art-3A10.1007-2Fs00707-013-0970-7

Acta Mech 225, 477492 (2014)DOI 10.1007/s00707-013-0970-7

Yu-Fu Ko Co Phung

Nonlinear static cyclic pushover analysis for flexural failureof reinforced concrete bridge columns with combineddamage mechanisms

Received: 15 June 2013 / Published online: 29 August 2013 Springer-Verlag Wien 2013

Abstract In a fixed connection of a reinforced concrete bridge column, experiments have shown that thelongitudinal reinforcing bars slip at the interface of the connection under cyclic seismic loading. The bond-slip(or strain penetration) of the longitudinal reinforcing bars causes a pinching effect in the columns hysteresiscurve. The bond-slip (or strain penetration) reduces the columns stiffness and increases its deformations duringan earthquake event, significantly affecting the performance of the column. Significant strength degradationhas also been observed after the column reaches its ultimate strength. This study is to model a reinforcedconcrete columns performance under cyclic pushover analysis with combined damage mechanisms includ-ing concrete cracking, concrete strength degradation due to concrete spalling, longitudinal reinforcing barsbuckling, and bond-slip between longitudinal reinforcing bars and concrete. Two multi-scale nonlinear finiteelement models with and without the bond-slip (or strain penetration) of a reinforced concrete bridge columnare proposed. The simulated columns hysteresis curves under nonlinear cyclic pushover are compared withavailable experimental data. The results show that the proposed models with bond-slip together with combineddamage mechanisms can effectively predict the seismically induced flexural failure behavior of the reinforcedconcrete bridge columns.

1 Introduction

Bridge columns are subjected to combined actions of axial force, shear force, torsion, and bending momentduring earthquakes, caused by spatially complex earthquake motions. Combined actions could create significanteffects on the strength and deformation capacity of reinforced concrete bridge columns, resulting in unexpectedlarge deformations and extensive damage that in turn influences the performance of bridges as vital componentsof transportation systems.

Unlike seismic design for buildings, seismic bridge design philosophies, current design codes and practicesallow damages to occur in the substructure during a maximum credible earthquake (MCE). However, thedamage must be under control and its location is usually in the bridge columns, which is pre-selected by theengineer to prevent collapse mechanisms. In seismic bridge design, the lateral deformation capacity of thebridge is usually limited to the flexural and shear strength of the columns. In addition, the bridge is designed

Y.-F. Ko (B) C. PhungDepartment of Civil Engineering and Construction Engineering Management,California State University, Long Beach, CA 90840-5101, USAE-mail: [email protected]: http://www.csulb.edu/colleges/coe/cecem/views/personnel/fulltime/ko.shtmlTel.: +1-562-9857884

C. PhungPublic Works, County of Orange, Santa Ana, CA 92703, USAE-mail: [email protected]

478 Y.-F. Ko, C. Phung

to have sufficient ductility to sustain the imposed deformation demand during an earthquake event. Therefore,accurately modeling the behavior of bridge columns under a cyclic loading is essential for the design andanalysis of a bridge structure under an extreme earthquake.

Researchers have strived to understand bridge behaviors and to improve bridge performances under seismicloadings through experimental works and numerical simulations.

In experimental works, for example, in order to reliably obtain seismic responses of as-built and repairedreinforced concrete bridge columns under near-fault ground motions, Chang et al. [1] performed pseudo-dynamic testing of two long/slender bridge columns with a reduced scale of 2/5. An identical specimen wastested under cyclic loading to estimate the basic properties of these columns, such as shear strength, flexuralstrength, and ductility, so that the seismic responses obtained from pseudo-dynamic tests can be thoroughlydiscussed. Bond-slip (strain penetration), longitudinal reinforcing bars buckling, and concrete spalling wereobserved during the cyclic loading tests. In addition, Haroun and Elsandadedy [2,3] conducted comprehensiveexperimental studies on scaled models of short/squat bridge columns repaired and retrofitted with advancedcomposite-material jackets. For short/squat bridge columns, the failure modes of bridge columns under seismicloadings are typically brittle due to shear forces. Brittle shear failure and shearflexure interaction effects needto be considered in the analytical models for this type of bridge columns.

In numerical simulations, Spacone et al. [4,5] proposed the force-based/flexibility-based fiber elementformulation based on force interpolation functions that strictly satisfy the equilibrium of bending momentsand axial force along the element. Their proposed algorithm demonstrated accurate and stable results even in thepresence of strength loss and is, thus, capable of tracing very well the highly nonlinear behavior of reinforcedconcrete members under the cyclic load combinations of bending moment and axial force. However, shearflexure interaction and bond-slip (strain penetration) effects are not integrated in their element formulation,and the built-in plane section assumption may not be appropriate for some members. Nevertheless, fiberanalysis remains the economic and accurate means to capture the seismic behavior of concrete structures.In addition, Lee and Billington [6] proposed a modified concrete constitutive model representing damageaccumulation from cyclic loading and implemented it for the fiber element analysis that incorporated changesto reloading behavior when moving from high tensile strain back to compression. Analysis using the modifiedconcrete constitutive model leads to improvements in the ability of the fiber element model to capture residualdisplacements of bridge columns under seismic loadings. However, the bond-slip (strain penetration) effectwas not considered in their numerical models.

Special attention is given to the failure modes of long/slender reinforced concrete bridge columns withbond-slip (strain penetration) effect and various combined damage mechanisms [7]. For long/slender reinforcedconcrete bridge columns, the failure modes of bridge columns under seismic loadings are typically ductile due toflexural bending moments if sufficient transverse reinforcing bars, i.e., hoops and ties, are adequately provided.Among the various damage mechanisms observed in ductile failure of long/slender reinforced concrete bridgecolumns, in particular, bond-slip (strain penetration) usually occurs along longitudinal reinforcing bars that arefully anchored into connecting concrete members, causing bar slips along a partial anchoring length and thuscreating additional member end rotations to the flexural members at the connection intersections. Capturingthe structural response and associated damage requires accurate modeling of localized inelastic deformationsoccurring at the member end regions as identified by shaded areas in Fig. 1. These member end deformationsconsist of two components: (1) the flexural deformation that causes inelastic strains in the longitudinal bars andconcrete, and (2) the member end rotation, as indicated by shaded areas in Fig. 1, due to reinforcement slip.The slip considered here is the result of strain penetration along a portion of the fully anchored bars into theadjoining concrete members (e.g., footings and joints) during the elastic and inelastic response of a structure.Therefore, the strain penetration behaviors increase both the local and globe deformations of the column andthe bridge. Ignoring the bond-slip (strain penetration) in linear and nonlinear analysis of concrete structureswill underestimate the deflections and member elongation, and overestimate the stiffness, hysteresis energydissipation capacities, strains and section curvature, thus leading to a conservative earthquake force.

Therefore, it is important and imperative to propose numerical models to rigorously consider the bond-slip(strain penetration) effect and various combined damage mechanisms observed through the experimental testsfor reinforced concrete bridge columns in order to more accurately capture the realistic behavior of flexuralfailure reinforced concrete bridge columns during seismic events.

The primary objective of this paper is to develop innovative numerical models to model the realisticnonlinear behavior of flexural failure reinforce concrete columns with combined damage mechanisms includingpinching behavior, strength deterioration as well as stiffness softening due to actions of axial force, shear forceand bending moment as a result of seismic loadings. Moreover, significant strength degradation occuring

Nonlinear static cyclic pushover analysis 479

Longitudinal

reinforcing bars

Bridge deck

Bridge column

Ground and

foundations

Fig. 1 Schematic representation of typical inelastic regions and bond-slip locations in a well-designed reinforced concrete bridgecolumn

after failures of longitudinal reinforcing bars at the plastic hinge zone was also observed by Chang et al. [1].Therefore, it is important to include the strength degradation in the models as well.

In the current study, we will propose analytical models that consider the combined damage mechanismsdue to bond-slip (strain penetration) between the concrete and the longitudinal reinforcing bars at the fixedend(s) of reinforced concrete bridge columns [810], buckling of the longitudinal reinforcing bars [1113],and concrete spalling [8,14,15]. The nonlinear seismic responses of long/slender reinforced concrete bridgecolumns due to cyclic loading reversals will be simulated by an open source finite element program, OpenSystem Earthquake Engineering Simulation (OpenSees), developed by Mazzoni et al. [16], using fiber elementformulation and nonlinear cyclic pushover analysis. The simulation results based on the proposed models willbe compared and calibrated with the hysteresis responses and experimental data of the specimens under cyclicloading by Chang et al. [1].

The proposed numerical models for long/slender reinforced concrete bridge columns will focus on inelas-tic flexural behavior due to bending moment and decouple with shear and torsion behaviors. For inelasticflexural behavior, the section analysis with a fiber model (with consideration of axialflexural interaction) ina one-dimensional stress field will be proposed to predict the ultimate strength and yielding displacement ofreinforced concrete bridge columns due to cyclic loading conditions during seismic events. In the proposedmodel, the damage mechanisms due to bond-slip (strain penetration) between the concrete and the longitudinalreinforcing bars at the fixed end(s) of the reinforced concrete bridge columns, buckling of the longitudinal rein-forcing bars, and concrete spalling will be considered. The nonlinear seismic responses of reinforced concretebridge columns due to cyclic loading reversals will be simulated by OpenSees using fiber element formulationand nonlinear cyclic pushover analysis. Simulated responses will be compared with the observed responsesfrom available experimental data at both global and local levels to validate the effectiveness of the models.

The remainder of the paper is organized as follows. In Sect. 2, we present theoretical backgrounds foranalytical modes for nonlinear finite element formulations at multi-scale levels. Subsequently, the descriptionand formulation of nonlinear finite element models are explained in detail in Sect. 3. In addition, two separatemodels are proposed to study the bond-slip (strain penetration) effect on the overall nonlinear cyclic pushoverbehavior of reinforced concrete bridge columns. In Sect. 4, comparisons between our numerical predictionsand available experimental results are given. We finally draw conclusions in Sect. 5.

2 Theoretical backgrounds for analytical models

The nonlinear seismic responses of long/slender reinforced concrete bridge columns due to cyclic loadingreversals will be simulated by an open source finite element program, OpenSees, using finite element method(FEM) with fiber-based finite element formulation as well as nonlinear cyclic pushover analysis. Nonlinearcyclic pushover analysis is a powerful tool for evaluating the inelastic and nonlinear seismic behavior ofstructures.


y

z 0.75m

0.6m

y, z: local coordinate of the cross

section with (32) #19 longitudinal

reinforcing bars.

Node j

Node p

Node i

X= x

Y = y

X, Y: global coordinate of the

finite element model

(a) (b)

Fig. 2 a Schematic representation of finite element models for reinforced concrete bridge bent/column; b cross-section ofreinforced concrete bride column

The innovative multi-scale research methodologies proposed in this paper will be divided into three scales:(1) Global structure scale: the effects of lateral load pattern in cyclic pushover analysis and geometric nonlin-earity on the structural dynamic behavior and responses of reinforced concrete bridge columns under seismicloadings can be studied at this scale; (2) Structural elements scale: the structure can be modeled as an assemblyof inter-connected finite elements with fiber element formulations at this scale. Plasticity theory and nonlinearconstitutive behavior are considered at the section scale of each element to simulate the hysteresis behaviorof the structural members; and (3) Microscopic scale: the effects of bond deterioration between concrete andlongitudinal reinforcements, buckling of longitudinal reinforcements, and concrete spalling on the overalleffective properties of reinforced concrete bridge columns can be investigated at this scale based on rigorousmicromechanics-based approaches.

2.1 Global structure scale

The effects of lateral load patterns in cyclic pushover analysis and geometric nonlinearity on the structuraldynamic behavior and responses of reinforced concrete bridge columns under seismic loadings can be studiedat this scale. In addition, determination of the onset and progression of plastic hinges and the related stiffnessdegradation due to bond-slip (strain penetration), buckling of longitudinal reinforcing bars, and concretespalling could be investigated. For example, the reinforced concrete bridge column could be modeled by finiteelement nodes, e.g., i, j, and p as shown in Fig. 2a. Additional nodes can be added between nodes j andp to refine the element length. A zero-length section element could be created at the base of the reinforcedconcrete bridge column between nodes i and j to capture the bond-slip (strain penetration) effect duringseismic loadings. Nonlinear beam-column fiber elements could be created between nodes j and p. Based onfiber element formulation, both the zero-length section element and sections within the nonlinear beam-columnfiber element will consist of the longitudinal reinforcement fibers, i.e., steel reinforcing bar fibers and concretefibers as shown in Fig. 2b. Material models describing the monotonic response and hysteresis rules will alsobe required and proposed for the steel reinforcing bar fibers and the concrete fibers.

A zero-length section element is a fiber discretization of the cross-section of a structural member. Zero-length section elements have been generally used for section analysis to calculate momentcurvature responsesin OpenSees. A method that uses a zero-length section element to capture the member end rotations resultingfrom the bond-slip (strain penetration) effect will be described as follows. The zero-length section element inOpenSees is assumed to have a unit length such that the element deformations (elongation and rotation) areequal to the section deformations (axial strain and curvature). Hence, the zero-length section element can beused to calculate rotation at a beam-column fiber element end under a moment. To incorporate a zero-length


section element in analysis, a duplicate node is required, i.e., the distance between nodes i and j is zero. Inaddition, the translational degrees of freedom of the nodes, i.e., between nodes i to j , will be constrained toeach other to prevent sliding of the beam-column fiber element under lateral loads because the shear resistanceis not included in the zero-length section element. It is noted that the combination of using the zero-lengthsection element and enforcing the plane section assumption at the end of the flexural member will impose highdeformations to the extreme concrete fibers in the zero-length section element.

2.2 Structural element scale

The structure can be modeled as an assembly of inter-connected finite elements with fiber element formulationsat this scale. Plasticity theory and nonlinear constitutive behavior are considered at the section scale of eachelement to simulate the hysteresis behavior of the structural members. Fiber-based finite element analysis hasbeen widely used to understand and predict the structures [4,5]. In the fiber-based analysis method, the flexuralmember is represented by unidirectional steel and concrete fibers, making the description of the correspondingmaterial models relatively easy. Because the steel and concrete fiber responses are specified in the directionof the member length, the fiber analysis concept is suitable for modeling flexural members regardless of thecross-sectional shape or the direction of the lateral load. In addition, Spacone et al. [4,5] proposed the force-based/flexibility-based fiber element formulation based on force interpolation functions that strictly satisfy theequilibrium of bending moments and axial force along the element. Their proposed algorithm demonstratedaccurate and stable results even in the presence of strength loss and is, thus, capable of tracing very wellthe highly nonlinear behavior of reinforced concrete members under the cyclic load combinations of bendingmoment and axial force. The force-based/flexibility-based fiber element formulation is also incorporated inOpenSees.

In fiber-based finite element analysis, the flexural member is represented by unidirectional steel rebar andconcrete fibers. The member stiffness and forces are obtained by numerically integrating the stiffness andforces of sections along the member length. The section deformation, e.g., displacement or rotation, is usedto obtain the strain in each fiber using the assumption that plane sections remain plane. The fiber stress andstiffness are updated according to the material models, followed by upgrading of the section force resultant andthe corresponding stiffness. Because the steel rebar and concrete fiber responses are specified in the directionof the member length, the fiber analysis can be used to model any flexural member regardless of its cross-sectional shape or the direction of the lateral load. The fiber analysis typically follows the direct stiffnessmethod, in which solving the equilibrium equation of the overall system yields the nodal displacements. Afterthe element displacements are extracted from the nodal displacements, the element forces are determined andthe member stiffness is upgraded, based on which the global stiffness matrix is assembled for the next timestep. The stiffness and forces of the fiber-based elements are obtained by numerically integrating the sectionstiffness and forces corresponding to a section deformation, i.e., axial strain and curvature . The sectiondeformation is calculated by interpolating the element end deformations, i.e., displacement and rotation, atthe integration points. From the section deformation, the strain in each fiber is obtained using the assumptionthat plane sections remain plane. The neutral axis position of the section at an integration point is determinedthrough an iterative procedure, which balances the force resultants at the section level as well as at the memberlevel [4,5,10].

2.3 Microscopic scale

The effects of bond deterioration between concrete and longitudinal reinforcing bars, buckling of longitudinalreinforcing bars, and concrete spalling on the overall effective properties of reinforced concrete bridge columnscan be investigated at this scale based on rigorous micromechanics-based approaches. Many studies havebeen published in the literature to predict the effective elastic moduli of random heterogeneous multiphasefiber/particle reinforced metal matrix composites and cement-based composites at microscopic level based onrigorous micromechanics approaches. We refer to Ju and Chen [17,18], Sun et al. [19,20], Liu et al. [21,22],Ju et al. [23,24], Ju and Ko [25], Ju et al. [26], Ju and Yanase [2731], Xu et al. [32], Ko and Ju [3335],Pan and Weng [36], and Shen and Li [37,38]. In particular, local bond-slip relationships (the bar stress vs.slip relationship) to model bond deterioration between concrete and longitudinal reinforcing bars as well asconstitutive law (stress vs. strain relationship) to model buckling of longitudinal reinforcing bars and concretespalling could be investigated at this scale.


Table 1 Loading sequences for cyclic loading test, Chang et al. [1]

Cycle number

Parameter 1,2 3,4 5,6 7,8 9,10 11,12 13,14 15,16 17,18 19,20Drift ratio (%) 0.25 0.50 0.75 1.00 1.50 2.00 3.00 4.00 5.00 6.00Displacement (mm) 8.125 16.25 24.38 32.50 48.75 65.00 97.50 130.0 162.5 195.0

In summary, the proposed innovative multi-scale research methodologies with parametric approach in thispaper together with nonlinear cyclic pushover analysis will enhance the confidence of simulation results. Prin-ciples and the main assumptions for the proposed nonlinear analysis techniques will provide solid foundationsfor future practical applications and model developments.

3 Description and formulation of nonlinear finite element models

Chang et al. [1] performed seismic cyclic pushover tests on as-built columns at a 2/5 reduced scale. Thespecimens were designed according to the 1995 version of the Taiwan Bridge Design Code which is basedon 1992 AASHTO Specifications and a standard design for the Taiwan Highway Bureau. The height of thecolumns was 3.25 m and their rectangular cross-section dimensions were 0.75 m 0.60 m. The longitudinalreinforcing bars consisted of 32 No. 6 (19 mm diameter) bars with a design yield strength fy = 420 MPa (actualyield strength from testing was 500 MPa) and were evenly distributed on all faces and throughout the height ofthe column with a constant concrete cover of 25 mm as shown in Fig. 2b. The concrete compressive strengthwas f pc = 21 MPa at 28 days (actual yield strength from testing was 23 MPa). The transverse reinforcing barswas made up of No. 3 (10 mm diameter) stirrups with a design yielding strength of fy = 280 MPa (actual yieldstrength from testing was 350 MPa) at a spacing of 100 mm. In addition, there are five confining crossties.The anchorage of the hoops and crossties at their two ends were 90 and 135, respectively. The longitudinalreinforcing bars ratio and the transverse reinforcing bars ratio are 1.95 and 1.04 %, respectively. In all of thetests, the axial load was taken to be 680 kN, which is often used by the Taiwan Highway Bureau to simulate atwo-lane bridge deck.

The induced displacements are shown in Chang et al. [1] and also listed in Table 1. The maximum driftratio was 6 % of the column height. Each drift ratio was repeated twice. The columns hysteresis responsesunder reversed cyclic loading are shown in Chang et al. [1]. The authors reported that a crack was initiated atthe bottom column at 0.25 % drift, and the concrete started to spall and plastic hinge started to form at 3 % drift.Longitudinal reinforcing bars buckled at 4 % drift and fractured at 5 % drift. Pinching and strength degradationwere also observed in the hysteresis loop. Slippage of the longitudinal reinforcing bars and closing of flexuralcracks in the plastic hinge zone of the columns might have caused the pinching behavior. As for the strengthdegradation, the authors explained that it might have been caused by the failure of the longitudinal reinforcingbars. Both behaviors reduced the energy dissipation per cycle of the tested columns.

Innovative multi-scale numerical models will be developed to model the realistic nonlinear behavior offlexural failure reinforce concrete columns including pinching behavior, strength deterioration as well asstiffness softening due to actions of axial force, shear force and bending moment as a result of seismicloadings. Proposed numerical simulation results will be compared with the experimental data recorded byChang et al. [1].

Progression of column yielding and damage is expected under strong ground motions, and thus, nonlinearfiber-based displacement-based beam finite elements will be used to represent the columns. In order to achievea more realistic representation of their responses, the proposed beam-column finite elements are endowedwith the ability to respond inelastically at every quadrature point. All fiber sections are assigned with theuniaxial material model tag of OpenSees. Three different constitutive rules are used simultaneously within across-section: (i) confined concrete, (ii) unconfined concrete, and (iii) longitudinal steel reinforcing bar. In thefollowing, detail aspects of column modeling, such as selecting material properties, modeling for bond-slip,and the type of nonlinear element will be explained in detail.

3.1 Material model for concrete fiber behavior

A material model describing the monotonic response and hysteresis rules is required for the concrete fibers. Theconcrete fibers in the zero-length element and nonlinear finite element was assumed to follow corresponding


Strain

tf

tsE

0E

pcf

psU

pcuf

0psc0E

Stress

Fig. 3 Stressstrain curve for uniaxial material Concrete02 (adapted from Mazzoni et al. [16])

hysteresis rules available in OpenSees through the material model known as Concrete02, where linear tensionsoftening (tension stiffening effect) is considered as shown in Fig. 3. The discretization schematic of the sectionconsisting of fibers is shown in Fig. 2b. The concrete fibers were divided into two separate regions of confinedand unconfined concrete as shown in dark gray and white color, respectively, in Fig. 2b to account for thecontribution of closed steel hoops (transverse reinforcing bars) to the concrete [39].

Concrete material (Concrete02) as shown in Fig. 3 is also an uniaxial material. To model the traction/tensilebehavior of the concrete, Concrete02 has been considered by presuming a brittle elastic trend. The elasticmodulus of the branch of traction/tensile behavior of concrete has been assumed equal to:

Ets = ft/0.002, (1)where ft = 0.70

f pc is the tensile strength of the concrete, and f pc is the concrete compressive strength at

28 days. Furthermore, the cyclic behavior of concrete has been characterized, referring to the experimental dataachieved. The model adopted for the concrete allows, if a cyclic analysis is performed, for an assessment of thecyclic degradation of resistance as well as the variation of the elastic modulus following the diffusion of crackswith expansion of the cycles. The compressive behavior of the concrete, Concrete02, considered the followingparameters as shown in Fig. 3: f pc, psc0, E0, f pcu, psU , and represent the concrete compressive strengthat 28 days, the concrete strain at maximum strength, the initial slope for compressive stressstrain curve, theconcrete crushing strength, the concrete strain at crushing strength, and the ratio between unloading slope atpsU and initial slope, respectively.

3.2 Material model for longitudinal and transverse reinforcing bars

The longitudinal steel reinforcing bar fibers in the zero-length section element and nonlinear finite element wasassumed to follow corresponding hysteresis rules available in OpenSees through the material model known asReinforcingSteel uniaxial material. Longitudinal reinforcing bars buckling effect is considered. An uniaxialmaterial called ReinforcingSteel in OpenSees was adopted to model the longitudinal reinforcing bars in thecolumn as shown in Fig. 4. This simulation is based on the Chang and Mander [40] uniaxial steel model. Thesimulation has incorporated additional reversal memory locations to better control stress overshooting. Thecycle counting method implemented in the routine achieves the same result as rainflow counting. The bucklingsimulations incorporated consist of a variation on Gomes and Appleton [41] and Dhakal and Maekawa [42]. Thebuckling and fatigue portions of this simulation are still being further enhanced and refined. Here, we adopteda buckling model of longitudinal reinforcing bars based on Gomes and Appleton [41] as shown in Fig. 4. The


uf

Stress

Strain

0.0r =

0.5r =

1.0r =

yf

b

Fig. 4 Stressstrain curve reinforced steel with buckling as modeled in ReinforcingSteel uniaxial material (adapted from Mazzoniet al. [16] and the Gomes and Appleton [41])

buckling model of longitudinal reinforcing bars proposed by Gomes and Appleton [41] is a modification ofthe Menegotto-Pinto [43] cyclic stressstrain steel relationship to take into account the effect of the inelasticbuckling of the longitudinal reinforcing steel bars. The buckling stressstrain path is simulated by a simplifiedmodel based on the equilibrium of a plastic mechanism of the buckled bar. In Gomes and Appleton [41],as shown in Fig. 4, the following factors are considered: amplification factor for the buckled stressstraincurve (), buckling reduction factor (r), buckling constant ( ), and slenderness ratio (lS R = Lu/db; Lu isthe unsupported length of the longitudinal reinforcing bar) of the longitudinal reinforcing bar. We proposedLu = L p in the calculation of lS R , where L p is the plastic hinge length. Once the plastic hinges formed at thebase of the bridge column, concrete spalling had already occurred. Thus, the longitudinal reinforcement lostthe supports from the surrounding concrete within L p.

Plastic hinge length (L p) is based on Caltrans Seismic Design Criteria [44] Sec. 7.6.2, Eq. (7.25) therein.That is,

L p ={

0.08L + 0.15 fyedbl 0.3 fyedbl (in, ksi)0.08L + 0.022 fyedbl 0.044 fyedbl (mm, MPa) (2)

where L is the member length from the point of maximum moment to the point of contra-flexure which is thecolumn height under current model, fye is the expected yield strength for longitudinal column reinforcement,and dbl is the nominal bar diameter of longitudinal column reinforcement.

The material stressstrain with buckled curve is shown in Fig. 4. fy and fu represents the yield strengthand ultimate strength of the material in tension, respectively. In addition,

b = fu b + 1 + ( fu ) ; b =

32

3lS Rs o , (3)

where b, o, and fu is the buckled stress, yield strain, and ultimate strength of the material in tension,respectively. Typically, in the analysis or modeling, the plastic hinge is lumped at the column base with theanalytical plastic hinge length (L p). The analytical plastic hinge length is the equivalent length of column overwhich the plastic curvature is assumed constant for estimating plastic rotation. However, in our proposed finiteelement models, instead of using lumped plastic hinges with the analytical plastic hinge length, displacement-based beam-column elements in OpenSees are employed which consider the spread of plasticity along theelement.

The effect and contributions of closed steel hoops (transverse reinforcing bars) will be considered in theincreased ductility confined concrete as stated in Sect. 3.1.


u

y

yS uS

K

bK

Bar Stress ( )

Loaded-End Slip ( )S

Fig. 5 Envelope curve for the bars stress versus loaded-end slip relationship as modeled in Bond_SP01 (adapted from Mazzoniet al. [16] and the Zhao and Sritharan [10])

3.3 Material model for bond-slip

Capturing the structural response and associated damage requires accurate modeling of localized inelasticdeformations occurring at the member end regions as identified by shaded areas in Fig. 1. These member enddeformations consist of two components: (1) the flexural deformation that causes inelastic strains in the longitu-dinal bars and concrete, and (2) the member end rotation, as indicated by arrows in Fig. 1, due to reinforcementslip. The slip considered here is the result of strain penetration along a portion of the fully anchored bars into theadjoining concrete members (e.g., footings and joints) during the elastic and inelastic response of a structure.Ignoring the strain penetration component may appear to produce satisfactory force-displacement response ofthe structural system by compromising strain penetration effects with greater contribution of the flexural actionat a given lateral load. However, this approach will appreciably overestimate the strains and section curvaturesin the critical inelastic regions of the member and thereby overestimate the structural damage. To capture thestrain penetration effects under multi-directional load, the slip that occurs to the longitudinal reinforcing barsat the wall base should be modeled on an individual basis. Using the zero-length section element availablein OpenSees [16], a bond-slip model to capture the strain penetration effects of fully anchored longitudinalrebar has been recently introduced [10]. In this element, the bar stress versus the slip response at the end ofthe flexural member is characterized using the stress versus slip function.

Bond-slip is modeled in OpenSees in Bond_SP01 which is developed by Zhao and Sritharan [10]. InFig. 5, monotonic bar stress versus slip response as modeled in Bond_SP01 is shown. This command isused to construct a uniaxial material object for capturing strain penetration effects at the column-to-footing,column-to-bridge bent caps, and wall-to-footing intersections. In these cases, the bond-slip associated withstrain penetration typically occurs along a portion of the anchorage length. This model can also be applied tothe beam end regions, where the strain penetration may include slippage of the bar along the entire anchoragelength, but the model parameters should be chosen appropriately. This model is for fully anchored steelreinforcement bars that experience bond-slip along a portion of the anchorage length due to strain penetrationeffects, which are usually the case for column and wall longitudinal bars anchored into footings or bridgejoints.

For the monotonic curve in Fig. 5,

={

K S, if S Sy (u y

)+ y, if S > Sy (4)

=S

S[(

1b

)Rc +(

SS

)Rc]1/Rc(5)


where = yuy is the normalized bar stress, S =

SSySy

is the normalized bar slip, = SuSySy is the ductilitycoefficient, b is the stiffness reduction factor, which represents the ratio of the initial slope of the curvilinearportion at the onset of yielding to the slope in the elastic region (K ). In addition, y and u are the yield andultimate strengths of the steel reinforcing bar, respectively. Sy and Su are the loaded-end slips when bar stressesare y and u , respectively. In addition,

Sy =

0.1

[db

4000Fy

f c(2 + 1)

]1/+ 0.013 (in, ksi)

0.4

[db4

Fyf c(2 + 1)

]1/+ 0.34 (mm, MPa)

(6)

In addition, extension of the monotonic bar stress versus slip response as modeled in Bond_SP01 is extendedto account for the hysteresis responses of bar stress versus slip [10]. The coefficient Rc with typical valuesin the range of 0.51.0 defined the shape of the reloading curve. Depending on the anchorage detail andthe corresponding mechanism, it is possible for a reinforcing bar with sufficient anchorage length to exhibitpinching hysteresis behavior in the bar stress versus slip response, especially when it is anchored into a joint.The coefficient Rc will permit the pinching characteristic to be accounted for in the analytical simulation ofthe flexural member. The lower end value of Rc will represent significant pinching behavior while a value of1.0 will render no pinching effect.

The zero-length section element available in OpenSees will be used to accurately model the strain pene-tration effects (or the fixed-end rotations shown in Fig. 1). Zero-length section elements have been generallyused for section analyses to calculate the moment corresponding to a given curvature. To model the fixed-endrotation, the zero-length section element should be placed at the intersection between the flexural member andan adjoining member representing a footing or joint as shown in Fig. 2a. A duplicate node is also requiredbetween a fiber-based beam-column element and the adjoining concrete element as shown in Fig. 2a. Thetranslational degree-of-freedom of this new node (i.e., node j in Fig. 2a) should be constrained to the othernode (i.e., node i in Fig. 2a) to prevent sliding of the beam-column element under lateral loads because the shearresistance is not included in the zero-length section element. The zero-length section element in OpenSees isassumed to have a unit length such that the element deformations (i.e., elongation and rotation) are equal to thesection deformations (i.e., axial strain and curvature). It is noted that the material model for the longitudinalreinforcing steel rebar fibers in the zero-length section element represents the bar slip instead of strain for agiven bar stress. The combination of using the zero-length section element and enforcing the plane sectionassumption at the end of a flexural member impose high deformations to the extreme concrete fibers in thezero-length element. These deformations would likely correspond to concrete compressive strains significantlygreater than the strain capacity stipulated by typical confined concrete models. Such high compressive strainsat the end of flexural members are possible because of additional confinement effects expected from the adjoin-ing members and because of complex localized deformation at the member end. Without further proof, it issuggested that the concrete fibers in the zero-length section element follow a concrete model in OpenSees(e.g., Concrete02).

3.4 Section properties

This command allows the user to construct a fiber section object. Each fiber section object is composed offibers, with each fiber containing a uniaxial material, an area, and a location within the local coordinate y-axisand z-axis as shown in Fig. 2b. All fiber sections are assigned with the uniaxial material model tag of OpenSees[16]. Three different constitutive rules are used simultaneously within a cross-section: (i) confined concrete,(ii) unconfined concrete, and (iii) steel rebar as described in previous sections.

3.5 Displacement-based beam-column element

We adopted displacement-based beam-column elements in OpenSees for the bridge column finite elementmodeling. This beam-column element is based on the displacement formulation and considers the spread ofplasticity along the element. In addition, the default integration along the element is based on the Gauss-Lobatto


(b)(a)

F

L = 3.25m

Fixed support

X= x



Y = y 0m

Zero-length

section element

Equate dx

Fixed support

F

L = 3.25m

X= x



Y = y

Fig. 6 a Reinforced concrete bridge columnmodel 1 (without bond-slip); b reinforced concrete bridge columnmodel 2 (withbond-slip)

quadrature rule (two integration points at the element ends). The global coordinate y-axis of the beam-columnelement was set to equal to the local coordinate y-axis as shown in Fig. 2b.

3.6 Flexural models (fiber-based models)

Model 1

The column was created using inelastic displacement-based beam-column elements, fixed at the base and freeat the top, where a force was applied in the direction of global y-axis to produce desire deflections as shownin Fig. 6a. Both the deflections and the loads were recorded by OpenSees. The cross-section and the columndimensions matched the physical model built and studied by Chang et al. [1]. An axial load of 680 kN wasapplied on the top of bridge column, followed by cyclic pushover analysis following the drift ratio defined inTable 1.

Concrete02, a uniaxial material, was employed for concrete as shown in Fig. 3. The following parameterswere adopted for the confined concrete: the concrete compressive strength at 28 days ( f pc = 23 MPa), theconcrete strain at maximum strength (psc0 = 0.003), the initial slope for the compressive stressstraincurve (E0 = 2 f pc/psc0 = 15,333.33 MPa), the concrete crushing strength ( f pcu = 0.4 f pc = 9.2 MPa),the concrete strain at crushing strength (psU = 0.01), the ratio between unloading slope at psU and initialslope ( = 0.1), the tensile strength of the concrete ( ft = 0.70

f pc = 23 (MPa) = 3.36 MPa), and the

tension softening stiffness (slope of the linear tension softening branch) (Ets = ft/0.002 = 1,680 MPa).On the other hand, the following parameters were adopted for unconfined concrete: the concrete compressivestrength at 28 days ( fupc = 23 MPa), the concrete strain at maximum strength (upsc0 = 0.003), theinitial slope for the compressive stressstrain curve (E0 = 2 fupc/upsc0 = 15,333.33 MPa), the concretecrushing strength ( fupcu = 0.0 fupc = 0.0 MPa), the concrete strain at crushing strength (upsU = 0.005),the ratio between unloading slope at psU and initial slope ( = 0.1), the tensile strength of the concrete( ft = 0.70

f pc = 23 (MPa) = 3.36 MPa), and the tension softening stiffness (slope of the linear tension

softening branch) (Ets = ft/0.002 = 1,680 MPa). It is emphasized that since shear failure does not governthe behavior of the column, section aggregated with elastic shear for concrete is not considered.

A uniaxial material called ReinforcingSteel in OpenSees was adopted to model the reinforcing steel inthe column as shown in Fig. 4. In this material, damage caused by buckling was modeled based on Gomesand Appleton [41] with lS R = Lu/db = L p/db = 0.26 m, where the plastic hinge length L p was calculatedaccording to Eq. (2), the amplification factor for the buckled stressstrain curve = 1.0, the buckling reduction


factor (r = 0.0), and the buckling constant ( = 0.5). The material stressstrain with buckled curve is shownin Fig. 4. The yield strength ( fy) of the material in tension was 420 MPa and the ultimate strength ( fu) of thematerial in tension was taken as 1.3 fy . In addition, the buckled stress b was calculated following Eq. (3).Typically, in the analysis or modeling, the plastic hinge is lumped at the column base with the analytical plastichinge length (L p). The analytical plastic hinge length is the equivalent length of column over which the plasticcurvature is assumed constant for estimating plastic rotation. However, in our model 1, instead of using alumped plastic hinge with the analytical plastic hinge length, a displacement-based beam-column element inOpenSees was employed which considers the spread of plasticity along the element.

Furthermore, after performing element refinement studies and convergence tests, we adopted sixdisplacement-based beam-column elements in our models for the bridge column finite element modeling.

An axial load of 680 kN was first vertically applied on the column top to simulate the two-lane bridge deckby the Taiwan Highway Bureau. Secondly, a cyclic pushover reference load was laterally applied to the columntop. Then, the cyclic pushover lateral displacements with maximum displacement at the top of the column ineach cycle were applied according to Table 1.

Model 2

In our proposed model 2, a zero-length section element (not zero-length element) was added at the fixedsupport of proposed model 1 as shown in Fig. 6b. Bond-slip was modeled in OpenSees in Bond_SP01, whichwas developed by Zhao and Sritharan [10]. In Fig. 5, monotonic bar stress versus slip response as modeledin Bond_SP01 is shown. This command is used to construct a uniaxial material object for capturing strainpenetration effects at the column-to-footing, column-to-bridge bent caps, and wall-to-footing intersections. Inthese cases, the bond-slip associated with strain penetration typically occurs along a portion of the anchoragelength. This model can also be applied to the beam end regions, where the strain penetration may includeslippage of the bar along the entire anchorage length, but the model parameters should be chosen appropriately.This model is for fully anchored steel reinforcement bars that experience bond-slip along a portion of theanchorage length due to strain penetration effects, which are usually the case for column and wall longitudinalbars anchored into footings or bridge joints.

For the monotonic curve in Fig. 5, we follow Eqs. (4)(6). The following material properties of Bond_SP01were adopted as = 0.4, Su = 30Sy , and b = 0.05. In addition, extension of the monotonic bar stress versusslip response as modeled in Bond_SP01 is extended to account for the hysteresis responses of bar stress versusslip [10]. The coefficient Rc with typical values in the range of 0.51.0 defined the shape of the reloading curve.Depending on the anchorage detail and the corresponding mechanism, it is possible for a bar with sufficientanchorage length to exhibit pinching hysteresis behavior in the bar stress versus slip response, especially whenit is anchored into a joint. The coefficient Rc will permit the pinching characteristic to be accounted for in theanalytical simulation of the flexural member. The lower end value of Rc will represent significant pinchingbehavior while a value of 1.0 will render no pinching effect. To match the significant pinching effect in theexperimental data reported by Chang et al. [1], Rc = 0.23 was employed.

Similarly, an axial load of 680 kN was vertically applied on the top of bridge column, followed by cyclicpushover analysis according to the drift ratio defined in Table 1. The concrete materials were the same in boththe zero-length section and inelastic beam-column elements. However, the reinforcing steel in the zero-lengthsection was replaced by another uniaxial material called Bond_SP01 material. The material was developed andcoded by Zhao and Sritharan [10] to capture the slippage of a fully anchored rebar in a column-to-footing fixedconnection. The Bond_SP01 yield and ultimate strength were the same as for the ReinforcingSteel material.It is emphasized that since shear failure does not govern the behavior of the column, section aggregated withelastic shear for concrete was not considered.

4 Comparisons of numerical simulations and experimental results

Finite element (FE) analyses were carried out under two-dimensional (2D) modeling schemes using theOpenSees platform. The available experimental data recorded by Chang et al. [1] were compared with oursimulation results based on our proposed model 1 and model 2. The investigations included: (1) historiesof applied lateral loads versus column displacement at bridge column top (including cracking, ultimate, andstrength degradation stages); (2) maximum strain distribution along bridge column height of an extreme barat the top-left corner of the cross-section during cyclic pushover up to 6 % drift ratio; and (3) maximum strain


Fig. 7 Comparison of experimental and analytical results: a hysteresis curve: lateral force versus displacement with model 1;b hysteresis curve: lateral force versus displacement with model 2

distribution along bridge column height of an extreme bar at the top-left corner of the cross-section at 1.6y .y is the yield lateral displacement. y = 38.4 mm is reported by Chang et al. [1].

4.1 Hysteresis force versus displacement curve at column top

Figure 7a shows the hysteresis force-displacement curves based on the model 1. It shows that the maximumlateral force was 446 kN based on experimental data compared with 400 kN obtained from OpenSees simu-lation results. At the lateral displacement of 200 mm, both the experiment and numerical simulation resultsyielded a similar lateral force of 300 kN. Our simulation results produced a very similar result compared tothe experimental data, except for the pinching effect. As the experimental data curve passed through zero dis-placement, the stiffness of the column approached zero. In our simulation the stiffness was non-zero. However,it decreased as the maximum displacement increased.

As shown on the lateral force versus displacement hysteresis curve in Fig. 7b, both experimental dataand our simulation results show an ultimate strength of 450 kN at cycles 1314. The maximum lateral forceremained relatively identical for the remaining cycles in OpenSees output, while it dropped drastically in theexperiment. However, our model captures well the maximum lateral force and the degradation pattern fromthe experimental data.

It is noted that our simulation results demonstrate the pinching effects and match with experimental datavery well as shown in Fig. 7b. However, some differences were observed. In our simulated hysteresis curve,decreased stiffness is observed at zero displacement starting from cycles 910. The stiffness at the intercept wasmaintained pretty much the same as the cyclic pushover proceeded. In contrast in the experimental curve, thepinching affect did not occur until the column passed its ultimate strength (after cycles 1314). The stiffness,as the curve passed though zero displacement, dropped drastically to decrease in the later cycles.

4.2 Strain distribution versus column height of an extreme longitudinal reinforcing bar

The top-left corner longitudinal bar of the cross-section of the reinforced concrete bridge was selected asthe monitoring fiber. Maximum strain distribution along bridge column height of an extreme longitudinalbar at the top-left corner of the cross-section during cyclic pushover up to 6 % drift ratio for both model1 and model 2 are rendered in Fig. 8a. The bar strain based on model 2 (with bond-slip/strain penetrationeffect considered) was smaller than model 1 (without bond-slip/strain penetration effect considered). Evenwithout experimental data available, according to Zhao and Sritharan [10], the model without bond-slip/strainpenetration effect considered usually overestimated the predictions of the bar strain. Our models also provedthe same observations and findings. In addition, the maximum strain distribution along the bridge columnheight of an extreme longitudinal bar at the top-left corner of the cross-section at 1.6y for both model 1 andmodel 2 are shown in Fig. 8b.

Sudden longitudinal reinforcing bar strain increments are observed in the calculated strain values near 54 cmin both Figs. 8a, b. This is the end node of the 1st beam-column element. This is because the longitudinal bar


0

50

100

150

200

250

300

350

Model 1 (w/o strain penetration)

Model 2 (w/ strain penetration)

Col

umn

Hei

ght

(mm

)

Bar Strain Bar Strain

0

50

100

150

200

250

300

350

0 0.02 0.04 0.06 0.08 0.1 0 0.005 0.01 0.015 0.02 0.025 0.03

Model 1 (w/o strain penetration)

Model 2 (w/ strain penetration)

Col

umn

Hei

ght

(mm

)

(b)(a)

Monitored bar

Loading

direction

y

z

Monitored bar

Loading

direction

y

z

Fig. 8 Maximum strain distribution along bridge column height of an extreme longitudinal bar at top-left corner of cross-section:a during cyclic pushover up to 6 % drift ratio (model 1 and model 2); b 1.6y (model 1 and model 2)

strain values above and below this node are calculated at Gauss integration points that belong to two beam-column elements. The interpolation algorithm in OpenSees does not guarantee consistency of fiber strains ofadjacent elements. The same observations are also noted by Zhao and Sritharan [10].

5 Conclusions

Based on multi-scale simulation approaches, i.e., global structure scale, structural elements scale, and micro-scopic scale, innovative analytical models are proposed to model and assess the reinforced concrete columnsperformance under cyclic pushover analysis with combined damage mechanisms including concrete cracking,concrete strength degradation due to concrete spalling, longitudinal reinforcing bars buckling, and bond-slip(strain penetration) between longitudinal reinforcing bars and concrete. Two multi-scale nonlinear finite ele-ment models with and without the bond-slip (or strain penetration) of a reinforced concrete bridge column werecreated using the Open System for Earthquake Engineering Simulation (OpenSees) utilizing the programsbuilt-in materials, displacement-based inelastic beam-column elements, and zero-length section elements.

Comparisons between the measured reinforced concrete bridge column top hysteresis lateral force versuslateral displacement and the numerical simulation results, which were obtained with and without the zero-lengthsection element to capture the bond-slip (or strain penetration) effect, strength degradations of concrete andlongitudinal reinforcing bars, and pinching behaviors were presented. In addition, the numerical simulationsof strain of the longitudinal reinforcing bars located at the corner of cross-section along the bridge columnheight with and without bond-slip (or strain penetration) effect were also presented. It was observed that thecolumn end rotation due to bond-slip (or strain penetration) effect could reduce longitudinal bar strains, thusstresses, in the column.

The proposed numerical models are capable to capture, advance, and improve the accuracy in the predictionsof the realistic nonlinear flexural failure behaviors of reinforced concrete bridge columns during seismic events.The importance of the proposed models in this paper is to advance and improve the accuracy in the predictionsof the realistic nonlinear behaviors of reinforced concrete bridge columns during seismic events.

It is noted that our proposed analytical models are limited to flexural failure governed of reinforced bridgecolumns. Axialflexural interaction, not axialflexuralshear interaction, is considered. However, our proposedanalytical models could provide accurate predictions in the assessment of performance of flexural failure


governed reinforced concrete columns during seismic events as well as for the applied research and industrialapplications.

Furthermore, the outcomes of current research will also pave the way for future research in the seismicresponse simulations for bridge columns retrofitting with advanced composites jackets [2,3,15,45], assessmentof post-earthquake condition of bridges [6], combination of shearflexuralaxial interaction effects [4649],the seismic response simulations of prototype reinforced concrete bridges [50], the short lap splices effects[51], assessment of seismic performance of squat reinforced concrete bridge columns considering inelasticshearflexural interaction by an ABAQUS User Element or OpenSees with new user elements, biaxial bendingeffects [52], improved damage models for concrete [53], improved bond-slip models with bond-slip stressdegradation, and performing more simulations for available experimental data for parameter characterizationsof our proposed analytical models.

Acknowledgments This work was in part sponsored by the 20132014 California State University at Long Beach Research,Scholarship and Creative Activity (RSCA) Awards for Assigned Time.

References

1. Chang, Y.S., Li, Y.F., Loh, C.H.: Experimental study of seismic behaviors of as-built and carbon fiber reinforced plasticsrepaired reinforced concrete bridge columns. J. Bridge Eng. ASCE 9(4), 391402 (2004)

2. Haroun, M., Elsanadedy, H.: Behavior of cyclically loaded squat reinforced concrete bridge columns upgraded with advancedcomposite-material jackets. J. Bridge Eng. ASCE 10(6), 741748 (2005)

3. Haroun, M., Elsanadedy, H.: Fiber-reinforced plastic jackets for ductility enhancement of reinforced concrete bridge columnswith poor lap-splice detailing. J. Bridge Eng. ASCE 10(6), 749757 (2005)

4. Spacone, E., Filippou, F., Taucer, F.: Fiber beam-column model for nonlinear analysis of R/C frames, part I: formula-tion. Earthq. Eng. Struct. Dyn. 25, 711725 (1996)

5. Spacone, E., Filippou, F., Taucer, F.: Fiber beam-column model for nonlinear analysis of R/C frames, part II: applica-tions. Earthq. Eng. Struct. Dyn. 25, 727742 (1996)

6. Lee, W.K., Billington, S.: Modeling residual displacements of concrete bridge columns under earthquake loads using fiberelements. J. Bridge Eng. ASCE 15(3), 240249 (2010)

7. Kim, T.H., Lee, K.M., Chung, Y.S., Shin, H.M.: Seismic damage assessment of reinforced concrete bridge columns. Eng.Struct. 27(4), 576592 (2005)

8. Park, R., Paulay, T.: Reinforced Concrete Structures. Wiley, New York (1975)9. Girard, C., Bastien, J.: Finite-element bond-slip model for concrete columns under cyclic loads. J. Struct. Eng.

ASCE 128(12), 15021510 (2002)10. Zhao, J., Sritharan, S.: Modeling of strain penetration effects in fiber-based analysis of reinforced concrete structures. ACI

Struct. J. 104(2), 133141 (2007)11. Gomes, A., Appleton, J.: Nonlinear cyclic stress-strain relationship of reinforcing bars including buckling. Eng.

Struct. 19(10), 822826 (1997)12. Tapan, M., Aboutaha, R.: Strength evaluation of deteriorated RC bridge columns. J. Bridge Eng. ASCE 13(3), 226236 (2008)13. Dhakal, R.P., Maekawa, K.: Modeling for postyield buckling of reinforcement. J. Struct. Eng. ASCE 128(9),

11391147 (2002)14. Paulay, T., Priestley, M.J.N.: Seismic Design of Reinforced Concrete and Masonry Buildings. Wiley, New York (1992)15. Priestley, M.J.N., Seible, F., Calvi, G.M.: Seismic Design and Retrofit of Bridges. Wiley-Interscience, New York (1996)16. Mazzoni, S., McKenna, F., Scott, M., Fenves, G.: Open system for earthquake engineering simulation user command-

language manualOpesnSees Version 2.0. Pacific Earthquake Engineering Research Center, University of California, Berke-ley, CA (2009)

17. Ju, J.W., Chen, T.M.: Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidalinhomogeneities. Acta Mech. 103, 103121 (1994)

18. Ju, J.W., Chen, T.M.: Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomo-geneities. Acta Mech. 103, 123144 (1994)

19. Sun, L.Z., Ju, J.W., Liu, H.T.: Elastoplastic modeling of metal matrix composites with evolutionary particle debonding. Mech.Mater. 35, 559569 (2003)

20. Sun, L.Z., Liu, H.T., Ju, J.W.: Effect of particle cracking on elastoplastic behaviour of metal matrix composites. Int. J. Numer.Methods Eng. 56, 21832198 (2003)

21. Liu, H.T., Sun, L.Z., Ju, J.W.: An interfacial debonding model for particle-reinforced composites. Int. J. DamageMech. 13, 163185 (2004)

22. Liu, H.T., Sun, L.Z., Ju, J.W.: Elastoplastic modeling of progressive interfacial debonding for particle-reinforced metalmatrix composites. Acta Mech. 181(1-2), 117 (2006)

23. Ju, J.W., Ko, Y.F., Ruan, H.N.: Effective elastoplastic damage mechanics for fiber reinforced composites with evolutionarycomplete fiber debonding. Int. J. Damage Mech. 15(3), 237265 (2006)

24. Ju, J.W., Ko, Y.F., Ruan, H.N.: Effective elastoplastic damage mechanics for fiber reinforced composites with evolutionarypartial fiber debonding. Int. J. Damage Mech. 17(6), 493537 (2008)

25. Ju, J.W., Ko, Y.F.: Micromechanical elastoplastic damage modeling of progressive interfacial arc debonding for fiberreinforced composites. Int. J. Damage Mech. 17, 307356 (2008)


26. Ju, J.W., Ko, Y.F., Zhang, X.D.: Multi-level elastoplastic damage mechanics for elliptical fiber reinforced composites withevolutionary complete fiber debonding. Int. J. Damage Mech. 18(5), 419460 (2009)

27. Ju, J.W., Yanase, K.: Elastoplastic damage micromechanics for elliptical fiber composites with progressive partial fiberdebonding and thermal residual stresses. Theor. Appl. Mech. 35(13), 137170 (2008)

28. Ju, J.W., Yanase, K.: Micromechanical elastoplastic damage mechanics for elliptical fiber-reinforced composites withprogressive partial fiber debonding. Int. J. Damage Mech. 18(7), 639668 (2009)

29. Ju, J.W., Yanase, K.: Micromechanics and effective elastic moduli of particle-reinforced composites with near-field particleinteractions. Acta Mech. 215(1), 135153 (2010)

30. Ju, J.W., Yanase, K.: Micromechanical effective elastic moduli of continuous fiber-reinforced composites with near-fieldfiber interactions. Acta Mech. 216(14), 87103 (2011)

31. Ju, J.W., Yanase, K.: Size-dependent probabilistic micromechanical damage mechanics for particle reinforced metal matrixcomposites. Int. J. Damage Mech. 20(7), 10211048 (2011)

32. Xu, B.W., Ju, J.W., Shi, H.S.: Progressive micromechanics modeling for pullout energy of hooked-end steel fiber in cement-based composites. Int. J. Damage Mech. 20(6), 922938 (2011)

33. Ko, Y.F., Ju, J.W.: Effects of fiber cracking on elastoplastic damage behavior of fiber reinforced metal matrix composites. Int.J. Damage Mech. 22(1), 4867 (2013)

34. Ko, Y.F., Ju, J.W.: New higher-order bounds on effective transverse elastic moduli of three-phase fiber reinforced compositeswith randomly located and interacting aligned circular fibers. Acta Mech. 223(11), 24372458 (2012)

35. Ko, Y.F., Ju, J.W.: Effective transverse elastic moduli of three-phase hybrid fiber reinforced composites with randomlylocated and interacting aligned circular fibers of distinct elastic properties and sizes. Acta Mech. 224(1), 157182 (2013)

36. Pan, H.H., Weng, G.J.: Study on the strain-rate sensitivity of cementitious composites. J. Eng. Mech. ASCE 136(9),10761082 (2010)

37. Shen, L., Li, J.: Effective elastic moduli of composites reinforced by particle or fiber with an inhomogeneous interphase. Int.J. Solids Struct. 40, 13931409 (2003)

38. Shen, L., Li, J.: Homogenization of a fiber/sphere with an inhomogeneous interphase for the effective elastic moduli ofcomposites. R. Soc. A 461, 14751504 (2005)

39. Saatcioglu, M., Razvi, S.R.: Strength and ductility of confined concrete. J. Struct. Eng. ASCE 118(6), 15901607 (1992)40. Chang, G., Mander, J.: Seismic energy based fatigue damage analysis of bridge columns: part Ievaluation of seismic

capacity. NCEER Technical Report 94-0006 (1994)41. Gomes, A., Appleton, J.: Nonlinear cyclic stress-strain relationship of reinforcing bars including buckling. Eng.

Struct. 19(10), 822826 (1997)42. Dhakal, R., Maekawa, K.: Modeling for postyield buckled of reinforcement. J. Struct. Eng. 128(9), 11391147 (2002)43. Menegotto, M., Pinto, P.: Method of analysis for cyclically loaded RC plane frames including changes in geometry and

non-elastic behavior of elements under combined normal force and bending. Symp. Resistance and ultimate deformabilityof structures acted on by well defined repeated loads, IABSE Reports 13, Lisbon (1973)

44. Caltrans Seismic Design Criteria, Ver. 1.6. California Department of Transportation, California (2010)45. Choi, K.K., Xiao, Y.: Analytical model of circular CFRP confined concrete-filled steel tubular columns under axial

compression. J. Compos. Constr. ASCE 14(1), 125133 (2010)46. Zhang, J., Xu, S.Y.: Seismic response simulations of bridges considering shear-flexural interaction of columns. Int. J. Struct.

Eng. Mech. 31(5), 545566 (2009)47. Zhang, J., Xu, S.Y., Tang, Y.C.: Inelastic displacement demand of bridge columns considering shear-flexure interac-

tion. Earthq. Eng. Struct. Dyn. 40(7), 731748 (2011)48. Xu, S.Y., Zhang, J.: Hysteretic shear-flexure interaction model of reinforced concrete columns for seismic response assess-

ment of bridges. Earthq. Eng. Struct. Dyn. 40(3), 315337 (2011)49. Xu, S.Y., Zhang, J.: Axial-shear-flexure interaction hysteretic model for RC bridge columns under combined actions. Eng.

Struct. 34(1), 548563 (2012)50. Johnson, N., Saiidi, M., Sanders, D.: Nonlinear earthquake response modeling of a large-scale two-span concrete bridge.

J. Bridge Eng. ASCE 14(6), 460471 (2009)51. Melek, M., Wallace, J.W.: Cyclic behavior of columns with short lap splices. ACI Struct. J. 101(6), 802811 (2004)52. Chang, S.Y.: Experimental studies of reinforced concrete bridge columns under axial load plus biaxial bending. J. Struct.

Eng. ASCE 136(1), 1225 (2010)53. Gopinath, S., Rajasankar, J., Iyer, N.R.: Nonlinear analysis of RC structures using isotropic damage model. Int. J. Damage

Mech. 21(5), 647669 (2012)

Nonlinear static cyclic pushover analysis for flexural failure of reinforced concrete bridge columns with combined damage mechanismsAbstract1 Introduction2 Theoretical backgrounds for analytical models2.1 Global structure scale2.2 Structural element scale2.3 Microscopic scale

3 Description and formulation of nonlinear finite element models3.1 Material model for concrete fiber behavior3.2 Material model for longitudinal and transverse reinforcing bars3.3 Material model for bond-slip3.4 Section properties3.5 Displacement-based beam-column element3.6 Flexural models (fiber-based models)Model 1Model 2

4 Comparisons of numerical simulations and experimental results4.1 Hysteresis force versus displacement curve at column top4.2 Strain distribution versus column height of an extreme longitudinal reinforcing bar

5 ConclusionsAcknowledgmentsReferences

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art-3A10.1007-2Fs00707-013-0970-7