1 Introduction

The use of fibres as shear reinforcement in Reinforced Concrete (RC) beams was mainly studied during the past three decades, even if the majority of these studied was carried out in the 2000s (Fig. 1). Fibres substantially enhance the shear strength and deformation capacity of structural elements [1, 2] in either vibrated [3,4,5,6,7,8,9] or self-compacting concrete [10, 11]; fibres also proved to be very effective in extruded concrete members [12]. Therefore, the effectiveness of fibres in enhancing the shear resistance is widely recognized in the scientific community. So far, the largest number of these studies dealt with steel fibres [3,4,5,6, 8,9,10,11,12] even if there was a recent growing interest on macro-synthetic ones [7].

Fig. 1
figure 1

Papers about shear behaviour on FRC beams

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Recent design guidelines allow to use fibres as shear reinforcement [13,14,15]. RILEM TC 162-TDF [13] proposed pioneer guidelines where fibre contribution to shear resistance was added to the concrete one as a separate term (based on post-cracking residual strengths). In other research studies, fibres and concrete contributions are generally not considered as additional terms since fibres markedly influence the typical mechanisms of shear transfer present in RC element (especially aggregate interlock); Fiber Reinforced Concrete (FRC) can be considered as a unique composite material characterized by significant toughness properties after cracking (due to the bridging effect of fibres). This concept was adopted in the recent Model Code 2010 (hereafter MC2010) [14], where the positive effect of fibres in shear is considered as an enhancement of concrete contribution. The latter was modelled in MC2010 by two different analytical equations (well-discussed in the following sections) based on the post-cracking residual strength provided by FRC, evaluated according to EN 14651 [16]. Consequently, a material mechanical characterization of FRC is required by MC2010.

Figure 1 shows the number of publications (indexed papers) with experimental results from shear tests on FRC elements in the last 30 years (between 1985 and 2016). It can be observed that the majority of experimental tests (72%) was carried out without providing any FRC mechanical characterization (only fibre type and amount are given), making difficult the use of these results by researchers. Only 28% of papers provides post-cracking mechanical properties of FRC, even if different characterization methods were applied. Less than half of them provides a mechanical characterization according to EN 14651 and, so far, only very few recent papers tried to link other tests to EN 14651 [17, 18]. Consequently, shear strength of elements reinforced by fibres is predicted by using formulations generally developed from a limited set of tests results [19, 20] or from literature-derived database [21,22,23,24], where most of them do not provide any FRC characterization [21, 22]. In fact, it is very difficult to merge experimental results from different researches when reported data are different [25, 26] or important parameters are not given.

In this context, this paper neither proposes a new formula to add to the already long list nor collects data from other Authors. The objective is to provide a comprehensive shear database to the scientific community, which allows both to improve exiting analytical models and to develop new design formulations in shear. The strength of this database is to have qualified and detailed compilation of data about FRC beams subjected to shear, merging the experiences carried out in the past decades at the University of Brescia and at the Universitat Politècnica de València. The geometry, reinforcement details, post-cracking residual strengths according to EN 14651 (fR,1 and fR,3) and the experimental results of 93 beams in FRC and their related 78 sample in RC (with or without web reinforcement) are presented. In addition, the database was analysed in order to capture the effect of the typical factors affecting the shear strength in case of FRC beams, as well as to study the increase in shear strength due to fibres addition. The shear formulations of both Eurocode 2 (hereafter EC2) [27] and MC2010 [14] for RC and FRC elements were compared against the database in order to highlight differences and inadequacy on shear strength predictions.

2 Database of FRC elements under shear loading

A comprehensive experimental database involving shear tests on beams reinforced by fibres (either steel and macro-synthetic one) is shown in Table 1. In addition, Table 2 lists the reference samples (RS) in RC made with or without web reinforcement of the beams reported in Table 1, allowing to evaluate the fibres influence on shear strength. This database was compiled from 12 experimental studies performed between 2006 and 2017 at the University of Brescia [28,29,30,31,32,33,34,35,36] and at the Universitat Politècnica de València [37,38,39]. The database contains test results of 171 beams (117 and 54 tested at the University of Brescia and Universitat Politècnica de València, respectively), where 93 are made with FRC (Table 1) and 78 are in RC made with or without web reinforcement (Table 2). The majority of the beams exhibited either a shear or web-shear failure mode. In case of FRC samples, 73 beams failed in shear and 20 in flexure, while for RC beams 65 exhibited a shear failure and 13 a flexure one. All members failing in flexure are anyway reported in this database since they might give data for further specific analysis and applications. However, these samples were not considered in the database analyses presented in Sect. 3. It should be also noticed that 9 FRC beams (over 93) and 2 RC beams (over 78) of database are prestressed.

Table 1 Database of FRC elements under shear loading (continued on next page)
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Table 2 Database of reference samples in RC (continued on next page)
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For the collection of a consistent and comprehensive database, the following parameters are showed for each beam:

  • order number (#);

  • beam ID;

  • web width (bw);

  • effective depth (d);

  • shear span to effective depth ratio (a/d);

  • longitudinal reinforcement ratio (ρ);

  • web reinforcement ratio (ρw);

  • cylinder concrete compressive strength (fc);

  • maximum aggregate size (dg);

  • compressive stress due to effective prestress (σp);

  • fibre type in terms of: material (S = steel, PP = polypropylene), length in mm/diameter in mm, shape (H = hooked ends, St = straight, C = crimped). As an example, the designation S30/0.6H refers to a steel fibres 30 mm long, with diameter of 0.6 mm and hooked ends.

  • fibres volume fraction (Vf);

  • residual flexural tensile strengths fR,1 and fR,3 (in accordance to EN 14651 [16]) corresponding to CMOD = 0.5 mm and CMOD = 2.5 mm, respectively;

  • failure mode (FM): S = shear; SY = shear after longitudinal rebar yielding; WS = web-shear, namely the failure due to diagonal tension developed before flexural cracking; F = flexure;

  • experimental shear strength (V u ).

The cross section of beams in this shear database is rectangular, except for samples in Refs. [37] and [38] that are characterized by I-shape. More details about flange dimensions can be found in [37] and [38], as well as in Ref. [11] a deep analysis on samples from 58 to 69 is given. Moreover, it is worth mentioning that both longitudinal and web reinforcement adopted in the database are made with the steel most commonly used in Europe, i.e. B500C [27].

3 Evaluation of the shear database

Figure 2 shows the distribution of the main parameters affecting the shear strength in the database considering the 73 specimens failed in shear. It can be observed that wide ranges of selected parameters are considered. d varies from 210 to 1440 mm with 70% of beams under 500 mm, while a/d ratio ranges from 2.5 to 3.5 for the majority of samples (80%). Typical ρ adopted in practice (up to 1.3%) can be observed for 40% of samples, while, in order to induce a shear failure, higher amount of longitudinal reinforcement was used elsewhere (up to 3.7%). Different concrete compressive classes were also studied, with fc ranging from 19 to 96 MPa (with 70% of samples characterized by a fc smaller than 45 MPa). Finally, the database incorporate FRCs characterized by a significantly wide range of residual mechanical properties (both softening and hardening materials under flexure): fR,3 varying from 1.3 to 10.6 MPa. In particular, 31 samples have a fR,3 smaller than 3.0, followed by 25 specimens in the range between 3 and 5 MPa and the remaining 17 beams with higher residual mechanical properties.

Fig. 2
figure 2

Distribution of parameters in the shear database for FRC elements (only shear failure)

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In these parameter ranges, the increase in shear strength due to fibre addition was analysed by comparing each FRC beam (fibre only, Table 1) to its reference sample (without web reinforcement, Table 2). In particular, the ratio between the shear strength of FRC beams (Vu,FRC) and the one of their reference samples (Vu,RS) was calculated and plotted as a function of fR,3 in Fig. 3. This allows to obtain the fibre contribution on shear strength, excluding the influence of all other affecting parameter (i.e. d, a/d, ρ, fc), which are equal between each FRC sample and its corresponding reference specimen. fR,3 was chosen since it characterizes the post-cracking performance given by different fibres (which differ in length, shape, material and amount) at ultimate limit state. Figure 3 clearly shows that an increase of fR,3 determines a significant increment of shear strength, resulting in a rather fundamental tool for engineers for both pre-design and verification phases against shear loading. For instance, a fR,3 of 2 and 5 MPa lead to a shear strength increment of about 50 and 100%, respectively. The parameter fR,3 was related to shear strength increment by using the method of least squares (imposing an intercept Vu,FRC/Vu,RS = 1 for fR,3 = 0) and it was observed that the best trend is a quadratic regression, even if it is characterized by a weak coefficient of determination (R2 = 0.35). The latter is due to the fact that for a given fR,3 the ratio Vu,FRC/Vu,RS showed a variability, as a consequence of the combined effect of both fR,3 (coefficient of variations ranging between 10 and 40% [40]) variability and shear strength dispersion (placing, compaction, beam geometry and fibre type can influence fibre distribution and orientation [41]). Moreover, the load at which the inclined shear cracking occurs is a function of the tensile strength of concrete, which in turn is a quite disperse material property. This also contributes, with the other factors as above-mentioned, to the shear strength variability of elements reinforced by fibres only.

Fig. 3
figure 3

Increase in shear strength due to effect of fibres

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The influence of fibres on well-known parameters affecting the shear strength was evaluated as well. In particular, Fig. 4 shows the influence of fibres on the shear strength as a function of effective depth (Fig. 4a) and longitudinal reinforcement ratio (Fig. 4b). In this figure, the normalized shear strength (vu/f (1/2)c ) is plotted as a function of the two parameters; moreover, FRC specimens were separated in three fR,3 ranges. It can be observed that, as already stated by Minelli et al. [32] and Shoaib et al. [6], the database confirms that FRC beams are affected by size effect (Fig. 4a). This influence is similar to RC beams without web reinforcement up to an effective depth of 1000 mm, after which the trend of the size effect law for FRC differs and it seems that a lower bound is reached (horizontal asymptote). Comparing the three performance ranges shown in Fig. 4a, it can be also observed that the decrease in the shear at failure is similar up to 1000 mm, underlining that the post-cracking performance of FRC does not significantly change the size effect law. Consequently, building codes could consider this positive effect of fibres in beams characterized by effective depth greater than 1000 mm, while classical RC size effect law can be applied for FRC elements with d ≤ 1000 mm. When considering the longitudinal reinforcement ratio, an increment of ρ causes a similar increase of ultimate shear strength in both RC and FRC elements, even for higher values of fR,3. This trend is also similar between the different FRC considered. Therefore, the longitudinal reinforcement influence on the shear strength seems not to clearly change from RC to FRC elements and its effects are comparable. The positive influence on the shear strength of the longitudinal reinforcement ratio is due to the increase in crack width for ρ reduction that causes a decrease in the shear transferred across the inclined crack by dowel action and aggregate interlock. Similarly to ρ, either f c or a/d influence resulted comparable both in RC and FRC samples. These database analysis underline that fibres mainly increase the shear strength of elements by transferring stresses across inclined shear cracks and by enhancing aggregate interlock mechanism [42,43,44].

Fig. 4
figure 4

Size effect trend for RC and FRC elements (a) and effect of fibres as the reinforcement ratio increases (b)

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4 Comparison with design equations of EC2 and MC2010

The shear strength predictions of two different analytical models recently included in MC2010 and developed for FRC beams were evaluated against the experimental results presented in the database. In addition, the provisions of their base models were also analysed against the reference samples listed in Table 2. In order to make this comparison more significant, the shear strength were calculated by assuming strength reduction factors (γc for EC2 and MC2010) equal to 1 and the mean values of the material mechanical properties. The mean value of tensile strength fct was evaluated according to Eq. 5.1-3a (\(f_{\text{ct}} = 0.3 \cdot \left( {f_{\text{ck}} } \right)^{2/3 }\)) and 5.1-3b (\(f_{\text{ct}} = 2.12 \cdot \ln \left[ {1 + 0.1 \cdot \left( {f_{ck} + 8\;{\text{MPa}}} \right)} \right]\)) of MC2010 for concrete characterized by \(f_{\text{ck}}\) ≤ 50 MPa and \(f_{\text{ck}}\) > 50 MPa, respectively. The characteristic value of the concrete compressive strength was calculated as \(f_{\text{ck}} = f_{\text{c}} - 8\) MPa according to Eq. 5.1-1 of MC2010. In case of MC2010 shear models, when required, the control section was considered at a location d from load point.

Table 3 summarize the shear formulations of these models for the prediction of the shear strength resistance (express in terms of SI units). Concerning FRC, the first model (Eq. (5) in Table 3) was built on the base formulation of EC2 (Eq. (1) in Table 3) [45] and it considers the fibre contribution by modifying the longitudinal reinforcement ratio. To the contrary, the second one (Eq. (7) in Table 3) was derived from the base formulation of MC2010 (Eq. (3) in Table 3) [46], which in turn is linked to the Modified Compression Field Theory [47]. Both formulations consider the fibre contribution not as a separate addendum but as an enhancement of the concrete contribution, as well as they adopt an ultimate residual tensile strength (fFtu). In case of Eq. (5), MC2010 suggests to estimate fFtu by a simplified linear model based on fR,1 and fR,3 (EN 14651), while in case of Eq. (7) it is suggested to evaluate it by direct tensile tests (without specifying the type of tensile tests). Since no direct tensile tests were carried out for any of the database series, the simplified linear model based on fR,1 and fR,3 (EN 14651) has been also applied in case of Eq. (7). Recent publications discussed also the possible influence of using the simplified linear model (fR,1 and fR,3) on Eq. (7) [17, 48]. Amin et al. [17] studying steel FRC showed that that MC2010 simplified linear model might overestimate fFtu, with direct consequence on the accuracy of the shear model [48]. Consequently, it should be noticed that the evaluation of Eq. (7) against database results could be also affected by the use of the simplified linear model for estimating fFtu.

Table 3 Design equations for the shear strength resistance of RC elements with and w/o fibres according to MC2010 [14] and EC2 [27]
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Figure 5 shows the comparison between the experimental and the ultimate shear strength predicted by the base models of EC2 (Eq. (1)) and MC2010 II level (Eq. (3)) for RC elements without web reinforcement. In this figure, specimens were ordered by increasing effective depth. It can be observed that both base models lead to good and very similar strength predictions. The mean value of the model safety factor Vu/Vu,code resulted in both case greater than the unity, i.e. 1.08 and 1.10 for EC2 and MC2010 model with coefficient of variation equal to 0.19. This confirms that both models are comparable, as well as they are mainly characterized by conservative or slightly unconservative predictions. In addition, it is worth mentioning that both models show a decrement of the ratio Vu/Vu,code when the effective depth increase: for d up to 250 mm this ratio is generally greater than 1, for 250 < d ≤ 1000 mm is around 1, while it is smaller than 1 (unconservative) for d > 1000 mm. Even if for d greater than 1000 mm only one samples is present in the database (H1500 PC), this general downward trend of Vu/Vu,code is clearly evident; this result underlines that EC2 and MC2010 could better take into account size effect in shear resistance of RC beams. To the contrary, no clear trend of Vu/Vu,code was observed varying ρ, a/d or fc. It is worth mentioning that both EC2 and MC2010 models resulted even more conservative in case of wide-shallow beams (W750 PC-1, W1000 PC-1 and 2, W430 PC-1 and 2, W770 PC-1 and 2) due also to the positive effect of width-to-effective depth ratio. More details about width-to-effective depth ratio influence on the shear strength can be found in [36].

Fig. 5
figure 5

Comparison against EC2 and Model Code 2010 shear model for beams without web reinforcement

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Likewise Figs. 5 and 6 shows the predictions of both EC2 [Eq. (1) + Eq. (2)] and MC2010 III level [Eq. (3) + Eq. (4)] model for beams with web reinforcement. In case of EC2 model, the shear strength was assumed as the sum of Vc and Vs with θ = 45°. It can be observed that, once again, both models are very similar and lead to good provisions, as well as they are generally conservative and characterized by a mean value of Vu/Vu,code close to 1.10. The prediction variability is 25% smaller as compared to beams without web reinforcement, as evidenced by the coefficient of variations. In addition, no variation of the ratio Vu/Vu,code is remarkable by varying d, ρ, a/d or fc increases.

Fig. 6
figure 6

Comparison against EC2 and Model Code 2010 shear model for beams with web reinforcement

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Figure 7 shows the comparison between Vu and Vu,code predicted by the two different shear formulations of MC2010 [Eqs. (5) and (7)] for elements with fibre only as shear reinforcement. In this figure, specimens were ordered by increasing the residual strength fR,3. It can be observed that Eq. (5) is characterized by a similar value of Vu/Vu,code as compared to its base model [Eq. (1)], even if the variability is 15% greater. To the contrary, Eq. (7) accuracy differs from the one of base model as demonstrated by the mean value of Vu/Vu,code, often smaller than 1, and by the greater coefficient of variation (+ 27%). Consequently, Eq. (7) leads to more variable and unconservative predictions than Eq. (5). In addition, it should be noticed that both models reduce their level of safety when fR,3 increases. In fact, for fR,3 > 3 MPa the ratio Vu/Vu,code resulted closer or even smaller than 1 as compared to samples with fR,3 ≤ 3. This underlines that both models require some modifications in order to better capture the increment of shear strength due to the effect of fibres. The reduction of the ratio Vu/Vu,code when the effective depth increase is also present as in the base models, even if, for d greater than 1000 mm, Eq. (7) resulted more unconservative than Eq. (5) (see samples H1500 FRC50 and H1500 FRC75 in Fig. 7). Equation (7) led also to more unconservative predictions for ρ > 0.02 and fc > 70 MPa, while Eq. (5) does not provide significant variation of Vu/Vu,code with increasing values of ρ or fc. However, it is should be noticed that, in Eq. (5), the shear strength is predicted by imposing a limitation on the longitudinal reinforcement ratio as in the base model, i.e. ρ ≤ 0.02. No influence of a/d was observed in both models.

Fig. 7
figure 7

Comparison against Model Code 2010 shear models for beams reinforced by fibres only

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Figure 8 shows the shear strength predictions of the two models [either Eq. (5) + Eq. (6) with θ = 45° or Eq. (7) + Eq. (8)] in case of combination of fibres and web reinforcement. It can be observed that both models resulted comparable and always unconservative; the mean value of Vu/Vu,code was of around 0.88 with a low CV. This evidence underlines that both models probably require modifications for improve their predictions in case of elements reinforced by both fibres and web reinforcement. However, since the available data concerning beams reinforced by both fibres and stirrups are limited (10 samples), further specific experiments should be carried out in order to confirm this model inadequacy.

Fig. 8
figure 8

Comparison against Model Code 2010 shear models for beams reinforced by both fibres and web reinforcement

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Finally, it can be underlined that, in case of prestressed elements, the equations listed in Table 3 are always conservative both in RC (Figs. 5, 6) and FRC elements (Figs. 7, 8). However, it should be reminded that only few prestressed specimens are available and the majority of them showed a web-shear failure.

5 Concluding remarks

Experimental tests on RC and FRC elements carried out in the last decade at both the University of Brescia and the Universitat Politècnica de València were collected and analysed in the present paper. The following conclusions might be drawn:

  1. (1)

    A comprehensive shear database of 171 elements (93 in FRC and their related 78 reference samples in RC with or without web reinforcement) is presented, defining the principal variables (d, a/d, ρ, ρw, fc, dg, σp) and the residual mechanical properties of FRC: fR,1 and fR,3. This database will allow further development and validation of suitable shear strength formulations for FRC elements;

  2. (2)

    a/d, ρ, and fc (related to tensile strength) similarly influence the shear strength both in RC and FRC elements. Instead, the beam size influence is different for d > 1000 mm since the horizontal asymptote is reached earlier in FRC members (for d ≤ 1000 mm classical RC size effect law can be applied also for FRC elements);

  3. (3)

    EC2 and MC2010 shear formulations lead to similar and good predictions of the shear resistance of RC elements with or without web reinforcement;

  4. (4)

    The two shear models provided by MC2010 (fFtu evaluated in both models according to MC2010 linear model, even if direct tensile tests are suggested by MC2010 for Eq. (7)) for elements reinforced by fibre only reduce their level of safety when fR,3 increases. In fact, for fR,3 > 3 MPa the model safety factor (Vu/Vu,code) resulted closer or even smaller than 1, as compared to samples with fR,3 ≤ 3;

  5. (5)

    The FRC shear strength equation of MC2010 based on EC2 model (Eq. 5 in Table 3) resulted more conservative than the one developed on the Modified Compression Field Theory (Eq. 7 in Table 3 with the assumption of evaluating fFtu from the MC2010 linear model). In particular, they mainly differ for specimens with d > 1000 mm, ρ > 0.02 and fc > 70 MPa.

  6. (6)

    The shear strength of elements reinforced by a combination of fibres and web reinforcement seems not to be well predicted by MC2010.

The Authors would like also to encourage all researchers to provide in their future works at least the information reported in the database in order to allow its extension. The results of FRC characterization tests are, with this respect, very important and needful.