Inverse Analysis Procedures for Determining the Tensile Stress Crack Openong Curve of Concrete

8/10/2019 Inverse Analysis Procedures for Determining the Tensile Stress Crack Openong Curve of Concrete

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Experimental determination of the stress-crack opening curve for concrete in tensionRILEM TC 187-SOC: Final report May 2007

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CHAPTER 4. INVERSE ANALYSIS PROCEDURES FOR

DETERMINING THE TENSILE STRESS-CRACK OPENING CURVE

OF CONCRETE

Prepared by Jos Luiz Antunes de Oliveira e Sousa(1)

, Ravindra Gettu(2)

and Yoshinori

Kitsutaka(3)

(1) Faculty of Civil Engineering, Universidade Estadual de Campinas, 13083-852 Campinas,

SP, Brazil(2) Indian Institute of Technology Madras, Department of Civil Engineering, Chennai

600036, India(3) Department of Architecture, Graduate School of Engineering, Tokyo Metropolitan

University, Tokyo, Japan

1. INTRODUCTION

The alternative to testing a specimen under uniaxial tension in order to obtain the stress-crack opening (-w) curve is the use of the experimentally-obtained response of a notchedspecimen (normally a notched beam or wedge-splitting specimen) to determine the -w curve

through inverse analysis. The specimen response is simulated with an analytical or numericalmodel (e.g., the model of Hillerborg et al. [1]) using a trial -w curve.The shape of the -wcurve is prescribed to be exponential, linear or bilinear, with the latter including drop-constant, drop-sloped and sloped-constant shapes. For plain concrete, the bilinear model has

been most widely used [2-8], along with exponential models [9-11]. For obtaining the -wcurve that permits the best simulation of the specimen response, a trial-and-error procedure oroptimization based on the minimization of the error in the fit [2, 12-15] is generally used.

A more versatile alternative is the construction of polylinear -w curves through theinverse analysis, where the shape of the curve is also free to vary [16-20]. This approach has

been included in two Japan Concrete Institute standards [21, 22] for determining the fractureenergy and the -w curves for plain and fiber reinforced concretes.

Most approaches inverse analyze the response of one test specimen to obtain the corresponding-w curve. However, some researchers have proposed the use of test data from complementarytests, such as the notched beam test response together with the tensile strength from the splittingtension or some other test [5, 23] or with the pre-peak data from an unnotched beam test [24]; testdata for different sizes of specimens [11]; or more than one data set from the same specimen [6],in order to improve the uniqueness of the inverse analysis solution.

2. ILLUSTRATION OF THE INVERSE ANALYSIS PROCEDURE

In order to illustrate a typical inverse analysis procedure, the steps involved are detailedfollowing the methodologies of Sousa and Gettu [14] and Kitsutaka [17] as applied to the data


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from tests of the notched beam. As is well-known, the notched beam configuration is convenientfor studying fracture behavior, and has been used extensively in the determination of the fracture

parameters of concrete [25-27]. Another popular specimen used for inverse analysis is the wedge-

splitting specimen [28], which has advantages such a relatively large ligament length to volumeratio and a negligible influence of the specimen weight on the test results [15, 29-33].

Once the test specimen and configuration are chosen, the first step is to obtain reliableexperimental data through tests performed under closed-loop control, which is essential forobtaining a stable response [34]. In the notched beam test, the load versus mid-span displacementcurve, as well as the load versus crack mouth opening displacement (P-CMOD) curve, can beobtained. However, the latter is much easier to determine since the CMOD is anyway used for thetest control whereas the deflection needs additional fixtures (see Fig. 1) for its measurement [35].The tests considered later on in this illustration were performed in a servohydraulic testingmachine with a digital closed-loop controller, under constant CMOD rates. The CMOD wasmeasured using a clip gage placed across the crack mouth. The specimen and the details of thetest configuration used are shown in Figure 1.

Figure 1: Notched beam test configuration

The response of the specimen (whose data is to be inverse analyzed) can be simulated usingappropriate analytical or numerical modeling techniques. Usually, such response is analyzed byconsidering the material outside the cracking or fracturing zone to be linear elastic and byrepresenting the fracture behavior (along the crack plane) through a discrete or smeared crack

model. The constitutive relation used in the fracture model is the unknown to be obtained throughthe analysis; i.e., the constitutive relation has to be determined from the inverse analysis.

In the approach of Sousa and Gettu [14], the discrete cohesive crack model [1] is used inconjunction with the cracked hinge model, which was presented originally by Ulfkjaer et al. [36],and further developed by Pedersen [37], and Stang and Olesen [7, 8]. The advantage of the Stang-Olesen cracked hinge model is that it yields closed-form analytical solutions for the entire load-crack opening curve. Other analytical or numerical methods, such as the finite element method, canalternatively be employed in the analysis [15, 29, 38, 39]. The purpose of the modeling is to obtain aP-CMOD curve for the given loading configuration and an assumed constitutive relation, i.e., anassumed -w curve. The analysis is repeated for different -w curves until the relation that gives the

Clip gage for CMOD

LVDT mounted on a yoke fordisplacement measurement

350 mm

175 mm

75 mm

175 mm

500 mm

250 mm 250 mm

Loading configuration End view Mid-section Clip gage

350 mm

175 mm

75mm

175 mm

500 mm

250 mm 250 mm

350 mm

175 mm

75 mm

175 mm

500 mm

250 mm 250 mm

Loading configuration End view Mid-section Clip gage


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P-CMOD response, which best fits the experimental data, is obtained. Note that the same procedurecan be adopted for the inverse analysis of the load-displacement response, if necessary. The best fitcan be determined through the minimization of an error function, such as the following [14]:

( ) ( )[ ] dvvvPvPv

numsqr =max

0

2exp )(

(1)

where is the crack mouth opening displacement (CMOD) obtained in the notched beam test,Pexp() and Pnum() are the experimental and numerical P-CMOD curves, respectively, and() is a weighting function, which can be introduced to give unequal importance to portionsof the fitting interval. Optimization algorithms are used to obtain a set of parameters that yieldthe best fit of the experimental results or minimum error in the fitting.

The shape of the -w curve is decided a priori in most inverse analysis approaches and,consequently, only the parameters that define the relation have to be determined. Several shapeshave been proposed and employed in the literature; some of the typical shapes or models are

given in Table 1, along with their parameters.Table 1: Softening models and corresponding parameters

a) Hordijk [9] with two parameters, ftand Gf

( ) ( ) 11 22

31

3

1c

ult

w

wc

ultt

ecw

we

w

wc

f

wult

+

+=

where c1=3.0, c2=6.93, andt

f

ultf

Gw = 136.5 w [mm]

[MPa]

ft

Gf

wult

b) Linear model with two parameters ftand Gf( )wa

f

w

t

11=

w [mm]

/ ft

1

Gf / ft

1

a1

c) Sloped-constant model with three parameters, ft, a1and b2

( )


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For plain concrete, it is sufficient to use a bilinear or exponential curve to represent the -wrelation. However, fiber reinforced concrete (FRC) may need a more complicated model in orderto represent the fiber pullout behavior at larger crack widths. In order to provide the user with a

range of choices, different shapes of the -w curve or softening models can be implemented in theinverse analysis software, for example, by having an object-oriented structure [14].

2.1 Example of application to plain concrete

The inverse analysis approach of Sousa and Gettu [14] is applied here to experimentaldata of plain concrete with a characteristic compressive strength of 30 MPa. The input dataconsists of the load versus crack mouth opening displacement curve obtained from a test of anotched beam [40] with dimensions of 150 mm 150 mm 600 mm and span of 500 mm,under center-point loading. A 25 mm deep notch was cut at the mid-span of the beam and thecorresponding CMOD was used for controlling the test in a closed-loop testing machine. Theexperimentally-obtained P-CMOD curve is shown in the inset of Figure 2.

0 0.04 0.08 0.12w [mm]

0

1

2

3

4

[MPa]

Uniaxial Tensile Tests

FIT3PB

0 0.1 0.2 0.3 0.4CMOD [mm]

0

4

8

12

16

Load[kN]

Experimental

FIT3PBDIANA

Load-CMOD Curve

Figure 2: -w curves for plain concrete exhibiting the P-CMOD response shown in the inset

For the inverse analysis, Hordijks model was prescribed for the -w curve (see Table 1).The optimized -w curve, with ft= 3.08 MPa and Gf= 0.075 N/mm, is shown in Figure 2,along with results obtained from three uniaxial tension tests [41] of notched cylinders of thesame concrete (with 150 mm diameter and 10 mm deep circumferential notches). The detailsof the tension tests are given elsewhere [42]. It can be seen that the -w curve obtainedthrough inverse analysis compares remarkably with the experimentally-obtained behaviour.The P-CMOD curves computed during the optimization process, using the final -w curve,through finite element analysis (denoted as DIANA after the software code used [43]) and thehinge model (denoted as FIT3PB after the program developed by Sousa and Gettu [14]), are


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shown in the inset of Figure 2, along with the experimental data. The finite elementsimulation was performed with the discrete cohesive crack modeled as interface elementsalong the predefined crack path, which enables the use of Hordijks model. It is seen that the

quasi-analytical solution based on the Stang-Olesen hinge model and the results from thefinite element analysis are in agreement with the experimental results.

The low scatter in the results of the inverse analysis can be observed in Figure 3, whichshows data from three different specimens. There is a marked variation in the ftvalue but thecurves as such do not vary significantly. Although no inverse analysis has been performed onthe uniaxial tension test results (dashed lines) to account for the nonuniformity of the stressdistribution on the cracked cross section, the results agree well with those of the notched

beam tests.

0 0.04 0.08 0.12 0.16cod [mm]

0

1

2

3

stress[MPa]

Beam 1: ft= 3.08 MPa

Gf= 0.075 N/mm


Gf= 0.083 N/mm


Gf= 0.095 N/mm

Uniaxial tension tests 1, 2, 3

averages and standard deviations (numerical):

ftavg = 2.63 MPa ft

std=0.338 MPa

Gfavg = 0.084 N/mm Gf

std= 0.0082 N/mm

Figure 3: -w curves from inverse analysis of beam data and from uniaxial tension tests

2.2 Example of application to fibre reinforced concrete

The analysis procedure of Sousa and Gettu [14] is applied here for determining the -wcurve of fiber reinforced concrete. In this example, the P-CMOD response of a steel fiberreinforced concrete (SFRC) beam (with dimensions of 150 mm 150 mm 600 mm, 500

mm span and mid-span notch of 25 mm) tested according to the RILEM TC 162Recommendations [40] is used. The concrete has the same mix as that of the plain concreteexample, with the addition of 40 kg/m3of hooked-ended steel fibers (with 60 mm length andaspect ratio of 65). The P-CMOD curve obtained in the test is given in Figure 4.

Two shapes of the -w curve have been used: the sloped-constant and the bilinear (seeTable 1). The optimized -w curves are shown in Figure 4 for the two models used; the

parameters of the sloped-constant curve are ft= 3.27 MPa, a1= 38.4 and b2= 0.403, and theparameters of the bilinear model are ft= 3.04 MPa, a1=26.9, b2= 0.313 and a2= -0.398. Notethat the tensile strength ft varies slightly with the type of the model since it is also one of the

parameters of the optimization process. The optimized bilinear -w curve compares well


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with the three curves from uniaxial tension tests of cylindrical cores extracted from beams ofthe same batch of concrete. The fits of the P-CMOD curve obtained with each model arecompared with the experimental data in the inset. In general, the analytical prediction with

the bilinear curve matches the experimental curve satisfactorily while the sloped-constantmodel leads to an average fit in the post-peak.

0 0.1 0.2 0.3 0.4w [mm]

0

1

2

3

4

[

MPa]

FIT3PB-Bilinear

FIT3PB-SlopedConst

Uniaxial Tension Tests

0 0.1 0.2 0.3 0.4CMOD [mm]

0

4

8

12

16

Load[kN]

FIT3PB-Bilinear

FIT3PB-SlopedConst

Experimental

Load-CMOD Curve

Figure 4: -w curves for SFRC and the corresponding P-CMOD curves in the inset

The experimental data of the FRC specimens were independently inverse-analysed byKitsutaka for obtaining the corresponding polylinear -w curves [16, 17, 44]. The resultobtained is compared with those of Sousa and Gettu [14] in Figure 5, where it can be seen thatthe bilinear model is in close agreement with the polylinear curve of Kitsutaka.


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-w curves - SFRC Beam 1

Fit3pb SlopeConst: ft

= 3.27 a1

= 38.4 b2

= 0.403

Fit3pb BiLin: ft= 3.04 a1= 26.9 b2= 0.313 a2= - 0.398

Kitsutakas polylinear

0 0.2 0.4 0.6 0.8 1cod [mm]

0

1

2

3

stress[MPa]

Figure 5: -w curves corresponding to results using hinge model and Kitsutakas approach

3. CONCLUSIONS

Inverse analysis is a powerful tool to obtain the stress-crack opening (-w) curve ofconcrete under tension using the experimental results of tests on notched beams and wedge-splitting specimens. Available optimization schemes can yield unambiguous results as seen inthe comparison of the polylinear analysis technique and a quasi-analytical approach using thehinge model. It is also evident that the values of the tensile strength and the fracture energyobtained from inverse analysis are dependent on the prescribed shape of the softening curve.

4. REFERENCES

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fracture mechanics tests,Eng. Fract. Mech., 35(1990) 117-126.[29] stergaard, L., Olesen, J. F., Stang, H. and Lange, D., A method for the fast and simple

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[35] Gopalaratnam, V.S. and Gettu, R., On the characterization of flexural toughness in fibrereinforced concretes, Cem. Concr. Composites, 17(1995) 239-254.

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[38] Sena Cruz, J. M., Barros, J. A. O., Fernandes, A. R., Azevedo, A. F. M. and Cames, A., Stress-crack opening relationship of enhanced performance concrete, Proc. 9th Portuguese Conferenceon Fracture (EST, Setbal, Portugal, 2004) 395-403.

[39] Barros, J. A. O., Cunha, V. M. C. F., Ribeiro, A. F. and Antunes, J. A. B., Post-crackingbehaviour of steel fibre reinforced concrete,Mater. Struct., 38(2005) 47-56.

[40] RILEM, Final recommendation: Bending test, Report of Technical Committee 162-TDF: Testand design methods for steel fibre reinforced concrete,Mater. Struct., 35(2002) 579-582.[41] RILEM, Recommendation: Uni-axial tension test for steel fibre reinforced concrete, Report of

Technical Committee 162-TDF: Test and design methods for steel fibre reinforced concrete,Mater. Struct., 34(2001) 3-6.

[42] Barragn, B., Gettu, R., Martn, M. A. and Zerbino, R. L., Uniaxial tension test for steel fibrereinforced concrete A parametric study, Cem. Concr. Composites, 25(2003) 767-777.

[43] Witte, F.C. and Kikstra, W.P., Eds., DIANA - Finite Element Analysis, User's Manual release7.2, Nonlinear Analysis (CD-ROM) (TNO Building and Construction Research, Delft, TheNetherlands, 2000).

[44] Kitsutaka, Y., Poly-linear inverse analysis program of tension softening diagram (www.jci-web.jp/committee_inv0001/TSDana/fmpana.html, 2001).

Inverse Analysis Procedures for Determining the Tensile Stress Crack Openong Curve of Concrete

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