Modeling for Postyield Buckling of Reinforcement

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Modeling for Postyield Buckling of ReinforcementRajesh Prasad Dhakal1 and Koichi Maekawa2

Abstract: Finite element microanalysis using fiber technique was carried out to study the buckling mechanism of reinforcing barfound that reinforcing bars under inelastic axial compression exhibit lateral deformation defined as buckling due to the geononlinearity. Further investigation revealed that the postbuckling average compressive stress is less than the local stress corresthe same strain due primarily to the different stiffness for loading and unloading fibers in the laterally deformed section. It wasthat the average compressive stress-strain relationship including the softening in the postbuckling range can be completely determs of the product of square root of yield strength and the slenderness ratio of the reinforcing bar. Moreover, a unique relbetween the average stress and average strain of reinforcing bars including the effect of buckling is established through variousanalyses. The comparison of the analytical results and proposed model with some experimental results showed good agreeverifying the reliability of the microanalysis and proposed computational model.

DOI: 10.1061/~ASCE!0733-9445~2002!128:9~1139!

CE Database keywords: Buckling; Deformation; Stress-strain relations; Reinforcement.

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Introduction

In reinforced concrete members, the reinforcing bars mightdergo high-compressive strain, which induces large lateral demation of reinforcing bars, hereafter referred to as buckling.cause of geometrical nonlinearity, average compressive scarried by reinforcing bar decreases in the postbuckling ranHowever, in tension, geometrical nonlinearity does not prevaino lateral deformation is induced even after yielding. Conquently, the average tensile stress-strain relationship overspecified control volume is exactly the same as the point wtensile stress-strain relationship. Therefore, using similar avestress-strain behavior for reinforcing bars in tension and compsion seems unconvincing.

Of course, the point wise stress-strain behavior of reinforcbars in compression is also the same as that in tension becaupoint wise relationships are not influenced by the change inometry~Dodd and Restrepo-Posada 1995!. Hence, using the poinwise stress-strain relationship can account for the lateral defortion of compressed longitudinal reinforcing bars provided thatcontrol volume, over which the average stress-strain relationis computed, is very small. However, in the finite element strtural analysis of larger scale, such small element size is notsible because of the long computation time and large memrequired. For finite element analysis of reinforced concrete st

1Research Fellow, School of Civil and Structural Engineering, Nyang Technological Univ., 50 Nanyang Ave., Singapore 639798;merly, Graduate Student, Univ. of Tokyo. E-mail: [email protected]

2Professor, School of Civil Engineering, Univ. of Tokyo, Hong7-3-1, Bunkyo-Ku, Tokyo 113, Japan. E-mail: [email protected]

Note. Associate Editor: C. Dale Buckner. Discussion open until Fruary 1, 2003. Separate discussions must be submitted for indivipapers. To extend the closing date by one month, a written requestbe filed with the ASCE Managing Editor. The manuscript for this pawas submitted for review and possible publication on April 28, 20approved on January 23, 2002. This paper is part of theJournal ofStructural Engineering, Vol. 128, No. 9, September 1, 2002. ©ASCEISSN 0733-9445/2002/9-1139–1147/$8.001$.50 per page.

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tures with substantial element size, an average stress-straintionship of reinforcing bars that takes into account the effectbuckling is needed. Hence, this research aims to formulate asatile average stress-strain relationship that can be applied tinforcing bars with any geometrical and mechanical propertie

To generate the data for model formulation, a parametric stbased either on experiment or analysis is necessary. Theresome experimental studies performed in the past to clarifybuckling mechanism of bare bar~Monti and Nuti 1992; Rod-riguez et al. 1999!. Tests are usually conducted within some fixranges of material properties, and in some cases we have tosider the boundary conditions, which is hardly reproduced intests. For obtaining widely applicable constitutive models,experiments should consist of test specimens that systematicover a wide range of geometrical as well as mechanical proties. Hence, analytical parametric study is preferred ahead oftensive experimental study, and some available experimentasults are used to justify the analytical method and also to vethe proposed constitutive equations.

Simulation of Bare Bar Buckling

Fiber Technique for Finite Element MethodMicroanalysis

A three-dimensional nonlinear finite element program calCOM3 @Concrete Model in 3D# ~Tsuchiya et al. 1999! is used forthe analytical parametric study. Nonlinear space frame elemanalyzed by fiber technique~Menegotto and Pinto 1973! are usedto model reinforcing bars. In fiber technique, each elemenrepresented using a single line coinciding with the centerlinethe member. The member cross section is divided into many cor subelements. The strain of each cell is calculated based onEuler-Kirchoff’s hypothesis, i.e., plane section remains plaafter bending. For each fiber strain along the axis of finite ement, response is calculated using the material constitutive mels representing the local behavior. The implementation of fitechnique is briefly illustrated in Fig. 1.

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In this study, isoparametric elements with three nodesused. In each node, six degrees of freedom are consideredthe displacements~u, v, andw! and rotations~ux , uy , anduz! inthree mutually perpendicular directions~local axesXl , Yl , andZl!. Displacement field for an element is calculated fromnodal displacements and the assumed quadratic shape funcNext, strain-displacement relationships are used to computeeralized strains, which include an axial component («0), twoshear components~gzy , gzx!, and three curvatures~fx , fy , andfz!. Based on these generalized strains, the average axial straeach fiber« f is computed. Based on this axial strain, fiber axstress is calculated from the point wise stress-strain relationsThese fiber stresses across the cross section are integratedtain the stress carried by the element.

The last two terms~second-order derivative terms! in the com-putation of average axial fiber strain includes the change inometry due to large displacement of the reinforcing bars durloading. In small strain problems, where the geometry ofstructure basically remains unchanged, these higher-order tcan be neglected. However, large lateral displacement of reinf

Fig. 1. Fiber technique adopted in finite element method micanalysis

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ing bars cannot be captured without giving due considerationgeometrical nonlinearity. This fact is analytically proved aftewards.

Preliminary Microanalysis

The reinforcing bar is modeled as a vertical column of lengthLand diameterD. The total lengthL is discretized into severasmall elements. Monotonic downward displacementD is appliedat the top of the reinforcement. To represent bare bar comprestests, the two ends of the reinforcing bars are represented bynodes; i.e., both displacement and rotation are not allowed toplace. An elastic-perfectly plastic point wise stress-strain relatiship with yield strength equal to 400 MPa and Young’s moduequal to 200 GPa is adopted.

First of all, a pair of analyses with and without considerigeometrical nonlinearity is performed, and the average respoare compared in Fig. 2~a!. If geometrical nonlinearity is not considered; i.e., the second-order terms are neglected in the comtation of fiber strain, the postyield buckling phenomenon canbe predicted, and the average response becomes the samepoint wise response. Next, mesh sensitivity analysis is conduto obtain the influence of mesh discretization in average respoA reinforcing bar of lengthL516 cm, diameterD516 mm, andyield strengthf y5400 MPa is analyzed with varying number oelements. The average load-displacement curves are comparFig. 2~b!. Using only one element to represent the bar rendethe average response similar to the assumed elastoplasticbehavior, and could not predict the softening of the average cpression in spite of considering geometrical nonlinearity. Aspected, the postbuckling average responses obtained usingelements exhibit increased softening, and the mesh dependnearly vanishes for more than four elements. Hence, at leastor more elements are used for further analyses.

Fig. 2. ~a! Effect of geometrical nonlinearity;~b! effect of mesh dicretization;~c! lateral displacement profile along length; and~d! variation ofaverage strain along length.

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Next, the lateral deformation profile and distribution of fibstrain along the bar length are explained. Three instants denby pointsA, B, and C in Fig. 2~b! are selected for comparisonPoint A refers to the instant just before the average stress sdecreasing, and pointsB and C lie in the softening phase. Thlateral displacement profiles plotted in Fig. 2~c! indicate that thebar deforms laterally only after pointA, and this lateral deformation is maximum at the midheight. Similarly, the average str~strain at the center of the cross-section! profiles plotted in Fig.2~d! reveal that the bar experiences uniform compression upoint A, i.e., the starting point of buckling. This uniformity is losafterwards and some portions, especially near the fixed supand around the midheight, experience continuous increascompressive strain while strains in other parts decrease. Duthe curvature induced by lateral deformation, different fibersthe same cross section undergo hardening and unloading deing on the position of the fibers. The difference between harding and unloading stiffness contributes to the decrease in theerage stress.

General Discussion on Finite Element MethodMicroanalysis

In the mesh for microanalysis, all nodes of all elements wperfectly aligned along the same vertical line and no initial iperfection was provided to induce lateral deformation. In spitethis fact, the analytical solution could capture the bucklimechanism. The two possible modes of deformation of aunder axial compression are axial mode representing longitudshortening and bending mode representing the usual buckshape. The axial mode always leads to point wise stress-sbehavior whereas the bending mode triggers the softening oerage stress provided that the material nonlinearity and pathpendency are considered in the constitutive model. Both of thmodes satisfy the requirements; i.e., the compatibility, equirium, and the constitutive equations. However, the energy assated with bending mode is smaller than that with axial moHence, if geometrical nonlinearity is considered and the interdiate nodes are free to deform laterally, the finite element soluselects the bending mode, as it requires less energy duringnonlinear iteration process.

For further understanding, the analysis of a bar under compsion was performed with geometrical nonlinearity but without apath dependency; i.e., stress reversal during cyclic loadingnot considered. In other words, once the compressive strainceeds yielding strain, the stress is kept constant regardless ostrain path~unloading or reloading!. As shown in Fig. 3, the av-erage stress-strain curve in this case is found to be the same a

Fig. 3. Effect of path dependency on average response

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point wise stress-strain relationship in spite of the presencelarge lateral deformation. Although high curvature~strain gradi-ent! across the section could be observed, the stresses at all fiwere equal because of the path-independent constitutive moConsequently, in spite of geometrical nonlinearity, average copressive stress did not decrease.

Euler’s theorem is commonly used to calculate the bucklload of axially compressed longitudinal bars. However, it is baon the assumption that the bar behaves elastically throughouloading. Hence, for short bars, the buckling load predictedEuler’s formula is usually higher than the yield load. But, becauof the elastoplastic nature of the reinforcing bars, the load canlinearly increase in the postyield range, and Euler’s theory islonger applicable. In this case, the FEM microanalysis predthat the buckling starts in the plastic region, where the axial stis slightly higher than or equal to yield stress but the strainsubstantially higher than the yielding strain. Needless to mentfor bars with higher-slenderness ratio, buckling starts in the etic range, and Euler’s formula gives a better prediction of buling load, which is smaller than the yield load.

The softening of average compressive stress in postbuckrange is due mainly to the difference between the stiffnesspostyield loading and unloading paths. The average responsepends on both geometrical nonlinearity and the path dependeand the influence of path dependency becomes crucial only wthe geometrical nonlinearity is taken into account. On the othand, the lateral deformation can be simulated only with gmetrical nonlinearity but the softening of the postbuckling avage stress cannot be captured without considering the path dedency in the nonlinear material model. The above discusssuggest that the average compressive behavior is not uniquedepends on the mechanical properties of the reinforcing bHowever, if unloading and hardening properties are knownadvance, the correct average constitutive model of reinforcbars with any type of point wise stress-strain relationship cananalytically obtained.

Verification and Parametric Study

Verification of Microanalysis

For verification of the analytical method, the computed resultscompared with experimental ones~Mander et al. 1984!. The ex-periment consists of direct compression tests of five shmedium-strength~f y5290 MPa, Es5200 GPa! reinforcing barswith different slenderness ratios~5.5, 6, 6.5, 10, and 15!. The

Fig. 4. Comparison with experimental results~Mander et al. 1984!


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normalized average compressive stress-strain curves arepared in Fig. 4. Note that the material model used in the vecation analysis represents the result of sample bar test, wexhibits reasonable hardening@«sh ~strain at hardeninginitiation!511.7«y ; Esh ~linear hardening stiffness!50.022Es ;sn ~maximum stress!5433 MPa#. The experimental and analytcal curves are compared for only three different slendernesstios, as the results corresponding to slenderness ratios 5.5, 66.5 are found to be very close to each other in experiment asas analysis, and only one representative case (L/D56) amongthese three was chosen. The results of FEM microanalysis agood agreement with the experimental curves giving ample pof the reliability of the analysis.

Next, the monotonic tests of Monti and Nuti~1992! areadopted for the verification of monotonic behavior. The expement consists of a series of monotonic tests on steel reinforbars with different slenderness ratios~5, 8, and 11!, and threedifferent bar diameters of 16, 20, and 24 mm were used for eslenderness ratio. Here, the average of the three responses~withdifferent diameters! for each slenderness ratio is used for coparison with FEM microanalysis. As the details of the mateproperties were not mentioned, the local behavior was fairlysumed to match the average response of the bar with slenderatio 5 because it was reported that the behavior of this shobar almost coincides with the local response. Hence, the folling values were adopted:@Es5200 GPa;f y5480 MPa;«sh ~strainat hardening initiation!52 «y ; Esh ~linear hardening stiffness!50.055Es ; su ~maximum stress!5690 MPa#. The comparativestress-strain curves are shown in Fig. 5. It can be observedthe results of FEM microanalysis are in fair agreement withexperimental results for all three cases of monotonic loadHowever, a small difference can be observed in the early haring range of the bar with slenderness ratio 5. This is due tofact that the nonlinear hardening of the bar with slenderness5 was represented by using an average linear hardening ranthe point wise stress-strain relationship. These verificationsample evidence that the coupled geometrical and path-depenmaterial nonlinear analysis can reliably predict the bucklmechanism and also the average stress-strain behavior of axcompressed reinforcing bars.

Effect of Length and Diameter

It is well known that the average behavior of the reinforcing bavery sensitive to its length and diameter. To elaborate more inregard, two sets of parametric analyses are conducted withstant diameter and different lengths as well as constant length

Fig. 5. Comparison with experimental results~Monti and Nuti 1992!


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different diameters. The normalized average responses arepared in Figs. 6~a and b!. The softening of average stress becomsteeper with the increase in the reinforcement length providedthe bar diameter is constant. Similarly, if the reinforcement lenis kept constant, the softening becomes milder with an increasbar diameter. Both of these give similar qualitative relationshi.e., the more slender the bar, the more severe the softeninaverage stress becomes. In order to obtain quantitative intertionship, two pairs of cases, so that the slenderness ratio~length-to-diameter ratio! is the same but the length and diameter adifferent, are compared in Fig. 6~c!. Irrespective of the differentvalues of length and diameter, the normalized average responsimilar for the same slenderness ratio. Finally, the normaliaverage responses for different slenderness ratios, assuelastic-perfectly plastic behavior andf y5400 MPa, are shown inFig. 6~d!. Agreeing with the conclusions of previous studies~Mauand El-Mabsout 1989; Monti and Nuti 1992; Dhakal anMaekawa 2000!, it is found that the buckling is delayed and thdegradation of postbuckling average stress is retarded withcrease in slenderness ratio of reinforcing bars.

Effect of Yield Strength

The writers believe that the strength of the reinforcing bar ainfluences the average behavior in compression. To exploreinter-relationship, two sets of sensitivity analyses are performwith constant slenderness ratio and different yield strengths.comparative normalized average stress-strain curves for slenness ratios 5 and 10 and yield strengths ranging from 100 to 1MPa are presented in Figs. 7~a and b!, respectively. The softeningof average stress becomes steeper with the increase instrength provided that the slenderness ratio is constant. Agrewith previous study~Mau and El-Mabsout 1989!, the averageresponse of the high-strength bar is found to become more brthan that of lower-strength bar. It was reported that the averresponse becomes nearly the same as the local response ifL/Dratio is equal to or smaller than 5~Mau 1990; Monti and Nuti1992!. However, the results suggested that this is not the reaand this critical slenderness ratio depends on the yield strengthe reinforcing bar. For example, the average response of thewith slenderness ratio 5 is very close to local response forf y

5100 MPa, but the difference between the local and averagesponses of a high-strength bar (f y51,600 MPa) with the sameslenderness ratio is very big. Hence, rather than specifyingrelationship in terms of slenderness ratio and yield strength srately, a combined term including both of these parameters semore suitable.

For this purpose, two pairs of special cases@one pair eachchosen from Figs. 7~a and b!# are compared with each other iFig. 7~c!. The first pair includes two cases, one withf y

5100 MPa; L/D510 and the other withf y5400 MPa; L/D55. Similarly, the second pair includes cases withf y

5400 MPa; L/D510 and f y51,600 MPa; L/D55. The twocases in each pair were intentionally chosen to have equal vof L/DA f y in spite of different values forL/D and f y . Interest-ingly, the normalized average responses of the two cases inpair were exactly similar to each other, suggesting that thereunique interrelationship between the parameterL/DA f y and theaverage response. Although some studies on modeling of avecompressive response of reinforcing bars have been performethe past~Mau and El-Mabsout 1989; Mau 1990; Monti and Nu1992!, this unique interrelationship between yield strength a

Fig. 6. Effect of length and diameter on average response

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average compressive response was not realized. Next, the noized average responses for different values ofL/DA f y , usingelastic-perfectly plastic point wise stress-strain relationship,shown in Fig. 7~d!. As expected, buckling happens earlier andpostbuckling average stress degrades faster with an increasevalue of this parameter.

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Reinforcing Bars with Strain Hardening

The earlier discussions were based on the microanalysis of rforcing bars of elastic-perfectly plastic material. In contrast, comonly used reinforcing bars exhibit hardening after a yield pteau in the postyield range, though the hardening stiffness

Fig. 7. Effect of yield strength on average response


Fig. 8. Average responses of reinforcing bars with hardening

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range may vary between the bars produced in different countTo cover all types of reinforcing bars, the parametric study heafter is conducted for linear hardening material, which along welastic-perfectly plastic material constitute the two oppositetremes of all possible hardening mechanisms. The assumedwise stress-strain relationship is shown in Fig. 8~a!, and the aver-age compressive stress-strain curves of bars with slendernessvarying between 5 and 15 are shown in Figs. 8~b–d! for yieldstrength equal to 200, 400, and 800 MPa, respectively. Aspected, the softening of average postbuckling compressive sfor the same slenderness ratio becomes steeper with the incin yield strength. Moreover, the starting point of negative slopedelayed, and the absolute value of normalized stress is also hthan that in elastoplastic bar.

Note that the negative slope of the average stress-strain cuin the later descending stage is nearly constant~approximatelyequal to 2% of Young’s modulus!, irrespective of the value of theparameterL/DA f y . Similarly, the average compressive strenearly becomes constant after it decreases to 20% of the ystrength. These behaviors were also seen in elastoplastic barscan greatly assist in the modeling of average compressive beior of reinforcing bars. As in elastoplastic bars, the averagesponse of these linear hardening bars also uniquely dependthe parameterL/DA f y , irrespective of separate values ofL, D,and f y . As the responses of bars with these two extreme mo~elastic-perfectly plastic and linear hardening! indicate similarinter-relationships between the slenderness ratio, yield strenand the average compressive response, the same should becable to reinforcing bars with any hardening mechanisms. Hethe limiting L/D ratio below which the average behavior is cloto the material model can be better approximated by a multipl(1/A f y) rather than a constant value regardless of bar strengtrecommended in previous studies.


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Average Stress-Strain Relationship

Through the analytical parametric study, various facts regardthe average behavior of reinforcing bars in compression arevealed. Some of them are:~1! the average compressive stresstrain relationship depends only on the product of slendernratio and the square root of yield strength;~2! the average com-pressive stress becomes smaller than the point wise stressthe initiation of buckling, and the starting point of average strdegradation also depends on the point wise stress-strain relaship; ~3! the stress degradation rate in the later stage of pbuckling average stress-strain curve is nearly constant witnegative slope approximately equal to 2% of the Young’s molus; and ~4! the average post-buckling compressive stresscomes constant after it decreases to 20% of the yield strengt

Guided by these unique inter-relationships, an average motonic compressive stress-strain model is proposed, the genlayout of which is sketched in Fig. 9. An intermediate poi~«* ,s* ! is established, after which a constant negative stiffnequal to 0.02Es is assumed until the average stress becomes eto 0.2f y . To represent the aforementioned mechanisms, the

Fig. 9. Schematic representation of proposed model

Fig. 10. Determination of coordinates of intermediate point

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Fig. 11. Comparison with monotonic and cyclic test results~Monti and Nuti 1992!


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As the framework for unloading and reloading behaviorstrain reversal, the authors adopt the Giuffre-Menegotto-Pmodel ~CEB 1996! assuming reduced reloading stiffness fropostbuckling compression state, as observed in experim~Monti and Nuti 1992; Suda et al. 1996!. To control the smooth-ness of the transition curves, the cyclic model uses a parameRcomputed from three constantsR0 , a1 , anda2 to be determinedexperimentally. Provided that the resulting value ofR is positive,the transition becomes smoother if the value ofa1 increases orthat of R0 anda2 decreases. As these constants do not influethe monotonic curve and small changes in their values marginaffect the transition shapes during strain reversals, the authave adoptedR0520; a1518.5; anda250.15 after extensivechecking. A complete path-dependent cyclic model relatingaverage stress with average strain can now be formulated by cbining the proposed monotonic compressive average stress-srelationship with tension envelope of the bare bar and the GiufMenegotto-Pinto model for the unloading/reloading loops.

Verification of Proposed Computational Model

First, the performance of the proposed model is comparedFEM microanalysis and experimental results of the monototests~Monti and Nuti 1992! in Fig. 11~a! As the point wise cyclicstress-strain relationship of the bar material was not reportedFEM microanalysis could not be carried out for cyclic tests, athe average cyclic responses observed in the experiment arepared only with those computed by the proposed model, as itrated in Figs. 11~b–d! for slenderness ratios 5, 8, and 11, resptively. As the hardening stiffness in the material modelsignificant (0.055Es) but lasts only for 8«y , the coefficient in theproposed model is assumed asa50.9 according to the earlierecommendation. As seen in the figure, the monotonic curvesdicted by the model for different slenderness ratios are closthe FEM microanalysis and experimental results. Moreover, italso be observed that the proposed model is in good agreewith the experimental results for all three cases of cyclic loadigiving ample evidence that the proposed monotonic average cpressive stress-strain relationship combined with GiuffMenegotto-Pinto cyclic model can be reliably used.

Fig. 12. Comparison of model with microanalysis~continuous linearhardening!


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Nevertheless, the verification for cyclic loading is conductwithin the range of 3% tensile and compressive strains;around 15 times yield strain for normal strength bars. Althou3% compressive strain is difficult to be reached, larger tenstrain can be induced in reinforcing bars inside RC structuduring strong earthquakes. Test data measuring much histrain are scarce, which might be due to the difficulty in measing higher strain with the commonly used strain gauges. Notethe average compressive stress of the most slender bar (L/D511) unloaded from large tensile strain starts to decrease bereaching the compressive strain zone because of the negslope at the target point. This behavior is qualitatively supportthe outcome of previous study~Rodriguez et al. 1999! that thecompressive stress degradation point~termed as buckling onsepoint! depends on the maximum tensile strain reached befostrain reversal. The effect of maximum tensile strain mightmore pronounced if the strain reversal is from much larger strand the proposed model needs to be verified for such large scyclic tests before drawing quantitative conclusions.

Next, the proposed model is compared with FEM microanasis for two extreme cases. Figs. 12 and 13 show the comparbetween the proposed model and FEM analysis for reinforcbars with continuous linear hardening and elastic-perfectly plamaterial models, respectively. The comparisons are performeddifferent values of the parameterFLD* 5L/DA( f y/100). As es-tablished earlier, the coefficienta in the proposed model is assumed as unity for the linear strain hardening case and 0.7elastic-perfectly plastic case. Average responses predicted bproposed model are very close to those by FEM microanalysisalmost all cases. As the model is valid for these two extrecases, the authors believe that it can represent the behavior ofwith any hardening range. The earlier verification of the modwith the test data of bars with reasonable hardening rangeadvocates this claim.

Conclusions

Microanalyses based on fiber technique using finite elementssmall size were conducted to predict the average behavior oinforcing bars in compression accompanying geometrically lanonlinearity. It was revealed that the average compressive beior depends only on one parameter; i.e.,L/DA f y ~product of slen-derness ratio and square root of yield strength!. It was also foundthat the conclusion from some previous studies, stating thataverage compressive response of reinforcing bars with slenness ratio equal to or less than 5 is the same as the local resp

Fig. 13. Comparison of model with microanalysis~elastic-perfectlyplastic!

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is not always true, especially in the case of high-strength bBased on the results of the parametric study, an average comsive stress-strain model is proposed that can be appliedreinforcing bars with any geometrical as well as mechanical prerties. Giuffre-Menegotto-Pinto model for cyclic loops is combined with the proposed model to yield a complete padependent cyclic model. The comparison of the microanalysiwell as the proposed model with some experimental resshowed good agreement and verified the reliability of propomodel for both monotonic and cyclic loading.

Acknowledgments

The writers gratefully acknowledge TEPCO Research Foundaand Grant-in-aid for scientific research No. 11355021 for proving financial support to accomplish this research.

Notation

The following symbols are used in this paper:D 5 diameter of reinforcing bar;

Es 5 Young’s modulus of reinforcing bar;Esh 5 hardening stiffness;f y 5 yield strength of reinforcing bar;L 5 test length of reinforcing bar;a 5 coefficient depending on local behavior of bar;

«, s 5 average strain and average stress;«* , s* 5 strain and stress at intermediate point;

«sh 5 strain at hardening initiation point;«y 5 yield strain of reinforcing bar;su 5 maximum stress carried by reinforcing bar; and

s l , s l* 5 local stresses at current strain and«* , respec-tively.

JO

.s-r

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Modeling for Postyield Buckling of Reinforcement

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