Shear Design of Strain-Hardening Fiber-Reinforced Concrete Beams

The tensile ductility of strain-hardening fiber-reinforced concrete (SH-FRC), such as ultrahigh-performance concrete (UHPC), constitutes a key performance metric that distinguishes its behavior from that of conventional concrete. Reinforced with a high dosage of discontinuous steel fibers, typical UHPC-class materials exhibit tensile strength exceeding 7.0 MPa (1.0 ksi) sustained well into the postcracking stress–strain regime. This behavior is shown to contribute significantly to the shear capacity of beams because fibers transmit forces across diagonal cracks, thus supplementing or even replacing the transverse reinforcement in resisting shear forces (Voo et al. 2006; Voo and Foster 2008; Baby 2012; Kamal et al. 2014; Pourbaba et al. 2018; El-Helou and Graybeal 2019; Hemstapat et al. 2020). Recent experimental investigations on UHPC beams and panel elements showed that, given the high compressive strength of UHPC, the shear failure is often controlled by the tensile behavior rather than compressive failure (Voo et al. 2010; Baby et al. 2014; Ahmad et al. 2019; Yap 2020; El-Helou and Graybeal 2022b). Although the design provisions for the shear behavior of conventional concrete are well established (ACI 2019; AASHTO 2020), they are not suitable for UHPC because they do not account for the material’s improved mechanical performance, leading to inefficient designs and uneconomical structures. This lack of national design provisions and the relatively high cost of the material compared to conventional concrete have hindered UHPC’s large-scale application for structural members in the United States. However, when design principles suited for the distinctive material behavior of UHPC are used, reductions in traditional steel reinforcement, member dimensions, and self-weight of the structure can be realized. These features can provide financial benefits associated with the reduction in material usage, load demands, member depth, labor, construction time, equipment needs, and shipping and handling costs, along with potentially eliminating beam lines or intermediate supports.

This article presents an analytical model to describe the stress–strain behavior of a cracked SH-FRC membrane element subjected to shear and axial stresses. The model is based on the force equilibrium and strain compatibility equations established over a rectangular SH-FRC membrane element, similar to the modified compression-field theory (MCFT) developed for conventional concrete by Vecchio and Collins (1986), and integrates new constitutive relationships that address behaviors specific to SH-FRC. The analytical results of the shear model are compared to the experimental data obtained from tests on UHPC panels subjected to pure shear performed by Yap (2020). The model formulation for membrane elements is then used to predict the shear behavior of the web of beams and formulate a design method to predict the shear capacity of SH-FRC beams. The proposed design approach for SH-FRC beams parallels the derivations of the simplified MCFT (Bentz et al. 2006), which is currently implemented in the AASHTO Load and Resistance Factor Design Bridge Design Specifications (AASHTO LRFD BDS) (AASHTO 2020). The capability of the method to predict the shear capacity of prestressed and non-prestressed UHPC beams is validated against experimental test results obtained by El-Helou and Graybeal (2022b) and Baby et al. (2014).

The shear model described in this study applies to SH-FRC materials, with an emphasis on UHPC materials exhibiting certain key performance characteristics. For the purpose of this work, UHPC is defined as a portland cement–based material composed of an optimized gradation of granular constituents, very low water-to-cementitious materials ratio, and high percentage of discontinuous internal steel fiber reinforcement. The mechanical properties include a minimum compressive strength of 124 MPa (18.0 ksi) and a minimum cracking strength of 5.0 MPa (0.73 ksi) that is sustained through a localization strain of at least 0.0025. The compressive properties are obtained from compression testing according to ASTM C1856 (ASTM 2017). The tensile characteristics are obtained from uniaxial tension testing according to AASHTO T397 (AASHTO 2022), which is based on the method developed by Graybeal and Baby (2013). In this test method, the stress–strain response of UHPC is obtained from a direct capture of the tensile load and associated strain applied over a known cross section and gage length during a fixed-end uniaxial displacement-controlled test. Fig. 1(a) shows examples of experimental stress–strain plots typical for the UHPC materials considered in this study. These trends can be separated into two classes of idealized tensile characteristic responses: (1) relatively constant postcracking stress with increasing strain, represented as an elastoplastic model shown in Fig. 1(b); and (2) strain hardening with continuous increase in postcracking stress, represented as a bilinear model shown in Fig. 1(c). In Figs. 1(b and c), $E$ is the modulus of elasticity, $f_{t,cr}$ is the stress at the occurrence of the first crack, and $f_{t,loc}$ and ${\epsilon }_{t,loc}$ are the localization stress and corresponding strain at onset of softening, which is used as the endpoint of the idealized stress–strain curves. More information on determining these key tensile parameters from experimental direct tension results can be found in El-Helou et al. (2022) and Haber et al. (2018).

A few design recommendation documents using various design principles to predict the shear capacity of UHPC beams have been published in other countries. For instance, the French National Addition to the Eurocode 2 (AFNOR 2016), which provides rules for UHPC design, calculates the total shear capacity, $V_{Rd,F}$, from the superposition of three resistance terms provided by the UHPC: $V_{Rd,c}$, the transverse reinforcement, $V_{Rd,s}$, and the fibers, $V_{Rd,f}$, as shown in

Three empirical relationships for $V_{Rd,c}$ are recommended depending on whether the section is prestressed, non-prestressed with transverse reinforcement, or non-prestressed without transverse reinforcement. These formulae depend on the UHPC’s compressive strength, effective shear depth (defined as the lever arm of internal forces corresponding to the applied bending moment), axial force in the cross section, width of the web, and height of the section.

The fiber contribution to the shear strength, $V_{Rd,f}$, is defined by Eq. (2), in which ${\sigma }_{Rd,f}$ is the mean value of the postcracking strength perpendicular to the shear crack of inclination, $\theta$; $b_w$ is the width of the web; and $z$ is the effective shear depth. The value of ${\sigma }_{Rd,f}$ is obtained from uniaxial tensile stress–strain results derived from a specified inverse analysis of the results of a flexural prism test, and includes a partial safety factor and an orientation factor. The inclination angle of the principal compression stress with respect to the longitudinal axis, $\theta$, is determined from an elastic stress transformation evaluated at the center of gravity of the beam at the expected force demand at the ultimate limit state. The value of $\theta$ is required to be equal to or greater than 30°. The French shear design methodology for beams concludes by requiring that the stress in the compression field be lower than the compressive resistance limit and that the flexural reinforcement be able to accommodate the increase in tensile force due to the applied shear force

In Switzerland, the recommendations for UHPC structural design (SIA 2016) include one term to represent the UHPC’s contribution, $V_{Rd,U}$, instead of a UHPC term and a fiber term, as recommended by the French specification document. The total capacity of the beam, $V_{Rd,S}$, is calculated as

$V_{Rd,U}$ is calculated according to Eq. (4), in which $f_{Uted}$ and $f_{Utud}$ are the uniaxial elastic (cracking) tensile strength and ultimate tensile strength of the UHPC, respectively. The tensile parameters are determined according to a direct tension method or calculated from inverse analysis of the results of a flexural prism test. Here, $V_{Rd,s}$ is the shear force contribution of the transverse steel reinforcement, as follows:

The Swiss recommendation specifies a minimum inclination angle, $\theta$, of 30° for most cases, 25° if the web is subjected to considerable compressive stresses, and 40° if the web is subjected to considerable tensile stresses or plastic deformations of the flanges. Similar to the French specifications, the Swiss recommendations include a limit on the stresses in the compression field and on the tensile stress in the longitudinal reinforcement.

The shear design recommendations in Canada are published in Annex A8.1 of the Canadian Highway Bridge Design Code (CSA 2019) and are applicable only for tension-softening fiber-reinforced concrete (TS-FRC). A model for SH-FRC, such as UHPC, referred to as tension-hardening fiber-reinforced concrete in CSA S6:19 (CSA 2019), is not provided. The recommendation for TS-FRC is based on the work of Foster and Agarwal (2018), who proposed a new equation to the existing MCFT model to account for the fiber contribution to the shear resistance. The factored shear resistance, $V_r$, is defined in Eq. (5), in which $V_c$ is the concrete contribution (without fibers), $V_s$ is the transverse steel contribution (using the same equations as for conventional concrete), $V_{fib}$ is the fiber contribution term defined in Eq. (6), and $V_p$ is the component of the prestress force in the direction of the shear force.

In Eq. (6), ${\gamma }_F$ = fiber orientation factor; ${\varphi }_F$ = material resistance factor applied to the postcracking tensile strength; $f_w$ = postcracking tensile strength derived as function of crack width; $b_v$ = web width; and $d_v$ = effective shear depth.

Although the existing models for SH-FRC can predict aspects of shear behavior, they rely on empirical approximations for the critical shear angle, do not correlate shear capacity as a function of the postcracking performance characteristics of SH-FRC, and assume that transverse steel reinforcement, when included, will always yield at shear failure. Based on these observations, there is a need to develop a mechanics-based shear design model that accounts for the postcracking strain capacity of SH-FRC and captures its influence on the critical shear angle and the transverse steel stress at failure. The model proposed in this article addresses this need.

The proposed model predicts the relationship between the average axial and shear stresses as a function of the average axial and shear strains of a cracked SH-FRC rectangular membrane element. The membrane element is assumed to mimic a portion of a structure with uniform thickness, such as the web of a beam, shear walls, shell structures, and so forth. The model formulation is founded on the principles of strain compatibility, in which stresses over strained sections are calculated based on known mechanical stress–strain models and summed to establish equilibrium. The proposed derivations parallel the approach followed in the MCFT developed by Vecchio and Collins (1986) for conventional reinforced concrete while engaging new material constitutive relationships, including pre- and postcracking tensile response, to complete the model for SH-FRC.

The capability of the membrane element model to simulate shear behavior was demonstrated herein by comparing the analytical results to the experimental data of four UHPC panels tested in pure shear in a shell element tester at the University of Toronto (Yap 2020). The UHPC panels contained the same 2% volume fraction of steel hooked-end and straight fibers. They had a nominal width and length of 1,626 mm (64.0 in.) and a thickness of 200 mm (7.9 in.), and were reinforced with conventional steel bars with different reinforcement ratios in the longitudinal and transverse directions.

Because the membrane element was subjected to pure shear ($f_x=f_y=0$), the model equations can be simultaneously solved to predict the applied shear stress, $v$, and strains, ${\epsilon }_x$, ${\epsilon }_y$, and ${\gamma }_{xy}$, by selecting a range of possible values for ${\epsilon }_1$. In analytically modeling the panel tests of Yap (2020), the input parameters for the steel bars ($E_s$ and $f_y$) and compressed UHPC ($E$ and $f_c^{\prime }$) were taken as equal to the reported average values obtained from material characterization tests performed on companion specimens. The reduction factors accounting for biaxial effects, ${\alpha }_{b1}$ and ${\alpha }_{b2}$, were set equal to 0.5 during the portion of the behavior after cracking of the UHPC (${\epsilon }_1>{\epsilon }_{t,cr}$) for all panels. In the absence of uniaxial tensile parameters reported from appropriate direct tension test methods on companion specimens, the reported principal tensile stress and strain values measured during the tests at the occurrence of the first shear crack and peak load were used as input parameters for the tensile constitutive relationships ($f_{t,cr}$, $f_{t,loc}$, and ${\epsilon }_{t,loc}$). By using these parameters, the analysis presented herein serves to demonstrate the appropriateness of the proposed modeling framework, not the individual tensile parameters used as inputs. It should be noted that Yap (2020) reported the tensile parameters obtained from an inverse analysis on the results of flexural prism tests companion to each panel except one panel (i.e., YS5), in which direct tension tests were carried out on dog-bone–shaped specimens in lieu of flexural prisms. The reported tensile parameters from prism tests were not used in the shear simulations because these values depend on the assumptions used in the inverse analysis technique, which has been shown to result in stress and strain values greater than those obtained from a direct tension test (Baby et al. 2012; Yap 2020). Also, results from the direct tension tests were not used because the tensile strains were not measured over a gauge length with a uniform strain field. The input parameters used in each panel simulation can be found in Yap (2020).

The trends of the simulated prepeak shear stress, $v$, versus shear strain, ${\gamma }_{xy}$, of two of the four panels are plotted atop their corresponding experimental data in Fig. 5 (with the unloading and reloading cycles omitted for clarity) to demonstrate the capability of the model to capture the overall shear behavior of UHPC panels. The simulated-to-experimental peak shear stress ratio for all four simulated panels ranged between 0.98 and 1.01, providing yet another verification that the proposed model can capture UHPC shear stress capacity when appropriate values of the uniaxial tensile properties are used as input parameters. Of note, the shear failures of all the panels corresponded to the first mode in which the peak shear stress occurred at the peak principal tensile stress. In addition, for the membrane element reinforced in both directions (i.e., YS1), the reported average strains along the $y$-direction, ${\epsilon }_y$, were lower than the yield strain of the steel bars in the same direction.

The experimental program of Yap (2020) included an additional panel not containing any conventional steel reinforcement. This panel was not incorporated in the simulation work in the current study because it failed at the onset of cracking and did not exhibit any postcracking strain capacity. This result may indicate that the material tested did not exhibit a strain-hardening behavior under direct tension (which is outside the scope of the proposed model). Additional experimental tests on unreinforced UHPC panels are required to assess the behavior. However, the current formulation is recommended for use only in structures in which at least one of the axial directions contains discrete steel reinforcement. In beam design, the proposed formulation is applicable for beams with unreinforced or reinforced webs only when steel flexural reinforcements are included along the longitudinal direction to resist flexural tensile loads, as recommended by El-Helou and Graybeal (2022a).

Similar to the MCFT for conventional concrete, the most accurate approach for SH-FRC beam shear analysis would be to incorporate the proposed material model to represent elements of the beam discretized into cross-sectional layers (Vecchio and Collins 1988) or into an array of biaxial membrane elements (Vecchio 1989). However, design formulations commonly use simplifications that enable the predictive model to be codified. This study derived a UHPC model for beams using a simplified approach, similar to the one adopted for the simplified MCFT (SMCFT) derived by Bentz et al. (2006) and incorporated into the AASHTO LRFD BDS (AASHTO 2020). The approach predicts the shear capacity of the beam section using one membrane element located within the web of the beam, as shown in Figs. 6(a and b), assuming that shear stress is constant along the effective shear depth of the beam, $d_v$, as defined in AASHTO LRFD BDS (AASHTO 2020). To simplify the presentation of the derivations, the beam’s shear reinforcement bars are positioned to align with the vertical direction ($y$-direction) and the prestressing force is assumed to align with the longitudinal direction ($x$-direction), as is the case, for example, with a straight tendon profile.

Following the assumptions adopted in the SMCFT (Bentz et al. 2006) and because the membrane element is used to model an area away from regions of the beam directly under concentrated loads, as shown in Figs. 6(a and b), the clamping stresses are assumed to be negligibly small ($f_y=0$). If the beam is reinforced with only vertical steel bars (i.e., ${\rho }_{sx}=0$, ${\rho }_{sy}={\rho }_v$, and $f_{sy}=f_s$), Eq. (11) can be rearranged to obtain the shear stress in the beam aswhere ${\rho }_v=A_v/b_{w}s$ is the transverse steel reinforcement ratio; $f_s$ = axial stress in the transverse steel; $A_v$ = area of transverse steel reinforcement within a distance $s$ in the longitudinal direction; and $b_w$ = minimum thickness of the beam’s web. Eq. (11) can also be derived from the free-body diagram of Fig. 6(c) by summing forces in the $y$-direction. The nominal shear capacity of the beam, $V_n$, can be calculated by multiplying the terms of Eq. (17) by the cross-sectional area contributing to the shear resistance, $b_w{d}_v$, and by setting $f_1$ equal to the UHPC localization stress, $f_{t,loc}$, which can be written asin whichand

The capability of the proposed model to predict the shear capacity of prestressed and non-prestressed UHPC beams is evaluated by comparing the model predictions to experimental data from a total of 13 beam shear tests performed at FHWA-TFHRC (El-Helou and Graybeal 2022b) and at the French Institute of Science and Technology for Transport, Development and Networks (IFSTTAR) (Baby 2012; Baby et al. 2014). The investigated parameters include the type of UHPC product, the number of prestressing strands or flexural reinforcing bars, and the presence and amount of transverse steel reinforcement in the beams. Relevant details of the experimental investigations at TFHRC and IFSTTAR can be found in the Appendix. The model simulations presented herein are considered validation analyses because they only relied on input mechanical property parameters obtained from material testing of specimens that accompanied each beam.

The predicted shear capacity for each girder was calculated by using the strength relationships of Eqs. (18)–(20) and the crack angle relationship of Eq. (24). The axial strain in the web, ${\epsilon }_x$, shown in Table 4, was taken as ${\epsilon }_s/2$, where ${\epsilon }_s$ is the strain in the tensile flexural reinforcement, determined according to Eq. (26) or Eq. (27). In determining ${\epsilon }_x$ and ${\epsilon }_s$: (1) the effective shear depth, $d_v$, was estimated by a strain compatibility analysis but not less than $0.9d_e$ or $0.72h$ (Tables 5 and 6), where $d_e$ is the depth of the centroid of tensile steel reinforcement [AASHTO LRFD BDS (AASHTO 2020)]; (2) the reduction factor for biaxial effects, ${\alpha }_{b1}$, was taken equal to 0.5; (3) $f_{po}$ was set equal to the strand jacking force [i.e., $0.7f_{pu}=\mathrm{1,303}\mathrm{MPa}$ or 189 ksi for girders tested at TFHRC, as recommended by AASHTO LRFD BDS (AASHTO 2020) for normal level of prestressing, and 1,133 MPa or 164 ksi for girders tested at IFSTTAR]; (4) $A_{ct}$ was taken equal to the area of UHPC within half of the section’s height on the flexural tensile side of the beam; (5) $E_s$ was set at zero because the girders did not contain any auxiliary steel reinforcement; (6) $E_p$ was taken equal to the reported values in each set of tests (i.e., 196.5 GPa or 28,500 ksi for girders tested at TFHRC and 195 GPa or 28,280 ksi for the prestressed girders tested at IFSTTAR); and (7) $N_u$ was taken as equal to zero.

The UHPC material parameters used in the shear model equations, namely $E$, $f_{t,cr}$, and ${\epsilon }_{t,loc}$, are shown in Table 4; they were obtained from uniaxial tests performed on cylindrical (for $E$) and prismatic (for $f_{t,cr}$, and ${\epsilon }_{t,loc}$,) specimens. The uniaxial tension specimens for the girders made at TFHRC were made from the same UHPC used to cast each of the girders, while the specimens for the girders tested at IFSTTAR were part of a parallel study on tensile behavior of UHPC that used the same UHPC type (Baby 2012; Graybeal and Baby 2013).

In determining the shear capacity of the tested girders, the shear demand, $V_u$, was set equal to the calculated capacity, $V_n$, obtained from Eq. (18), because the model is used to predict the shear failure of the beam. However, the demand moment at critical shear location, $M_u$, was calculated at a distance $d_v$ from the applied point load within the constant shear region. Because the value of ${\epsilon }_x$ depends on $V_n$, which also depends on ${\epsilon }_x$, iterations were performed to find the solution for $V_n$. The predicted results for ${\epsilon }_x$, $f_s$, $\theta$, and $V_n$ for each of the girders are presented in Table 4. The values of critical shear angle, ${\theta }_{simp}$, determined based on the simplified method, are also shown in Table 4 for comparison.

This article presents an analytical model for the analysis of an SH-FRC cracked membrane element subjected to shear and axial stresses. The capability of the model to capture the full shear stress–strain behavior is demonstrated by comparing the analytical results to experimental tests performed by Yap (2020) on UHPC panels subjected to pure shear. The membrane element is then used to develop a design method for the shear capacity of beams made with SH-FRC. The beam shear capacity is calculated from the contribution of the SH-FRC and the reinforcing steel as a function of the inclination angle of the critical crack with the longitudinal axis. The model accounts for the various parameters influencing the capacity, including the ultimate tensile stress, crack localization strain, axial strain in the web, section geometry, and transverse steel reinforcement ratio. The results of the design method for beams are in good agreement with the experimental data from 13 prestressed and non-prestressed beam shear tests performed by two different testing labs on four UHPC products. These results also show that the failure of UHPC components reinforced with conventional steel can occur before the yielding of the reinforcing bars.

The shear design formulation presented herein was developed to extend provisions of the AASHTO LRFD BDS (AASHTO 2020) to allow for design of UHPC components. The model was derived by combining engineering mechanics and concepts underlying the MCFT with constitutive relationships for UHPC, the most common type of SH-FRC, derived directly from material testing. As such, this work constitutes a fundamental step in ongoing efforts to develop UHPC structural design guidance.

The following symbols are used in this paper:

$A_{ct}$

area of UHPC within the half of the section’s height on the flexural tensile side of the beam, as defined in AASHTO (2020);

$A_{ps}$

total area of prestressing steel on the flexural tension side of the section;

$A_s$

total area of non-prestressed steel on the flexural tension side of the section;

$A_v$

area of transverse reinforcement within a distance $s$;

$a$

shear span of UHPC beams tested in shear;

$a_c$

longitudinal distance at midheight of the beam between the support and the location of the critical shear crack;

$a_g$

maximum size of aggregate in conventional concrete mix;

$b_v$

minimum thickness of the web of beams, as defined in CSA (2019);

$b_w$

minimum thickness of the web of beams;

$d_e$

depth of the centroid of tensile steel reinforcement, as defined in AASHTO (2020);

$d_v$

effective shear depth, as defined in CSA (2019) and AASHTO (2020);

$E$

modulus of elasticity of UHPC;

$E_{c,p}$

secant compression stiffness peak shear stress in the UHPC panels tested by Yap (2020);

$E_{c2}$

compression stiffness before cracking in the UHPC panels tested by Yap (2020);

$E_p$

modulus of elasticity of prestressing steel;

$E_s$

modulus of elasticity of steel reinforcement in the transverse or longitudinal directions of reinforced UHPC beams;

$f_c^{\prime }$

compressive strength of UHPC or conventional concrete;

$f_{cx}$

stress in concrete in $x$-direction;

$f_{cy}$

stress in concrete in $y$-direction;

$f_{po}$

a parameter taken as modulus of elasticity of prestressing tendons multiplied by the locked-in difference in strain between the prestressing tendons and the surrounding concrete, as defined in AASHTO (2020);

$f_{pu}$

specified ultimate tensile stress of prestressing steel;

$f_s$

stress in vertical steel reinforcement of UHPC beams ($y$-direction);

$f_{sx}$

stress in steel reinforcement in $x$-direction;

$f_{sxcr}$

local stress in steel reinforcement in $x$-direction at diagonal shear cracks;

$f_{sy}$

stress in steel reinforcement in $y$-direction;

$f_{sycr}$

local stress in steel reinforcement in $y$-direction at diagonal shear cracks;

$f_{t,cr}$

effective cracking strength, taken as the stress at the intercept between the uniaxial stress–strain curve and a line with a slope equal to the elastic modulus and a strain offset of 0.02%;

$f_{t,loc}$

localization stress when the uniaxial tensile stress is continuously decreasing with increasing strain;

$f_{Uted}$

design value of cracking strength of UHPC, as defined in SIA (2016);

$f_{Utud}$

design value of the ultimate tensile strength, as defined in SIA (2016);

$f_w$

postcracking tensile strength derived as function of crack width, as defined in CSA (2019);

$f_x$

stress in the $x$-direction;

$f_y$

stress in the $y$-direction;

$f_{yx}$

yield stress of steel in the $x$-direction;

$f_{yy}$

yield stress of steel in the $y$-direction;

$f_1$

principal tensile stress;

$f_2$

principal compressive stress;

$f_{2,u}$

principal compressive stress in the webs of UHPC beams at shear failure;

$h$

overall height of beam;

$M_u$

factored moment in the section;

$N_u$

factored axial force, taken as positive if tensile and negative if compression;

$s$

longitudinal spacing of transverse reinforcement;

$s_x$

spacing of diagonal shear cracks along the $x$-direction in conventional concrete;

$s_y$

spacing of diagonal shear cracks along the $y$-direction in conventional concrete;

$s_{\theta }$

crack spacing used to determine the crack opening in conventional concrete subjected to shear cracks;

$V_c$

shear capacity of concrete contribution without fibers, as defined in CSA (2019);

$V_{exp}$

experimental value of the shear capacity of UHPC beams tested in shear;

$V_{fib}$

shear capacity provided by the fibers, as defined in CSA (2019);

$V_n$

nominal shear resistance of UHPC beams at failure, calculated according to the proposed shear methodology;

$V_p$

component of prestress force in the direction of the shear force;

$V_{Rd,F}$

total shear capacity of UHPC beams, as defined in AFNOR (2016);

$V_{Rd,S}$

total shear capacity of UHPC beams, as defined in SIA (2016);

$V_{Rd,c}$

shear capacity provided by UHPC, as defined in AFNOR (2016);

$V_{Rd,f}$

shear capacity provided by the fibers, as defined in AFNOR (2016);

$V_{Rd,s}$

shear capacity provided by transverse reinforcement, as defined in AFNOR (2016) and SIA (2016);

$V_{Rd,U}$

shear capacity provided by the UHPC, as defined in SIA (2016);

$V_{\mathrm{UHPC}}$

shear resistance provided by the UHPC at failure;

$V_r$

factored shear resistance of UHPC beams;

$V_s$

shear resistance provided by the transverse reinforcement at failure;

$V_u$

factored shear force demand at the section of the beam under consideration;

$v$

shear stress in $xy$-plane;

$v_{ci}$

shear stress at diagonal cracks of conventional concrete;

$v_{exp,p}$

peak shear stress in $xy$-plane, as obtained from UHPC panels tested by Yap (2020);

$v_{\max }$

maximum shear stress at which crushing of the compression strut occurs;

$v_{sim,p}$

simulated peak shear stress in $xy$-plane using the proposed formulation of cracked UHPC membrane elements;

$w$

shear crack width of conventional concrete;

$z$

effective shear depth, as defined in AFNOR (2016) and SIA (2016);

${\alpha }_{b1}$

reduction factor to account for biaxial effects on the secant stiffness of compressed UHPC;

${\alpha }_{b2}$

reduction factor to account for biaxial effects on the compressive strength of UHPC;

${\epsilon }_{cu}$

uniaxial strain value corresponding to the compressive strength of UHPC;

${\epsilon }_s$

strain in longitudinal steel reinforcement of UHPC beams;

${\epsilon }_{t,cr}$

strain at effective cracking stress taken as $f_{t,cr}/E$;

${\epsilon }_{t,loc}$

localization strain when the uniaxial tensile stress is continuously decreasing with increasing strain;

${\epsilon }_x$

strain in the $x$-direction;

${\epsilon }_y$

strain in the $y$-direction;

${\epsilon }_1$

principal tensile strain;

${\epsilon }_2$

principal compressive strain;

${\gamma }_F$

fiber orientation factor, as defined in CSA (2019);

${\gamma }_{xy}$

shear strain in $xy$-plane;

${\varphi }_c$

resistance factor for axial resistance;

${\varphi }_F$

shear resistance factor applied to the postcracking tensile strength of TS-FRC, as defined in CSA (2019);

${\varphi }_f$

resistance factor for flexural resistance;

${\varphi }_v$

resistance factor for shear resistance;

${\rho }_{sx}$

steel reinforcement ratio in $x$-direction;

${\rho }_{sy}$

steel reinforcement ratio in $y$-direction;

${\rho }_v$

vertical steel reinforcement ratio in UHPC beams ($y$-direction);

${\sigma }_{Rd,f}$

mean value of the postcracking strength of UHPC, as defined in AFNOR (2016);

$\theta$

inclination of principal compressive stress with respect to longitudinal axis; inclination angle of critical shear crack in UHPC beams;

${\theta }_{exp}$

experimental value of the inclination of the critical shear crack with respect to the longitudinal axis of UHPC beams tested in shear; and

${\theta }_{simp}$

predicted value of the inclination of the critical crack using simplified method.

Some data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request. Some data, models, or code generated or used during the study are proprietary or confidential in nature and may only be provided with restrictions.

The research presented in this paper was funded by FHWA. The publication of this paper does not necessarily indicate approval or endorsement of the findings, opinions, conclusions, or recommendations either inferred or specifically expressed herein by FHWA or the United States Government. This research could not have been completed without the dedicated support of the technical professionals associated with the FHWA Structural Concrete Research Program.