Precast steel–UHPC lightweight composite bridge for accelerated bridge construction

Abstract

In this study, a fully precast steel–ultrahigh performance concrete (UHPC) lightweight composite bridge (LWCB) was proposed based on Mapu Bridge, aiming at accelerating construction in bridge engineering. Cast-in-place joints are generally the controlling factor of segmental structures. Therefore, an innovative girder-to-girder joint that is suitable for LWCB was developed. A specimen consisting of two prefabricated steel–UHPC composite girder parts and one post-cast joint part was fabricated to determine if the joint can effectively transfer load between girders. The flexural behavior of the specimen under a negative bending moment was explored. Finite element analyses of Mapu Bridge showed that the nominal stress of critical sections could meet the required stress, indicating that the design is reasonable. The fatigue performance of the UHPC deck was assessed based on past research, and results revealed that the fatigue performance could meet the design requirements. Based on the test results, a crack width prediction method for the joint interface, a simplified calculation method for the design moment, and a deflection calculation method for the steel–UHPC composite girder in consideration of the UHPC tensile stiffness effect were presented. Good agreements were achieved between the predicted values and test results.

Graphical abstract

Keywords

accelerated bridge construction / ultrahigh-performance concrete / steel–UHPC composite bridge / UHPC girder-to-girder joint

Introduction

Accelerated bridge construction (ABC) is a bridge design and construction method that takes full advantage of prefabrication. ABC is particularly advantageous for the construction of bridges in modern cities where massive cast-in-situ operations are generally unsuitable because of strict requirements on constructional duration and operation space.

Ultrahigh-performance concrete (UHPC) is a class of cement-based materials, and previous studies have demonstrated that it has excellent anti-cracking, anti-freeze, and anti-corrosion properties [1–4]. UHPC has been widely used in bridge engineering to accelerate bridge construction [5–7]. The use of UHPC can reduce bridge weights by 37%–54% [8,9].

In the present study, the authors replaced the normal-strength concrete deck of a steel–concrete composite bridge with ultra-high performance UHPC. Then, a fully precast steel–UHPC lightweight composite bridge (LWCB) scheme was proposed. The LWCB contained several fully precast steel–UHPC composite girder components, and each component consisted of one UHPC panel and two I-shaped steel girders. The component was prefabricated monolithically. During bridge construction, the prefabricated components can be hoisted using conventional equipment, and the entire bridge eliminates the pre-stressing system. This approach considerably accelerates on-site bridge construction.

In general, continuous bridge design focuses on the concrete deck in the negative moment region. The most critical part is the girder-to-girder joint located at the top of the pier.

Numerous studies have been conducted on the successful application of UHPC in bridge deck-to-deck joints. Graybeal [10] developed a structural test for a full-scale pi-girder fabricated with a longitudinal joint. In the following year, he completed a series of tests on high-performance concrete panels with on-site cast UHPC connections [11]. In 2013, Aaleti et al. [12] published a design guide for precast UHPC waffle deck panel systems and connections. Perry provided the design and details of the narrow joints used in Mackenzie River Twin Bridges in 2013 [13]. Graybeal [14] published a report that provided detailed guidance for the design and deployment of field-cast UHPC connections. In China, Guan [15] conducted a full-scale UHPC longitudinal joint test in Hunan University and showed that the cracking strength of the joint could meet the design requirement.

Meanwhile, only a few studies have focused on UHPC girder-to-girder joints that make bridges continuous. To address this issue, the present study developed an innovative design of a girder-to-girder joint that is suitable for LWCB. A flexural test was conducted under a negative bending moment. Based on the test results, the calculation methods for interface crack width, design moment and deflection of critical points, and ultimate flexural capacity were presented. In addition, the fatigue performance of the UHPC deck was discussed and assessed.

The first part of the paper introduces the design concept of the background bridge—Mapu Bridge, which is the first small–middle span steel–UHPC LWCB in China. The second part describes the flexural test conducted on the proposed girder-to-girder joint of Mapu Bridge. The last part discusses the test results in detail.

Background: Mapu Bridge

Design aspects

Mapu Bridge (Fig. 1) is an overpass that crosses Huiqing Expressway, and it is the first small–middle span steel–UHPC LWCB in China. The span of the bridge is 4 × 25 m. Three prefabricated steel–UHPC composite girder components are arranged with 2.8 m spacing in the transverse direction, whereas four spans are arranged along the longitudinal direction. The bridge has a total of 12 components. The height of the girder is 101 cm, which consists of one UHPC deck with a thickness of 13 cm and two I-shaped steel girders with a height of 88 cm. The two parts are connected by studs. The bridge details are shown in Fig. 1.

Figure 2 displays the internal force distribution of Mapu Bridge. The negative bending moment of the girder reaches the maximum at the top of the pier and gradually decreases from the peak region to the two sides. The interfacial strength between the poured UHPC and the prefabricated part is low due to the discontinuity of the steel fiber. Thus, this study proposed a joint with a lengthened top (called “T-shaped joint”) to move the weak interface position away from the tensile stress peak region and reduce the risk of interface cracking. In the design of Mapu Bridge, the length of the joint top portion is 5 m. At the edge of the joint, the design tensile stress σ₀ should be less than the allowable stress (σ). A schematic of the joint is presented in Fig. 3. Figure 3(a) shows the external of the T-shaped joint. From the side view, the joint is shaped like the letter “T”. In this part, the UHPC is cast on-site. Figure 3(b) displayed the internal part of the T-shaped joint. The internal part consists of two ends of the steel–UHPC composite girder components, and is prefabricated together with the composite girder components in the factory. The interior of the joint includes six steel stiffeners and two steel diaphragms. Headed studs are welded to the surface of these steel stiffeners and diagrams, which can effectively fix the cast-in-site UHPC. The longitudinal reinforcement protruding from the upper part prefabricated UHPC panel, which can ensure the anchorage length, and also resist the negative bending moment together with the connecting reinforcement arranged on site.

The conventional method of constructing steel–concrete composite bridge connections involves strengthening by using a large amount of reinforcement or applying a post-tensioning system. By contrast, the proposed joint decreases the reinforcement ratio and eliminates the post-tension system. In construction site, only the joint connection steel bars and post-cast UHPC need to be applied after erecting the prefabricated components. This approach significantly reduces the construction time on site. Crack resistance and durability are also greatly improved due to the ultrahigh performance of UHPC. In the following section, the design load of Mapu Bridge is analyzed to verify the feasibility of the proposed scheme.

Analysis of Mapu Bridge

According to the specifications of Refs. [16,17], the flexural capacity of the steel–concrete composite bridge should be calculated by using the linear elastic model with the fundamental combination of actions in ultimate limit states. According to Ref. [18], the frequent combination of actions in serviceability limit states should be utilized in structural crack resistance verification. The vehicle lane load without an impact factor should be applied to check the structural deflection, and the value should be less than L/500.

The method of finite element analysis (FEA) is often used to calculate design stress in modern engineering. The precise mathematical model in FEA software can effectively solve mechanical problems [19,20]. Thus, the FEA software Midas was employed in the current calculations. The design stress of Mapu Bridge was calculated by using the linear elastic model. The FE model consists of 2270 nodes and 1889 beam elements. The steel bar, I-shaped steel girders, and UHPC panel in the model are considered as ideal elasticity. The elastic modulus and the Poisson ratio of the steel are 206 GPa and 0.3, respectively [17], and those of UHPC are 42.6 GPa and 0.2, respectively [21]. The slip between the UHPC panels and I-shaped steel girders was ignored. In addition to self-weight, the applied loads included secondary dead load (the weight of the pavement and guardrail), uneven support settlement, shrinkage, creep, temperature effect, and vehicle load (with impact factor). The calculation results for the two-stage (i.e., simply supported stage I and continuous stage II) combined stresses of the bridge are shown in Table 1.

In Table 1, the fundamental combination of actions is

(1)S_{\mathrm{ud}}=\mathrm{1.1}\times S\left({\displaystyle \sum _{i=\mathrm{1}}^m\mathrm{1.2}{G}_{i\mathrm{k}}}+{\mathit{\gamma }}_{\mathrm{Q1}}{Q}_{\mathrm{1k}}+\mathrm{0.75}{\displaystyle \sum _{j=\mathrm{2}}^n\mathrm{1.4}{Q}_{j\mathrm{k}}}\right).

The frequent combination of actions is

(2)S_{\mathrm{fd}}=S\left({\displaystyle \sum _{i=\mathrm{1}}^m{G}_{i\mathrm{k}}}+\mathrm{0.7}{Q}_{\mathrm{1}\mathrm{k}}+{\displaystyle \sum _{j=\mathrm{2}}^n{\psi }_{\mathrm{q}j}{Q}_{j\mathrm{k}}}\right),

where G_ik denotes dead loads, Q_lk pertains to vehicle actions, Q_jk refers to other actions (e.g., wind and snow loads) except for vehicle actions, and γ_Q1 and y_qj are partial safety factors. For the fundamental combination of actions, the partial safety factors of bearing settlement, temperature action, and shrinkage and creep are 0.5, 1.05, and 1, respectively; for the frequent combination of actions, the respective coefficients are 1, 0.8, and 1.

Q345 is a type of steel in China. In reference to a standard presented in Ref. [17], the yield strength of Q345 is 345 MPa. The calculation results in Table 1 reveal that the tensile and compressive stresses of the I-shaped steel girders can meet the design requirement. By contrast, the typical compressive strength of UHPC material is higher than 120 MPa [22,23]. Thus, the compressive strength of the UHPC panel can also meet the design requirement. The deflection of the bridge is 31.33 mm, which is less than L/500 (25000/500= 50 mm). Therefore, the tensile strength of the girder-to-girder joint at the bridge pier top should be paid considerable attention.

The fatigue performance of the UHPC deck in the negative moment region was also considered in this work. The design fatigue amplitude was determined in accordance with the fatigue model I specified in Ref. [17], that is, the equivalent lane load with a concentrated load of 0.7P_k (0.7 × 310 kN) and a uniform load of 0.3  q_k (0.3 × 10.5 kN/m). Therefore, the maximum fatigue amplitude at the bridge pier top and the joint interface were 3.26 and 1.96 MPa, respectively.

Large-scale model test of the girder-to-girder joint

Specimen configuration

The specimen was reduced to a scale of half the size of Mapu Bridge, as displayed in Fig. 4. The specimen consisted of two prefabricated steel–UHPC composite girder parts and one post-cast joint part. The length of the specimen was 6.4 m, and the width was 0.7 m. The reinforcement layout is provided in Fig. 4(b), where “LR” and “LCR” denote longitudinal reinforcement and longitudinal connection reinforcement in the joint part, respectively. Both of them have a diameter of 16 mm. “TR” stands for transverse reinforcement with a diameter of 12 mm. “VR” refers to the transverse steel bar with a diameter of 8 mm. The reinforcement ratio and the joint part are the similar to those of Mapu Bridge.

The composition of the UHPC material was similar to that reported in a past work [24]. Two types of steel fibers, namely, F 0.2 × 13 mm straight and F 0.2 × 13 mm hook-end fibers, existed in the material. The volume content was slightly different for the prefabricated UHPC deck and the joint. The proportions of the straight and hook-end fibers in the joint part were 1% and 1.5%, respectively. Meanwhile, the volume fractions of the two fiber types for the two prefabricated parts were both 1.0%.

The main fabrication processes were as follows. First, the two prefabricated parts were cast 15 d before the joint part was cast and were cured with steam (98 °C for 48 h). Second, the joint part was cast after surface treatment and applied with natural curing. The interface was chiseled by high-pressured water. The surface roughness is shown in Fig. 5.

Mechanical property tests of UHPC were conducted before the flexural test. The quantity, dimensions, and test methods were in reference to the standard presented in Ref. [25]. The results are listed in Table 2. The specimen was placed upside down to produce a negative bending moment and supported on the bottom and loaded from the top (Fig. 6). The deformation, strain, and cracking development were recorded during the test.

Test results

Specimen deflection development

As shown in Fig. 7, the loading process is divided into five stages. Stage I is from the beginning of the test to 336.6 kN. At this stage, the joint interface cracked, the width reached 0.05 mm (hereinafter referred to as visible crack width) at Point 1, and the UHPC matrix crack reached the visible level at Point 2. Stage II is from 336.6 to 782 kN. At this stage, the stiffness decreased slightly, and small and dense cracks propagated in the UHPC deck. The main crack (the crack that is more extensive and developed faster than the rest) gradually appeared at the joint interface, and the lower flange of the I-shaped steel girder reached its yield strength at Point 3. Stage III is from 782 to 815.2 kN. At this stage, the tensile strain of UHPC reached the ultimate value (according to Ref. [21], the ultimate strain of plain UHPC with 2% steel fiber is 765 με) at Point 4. Stage IV is from 815.2 to 977.9 kN. At this stage, the stiffness of the specimen decreased significantly, and the deflection and main crack width increased drastically. When the applied load reached 977.9 kN, the reinforcement in the UHPC panel reached the yield strength. Stage V is from 977.9 to 1033.9 kN. At this stage, the remaining parts of the I-shaped steel girder gradually yielded until 1033.9 kN (Point 6). The lower flange of the I-shaped steel girder bucked close to the loading point, and the test was terminated.

Crack development

Figure 8 displays the crack distribution on the UHPC panel. A small crack was first found at Interface C-1 due to the lower strength of the successively poured UHPC interface, but this crack was almost invisible to the naked eye. Subsequently, some diagonal cracks appeared on the left side of the specimen, which was caused by unevenness of the left support. With the load increases, a large number of dense and small cracks appeared on the UHPC surface. Among them, under the effect of stress concentration, the cracks under the loading points (Matrix C-1 and C-2) developed wider and longer rapidly.

Figure 9 plots the development of four main cracks (Interface C-1 and C-2, Matrix C-1 and C-2). As shown in the figure, the interface crack developed faster than the matrix crack did. When the matrix crack width was less than 0.1 mm, the crack expansion was faster than that after 0.1 mm.

Figure 10 shows the cracking characteristics of three different substrates. Figure 10(a) exhibits the crack in the joint interface. The joint interface composed of pre-cast and fore-cast UHPC. Although the pre-cast UHPC is roughened before pouring the fore-cast UHPC, it is still apparent that the steel fiber distribution in the joint interface is disorderly, and the quantity is much smaller than that in the matrix crack in Fig. 10(b). Figure 10(b) shows that more fibers were present in the matrix crack than in the interface crack, and the fibers were longitudinally distributed, which is beneficial for resisting tensile stress. A considerable amount of fiber was observed in the plain UHPC specimen crack in Fig. 10(c), and, the fibers showed prominent longitudinal distribution characteristics. Compared to the UHPC matrix in Fig. 10(b), the plain UHPC has no steel bars, so the distribution of steel fibers is more uniform, which can better resist tensile stress.

Thus, the following conclusions can be obtained. First, at the joint interface, the fibers were few and irregularly distributed. Hence, the strength was lower than that of the UHPC matrix. Second, reinforcement of the UHPC matrix may lead to an uneven fiber distribution, which further resulted in the initial crack strength being lower than that of plain UHPC.

Strain development in the I-shaped steel girder

Figure 11 shows the strain development in the lower flange of the I-shaped steel girder. The curve deflected at 782 kN as the load increased. This behavior indicated that the lower flange remained elastic before 782 kN. The corresponding strain reached 1672–1908 με. The standard in Ref. [17] indicates that the yield strength of Q345 is 345 MPa, the Young’s modulus is 206 GPa, and the corresponding yield strain is 1675 με. Thus, the lower flange of the I-shaped steel girder gradually yielded after 782 kN.

Discussion

Nominal stress calculation

In this section, the nominal stress of the critical parts were calculated and compared with the design stress given in Table 2 to check the validity of the design of Mapu Bridge.

According to previous research [26], when the maximum crack width of UHPC does not exceed 0.05 mm, the crack does not influence the durability of UHPC. Thus, this durability-based design criterion can be applied [26]. In the present work, maximum crack width≤0.05 mm was used as the limit of serviceability limit states.

Nominal stress was calculated by using \sigma =M_{\mathrm{d}}y/\left({\alpha }_{\mathrm{E}}I\right), where M_d denotes the applied moment, y denotes the distance from the neutral axis to the cracked surface, a_E = E_s/E_UHPC, and I denotes the second-moment area of conversion of the composite section. The applied load and the nominal stress of the critical sections (Fig. 12) are listed in Table 3. Given that the crack width in the mid-span was minimal, the nominal stress was calculated with the limit value (1033.9 kN).

A review of the design stress in Table 1 indicated that the maximum tensile stress of the UHPC deck was 9 MPa in the pier top and 5.08 MPa in the joint interface. The test results in Table 3 showed that all the sections could meet the design requirements.

Fatigue assessment of the UHPC deck

At present, many studies dealing with the fatigue assessment of conventional concrete panels are available. However, further research on UHPC decks is still required. In Ref. [27], UHPC panel fatigue assessment employed the formula of steel bridge fatigue evaluation (Eq. (3)). According to the formula, the number of fatigue load cycles is inversely proportional to the cube of the stress amplitude [17,28]. The fatigue assessment of the UHPC deck in LWCB in the current study was conducted based on this method.

(3)N_i\propto \mathrm{1}/{\left(\mathrm{\Delta }{\sigma }_{\mathrm{R}i}\right)}^{\mathrm{3}}.

The formula can be expressed as

(4){\displaystyle \frac{N_{\mathrm{1}}}{N_{\mathrm{2}}}}\propto {\displaystyle \frac{{\left(\mathrm{\Delta }{\sigma }_{\mathrm{R2}}\right)}^{\mathrm{3}}}{{\left(\mathrm{\Delta }{\sigma }_{\mathrm{R1}}\right)}^{\mathrm{3}}}},

where N_i denotes the fatigue load cycle and Δσ_Ri is the fatigue stress amplitude.

According to the test results of Ref. [29], when the maximum stress amplitude is 0.9 times the service load, the maximum crack width of the UHPC panel is less than 0.05 mm when the load cycles reach 1000000. For a joint test, the load is 1.0 times the service load, and the corresponding number of cycles is 1000000. In reference to these results, a fatigue assessment was performed on the mid-span and joint interface of the deck in the current study. Table 4 lists the fatigue stress amplitude and static test results, and the calculation results are shown in Table 5.

According to the results in Table 5, the minimum number of fatigue cycles of the UHPC deck in LWCB was N = 3482 × 10⁴, which is far more than 2000000 cycles. Thus, the fatigue performance of the UHPC deck can meet the design requirement.

Loading process analysis

The following sections discuss the five stages mentioned above separately. The crack width prediction method for the joint interface, the simplified calculation method for the design moment, and the deflection calculation method for the steel–UHPC composite girder in consideration of the UHPC tensile stiffness effect are presented.

Stage I

Stage I is from the beginning of the test to the appearance of a visible crack in the prefabricated UHPC panel. At this stage, the crack width of the joint interface reached 0.05 mm at Point 1. The following text is divided into three parts: 1) crack width prediction method for the joint interface, 2) simplified calculation method for the design moment, and 3) deflection calculation method for the steel–UHPC composite girder in consideration of the effect of UHPC tensile stiffness.

(1) Prediction of joint interface crack width

Luo et al. [30] tested 40 steel–UHPC composite plates and eight steel–UHPC composite beams. The cracking characteristics in the transverse and longitudinal directions were explored, and a modified formulation based on MOHURD [31] was proposed to predict the maximum crack width in a composite (steel+UHPC) lightweight deck system:

(5)w_{\max }={\alpha }_{\mathrm{cr}}\psi {\displaystyle \frac{{\sigma }_{\mathrm{s}}}{E_{\mathrm{s}}}}{l}_{\mathrm{cr}},

(6){\alpha }_{\mathrm{cr}}={\tau }_{\mathrm{l}}{\tau }_{\mathrm{s}}\beta {\alpha }_{\mathrm{c}},

(7)l_{\mathrm{cr}}=\mathrm{1.06}{c}_{\mathrm{s}}+\mathrm{0.152}{\displaystyle \frac{d_{\mathrm{eq}}}{{\rho }_{\mathrm{te}}}},

(8)\psi =\mathrm{1.1}-\mathrm{0.12}{\displaystyle \frac{f_{\mathrm{tk}}}{{\rho }_{\mathrm{te}}{\sigma }_{\mathrm{s}}}}.

For a detailed discussion of the equations, readers can refer to the study of Luo et al. [30]. Notably, Luo’s formula applies to composite (steel+UHPC) lightweight deck systems, the steel–UHPC LWCB proposed in this work is slightly different.

τ_l is a coefficient related to the long-term effects of loads or actions. For a prefabricated UHPC panel, Luo et al. [30] omitted this coefficient due to the extremely low shrinkage and creep of UHPC after steam curing. In the present study, the recommended value of τ_l =1.5 for the joint interface was used in reference to MOHURD [31]; σ_s is the reinforcement stress. Luo et al. [30] stated that the tensile strength of UHPC should be considered when calculating the stress of the reinforcing bar due to the bridging effect of steel fibers. For the interface position of successively poured UHPC joints in the present study, the cracked section could effectively predict the stiffness of the specimen in the serviceability limit states.

In addition, the crack width of the centroid of longitudinal reinforcement (w_max) should be converted to that of the top surface (w_smax) as follows:

(9)w_{\mathrm{smax}}=w_{\max }{\displaystyle \frac{h-x}{h-x-c}},

where h is the height of the beam, x is the distance from the neutral axis of the composite beam to the top surface of the UHPC deck, and c is the distance from the centroid of longitudinal reinforcement to the top surface of the UHPC deck. The test value and calculation results are shown in Fig. 13.

The prediction value is in agreement with the test results. A small deviation was observed in the beginning of the prediction because the formulas ignored the tensile strength of the interface when calculating the reinforcement stress; they only accounted for the section of steel bars and girders.

Therefore, the cracking bending moment of the joint interface can be inversed based on Eqs. (5)–(9) as follows:

(10)M_{\mathrm{c}\mathrm{a}}={\displaystyle \frac{(h-x-c)}{(h-x)}}\frac{0.05E_{\mathrm{s}}^2{I}_{\mathrm{c}\mathrm{r}}}{{\alpha }_{\mathrm{c}\mathrm{r}}\psi l_{\mathrm{c}\mathrm{r}}{y}_{\mathrm{r}}}\mathrm{.}

(2) Design moment of Point 2

The crack width of the prefabricated UHPC panel had just reached 0.05 mm at Point 2. The corresponding strain was defined as visible initial crack strain ε_ca. Then, the internal force was calculated in accordance with Fig. 14(d) as follows:

(11){\sigma }_{\mathrm{c1}}={\epsilon }_{\mathrm{ca}}{E}_{\mathrm{c}},

(12){\sigma }_{\mathrm{r1}}=y_{\mathrm{r}\mathrm{0}}/\left(h-y_{\mathrm{0}}\right){\epsilon }_{\mathrm{ca}}{E}_{\mathrm{s}},

(13){\sigma }_{\mathrm{t1}}=\left(h_{\mathrm{s}}-y_{\mathrm{0}}\right)/\left(h-y_{\mathrm{0}}\right){\epsilon }_{\mathrm{ca}}{E}_{\mathrm{s}},

(14){\sigma }_{\mathrm{l1}}=y_{\mathrm{0}}/\left(h-y_{\mathrm{0}}\right){\epsilon }_{\mathrm{ca}}{E}_{\mathrm{s}},

(15)M_{\mathrm{c1}}=\left({\sigma }_{\mathrm{c1}}+{\sigma }_{\mathrm{t1}}\right)h_{\mathrm{c}}{b}_{\mathrm{c}}{y}_{\mathrm{c0}},

(16)M_{\mathrm{r1}}={\sigma }_{\mathrm{r1}}{A}_{\mathrm{r}}{y}_{\mathrm{r0}},

(17)M_{\mathrm{s1}}={\sigma }_{\mathrm{t1}}{h}_{\mathrm{t}}{b}_{\mathrm{t}}\left(h_{\mathrm{s}}-y_{\mathrm{0}}-h_{\mathrm{t}}/\mathrm{2}\right)+\left[{\sigma }_{\mathrm{t1}}{\left(h_{\mathrm{s}}-y_{\mathrm{0}}-h_{\mathrm{t}}\right)}^{\mathrm{2}}+{\sigma }_{\mathrm{l1}}{\left(y_{\mathrm{0}}-h_{\mathrm{l}}\right)}^{\mathrm{2}}\right]b_{\mathrm{f}}/\mathrm{3}+{\sigma }_{\mathrm{l1}}{h}_{\mathrm{l}}{b}_{\mathrm{l}}\left(y_{\mathrm{0}}-h_{\mathrm{l}}/\mathrm{2}\right),

(18)M_{\mathrm{1}}=M_{\mathrm{c1}}+M_{\mathrm{r1}}+M_{\mathrm{s1}},

where E_c and E_s are the Young’s modulus of UHPC and steel, respectively. y_r0, y_c0, and y₀ are the distances from the neutral axis to the reinforcement center, UHPC deck center, and bottom of the I-shaped steel girder, respectively. h, h_c, and h_s are the heights of the section in Fig. 14(a), the UHPC deck, and the I-shaped steel girder, respectively. b_c, b_l, b_f, and b_t are the widths of the UHPC deck, top flange, web, and lower flange of the I-shaped steel girder, respectively. σ_c1, σ_r1, σ_t1, and σ_l1 denote the stress of the UHPC top surface, reinforcement, and top and lower flanges of the I-shaped steel girder, respectively. A_r is the area of the longitudinal reinforcement bars. M_c1, M_r1, and M_s1 are the calculation moments of the UHPC deck, reinforcement, and I-shaped steel girder, respectively. M₁ pertains to the calculation design moment in Stage I.

According to our previous research [21], the elastic limit strain of UHPC with 2% steel fiber content is 196 με. Thus, we set ε_ca = 196 με in the current study. After calculation, the design moment of Point 2 was determined to be M_l = 213.36 kN·m. According to the crack width development illustrated in Fig. 9, when the crack width of the UHPC matrix reached 0.05 mm, the corresponding load was in the range of 312.3–378.6 kN (197.53–239.46 kN·m). The analytical value was within this range, indicating that the calculation result is reasonable. Moreover, the test value of Point 2 was 212.90 kN·m, which is consistent with the analytical value.

(3) Calculation method for the deflection of the steel–UHPC composite girder

Several methods consider the effect of concrete tensile stiffness when calculating deflection for concrete members under tension. For example, GB50010-2010 [31] uses the stiffness analysis method, and CEB-FIP [32] adopts the modification method of tensile stiffness. The method of the effective moment of inertia is utilized in ACI 318M-89 [33]. Meanwhile, ACI 318M-89 stipulates that when calculating the deflection of reinforced concrete members, the effective moment of inertia of the section should be used as follows:

(19)I_{\mathrm{eff}}={\left({\displaystyle \frac{M_{\mathrm{cr}}}{M}}\right)}^{\mathrm{3}}{I}_{\mathrm{0}}+\left[\mathrm{1}-{\left({\displaystyle \frac{M_{\mathrm{cr}}}{M}}\right)}^{\mathrm{3}}\right]I_{\mathrm{cr}}\le I_{\mathrm{0}},

where M denotes the bending moment calculated on the basis of elastic theory, M_cr denotes the cracking moment, I₀ denotes the moment of inertia of the original section, and I_cr is the moment of inertia of the cracked section.

For the steel–UHPC composite girder under a negative moment, we adopted I₀ as the moment of inertia of the original section of the steel–UHPC composite girder. I_cr is the moment of inertia of the cracked section, which consists of the reinforcement and I-shaped steel girder. Cracking moment M_cr refers to the bending moment when the UHPC deck initially cracks. The calculation formula is

(20)M_{\mathrm{cr}}={\displaystyle \frac{f_{\mathrm{ct}}{I}_{\mathrm{0}}{\alpha }_{\mathrm{E}}}{y_{\mathrm{cr}}}},

where f_ct is the initial crack strength, α_E =  E_s/E_c, and y_cr is the distance from the neutral axis to the bottom of the I-shaped steel girder in the cracked section.

The following formula can be used for the mid-span deflection of the specimen in the present study.

(21)D={\displaystyle \frac{M}{E_{\mathrm{s}}{I}_{\mathrm{eff}}}}\left(\mathrm{3}{L}^{\mathrm{2}}-\mathrm{4}{a}^{\mathrm{2}}\right).

According to Zhang et al. [21], the initial crack strength of UHPC with 2% fiber content is 6.67 MPa. Hence, the deflection of Point 1 is D₁ = 13.02 mm. The test value of Point 1 is 12.32 mm, which is in good agreement with the calculation result.

Stage II

Stage II begins at Point 2 and ends at Point 3. At this stage, the stiffness of the specimen decreased slightly, and the section characteristics could be calculated using the same section as that used in Stage I. The lower flange of the I-shaped steel girder had just begun to yield at Point 3. Therefore, the calculation assumed that the lower flange reached the yield strength, i.e., σ_l = f_ys. The calculation diagram is given in Figs. 14(a)–14(d). In this case, Eqs. (12)–(17) can be changed to

(22){\sigma }_{\mathrm{r2}}=f_{\mathrm{ys}}{y}_{\mathrm{r0}}/y_{\mathrm{0}},

(23){\sigma }_{\mathrm{t2}}=f_{\mathrm{ys}}\left(h_{\mathrm{s}}-y_{\mathrm{0}}\right)/y_{\mathrm{0}},

(24){\sigma }_{\mathrm{l2}}=f_{\mathrm{ys}},

(25)M_{\mathrm{c2}}=f_{\mathrm{ca}}{h}_{\mathrm{c}}{b}_{\mathrm{c}}{y}_{\mathrm{c0}},

(26)M_{\mathrm{r2}}={\displaystyle \frac{y_{\mathrm{r0}}}{y_{\mathrm{0}}}}{f}_{\mathrm{ys}}{A}_{\mathrm{r}},

M_{\mathrm{s2}}={\displaystyle \frac{h_{\mathrm{s}}-y_{\mathrm{0}}}{y_{\mathrm{0}}}}{f}_{\mathrm{ys}}

(27)\left[A_{\mathrm{t}}{y}_{\mathrm{t}0}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{b}_{\mathrm{f}}{\left(y_{\mathrm{t}0}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{h}_{\mathrm{t}}\right)}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{b}_{\mathrm{f}}{\left(y_{\mathrm{0}}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{h}_{\mathrm{l}}\right)}^{\mathrm{2}}+A_{\mathrm{l}}{y}_{\mathrm{l0}}\right],

where f_ys is the yield strength of the I-shaped steel girder; f_ca is the ultimate elastic strength of UHPC, namely, f_ca= E_caε_ca; A_t and A_l are the areas of the top and lower flanges, respectively; and y_t0 and y_l0 are the distances from the neutral axis to the center of the top and lower flanges, respectively.

The calculation design moment of Point 3 was M₂ = 495.90 kN·m. The corresponding deflection, D₂ = 35.03 mm, was obtained by using Eq. (21). According to the test results for the lower flange strain (Fig. 11), the lower flange reached the yield strength at 782 kN (494.62 kN·m), which is consistent with the calculation result.

Stage III

Stage III is from Point 3 to Point 4. At this stage, the lower flange of the I-shaped steel girder reached the yield strength. The strain of the prefabricated UHPC panel reached the ultimate value at Point 4. Figure 14(e) can be used as the stress calculation diagram, and the height of the elastically compressed part of the web (y_t3) needs to be calculated.

(28)y_{\mathrm{t3}}=f_{\mathrm{ys}}/[E_{\mathrm{s}}·{\epsilon }_{\lim }\left(h-y_{\mathrm{0}}\right)],

(29)M_{\mathrm{c3}}=f_{\mathrm{ca}}{h}_{\mathrm{c}}{b}_{\mathrm{c}}{y}_{\mathrm{c0}},

(30)M_{\mathrm{r3}}={\displaystyle \frac{y_{\mathrm{r0}}^{\mathrm{2}}}{h-y_{\mathrm{0}}}}{\epsilon }_{\lim }{E}_{\mathrm{s}}{A}_{\mathrm{r}},

(31)M_{\mathrm{s3}}={\displaystyle \frac{h_s-y_{\mathrm{0}}}{h-y_{\mathrm{0}}}}{\epsilon }_{\lim }{E}_{\mathrm{s}}\left[A_{\mathrm{t}}{y}_{\mathrm{t0}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{b}_{\mathrm{f}}{\left(h_{\mathrm{s}}-y_{\mathrm{0}}-h_{\mathrm{t}}\right)}^{\mathrm{2}}\right]+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{f}_{\mathrm{ys}}{b}_{\mathrm{f}}{y}_{\mathrm{t3}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{f}_{\mathrm{ys}}{b}_{\mathrm{f}}\left[{\left(y_{\mathrm{0}}-h_{\mathrm{l}}\right)}^{\mathrm{2}}-y_{\mathrm{t3}}^{\mathrm{2}}\right]+f_{\mathrm{ys}}{A}_{\mathrm{l}}{y}_{\mathrm{l0}}.

According to Zhang et al. [21], the ultimate tensile strain of UHPC with 2% steel fiber content is 765 με. Thus, the calculation design moment of Point 4 is M₃ = 514.93 kN·m, and the deflection is D₃ = 36.43 mm.

Stage IV

Stage IV begins with Point 4 and ends with Point 5. At this stage, the top and lower flanges, and part of the web of the I-shaped steel girder reached their yield strength. The reinforcement in the UHPC deck reached its yield strength at Point 5, and the UHPC reached the ultimate strain. Thus, the cracked section (Fig. 15(a)) should be used. The calculation diagram is shown in Fig. 15(c). Similarly, the height of the elastic part of the web should be calculated first.

(32)y_{\mathrm{t4}}=f_{\mathrm{ys}}/f_{\mathrm{yr}}{y}_{\mathrm{r-cr}},

(33)M_{\mathrm{c4}}=f_{\mathrm{ca}}{A}_{\mathrm{c}}{y}_{\mathrm{c-cr}},

(34)M_{\mathrm{r4}}=f_{\mathrm{yr}}{A}_{\mathrm{c}}{y}_{\mathrm{r-cr}},

(35)M_{\mathrm{s4}}=f_{\mathrm{ys}}{A}_{\mathrm{t}}{y}_{\mathrm{t-cr}}+f_{\mathrm{ys}}{b}_{\mathrm{f}}{y}_{\mathrm{t4}}\left({\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{h}_{\mathrm{s}}-{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{y}_{\mathrm{cr}}+{\displaystyle \frac{\mathrm{11}}{\mathrm{12}}}{y}_{\mathrm{t4}}\right)+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{f}_{\mathrm{ys}}{b}_{\mathrm{f}}\left(y_{\mathrm{cr}}-y_{\mathrm{t4}}-h_{\mathrm{l}}\right)\left(y_{\mathrm{cr}}+y_{\mathrm{t4}}\right)+f_{\mathrm{ys}}{A}_{\mathrm{l}}{y}_{\mathrm{l-cr}},

where y_r-cr, y_c-cr, y_t-cr, and y_l-cr are the distances from the neutral axis to the center of the reinforcement, UHPC deck, top flange, and lower flange of the I-shaped steel girder in the cracked section, respectively (Fig. 15(a)).

Accordingly, the calculation design moment of Point 5 is M₄ = 620.01 kN·m, and the defection is D₄ = 44 mm.

Stage V

Stage V is from Point 5 to Point 6. The specimen reached the ultimate load at Point 6. The calculation diagram is shown in Fig. 15(e). With the increase in load, the height of the elastic part of the web (y_t) decreased, and the full cross-section of the I-shaped steel girder yielded. The calculation in this work simplifies this process, and the calculation diagram is shown in Fig. 15(e). The ultimate bearing capacity can be obtained as

(36)M_{\mathrm{u},\mathrm{U}\mathrm{L}\mathrm{S}}=f_{\mathrm{c}\mathrm{a}}{A}_{\mathrm{c}}{y}_{\mathrm{c}\mathrm{-cr}}+f_{\mathrm{y}\mathrm{r}}{A}_{\mathrm{r}}{y}_{\mathrm{r}\mathrm{-cr}}+f_{\mathrm{y}\mathrm{r}}\frac{y_{\mathrm{s}\mathrm{-cr}}}{y_{\mathrm{r}\mathrm{-cr}}}{A}_1{y}_{1\mathrm{-cr}}+\frac{1}{3}{f}_{\mathrm{y}\mathrm{r}}{b}_{\mathrm{f}}\left(\frac{y_{\mathrm{s}\mathrm{-cr}}}{y_{\mathrm{r}\mathrm{-cr}}}{y_{\mathrm{f}\mathrm{t}}}^2+\frac{y_{\mathrm{cr}}}{y_{\mathrm{r}\mathrm{-cr}}}{y_{\mathrm{f}\mathrm{c}}}^2\right)+f_{\mathrm{y}\mathrm{r}}\frac{y_{\mathrm{cr}}}{y_{\mathrm{r}\mathrm{-cr}}}{A}_{\mathrm{t}}{y}_{\mathrm{t}\mathrm{-cr}},

where y_s-cr is the distance from the centroid of the I-shaped steel girder to the neutral axis of the cracked section, y_ft is the distance between the centroid of the tension part of the web and the neutral axis of the composite girder, and y_fc is the distance between the centroid of compression part of the web and the neutral axis of the composite girder.

The calculation design moment of Point 6 is M₅ = 650.07 kN, and the deflection is D₅ = 74.39 kN; the test values are 653.94 kN·m and 65.43 mm, respectively. The calculation result is slightly conservative.

The calculation of critical points in the five stages was completed, and a comparison of the calculation and experimental results is plotted in Fig. 16. Points 2–4 are in good agreement with the experimental values, whereas Points 5–6 are slightly different from the test results because the equations simplified the actual loading process. The calculation results for the ultimate bearing capacity and deflection of the structure are conservative but acceptable.

Conclusions

A fully precast steel–UHPC LWCB and the corresponding design of the negative moment region were proposed. The critical design of the negative moment region was verified through experiments. On the basis of the test results, the following conclusions were obtained.

1) An FE analysis was performed with the first small–middle span LWCB in China. The design stress was calculated and compared with the test results. The strength of the steel girders and UHPC panel could meet the design requirements.

2) The fatigue performance of the proposed UHPC joint was assessed in reference to past research, and the results showed that the fatigue performance could meet the design requirements.

3) The equations for the maximum crack width at the joint interface were presented based on past research. The prediction and experimental results showed good agreement.

4) A simplified calculation method for the critical stages of the entire loading process was developed. The formulas of design moment and deflection in consideration of the UHPC tension stiffness effect were presented. The comparisons with the test results revealed that the proposed methods can effectively predict the behavior of steel–UHPC composite girders under a negative bending moment.

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