It is inevitable that concrete structures are subjected to tensile stress during their service lives whether this arises from shrinkage or from flexural bending and shear [1]. Consequently, there is a need to provide the structure with shear strength, flexural strength, ductility, and crack resistance. Because concrete does not have good properties inherently in each regard it needs to be reinforced with an appropriate material in order to render an otherwise weak cementitious structure with the performance required [2]. Steel has always been used and still is the most promising reinforcement as it combines a high tensile strength with a high Young’s modulus. High tensile strength steel reinforcement is available as rebar, mesh and fibres [3]. The latter, i.e. short strand steel fibres, are available in different size, shape and alloy type. Of these, hooked end steel fibres have recently received greatest interest among engineers and researchers [4] because of their resistance to being pulled out under tensile loading.

The main contribution from a fibre reinforcement is to absorb tensile without brittle fracture. Tensile stress induced in the cementitious matrix are transferred to the steel fibre by the durable bond characteristics between both basic materials [5]. A fibre, mechanically pre-deformed with hooked ends, bonds to the matrix sufficiently well to resist the pull-out loading. A spread of randomly oriented and distributed fibres can resist micro-cracking at low loads more effectively [6]. For an individual fibre bridging a micro-crack, debonding starts along its short, 50 mm straight length under low loads. With increased loading a full debonding occurs [7]. There follows further micro-cracking leading, eventually, to the formation a macro-crack. The bond mechanisms involved between fibre and matrix have been investigated to improve the mechanical performance of steel fibre reinforced concrete (SFRC) [8]. Here the interaction between interfacial bonds across many fibres in a cementitious matrix becomes active in providing increased resistance by prolonging the micro-cracking stage [9]. It follows that an SFRC with a large number of fibres contributes more to improving the tensile resistance than another with a lower volume fraction within a similar grade of matrix material.

Numerous analytical and experimental studies have been carried out to investigate the fibre-matrix interfacial properties [10,11,12,13,14,15]. The majority of these investigations have been devoted to study the bond characteristics when fibres are aligned with the loading direction [16,17]. Relatively few studies have been carried out on the bond behaviour of misaligned fibre reinforcement, which constitutes a more representative practical structure. It appears that the mechanisms governing pull-out behaviour of misaligned fibres differ from those verified experimentally for aligned fibres [18]. In addition to those mechanisms governing pull-out of aligned fibre, i.e. initial de-bonding followed by frictional pull-out, the misaligned fibre introduces fibre bending, matrix spalling and local friction effects.

Some recent studies [19,20,21,22] have investigated the mechanisms governing pull-out behaviour of inclined steel fibres in a cementitious matrix. It would appear from these that existing theory is complicated by the many empirical parameters employed to match specific experimental data. Thus, the predictions to a real pull-out condition are questionable across a wide variation in testing procedures that have been adopted in the absence of a standard test. In order to admit this general practical concern theoretical approaches have abounded on the specifics.

Soetens et al. [19] developed a semi-analytical model to simulate the pull-out behaviour of inclined fibres. The accounted for variations in mechanical and geometrical characteristics of the fibre, as well as the influence of fibre orientation

Lee et al. [20] developed an analytical model to predict the pull-out response of inclined hooked-end fibre embedded in an ultra-high strength cementitious matrix. An apparent shear strength and slip coefficient were introduced to express the variation of peak pull-out load and peak slip.

Laranjeira et al. [21] proposed an analytical model to predict pull-out behaviour of the inclined straight and hooked end fibres. A multilinear diagram based on a set of key-points was used to describe the pull-out-slip response. Their model takes account of debonding, friction effects and matrix spalling due to a variation in pull-out inclination.

Zhan et al. [22] developed an analytical model, similar to [21], in which the effect of the hooked-end is captured by a series of key stages in the force versus displacement response during pull-out. The force, calculated for each stage, augments the theoretical force required for the pull out of a straight fibre embedded with similar inclination.

Building upon the investigations reviewed above, it is recognised that relatively few analyses have been made upon the effects of fibre inclination angle upon pull-out force. Hence, the main intention of this paper is to offer an alternative global explanation of the pull-out behaviour that has been observed for inclined, hooked-end steel fibres. The bounding forces found from the present model are validated against new laboratory test results and other published data.

Figure 1a shows a single hooked-end, inclined steel fibre embedded in a concrete matrix. When the fibre orientation lies with an inclination $\theta$ to an externally applied tensile force $F$ the stress state within the fibre is not unidirectional, consisting of axial, transverse and shear components (${\sigma }_{11},{\sigma }_{22},{\sigma }_{12}$) aligned with the fibre co-ordinates (see Figure 1b). Laboratory testing of an inclined fibre provides the pull-out force $F$ applied horizontally to half the length of a single-hooked-end, embedded fibre for a useful inclination range: $0\le \theta \le {90}^{°}$ in the manner of Figure 2a. It is expected that the pull-out force first breaks the interface bond under a complex stress state in a similar manner to that explained above for a straight fibre. Then, with the fibre freed from the matrix, the fibre shank’s stress components shown (enlarged in Figure 1b) are provided by a simpler off-axis stress transformation under the uniaxial loading [23]. where σ = F/A is the applied stress, found from the fibre section area $A$ over which$F$ is distributed uniformly at its half-length entry position in Figure 2a.

Figure 8 shows the effect of inclination angle on the maximum pull-out load (F_θ) of inclined hooked end fibres with the following inclinations: 0˚, 15˚, 30˚, 45˚ and 60˚. Those mechanisms governing the pull-out behaviour of inclined hooked end fibre are taken to be similar to those found for aligned straight fibres. These are: initial de-bonding, followed by unbending then frictional pull-out of the straightened fibre. First the force applied must overcome a chemical bonding and frictional resistance from the interface to the fibre. Slip occurs when an increased force releases the mechanical anchorage provided by the hooked end. The unbending required for this release occurs involves plastic deformation within hinges that form successively at each bend of the fibre hook. When straightened the completed pull out of the fibre occurs under a much lower force that equates to any remaining sliding friction. The mechanical anchorage contribution provided by a hooked end fibre increases the pull-out load after de-bonding significantly. In contrast, following debonding of an embedded straight fibre only frictional resistance remains to be overcome under a falling load. The pull-out load versus slip responses shown in Figure 8 reveal the load variations found from separate pull out tests upon hooked-end fibres at five orientations.

The pull-out plots are often presented as an average of a number of tests conducted under similar controlled test conditions of temperature, straining rate, loading alignment and consistency of material composition. Though the averaging has removed vibrational scatter in the loading an irregularity remains in each plot which can defy an accurate prediction. Despite this, most reports on pull-out testing continue to identify a sequence of failure events within the plot of pull-out load versus fibre slip [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. The first event is the region along a rising curve within which fibre de-bonding has occurred. Thereafter, each flat load plateau is identified [20] with the bending required to straighten the fibre hook prior to its complete removal from the matrix at a slip displacement, which equals 30 mm as a straightened embedded length (see Figure 8). The anchorage offered by the hooked end raises the load to the maximum observed as required for straightening the fibre at its first bend. Thereafter, lower load plateaus have been associated, successively, with bending at the trailing ends of the hook as necessary for the fibre to slip further in the straight region of the matrix tunnel. Some erosion of that straight region occurs with spalling in the matrix around the fibre at its exit surface. Therein, with some loss of fibre orientation, further load plateaus appear less distinct, where the number of these depends the hook design. Spalling becomes more severe in fibre with larger inclination to the pulling force. A bending of fibre in a spalled surface cavity leads to premature failure [18,19,20,21] of the fibre before its complete removal. Such behaviour has motivated the present bounding approach to admit steeper fibre inclinations under loading where fibre fracture interrupts its complete withdrawal. Invoked by the bounding approach the critical load for an incomplete pull-out, i. e. from fibre fracture, lends itself to a division between its upper and lower bound limits. It is also suggested here that a further intermediate bound be placed upon the tensile strength of the concrete at the interface region where spalling occurs. The ratio of tensile strength of concrete is a small fraction of that for the steel fibre. Consequently, spalling would be expected to occur around the fibre’s entry position to a freed crack surface under a relatively low pull-out load. This behavior arises in practice where a randomly orientated fibre distribution serves to bridge free crack surfaces that arise in a concrete matrix under service loading conditions, i.e. the primary role of reinforcement. In serving that role a bridging fibre(s) with orientation would be more likely to create a spalling cavity than an aligned fibre. Pulling on a mis-aligned fibre creates a bending moment at entry to the matrix. Quite apart from its contribution to fibre failure, this bending imparts both tension and compression to the surrounding matrix, with the concrete on its tensile side being most likely to spall. There may appear opposing force at work here! While spalling acts to lessen resistance to pull out the shank inclination increases that resistance in a lower bound mode (see Figure 6a-d). Conversely, the upper bound mode shows that a shank failure stress is reduced as the plane stress state (Figure 2a) within it is accentuated by the fibre’s orientation. The interplay between such influences upon the failure stress (i.e. the ‘pull-out force per unit area) depends upon orientation in a manner suggested from bounding failures: a lower bound failure from complete pull-out of slightly mis-alignment fibre; an upper bound shank failure for misaligned fibre beyond a critical orientation. In Figs 6a-d the unit ordinate cuts off a l. b. failure i.e. when the applied stress attains the ultimate strength of the fibre. Experiment shows that this condition is never attained. Thus, with the straightening of the hook end, preceding its complete withdrawal, the applied stress is found to be less than the ultimate strength. Fibre pull-out would appear to be most consistent with an aligned stress condition. The implication drawn from Figure 6 is that a l. b. fibre pull-out applies only to low range of ‘misaligned’ fibre. Figure 6 suggests safely that a range of θ < 10˚ applies to a l. b. pull-out plot before it is intercepted by an upper-bound fibre failure. The latter requires less loading to be applied for failure in fibre with increasing orientation as the stress components seen within the plane stress state of Figure 1b increase. Thus, a combined stress failure in an off-axis fibre occurs under an applied load less that the load required for a uniaxial tensile failure of an aligned fibre. The combined loading may also involve flexure of a fibre within a spalled cavity and further surface shear from any fibre slip from within the surrounding matrix. Again, each influence would contribute to combined stress yielding under a lower applied load compared to the load necessary for uniaxial yielding to occur. Experimental data lends support to the yield criterion. It can be seen from Figure 9 that the maximum pull-out load at 15° and 30° fibre shank inclinations are 3.2% and 5.3% higher than that for an aligned fire shank (0°). At 45˚ and 60˚ inclinations, the maximum pull-out load is 20.2 % and 36.8 % less than that of the aligned fibre. The maximum pull-out load of hooked end fibre increased with inclination angle up to 30° in accordance with the lower bound Equation (6b). At greater inclinations 45°and 60° an upper bound failure occurs assisted by spalling. Here with a fibre failure the full deformation is reached in the shank at ultimate strength, before the end can straighten to facilitate pull-out. Despite the change in failure mode it is seen that the upper bound loading remains high within the orientation dependent strength of cold drawn steel fibre. Lower bound loads do involve plastic hinging under flexure but at lower loading than that required to produce an off-axis tensile shank failure.

The maximum pull-out load appears for a fibre inclination of 30˚. This has been explained in a change in deformation mode from plastic unhinging in the hook to shank plasticity, the latter reminiscent of triaxial necking in a tension test. Adding to the complexity of pull-out plasticity, the applied force must overcome the frictional resistance at the fibre bend when interfaced with coarse aggregate and also account for any relief in that force from the freed interface surface arising from the spalling that occurs. In combination it would appear that the respective changes to the force amount to a cancellation of such spurious influences upon it, consistent with the force bounding estimates given here.

In order to ascertain the reliability of the proposed approach to describe the failure of inclined hooked end fibres, various experimental results from data published in the literature [26,27,28] have been compared with the bounds found from Equations (3) and (6b). The experimental data have been chosen primarily to enable an account of the embedded strength of hooked-end fibre with different shank orientation. Other variables include fibre diameter, the tensile strength of the fibre and the compressive strength of the concrete. In the experiments the embedded shank half-length has varied together with its hook end geometry. The upper bound (u. b.) does not require these specific details explicitly only that the hook should provide sufficient anchorage for a shank failure to occur. Diameter appears in the calculation for the nominal stress σ_θ under the applied force and diameter is also maintained in testing fibre for the tensile and shear strength assessments described above. Concrete strength is important to an u. b. estimate, the latter being lowered by bending when spalling occurs in a weaker matrix composition. Embedded length influences any slip that occurs but does not contribute to the shank failure load or the σ_θ calculation. The normalised unit ordinate in each of Figs 6a-d indicates that if a shank failure should occur in aligned failure it would do so at a force corresponding to the fibre’s UTS. Hence the normalised ordinate intercept of unity shown marks the start of all u. b. plots. This intercept refers to a pure (unlikely) tensile failure of aligned fibre and is therefore independent of the matrix composition. With increasing fibre mis-alignment to the force the u. b. plot of transversely isotropic fibre depends upon orientation in a manner controlled by its two strength ratios. Figure 10 examines the upper bound estimates in more detail. Five combinations of pairs of strength ratios are sufficient to show that the experimental data are captured by these u. b. plots. It is noted that u. b. plots may rise or fall from its unit intercept in the manner shown depending upon the strength combination chosen in Figure 10. This figure presents from Equation (3) upper bounds to the strength dependency of transversely isotropic fibre with embedded orientation. In Figure 10 anisotropic, off -axis plots apply to strength ratios typical of cold drawn high strength metallic fibre. Specific strength ratios that appear are taken from the ranges:

Overlaid upon the five u. b. plots in Figure 10 are the available experimental data. The five upper bound predictions shown apply to their two strength ratios lying in the ranges:

Overall, the experimental data agrees mostly with a strength ratio combination of $\frac{{\sigma }_{1f}}{{\sigma }_{2f}}=1\frac{1}{2}$ and $\frac{{\sigma }_{1f}}{{\sigma }_{12f}}=\frac{3}{4}$. For the new data (o) the reduction to upper bound strength at and beyond the 30° inclination angle has been attributed to matrix spalling during the pull-out process. With the change from lower to upper bound the fibre fails from having attained its off-axis combined strength under the applied force.

Considerably more experimental data [31] are available for aligned fibre where lower bound force/stress estimates apply to a complete pull-out. Further data [31,32,33] suggests that fibres will pull out when embedded with a slight mis-alignment to the force. In Fig 6, for example, the lower bound peak force attained has been converted to a nominal stress and then normalised with the fibre’s axial tensile strength (see Equations 6a, b). Results (which include aligned fibre, θ = 0˚), show that the ultimate tensile strength of the fibre material is never reached under the peak load required to straighten the hook for complete pull out to occur. Therefore, the aligned fibre results indicate an intercept value of less than unity to be placed upon the ordinate in Figure 10. This marks that start of the lower bound plot that rises in proportion to 1/cosθ for a narrow orientation range. Here, Figs 6a-d have shown the range that applies to l. b plots for four composites in which each aligned fibre-cement combination has a different intercept on the stress axis. Hence the intercepts for each u .b. plot depend upon the fibre-cement combination[34].. In contrast each u. b plot in Figure 10 has a common intercept of unity at the UTS of the fibre.

In this paper a new bounding approach has been developed to predict the pull-out response of misaligned hooked-end steel fibre embedded in a concrete matrix. An upper bound estimate to the pull-out strength is based upon a transversely isotropic failure condition arising in the straight length of the inclined fibre shank. That bound accounts for orientation and strength properties of the deep drawn fibre. The transversely isotropic failure criterion applies to deep drawn wire having grains elongated by multiple reductions to its final diameter. Metallography revealed that elongated grains, with an aspect ratio exceeding five, were aligned with the fibre’s length direction. An upper bound to the interrupted pull-out failure of an embedded, inclined fibre accounts for its anisotropic strength under the off-axis plane stress state that exists within its shank. Fibre strength tests have been proposed to measure the transverse strength and the shear strengths required for setting an upper bound limit curve. A lower bound limit is based upon the complete pull-out of an aligned, hooked end fibre embedded in a cementitious matrix. There the peak force is associated with that force required to straighten the hook end. Therein experimental data are used to set the peak nominal stress (force/area) limit initially for an aligned fibre. For inclined fibre that stress increases with it being inversely proportional to the cosine of an inclination angle. Normally, in a strong matrix the lower bound mode is restricted to a range 0 – 30. Beyond this range fibre failure occurs in the upper bound mode in which the shank yields. However, a lower bound mode range may be extended where matrix spalling allows the fibre to bend in closer alignment with the pulling force. This applies provided such bending does not result in fibre breakage where then it has contributed to an upper bound failure. It is suggested here that a further intermediate bound be placed upon the strength of the concrete mix in the interface region where spalling occurs. Consideration of the u.b. and l.b. modes of failure proposed here and the division arising between them has provided, this far, an explanation of all available experimental data. Finally, the motivation for the present study has been to admit a two-mode failure analysis of misaligned hooked-end steel fibre. Such behavior is relevant to concrete reinforcement with short fibre having random orientation as for pre-cast concrete sections say for tunnels and a viaduct. Where loading is complex and the random fibre orientation ensures some alignment with varying principal stress directions. In contrast, the r-bar reinforcement of beams and columns is aligned with the unvarying direction of maximum tension.