Engineering Structures 104 (2015) 174–192

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Engineering Structures

journal homepage: www.elsevier .com/locate /engstruct

Nonlinear line-element modeling of flexural reinforced concrete walls

http://dx.doi.org/10.1016/j.engstruct.2015.08.0370141-0296/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 206 685 2563; fax: +1 206 543 1543.E-mail address: lowes@uw.edu (L.N. Lowes).

Joshua S. Pugh a, Laura N. Lowes b,⇑, Dawn E. Lehman b

a EDG, Inc., Houston, TX, United StatesbCivil and Environmental Engineering, University of Washington, Seattle, WA, United States

a r t i c l e i n f o

Article history:Received 26 February 2015Revised 14 July 2015Accepted 28 August 2015Available online 19 October 2015

Keywords:AnalysisNonlinear analysisModelingConcreteConcrete wallEarthquake engineering

a b s t r a c t

The research presented here developed a model for simulating the nonlinear cyclic response of flexure-controlled concrete walls, which meets the dual objectives of accuracy and computational efficiency. Theproposed model represents a significant advancement in that it provides accurate simulation of the dom-inate failure mechanism exhibited by flexural walls in the laboratory and field: compression-controlledfailure characterized by simultaneous crushing of concrete and buckling of longitudinal reinforcement.The first steps in the model development effort comprised assembly of an experimental database andreview of current modeling approaches for walls (e.g., lumped plasticity, distributed plasticity, andcontinuum elements). Model evaluation indicated that the most viable option to achieve accuracy andefficiency was the use of beam–column line elements with fiber-type cross-section models at the integra-tion points. Initially, both displacement-based and force-based element formulations were evaluated;however, the displacement-based formulation resulted in an inaccurate representation of the axial forcedistribution along the length of the element. Therefore, only the force-based formulation was chosen forfurther study. The basic model included standard 1D constitutive models for confined concrete, plain con-crete and reinforcing steel. Comparing simulated and measured response data showed that the concreteand steel material models must be regularized using a mesh-dependent characteristic length and amaterial-dependent post-yield energy to enable accurate, mesh-objective simulation of strength lossdue to compression failure. The post-yield energy values were determined using relevant experimentaldata, an important but missing component of prior research on material regularization. The results of thisstudy show that use of the regularized constitutive models significantly improved the accuracy ofresponse predictions.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Reinforced concrete walls are used commonly in mid- andhigh-rise buildings to resist earthquake loads. As such, numericalmodels are required that provide accurate prediction of theresponse of these walls under earthquake loading, including accu-rate prediction of the stiffness, strength, drift capacity and hys-teretic response. Engineers require these models to enableperformance-based design of walled buildings, and researchersrequire these models to investigate the behavior and performanceof walls as well as to advance seismic design procedures for walls.

The primary objective of the research present herein was todevelop practical recommendations for modeling slender walls,which respond primarily in flexure. Computationally efficientbeam–column elements were selected; these elements are read-ily available in nonlinear structural analysis software packages.

Therefore a secondary objective was to identify the domain forwhich these models provide accurate prediction of response.

Performance-based earthquake engineering requires accurateassessment of all aspects of the response, including strengthdegradation. To better understand the response and failure modeof flexural walls, a brief review of experimental investigation ofwall behavior is provided. The review indicates that even whendesigned to meet codified provisions for tension-controlledresponse, the most common failure mode for slender flexural wallsis compression, which includes crushing of core concrete and barbuckling. The predominance of this failure mode is confirmed bypost-earthquake reconnaissance. Less common is a true tension-controlled failure, with direct or low-cycle fatigue induced fractureof the bars. Thus, the review indicates that nonlinear wall modelsmust be capable of simulating strength loss due to compressionfailure.

Commonly used elements for wall modeling are reviewed withan emphasis on line elements, which can provide computationallyefficient simulation of response. Current line-element formulations

Nomenclature

Acv shear area of the wall defined equal to 5/6Ag

Ag gross cross-sectional area of the wallB hardening ratio for reinforcing steelBR buckling rupture failure modeCB compression buckling failure modeEc elastic modulus of concrete defined per ACI 318 (2011)f 0c concrete compressive strengthfy yield strength reinforcementfu ultimate strength reinforcementGc elastic concrete shear modulusGeff effective concrete shear modulus, which is defined to be

less than elastic shear modulus to account for damageGf concrete fracture energyGfc unconfined concrete crushing energyGfcc confined concrete crushing energyGs steel hardening energyheff height at which base shear is applied in the laboratoryhwall wall heightH wall heightIP integration point for distributed-plasticity beam–

column element at which a fiber-section is used todetermine flexural response

ks shear form factor, taken as 5/6 for planar wallKc ratio of confined to unconfined concrete crushing

energyK ratio of confined to unconfined concrete compressive

strengthlw wall lengthLgage gage length used to measure steel stress–strain proper-

tiesLIP length associated with an integration point (IP), as

determined by the quadrature rule used in the elementformulation

M moment at the base of the wall (Table 1)Mmax maximum base moment developed during the testMn nominal flexural strength of the wall computed per ACI

318 (2011) using measured concrete and steel strengths

My yield moment is defined by first yield of the boundaryelement longitudinal reinforcement

N wall axial loadR bar rupture failure modet wall thicknessV shear force at the base of the wall (Table 1)Vbase shear force at the base of the wallVmax maximum base shear developed during the testVn shear strength of the wall computed per ACI 318 (2011)

using measured concrete and steel strengthc shear strainDu drift capacity = drift at which the lateral load carrying

capacity of the wall dropped to 80% of the historic max-imum, for drift demands in excess of historic drift de-mands

Dv lateral displacement at the top of the wall due to sheardeformation

Dy measured drift at computed yield moment, My

e0 concrete strain at maximum compressive strengthe20u regularized concrete strain at 80% compressive strength

losses computed maximum steel strain at nominal flexural

strength, Mn

eu_exp measured steel strain at ultimate strengtheu regularized steel strain at ultimate strengthey steel yield strainqlong gross longitudinal reinforcement ratio computed using

the total wall area and total area of longitudinal rein-forcement

qh horizontal reinforcement ratio within the web of thewall

qvol volumetric reinforcement ratio for confined boundaryelements

/ fiber section curvature

J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192 175

use conventional 1D constitutive models for concrete andreinforcing steel and fiber-type section models; these models areused commonly for reinforced concrete beams and columns. Theseconventional approaches are evaluated using data from 21 slenderwall tests. The results show that for walls exhibiting compression-controlled response, conventional models fail to converge to asingle solution. An alternative approach is developed using regu-larized constitutive models. The proposed modeling approach iscompared with experimental results and shown to provide accu-rate simulation of the nonlinear response of slender flexural walls.

1.1. Experimental behavior of slender concrete walls

Experimental testing indicates that slender walls withshear-span ratios (distance between the heights at which the wallhas zero and maximum moment divided by the length of the wall)greater than 2.0 typically respond in flexure [38]. Walls with shear-span ratios exceeding 2.0 are common and therefore accurate mod-eling of their response, including strength degradation, is neededfor seismic design and assessment of buildings that have structuralwalls as their primary lateral load system.

As part of a companion research effort, Birely [8] evaluated over60 slender (shear-span ratio greater than 2.0) experimental wallspecimens. Study of the tests indicated that failure, i.e. degradationin the lateral load carrying capacity, of a flexural wall is a result of

one or more damage modes: (1) compression damage whichincludes crushing of boundary-element concrete and buckling ofboundary element reinforcing steel (Fig. 1a), (2) low-cycle fatigueof the reinforcement, which includes buckling and subsequentfracture of longitudinal steel, and (3) rupture of longitudinal steel(Fig. 1b). For some walls, the primary failure mode is shear; dataanalysis suggests that this mode is exhibited for high sheardemands in particular if the ratio of the shear demand (Vu) toACI (ACI 318 2011) capacity (Vn) exceeds 1. Fig. 2 shows theobserved damage modes of the slender-wall database. As indicatedin the plot, the majority of the walls exhibited a compression-controlled failure mode (i.e., boundary element crushing). It is ofnote that all of the walls that exhibited compression-controlledfailure satisfied the ACI tension-controlled design limit (ACI 3182011). Therefore, simply designing a wall to meet the tension-control criterion does not necessarily result in tension-controlledfailure.

Although slender walls may not exhibit typical shear damage(e.g., diagonal cracking or crushing along a diagonal line), experi-mental research indicates that the impact of shear on wallresponse is not negligible. For example, consider experimentaltests on well-confined planar walls by Lowes et al. [23]. In thisstudy, two of the tests were designed to be nominally identicalwith different level of shear demand (the plastic shear demandwas increased by decreasing the moment applied to the top of

Fig. 1. Typical damage modes in slender walls (a) compressive damage and (b)tension damage.

Fig. 2. Percentage of slender rectangular wall specimens exhibiting a specificdamage mode.

176 J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192

the specimen and thereby decreasing the effective height).Although neither specimen exhibited a shear failure mode, thespecimen with the higher shear demand had a lower drift capacity.Additionally, these flexural walls sustained shearing deformationsthat accounted for as much as 30% of total wall deformation [23].Both results indicate that including shear in wall models isimportant.

1.2. Prior research studies on simulation of walls

There are a wide range of modeling approaches that have beenused to simulate the behavior of concrete walls. In engineeringpractice, it is typical to model part or all of the building structure;therefore, computationally efficient models are used. Mostnonlinear analysis software used in the design office includes lineelements. Within the category of line-element models, a range of

formulations exist including concentrated hinge models,fiber-section lumped plasticity models and distributed plasticitybeam–column elements. Previous research efforts have also stud-ied the use of line-element models, as described below. Howeverfew programs have conducted a systematic evaluation of thisapproach using a substantial experimental database, and thereforeit is difficult to determine their accuracy solely using the results ofprior research.

Other researchers have used continuum models to simulatewalls. This category of modeling typically includes shell elements,fiber shell elements, and solid elements. These models are advan-tageous in that nonlinear shear response and flexure–shear inter-action are captured by the model. However, continuum modelingapproaches are computationally expensive and therefore are notcommonly used in practice. Some of the relevant research usingboth line-element and continuum-element models are summa-rized below. Additional information may be found in Pugh [38].

One of the simplest and most basic approaches to simulatingthe nonlinear response of walls is to use a lumped-plasticity line-element model. The lumped-plasticity model locates a flexuralhinge at the critical section with an elastic beam–column elementused to model the remainder of the wall. Since the concrete andsteel materials are not modeled explicitly, a multi-linearmoment–rotation response curve is used. The only standardizedcurve for walls is specified in ASCE/SEI [2,15]. ASCE 41 provides abackbone curve; model parameters for the cyclic response arenot provided. The parameters of the backbone curve depend onthe axial load ratio, shear stress demand and confinement.Although the approach is computationally efficient, its accuracyis limited to the calibrated parameter values. In addition, themoment–rotation response of the hinge must be defined prior tothe analysis; therefore the model cannot account for the impacton response of variation in axial or shear load. If hinging occursaway from the assumed critical section, multiple analyses arerequired in which additional hinges are introduced. Research indi-cates that the current ASCE 41 parameters may underestimate thestrength and deformability of more slender walls [26]. Because ofits simplicity, computational efficiency and robustness, this model-ing approach is used commonly in practice and by researchers toinvestigate the seismic response and design of walled buildings[36,10,39].

More sophisticated lumped-plasticity models simulate theflexural response of the wall cross section using a fiber-typediscretization model. This approach permits computation of themoment–curvature response of the critical section; a plastic-hinge length is commonly used to compute the moment–rotationresponse. The fiber discretization of the section comprises concreteand steel ‘‘fibers” for which stress–strain response is defined byone-dimensional constitutive models. The curvature and averageaxial strain imposed on the section determine the axial strain ofthe individual fibers; the moment and axial loads are computedfrom the fiber stresses and areas. This type of lumped-plasticitymodel simulates the impact of axial load on flexural response.The cyclic response of the section and component are also modeleddirectly if cyclic material constitutive models are employed. Thereare disadvantages to this modeling approach. Thesemodels employthe assumption of a linear strain field, which may introduce errorfor some wall configurations (e.g. long planar walls and walls withlong flanges). Flexure–shear interaction is also neglected. Finally,the value and static nature of the plastic hinge length mayintroduce errors. As discussed above, multiple analyses withupdated models may be required to accurately simulate thedistribution of nonlinearity along the wall height. Although a viableapproach, no studies were found in the literature validating it forwall modeling.

J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192 177

An alternative to the lumped-plasticity element is thedistributed-plasticity beam–column element. Typically, a fibercross section is incorporated into these elements. Displacement-based elements employ the traditional assumptions of a linear cur-vature field and constant average axial deformation field along thelength of the element. Force-based elements employ the assump-tions of a linear moment and constant axial force distributionsalong the length of the element. Because these elements are usedwith a displacement-based finite element analysis, in whichcompatibility between elements is satisfied apriori and nodal dis-placements are determined to satisfy nodal equilibrium, the useof force-based elements requires an intra-element solution todetermine the element force distribution that produces the nodaldisplacements that satisfy compatibility between elements [41].

Distributed-plasticity elements include multiple nonlinearfiber-type sections along the length of the element. This enablesexplicit simulation of nonlinear action at multiple locations upthe height of the wall, overcoming a significant shortcoming oflumped-plasticity models. Some of the limitations noted with thefiber cross-section lumped-plasticity element are shared by thedistributed-plasticity elements including (i) the assumption of alinear strain field across the section and (ii) the decoupling of flex-ure and shear response. Additionally, a relatively fine mesh of tra-ditional two-node displacement-based elements and/or the use ofhigher-order displacement-based elements, with three or morenodes, may be required to accurately simulate response atlocations up the height of the wall where inelastic action issignificant. Further, force-based elements may exhibit conver-gence problems when strength degradation occurs. However,distributed-plasticity elements are computationally efficient, typi-cally numerically robust, and represent a potentially powerful toolfor nonlinear analysis of walled buildings. Boivin and Paultre [9]employ a fiber-section model and the distributed-plasticity force-based beam–column element implemented in OpenSees to investi-gate shear and moment demands in slender walls subjected toearthquake loading.

A number of variations on the fiber-section hinge model and thefiber-section distributed-plasticity beam–column element havealso been employed and implemented in research and commercialsoftware to simulate wall response. For example, Orakcal et al. [32]employ the multiple-vertical-line-element model (MVLEM) pro-posed by Vulcano et al. [46] to simulate the nonlinear flexuralresponse of slender walls. The MVLEM is essentially a finite-length fiber-section model combined with a horizontal spring thatsimulates the shear flexibility of the wall; Orakcal et al. employstandard one-dimensional cyclic constitutive models to definethe response of concrete and steel fibers and an elastic shear-response model. Orakcal and Wallace [31] show that the MVLEMcan provide accurate simulation of wall response for walls forwhich the assumptions of plane sections remain plane, elasticshear response, and decoupling of flexure and shear response arevalid. Ghobarah et al. [18] and Galal [17] simulated the responseof a five-story wall subjected to dynamic shake-table loading[13] using a model in which fiber-type sections were placedbetween elastic two-dimensional plane-stress elements (a plane-sections-remain-plane constraint was imposed at the interfacebetween the fiber-section and the plane-stress elements); themodel provided reasonably accurate simulation of strength andhysteretic response under dynamic loading. Finally, the fiber shellwall element implemented in Perform (http://www.csi.berkeley.edu) may be considered a variation of the fiber-type section modelas flexural response is determined by the stress–strain response ofvertical fibers with shear response determined by an independentone-dimensional shear model.

Multiple researchers have sought to improve simulation ofwalls using fiber-section distributed-plasticity elements by incor-porating simulation of flexure–shear interaction. In these models,fiber response is typically defined by a two-dimensional strainfield, the assumption that fibers are in a state of plane stress, anda two-dimensional constitutive model. Examples of this type ofmodel include that proposed by Jiang and Kurama [21] and byPetrangeli [37], both of which use the ‘‘microplane” model [3,5,4,33]to define multi-dimensional concrete response. Response 2000[7] employs a similar approach with multi-dimensional concreteresponse defined using the Modified Compression Field Theory[45]. Jiang and Kurama [21] show that this modeling approachenables simulation of nonlinear shear response, interaction of flex-ure and shear mechanism and reasonably accurate simulation ofobserved response for a limited number of wall specimens. How-ever, while these models provide the potential for improved simu-lation of response, the flexure–shear fiber-section elements aremore computationally demanding and less numerically robustthan the models in which flexure and shear response are decou-pled. Additionally, these models are limited by the assumptionabout the shear strain distribution on the section. Finally, thisapproach is limited by the difficulty of using line elements tomodel complex wall geometries.

Analysis using continuum elements (i.e. shell and solidelements) has the potential to provide more accurate simulationof nonlinear wall response than the models discussed above. Con-tinuum models do not require simplifying assumptions about thestrain or stress field at the section level, inherently simulate theinteraction of nonlinear flexure, shear and torsional responsemodes, and facilitate representation of complex wall configura-tions (e.g., C, T, and H cross sections). For isolated planar and, insome cases, non-planar walls, two-dimensional plane-stress ele-ments can provide accurate results. For non-planar walls, fibershell elements (where fibers are two-dimensional and concretefiber response is determined using a two-dimensional plane-stress concrete constitutive model) or three-dimensional brick ele-ments can provide accurate simulation of response. The challengewith continuum analyses is that they are computationallydemanding; this typically precludes their use in standard practicein particular when full building analyses subjected multipleground motions are required. Additionally, implicit solution algo-rithms are typically used and are often plagued by convergenceissues, especially when strength loss initiates. Research effortshave addressed the use of continuum elements. Palermo and Vec-chio [35] employed VecTor2, which utilizes two-dimensional planestress elements and a variant of the two-dimensional modifiedcompression field theory (MCFT) reinforced concrete constitutivemodel, to achieve satisfactory simulation of walls tested byPalermo and Vecchio [34]. Continuum-type models offer thepotential for accurate simulation of both local response quantitiesand response mechanisms beyond flexure; however, the high com-putational demands associated with these models makes themimpractical for use in practice or in research addressing the designand performance of structural systems.

The primary limitation of the prior studies is that a very limiteddata set was used to validate the models; typically each previouslystudied modeling approach was validated using data from 1 to 4test specimens. The literature is rich with test data for structuralwalls, and a more thorough evaluation of wall models is warranted.The research presented herein evaluated commonly used line-element models using data for slender walls with a range of designcharacteristics tested using varying procedures and protocols. Thefollowing sections describe the data set developed for the studyand the evaluation procedure and results.

178 J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192

2. Evaluation of line elements seismic analysis of walledbuildings

2.1. Experimental data set

Previously tested rectangular slender wall specimens were usedto evaluate the modeling approaches discussed above. Thespecimens were selected based on the following criteria:

� Planar (rectangular) wall specimen subjected to in-planeflexure, shear and axial loading.

� Specimen failure resulted from deteriorating flexuralresponse. This included tension and compression controlledresponse (e.g., bar fracture or concrete crushing and barbuckling).

� Wall specimen thickness in excess of 76 mm (3 in.). Wallspecimens thinner than this were not considered to exhibitbehavior representative of full-scale reinforced concretewalls.

� Data required to fully define a numerical model were provided.Required data included concrete compressive strength,reinforcing steel stress–strain response, specimen geometry,reinforcement layout, and test specimen boundary conditionsin the laboratory.

� Data required to evaluate simulation results were provided.These included global load–displacement response and theobserved failure mechanism.

The assembled data set included 21 reinforced concrete wallspecimens from 7 experimental research programs (Table 1).Quantities in Table 1 are defined as follows:

Geometry

� Scale = t/305 mm where t is the thickness of the wall in mm.� Cross-sectional aspect ratio = lw/t, where lw is the length of thewall.

� Vertical aspect ratio = h/lw, where h is the height of the wall.

Material properties

� f 0c = concrete compressive strength.� fyBE = yield strength of boundary element longitudinalreinforcement.

Reinforcement

� qlong = gross longitudinal reinforcement ratio computedusing the total wall area and total area of longitudinalreinforcement.

� qh = horizontal reinforcement ratio within the web of the wall.� qvol = volumetric reinforcement ratio for confined boundaryelements.

Loading

� Shear span ratio =M/(Vlw), whereM is the moment developed atthe base of the wall, V is the shear developed at the base of thewall and lw is the length of the wall. Note that the shear spanratio equals the vertical aspect ratio if zero moment is appliedat the top of the wall.

� Axial load ratio = N/Agfc, where N is the axial load at the base ofthe wall (including self-weight of the specimen computedassuming a unit weight of 23.6 kPa (150 lb/ft3)), Ag is thegross area of the wall and fc is the concrete compressivestrength.

Measured strength

� Shear stress demand = Vmax=Acv

ffiffiffiffif 0c

q, where Vmax is the maxi-

mum base shear developed during the test, Acv is the shear area,taken equal to 5/6Ag for a rectangular section, and fc is theconcrete compressive strength.

� Shear demand-capacity ratio = Vmax/Vn, where Vmax is the max-imum base shear and Vn is the shear strength computed per ACI318 (2011) using measured concrete and steel strengths.

� Flexural strength ratio =Mmax/Mn, where Mmax is the maximumbase moment developed during the test and Mn is the nominalflexural strength of the wall computed per ACI 318 (2011) usingmeasured concrete and steel strengths.

Response

� Dy = measured drift at computed yield moment, My, where thecomputed yield moment is defined by first yield of the bound-ary element longitudinal reinforcement.

� Du = drift capacity = drift at which the lateral load carryingcapacity of the wall dropped to 80% of the historic maximum,for drift demands in excess of historic drift demands.

� es = computed maximum steel strain at nominal flexuralstrength, Mn. Note that per the ACI Code, a tension-controlledflexural section has es > 0.005.

� Failure mode indicates the primary mechanism causing loss oflateral load carrying capacity: concrete crushing and bucklingof longitudinal steel (CB), buckling followed by rupture of longi-tudinal steel (BR), or rupture of longitudinal steel (R).

A few important aspects of the database are of note. The aspectratio of the walls was generally greater than 2.0 (two had ratios of1.6) and the shear-demand capacity ratio was generally less than1.0 (one has a ratio of 1.1). Both of these characteristics areexpected for walls that exhibit a flexural response mode.

Nine (9) of the specimens sustained bar fracture; the remainingtwelve (12) specimens sustained a compression failure mode. Mostof the specimens were subjected to modest axial load ratios, withthe largest axial load ratio being 0.13. While a compression-controlled failure is not commonly expected for a wall with a mod-est axial load, prior research by Birely [8] and Pugh [38] indicatesthat compression-controlled response is common for modernwalls, even those designed to meet the tension-controlled flexuralresponse criteria laid out in ACI 318-11. The data in Table 1 (et)shows that for all of the specimens, including those sustainingcompressive failure modes, steel tensile strains greatly exceedthe ACI limit for tension-controlled flexure response of 0.005.

All of the specimens in the database were used to evaluate thebasic and proposed modeling approaches. Detailed analysis resultscannot be presented for each specimen due to length restrictions.Instead, results for the entire data set are tabulated where appro-priate and a selected group of specimens are used to illustratethe results. Specifically four specimens were selected for presenta-tion in the paper; these specimens span the range of failure modesidentified above as well as a range of shear demands. These spec-imens include: WSH4, which failed due to compression/buckling(CB); WSH1, which failed due to bar rupture (R) and had a rela-tively low shear demand; RW1, which failed due to bar bucklingfollowed by bar rupture (BR), and S6, which failed due to compres-sion/buckling (CB) and had a relatively large shear demand.

3. The basic model

Fiber-type beam–column elements have distinct advantages forsimulating the nonlinear response of reinforced concrete and steel

ble1

perimen

talda

taset.

Specim

enID

Author

Scale

l w/t

h/l w

f0 cf yBE

qv

qvo

lqh

M/(Vl w)

P/(A

gf0 c)

Vmax/(Acvpf0 c)

Vmax/V

nM

max/M

nD

yD

ue s

atM

nFa

ilure

mod

e

(MPa

)(psi)

(MPa

)(ksi)

(%)

(%)

(%)

(MPa

)(psi)

(%)

(%)

WSH

42Daz

ioet

al.[14

]0.49

13.3

2.02

40.9

5932

576

83.5

0.85

0.00

0.25

2.28

0.06

0.23

2.8

0.62

1.06

0.29

1.60

0.01

8CB

WSH

6Daz

ioet

al.[14

]0.49

13.3

2.02

45.6

6613

576

83.5

0.85

1.44

0.25

2.26

0.11

0.30

3.6

0.83

1.11

0.31

2.04

0.01

1CB

W1

Liu[22]

0.65

6.07

3.13

33.1

4801

458

66.4

1.24

2.26

0.40

3.13

0.08

0.19

2.3

0.46

58.8

0.64

2.98

0.01

7CB

PW2

Lowes

etal.[23

]0.50

20.0

1.20

40.3

5843

579

84.0

1.35

1.24

0.28

2.08

0.13

0.44

5.3

1.11

1.25

0.45

1.50

0.01

1CB

PW3

Lowes

etal.[23

]0.50

20.0

1.20

34.3

4980

354

51.3

1.68

1.37

0.28

2.00

0.10

0.37

4.4

0.88

1.51

0.24

1.22

0.01

1CB

PW4

Lowes

etal.[23

]0.50

20.0

1.20

29.5

4272

463

67.1

1.35

1.24

0.28

2.00

0.12

0.38

4.6

0.88

1.19

0.40

1.01

0.01

4CB

RW

2Th

omsenet

al.[43

]0.33

12.0

3.00

34.0

4925

434

63.0

1.15

1.17

0.33

3.13

0.09

0.22

2.7

0.52

1.16

0.55

2.35

0.01

6CB

S51

Vallenas

etal.[44

]0.37

21.2

1.26

34.5

5004

482

69.9

1.73

0.84

0.55

1.60

0.05

0.57

6.8

0.85

1.18

0.31

1.47

0.02

0CB

S62

Vallenas

etal.[44

]0.37

21.2

1.26

27.8

4033

482

69.9

1.73

0.84

0.55

1.60

0.05

0.53

6.4

0.80

1.12

0.32

1.65

0.02

0CB

WR20

Ohet

al.[30

]0.66

7.50

1.33

34.2

4960

449

65.1

0.62

1.43

0.28

2.00

0.10

0.25

3.0

0.76

1.08

0.35

2.82

0.01

6CB

WR10

Ohet

al.[30

]0.66

7.50

1.33

36.2

5250

449

65.1

0.62

2.85

0.36

2.00

0.10

0.24

2.9

0.65

1.08

0.47

2.82

0.01

7CB

WR0

Ohet

al.[30

]0.66

7.50

1.33

32.9

4772

449

65.1

0.62

0.00

0.28

2.00

0.11

0.25

3.0

0.74

1.17

0.52

2.14

0.01

5CB

WSH

2Daz

ioet

al.[14

]0.49

13.3

2.02

40.5

5874

583

84.6

0.56

1.01

0.25

2.28

0.06

0.19

2.3

0.53

1.12

0.27

1.75

0.02

3BR

WSH

3Daz

ioet

al.[14

]0.49

13.3

2.02

39.2

5685

601

87.2

0.85

0.96

0.25

2.28

0.06

0.24

2.9

0.67

1.1

0.32

2.07

0.01

8BR

WSH

5Daz

ioet

al.[14

]0.49

13.3

2.02

38.3

5555

584

84.7

0.40

0.75

0.25

2.28

0.14

0.23

2.8

0.59

1.08

0.20

1.52

0.01

1BR

W2

Liu[22]

0.65

6.07

3.13

70.7

1024

545

866

.41.24

2.26

0.47

3.13

0.04

0.14

1.7

0.37

62.1

0.55

2.91

0.02

7BR

RW

12Th

omsenet

al.[43

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structures in engineering practice. These types of models are desir-able as they offer (i) model definition on the basis of simple 1Dmaterial response models and structure geometry, (ii) the potentialfor accurate simulation of system response (for flexure-controlledsystems), and (iii) more moderate computational demands thancontinuum elements. The primary focus of the first part of thisstudy was to evaluate the accuracy with which this conventionalmodeling approach simulates the response of flexural walls.

3.1. Formulations

The OpenSees platform (http://opensees.berkeley.edu) and thedistributed-plasticity beam–column element formulations avail-able in that platform were employed for the current study. Boththe displacement- and force-based element formulations werestudied. Fig. 3 shows schematics of a model of a cantilever wall,one employing force-based elements and one employingdisplacement-based elements. For both element formulations, theGauss–Labotto numerical integration rule was used. This methodincludes integration points, IPs, located at the ends of the elementand distributed along the length of the element at locationsdefined by the integration method. At these IPs, flexural responseis modeled using a fiber-type section model and 1D concrete andsteel constitutive models. For both element formulations a linearshear response model was incorporated into the model. A meshrefinement study by Pugh [38] indicated that a section modelemploying 32 concrete fibers within the boundary element andsimilarly sized fibers within the web of the wall was required toproduce a converged solution for the section. Results are presentedfor different levels of mesh refinement at the element level (i.e.number of elements and number of fiber sections per element).

For the fiber sections, nonlinear material response was definedusing typical 1D concrete and steel response models (Fig. 4). Con-crete response was defined using the Yassin model [49] which isimplemented in OpenSees as Concrete02. Using this model, (i)pre-peak response in compression is defined by the Hognestad[19], (ii) post-peak response in compression is linear to a residualcompressive strength that is 20% of maximum strength, (iii) tensileresponse is bilinear to a residual tensile strength of zero, (iv) theunloading path from the compression envelope is bi-linear andfrom the tension envelope is linear, (v) reloading paths are linear,and (vi) reloading in tension and compression occurs immediatelyupon unloading to a state of zero stress. To define parameters inFig. 4 for a specific test specimen, (i) the reported concrete com-pressive strength (fc) was used, (ii) concrete strain at peak strength(e0) was defined such that the concrete modulus at zero strain (Ec)was equal to that defined by ACI 318 (2011), (iii) for confined con-crete, parameters K and e20 in Fig. 4 were determined using the rec-ommendations of Saatcioglu and Razvi [40], (iv) for unconfinedconcrete, e20 = 0.008, (v) concrete tensile strength was defined

equal to 0:33ffiffiffiffif 0c

qMPa 4

ffiffiffiffif 0c

qpsi

� �per Wong and Vecchio [48],

and (vi) concrete post-peak stiffness in tension was defined equalto 0.05 Ec per Yassin [49].

The OpenSees Steel02 material model [16] was used to simulatethe cyclic response of reinforcing steel. This model employs a bilin-ear envelope with unload–reload paths defined using Menegottoand Pinto [27] curves. Fig. 4 shows a cyclic strain history. Parame-ters defined in Fig. 4 were taken directly from the reported mate-rials properties for each specimen in the database, with b definedby the secant to the measured stress–strain curve from yield toultimate strengths. The OpenSees MinMax material model wasused with the Steel02 model to simulate loss of tensile strengthwhen steel tensile strain exceeded the measured rupture strainand to simulate loss of compressive strength when steel compres-

Fig. 3. Typical simulation model of test wall.

180 J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192

sive strain exceeded that resulting in 80% strength loss forsurrounding concrete (e20 in Fig. 4a).

Because the fiber-type section model does not simulate defor-mation due to shear, a shear response model was incorporated intothe force- and displacement-based element models. The shearresponse is incorporated using two different approaches, depend-ing on the type of element formulation. The displacement-basedbeam–column element formulation assumes Bernoulli–Euler beamtheory, and a shear spring was introduced in series with the beam–column elements at the bottom of the wall (Fig. 3). For theforce-based beam–column element, a shear-response model wasincorporated at the section level per Marini and Spacone [25].The shear response of the wall was assumed to be linear elastic,with V = cGeffksAcv where V is the shear force on the section, c isthe shear strain on the section, Geff is the effective shear modulusof the section, ks the shear form factor taken as 5/6 for rectangularwalls, and Acv is the shear area of the section. For the current study,Geff = 0.1Gc = 0.04Ec, where Ec is the elastic modulus of concretedefined per ACI 318 (2011) [24]. For the force-based element, shearresponse at the section is defined by the V–c relationship. For thedisplacement-based element, a shear load versus lateral displace-ment (V–Dv) relationship is required for the shear spring placedat the wall base; this was defined on the basis of Dv = cH whereH is the height of the specimen.

Fig. 4. Constitutive models used for basic model (a) 1D concrete model by Yassin[49] and (b) 1D steel model by Filippou et al. [16].

3.2. Evaluation

For each specimen listed in Table 1, response was simulatedusing the basic model described in Section 3.1. Three levels ofmesh refinement were employed each for models comprisingdisplacement-based elements and force-based elements. Analyseswere conducted using displacement-control, both for models com-prising displacement-based as well as force-based elements; thedisplacement history imposed in the simulations was approxi-mately the same as that imposed in the laboratory up to the pointof failure. For lower resolution meshes, i.e. meshes comprising ele-ments with fewer integration points (fiber sections), simulateddrift capacity exceeded drift capacity measured in the laboratory.For these simulations, the simulated displacement history includedadditional displacement cycles to displacement demands exceed-ing those imposed in the laboratory to enable determination ofsimulated drift capacity.

Analysis results were evaluated on the basis of the accuracywhich the following quantities were simulated: (1) secant stiffnessto the theoretical yield force, (2) maximum base shear force, and

(3) drift capacity, defined as the drift corresponding to loss of20% of the lateral-load carrying capacity. Table 2 presents thesedata, with simulated results normalized by measured results, forall of the specimens in the database. Fig. 5 shows measured andsimulated response for the four selected specimens. In Fig. 5,response was simulated using the force-based elements, shearforce is normalized by the ACI shear strength (Vn) [1], and drift isdefined as the lateral displacement at the top of the specimen nor-malized by the specimen height.

The mesh refinement study was conducted to evaluate thenumber of elements and integration points needed to provide a

Table 2Results for (a) R specimens, (b) BR specimens and (c) CB specimens.

Specimen Failure mode Displacement-based Force-based

ky,sim/ky,obs Vbm,sim/Vbm,obs Du,sim/Du,obs ky,sim/ky,obs Vbm,sim/Vbm,obs Du,sim/Du,obs

4 EL 8 EL 16 EL 4 EL 8 EL 16 EL 4 EL 8 EL 16 EL 3 IP 5 IP 7 IP 3 IP 5 IP 7 IP 3 IP 5 IP 7 IP

WSH4 CB 0.98 0.98 0.97 1.05 1.00 0.97 1.54 0.94 0.62 0.99 0.97 0.97 0.93 0.94 0.94 1.27 0.57 0.44WSH6 CB 0.89 0.89 0.89 0.98 0.95 0.93 2.68 1.74 1.19 0.88 0.89 0.89 0.89 0.91 0.91 2.58 0.95 0.65W1 CB 1.22 1.18 1.17 1.11 1.06 1.05 3.86 2.69 2.18 1.12 1.16 1.17 0.98 1.05 1.04 2.69 1.62 1.64PW2 CB 1.01 1.00 1.00 1.04 1.01 0.99 3.00 1.89 1.38 0.99 1.00 1.00 0.96 0.98 0.98 2.33 1.13 1.00PW3 CB 0.87 0.87 0.87 1.02 1.00 0.99 4.51 3.43 2.87 0.87 0.87 0.87 0.96 0.98 0.98 4.10 2.46 2.36PW4 CB 1.05 1.05 1.05 1.28 1.24 1.23 4.93 3.77 3.24 1.04 1.04 1.04 1.20 1.20 1.21 3.48 2.88 2.81RW2 CB 1.17 1.13 1.12 0.94 0.92 0.92 3.19 2.25 1.76 1.06 1.10 1.11 0.84 0.92 0.91 3.07 1.38 1.38S5 CB 0.84 0.84 0.84 1.12 1.09 1.07 2.25 1.78 1.53 0.84 0.84 0.84 1.05 1.05 1.06 1.77 1.44 1.43S6 CB 0.88 0.88 0.88 1.18 1.15 1.14 3.34 2.66 2.26 0.88 0.88 0.88 1.11 1.11 1.12 2.62 2.13 2.13WR20 CB 1.05 1.01 1.01 0.97 0.93 0.92 1.37 0.80 0.52 1.01 1.00 1.01 0.89 0.90 0.90 1.25 0.47 0.31WR10 CB 1.15 1.09 1.09 0.99 0.95 0.94 1.93 1.35 0.80 1.06 1.08 1.08 0.91 0.92 0.92 1.86 0.79 0.46WR0 CB 1.18 1.16 1.12 0.99 0.96 0.94 0.92 0.56 0.39 1.09 1.11 1.10 0.91 0.92 0.92 0.76 0.34 0.27

Mean CB 1.02 1.01 1.00 1.06 1.02 1.01 2.79 1.99 1.57 0.99 0.99 0.99 0.97 0.99 0.99 2.31 1.35 1.24COV CB 0.13 0.12 0.11 0.09 0.09 0.09 0.45 0.51 0.60 0.10 0.11 0.11 0.11 0.10 0.10 0.42 0.60 0.70

WSH2 BR 1.00 1.00 1.00 0.98 0.95 0.94 3.42 2.00 1.40 1.00 0.98 1.00 0.88 0.93 0.93 2.57 0.98 0.84WSH3 BR 0.97 0.96 0.96 1.00 0.96 0.95 2.67 1.70 1.20 0.99 0.95 0.95 0.90 0.93 0.94 2.42 0.92 0.72WSH5 BR 0.82 0.81 0.81 1.01 0.96 0.94 1.75 1.14 0.72 0.78 0.80 0.80 0.90 0.91 0.91 1.66 0.66 0.47W2 BR 1.20 1.16 1.14 1.10 1.07 1.07 4.12 3.38 2.71 1.08 1.11 1.12 0.98 1.07 1.06 2.75 2.01 2.11RW1 BR 1.12 1.11 1.10 1.02 0.99 0.99 2.85 1.98 1.48 1.06 1.08 1.09 0.91 0.98 0.97 2.76 1.11 1.10R1 BR 0.96 0.97 0.96 1.13 1.10 1.09 3.48 3.48 3.22 0.88 1.00 0.94 1.04 1.05 1.05 3.47 2.93 2.82R2 BR 1.17 1.14 1.13 1.17 1.14 1.14 2.77 2.77 2.77 1.09 1.11 1.12 1.10 1.11 1.11 2.77 2.71 2.68

Mean BR 1.03 1.02 1.01 1.06 1.03 1.02 3.01 2.35 1.93 0.98 1.00 1.00 0.96 1.00 1.00 2.53 1.62 1.54COV BR 0.13 0.12 0.12 0.07 0.07 0.08 0.25 0.38 0.50 0.12 0.11 0.12 0.09 0.08 0.08 0.21 0.57 0.64

WSH1 R 1.00 1.00 1.00 1.05 1.01 1.00 3.36 1.95 1.39 0.98 0.99 1.00 0.94 0.97 0.97 2.56 1.08 0.79PW1 R 0.99 0.99 0.99 1.06 1.04 1.03 2.88 1.95 1.53 0.98 0.99 0.99 1.00 1.01 1.01 2.22 1.31 1.20

Mean R 0.99 1.00 0.99 1.06 1.02 1.01 3.12 1.95 1.46 0.98 0.99 0.99 0.97 0.99 0.99 2.39 1.20 1.00

Mean ALL 1.02 1.01 1.00 1.06 1.02 1.01 2.90 2.11 1.67 0.98 1.00 1.00 0.97 0.99 0.99 2.43 1.42 1.31COV ALL 0.12 0.11 0.11 0.08 0.08 0.08 0.36 0.43 0.53 0.10 0.10 0.10 0.09 0.08 0.09 0.33 0.56 0.65

J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192 181

converged solution. To study the effects of mesh refinement ondisplacement-based element models, meshes of 4, 8 and 16elements were used to model the full height of the specimen. Five(5) fiber sections (i.e., integration points) were used for eachdisplacement-based element. For the force-based element, meshescomprising a single force-based element including 3, 5 and 7 fibersections (i.e. integration points) were used to model the full heightof the specimen.

Table 2 presents the results of the mesh refinement study for allof the wall specimens; specimen data are organized on the basis offailure mode, which include compression failure (CB), bar rupturefollowing buckling (BR) and bar rupture prior to buckling (R). Theresults differ depending on the mesh refinement and the elementformulation. The table shows the predicted values for thedisplacement-based formulation element model in the leftmostcolumns and the force-based formulation model in the rightmostcolumns. Both are evaluated below.

The results indicate that accurate simulation of the yield stiff-ness and strength was achieved regardless of the mesh size. Forthe different element formulations and levels of mesh refinement,average ratios of simulated to measured yield stiffness ranged from0.98 to 1.02 and ratios of simulated to measured strength rangedfrom 0.96 to 1.06; coefficients of variation did not exceed 13%.For the displacement-based element model, increased mesh refine-ment did improve the accuracy of simulated response. For theforce-based element, improvement in accuracy was observed asthe number of integration points increased from 3 to 5, but meshesemploying 5 and 7 integration points generated essentially thesame results.

The data in Table 2 shows that both element formulations exhi-bit significant mesh sensitivity in prediction of drift capacity; theresponse histories for the selected specimens further demonstrate

this for the force-based element formulation. These data show thatmore highly refined meshes, with more integration points (i.e. fibersections) or more elements, predicted reduced drift capacity andmore rapid strength loss.

To investigate the observed mesh sensitivity, section behaviorwas considered for the force-based element (Fig. 6). Fig. 6a and bshows, for the force-based element, the simulated curvature pro-files up the height of the wall for three levels of mesh refinementat two drift demand levels: just beyond the yield of the longitudi-nal reinforcement (Fig. 6a) and just following the onset of strengthloss (Fig. 6b). Fig. 6c and d shows the simulated force versusdrift history up to the point at which the curvature profiles aresimulated.

The data in Fig. 6 shows that mesh sensitivity in simulated driftcapacity results from localization of deformations at a single criti-cal section. Comparing Fig. 6a and b shows that this localization isunique to the softening regime. For the hardening portion of thecurve (Fig. 6a and c), mesh refinement simply results in a moreaccurate representation of the curvature profile up the height ofthe wall. However, for the softening portion of the curve(Fig. 6b and d), mesh refinement results in larger curvaturedemands occurring over a shorter height of the column. Strengthloss at a single critical section results in increased deformation atthat softening section and elastic unloading, and thus reduceddeformation, at the other sections. For a hardening system,increased deformation at the critical section is accompanied byincreased strength, which results in the spread of yielding andincreased load and deformation at other sections.

Reinforced concrete walls typically exhibit a softening-typeresponse at larger drift demands because response is deter-mined by concrete crushing and/or buckling of longitudinalreinforcement (Fig. 1a). The data in Fig. 2 shows that almost 60%

182 J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192

of the wall test specimens surveyed by Birely [8] exhibited com-pression damage modes. In the database assembled for this study(Table 1), 20 of the 22 specimens are impacted by compressiveresponse of concrete and steel and 12 specimens exhibitcompression-controlled failure. Thus, structural reinforced con-crete walls that are part of the lateral load resisting system maybe expected to exhibit a softening-type response. Further, it shouldbe expected that traditional analysis methods will result in simu-lation of wall response that is mesh-sensitive, with drift capacityand the rate of strength loss determined entirely by the level ofmesh refinement employed in the model.

In addition to demonstrating mesh-sensitive drift capacityresults, the data in Table 2 also shows that maximum strength aspredicted using the displacement-based element formulation con-verges relatively slowly with increasing number of elements andthat a relatively large number of elements are required to accu-rately simulate laboratory results. This is unexpected given thatmodel strength is defined by the fiber-section flexural response,which is identical for all models and meshes considered. A closerlook at the results reveals the extreme variability in the axial loaddistribution within the displacement-based element model (Fig. 7).These data show that at a low level of mesh refinement (4 ele-ments) simulated axial load at a section varies between 40% and160% of the applied constant axial load and that even for a highly

Fig. 5. Simulated response histories for selected specimens with basic force-based elemRW1, BR failure and (d) S6, CB failure, high shear stress demand.

refined mesh (32 elements) simulated axial load at a section variesbetween 75% and 110% of the applied constant axial load. For thedisplacement-based element formulation, this variation in axialload at the section occurs despite the fact that the resultant axialload for the element is equal (within the solution tolerance) tothe globally applied axial load. This is due to the fact thedisplacement-based element formulation assumes a constant axialstrain at each section along the length of the element and thatequilibrium is satisfied on average within the element. Since axialload at the section level affects the flexural strength of the section,the variation in axial load along the length of the element results invariation in element flexural strength. While increasing the num-ber of elements reduces the section-level variation in axial load,axial load variation for practical levels of mesh refinement (i.e.12–16 elements) was found to be sufficiently large that it affectedmodeling recommendations. For these reasons, the displacement-based element formulation was not used in the remainder of theevaluations or model development study.

The potential for distributed plasticity force-based beam–column elements to exhibit mesh-sensitivity due to localizationof deformation at a single softening section has been identifiedpreviously [12]. However, to date, distributed plasticity elementshave been used primarily to simulate the response of concreteand steel beams and columns that exhibit a hardening-type

ent model (a) WSH4, CB failure, low shear stress demand, (b) WSH1, R failure, (c)

Fig. 6. Local deformation and global response of WSH4 using force-based formulation (a) / diagram (D = 0.30%), (b) / diagram (D = 1.73%), (c) drift history (D = 0.30%) and (d)drift history (D = 1.73%).

J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192 183

response out to large drift demands. Typically, failure of thesecomponents is defined via post-processing of analysis results, oftenusing empirically derived drift-capacity models that define thedrift at which various failure models (e.g. reinforcement bucklingand fracture in concrete columns) results in loss of lateral load car-rying capacity. However, this approach is not acceptable for fullbuilding analyses for which realistic modeling of internal loadredistribution or collapse analysis is of interest. Therefore anexplicit method for evaluating the drift capacity of these importantcomponents is required.

4. Material model regularization to limit mesh sensitivity

Prior research has identified the potential for mesh-sensitivesimulation results such as those presented above [12]. However,it is not standard practice to regularize material response modelsin simulation of RC components using line-element type models.There are several reasons for this. The most important is the expec-tation that reinforced concrete components designed to meet cur-rent ACI Code requirements will exhibit a hardening-type responseout to large drift demands. However, experimental data [8,38]show that most RC walls do not exhibit a hardening-type responseout to large drift but instead exhibit softening at moderate drifts.

To limit mesh sensitivity, it is common practice in continuumanalysis of plain and reinforced concrete components to regularizethe concrete post-peak tensile stress–strain response model usingthe concrete fracture energy (Gf) in combination with a mesh-

dependent characteristic length. Regularization of concrete com-pressive response is also common in continuum analysis, thoughmethods and energy values used in the process vary. Colemanand Spacone [12] extended the regularization approach used forcontinuum analysis to line-element analysis, recommending regu-larization of the 1D constitutive model used to simulate concretecompressive response in the fiber-section model for componentsthat exhibit a softening moment–curvature response at the sectionlevel. Coleman and Spacone [12] applied the proposed regulariza-tion approach to model the response of a cantilever column sub-jected to lateral and axial loading by Tanaka and Park [42]. Thecolumn was modeled using a single FBBC element and mesheswith 4, 5 and 6 fiber sections. Bilinear steel response with 1% hard-ening was used for the reinforcement and the concrete materialresponse was modeled using a Park-Kent model with material reg-ularization. Unconfined confined concrete fibers were assigned acrushing energy of 25 N/mm based on recommendations by Jansenand Shah [20] and the confined core concrete was assigned a valuesix times larger than this (150 N/mm) to account for the enhancedresponse provided by the transverse reinforcement. Resultspresented by Spacone and Coleman (Fig. 8) shows the proposedmaterial regularization procedure significantly reduced the meshdependent softening response and enabled objective numericalsimulation of the post-peak response.

For simulation of structural walls, softening section responseresults from material softening of plain concrete subjected tocompression, confined concrete subjected to compression, and

Fig. 7. Normalized axial load (N) ratio as function of wall height.

Fig. 8. Objective vs. non-objective response (image from [12]).

Fig. 9. Regularized material constitutive models (a) unconfined concrete compres-sion response, (b) confined concrete compression response and (c) steel response.

184 J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192

reinforcing steel subjected to compression or tension. Fig. 9 showsthe approach adopted in this study for regularizing each of these aswell as the definitions of the unconfined concrete crushing energy(Gfc), confined concrete crushing energy (Gfcc) and steel hardeningenergy (Gs) used in the regularization process. In Fig. 9, LIP is thelength associated with the fiber section (i.e. integration point) forwhich the material model is used. Given energy values Gfc, Gfcc

and Gs as well as LIP, the concrete and steel stress–strain responseenvelopes can be regularized. Regularization of concrete tensileresponse is not required for analysis of reinforced concretecomponents using fiber-type beam–column elements because(i) concrete softening in tension does not result in softening ofthe reinforced concrete section and thus localization of deforma-tions and (ii) concrete tensile response has minimal impact onresponse once softening occurs due to flexural failure.

For concrete response in compression, there are no currentstandard practices for experimental testing to determine Gfc or Gfcc

and few empirical models are found in the literature. Two studiesaddressing crushing energy are: Jansen and Shah [20] who recom-mend a value of 25 N/mm for normal-weight concrete, and Naka-mura and Higai [28] who define crushing energy for unconfinedconcrete to be a function of compressive strength such thatnormal-weight concrete has Gfc = 80 N/mm. Coleman and Spacone[12] employed the Gfc value recommended by Jansen and Shah forunconfined concrete and recommend a value of Gfcc = 6Gfc forconfined concrete; however, these values are not verified throughcomparison of simulated and measured response for typical rein-forced concrete components.

Steel exhibits a hardening-type response. Thus, localization ofdeformation would be not expected in a fiber-type beam–columnelement model of a steel beam or column; instead material

hardening would ensure distributed yielding along the length ofthe beam–column element. However, for a reinforced concrete sec-tion in which concrete crushing results in softening of the criticalsection, concrete and reinforcing steel deformations do localize.Thus, steel strain demands at the critical section are affected bymesh refinement, and regularization of the steel material responseis required to achieve mesh objective results; this is demonstratedbelow and in Pugh [38]. Given the hardening energy (Gs) and basicmaterial response parameters as well as LIP, a post-yield hardeningmodulus can be defined for reinforcing steel. Chiaramonte [11]addresses regularization of steel response for analysis usingdistributed plasticity elements; Coleman and Spacone [12] do notaddress regularization of steel response.

The following sections summarize the energy values and themethods used in this study to determine these values as well ascompare results generated using the basic model with thosegenerated using regularized material models.

4.1. Determination of steel post-yield energy and definition of steelmaterial response

For RC components that exhibit softening at the section level,regularization of the steel as well as concrete material responseis required. This is because section softening results in localizationof inelastic deformation at a single critical section, and strain

Fig. 10. Evaluation of unconfined concrete crushing energy.

J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192 185

demands for both steel and concrete fibers are affected by thislocalization of deformation demands.

To determine a post-yield energy for use in regularizing rein-forcing steel response, an approach often employed for continuumanalysis of unconfined concrete responding in compression wasadopted. For the reinforcing steel, the post-yield energy, Gs, wasdefined to be equal to the area under the experimentally deter-mined post-yield stress–strain history multiplied by the lengthover which inelastic deformation localizes, which was assumedequal to the gage length, Lgage, used in the laboratory. This isillustrated in Fig. 9c. Thus,

Gs ¼ 12eu exp � ey� �ðf u þ f yÞLgage ð1Þ

where eu exp is the strain at ultimate strength, f u, as determinedfrom laboratory testing and all other parameters are as previouslydefined. To determine the strain at ultimate strength, eu, used indefining the regularized model, post-yield energy is assumed tobe dissipated over the integration length used in the model(Fig. 9c):

eu ¼ ey þ 2ðf u þ f yÞ

Gs

LIP¼ ey þ Lgage

LIP

� �eu exp � ey� � ð2Þ

where all variables are as defined previously. Thus, as the meshbecomes more highly refined and LIP is reduced, the hardeningmodulus for the reinforcing steel is also reduced and larger strainsare required to achieve a post-yield stress level.

It should be noted that the gage length used in steel materialtesting was not available for all of the test specimens in the dataset. However, sensitivity studies performed on specimens experi-encing crushing failures suggest results are not particularly sensi-tive to reasonable variations in the gage length used to regularizesteel response. Analyses of crushing specimens were performedconsidering gage lengths between 100 and 760 mm. Over thisrange of gage lengths, simulated system response quantities (stiff-ness, peak strength, drift capacity) varied by less than 5% [38]. Forthe analyses in which a gage length was not reported, a length of200 mm (8 in.) was used for numerical modeling of wall speci-mens. This length is the gage length required per ASTM A370(Methods for Testing Steel Reinforcing Bars).

For the regularized steel material model, the OpenSees MinMaxmaterial was used to simulate strength loss due to fracture undertensile loading and buckling under compressive loading. The regu-larized steel tensile strain at onset of strength loss was computedusing an approach identical to that presented in Eq. (2). The regu-larized steel compressive strain at onset of compression strengthloss was defined equal to the regularized concrete strain at 80%of concrete compression strength loss, e20u per Eq. (3) below.

Fig. 11. Evaluation of energy coefficient (Kc) for confinement concrete.

4.2. Determination of unconfined and confined concrete crushingenergy and definition of concrete material response

The response of a wall with poorly confined concrete will bedetermined by the compressive response of the unconfined con-crete. Since standardized tests do not exist for determining thecrushing energy, Gfc, required to regularize the concrete constitu-tive model and define unconfined concrete compressive response,Gfc was determined using laboratory data from tests of walls withpoorly confined concrete exhibiting compression-controlledfailure. Specifically, load–displacement data for Specimen WSH4and Specimen WR0 were used to determine the crushing energyfor unconfined concrete. These were the only wall test specimensin the data set (Table 1) constructed entirely of unconfinedconcrete and exhibiting a compression-controlled failure.

To determine Gfc, a series of analyses were run of the WSH4 andWRO specimens using the basic force-based element modeldescribed above with material regularization. In these analyses,the unconfined concrete crushing energy, Gfc, was varied with theresult that the strain at onset of residual compressive strength,e20u, varied:

e20u ¼ Gfc

0:6f cLIP� 0:8f c

E0ð3Þ

Fig. 10 shows a plot of simulated to observed drift capacity as afunction of Gfc/fc. On the basis of these data, the crushing energyrecommended for use with the force-based element (FBE) wasdetermined to be

Gfc ¼ 2f 0c N=mm ð4Þ

with f 0c defining concrete compressive strength in MPa. A similarapproach was used to determine an appropriate confined concretecrushing energy, Gfcc. A series of analyses were done of the nine(9) walls from the database with confined concrete that exhibitedcompression-controlled failure under cyclic loading. The basicforce-based element model with material regularization was used.Unconfined concrete material response was regularized using Eq.(4) for theses analyses. A range of confined concrete crushing ener-gies were used.

Fig. 11 shows the results of these analyses, with the ratio of sim-ulated to predicted drift capacity plotted versus Kc = Gfcc/Gfc withdata grouped by boundary element confinement configuration.The data in Fig. 11 shows that for all cases, except the PW4 speci-men, an increase in confined concrete crushing energy relative tounconfined concrete crushing energy is required to accurately sim-ulate drift capacity. The data in Fig. 11 shows also that, with theexception of PW4, which was deemed an outlier, Kc = Gfcc/Gfc

increases with increasing confinement. Walls with rectangular

186 J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192

boundary elements and crossties restraining all longitudinalreinforcement have the largest Kc values (average Kc = 2.30) withconfinement providing a significant enhancement of concretepost-peak strain capacity, while a wall with a rectangular bound-ary element with no crossties has the smallest Kc value

Fig. 12. Comparison of two specimens simulated using basic constitutive models, concreno material regularization, (b) WR0, basic model, no material regularization, (c) WSH4, coregularized model, concrete and steel and (g) WR0, regularized model, concrete and ste

(Kc = 1.15) with confinement providing minimal enhancement ofconcrete post-peak strain capacity. Walls with square boundaryelements fall in the middle of this range with an average Kc of1.45. On the basis of the data presented in Fig. 11, confinedconcrete crushing energy was defined as follows:

te regularization only and steel and concrete regularization (a) WSH4, basic model,ncrete material regularization, (d) WR0, concrete material regularization, (e) WSH4,el material regularization.

Fig. 13. Flexural strength ratio at analysis step prior to 20% strength loss.

J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192 187

Gfcc ¼ 1:70Gfc ð5ÞWhile the data in Fig. 11 suggest that Gfcc is a function of

confinement configuration, the data in Fig. 11 were deemed to beinsufficient to develop a predictive model.

4.3. Comparison of the basic and regularized models

Fig. 12 shows measured and simulated results for twospecimens that responded in a compression buckling (CB) mode,Specimens WSH4 and OhWR0. The response of each specimenwas simulated using a single force-based element with five,seven and nine fiber sections; for each level of mesh refinement,three different sets of constitutive models were employed.Fig. 12a and b shows simulated results with no material regulariza-tion. Fig. 12c and d shows results for concrete material regulariza-tion in compression and with no regularization of reinforcing steelresponse. Fig. 12e and f shows results with concrete and steelmaterial regularization. On the basis of these results, the followingconclusions and recommendations are made:

1. For a component that exhibits softening response, refining themesh without material regularization results in more rapiddegradation of the post-peak response.

2. Regularization of the steel and concrete compression models(confined and unconfined) is required to achieve objectiveresponse prediction of softening specimens.

4.4. Prediction of softening response and identification of whenregularization of material response is required

For softening systems, inelastic deformation localizes at a singlesection of the beam–column element model, and regularization ofmaterial response is required to provide accurate, mesh-objectivesimulation of response. For hardening systems, however, inelasticdeformation does not localize, and regularization of materialresponse is not required. Review of flexural wall tests in the labo-ratory [38,6] as well as continuum-type analyses of flexural planarwalls with a wide range of design configurations and subjected to awide range of axial and shear loads [47] suggests that most wallsdesigned for building construction will exhibit a softening typeresponse. Specifically previous data suggest that walls with at leastlight axial load and moderate longitudinal reinforcement ratioswill exhibit a softening type response, while only walls with verylight to zero axial loads and/or very low longitudinal reinforcementratios will exhibit a hardening response. For example, in the cur-rent study, only specimens R1 and R2 exhibit a hardening-typeresponse; these are the only specimens in the data set with zeroaxial load, and specimen R1 has one of the smallest longitudinalreinforcement ratios in the data set. However, while most rein-forced concrete walls can be expected to exhibit a compression-controlled softening-type response; not all do. Thus, accurate sim-ulation of response requires a method for identifying hardeningversus softening response. Comparing results for specimens R1and R2, which exhibit a hardening type response, with those forthe other specimens in the data set, which exhibit a softening typeresponse, two different approaches for identifying softening versushardening response are proposed:

1. For a softening system, a moment–curvature analysis using reg-ularized material models will predict gradual softening prior toreaching the 20% strength loss (base moment is 80% of maxi-mum base moment) used to define failure and drift capacity.For a hardening system, hardening is observed up to the pointat which bar fracture results in catastrophic strength loss.Fig. 13 shows, for the 21 specimens in the data set, the ratio

of moment capacity for the analysis step prior to that at which20% strength loss is observed normalized by maximummomentcapacity. A ratio of 1.0 is computed for specimens R1 and R2,indicating that the specimen does not exhibit significantstrength loss prior to catastrophic failure; a ratio less than 1.0is computed for all other specimens.

2. For softening specimens, drift capacity simulated using basic,unregularized material models exhibits mesh-sensitivity withincreasing mesh refinement resulting in reduced drift capacity,while regularization of material response results in mesh-objective simulation of response with increasing mesh refine-ment resulting in convergence to a predicted drift capacity.The opposite of this is true for hardening systems. Specifically,for hardening systems, results for simulated drift capacity con-verge with increasing mesh refinement when unregularizedmaterial models are used and results diverge with increasingmesh refinement when regularized material models are used.Comparing data in Tables 2 and 4 for specimens R1 and R2 withthose for other specimens show this though it should be notedthat drift capacities for specimens R1 and R2 predicted usingunregularized models are inaccurate as the unregularizedmodels do not simulate bar fracture due to low-cycle fatigueassociated with buckling.

Finally, evaluation of steel tensile strain at nominal flexuralstrength (et in Table 1) suggests that this parameter could also beused to predict hardening response. Specimens R1 and R2 have etvalues of 0.071 and 0.061, respectively. These strain values greatlyexceed the et values of the other specimens in the data set, whichhave a maximum of 0.027 and an average of 0.017. Large steelstrains at nominal strength indicate a system for which flexuralresponse is highly tension-controlled; flexural failure of tension-controlled components is typically characterized by bar fracturefollowing buckling, though bar fracture prior to bar buckling maybe observed if the strain capacity of the reinforcing steel is low.However, given the small size of the data set employed in thisstudy, identification of a limit on et beyond which hardeningresponse could be expected is not appropriate. Future researchaddressing this issue is recommended.

The above methods for determining if regularization of materialresponse is required employ the results of section analysis, andthus require that the axial load applied to the section be specified.For analyses of building systems subjected to earthquake loading,axial load will vary during the analysis. Given that most wallscan be expected to exhibit a softening type response, it is recom-mended that softening type response be assumed initially and thatmaterial response be regularized to provide accurate simulation.Following the nonlinear analysis, wall sections may be checked

Table 3Recommendations for regularized material models for RC wall simulation.

Material Regularization values

Longitudinal reinforcing steel Eqs. (1) and (2)Concrete subjected to tension Not requiredUnconfined concrete subjected to compression Gfc = 2f 0c (N/mm)Confined concrete subjected to compression Gfcc = 1.7 Gfc

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for hardening- versus softening type response by doing a sectionanalysis using two axial load levels that likely bound the criticalrange: (i) axial load due to gravity and (ii) the maximum axial loaddeveloped during the earthquake analysis. Following are possibleoutcomes:

i. Regularization of material response is appropriate as thesection exhibits softening under axial load associated withgravity loading and with gravity plus earthquake loading.

ii. Regularization of material response is not appropriate as thesection exhibits hardening under axial load associated withgravity loading and with gravity plus earthquake loading.

iii. Additional investigation is required as the section exhibitshardening under axial load associated with gravity loadingand softening under axial load associated with gravity plusearthquake loading. Here additional section analyses shouldbe performed to consider whether section response is soft-ening or hardening under the axial loads applied at the timethe maximum moment demand develops.

Table 4Comparison of simulated and observed response quantities for regularized model.

Specimen Failure mode Force-based

Stiffness ratio Strength ratio

3 IP 5 IP 7 IP 3 IP 5 I

WSH4 CB 0.96 0.96 0.96 0.95 0.9WSH6 CB 0.95 0.95 0.95 0.91 0.9W1 CB 1.16 1.17 1.18 1.02 1.0PW2 CB 1.06 1.06 1.06 0.97 0.9PW3 CB 0.95 0.95 0.95 0.93 0.9PW4 CB 1.11 1.11 1.11 1.14 1.1RW2 CB 1.14 1.15 1.15 0.90 0.8S5 CB 0.87 0.87 0.87 0.98 0.9S6 CB 0.91 0.91 0.91 1.03 1.0WR20 CB 1.02 1.03 1.03 0.89 0.9WR10 CB 1.05 1.08 1.07 0.92 0.9WR0 CB 1.09 1.11 1.11 0.91 0.9

Mean CB 1.02 1.03 1.03 0.96 0.9COV CB 0.09 0.10 0.10 0.07 0.0

WSH2 BR 1.02 1.02 1.01 0.93 0.9WSH3 BR 0.98 0.98 0.98 0.93 0.9WSH5 BR 0.84 0.84 0.84 0.92 0.9W2 BR 1.08 1.11 1.10 1.05 1.0RW1 BR 1.12 1.12 1.12 0.97 0.9R1a BR 1.06 1.06 1.06 1.09 1.0R2a BR 0.88 0.88 0.88 1.13 1.1

Meana BR 1.01 1.01 1.01 0.96 0.9COVa BR 0.11 0.11 0.11 0.06 0.0

WSH1 R 1.03 1.02 1.02 0.97 0.9PW1 R 1.06 1.06 1.06 1.01 1.0

Mean R 1.05 1.04 1.04 0.99 1.0

Mean CB 1.02 1.03 1.03 0.96 0.9COV CB 0.09 0.09 0.09 0.07 0.0

a Specimens R1 and R2 can be identified as exhibiting hardening up to the point of catafor these walls and results in inaccurate, mesh–sensitive simulation of drift capacity. Resphowever, results for these walls are not included in calculated means and COVs for spe

5. Evaluation of the model using proposed regularizedconstitutive models

To evaluate the modeling approach described above, simulatedand measured base shear versus drift histories were compared forall of the walls listed in Table 1. Note that specimens exhibiting CBfailure modes (12 of 22 specimens) were used to calibrate themodel. Simulations were conducted using the OpenSees platform.Although in the full study, two models were used to simulate eachof the cantilever wall specimens, one comprising displacement-based elements and one comprising force-based elements,only data for the force-based element models are presentedhere because of the issues noted with the axial load variation.Pugh [38] presents results for the displacement-based elementmodels.

For the force-based elements, a three-element mesh was usedwith 3, 5 and 7 fiber sections along each element. Each fiber-typesection represents an integration point (IP) for the Gauss–Labottonumerical integration rule that was employed for computing ele-ment resultant quantities. Each fiber-section includes employed32 fibers within the confined boundary elements and fibers of sim-ilar thickness within the web of the walls. Unconfined and confinedconcrete material response was regularized using the recom-mended regularization values provided in Table 3. Analyses wereconducted under displacement control using the displacementhistory imposed the laboratory. The boundary conditions and loadpatterns applied in the laboratory were represented in thesimulations, with one exception. Walls that were subjected to a

Drift ratio Failure mode

P 7 IP 3 IP 5 IP 7 IP 3 IP 5 IP 7 IP

5 0.95 1.05 1.05 1.13 C C C2 0.92 0.80 0.70 0.70 C C C1 1.02 1.00 1.00 1.00 C C C6 0.96 0.82 0.87 0.92 C C C3 0.93 0.95 1.00 1.08 C C C1 1.10 1.32 1.45 1.56 C C C8 0.87 1.13 1.21 1.32 C C C7 0.97 0.70 0.86 1.05 C C C0 0.99 1.16 1.23 1.30 C C C0 0.90 0.99 0.90 0.83 C C C3 0.93 1.01 0.99 0.96 C C C2 0.92 0.94 0.90 0.88 C C C

6 0.96 0.99 1.01 1.07 – – –7 0.06 0.17 0.20 0.23 – – –

2 0.91 0.93 1.15 1.27 R R R3 0.93 0.89 1.00 1.00 C C C3 0.93 0.83 0.72 0.72 C C C3 1.02 1.58 1.37 1.37 C C C5 0.94 1.06 1.17 1.17 C C C8 1.06 1.67 2.54 3.10 R R R1 1.09 1.41 2.09 2.47 R R R

5 0.95 1.06 1.08 1.11 – – –5 0.05 0.29 0.22 0.23 – – –

8 0.98 1.07 1.17 1.20 R R R1 1.00 1.01 1.05 1.15 C C C

0 0.99 1.04 1.11 1.18 – – –

6 0.96 1.01 1.04 1.08 – – –6 0.06 0.20 0.19 0.21 – – –

strophic strength loss; as such, regularization of material response is not appropriateonse data generated using regularized models of these walls are provided in Table 4;cimens exhibiting the BR failure mode.

J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192 189

distributed lateral loading in the laboratory were subjected to asingle lateral load at the top of the wall in the analysis. Results ofa preliminary study comparing models in which the distributedload was simulated exactly and was approximated using a singlepoint load at the top of the wall and indicated that this aspect ofthe model had essentially no impact on critical response quantities[38].

The results of the study are presented in tabular form in Table 4.Fig. 14 presents the simulated and measured results of the valida-tion study for the four selected specimens, which vary in damagemode and shear demand-capacity ratio. Fig. 15 shows the ratio ofsimulated to observed drift capacity for the basic and regularizedforce-based element models. The data in Table 4 and Figs. 14 and15 support the following observations:

1. Peak base shear strength of all walls was accurately simulatedby all force-based element meshes considered. The mean simu-lated to observed peak strength ratio was 0.96 with a coefficientof variation of 0.0 for compression controlled wall specimensand 0.95 with a coefficient of variation of 0.05 for tensioncontrolled specimens.

2. Secant stiffness to yield was accurately simulated by allforce-based element meshes considered. For all walls, the meansimulated to observed yield stiffness ratio was 1.02 with acoefficient of variation of 0.09.

Fig. 14. Comparison of regularized model and measured response for selected specimenfailure and (d) S6, CB failure, high shear stress demand.

3. Drift capacity data for walls failing due to flexural compression(CB) show a small dependency on the number of integrationpoints (Fig. 15). The ratio of simulated to observed drift capacityfor models using three (3) fiber sections was 0.99 with a coeffi-cient of variation of 0.17. The mean simulated to observed driftcapacity ratio for models using seven (7) fiber sections was 1.07with a coefficient of variation of 0.23. However, as indicated inFig. 15 and comparing the data presented in Tables 2 and 4, thisis a significant improvement over the basic model.

4. For the two (2) specimens failing due to tension rupture prior tosignificant buckling (R), drift capacity was slightly overesti-mated by the model, with a mean simulated to observed driftcapacity ratio of 1.11 for the five fiber-section model. Driftcapacity data for these specimens also show some mesh depen-dency; however, the mesh dependency is substantially less thanthat observed for the basic model.

5. For the five (5) specimens failing due to tension rupture aftersignificant buckling (BR) and identified as softening systems,drift capacity was again slightly overestimated (ratios of simu-lated to observed drift capacity ranged from 1.06 to 1.11 formodels with three, five and seven integration points), the coef-ficient of variation for these ratios was relatively high (ratiosranged from 0.29 to 0.22), and some mesh dependency wasobserved. The overestimation of and variation in simulated driftcapacity likely results from the simplicity with which strength

s (a) WSH4, CB failure, low shear stress demand, (b) WSH1, R failure, (c) RW1, BR

Fig. 15. Comparison of mean drift capacity ratios for both basic and regularizedmodeling approaches (a) specimens with CB failure and (b) specimens with BRfailure.

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deterioration due to buckling of reinforcing steel was simu-lated; however, these results are likely acceptable for mostapplications.

6. For the two (2) specimens failing due to tension rupture aftersignificant buckling (BR) and identified as hardening systems,simulation results are highly inaccurate and exhibit significantmesh dependency. These results demonstrate that regulariza-tion of material response is not appropriate for systems inwhich hardening is observed up to the point of catastrophicstrength loss.

5.1. Use of simulated response quantities to assess performance

Regularization of material response provides a means of limit-ing mesh sensitivity and enabling accurate simulation of driftcapacity for walls, and other reinforced concrete components, forwhich material softening produces section softening and, ulti-mately, component strength loss. However, material regularizationdoes limit the manner in which simulated material response datacan be used to assess component performance. Specifically, simu-lated concrete and steel strain data cannot be directly comparedwith material test data; a maximum simulated unconfined con-crete compressive strain of 0.003 mm/mm or a maximum reinforc-ing steel tensile strain of 0.2 mm/mm is not relevant and must bereassessed.

However, simulated concrete and steel stress–strain data can beevaluated in relative terms to assess performance. In this study, theregularized concrete strain resulting in an 80% loss in concretecompressive strength at the critical fiber section was used to definea compression failure and a maximum steel tensile strain equal to100% of the regularized rupture strain was used to define a strainlimit characterizing steel rupture.

For each specimen model, the extreme fiber concrete compres-sive strain (negative value) and the maximum extreme fiber steelstrains strain were determined at the load step for which the sim-ulated drift capacity was reached. The peak strain values were thencompared to the strain limit values. The computed strain was nor-malized to the limiting strain values. If the compressive strain ratioexceeded 1.0, a crushing failure mode was identified; if the tensilestrain ratio exceeded 1.0, a rupture failure mode was identified.

Table 4 presents the observed and simulated failure mode. Fail-ure in the laboratory was classified as simultaneous concretecrushing and buckling (CB), buckling followed by bar rupture(BR) or bar rupture prior to buckling (R). However, as discussedabove, simulated failure modes were limited to CB and R. Simula-tion of the BR failure mode using a fiber type model is challengingand requires the use of either an empirically calibrated steel con-stitutive model that simulates strength loss due to buckling or anempirically calibrated fracture strain for buckling bars; simulationof BR failure was not addressed in this study. The data in Table 4support the following observations and conclusions.

1. For the twelve specimens that exhibited CB failures in the lab-oratory, the laboratory failure mode was correctly simulated asCB.

2. For the five non-hardening specimens (use of regularized mate-rial models is not appropriate for hardening systems) thatexhibited a BR failure in the laboratory, four were predictedto exhibit a CB failure and one was predicted to exhibit a R fail-ure. For these five specimens, the error is simulated drift capac-ity ranged from 0% to 40%, which is not excessively large. Thissuggests that once compression damage initiates, whether fail-ure occurs due to concrete crushing and bar buckling or fractureof previously buckled bars has limited impact on drift capacity.

3. Two specimens exhibited R failure in the laboratory. The R fail-ure was correctly predicted for specimen WSH1; this is signifi-cant as the WSH1 specimen was constructed using reinforcingsteel with extremely low strain capacity (0.05 mm/mm) suchthat an R failure could be expected. The R failure was not cor-rectly predicted for specimen PW1; this is considered to be lesssignificant as a splice at the base of the PW1 specimen likelycontributed to development of the R failure mode.

Ultimately, the data in Table 4 demonstrate that regardless offailure mode, on average the proposed model simulates driftcapacity with acceptable accuracy and precision (average error of8% with a cov of 21%) for hardening systems. Additionally, themodel accurately predicts CB failures and R failures, for systemswith low strain capacity steel that are vulnerable to this relativelyuncommon failure mode. Finally, despite the fact that the modeldoes not simulate the BR failure mode, drift capacity predictedbased on a CB failure is still acceptably accurate on average (aver-age error of 11%).

6. Summary and conclusions

Reinforced concrete walls are one of the most common lateralload resisting systems used in regions of high seismicity. However,prior test results and post-earthquake reconnaissance efforts indi-cate that even walls that meet the ductile detailing and tension-controlled response requirements according to ACI 318-11 [1]can exhibit a softening response mode and compression-controlled failure. Therefore, performance evaluation of buildingsthat include these components requires computationally efficientnonlinear models that accurately capture this softening behavior.

The research presented here considered standard modelingtechniques, evaluating and ultimately improving the accuracy with

J.S. Pugh et al. / Engineering Structures 104 (2015) 174–192 191

which these models simulate the response of slender RC walls. Themodels were evaluated using a dataset of 21 slender walls tested inthe laboratory by 7 research groups; wall specimens exhibited arange of failure modes. Two nonlinear beam–column element for-mulations were evaluated, a displacement-based formulation anda force-based formulation; both element formulations employedfiber-type sections models with standard 1D constitutive modelsfor concrete and steel. Both element formulations were found toprovide accurate and precise simulation of stiffness and strength.However, both formulations were also found to exhibit mesh-sensitivity in simulating drift capacity, with increasing meshrefinement resulting in reduced drift capacity and increasinglyrapid strength loss. Additionally, the displacement-based formula-tion was found to produce significant errors in the axial load distri-bution along the length of the element and, thus, was regarded asinherently inaccurate.

To minimize mesh-sensitivity and provide accurate simulationof drift-capacity, the 1D concrete and steel constitutive modelswere regularized. Energy values used for regularization of uncon-fined and confined concrete compression response were developedusing data from a limited set of prior wall tests. Energy values usedfor regularization of reinforcing steel response were developedfrom material test data. Results show that the forced-basedbeam–column element, with the regularized material models, pro-vides accurate simulation of stiffness, strength and drift capacitywith minimal mesh-sensitivity.

Acknowledgements

The research presented here was funded by the NationalScience Foundation through the Network for Earthquake Engineer-ing Simulation Research Program, Grant Nos. 0421577 and0829978, Joy Pauschke, program manager and by the CharlesPankow Foundation.

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