Experimental and Numerical Investigation on Flexural Behaviors of a 30 m Full-Scale Prestressed UHPC-NC Composite Box Girder

Abstract

Ultra-high-performance concrete (UHPC) exhibits significantly superior compressive and tensile properties compared to conventional concrete, demonstrating substantial application potential in bridge engineering. This study conducted full-scale bending tests on a 30 m prestressed UHPC-NC composite box girder within an actual engineering context. The testing flexural capacity $M_t=\mathrm{34,469.2} \mathrm{k}\mathrm{N}·\mathrm{m}$ exceeded the design requirement $M_d=\mathrm{18,138.0} \mathrm{k}\mathrm{N}·\mathrm{m}$, with $M_t/M_d=1.90$. Finite element modeling (FEM) was employed to analyze and predict experimental outcomes, revealing a simulated flexural capacity of approximately 37,597.1 $\mathrm{k}\mathrm{N}·\mathrm{m}$. The finite element models further explored failure mode transitions governed by the loading position while the concentrated load-to-support distance exceeds 9.62 m (shear span to effective depth ratio λ = 6.3), and the box girder fails in flexure; while less than 9.62 m, the box girder fails in shear. The flexural capacity of the test girder was also estimated using Response-2000 software and the recommended formulas from the Chinese code T/CCES 27-2021 (Technical specification for ultra-high-performance concrete girder bridge). The Response-2000 software yielded a flexural capacity estimate of $M_r=\mathrm{30,816.1} \mathrm{k}\mathrm{N}·\mathrm{m}$. The technical specification provided two estimating results: (with safety factors) $M_{u1}=\mathrm{25,414.4} \mathrm{k}\mathrm{N}·\mathrm{m}$ and (without safety factors) $M_{u2}=\mathrm{33,810.9} \mathrm{k}\mathrm{N}·\mathrm{m}$. All estimated values of Response-2000 and Chinese code were rationally conservative ($M_r,{ M}_{u1}, M_{u2}<M_t$). Comparative analysis demonstrates that Abaqus FEM accurately simulates the flexural behavior of the prestressed UHPC-NC composite box girders. Both Response-2000 calculations and the Chinese code T/CCES 27-2021 provide critical references for similar applications of prestressed UHPC-NC composite box girders.

1. Introduction

Due to the ongoing economic development and consequent investment in infrastructure construction, reinforced concrete structures encounter formidable challenges arising from long-term aging and harsh environmental conditions, with instances of structural failure posing substantial risks to traffic and travel safety and operational reliability. These deficiencies necessitate costly rehabilitation measures, extended construction timelines, and inevitable disruptions to urban mobility and commercial activities. To address the limitations of conventional reinforced concrete structures, an increasing number of innovative concrete materials and structural configurations have been developed [1,2,3]. Due to its exceptional mechanical properties, such as its ultra-high compressive strength and high tensile strength, as well as its excellent cracking strength and strain-hardening characteristics under tensile loading [4,5,6]—which are largely caused by the fact that the mixing ratio typically contains about 2% steel fiber admixture [7]—UHPC is regarded as a revolutionary and innovative engineering material [8,9,10]. Consequently, UHPC has found extensive use in engineering [11], particularly in the field of innovative bridge structures, key correction of precast concrete elements, and structure rehabilitation of bridges.

Web of Science records indicate that the first UHPC article was published in 1999. From 1999–2008, annual publications remained stable (several to over a dozen). During 2009–2015, output grew steadily to dozens yearly. Since 2016, publications have surged dramatically (Figure 1), demonstrating UHPC’s considerable recent attention.

In 1999, Reda, MM. et al. [12] published the first seminal study on ultra-high-performance concrete (UHPC), with a focus on its microstructural properties. Their work identified that UHPC possesses a remarkably dense microstructure relative to both conventional and high-performance concrete, establishing a fundamental characteristic that has informed subsequent research into this material.

In subsequent years, numerous researchers pursued investigations into UHPC’s application domains and material properties. Attar et al. [13] explored the use of UHPC in fabricating molds for blow molding processes. Their experimental results indicated that UHPC-based composites show strong potential for such applications due to favorable thermal characteristics, high dimensional accuracy, and reduced fabrication time. Concurrently, Schrefler et al. [14] developed integrated physical, mathematical, and numerical models to simulate the thermal, hydraulic, and mechanical responses of UHPC structures under heating conditions. These models incorporated previously unknown behavioral data obtained from empirical studies on UHPC’s performance. In a more applied direction, Graybeal [15] conducted full-scale tests on AASHTO Type II girders fabricated without mild steel reinforcement, demonstrating that UHPC’s inherent fiber reinforcement could adequately serve in highway bridge girders. This study also included extensive material characterization, leading to proposed design principles for UHPC structural members. Further advancing constitutive modeling, Gruenberg et al. [16] derived essential parameters for calibrating a three-phase model under static loading conditions and the parameters for an anisotropic damage model are determined for fatigue loading.

Over the past 15 years, research on ultra-high-performance concrete (UHPC) has substantially expanded, with a notable increase in publications focusing on experimental validation and design guidelines for practical engineering applications. This period has also witnessed enhanced scrutiny of its mechanical properties. Shi, Caijun et al. [17] reviewed the theoretical principles, raw materials, mixture design methods, and preparation techniques for UHPC, highlighting four fundamental principles for UHPC design: reduction in porosity, improvement in microstructure, enhancement in homogeneity, and increase in toughness. Wang, Dehui et al. [18] further reviewed the theoretical principles, raw materials selection, mixture design, and preparation techniques for UHPC, specifically addressed hydration, microstructure, mechanical properties, dimensional stability, and durability. Wu, ZM et al. [7] investigated the effects of three steel fiber shapes (straight, corrugated, and hooked-end) with different volume fractions (Vf = 0, 1%, 2%, and 3%) on the mechanical properties of UHPC. Their results showed that increased fiber content and the use of deformed fibers reduce UHPC flowability, while significantly enhancing compressive and flexural properties. In comparison with straight fibers, hooked-end and corrugated fibers increased compressive strength by 48% and 59%, respectively. Although fiber content had limited effects on first-crack strength and deflection, it substantially increased peak load. Shafieifar, M et al. [19] determined the tensile and compressive behavior of UHPC and conducted a comparison with normal strength concrete to facilitate the development of a numerical model for simulating UHPC behavior using the finite element method. The study provided mechanical properties of UHPC applicable for FE software using the concrete damage plasticity model, offering a useful reference for simulating UHPC in the absence of sufficient experimental data. The numerical results generally showed good agreement with experimental findings.

Regarding novel bridge structures, the Sherbrooke Pedestrian Bridge in Canada [20], the first prestressed UHPC structural bridge in the world, the Mars Hill Bridge completed in Iowa in 2006, the first UHPC highway bridge in the United States, the first hoopless prestressed UHPC girder bridge built in China in 2018 by Zhonglu Dura International Engineering Co., Ltd., Guangzhou, Guangdong, China and the first steel-UHPC combination truss in China in Zengcheng, Guangzhou are notable examples of its application in bridge engineering [21,22].

However, although UHPC has been successfully applied in multiple bridge projects and maintained safe operation for several years, further research is still necessary to accurately predict its mechanical properties. Given the relatively high cost of UHPC [8,23], precise control over material usage becomes crucial to improve cost-effectiveness in engineering applications. In 2018, Chen et al. [24] tested four rectangular cross-section beam specimens with different reinforcement ratios and found that the presence of shear stress enhanced the flexural capacity of rebar-reinforced UHPC beams. Nguyen et al. [25] developed a 3D simulation model of a 60 m long prestressed UHPC box girder using the concrete damage plasticity (CDP) model, demonstrating that the proposed model accurately captures structural behavior and predicts load-carrying capacity. Cheng et al. [26] conducted bending tests on precast UHPC diaphragm slabs and concluded that the slabs exhibited excellent mechanical performance and deformation capacity. Their study also indicated that theoretical formulas, Response-2000 (v1.0) software, and Abaqus finite element models can effectively predict the flexural capacity of UHPC slabs. Lin et al. [27] manufactured prestressed UHPC-NC composite beams with varying NC layer heights and performed static bending tests. Jiang et al. [5] designed and fabricated eight UHPC-rehabilitated normal concrete (NC) overlay specimens simulating the negative moment region of bridge decks, conducting four-point bending tests to study cracking behavior and flexural performance. The results revealed that the UHPC layer increased crack resistance by 150–275% and significantly improved flexural bearing capacity. Luo et al. [28] carried out tests on a 1:2 scale model of a UHPC box girder bridge deck system to investigate mechanical performance. They observed that, due to the lower support strength of stiffening ribs compared to the web, the UHPC deck should be treated as a two-way rather than one-way slab in the elastic stage. Although the test loading location was unfavorable and initial cracking occurred in the UHPC deck, the final failure mode was bending failure of the transverse rib. Zhu et al. [19] focused on the negative moment region of a three-span continuous small-box girder bridge, using it as the basis for designing a stepped wet joint girder with UHPC partially cast on the top flange. Through three-point bending tests, the study compared the mechanical performance of joint regions cast with normal concrete (NC) and UHPC.

The results show that the contribution of the top plate UHPC to tensile stress cannot be ignored. While predictive tools for UHPC mechanical properties are maturing with improved validation, significant gaps persist in full-scale verification for large-volume structural members in practical engineering applications. This study bridges the critical validation gap through experimental and numerical analysis of a 30 m full-scale UHPC-concrete composite box girder, providing essential reference data for design standardization and broader implementation of UHPC structural elements.

This study is based on the Songze Elevated Road Western Extension Project in Shanghai. The Shanghai Urban Construction Design and Research Institute (Group) Co., Ltd., Shanghai, China developed and designed two 30 m pretensioned U-shaped ultra-high-performance concrete (UHPC) composite box girders as shown in Figure 2, which were installed on a ramp bridge with a span of 30 m and width of 8.5 m. The cross-section arrangement consists of two box girders spaced at 4.1 m intervals with 400 mm wide cast-in-place joints. The design service life is 100 years under City-A level loading. UHPC offers superior mechanical properties, making it a suitable alternative when conventional concrete cannot meet design requirements under height constraints [5]. To ensure proper connection with adjacent spans, the composite box girders maintain a uniform height of 1.6 m. Post-tensioning requires larger duct dimensions, which conflicts with the goal of reducing structural thickness in UHPC applications. Therefore, pretensioning was adopted for the main UHPC girders as a more efficient solution. The design combines UHPC and normal concrete (NC) to optimize cost efficiency: the webs and bottom slab use UHPC, while the top slab, located in the compression zone, employs C60 concrete. On 10 November 2021, these composite girders were successfully installed at the construction site of the Songze Elevated Road Western Extension Project (Section 2), as shown in Figure 3. To further investigate the flexural performance of these girders, an identical full-scale test beam was fabricated for full-scale bending and shear tests. This study focuses exclusively on the flexural experiments.

To understand the structural behavior of UHPC girders in real-world applications, obtaining key data through flexural failure testing is essential. Although full-scale beam capacity tests present challenges due to large specimen dimensions and high costs, they remain the most reliable method for evaluating true structural performance, verifying the appropriateness of bridge design [26,29,30]. This study focuses on flexural testing of 30 m pretensioned prestressed UHPC-concrete composite box girders with simply-supported conditions, aiming to enhance the performance of conventional prestressed concrete box girders while expanding UHPC applications in medium-span bridges to optimize mechanical behavior, durability, and cost-effectiveness. During the design process, multiple load effects were considered, including self-weight, prestressing forces, vehicle loads (City-A level), temperature effects, shrinkage, and creep. The load combinations were calculated according to the JTG 3362-2018 (Specifications for Design of Highway Reinforced Concrete and Prestressed Concrete Bridges and Culverts) [31], resulting in a maximum design moment at midspan of $M_d=\mathrm{18,138.0} \mathrm{k}\mathrm{N}·\mathrm{m}$ for the ultimate limit state, which was calculated using the finite element analysis software Midas Civil (v8.32). This design moment serves as the baseline for evaluating the test beam’s actual capacity against code requirements.

In summary, to verify the safety margins and determine the specific flexural capacity of the 30 m pretensioned prestressed UHPC-NC composite box girder, an identical full-scale test beam was additionally fabricated and subjected to full-scale testing. The flexural capacity of UHPC girders was predicted using three analytical methods: Abaqus (v2022) finite element simulation, Response-2000 software calculating, and calculation formulas from the Chinese code T/CCES 27-2021 (technical specification for ultra-high-performance concrete girder bridge) [32]. These tools were systematically evaluated for their predictive accuracy regarding the flexural capacity of prestressed UHPC-NC composite box girders.

2. Experimental Program

2.1. Details of Tested UHPC-NC Beams

The full-scale prestressed UHPC-NC composite box girder test specimen was constructed by combining a precast prestressed U-shaped UHPC girder measuring 2500 mm in width and 1360 mm in height with a cast-in-place normal concrete deck slab measuring 3950 mm in width and 260 mm in thickness, forming a composite structure with overall dimensions of 3950 mm in width and 1620 mm in height as shown in Figure 4. The U-shaped UHPC girder features a 120 mm thick web and 200 mm thick bottom slab, with a specially designed 300 mm × 337 mm enlarged section at the top flange of the U-shaped girder to accommodate the embedded connection rebars to connecting the deck slab. The reinforcement system incorporates 24 pieces of 36 mm diameter steel rebars and 36 strands of 17.8 mm diameter prestressing tendons at the bottom of the box girder, with the main reinforcement protected by 50 mm concrete cover, 36 mm diameter vertical stirrups were spaced at 200 mm intervals, while the prestressing tendons maintain a 40 mm cover to provide full-section prestress to the hollow-core girder, with the reinforcement layout and dimensional details illustrated in Figure 5.

2.2. Experimental Setup, Instrumentations, and Loading Protocol

The loading configuration for this bending test is illustrated in Figure 6, utilizing a reaction frame equipped with two 300-ton high-precision hydraulic jacks to apply loads to the full-scale beam. The applied load values were determined through conversion of pressure gauge readings from the hydraulic control pump. Two rubber bearings were positioned on each side of the test beam at locations 500 mm from both ends. To comprehensively monitor deflection behavior during loading, deflector measurement points were established at: (1) both support locations, (2) the midspan loading point, (3) a position 6220 mm left of the loading point centerline, and (4) a position 5280 mm right of the loading point centerline. These deflector measurement points consisted of total station targets mounted on the beam surface. Simultaneously, electrical resistance strain gauges were affixed to the beam surface to capture strain responses, with data acquisition performed by a dedicated strain collection system. The field test setup is documented in Figure 7.

The strain measurement system comprised a total of 17 strain gauges strategically installed at critical locations surrounding the loading center section, as illustrated in Figure 8, Figure 9 and Figure 10. The gauges were distributed across three key regions: the top surface, left side, and right side of web of the test beam. The technical specifications of the strain gauges are listed in

Table 1. Strain gauge specification sheet.

Parameter	Unit	Technical Specifications
Model		120–80AA
Gauge Length × Width	mm	85.8 × 4.7
Grid Length × Count	mm	80.0 × 3.0
Base Material		Epoxy Phenolic
Grid Material		Constantan
Nominal Resistance Tolerance	Ω	120 ± 1 Ω
Average Resistance Deviation	Ω	≤±0.1%
Operating Temperature		−30 °C to 60 °C
Gauge Factor		2.0–2.2
Gauge Factor Deviation		≤±1%
Strain Limit	%	2.0 ± 1%
Fatigue Life (±1000 με)	cycles	1,000,000
Thermal Output Coefficient	μm/m/°C	≤2
Thermal Output Deviation	±μm/m	≤30
Insulation Resistance	MΩ	10,000
Room Temp Strain Limit	μm/m	20,000
Mechanical Hysteresis	μm/m	1.2
Self-Temperature-Compensation Number		11

.

This flexural test comprised two loading phases conducted consecutively on the same test beam. The first loading test aimed to verify whether the test beam could withstand the designed maximum bending moment of 18,138.0 kN·m, corresponding to an applied load of approximately 2042.6 kN. Considering enhancing the safety margin of 1.2 times the design moment (21,765 kN·m), the corresponding test load was terminated at approximately 2600 kN. The loading procedure employed incremental stages of 130 kN (${\displaystyle \frac{1}{20}}$% of prediction of flexural bearing capacity), progressing from 0 kN to the target load of 2600 kN, followed by complete unloading upon test completion.

The second loading test was performed to determine the ultimate flexural capacity of the test beam. This phase utilized larger load increments of 200 kN (${\displaystyle \frac{1}{20}}$ of prediction of flexural bearing capacity) per stage, with loading commencing from 0 kN until reaching the maximum load-bearing capacity. In this test, the beam was successfully loaded to 4400 kN, representing its final flexural resistance. It should be noted that the experiment was terminated at 4400 kN due to the insufficient loading capacity of the reacting frame.

3. Experimental Results

Due to the influence of the first loading test, visible cracks had already developed in the test beam prior to the commencement of the second test, with the pre-existing crack pattern documented in Figure 11. In the first test, when the load reached 390 kN, the first crack initiated in the midspan zone of the UHPC, with a width of 0.04 mm. Thereafter, the crack persisted in propagation. Eventually, at a load of 2600 kN (corresponding to 1.2 times the design bending moment), the crack width reached its maximum value of 0.26 mm, which also occurred at the bottom of the midspan beam.

The test results demonstrate the structural behavior under progressively increasing loads. During the first loading test, when the total load (sum of both jack readings) reached 380 kN, flexural cracks initially appeared at the beam bottom. At the maximum load of 2600 kN, a maximum midspan moment of 22,000 kN·m (including self-weight effects) was produced. The test was terminated at this stage without observing failure modes.

In the second loading test, when the load reached 4400 kN, a maximum moment of 34,469.2 kN·m (including self-weight effects) was achieved. No failure modes were observed.

The maximum midspan moments of both 22,000 kN·m and 34,469.2 kN·m were calculated based on an idealized two-dimensional model following elastic theory. The theoretical model is illustrated in Figure 12. The applied concentrated loads were 1300 kN and 2200 kN. Self-weight was modeled as a uniformly distributed load, calculated as: $1.561 {\mathrm{m}}^2\times 2.5 \mathrm{t}/{\mathrm{m}}^3\times 9800 \mathrm{N}/\mathrm{t}=\mathrm{38,248} \mathrm{N}/\mathrm{m}$.

3.1. Load–Displacement Curves

Displacement Data Processing

Figure 13 presents the load–displacement curves at the loading point (with support displacements subtracted). The model curve was obtained from finite element simulation of the test beam using Abaqus software (detailed in Section 4). The F-Test curve represents results from the first loading test, showing a loading point displacement of 110 mm at the maximum load of 2600 kN. The S-Test curve corresponds to the second loading test, where the loading point reached 228 mm displacement at the ultimate load of 4400 kN. All test curves demonstrate linearly increasing loading point deflection with load increment. Figure 14 displays the progressive displacement measurements at various monitoring points during the first loading test, while Figure 15 shows the corresponding displacement data from the second loading test. For Figure 14 and Figure 15, the same curve represents the vertical displacements at different positions of the beam under a load.

As shown in Figure 8, the initial slope of the F-Test curve is significantly steeper than that of the S-Test curve. This is because the UHPC cracked during the first loading, pre-cracking stiffness approximately $6.5\times {10}^7 \mathrm{N}/\mathrm{m}$, resulting in reduced resistance from the UHPC and decreased beam stiffness during the second test, approximately $1.96\times {10}^7 \mathrm{N}/\mathrm{m}$, reducing to 30.2% of the first test’s value. Beyond this point, the F-Test and S-Test curves gradually converge, with the F-Test curve ultimately rejoining the S-Test curve toward its end.

3.2. Strain Data Analysis

Data Processing

As shown in Figure 16, when the load reached 4400 kN, the maximum average compressive strain at the beam top (Points 9–16) was 1056.1 × 10⁻⁶, while the second layer (Points 1, 3, 5, 7) reached a maximum average compressive strain of 559.0 × 10⁻⁶. The load–strain curves in Figure 16 demonstrate that the average strains at both the beam top and second layer increased linearly with the applied load. The third layer (Points 2, 4, 6, 8) exhibited excellent linear behavior up to 3400 kN. However, at 3600 kN, crack propagation into the beam web caused a sudden strain increase.

Figure 17 shows the cross-sectional strain distribution. It can be seen that before reaching 3400 kN, the beam section satisfied the plane-section assumption, with the neutral axis located at 0.9 m of the beam height from the bottom.

4. ABAQUS Numerical Simulation

4.1. Finite Element Model Development

4.1.1. Simulation Background

The experimental testing of full-scale UHPC beams incurs prohibitively high costs, which has substantially limited further investigation into the flexural behavior of this box girder configuration. As noted previously, the current study tested only a single full-scale UHPC test beam, and equipment malfunction prevented loading the specimen to complete failure during the full-scale testing. To extend the full-scale beam loading investigation and further analyze the structural performance of this test beam, finite element models were developed using Abaqus software for numerical simulation, with the objective of determining the peak load capacity, displacement characteristics, and additional flexural performance parameters.

4.1.2. Element Types and Mesh Generation

To thoroughly investigate the flexural performance of prestressed UHPC-NC composite beams, three-dimensional finite element analysis models were established using Abaqus software. The model incorporates the normal concrete (NC) top slab, UHPC box girder, bottom prestressing tendon, bottom reinforcement, top reinforcement, longitudinal reinforcement, and reinforcement in the connection zones. The model dimensions match those of the test beams. A three-point bending setup was implemented, with loads applied at midspan zone and reactions at both ends transferred through two rectangular loading plates. The model employs solid elements 8-node linear brick elements with reduced integration (C3D8R) [33,34] to simulate various components, including the UHPC-NC composite beam, reinforcing steel, loading plates, and supporting plates, as illustrated in Figure 18. Compared with fully integrated elements, these reduced integration elements offer faster computational speed while preventing shear locking issues under bending loads. Additionally, reduced integration elements demonstrate less sensitivity to element shape variations in terms of computational accuracy. For the reinforcement steel, truss elements (T3D2) were selected for simulation purposes.

This study employs “Tie” constraints to simulate the connections between the precast UHPC beam and NC deck slab, between the main girder and supports, and between the main girder and loading plates. This is attributed to the stirrups simultaneously engaging both the UHPC and NC components, with no interfacial slip phenomena observed during testing [35,36]. The “Embedded region” constraint is adopted to define the bond relationship between reinforcement/prestressing tendons and concrete, with the steel reinforcement serving as the embedded element and the concrete beam as the host element. In the finite element model, loading plates are established at both loading and support locations. A reference point (RP1) is created on the top loading plate, followed by coupling between the loading plate’s top surface and RP1. Displacement is then applied at RP1 to simulate the loading process, thereby obtaining the ultimate flexural and shear capacities of the concrete beam. Furthermore, the support constraints at the bottom of the test beam are configured according to simply-supported beam boundary conditions to accurately replicate its actual engineering loading scenario.

This study adopts the concrete damaged plasticity (CDP) model for finite element analysis of concrete [37]. The model is established based on the damage theory proposed by Lubliner, Lee, and Fenves [38,39], which considers the distinct material behaviors under tension and compression, with the fundamental assumption that concrete failure occurs primarily through tensile cracking and compressive crushing [1]. In the ABAQUS implementation, the CDP model requires a comprehensive definition of concrete’s plastic parameters, including the yield function, flow rule, and related coefficients, as well as the characterization of relationships between tensile plasticity and damage, and between compressive plasticity and damage. The specific parameter values adopted for concrete plasticity in the CDP model are summarized in

Table 2. Parameter values for concrete plasticity model.

Expansion Angle $\mathsf{\phi }$	Eccentricity $\mathsf{\epsilon }$	${\mathsf{\sigma }}_{\mathbf{b}0}/{\mathsf{\sigma }}_{\mathbf{c}0}$	${\mathbf{K}}_{\mathbf{c}}$	Bonding Coefficient
$40°$	0.1	1.16	0.667	0.004

.

Since standardized material testing was not performed during the experimental phase, the material properties in the model primarily adopted reference values with targeted modifications to improve simulation accuracy. The tensile constitutive relationship of UHPC demonstrated significant influence on the load–displacement curve in this simulation, making it the most critical calibration parameter. Through extensive adjustments to the tensile stress–strain behavior, the current model achieves excellent agreement with experimental data.

The uniaxial compression stress–strain relationship of UHPC adopts the formula recommended by Ref. [40]:

$${\sigma }_{\mathrm{c}}\left(\epsilon \right)=\left\{\begin{array}{c}{f}_{\mathrm{c}}{\displaystyle \frac{n\xi -{\xi }^2}{1+\left(n-2\right)\xi }} \epsilon ⩽{\epsilon }_0\\ f_{\mathrm{c}}{\displaystyle \frac{\xi }{2(\xi -1{)}^2+\xi }} \epsilon >{\epsilon }_0\end{array}\right.$$

(1)

where $f_{\mathrm{c}}$ represents the axial compressive strength of UHPC, taken as 135 MPa; $\epsilon$ is the compressive; ${\sigma }_{\mathrm{c}}$ is the compressive stress; ${\epsilon }_0$ is the peak strain of UHPC, taken as $3375\times {10}^{-6}$; $E_c$ is the initial elastic modulus, taken as 42 GPa; and $E_s$ is the secant modulus. $\xi ={\displaystyle \frac{\epsilon }{{\epsilon }_0}}; n=E_c/E_s$.

The uniaxial compression stress–strain relationship of UHPC adopts the formula recommended by Ref. [41]:

The tensile constitutive model of UHPC is expressed in terms of strain, including the elastic stage, the linear softening stage, and the exponential softening stage. The elastic branch and linear softening branch are defined by linear interpolation between the origin point, tensile strength point, and residual tensile strength point. The tensile stress–strain relation at the exponential softening stage is fitted in the form of an exponential function, as shown in Figure 19.

$${\sigma }_{ct}=\left\{\begin{array}{cc}{\displaystyle \frac{f_t}{{\epsilon }_{cte}}}{\epsilon }_{ct}\hfill & \hfill 0<{\epsilon }_{ct}<{\epsilon }_{cte}\\ f_{ct}+{\displaystyle \frac{f_{ctr}-f_{ct}}{{\epsilon }_{ctr}-{\epsilon }_{cte}}}·\left({\epsilon }_{ct}-{\epsilon }_{cte}\right)\hfill & \hfill {\epsilon }_{cte}<{\epsilon }_{ct}<{\epsilon }_{ctr}\\ f_{ct}\times \left[1+c_1{\left({\displaystyle \frac{{\epsilon }_{ct}-{\epsilon }_{cte}}{{\epsilon }_{ct,max}}}\right)}^3\right]e^{-c_2{\displaystyle \frac{{\epsilon }_{ct}-{\epsilon }_{cte}}{{\epsilon }_{ct,max}}}}\hfill & \hfill {\epsilon }_{ctr}<{\epsilon }_{ct}\end{array}\right.$$

(2)

where $f_{\mathrm{c}\mathrm{t}}$ represents the tensile strength of UHPC, taken as 8 MPa; ${\epsilon }_{\mathrm{c}\mathrm{t}\mathrm{e}}$ is the tensile strain at the tensile strength, taken as $580\times {10}^{-6}$; $f_{\mathrm{c}\mathrm{t}\mathrm{r}}$ represents the residual tensile strength of UHPC, taken as 5.34 MPa; ${\epsilon }_{\mathrm{c}\mathrm{t}\mathrm{r}}$ is the characteristic strain at residual tensile strength, taken as $1130\times {10}^{-6}$; and ${\epsilon }_{\mathrm{c}\mathrm{t},\mathrm{m}\mathrm{a}\mathrm{x}}$ represents the maximum tensile strain, taken as 0.0325. $c_1$ and $c_2$ are the curve-fitting parameters, taken as $c_1=100$ and $c_2=15$.

The constitutive relationship of UHPC obtained from the above equations is shown in Figure 20.

The constitutive relationship of the steel reinforcement follows an ideal elastic–plastic model. The test specimens employed HRB400 and HRB500 rebars along with Standard 1860 prestressing strands, with material properties specified according to Ref. [4]. The yield strengths $f_y$ are 400 MPa for HRB400, 500 MPa for HRB500, and 1860 MPa for the steel strands, with a common elastic modulus $E_s$ of 200 GPa and Poisson’s ratio ${\upsilon }_g$ of 0.3 for all reinforcement types. As required by the specifications, the minimum total elongation at fracture must exceed 7.5%.

4.1.3. Prestress Simulation Methodology

The simulation of external prestressing tension process refers to Renta et al. [42], adopting a thermal expansion coefficient of $1.2\times {10}^{-5}$ for the prestressing strands. The initial prestress was applied to the tendons using the “temperature reduction method”. In this study, the applied prestressing stress was 1260 MPa. The relationship between the prestress magnitude and the applied temperature variation is calculated by the following formula:

$$\Delta T={\displaystyle \frac{F}{\alpha EA}}={\displaystyle \frac{\sigma }{\alpha E}}={\displaystyle \frac{1260}{1.2\times {10}^{-5}\times 195}}=538.46 °\mathrm{C}$$

(3)

• $\Delta T$—Temperature variation applied (°C)
• $F$—Initial prestressing force (N)
• $\alpha$—Thermal expansion coefficient
• $E$—Elastic modulus 195 GPa
• $A$—Cross-sectional area (mm²)
• $\sigma$—Initial prestress value (MPa)

4.2. Finite Element Analysis Results

4.2.1. Load–Displacement Curve Analysis

Figure 12 shows the displacement–load curve from the finite element model’s bending test, where “Model” represents the simulated load–displacement curve, “F-Test” denotes the displacement–load curve from the first bending test, and “S-Test” corresponds to the displacement–load curve from the second bending test. During the initial loading phase, the simulation curve shows excellent agreement with the first test results, with all data points falling within a small error margin. The simulated curve also closely matches the second test results, demonstrating that the finite element model can accurately predict the experimental outcomes. However, the simulation results deviate more significantly after 4400 kN in the second test due to localized damage caused by the first, which led to stiffness degradation.

The finite element analysis results demonstrate that the ultimate flexural load capacity of the structural model reaches 4760 kN, which corresponds to a bending moment capacity of 39,100 kN·m (including self-weight). This represents a significant 115.6% safety margin above the design ultimate moment capacity of $M_d$ 18,138.0 kN·m.

4.2.2. Strain Localizations Analysis

To further investigate the mechanical behavior of the model beam, logarithmic strain data from the UHPC at the beam bottom were obtained at four critical stages. These stages correspond to UHPC cracking (reach maximum tensile stress), bottom longitudinal reinforcement yields, prestressed steel tendon yields, and attainment of ultimate load, as shown in Figure 21.

Observation of the curves reveals that strain continuity remained intact immediately before UHPC cracking. As loading progressed to the yielding of longitudinal reinforcement, pronounced fluctuations emerged in the midspan region with a wavelength measuring approximately 1 m. Subsequent loading stages including yielding of prestressing tendons and ultimate load exhibited significant transformations in curve peaks while troughs remained virtually unchanged. Comparative analysis with tensile damage contours at ultimate load demonstrated direct spatial and quantitative correspondence between strain peaks and tensile damage concentration zones. This confirms marked strain localization phenomena. After tensile damage concentration zones formed, strain accumulation concentrated predominantly within these zones corresponding to curve peaks. Meanwhile, regions between damage zones maintained strains consistent with initial UHPC cracking levels corresponding to curve troughs.

4.2.3. Plastic Hinge Formation, Rotation Capacity, and Curvature Ductility

When investigating the flexural behavior of test beams, their response during the elastoplastic phase is equally critical. Relevant research can be approached from perspectives of plastic hinge formation, rotation capacity, and curvature ductility. However, the full-scale testing are data insufficient for such studies. Consequently, further investigation into the elastoplastic flexural behavior necessitates analysis of finite element model data.

Figure 22 presents the cross-sectional curvature of the model beam at four stages: UHPC cracked, bottom longitudinal reinforcement yields, prestressed steel tendon yields, and ultimate load. It is evident that the cross-sectional curvature exhibits symmetry about the loading point and approximately increases linearly as the distance to the loading point decreases. Furthermore, it escalates with increasing load, reaching its maximum at the loading point (midspan), while curvature at all locations along the beam continuously increases. Additionally, periodic fluctuations are observed in the curves simultaneously, which likely correlate with strain localization phenomena. The curvature changes accelerated at strain concentration zones, but progressed slower in other regions.

Figure 23 displays the relationship between midspan curvature and load for the model beam. Before the UHPC cracks, the midspan cross-sectional curvature increases at the slowest pace; after the UHPC cracks, the change rate accelerates somewhat. When the bottom longitudinal reinforcement yields, the curvature change rate further increases. Until the prestressed steel tendons yield, the curvature change rate reaches its highest, while simultaneously, the load continues increasing until peak load.

After the prestressed tendons yield, the midspan section theoretically fails. At this moment, the section bending moment is 37,271.1 kN∙m, reaching elastic-theoretical ultimate load, yet the load still persistently increases approximately 249 kN, and the section bending moment maximally reaches 39,100.5 kN∙m. This indicates that approximately 249 kN structural reserve capacity remains during the elastoplastic stage.

The midspan section forms a plastic hinge at 4150 kN load, the curvature being 0.00218 m⁻¹. At 4760 kN, the plastic hinge fails, the curvature being 0.00337 m⁻¹ and the curvature ductility being 0.353. The calculation process is as follows, where ${\varphi }_y$ is yield curvature and ${\varphi }_u$ is ultimate curvature:

$${\mu }_{\varphi }={\displaystyle \frac{{\varphi }_u-{\varphi }_y}{{\varphi }_u}}={\displaystyle \frac{0.00337-0.00218}{0.00337}}=0.353$$

(4)

Furthermore, through analysis of strain data, the neutral axis height at the midspan cross-section (measured from the bottom) was determined to be 1.30 m at the moment of the bottom longitudinal reinforcement yielding, 1.33 m upon yielding of the prestressed steel tendons, and 1.38 m at ultimate load. In the test, the neutral axis height remained approximately 0.9 m until the load reached 3400 kN. A pronounced upward migration of the neutral axis position was observed.

4.2.4. Simulation of Flexural and Shear Failure Modes

Table 3. Numerical simulation results under different shear span-to-depth ratios.

Loading Point-to-N-Support Distance $\mathbf{L}\mathbf{/}\mathbf{m}$	Shear Span Ratio $\mathit{\lambda }=\frac{\mathit{L}}{{\mathit{h}}_{\mathbf{0}}}$	Load $\mathit{F}\mathbf{/}\mathbf{k}\mathbf{N}$	Left Support Reaction ${\mathit{R}}_{\mathit{l}}\mathbf{/}\mathbf{k}\mathbf{N}$	Right Support Reaction ${\mathit{R}}_{\mathit{r}}\mathbf{/}\mathbf{k}\mathbf{N}$	Maximum Moment $\mathit{M}\mathbf{/}\mathbf{(}\mathbf{k}\mathbf{N}\mathbf{·}\mathbf{m}\mathbf{)}$
5.42	3.6	5355	1561.0	4941.5	26,157.3
6.71	4.4	5030	1501.0	4676.4	24,720.5
8	5.3	5084	1957.1	4274.3	32,877.1
9.62	6.3	5247	2290.6	4103.8	37,597.1
11.24	7.4	5040	2500.6	3686.9	38,893.6
14.40	9.5	4760	2905.1	3002.3	39,100.5

Note: $h_0=1520 \mathrm{m}\mathrm{m}$.

presents the Loading Point-to-N-Support Distance, Load, Left Support Reaction, and Maximum Moment of the model under different Shear Span Ratios. Figure 24 clearly reveals distinct post-peak behaviors corresponding to different shear span-to-depth ratios (λ). For specimens with higher λ values (5.3, 6.3, 7.4, and 9.5), the load–displacement curves exhibit relatively gentle post-peak slopes with prolonged tail portions, indicating more ductile structural responses and gradual capacity degradation. In contrast, specimens with lower λ values (3.6 and 4.4) demonstrate significantly different behavior, characterized by steeper post-peak declines and substantially shortened tail portions, with rapid loss of load-carrying capacity after reaching peak resistance.

When the shear span-to-depth ratios λ = 3.6, λ = 4.4, and λ = 5.3, the models reached their maximum loads at shear failure. As observed in Figure 25a, continuous yielding zones in stirrups are visible from the supports to the loading points.

When the shear span-to-depth ratio λ = 6.3, λ = 7.4, and λ = 9.5, the models initially experienced shear failure, but continued to sustain increasing loads until reaching maximum load capacity at flexural failure. As observed in Figure 25b, continuous yielding zones in stirrups extending from supports to loading points remain visible.

As shown in

Table 4. Reinforcement yielding progression in shear failure models.

Shear Span Ratio	λ = 3.6	λ = 4.4	λ = 5.3
Load (kN)	5355	5030	5084
Stirrup stress (MPa)	400	400	400
Longitudinal rebar stress (MPa)	280	355	489
Prestressing tendon stress (MPa)	1465	1588	1732

and

Table 5. Reinforcement yielding progression in flexural failure models.

λ = 6.3
Load (kN)	4658	4882	5218	5247
Stirrup stress (MPa)	400	400	400	400
Longitudinal rebar stress (MPa)	473	500	500	500
Prestressing tendon stress (MPa)	1696	1742	1860	1860
Reserve capacity	11.2%	7.0%	0.6%	0%
λ = 7.4
Load (kN)	4279	4487	4847	5040
Stirrup stress (MPa)	400	400	400	400
Longitudinal rebar stress (MPa)	466	500	500	500
Prestressing tendon stress (MPa)	1689	1731	1860	1860
Reserve capacity	15.1%	11.0%	3.8%	0%
λ = 9.5
Load (kN)	4150	4150	4511	4760
Stirrup stress (MPa)	400	400	400	400
Longitudinal rebar stress (MPa)	500	500	500	500
Prestressing tendon stress (MPa)	1741	1741	1860	1860
Reserve capacity	12.8%	12.8%	5.2%	0%

, models with shear span-to-depth ratios λ = 3.6, λ = 4.4, and λ = 5.3 immediately reached peak load after stirrup yielding. In contrast, models with λ = 6.3, λ = 7.4, and λ = 9.5 exhibited continued load-bearing capacity growth post-stirrup yielding, ultimately achieving ultimate load only after prestressing tendon yielding. This behavioral divergence serves as the definitive indicator for classifying the former as shear failure and the latter as flexural failure. This confirms λ = 6.3 as the critical threshold between shear and flexural failure modes: when the concentrated load-to-support distance exceeds 9.62 m (λ = 6.3), the box girder fails in flexure; when less than 9.62 m (λ = 6.3), the box girder fails in shear. In contrast, Figure 25c shows no continuous yielding zones in stirrups between supports and loading points. The load–vertical displacement curves of the model beam exhibit two distinct turning points. The first turning point corresponds to the reduction in bottom slab stiffness following UHPC cracking, while the second turning point occurs when both the bottom longitudinal reinforcement and prestressing tendons reach their yield strength, leading to flexural failure of the entire box girder. The finite element simulation results indicate that a conservative estimate of the ultimate flexural capacity of this box girder is approximately 37,597 kN·m. Figure 25c shows the failure mode and stress distribution of the stirrups and the bottom longitudinal reinforcement when λ = 9.5.

The Abaqus software enables viewing component data throughout the simulation steps. By examining the maximum stresses in the HRB400 stirrup, HRB500 longitudinal rebar, and Standard 1860 prestressing tendon, component stresses and load levels at key steps were obtained, as presented in Table 4 and Table 5.

Table 4 presents the finite element simulation results for shear span-to-depth ratios λ = 3.6, 4.4, and 5.3, detailing the stress development in stirrups, longitudinal reinforcement, and prestressing tendons near failure. At peak load, stirrup stresses reach the ultimate value of 400 MPa, while neither longitudinal reinforcement nor prestressing tendons yield, indicating pure shear failure in the test beam. Table 5 shows simulation results for λ = 6.3, 7.4, and 9.5. As loading progressively increases, stirrup stresses first reach 400 MPa and yield, followed by longitudinal reinforcement stresses attaining 500 MPa. Then, prestressing tendon stresses achieve 1860 MPa. Finally, load reaches maximum value.

4.3. Results of ABAQUS Numerical Simulation

Based on the comprehensive analysis, we derive the following conclusions:

• The finite element models effectively capture the mechanical behavior of the test beam.
• Pronounced strain localization phenomena are observed in the model beam.
• The model beam enters the elastic-plastic stage at a midspan load of 4608 kN, developing plastic hinges at midspan with a curvature coefficient of 0.353.
• Flexural failure governs when the shear span-to-depth ratio λ ≥ 6.3, while shear failure occurs when λ < 6.3.
• The maximum sectional bending moment sustained by the test beam is predicted as 37,597.1 kN·m.

5. Response-2000 Flexural Capacity Analysis

The flexural capacity of the test beam’s normal section was calculated using the software Response-2000 [43]. The computational model employed a simplified equivalent I-section configuration with a 331 mm web thickness, comprising the following components: (1) C60 concrete top slab, (2) UHPC girder body, (3) 24 HRB500 longitudinal rebars at the bottom flange, and (4) 36 prestressing strands at the soffit. Material properties were assigned as per experimental references: UHPC exhibited maximum compressive strength of 150 MPa and tensile strength of 6 MPa; HRB500 rebars demonstrated yield strength of 500 MPa and ultimate tensile strength of 750 MPa; and prestressing strands achieved ultimate tensile strength of 1724 MPa. This simplification adheres to Saint-Venant’s principle for global structural response analysis, which is shown in Figure 26.

Figure 27 shows that the calculated flexural capacity of the test beam’s normal section by Response-2000 software is 30,816.1 kN·m. After subtracting the bending moment caused by self-weight, the calculated total loading capacity of the jacks is 3872.7 kN. During the test, the beam did not fail at this load level and could sustain additional loading, demonstrating that Response-2000’s calculation underestimates the actual flexural capacity of this test beam.

6. Code-Based Flexural Capacity Calculation

6.1. Calculation

The flexural capacity verification was conducted according to the T/CCES 27-2021 (technical specification for ultra-high-performance concrete girder bridge) [32]. In accordance with Clause 5.2.4 of the specification, the compression zone height range of the section was determined, with consideration given to the tensile contribution of the UHPC bottom flange:

$$\begin{gathered} {\mathrm{f}}_{\mathrm{s}\mathrm{d}}{\mathrm{A}}_{\mathrm{s}}+{\mathrm{f}}_{\mathrm{p}\mathrm{d},\mathrm{i}}{\mathrm{A}}_{\mathrm{p},\mathrm{i}}+{\mathrm{f}}_{\mathrm{p}\mathrm{d},\mathrm{e}}{\mathrm{A}}_{\mathrm{p},\mathrm{e}}+0.5{\mathrm{f}}_{\mathrm{t}\mathrm{d}}\mathrm{b}\left(\mathrm{h}-{\mathrm{h}}_{\mathrm{f}}^{\prime }/\mathsf{\beta }\right) \\ \le {\mathrm{f}}_{\mathrm{c}\mathrm{d}}{\mathrm{b}}_{\mathrm{f}}^{\prime }{\mathrm{h}}_{\mathrm{f}}^{\prime }+{\mathrm{f}}_{\mathrm{s}\mathrm{d}}^{\prime }{\mathrm{A}}_{\mathrm{s}}^{\prime }+\left({\mathrm{f}}_{\mathrm{p}\mathrm{d},\mathrm{i}}^{\prime }-{\mathsf{\sigma }}_{\mathrm{p},\mathrm{i}0}^{\prime }\right){\mathrm{A}}_{\mathrm{p},\mathrm{i}}^{\prime } \end{gathered}$$

(5)

Calculation result of the left side of the equation in Clause 5.2.4 of the code, where the ratio β between the rectangular stress block height x in the compression zone of the flexural member and the neutral axis depth (actual compression zone height) ${\mathrm{x}}_0$ is taken as 0.80 according to Clause 5.1.4 of the code:

$$\begin{array}{c}{f}_{sd}{A}_s+f_{pd,i}{A}_{p,i}+f_{pd,e}{A}_{p,e}+0.5f_{td}b\left(h-{\displaystyle \frac{h_f^{\prime }}{\beta }}\right)+0.5f_{td}{b}_t{h}_t\hfill \\ =415\times 24\times 804+36\times 191\times 1260+0.5\times 4.41\times 240\times \left(1620-{\displaystyle \frac{200}{0.80}}\right)\hfill \\ +0.5\times 4.41\times \left(200\times 960\right)\hfill \\ =\mathrm{17,819.9} \mathrm{k}\mathrm{N}\hfill \end{array}$$

(6)

According to Clause 5.2.3 of the code, the height of the compression zone x is calculated considering the tensile contribution of the UHPC bottom flange, determined by:

$$\begin{gathered} {\mathrm{f}}_{\mathrm{s}\mathrm{d}}{\mathrm{A}}_{\mathrm{s}}+{\mathrm{f}}_{\mathrm{p}\mathrm{d},\mathrm{i}}{\mathrm{A}}_{\mathrm{p},\mathrm{i}}+{\mathrm{f}}_{\mathrm{p}\mathrm{d},\mathrm{e}}{\mathrm{A}}_{\mathrm{p},\mathrm{e}}+0.5{\mathrm{f}}_{\mathrm{t}\mathrm{d}}\mathrm{b}\left(\mathrm{h}-\mathrm{x}/\mathsf{\beta }\right)+0.5{\mathrm{f}}_{\mathrm{t}\mathrm{d}}{\mathrm{b}}_{\mathrm{t}}{\mathrm{h}}_{\mathrm{t}}={\mathrm{f}}_{\mathrm{c}\mathrm{d}}{\mathrm{b}}_{\mathrm{f}}^{\prime }\mathrm{x}+{\mathrm{f}}_{\mathrm{s}\mathrm{d}}^{\prime }{\mathrm{A}}^{\prime } \\ =24.4\times 3950\times 200+52\times 201\times 330=\mathrm{22,725.2} \mathrm{k}\mathrm{N} \end{gathered}$$

(7)

According to Clause 5.2.4 of the code, the flexural capacity of the section is calculated as follows: where ${\mathrm{x}}_{\mathrm{t}}=1360\left(\mathrm{m}\mathrm{m}\right)$, then

$$\begin{array}{l}{\mathrm{M}}_{\mathrm{u}}={\mathrm{f}}_{\mathrm{c}\mathrm{d}}{\mathrm{b}}_{\mathrm{f}}^{\prime }\mathrm{x}\left({\mathrm{h}}_0-{\displaystyle \frac{\mathrm{x}}{2}}\right)+{\mathrm{f}}_{\mathrm{s}\mathrm{d}}^{\prime }{\mathrm{A}}_{\mathrm{s}}^{\prime }\left({\mathrm{h}}_0-{\mathrm{a}}_{\mathrm{s}}^{\prime }\right)+\left({\mathrm{f}}_{\mathrm{p}\mathrm{d},\mathrm{i}}^{\prime }-{\mathsf{\sigma }}_{\mathrm{p},\mathrm{i}0}^{\prime }\right){\mathrm{A}}_{\mathrm{p},\mathrm{i}}^{\prime }\left({\mathrm{h}}_0-{\mathrm{a}}_{\mathrm{p},\mathrm{i}}^{\prime }\right)-0.5{\mathrm{f}}_{\mathrm{t}\mathrm{d}}\mathrm{b}{\mathrm{x}}_{\mathrm{t}}\left({\displaystyle \frac{{\mathrm{x}}_{\mathrm{t}}}{2}}-\mathrm{a}\right)\\ =24.4\times 3950\times 144\times \left(1541-144÷2\right)+357\times 52\times 204\times \left(1541-100\right)\\ -0.5\times 4.39\times 240\times 1360\times \left(1360÷2-79\right)=\mathrm{25,414.4}\times {10}^6 \mathrm{N}·\mathrm{m}\mathrm{m}=\mathrm{25,414.4} \mathrm{k}\mathrm{N}·\mathrm{m}\end{array}$$

(8)

$$\begin{array}{l}{\mathrm{M}}_{\mathrm{u}}={\mathrm{f}}_{\mathrm{c}\mathrm{k}}{\mathrm{b}}_{\mathrm{f}}^{\prime }\mathrm{x}\left({\mathrm{h}}_0-{\displaystyle \frac{\mathrm{x}}{2}}\right)+{\mathrm{f}}_{\mathrm{s}\mathrm{k}}^{\prime }{\mathrm{A}}_{\mathrm{s}}^{\prime }\left({\mathrm{h}}_0-{\mathrm{a}}_{\mathrm{s}}^{\prime }\right)+\left({\mathrm{f}}_{\mathrm{p}\mathrm{d},\mathrm{i}}^{\prime }-{\mathsf{\sigma }}_{\mathrm{p},\mathrm{i}0}^{\prime }\right){\mathrm{A}}_{\mathrm{p},\mathrm{i}}^{\prime }\left({\mathrm{h}}_0-{\mathrm{a}}_{\mathrm{p},\mathrm{i}}^{\prime }\right)-0.5{\mathrm{f}}_{\mathrm{t}\mathrm{k}}\mathrm{b}{\mathrm{x}}_{\mathrm{t}}\left({\displaystyle \frac{{\mathrm{x}}_{\mathrm{t}}}{2}}-\mathrm{a}\right)\\ =34.1\times 3950\times 144\times \left(1541-144÷2\right)+400\times 52\times 204\times \left(1541-100\right)\\ -0.5\times 8.12\times 240\times 1360\times \left(1360÷2-79\right)=\mathrm{33,810.9}\times {10}^6\mathrm{N}·\mathrm{m}\mathrm{m}=\mathrm{33,810.9} \mathrm{k}\mathrm{N}·\mathrm{m}\end{array}$$

(9)

6.2. Results of Code-Based Flexural Capacity Calculation

Based on Equations (8) and (9), the calculations yield the following results:

• ${\mathrm{M}}_{\mathrm{u}1}$ represents the design flexural capacity, calculated by incorporating all relevant safety factors, reduction coefficients, and the design tensile strength of prestressing strands $f_{pd}=1260 \mathrm{M}\mathrm{P}\mathrm{a}$, yielding the design flexural capacity ${\mathrm{M}}_{\mathrm{u}1}$.
• Design flexural capacity at loading section: ${\mathrm{M}}_{\mathrm{u}1}=\mathrm{25,414.4} \mathrm{k}\mathrm{N}·\mathrm{m}$. Corresponding test load: ${\mathrm{F}}_{\mathrm{u}1}=3092.9 \mathrm{k}\mathrm{N}$.
• ${\mathrm{M}}_{\mathrm{u}2}$ represents the ultimate flexural capacity, calculated without considering safety factors or reduction coefficients, using the characteristic tensile strength of strands $f_{pk}=1860 \mathrm{M}\mathrm{P}\mathrm{a}$ to determine the ultimate moment capacity ${\mathrm{M}}_{\mathrm{u}2}$.
• Ultimate flexural capacity at the loading section ${\mathrm{M}}_{\mathrm{u}2}=\mathrm{33,810}.9 \mathrm{k}\mathrm{N}·\mathrm{m}$. Corresponding test load of ${\mathrm{F}}_{\mathrm{u}2}=4304.8 \mathrm{k}\mathrm{N}$.

7. Comprehensive Analysis of Flexural Capacity

Table 6. Comparison of simulated and calculated flexural capacity values obtained by different methods.

Methods	ID	Ultimate Flexural Capacity $\mathbf{/}\mathbf{k}\mathbf{N}·\mathbf{m}$	Corresponding Test Load $\mathbf{/}\mathbf{k}\mathbf{N}$	${\mathit{M}}_{\mathit{i}}\mathbf{/}{\mathit{M}}_{\mathit{d}}$
Full-scale test	$M_t$	34,469.2	4400	1.90
Abaqus finite element simulation	$M_a$	37,597.1	4851.5	2.07
Response-2000 software	$M_r$	30,816.1	3872.7	1.70
Chinese code T/CCES 27-2021	$M_{u1}$	25,414.4	3092.9	1.40
	$M_{u2}$	33,810.9	4304.8	1.86

Note: maximum design moment $M_d=\mathrm{181,38.0} \mathrm{k}\mathrm{N}·\mathrm{m}$, corresponding test load $F_d=2042.6 \mathrm{k}\mathrm{N}; M_i=M_a, M_t, M_r, M_{u1}, M_{u2}$.

lists the experimental and predicted values of the flexural capacity for the test beam mentioned previously. $M_t$ represents the flexural capacity obtained from the full-scale test results. Since the test beam did not fail, $M_t$ does not represent the actual ultimate flexural capacity, but indicates that the true ultimate flexural capacity should exceed $\mathrm{34,469} \mathrm{k}\mathrm{N}·\mathrm{m}$. $M_a$ is the ultimate flexural capacity obtained through Abaqus finite element simulation. Its load–displacement curve at the loading point shows good agreement with the test results, making it a reliable reference for the ultimate flexural capacity in this bending test. $M_r$ is the predicted value of ultimate flexural capacity calculated by Response-2000 software. $M_{u1}$ and $M_{u2}$ are the predicted values of design flexural capacity and ultimate flexural capacity, respectively, calculated according to the equations in the T/CCES 27-2021 (technical specification for ultra-high-performance concrete girder bridge). $M_{u1}$ conservatively considers all safety factors, reduction coefficients, and the design tensile strength of strands ($f_{pd}=1260 \mathrm{M}\mathrm{P}\mathrm{a}$), while $M_{u2}$ does not. Since Response-2000 software does not account for the contribution of steel fibers to the tensile strength in UHPC, $M_r$ is lower than $M_{u2}$.

As shown in Table 6, the ranking $M_a>M_{u2}>M_r>M_{u1}>M_d$ demonstrates that the beam possesses an adequate safety margin in flexural capacity to meet engineering requirements. The ratio $M_i/M_d$ represents the capacity-to-demand ratio for flexural resistance. The flexural capacity $M_{u1}$ obtained from the T/CCES 27-2021 (technical specification for ultra-high-performance concrete girder bridge) yields $M_{u1}/M_d=1.4$, indicating a built-in safety margin at the design stage. The ultimate flexural capacity $M_{u2}$ from the same specification shows $M_{u2}/M_d=1.86$, representing a 32% increase compared to $M_{u1}/M_d$. The experimentally obtained ultimate capacity $M_t$ gives $M_t/M_d=1.90$, which is 2% higher than $M_{u2}/M_d$. Since the test beam did not fail at $M_t$, this confirms that the code-specified $M_{u2}$ remains conservative. The simulated capacity $M_r$ from Response-2000 software results in $M_r/M_d=1.7$, while the finite element prediction $M_a$ from Abaqus shows $M_a/M_d=2.07$. This suggests that the actual flexural capacity may be twice the designed maximum moment, confirming the beam’s substantial reserve capacity. Both Response-2000 and the code formula provide conservative estimates of the beam’s true flexural performance.

8. Conclusions

This experimental study investigated the flexural performance of a 30 m full-scale prestressed UHPC-NC composite beam and established a finite element model using Abaqus for extended parametric analysis. The flexural capacity was predicted through experimental testing, finite element analysis, Response-2000 software, and theoretical calculations, yielding the following principal conclusions:

(1)

The 30 m prestressed full-scale UHPC-NC composite box girder demonstrated excellent load-bearing capacity, withstanding a maximum load of 4400 kN without experiencing either shear or flexural failure, significantly exceeding the design load of 2042.6 kN.

(2)

The test beam was modeled and analyzed using Abaqus finite element software. The analysis indicates the following: when the concentrated load-to-support distance exceeds 9.62 m (λ = 6.3), the box girder fails in flexure; and when less than 9.62 m (λ = 6.3), the box girder fails in shear. The calculated flexural strength is 37,597 kN·m, corresponding to a test load of 4851.5 kN.

(3)

The flexural strength of the full-scale beams predicted by Response-2000 software yielded a calculated flexural strength of 30,816 kN·m, corresponding to a load of 3508.8 kN. It demonstrates that Response-2000’s calculation of normal section flexural capacity underestimates the actual flexural capacity of this test beam.

(4)

The ultimate flexural capacity calculated according to the Chinese code T/CCES 27-2021 (technical specification for ultra-high-performance concrete girder bridge) is 33,810 kN·m, corresponding to a load of 4304.8 kN. The calculation method in the specification ultimately underestimates the actual flexural capacity of this test beam.

(5)

Both the calculation results from the Chinese code T/CCES 27-2021 (technical specification for ultra-high-performance concrete girder bridge) ($M_{u1}/M_d =1.4$, $M_{u1}/M_d =1.86$) and the Response-2000 software ($M_r/M_d =1.7$) provide conservative estimates of flexural capacity ($M_r, M_{u1}, M_{u2}<M_t$), which remain applicable for the design of prestressed UHPC-NC composite beams.

(6)

Based on comprehensive analysis, we recommend adopting the Chinese code T/CCES 27-2021 (technical specification for ultra-high-performance concrete girder bridge) for projects requiring more conservative design approaches.

Regarding Conclusions 1–4, these pertain specifically to specimen-dependent data derived from the test beam, and are inherently tied to this particular structural configuration. In contrast, Conclusions 5–6 exhibit general applicability to analogous structural systems, representing broader engineering principles.