Finite element analysis of flexural behavior in ultra-high-strength concrete beams with BFRP and hybrid BFRP-steel reinforcement

Abstract

Three-dimensional finite-element (FE) simulations were developed to assess the flexural response of ultra-high-strength concrete (UHSC) beams reinforced with basalt-FRP (BFRP) bars and with BFRP–steel hybrids. The models were calibrated against published tests and reproduced midspan force–deflection curves, reinforcement strains, and crack development with good fidelity. A parametric study examined (i) the BFRP reinforcement ratio relative to a balanced level and (ii) the BFRP-to-steel area ratio in hybrid sections. Results show that increasing the BFRP ratio raises ultimate flexural capacity but reduces deformability and energy dissipation. Hybrid reinforcement recovers inelastic reserve while preserving the durability advantages of FRP, yielding a more favorable balance between strength and serviceability. A yield-anchored modified ductility index tailored for hybrid sections is formulated to quantify post-yield rotational capacity. The findings provide design guidance for proportioning UHSC beams in conventional and aggressive environments: BFRP reinforcement ratios of ρ_f/ρ_fb ≥ 1.4 are suitable where strength and durability dominate, whereas hybrid layouts are recommended when controlled deflection and ductility are also required.

Introduction

Fiber-reinforced polymer (FRP) bars are increasingly used as a substitute for steel in reinforced-concrete members because they are immune to corrosion, possess high specific tensile strength, and perform well in harsh environments^1–5. FRP reinforcement is being actively explored for application in deep beams, compression members (CFFTs), and beam–column joints, as well as in recycled and seawater–sea-sand concretes^6–10. Among available FRP types, basalt FRP (BFRP) offers an attractive compromise: it provides better chemical stability than E-glass FRP at a comparable cost, and it is substantially less expensive than carbon FRP while maintaining satisfactory mechanical capacity^11,12. A recognized drawback, however, is the essentially linear-elastic response up to rupture and the comparatively low modulus, which can result in larger service deflections and crack widths and a sudden, non-ductile failure mode.

Material-level strategies to mitigate this brittleness include hybrid rebars that combine different FRP types or incorporate an FRP jacket over a steel core^13–15. These composite bars can deliver higher stiffness, improved post-yield behavior, and steel corrosion protection, while FRP–FRP hybrids can be engineered for pseudo-ductility to delay catastrophic failure.

Ultra-high-strength concrete (UHSC) has also gained prominence for its exceptional compressive strength, enhanced durability, and dense microstructure^16,17, making it attractive for long-span bridges, high-rise buildings, and infrastructure in severe environments. However, UHSC’s inherent brittleness often requires large volumes of steel reinforcement, leading to congestion, construction difficulties, and honeycombing risk. Integrating FRP reinforcement into UHSC can enhance corrosion resistance and extend service life, yet UHSC’s high stiffness and strength significantly influence cracking, serviceability, and reinforcement–concrete interaction.

Despite extensive research on FRP-reinforced beams and on UHSC performance separately, there is limited understanding of how BFRP reinforcement ratio and hybridization with steel affect the flexural response, serviceability, and ductility of UHSC beams. Addressing this gap is essential for developing design approaches that exploit UHSC’s material advantages while overcoming its brittleness through advanced reinforcement strategies.

A common means of improving the behavior of members reinforced with FRP bars is to increase the BFRP reinforcement ratio—consistent with ACI 440.1R guidance—to restrain deflection and cracking, although serviceability often remains the governing design limit¹⁸. At the section level, hybrid reinforcement that combines FRP and steel exploits the complementary benefits of both materials: steel contributes stiffness and ductility, whereas FRP provides corrosion resistance and high tensile strength^19,20. Experimental and analytical studies report that such hybrid RC beams exhibit higher post-cracking stiffness, improved deformability, and better crack control than beams reinforced only with FRP bars^21–23.

Ductility is a key performance metric for structural safety and resilience. In conventional steel-reinforced concrete beams, it is commonly quantified as the deformation at ultimate capacity normalized by the deformation at first yield of the steel reinforcement²⁴, yielding a non-dimensional measure that may be expressed in terms of curvature, rotation, or displacement. FRP RC beams, lacking a yield point, require alternative indices such as energy-based measures, deformation ratios at specific loads, or pseudo-yield points from bilinear approximations³. In hybrid FRP–steel beams, the steel yields first, enabling the use of conventional ductility indices while capturing FRP’s role in sustaining loads beyond yield. Such indices are critical for assessing serviceability, energy dissipation, and failure warning capacity in both FRP-only and hybrid designs.

Finite-element (FE) models were developed to evaluate the bending response of ultra-high-strength concrete beams reinforced with BFRP bars and with BFRP–steel hybrids. The parametric program assessed the influence of BFRP reinforcement ratio, the extent of steel substitution, and reinforcement configuration on flexural capacity and ductility, providing guidance for the design of durable high-performance members for both conventional and corrosive environments.

Finite element modeling

Three-dimensional finite-element (FE) models (Fig. 1) were developed for UHSC beams reinforced with BFRP bars, with and without partial steel substitution, to quantify the effects of reinforcement ratio and BFRP–steel hybridization. The numerical framework was verified against published test data. Simulations were executed in ATENA 3D Engineering v5.9²⁵, using element formulations, constitutive laws, and mesh densities selected to reproduce the governing response mechanisms. To reduce computational cost, only half of each beam was modeled by exploiting geometric symmetry; the symmetry plane was restrained against out-of-plane motion (u_x = 0). A roller support was modeled by applying a vertical displacement constraint (u_z=0) along the midline of the support steel plate to allow for beam-end rotation, rather than constraining the entire surface. The load was applied vertically (z-axis) with the displacement control mode at the nodes located in the middle of the loading steel plates, corresponding to the experimental loading points. This approach was selected because it has been shown in the literature^26,27 to better reproduce the complete load-deflection behavior, including the post-peak softening response, compared to force-controlled methods. This is particularly important for capturing the failure mechanisms in both ductile (steel-reinforced) and brittle (FRP-reinforced) beams. The specific magnitude of the prescribed displacement was not a pre-defined single value but was applied incrementally until structural failure occurred. This is inherent to the nonlinear analysis procedure employed a Newton–Raphson solution scheme with convergence checked at each displacement increment. Elements in both the concrete and steel meshes were set to a nominal size of 20–25 mm.

Fig. 1.

Typical 3D finite element model.

Concrete modeling

In this study, concrete was modeled using the 20-node isoparametric brick element, as illustrated in Fig. 1. This element is a higher-order, three-dimensional finite element designed to model complex stress states in nonlinear concrete structures. It employs Gaussian integration to compute stiffness and internal force matrices with high numerical accuracy. Typically, a 3 × 3 × 3 integration scheme (27 Gauss points) is used, which is particularly effective for nonlinear analyses involving advanced material models. This integration approach ensures accurate capture of stress and strain distributions within the element, especially in regions with steep gradients due to cracking, crushing, or other localized nonlinearities²⁸. The nonlinear behavior of concrete was simulated using the internal material model named “CC3DNonLinCementitious2”, which is part of the ATENA Engineering software suite developed by Cervenka Consulting. This constitutive model combines both fracture mechanics and plasticity theory to capture the complex nonlinear behavior of concrete underloading. The fracture part of the model is based on a smeared crack approach utilizing the Rankine failure criterion to determine crack initiation. For modeling concrete crushing under compressive stresses, plastic behavior is represented using the Menétrey-Willam failure surface, which is a three-parameter failure criterion particularly suited for concrete and similar quasi-brittle materials²⁹. This fracture-plastic approach allows simultaneous simulation of tensile cracking and compressive crushing in a unified framework. The shear behavior of cracked concrete is modeled using the Modified Compression Field Theory (MCFT), originally proposed by Vecchio and Collins³⁰. This approach accounts for the interaction between the normal and shear stresses in cracked regions and allows for realistic modeling of post-cracking shear resistance. In addition, the model considers compressive strength reduction in directions parallel to crack planes, reflecting the degradation of material properties after cracking. Another significant feature of the CC3DNonLinCementitious2 model is the inclusion of tension stiffening, which represents the contribution of the concrete between cracks to the overall stiffness of reinforced concrete members. This effect is critical for capturing the post-cracking load-deformation behavior, particularly in tension-dominated regions.

The key parameters for the concrete constitutive model were defined directly within ATENA, with the default values generated based on the concrete’s cube compressive strength (). The following specific parameters and criteria were used:

• The cylinder compressive strength was calculated as .

• The mean tensile strength was initially set to .

• The initial elastic modulus was defined as .

• Poisson’s ratio was set to .

• The critical compressive displacement for softening was .

• The compressive strength reduction factor in cracked concrete was .

• The plastic strain at compressive strength was .

• The fracture energy was .

• Exponential tension stiffening based on the fracture energy was implemented.

Given the significant influence of tensile strength on the cracking load and the overall structural stiffness, a lower-bound value was adopted to ensure a conservative and realistic response. In accordance with Eurocode 2, which defines the characteristic tensile strength () as the lower 5% fractile, the tensile strength in the model was taken as . Finally, a crack shear stiffness factor of was used, and aggregate interlock was enabled within the constitutive model.

Reinforcement modeling

In this study, reinforcement bars, including longitudinal reinforcement and stirrups, were modeled using discrete truss (1D bar) elements embedded within the three-dimensional concrete elements. This discrete approach is advantageous for accurately capturing the mechanical response of individual reinforcing elements, especially in regions with concentrated stresses such as beam steps, load introduction zones, and supports. FRP and steel reinforcements were assumed to be perfectly bonded to the concrete, with no slip at the interface²⁸. The truss element used in ATENA is a one-dimensional finite element designed to carry axial forces only, without bending or shear deformation. Each element connects two nodes and simulates axial tension or compression based on the defined cross-sectional area and constitutive law. This makes the truss element particularly suitable for modeling both steel and FRP reinforcement, which primarily act in tension or compression in reinforced concrete members.

For steel reinforcement, a bilinear elasto-plastic material model with strain hardening was used. The stress-strain relationship was defined by the modulus of elasticity (E_s), yield strength (f_y), and hardening modulus, reflecting the steel’s post-yield behavior. Default hardening parameters from ATENA’s material database were used unless experimental data were available. This model permits stress to increase after yielding, effectively simulating the ductile behavior of conventional steel reinforcement³¹. For FRP rebars, a linear elastic–brittle material model was adopted, as FRP materials exhibit no yielding and fail suddenly once their ultimate tensile strength is reached. The constitutive behavior was defined by the modulus of elasticity (E_f) and the ultimate (f_fu) tensile strength, based on manufacturer data or established literature. In ATENA, this behavior can be modeled using a linear stress-strain law with failure at a prescribed strain limit. Since FRP is anisotropic and generally brittle, no strain hardening was included.

Modeling of loading and support plates

In the finite element model, both the support and loading plates were represented using steel plates with a width of 50 mm, extending across the full width of the beam to ensure uniform stress distribution. To allow beam-end rotation and better reflect actual structural behavior, vertical displacement constraints were applied only along the midline of the support plate, rather than across its entire surface. The loading plate was modeled similarly, with prescribed vertical displacement applied at its center to simulate the experimental loading conditions. Both plates were assigned an elastic modulus of 200 GPa and a Poisson’s ratio of 0.3, corresponding to typical structural steel. The interfaces between the plates and the concrete beam were not assumed to be perfectly bonded. Instead, contact elements were introduced to simulate potential sliding. A frictional contact model was employed with a normal stiffness of 5 N/mm³ and a coefficient of friction of 0.4, representing common steel–concrete interface conditions^32,33. This modeling strategy allowed for relative slip under load while maintaining numerical stability and physical realism.

Validation of the analytical model

A total of six specimens were selected from various literature sources to validate the numerical analysis. Three of these specimens were cast using ultra-high-strength concrete and reinforced with varying reinforcement ratios³⁴. The remaining three were reinforced with different types of FRP and fabricated using concrete of varying strengths. All beams were tested under four-point bending with differing simply supported clear spans. Some specimens included stirrups and compression reinforcement in the shear span region only. Table 1 provides detailed information on the selected specimens. Properties of the tensile reinforcement of the modeled beams are listed in Table 2. Additional details can be found in the referenced sources^3,35,36.

Table 1.

Details of specimens used for validation.

Spec.	References	fcu` (MPa)	Concrete dimensions (mm)		Reinforcement
			L./CL./Sh.*	Cross-section	Tensile reinf.	Comp. reinf.	Stirrups
B2-NF	Hasgul et al.34	133	2500/2100/700	150 × 250	2Φ20-St.	2Φ8-St. †	Φ8@100 mm-U
B3-NF	Hasgul et al.34	135	2500/2100/700	150 × 250	2Φ24-St.	2Φ8-St. †	Φ8@100 mm-U
B4-NF	Hasgul et al.34	146	2500/2100/700	150 × 250	4Φ20-St.	2Φ8-St. †	Φ8@100 mm-U
B-30	Wang et al.35	36.26	2600/2000/600	200 × 250	5Φ25-G.	2Φ8-G.	Φ8/10@60 mm††
P4C1	Wang & Belarbi3	56.47	2032/1828/609	178 × 229	2#4-C.	--	#3@89 mm-U
B2	Qu et al.36	30.95	2100/1800/600	180 × 250	4Φ12.7-G.	2Φ10-St. †	Φ10@100 mm

L./CL./Sh. ^* = Total length/ Clear span/ Shear span; st. = Steel; G. = GFRP; C. = CFRP; U = U-shape; ^† = Located in shear span only; ^†† = Arranged alternately.

Table 2.

Mechanical properties of the tensile reinforcement Bars.

Specimen	References	Tensile Reinf.	Type	Yield Strength (Mpa)	Tensile/Ultimate strength (Mpa)	Modulus of elasticity (Gpa)
B2-NF	Hasgul et al.34	2Φ20-St.	Steel	462	654	200*
B3-NF	Hasgul et al.34	2Φ24-St.	Steel	491	606	200*
B4-NF	Hasgul et al.34	4Φ20-St.	Steel	468	661	200*
B-30	Wang et al.35	5Φ25-G.	GFRP	--	884	43.3
P4C1	Wang & Belarbi3	2#4-C.	CFRP	--	2069	124
B2	Qu et al.36	4Φ12.7-G.	GFRP	--	782	45.0

^* Commonly adopted value for the elastic modulus of steel.

The load–mid-span deflection curves in Fig. 2 illustrate the experimental and finite element (FE) behavior of both UHSC-steel and FRP-reinforced concrete specimens. For the UHSC-steel specimens Fig. 2a, the FE models capture the initial stiffness, peak load, and post-peak softening behavior with high accuracy. A slightly stiffer post-cracking response in the FE results compared to the experimental curves may be attributed to the perfect bond assumption adopted at the concrete–steel interface in the numerical model³⁷. The numerical curves exhibit close alignment with the experimental data across all three specimens, indicating that the model effectively simulates both the elastic and plastic responses, as well as the ductile behavior associated with steel reinforcement.

Fig. 2.

Load vs. mid-span deflection curves, test and FE prediction.

In contrast, the FRP-reinforced specimens, Fig. 2b, show a noticeably different structural response due to the linear-elastic nature of FRP materials until failure. The numerical predictions generally follow the experimental trends well, particularly in the elastic range. However, the FE models slightly overestimate the stiffness in some specimens, such as B-30 and B2, possibly due to challenges in accurately modeling bond-slip effects, concrete-FRP interaction, or local cracking phenomena. Despite these differences, the FE curves remain within an acceptable range, demonstrating the model’s ability to reasonably simulate the behavior of FRP-reinforced members with different reinforcement types and concrete strengths.

Figure 3 presents the mid-span concrete and tensile reinforcement strains for the investigated specimens. In some subfigures, finite element (FE) results are shown without corresponding experimental curves due to the absence of such data in the literature. It is evident that the concrete strain in all six specimens-both experimental and numerical-exceeded the theoretical ultimate compressive strain of 0.003. The load–concrete strain response of the FE models closely follows the experimental trends, demonstrating good agreement. In the case of tensile reinforcement, the FE-predicted steel strains continued to rise beyond the experimental values up to failure. This discrepancy may be attributed to the limited capacity of strain gauges commonly used in experimental setups. Similarly, concrete strains in the FE simulations tended to be slightly higher than their experimental counterparts. Both the experimental and FE steel-reinforced specimens exhibited a distinct plastic plateau in the tensile strain response after yielding, reflecting typical ductile behavior. In contrast, the FRP-reinforced specimens showed a linear-elastic response up to failure, as expected for brittle reinforcement materials. The FRP strains remained below their ultimate capacity in both experimental and FE results, indicating an over-reinforced failure mode governed by concrete crushing rather than FRP rupture. Overall, the FE model effectively captures the strain development in both the compression and tension zones under flexural loading. Figure 3 highlights the model’s ability to simulate key structural responses, including concrete crushing and steel yielding, thus validating its predictive capability.

Fig. 3.

Concrete and Tensile Reinforcement Strains at Mid-Span for the Tested Specimens. (a) Concrete Strain in UHSC-Steel Reinforced Specimens (b) Steel Strain in UHSC-Steel Reinforced Specimens (c) Concrete Strain in FRP-reinforced specimens (d) FRP Rebar Strain in FRP-reinforced Specimens.

Figure 4 presents the cracking patterns observed in specimens B2-NF, B3-NF, B4-NF, and B-30 together with the corresponding predictions from the finite-element (FE) models. The cracking patterns for specimens P4C1 and B2 are not shown because their experimental failure modes were not reported in the literature. Overall, the agreement between experiment and simulation is strong for the investigated specimens. In both the laboratory tests and the FE simulations of B2-NF, B3-NF, and B4-NF, distinct flexural cracks formed along the span, with higher crack densities near mid-span—consistent with flexural behavior under four-point bending. The FE model not only reproduces the global crack layout but also captures localized concrete crushing at the top surface beneath the loading point, coinciding with the region of maximum compressive stress noted experimentally. For specimen B-30, which was over-reinforced with GFRP bars, digital image correlation (DIC) was used to document crack propagation at the pure-bending region; the simulated and measured crack formations show close correspondence. These observations indicate that the model realistically represents the initiation and progression of flexural cracking as well as concrete crushing.

Fig. 4.

Experiments versus FE model predictions of concrete cracking in: (a) B2-NF, (b) B3-NF, (c) B4-NF, (d) B-30.

Table 3 presents a comparison between experimental and finite element (FE) results for load capacity and maximum deflection across six specimens. For the UHSC-steel specimens, the FE model demonstrated strong agreement with the experimental results, with load capacity differences ranging from 3.73% to 7.08% and deflection discrepancies below 2.12%, indicating reliable prediction of both strength and stiffness. The FRP-reinforced specimens also showed good correlation, with load capacity differences remaining within 2.26% to 3.62% for most cases and deflection deviations generally under 8.6%. These results highlight the model’s robustness in capturing the behavior of both conventional UHSC-steel beams and more complex FRP-reinforced systems with varying material properties.

Table 3.

Comparison of experimental and numerical results for the validated Specimens.

Specimen	Load capacity (kN)			Maximum deflection (mm)
	Exp.	Num.	Diff. (%)	Exp.	Num.	Diff. (%)
B2-NF	187.2	174.9	6.57	52.13	52.30	0.33
B3-NF	275.3	255.8	7.08	53.65	53.76	0.21
B4-NF	318.9	307.0	3.73	35.01	42.42	2.12
B-30	185.6	181.4	2.26	22.48	20.55	8.58
P4C1	167.8	169.0	0.72	30.0	31.50	5.00
B2	146.3	141.0	3.62	28.62	27.10	5.31

The validation results confirm the accuracy and reliability of the developed finite element model in predicting the flexural behavior of UHSC beams with varying reinforcement ratios. The numerical simulations closely matched the experimental results in terms of ultimate load, mid-span deflection, strain development, and cracking patterns. Overall, the model demonstrated a strong capability to replicate both global structural responses and localized failure mechanisms, reinforcing its suitability for advanced nonlinear analysis.

Parametric study

The primary objective of this study is to optimize the flexural performance of UHSC beams through longitudinal-reinforcement hybridization. Using a validated finite-element (FE) model, the author conducted a comprehensive parametric study to identify effective configurations of longitudinal tensile reinforcement for UHSC members. The analysis focuses on tensile-reinforcement variables-namely the FRP reinforcement ratio (ρ_f) and the basalt-FRP-to-steel hybridization ratio (A_f/A_s). Transverse reinforcement is kept constant (to satisfy shear capacity) and is not treated as a study parameter. In the shear spans, 8 mm steel stirrups at 80 mm spacing were used in all beams. Two 12 mm steel bars were kept constant as the top (compression) reinforcement, while the bottom (tension) reinforcement varied by specimen. All specimens had a total length of 2500 mm, a clear span of 2100 mm, and a rectangular cross-section of 150 mm (width) × 250 mm (depth). Beams were tested in four-point bending with a shear span of a = 700 mm. The concrete compressive strength was set to f’c = 113 MPa, corresponding to the reference beam B2-NF from the validated experimental program of Hasgul et al.³⁴. The BFRP bars were modeled with a modulus of 48 GPa and an ultimate tensile strength of 1100 MPa, consistent with the material properties reported by Hussein et al.¹¹. Steel reinforcement, where present, was assigned a yield strength of 462 MPa and an elastic modulus of 200 GPa, as characterized in Hasgul et al.³⁴.

FRP reinforcement ratio

This section investigates the influence of the longitudinal reinforcement ratio, ρ_f = A_f/(b⋅d), of basalt fiber-reinforced polymer (BFRP) bars on the flexural performance of UHSC-FRP beams. The balanced reinforcement ratio, ρ_fb​, which represents the condition at which FRP rupture and concrete crushing occur simultaneously, was calculated based on sectional force equilibrium in accordance with ACI 440.1R¹⁸, Eq. (1).

1

where f_fu​ and E_f denote the ultimate tensile strength and elastic modulus of the FRP bars, respectively, and β₁ is calculated as in steel-reinforced sections.

Five reinforcement configurations were analyzed based on the reinforcement ratio ρ_f/ρ_fb, including two under-reinforced beams (ρ_f/ρ_fb = 0.6 and 0.8) and three over-reinforced beams (ρ_f/ρ_fb = 1.4, 2.0, and 3.0). According to ACI 440.1R¹⁸, FRP-reinforced concrete beams should be designed as over-reinforced sections with ρ_f/ρ_fb ≥ 1.4 to ensure a more ductile failure mode governed by concrete crushing. Therefore, the beam with ρ_f/ρ_fb = 1.4 is used as the reference specimen in this study.

Figure 5 and Table 4 summarize the numerical results, including load-deflection curves, ultimate capacity, post-cracking stiffness, and service load performance.

Fig. 5.

Effect of FRP reinforcement ratio on the behavior of FRP-UHSC beams: (a) load mid-span deflection; (b) ultimate load & ultimate defection; (c) post-cracking stiffness & service load; (d) service load.

Table 4.

Summary of numerical results on the effect of reinforcement ratio on the behavior of FRP-RC Beams.

Specimen	ρf/ρfb	Ultimate load (kN)	Max. def. (mm)	Max. FRP stress (Mpa)	Post-cracking stiffness (kN/mm)	Service Load (kN)	Failure mode
ρf/ρfb-3	3	266	50.9	657	4.87	46.6	Concrete crushing
ρf/ρfb-2	2	226	56.6	779	3.65	40.5	Concrete crushing
ρf/ρfb-1.4	1.4	193	64	1017	2.73	34	Concrete crushing
ρf/ρfb-0.8	0.8	123	59.2	1060	1.77	28.6	FRP rupture
ρf/ρfb-0.6	0.6	96	55.5	1060	1.42	25.6	FRP rupture

As shown in Fig. 5a, FRP-reinforced UHSC beams exhibited a bilinear load-deflection response comparable to that of beams with normal-strength concrete. The response includes a linear elastic segment before cracking, followed by a segment of reduced stiffness after cracking. The pre-cracking behavior was nearly identical across all beams. Once cracking occurred, a sudden deflection jump appeared—less pronounced with higher reinforcement ratios—followed by a continued increase in deflection at reduced stiffness.

Figure 5b illustrates that increasing ρ_f/ρ_fb leads to higher ultimate load capacity, although the rate of increase diminishes beyond the balanced ratio. For over-reinforced beams, increasing ρ_f/ρ_fb from 1.4 to 2.0 (a 43% increase) resulted in a 17% gain in ultimate load. However, a further increase of 50% (from 2.0 to 3.0) led to only a 20.8% gain. In contrast, for under-reinforced beams, increasing ρ_f/ρ_fb from 0.6 to 0.8—a relatively modest 33.3% increase—produced a substantial 28% improvement in ultimate load. These findings highlight the more efficient reinforcement utilization in under-reinforced beams, where the structural response is more sensitive to changes in reinforcement ratio. This diminishing return in strength suggests that an upper limit for the reinforcement ratio should be established to optimize material usage and cost efficiency. FRP stress further reinforces this conclusion. For ρ_f/ρ_fb = 1.4, the FRP bars reached 92% of their tensile capacity, while for ρ_f/ρ_fb = 3.0, only 60% was utilized, indicating underuse of the material at higher ratios.

In terms of deflection behavior, increasing the reinforcement ratio above the balanced value reduced the maximum deflection, as shown in Fig. 5b. This trend aligns with traditional behavior observed in steel-reinforced concrete sections.

The slope of the post-cracking portion of the load–deflection curve—referred to as post-cracking stiffness—is clearly influenced by the reinforcement ratio (Fig. 5a and c). This stiffness increases almost linearly with ρ_f/ρ_fb. Compared to the reference beam (ρ_f/ρ_fb = 1.4), the post-cracking stiffness increased by 78% for the ρ_f/ρ_fb = 3.0 beam, while it decreased by 48% for the ρ_f/ρ_fb = 0.6 beam.

In this study, the service load was defined as the load corresponding to a mid-span deflection reaching the ACI-prescribed limit of span/360. Based on this criterion, the service load demonstrated a strong dependency on the reinforcement ratio (Fig. 5c and d), increasing nearly linearly with ρ_f/ρ_fb. Compared to the reference beam, the service loads of the ρ_f/ρ_fb = 0.6 and 0.8 beams were 25% and 16% lower, respectively, while those of the ρ_f/ρ_fb = 2.0 and 3.0 beams were 19% and 37% higher, respectively. It is also worth noting that, for the over-reinforced beams investigated in this study, the service load did not exceed 18% of their ultimate load, while for the under-reinforced beams, it remained below 27%. These values are noticeably lower than the typical ranges reported in the literature, which estimate service loads to fall between 25% and 35% of the nominal strength^4,39. These results emphasize that BFRP-reinforced beam design is largely governed by serviceability rather than ultimate strength. Excessive deflections are a common issue in practice, particularly in steel-reinforced members. If conditions allow, introducing an initial camber during construction can help mitigate excessive deflections under service loads³⁸.

Using the strain-compatibility concept, the predicted nominal capacity (M_n) of each beam was determined as per ACI 440.1R¹⁸ using Eqs. (2) & (3).

2

3

Where β₁ is the concrete stress-block factor, and the maximum concrete strain ε_cu is taken as 0.003. A_f represents the cross-sectional area of the FRP reinforcement in mm², and d is the distance from the centroid of the tension reinforcement to the extreme compression surface of the concrete beam, in mm. f_f denotes the stress in the FRP bars (in MPa), while E_f is the elastic modulus of the FRP bars, also expressed in MPa. f_fu refers to the design tensile strength of the FRP bars in MPa. b is the beam width in mm, and f′c is the specified compressive strength of concrete in MPa. The variable c represents the distance from the extreme compression fiber to the neutral axis, in mm.

Figure 5b shows a comparison between the predicted ultimate loads obtained using ACI equations and those derived from the FE simulation. Both methods exhibited a consistent trend of increasing ultimate load with higher reinforcement ratios. The predictions were closely aligned for under-reinforced beams, while slight differences were observed in the over-reinforced range. The percentage differences in ultimate load between the two methods for ρ_f/ρ_fb = 3.0, 2.0, 1.4, 0.8, and 0.6 were 8.65%, 9.29%, 9.33%, 1.63%, and 5.21%, respectively. An average difference of 6.82% indicates generally good agreement between the ACI-based predictions and the FE simulation results.

Due to the lower modulus of elasticity, brittle-elastic behavior, and distinct bond characteristics of FRP bars, ACI 440.1R requires the use of a direct method of deflection control. This method employs a modified expression for the effective moment of inertia, Ie, as given by Eq. (4), to account for the reduced tension stiffening in FRP-reinforced members compared to steel-reinforced beams.

4

where M_cr is the cracking moment; M_a is the service moment; I_g is the gross moment of inertia; and I_cr is moment of inertia of transformed cracked section. β_d is a reduction coefficient which accounts for the bond properties and modulus of elasticity of FRP bars.

Figure 5.b presents a comparison between the maximum mid-span deflections obtained using the ACI 440.1R direct method and those predicted by finite element (FE) analysis. Both methods exhibit similar trends. In the under-reinforced region, the predicted deflections show close agreement. However, in the over-reinforced region, the discrepancy between the two methods increases significantly. The figure also shows that deflection increases with the reinforcement ratio in the under-reinforced domain but begins to decrease as the ratio exceeds the balanced condition. This reduction in deformability in heavily reinforced members reflects the increasing brittleness of the system.

BFRP-steel hybridization ratio

A promising approach to mitigate the brittle failure of FRP–UHSC flexural members is the use of hybrid tensile reinforcement, which combines steel and FRP bars. Hybrid reinforcement leverages the complementary properties of both materials, addressing the limited ductility of FRP bars while enhancing beam ductility and stiffness through the superior plastic behavior of steel. In this section, In this study, seven numerical simulations were conducted using BFRP-to-steel hybridization ratios (A_f/A_s) of 0.25, 0.33, 0.5, 1.0, 2.0, 3.0, and 4.0, while maintaining a constant total reinforcement area across all beams. A fully steel-reinforced beam (B-S) and a fully BFRP-reinforced beam (B-F) were also analyzed and used as reference models for comparison. Table 5 presents the properties and key results of the investigated beams, including the effective reinforcement ratio ρ_{sf, s} and the mechanical reinforcement ratio ρ_{sf, f}, defined by Eqs. (5) and (6), respectively. These are expressed relative to the yielding reinforcement ratio ρ_{s, b} and the critical reinforcement ratio ρ_{f, b} as defined in Eqs. (7) and (8).

5

6

7

8

To ensure a desirable failure mode—characterized by steel yielding followed by concrete crushing without FRP rupture—the section should satisfy the conditions ρ_{sf, s} ≤ _{s, b} and ρ_{sf, f} ≥ ρ_{f, b}, recommended by Lei Pang et al.³⁹. The corresponding load–deflection responses for the simulated beams are illustrated in Fig. 5.

Figure 6 shows that hybrid-reinforced sections exhibit a tri-linear load–deflection response, characterized by three distinct stages: cracking, steel yielding, and either concrete crushing or FRP rupture, depending on the reinforcement configuration. This behavior differs from the typical bi-linear response observed in purely FRP-reinforced beams. At low hybridization ratios (A_f/A_s =0.25 to 0.5 Fig. 6a), the response is relatively ductile, with significant post-yield deformation before failure. However, as the FRP content increases beyond this range, the load capacity after the peak drops more sharply, indicating a more brittle failure mode. For higher hybridization ratios (A_f/A_s = 1 to 4 Fig. 6b), the beams demonstrate increased ultimate load capacity but decreased post-peak ductility. The curves become stiffer, and the failure occurs more abruptly, consistent with the linear-elastic behavior of FRP up to rupture. Although ρ_{sf, f} exceeded ρ_{f, b} for all the tested beams, some specimens failed by steel yielding followed by FRP rupture. This occurred specifically in beams where ρ_{sf, f} was less than 1.4 times ρ_{f, b} as confirmed by the maximum FRP stress values reported in Table 5. In all beams, steel yielding was observed, as the condition ρ_{sf, s} ≤ ρ_{s, b} was satisfied. Notably, Lei Pang et al.³⁹ recommended two reinforcement conditions to ensure a ductile failure mode: ρ_{sf, s} ≤ 0.75_{ρs, b} and ρ_{sf, f} ≥ 1.4ρ_{f, b}.

Fig. 6.

Effect of Hybrid Reinforcement Ratio on Beam Performance: (a) Under-reinforced beams; (b) Over-reinforced beams.

At the service deflection limit (span/360), increasing the hybrid reinforcement ratio A_f/A_s leads to a noticeable improvement in beam stiffness, thereby enhancing the service load capacity, as illustrated in Fig. 7. For example, at the allowable deflection limit, the service load increases from 40.5 kN for the fully FRP-reinforced beam (B-F) to 74.3 kN for the fully steel-reinforced beam (B-S), confirming the beneficial effect of higher steel content on flexural stiffness under service conditions.

Fig. 7.

Influence of BFRP-to-Steel Hybrid reinforcement ratio on beam behavior under service load conditions.

Figure 8 illustrates the variation of ultimate load, yield load, and service load with respect to the BFRP-to-steel hybrid reinforcement ratio (A_f/A_s). The figure reveals that the rate of change in all three load parameters is more pronounced for A_f/A_s less than 1.0, while the curves tend to flatten beyond A_f/A_s of approximately 2.0. This indicates that the reinforcement ratio has a greater influence on structural behavior in the under-reinforced range (i.e., steel-dominated configurations), where even modest changes in A_f/A_s result in significant improvements in yield and service loads.

Fig. 8.

Variation of Service, Yield, and Ultimate Loads with BFRP-to-Steel Hybrid Reinforcement Ratio.

Decreasing the hybrid reinforcement ratio (A_f/A_s) below 1.0 leads to a notable increase in both the service load and yield load, primarily due to the higher proportion of ductile steel reinforcement. However, this adjustment also results in a slight reduction in ultimate load capacity, as FRP—despite its brittle nature—contributes significantly to the load-bearing capacity at higher stress levels. These findings suggest that maximizing ultimate load alone may not be an optimal design strategy, especially if it compromises serviceability and ductility. Instead, a more desirable approach involves achieving higher service and yield loads with a moderate ultimate capacity and a ductile failure mode. Ahmed Hussain et al. reported that the design of hybrid reinforced concrete (HRC) beams is predominantly governed by serviceability limits due to the low axial stiffness of BFRP bars. Based on both experimental and numerical results, they concluded that the hybrid reinforcement ratio (A_f/A_s) should be maintained below 1.9 to satisfy service deflection requirements.

As shown in Fig. 8, increasing the hybrid reinforcement ratio (A_f/A_s) enhances the ultimate load capacity, which gradually approaches the capacity of the fully FRP-reinforced beam (B-F) as the ratio exceeds 2.0. However, this marginal gain in strength beyond A_f/A_s = 2.0 indicates a diminishing return, making such configurations structurally excessive and economically inefficient. This trend highlights the contribution of FRP bars to improving flexural strength at the ultimate limit state.

Conversely, both the service load and yield load decrease with increasing A_f/A_s, due to the reduction in steel content, which plays a critical role in controlling stiffness and yielding behavior under service conditions.

In summary, Fig. 8 underscores the importance of selecting a balanced hybrid reinforcement ratio that effectively leverages the strengths of both materials to achieve an optimal combination of strength, ductility, and serviceability. The results suggest that maintaining A_f/A_s below 1.0 is more desirable for ensuring high performance at the service stage while still achieving adequate ultimate strength.

Ductility of FRP and hybrid FRP–steel reinforced beams

Because FRP bars exhibit linear-elastic behavior up to rupture without a distinct yield point, the conventional ductility index used for steel-reinforced concrete beams (ultimate deflection divided by yielding deflection) is not directly applicable. To address this limitation, several alternative ductility and deformability models have been proposed^40–43. Three of the most widely cited are the models by Naaman and Jeong (1995)⁴⁰, Mufti et al. (1996)⁴¹, and Abdelrahman et al. (1995)⁴². The three models will be implemented in this study to evaluate the ductility of BFRP and BFRP-steel reinforced beams. Furthermore, a new ductility index tailored specifically for hybrid-reinforced beams is proposed herein.

Naaman and Jeong⁴⁰ proposed an energy-based ductility index calculated as the ratio of the total energy absorbed up to failure (E_tot) to the elastic energy (E_el) stored in the member. This method relies on integrating the area under the load–deflection (or moment–curvature) curve up to failure and separating the elastic portion from the total energy. The calculation of the ductility index (µ_e) based on Naaman and Jeong method is shown in Eq. (9).

9

Mufti et al.⁴¹ introduced a deformation-based index to assess the deformability of FRP-RC members by comparing the moment–curvature state at ultimate to a “service-level” state, typically defined at a concrete top-fiber strain of 0.001. The deformability factor, J, is calculated as:

10

where M_u and Ψ_u are the moment and curvature at ultimate, and M_c and Ψ_c are the moment and curvature at the service-level strain. For rectangular FRP-RC sections, the deformability index J should be greater than or equal to 4.0 to ensure a reasonable non-brittle failure mode¹¹.

Abdelrahman et al.⁴² proposed an equivalent deformability factor µ defined as the ratio between the actual deformation at the ultimate state of the member Δ_u and the equivalent deformation of the uncracked section corresponding to the same ultimate moment level Δ_I.

In hybrid FRP–steel beams, the onset of steel yielding provides a clear mechanical transition that FRP-only members inherently lack. Energy-based ductility measures (e.g., Naaman and Jeong⁴⁰ are well established but require numerical integration of the full load–deflection curve and are sensitive to local noise near peak and softening; moreover, they were originally formulated for FRP-prestressed/FRP-RC systems without a defined yield plateau. Deformability-based indices (e.g., Mufti et al.⁴¹) replace “yield” with a service-state reference (often ) and are widely cited for all-FRP members and codified in ACI 440.1R, but their value depends on the chosen service criterion rather than a material transition. Abdelrahman et al.⁴² further framed deformability for FRP-prestressed members by relating ultimate deformation to the elastic (uncracked) response at the same ultimate load, which is informative for serviceability but does not capture post-yield reserve that governs warning and energy absorption in hybrids. To capture the specific mechanics of hybrid sections—steel yielding followed by FRP-governed hardening—the author adopts a modified ductility index, Eq. (11),

11

where P_u is the ultimate load and P_y is the yielding load; and Δ_u, and Δ_y are the corresponding deflections respectively.

In hybrid beams, steel yields first while FRP remains linear; therefore, a yield-anchored index is most representative. Equation (11) reduces to the familiar steel RC ductility but penalizes high post-yield stiffness via , capturing the reduced inelastic reserve characteristic of FRP-rich hybrids. This directly reflects the design intent of hybridization reported in the hybrid-RC literature. From practical and metrological advantages, Eq. (11) uses four directly observable quantities and does not require curve integration (Eq. 9) nor a code-chosen service strain (Eq. 10); Because is identified from steel yielding—a distinct, instrumentally robust event— Eq. (11) is less sensitive to noise than energy-based indices.

Figure 9 illustrates the variation of the different ductility and deformability indices as a function of ρ_f/ρ_fb for FRP-reinforced beams and A_f/A_s for hybrid-reinforced beams.

Fig. 9.

Variation of ductility and deformability indices with: (a) fρ_f/ρ_fb for FRP-reinforced beams, and (b) A_f/A_s for hybrid-reinforced beams.

The results in Fig. 9 show that all indices generally decrease with increasing ρ_f/ρ_fb or A_f/A_s, except for the Mufti et al.⁴¹ index, which increases for under-reinforced sections due to its formulation for beams failing by concrete crushing²⁴. Figure 9a shows that increasing ρ_f/ρ_fb beyond 2 results in only minor changes in the deformability indices; therefore, increasing the reinforcement ratio beyond this value offers no significant benefit. Similarly for hybrid reinforced beams, increasing A_f/A_s more than 2 results in a minor change in the ductility and the ductility increases for under reinforced sections. The Abdelrahman et al.⁴² model generally produces higher values compared to the other indices, exceeding 10 for FRP-reinforced beams and 15 for hybrid-reinforced beams. Across the hybrid series, Eq. 11 monotonically tracks the expected decline in ductility as FRP content grows: the normalized ductility drops from 0.58 to 0.39 relative to the steel control, flagging the loss of inelastic reserve when A_f/A_s exceeds ≈ 1–2. Trends from Eq. 11 match the Naaman–Jeong⁴⁰ energy metric in ranking configurations (both decrease with A_f/A_s and plateau beyond ≈ 2), but Eq. 11 avoids the integration sensitivity near peak/softening. In general, the ductility of hybrid-reinforced beams, based on the proposed model, is lower than that of the reference steel beam (B-S). The ratios of the ductility index for A_f/A_s values of 0.25, 0.33, 0.5, 1, and 2 are 0.58, 0.55, 0.52, 43, and 0.39, respectively. These trends are specific to the parameters investigated in this study, and variations in factors such as concrete strength, FRP type, and beam dimensions may influence the results.

The proposed ductility index, µ_mod, provides an effective means of quantifying the post-yield ductility of hybrid BFRP-steel UHSC beams within the specific parametric framework of this study. It is important to note that the index’s formulation and the resulting values are specific to the material properties (UHSC, BFRP) and geometries analyzed herein. Therefore, a comprehensive investigation into the index’s sensitivity to a wider range of parameters—such as concrete strength, FRP type and modulus, and beam dimensions—is a prerequisite for its potential future adoption in design guidelines.

Design recommendations

Based on the finite element analysis and parametric study presented in this research, the following design recommendations are proposed for the flexural design of ultra-high-strength concrete (UHSC) beams reinforced with BFRP bars and hybrid BFRP-steel reinforcement. These guidelines aim to help designers balance the competing demands of strength, serviceability, and ductility.

Beams reinforced with BFRP bars only

For applications where corrosion resistance is paramount and some loss of deformability is acceptable, such as in aggressive environmental conditions:

• Reinforcement ratio: To prevent sudden brittle failure due to FRP rupture, sections should be designed as over-reinforced. A BFRP reinforcement ratio of ρ_f/ρ_fb ≥ 1.4 is recommended, as per ACI 440.1R. This ensures a failure mode governed by concrete crushing.

• Optimal upper limit: While increasing ρ_f/ρ_fb enhances ultimate capacity and service load stiffness, the rate of return diminishes significantly beyond ρ_f/ρ_fb = 2.0. Furthermore, FRP material utilization drops considerably (e.g., to ~ 60% at ρ_f/ρ_fb = 3.0). Therefore, exceeding this ratio is structurally inefficient and not economically justified. A practical upper limit of ρ_f/ρ_fb ≤ 2.0 is suggested.

• Ductility and deformability: For BFRP-only beams designed within the recommended range (1.4 ≤ ρ_f/ρ_fb ≤ 2.0), the deformability factor J as per Mufti et al.⁴¹ ranges from 8.87 to 9.38. This comfortably exceeds the referenced¹¹ recommendation of J ≥ 4.0, ensuring a sufficient deformation capacity and a non-brittle failure mode.

Beams with hybrid BFRP-steel reinforcement

For applications where a combination of durability, strength, and controlled ductility is required:

• Optimal hybridization ratio: To achieve a favorable balance of high service load, good yield capacity, and adequate ultimate strength with ductile behavior, it is recommended to maintain A_f/A_s ≤ 1.0. Configurations within this range leverage the stiffness of steel for serviceability and the strength of BFRP for ultimate capacity.

• Failure mode control: To ensure a ductile failure mode characterized by steel yielding followed by concrete crushing (without FRP rupture), the following conditions should be met: ρ_{sf, s} ≤ 0.75ρ_{s, b} (to ensure steel yields) ρ_{sf, f} ≥ 1.4ρ_{f, b} (to prevent FRP rupture before concrete crushing) These conditions provide a safety margin beyond the theoretical balanced failure.

• Avoiding excessive FRP content: Hybrid ratios exceeding A_f/A_s = 2.0 offer only marginal gains in ultimate strength while significantly compromising service load capacity, yield load, and ductility. Such configurations should generally be avoided as they approach the brittle characteristics of FRP-only beams.

• Ductility: To ensure adequate inelastic deformation capacity, the modified ductility index should satisfy µ_mod ≥ 0.5µ_steel, where µ_steel is the ductility index of an equivalent steel-reinforced beam. Maintaining a hybrid ratio of A_f/A_s ≤ 1.0 demonstrably achieves this criterion.

Conclusions

This study examined the flexural performance of UHSC beams reinforced with BFRP bars and with hybrid BFRP–steel configurations using validated 3D finite-element models. The main findings are:

• The developed FE model reproduced the experimental beams with small errors—ultimate capacities within ≈ 5–8% and deflections generally within ≤ 9%. In addition, ACI-440 strength predictions and the FE results agreed with an average difference of ≈ 6.8%, confirming model reliability.

• For BFRP-only beams, increasing the reinforcement ratio raises ultimate load but with diminishing returns beyond ~ 2.0, and reduces deformability; FRP stress utilization drops markedly at higher ratios. Consistent with ACI 440, over-reinforced design with a ratio of ≥ 1.4 is preferred to prevent brittle failure.

• For hybrid BFRP–steel beams, capacity increases with but gains are marginal beyond ~ 2.0, while service and yield loads decline; the best balance of strength, stiffness, and warning occurs near .

• A yield-anchored modified ductility index was formulated for hybrids. In our dataset, the normalized hybrid ductility declined from ~ 0.58 to ~ 0.39 as increased from 0.25 to 2.0, and the ranking of configurations aligned with the Naaman–Jeong energy metric while avoiding sensitivity to post-peak curve integration.

Author contributions

Nageh M. Ali: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing - Original Draft, Writing - Review & Editing, Visualization, and Supervision.

Funding

This work was supported by the Deanship of Postgraduate Studies and Scientific Research at Majmaah University (Project No. R-2025-2199).

Data availability

The data that support the findings of this study are available from the corresponding author, Dr. Nageh M. Ali (email: Nageh82@mu.edu.sa), upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.