Determining the Tensile Stress-Crack Opening Curve of Concrete by Inverse Analysis

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Determining the Tensile Stress-Crack Opening Curveof Concrete by Inverse Analysis

José Luiz Antunes de Oliveira e Sousa1 and Ravindra Gettu2

Abstract: The determination of the fundamental stress versus crack opening ��-w� response of concrete under uniaxial tension isperformed in this study through inverse analysis using data from notched beam tests. The procedure used for optimizing the parametersof the �-w relation using the load versus crack mouth opening displacement response of the notched beam is described. Satisfactorycomparisons have been obtained between the �-w curves obtained through the inverse analysis and those directly measured in uniaxialtension tests. The use of weighting functions in the inverse analysis may be necessary when large crack widths are to be considered.

DOI: 10.1061/�ASCE�0733-9399�2006�132:2�141�

CE Database subject headings: Tensile stress; Cracking; Concrete; Tests.

Introduction

The structural behavior of concrete during cracking can be de-scribed by nonlinear fracture mechanics models such as thefictitious crack model proposed by Hillerborg et al. �1976�. Thecrack propagation is represented in this approach by a fictitiouscrack, which consists of the actual stress-free crack plus aninelastic fracture process zone, in which the cohesive stressesare modeled using an appropriate stress-crack opening ��-w�relationship. The application of Hillerborg’s approach or thealmost equivalent crack band model �Bazant and Planas 1998�requires the knowledge of the characteristic �-w curve ofthe concrete. Under ideal conditions, this relation should be ob-tained from uniaxial tension tests of the concrete. Accordingly,several researchers have successfully used uniaxial tension testconfigurations to obtain the constitutive tensile behavior �e.g.,Gopalaratnam and Shah 1987; Rossi 1995; Mechtcherine andMüller 1998; Stang and Bendixen 1998; Stang and Olesen 1998;Shi and van Mier 2001; Barragán et al. 2003�. Nevertheless, suchtests are complicated to perform since it is difficult and/or time-consuming to prepare the specimen, impose unambiguous bound-ary conditions in the setup, and control the test in a stable manner�Dupont and Vandewalle 2002�.

The alternative that is currently being studied exhaustively isthe use of the experimentally obtained response of a notchedbeam to determine the �-w curve of the corresponding material

1School of Civil Engineering, Univ. Estadual de Campinas, CaixaPostal 6021, 13083-852 Campinas, SP, Brazil �corresponding author�.E-mail: [email protected]

2Dept. of Civil Engineering, Indian Institute of TechnologyMadras, Chennai 600036, India; formerly, School of Civil Engineering,Univ. Politècnica de Catalunya, Barcelona, Spain. E-mail:[email protected]

Note. Associate Editor: Yunping Xi. Discussion open until July 1,2006. Separate discussions must be submitted for individual papers. Toextend the closing date by one month, a written request must be filed withthe ASCE Managing Editor. The manuscript for this paper was submittedfor review and possible publication on December 12, 2003; approved onMarch 7, 2005. This paper is part of the Journal of Engineering Me-chanics, Vol. 132, No. 2, February 1, 2006. ©ASCE, ISSN 0733-9399/
2006/2-141–148/$25.00.
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through inverse analysis. The beam response is simulated withan analytical or numerical model �e.g., Hillerborg et al. 1976�using a trial �-w curve. The shape of the �-w curve is prescribedto be exponential, linear, or bilinear, with the latter includingdrop-constant, drop-sloped, and sloped-constant shapes. Forplain concrete, the bilinear model has been most widely used�Roelfstra and Wittmann 1986; Alvaredo and Torrent 1987; Witt-mann et al. 1987; Guinea et al. 1994; Bolzon and Maier 1998;Stang and Olesen 1998; Kooiman et al. 2000; Stang and Olesen2000�, along with exponential models �Hordijk 1991; Duda andKönig 1992; Gettu et al. 1998�. For obtaining the �-w curve thatpermits the best simulation of the beam response, a trial-and-errorprocedure or optimization based on the least-squares approach�Roelfstra and Wittmann 1986; Kooiman et al. 2000� is generallyused. Another more recent alternative is the use of procedures thatconstruct polylinear �-w curves during the inverse analysis,where the shape of the curve is also free to vary �Kitsutaka 1995;Uchida et al. 1995; Nanakorn and Horii 1996; Kitsutaka 1997;Kitsutaka and Oh-oka 1998; Kitsutaka et al. 2001�. Most ap-proaches use the response of one test specimen to obtain thecorresponding �-w curve. However, some researchers have pro-posed the use of complementary test data, such as the tensilestrength obtained in the splitting tension test �Guinea et al. 1994�,test data for different sizes of specimens �Gettu et al. 1998�, ormore than one data set from the same specimen �Bolzon andMaier 1998�, in order to improve the uniqueness of the inverseanalysis solution.

The present methodology for the quasi-automatic deter-mination of the cohesive �-w relation uses the load versus crackmouth opening displacement �P-CMOD� curves obtained fromnotched beam tests. During the inverse analysis, a P-CMODcurve is obtained for each trial �-w curve, defined by a set ofparameters, and compared to the corresponding experimentalresult, in the least-squares sense. Optimization algorithms areused to obtain a set of parameters that yield the best fit of theexperimental results. The methodology has been implementedin a software program that takes advantages of object oriented

++
programming in C .
L OF ENGINEERING MECHANICS © ASCE / FEBRUARY 2006 / 141

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Procedure

Details of Input Data and Numerical Implementationof the Procedure

As explained earlier, the objective of the present work is thedetermination of the relation between the stress �� and the crackopening �w� of concrete subjected to uniaxial tension fromnotched beam test data. The data sets that are normally obtainedin a notched beam test are the load-deflection or the load-crackmouth opening displacement �P-CMOD� curves. The P-CMOD isgenerally preferred for the analysis since the deflection may beaffected by extraneous deformations and needs sensors additionalto that needed for controlling the test, which is normally rununder CMOD control. Note that for notched specimens, the gaugelength of the CMOD sensor does not influence the results as longas a clip gauge or other suitable sensor is used. The test itself canbe performed under four-point bending �i.e., loaded at third-span�or three-point bending �i.e., loaded at midspan�. In the presentwork, three-point bend �3PB� specimens are considered thoughthe approach is also valid for four-point bending. Moreover, onlyone data set from one specimen is used in the inverse analysisperformed here since this leads directly to the characterization ofthe material in the specimen, and facilitates the evaluation of theinherent variability of test results and distortions arising frommalfunctioning of the test setup.

The inverse analysis procedure used here has been imple-mented in a software program, called FIT3PB, the details ofwhich are given in the Appendix. The user chooses the shapeof the �-w curve from various options �as discussed later�, andsupplies seed values for the parameters, along with the appropri-ate bounds. The program first linearizes, if needed, the beginningof the P-CMOD curve to eliminate any spurious nonlinearity dueto settling of the beam. Then, the initial linear part of the curve isused to determine the modulus of elasticity of the concrete. �Al-ternatively, the modulus of elasticity can be given as part of theinput data.� Using the assumed �-w curve, the P-CMOD curve isdetermined through an appropriate analytical/numerical module.For evaluating the fit, the error is computed as the integral of thesquare of the difference between the experimental and numericalcurves over the prescribed CMOD interval. An optimization algo-rithm is used to determine the parameters of the �-w curve thatlead to a minimum error. The final �-w curve and the correspond-ing prediction for the P-CMOD curve are provided as output,along with a nondimensional fitting error defined with respect tothe area under the P-CMOD curve. Weighting functions can beprovided to emphasize the importance of certain regimes of theP-CMOD curve within the optimization process.

The software program can employ user-supplied modulesfor calculating the modulus of elasticity of the concrete andthe P-CMOD curve for a given �-w curve. In the present work,the module employed is based on the analytical formulation ofStang and Olesen �Stang and Olesen 1998, 2000�, which uses acracked hinge that is modeled as a layer of independent springelements, the behavior of which are governed by the constitutiverelations of the concrete in the uncracked and cracked states. Thesprings are attached to the rigid boundaries of the hinge, whichhas a length equal to half the beam depth. Beyond these bound-aries, the classical elastic beam theory is applied, limiting thenonlinearity provoked by the crack to the hinge. The idea of acracked hinge was presented originally by Ulfkjaer et al. �1995�,and further developed by Pedersen �1996�; and Stang and Olesen
�1998, 2000�. The advantage of the cracked hinge model is that it
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J. Eng. Mech. 2006.

yields closed-form analytical solutions for the entire load-crackopening curve. This feature allows the development of simpleroutines that can replace time-consuming finite element modeling.

Shape of the �-w Curve

The object-oriented structure of FIT3PB allows the implemen-tation of different shapes of the �-w curve or softening models.In general, the softening models are defined with reference tothe tensile strength �f t�. Also, for all the models, w�0 and��w��0. The following models have been implemented in theprogram, one of which has to be chosen by the user for fitting the�-w curve:• The Hordijk’s model �Hordijk 1991� with two parameters, f t

and Gf �see Fig. 1�a��

��w�f t

= �1 + �c1w

wult�3�e�−c2w/wult� −

w

wult�1 + c1

3�e−c2 �1�

where c1=3.0; c2=6.93; and wult=5.136�Gf / f t.• A sloped-constant model with three parameters, f t, a1, and b2

�see Fig. 1�b��

��w�f t

= �1 − a1w if w � w1

b2 if w � w1� �2�

• A bilinear model, for plain concrete or steel fiber reinforcedconcrete �SFRC�, with four parameters, f t, a1, a2, and b2 �seeFig. 1�c��

��w�f t

= �1 − a1w if w � w1

b2 − a2w if w � w1� �3�

• A trilinear model with six parameters, f t, a1, a2, b2, a3, and b3

Fig. 1. Softening models and corresponding parameters

�see Fig. 1�d��

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��w�f t

= 1 − a1w if w � w1

b2 − a2w if w1 � w � w2

b3 − a3w if w � w2 �4�

Measure of the Fitting Error

As mentioned earlier, the optimization of the �-w curve is per-formed by minimizing the error in the fitting of the P-CMODcurve. The fitting error function used in this process is defined as

�sqr =�0

�max

�Pexp�� − Pnum�� 2��d� �5�

where �=crack mouth opening �CMOD� in the target experimen-tal curve; ��=weighting function introduced to permit unequalimportance of different portions of the fitting interval, accordingto the desired application; Pexp�� and Pnum��=respectively, theexperimental and numerical values of the load corresponding tothe same abscissa.

The shape of the error function �sqr is presented in Fig. 2 for anexample using Hordijk’s model �Hordijk 1991�, described by thetwo parameters f t and Gf. The shape indicates clearly the exis-tence of a minimum, which can be located using optimizationtechniques. Therefore, this error function seems to be appropriatefor being used as the objective function in the optimization pro-cess. Although the plots are limited to functions of two variables,similar behavior is expected in the cases with more variables.

For the comparison of the solutions obtained with differentsoftening models and for evaluating if a fit is satisfactory, analternative error measure is more appropriate

�abs =

�0

�max

�Pexp�� − Pnum��d�

�0

�max

Pexp��d�

�6�

The present approach has also been validated for the case ofthe inverse analysis of the load-deflection curve, instead of theP-CMOD curve; in this case, � in Eqs. �5� and �6� is the load linedeflection. Moreover, when the load-deflection response is usedas the input, the error definition in Eq. �6� has a physical meaningsince it then indicates an upper bound of the fraction of thefracture energy that corresponds to the difference betweenthe experimental curve and the fit. Nevertheless, it has been es-tablished in several studies that there is an almost proportional

Fig. 2. Plots of the fitting er

relation between CMOD and deflection �e.g., Gopalaratnam et al.

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1995; Stang 2002�, indicating that this interpretation would beadequate by approximation.

Fitting Procedure

The minimization of the objective function �sqr is performedthrough optimization techniques in n-dimensional space, wheren=number of parameters that are free to vary. The minimizationalgorithms are based on gradient methods combined with line-search techniques �Luenberger 1984�. Numerical techniques areused to compute the partial derivatives of the error function �sqr

�Eq. �5�� with respect to the softening curve parameters. Secondpartial derivatives, represented by the Hessian matrix, may alsobe used. When the second partial derivatives are not computed orthe Hessian matrix is not positive definite, the steepest descentmethod is used. When the Hessian matrix is positive definite, aNewton–Raphson algorithm can be successfully applied, therebyimproving the convergence rate.

For evaluating the adequacy of the fit, the values obtained for�abs �Eq. �6�� with different models can be compared in order tochoose the most satisfactory result. Generally, a �abs value of lessthan 2% indicates a good fit.

Fitting Interval and Weighting Function

In principle, an adequate model should be able to describe thematerial behavior under all circumstances. However, this mayonly be possible through the use of complex models and inverseanalysis procedures, and/or the use of data from additional tests.On the other hand, the material is required to perform adequatelyin a structure over a specific range that is limited by serviceabilitycriteria and economics. Therefore the most adequate approachseems to be the use of a model with few parameters, preferablywith physical significance. For maintaining an acceptable degreeof simplicity in the model and in the analysis procedure, the errorfunction �sqr, which is the objective function in the error minimi-zation process, requires the definition of an integration interval�0−�max . Furthermore, although not essential for characterizingthe least-squares error, a weighting function �� may be neededin Eq. �5� to assign unequal weights to different regimes of theP-CMOD curve. For plain concrete, it appears that these issuesare not important, as seen later in the example applications. How-ever, for steel fiber reinforced concrete, the toughening responseof the material necessitates the choice of a fitting interval and aweighting function in order to adequately simulate the behavior

qr as a function of f t and Gf
ror �s
with a simple model.


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Example Applications

Plain Concrete

The inverse analysis approach is applied here to experimentaldata of plain concrete with a characteristic compressive strengthof 30 MPa �see mix proportions in Table 1�. The input data con-sists of the load versus crack mouth opening displacement curveobtained from a test of a notched beam with dimensions of150 mm�150 mm�600 mm and a span of 500 mm, undercenter-point loading. A 25 mm deep notch was cut at the midspanof the beam and the corresponding CMOD was used for control-ling the test in a closed-loop testing machine. The experimentallyobtained P-CMOD curve is shown in Fig. 3�a�.

For the inverse analysis, the model of Hordijk was prescribedfor the �-w curve �see Eq. �1�; Fig. 1�a��. The seed values usedfor the two parameters that have to be optimized were determinedas follows. For the tensile strength, f t, the maximum loadobserved in the test �approximately equal to 14 kN in Fig. 3�a��is used to calculate the maximum tensile stress using the liga-ment depth and elastic beam theory �RILEM 2002�, and takenas the corresponding seed value; that is, the seed value of f t

is taken as 1.5� �14 kN�� span=500 mm�� width=150 mm�� ligament depth=125 mm�2=4.5 MPa.

For the fracture energy, Gf, the area under the P-CMODcurve until the termination of the test is determined �in Fig. 3�a�,the area under the curve is approximately equal to 1,600 N mm�,and divided first by 1.18 �an empirically calibrated factorobtained for the present geometry by comparing areas underthe P-CMOD and load-deflection curves� to get an approxi-mate value of the corresponding area under the load-displacementcurve �RILEM 2002� and then by the ligament area to getthe energy per unit area; that is, the seed value of Gf

Table 1. Components of Concrete

Components �kg/m3�

Cement �Type CEM I 42.5R� 385

Crushed limestone gravel �5–12 mm� 183

Crushed limestone gravel �12–25 mm� 773

Crushed limestone sand �0–5 mm� 851

Water added 168

Plasticizer 4.6 L

Fig. 3. �-w curves for plain concrete and the correspondingP-CMOD curves in the inset


J. Eng. Mech. 2006.

is taken as �1,600 N mm� /1.18/ �ligament area=width� ligamentdepth=150�125 mm2�=0.072 N/mm. The optimized �-wcurve, with f t=3.08 MPa and Gf =0.075 N/mm, is shown inFig. 3�b�, along with results obtained from three uniaxial tensiontests of notched cylinders of the same concrete. The details ofthe uniaxial tension tests are given elsewhere �RILEM 2001;Barragán 2002; Barragán et al. 2003�. It can be seen that the�-w curve obtained through inverse analysis compares satis-factorily with the experimentally obtained curves. The P-CMODcurve computed during the optimization process, using the final�-w curve, is shown in Fig. 3�a�.

Fiber Reinforced Concrete

The inverse analysis procedure has also been used for de-termining the �-w curve of fiber reinforced concrete. In thisexample, the P-CMOD response of a steel fiber reinforcedconcrete �SFRC� beam �with dimensions of 150 mm�150 mm�600 mm, 500 mm span and midspan notch of 25 mm� testedaccording to the RILEM TC 162 recommendations �RILEM2002� is used. The concrete has the same mix as that of the plainconcrete example �see Table 1 for mix proportions�, with the ad-dition of 40 kg/m3 of Dramix RC 65/60 BN hooked-ended steelfibers �see Table 2 for details of the fiber�. The P-CMOD curveobtained in the test is given in Fig. 4�a�.

Two shapes of the �-w curve have been used: the sloped-constant and the bilinear �Eqs. �2� and �3�, respectively�. For ob-taining the seed value of f t the same procedure is used as in theplain concrete example. The seed values for the other parametersof the sloped-constant model were a1=30 and b2=0.5. For thebilinear model the seed values were the parameters optimizedpreviously with the sloped-constant model, along with a2=−1.0.The optimized �-w curves are shown in Fig. 4�b� for the twomodels used; the parameters of the sloped-constant curve are

Table 2. Properties of Fibers

Properties

Length �mm� 60

Aspect ratio �length/diameter� 65

Tensile strength �MPa� 1,000

Minimum value of maximum deformation �%� 0.8

Fig. 4. �-w curves for SFRC and the corresponding P-CMOD curvesin the inset

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f t=3.27 MPa, a1=38.4, and b2=0.403, and the parameters ofthe bilinear model are f t=3.04 MPa, a1=26.9, b2=0.313, anda2=−0.398. Note that the tensile strength f t varies slightly withthe type of the model since it is also one of the parameters of theoptimization process. The optimized bilinear �-w curve compareswell with the three curves from uniaxial tension tests of cylindri-cal cores extracted from beams of the same batch of concrete. Thefits of the P-CMOD curve obtained with each model are com-pared with the experimental data in Fig. 4�a�. In general, theanalytical prediction with the bilinear curve matches the experi-mental curve satisfactorily while the sloped-constant model leadsto an average fit in the postpeak.

Effect of Interval Choice and Weighting Function

The error function �sqr in Eq. �5� integrates the square of the errorin the fitting of the load over the prescribed CMOD interval. Anappropriate weighting function ��CMOD� can be used to givemore importance to one regime than another or the rest of thecurve. In Fig. 5, such a weighting function is shown, where thereis a relatively higher weight �i.e., 10� for the peak, and a linearincrease in the prepeak regime and a symmetric decrease in thepostpeak regime, followed by a constant value of 1. The Hordijk’smodel is fitted through the inverse analysis of the P-CMOD datafor the notched plain concrete beam considered earlier. The �-wcurves obtained with and without weighting are shown in theinset and the numerical predictions of P-CMOD in the main plotof Fig. 5. It can be seen that there is practically no difference inthe fitting of the curve for this case. The variations in the �-wcurve due to the weighting are also negligible. This and othersimulations have confirmed that weighting is not needed for theinverse analysis of plain concrete beams.

Weighting becomes more important in the case of behaviorthat needs more complex models. For example, in the inverseanalysis of a notched SFRC beam considered earlier with a tri-linear model for the �-w curve, the importance of the weightingfunction is related to the fitting interval �of CMOD� and the ac-ceptable error in certain regimes of the P-CMOD response. Fig. 6shows the predicted and experimental P-CMOD curve in theCMOD interval of �0.0–1.0 mm with the optimized �-w curvesfrom the inverse analysis performed with and without weighting.The weighting function is shown with dashed lines in the figure;the function used has a peak weight of 10 and a plateau valueof 3. The two optimized �-w curves are shown in the inset. It can

Fig. 5. Effect of weighting function for plain concrete

be seen that the differences in both the �-w curve and the fit of

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J. Eng. Mech. 2006.

the P-CMOD curve due to weighting are small. This implies thatweighting does not have to be used when a small CMOD range isof interest.

However, when the fitting interval is larger, say �0.0–2.0 mm ,as in Fig. 7, the fits of the P-CMOD curve vary significantly,especially around the peak, when the predictions are made withinverse analyses performed with and without weighting. The maindifferences are in the first �see inset� and final parts of the curve.It is clear that with a higher weight for the initial part of theP-CMOD curve a better fit of the peak load is obtained whileequal weighting of all the regimes leads to a better fit of the latterregimes. In the corresponding �-w curves shown in Fig. 8, it canbe seen that equal weighting tends to give a higher tensilestrength that leads to the overestimation of the peak load.

The use of weighting functions has the objective of givingunequal importance to certain ranges of the data based on thereliability of the measurements or the relative precision neededin the fits of the different ranges. Here, weighting is used for thecase of SFRC to ensure a good fit of the peak though the shapeof the �-w curve is governed by the postpeak response. In thismanner, the �-w curve represents better the failure of the matrixand the peak load as well as the pullout of the fibers that governtoughening. The discussion on weighting functions and choice

Fig. 6. Effect of weighting function on the P-CMOD and �-w curvesfor SFRC, performing inverse analysis in the interval �0.0;1.0 mm

Fig. 7. Effect of weighting function on the P-CMOD curve forSFRC, performing inverse analysis in the interval �0.0;2.0 mm


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of fitting interval presented here is intended solely to show thatthese issues have nonnegligible effects on the resulting param-eters, especially for SFRC.

Conclusions

The development of a system of algorithms for obtaining thestress-crack opening curve under tension through inverse analysisusing the P-CMOD response of notched beams has been pre-sented. In the present implementation, the structural analysis isbased on a cracked hinge formulation, which uses a trial �-wcurve to obtain the numerical P-CMOD response necessary forthe optimization process that yields a set of parameters that bestfit the experimental data. The examples treated here illustrate theapplication of the approach for plain and fiber reinforced con-cretes. It is observed that a weighting function is needed when thelarge range of crack width has to be considered in the analysis, forexample, in fiber concretes.

Although the tensile strength and the fracture energy are con-sidered to be material properties, these two parameters cannot betaken independently from the relationship that models the entiresoftening curve. The values of the parameters may differ slightlywhen different models are considered.

Acknowledgments

The writers are grateful for the financial support from FAPESP-Fundação de Amparo à Pesquisa do Estado de São Paulo�Brazil�, through Grant 00/10616-9, from the Spanish Ministryof Science and Technology �MCYT� Grants PB98-0298 andMAT2005-5530, and from CAPES-Fundação de Aperfeiçoamentode Pessoal de Nivel Superior, a human resources funding agencyof the Brazilian Ministry of Education. The work on the SFRCanalysis was part of the Commission of the European Communi-ties Contract BRPR.CT98.0813 for the project Test and designmethods for steel fibre reinforced concrete; the other partnersof the project were BEKAERT �coordinator�, Balfour Beatty,Belgian Building Research Institute, University of Wales-Cardiff,Catholic University of Leuven, FCC Construcción, Ruhr-University of Bochum, SIBO-Gruppe GmbH and Technical Uni-

Fig. 8. Effect of weighting function on the �-w curve for SFRC,performing inverse analysis in the interval �0.0;2.0 mm

versity of Braunschweig. Dr. Bryan E. Barragán is thanked for


J. Eng. Mech. 2006.

providing experimental results. Professor Henrik Stang is thankedfor providing background information and the structural analysismodule used in this work.

Appendix. Description of the Inverse AnalysisProgram

The FIT3PB program has the structure of classes described inFig. 9, with the following main classes:1. Object: Root class, with general capabilities, superclass of

all other classes in the system.2. PDelta: Corresponds basically to the analytical formulation

developed by Stang and Olesen �1998, 2000�, using the Ccode with some adaptations to create a C++ class. The basicmember functions are• Compute(): Generates the P-CMOD data for the current

set of softening parameters.• FitYoungModulus(): Determines the modulus of elasticity

based on the initial linear portion of the experimentaldata, using the formulation of Stang and Olesen �1998,2000�.

3. SoftModel: Abstract class to handle softening model infor-mation. Each derived subclass corresponds to a specificmodel. For example, SigwHordijk corresponds to Hordijk’smodel �see Eq. �1��, SigwSlopeConstant corresponds to thesloped-constant model �see Eq. �2��, and SigwBiLin corre-sponds to the bilinear model �see Eq. �3��.

The instance variables for each of the SoftModel sub-classes are the softening parameters of the correspondingmodel. The basic member functions are• sigw(): returns the cohesive stress � for a given crack

opening displacement �w�.

Fig. 9. Structure of classes in FIT3PB

• sigww(): returns w�sigw��.

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4. OptMngr: Class that handles the error minimization. Thebasic member function is• MinimizeError(): Handles algorithms to determine partial

derivatives of the error objective function, and the opti-mization algorithms �i.e., search direction and line searchminimization�.

Several complementary nonmember functions are imple-mented to handle the algorithms for the first and second partialderivatives, conjugate gradient, and line search. The derived sub-classes handle the different softening models; OptMngrHordijkhandles Hordijk’s model and so on.

The instance variables are the vector space dimension for theoptimization process �i.e., number of parameters considered asvariables in the fitting process�, the corresponding instances ofthe SoftModel subclass and the target experimental data �Poly-Func class�.

Auxiliary Classes:1. BeamData: Handles specimen geometry and physical data.2. PolyFunc: Handles description of functions by a polylinear

function.Instance variables:• Number and list of points.Member functions:• DiffSqr(): Integrates the squared difference between cur-

rent and another instance.• DiffAbs(): Integrates the absolute value of the difference

between current and another instance.• EvaluateAt(): Evaluates the function for a given abscissa.• LineFit(): Fits a line through points selected via a bound-

ing box.3. Vector, Point, Line: Handle operations in n-dimensional

vector space.4. WorkSheet, Summary: Handle data collection for output

during program execution.5. CommandMngr: Implements a parser that allows execution

through command lines or a graphical user interface, whichconsists basically of a dialogue box to handle these commandlines and a window to follow the fitting convergence.

Notation

The following symbols are used in this paper:a1 ,a2 ,a3 ,b2 ,b3

line coefficients for describing a �-w curve;f t tensile strength;

Gf fracture energy;P applied load;

Pexp value of P from laboratory test;Pnum value of P from numerical analysis;

v alternative symbol for CMOD;vmax maximum value for v in the interval;

w crack opening displacement;w1 ,w2 ,wult limiting crack opening displacement values

for describing �-w curves;�abs error function in terms of integral of absolute

values of differences;�sqr error function in terms of integral of squared

differences;� tensile cohesive stress; and

��v� weighting function.

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Determining the Tensile Stress-Crack Opening Curve of Concrete by Inverse Analysis

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