Comparisons Between Pull-Out Behaviour of Various Hooked-End Fibres in Normal–High Strength Concretes

Abstract

In this paper an elastic–plastic response has been developed to predict the pull-out 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 pull-out. The proposed formula has been incorporated into a frictional pulley force analysis in order to predict the applied loading at each stage of pull-out. 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 pull-out 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 so-called post-cracking behaviour (Abdallah et al. 2016b; El-Mal 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 micro-crack, preventing the crack from growing.

The post-cracking 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 pull-out 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 load-carrying 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 crack-bridging 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 pull-out (Abdallah and Fan 2017). Fibre with more a complex pre-deformed geometry place a higher complexity on the analysis of its pull-out response (Abdallah et al. 2017c, d). As the number of hook bends increases, both the maximum pull-out load and pull-out work increase significantly. Numerous experimental investigations on the pull-out behaviour of pre-deformed fibres have been carried out to clarify the influence of fibre geometry on the bond-slip response (Isla et al. 2015; Tuyan and Yazici 2012; Soetens et al. 2013b; Robins et al. 2002). It has been shown from the pull-out test that a hooked end fibre is an effective method for improving the bond-slip resistance (Abdallah et al. 2017a).

Over the past four decades several analytical models have been proposed to predict the pull-out behaviour of steel fibres with various shapes (Lee et al. 2010; Nammur and Naaman 1989; Soetens et al. 2013a; Georgiadi-Stefanidi et al. 2010; Naaman et al. 1991a, b; Deng et al. 2018). The first attempts to predict the pull-out 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 pull-out. Sujivorakul et al. (2000) extended the straight fibre pull-out analysis developed by Naaman et al. (1991a) by adding a non-linear 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 pull-out response of crimped and hooked-end steel fibres. Their model is based both on the amount of plastic work required to straighten the fibre during pull-out and frictional resistance in the curved ducts. Won et al. (2015) extended this to the work required in straightening arch-type 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 semi-empirical pull-out 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 fibre-matrix conditions for a pull-out force where a full straightening of the fibre was observed. This paper investigates the conditions when the hook is not fully straightened during the pull-out 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 pull-out 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 normal-high 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.

Fig. 1

Geometrical properties of hooked end steel fibres.

Table 1 The measured geometric and mechanical properties of hooked-end fibres.

Fig. 2

Stress–strain curves from fibres tensile tests.

Pull-out 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 pull-out 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.

Table 2 Mix design of mixtures.

2.2 Test Setup

The pull-out 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 pull-out 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.

Fig. 3

Pull-out test setup.

2.3 Experimental Results

The average pull-out-slip curves of (3DS, 4DS and 5DS) straight fibres pulled-out from a NSC, MSC and HSC is shown in Fig. 4a–c. These show that the pull-out behaviour of all straight behaviour is characterized by a rapid increase up to the peak load, followed by a sudden drop in the pull-out load, indicating full debonding of fibre/matrix interface. After that, the pull-out behaviour is controlled entirely by dynamic frictional resistance, where pull-out load gradually decreases with the increase in slip. For the same concrete grade, the maximum pull-out 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 pull-out 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.

Fig. 4

Average pull-out-slip response of straight steel fibres. a NSC, b MSC and c HSC.

Fig. 5

Maximum pull-out load (P_max) of 3DS, 4DS and 5DS straight fibres.

The experimental pull-out-slip curves of (3DH, 4DH and 5DH) hooked end fibres pulled-out from a NSC, MSC and HSC matrix are presented in Figs. 6, 7 and 8. It can be seen that the initial pull-out 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 ‘straight-fibre’ 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 pull-out load after de-bonding 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 pull-out 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 pull-out, 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 pulled-out 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.

Fig. 6

Pull-out-slip curves of 3DH hooked-end steel fibres. a NSC, b MSC and c HSC matrix.

Fig. 7

Pull-out-slip curves of 4DH hooked-end steel fibres. a NSC, b MSC and c HSC matrix.

Fig. 8

Pull-out-slip curves of 5DH hooked-end steel fibres. a NSC, b MSC and c HSC matrix.

Fig. 9

Maximum pull-out load (P_max) of 3DH, 4DH and 5DH hooked end fibres.

Fig. 10

Deformation and straightening of the hook after pull-out test.

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 pull-out 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.

Fig. 11

Stress distribution of the steel fibre circular section.

The moment $M_{E}$ refers to a limiting condition at which the elastic theory of bending applies:

$$ \frac{{M_{E} }}{I} = \frac{{E_{f} }}{{\rho_{E} }} = \frac{{\sigma_{y} }}{{r_{f} }} $$

(1)

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 x-axis. The following relationships apply from re-arranging Eq. (1) when $I = \frac{{\uppi r_{f}^{4} }}{4}$ for the fibre’s circular section:

$$ M_{E} = \frac{{E_{f} I}}{{\rho_{E} }} = \frac{{\sigma_{y} I}}{{r_{f} }} $$

(2)

$$ \rho_{E} = \frac{{E_{f} I}}{{M_{E} }} = \frac{{E_{f} r_{f} }}{{\sigma_{y} }} $$

(3)

$$ \therefore M_{E} = \frac{{\pi E_{f} r_{f}^{4} }}{{4\rho_{E} }} = \frac{{\pi \sigma_{y} r_{f}^{3} }}{4} $$

(4)

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:

$$ M_{ep} = M_{e} + M_{p} $$

(5)

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:

$$ M_{e} = \frac{{\sigma_{y} I_{e} }}{{r_{f} - h}} = \frac{{\sigma_{y} \times \pi (r_{f} - h)^{4} }}{{4\left( {r_{f} - h} \right)}} = \frac{{\pi \sigma_{y} }}{4} (r_{f} - h)^{3} \;\rho_{e} = \frac{{E_{f} \left( {r_{f} - h} \right)}}{{\sigma_{y} }} = \frac{{E_{f} r_{f} \left( {1 - \frac{h}{{r_{f} }}} \right)}}{{\sigma_{y} }} $$

(6)

Fig. 12

The elastic and plastic zone areas of the steel fibre circular section.

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 $2{{y}}$:

$$ \delta M_{p} = 2\sigma _{y} \delta Ay = 2\sigma _{y} \left[ {{2( {r_{f}^{2} - y^{2} }})^{1/2}} \delta y \right] \times y $$

where $\delta A$ is the bracketed quantity. Integrating between the limits of $y$ as:

$$ M_{p} = 4\sigma_{y} \mathop \int \limits_{{\left( {r_{f} - h} \right)}}^{{r_{f} }} y (r_{f}^{2} - y^{2} )^{{1/2}} dy = - \frac{4}{2}\sigma_{y} \mathop \int \limits_{{\left( {r_{f} - h} \right)}}^{{r_{f} }} - 2y(r_{f}^{2} - y^{2} )^{{1/2}} dy $$

$$ M_{p} = \frac{4}{3}\sigma_{y} [h\left( {2r_{f} - h} \right)]^{3/2} $$

(7)

The sum of Eqs. (6) and (7) provides the elastic–plastic moment required for plastic penetration of each zone to depth $h$:

$$ \therefore M_{ep} = \frac{{\pi \sigma_{y} }}{4} (r_{f} - h)^{3} + \frac{{4\sigma_{y} }}{3} [h(2r_{f} - h)]^{{3/2}} \;\left\{ {\begin{array}{*{20}l} {first\,yield\,{\kern 1pt} h = 0,M_{E} = \frac{{\pi \sigma_{y} r_{f}^{3} }}{4}} \\ {fully\,plastic\,{\kern 1pt} h = r_{f} ,M_{P} = \frac{{4\sigma_{y} r_{f}^{3} }}{3}} \\ \end{array} } \right. $$

(8)

When Eq. (8) is normalised with the fully elastic moment in Eq. (1) it becomes:

$$ \frac{{M_{ep} }}{{M_{E} }} = \frac{{\left( {r_{f} - h} \right)^{3} }}{{r_{f}^{3} }} + \frac{16}{{3\pi r_{f}^{3} }} [h(2r_{f} - h)]^{{3/2}} $$

which is written in a convenient dimensionless form:

$$ \frac{{M_{ep} }}{{M_{E} }} = \left( {1 - \frac{h}{{r_{f} }}} \right)^{3} + \frac{16}{3\pi }\left[ {\frac{h}{{r_{f} }}\left( {2 - \frac{h}{{r_{f} }}} \right)} \right]^{{3/2}} $$

(9)

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:

$$ M_{P} = \frac{16}{3\pi } \times \frac{{\pi \sigma_{y} r_{f}^{3} }}{4} = \frac{{4\sigma_{y} r_{f}^{3} }}{3} $$

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 $\rho_{e}$:

$$ \frac{{M_{ep} }}{{M_{E} }} = \left( {\frac{{\rho_{e} \sigma_{y} }}{{E_{f} r_{f} }}} \right)^{3} + \frac{16}{3\pi }\left[ {\left( {1 - \frac{{\rho_{e} \sigma_{y} }}{{E_{f} r_{f} }}} \right)\left( {1 + \frac{{\rho_{e} \sigma_{y} }}{{E_{f} r_{f} }}} \right)} \right]^{{3/2}} = \left( {\frac{{\rho_{e} \sigma_{y} }}{{E_{f} r_{f} }}} \right)^{3} + \frac{16}{3\pi }\left[ {\left( {1 - \frac{{\rho_{e} \sigma_{y} }}{{E_{f} r_{f} }}} \right)^{2} } \right]^{{3/2}} $$

(10)

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:

$$ \frac{{M_{e} }}{{M_{E} }} = \frac{{\sigma_{y} I_{e} }}{{\left( {r_{f} - h} \right)}} \times \frac{{r_{f} }}{{\sigma_{y} I_{E} }} = \frac{{I_{e} }}{{I_{E} }}\left( {\frac{{r_{f} }}{{r_{f} - h}}} \right) = \frac{{I_{e} }}{{I_{E} }}\left( {\frac{1}{{1 - \frac{h}{{r_{f} }}}}} \right) $$

(11)

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:

$$ \frac{{A_{p} }}{{A_{f} }} = R; where \, A_{f} = \pi r_{f}^{2} $$

in which the geometry of Fig. 12 shows:

$$ \frac{{A_{p} }}{2} = \frac{1}{2}r_{f}^{2} \left( {180 - 2\theta_{e}^{^\circ } } \right)\frac{\pi }{180} - r_{f} \cos \theta_{e} \left( {r_{f} - h} \right) $$

$$ A_{p} = r_{f}^{2} \left( {1 - \frac{{\theta_{e} }}{90}} \right)\pi - 2r_{f}^{2} \left( {1 - \frac{h}{{r_{f} }}} \right)\cos \theta_{e} $$

Hence

$$ \frac{{A_{p} }}{{A_{f} }} = R = \left( {1 - \frac{{\theta_{e} }}{90}} \right) - \frac{2}{\pi }\left( {1 - \frac{h}{{r_{f} }}} \right)\cos \theta_{e} $$

in which

$$ h = r_{f} - r_{f} \sin \theta_{e} ,\; \therefore \frac{h}{{r_{f} }} = 1 - \sin \theta_{e} ; \;\therefore 1 - \frac{h}{{r_{f} }} = \sin \theta_{e} $$

(12)

Therefore,

$$ R = \left( {1 - \frac{{\theta_{e} }}{90}} \right) - \frac{2}{\pi }\sin \theta_{e} \cos \theta_{e} $$

giving an equation between $R$ and $\theta_{e}$

$$ \therefore R = \left( {1 - \frac{{\theta_{e} }}{90}} \right) - \frac{1}{\pi }\sin 2\theta_{e} $$

(13)

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.

Fig. 13

Curvature for elastic–plastic bending.

At the interface position $r_{f} - h$ between the two zones:

$$ \frac{{\sigma_{y} }}{{\left( {r_{f} - h} \right)}} = \frac{{M_{e} }}{{I_{e} }} = \frac{{E_{f} }}{{\rho_{e} }} $$

(14)

which gives an elastic curvature $\rho_{e}$ for the neutral axis (N.A) as shown:

$$ \rho_{e} = \frac{{E_{f} I_{e} }}{{M_{e} }} = \frac{{E_{f} \left( {r_{f} - h} \right)}}{{\sigma_{y} }} $$

From Eq. (14)

$$ \sigma_{y} \rho_{e} = E_{f} \left[ {r_{f} - r_{f} \left( {1 - \sin \theta_{e} } \right)} \right] $$

$$ \frac{{\sigma_{y} \rho_{e} }}{{E_{f} }} = r_{f} \sin \theta_{e} $$

$$ \therefore \rho_{e} = \frac{{E_{f} r_{f} \sin \theta_{e} }}{{\sigma_{y} }} $$

(15)

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:

Fig. 14

Bend angle and centre line curvature.

$$ \sin \frac{\beta }{2} = \frac{l}{{2\rho_{e} }} $$

(16)

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,

$$ \theta_{e} = 2.191^\circ $$

$$ R = \left( {1 - \frac{2.191}{90}} \right) - \frac{1}{\pi }\sin 4.382 = 0.951 $$

and from Eq. (12)

$$ h = 0.45\left( {1 - \sin 2.191} \right) = 0.433 $$

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 $\varepsilon_{R} = \frac{y}{{\rho_{R} }}$.

Fig. 15

Curvatures for applied moment release.

Thus at a given depth $y$ between applying then releasing $M_{ep}$ in Fig. 15a, b.

$$ \frac{1}{{\rho_{R} }} = \frac{1}{{\rho_{e} }} - \frac{1}{{\rho_{E} }} = \frac{{\rho_{E} - \rho_{e} }}{{\rho_{E} \rho_{e} }} $$

$$ \rho_{R} = \frac{{\rho_{e} \rho_{E} }}{{\rho_{E} - \rho_{e} }} $$

(17)

where the respective elastic–curvatures in Fig. 15a, b are:

$$ \rho_{e} = \frac{{E_{f} \left( {r_{f} - h} \right)}}{Y} = \frac{{E_{f} I_{e} }}{{M_{e} }} $$

$$ \rho_{E} = \frac{{E_{f} y}}{\sigma } = \frac{{E_{f} I}}{{M_{ep} }} $$

Substituting each elastic curvature $\rho_{e}$ and $\rho_{E}$ in Eq. (17):

$$ \rho_{R} = \frac{{\left( {\frac{{E_{f} I_{e} }}{{M_{e} }}} \right)\left( {\frac{{E_{f} I}}{{M_{ep} }}} \right)}}{{\left( {\frac{{E_{f} I}}{{M_{ep} }} - \frac{{E_{f} I_{e} }}{{M_{e} }}} \right)}} = \frac{{E_{f} II_{e} }}{{M_{e} M_{ep} \left( {\frac{I}{{M_{ep} }} - \frac{{I_{e} }}{{M_{e} }}} \right)}} = \frac{{E_{f} II_{e} }}{{IM_{e} - I_{e} M_{ep} }} $$

$$ \frac{{\rho _{R} }}{{\rho _{E} }} = \frac{{\left( {\frac{{I_{e} }}{I}} \right)}}{{\left( {\frac{{M_{e} }}{{M_{E} }}~ - \frac{{~I_{e} }}{I}\left( {\frac{{M_{{ep}} }}{{M_{E} }}} \right)} \right)}} $$

(18)

Substituting Eqs. (9) and (11) in to Eq. (18)

$$ \frac{{\rho_{R} }}{{\rho_{E} }} = \frac{{\left( {\frac{{I_{e} }}{I}} \right)}}{{\frac{{I_{e} }}{{I_{E} }}\left( {\frac{1}{{1 - \frac{h}{{r_{f} }}}}} \right) - \frac{{I_{e} }}{I}\left\{ {\left( {1 - \frac{h}{{r_{f} }}} \right)^{3} + \frac{16}{3\pi }\left[ {\frac{h}{{r_{f} }}\left( {2 - \frac{h}{{r_{f} }}} \right)} \right]^{{3}/ {2}}} \right\}}} $$

Now $I = I_{E}$ with an elastic recovery across the full section and therefore:

$$ \frac{{\rho_{R} }}{{\rho_{E} }} = \frac{{1 - \frac{h}{{r_{f} }}}}{{1 - \left\{ {\left( {1 - \frac{h}{{r_{f} }}} \right)^{4} + \left( {1 - \frac{h}{{r_{f} }}} \right)\frac{16}{3\pi }\left[ {\frac{h}{{r_{f} }}\left( {2 - \frac{h}{{r_{f} }}} \right)} \right]^{{{3/2}}} } \right\}}} $$

(19)

Setting $y = \frac{{\rho_{R} }}{{\rho_{E} }}$ and $x = \frac{h}{{r_{f} }}$, Eq. (19) becomes:

$$ y = \frac{1 - x}{{1 - \left\{ {(1 - x)^{4} + \frac{16}{3\pi }\left( {1 - x} \right)\left[ {x\left( {2 - x} \right)]^{3/2} } \right.} \right\}}} $$

(20)

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 seven-steps 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 cross-section for a given fibre geometry as shown in Table 4.

Table 3 x and y values of Eq. (20).

Fig. 16

Normalized curvature (y) versus penetration depth ratio (x).

Table 4 Seven-steps enable an estimate of the extent of plasticity within the equivalent bend radius of a hooked end fibre.

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 pull-out for all hooked end fibres. From static force and moment equilibrium, the pull-out force ${{P}}$ in each pull-out 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).

$$ \Delta P^{\prime} = \frac{{F_{PH} }}{1 - \mu \times \cos \beta } $$

(21)

$$ \Delta P^{\prime\prime} = \frac{{2F_{PH} \left[ {1 + \frac{\mu \times \cos \beta }{1 - \mu \times \cos \beta }} \right]}}{1 - \mu \times \cos \beta } $$

(22)

$$ \Delta P^{\prime\prime\prime} = \frac{{F_{PH } \left[ {3 + \left( {\frac{2\mu *\cos \beta }{1 - \mu *\cos \beta }} \right)\left[ {2\left( {1 + \frac{\mu *\cos \beta }{1 - \mu *\cos \beta }} \right) + 1} \right]} \right]}}{{\left( {1 - \mu *\cos \beta } \right)}} $$

(23)

$$ \Delta P^{''''} = \frac{{F_{PH } \left[ {4 + \left( {\frac{2\mu *\cos \beta }{1 - \mu *\cos \beta }} \right)\left[ {3 + 2\mu *\cos \beta \left[ {2\left( {1 + \frac{\mu *\cos \beta }{1 - \mu *\cos \beta }} \right) + 1} \right] + 2\left( {1 + \frac{\mu *\cos \beta }{1 - \mu *\cos \beta }} \right) + 1} \right]} \right] }}{{\left( {1 - \mu *\cos \beta } \right)}} $$

(24)

where $\Delta P^{\prime},\Delta P^{\prime\prime},\Delta P^{\prime\prime\prime}, \Delta P^{''''}$ represent the pull-out force due to plastic deformation contribution of one, two, three and four plastic hinges, respectively (Abdallah et al. 2016a). By adding the pull-out force at the onset of complete debonding $P_{1}$ to Eqs. (21)–(24), the pull-out force $P$ as a function of fibre slip $\Delta$ in all main stages can be obtained as follows:

5.1 For 3DH fibre

$$ P = \left\{ {\begin{array}{*{20}c} {P_{2} = P_{1} + \Delta P^{\prime\prime}\;\left( {\text{Eq.}\,(18)} \right)} \\ {P_{3} = P_{1} + \Delta P^{\prime}\;\left( {\text{Eq.}\,(17)} \right)} \\ \end{array} } \right\} $$

(25)

5.2 For 4DH fibre

$$ P = \left\{ {\begin{array}{*{20}c} {P_{2} = P_{1} + \Delta P^{\prime\prime\prime}\;\left( {\text{Eq.}\,(19)} \right)} \\ {P_{3} = P_{1} + \Delta P^{\prime\prime}\;\left( {\text{Eq.}\,(18)} \right)} \\ {P_{4} = P_{1} + \Delta P^{\prime}\;\left( {\text{Eq.}\,(17)} \right)} \\ \end{array} } \right\} $$

(26)

5.3 For 5DH fibre

$$ P = \left\{ {\begin{array}{*{20}c} {P_{2} = P_{1} + \Delta P^{''''}\; \left( {\text{Eq.}\,(20)} \right)} \\ {P_{3} = P_{1} + \Delta P^{\prime\prime\prime}\;\left( {\text{Eq.}\,(19)} \right)} \\ {P_{4} = P_{1} + \Delta P^{\prime\prime}\;\left( {\text{Eq.}\,(18)} \right)} \\ {P_{5} = P_{1} + \Delta P^{\prime}\;\left( {\text{Eq.}\,(17)} \right)} \\ \end{array} } \right\} $$

(27)

The ${\text{P}}_{1}$ value can be predicted using Eq. (28) of straight fibre developed by Naaman et al. (1991a).

$$ P_{1} = \psi \tau_{fd} \left( \Delta \right) \times \left( {l - \Delta } \right) $$

(28)

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 pull-out 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 pull-out for all hooked end fibres are summarized in Table 5. Here, the continuous pull-out 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).

Table 5 The predicted pull-out forces for all hooked end fibres.

Fig. 17

Comparison of the predicted and experimentally measured pull-out curves for 3DH fibre. a NSC, b MSC and c HSC matrix.

Fig. 18

Comparison of the predicted and experimentally measured pull-out curves for 4DH fibre. a NSC, b MSC and c HSC matrix.

Fig. 19

Comparison of the predicted and experimentally measured pull-out curves for 5DH fibre. a NSC, b MSC and c HSC matrix.

The respective figures show the comparison between the predicted and experimental pull-out 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 pull-out load-slip curve. It is clear that the proposed model is able to capture the main features of pull-out behaviour and to predict accurately the pull-out load-slip 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 pull-out 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 pull-out. 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 pull-out. 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 pull-out-slip response for all hooked end fibres-concrete combination realistically. Predictions show greatest accuracy in matching the pull-out-slip behaviour of hooked end fibres embedded in MSC and HSC. This model was also able to take into account matrix damage and fibre rupture.