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Comparisons Between PullOut Behaviour of Various HookedEnd Fibres in Normal–High Strength Concretes
International Journal of Concrete Structures and Materials volume 13, Article number: 27 (2019)
Abstract
In this paper an elastic–plastic response has been developed to predict the pullout behaviour of various hooked end fibres embedded in normal–high strength concretes. An elastic–plastic moment expression has been proposed to represent the partially plastic hinge formed during pullout. The proposed formula has been incorporated into a frictional pulley force analysis in order to predict the applied loading at each stage of pullout. This prediction accounts for the variation of geometrical and tensile properties of the fibres as well as concrete strength. The proposed model is validated against experimental pullout results of various hooked end fibres.
1 Introduction
The use of steel fibre reinforced concrete (SFRC) as an alternative to traditional rebar or mesh reinforcement has highly influenced the design of modern structures. Plain concrete is characterised by its brittle behaviour in tension (Zendaoui et al. 2016; Abdallah et al. 2018a, b). Steel fibres incorporated are dispersed within the concrete in order to improve its ductility, tensile behaviour and impact resistance (Tadepalli et al. 2013; Adjrad et al. 2016; Abdallah et al. 2018a, b). Of these the greatest advantage of including steel fibres in the concrete is to impart ductility to an otherwise brittle material. Steel fibres enable the concrete to continue carry load after cracking, the socalled postcracking behaviour (Abdallah et al. 2016b; ElMal et al. 2015). A further benefit of the addition of steel fibres is to impart extremely good crack resistance. High strength steel fibres when distributed homogeneously throughout the concrete matrix provide very effective crack control as there will always be a fibre in the neighbourhood of a microcrack, preventing the crack from growing.
The postcracking behaviour of SFRC is closely related to the bond characteristics between the fibre and the matrix (Won et al. 2013; Abdallah et al. 2017c, d). The efficiency of fibres in bridging cracks depends upon the bond mechanisms associated with the pullout behaviour (Islam and Alam 2013; Li and Liu 2016; Abdallah et al. 2017b). Tensile stress in the concrete matrix is transferred to the fibre through a characteristic durable bond at their interface. In order to increase the loadcarrying capacity of SFRC, steel fibres with a variety of different shape and size are introduced. Hooked ends, enlarged ends, spiral, twisted and crimped are among the different designs available. Of these, Dramix hooked end fibres of 4D (double bend) and 5D (triple bend) geometries were developed specifically to increase the crackbridging ability of SFRC. The high mechanical anchorage effect of these fibres was demonstrated in a number of prior studies showing that their strong bonding to the matrix resulted in a high resistance to pullout (Abdallah and Fan 2017). Fibre with more a complex predeformed geometry place a higher complexity on the analysis of its pullout response (Abdallah et al. 2017c, d). As the number of hook bends increases, both the maximum pullout load and pullout work increase significantly. Numerous experimental investigations on the pullout behaviour of predeformed fibres have been carried out to clarify the influence of fibre geometry on the bondslip response (Isla et al. 2015; Tuyan and Yazici 2012; Soetens et al. 2013b; Robins et al. 2002). It has been shown from the pullout test that a hooked end fibre is an effective method for improving the bondslip resistance (Abdallah et al. 2017a).
Over the past four decades several analytical models have been proposed to predict the pullout behaviour of steel fibres with various shapes (Lee et al. 2010; Nammur and Naaman 1989; Soetens et al. 2013a; GeorgiadiStefanidi et al. 2010; Naaman et al. 1991a, b; Deng et al. 2018). The first attempts to predict the pullout behaviour of hooked end fibres were advanced by (Chanvillard 1999) and (Alwan et al. 1999). Chanvillard model’s is based on the concept of virtual work dividing the hook into distinct straight and curved parts. Alwan et al. (1999) used the frictional pulley analogue to determine the anchoring forces provided by the fibre hook. The latter’s contribution is considered a function of the work needed to straighten the fibre during pullout. Sujivorakul et al. (2000) extended the straight fibre pullout analysis developed by Naaman et al. (1991a) by adding a nonlinear spring to the end of the fibre to simulate the mechanical anchorage contribution.
In recent work, (Laranjeira et al. 2010; Lee et al. 2010; Ghoddousi et al. 2010) proposed alternatives to the pulley model of Alwan et al. (1999). Zīle et al. (2013) developed an analytical model to simulate the mechanical contribution from fibre geometry to the pullout response of crimped and hookedend steel fibres. Their model is based both on the amount of plastic work required to straighten the fibre during pullout and frictional resistance in the curved ducts. Won et al. (2015) extended this to the work required in straightening archtype steel fibres.
The mechanics as explained in these theories involves a large number of parameters which are mostly restricted to a set of experimental boundary conditions and material properties. The rising applications of 4D and 5D hooked end fibres in modern structural applications have created the need to simplify theoretical analyses. For example, a semiempirical pullout analysis of these fibres was proposed by the authors to match observed behaviour satisfactorily (Abdallah et al. 2016a). This model adopted Alwan’s frictional pulley (Alwan et al. 1999) to account for variations in mechanical and geometrical properties of the fibres as well as the matrix strength. Previously the authors (Abdallah et al. 2016a) examined fibrematrix conditions for a pullout force where a full straightening of the fibre was observed. This paper investigates the conditions when the hook is not fully straightened during the pullout as has been often observed. The elastic–plastic moment expression was advanced to represent the actual the plastic penetrations involved in the unbending required. Here again when taken with the frictional pulley concept (Abdallah et al. 2016a) this paper shows how plasticity explains the pullout behaviour of 3D, 4D and 5D fibres embedded in a specific concrete matrix. It is shown how this approach is applied to experimental results for each hooked end fibre and three concrete matrices in normalhigh strength range.
2 Experimental Program
2.1 Materials and Specimen
Three types of commercially available Dramix hooked end steel fibres, namely: 3DH (single bend), 4DH (double bend) and 5DH (triple bend) fibres were used in this study. These fibres have exactly the same length (60 mm), diameter (0.90 mm) and aspect ratio (l_{f}/d_{f} = 65) but differ in their hook geometry and tensile strength. The measured geometrical and mechanical properties of these fibres (stacked) are depicted in Fig. 1 and detailed in Table 1. The average stress–strain curves of each fibre is shown in Fig. 2. To investigate the mechanical anchoring effect provided by each fibre hook, straight fibres (3DS, 4DS and 5DS) were obtained from the same hooked end fibres (3DH, 4DH and 5DH) by removing their hook ends.
Pullout specimens were produced in three different concrete strengths, namely normal (NSC), medium (MSC) and high (HSC). Table 2 summarizes the materials used and mix proportions for all mixtures. Cubic specimens with a side dimension of 100 mm were used for single pullout tests. Each cube contained three fibres embedded carefully to a depth of a half fibre length i.e. 30 mm. For each mix, five cubic specimens were also prepared for compressive strength tests. During casting, the dry materials (cement, silica fume, sand and crushed granite) were firstly mixed for approximately 1 min before water and superplasticizer (for the HSC) were added. This was then mixed for roughly 11 min. After casting and vibrating, the specimens were covered with a plastic sheet to avoid moisture loss and left for 24 h at room temperature. After that, they were demoulded and left to cure for a further 28 days in a conditioning chamber at 20 ± 2 °C and 96 ± 4% RH.
2.2 Test Setup
The pullout tests were performed using a specially designed grip system, as illustrated in Fig. 3, which was attached to an Instron 5584 universal testing machine. The grips were designed such that the forces applied to the fibre would represent that in a fibre bridging a crack. The body of the gripping system was machined in a lathe using mild steel and had a tapered end to allow the insertion of four M4 grub screws (Fig. 3). These were then tightened around the steel fibre to an equal torque for an even distribution of gripping pressure to minimise the deformation of the fibre ends and avoid breakage at the tip. Two linear variable differential transformer (LVDT) transducers were used to measure the distance travelled by the steel fibre relative to the concrete face during testing (i.e. the pullout distance). They were held in place using aluminium sleeves on either side of the main grip body (Fig. 3). The LVDT probes had ball bearings at their tips for accuracy in measurements taken from the top datum face. The sample was secured to the Instron base using clamps with riser blocks and M16 studs. The base rested on a round brass disc to retain flatness under test at a displacement rate of 10 µm/s.
2.3 Experimental Results
The average pulloutslip curves of (3DS, 4DS and 5DS) straight fibres pulledout from a NSC, MSC and HSC is shown in Fig. 4a–c. These show that the pullout behaviour of all straight behaviour is characterized by a rapid increase up to the peak load, followed by a sudden drop in the pullout load, indicating full debonding of fibre/matrix interface. After that, the pullout behaviour is controlled entirely by dynamic frictional resistance, where pullout load gradually decreases with the increase in slip. For the same concrete grade, the maximum pullout load (P_{max}) of 3DS, 4DS and 5DS fibres was quite similar, as expected. This is mainly due to the fact that straight fibres (without hooks) have the same geometry, diameter, and embedded length and hence the pullout behaviour would remain almost the same. On the other hand, Fig. 4a–c show that P_{max} for all straight fibres increases significantly as the compressive strength of the concrete increase. Compared to the NSC, the percent increase in the P_{max} in the case of MSC and HSC is 40% and 98%, respectively as shown in Fig. 5.
The experimental pulloutslip curves of (3DH, 4DH and 5DH) hooked end fibres pulledout from a NSC, MSC and HSC matrix are presented in Figs. 6, 7 and 8. It can be seen that the initial pullout response of hooked end fibres is again governed by a combination of two different mechanisms: debonding of the fibre–matrix interface and frictional slip of the fibre. However, in addition to these ‘straightfibre’ mechanisms, mechanical interlock is introduced by the plastic deformation of the fibre hook. In contrast to a straight fibre, the mechanical anchorage contribution provided by a hooked end fibre increases the pullout load after debonding significantly. It can be observed that P_{max} for all hook end fibres increases with an increase in matrix strength. It is also evident that an increase in number of hook bends increases P_{max} dramatically. The 5DH (triple bend) fibres showed considerably higher values of P_{max} in all concrete matrices (Fig. 9). In addition, the pullout work of 5DH fibre was significantly higher than that of the 3DH (single bend) and 4DH fibres (double bend). The higher performance for 5DH fibre is due to the high energy needed to deform and straighten the hook bends. However, despite the increased energy consumed during the pullout, the full straightening of 4DH and 5DH hook bends was not observed (Fig. 10). However, the straightening of the hook increases significantly as the matrix strength increases e.g. a hooked end fibre pulledout from the HSC showed higher levels of hook straightening that those of the NSC and MSC (Fig. 10). This indicates consistently that concrete matrix with higher strength is needed to ensure the hook bends are almost completely straightened. The benefit of this feature of fibre reinforcement is worth evaluating in the following section.
3 Elastic–Plastic Moment Expression
An estimate of the spread of plasticity within the hook bends is a necessary requirement when predicting the force required to pullout an embedded fibre from a concrete matrix. What follows is based upon a wire bend of uniform curvature ‘equivalent’ to the various hook geometries i.e. 3D, 4DH and 5D fibres.
Consider a circular wire fibre section of radius \( r_{f} \) subjected to a uniform hogging bending moment \( M \) as shown in Fig. 11. A fully elastic moment \( M_{E} \) refers to that stress distribution given in which the yield stress \( \sigma_{y} \) has been reached at the top and bottom positions for the section in tension and compression respectively.
The moment \( M_{E} \) refers to a limiting condition at which the elastic theory of bending applies:
in which \( E_{f} \) is the fibre modulus, \( \rho_{E} \) is the fully elastic curvature of the unstressed neutral axis (N.A) and \( I = \smallint_{A} y^{2} dA \) is the second moment of the section’s area about the xaxis. The following relationships apply from rearranging Eq. (1) when \( I = \frac{{\uppi r_{f}^{4} }}{4} \) for the fibre’s circular section:
When \( M > M_{E} \) plastic zones penetrate inwards from the top and bottom to depth \( h \) as shown in Fig. 12. The new beam curvature \( \rho_{e} \) (Fig. 11) applies to the inner region of elastic material with reduced second moment of area \( I_{e} \). The elastic–plastic moment \( {{M}}_{ep} \) is spilt with two contributions:
where the elastic contribution, \( M_{e} \) applies bending theory to an elastic core. Given the small diameter of wire this core is assumed to be circular of radius \( r_{f}  h \) with a simplified second moment of area \( I_{e} = \frac{{\pi (r_{f}  h)^{4} }}{4} \). Here, \( M_{e} \) and the elastic curvature \( \rho_{e} \) are given as:
The plastic contribution \( M_{p} \) assumes that \( \sigma_{y} \) is uniformly distributed within each penetration area. For the strip area \( \delta A \) shown in the first diagram of Fig. 11, \( \delta M_{p} \) becomes the couple produced from the opposing parallel forces \( \sigma_{y} \delta A \) separated by \( 2{{y}} \):
where \( \delta A \) is the bracketed quantity. Integrating between the limits of \( y \) as:
The sum of Eqs. (6) and (7) provides the elastic–plastic moment required for plastic penetration of each zone to depth \( h \):
When Eq. (8) is normalised with the fully elastic moment in Eq. (1) it becomes:
which is written in a convenient dimensionless form:
So enabling the following checks: when \( \frac{h}{{r_{f} }} = 0 \), \( M_{ep} = M_{E} \), at full elasticity; when \( \frac{h}{{r_{f} }} = 1 \), \( \frac{{M_{P} }}{{M_{E} }} = \frac{16}{3\pi } \) at full plasticity giving:
Substituting from Eq. (6): \( 1  \frac{h}{{r_{f} }} = \frac{{\rho_{e} \sigma_{y} }}{{E_{f} r_{f} }} \), the moment ratio in Eq. (9) is expressed in terms of \( \rho_{e} \):
in which the first elastic term may take an alternative form using \( I_{e} \) for the interior elastic region and \( I_{E} \) for the fully elastic section:
3.1 Plastic Penetration Section Geometry
The plastic zone areas \( \frac{{A_{p} }}{2} \) in the Fig. 12 are in the ratio with the fibre section area \( A_{f} \) as:
in which the geometry of Fig. 12 shows:
Hence
in which
Therefore,
giving an equation between \( R \) and \( \theta_{e} \)
3.2 Curvature–Plastic Penetration Relationship
Elastic bending theory applies under the elastic contribution (\( M_{e} \)) to the applied moment (\( M_{ep} \)) shown in Fig. 13.
At the interface position \( r_{f}  h \) between the two zones:
which gives an elastic curvature \( \rho_{e} \) for the neutral axis (N.A) as shown:
From Eq. (14)
Equation (15) enables the curvature to be connected to the penetration area ratio. That is, from Eq. (13) \( R = \frac{{A_{p} }}{{A_{f} }} \) is found for a given \( \theta_{e} \).
For example, with \( R = 0.3\left( {30\% } \right) \) and \( r_{f} = 0.9 \) mm, \( \theta_{e} = 35.8^\circ \) is found from Eq. (13) by trial. Then from Eq. (12) \( h = 0.187 \) mm when Eq. (8) gives \( M_{ep} = 0.0792\sigma_{y}. \)
The bend angle (Fig. 14) \( \beta = 45^\circ \) applies to the 3DH fibre. Taking the bend length \( l \) to be a chord of a circle of curvature \( \rho_{e} \) shows:
Hence, for 3D fibres, where \( \beta = 45^\circ \), Eqs. (10) and (11) give: \( \sin \theta_{e} = \frac{{\rho_{e } \sigma_{y} }}{{E_{f} r_{f} }} = \frac{{\sigma_{y} \times l}}{{2E_{f} r_{f} \sin \left( {\frac{\beta }{2}} \right)}} = \frac{1090 \times 2.5}{{2 \times 207 \times 10^{3} \times 0.45 \times \sin \left( {\frac{45}{2}} \right)}} = 0.0382 \) when from Eq. (13) correspondingly,
and from Eq. (12)
It appears that this initial estimate reveals too great a spread of plasticity (95.1%) when based solely upon an applied moment. Of course, when the latter is released the curvature \( \rho_{e } \) is altered by a partial recovery of elastic strain i.e. ‘springback’. Therefore the analysis requires an account of springback to provide the residual (initial) wire curvature \( \rho_{R } \) more accurately.
3.3 Account of Springback
In Fig. 15a the elastic curvature \( \rho_{e} \) under \( M_{ep} \) is that of the N.A within the central elastic zone. When \( M_{ep} \) is released an elastic springback occurs over the whole section for which the residual strain \( \varepsilon_{R} =\varepsilon_{e} \varepsilon_{E} \) connects to the required curvature \( \rho_{R } \) in a reciprocal relationship \( \varepsilon_{R} = \frac{y}{{\rho_{R} }} \).
Thus at a given depth \( y \) between applying then releasing \( M_{ep} \) in Fig. 15a, b.
where the respective elastic–curvatures in Fig. 15a, b are:
Substituting each elastic curvature \( \rho_{e} \) and \( \rho_{E} \) in Eq. (17):
Substituting Eqs. (9) and (11) in to Eq. (18)
Now \( I = I_{E} \) with an elastic recovery across the full section and therefore:
Setting \( y = \frac{{\rho_{R} }}{{\rho_{E} }} \) and \( x = \frac{h}{{r_{f} }} \), Eq. (19) becomes:
in which \( \rho_{E} = \frac{{E_{f} r_{f} }}{{\sigma_{y} }} \) giving \( y = \frac{{\rho_{R} \sigma_{y} }}{{E_{f} r_{f} }} \) for \( \rho_{R} = \frac{l}{{2\sin \left( {\frac{\beta }{2}} \right)}} \)
The following sevensteps enable an estimate of the extent of plasticity within the equivalent bend radius of a hooked end fibre.
4 Method

1.
The hook’s bend angle \( \beta \) and straight length \( l \) approximate to an ‘initial’ curvature \( \rho_{R} \)
$$ \rho_{R} = \frac{l}{{2\sin \left( {\frac{\beta }{2}} \right)}} $$ 
2.
The known yield strength \( \sigma_{y} \), modulus \( E_{f} \) and fibre radius \( r_{f} \) provide \( y:\)
$$ y = \frac{{\rho_{R} \sigma_{y} }}{{E_{f} r_{f} }} $$ 
3.
Find \( x \) from \( y = f\left( x \right) \) in Eq. (20) above

4.
Find \( \theta_{e} \) from Eq. (12): \( 1  x = \sin \theta_{e} \)
$$ \theta_{e} = \sin^{  1} \left( {1  x} \right) $$ 
5.
Find \( R = \frac{{A_{p} }}{{A_{t} }} \) from Eq. (13)
$$ R = \left( {1  \frac{{\theta_{e} }}{90}} \right)  \frac{1}{\pi }\sin 2\theta_{e} $$ 
6.
Find \( h \) from Eq. (12)
$$ h = r_{f} \left( {1  \sin \theta_{e} } \right) $$ 
7.
Find \( M_{ep} \) from Eq. (8)
$$ M_{ep} = \frac{{\pi \sigma_{y} }}{4} (r_{f}  h)^{3} + \frac{{4\sigma_{y} }}{3} [h(2r_{f}  h)]^{3/2} $$
The graph of Table 3 entries in Fig. 16 enables \( x \) to be read from \( y \) according to step 3. All fibre geometries lie within the highlighted region shown. Thereafter the calculation of \( \theta_{e} \) and \( R \) determine the spread of plasticity within the crosssection for a given fibre geometry as shown in Table 4.
5 Incorporating the Elastic–Plastic Moment Expression Into Frictional Pulley Model
The elastic–plastic moment expression proposed above (step 7) has been incorporated into frictional pulley model (Abdallah et al. 2016a) in order to predict the forces at each stage of pullout for all hooked end fibres. From static force and moment equilibrium, the pullout force \( {{P}} \) in each pullout stage has been determined (Abdallah et al. 2016a) as given by Eqs. (21)–(24). The present mathematical interpretation of frictional pulley model has been explained in detail by Abdallah et al. (2016a).
where \( \Delta P^{\prime},\Delta P^{\prime\prime},\Delta P^{\prime\prime\prime}, \Delta P^{''''} \) represent the pullout force due to plastic deformation contribution of one, two, three and four plastic hinges, respectively (Abdallah et al. 2016a). By adding the pullout force at the onset of complete debonding \( P_{1} \) to Eqs. (21)–(24), the pullout force \( P \) as a function of fibre slip \( \Delta \) in all main stages can be obtained as follows:
5.1 For 3DH fibre
5.2 For 4DH fibre
5.3 For 5DH fibre
The \( {\text{P}}_{1} \) value can be predicted using Eq. (28) of straight fibre developed by Naaman et al. (1991a).
where \( \uppsi \) is the fibre perimeter, \( \tau_{{fd}} \) (∆) is the frictional shear stress function for a slip ∆, and (l − ∆) is the length of fibre remaining embedded.
6 Model Validation
The frictional pulley model predictions using the proposed elastic–plastic moment expression presented above was validated against experimental pullout results for various hooked end fibres. All the material properties of the fibres (fibre geometry and tensile strength) as well as the concrete properties were used as the input parameters in this model (Tables 1 and 2). The predicted forces (\( P_{1} , P_{2} \ldots P_{5} \)) at each stage of pullout for all hooked end fibres are summarized in Table 5. Here, the continuous pullout curves shown in Figs. 17, 18 and 19 were fitted numerically to \( P_{1} , P_{2} \ldots P_{5} \) using a fifth degree polynomial function (Abdallah et al. 2016a).
The respective figures show the comparison between the predicted and experimental pullout curves of all hooked end fibres embedded in NSC, MSC and HSC. For each hooked end geometry, the model predictions, exhibited in Figs. 17, 18 and 19, can reasonably well simulate the experimental pullout loadslip curve. It is clear that the proposed model is able to capture the main features of pullout behaviour and to predict accurately the pullout loadslip curve, irrespectively of fibre geometry and tensile strength. The model takes also into account the mechanical anchorage effect provided by the fibre hook as well as the variation of the concrete properties. It can be observed that with increasing the matrix strength from NSC to MSC and HSC, the straightening of the hook becomes progressively increased for all fibres (Fig. 10). Therefore, the accuracy of the model predictions is increasingly improved for all fibres.
7 Conclusion
In this paper an elastic–plastic response model has been advanced to predict the pullout behaviour of various hooked end steel fibres embedded in normal, medium and high strength concretes. The model accounts for the amount of deformation and straightening of the hook during the pullout. The spread of plasticity within hook bends was found to be closely related to matrix strength. The elastic–plastic moment expression developed has been incorporated into frictional pulley model to predict the forces at each stage of pullout. The input parameters to the model were the mechanical and geometrical properties of the fibres together with the ultimate strength of the concrete grades.
The proposed model has been validated against experimental results for each combination between three hooked end fibres embedded in normal, medium and high strength concretes. The results show that the method proposed can predict the pulloutslip response for all hooked end fibresconcrete combination realistically. Predictions show greatest accuracy in matching the pulloutslip behaviour of hooked end fibres embedded in MSC and HSC. This model was also able to take into account matrix damage and fibre rupture.
Abbreviations
 P :

pullout force
 ∆ :

relative slip of the fibre
 ∆P :

pullout force due to plastic hinges contribution
 PH :

plastic hinge
 F _{ PH } :

rotational friction component
 I :

second moment area
 M _{ E } :

elastic moment
 M _{ ep } :

elastic–plastic moment
 M _{ p } :

plastic moment
 δA :

incremental strip area
 E _{ f } :

fibre modulus
 I :

the second moment of the section’s area
 δy :

incremental strip depth
 µ :

friction coefficient
 R :

penetration depth
 I _{ E } :

the second moment of the fully elastic
 r _{ f } :

fibre radius
 σ _{ y } :

fibre yield strength
 h :

plastic depth
 E _{ f } :

modulus of elasticity of fibre
 θ _{ e } :

angular penetration measure
 A _{ f } :

total area of the steel fibre circular section
 A _{ p } :

plastic penetration area
 ρ _{ E } :

fully elastic curvature
 ρ _{ e } :

curvature of inner elastic zone
 ρ _{ R } :

initial curvature
 ε _{ R } :

residual strain
 ε _{ e } :

elastic strain
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Both authors contributed equally to this work. Both authors read and approved the final manuscript.
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The first author is grateful for the financial support provided by the Ministry of Higher Education and Scientific Research of the Iraqi Government.
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Abdallah, S., Rees, D.W.A. Comparisons Between PullOut Behaviour of Various HookedEnd Fibres in Normal–High Strength Concretes. Int J Concr Struct Mater 13, 27 (2019). https://doi.org/10.1186/s4006901903370
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DOI: https://doi.org/10.1186/s4006901903370