FEA Implementation of Smeared Cyclic Bond Slip-Based

92 ACI Structural Journal/January-February 2010

ACI Structural Journal, V. 107, No. 1, January-February 2010.MS No. S-2009-004.R1 received April 20, 2009, and reviewed under Institute publication

policies. Copyright © 2010, American Concrete Institute. All rights reserved, including themaking of copies unless permission is obtained from the copyright proprietors.Pertinent discussion including author’s closure, if any, will be published in the November-December 2010 ACI Structural Journal if the discussion is received by July 1, 2010.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

A reinforced concrete (RC) material model that includes frictionalbond-slip behavior is proposed for use in a two-dimensional (2D)total strain-based finite element analysis (FEA). The initialmotivation for this model is to improve tension stiffening behaviorin total strain-based models. The proposed material model isbased on a simplified mechanistic concept and is capable ofmodeling these mechanistic behaviors under cyclic loading in asmeared (average) manner. It is validated with shear wall testresults available herein. Another motivation for this model is tounderstand the significance of crack strains so that the model canbe extended in the future to include shear friction behavior forcracked concrete. This paper presents an FEA implementation ofthe proposed RC material model in the OpenSees framework.

Keywords: bond slip; cyclic; FEA implementation; friction; reinforcedconcrete; shear wall; two-dimensional membrane.

INTRODUCTIONTotal strain-based membrane models, such as the modified

compression field theory (MCFT) for finite element analysis(FEA) of reinforced concrete (RC), have been used successfullyto analyze a variety of concrete structures.1 A major advantageof the total strain approach is that consideration of crack widthsand crack spacing can be avoided by computing only theaverage strain and stress in the concrete and smeared steel, thatis, uniaxial steel constitutive relationship (steel strain versusaverage steel stress, which equals the total force developedin the reinforcement divided by the gross concrete area). Inimplementing a total strain-based membrane model, either afixed2 or rotating3 crack model has been used to build aconcrete stress-strain relationship. Additional refinements,such as the reduction in concrete compressive strength as afunction of transverse tensile strain and the plasticexpansion4 of concrete due to transverse compressive stress,are important components of these models.

Many FEA models assume a perfect bond between thereinforcement bars and concrete elements; however, a cyclicresponse of RC structures, beyond yielding of the steelreinforcement, includes bond-slip behavior.5-7 Many bond-slipmodels have been developed to better predict the cyclicresponse of concrete structures. The early experimental workon cyclic bond-slip was largely conducted by Eligehausen,Popov, and Bertero.5 Lowes6 has successfully modeledbond-slip behavior of beam-column joints by using a zero-length interface element between a two-dimensional (2D)concrete element and a steel truss element. Bentz7 proposedthat a tension stiffening model for monotonic loading bebased on bond characteristics of the reinforcement. Herein, asimplified mechanistic material model that includes bond-slip behavior based on frictional resistance has been developedand implemented for use in a 2D total strain-based FEA of

RC shear wall structures under cyclic loading (for example,seismic loading).

Tension stiffening behaviorTotal strain-based models use tension stiffening models to

represent cracked concrete in the tensile region. With thisapproach, concrete retains tensile strength after cracking torepresent the average tensile stress between cracks. Theeffective yield strength of the reinforcement must be reducedby the amount attributed to the residual concrete tensilestrength at a strain value corresponding to tensile yielding ofthe reinforcement. In a previous study,8 the experimentallydetermined tensile stiffness behavior for RC panels9 wasfound to be more complex than the behaviors that existinganalytical models are able to represent, particularly undercyclic loading. The previously studied 2D bond-slip-basedmembrane model divides a 2D element into bonded and slipregions by the factor λ (refer to Fig. 1), which is defined asthe percentage of unbonded length due to concrete crackswith respect to the total element length. The value of λ iscomputed to be consistent with the tension stiffening modelin the direction perpendicular to the tensile crack orientation.It is essentially a friction model that evaluates maximumfrictional stress developed in a frictional block between theconcrete and steel reinforcement in the slip region. With thismodel, the stresses in the steel and concrete are different inthe bonded and slip regions, as shown in Fig. 1. This bond-slip model can be used to predict total stress histories from

Title no. 107-S10

FEA Implementation of Smeared Cyclic Bond Slip-Based Two-Dimensional Membrane Modelby Migeum So, Thomas G. Harmon, and Shirley Dyke

Fig. 1—Bonded and slip regions.

93ACI Structural Journal/January-February 2010

strain histories for panels subjected to uniform cyclic shearstress.9 Figure 2 shows two of the panels with either 0-90 or45-135 degree reinforcement. The total strain-based residualconcrete stresses can be computed from the total stresses bysubtracting the steel stresses that are functions of the totalstrain. Figure 2 compares residual total concrete stresses inthe x-direction, σcx, obtained from the experimental andpredicted results of the bond-slip model as a function of thetotal concrete strain in the x-direction, εcx. The modelcaptures much more of the complex behavior than existingtension stiffening models.8

Therefore, in this study, the bond-slip behavior is indirectlyimplemented using an equivalent average tension stiffeningmodel. In a typical concrete tension model, the averageconcrete stress in tension reaches a maximum positive valueat cracking and then decays according to some function ofthe tensile strain. This reduction in average tensile stress isdue to loss of bond between the concrete and steel. Thedecay function for monotonic loading is experimentallydetermined.10 Difficulties arise, however, when consideringcyclic loading because there is no mechanistic conceptbehind the tension stiffening model to guide the development ofa cyclic model, particularly with respect to crack closing. Avariety of approaches to loading and unloading and crackclosing and opening have been proposed2,4 to provide aconstitutive model under cyclic loading. These methodsbasically connect empirically determined tension andcompression envelope curves without providing any mechanisticmodel. Herein, the mechanistic bond-slip model is used toderive a consistent total strain-based model that includesbond slip so that material level iteration8 can be avoided inFEA. In addition, a simple method is developed to determinecrack strains from the total strains so that shear frictionbehavior11 can be implemented in the future.

OpenSeesThis paper discusses the development of a bond-slip-based

concrete material model in the OpenSees framework (http://opensees.berkeley.edu). The OpenSees software, which isdeveloped as the simulation tool for the Network forEarthquake Engineering Simulation (NEES), is used toimplement the proposed material model so that the sourcecode is openly available to the structural engineering researchcommunity to evaluate and modify. The hierarchical nature of theOpenSees software architecture allows new material models to beseamlessly added to the framework by keeping elementand material implementations separate.12

RESEARCH SIGNIFICANCEThe 2D nonlinear bond-slip-based material model

described herein is the first step toward implementing a friction/dilatancy model13 for cyclic shear behavior that appropriatelymodels shear friction behavior predicted from crack slip and

dilatancy, and to include additional features attributed tononorthogonal secondary cracks. This aspect is beingstudied for an FEA implementation and will be considered ina subsequent paper. It is also intended that all progress willbe implemented within the OpenSees platform for ease ofevaluation, improvement, and sharing of ideas.

FEA FORMULATIONOrthogonal fixed crack model

A simplified fixed crack model was used in implementinga 2D nonlinear RC material model. The first crack (principalstress) direction is saved and is used to build the concretestress-strain relationship for the remainder of the analysis. Asecond crack may form perpendicular to the first crack. Thedeviation angle between the first crack and concrete principalstress directions is neglected in this study, but will be consideredin the future.

Concrete material tangent stiffnessThe total element strains εx, εy, and γxy in the global

x-y-coordinate system must be transformed to the first crack(d-r) coordinate system by the transformation angle, θdr, touse the concrete constitutive model. The concrete strainvector and incremental stress vector in the first crack coordinatesystem are determined by Eq. (1) and (2), respectively,where T is a transformation matrix. The concrete materialtangent stiffness matrix, [Dc], is the 3 x 3 matrix presented inEq. (2), and the stress-strain relationship (partial derivatives)and Gc in Eq. (2) are defined in the later section, concrete model.

(1)εd

εr

γdr

T θdr( )[ ]εx

εy

γxy

=

ACI member Migeum So is a Graduate Engineer of the Structural Diagnostics Groupfor Walter P Moore and Associates. She received her PhD from Washington University,St. Louis, MO. Her research interests include composite materials, nonlinear FEA/repair of concrete structures, and sustainable design as a LEED AP.

ACI member Thomas G. Harmon is the Clifford Murphy Professor of Civil Engineering atWashington University. His research interests include reinforced concrete structures,confined concrete, precast and prestressed concrete, and composite materials.

Shirley Dyke is a Professor of Mechanical and Civil Engineering at Purdue University,West Lafayette, IN. Her research interests include structural dynamics, structuralhealth monitoring, and structural control systems.

Fig. 2—Equivalent tension stiffening models.8


(2)

Steel material tangent stiffnessThe total strains in the global x-y-coordinate system are

transformed to the reinforcement steel (L-T) directions by thetransformation angle, θLT. The steel strain and incremental stressvectors in the L-T-coordinate system are determined by Eq. (3)and (4), respectively, given that the L and T directions areperpendicular to each other. The steel material tangentstiffness matrix (Ds) is the 3 x 3 matrix presented in Eq. (4).The variables ρL and ρT are the reinforcement ratios in the Land T directions, and the stress-strain relationship is definedin the later section, analytical reinforcing steel model.

(3)

(4)

Total material stiffness matrix [Dxy]The combined RC material tangent stiffness matrix [Dxy],

formulated in the global x-y-coordinate system, is determinedby Eq. (5).

(5)

This material stiffness matrix is used to determine theelement stiffness matrix. It may be modified in the future toprovide an improved material model.

MATERIAL MODELSOpenSees FEA framework

Many multi-dimensional nonlinear material models havebeen developed to analyze soil structures in the OpenSeesframework; however, a 2D nonlinear material model foranalyzing RC shear wall structures has not been madeavailable through the OpenSees Web site. The proceduresused to adopt a uniaxial material model12 were closelyfollowed to develop a 2D nonlinear material model in theOpenSees framework.

The proposed total strain-based bond-slip material modelintegrates the steel and concrete behaviors in a single sourcecode and includes bond-slip behavior indirectly in the tensileregime and 2D concrete softening effects in the compressiveregime as discussed in the following.

Effect of bond slip on behavior of RC shear wallA simply supported unit-dimension four-node quad

element was analyzed with two different RC materialmodels, referred to as Model NS (no slip) and Model BS(bond slip). Model NS simply adopts an existing uniaxialmodel (Concrete 02, OpenSees) and modifies the compressiveregime as discussed in the following (refer to Fig. 3). TheConcrete 02 Model was developed by OpenSees developersand is publicly available on the OpenSees Web site. For theproposed bond-slip-based material model (Model BS), boththe tensile and compressive regimes were modified asdescribed below (refer to Fig. 4). The element was subjectedto uniform cyclic x-direction strain history. The y-directionstrain was one half the x-direction strain and of oppositesign. The x-direction stress-strain relationships generated bythis analysis are shown in Fig. 5 and 6 including the concrete,steel, and combined stress-strain relationships.

Concrete model—compressive regimeThe uniaxial concrete material model Concrete 02, available in

OpenSees, was modified to include reduced compressivestrength of cracked concrete under biaxial loading (refer tothe dashed curve in Fig. 3 and 4). The d-direction stress, σd,envelope curve in the compression region is revised to be a

Δσd

Δσr

Δτdr

∂σd

∂εd

---------∂σd

∂εr

--------- 0

∂σr

∂εd

--------∂σr

∂εr

-------- 0

0 0 Gc

Δεd

Δεr

Δγdr

=

εL

εT

0

T θLT( )[ ]εx

εy

γxy

=

ΔσL

ΔσT

0

ρLdσL

dεL

--------- 0 0

0 ρTdσT

dεT

--------- 0

0 0 0

ΔεL

ΔεT

0

=

Dxy[ ] T θdr

–( )[ ] Dc[ ] T θdr

( )[ ] T θLT

–( )[ ]+ Ds[ ] T θLT

( )[ ]=

Fig. 3—Concrete Model NS: (a) tensile stiffening model.Fig. 4—Concrete Model BS: (a) tensile stiffening model;and (b) crack closing model.

95ACI Structural Journal/January-February 2010

function of tensile strain in the orthogonal r-direction, εr, andvice versa to represent the fact that tensile strain in one direc-tion reduces the compressive strength in the other direction.Herein, the equations used for the uniaxial concrete model aremodified to include this softening effect, as shown in Eq. (7)and (8), using the softening parameter ζ, determined by Eq. (6).The softening parameter was proposed by Vecchio based onexperimental results.3 It is noted that Eq. (7) and (8) do not applyto the biaxial compression case; however, a separate equationmay be easily introduced in the proposed concrete model.

(6)

(Reference 3)

(7)

(References 2 and 3)

(8)

(References 2 and 3)

Concrete model—tensile regimeThe proposed tension stiffening model simulates the

phenomenon where a cracked concrete element can carrysome tensile forces across cracks due to bond between steelreinforcement and concrete. A crack closing model represents thebehavior when a cracked concrete element takes compressionthat is not carried by steel reinforcement as the cracks areclosed. Thus, the estimated concrete stress should bedependent on the steel reinforcement ratio ρ and bardiameter db, which determines the frictional contact areawhich, in turn, affects the surrounding concrete behavior.Figure 3 shows that zero friction is provided in the tensileregime for Model NS; however, the tension stiffening(monotonic) curve proposed by Stevens et al. and representedby Eq. (9)10 is adopted for Model BS. Herein, the concretetensile stress fc is limited to the maximum stress that can bedeveloped in the concrete due to friction between the steeland concrete with Ct set to 10 mm (0.394 in.). This frictionstress approaches Ctρ/db in Eq. (9) and is referred to as “friction”in this paper (refer to Fig. 4), where εc and εt are totalconcrete strain and tensile cracking strain, respectively.

(9)

Figure 3 shows that Model NS does not provide a rationalcrack closing model as reloading/unloading in tension issimply modeled with a single line. To develop a completestress-strain relationship under cyclic loading, not onlyshould the monotonic envelope curve be provided but alsothe unloading and crack closing models that represent thefrictional resistance. Relatively stiff response is expectedwhen the load reverses from the tension stiffening curvebecause the friction has to be overcome before the reinforce-

ment bars slide in the opposite direction. In this study, theinitial stiffness Et is used to provide numerical stability forunloading from the tension stiffening curve, as presented inFig. 4.

Once the friction is overcome, the reinforcement bars slidein the opposite direction until the crack surfaces come intocontact. Concrete elements subjected to biaxial tension-compression loading expand not only because of the tensilecrack strains but also because of compressive stress appliedin the direction perpendicular to the tension direction.Concrete crack strains should be evaluated based on tensilecrack strains as well as concrete expansion strains due to thecompressive stress in the direction perpendicular to thetensile direction. A portion of the concrete expansion strainsare permanent so that it allows concrete to take the compressivestress although the total strains are positive.8

A quadratic stress-strain relationship is proposed toprovide a crack closing model (Points 1 through 2), aspresented in Fig. 4. Point 1 in Fig. 4 is the state where thecrack strain (discussed later in this paper) becomes zero and,thus, the concrete takes compression even though theaverage strain is positive. As the strain approaches zero(Point 2), the concrete takes compression and continues withreloading in compression (Point 3). The multiplier, S, and εC,shown in Fig. 4, are proposed to be 0.18 and 8Sεmax in thisstudy to best match the experimental results but should befurther refined based on experimental studies. Once the threepoints Point 1(Sεmax, friction), Point 2(0, σε0), andPoint 3(εC, EcεC) are determined, they are used to computethe three parameters that constitute Eq. (10). In addition, thisequation provides a smooth transition between crack closingand compression reloading. The parameters A, B, and C areproposed to be empirically determined based on the threepoints defined previously

(10)

where

Concrete model—shear retention factorA constant shear modulus Gc, presented in Eq. (2), is

defined by Eq. (11).

(11)

The shear retention factor β can range from a very smallvalue, say 0.001 to 1.0. It indicates the retained concreteshear stiffness, relative to the initial uncracked concreteshear stiffness, after the concrete cracks. This approach willbe replaced with a shear friction model14 in the future.

Analytical reinforcing steel modelThere are many available reinforcing steel models that

include the classical reinforcing steel behavior. Steel modelsproposed by Seckin14 and Yokoo et al.15 have been used inimplementing total strain-based membrane models such asMCFT and Cyclic Softening Membrane Model (CSMM),2

ζ 1

0.8 0.34 εr

εc0

-------–

-------------------------------=

σd ζfc′ 2 εd

ζεc0

----------⎝ ⎠⎛ ⎞ εd

ζεc0

----------⎝ ⎠⎛ ⎞

2

– 0 εd ζεc0≤ ≤=

σd fcu ζfc′–( )εd ζεc0–

εcu ζεc0–----------------------- fc′ ζ+ ζε0 εd εcu≤<=

fc 1 Ctρdb

-----–⎝ ⎠⎛ ⎞ e

270

Ctρ

db-----

----------------– εc εt–( )

Ct+ ρdb

-----=

fc Aεc2 Bεc C+ +=

Aεmax EcεC σε0–( ) εC+ σε0 friction–( )

εCεmax εC S2εmax–( )---------------------------------------------------------------------------------------------; =

B AεC=σε0

εC

-------- Ec; C σε0=–+

Gc βEc

2 1 v+( )--------------------=


respectively. Hoehler and Stanton’s steel model16 was usedfor the previous bond slip membrane model study.8

In this study, the OpenSees Hysteretic Uniaxial MaterialModel is selected instead of the Chang and Mander’smodel,17 adopted as the Reinforcing Steel Model inOpenSees, to represent the smeared reinforcing steel materialbehavior because it provides better convergence, eventhough it simplifies the steel material behavior as shown inFig. 5(b) and 6(b). Belarbi and Hsu18 proposed that theclassical reinforcing steel envelope curve be reduced as a

function of the steel reinforcement ratio to better representthe behavior of smeared reinforcing steel (refer to Fig. 7(a)).A similar approach is used for this study. In theory, the totalstress (sum of the average concrete and steel stresses) at anopen crack cannot exceed the yield stress and, therefore, thebare steel stress for use in a 2D membrane element must bereduced to represent the average steel stress. The uniaxialenvelope curve of bare steel is reduced by an R factor and isfurther reduced (refer to Fig. 7(b)) by the amount of average

Fig. 6—Analysis results of single element using MaterialModel BS.

Fig. 5—Analysis results of single element using MaterialModel NS.

ACI Structural Journal/January-February 2010 97

concrete frictional stress evaluated by Eq. (9). Therefore, theproposed steel envelope curve is dependent upon themaximum friction developed in concrete which, in turn, is afunction of the reinforcement ratio and bar size. In this study,the R factors are determined as a function of ρ (reinforcementratio) by interpolating the reduction in the steel stress inFig. 7(a). The R values (for ρ ≈ 3%) used in this paper arepresented in Fig. 8(b) through (d) in conjunction with theanalysis results. As shown in Fig. 7(b), a 2% slope isprovided (1.02fy) to prevent a zero slope, and the slope of Esh(= 1.02Es) is used for strain values greater than εsh (= 8εy) todetermine the steel stress.

Proposed RC modelThe concrete and steel material models described previously

are included in a single source code (C++) so as to allow thecoupling of concrete and steel behavior. Provided that theinitial crack is formed in the x- or y-direction, Fig. 5(c) and6(c) show the combined stress-strain relationships in thex-direction for Models NS and BS, respectively.

Significance of separating crack strainA secondary, but important, objective of this paper is to

investigate the significance of separating crack strain fromtotal strain to implement a shear friction model. The fundamentalconcept of a shear friction model is enforcing a relationshipbetween crack slip and crack separation when cracks are incontact and subjected to shear.13 Crack separation due to slipcauses additional steel tensile stress, which must be equilibratedby compressive crack surface contact stress. This, in turn,increases the level of shear stress needed to overcome frictionand dilantancy to allow crack slip. The simplest crack slipversus crack separation strain relationship that can be developed islinear (γcr = aεcr), where γcr is the crack shear strain, εcr isthe crack normal tensile strain, and a is a constant; however,even this simplest possible relationship requires materiallevel iteration when implemented in an FEA. Although notas accurate, the material level iteration can be eliminated byimplementing the crack slip relationship as a total strainrelationship (γ = aε).

Concrete shear deformation is typically very small foruncracked concrete and even some cracked concrete asevidenced by the insensitivity of analytical results to the

value of the shear retention factor β. Once cracks are wide,however, shear behavior can be important and may be poorlypredicted by constant shear retention factors. This is themotivation for replacing the constant shear retention modelwith a new shear friction model13 in future work. It alsoseems reasonable to expect that the linear relationship (γ = aε)will give sufficiently accurate results because crack strainswill dominate the total strain when either shear or tensilestrains are large. This issue is discussed in the following inlight of analytical results for shear walls.

STUDIED RC SHEAR WALLAn RC shear wall tested by Pilakouta and Elnashai19,20 was

selected to demonstrate the analysis capability of the twostudied material models. The shear wall, SW4, was tested asan isolated cantilever. Table 1 provides a summary of thematerial properties and reinforcement details and Fig. 9 illus-trates the reinforcement details for the studied shear wall.

ANALYTICAL PROCEDUREBuilding Tcl input file for RC shear wall model

OpenSees input files are written with the Tool CommendLanguage (Tcl). Therefore, a Tcl input file is constructed tobuild an RC shear wall model for FEA using the NS and BSmaterial models developed in the OpenSees framework. Ashear wall model was built using 163 four-node quad-planestress elements, as shown in Fig. 10. The mesh was dividedinto four regions: the web, boundary flanges, top slab, andbottom slab, as indicated by different levels of shading. Thebottom slab was fixed at the base. A cyclic displacement

Fig. 7—Reinforcing steel stress-strain relationship: (a) average reinforcing steel stress-strain curves for varying percentage ofsteel suggested by Belarbi and Hsu16; and (b) comparison of steel envelope curves. (Note: 1 ksi = 6.9 MPa; 1 psi = 6.9 KPa).

(a) (b)

Table 1—Material properties used for SW4 analysis

Specimen Zone Concrete

Reinforcement

ConfiningHorizontal Vertical

Properties —fc′ ,

MPa (ksi)

Ec,MPa (ksi)

ρ, %

fy-type,MPa (ksi)

ρ, %

fy-type,MPa (ksi)

ρ, %

fy ,MPa (ksi)

SW4

Web 36.9(5.35)

35,240(5111) 0.39

550-HT6

(79.77)0.31

550-HT6

(79.77)— —

Flange 36.95.35

35,240(5111) 0.78

550-HT6

(79.77)2.83

500-HT12

(72.52)0.43 545

(79)


history was applied to the center eight nodes of the top slabin the horizontal direction and the corresponding horizontalforces were recorded. The shear retention factor, β = 0.012,was used to analyze SW4.

FEAThe Tcl input file also includes the material model assignment

for each element, analysis method, and nonlinear solutionalgorithm. The Newmark time-stepping method that uses threequadratic basis functions in time to solve the equation of motion(with 5% damping) was selected for a transient analysis withγ = 0.5 and β = 0.25. The Newton Raphson method wasselected as the nonlinear solution algorithm.

COMPARISON OF PREDICTIONSAND EXPERIMENTAL RESULTS

The global force-displacement response of the shear wallmodel is presented in Fig. 11 and 8. Figure 11 includes thepushover analysis results produced using the two studiedmaterial models, and Fig. 8(b) and (c) show the cyclic analysisresults produced using materials Models NS and BS, respectively,in comparison with the experimental results (refer to Fig. 8(a)).

Model NS yields surprisingly good force-displacement response.A slightly improved reloading force-displacement relationshipis produced by using material Model BS, as indicated by thearrow A in Fig. 8(c). Furthermore, the strength reduction at eachcycle is reasonably modeled. The strength seems to be overpre-dicted, however, by providing the friction in the concrete model

Fig. 9—Details of SW419: (a) dimensions; (b) reinforcementlayout (elevation); and (c) Section A-A′.

Fig. 10—FEA model (SW4).

Fig. 8—Cyclic analysis results (SW4).

ACI Structural Journal/January-February 2010 99

without further reducing the steel envelope curve. Thus, the Rfactor used in the steel model (refer to Fig. 7(b)) is reduced tolower the envelope curves as shown in Fig. 8(d).

Computed crack strainsFigures 12(a) through (c) present the local stress-strain

behavior for three elements (refer to Fig. 10) corresponding

to the force-displacement response shown in Fig. 8(c). Onlythe y-direction relationships are shown because the x-directionstress-strain relationships are linear.

Figures 12(d) through (f) show the relationships betweentotal normal strain and crack normal strain for the same threeelements. The crack strains εcr were computed bysubtracting the strain in the solid concrete (which equals theconcrete stress, σc, divided by the elastic modulus ofconcrete, Ec) from the total strain, ε.

(12)

As expected, the total strains are almost exactly crackstrains except for compressive strains and very low values oftensile strains. Therefore, it is suggested that the linear crackslip-total strain relationship, γ = aε, be used in developing atotal strain-based model for shear friction.

DISCUSSIONIn this study, a constant bond-slip discrete frictional force is

assumed between concrete and steel; however, experimentally

εcr εσc

Ec

-----–=

Fig. 11—Pushover analysis results (R = 0.95).

Fig. 12—Element stress-strain response for Model BS and crack strain.


observed bond stress-slip relationships are more compli-cated.5,6 Therefore, better results could be obtained with arefined model that includes bond stress, that is distributedalong the length of the unbonded region and varies in intensity.

In this paper, an orthogonal crack model is adopted toanalyze a shear wall structure that does not exhibit shear-critical behavior. Secondary cracks and shear stressdeveloped along the crack surfaces, however, should befurther studied in conjunction with the proposed model toanalyze shear-critical structures.

The convergence for FEA of the shear wall model SW4 isobserved to be sensitive to the value of the shear retentionfactor. There are two major problems with the reduced shearmodulus approach. The first is that the value of β is arbitrary.The second is that there is no limit to the shear strengthaccording to this model. A limit to the crack shear stress can,of course, be imposed. But it is well known that the amountof shear that can be transmitted across a concrete crackdepends on the amount of steel crossing the crack and thenormal stress on the crack surface due to external forces. Theshear-friction design method of ACI 318-0511 captures thisfundamental behavior of shear across cracks in RC. The FEAimplementation will be discussed in a separate paper.

CONCLUSIONSThe bond-slip friction behavior is implicitly embedded in

the proposed concrete material model for total strain-basedFEA. Based on the results of this study, the followingconclusions are drawn:

1. A 2D nonlinear material model that is capable ofrepresenting frictional resistance and bond-slip behavioris developed for use in a 2D total strain-based FEA in theOpenSees framework.

2. One benefit of the proposed material model herein isthat it is better able to model concrete compressive stressesfor crack closing than prior models which, in turn, improvesthe reloading portion of the force-displacement response.

3. The RC shear wall analysis results show reasonableagreement with the experimental results although theysuggest that the concrete crack model and the shear stress-strain relationship should be further improved.

4. The yield level of the steel model must be reduced toaccount for the bond stress of the tension stiffening model.The bond stress may be estimated based on available testresults5,6 although more experimental results are needed.

5. Total strains can be substituted for crack strains indeveloping a total strain-based shear-friction model.

ACKNOWLEDGMENTSThe research was supported by Grant No. CMMI-0625640 from the

National Science Foundation.

NOTATIONEc = elastic modulus of concreteEs = elastic modulus of steelEsh = modulus of strain hardeningfc ′ = cylinder compressive strength of concretefc = concrete stressfcu = crushing stress of concreteft = tensile cracking stress of concretefy = yield stress in bare steel barsR = steel stress reduction factors = crack spacingεc = concrete strainεco = strain at peak compressive stress in concrete cylinderεcu = crushing strain of concreteεmax = maximum (experienced) total strain

εmin = minimum (experienced) total strainεs = steel strainεsh = strain corresponding to initial strain hardeningεt = tensile cracking strain of concreteεy = steel yield strainγ = debonded length factorρ = reinforcement ratioσc = concrete stressσc

b = concrete stress in bond regionσc

sl = concrete stress in slip regionσs = steel stressσε0 = stress in concrete at zero strain, ε0

REFERENCES1. Palermo, D., and Vecchio, F. J., “Simulation of Cyclically Loaded

Concrete Structures Based on the Finite-Element Method,” Journal ofStructural Engineering, ASCE, V. 133, No. 5, May 2007, pp. 728-738.

2. Mansour, M., and Hsu, T. T. C., “Behavior of Reinforced ConcreteElements under Cyclic Shear. II: Theoretical Model,” Journal of StructuralEngineering, ASCE, V. 131, No. 1, Jan. 2005, pp. 54-64.

3. Vecchio, F. J., and Collins, M. P., “The Modified Compression-FieldTheory for Reinforced Concrete Elements Subjected to Shear,” ACIJOURNAL, Proceedings V. 83, No. 2., Mar.-Apr. 1986, pp. 219-231.

4. Palermo, D., and Vecchio, F. J., “Compression Field Modeling ofReinforced Concrete Subjected to Reversed Loading: Formulation,” ACIStructural Journal, V. 100, No. 5, Sept.-Oct. 2003, pp. 616-625.

5. Eligehausen, R.; Popov, E. P.; and Bertero, V. V., “Local Bond Stress- SlipRelationships of Deformed Bars under Generalized Excitations,” Report UCB/EERC-83/23, EERC, University of California, Berkeley, CA, 1983, 169 pp.

6. Lowes, L. N.; Moehle, J. P.; and Govindjee, S., “Concrete-Steel BondModel for Use in Finite Element Modeling of Reinforced Concrete Struc-tures,” ACI Structural Journal, V. 101, No. 4, July-Aug. 2004, pp. 501-511.

7. Bentz, E. C., “Explaining the Riddle of Tension Stiffening Models ofShear Panel Experiments,” Journal of Structural Engineering, ASCE,V. 131, No. 9, Sept. 2005, pp. 1422-1425.

8. So, M.; Harmon, T. G.; Yun, G. J.; and Dyke, S., “Inclusion of SmearedCyclic Bond-Slip Behavior in Two-Dimensional Membrane Elements,” ACIStructural Journal, V. 106, No. 4, July-Aug. 2009, pp. 466-475.

9. Mansour, M., and Hsu, T. T. C., “Behavior of Reinforced ConcreteElements under Cyclic Shear. I: Experiments,” Journal of StructuralEngineering, ASCE, V. 131, No. 1, Jan. 2005, pp. 44-53.

10. Stevens, N. J.; Uzumeri, S. M.; Collins, M. P.; and Will, G. T.,“Constitutive Model for Reinforced-Concrete Finite-Element Analysis,”ACI Structural Journal, V. 88, No. 1, Jan.-Feb. 1991, pp. 49-59.

11. ACI Committee 318, “Building Code Requirements for StructuralConcrete (ACI 318-05) and Commentary (318R-05),” American ConcreteInstitute, Farmington Hills, MI, 2005, pp. 167-170.

12. Scott, M. H., and Fenves, G. L., “How to Introduce a New Materialinto OpenSees,” PEER Report, V. 1.1, University of California, Berkeley,CA, Aug. 2001, pp. 1-30.

13. So, M., “Total-Strain Based Bond/Slip and Shear/Friction MembraneModel for Finite Element Analysis of Reinforced Concrete,” PhDdissertation, Washington University, St. Louis, MO, 2008, 240 pp.

14. Seckin, M., “Hysteretic Behavior of Cast-in-Place Exterior Beam-Column Sub Assemblies,” PhD thesis, University of Toronto, Toronto, ON,Canada, 1981, 266 pp.

15. Yokoo, Y., and Nakamura, T., “Nonstationary Hysterectic UniaxialStress-Strain Relations of a Steel Bar,” Transactions of the ArchitecturalInstitute of Japan, Tokyo, Japan, No. 260, 1977, pp. 71-80.

16. Hoehler, M. S., and Stanton, J. F., “Simple Phenomenological Modelfor Reinforced Steel under Arbitrary Load,” Journal of StructuralEngineering, ASCE, V. 132, No. 7, July 2006, pp. 1061-1069.

17. Chang, G. A., and Mander, J. B., “Seismic Energy Based FatigueDamage Analysis of Bridge Columns: Part I—Evaluation of SeismicCapacity,” Technical Report NCEER-94-0006, NCEER, SUNY, Buffalo,NY, 1994.

18. Belarbi, A., and Hsu, T. T. C., “Constitutive Laws of SoftenedConcrete in Biaxial Tension-Compression,” ACI Structural Journal, V. 91,No. 4, July-Aug. 1994, pp. 465-474.

19. Pilakoutas, K., and Elnashai, A., “Cyclic Behavior of ReinforcedConcrete Cantilever Walls, Part I: Experimental Results,” ACI MaterialsJournal, V. 92, No. 3, May-June 1995, pp. 271-281.

20. Pilakoutas, K., and Elnashai, A., “Cyclic Behavior of ReinforcedConcrete Cantilever Walls, Part II: Discussions and Theoretical Comparisons,”ACI Materials Journal, V. 91, No. 2, May-June 2002, pp. 1-11.

FEA Implementation of Smeared Cyclic Bond Slip-Based

Documents

Transcript of FEA Implementation of Smeared Cyclic Bond Slip-Based