Crack Width Calculation Method for Concrete in Hogging Moment Region of Steel–UHPC–NC Composite Girder with Integrated Piers

Abstract

The application of ultra-high performance concrete (UHPC) in the hogging moment region significantly enhances the crack resistance of concrete slabs of composite girders with integrated piers, while also providing economic benefits. To investigate the crack resistance performance and develop a calculation method for crack width in hogging moment region of steel–UHPC–normal concrete (NC) composite girders, a full-scale bending test was conducted. Based on the test results, the post-cracking residual tensile strength of UHPC was determined according to the energy equivalence principle. A calculation method for reinforcement stress incorporating the tensile contribution of UHPC at a cracked section was proposed and then the applicability for current design codes for crack width calculation was evaluated. For the UHPC–NC interface, a corresponding crack width calculation method was developed. The results indicate that cracks initiated on the surface of the NC layer beneath the UHPC overlay at the cantilever root. Then cracks developed in sequence at the top surface of the UHPC layer cantilever root, the UHPC–NC interface, and the mid-plane of the girder-to-pier joint. Ultimately, UHPC cracks exhibited a “numerous and closely spaced” distribution, whereas NC cracks were “few and widely spaced.” When the residual tensile strength of UHPC at cracked section was considered, the mean value and average coefficient of variation in the ratios of calculated to measured reinforcement stresses for different sections were 1.07 and 0.10, respectively, which can be further used for crack width calculation. The mean ratios of code-predicted to measured UHPC crack widths for different sections using the Chinese code, French code, and European code were 1.10, 0.98, and 1.13, respectively, with corresponding average coefficients of variation of 0.25, 0.33, and 0.28; the Chinese code is recommended for UHPC crack width prediction. For the UHPC–NC interface, an expression for crack width calculation was derived using the comprehensive theory, and the mean ratio of calculated to measured values and the coefficient of variation were 1.08 and 0.18, respectively, demonstrating good predictive accuracy.

1. Introduction

Prefabricated steel–concrete composite girders have been widely used in modern short- and medium-span bridge engineering due to notable advantages such as construction convenience and quality controllability [1]. However, in continuous composite girders, the concrete slab in the hogging moment region is in tension and the steel girder in compression, which fails to exploit the complementary strengths of the two materials, and renders the concrete slab prone to cracking. Such cracks not only reduce sectional stiffness but also accelerate reinforcement corrosion as well as concrete deterioration, thereby seriously compromising the long-term durability and service life of the structure [2,3].

UHPC, an innovative fiber-reinforced cementitious composite with excellent mechanical properties and durability [4,5,6], has undergone rapid development in recent years, with widespread application in bridge engineering. Shao [7] and Li [8] studied the flexural performance and cracking characteristics of steel–UHPC lightweight composite slabs, and summarized the influence of reinforcement ratio and cover thickness on crack development. Liu [9] and Zhang [10] investigated the flexural performance of steel–UHPC I-shaped composite girders in sagging and hogging moment regions, and identified the typical flexural failure modes. Zhu [11], Yoo [12], and Zhang [13] investigated steel–UHPC waffle slabs and examined the influence of structural parameters on their failure modes and bearing capacity. Existing studies show that compared with steel–NC composite girders, steel–UHPC composite girders feature thinner slabs, greater spanning capacity, and superior durability and fatigue performance.

However, the high cost of UHPC limits its large-scale application. Therefore, exploring a relatively economical approach to incorporating UHPC into steel–concrete composite girders at critical locations constitutes an important design principle for optimizing steel–UHPC composite structures. Liu [14] found that steel–UHPC–NC composite girders exhibit lower material cost and higher flexural efficiency compared with steel–UHPC composite girders. Men [15] found that steel–UHPC–NC composite girders showed a 67% higher cracking load than steel–NC composite girders, and only a 5–6% lower flexural capacity than steel–UHPC composite girders. Push-out tests conducted by Yang [16] demonstrated that the ultimate shear strengths of 19 mm and 22 mm diameter studs in UHPC–NC were only 3.8% and 2% lower, respectively, than those of studs embedded in UHPC. These studies confirm that steel–UHPC–NC composite girders achieve a favorable balance between structural performance and material cost.

Nevertheless, the mechanical behavior of the UHPC–NC interface—particularly the initiation and propagation of cracks—is critical to the durability and long-term serviceability of such composite girders. The interfacial bond between UHPC and NC is primarily composed of chemical adhesion, mechanical interlocking, and dowel action of the reinforcing bars [17]. Mohammed [18] and He [19] found that the substrate surface treatment has a significant effect on the bond strength of the UHPC–NC interface. Hussein [20,21] investigated the cohesive and frictional behavior of unreinforced UHPC–NC interfaces under different degrees of NC surface roughness and developed a traction–separation model. Tayeh [22] conducted slant shear and splitting tensile tests and indicated adequate short-term bond strength, characterized by failure predominantly occurring within the NC. Yang [23] demonstrated that the combined action of mechanical interlocking and dowel effect effectively restrains crack propagation, thereby improving shear resistance of the UHPC–NC interface. While these studies have advanced the understanding of interfacial bond performance, research on crack width calculation for UHPC–NC interfaces remains limited. A comparison of existing UHPC design codes further shows that, although the Chinese code [24] introduces a modification factor to account for the influence of steel fibers, and both the French code [25] and European code [26] incorporate the post-cracking residual tensile strength of UHPC into their crack width formulas, none of these codes can directly reflect the effect of the UHPC–NC interface on crack width.

This study proposes a steel–concrete composite girder with integrated piers, incorporating UHPC in the hogging moment region. A flexural loading test was carried out on a full-scale segmental model to investigate the crack initiation and propagation. The residual tensile strength of UHPC was determined based on a simplified uniaxial tensile constitutive model. Subsequently, a reinforcement stress calculation method that accounts for the tensile contribution of UHPC at cracked sections was proposed, and the applicability of crack width calculation methods in current design codes was evaluated. For the UHPC–NC interface, a corresponding crack width formula was derived based on the comprehensive theory, providing a reference for the design of similar engineering.

2. Test Overview

2.1. Background Project

The background project is a 3 × 30 m composite girder bridge. The width of a single carriageway is 16.5 m, consisting of four π-shaped composite girders; the standard cross-section is shown in Figure 1a. A 6 m-long and 10 cm-thick UHPC layer is cast on the top surface of the pier, as shown in Figure 1b.

The construction procedure of the background project is as follows: the steel girders were fabricated in a steel factory and transported to the precast yard, where the concrete slab is cast to form π-shaped composite girders. The girders were then transported to the bridge site as an integral unit and erected using a launching gantry. After the main girders were positioned, the longitudinal wet joints between adjacent π-shaped composite girders in the transverse direction were casted first, followed by casting the concrete crossbeam at the intermediate support, and finally casting the UHPC layer on the top surface of the pier.

2.2. Specimen Design and Fabrication

The length of the specimen was determined based on the ratio of bending moment to shear force under dead and lane loads. The total length was taken as 9 m. The specimen configuration is shown in Figure 2. The concrete slab thickness is 25 cm, thickened to 40 cm at the steel girder. A UHPC layer with a thickness of 100 mm and a length of 6 m is cast on the top surface of the pier.

Three layers of longitudinal reinforcement were arranged in the concrete slab. The top-layer bars are 20 mm in diameter, while the remaining longitudinal bars are 25 mm in diameter. The transverse reinforcement diameters in the precast slab and the cast-in-place UHPC layer are 16 mm and 20 mm, respectively. The stud connectors at the top and bottom flanges of the steel girder are 22 × 220 mm, and those welded to the end plate are 19 × 180 mm.

The fabrication procedure of the test specimen was consistent with the construction process of the prototype bridge: (1) the steel girder was fabricated in the factory and transported to the construction site; (2) the concrete slab and the substructure of the composite girder were casted separately, and after the concrete reached the required strength, the composite girder and the substructure were assembled, followed by casting the pier-to-girder joint; (3) the top-layer reinforcement was tied, the UHPC topping layer was cast, and the specimen was then cured outdoors under natural conditions for 28 days after completion. The main fabrication steps are shown in Figure 3. In order to simulate the most unfavorable conditions of the actual bridge, no special treatment was applied to the UHPC–NC interface.

2.3. Loading and Measurement Scheme

The loading setup of the specimen is shown in Figure 4. Two hydraulic jacks were arranged on both sides of the specimen to perform synchronous monotonic loading. The load was applied to the specimen through a distribution girder positioned between the hydraulic jacks and the specimen. Before the formal loading, a pre-load was carried out to ensure that the loading setup and measuring instruments were working normally and to compact the possible gaps between the components. During formal loading, the load was applied in 50 kN increments up to 60% of the cracking load; beyond 60% of the cracking load, the increment was reduced to 20 kN until concrete cracking occurred in the hogging moment region, after which loading continued in 50 kN increments until specimen failure. The failure criterion was defined as yielding of the reinforcement or the steel plate.

As shown in Figure 5, a total of 7 representative sections along the longitudinal direction were selected for strain monitoring. The displacement at the two loading points was measured. During testing, the development of crack width was recorded using a crack width gauge with an accuracy of 0.01 mm.

2.4. Material Property Tests

The Q355D steel type was used for the steel plates, and the reinforcing bars were HRB400. The manufacture was China Baowu Steel Group Corporation Limited. The mechanical properties of the steel plates and reinforcing bars are listed in

Table 1. Material properties of steel.

Material	Diameter/Thickness (mm)	Yield Strength (MPa)	Tensile Strength (MPa)
Reinforcement	20	460	642
	25	415	622
Steel plate	16	428	532
	22	433	543

.

All concrete material specimens were cured under the same conditions as the prototype bridge. Material tests were conducted after 28 days of outdoor natural curing. For the C50 concrete, 150 × 150 × 150 mm cubes and 150 × 150 × 300 mm prisms were used to measure the cube compressive strength and elastic modulus, respectively. For UHPC, 100 × 100 × 100 mm cubes, 100 × 100 × 300 mm prisms, and 50 × 100 mm dog-bone-shaped specimens [27] were used to measure the cube compressive strength, elastic modulus, and uniaxial tensile strength, respectively. The concrete material test properties are listed in

Table 2. Material properties of concrete.

	Tensile Strength (MPa)	Cube Compressive Strength (MPa)	Elastic Modulus (GPa)
UHPC	7.81	121.3	45.1
C50	/	52.7	34.2

.

Table 3. Material compositions of 1 m³ UHPC (unit: kg).

Cement	Fly Ash	Silica Fume	Quartz Sand	Admixture	Water	Steel Fibers	Liquid Superplasticizer
600	150	100	1275	375	207	188	26

lists the material compositions of 1 m³ UHPC. The volume fraction of steel fibers in UHPC is 2.5%, with fiber dimensions of 13 mm in length and 0.2 mm in diameter.

3. Test Results and Discussion

3.1. Overall Failure Process

The structural behavior of the specimen during loading can be divided into two stages.

Stage I: Elastic stage. In this stage, the UHPC–NC slab remained intact with no visible cracking, and the structure behaved in a linear-elastic manner.

Stage II: Crack propagation stage. When the load reached 500 kN, cracks first appeared on the side surface of the NC layer at the cantilever root section, and then extended to the top surface of the UHPC layer along the thickness direction. When the load reached 750 kN, cracking occurred at the UHPC–NC interface. When the load reached 1050 kN, the maximum crack width on the top surface of the UHPC layer reached 0.2 mm. At this point, both the reinforcement and the steel girder remained elastic. When the load increased to 2058.7 kN, the top-layer longitudinal reinforcement at the cantilever root section just reached tensile yielding. The test was terminated at this load level, with a corresponding vertical displacement of approximately 18 mm.

During loading, no obvious slip or separation between the I-shaped steel girder and the concrete slab was observed.

3.2. Load–Displacement Curve

The load–displacement curves at both sides are shown in Figure 6. As can be seen, the specimen maintained good stiffness throughout the entire loading process. Before the load reached 500 kN, the load–displacement relationship exhibited a linear trend; when the load exceeded 500 kN, the response became nonlinear. This is mainly because the cracks appeared on the side surface of the NC layer beneath the UHPC overlay at approximately 500 kN. With further loading, cracks developed successively at the top surface of the UHPC layer and at the UHPC–NC interface. As the load increased, cracks in these regions continuously propagated, leading to a reduction in structural stiffness. In addition, the load–displacement curves at the two ends of the specimen did not completely coincide, probably due to manufacturing errors.

3.3. Strain Distribution

The strain distributions along the depth of the girder at different load levels for each section are shown in Figure 7, and the bottom plate of the steel girder was taken as a zero reference point. As shown, in the initial loading stage, the sectional strain distribution varied linearly with the section depth. With continued loading, the longitudinal reinforcement strain increased abruptly. This is mainly because the tensile stress is carried by both the reinforcement and the concrete before cracking; once cracks occurred, either in the UHPC layer or in the NC layer, the concrete slab loses part of its tensile capacity, and the corresponding tensile stress is then transferred to the reinforcement.

Based on the load levels at which the reinforcement strain shows abrupt increases, the cracking loads of different sections can be approximately inferred. The cracking loads corresponding to Sections 1, 2, and 3 were 750 kN, 1050 kN, and 500 kN, respectively, which agree well with the cracking loads observed in the test. As the load further increased, the concrete slab became fully cracked and almost completely ceased to contribute, causing the neutral axis to shift downward. When the load reached approximately 2058.7 kN, the strain of the top-layer longitudinal reinforcement at the cantilever root section (Section 3) exceeded 2233 με, indicating that the reinforcement reached tensile yielding.

3.4. Crack Development in the Concrete Slab

At each specified loading stage, the crack widths were measured under the certain maintained load. The widest crack in the critical section was selected visually, and three independent readings were taken at the location of maximum opening and averaged.

The crack development of the test girder is shown in Figure 8. Cracks first occurred on the surface of the NC layer at the cantilever root sections (Sections 3 and 4). The corresponding cracking load was 500 kN, and the initial crack width was 0.03 mm. Immediately thereafter, the side crack at the cantilever root propagated along the thickness direction into the UHPC layer. When the load reached 600 kN, the maximum crack width on the UHPC top surface at the cantilever root reached 0.05 mm.

With further loading, cracks appeared earlier outside the pier-to-girder joint than within the joint. When the load reached 750 kN, cracks formed at the UHPC–NC interface. When the load reached 1050 kN, the maximum crack width on the top surface of the UHPC layer reached 0.2 mm. In the later loading stage, frequent sounds associated with pulling of steel fiber were heard in the UHPC layer, and cracks within pier-to-girder joint developed more rapidly. For safety considerations, crack observation was stopped at 1650 kN; at this load level, the maximum crack width on the UHPC top surface was 0.4 mm. As shown in Figure 8d, UHPC cracks exhibited a “numerous and closely spaced” distribution, whereas NC cracks were “few and widely spaced.”

Figure 9 presents the load–maximum crack width for representative sections. The locations of points A–D are marked in Figure 8a. From the starting points of each curve, it can be observed that cracking occurred earliest on the side surface at the cantilever root section, followed by the top surface at the cantilever root section, then the UHPC–NC interface, and finally the mid-plane of the pier-to-girder joint.

Throughout the test, crack width growth on the side surface at the cantilever root and at the UHPC–NC interface was relatively slow; when the load reached 1650 kN, the maximum crack widths at these locations reached 0.16 mm and 0.10 mm, respectively. In the early loading stage, crack development on the top surface at the cantilever root and on the mid-plane of the pier-to-girder joint was also relatively slow; however, once the crack width reached 0.10 mm, crack growth at each section accelerated. At a load of 1650 kN, the maximum crack widths at these two locations reached 0.40 mm and 0.20 mm, respectively.

4. Evaluation and Analysis of Existing Crack Width Calculation Methods

4.1. Existing Crack Width Calculation Methods

Although the UHPC crack width formulas in current design codes domestically and internationally differ slightly, the parameters involved are essentially consistent.

4.1.1. Chinese Code

China’s Technical Specification for Highway Ultra-High Performance Concrete (D60-02-2023) introduces a fixed modification factor, based on the Specifications for Design of Highway Reinforced Concrete and Prestressed Concrete Bridges and Culverts (JTG 3362-2018) to account for the influence of steel fibers on crack width.

$$w_{\max }=C_1{C}_2{C}_3{\displaystyle \frac{{\sigma }_{\mathrm{sr}}}{E_{\mathrm{s}}}}({\displaystyle \frac{c+d_{\mathrm{eq}}}{0.3+1.4{\rho }_{\mathrm{te}}}})(1-{\beta }_{\mathrm{w}}{\lambda }_{\mathrm{f}})$$

(1)

In the equation, w_max is the maximum crack width; σ_sr is the reinforcement stress; E_s is the elastic modulus of steel; c is the concrete cover thickness; d_eq is the equivalent diameter of longitudinal reinforcement; C₁ is the shape coefficient of reinforcement (taken as 1); C₂ is the long-term effect influence coefficient (taken as 1); and C₃ is the force characteristic coefficient, taken as 1.2 for axial tension. β_w is the steel fiber influence coefficient (0.4 for ultra-high performance reactive powder concrete); and λ_f is the characteristic coefficient of steel fiber content, defined as the product of the steel fiber volume fraction and its aspect ratio.

4.1.2. French Code

The French code provides a UHPC crack width expression based on the comprehensive theory and considers the influence of steel fiber orientation on crack width.

$$w_{\max }=s_{\mathrm{r},\mathrm{m}}({\epsilon }_{\mathrm{sm}}-{\epsilon }_{\mathrm{cm}})$$

(2)

$${\epsilon }_{\mathrm{sm}}-{\epsilon }_{\mathrm{cm}}={\displaystyle \frac{{\sigma }_{\mathrm{sr}}}{E_{\mathrm{s}}}}-{\displaystyle \frac{f_{\mathrm{ctfm}}}{K_{\mathrm{global}}{E}_{\mathrm{cm}}}}-{\displaystyle \frac{1}{E_{\mathrm{s}}}}\left[k_{\mathrm{t}}(f_{\mathrm{ctm},\mathrm{el}}-{\displaystyle \frac{f_{\mathrm{ctfm}}}{K}})({\displaystyle \frac{1}{{\rho }_{\mathrm{te}}}}+{\displaystyle \frac{E_{\mathrm{s}}}{E_{\mathrm{cm}}}})\right]$$

(3)

$$s_{\mathrm{r},\mathrm{m}}=2.55(l_0+l_{\mathrm{t}})$$

(4)

$$l_0=1.33\frac{c}{\delta }$$

(5)

$$l_{\mathrm{t}}=2\left[0.3k_2(1-{\displaystyle \frac{f_{\mathrm{ctfm}}}{K_{\mathrm{global}} f_{\mathrm{ctm},\mathrm{el}}}}){\displaystyle \frac{1}{\delta \eta }}\right]{\displaystyle \frac{d_{\mathrm{eq}}}{{\rho }_{\mathrm{te}}}}\ge {\displaystyle \frac{l_{\mathrm{f}}}{2}}$$

(6)

$$\delta =1+0.4({\displaystyle \frac{f_{\mathrm{ctfm}}}{{K^{\prime }}_{\mathrm{global}} f_{\mathrm{ctm},\mathrm{el}}}})\le 1.5$$

(7)

In the equation, σ_r,m is the crack spacing; ε_sm and ε_cm are the mean strain difference in the reinforcement and the concrete, respectively, within the crack spacing; f_ctm,el and f_ctfm are the elastic-limit tensile strength of UHPC and the mean post-cracking residual tensile strength, respectively; ρ_te is the longitudinal reinforcement ratio calculated based on the effective tensile area; E_cm is the elastic modulus of UHPC; l_f is the steel fiber length; K is the fiber orientation factor; K_global and K^′_global are the fiber orientation factors associated with the longitudinal and transverse directions, respectively; k_t is the load characteristic factor (0.6 for short-term loading); k₂ is the strain distribution factor at the cracked interface (1.0 for axially tension); and in the absence of prestress, η is taken as 2.25.

4.1.3. European Code

The European code also provides a crack width expression for SFRC based on the comprehensive theory and takes into account the effect of SFRC shrinkage strain.

$$w_{\max }=s_{\mathrm{r},\mathrm{m}}({\epsilon }_{\mathrm{sm}}-{\epsilon }_{\mathrm{cm}}-{\epsilon }_{\mathrm{cs}})$$

(8)

$${\epsilon }_{\mathrm{sm}}-{\epsilon }_{\mathrm{cm}}-{\epsilon }_{\mathrm{cs}}={\displaystyle \frac{{\sigma }_{\mathrm{sr}}-{\beta }_1{\sigma }_{\mathrm{sr}}^{\prime }}{E_{\mathrm{s}}}}-{\eta }_{\mathrm{r}}{\epsilon }_{\mathrm{sh}}\ge 0.4{\displaystyle \frac{{\sigma }_{\mathrm{sr}}}{E_{\mathrm{s}}}}$$

(9)

$${\sigma }_{\mathrm{sr}}^{\prime }={\displaystyle \frac{f_{\mathrm{ctm}}-f_{\mathrm{Ftsm}}}{{\rho }_{\mathrm{te}}}}(1+{\alpha }_{\mathrm{E}}{\rho }_{\mathrm{te}})$$

(10)

$$s_{\mathrm{r},\mathrm{m}}=2\left[kc+{\displaystyle \frac{1}{4}}{\displaystyle \frac{(f_{\mathrm{ctm}}-f_{\mathrm{Ftsm}})}{{\tau }_{\mathrm{bm}}}}{\displaystyle \frac{d_{\mathrm{eq}}}{{\rho }_{\mathrm{te}}}}\right]$$

(11)

In the equation, f_ctm and f_Ftsm are the ultimate tensile strength of SFRC and the mean post-cracking residual tensile strength, respectively; α_E is the elastic modulus of steel to that of SFRC; τ_bm is the mean bond stress between the reinforcement and SFRC; k is an empirical coefficient accounting for the influence of concrete cover thickness (taken as 1.0); b₁ is an empirical coefficient reflecting load characteristics (taken as 0.6); and η_r is a coefficient accounting for shrinkage contribution, which is taken as 0 for short-term loading.

4.2. Reinforcement Stress Calculation

In existing crack width calculation formulas, reinforcement stress is a key parameter. For UHPC structures, the bridging effect of steel fibers enables the UHPC matrix to sustain a certain tensile stress after cracking, thereby effectively reducing reinforcement stress. Therefore, when calculating the reinforcement stress at cracked sections, the contribution of the post-cracking residual tensile strength of UHPC should be considered. Throughout the test, good composite action was maintained between the steel girder and the concrete slab, with no obvious slip or separation observed.

According to the following basic assumptions, the reinforcement stress at cracked sections can be calculated: (1) the cross-section of the composite girder satisfies the plane-section assumption; (2) the tensile strength of normal concrete is neglected; (3) the “bridging effect” of steel fibers after UHPC cracking is considered, i.e., the post-cracking residual tensile strength contribution of UHPC is included.

Therefore, the cracked section of a steel–UHPC–NC composite girder consists of UHPC, longitudinal reinforcement, and the steel girder. As shown in Figure 10, the moment–curvature relationship of the section can be obtained from the axial force equilibrium Equation (12) and the moment equilibrium Equation (13). Then, according to Equation (14), the reinforcement stress in the UHPC layer under different load levels can be determined.

A_uλ_tf_ct,UHPC + A_rus_su + A_rcs_sc + N_tt + N_tw = N_cw + N_cb

(12)

M = M_tt + M_tw + M_cw + M_cb + A_uλ_tf_ct,UHPCy_u + A_ruσ_suy_ru + A_rcσ_scy_rc

(13)

σ_su = E_sϕy_ru

(14)

where A_u is the UHPC area; f_ct,UHPC is the ultimate tensile strength of UHPC; λ_t is the ratio of the post-cracking residual tensile strength of UHPC to its ultimate tensile strength; A_ru is the area of longitudinal reinforcement in the UHPC layer; σ_su is the stress of the longitudinal reinforcement in the UHPC layer; A_rc is the area of longitudinal reinforcement in the normal concrete; σ_sc is the stress of the longitudinal reinforcement in the normal concrete; N_tt and M_tt are the tensile force and bending moment carried by the top plate of the steel girder, respectively; N_tw and M_tw are the tensile force and bending moment carried by the web in the tension zone of the steel girder, respectively; N_cb and M_cb are the compressive force and bending moment carried by the bottom plate of the steel girder, respectively; N_cw and M_cw are the compressive force and bending moment carried by the web in the compression zone of the steel girder, respectively; M is the sectional bending moment; f is the sectional curvature; E_s is the elastic modulus of steel; y_t is the distance from the neutral axis of the cracked section to the top surface of the steel girder; y_c is the distance from the neutral axis of the cracked section to the bottom surface of the steel girder; y_rc is the distance from the neutral axis of the cracked section to the centroid of longitudinal reinforcement in the normal concrete layer; y_ru is the distance from the neutral axis of the cracked section to the centroid of longitudinal reinforcement in the UHPC layer; and y_u is the distance from the neutral axis of the cracked section to the centroid of the UHPC.

The constitutive model for the softening branch of UHPC adopted the bilinear tensile stress–crack width model recommended by JSCE [28], as shown in Figure 11. In this model, the maximum crack width w_max is assumed to be half of the steel fiber length l_f (13 mm). During the softening stage, localized crack concentration occurs in UHPC, and the bridging effect of steel fibers prevents a sharp drop in tensile stress after cracking. The post-cracking residual tensile strength of UHPC was determined based on the energy equivalence principle, and is defined as the average stress within the range where the crack width is less than 0.2 w_max [29]. Based on this constitutive model, according to S_OABCD = S_OEFD, the ratio of the post-cracking residual tensile strength to the ultimate tensile strength of UHPC λ_t was calculated to be 0.96.

During the specimen fabrication, some strain gauges on the reinforcement were damaged; the calculated and remaining measured longitudinal reinforcement stresses at different sections were compared in Figure 12.

Table 4. The mean values and coefficient of variations in the ratio of the calculated values to the measured values of reinforcement stress at different sections.

Section 3			Section 2
Label	Mean	Coefficient of Variation (COV)	Label	Mean	Coefficient of Variation (COV)
L3-4	1.14	0.06	L2-3	1.10	0.14
L3-4	1.12	0.07	L2-4	1.02	0.11
L3′-2	1.02	0.10	L2′-4	1.13	0.13
L3′-4	0.89	0.10	L2′-5	1.15	0.07
Average	1.04	0.08	Average	1.10	0.11

lists the mean value and coefficient of variation (COV) of the ratios of calculated to measured reinforcement stresses for different sections. As shown in the table, the average mean ratio and the average COV for Section 3 are 1.04 and 0.08, respectively, while those of Section 2 are 1.10 and 0.11, indicating good predictive accuracy, and the results can be further used for crack width calculation.

4.3. Comparison of Existing Crack Width Calculation Methods

As shown in Figure 13, the crack widths on the UHPC surface of Sections 2 and 3 under different load levels were calculated using the Chinese code, the French code, and the European code, respectively. For Section 2, compared with the French code, the predictions from the Chinese code and the European code were closer to the measured values. For Section 3, when the crack width was less than 0.2 mm, the predictions were conservative; once the crack width exceeded 0.2 mm, the calculated values were all lower than the measured values, and the discrepancy increased with increasing load. It is therefore recommended that the codes examined in this study should be limited to predicting crack widths below 0.2 mm.

Table 5. Comparison of the calculated and measured values of the crack spacing at the UHPC–NC interface.

	Crack Spacing (mm)	Relative Deviation (%)
Measured value	170	/
Chinese code	/	/
French code	568	234.1
Europe code	207	21.8
Method proposed	156	8.2

presents the mean values and COVs of the ratios of the code-predicted to measured crack widths, considering only crack widths below 0.2 mm. The average mean ratios for UHPC crack widths at different sections predicted by the Chinese, French, and European codes were 1.10, 0.98, and 1.13, respectively, with corresponding average COVs of 0.25, 0.33, and 0.28. Therefore, the Chinese code is recommended for crack width calculation.

5. Crack Width Calculation for UHPC–NC Interface

Current design codes have not yet provided a method for calculating crack width at the UHPC–NC interface. In this study, a crack width calculation formula for the UHPC–NC interface is derived based on the comprehensive theory.

In bond–slip theory, the crack width at the reinforcement is determined by the difference between the extensions of the reinforcement and the concrete between adjacent cracks, as expressed in Equation (15), where S_r,m is the crack spacing, and ε_sm and ε_cm are the mean strain difference in the reinforcement and the concrete, respectively, within the crack spacing.

$$w_{\mathrm{m}}=s_{\mathrm{r},\mathrm{m}}({\epsilon }_{\mathrm{sm}}-{\epsilon }_{\mathrm{cm}})$$

(15)

5.1. Cracking Space

Existing studies have shown that surface roughness is the most critical factor governing the bond strength between UHPC and NC [18,19]. Due to the segmented casting procedure and the unroughened NC substrate, the UHPC at the joint is assumed to contribute neither tensile resistance nor interfacial bond strength in the crack width calculation. That is, once cracking initiates, the UHPC layer provides no restraint to the NC tension zone, and the predicted crack width represents a conservative upper-bound estimate.

At the initial cracking stage, cracks first occurred at the region with deficiency on the UHPC structure. Because of the presence of bond stress, over a certain length along the reinforcement the concrete tensile stress increases to its tensile strength. The corresponding bond stress transfer length, l_tr, is regarded as the theoretical minimum crack spacing. When the spacing between two cracks is less than 2l_tr, the available bond transfer length is insufficient, and the bond stress between the reinforcement and concrete will not make the concrete stress reach its tensile strength; therefore, no new crack will form. In this case, the maximum crack spacing is 2l_tr. Hence, the crack spacing falls within the range of l_tr~2l_tr.

Considering the large dispersion in the spacing of major cracks, the average crack spacing is taken as 1.5l_tr [28]. Taking into account the influence of the concrete cover thickness c, the crack spacing can be calculated using Equation (16).

$$s_{\mathrm{r},\mathrm{m}}=1.5(c+l_{\mathrm{tr}})$$

(16)

As shown in Figure 14, the UHPC within the bond transfer length range l_tr1 on the right side is taken as a free body. From force equilibrium, it can be obtained that:

$$A_{\mathrm{u}}(f_{\mathrm{ct},\mathrm{UHPC}}-0)=m\pi d_{eq}{l}_{\mathrm{tr}1}{\tau }_{\mathrm{bm}1}$$

(17)

$${\tau }_{\mathrm{bm}1}=k_{\mathrm{b}1}{f}_{\mathrm{ct},\mathrm{UHPC}}$$

(18)

where A_u is the UHPC area; m is the number of reinforcing bars; τ_bm1 is the mean bond stress between the reinforcement and UHPC, which is assumed to be linearly related to the ultimate tensile strength of UHPC, and k_b1 is taken as 1.8 [30]. Substituting Equation (18) into Equation (17), the load transfer length in UHPC, l_tr1 can be obtained as:

$$l_{\mathrm{tr}1}={\displaystyle \frac{d_{\mathrm{eq}}}{4k_{\mathrm{b}1}{\rho }_{\mathrm{te}}}}$$

(19)

where ρ_te is the longitudinal tensile reinforcement ratio calculated based on the effective tensile area.

Therefore, the crack-spacing expression on the UHPC side is given as follows:

$$s_{\mathrm{r},\mathrm{m}}=1.5(c+l_{\mathrm{tr}1})=1.5(c+{\displaystyle \frac{d_{\mathrm{eq}}}{4k_{\mathrm{b}1}{\rho }_{\mathrm{te}}}})$$

(20)

Following the above procedure, the normal concrete within the bond stress transfer length range l_tr2 on the left side is taken as a free body. From force equilibrium, the crack-spacing expression on the normal concrete side can be obtained as:

$$s_{\mathrm{r},\mathrm{m}}=1.5(c+l_{\mathrm{tr}2})=1.5(c+{\displaystyle \frac{d_{\mathrm{eq}}}{4k_{\mathrm{b}2}{\rho }_{\mathrm{te}}}})$$

(21)

where k_b2 is the ratio of the mean bond stress between the reinforcement and normal concrete to the ultimate tensile strength of normal concrete, taken as 2.5 [31].

5.2. Strain Difference over the Transfer Length

At the initial cracking stage, cracks first occur at the weak region of the UHPC structure, and the strain difference exists only within the load transfer zone near the crack. Based on Leutbecher’s theoretical model, the mean strain difference $\Delta \epsilon$ can be expressed as:

$${\epsilon }_{\mathrm{sm}}={\epsilon }_{\mathrm{s}}^{\mathrm{II}}-\beta ({\epsilon }_{\mathrm{s}}^{\mathrm{II}}-{\epsilon }_{\mathrm{s}}^{\mathrm{I}})$$

(22)

$${\epsilon }_{\mathit{cm}}=\beta {\epsilon }_c^{\mathrm{I}}$$

(23)

$$\Delta \epsilon ={\epsilon }_{\mathit{sm}}-{\epsilon }_{\mathit{cm}}=(1-\beta )\frac{{\sigma }_{\mathit{sr}}}{E_s}$$

(24)

where ${\epsilon }_s^{\mathrm{II}}$, ${\epsilon }_s^{\mathrm{I}}$ and ${\epsilon }_c^{\mathrm{I}}$ denote the reinforcement strain at the cracked section, the reinforcement strain, and concrete strain at the uncracked section, respectively; σ_sr is the reinforcement stress at the cracked section; and β is the coefficient associated with the equivalent rectangular strain distribution. When the strain distribution of reinforcement or concrete is assumed to be parabolic, β is taken as 0.6 [29].

For the stabilized cracking stage, as shown in Figure 15b, the mean strain difference can be expressed as:

$$\begin{gathered} {\epsilon }_{\mathrm{sm}}={\epsilon }_{\mathrm{s}}^{\mathrm{II}}-\beta \Delta \epsilon ={\displaystyle \frac{{\sigma }_{\mathrm{sr}}}{E_{\mathrm{s}}}}-\beta {\displaystyle \frac{\Delta {\sigma }_{\mathrm{s}}}{E_{\mathrm{s}}}}={\displaystyle \frac{{\sigma }_{\mathrm{sr}}}{E_{\mathrm{s}}}}-\beta {\displaystyle \frac{A_{\mathrm{s}}\Delta {\sigma }_{\mathrm{s}}}{A_{\mathrm{s}}{E}_{\mathrm{s}}}} \\ ={\displaystyle \frac{{\sigma }_{\mathrm{sr}}}{E_{\mathrm{s}}}}-\beta {\displaystyle \frac{A_{\mathrm{s}}\Delta {\sigma }_{\mathrm{c}}}{A_{\mathrm{s}}{E}_{\mathrm{s}}}}={\displaystyle \frac{{\sigma }_{\mathrm{sr}}}{E_{\mathrm{s}}}}-\beta {\displaystyle \frac{(1-{\lambda }_{\mathrm{t}})f_{\mathrm{ct}}}{{\rho }_{\mathrm{te}}{E}_{\mathrm{s}}}} \end{gathered}$$

(25)

$${\epsilon }_{\mathrm{cm}}=\beta {\displaystyle \frac{(1-{\lambda }_{\mathrm{t}})f_{\mathrm{ct}}}{E_{\mathrm{c}}}}=\beta {\displaystyle \frac{(1-{\lambda }_{\mathrm{t}})f_{\mathrm{ct}}}{E_{\mathrm{s}}}}{\alpha }_{\mathrm{E}}$$

(26)

$$\Delta \epsilon ={\epsilon }_{\mathrm{sm}}-{\epsilon }_{\mathrm{cm}}={\displaystyle \frac{{\sigma }_{\mathrm{sr}}}{E_{\mathrm{s}}}}-\beta {\displaystyle \frac{(1-{\lambda }_{\mathrm{t}})f_{\mathrm{ct}}}{E_{\mathrm{s}}}}({\displaystyle \frac{1}{{\rho }_{\mathrm{te}}}}+{\alpha }_{\mathrm{E}})$$

(27)

For the UHPC side, f_ct in the equation denotes the ultimate tensile strength of UHPC, and α_E is the ratio of the elastic modulus of steel to that of UHPC. For the normal concrete side, f_ct denotes the ultimate tensile strength of normal concrete, and α_E is the ratio of the elastic modulus of steel to that of normal concrete. Therefore, the final expression for the mean strain difference is given as follows:

$$\Delta \epsilon =\max \left\{(1-\beta ){\displaystyle \frac{{\sigma }_{\mathrm{sr}}}{E_{\mathrm{s}}}},{\displaystyle \frac{{\sigma }_{\mathrm{sr}}}{E_{\mathrm{s}}}}-\beta {\displaystyle \frac{(1-{\lambda }_{\mathrm{t}})f_{\mathrm{ct}}}{E_{\mathrm{s}}}}({\displaystyle \frac{1}{{\rho }_{\mathrm{te}}}}+{\alpha }_{\mathrm{E}})\right\}$$

(28)

Based on the above analysis, the crack width at the reinforcement at the UHPC–NC interface, w_m, can be expressed as follows. In the calculation, the crack spacing and the mean strain difference over the load transfer length should first be determined separately for the UHPC side and the normal concrete side, and then the larger of the two products should be adopted.

$$w_{\mathrm{m}}=\max \left\{(1-\beta ){\displaystyle \frac{{\sigma }_{\mathrm{sr}}}{E_{\mathrm{s}}}}{s}_{\mathrm{r},\mathrm{m}},{\displaystyle \frac{{\sigma }_{\mathrm{sr}}}{E_{\mathrm{s}}}}-\beta {\displaystyle \frac{(1-{\lambda }_{\mathrm{t}})f_{\mathrm{ct}}}{E_{\mathrm{s}}}}({\displaystyle \frac{1}{{\rho }_{\mathrm{te}}}}+{\alpha }_{\mathrm{E}})s_{\mathrm{r},\mathrm{m}}\right\}$$

(29)

The crack width at the surface location of the joint w_s is converted from w_m as follows:

$$w_{\mathrm{s}}=w_{\mathrm{m}}{\displaystyle \frac{h-x^{\prime }}{h-h_{\mathrm{r}}-x^{\prime }}}$$

(30)

where h_u is the section depth; h_r is the distance from the centroid of the reinforcement to the top surface of the concrete slab; and x′ is the depth of the uncracked tensile zone.

Figure 16 compares the code-predicted and measured crack widths at the UHPC–NC interface. Table 5 presents a comparative analysis of the calculated and measured crack spacings obtained using different codes. The crack spacing predicted by the French code shows a relatively large deviation from the measured values, which leads to crack widths much larger than the test results; moreover, the discrepancy increases with increasing load. The crack spacings and crack widths predicted by the Chinese code and the European code are both slightly larger than the measured values, indicating conservative predictions. Based on the comprehensive theory, this study modifies the crack-spacing expression; the resulting crack spacing deviates from the measured values by less than 10%. The proposed method yields the smallest mean value and average COV of the ratios of the calculated to measured crack widths, which are 1.08 and 0.18, respectively, and can predict the crack width at the UHPC–NC interface with good accuracy.

6. Conclusions

This study experimentally investigated the static performance of the hogging moment region of the steel–UHPC–NC composite girder with integrated piers. Based on the test results, a method was proposed to calculate reinforcement stresses accounting for the post-cracking residual tensile strength of UHPC at cracked sections. The existing design codes for UHPC crack width calculation was evaluated, and a crack width formula specifically for the UHPC–NC interface was developed. The main conclusions are as follows:

(1)

Crack development followed a distinct sequential pattern. Cracks first appeared on the side surface of the NC layer beneath the UHPC layer at the cantilever root section. Subsequently, cracks occurred on the top surface of the UHPC layer, then at the UHPC–NC interface, and finally on the mid-plane of the pier-to-girder joint. Notably, UHPC exhibited a “numerous and closely spaced” crack distribution, whereas NC showed a “few and widely spaced” pattern.

(2)

A reinforcement stress calculation method that incorporates the tensile contribution of UHPC at cracked sections was proposed. The mean value and average coefficient of variation in the ratios of the calculated to measured reinforcement stresses for different sections were 1.07 and 0.10, respectively, indicating good predictive accuracy.

(3)

All three evaluated design codes are recommended for predicting crack widths below 0.2 mm. The mean ratios of the predicted to measured UHPC crack widths for different sections using the Chinese code, French code, and European code were 1.10, 0.98, and 1.13, respectively, with corresponding average coefficients of variation of 0.25, 0.33, and 0.28; the Chinese code is recommended for UHPC crack width prediction.

(4)

For the UHPC–NC interface, a crack width expression was derived based on the comprehensive theory. The mean ratio of calculated to measured values and the coefficient of variation were 1.08 and 0.18, respectively, indicating good agreement with the experimental results.

(5)

Due to the limited number of full-scale test girders, the available crack width data are relatively limited. Further experimental studies are recommended to validate the proposed crack width formula for the UHPC–NC interface.