Study on the flexural performance and deflection calculation method of UHPC-NC hybrid beams with different parameters

Abstract

Based on the excellent mechanical properties of ultra high performance concrete (UHPC) such as high strength and good toughness, this study proposed an ultra high performance concrete-normal concrete (UHPC-NC) hybrid beams, in which UHPC was used to replace part of the NC in the tension or compression zone along the beams. The flexural behavior of 4 UHPC-NC hybrid beams and 2 control beams at different UHPC height was investigated. Based on the flexural experiments of UHPC-NC hybrid beam, a finite element model was established using ABAQUS to investigate the effects of UHPC height, reinforcement ratio, NC compressive strength and UHPC tensile strength on the flexural behavior of the hybrid beams. In addition, the differences in flexural performance and force mechanisms between the hybrid beams and the control beams were compared. Finally, based on the immediate stiffness and effective cross-sectional moment of inertia of the hybrid beams, a model for the mid-span deflection calculation of the UHPC-NC hybrid beam at a reasonable UHPC height was proposed. The results show that the load carrying capacity, energy absorption capacity and stiffness of the hybrid beams increase with the UHPC height, while the ductility initially increases and then decreases. The UHPC height has a significant effect on the flexural performance of UHPC-NC hybrid beams. The reinforcement ratio has positively improved the load carrying capacity of the hybrid beams, although specimens with larger reinforcement ratios exhibit lower ductility. The mid-span deflection calculation model for UHPC-NC hybrid beams was proposed in this paper is in good agreement with the existing experimental results, the calculation process of the calculation model is relatively simple and can provide a theoretical basis for engineering design.

Keywords

UHPC-NC hybrid beams load-deflection curves flexural performance damage analysis deflection calculation

Introduction

In recent years, there has been an increasing focus on the development of building structures that are lightweight, have large spans, high load-bearing capacity, and enhanced durability, among other features. NC (Normal Concrete) is not suitable for many complex buildings due to its limitations, such as heavy weight, low strength, and poor crack resistance. UHPC (Ultra-High Performance Concrete) has been widely used in nuclear power plants, bridges, and other projects due to its exceptional qualities, including high bearing capacity (Liang et al., 2019a), good ductility (Kaan et al., 2019), and excellent durability. These qualities also make UHPC highly promising for reducing structural deadweight, streamlining reinforcement, enhancing structural durability, and optimizing assembly connections (Abdal et al., 2023; Wei et al., 2023; Xue et al., 2020; Zhou et al., 2018). However, the use of UHPC in practical engineering is limited by its high cost.

UHPC can form better bond strength with NC at the interface (Yang et al., 2019). The high corrosion resistance and low permeability of UHPC (Alkaysi et al., 2016; Matos et al., 2021) help better prevent structural deterioration and increase the service life of structures. An increasing number of researchers are combining UHPC with the bottom tensile layer of girders to develop hybrid girders that balance mechanical properties and cost. When UHPC was placed in the tensile zone of beams for four-point flexural tests by Liu et al. (2020) and Tayeh et al. (2013), it was found that the reinforcement primarily contributed to the flexural capacity of the UHPC-NC hybrid beams. There was no relative slippage and a stronger connection between UHPC and NC, demonstrating that the interface could meet real technical requirements and provide strong bond performance after chiseling treatment. The load-carrying capacity of the hybrid beams greatly increased, and the formation of mass cracking was prevented as the height of the UHPC in the tension zone increased (Yuan et al., 2022). Furthermore, research has shown that UHPC-NC hybrid beams exhibit significantly improved stiffness, flexural capacity, and ductility, with no noticeable decrease in ductility (Turker and Torun, 2020). Additionally, hybrid beams made with UHPC in both the compression and tension zones exhibit sufficient load-carrying capacity, stiffness, and ductility, but they have wider crack widths compared to pure UHPC beams (Yang et al., 2019). Moreover, the UHPC height played a crucial role in determining the stiffness of the hybrid beams, helping to inhibit the formation of local cracks and improving their durability (Zhou and Sheng, 2022).

Furthermore, many researchers have begun using UHPC as a concrete casting formwork as their research progressed. Li et al. (2019) proposed UHPC-NC hybrid beams, which consist of internal cast-in-place concrete and precast double I-beam UHPC slabs. By strengthening the tensile strength of UHPC and the main reinforcement of the beams, the ultimate load-carrying capacity of the hybrid beams can be greatly increased. Hybrid beams can also be made more cost-effective by increasing the area ratio of NC to UHPC (Zhang et al., 2022). Meanwhile, the yield load and ultimate load of UHPC-NC hybrid beams increased by approximately 10%, and the cracking load of beams without demoulding increased by nearly 50% compared to NC beams (Liang et al., 2019b). Useful theoretical calculations and analyses can be performed based on a large number of tests. For instance, the flexural moment-curvature relationship can be obtained through cross-section analysis using Open Sees (Yuan et al., 2022), UHPC-NC hybrid beams under flexure can be analyzed numerically (Zhu et al., 2022), the load-deflection curves of the hybrid beams can be derived from theory, and reasonable reinforcement ratios and high span ratios can be calculated to effectively improve the flexural capacity of UHPC-NC hybrid beams. Structures inevitably suffer damage as their service lives extend, and UHPC-reinforced damaged structures have emerged as a prominent area of contemporary research. Superior results have been achieved with reinforcement techniques such as the reinforced mortar layer (Alharthi et al., 2021), prestressed UHPC layer (Zhang et al., 2023a), and hybrid steel plate and UHPC (SP-UHPC) (Zhang et al., 2023b) for repairing damaged reinforced concrete (NC) beams. Therefore, UHPC-NC hybrid beams not only show good cost effectiveness, but have excellent mechanical properties, which can be widely used in practical projects such as reinforcing damaged structures and repairing buildings.

In summary, the addition of a UHPC to NC beams can significantly improve the flexural performance of hybrid beams in terms of load-carrying capacity, crack resistance, stiffness, and ductility. This indicates that UHPC-NC hybrid beams have a wide range of potential applications in building structures. This hybrid approach provides a new avenue for advancing the development and utilization of UHPC in construction. However, due to limitations in testing costs and technology, there are few test specimen parameters available in existing studies, which cannot systematically reflect the flexural performance of hybrid beams. As is well known, a sufficient number of experimental samples is necessary to study the flexural behavior of hybrid beams. A small number of experimental studies are often anecdotal, and the multiple influencing factors of UHPC-NC hybrid beams must be considered to improve the reliability and authority of research on their flexural performance. However, research on the influence of multiple factors on the flexural behavior of hybrid beams remains limited. Meanwhile, as a typical bending member, the deflection of UHPC-NC hybrid beams affects the normal use of the structure, so the mid-span deflection of the beams during normal use should also be emphasized. However, there are few studies on the deflection of this type of structure to guide the use in actual projects. Therefore, it is important to propose a convenient and accurate method for calculating the mid-span deflection of UHPC-NC hybrid beams. To address this, based on bending experiments of UHPC-NC hybrid beams, a finite element model of a UHPC-NC hybrid beam subjected to four-point bending was established using ABAQUS. The effects of UHPC height, reinforcement ratio, NC compressive strength, and UHPC tensile strength on the flexural performance of UHPC-NC hybrid beams were investigated. Additionally, the differences in force mechanisms between NC and UHPC beams are discussed. Finally, the mid-span deflection calculation model of UHPC-NC hybrid beams was established based on the immediate stiffness and effective cross-sectional moment of inertia of the hybrid beams, with a view to providing theoretical guidance for practical engineering.

Experimental study

Specimen detail

The cross-sectional width, height, and length of the test beams are 150 mm, 250 mm, and 2000 mm, respectively, with a clear span of 1800 mm and a purely curved area of 600 mm. Four UHPC-NC hybrid beams (UN-B-50, UN-B-100, UN-T-50, UN-T-100) and two control beams (N-1, U-1) were designed to study the effects of different UHPC heights on the cracking resistance, bearing capacity, crack distribution, and failure mode of UHPC-NC hybrid beams. All beams had the same reinforcement ratio (2Φ10 for compression reinforcement, 2Φ18 for tension reinforcement, and Φ8 for stirrups). The distance between the stirrups in the flexural shear section and the pure flexural section of the beam is 100 mm and 150 mm, respectively. The detailed size and section form are shown in Figure 1. In this study, UN-B indicates that UHPC is placed in the tension area at the bottom of the beam, and UN-T indicates that UHPC is placed in the compression zone at the top of the beam. The concrete for all specimens was C40, and the ultra-high-performance concrete was proportioned according to the research group’s guidelines (Wang, Liang and Shi, 2024). UHPC is mainly composed of silicate cement, silica fume, quartz powder, quartz sand, water reducing agent and steel fiber. The steel fiber is mixed with end straight fiber and end hook fiber, and the mixing ratio is 1:1, in which the length of end straight steel fiber and end hook steel fiber are 13 mm and 16 mm, and the diameter is 0.2 mm. The mechanical properties of NC and UHPC materials were obtained according to GB/T 50081-2019 (2019), T/CCPA 7-2018 (2018), and the mechanical properties of steel were determined according to GB/T228.1-2010 (2011). The mechanical properties of the materials and the design parameters for each specimen are shown in Table 1.

Figure 1.

Geometric dimensions of test specimens (unit: mm). (a) N-1 (b) UN-B-50 (c) UN-B-100, (d) UN-T-50 (e) UN-T-100 (f) U-1.

Table 1.

Characteristics of tested beams.

Specimens	h (mm)	NC(MPa)		UHPC(MPa)		H (mm)	Internal reinforcement
Specimens	h (mm)	f _cu	f _t	f _cu	f _t	H (mm)	Tensile	f_y (MPa)	Compression	f_y (MPa)
N-1	0	38.0	3.0	/	/	250	2Φ18	443	2Φ10	428
U-1	250	/	/	124.6	8.3	250	2Φ18	443	2Φ10	428
UN-B-50	50	38.0	3.0	124.6	8.3	250	2Φ18	443	2Φ10	428
UN-B-100	100	38.0	3.0	124.6	8.3	250	2Φ18	443	2Φ10	428
UN-T-50	50	38.0	3.0	124.6	8.3	250	2Φ18	443	2Φ10	428
UN-T-100	100	38.0	3.0	124.6	8.3	250	2Φ18	443	2Φ10	428

Note: H and h are hybrid beam height and UHPC height, respectively.

Figure 2 illustrates the steps involved in the preparation and upkeep of the beams. There are as follows: (a) the wood mold was filled with the reinforcement cage with strain gauge in the design position; (b) UHPC was poured into the mold to the appropriate design height, vibration compaction was applied to smooth out the surface, and then NC concrete is poured to create a hybrid beam; (c) The procedure complies with step (b); and (d) the test beams were left in an appropriate setting for a duration of 28 days, which they were anticipated to experiment.

Figure 2.

Preparing procedures of test specimens (a) Reinforcement cage into mold; (b) Pouring of sub-base UHPC; (c) Pouring the top UHPC; (d) Maintenance and demolding.

Test setup, instrumentation, and loading protocol

The four-point flexural tests on the hybrid beams were conducted using a YAW-5000 microcomputer-controlled electro-hydraulic servo pressure testing machine. The loading schematic, experimental setup, and reinforcement strain gauge arrangement are respectively illustrated in Figure 3(a)–(c). Strain and displacement data from both concrete and steel reinforcement were recorded using a TDS-602 static data acquisition system. All flexural performance evaluations were performed in accordance with GB/T 50152-2012 (2012). Prior to formal testing, preliminary pre-loading was implemented at a rate of 0.2 mm/min to verify proper instrumentation functionality. This pre-loading phase was carefully controlled not to exceed 5% of the predicted ultimate load or surpass the anticipated cracking threshold. The main test employed displacement-controlled loading at a rate of 0.5 mm/min, with termination criteria set at either 85% post-peak load degradation or severe structural failure. Throughout the experiment, loading rates were periodically adjusted as needed, while crack propagation patterns were systematically documented using high-intensity flashlight illumination and corresponding observational records.

Figure 3.

Test setup and measuring point arrangement. (a) Beam set-up and instrumentation (unit: mm). (b) Laboratory apparatus and instrumentation (c) Strain gauge arrangement.

Test results and discussion

Crack development and failure mode

Figure 4 illustrates the failure modes and crack progression across three characteristic stages (pre-cracking, crack propagation, and post-yield) of the hybrid beams. All cracks originated from the tensile zone at the beam bottom within the pure bending region. The specimens universally exhibited three distinct failure phases: initial cracking, stable crack development, and rapid crack extension prior to failure. During the initial cracking stage, UN-B hybrid beams demonstrated significantly more microcracks than UN-T counterparts. However, with UHPC positioned at the beam’s tensile face, UN-B specimens ultimately developed fewer macrocracks at failure compared to UN-T beams. This crack pattern correlation has been previously documented by Yin et al. (2017). Furthermore, Yuan et al. (2022) quantitatively established an inverse relationship between UHPC layer thickness and the number of dominant cracks observed.

Figure 4.

Crack development of hybrid beams. (a) N-1. (b) U-1. (c) UN-B-50. (d) UN-B-100. (e) UN-T-50 (f) UN-T-100.

In the tension zone, vertical cracks propagated and deepened as the tensile reinforcement in the beam yielded. This shifting of the neutral axis position upward gradually reduced the compression zone height. Ultimately, failure occurred through either crushing of the normal concrete (NC) layer in the compression zone or crushing of the UHPC layer. All test beams exhibited reinforcement-governed failure mechanisms. For UN-B-50 and UN-B-100 hybrid beams, the UHPC layer at the beam soffit effectively restrained crack initiation and propagation, with failure characterized by a single dominant through-thickness crack (Figure 4(c), and (d)). In contrast, UN-T-50 and UN-T-100 hybrid beams failed through the formation of multiple closely spaced primary cracks within the flexural region (Figure 4(e), and (f)). During continued midspan deformation under loading, the reduced compression area in the upper section attributable to UHPC’s superior compressive strength accelerated failure progression.

Crack resistance effect of UHPC

The upward expansion of cracks weakened with the height of the UHPC increased, the UHPC at the top of the beams significantly prevented crack penetration and the ductility of the hybrid beams was significantly increased. As shown in Figure 5, UHPC replaced a specific amount of NC has a significant effect on the creation and development of cracks. Both N-1 and U-1 beams formed through main cracks at the time of damage, the steel fibers significantly inhibited crack expansion when the UHPC was located at the bottom of the beam, which resulting in significantly smaller crack widths. All the steel fibers of the UN-B hybrid beams were pulled out, only a small amount of steel fibers of the UN-T hybrid beams was pulled out. The main reason is that the main crack in the UN-B hybrid beam was wider and the steel fibers were pulled out, while the cracks of the UN-T hybrid beams are smaller and the steel fibers are subjected to less force, which ensures that they are not pulled out.

Figure 5.

Damage detail of the test specimens. (a) N-1. (b) U-1. (c) UN-B-50. (d) UN-B-100. (e) UN-T-50. (f) UN-T-100.

Load-deflection curves

The load-deflection curves of the UHPC-NC hybrid beams and control beams are shown in Figure 6, the deflection of the beams is Δ = d₃-(d₁+d₅)/2, where d₁, d₃ and d₅ are the values of the displacement gauges D1, D3 and D5. In Figure 6, the N-1 beam exhibited a sudden load drop after reaching its ultimate load, while both the UHPC-NC hybrid beams and the U-1 beam demonstrated good ductility. The deflection of the UN-T hybrid beams was significantly larger than that of the UN-B hybrid beams, while the UN-B beams had better load-carrying capacity compared to the UN-T hybrid beams. The cracking load of the UN-B-50 and UN-B-100 beams increased by 84% and 131% compared to N-1 beams, respectively. On the other hand, the cracking load of the UN-T hybrid beam was nearly identical to that of the N-1 beam. The ultimate load of the UN-B-50, UN-B-100, UN-T-50, and UN-T-100 hybrid beams increased by 20.3%, 26.9%, 15.8%, and 16.8%, respectively, when the height of the UHPC was increased from 0 mm to 50 mm and 100 mm. The results indicate that placing the UHPC at the bottom of the beam is significantly more effective than placing it at the top in terms of increasing the ultimate load of the hybrid beams.

Figure 6.

Load-deflection curves of test specimens. (a) UN-B hybrid beam (b) UN-T hybrid beam.

Finite element analysis

Establishment of the finite element model

Material constitutive model

In this paper, the NC material model is selected from the uniaxial tensile and compressive stress-strain principal structure of concrete provided by the GB50010-2010 (2011). The compressive and tensile stress-strain curve can be calculated by the following equation (1), and (2).

σ = {\begin{cases} \frac{f_{c} n x}{n - 1 + x^{n}} & x \leq 1 \\ \frac{f_{c} x}{α_{c} {(x - 1)}^{2} + x} & x > 1 \end{cases}

(1)

σ_{t} = {\begin{cases} f_{t} (1.2 x - 0.2 x^{6}) & x \leq 1 \\ f_{t} \frac{x}{α_{t} {(x - 1)}^{1.7} + x} & x > 1 \end{cases}

(2)

Where,

x = ε / ε_{c}

f_{c}

ε_{c}

are the axial compressive strength and peak compressive strain of concrete,

ε_{c} = (172 \sqrt{f_{c}} + 700) \times 10^{- 6}

α_{c}

is the parameter value of the descending section of the curve,

α_{c} = 0.157 f_{c}^{0.875} - 0.905

f_{t}

ε_{t}

are the axial tensile strength and peak tensile strain of concrete,

ε_{t} = 65 f_{t}^{0.54} \times 10^{- 6}

α_{t}

is the parameter of the descending section of curve,

α_{t} = 0.31 f_{t}^{2}

The properties of UHPC materials will vary greatly due to differences in manufacturing, and there are also certain differences in the constitutive model of UHPC. Therefore, the constitutive model for the fit test should be selected as much as possible during the selection process. Through the analysis and study of different UHPC compressive constitutive models, this paper selects the UHPC compressive stress-strain constitutive model proposed in document (Singh et al., 2017), with the following:

σ = {\begin{cases} f_{c} [\frac{(E_{c} / E_{sc}) (ε / ε_{c}) - {(ε / ε_{c})}^{2}}{1 + (E_{c} / E_{sc} - 2) (ε / ε_{c})}] & 0 < ε < ε_{c} \\ [\frac{f_{c}}{1 + 1 / 4 {[(ε / ε_{c} - 1) / (ε_{cu} / ε_{c} - 1)]}^{1.5}}] & ε_{c} \leq ε < ε_{cu} \end{cases}

(3)

Where,

ε_{c}

f_{c}

are the peak compressive strain and axial compressive strength of the UHPC,

ε_{c} = (9.31 f_{c} + 2269) \times 10^{- 6}

E_{c}

E_{sc}

are the UHPC modulus of elasticity and cut line modulus,

E_{sc} = f_{c} / ε_{c}

ε_{cu}

is the UHPC ultimate compressive strain.

According to the results of UHPC axial tensile test, the tensile stress-strain curve of UHPC mainly includes elastic stage, strain hardening stage and strain softening stage. Therefore, according to the actual stress-strain curve, the stress-strain relationship proposed by Zhang et al. (2015) is selected for the elastic stage and strain-hardening stage. The strain softening section adopts the constitutive model in Code (AFGC and SETRA, 2013), as shown in equation (5).

σ_{t} = {\begin{cases} \frac{f_{t}}{ε_{t 0}} ε & 0 < ε \leq ε_{t 0} \\ f_{t} & ε_{t 0} \leq ε \leq ε_{tu} \end{cases}

(4)

σ_{t} = f_{t} \frac{1}{{(1 + w / w_{p})}^{p}}

(5)

Where,

ε = f_{t} / E_{c} + w / l_{c}

f_{t}

is the average stress of UHPC at the strain hardening stage.

ε_{t 0}

and

ε_{tu}

are the elastic tensile strain and ultimate tensile strain of UHPC.

P

is the fitting parameter,

P = 0.95

w_{p}

is the crack width on the curve when the stress decreases to

2^{- p} f_{t}

w_{p} = 1 mm

l_{c}

is the characteristic length of the section.

The concrete model was selected from the Concrete Damage Plasticity (CDP) model available in ABAQUS. Recommended values were used for the relevant parameters required for the plastic damage model (Sidoroff, 1981). The damage evolution parameters for uniaxial compression and tension of concrete in the design specification are derived based on an elastic damage model, whereas the CDP model is a damage model based on a plastic continuous medium, which utilizes isotropic damage elasticity and isotropic tensile and compressive plasticity in all directions to compute the nonlinear behavior of concrete, and the damage factor d in the simulation needs to take into account the plastic strain. The compression and tensile damage factors are denoted as $d_{c}$ and $d_{t}$ , respectively.

{\begin{cases} d_{c} = \frac{(1 - β_{c}) {ε_{c}}^{in} E}{σ_{c} + (1 - β_{c}) {ε_{c}}^{in} E} \\ d_{t} = \frac{(1 - β_{t}) {ε_{t}}^{in} E}{σ_{t} + (1 - β_{t}) {ε_{t}}^{in} E} \end{cases}

(6)

Where,

{ε_{c}}^{in}

and

{ε_{t}}^{in}

are inelastic strains in compression and tension respectively,

{ε_{c}}^{in} = ε_{c} - σ_{c} / E

{ε_{t}}^{in} = ε_{t} - σ_{t} / E

β_{c}

and

β_{t}

are the correlation coeffcients in compression and tension respectively,

β_{c} = 0.4

β_{t} = 0.5

. E is the initial elastic modulus of concrete.

The strain intensification properties of reinforcement after yield are not considered, the reinforcement principal structure in this paper is modeled as a bilinear model with a slope of 0.01 E_s for the second segment. Reinforcement, concrete and UHPC material models are shown in Figure 8.

Interface contact methods

For the interface simulation of UHPC and NC hybrid beams, the main interface simulation methods are (1) Complete bond (Tie); (2) Coulomb function and (3) Cohesion model. A large number of tests have concluded that UHPC and NC have good bonding properties and there not slip relative to each other when working together (Kadhim et al., 2021; Tayeh et al., 2013). In this paper, the above three simulation methods were selected to verify the finite element of the hybrid beams. Figure 7(b) shows that complete bonding (Tie) assumes no relative slip between the two contact surfaces, and the bearing capacity and rigidity of the simulation results are slightly greater than the test values. Coulomb friction uses default values for normal hard contact and tangential friction forces to simulate the UHPC-NC interface behavior. This method ignores the role of interface cohesion and only considers the effect of friction coefficient on interface properties, resulting in low interface bond strength, so the simulated value is much smaller than the experimental value. The cohesion model takes into account the cohesion and friction coefficient of the hybrid interface, and the simulation results are more consistent with the experimental values. Figure 7 illustrates the constitutive relationship of the cohesion model, $K_{n}$ , $K_{s}$ and $K_{t}$ is the normal stiffness and the two tangential stiffness of the interface, respectively. $t_{n}^{0}$ is the maximum normal stress perpendicular to the interface. $t_{s}^{0}$ and $t_{t}^{0}$ is the maximum tangential stress parallel to the interface. $σ_{m}^{f}$ is the effective separation at which the interface fails. Based on experimental studies and finite element analysis, Hussein et al. (2017) provided the values of the relevant parameters of the cohesive model for interfaces with different roughness (shown in Table 2). Duo to the UHPC and NC interfaces of the hybrid beam are roughened, the cohesion parameters for a rough interface are selected to simulate the UHPC-NC interface behavior in this paper.

Figure 7.

Comparison of simulation methods for UHPC-NC hybrid interface. (a) Cohesion model. (b) Interface simulation comparison.

Table 2.

The value of parameters.

Property	Smooth	Mid-rough	Rough
$K_{n}$ (N/mm³)	1358	1358	1358
$K_{s}$ and $K_{t}$ (N/mm³)	20358	20358	20358
$t_{n}^{0}$ , $t_{s}^{0}$ and $t_{t}^{0}$ (MPa)	3.02	5.01	5.63
$σ_{m}^{f}$	0.018	0.117	0.241
Stabilization	0.001	0.001	0.001

Element type and boundary conditions

Concrete, loading pads and bearing pads use three-dimensional solid linear reduction integral unit (C3D8R). Reinforcements are simulated using spatial two-node truss units T3D2. The finite element model adopts four-point flexural loading. In order to prevent deformation of the loading block during the analysis process, and stiffness of loading block is set to be much higher than the stiffness of the beam. The finite element uses a cohesion model to consider the interaction between the UHPC-NC interface, and the contact surface between the loading block and the beam was connected by Tie. The reinforcement cage formed by the longitudinal reinforcement and stirrup was embedded into the beam, and the bond slip between the reinforcement and concrete was ignored. The left end of the finite element model was set as the fixed hinge support, and the right end was set as the sliding hinge support. In order to take into accounting the cost and accuracy of calculation, the mesh size of this finite element model is 20 mm. The finite element model is shown in Figure 8.

Figure 8.

The model of three-dimensional finite element.

Model validation

Previous studies and experimental results in this paper show that there is no bond slip at the UHPC-NC interface, so this paper selects the cohesion parameter of rough interface to simulate the UHPC-NC interface behavior. To verify the reliability of the finite element simulation method and the accuracy of parameter values, the failure mode and load-displacement curves are compared in Figures 4 and 9. The results in Figure 4 show that the crack extension of numerical simulation is similar to the test results, the main cracks of beam UN-T-50 is more numerous and dispersed. The UHPC significantly inhibits the generation and development of the cracks, which leads to main cracks of beams UN-B-50 are more concentrated. The results of numerical simulation agree with the failure mode of the test beam. However, it is limited by the fact that there is always a difference between the finite element simulation and the experimental conditions. It can be seen from the finite element simulation results and experimental results that when the UN-T hybrid beam under test failed, the cracks in the NC layer extended into the UHPC layer, while the cracks in the finite element model stopped at the UHPC-NC interface and did not extend into the UHPC. This kind of crack propagation law is also exhibited in the failure mode of UN-B hybrid beams. Figure 9 shows that the stiffness and bearing capacity of the hybrid beams obtained by the finite element simulation are slightly greater than the test values, but the simulation results of the load-displacement curve of each test beam are in good agreement with the test results. It shows that the selection of the material constitutive model, mesh division and other modelling processes are reasonable, which can accurately simulate the flexural performance of the test beams in the flexural process.

Figure 9.

Experimental and numerical load-deflection curves of the test beams. (a) N-1. (b) U-1. (c) UN-B-50. (d) UN-B-100. (e) UN-T-50. (f) UN-T-100.

Analysis of flexural performance of hybrid beams

Specimens design

In this paper, the flexural properties of four hybrid beams and two reference beams were tested through four-point flexural tests. However, due to typical test parameters, few samples, and test errors, the conclusions obtained under this situation are not reliable and authoritative. The test results are fortuitous and cannot accurately reflect the influence law of UHPC height on the flexural performance of the hybrid beam, and it is impossible to systematically study the hybrid performance of this hybrid beams. To fully investigate the effect of test parameters on the flexural performance of UHPC-NC hybrid beams. On the basis of accurate finite elements, the effects of UHPC height(H), longitudinal reinforcement ratio(ρ), NC compression strength (f_c) and UHPC tensile strength (f_t) on the flexural performance of UN-B and UN-T hybrid beams are considered, and the detailed parameters of the specimens are listed in Table 3.

Table 3.

The parameter design of specimens.

Specimen number	Dimensions (mm³)	Specimens parameter				Location of UHPC
Specimen number	Dimensions (mm³)	H (mm)	ρ(%)	f_c (MPa)	f_t (MPa)	Location of UHPC
N-250	150 × 250 × 2000	0	0.74	40	7.0	/
UN-B-50	150 × 250 × 2000	50	0.74	40	7.0	Bottom
UN-B-100	150 × 250 × 2000	100	0.74	40	7.0	Bottom
UN-B-150	150 × 250 × 2000	150	0.74	40	7.0	Bottom
UN-B-200	150 × 250 × 2000	200	0.74	40	7.0	Bottom
UN-T-50	150 × 250 × 2000	50	0.74	40	7.0	Top
UN-T-100	150 × 250 × 2000	100	0.74	40	7.0	Top
UN-T-150	150 × 250 × 2000	150	0.74	40	7.0	Top
UN-T-200	150 × 250 × 2000	200	0.74	40	7.0	Top
U-250	150 × 250 × 2000	250	0.74	40	7.0	/
UN-B-100-0.74	150 × 250 × 2000	100	0.74	40	7.0	Bottom
UN-B-100-0.91	150 × 250 × 2000	100	0.91	40	7.0	Bottom
UN-B-100-1.10	150 × 250 × 2000	100	1.10	40	7.0	Bottom
UN-B-100-1.42	150 × 250 × 2000	100	1.42	40	7.0	Bottom
UN-T-100-0.74	150 × 250 × 2000	100	0.74	40	7.0	Top
UN-T-100-0.91	150 × 250 × 2000	100	0.91	40	7.0	Top
UN-T-100-1.10	150 × 250 × 2000	100	1.10	40	7.0	Top
UN-T-100-1.42	150 × 250 × 2000	100	1.42	40	7.0	Top
UN-B-100-C30	150 × 250 × 2000	100	0.74	30	7.0	Bottom
UN-B-100-C40	150 × 250 × 2000	100	0.74	40	7.0	Bottom
UN-B-100-C50	150 × 250 × 2000	100	0.74	50	7.0	Bottom
UN-T-100-C30	150 × 250 × 2000	100	0.74	30	7.0	Top
UN-T-100-C40	150 × 250 × 2000	100	0.74	40	7.0	Top
UN-T-100-C50	150 × 250 × 2000	100	0.74	50	7.0	Top
UN-B-100-U5	150 × 250 × 2000	100	0.74	40	5.0	Bottom
UN-B-100-U7	150 × 250 × 2000	100	0.74	40	7.0	Bottom
UN-B-100-U10	150 × 250 × 2000	100	0.74	40	10.0	Bottom
UN-T-100-U5	150 × 250 × 2000	100	0.74	40	5.0	Top
UN-T-100-U7	150 × 250 × 2000	100	0.74	40	7.0	Top
UN-T-100-U10	150 × 250 × 2000	100	0.74	40	10.0	Top

Load-mid span deflection curves

The load-deflection curves of the UHPC-NC hybrid beams are shown in Figure 10(a)–(h), and the characteristic load-deflection curve of the hybrid beam is shown in Figure 10(i). These load-deflection curves show that there are three working stages, that is, pre-cracking phase (before cracking), crack development stage, and post-yield stage. At the first pre-cracking phase (i.e., curve OA), the deflection of the specimen is very small, and the hybrid beams is almost undeformed as shown in Figure 10(i). After the first stage, in the second crack development stage (curve AB in Figure 10(i)), the flexural load of specimens increases almost linearly with the applied displacement. At the end of this stage, all specimens cracks develop sufficiently and the specimen enters the yield state (point B in Figure 10(i)). During the yield stage, the specimens start to perform nonlinearly due to the cracking of the UHPC and nonlinear compressive behavior of the hybrid beams. The flexural load of the hybrid beams decreases rapidly with the applied displacement increases. At post-yield stage (curve BC in Figure 10(i)), the flexural load of the specimens decreases slightly, while the deflection in the span of the hybrid beams increases sharply. It can also be noticed that all UHPC-NC hybrid beams show better ductility during the bending process.

Figure 10.

Load-deflection curves of test specimens. (a)(b) Effect of H on UN-B and UN-T hybrid beams. (c)(d) Effect of ρ on UN-B and UN-T hybrid beams. (e)(f) Effect of f_c on UN-B and UN-T hybrid beams. (g)(h) Effect of f_t on UN-B and UN-T hybrid beams. (i) Characteristic curve of specimens.

Due to the different parameters of the specimens, it is not possible to compare and analyze them directly. In order to analyze the characteristic load values of the UHPC-NC hybrid beams, the crack loads $(f_{cr})$ , yield loads $(f_{y})$ and ultimate loads $(f_{u})$ are normalized as shown in Figure 11.

Figure 11.

Normalized curves of beams. (a) Effects of UHPC height. (b) Effects of reinforcement ratio. (c) Effects of NC compressive strength. (d) Effects of UHPC tensile strength.

In Figure 11(a), with the UHPC height increases from 0 mm to 50 mm, 100 mm, 150 mm, 200 mm and 250 mm, the cracking loads of UN-B hybrid beams increased significantly due to the placement of the UHPC layer at the bottom of the beams, by 76%, 116%, 124%, 124% and 127%, respectively, while the cracking load of the UN-T hybrid beams is consistent with N-1 (0 mm). UHPC height has a positive effect on the yield load and ultimate load of UN-B and UN-T hybrid beams, but the improvement of UHPC on the flexural capacity of UN-B hybrid beams is greater than the UN-T hybrid beams. It indicates that the UHPC placed at the bottom has a positive effect on the flexural performance of the hybrid beams. The main reasons are as follows: (1) UHPC tensile strength is significantly higher than the NC tensile strength, so that the cracking load increases significantly; (2) the better cooperation between UHPC and reinforcement significantly improves the bearing capacity of the hybrid beams, while the UHPC placed on the top of the beam has almost no effect on the flexural capacity.

As shown in Figure 11(b), the reinforcement ratio has no effect on the cracking load of the hybrid beams, and the yield and ultimate loads increase linearly and positively with the reinforcement ratio. The enhancement effect of reinforcement ratio on yield and ultimate load of UN-T beams is greater than UN-B beams, which is mainly due to the UHPC of UN-T beams is located in the compression zone, and increasing the reinforcement ratio can give full play to the high-strength material properties of UHPC. Figure 11(c) shows that the yield load and ultimate load of the UN-T hybrid beam decrease slightly with the increase of NC strength, but the decrease is slight. This may be due to the increase of NC brittleness with the increase of strength. Once the NC of the UN-T hybrid beam cracks in the tension area, the crack immediately spreads to the top UHPC layer. At this time, the NC exits work due to all cracking, causing the hybrid beam to move upward in the neutral axis, and the beam enters the yield state and the limit state earlier. However, the upward movement of the neutral axis of the hybrid beam makes the hybrid beam fail to make full use of the high compressive strength of UHPC in the compression zone. The final failure mode of the hybrid beams is that the steel bars snap and the bearing capacity is lost. Figure 11(d) shows that increasing the UHPC tensile strength only improves the cracking, yield and ultimate loads of the UN-B hybrid beams with the UHPC located at the bottom of the beams, and has no effect on the characteristic load values of the UN-T beams. It shows that UHPC with higher tensile strength can be used for UN-B hybrid beams to improve flexural properties, while UN-T hybrid beams with higher UHPC tensile strength will increase the total cost.

Flexural ductility

The ductility is defined the ratio of the ultimate load displacement ∆_u to the yield displacement ∆_y (Park, 1989). The effect of UHPC height, reinforcement ratio, NC and UHPC strength on the displacement ductility of the hybrid beams are shown in Figure 12.

μ = \frac{Δ_{u}}{Δ_{y}}

(7)

Figure 12.

Comparison of the ductility of each hybrid beams. (a) Effects of UHPC height (b) Effects of reinforcement ratio. (c) Effects of NC compressive strength (d) Effects of UHPC tensile strength.

Figure 12 show that the ductility of the UN-T hybrid beams was better than the UN-B hybrid beams. (1) The ductility of UN-B hybrid beams decreases with increasing UHPC height, and the UN-T hybrid beam with a UHPC height of 50 mm has the best ductility. The lower ductility of UN-B hybrid beams can also be derived from the small deflection when they fail under flexural load (Figure 12(a)). (2) The increase in reinforcement ratio weakened the ductility of the hybrid beams, resulting in a decreasing trend in ductility for both types of beams (in Figure 12(b)). (3) The ductility of the UN-T hybrid beams was substantially improved by increasing the NC compressive strength in the tensile region, which NC strength increased from 30 MPa to 40 MPa and 50 MPa, and the ductility was improved by 54.5% and 67.3% (in Figure 12(c)). (4) The effect of UHPC tensile strength on the ductility of UN-B and UN-T hybrid beams can be ignored (Figure 12(d)).

The above research results show that UN-T hybrid beams exhibit better ductility than UN-B beams. The main reason is that the yield displacements ∆_y of UN-T and UN-B hybrid beams are similar, the ultimate displacement ∆_u of the UN-B hybrid beam is greatly reduced due to the strengthening effect of UHPC located at the bottom of the beam. As a result, the UN-T hybrid beam has a larger ultimate displacement ∆_u than the UN-B hybrid beam. Therefore, the ductility index of the UN-T hybrid beam obtained by using the above ductility calculation method is significantly greater than that of the UN-B hybrid beam. Meanwhile, from the crack extension patterns of the UN-B and UN-T hybrid beams demonstrated in Figure 4(c)–(f), it can be seen that the ductility of specimen and cracks present a high correlation. For the UN-B hybrid beams, the UHPC placed at the bottom of the beams inhibited the emergence and extension of cracks, resulting in fewer cracks at the bottom of the beams when they were damaged, wider main cracks, and poorer ductility, When the UN-T hybrid beam fails, the crack at the bottom expands fully, so it shows excellent ductility.

Energy absorption capacity

The energy absorption capacity of the specimen is defined as the area of the envelope under the load-deflection curve,and the energy absorption of each hybrid beams is shown in Figure 13. UHPC height and reinforcement ratio have a significant effect on the energy absorption capacity of the specimens. The energy absorption capacity of UN-T hybrid beams (UN-T-50, UN-T-100) is 95% higher than the N-1 beam, and even the energy absorption capacity of both is better than that of full UHPC beam U-1. Figure 13(a) shows that the maximum energy absorption capacity of the UN-B hybrid beam occurred in specimens with a UHPC height of 100 mm. With the increase of reinforcement ratio, the energy absorption capacity of UN-B hybrid beams slowly increases, while the performance of UN-T hybrid beams is weakened. The energy absorption capacity of the hybrid beams is hardly affected by the strength of NC and UHPC (Figure 13(c),and (d)). In a word, the UN-T hybrid beams exhibited superior flexural performance, the main reason is that main cracks were dispersed and numerous in the UN-T hybrid beams, resulting in a larger deflection of the beams during loading and a significant increase in their flexural energy absorption capacity. In contrast, the cracks of UN-B hybrid beams are relatively few and eventually expanded into through main crack.

Figure 13.

Energy absorption capacity of test beams. (a) Effects of UHPC height (b) Effects of reinforcement ratio. (c) Effects of NC compressive strength (d) Effects of UHPC tensile strength.

Flexural stiffness and damage analysis

Flexural stiffness

Figure 14 demonstrates the stiffness of some of the combined beams at the yield and ultimate loads, where the stiffness of the specimens decreases gradually with increasing load. Relative stiffness was used to evaluate the flexural stiffness degradation law of the combined beams, which is defined as the ability to resist load/displacement. This study discusses the influence of UHPC height, reinforcement ratio, NC compressive strength and UHPC tensile strength on the relative stiffness of UHPC-NC hybrid beams. The percentage of relative stiffness of hybrid beams is defined as the ratio of the cut-line stiffness of the hybrid beams to the cut-line stiffness of beam N-1 at yield loads (point B in Figure 15) and ultimate loads (point E in Figure 15).

K_{rs} = \frac{K_{NU} - K_{N}}{K_{N}} \times 100 %

(8)

Where,

K_{rs}

is the relative stiffness of hybrid beams; and

K_{NU}

are the cut-line stiffness of the NC (N-1 beam) and UHPC-NC hybrid beams, respectively (Figure 15).

Figure 14.

Stiffness degradation of the hybrid beam. (a) N-250. (b) U-250. (c) UN-B-50. (d) UN-B-100. (e) UN-T-50, (f) UN-T-100.

Figure 15.

Concept of damage variable.

The variation in the relative stiffness percentage (k) of each beam at the yield load and ultimate load is shown in Figure 16. (1) The effects of UHPC height, reinforcement ratio, NC, and UHPC strength on the stiffness of the hybrid beams are negligible at the yield load. In Figure 16(b), only the specimen UN-T-100-14.2 shows a significant increase in stiffness, with a 132% rise in relative stiffness. The UHPC tensile strength located at the bottom of the beam was increased from 5 MPa to 10 MPa, resulting in a 22% increase in the stiffness of the hybrid beams (Figure 16(d)). (2) When the UHPC height was increased from 0 mm to 50 mm, 100 mm, 150 mm, and 200 mm, the stiffness of the UN-B hybrid beams increased by 22%, 34%, 38%, and 40%, respectively, while the stiffness of the UN-T hybrid beams gradually decreased. Increasing the reinforcement ratio to 1.42% resulted in a 47% and 123% increase in the stiffness of the UN-B and UN-T hybrid beams, respectively (Figure 16(b)). Since the hybrid beams have already cracked at the ultimate state, the concrete strength has no significant effect on the stiffness of the hybrid beams. The stiffness of the UN-T hybrid beams decreases significantly as the NC strength increases. This is due to the increase in NC strength, which leads to greater brittleness and more extensive crack propagation before failure, thereby reducing the stiffness of the hybrid beam.

Figure 16.

Comparison of the flexural stiffness of each hybrid beams. (a) Effects of UHPC height. (b). Effects of reinforcement ratio. (c) Effects of NC compressive strength. (d) Effects of UHPC tensile strength.

Damage analysis

Figure 17 illustrates the damage of some hybrid beams at yield and ultimate loads. The cloud diagram shows that increasing the load aggravates the damage degree of the specimen. The damage of the UN-B hybrid beams spreads from the UHPC layer to the NC layer. The damage to the NC layer at the bottom of the UN-T c hybrid beams is more serious. In order to quantify the damage law of hybrid beams during flexural, the damage degree D is used to quantitatively describe the flexural damage process of the hybrid beams (Roufaiel and Meyer, 1987), which is expressed as shown in equation (9). The beams are in the non-damage at the beginning of loading (D = 0). As the load increases, the beams show damage and the accumulation of damage gradually aggravates. The whole specimens are considered to have lost bearing capacity when the concrete in the compression zone of the UHPC-NC hybrid beams were completely crushed (D = 1.0).

D = \frac{K_{u} (K_{0} - K_{i})}{K_{i} (K_{0} - K_{u})}

(9)

Where,

K_{0}

is the initial elastic stiffness;

K_{i}

and

K_{u}

are the cut-line stiffnesses at any point and boundary point on the load-deflection curve, the damage variables are shown in Figure 15.

Figure 17.

Damage evolution of the hybrid beam. (a) N-250. (b) U-250. (c) UN-B-50. (d) UN-B-100. (e) UN-T-50, (f) UN-T-100.

The damage pattern of the hybrid beam is shown in Figure 18. The damage to the hybrid beams gradually increased with the deflection. The UN-B hybrid beams sustained more damage than the UN-T hybrid beams for the same deflection. In Figure 18(a), the slope of the damage curves becomes steeper as the UHPC height increases, indicating that increasing the UHPC height aggravates the damage to the hybrid beams. Without affecting the flexural performance, the UHPC height can be appropriately reduced to delay the damage and improve the cost-effectiveness of the hybrid beams. Figure 18(b) shows that increasing the reinforcement ratio reduces the damage to UN-B hybrid beams but exacerbates the damage to UN-T hybrid beams. When the specimen entered the yield stage, the damage levels of the UN-B and UN-T hybrid beams were [0.21, 0.25] and [0.11, 0.24], respectively. From Figure 18(c), and (d), it can be observed that NC and UHPC strength have no significant effect on the damage to the hybrid beams. However, when the NC strength was reduced to 30 MPa, the damage to the UN-T-100-C30 specimen was notably higher. At the yield stage, its damage level increased by 116% compared to the UN-T-100-C50 hybrid beams (the damage level increased from 0.12 to 0.26). The research suggests that design parameters with a relatively small reinforcement ratio and a UHPC height with 100 mm should be selected in structural design to minimize the damage to hybrid beams.

Figure 18.

Damage analysis of UHPC-NC hybrid beams. (a) Effects of UHPC height (b) Effects of reinforcement ratio. (c) Effects of NC compressive strength (d) Effects of UHPC tensile strength.

Cost effectiveness

Based on the prices of various materials, the total prices of hybrid beams with different UHPC heights were estimated (Qasim, 2019). However, it is unfair to only assess the cost of each beam without counting the mechanical properties of the specimens. The further evaluations were done by considering the coupled functional unit consisting of the cost and the load carrying capacity. This assessment is undertaken by comparing the cost-to-ultimate load ratio for each beam or referring to the term ‘unit cost’, as shown in Figure 19. It can be seen that due to the high price of UHPC, the cost of UHPC-NC hybrid beams increases with the increase of UHPC height. The ultimate load of hybrid beams with UHPC located at the bottom of the beam and height of 100 mm and 150 mm can reach 91.3% and 96.4% of the full UHPC beam, while the ‘unit cost’ is significantly reduced by 54.3% and 32.4% (See Figure 19(a)). Similarly, it can be seen from Figure 19(b) that the loads of hybrid beams with UHPC located at the top of the beam and heights of 50 mm and 100 mm can reach 79.1% and 81.9% of the UHPC beams, ‘unit cost’ is reduced by 49.6% and 38.0%.

Figure 19.

Cost effectiveness of the hybrid beams. (a) UHPC at the bottom (b) UHPC at the top.

The above results show that the UHPC is located at the bottom of the beam to form a UN-B hybrid beam, which exhibits better cost effectiveness at a height of 100 mm ∼ 150 mm. When the UHPC height of the UN-T hybrid beam is 50 mm, the cost is reduced by 50% under the condition of objective bearing capacity. At this time, it also shows high cost performance.

Deflection calculation of UHPC-NC hybrid beams

The deflection calculation model of UHPC-NC hybrid beams is derived based on the GB50010−2010 (2011) and the knowledge of structural mechanics. Firstly, the cross-section flexural stiffness of the hybrid beam is calculated based on the effective cross-section moment of inertia method, and then the deflection of the hybrid beam is obtained with the knowledge of structural mechanics, and finally, the applicability of the deflection calculation model is verified by the experimental data and finite element results.

UHPC reasonable height

The above research shows that the UHPC height has the most significant effect on the flexural properties of hybrid beams, and the UHPC height will also change the neutral axis of hybrid beams. When the UHPC height is high, the neutral axis of the beam may be located in the UHPC, and UHPC height is low, the neutral axis is outside the UHPC. The location of the neutral axis must be specified to derive the deflection calculation model for a hybrid beam based on the effective section moment of inertia method. Figures 11(a), 12(a) and 13(a) show that equation (1) the flexural load capacity of the UN-B hybrid beams is slightly increased when the UHPC height exceeds 150 mm, whereas the ductility and energy-absorbing capacity decrease dramatically; (2) the UHPC height does not have a significant effect on the load carrying capacity of the UN-T hybrid beams, and the ductility and energy-absorbing capacity reaches its optimum at the UHPC height is 50 mm. Meanwhile, the literature (Wang et al., 2024b) shows that the UHPC boundary heights of the UN-B and UN-T hybrid beams are 83.3 mm ∼ 150 mm, and 28.6 mm ∼ 51.3 mm, it shows that the theoretical calculation is consistent with the finite element results. Therefore, the deflection calculation model of the hybrid beam is derived based on the UHPC boundary height, the neutral axis of the hybrid beam is located outside the UHPC.

Calculation of immediate stiffness

UN-B hybrid beam

When the UHPC height h_i≤150 mm, the UN-B hybrid beam section is shown in Figure 20. Before the hybrid beam cracks, the full-section concrete is subjected to stress. The height of the compression zone of the concrete section $x_{0}$ is determined by the equal area moments of the tension and compression zones of the section (Figure 20(b)).

\frac{1}{2} b x_{0}^{2} + n_{s} A_{s}^{'} (x_{0} - a_{s}^{'}) = \frac{1}{2} b {(h - h_{i} - x_{0})}^{2} + n_{u} A_{u} (h - \frac{1}{2} h_{i} - x_{0}) + n_{s} A_{s} (h_{0} - x_{0})

(10)

Figure 20.

Transformed section of the UN-B hybrid beams.

The height of the concrete compression zone $x_{0}$ is shown in equation (11). The moments of inertia of the converted sections of the UN-B hybrid beams before crack can be obtained.

x_{0} = \frac{b {(h - h_{i})}^{2} + n_{u} A_{u} (2 h - h_{i}) + 2 n_{s} (A_{s}^{'} a_{s}^{'} + A_{s} h_{0})}{2 [b (h - h_{i}) + n_{s} (A_{s}^{'} {+ A}_{s}) + n_{u} A_{u}]}

(11)

I_{0} = \frac{1}{3} b [x_{0}^{3} + {(h - h_{i} - x_{0})}^{3}] + n_{u} A_{u} {(h - \frac{1}{2} h_{i} - x_{0})}^{2} + n_{s} A_{s} {(h_{0} - x_{0})}^{2} + n_{s} A_{s}^{'} {(x_{0} - a_{s}^{'})}^{2}

(12)

Where, h,

h_{i}

are the height of the hybrid beam and UHPC.

A_{s}

A_{s}^{'}

are the cross-sectional areas of tension and compression reinforcement.

n_{s}

is the ratio of the modulus of elasticity of the reinforcement to concrete,

n_{s} = E_{s} / E_{c}

n_{u}

is the ratio of the modulus of elasticity of the UHPC to concrete UHPC,

n_{u} = E_{u} / E_{c}

A_{u}

is the cross-sectional area of the UHPC.

After cracking of the UN-B hybrid beam, the UHPC matrix in the tension zone gradually withdrew from work due to crack development. The bridging effect of steel fibers allows UHPC to still provide tensile force in the tension zone. Therefore, this paper considers that all UHPC and NC in the tension zone have been withdrawn from work, and the steel fiber in the UHPC in the tension zone has been converted into the corresponding equivalent concrete area. The equivalent cross-section of steel fibers after cracking of the hybrid beam is shown in Figure 20(c), and the cross-sectional moment of inertia is given in equation (13).

\frac{1}{2} b x_{cr}^{2} + n_{s} A_{s}^{'} (x_{cr} - a_{s}^{'}) = n_{f} α V_{f} b h_{i} (h - \frac{1}{2} h_{i} - x_{cr}) + n_{s} A_{s} (h_{0} - x_{cr})

(13)

Based on the height of the concrete compression zone at the cracked section $x_{cr}$ , the moment of inertia at the cracked section of the UN-B hybrid beam can be obtained as follows.

x_{cr} = \frac{1}{b} {\begin{cases} \sqrt{β_{f} b^{2} (β_{f} h_{i}^{2} + 2 h h_{i} - h_{i}^{2}) + 2 β_{f} b n_{s} h_{i} (A_{s}^{'} {+ A}_{s}) + 2 n_{s} b (A_{s}^{'} a_{s}^{'} + A_{s} h_{0}) + n_{s}^{2} {(A {}_{s}^{'}+ A_{s})}^{2}} \\ - n_{s} (A_{s}^{'} {+ A}_{s}) - β_{f} b h_{i} \end{cases}}

(14)

I_{cr} = \frac{1}{3} b x_{cr}^{3} + n_{f} α V_{f} b h_{i} {(h - \frac{1}{2} h_{i} - x_{cr})}^{2} + n_{s} A_{s} {(h_{0} - x_{cr})}^{2} + n_{s} A_{s}^{'} {(x_{cr} - a_{s}^{'})}^{2}

(15)

where,

β_{f}

is the conversion area factor of steel fibers,

β_{f} = n_{f} α V_{f}

n_{f}

is the ratio of the modulus of elasticity of the steel fiber to concrete,

n_{f} = E_{f} / E_{c}

V_{f}

is the volume fraction of steel fibers.

α

is the effective orientation factor of steel fibers,

α = 0.405

(Zhao, 2005).

I_{cr}

is the moment of inertia of the section of the hybrid beam section after cracking.

UN-T hybrid beam

When the UHPC height h_i ≤ 50 mm, the UN-B hybrid beam section is shown in Figure 21. Before the hybrid beam cracks, the full-section concrete is subjected to stress (Figure 21(b)), and the height of the compression zone of the concrete section $x_{0}$ is as follows.

\frac{1}{2} b {(x_{0} - h_{i})}^{2} + n_{s} A_{s}^{'} (x_{0} - a_{s}^{'}) + n_{u} A_{u} (x_{0} - \frac{1}{2} h_{i}) = \frac{1}{2} b {(h - x_{0})}^{2} + n_{s} A_{s} (h_{0} - x_{0})

(16)

Figure 21.

Transformed section of the UN-T hybrid beams.

The height of the concrete compression zone $x_{0}$ is shown in equation (18). The moments of inertia of the converted sections of the UN-T hybrid beams before crack can be obtained.

x_{0} = \frac{b (h^{2} - h_{i}^{2}) + n_{u} A_{u} h_{i} + 2 n_{s} (A_{s}^{'} a_{s}^{'} + A_{s} h_{0})}{2 [b (h - h_{i}) + n_{s} (A_{s}^{'} + A_{s}) + n_{u} A_{u}]}

(17)

I_{0} = \frac{1}{3} b [{(x_{0} - h_{i})}^{3} + {(h - x_{0})}^{3}] + n_{u} A_{u} {(x_{0} - \frac{1}{2} h_{i})}^{2} + n_{s} A_{s} {(h_{0} - x_{0})}^{2} + n_{s} A_{s}^{'} {(x_{0} - a_{s}^{'})}^{2}

(18)

After cracking of the UN-T hybrid beams, the concrete in the tension zone quits working. Therefore, the interface has only steel reinforcement to provide tensile force after cracking, and the contribution of steel fibers in the UHPC in the compression zone to the compressive force is not considered in this paper. The cross-section of the hybrid beam after cracking is shown in Figure 21(c), and the moment of inertia of the cross-section is shown in equation (19).

\frac{1}{2} b {(x_{cr} - h_{i})}^{2} + n_{s} A_{s}^{'} (x_{cr} - a_{s}^{'}) + n_{u} A_{u} (x_{cr} - \frac{1}{2} h_{i}) = n_{s} A_{s} (h_{0} - x_{cr})

(19)

The moment of inertia at the cracked section of the UN-T hybrid beam can be obtained from the height of the concrete compression zone at the cracked section $x_{cr}$ .

x_{cr} = \frac{1}{b} {\begin{cases} \sqrt{2 b n_{s} (A_{s} h_{0} - A_{s} h_{i} + A_{s}^{'} a_{s}^{'} - A_{s}^{'} h_{i}) + {(n_{u} A_{u} + n_{s} A_{s}^{'})}^{2} + n_{u} A_{u} (2 n_{s} A_{s} - b h_{i}) + n_{s}^{2} {(A_{s}^{2} + 2 A_{s} A_{s}^{'})}^{2}} \\ + b h_{i} - [n_{s} (A_{s}^{'} + A_{s}) + n_{u} A_{u}] \end{cases}}

(20)

I_{cr} = \frac{1}{3} b {(x_{cr} - h_{i})}^{3} + n_{u} A_{u} {(x_{cr} - \frac{1}{2} h_{i})}^{2} + n_{s} A_{s} {(h_{0} - x_{cr})}^{2} + n_{s} A_{s}^{'} {(x_{cr} - a_{s}^{'})}^{2}

(21)

The effective moment of inertia I_e of the hybrid beam section is calculated according to the Code (ACI, 2014), the expression for the effective moment of inertia of the section is follow.

I_{e} = {(\frac{M_{cr}}{M})}^{3} I_{0} + [1 - {(\frac{M_{cr}}{M})}^{3}] I_{cr}

(22)

In summary of the above analysis, the immediately flexural stiffness of the UHP-NC hybrid beams based on the effective section moment of inertia method is calculated as follows.

B_{s} = {\begin{cases} E_{c} I_{0} & M \leq M_{cr} \\ E_{c} I_{e} & M_{cr} \leq M \end{cases}

(23)

Deflection calculation

According to the knowledge of structural mechanics, the deflections of reinforced concrete beams loaded symmetrically at two points under immediate flexural moments are as follows.

f = \frac{P a}{48 B_{s}} (3 l^{2} - 4 a^{2})

(24)

Where,

f

is the deflection of the hybrid beam.

P

is the section load.

a

is the distance from the loading point to the support.

l

is the calculated span of the hybrid beam.

B_{s}

is the immediately flexural stiffness of the UHP-NC hybrid beams.

Model validation

In this paper, based on the immediate stiffness and deflection calculation method of the hybrid beams, that is, equations (10) to (24), the mid-span deflection values corresponding to the crack load and yield load of the UHPC-NC hybrid beams are obtained. The comparison of calculated and tested values (Alharthi et al., 2021; Hu et al., 2022; Safdar et al., 2016; Wang et al., 2025; Yu et al., 2022; Yuan et al., 2022) of mid-span deflection of UN-B and UN-T hybrid beams are shown in Figures 22 and 23. From Figure 22, it can be seen that the error between the calculated deflection values and the test values of the UN-B hybrid beams is in the range of [-15%, +15%], which indicates that the calculated values $(f_{Cal})$ of the mid-span deflection of the UN-B hybrid beams under crack and yield loads are in better agreement with the test values $(f_{Exp})$ , and the mean values of $f_{Cal} / f_{Exp}$ are 0.97 and 0.94, respectively. Under yield load, the calculated mid-span deflection value of the UN-B hybrid beams under yield load is larger than the experimental value, which is mainly due to the fact that the bridging effect of the steel fibers after cracking of the UN-B hybrid beams has good suppression effect on the cracks, which makes the real cross-section stiffness of the combined beams larger than that obtained from the theoretical calculation. Therefore, the theoretically calculated deflection values are generally larger than the test values. Figure 23 shows that the error between the calculated deflection values and the test values for the UN-T hybrid beams is in the range of [-10%, +10%] under crack load. The error between the calculated deflection value and the test value of the hybrid beam under yield load is within 15%, and the calculated value is smaller than the test value, which is the fact that the effect of diagonal cracks in the flexural-shear section on the beam deformation is neglected when calculating the deflection of the hybrid beam. Therefore, the mid-span deflection calculation method of UHPC-NC hybrid beams based on the effective moment of inertia has good calculation accuracy. The mid-span deflection of hybrid beams under normal service conditions can be accurately predicted. The calculation method can provide a theoretical reference for the design of UHPC-NC hybrid beams in the use stage.

Figure 22.

Comparison between calculated and experimental values of UN-B hybrid beams. (a) Cracking deflection (b) Yield deflection.

Figure 23.

Comparison between calculated and experimental values of UN-T hybrid beams. (a) Cracking deflection (b) Yield deflection.

Conclusion

In this paper, the effects of UHPC height, reinforcement ratio, NC and UHPC tensile strength on the flexural performance (load-bearing capacity, ductility, energy absorbing capacity, stiffness, etc.) of UHPC-NC hybrid beams were investigated. The mid-span deflection calculation model of the hybrid beam is proposed based on the effective section bending stiffness and the knowledge of structural mechanics. The following main conclusions are drawn:

1. All specimens exhibited ductile failure modes under four-point flexural loads. The UN-B hybrid beams developed damage through the main crack because the UHPC prevented the development of cracks. The UN-T hybrid beams were damaged by the formation of multiple intense main cracks in the flexural section. The steel fibers significantly inhibited the expansion of the cracks, which resulted in significantly smaller crack widths in the combined beams.

2. The UHPC height has a significant effect on the load carrying capacity, ductility, energy absorption capacity and stiffness of UHPC-NC hybrid beams. UHPC height was increased from 0 mm to 100 mm, the crack and yield loads of UN-B hybrid beams increased by 116% and 32%, respectively. The ductility, energy absorption capacity, and stiffness of the UN-T hybrid beam increased by 36.8%, 95%, and 34%, respectively. Replacing part of NC with UHPC can significantly improve the performance of beams, and it can be used in engineering as a new structure.

3. The load capacity of the hybrid beams increases linearly with the increase in reinforcement ratio. The reinforcement ratio increased from 0.74% to 1.42%, the ductility of the hybrid beams showed a decreasing trend. In this case, the UN-T hybrid beams ductility decreased by 57%. NC and UHPC strengths have a large effect on the load carrying capacity and damage of the hybrid beams, but not on the other flexural properties.

4. The UHPC-NC hybrid beams offer considerable cost effective. 91.3% of the load carrying capacity of the UHPC beams can be achieved at a UHPC height of 100 mm for the UN-B hybrid beams, and the cost is reduced by 54.3%. The UN-T hybrid beams with a UHPC height of 50 mm have a cost reduction of 49.6%. Therefore, this paper suggests the optimum UHPC heights of 100 mm and 50 mm for UN-B and UN-T hybrid beams.

5. Based on the immediate stiffness and effective cross-sectional moment of inertia of the hybrid beams, a model for the mid-span deflection calculation of the UHPC-NC hybrid beam at a reasonable UHPC height was proposed. The mid-span deflection calculation model is in good agreement with the existing experimental results. For most of the specimens, the relative error between the values of calculated and tested is less than 15%. Additionally, the calculation process of the calculation model is relatively simple, making it suitable for engineering design applications.

Footnotes

ORCID iDs

Qiuwei Wang

Biao Liu

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the Independent Research Project Fund of State Key Laboratory of Green Building (LSZZ-K202508) and Shaanxi Natural Science Basic Research Program (2025JC-YBMS-464). Their supports are sincerely appreciated.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Abdal

Mansour

Agwa

, et al. (2023) Application of ultra-high-performance concrete in bridge engineering: current status, limitations, challenges, and future prospects. Buildings 13(1): 185.

ACI (2014) Building code requirements for structural concrete (ACI 318-14). Farmington Hills, Michigan: American Concrete Institute.

AFGC and SETRA (2013) Ultra High Performance fiber-reinforced Concrete. Bagneux: AFGC-SETRA Working Group.

Alharthi

Emara

Elamary

, et al. (2021) Flexural response and load capacity of reinforced concrete beams strengthened with reinforced mortar layer. Engineering Structures 245: 112884.

Alkaysi

El-Tawil

Liu

, et al. (2016) Effects of silica powder and cement type on durability of ultra high performance concrete (UHPC). Cement and Concrete Hybrids 66: 47–56.

GB/T 50081-2019 (2019) Standard for Test Methods of Concrete Physical and Mechanical Properties. Beijing: China Architecture&Building Press.

GB/T 50152-2012 (2012) Standard for Test Methods for Concrete Structures. Beijing: standards press of China.

GB/T228.1-2010 (2011) Metallic materials-tensile testing-part1: Method of Test at Room Temperature. Beijing: Standards Press of China.

Wang

, et al. (2022) Experimental study on the flexure of UHPC beams reinforced with high-strength steel bars. Protective Engineering 44(03): 14–21.

10.

Hussein

Walsh

Sargand

, et al. (2017) Modeling the shear connection in adjacent box-beam bridges with ultra high-performance concrete joints. I: Model calibration and validation. Journal of Bridge Engineering 22(8): 04017043.

11.

Kaan

Umut

Tamer

, et al. (2019) Hybrid fiber use on flexural behavior of ultra high performance fiber reinforced concrete beams. Hybrid structures 229: 111400.

12.

Kadhim

MMA

Jawdhari

Peiris

(2021) Development of hybrid UHPC-NC beams: a numerical study. Engineering Structures 233: 111893.

13.

Zhao

Zhu

, et al. (2019) Test and finite element analysis on flexural performance of UHPC-NC hybrid structure. Highway Engineer 44(02): 194–200.

14.

Liang

Wang

, et al. (2019a) Investigation on flexural capacity of reinforced ultra-high performance concrete beams. Engineering Mechanics 36(5): 110–119.

15.

Liang

Wang

, et al. (2019b) Flexural behavior and capacity analysis of RC beams with permanent UHPC formwork. Engineering Mechanics 36(9): 95–107.

16.

Liu

Sun

Zou

(2020) Experimental study on flexural performance of ultra high performance concrete-normal concrete hybrid simply supported beam. Journal of Tongji University 48(5): 664–672.

17.

Matos

Nunes

Costa

, et al. (2021) Durability of an UHPC containing spent equilibrium catalyst. Construction and Building Materials 305: 124681.

18.

Park

(1989) Evaluation of ductility of structures and structural assemblages from laboratory testing. Bulletin of the New Zealand National Society for Earthquake Engineering 22: 155–166.

19.

Qasim

(2019) Comparative study between the cost of normal concrete and reactive powder concrete. IOP Conference Series: Materials Science and Engineering 518(2): 022082.

20.

Roufaiel

MSL

Meyer

(1987) Analytical modeling of hysteretic behavior of RC frames. Journal of Structural Engineering 113(3): 429–444.

21.

Safdar

Matsumoto

Kakuma

(2016) Flexural behavior of reinforced concrete beams repaired with ultra-high performance fiber reinforced concrete (UHPFRC). Hybrid Structures 157: 448–460.

22.

Sidoroff

(1981)Description of anisotropic damage application to elasticity. Physical non-linearities in structural analysis. Heidelberg: Springer Berlin Heidelberg.

23.

Singh

Sheikh

Ali

, et al. (2017) Experimental and numerical study of the flexural behavior of ultra-high performance fiber reinforced concrete beams. Construction and Building Materials 138: 12–25.

24.

T/CCPA 7-2018 (2018) Fundamental Characteristics and Test Methods of Ultra High Performance Concrete. Beijing: Standards Press of China.

25.

Tayeh

Bakar

Megat

, et al. (2013) Flexural strength behavior of hybrid UHPFC-Existing concrete. Advanced Materials Research 701: 32–36.

26.

Turker

Torun

(2020) Flexural performance of highly reinforced composite beams with ultra-high performance fiber reinforced concrete layer. Engineering Structures 219: 110722.

27.

Wang

Liang

Shi

(2024a) Mechanical properties and constitutive model of ultra high performance concrete with hybrid steel fiber under axial tension. Acta Materiae Compositae Sinica 41(01): 383–394.

28.

Wang

Liu

Shi

, et al. (2024b) Study on flexural performance and calculation method of UHPC-NC hybrid beams. Engineering Mechanics 233: 1–13.

29.

Wang

Liu

Shen

, et al. (2025) Flexural response and load capacity of reinforced concrete beams strengthened with reinforced mortar layer. Journal of Basic Science and Engineering 33(02): 398–408.

30.

Wei

Guo

, et al. (2023) Materials, structure, and construction of a low-shrinkage UHPC overlay on concrete bridge deck. Construction and Building Materials 406: 133353.

31.

Xue

Briseghella

Huang

, et al. (2020) Review of ultra high performance concrete and its application in bridge engineering. Construction and Building Materials 260: 119844.

32.

Yang

, et al. (2019) Experimental study on flexural behavior of precast hybrid UHPC-NSC beams. Journal of Building Engineering 70: 106354.

33.

Yin

Teo

Shirai

(2017) Experimental investigation on the behaviour of reinforced concrete slabs strengthened with ultra-high performance concrete. Construction and Building Materials 155: 463–474.

34.

Wang

, et al. (2022) Flexural behavior of ultra-high performance concrete functionally graded hybrid beams. Journal of the China Railway Society 44(10): 161–170.

35.

Yuan

Liu

Tong

, et al. (2022) Experimental, analytical, and numerical investigation on flexural behavior of hybrid beams consisting of ultra-high performance and normal-strength concrete. Engineering Structures 268: 114725.

36.

Zhang

Shao

, et al. (2015) Axial tensile behavior test of ultra high performance concrete. China Journal of Highway and Transport 28(08): 50–58.

37.

Zhang

Yang

Xie

, et al. (2022) Flexural behaviour and cost effectiveness of layered UHPC-NC composite beams. Engineering Structures 273: 115060.

38.

Zhang

Huang

Liu

, et al. (2023a) Flexural behavior of damaged RC beams strengthened with prestressed UHPC layer. Engineering Structures 283: 115806.

39.

Zhang

Wang

Qin

, et al. (2023b) Experimental and analytical studies on the flexural behavior of steel plate-UHPC composite strengthened RC beams. Engineering Structures 283: 115834.

40.

Zhao

(2005) Advance Reinforced Concrete Structure Theory. Beijing: China Machine Press 442-446.

41.

Zhou

Sheng

(2022) Experimental study of flexural performance of UHPC-NC laminated beams exposed to fire. Materials 15(7): 2605.

42.

Zhou

Song

, et al. (2018) Application of ultra high performance concrete in bridge engineering. Construction and Building Materials 186: 1256–1267.

43.

Zhu

Qin

Shi

(2022) Calculation method for flexural capacity of prestressed UHPC-NC hybrid beams. Journal of Central South University 53(10): 3989–4000.