COMPARISON BETWEEN INVERSE ANALYSIS PROCEDURE RESULTS AND EXPERIMENTAL MEASUREMENTS OBTAINED FROM UHPFRC FOUR-POINT BENDING TESTS

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1 COMPARISON BETWEEN INVERSE ANALYSIS PROCEDURE RESULTS AND EXPERIMENTAL MEASUREMENTS OBTAINED FROM UHPFRC FOUR-POINT BENDING TESTS J.Á. López (1), P. Serna (1), J. Navarro-Gregori (1) and H. Coll (2) (1) ICITECH, Universitat Politècnica de València, Spain (2) Research & Development Concretes S.L., Spain Abstract Determination of the tensile properties of such a deflection hardening response material as UHPFRC is still a serious challenge for both researchers and designers. This process involves many factors, such as specimen size, fibre orientation or test typology. The so-called inverse analysis is used to obtain the tensile constitutive properties that are consistent with the specimen response in a bending test. This work presents an inverse analysis methodology developed to be applied over fourpoint bending tests. This procedure is based on a deflection to curvature transformation and on a point-by-point formulation. A set of UHPFRC specimens with different heights but with the same slenderness have been tested. The deflection to curvature transformation used as input for the inverse analysis procedure has been validated by its comparison to the experimental average curvature within the constant moment area. A clear scale-effect in the stress-strain relationship is derived from these tests after macrocrack starts growing. A strain-crack opening transformation as a function of the specimen height is proposed for four-point bending tests setup. It permits avoiding any size influence in the post-peak behaviour, showing that stress-crack opening relationship is not dependant on the specimen height. 1. INTRODUCTION Characterization of tensile behaviour of UHPFRC is still a challenge for researchers and there is no agreement yet either in the standard test set-up or in the advisability of a notch. Bending tests are easy to conduct if compared to direct tensile tests, but they require a sophisticated post-process to obtain the tensile behaviour of the specimen: the so-called inverse analysis procedure. On the other hand, direct tensile tests may be more appropriate since they directly provide the uniaxial tensile behaviour of the specimen. However, these tests are challenging to perform as they are very sensitive to several factors, such as specimen imperfections, loading machine stiffness, shrinkage, boundary conditions, stress concentrations at the fixation points, or the non-uniformity of the material. 185

2 With regard to the advisability of a notch, notched bending tests allow a better analysis once macro-crack appears. However, in such a deflection hardening material as UHPFRC, these tests require a previous determination of the cracking strength, which has to be determined from an unnotched test according to [1]. Moreover, and contrary to notched tests, unnotched tests allow determining the strain-hardening behaviour that characterises UHPFRC tensile behaviour. However, post-cracking behaviour is more difficult to analyse. In the case of deflection hardening materials without pronounced discontinuities on its equivalent bending strength-deflection relationship, notched four point bending tests (FPBT) arise as a suitable test to characterize both the stress-strain response up to crack localization point, and the stress-crack opening relationship just after it. The only requirement for it is the establishment of a complete stress-strain curve. Since macro-crack formation is a discrete phenomenon, the strain beyond crack localization point has to be referred to a certain length. In the case of a FPBT, this length has to be the central one-third of the specimen. Inside this area, bending moment remains constant and either curvature or strain at most tension fibre can be averaged over to perform the inverse analysis. If this procedure were followed, definition of the uniaxial tensile behaviour of deflection hardening materials could be obtained from only one test. The more accurate the inverse analysis procedure is, the more realistic the tensile parameters will be. 2. INVERSE ANALYSIS PROCEDURE The inverse analysis procedure proposed herein is based on the determination of the average curvature in the central one-third from displacement measured at mid-span. Since the only parameter which must be recorded is the displacement at mid-span, the test setup is easy to perform. For this purpose, the deflection at mid span to curvature transformation in [2] is used. This transformation assumes a linear elastic distribution of the curvature up to 70% of the maximum load, approximately, and a logarithm hypothesis from this point on, even in the unloading branch. According to [2], the average curvature at central one-third can be obtained as a function of the displacement at mid span, following Eq. (1), where is the total applied load, is the elastic modulus, and, are the specimen width and height, respectively. This expression assumes that each span length of the FPBT is one-third of the specimen length between supports ( Applying this transformation requires the use of the elastic modulus. Following the linear elastic theory and taking into account the shear deflection, elastic modulus can be obtained following Eq. (2), from one point, in the linear ascending loading curve Eqs. (1) and (2) allow determining the load-average curvature curve. Once this curve has been obtained, there are two main options to conduct the inverse analysis: a close-form 186

3 moment-curvature formulation [3]; or a point-by-point inverse analysis [4]. In this work, a point-by-point analysis has been used to determine the whole stress-strain relationship without imposing any restrictions to its shape. Despite the fact that method in [4] is based on the measurement of the strain at most tension fibre, method proposed herein uses as input the average curvature. 3. EXPERIMENTAL PROGRAM Three different square cross-section types of specimens were prepared with a variable depth of 50, 100 and 150 mm, to investigate the size effect and the accuracy of the displacement to curvature transformation. A total of eight 50 and 100-mm specimens and four 150-mm specimens were made. In a 25% of the specimen tested the macro-crack appeared out of the central one-third. The ratio was constant for all of them. Despite EN :2009 proposes a ratio equals to 3, a decision to use a value of 4.5 was made to dispose of a larger constant moment area to improve the determination of multi-micro cracking stage and average crack spacing. A 2% in volume of smooth-straight (13/0.20) steel fibres were used in a Ultra-High-Performance cementitious matrix with a compressive strength of 170 MPa. The geometry of the specimens and the FPBT set-up are shown in Fig. 1. A couple of LVDT s where used in the back side of the specimen to obtain the experimental average curvature. On the front side, another LVDT was used to record the displacement at mid span. Figure 1: Test set up for the three types of specimens Eq. (2) can be rewritten as in Eq. (3) and (4), showing that results from FPBT can be expressed as a function of two normalised parameters. If the ratio keeps constant, the load-deflection at mid span curve can be expressed as normalised bending strength versus normalised displacement at mid span, which is not dependant on the height specimen. Thus, variation of this normalised curve for different specimen heights shows the size-effect on the flexural response

4 Normalised bending strength versus normalised displacement at mid span curves of the test series with 50, 100 and 150-mm square cross-section are shown in Fig 2a, 2b and 2c. The average curves are depicted in bold black line, and they are compared in Fig 2d. From the curves in Fig. 2 a clear size effect in deformation can be deduced after peak, since the unloading slope is lower as the specimen height decreases. Fig. 2d shows that the three average curves for the different specimen height offer close values for the maximum flexural strength at normalised displacement at peak, which is unexpected since a better fibre orientation is assumed for smaller specimens. However, favourable orientation of fibres cannot be concluded from these results, and it does not seem to be a size effect up to peak load neither in deformations nor in strength. Figure 2: Normalised curves for the test series with 50 (a), 100 (b), 150 (c) mm square cross-section; average curves in (d). 3. DISPLACEMENT TO CURVATURE VALIDATION According to the test setup in Fig. 1, the experimental average curvature at central onethird can be obtained following Eq. (5). Moreover, according to [2], the average curvature at this area can be obtained from the experimental displacement at mid span in a FPBT, following Eq. (1). As the point-by-point inverse analysis method used herein is based on the average curvature determination from Eq. (1), the closer these two curves are, the better the accuracy of the tensile parameters determined by the inverse analysis procedure will be. 188

5 5 Fig. 3 shows the comparison between experimental average curvature curve and theoretical one derived from deflection at mid span measurement using Eq. (1) for two specimens of the test series with 50, 100 and 150 mm. The average curvature has been normalised by multiplying it by the specimen height. Fig. 3 shows that formulation proposed in [2] offers a very accurate a simply way to obtain the average curvature from displacement at mid span. Fig. 3 also permits to appreciate the size-effect phenomenon which is directly related to macro-crack propagation and only affects the unloading branch. In Table 1, the sum of squares of the difference between experimental and theoretical curvatures values for all specimens are shown, except for those in which the macro-crack appeared out of the central one-third. In four specimens the top LVDT did not measure properly and experimental curvature could not be determined. These specimens are marked with an x in Table 1. Specimen number 3 (50-mm height) has the maximum error. Fig. 3 shows that even in this case theoretical curve is very close to experimental one. Even though formulation in [2] does not consider macro-crack position along the central one-third results can be considered accurate enough for being used as input for the inverse analysis procedure. Figure 3: Comparison between theoretical and experimental curves for the test series with 50, 100, 150 mm square cross-section 189

6 Table 1: Difference between theoretical and experimental curvature values Specimen height 50 mm 100 mm 150 mm Specimen number Sum of squares (m -2 x x x x ) 4. POINT-BY-POINT INVERSE ANALYSIS In order to obtain the tensile parameters of the UHPFRC specimens performed, a point-bypoint inverse analysis using the same formulation proposed in [3], but using the average curvature as input instead of strain at most tension fibre, has been used. Formulation details can be found in [4]. Fig. 4 (left) shows a typical curve obtained after a point-by-point inverse analysis. It can be noted that this curve is very difficult to work with. Results from the inverse analysis procedure have shown that a quadrilinear assumption in tension could reproduce the tensile behaviour of UHPFRC accurately. That is why a quadrilinear tensile law has been proposed and has been fitted to the results from inverse analysis, keeping the same area under the curve. Fig 4 (right) shows the constitutive law assumed, along with the notation of the seven parameters used to describe it: elastic modulus ; first cracking tensile strength ; ultimate tensile strength, and its ultimate strain,, strength and strain at the end of the first unloading branch,,, ; and maximum strain at zero strength,. According to Fig. 4 (right), and following the criteria found in [5], the crack opening relationship can be derived from stress-strain law following Eq. (6). Initial crack opening, obtained as the product between, and average crack spacing, can be neglected in crack opening determination as its value is around 20 μm, too small in comparison to and values. The strain-crack opening relationship also depends on the stress variation ( 0) and on the unloading modulus ( ). Previous tests performed advice using an unloading modulus of 19% of the elastic modulus for this type of concrete. This value agrees with that proposed by [6] for a UHPFRC with a 2% in volume of fibres. Figure 4: Inverse analysis results for specimen 5 (100mm-height) and quadrilinear law adjustment (left); stress-strain law assumed and notation used (right),

7 Fig. 5 depicts the quadrilinear stress-strain laws derived from inverse analysis procedure proposed for the different test series. Although a better fibre orientation in smaller specimens cannot be deduced from bending results, stress-strain results show that first cracking strength ( ) is higher the smaller the specimen size is. This fact can be explained due to a better fibre orientation. No scale-effect is detected at peak, neither in stress nor in strain. A strong scaleeffect at the unloading branch after crack localization can be concluded from Fig. 5 (d), as it had been already predicted from Figs. 2 and 3. In order to avoid scale-effect in tension, a tensile characterization of strain-hardening materials with a stress-crack opening relationship is completely necessary after crack localization takes place. On the other hand, as strain-hardening behaviour does not seem to be affected by scale-effect and crack opening is too small to be included in a stress-crack opening relationship, a stress-strain relationship with the average crack spacing measurement could be more appropriate for the ascending tensile branch. With this criterion, uniaxial tensile law for strain-hardening materials should look like Fig. 6. Following this procedure, both hardening and crack opening stages can be determined from the same unique test. Figure 5: Stress-strain inverse analysis results obtained from curves for the test series with 50 (a), 100 (b), 150 (c) mm square cross-section; averages curves in (d) 6. CONCLUSIONS From the experimental program carried out some conclusions concerning the inverse analysis procedure proposed and scale-effect phenomenon can be drawn. It has been checked that displacement at mid-span to curvature transformation proposed in [2] offers an accurate approximation in comparison with experimental measurements. Using this transformation and 191

8 point-by-point inverse analysis procedure proposed in [3], a complete stress-strain law can be derived from the displacement at mid-span measurement and test set-up. As stress-strain relationship is size-dependant after macro-crack onset, a strain-crack opening transformation is proposed to avoid scale-effects. Finally, a stress-strain relationship up to crack localization with the average crack spacing determination, and a stress-crack opening relationship from this point on, are proposed to characterize strain-hardening material tensile properties (Fig. 6). Figure 6: Material uniaxial tensile properties characterisation for the test series with 50, 100, 150 mm square cross-section. Stress-strain and average crack spacing up to peak (left); stresscrack opening after crack localization. ACKNOWLEDGEMENTS This work forms part of the FISNE research projects, with reference BIA , supported by the Spanish Ministry of Economy and Competiveness and the FEDER fund. Support for this project is gratefully acknowledged. We also wish to thank the Spanish Ministry of Education, Culture and Sport for its FPU scholarship programme. REFERENCES [1] Association Française de Genie Civil (AFGC), Ultra High Performance Fiber Reinforced Concretes. Reccomendations. [2] Lopez, J.A., Serna, P., Navarro-Gregori, J., Camacho,E, An inverse analysis method based on deflection to curvature transformation to determine the tensile properties of UHPFRC. Materials and Structures. DOI /S [3] Soranakom, C. and Mobasher, B., Closed-Form Moment-Curvature Expressions for Homogenized Fiber-Reinforced Concrete. ACI Materials Journal, 104(4): [4] Baby, F., Graybeal, B., Marchand, and P. Toutlemonde, F., Proposed Flexural Test Method and Associated Inverse Analysis for Ultra High Performance Fiber Reinforced Concrete. ACI Materials Journal, 109(5): [5] Spasojevic, A., Structural Implication of Ultra High Performance Fibre Reinforced Concrete in Bridge Design. PhD-Thesis, École Polytechnique Fédérale de Lausanne, April. [6] Wille, K., El-Tawil, S., Naaman, A.E. Properties of strain hardening ultra high performance fiber reinforced concrete (UHP-FRC) under direct tensile loading. Cement & Concrete Composites. Vol pp