Acta Polytechnica https://doi.org/10.14311/AP.2021.61.0740 Acta Polytechnica 61(6):740–748, 2021 © 2021 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague DETERMINATION OF BOND MODEL FOR 7-WIRE STRANDS IN PRETENSIONED CONCRETE BEAM Vadzim Parkhats∗, Rafał Krzywoń, Jacek Hulimka, Jan Kubica Silesian University of Technology, Faculty of Civil Engineering, Department of Structural Engineering, Akademicka 5, 44-100 Gliwice, Poland ∗ corresponding author: vadzpar036@student.polsl.pl Abstract. A correct choice of a bond model for prestressing tendons is crucial for the right modelling of a structural behaviour of a pretensioned concrete structure. The aim of this paper is the determination of an optimal bond model for 7-wire strands in a prestressed concrete beam produced in a precast concrete plant of Consolis Poland. ATENA 3D is used to develop finite element models of the beam that differ only in a bond stress-slip relationship of tendons. The bond stress-slip relationships for modelling are taken from the results of bond tests carried out by different researchers in previous years. Moreover, for comparison purposes, a simplified 2D model of the beam is created in Autodesk Robot. The strain distribution at the time of the strand release is found for each of the finite element models. The determined strain distributions are compared with the strain distribution in the beam established by an experimental test using a measuring system based on a digital image correlation. On the basis of the comparison results, the most appropriate bond models for 7-wire strands used in the beam are identified. Keywords: Bond stress-slip relationship, digital image correlation, end zone, prestressed concrete, pretensioned concrete beam, strand release. 1. Introduction As is known, the prestressing force is transferred to the concrete of a pretensioned member by the bond between the concrete and the prestressing steel [1]. Therefore, an accurate definition of a bond model for prestressing tendons in finite element modelling of pretensioned concrete structures is key for the deter- mination of correct results. Unfortunately, a bond stress-slip relationship for a prestressing tendon is omitted from most design standards. Model Code 2010 [2] contains bond models for ribbed and plain reinforcing bars, but not for prestressing strands. Digital image correlation (DIC) is a non-contact optical technique for measuring strain and displace- ment [3]. In recent years, this measurement technique has been used more and more often in various fields of study: civil engineering [4], applied mechanics [5], biology [6], aerospace engineering [7] and others [8, 9]. In particular, it should be noted that the DIC tech- nique is nowadays used in the research of prestressed concrete [10–15]. In this paper, a review of bond models for tendons found in the literature is done. Moreover, an experi- mental test carried out with the help of a DIC mea- surement system and consisting in a determination of strain distribution in the end zone of a preten- sioned concrete beam at the time of the strand release is described. Bond models found in the literature are applied in finite element modelling of the beam used in the experimental test to establish the most appropriate ones. The appropriateness of the bond models is evaluated by the comparison of the strain distributions determined in specific points on the side surface of the beam by means of DIC measurements and finite element modelling. The best fitting bond models will be additionally verified on other preten- sioned concrete structures of the same manufacturer in the future. Furthermore, it is also planned to carry out bond tests for deducing our own bond model for the used 7-wire strands. 2. Literature review of bond models for strands Bond models for 7-wire strands found in the literature are presented in this section. Bond models for other types of tendons are omitted, since only 7-wire strands were used in the beam analysed in the experimental test. Balazs [16] presented the bond stress-slip relation- ship 1 based on results of pull-out tests with 7-wire strands with a diameter of 12.8 mm. The specified concrete strength at transfer was f′ci = 40 MPa. τ = ψc(f′ci) 0.5(S/db)b (1) where: τ is the bond stress [MPa]; S is the slip [m]; db is the strand diameter [m]; ψ is the factor [–] for the upper bound (ψ0.95 = 1.35), the mean value (ψm = 1.00) and the lower bound (ψ0.05 = 0.65) of bond stresses; c and b are the experimental constants; for db = 12.8 mm: c = 2.055 MPa0.5 and b = 0.25. Oh et al. [17] carried out bond tests for 12.7 mm and 15.2 mm strands using concrete with the strength 740 https://doi.org/10.14311/AP.2021.61.0740 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 61 no. 6/2021 Determination of bond model for 7-wire . . . (a). (b). Figure 1. Bond stress-slip relationship [20]: (A) for ribbed bars in the case of a pull-out failure according to Model Code 2010, (B) for 7-wire smooth strands according to the Equations 8–12. f′ci = 32.71–35.50 MPa. The following equation was obtained: τ = C(S/db)b (2) where: C and b are experimental constants; for db = 12.7 mm: C = 13.787 MPa and b = 0.3301, for db = 15.2 mm: C = 9.331 MPa and b = 0.2688. Lim et al. [18] presented a bond stress-slip rela- tionship obtained by a measurement of strains in pretensioned members with the help of strain gauges. Their relationship for 15.2 mm strands is identical to the Equation 1 with ψ = 1.00, but the values of the experimental constants are different: c = 10.7 MPa0.5 and b = 0.27. Orr et al. [19] carried out pull-out tests for both unstressed and stressed 15.2 mm strands. The speci- fied concrete strength at the transfer was f′ci = 54.2 MPa. Their model is based on the bond stress-slip re- lationship 3–6 for ribbed bars in the case of a pull-out failure presented in Model Code 2010 [2] (Figure 1a). τ = τmax(S/S1)α for 0 ≤ S ≤ S1 (3) τ = τmax for S1 ≤ S ≤ S2 (4) τ = τmax − (τmax − τf )(S − S2)/(S3 − S2) for S2 ≤ S ≤ S3 (5) τ = τf for S > S3 (6) The following values of the parameters of their model were proposed: α = 0.5; S1 = S2 = S3 = 0.1 mm for stressed strands and S1 = S2 = S3 = 2 mm for unstressed strands. The Equation 7 is stated for the determination of the bond stress: τmax = τf = δ1 δ2 0.70 (fcm)0.5 (7) where: δ1 accounts for the reduction in the bonded perimeter in specimens with reduced cover [–]; δ2 accounts for the confinement from cover or transverse reinforcement [–]; fcm is the mean concrete cylinder strength of the specimen [MPa]. As opposed to the above-mentioned researchers, Khalaf and Huang [20] developed an analytical bond model for both 3-wire and 7-wire strands. It was vali- dated by a comparison of its results and the results of experimental tests. The model considers the surface condition of a strand, the geometry and the number of wires, the concrete parameters, and the influence of elevated temperatures. The model is based on the Model Code 2010 relationship (Eq. 3–6), but is modi- fied in the case of 7-wire smooth strands (Figure 1): τ = τmax(S/S1)α for 0 ≤ S ≤ S1 (8) τ = τmax − (τmax − τ2)(S − S1)/(S2 − S1) for S1 ≤ S ≤ S2 (9) τ = τ2 + (τmax − τ2)(S − S2)/(S3 − S2) for S2 ≤ S ≤ S3 (10) τ = τmax − (τmax − τ4)(S − S3)/(S4 − S3) for S3 ≤ S ≤ S4 (11) τ = τf for S > S4 (12) The following parameters are considered in the case of 7-wire smooth strands: S1 = 0.25 mm; S2 = 0.5 mm; S3 = 3.5 mm; S4 = 8 mm; τ2 = 0.75 τmax; τ4 = 0.35 τmax. The bond stress is calculated in ac- cordance with the Equation 13: τmax = Tb/Ab (13) 741 V. Parkhats, R. Krzywoń, J. Hulimka, J. Kubica Acta Polytechnica Cement CEM I 52,5R Lafarge 414 Limestone powder Cemex Rudniki 58 Sand (0–2 mm) 681 Crushed granodiorite (2–8 mm) 474 Crushed granodiorite (8–16 mm) 664 Water 116.6 Admixture Sika 34RS 3.68 Table 1. Composition of concrete [kg/m3]. Property Value Standard deviation Cube compressive strength [MPa] 56.6 0.50 Cylinder compressive strength [MPa] 48.32 1.09 Mean tensile strength [MPa] 3.98 – Modulus of elasticity [GPa] 25.81 0.27 Poisson’s ratio [–] 0.19 0.016 Strain at failure [‰] 2.72 0.20 Plastic part of strain [‰] 0.85 0.15 Tensile strain at failure [‰] 0.154 – Density [kg/m3] 2444 – Table 2. Average properties of concrete at the time of strand release. where: Tb is the maximum bond force [MN] found according to the Equation 15; Ab is the contact area between the strand and the concrete [m2]; Ab = π db Lb (14) where Lb is the embedded length of the strand [m]. Tb = [µvc dw lw n + 0.6 π dw lw n (C′ + µσn)]/ cos θ (15) where: dw, lw, n, θ = 9° are the diameter [m], the length [m], the number [–], and the pitch angle of the outer wires, respectively; µ is the coefficient of friction between the concrete and the steel [–]; C′ is the cohesion between the concrete and the steel [–]; for 7-wire smooth strands µ = 0.4 and C′ = 1.3; σn is the normal stress perpendicular to the strand axes [MPa]; vc is the shear strength of the shear keys in the concrete mass [MPa]; vc should not be greater than 0.2 f′c, where f′c is the concrete compressive strength [MPa]; for the pull-out bond (σn = 0) vc is calculated according to the Equation 16; vc = ft [f′c/ft + 2 − 2 (1 + f ′ c/ft) 0.5]0.5 (16) where ft is the concrete tensile strength [MPa]. 3. Experimental test In this section, the experimental test carried out in a precast concrete plant of Consolis Poland is de- scribed. It consisted in the measurement of strains in the end region of the prestressed concrete beam at the time of the strand release by a DIC system. The test results are necessary to verify strain distributions in the finite element models of the beam. 3.1. Specimen description The 11.66-metre long pretensioned concrete beam (Figures 2 and 3) was used for the test. The cross section was I-shaped in the central part, whereas the anchorage zone was equipped with an end block. Three horizontal openings in the central part and four vertical openings in the end zones were provided. Twenty prestressing tendons with a diameter of 15.7 mm made of steel Y1860S7 were used in the beam. Their tension was 1250 MPa. It should be noted that two of them had a shielding two metres long. The strands were cut with an acetylene torch in a sequence shown in Figure 2b. The stirrups, 8 and 10 mm in diameter, were made of steel B500. Their spacing in the end region was not greater than 95 mm. The beam was made of concrete of the strength class C50/60. Its composition is presented in Table 1. The mechanical properties at the time of the strand release (44 hours after the pouring) were determined with the help of additional tests on the cubic and cylindrical concrete specimens and are summarised in Table 2. 3.2. Test procedure The side surface of the beam was covered with paint and recorded during the strand release by a DIC mea- surement system. On the basis of the obtained images, strain distribution was established in the 2.4-metre- long end region. The GOM ARAMIS system equipped 742 vol. 61 no. 6/2021 Determination of bond model for 7-wire . . . (a). (b). (c). Figure 2. Pretensioned concrete beam: a) a side view of the end region and the location of virtual extensometers, b) the arrangement of prestressing steel and the order of the strand release, c) the reinforcing steel in the end zone. Figure 3. Sections 1 – 1 and 2 – 2 in the end region (see also Figure 2c). 743 V. Parkhats, R. Krzywoń, J. Hulimka, J. Kubica Acta Polytechnica Figure 4. Finite element models created in Autodesk Robot (on the left) and ATENA 3D (on the right). with two prime-lens 6-megapixel cameras (24 mm fo- cal length) was used. The recording frequency was 4 Hz. 4. Finite element modelling ATENA 3D and Autodesk Robot are used for the finite element modelling. Because of the symmetry of the specimen, half of the beam is modelled (Figure 4). Seven finite element models created in ATENA 3D consider both the prestressing and the reinforcing steel, openings in the beam, and the order of the strand release. The concrete is defined by the material “3D Nonlinear Cementitious 2”. The mesh size is 5 cm. The models differ only in a bond stress-slip relation- ship for prestressing tendons. The relationships for strands deduced by [16, 17, 19, 20] are applied. Addi- tionally, the Model Code 2010 relationship (3–6) for ribbed bars in the case of a pull-out failure is used to evaluate its appropriateness for 7-wire strands. The relationship of Lim et al. [18] is omitted, because it leads to questionable results. One simplified 2D finite element model based on linear elastic properties of concrete is developed in Autodesk Robot. The reinforcement and openings in the beam are omitted. Prestressing is simulated by a linear increase of negative temperature over the transmission length of the beam (i. e., approximately 780 mm). The value of the transmission length is calculated with the help of the simplified method presented in [21]. 5. Results and discussion The small values of strains during the strand release lead to difficulties in interpretation of the results ob- tained by ARAMIS. Firstly, instead of using deformation maps, the re- sults are presented with the help of virtual extensome- ters 200 mm long. The location of the longitudinal ex- tensometers is shown in Figure 2a. They are situated on the side surface of the beam in the characteristic points of strain variation, namely on the level of the bottom row of strands (points 1, 2, 3, 4, and 5). The location of the extensometers is chosen so that they evenly cover the transmission length. The other problem are great fluctuations of the values in the strain distribution diagrams. Therefore, the weighted moving average method is used to de- crease the fluctuations: the strains in the diagrams are averaged out for seven previous values and seven subsequent ones. The transverse strain distributions in the end block are not presented in the paper, because they are char- acterised by such significant fluctuations that the veri- fication of the models using these results seems point- less. The longitudinal strain distributions in the charac- teristic points of the beam (Figure 2a) obtained by ARAMIS and finite element modelling are shown in Figure 5. The horizontal axis corresponds to the space of time when the strand release was carried out. It is observed that, with the increase of the distance from the end of the beam, all the finite element models overestimate the strains in comparison with the values obtained by ARAMIS approximately until the time when the bottom row of strands starts to be released (between 125 and 150 s, especially at the point 5, a characteristic leap is seen). After this moment, the values obtained by the modelling and the experimental test are close. The possible cause is that between 125 and 150 s, a detachment of the concrete from the formwork happens, so the structural behaviour of the beam is changed. Therefore, in the finite element modelling, the bottom surface of the beam should be restrained from moving in a vertical direction over the full length of the beam until the moment of the detachment. After the detachment, only the place of junction of the end and bottom surfaces should be restrained in this way (as is done in the analysed finite element models – see Figure 4). This explanation 744 vol. 61 no. 6/2021 Determination of bond model for 7-wire . . . Figure 5. Longitudinal strains in the end zone during strand release (location of the points is shown in Figure 2a). is confirmed by the longitudinal strain distributions presented in Figure 6, where the strains obtained by ARAMIS are compared with the strains obtained with the help of the finite element model restrained from moving in a vertical direction over the full length. It can be seen that the values in the diagrams are similar until the time between 125 and 150 s, i. e., until the moment of the detachment. Coefficients of determination between the strains according to ARAMIS and the predicted strains are presented in Tables 3 and 4. On the basis of the strain distributions (Figures 5, 7) and the coefficients of correlation (Tables 3, 4), it is established that the finite element models using the bond stress-slip relationships of Model Code 2010 [2] for ribbed bars, Orr et al. [19] for unstressed strands, and Khalaf and Huang [20] give results that are significantly different than those obtained with the help of ARAMIS. The finite element model based on the Model Code 2010 relationship overestimates the values of the strains, whereas the models using the relationships of Orr et al. [19] for unstressed strands and Khalaf and Huang [20] underestimate them. In the case of the model based on the Model Code 2010 relationship, the difference has been expected, since this relation- ship is developed for ribbed reinforcing bars, but not for prestressing tendons. The inappropriateness of the model using the relationship of Orr et al. [19] for unstressed strands indicates that the prestressing of strands in bond tests is crucial for deducing a realistic bond stress-slip relationship. Concerning the model based on the relationship of Khalaf and Huang [20], it 745 V. Parkhats, R. Krzywoń, J. Hulimka, J. Kubica Acta Polytechnica Figure 6. Longitudinal strains in the end zone during strand release – moment of the detachment of the concrete from the formwork. Figure 7. Longitudinal strains along the beam length on the level of the bottom row of strands after release of all the tendons. Bond model Points 1 2 3 4 5 Autodesk Robot (2D Model) 0.505 0.978 0.962 0.974 0.971 Orr et al. (stressed strand) 0.538 0.975 0.944 0.953 0.959 Orr et al. (unstressed strand) 0.546 0.974 0.942 0.954 0.959 Oh et al. (equation for db = 15.2 mm) 0.546 0.975 0.946 0.957 0.960 Oh et al. (equation for db = 12.7 mm) 0.548 0.974 0.945 0.956 0.960 Balazs ψ = 1.00 0.549 0.973 0.943 0.956 0.961 Model Code 2010 (ribbed bar) 0.552 0.968 0.942 0.957 0.961 Khalaf and Huang 0.545 0.977 0.949 0.958 0.960 Table 3. Coefficient of determination between the strains according to ARAMIS and the predicted strains (for the strains shown in Figure 5). Bond model Coefficient of determination Autodesk Robot (2D Model) 0.954 Orr et al. (stressed strand) 0.980 Orr et al. (unstressed strand) 0.937 Oh et al. (equation for db = 15.2 mm) 0.966 Oh et al. (equation for db = 12.7 mm) 0.976 Balazs ψ = 1.00 0.971 Model Code 2010 (ribbed bar) 0.896 Khalaf and Huang 0.924 Table 4. Coefficient of determination between the strains according to ARAMIS and the predicted strains (for the strains presented in Figure 7). 746 vol. 61 no. 6/2021 Determination of bond model for 7-wire . . . is difficult to explain the cause of the different results. It is interesting to note that this is the only analyti- cally developed relationship in the literature review, as opposed to the others determined by experimen- tal tests. Experimental tests on more pretensioned concrete structures with different design parameters should be carried out to draw definitive conclusions about appropriateness of the analysed bond stress-slip relationships for the strands and concrete used. 6. Conclusions In this paper, bond stress-slip relationships for 7- wire strands of different researchers are analysed to evaluate their appropriateness for the use in finite element modelling of the pretensioned concrete beam made in a precast concrete plant of Consolis Poland. The assessment is done by a comparison of the strain distributions in the beam found by ARAMIS and the finite element modelling. The following conclusions are drawn: (1.) The longitudinal strain distributions in the finite element models based on the bond stress-slip rela- tionships of Balazs [16], Orr et al. [19] (for stressed strands), and Oh et al. [17] are fairly similar to the results of the DIC measurements. Moreover, using the simplified model developed in Autodesk Robot gives satisfactory results as well. However, it is worth noting that the relationships of Oh et al. [17] do not consider the concrete strength at the transfer, so that the satisfying similarity might be accidental. Neglecting the concrete strength at the transfer restricts their applicability to the finite element modelling of pretensioned concrete structures. In addition, it is found that the finite element models based on the bond relationships of Orr et al. [19] for unstressed strands and Khalaf and Huang [20] underestimate the strains, whereas the model using the Model Code 2010 relationship for ribbed bars overestimates them. However, these findings have to be additionally verified on the basis of experimental tests on other pretensioned concrete members that are planned in the future. Besides, bond tests utilising the same prestressing strands and concrete mix are proposed as a direction for a future research. (2.) The strain distributions obtained by ARAMIS are characterised by great fluctuations that complicate the analysis of the results. It concerns the transverse strain distributions. In future tests, the scanned region is planned to be reduced to increase the resolution and make the results more legible. (3.) It is established that the structural behaviour of a pretensioned concrete beam might change during the strand release because of the detachment of the concrete from the formwork. Thus, a finite ele- ment model of a pretensioned concrete beam should be restrained from moving in a vertical direction differently before and after the detachment. References [1] C. W. Dolan, H. R. Hamilton. 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Stowarzyszenie Producentów Cementu, Kraków, 2008. 748 https://doi.org/10.1016/j.engstruct.2019.05.030 https://doi.org/10.1016/j.compstruct.2020.112223 https://doi.org/10.1016/j.conbuildmat.2019.117687 https://doi.org/10.1016/j.conbuildmat.2019.117687 https://doi.org/10.15554/pcij.11011992.60.71 https://doi.org/10.22636/JKCI.2000.12.6.3 http://www.framcos.org/FraMCoS-8/p348.pdf https://doi.org/10.1016/j.engstruct.2017.01.061 https://doi.org/10.1016/j.conbuildmat.2016.10.016 https://doi.org/10.1016/j.conbuildmat.2016.10.016 Acta Polytechnica 61(6):740–748, 2021 1 Introduction 2 Literature review of bond models for strands 3 Experimental test 3.1 Specimen description 3.2 Test procedure 4 Finite element modelling 5 Results and discussion 6 Conclusions References