vm)

hardening reference point(m,Vm)

previous peak (p,Vp)

pinching reference point (p,Vp)

G

H

Figure 3. Illustration of the shear hysteretic model by Ozcebe and Saatcioglu [6].

3. Unloading stiffness of post-yield unloading branches above the cracking load (e.g. IJ ;OP) is given by Equation (2) and below (e.g. JK ; PQ) by Equation (3). Thereference stiffness k2 is the slope connecting the yield point to the crack point on the oppositeside, as shown in Figure 5(b). The normalized stiffness degrading with respect to ductilitylevel is shown as the solid lines in Figure 5(a).

kuld1 = k2(10.05 /y) (2)kuld2 = 0.6 k2(10.07 /y) (3)

4. Reloading in a direction where its cracking load has never been exceeded, the reloadingbranch aims at its crack point (e.g. BC ; KL; QR).

5. Reloading goes back to the onset point of unloading branch (e.g. HG) if the precedingunloading branch is completed before it reaches the zero shear force level (e.g. GH ).

6. Cracked reloading branches up to the cracking load (e.g. DE ; MN ; TU ) point tothe pinching reference point, whose shear force (V p) is a fraction of the peak shear of theprevious unloading branch (i.e. previous peak, Vp) and is given by Equation (4).

V p=Vp e[0.82(N/N0)0.14](p/y) (4)where N/N0 is the ratio of applied load to nominal compressive capacity of column and0.82 (N/N0)0.14


max peak

pinching reference point

previous peak

shear

deflection

hardening reference pt

max peak

pinching reference pointprevious peak

shear

deflection

Vy

VcrVcr

shear

deflection

Vcr

deflection

Vcr

shear

(a) (b)

(c) (d)

Figure 4. Examples of model defects in Ozcebe and Saatcioglus shear hysteretic model [6]: (a) negativeunloading stiffness at large ductility level; (b) negative residual in a positive unloading branch; (c) nearly

zero pinching stiffness; and (d) negative hardening stiffness.

the ductility level is equal to or greater than 14.29 (Figure 4(a)). These equations for unloadingstiffness must be revised first to allow for possible larger ductility levels. In addition, dependingon the shape of the primary curve and the locations of crack and yield points on it, it is possible inthe model that the residual displacement of a positive unloading branch turns out to be negative,and vice versa, due to an inadequate small unloading stiffness (Figure 4(b)). Moreover, when theelement is reloaded previously from the opposite side, the pinching stiffness is controlled by areference point (V p) as a function of the previous peak shear (Vp) given by Equation (4). If thisprevious peak is very small, the calculated pinching stiffness will be very close to zero and thus notreasonable (Figure 4(c)). Finally, for the cases when the pinching reference point falls below thecracking shear level, resulting in a very large crack closing displacement, the hardening stiffnessmight become negative and cause serious errors (Figure 4(d)). All of these model defects (asdepicted in Figure 4) will induce convergence problems in the FE programs and fail the analysesaccounting for SFI.

2.4. Improved shear and flexure hysteretic models

The problem with Ozcebe and Saatcioglus shear reversal model of yielding a zero or negativestiffness when ductility level is equal to or greater than 14.29 is fixed by introducing a new setof stiffness degrading equations. For shear model, the revised unloading stiffness above the crackshear level is given by Equation (6), and below the crack shear level by Equation (7). The newequations extend the maximum allowable ductility level to 50.0, while maintaining the stiffnessvery close to Ozcebe and Saatcioglus original model under low ductility level. Comparisons of thenormalized unloading stiffness between revised and original equations are displayed in Figure 5(a).

kuld1 = k2 1.4 e0.35(/y)0.01 (10.02 /y)3.5 (6)

kuld2 = 0.6 k2 1.3 e0.35(/y)0.01 (10.02 /y)5.5 (7)



My, Vy

Mcr, Vcr

k1

k2 k3

k4

deflection

onset point of unloading branchforce

k5

kuld1

kuld2

kp

0 2 4 6 8 10 12 14 16 18-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Nor

mal

ized

Unl

oadi

ng S

tiffn

ess,

k/k

2 above Vcr, Ozcebe'sbelow Vcr, Ozcebe'sabove Vcr, proposed

below Vcr, proposed

(a) (b)

Figure 5. (a) Comparison between revised and original unloading stiffness and (b) reference stiffnesses.

Similar equations for flexure reversal model based on Ozcebe and Saatcioglus shear model arealso proposed and calibrated with over 20 static cyclic column tests data. For unloading above thecrack moment level, the stiffness is given by Equation (8), and below by Equation (9). Since thereis no significant pinching behavior in the flexural response, the pinching stiffness in the originalshear model is replaced by Equation (10) (the reloading stiffness from M =0 to M =Mcr] in theproposed flexure reversal model, subjected to a minimum stiffness k5. As for the reloading branchabove Mcr, the reloading reference point (M m), accounting for the strength deterioration due toloading cycles, is a fraction of the maximum peak point (Mm), and is given by Equation (11).

kuld1 = k2 1.2 e0.125(/y)0.25 (10.016 /y)3.5 (8)

kuld2 = 0.70 k2 1.2 e0.125(/y)0.35 (10.020 /y)4.5 (9)

kp = 0.56 k2 1.2 e0.125(/y)0.35 (10.020 /y)4.5 (10)

M m = Mm e[0.002n

m/y0.010

n(m/y)] (11)

where n is the number of cycles in one direction within its max rotation range, mcr, and Mmis the bending moment on flexural primary curve corresponding to m.

Minimum unloading stiffnesses, k3 and k4 (see Figure 5(b)) applied to both hysteretic modelsand k5 applied to the flexural model, are used to prevent the extraordinary flat unloading slopes.The stiffness k3 is the slope connecting the onset point of unloading branch to the crack point onthe opposite side; the stiffness k4 is the slope connecting the point on the current unloading branchat the crack force level to the crack point on the opposite side; and the stiffness k5 is the slopeconnecting the point on the current unloading branch at the zero force level to the crack point onthe opposite side. Unlike k1 and k2, which are constants, k3, k4, and k5 are variables dependingon the location of the onset of current unloading branch, and none of the k3, k4, or k5 should betaken larger than k1 which might take place when unloaded from points above the primary curveat small ductility.

To prevent the error of nearly zero pinching stiffness in the shear hysteretic model from occurring,the improved model checks the situations when the peak shear of previous unloading branch issmaller than the crack shear, under which the last peak point whose shear is larger than the crackshear level should be used as the previous peak instead. Regarding the fourth source of theaforementioned model defects (Figure 4(d)), a make-up rule is applied that should the pinchingreference point fall below the crack shear level, the previous peak is enforced to serve as thepinching reference point. This additional make-up rule is appropriate, because in around 95% of



last peak whose V>Vcr

pinching reference point

shear

deflection

hardening reference point

Vcr

max peak

pinching reference pointprevious peak serves as

shear

deflection

hardening reference point

Vcr

(a) (b)

Figure 6. Remedies for the defects in shear hysteretic model: (a) fix for nearly zero pinching stiffnessand (b) fix for negative hardening.

the cases it takes place at the locations where the unloading shear forces (i.e. Vp in Equation (4))barely exceed the crack load. For the rest 5% of cases in which the column is seriously damaged(e.g. low cycle fatigue at large displacement ductility level) or it undergoes considerable softeningso that the pinching reference point falls below the crack shear level, the make-up rule should notbe applied and can be opted out in the program. The FE program, however, in such cases will failvery soon due to convergence difficulty, and it can be perceived as the failure of the RC column.Figure 6 schematically illustrates the remedies for these two model defects.

3. IMPLEMENTATION AND CALIBRATION OF THE PROPOSED MODEL

3.1. Model implementation

The hysteretic models and interactive scheme described above have been implemented as an UELin the commercial FE analysis software, ABAQUS, as illustrated in Figure 7 (the linear axial springis present but not shown in Figure 7 since it is not part of the UEL). The realistic and idealizedconfigurations of bridge deck-column are displayed in Figures 7(a, b). The shearflexure interactionuser-element (SFI-UEL) (Figure 7(c); j and j is the same node) comprises two springs, F-UELand S-UEL, controlling the flexural and shear responses of the cantilever column, respectively. Theelement allows a displacement-controlled analysis with an input displacement time history, whichin the displacement-based FE program is a trial vector consisting of four global displacementdegree-of-freedoms (DOFs) (as shown in Figure 7(d)) passed in from the main program duringevery time increment. The task of the SFI-UEL is to return to the main program the correspondingforce vector and tangent stiffness matrix, subject to the past deformation history. There are onlytwo effective DOFs in the SFI-UEL, the relative nodal displacement and rotation (u and ),and the incremental forcedisplacement relationship of SFI-UEL is given by:{

V (t)

M(t)

}=

[dV/du dV/d

dM/du dM/d

]{u(t)

(t)

}(12)

The diagonal terms (i.e. dV/du and dM/d) in Equation (12) are time-history dependent and aredetermined by the proposed shear and flexure hysteretic models, respectively. The off-diagonalterms (i.e. dV/d and dM/du) which couple together the flexural and shear responses, on the otherhand, are derived from the assumption that in a cantilever column, the end moment is equal to theshear force times the column height (i.e. M =V h), and given by dV/d=d(M/h)/d= (dM/d)/hand dM/du=d(V h)/du= (dV/du)h. This assumption works fine in the static analyses. However,it is found to yield incorrect solutions in the transient time-history analyses if there are massesdistributed along the column height. In a transient time-history analysis, due to the inertia forces



i

j

1

4

3

2

SFI-UEL

Rigidcolumn

Inflection point

Deck

F-UEL

S-UELi

j

Deck

j'

(a) (b) (c) (d)

Figure 7. Implementation of proposed shearflexure interaction model: (a) realistic configuration; (b)idealized configuration; (c) SFI User Element (SFI-UEL); and (d) effective DOFs in SFI-UEL.

contributed from the distributed masses, there are additional shear forces acting on the cantilever.As a result, the end moment of a cantilever column becomes:

M =V h+(mi ai yi ) (13)Equation (13) is a key to capture the SFI effects, for it provides the relationship between the

flexural and shear responses. Since the assumption of M =V h does not hold any more, the SFI-UELbecomes quite unstable so that sometimes it fails to converge whereas at other times it convergesat a higher mode of deformation in which the flexure spring rotates in the opposite direction tothat of the stretched shear spring. For modeling techniques adopting the concentrated plastic hingeapproach, it is very difficult to account for the second term in Equation (13) and consequentlydifficult to couple together the flexural and shear responses directly in the tangent stiffness matrix.

To resolve this issue, in this study the masses of columns are lumped to the ends (i.e. nodej in Figure 7) such that off-diagonal terms derived according to M =V h can still be valid. Inother words, the accuracy of the SFI-UEL under transient analysis is slightly sacrificed to ensurethe column to be deformed in the first vibration mode generating a conservative estimation ondisplacement demand and to capture the SFI at the element level through the tangent stiffnessmatrix. It is later noticed that with the simplification of lumping the column mass to the ends,the requirements of local and global equilibrium alone are sufficient to enforce the relationshipthat M =V h to be hold, making the entire system to converge at the same solution regardless theoff-diagonal terms being exist or not. In fact, the FE program took only two to three iterationsto achieve the convergence when the off-diagonal terms are removed, compared to four to fiveiterations when those terms are added. Therefore, in the final SFI-UEL formulation, only the twodiagonal terms are used.

The shear and flexural primary curves obtained by the approach illustrated in Figures 1 and 2are input into the UEL along with the constant axial force ratio as well as the critical pointsfor cracking (Vcr and Mcr) and yielding (Vy and My). The cracking load is defined as the pointon primary curve where the strain of the outermost tensile concrete fiber exceeds the concretecrack strain, and the yielding load is the point where the outermost stretching rebars first yield.Nevertheless, considering the fact that the ascending branch of total primary curve defines thereference stiffnesses, k1 and k2, and therefore affects the hysteretic responses very much, in theseverifications the envelops of experimental hysteretic loops are used directly as total primary curvesof the column specimens to filter out the possible error introduced by the inaccurate prediction ofanalytical primary curves.

3.2. Model calibration

To validate the UEL, comparisons of the hysteretic loops under both static cyclic pushover testsand dynamic shake table tests have been made. Table I summarizes the geometry, reinforcement,material properties, and applied axial load of the examined column specimens. The first 13 speci-mens are from static cyclic pushover experiments whereas the rest three are from shaking table tests



-400 -300 -200 -100 0 100 200 300 400-4000

-2000

0

2000

4000

Displacement (mm) Displacement (mm)

She

ar (

kN)

Test NIST-Shear

Flex. spring

-60 -40 -20 0 20 40 60-150

-100

-50

0

50

100

150

She

ar (

kN)

Test TP-021

Flex. spring

-300 -250 -200 -150 -100 -50 0 50 100-100

-50

0

50

100

Displacement (mm)

She

ar (

kN)

Test PEER-158

Flex. spring

-600 -400 -200 0 200 400 600-1500

-1000

-500

0

500

1000

1500

Displacement (mm)S

hear

(kN

)

Test NIST-Flex

Flex. spring

Stone and Cheok [30]

(PEER-54 [NIST-Shear])

H/D=3.0

Yoneda et al. [31]

(TP-021)

H/D=3.375

Hamilton et al. [32]

(PEER-158)

H/D = 4.56

Cheok and Stone [33]

(PEER-53 [NIST-Flex])

H/D=6.0

(a) (b)

(c) (d)

Figure 8. Performance of proposed hysteretic flexure model for flexure-dominated columns.

-30 -20 -10 0 10 20 30-200

-100

0

100

200

Displacement (mm)

She

ar (

kN)

Test PEER-62Shear spring

-20 -15 -10 -5 0 5 10 15 20-600

-400

-200

0

200

400

600

Displacement (mm)

She

ar (

kN)

Test PEER-105

Shear spring

-30 -20 -10 0 10 20 30-500

0

500

Displacement (mm)

She

ar (

kN)


-15 -10 -5 0 5 10 15-200

-100

0

100

200

Displacement (mm)

She

ar (

kN)


BRI [34]

(PEER-62)

H/D=1.0

Priestley and Benzoni [35]

(PEER-105)

H/D=1.5

McDaniel [36]

(PEER-163)

H/D = 2.0

Hamilton et al. [32]

(PEER-160)

H/D=2.578

(a) (b)

(c) (d)

Figure 9. Performance of proposed hysteretic shear model for columns with significant SFI.

(UNR-9F1 and UNR-9S1) and a prototype bridge (Bridge #4), respectively. Majority of the cycliccolumn specimens are selected from the PEER column database (http://nisee.berkeley.edu/spd/)[29]. Experimental results are consistently plotted in solid (blue) lines and simulated results indotted (red) lines for all comparisons (Figures 814).



-80 -60 -40 -20 0 20 40 60 80-80

-60

-40

-20

0

20

40

60

80

Column Total Drift (mm)

-80 -60 -40 -20 0 20 40 60 80-80

-60

-40

-20

0

20

40

60

80s t ; f t


-80 -60 -40 -20 0 20 40 60 80-80

-60

-40

-20

0

20

40

60

80s t ; f t


Rea

ctio

n F

orce

at G

roun

dN

ode

(kN

)R

eact

ion

For

ce a

t Gro

und

Nod

e (k

N)

Rea

ctio

n F

orce

at G

roun

dN

ode

(kN

)R

eact

ion

For

ce a

t G

roun

dN

ode

(kN

)

-80 -60 -40 -20 0 20 40 60 80-80

-60

-40

-20

0

20

40

60

80s t ; f t (rigid)


Test PEER-93

ABAQUS UEL

Test PEER-93

ABAQUS UEL

Test PEER-93

ABAQUS UEL

Test PEER-93

ABAQUS UEL

s t ; f t

(a) (b)

(c) (d)

Figure 10. Hysteretic responses of PEER-93 assuming different shear-to-total displacement ratios.

-80 -60 -40 -20 0 20 40 60 80-200

-150

-100

-50

0

50

100

150

200

Column Tip Displacement (mm)

She

ar F

orce

(kN

)

Test TP-031ABAQUS UEL

-80 -60 -40 -20 0 20 40 60 80-200

-150

-100

-50

0

50

100

150

200

Column Tip Displacement (mm)

She

ar F

orce

(kN

)

Test TP-032ABAQUS UEL

(a) (b)

Figure 11. Prediction of cyclic responses of columns under different axial loads: (a) TP-031, axialload=12.8% compression and (b) TP-032, axial load = 4.6% tension.

3.2.1. Cyclic tests of columns Figure 8 compares the experimental and simulated results of fourcyclic column tests [3033] as listed in Table I whose behavior is dominated by flexural responsewithout significant SFI. On the other hand, Figure 9 compares the experimental and simulatedresults of the other four cyclic column [32, 3436] tests whose behavior is dominated by shearresponse showing significant pinching behavior. As the experimental envelope curves are input asthe primary curves of the numerical model, the good agreement in hysteresis loops validates theproposed hysteretic flexure and shear models.

As aforementioned in the methodology section, the shear and flexural primary curves for theproposed SFI scheme can be theoretically derived, as demonstrated in Figures 1 and 2. Experimental



-150 -100 -50 0 50 100 150-600

-400

-200

0

200

400

600


Rea

ctio

n F

orce

at G

roun

d N

ode

(kN

)

Test PEER-121

ABAQUS UEL

-800 -600 -400 -200 0 200 400 600-250

-200

-150

-100

-50

0

50

100

150

200

250


Rea

ctio

n F

orce

at G

roun

d N

ode

(kN

)

Test PEER-122

ABAQUS UEL

(a) (b)

Figure 12. Predicted cyclic responses of columns with different aspect ratio: (a) PEER-121, aspect ratio=3and (b) PEER-122, aspect ratio=8.

0 10 20 30 40-0.1

-0.05

0

0.05

0.1

disp

lace

men

t (m

)

UNR Test

ABAQUS UEL

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06-150

-100

-50

0

50

100

150

shea

r (k

N)

UNR Test

ABAQUS UEL

0 10 20 30 40-150

-100

-50

0

50

100

150

time (sec)

shea

r (k

N)

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06-150

-100

-50

0

50

100

150

displacement (m)

shea

r (k

N)

(a) (b)

(c) (d)

Figure 13. Predicted dynamic responses of column 9F1 subject to 2.5 1940 El Centro earthquake:(a) displacement time history; (b) hysteretic response at current stage; (c) shear force time history; and

(d) accumulated hysteretic response.

evidences and Figure 2(c) both indicate that the shear-to-total displacement ratio increases as thedisplacement ductility increases. In practice, however, it is desired to directly break out the shearprimary curve and flexural primary curve from the total primary curve by specifying a constantshear-to-total displacement ratio over the entire curve. This simplification is especially useful whenthe total primary curve is obtained through real column test programs.

Figure 10 compares the hysteretic loops of PEER-93 column using the proposed shear andflexure plastic hinge models and SFI scheme, presuming the shear-to-total displacement ratioequal to 7, 20, 40, and 100%, respectively. The results show that with shear-to-total displacementratio increasing, the pinching behavior becomes more and more significant. However, even if thepresumed shear-to-total displacement ratio is increased from 7 to 40%, the difference between



10 15 20 25 30 35 40-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

disp

lace

men

t (m

)

UNR Test

ABAQUS UEL

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-400

-200

0

200

400

shea

r (k

N)

UNR Test

ABAQUS UEL

10 15 20 25 30 35 40-400

-200

0

200

400

time (sec)

shea

r (k

N)

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-400

-200

0

200

400

displacement (m)sh

ear

(kN

)

(a) (b)

(c) (d)

Figure 14. Predicted dynamic responses of column 9S1 subject to 2.5 1940 El Centro earthquake:(a) displacement time history; (b) hysteretic response at current stage; (c) shear force time history; and

(d) accumulated hysteretic response.

predicted loops is still insignificant, given the same total primary curve, crack, and yield points. Itimplies that this ratio, as long as remaining within a reasonable range, is a less important factor asit merely affects the pinching behavior of the response. Therefore, for the rest calibration resultsshown, the shear and flexural primary curves are generated by assuming the shear and flexuraldisplacements accounting for 7 and 93% of the column total lateral deflection, respectively. (Inthe case of the PEER-93 specimen, the shear-to-total displacement ratio is changing with lateralductility but in general less than 6% (see Figure 2(c)) according to MCFT.) This is consistent withthe previous finding that the contribution of the shear deformation is less than 10% of the totaldeformation for a properly designed column [37]. However, shear deformation can be significantfor older columns with poor reinforcement details and it can increase to approximately 40% of totaldisplacement at larger displacement ductility level [25]. The presumed shear-to-total displacementratio (7%) can be changed easily under the proposed analytical scheme, and its effect is minor forthe given total primary curve as has been observed in Figure 10.

The specimens TP-031 and TP-032, tested by Sakai and Kawashima [38], have identical geom-etry and reinforcement details but with different axial loads (12.8% compression for TP-031 and4.6% tension for TP-032, respectively). The aspect ratio is about 3.375 for these two specimensindicating moderate SFI. Figure 11 compares the computed cyclic shear force-column tip displace-ment loops of column tests TP-031 and TP-032 using the SFI-UEL with the experimental loops.The experimental results indicate that the variation in axial force has a significant effect on thelateral hysteretic response of RC columns. The small tension force considerably reduces the ulti-mate capacity but increases the lateral deformation of the columns. The good comparison betweenthe computed and experimental results shows that the developed analytical approach is able toaccurately model the nonlinear response as well as the strength degradation and pinching behaviordue to the cyclic loading with either compressive or tensile axial loads.

Figure 12 compares the computed cyclic shear forcetotal displacement loops of column testsPEER-121 and PEER-122 [39] using the UEL with the experimental loops. These two columns areessentially identical to each other except the aspect ratio (i.e. height/diameter ratio). The aspectratio of the former is 3, which will demonstrate higher level of SFI, and the latter is 8, which



is primarily flexure dominant. The good agreement between experimental and numerical resultsshown in Figure 12 validates that the implemented UEL is able to capture the cyclic responses ofeither shearflexure- or flexure-dominant columns corresponding to the different aspect ratios.

3.2.2. Dynamic shake table tests of columns. Dynamic validation of the UEL is demonstratedin Figure 13 by comparing the predicted displacement and shear force time history as well ashysteretic loops of the column 9F1 with the experimental data obtained in a shake table test programconducted at the University of Nevada, Reno [40]. In the test program, the column specimen 9F1is subject to multi-event of earthquakes with increasing motion intensity ranging from 0.33 to4.0 times the ground motion of the 1940 El Centro earthquake record. The simulation was alsoconducted sequentially with the increasing intensity of the earthquake input motion. The numericalpredictions at each intensity level are very close to the experimental results. The comparisonpresented here is the sixth stage of the shake table test whose intensity is 2.5 times the originalEl Centro earthquake record.

Figure 14 plots the predicted displacement and shear force time history as well as hystereticloops of a similar column shake table test program, the column specimen 9S1 conducted by thesame research team [41], together with the experimental results. Column specimen 9S1 has thesame diameter as that of 9F1 but only two-thirds of its height, thus it exhibits a higher levelof SFI than 9F1. The column is again tested sequentially under multi-event of earthquakes withincreasing motion intensity ranging from 0.25 to 3.25 times the ground motion of the 1940 ElCentro earthquake record, and so is the numerical simulation. The results shown in Figure 14 arethe sixth stage of the shake table test with input ground motion intensity equal to 2.5 times theEl Centro earthquake record. It is seen that the proposed model is able to predict the peak displace-ment and shear force responses reasonably well although the computed hysteresis loops show somevisible difference with the experimental results. The acceptable comparisons again validate thatthe developed user element is capable of modeling the dynamic SFI behavior of columns.

4. FACTORS AFFECTING SFI OF COLUMNS

4.1. Influences of column aspect ratio on total primary curve

Columns having the same cross section but with different aspect ratios demonstrate very inter-esting self-similarity, which can be clearly identified through their normalized primary curves. Forexample, by normalizing Equations (4) and (5) with respect to Vy and Equation (6) with respect toky (defined as Vy/y), Equations (4)(6) can be re-written in dimensionless forms as shown below:

V p V pVy

= VpVy

e[0.82(N/N0)0.14](p/y)= Vp e[0.82(N/N0)0.14](p/y) (14)

V m V mVy

= VmVy

e[0.014n

m/y0.010

n(m/y)]

= Vm e[0.014n

m/y0.010n(m/y)] (15)

kuld1 kuld1ky

= kuld1Vy/y

= (Vcr+Vy)/Vy(cr+y)/y 1.4 e

0.35(/y)0.01 (10.02 /y)3.5

= k2 1.4 e0.35(/y)0.01 (10.02 /y)3.5 (16)Equations (14)(16) imply that if two columns share the same normalized primary curve (i.e.

V and are normalized with respect to Vy and y, respectively) and normalized critical points,they will produce identical dimensionless unloading and pinching stiffness on the dimensionlesscoordinates (i.e. V/Vy vs /y) when subject to the same ductility history. Figure 15(a) shows



0 2 4 6 8 10 120

50

100

150

200

250

Column Tip Drift (mm)

She

ar (

kN)

H/D=1.249 (M/V=0.381)

H/D=2.249 (M/V=0.686)

H/D=3.249 (M/V=0.991)

H/D=4.249 (M/V=1.296)

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Nor

mal

ized

She

ar F

orce

(V

/Vy)

H/D=1.249 (M/V=0.381)

H/D=2.249 (M/V=0.686)

H/D=3.249 (M/V=0.991)

H/D=4.249 (M/V=1.296)

(a) (b)

Figure 15. Variation of primary curve with aspect ratio and the normalized primary curves of column:(a) primary curves for different aspect ratios and (b) normalized primary curves.

the total primary curves of columns having same cross section as the specimen PEER-93 [24] butwith various heights, derived theoretically using the approach introduced in Figure 1. The crackloads are marked with circles and yield loads with squares on the corresponding primary curves. Itcan be observed that columns with smaller aspect ratio are relatively stronger (i.e. higher ultimatecapacity) and stiffer (i.e. smaller lateral displacement) than those with larger aspect ratios, andthat the curve changes drastically when aspect ratio changes. However, if the vertical axes of thesecurves are normalized by their own yielding shear forces and the horizontal axes by their ownyielding displacements, the ascending branches as well as the critical points of the normalizedprimary curves become almost identical, regardless of their aspect ratios, as shown in Figure 15(b).This interesting similarity indicates that for columns with the same cross section and under thesame axial load, they will have almost identical normalized primary curves and critical points, andthus demonstrate very similar hysteretic behaviors on the dimensionless coordinates. It explainswhy the equations can work for both shear-dominant and flexure-dominant columns, as is thecomparison of column tests PEER-121 and PEER-122 shown in Figure 12, and it justifies the useof a single set of unloading and reloading rules to simulate the responses of a wide spectrum ofRC columns. Nevertheless, the similarity among columns does not eliminate all the diversities inan individual column. For example, columns with smaller aspect ratios will have larger normalizedstrength (i.e. larger Vm in Equation (15), as displayed in Figure 15(b)), and consequently accordingto the hysteretic shear model they will have larger dimensionless hardening stiffness, demonstratingmore significant pinching behavior.

4.2. Effects of axial load variation on total primary curve

Similar to aspect ratio, axial load variation also has substantial effects on total primary curve.Recalling the comparisons presented in Figure 11, the difference in axial load considerably influ-ences the lateral hysteretic responses of RC columns in terms of strength and stiffness change aswell as the significance of pinching phenomena. Figure 16(left) further displays the theoreticallyderived total primary curves of PEER-93 subjected to different level of axial loads ranging from 5%tension to 20% compression of its nominal axial capacity. As the compressive axial load increase,lateral resistance of the column becomes stronger and stiffer. Unlike the effects of aspect ratio,normalization of primary curves under different axial load levels does not conclude a unique curve,as shown in Figure 16(right). It can be observed that under larger tensile axial force, the crackpoint on the primary curve is lower. It is then concluded that the tensile axial load is detrimentalto RC columns, making the columns weaker (lower ultimate shear capacity), softer (larger columndrift), and easier to be damaged (lower cracking shear force).

The effects of above two factors, namely, the aspect ratio and axial load level unveil why thetraditional modeling techniques with fixed momentcurvature relationship unable to simulate well



0 5 10 15 20 25 30 350

10

20

30

40

50

60

70

80

Column Tip Drift (mm)

She

ar (

kN)

P/P0=-5% (T)P/P0=-2% (T)P/P0=0 (-)P/P0=5% (C)P/P0=10% (C)P/P0=20% (C)

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

Nor

mal

ized

She

ar F

orce

(V

/Vy)

P/P0=-5% (T)P/P0=-2% (T)P/P0=0 (-)P/P0=5% (C)P/P0=10% (C)P/P0=20% (C)

(a) (b)

Figure 16. Primary curves of PEER-93 subjected to different level of axial loads: (a) primary curves fordifferent axial loads and (b) normalized primary curves.

the behaviors of RC columns. To sum up, consideration of nonlinear SFI is not simply replacingthe linear shear spring with a nonlinear one. The most important point is that the adopted methodmust be able to adequately reflect the changes in the primary curve (including the ultimate capacity,crack and yield points, and softening branch) at the columns critical section due to the complexaxialshearflexure interaction, and to account for the accumulation of material damage (includingstrength deterioration, stiffness degrading, and pinching) induced by cyclic loading reversals.

5. SEISMIC RESPONSE ASSESSMENT OF BRIDGES CONSIDERING SFI

The developed UEL has been successfully applied to evaluate the seismic responses of threeprototype bridges [10] using the commercial FE software, ABAQUS. The bridge system simulationsconsider the criteria for selection of ground motions (representative of the possible future excitationscaused by the dominant adjacent faults of the site), the complex soilstructural interaction effects atthe bridge abutments and foundations, the modeling of the whole bridge systems, and the modelingof nonlinear behavior of column elements. Numerical model and structural characteristics of oneof the prototype bridges, Bridge #4, whose linear elastic model was originally reported in FHWApublication [42], is illustrated and summarized in Figure 17. This prototype bridge represents atypical old design bridge with fundamental structural period of 0.8 s. Details of the column crosssection in bent 1 of the bridge are listed in Table I. Transient time history analyses are conductedwith all three directions of input ground motions (two horizontals + one vertical).

Figure 18 compares the transverse and longitudinal column drift and section force time historiesof the column in bent 1 of Bridge #4, using the developed user element (UEL model) and thenonlinear M model. Nonlinear M model in ABAQUS adopts the 3D Timoshenko beamelement theory with user-assigned nonlinear momentcurvature relationship and constant transverseshear stiffness. Comparison of the hysteretic loops of the nonlinear M and the UEL models aredemonstrated in Figure 19. The results show that the UEL in general yields larger column driftand smaller section forces than that of the nonlinear M model. The differences in responsesare largely due to the distinctive hysteretic rules employed by these two models. Furthermore,the difference between these two models becomes more significant at larger displacement levelas can be observed from the longitudinal direction in this case. It is noteworthy that in thisdirection, the nonlinear M model by applying the traditional plasticity theory overestimates theresidual displacement during the major shock of earthquake due to failure of capturing the stiffnesssoftening in unloading branches, resulting in an even larger absolute displacement in the followingaftershock, whose effective amplitude is merely one-third to one-half of that of the major shock.



- 320 ft long 30 skewed bridge with 3 continuous spans - Pier Type: two-column integral bent, monolithic at

column top while pinned at base - Foundation Type: spread footing - Expansion Joints: expansion bearings & girder stops - Force Resisting Mechanism:

[Longitudinal] intermediate bent columns & free longitudinal movement at abutments [Transverse] intermediate bent columns & abutments

Figure 17. Numerical model in ABAQUS and structural details of Bridge #4.

10 15 20 25

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100

200

Transverse

Col

umn

Drif

t (m

m)

10 15 20 25

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Longitudinal

Col

umn

Drif

t (m

m)

UEL

10 15 20 25

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1000

2000

Sec

tion

For

ce (

kN)

Time (sec)

10 15 20 25

-2000

-1000

0

1000

2000

Sec

tion

For

ce (

kN)

Time (sec)

(a) (b)

(c) (d)

Figure 18. Comparison of column drift and section force time histories in Bent 1 of Bridge #4, usingnonlinear M (Timoshenko beam) model and UEL model under 1987 Whittier Narrows earthquake.

This observation discloses the fact that the traditional Timoshenko beam theory may not predictwell even the responses of RC columns without significant SFI (H/D=5.0 in this bridge).

Figure 20 plots separately the shear and flexural responses of the same column using the UELmodel, including the time histories of shear displacement, shear force, rotation, bending moment,and the correspondent hysteresis loops as the bridge is subject to the ground motions of the1987 Whittier Narrows Earthquake recorded at ComptonCastlegate St station. It is seen thatthe proposed SFI scheme and the implemented UEL successfully model the nonlinear shear andflexural responses of the bridge (a MDOF system) under seismic loadings.

As the axial load variation has noticeable effects on the SFI behavior of columns and verticalground motion component is present during earthquakes, the effect of axial load variation shouldbe considered carefully in the seismic evaluation of RC bridges. Although the proposed SFIscheme and the developed UEL in this paper have included the influence of axial load on lateralresponses, they are not able to account for axial load fluctuation in a transient time history analysis.An axialshearflexure interaction model is currently under development to account for the axialload variation through parameterized primary curves and consistent damage indexes to enable thetransition of loading and unloading branches between different primary curves corresponding tovariable axial load levels.



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She

ar (

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Transverse

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She

ar (

kN)

Longitudinal

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2000

Column Drift (mm)

She

ar (

kN)

UEL

-200 -100 0 100 200-2000

-1000

0

1000

2000

Column Drift (mm)

She

ar (

kN)

UEL

(a) (b)

(c) (d)

Figure 19. Hysteretic loops in bent 1 of Bridge #4 adopting nonlinear M and UEL models.

10 15 20 25-10

0

10

10 15 20 25-2000

0

2000

V (

kN)

10 15 20 25-0.02

0

0.02

0.04

10 15 20 25-1

0

1x 10

4

M (

kN-m

)

Time (sec)

-10 -5 0 5 10-1500

-1000

-500

0

500

1000

1500

She

ar F

orce

(Lo

ngi.)

, V (

kN)

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

Ben

ding

Mom

ent,

M (

kN-m

)

Figure 20. Shear and flexural responses in bent 1 of Bridge #4 using UEL model (in longitudinal direction).

6. CONCLUSIONS

This paper presents a coupled hysteretic SFI model to efficiently simulate the nonlinear behavior ofRC columns under combined axial, shear, and bending moments for seismic assessment of bridges.The important SFI is introduced through primary curves at the section level with considerations



of combined loadings and enforced by the local and global equilibrium requirements during thetime history analysis. The hysteretic flexure and shear springs are coupled at the element level andare implemented as a single UEL in a displacement-based finite element program, ABAQUS. Theimproved hysteretic rules are developed for better numerical implementations and can realisticallycapture the pinching, stiffness softening, and strength deterioration of columns due to cyclic loadreversals. The proposed model was calibrated against column specimens tested under static cyclicand dynamic loadings. The good agreement between the numerical prediction and experimentaldata can be observed for both flexure- and shear-dominant columns. The proposed model hasbeen successfully applied to evaluate the seismic response of a realistic prototype bridge. It isshown that considering nonlinear SFI in general yields larger column drifts and smaller sectionforces than the case when only nonlinear flexural behavior is considered. Two factors, namely theaspect ratio and axial load level, aside from the section properties are found to affect significantlythe SFI behavior of columns. In summary, the proposed model accounts for the SFI behavior ofcolumns in a simple and computationally efficient way. Future study will extend to include theeffects of variable axial load due to vertical ground motions and to improve the analytical methodto derive total primary curve of columns with consideration of axialshearflexure interaction andbond-slip.

NOTATION

average shear strain on column cross section rotation angle in the flexure springm maximum rotation angle experienced in the flexure springy yielding rotation level defined on the flexural primary curve curvature of column cross section shear displacement in the shear springcr cracking displacement level defined on the shear primary curvef flexural contribution to total displacementp displacement on previous unloading branch in the shear spring corresponding to Vps shear contribution to total displacementt total displacement or total column tip driftm maximum displacement experienced in the shear springy yielding displacement level defined on the shear primary curveai acceleration of i th column elementh height of cantilever columnmi mass of i th column elementn number of cycles in one direction at its maximum deformation range mcr or

mcryi distance from reflection point (or from the free end of cantilever column) to i th column

elementM moment in the flexure springMm the bending moment on flexural primary curve corresponding to mM m reference bending moment for reloading above cracking loadMy yielding moment level defined on the flexural primary curveN applied axial loadN0 nominal axial compressive capacity of the column sectionV shear force in the shear springVm the shear force on shear primary curve corresponding to mV m hardening reference shear forceVp peak shear force on previous unloading branch in the shear springV p pinching reference shear forceVy yielding shear level defined on the shear primary curve



ACKNOWLEDGEMENTS

The research presented here was funded by National Science Foundation (NSF) through the Networkfor Earthquake Engineering Simulation Research Program, grant CMMI-0530737, Joy Pauschke, programmanager. Any opinions, findings, and conclusions or recommendations expressed in this paper are those ofthe author(s) and do not necessarily reflect the views of the NSF. The authors thank Dr. David H. Sandersfor providing the shaking table test results for model calibration. We greatly appreciate two anonymousreviewers for their valuable comments that helped to improve the paper.

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