Analytical and experimental shear evaluation of GFRP-reinforced concrete beams

Reinforced Concrete (RC) technology is advancing towards new frontiers enhancing its sustainability and durability through innovative materials. In particular, the application of Glass Fiber Reinforced Polymer (GFRP) bars, in lieu of steel reinforcement, shows excellent performance, especially in aggressive environments. Nevertheless, current international design guidelines and standards tend to be rather conservative, especially concerning shear reinforcement. This element hinders the technology’s competitiveness, not only in terms of material consumption but also in construction efficiency. This research aims to conduct an analytical comparison and experimental validation of the formulations found in some international standards pertaining to shear capacity in a specific case. The focus is on scenarios involving reduced shear reinforcement and cases where the number of stirrups falls below the minimum recommended by these standards. In the sample beam tests, two distinct flexural GFRP reinforcement ratios were employed to evaluate their influence on shear capacity, leading to diverse failure mechanisms: rupture of longitudinal GFRP bars and concrete crushing. The experimental results were used to compare the North American ACI, French AFGC, and Italian CNR shear capacity design approaches in the case of reduced transversal reinforced ratio. Analytical capacity expressions of the standards above are discussed with some remarks aiming at structural optimization.

Factor is taken as 0.85 for concrete strength fc 0 up to and including 28 MPa.For strength above 28 MPa, this factor is reduced continuously at a rate of 0.05 per every 7 MPa of strength over 28 MPa but is not taken less than 0.65.As utilized in this paper, the symbols and equations from ACI, AFGC, and CNR [1][2][3][4] have been harmonized to facilitate a more effective comparison of the standards and enhance text readability.They are summarized above.

Introduction
Composites gradually entered the history of concrete construction, first as retrofit reinforcements and later as reinforcing bars for new constructions [5].Fiber Reinforced Polymers (FRP) bars are particularly interesting for structures exposed to aggressive environmental conditions, such as coastal structures, ports, bridges, and industrial facilities [6][7][8][9].The Glass Fiber Reinforced Polymer (GFRP) technology is gaining increasing interest, driven by the rising cost of steel and the improvements in life cycle performance that composites can offer, coupled with ongoing enhancements in the production chain.In the history of RC structures, the analysis of shear capacity evolved with the following main steps.Following Jourawsky's theory (for homogeneous, elastic, and uncracked beams, with the limitation of the shear stress), the first design method for the cracked beam is represented by the truss-model, well-known as Ritter-Mo ¨rsh (R-M) trellis theory.In this, the check of tensile action in stirrups (V s ) and compressive action in concrete struts (V c,max ) determines the capacity of the members.This model does not assign any shear resistance to the compressed concrete area (area of the horizontal strut of a RC beam).An important parameter in the R-M model is the inclination of the compressed strut, usually indicated as h, that identifies the trellis and consents to consider it statically determined together with the hypothesis of uniform plasticity of the stirrups.With this hypothesis, R-M model also takes the name of the ''simplified truss model'' or ''plastic truss analogy''.
A further step actually used in ACI and most modern code is represented by Eq. ( 1): In this, the contribution of the concrete V c in compression is summed up to the contribution of the stirrups V s .In the beginning, the expression of V c was experimentally derived as an extrapolation of experimental data while the contribution of stirrups V s was derived by the R-M model, with the limitation of h = 45°.The progress obtained with experimentation consented to express V c as a function of the mechanical parameter of the concrete sections.In this path the check of the resistance of the compressed struts is implicitly considered in a reduction of V c and the coexistence of V c and V s in a unique expression is obtained by a limitation of tensile stress and strain in the stirrup as well as considering h = 45°.Other methods are based on the Compression Field and Shear Friction Theory.Compression Field Theory is based on the first work of Collins and Mitchel [10].This constitutes the basis of the first version of the Canadian Concrete Code and had some conceptual modifications from 1984 to 2004.The Shear Friction theory focuses on the shear friction of slender elements.Except for the R-M model, all these methods are based on Eq. ( 1) and present a different expression for V c .
Equation ( 1), now applied in ACI for ordinary concrete members, is modified to calculate the resistance of RC-FRP members considering the different mechanical properties of FRP with respect to steel.As best explained in Sect. 2 the expression for V c usually derives from the one used for ordinary RC members with some modification to consider the different mechanical characteristics of FRP bars.In FRP-RC, as for conventional steel-RC, the concrete contribution to the shear strength (V c ) includes, directly or indirectly, uncracked concrete and aggregate interlock.Dowel action, developed by the longitudinal reinforcement, is usually neglected because of the orthotropic behavior of FRP bars [11,12].The contribution of stirrups or of the transverse reinforcement (V s of steel becomes V f for FRP members) is typically estimated using the truss analogy with h = 45°and is summed up to V c .This except for the French standards that consent to consider the maximum value between V c and V f , depending, the latest, on the inclination of the compressed strut h.In this case, V f assumes the following value: So in AFGC the term V f can be increased up to 2.5 by the terms cot(h) but is penalized by the reduction of resistance of stirrups f fr as best expressed in the following Table 3.
Even though research on the shear performance of FRP-RC members is an active topic of investigation, a comprehensive list of works preceding 2015 was published by Razaqpur and Spadea [13] and, before them, by Tureyen and Frosch 2003 [14].This list includes more than 300 shear test results for FRP-RC beams, many of which were conducted on small-scale rectangular cross-section specimens.Fico et al. (2007) in ''Assessment of Eurocode-like design equations for the shear capacity of FRP RC members'' [15] presents an extensive comparison of Canadian CSA S-802, American ACI 440.1R,Italian CNR-DT 203, and Japanese JSCE expression for shear capacity for FRP-RC beams.The analysis is based on a wide database of beams with and without shear reinforcement.The paper analyzes the key points of the shear design of FRP prototypes and is a milestone in the state of the art presenting the assessment of Eurocode-like equations and introducing the actual basis of Italian standard.[17].The model proposed in the aforementioned works exhibits the lowest coefficient of variation among all the analyzed models.This holds significant importance because this model is grounded in the principles of structural mechanics.Mukhtar and Deifalla [18] proposed some modifications of the Critical Shear Crack Theory (CSTC) based on the analysis of 420 experimental tests of FRP-RC deep beam without stirrups.This results in two versions of CSTC called Modified Critical Shear Crack Theory (MCSTC).MCSTCI and MSCTCII are hybrid models, based on a well-known physical based model with an extensive empirical fitting.It is interesting to emphasize that shear capacity is expressed here as a function of slenderness ratio (a/d) besides elastic modulus ratio (Es/Ef).These models are validated by the authors on a blind set of experimental data and compared with the results furnished by international standards producing interesting results.In terms of shear capacity prediction, empirical expressions for the concrete contribution to shear were proposed and adjusted to account for the elastic modulus ratio between steel and FRP bars [19,20].In other instances, experimental evidence of the shear performance of FRP-RC beams was used to develop empirical or semi-empirical expressions using tailored parameters [21,22].Other works have considered mechanistic models based on the modified compression field theory for the prediction of the shear strength of FRP-RC members providing a more rational approach to design [23,24].Finally, more recent works have implemented optimization approaches to predict the shear strength of FRP-RC elements.For example, Nehdi et al. [17] and Ebid et al. [25] produced equations that include optimal parameters to minimize errors.Similarly, other studies have focused on machine learning [26] and deep learning [27], creating models based on large datasets to account for a wide variety of parameters.These models, although very accurate for predictive use, do not appear to have immediate application for design purposes.
It is well known that the post-cracking behavior of beams reinforced with FRP bars differs from that of beams reinforced with traditional steel bars [28].This is essentially due to the linear-elastic behavior of FRP bars, their anisotropy, reduced axial stiffness [29], and their bond behavior [30,31].The FRP-RC beams exhibit larger and deeper cracks compared to steel-RC beams.While there is substantial agreement on bending capacity in the technical literature, the expressions for shear capacity are rather conservative and produce scattered results [32].For this reason, this research focuses on the shear behavior of GFRP-RC beams.The two sample beams designed herein are characterized by a limited stirrups amount and two salient bending reinforcement ratios.A highlight of the scope and aim of the paper is therefore presented here: -Experimental tests are carried out on beams with a reduced stirrups amount and with different bending reinforcements (under-reinforced and overreinforced), --ACI, AFGC, and CNR standards expressions are applied to the case study presented here to assess their predictive capabilities.It's important to note that, since the number of stirrups falls below the minimum allowable, both prototypes mentioned should be analyzed as beams without explicit shear reinforcement.The question is whether these models can accurately predict the experimental results.-The position [cot(h) = 1] in ACI and CNR, typically assumed when considering concrete strength in conjunction with stirrups strength, along with the ''strain limitation'' (referred to in this work as coefficient a E ), is compared to the experimental results; -A parametric analysis of the expression of ACI, AFGC, and CNR is carried out providing enhancing insight into the current performance requirements, especially the strain limitation in stirrups, -A fem modeling is tuned on the experimental test carried out to contribute to a better understanding of the shear behavior of FRP-RC prototypes.
Specifically, the two main types of failure obtained (FRP rupture and concrete crushing) are investigated considering the sectional approach of four selected North American and European standards; analytic shear capacity expressions of the ACI 440.1 R15 [1] and ACI 440-22 [2], AFGC [3] and CNR, [4] are examined and compared to the outcomes of experimental tests.

Capacity models
International standards offer multiple expressions to assess the capacity of FRP RC beams.Fib Bulletin 40 [33] provides a comprehensive discussion of the shear capacity of concrete beams reinforced with FRP bars.In ''FRP Reinforcement of RC Structures,'' the authors extensively review key contributions in this field.They highlight how formulations from various international standards, such as the Japanese JSCE, British design codes BS8110, American ACI, Canadian CSA, and Italian CNR, initially tailored for ordinary steel-reinforced concrete, were modified to account for the unique rheological behavior of FRP bars, resulting in current expressions.Notably, Guadagnini et al. introduced a design approach that suggested increasing the deformation limit of stirrups to 0.0045 for both shear and flexural reinforcements (originally assumed to be 0.002, corresponding to steel yielding).Today, this recommendation has been incorporated into some current standards, with ACI and AFGC now proposing a limit of 0.005.Some other interesting remarks are worthy of attention: the approach of variable strut angle, the only shear design method adopted in Eurocode 2, thus ignoring the concrete contribution for members with shear reinforcements, is adopted from the French AFGC standard joined with a deformability limit of 0.005.From the other side, FIB Bulletin 40 recommends the simplified fixed strut angle, with cot(h) = 1.0, in the case the concrete strength is considered together with transverse reinforcements.
The different approaches were investigated by applying them to both the sample beams tested here.In this paper, the focus is on the failure caused by the combination of tension in the stirrups and the shear failure of the compressed concrete zone shown in Fig. 4. The Shear capacity of RC-FRP elements V is obtained by the sum of tangential stress acting on the concrete compressive section V c added to the contribution of the FRP stirrups in tension V f , as in Eq. ( 3).
As anticipated in the introduction, whereas North American and Italian shear capacity expression [1,2,4] considers the sum of V c and V f , The French AFGC [3] does not admit to summing together these two values but, unlike the other standards, takes into account the inclination of the strut h in V f .The reduced resistance of the FRP stirrups (f fr ) derives from the tensile strength of the straight bar, which is penalized to consider the bent effect, to limit the crack width, and to reduce the effects of fatigue in GFRP stirrups (especially for bridges).The expressions for f fr , V c, and V f are listed in Table 1.
To simplify the comparison of the equations in Table 1 the unified symbols indicated in the   4. It's worth noting that there are distinct differences in the percentile values of characteristic strength distributions between the American ACI and the European AFGC and CNR standards (''Clarification'' section ''Appendix'').The strength of the GFRP bars depends on the diameter as well.
In CNR V c is a function of s r , and s r = 0.25 f ct (being f ct = 0.3*fc^(2/3)), while in the other technical standard V c is a function of concrete compressive strength f c .The forthcoming ACI 440-22 proposes two different expressions for V c , depending on the longitudinal reinforcement ratio.V f is calculated as the min of f fr = a E E f and f fr = a b f f in ACI and AFGC.The term a E assumes the value 0.004 for ACI 440.1R-15 and 0.005 for ACI 440.11-22 and AFGC.The term a b , which introduces the weakening effect due to the bending, can be determined with Eq. (4) [34], in which r b is the bending radius and d b is the bar's diameter.The AFGC is the only one that introduces cot(h) in V f expression, according to Eurocode 2 [35] for ordinary RC members.The inclination of h is limited to 33°( cot(h) = 1.5) according to the indication of AFGC (with the scope to reduce the effect of fatigue in the stirrups) whereas for ACI and CNR cot(h) = 1.0 as also indicated in Table 1.
The Italian CNR expresses f fr recommending the use of experimental results or with a ''safety factor'' c f,U = 2, while there are no restrictions on strain limitation (a E ).
Regarding safety factors (s.f.): -The American approach considers a single coefficient (A \ 1) that affects nominal capacity, so the design value of shear capacity is obtained by the nominal value, calculated with the characteristics value of strength, multiplied by A. -The European approach, according to EC2 [35], considers several coefficients (c i [ 1) that affect strength, and thus the design value of shear capacity is obtained from the expression of Table 3 by dividing the respective characteristic strengths by c i .
The values of A and c are summarized in Table 2, where expected (e) and design (d) values of concrete compressive strength f c , shear strength-calculated according to Italian standard-s r , tensile strength of the straight FRP bars f f and tensile strength of the bent FRP bars f fr are mentioned (Table 3).
3 Experimental evidence

Design
The experimental activities are focused on four-point bending tests on beams with constant stirrups spacing equal to 0.8 times the height of the section h.The ongoing approaches on shear capacity are based on Eq. ( 3), where V c is essentially related to the concrete resistance and longitudinal reinforcement ratio.To evaluate the effect of the latter on the shear capacity, two different beams were designed with a crosssection of 200 9 400 mm, equal stirrups spacing, and two different longitudinal reinforcement ratios.
As is commonly known, the quantity of longitudinal reinforcement plays a critical role in determining two distinct types of bending failure.In beams with limited bending reinforcement, often referred to as under-reinforced (U-R), failure occurs primarily due to the yielding and collapse of the tensile bars.Conversely, in sections with a substantial amount of reinforcement, known as over-reinforced (O-R), failure is typically associated with concrete crushing.These dual-limit behaviors are depicted in Fig. 1, which also delineates the transition zone between U-R and O-R.The sample beams in question were intentionally designed to exhibit U-R and O-R behaviors.However, it's important to note that both of these specimens were engineered to induce shear failure rather than bending failure, as elucidated below.The mechanical properties of concrete and GFRP bars are listed in Table 4.For the concrete, four compression tests, four indirect tensile tests, and two tests for elastic modulus are carried out.The value of tensile strength calculated with the Italian standard ( §11.2.10.2NTC2018) [36] is also presented, being the same value used to calibrate the capacity equation in CNR [4].GFRP bar properties of straight bars were adopted from Kocaoz et al. [37], being in good agreement (but less prudential) with those reported by the Characteristics values, s.f.en expression according to the respective standard (Table 3) Fig. 1 Limit of reinforcement ratio q fb vs strength reduction factors [1] and beam sections manufacturer and in ASTM D7957 [38].The reduced strength of the bent bars is experimentally calculated on six stirrups specimens according to ASTM D7914/ D7914M-14.
O-R and U-R beams are based on ACI 440.1 R15 [1].In particular the balanced failure conditions are furnished by (5): where the symbols are listed in Notations.Equation (5) allows defining the so-called transition zone: when q f \ q fb the flexural failure is controlled by the reinforcement rupture, but when q f [ 1.4q fb, it is controlled by the concrete crushing.Two different flexural reinforcement ratios are selected (Fig. 1) to investigate their effect on shear capacity.
The two reinforcement ratios, smaller and larger than the balanced failure limit, are computed considering nominal diameters and are: Taken the flexural capacity of the U-R beam (M min ) and the shear arm of the beam being twice the height of the sections (e = 800 mm in Fig. 2), the shear capacity V, in order to obtain shear failure, must be: From this, the transverse reinforcement area A sw- = 62.3 mm 2 , with a spacing of 320 mm (A sw / s = 0.190 mm 2 / mm) is chosen for both beams as depicted in (Fig. 1).The shear reinforcement is the same for both the beams thus, the only difference in q f .It is observed that these values of A sw /s w are below the limits (Table 5) specified by the selected standards [1][2][3][4].The execution of the beams and the experimental tests were conducted at the Material Testing Laboratory of the University of Cagliari.
With #1 two arms stirrups the minimum spacings to be considered in the design, according to AFGC, is 172 mm.The spacing used in this experimental campaign is 1.86 times the minimum value required by AFGC and slightly higher than the value provided by CNR (310 mm).With this spacing (s = 320 mm) shear capacity should be calculated as the only

Experimental setup and test results
The test setup is described in Fig. 2. Six LVDT transducers were placed at sections C, M, and D, at the intrados of the beam, three on each side.The load (P) was applied using a hydraulic jack.Load application points were placed with a shear arm e = 800 mm (Fig. 3) having in a ratio e/d = 2.2.
The load P from the hydraulic jack was applied until the failure was reached.The relative V-d (V = P/ 2) curve is presented in Fig. 8.The displacement (d) in section M (the average from the front and rear displacements) is used in these plots.It should be noted that the shear crack at collapse intersected at least one stirrup (Fig. 4).A different inclination of the compressed strut for the two specimens is observed.The first crack (numbers from 1 to 3 in Fig. 3) presents an inclination of about 60°for the UR and 50°for the OR beam.Then, after the formation of progressive sub-horizontal cracking due to loss of adherence, the compressed strut re-orients itself involving the adjacent stirrups in the failure mechanism.The geometry of the systems limits the terms cot(h)Ád/s to the maximum value 2, being 2, the maximum number of stirrups that can be involved in the failure mechanism.
So, the initial diagonal cracks (1 in Fig. 3) are joined with a pseudo horizontal one (2 in Fig. 3, more developed in the case of OR, as expected) that highlights loss in adhesion of horizontal reinforcements.It should be noted that the shear crack at collapse intersected at least one stirrup (Fig. 4).The evidence that diagonal cracks are joined with a pseudo-horizontal one (more developed in the case of OR, as expected) highlights a loss in adhesion of horizontal reinforcements.For the O-R beam, after the opening of a large diagonal crack, collapse is obtained for the crushing of the compressive concrete area (Fig. 4).In the U-R beam, final failure occurred due to the rupture of the longitudinal bars, which happened with the complete cracking of the section, the stirrup failure, and 24 mm of deflections.In the case of U-R beams the final collapse is obtained with the failure of longitudinal reinforcement, which happened after the Fig. 3 Reinforcement of the two specimens and test setup.The red line highlights the path of the shear cracking and the numbers from 1 to 3 its evolution during the experimental test, involving two stirrups for O-R and one stirrup for the U-R specimen complete cracking of the section, stirrup failure (Fig. 4), and 45 mm of deflections (Fig. 8).In Fig. 5 the shear capacities V are calculated according to the mentioned standards.V e is the expected value, calculated without the safety coefficient and with the average value of strength, whereas V d is the design value, calculated with the safety coefficient (Table 2) and the characteristics values.V exp is the experimental value.It should also be highlighted that the use of f fr = a b Áf fb , with a b from Eq. (4) [34], furnished in this case a b = 0.7 with and f fr = 705 MPa, is close to the value experimentally determined.Table 4 shows a different ratio of V c /V, V f / V and V/V exp for the analyzed standard (Table 6).
The comparison between the bending capacity expression is presented in Fig. 6.The examined standards agree on the bending, showing a reduced variation of M. It is observed that the moment M for which shear failure occurred (green line in Fig. 2) is less than the expected value from the analytic formulation.This is consistent with the design assumptions/hypothesis to obtain shear failure for both specimens.The position of the y-axis of balanced failure is used in the case of U-R and the iterative procedure is used to calculate y for the O-R specimen.In both cases, the upper reinforcement is neglected in the calculation for safety.

Numerical model
The experimental campaign was studied with sophisticated 3D nonlinear finite element models using the programs STKO [39] and OpenSees [40]. Figure 5a shows the mesh discretization and the boundary conditions.The model uses the condition of symmetry to reduce the number of degrees of freedom.The beams are discretized with four node tetrahedron elements and the rebars with displacement beam columns with fiber sections.Concrete is modeled with the ASDConcrete3D constitutive low using an IMPL-EX integration scheme [41,42] and the longitudinal rebars and stirrups with a linear elastic material.The bond is modeled according to experimental data of G-FRP from Aiello et all.[30] that carried out bond tests on Glasspree type bars (Sireg, Italy); the same bars, and with the same surface finishing, were used in the experimental campaign.The material properties are those listed in Tab 4. The concrete tensile fracture energy is calculated using CEB FIB Model Code 2010 [43], and the compressive fracture energy is assumed to be 250 times the tensile fracture energy.The comparison between the numerical and experimental load-displacement curves shows very good agreement as shown in Fig. 6.The numerical failure mode of the two beams confirms the experimental findings (Fig. 5b and c).The numerical model shows that the O-R beam failed because of excessive stresses in the straight part of the stirrups and concrete crushing, and the U-R failed because of the longitudinal bar tensile failure [41] (Figs. 7, 8).  7.
Bending capacity M presents a reduced percentage deviation (up to 21% for U-R and from 4 to 23% for O-R).Shear capacity V presents a greater percentage deviation (from 9% up to 30 for U-R and from 9 to 23% for O-R).The more significant variation depends on the different assumptions on strength and the safety factors used by the different standards and is obtained in the case of design values.The greatest dispersion is found in the case of design, produced by the incidence of the safety coefficient and in the different assumptions in the calculation V c and V f .The variation of V f is significantly related to the different assumptions made for f fr (as also shown by the fact that V fe , calculated    4-presents a more contained variability).Specifically, for the expected value, the dispersion of V c is up to 27% (for U-R) and up to 22% for the V f (for U-R).A different scenario is obtained in the case where the design values are considered.The incremental longitudinal reinforcement ratio from UR to OR increases the height of the neutral y-axis position (compressed concrete section) which leads to an increase in V c strength.This is in good agreement with experimental evidence.Compared to a traditional steel reinforced section, with an equivalent bending reinforcement area, a section with FRP reinforcement presents a shallower neutral y-axis position because of the lower axial stiffness of FRP reinforcement.The crack width and extension are correspondingly greater.This element seems to encourage the use of FRP reinforcements joined to precompression.The technical standards analyzed consider the shearbending interaction differently.ACI 440 R.15 and ACI 440-22 consider it through the k(n fc , q f ) and k r (n fc , q f ) coefficient respectively, as shown in the ''Appendix''.Specifically, the incoming version of ACI 440-22 introduces the following changes (Table 1): • Two V c expressions are given, with the indication to take the maximum value.• k s coefficient is introduced (in line with the European approach).This penalizes the higher cross-section.
AFGC and CNR made q f explicit in the expression of V c .In AFGC, shear capacity depends on C rd , k, q f and n fs while in CNR it depends on k, q f and n fs .The different V c expressions of Table can be expressed as follows: where s usually depends on concrete compressive strength f c , on longitudinal reinforcement ratio q f , FRP modulus E f and effective depth d (Eq.9): Therefore, s takes the role of the average tension virtually acting on the area (b w •d), Eq. ( 10).
Considering the geometry of the sample beams of this paper s is plotted in Fig. 9 versus f c and q f .CNR has proved to be the least conservative technical standard and the one that most encourages   the use of more performative concrete.The same can be observed by analyzing the V c versus q f graph.In CNR V c depends on s r (f ct ) while in ACI and AFGC V c depends on f c .Here the value of f ct derived from the Italian standard [f ct = 0.3*f c ^(2/3)] was used, in agreement with the experimental calibration of CNR.It seems appropriate to perform further experimental investigations to improve the predictive reliability of V c .This is of remarkable interest especially for beams with a reduced shear reinforcement A sw /s, as for the sample beams of this work.

GFRP stirrups strength V f
While the ACI and AFGC limited stress in transverse reinforcement to reduce crack width, the CNR only posited a mechanical limitation (f fr = 0.5 f f ).This allows in CNR the use of fewer stirrups (and less restrictive limitation in stirrups spacing, Table 5).The f fr /f f ratio is graphed in Fig. 10  3 -f fr /f f ratio is calculated using average values for f f and E f bars, respectively.The strain limitation on f fr (Table 1) produces a quite different value of f fr /f f ratio, having in practical application different rates of strength use.The strength reduction of Fig. 10 is calculated using the bending coefficient a b of Eq. ( 5).For E f and f f average values from the literature are considered.
In the light of above findings, it is of significant interest to propose a synthesis approach between the two which is considered in the following paragraph.

Overall shear strength capacity
Looking at the histograms of Fig. 5 it appears that CNR produces the best estimation of the shear V exp, while AFGC and ACI underestimate it.Since V c and V f acted in parallel (Eq.11), a strain limitation on f fr is due to guarantee of the effective simultaneity of the two contributions of V c and V f .However, the current one (f fr = 0.004 or 0.005 E fb ) appears quite conservative for the specimens analyzed in this work.On the other hand, the experimental campaign on bent bars carried out by Imjai et al. in 2017 [44] shows that the resistance of bent bars varies from 39 to 80% of the resistance of the straight bars (with a bending ratio r b / d C 3, both for thermoplastic rather than for thermoset composites).So, the ratios from 18% up to 30% appear quite restrictive, in particular, if joined to the limitation h = 45°.A further release of the international standards should consider providing less scattered values both for V c and V f and a formally unitary analytical approach.Considering the limits above, the R-M approach with the current ones of ACI and CNR are compared.In R-M is excluded the limitation on the deformability of the stirrups since the term V c is neglected.These two approaches are indicated in the following with subscripts 1 and 2. Considering Eq. ( 6) for V c the expression V 1 and V 2 are: with f fr1 is the experimental resistance of stirrups that can be in general expressed as a ratio of the resistance of the straight bar f fr1 = a b f f .
with f fr2 = min (a E ÁE f ; a b Áf f ).Equations ( 11) and ( 13) are shoved in Fig. 11.The shear capacity V i is normalized concerning V c .The V i / V c -(A fw /s) lines are graphed considering in V c s = 0.6 MPa (for q f = 0.01) and s = 1.0MP for (q f- = 0.04), f c = 40 MPa.In V 1 a b = 0.5 (as in the approach of CNR) is chosen.The experimental value of a b is 0.7.
Matching Eqs. ( 11) and (12), neglecting the terms a E with respect to a b •cot(h), and explicating A fw /s is obtained: A fw s (A fw /s) * , graphically represented by the intersection of the yellow and the shades of blue lines, is the stirrups amount from which V 2 produced a smaller value concerning V 1 .It is also possible to calculate a defined stirrup amount that determines which one of the two methods is less conservative.(A fw /s) * in the considered experimental campaign assuming the value 0.118 mm 2 /mm in case of s = 0.6 MPa and 0.281 mm 2 /m in case of s = 1.0 MPa, being 0.196 mm 2 /mm the stirrups amount of the sample beams.The anticipated shear capacity of the specimens under consideration, as calculated using Eq. ( 11) with cot(h) = 1.5, assumes the value V e = 76.1 kN (V e / V exp = 82% for OR and 88% for UR specimens).It's noteworthy that this value is slightly higher than those obtained by ACI and AFGC for UR cases and nearly identical to those obtained for OR (Table 7).So, if A fw /s C (A fw /s) * the approach (1), although based on prudent values of f fr1 , is less conservative and may be used for better material utilization (even neglecting the V c terms).This is also allowed in practical cases by the corrosion resistance of FRP reinforcement that consents to defer the limitation on crack width to the serviceability limit states design scenarios, without implicating V c and f fr2 reduction at the Ultimate Limit State.Considering the minimum shear reinforcement of Table 2 (0.172 mm 2 /m) it turns out that approach (2) produces, in general, conservative results with respect to approach (1).This produces practical implications in structural design.

Conclusion
In this paper, the results of an experimental campaign carried out on GFRP-reinforced concrete beams, with a reduced amount of shear reinforcement, were compared with the shear capacity formulations of the American ACI (version of 2022 and previous), the French AFGC, and the Italian CNR.The comparison between the three analyzed standards has shown that: -The bending capacity expression M of the analyzed standards produced consistent results, with a reduced deviation, under the 4% in the case that the expected values are considered (and up to 23% in case of design value).-Despite the limited quantity of stirrups used in the current case studies, following standard guidelines, they should be analyzed as beams without explicit shear reinforcement.However, the experimental tests reveal a failure mechanism and a corresponding failure load that is characteristic of shear-reinforced beams.-ACI, AFGC, and CNR standards expressions are applied to the illustrated case of study, and it is shown that the current standard formulae always underestimate the shear capacity of the prototypes (min 25%, max 6%); -The shear capacity expression of the analyzed standards produced a deviation from 9% (in case of expected value) to 30% (in case of design values).
The design values of shear capacity V are between 9 and 30% for U-R and between 9 and 23% for O-R.The CNR code produces a major estimation of shear capacity both for the design and for expected values.-The calculation of V c is carried out with different formulas leading to scattered results (up to 27%).This indicates that further investigations should be pursued, particularly for beams without or with reduced shear reinforcement.
The parametric analysis carried out joined to the experimental evidence on the two cases of study, consent to highlights the following points: -Specimen failure modes and FEM models show that, after the first shear crack is formed, the strut will reorient (even with widely spaced stirrups and for under-reinforced beams) as long as an adequate bond length is ensured for the bending reinforcement.-Considering the inclination h of the compressed strut (as in AFGC), could improve the predictive ability of the V f , expression.In ACI V f is penalized by the reduction of f fr (f fr = 0.005 E f means f fr & 0.2 f fb in case of GFRP).Taking into account the inclination h of the compressed strut and following the physical-based R.M. approach (also confirmed from the experimental evidence), can be useful to obtain less conservative V f estimation.This can be useful, especially for prestressed structures and in the case of thermoplastic GFRP bars where the reduction of strength in stirrups due to bent effect is less relevant (and as a consequence, a strain limitation appears quite restrictive).-Assuming h = 45°as in ACI and CNR and subsequently performing reverse calculations for a E the obtained values fall within the range of 0.017-0.022,all of which are greater than 0.005.So, considering the data of the presented tests, a value of a E = 0.010, may provide precautionary but less conservative results.This finding is in line with the proposal of the precedent researchers [45][46][47] and should be evaluated by analyzing the data of a parametric experimental campaign.-It may be useful for structural optimization to provide a different approach depending on the percentage of shear reinforcement, based on the limit value of (A fw /s) * (Eq.14 in Sect.4.3).This is of particular interest in the case of prestressed structures (where h \ 45°can be affordably assumed).-FEM models with the programs SKTO developed by ASDEA and Opensees, tuned on experimental data, are able to accurately reproduce the experimental behavior and can be used in structural optimization.
This element represents an enhancing step between a better optimization of the material and a more physical and mechanical analytical expression.
The discussion of safety factors, which do vary among the different codes analyzed, falls outside the scope of this work.Instead, the proposal here is to consider adopting a distinct reduction coefficient that can more effectively account for the reduced resistance of GFRP stirrups (f fr ) in the studied case.With this approach, it becomes possible to reconsider the design capacity by utilizing Eqs. ( 11) and ( 13), aiming for material optimization as elaborated in Sect.4.3.
It's worth highlighting the significance of this research, as underscored by the observed variability in the analytical approaches presented here, especially concerning V f .In this regard, further experiments and parametric investigation are being conducted to explore the diverse shear resistance mechanisms in prototypes reinforced with varying percentages of longitudinal reinforcement and different stirrup area and spacing percentages.
Acknowledgements The authors would like to express their special thanks to SIREG GEOTECH (Italy) for providing technical data and the Glasspree GFRP bars used for testing and to the technical staff of the Laboratory of Materials and Construction of the University of Cagliari, and its spin-off Secured Solutions, for the special support in managing the experimental test.
Funding Open access funding provided by Universita `degli Studi di Cagliari within the CRUI-CARE Agreement.

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The strength of concrete is characterized by experimental tests.The strength G-FRP bars used are statistically characterized in [37] by furnishing the average (X a ) and the standard deviation (r).
From this, the characteristic values (X c ) are calculated.
X c corresponds to the 5.0th normal distribution for the European (Italian and French) standard and the 0.1th normal distribution for the North American Standard.So, being r the standard deviation value, according to the European approach: While, according to the North American approach: About the subscript can be useful to clarify: e expected (average material strength, no safety factors).n nominal -from ACI -(characteristic material strength, no safety factors).d design (characteristic material strength, and safety factors).

Equations
The expressions not explicitly illustrated in the paper can be found below.The value of k in ACI, AFGC, and CNR assumes a different expression: ACI 440-22 [2]

e
Eccentricity-horizontal distance by point of load and the axis of the restraints in the test setup E i Young modulus of the i-th material H Height of the section M Bending capacity n fs FRP/steel young modulus ratio = E f /E s n fc FRP/concrete young modulus ratio = E f /E c f c Concrete compressive strength f ct Concrete tensile strength, f ct = 0.3*fc ^(2/3) is the concrete tensile strength according to Italian Standard for Construction (NTC 2018) f f FRP tensile strength of straight bars f fr FRP reduced tensile strength of bent bars r b Bending radius of the FRP stirrups s Stirrups spacing s R Shear strength V Shear capacity of the section V c Shear capacity of the concrete compressive section V c,max Maximum shear capacity of the concrete compressive section due to the failure of the strut V f Shear capacity of the FRP stirrups V s Shear capacity of the steel stirrups a b Strength reduction factor for bending effect a E Strength reduction factor of elastic modulus E (also indicated as ''strain limitation'' in literature) b 1 c i Safety factor according to AFGC, CNR c f,A Safety factor for stirrups according to CNR DT203/2006 (assumed equal to 0.5) d Vertical displacement e cu Ultimate deformation of concrete in compression h Inclination of the strut k s Factor according to ACI 440.11-22 (see ''Equation'' section in ''Appendix'') q f A f ?/(b w d)-Longitudinal reinforcement ratio q fb Longitudinal reinforcement ratio (in tension) producing balanced failure s R 0.25f ct -Shear resistance according to CNR DT203/2006 s Average shear strength, Average shear strength of the concrete section calculating as V/(b w *d) A Safety factor according to ACI authors d Design Values, according to a certain standard, are based on characteristic values of strength with the application of safety factors.e Expected Values are based on the average value of strength without safety factors.Expected values are used to interpret experimental (exp) data.
Fico et al. proposed a model that considers strain limits for stirrups contribution (0.0035 for CFRP, 0.0070 for AFRP, and 0.0085 for GFRP).Oller et al. [16] developed a simple and rational model for the shear capacity of FRP-RC beam.The model is based on the coexistence of the following three mechanisms: shear force in the uncracked concrete chord, tensile strength along the crack length, and shear strength of the transverse reinforcements.The expression of Oller et al. is applied to a database of 112 shear tests on FRP-RC beams producing interesting results considering the mean value (1.08) and the relative coefficient of variation (19.5%).The paper also presents a wide comparison of the actual standards including, but not limited to, the model proposed by other authors like Fico et al.Hegger et al. and Nehdi et al.
, V c and V f contributions are not considered as acting together b a E assumes the value 0.004 in ACI 440.1R-15 and 0.005 in ACI 440.11-22 and AFGC Symbols in the table are unified according to the list of symbols presented above appendices were used.The average and characteristic values of the compressive strength of concrete f c, the tensile strength of concrete f t , and the shear strength of concrete s r are indicated in Table The first one corresponds to Under-Reinforced (U-R) section with two #4 bars (M13).The second one corresponds to Over-Reinforced (O-R) section with four #4 bars (M13) as tensile reinforcement.In both cases, the compression reinforcement consists of two #4 (M13) bars.Figure 1 also shows the s.f.U that is used with the nominal bending resistance (M n ) to obtain the design resistance (M d ) according to [1].U assumes two distinct values for U-R (0.55) and O-R (0.65) beams.

Fig. 2
Fig. 2 Layout of the test and main dimension of the sample beams (mm)

Fig. 4
Fig. 4 Failure of O-R (a, c) and U-R (b, d) beams and detail of the respective intrados (c and d).Red arrows and circles indicate the failed stirrups

Fig. 5
Fig. 5 Shear capacity for U-R section, Expected (a) and Design (b) values, and for O-R section, Expected (c) and Design (d) values.The green line represents experimental results

Fig. 6
Fig. 6 Bending capacity of U-R (a) and O-R (b) beam.The horizontal line represents the bending value corresponding to the shear collapse

Fig. 7
Fig. 7 Numerical model, mesh discretization and boundary conditions (a), comparison of numerical and experimental failure mode of O-R (b) and U-R specimens (c)

Fig. 8 VTable 7
Fig. 8 V-d curve of the O-R and U-R beam specimens: experimental and numerical data

Fig. 9 s 5 Fig. 10 f
Fig. 9 s versus f 0 c (a) and a versus q f (b).Expected values

Fig. 11
Fig.11V i /V c versus A sw /s in case of q f = 0.01 (a) and q f = 0.04 (b)

Table 3
Expected and design values of strength Expected Design f c Average value without s.fCharacteristics values and s.f.according to the respective standard s r 0.25 f cta without s.f0.25 f ctd with s.f f f Average value without s.fCharacteristics values and s.f.according to the respective standard f fr Average value of the resistance of the stirrups (