Introduction

The incremental launching method is a common construction technique widely used in the construction of beam bridges, cable-stayed bridges, and arch bridges1,2,3,4. This method offers advantages such as minimal disruption to traffic and reduced construction site requirements, making it one of the preferred techniques for crossing existing roads and canyon bridges. The use of temporary cable tower structures further extends the span capabilities of bridges constructed using the incremental launching method5.

The cable force of the temporary cable is very important to the safety of the structure. The stress distribution in a cable-stayed auxiliary bridge constructed using the incremental launching method differs from that of a conventional cable-stayed bridge. As the bridge advances through various incremental launching states, both the position and support of the cable-stayed auxiliary structure change. This results in the bridge functioning as a separate stress system in each launching state, with distinct stress modes. Consequently, the structural stress and deformation in different construction states cannot be accumulated. Improper cable force during this process may compromise the safety of the bridge, particularly when the structure reaches the maximum cantilever state of the largest span. Additionally, the calculation method for the cable force in a cable-stayed bridge is typically designed to ensure reasonable stress in the completed bridge state. However, unlike cable-stayed bridges, the temporary cables used during construction are removed once the bridge is completed. This difference necessitates distinct calculation methods for the temporary cable forces compared to those used for permanent cable-stayed bridges. These factors not only increase the complexity of calculating cable forces but also raise the requirements for the accuracy and applicability of the cable force calculations.

Many scholars have conducted research on the bridge construction method using the incremental launching technique. Tian et al.6 employed the finite element method to analyze the impact of parameters such as the stiffness, length, and weight of the guide beam on the internal forces of the main beam. Their research led to the determination of an optimal configuration for the guide beam during the launching process. Xie et al.7 used the finite element method to analyze the steel box girder incremental launching construction process and found that the length and weight of the guide beam significantly affect the stress and deformation during construction. Ren et al.8 investigated the impact of guide beam stiffness, weight, and auxiliary pier position on the mechanical performance of the main beam using the finite element method. They discovered that the weight of the guide beam had a greater impact on the main beam's mechanical performance than its stiffness. Chen et al.9 examined the optimal layout of temporary piers for bridge incremental launching construction and proposed measures to optimize the stress on these temporary piers. Zhao et al.5 summarized research on overall stress analysis, local stress analysis, and construction control associated with the incremental launching method in bridge construction. Luo et al.10 studied and analyzed the stress on temporary members of steel–concrete composite continuous beam bridges during incremental launching construction. Ding et al.11 investigated the incremental launching construction technology for suspension continuous steel beam bridges and proposed using elastic supports to mitigate the adverse effects of height differences in the steel beams. Zhang et al.12 analyzed the most critical stress positions of the structure and examined the local stress characteristics through overall stress analysis of long-span cable-stayed bridges. Meng et al.13 investigated the monitoring technology for the entire construction process of multi-span long arch steel girder bridges using a combination of numerical calculations and field tests. Xie et al.14, focusing on self-anchored steel box girder suspension bridges, analyzed the impact of design parameters of the guide beam on the stress and deformation of the steel box girder, as well as its local stability during the construction process. Fontan et al.15 optimized and improved the design formula for the guide beam of prestressed concrete beam bridges. Jung et al.16 verified the construction safety of prestressed concrete box girder bridges with corrugated steel webs. The results demonstrated that this type of bridge can be successfully constructed using the incremental launching method. Wang et al.17 addressed the issue of low stiffness in U-shaped steel beams by adopting a local stress control strategy to prevent local yielding of the bottom plate, thereby enabling the incremental construction of U-shaped steel beams. The research reviewed above indicates that many scholars have extensively studied stress analysis and design of guide beams, the impact of parameters on bridge stress, bridge alignment control, and local stress analysis of bridges. However, there is comparatively less research on the calculation of cable forces in temporary tower structures, with only a few studies focusing on methods for calculating cable forces. Wang et al.18 used finite element software to perform trial calculations and analysis of cable forces in steel box girder incremental launching construction. They obtained reasonable values for cable forces, as well as stress and displacement results during the incremental launching process. Wang et al.19 analyzed the impact of cable forces on steel beam alignment using the finite element method but did not specify a calculation method for cable forces.

To ensure the structural safety, calculating temporary cable forces presents the challenge of adapting to the stress changes in the bridge at different construction stages. This study proposes a calculation method for adaptive cable forces based on changes in the unstressed cable length within an adaptive structural system.

This paper focuses on the incremental launching construction technology for the steel box girder of the Fenshui River Bridge. Firstly, based on the maximum cantilever state of the structure, the influence matrix method is used to determine the cable force, with the vertical deformation of the front end of the guide beam and the horizontal deformation of the top of the tower as control objectives. The unstressed cable length is then calculated based on the mechanical relationship between cable deformation and cable force. An adaptive cable force calculation method is proposed, utilizing changes in the unstressed cable length within an adaptive structural system. The validity of this calculation method is confirmed through comparison with measured values of cable force, deformation, and stress during bridge construction. Additionally, the layout of the tower was analyzed, leading to the proposal of a reasonable positioning for the tower.

Cable force calculation method

The core of the temporary cable force calculation method consists of the following three components:

  1. (1)

    Selection of the maximum cantilever state: The vertical deformation of the front end of the guide beam and the horizontal deformation of the top of the tower are used as control objectives. The influence matrix method is employed to calculate the optimal cable force for the structure.

  2. (2)

    Calculation of unstressed cable length: This is determined based on the mechanical relationship between cable deformation and cable force.

  3. (3)

    Adaptation to structural changes: The extension and shortening of the unstressed cable length are used to adjust to changes in the structural system of the steel box girder throughout the construction process. If the deformation of the front end of the guide beam or the top of the tower exceeds acceptable limits during construction, the limit values are adjusted, and recalculations are performed.

In the incremental launching construction of steel box girders, the maximum cantilever state represents the most critical and hazardous stress condition, making it essential to ensure that the construction state meets the required standards. Additionally, as the construction progresses, the structural system continuously changes, including variations in the length and position of the bridge, which significantly alters the structural stress mode. To address these changes, the method proposed in this study introduces the use of unstressed cable length to adapt to the evolving structural system. These two aspects represent the innovative contributions of the cable force calculation method presented in this study.

Based on the number of cables used in similar projects10,19, most bridges employ either 2 or 4 groups of temporary cables. The following describes the calculation method for cable force using 4 groups of cables as an example. The calculation method is applicable regardless of the number of cables used.

Step 1 Establish the finite element model of the entire steel box girder incremental launching construction process based on the construction scheme.

Step 2 Select the finite element model corresponding to the maximum cantilever stage of the structure. At this stage, the front end of the guide beam will span the maximum distance, representing the maximum cantilever state of the structure. When calculating the cable force, it is essential to ensure that the vertical displacement at the front end of the guide beam are accounted for \({\text{d}} \le \left[ {\text{d}} \right]\), and the horizontal displacement at the top of the tower meet the requirement of \({\text{x}} \le \left[ {\text{x}} \right]\). The horizontal inclination angles of cables C1, C2, C3 and C4 are denoted as \(\alpha_{1}\), \(\beta_{1}\), \(\beta_{{2}}\) and \(\alpha_{{2}}\) respectively. The geometric information for the maximum cantilever state of the structure is illustrated in Fig. 1.

Fig. 1
figure 1

Geometric information.

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Step 3 Input the initial cable forces for cables C1, C2, C3, and C4 into the finite element model for the maximum cantilever stage. The cable forces are denoted as T1, T2, T3 and T4. Let \({\text{T}}_{1} = {\text{T}}_{1}^{\prime }\) and \({\text{T}}_{2} = {\text{T}}_{2}^{\prime }\). \(T_{4} = T_{1}^{\prime } \cos \alpha_{1} {/}\cos \alpha_{2}\) and \({\text{T}}_{3} { = }T_{2}^{\prime } \cos \beta_{1} {/}\cos \beta_{2}\) can be obtained from \(T_{1} \cos \alpha_{1} = T_{4} \cos \alpha_{2}\) and \(T_{2} \cos \beta_{1} = T_{3} \cos \beta_{2}\). \({\text{T}}_{1} = {\text{T}}_{1}^{\prime }\), \({\text{T}}_{2} = {\text{T}}_{2}^{\prime }\), \(T_{4} = T_{1}^{\prime } \cos \alpha_{1} {/}\cos \alpha_{2}\) and \({\text{T}}_{3} { = }T_{2}^{\prime } \cos \beta_{1} {/}\cos \beta_{2}\) are substituted into the finite element model of the maximum cantilever stage.

Step 4 Run the finite element model to obtain the vertical displacement \({\text{d}}\) at the front end of the guide beam and the horizontal displacement \({\text{x}}\) at the top of the tower. \(\Delta {\text{d}}\) and \(\Delta {\text{x}}\) are obtained by \(\Delta {\text{d = }}\left[ {\text{d}} \right] - {\text{d}}\) and \(\Delta {\text{x = }}\left[ {\text{x}} \right] - {\text{x}}\).

Step 5 According to formulas (1) and (2), the variation of cable force \(\Delta {\text{T}}_{1}\), \(\Delta {\text{T}}_{{2}}\), \(\Delta {\text{T}}_{{3}}\), \(\Delta {\text{T}}_{{4}}\) can be obtained.

$$\left[ {\begin{array}{*{20}c} {{\text{d}}_{{1}} } & {{\text{d}}_{2} } \\ {{\text{x}}_{{1}} } & {{\text{x}}_{{1}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\Delta T_{1} } \\ {\Delta T_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\Delta {\text{d}}} \\ {\Delta {\text{x}}} \\ \end{array} } \right],$$
(1)
$$\Delta T_{1} \cos \alpha_{1} = \Delta T_{4} \cos \alpha_{2} ,\,\,\Delta T_{2} \cos \beta_{1} = \Delta T_{3} \cos \beta_{2} .$$
(2)

Formula (1) employs the influence matrix method, which is commonly used to determine cable forces in cable-stayed bridges. In the maximum cantilever state, the steel box girder can be approximated as a cantilever structure system. The primary stress characteristics of this system are the significant vertical deformation at the front end of the guide beam and the substantial horizontal deformation of the tower. When the structural control objectives are well-defined, the influence matrix method provides an efficient means to calculate the cable forces.

Step 6 Obtain the cable force of the structure at the maximum cantilever stage.

\({\text{T}}_{1} = {\text{T}}_{1}^{\prime } + \Delta {\text{T}}_{1}\), \({\text{T}}_{2} = {\text{T}}_{2}^{\prime } + \Delta {\text{T}}_{2}\), \(T_{4} = T_{1}^{\prime } \cos \alpha_{1} {/}\cos \alpha_{2} + \Delta {\text{T}}_{4}\) and \({\text{T}}_{3} { = }T_{2}^{\prime } \cos \beta_{1} {/}\cos \beta_{2} + \Delta {\text{T}}_{3}\).

Step 7 According to the obtained cable force T, calculate the unstressed length of the buckle cable according to formula (3):

$$L_{{\text{i}}} = L - \Delta L_{e} ,$$
(3)

where \(L_{i}\)—the unstressed length of the ith cable under the maximum cantilever state. \(L\)—geometric length after taking into account the deformation of both ends of the cable. \(\Delta L_{e}\)—elastic elongation of cable generated by cable force T, \(\Delta L_{e} = TL/EA\).

Step 8 After obtaining the unstressed cable length \(L_{i}\), input the unstressed cable length into the finite element model of the whole process of steel box girder construction. After recalculation, the initial tension force T0 at the tension stage can be obtained. Similarly, the cable forces at different construction stages can be obtained.

Step 9 Examine the deformation of the front end of the guide beam and the top of the tower at various construction stages in the finite element model. If the deformations meet the specified limits, then the calculated cable forces are considered satisfactory. If the deformation at the front end of the guide beam or the top of the tower exceeds the allowable limits, adjust the limit values and repeat steps 2 through 8 until the requirements are met.

The flowchart of the cable force calculation for temporary cables is shown in Fig. 2.

Fig. 2
figure 2

Flow chart of cable force calculation.

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Project introduction and finite element model

Project introduction

The Fenshui River Bridge is a steel–concrete composite continuous beam bridge with a main span of 90 m and side spans of 60 m, situated between Wuxi and Kaizhou in Chongqing, China. The bridge features two separate steel box girders connected by crossbeams, with the upper part consisting of a concrete slab. The steel boxes are made from Q355 steel, and the concrete slab is C50. The steel box girder is segmented into 23 beam sections along the bridge's longitudinal direction. The Fenshui River Bridge is illustrated in Fig. 3, and its cross-section is shown in Fig. 4.

Fig. 3
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Fenshui river bridge.

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Fig. 4
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Cross sectional view.

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Construction method of steel box girder

The steel box girder of the Fenshui River Bridge has a total length of 210 m and a longitudinal slope of 2.6%. It is constructed using the incremental launching method, with the assembly area positioned between piers P16 and P18. Steel pipe supports (PZ1-PZ6) are placed between the bridge piers to serve as installation platforms for the guide beams and steel box girder. The layout of the incremental launching construction for the steel box girder is depicted in Fig. 5, and the construction site diagram is shown in Fig. 6.

Fig. 5
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The layout of steel box girder incremental launching construction.

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Fig. 6
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Construction site picture.

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The guide beam, as shown in Fig. 7, has a total length of 50 m and is constructed from truss-type variable cross-section "H" steel. It is horizontally divided into two sections, connected by steel pipe trusses. For ease of installation and disassembly, the guide beam is segmented into five sections with lengths of 7000 mm, 12,000 mm, 7000 mm, 11,800 mm, and 12,200 mm, respectively. Both the guide beam and steel pipe trusses are made of Q355 steel. To facilitate crossing over the bridge pier, a nose bridge with a height of 750 mm is installed at the front end of the guide beam.

Fig. 7
figure 7

Guide beam.

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Temporary tower structures are used to arrange cables and improve the stress distribution of steel box girders during incremental launching construction. The tower is 30 m high and consists of four vertical steel pipes, each with a diameter of φ630 × 10 mm. These vertical steel pipes are connected by horizontal and inclined steel pipes, which have a diameter of φ320 × 8 mm. To enhance the stress distribution at the base of the tower, additional inclined steel pipes are included at the bottom, with a diameter of φ630 × 10 mm. The tower is depicted in Fig. 8.

Fig. 8
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Tower.

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The tower design takes into account various factors, including the forces on the guide beam and steel box girder, the inclination angle of the cables, the tower's position, and the ease of installation and removal. The center of the tower is positioned 40 m from the front end of the steel box girder, ensuring that the tower is directly above the P15 bridge pier when the guide beam reaches its maximum cantilever. This positioning optimizes the effectiveness of the tower and cables. To facilitate quick installation and removal, high-strength bolts are used to connect the bottom of the tower to the top plate of the steel box girder, rather than welding. Additionally, local reinforcement is applied inside the steel box girder at the base of the tower to enhance stress distribution.

Based on the number of cables used in similar projects10,19 and considering the stress conditions of the Fenshui River Bridge, four sets of cables are employed for the bridge, as illustrated in figures C1 to C4 in Fig. 5. C1 and C2 are positioned on the guide beam, while C3 and C4 are located on the steel box girder. Each cable consists of steel strands with a tensile strength of 1860 MPa. The cross-sectional area of each steel strand is 139 mm2, with 8 steel strands used for C1 and C4, and 5 steel strands used for C2 and C3.

The steel box girder incremental launching construction process is divided into stages CS1 through CS13. The CS1 stage involves the installation of the guide beam and steel box girders 1# to 7# using a truck crane. In CS2, the steel box girder is pushed 36.7 m. The CS3 stage covers the installation of girders 8# to 11#, followed by CS4, which includes the installation of towers and cable tensioning. During CS5, the girder is pushed by 18.35 m, and in CS6, girders 12# to 13# are installed. CS7 involves pushing the girder by 34.95 m, reaching the maximum cantilever state, with support on pier 14. In CS8, the guide beam is supported on pier 14. The CS9 stage includes the installation of girders 14# to 17#, while CS10 involves the removal of towers and cables. CS11 covers pushing the girder by 34.2 m and installing girders 18# to 21#, and CS12 involves pushing the girder by 45 m and installing girders 22# to 23#. Finally, CS13 marks the completion of the incremental launching construction and the removal of the guide beam. The construction scheme for stages CS1, CS5, CS7, CS8, CS10, and CS13 is illustrated in Fig. 9.

Fig. 9
figure 9

Incremental launching construction process of steel box girder.

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Finite element model

Midas Civil is a three-dimensional finite element software designed to simulate complex bridge construction processes20,21. It utilizes cable elements to represent cables and beam elements to model steel box girders. The software allows for the simulation of the construction process by adjusting support positions as the construction progresses.

The finite element model of the Fenshui River Bridge's incremental launching construction is established using Midas Civil, as shown in Fig. 10. The simulation of the bridge's incremental launching process is performed by adjusting the support positions2. The model accounts for the longitudinal slope of the bridge and the manufacturing camber of the steel box girder. It incorporates cable elements for cables and beam elements for the rest of the structure, comprising 678 nodes and 609 elements. Supports include the pushing device and steel pipe supports, modeled with compression-only boundaries. The guide beam and steel box girder are made from Q355 steel, with an elastic modulus of 2.06 × 105 MPa and a unit weight of 105.3 kN/m3. The tower is constructed from Q235 steel, with the same elastic modulus but a unit weight of 84.6 kN/m3. The cable is composed of steel strands, featuring an elastic modulus of 1.95 × 105 MPa and a unit weight of 78.5 kN/m3. The finite element simulation ensures that the weight distribution matches the actual weight of the bridge.

Fig. 10
figure 10

Finite element model.

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To ensure the safety of the steel box girder throughout the incremental launching construction process, the stresses and deformations must comply with the relevant specifications22,23. The stress in the tower (made of Q235) must not exceed 215 MPa, while the stress in the steel box girder and guide beam (made of Q355) must be below 305 MPa. The horizontal deformation of the tower should be less than h/400, where h is the height of the tower. For a 30-m-high tower, this means the horizontal deformation should be under 75 mm. Additionally, to ensure that the front end of the guide beam can smoothly pass through the 14# pier during the maximum cantilever state, its deformation should be less than 800 mm, considering the height of the nose beam and the extension range of the jack.

Engineering monitoring

To ensure safety during the bridge construction, both the deformation and stress of the structure were continuously monitored throughout the incremental launching of the steel box girder. Deformation measuring points are strategically placed at key locations on the guide beam, steel box girder, and tower, and are measured using a total station, as depicted in Fig. 11. The total station model used is the GM-52, which has an accuracy of 2". Measuring points D1 to D10 are used to monitor the vertical deformation of the guide beam and steel box girder. Specifically, D1 and D2 are positioned on the top of the guide beam, D3 is placed at the joint between the guide beam and the steel box girder, and D4 to D10 are arranged along the steel box girder. The TX1 measuring points are placed on the top of the tower to monitor its horizontal deformation.

Fig. 11
figure 11

Structural deformation monitoring. (a) General arrangement. (b) Horizontal deformation measuring point. (c) Vertical deformation measuring point. (d) Measuring instrument.

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Stress sensors are positioned at critical stress points during the incremental launching construction of the steel box girder, as illustrated in Fig. 12. For the guide beam, stress sensors are installed on the top and bottom plates of the section (see Fig. 12b). In the steel box girder, sensors are placed on the top and bottom plates of the girder (see Fig. 12c). The tower stress sensors are located on the vertical steel pipe at the base of the tower (see Fig. 12d). The strain gauges used are of model WY-B150B, with an accuracy of 0.1 με.

Fig. 12
figure 12

Structural stress monitoring.

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A pressure sensor is employed to monitor the cable force during the incremental launching construction of the steel box girder. The sensor model JMZX-3108HAT is used, with a force measuring range of 1100 kN. This sensor is installed at the tensioning section of the C1 cable, located at the lower end of the cable, as shown in Fig. 13. During cable tensioning on-site, a lifting jack is used to apply tension to the steel beam.

Fig. 13
figure 13

Cable force monitoring.

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Engineering verification

According to the cable force calculation method described in Sect. "Cable force calculation method", the calculation process is as follows:

Step 2 The horizontal inclination angles of cables C1, C2, C3 and C4 are 25.367°, 32.886°, 30.378°and 27.654°respectively.

Step 3 Let \({\text{T}}_{1}\) = 100 kN and \({\text{T}}_{2}\) = 100 kN. \(T_{4}\) = 102.01 kN and \({\text{T}}_{3}\) = 97.339 kN can be obtained from \(T_{1} \cos \alpha_{1} = T_{4} \cos \alpha_{2}\) and \(T_{2} \cos \beta_{1} = T_{3} \cos \beta_{2}\).\({\text{T}}_{1}\) = 100 kN, \({\text{T}}_{2}\) = 100 kN, \(T_{4}\) = 97.339 kN and \({\text{T}}_{3}\) = 102.01 kN are substituted into the finite element model of the maximum cantilever stage (CS7).

Step 4 Run the finite element model obtain the vertical displacement \({\text{d}}\) = − 2875.369 mm at the front end of the guide beam and the horizontal displacement \({\text{x}}\) = − 281.899 mm at the top of the tower. Fenshui River Bridge requires that the vertical displacement of the front end of the guide beam \(\left[ {\text{d}} \right] \le\)− 800 mm and the horizontal displacement of the top of the tower \(\left[ {\text{x}} \right] \le\)-75 mm. For safety reasons, let \(\left[ {\text{d}} \right] =\)− 600 mm and \(\left[ {\text{x}} \right] =\)− 60 mm.\(\Delta {\text{d}}\) = 2275.36 mm and \(\Delta {\text{x}}\) = 221.899 mm are obtained by \(\Delta {\text{d = }}\left[ {\text{d}} \right] - {\text{d}}\) and \(\Delta {\text{x = }}\left[ {\text{x}} \right] - {\text{x}}\).

Step : Bring \(\left[ {\begin{array}{*{20}c} {{\text{d}}_{{1}} } & {{\text{d}}_{2} } \\ {{\text{x}}_{{1}} } & {{\text{x}}_{{1}} } \\ \end{array} } \right]{ = }\left[ {\begin{array}{*{20}c} {{0}{\text{.307}}} & {{0}{\text{.3154}}} \\ {{0}{\text{.1641}}} & {{0}{\text{.151}}} \\ \end{array} } \right]\) into formula (1) to obtain \(\Delta T_{1}\) = 769 kN and \(\Delta T_{{2}}\) = 407 kN. Then use formula (2) to obtain \(\Delta T_{3}\) = 396.661 kN and \(\Delta T_{4}\) = 790.161 kN.

Step 6 At CS7 stage, the cable forces \({\text{T}}_{1}\), \({\text{T}}_{2}\), \({\text{T}}_{3}\) and \({\text{T}}_{4}\) are 869 kN, 507 kN, 494 kN and 887.5 kN respectively.

Step 7 The length L of cables C1, C2, C3 and C4 in CS7 after deformation is 64,841.8588 mm, 51,888.4506 mm, 52,989.5368 mm and 65,999.8831 mm respectively. The elastic modulus is 195,000 MPa, and the cross-sectional areas are 1112 mm2, 695 mm2, 695 mm2 and 1112 mm2 respectively. Then the elastic elongation of cables generated by cable forces T1, T2, T3 and T4 are 259.458 mm, 193.6488 mm, 192.3371 mm and 269.0856 mm respectively. The unstressed length L1, L2, L3 and L4 of the cable obtained by formula (3) are 64,582.4 mm, 51,694.8018 mm, 52,797.1997 mm and 65,730.7975 mm respectively.

Step 8 Input the unstressed cable lengths L1, L2, L3, and L4 into the finite element model of the steel box girder launching process. After recalculation, the initial tensioning forces for cables C1, C2, C3, and C4 are 528.6 kN, 298.3 kN, 316.2 kN, and 644.1 kN, respectively. Similarly, cable forces for different construction stages can be determined.

Step 9 Check the finite element model. The maximum deformation of the front end of the guide beam at different pushing stages is − 586 mm, and the horizontal deformation of the tower in the longitudinal direction is -58 mm, both of which meet the limit requirements.

To illustrate the influence of cables on the mechanical behavior of long-span bridges without auxiliary piers during incremental launching construction, the vertical deformation at D1 and stress at S2, calculated with and without cables, are compared. The deformation comparison is shown in Fig. 14, and the stress comparison is shown in Fig. 15.

Fig. 14
figure 14

Comparison diagram of D1 deformation.

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Fig. 15
figure 15

S2 stress comparison diagram.

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Figures 14 and 15 show that during the incremental launching construction of the steel box girder, the deformation behavior at D1 is consistent under both scenarios. However, without cables, the deformation value at D1 reaches 3234 mm in the maximum cantilever state, which is significantly larger than the value with cables. This excessive deformation at D1 would prevent the guide beam from passing through pier 14. Similarly, the stress variation at S2 follows the same trend in both scenarios. However, without the cables, the stresses at the upper and lower edges of the section at S2 are 339 MPa and − 328 MPa, respectively, in the maximum cantilever state. These stress values exceed the specification limit of 305 MPa, indicating non-compliance with the required standards. Therefore, when constructing large-span bridges using the incremental launching method, it is essential to use cables and towers to address structural deformation and ensure safety.

The unstressed cable lengths for C1, C2, C3, and C4 are 259.458 mm, 193.6488 mm, 192.3371 mm, and 269.0856 mm, respectively. By inputting these unstressed cable lengths into the finite element model, the variation in cable force during construction can be determined. During actual construction, the unstressed length of the cable is managed by adjusting the pull-out amount of the anchor head during cable tensioning. The initial tensions of the cables are 528.6 kN, 298.3 kN, 316.2 kN, and 644.1 kN, respectively.

The comparison between the calculated and measured values of cable force is illustrated in Fig. 16. As shown in Fig. 16, the cable force initially decreases, then increases, and subsequently decreases again. These fluctuations in cable force are associated with the varying construction states of the steel box girder, with the cable force peaking during the maximum cantilever state of the structure. This behavior is due to the cables supporting the guide beam and steel box girder. When the structure reaches its maximum cantilever state, the extensive cantilever results in significant downward deformation. The cables, in turn, prevent this downward deformation and stretch passively, which causes the cable force to increase. The calculated value of the C1 cable force is in good agreement with the measured value. At the maximum cantilever state of the structure, the calculated and measured values for C1 cable force are 868.5 kN and 903 kN, respectively, with a difference of 3.96%. This alignment not only verifies the effectiveness of the cable force calculation method presented in this paper but also provides a valuable reference for cable force calculations in similar bridge construction projects. It underscores the critical role of cables in ensuring the structural stress safety of such constructions.

Fig. 16
figure 16

Cable force comparison diagram.

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During the incremental launching construction of the steel box girder, the comparison between theoretical and measured deformation values of key sections is shown in Fig. 17. As depicted, the deformation of different sections of the guide beam and steel box girder fluctuates, but the theoretical values closely match the measured values. At the CS7 stage, which represents the maximum cantilever state of the structure, both the deformation of the front end of the guide beam and the top of the tower are at their peak, making it the most critical phase for stress. At this stage, the theoretical vertical deformation of D1 at the front end of the guide beam is 586.3 mm, while the measured value is 624.4 mm, resulting in a difference of 6.4%. Similarly, the theoretical horizontal deformation of TX1 at the top of the tower is 59.7 mm, compared to a measured value of 62.5 mm, a difference of 4.8%. The deformation at D1 is within the required limits for the guide beam to smoothly cross the pier, and the deformation of the tower also adheres to specification requirements.

Fig. 17
figure 17

Deformation comparison diagram.

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During the incremental launching construction of steel box girder, the comparison between the calculated and measured deformation values of key structural sections D1, D2, D3 and D5 is shown in Table 1.

Table 1 Deformation comparison table (mm).
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Figure 18 displays the comparison between the theoretical and measured stress values at key sections of the structure. From Fig. 18, it is evident that the stress at the bottom of the guide beam, steel box beam, and buckle tower exhibits a consistent trend throughout the construction process. The theoretical stress values closely match the measured values. Notably, significant stress variations are observed at S1 and S2 in the structure. The section stress at S1, located at the root of the guide beam, experienced the most significant change during the removal of the buckle tower (CS10). The calculated maximum tensile stress was 127 MPa, while the measured value was 120 MPa, resulting in a difference of 5.5%. The calculated maximum compressive stress at S1 is − 132 MPa, while the measured value is − 126.8 MPa, resulting in a difference of 3.9%. At S2 of the steel box girder, which experienced the most significant stress changes in CS11, the calculated maximum tensile stress is 107 MPa compared to the measured 101.2 MPa, showing a 5.4% difference. The calculated maximum compressive stress at S2 is − 111 MPa, whereas the measured value is − 103.7 MPa, with a 6.6% difference. These results confirm that the structural stress meets the specification requirements.

Fig. 18
figure 18

Stress comparison diagram.

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During the incremental launching construction of steel box girder, the stress changes greatly at S1 and S2 of the structure, and the comparison between the calculated and measured stress values is shown in Table 2.

Table 2 Stress comparison table (MPa).
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Analysis of tower layout position

The function of the tower and cables is to ensure the safety of the structure during the incremental launching construction, particularly in the most unfavorable cantilever state. When the structure is in the maximum cantilever state, with the front end of the guide beam supported on pier 14, the tower is set on pier 15 (Scheme 1) in this paper. To explore the rationality of the tower location, the distance between the tower and pier 15 is changed to 20 m, recorded as Scheme 2. The stress of the steel box girder section at the tower position, with the guide beam supported at pier 14, is compared for the two schemes, as shown in Fig. 19. It can be seen from Fig. 19 that the stresses in the steel box girder section are significantly different between the two schemes. In scheme 1, with the tower located on pier 15, the upper and lower edge stresses are 55.2 MPa and -65.6 MPa, respectively. In contrast, in scheme 2, where the tower is placed 20 m from pier 15, the stresses increase to 123 MPa and − 128 MPa, respectively. This indicates that the stresses in scheme 2 are approximately double those in scheme 1. In addition, the stress signs of the upper and lower edges of the steel box girder sections in the two schemes have changed. This is because the tower in Scheme 1 is located above the pier, allowing the cable force's effect on the tower to be transferred to the pier, which reduces the stress on the steel box girder.

Fig. 19
figure 19

Stress comparison.

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If the tower is not positioned on the pier, the cable force acts directly on the steel box girder. This results in excessive stress on the steel box girder, causing compression on the upper edge of the section and tension on the lower edge. Therefore, when the guide beam spans the maximum distance, the tower should be positioned directly above the pier to maximize the effectiveness of both the tower and the cables.

Conclusion

In view of the challenges faced in constructing the steel box girder of the Fenshui River Bridge by the incremental launching method without auxiliary piers, this paper presents a method using cables and towers for auxiliary construction and reaches the following main conclusions:

  1. (1)

    First, based on the maximum cantilever state of the structure, with the vertical deformation of the front end of the guide beam and the horizontal deformation of the top of the tower as the control objectives, the cable force is calculated using the influence matrix method. The unstressed cable length is then determined according to the mechanical relationship between cable deformation and cable force. An adaptive cable force calculation method is proposed through adjustments to the unstressed cable length to accommodate changes in the structural system.

  2. (2)

    During the incremental launching construction of the steel box girder, the cable force fluctuates significantly, reaching its maximum value when the structure is in the maximum cantilever state. The cable forces calculated using this method are in good agreement with the measured values. For instance, at the maximum cantilever state of the structure, the calculated and measured values of the C1 cable force are 868.5 kN and 903 kN, respectively, resulting in a difference of 3.96%. Additionally, the calculated deformation and stress values of key sections are consistent with the measured values, with maximum deviations of 6.4% and 6.6%, respectively.

  3. (3)

    To maximize the effectiveness of the tower and cable, the tower should be positioned directly above the pier when the guide beam crosses the maximum span. This arrangement allows the cable's force on the tower to be efficiently transferred to the pier.

The results of this paper can serve as a valuable reference for the construction of similar types of bridges. Further studies will focus on optimizing the length and stiffness of the guide beam, as well as exploring methods to enhance the overturning safety of the steel box girder during the construction process.