Research on Finite Element Analysis Method of Curved Beam Walking Incremental Launching Construction

Abstract

The “direct method” is commonly employed to establish analytical models for assessing the stress state of curved beam bridges during incremental walking-launch construction. However, this approach often involves cumbersome mathematical derivations for curved elements and entails high computational costs. To overcome these limitations, this study proposes a “straight-line substitution method” and examines its applicability for analyzing the mechanical behavior of a composite system consisting of steel box girders and steel guide beams during the curved beam walking-launch process. Using a curved river-crossing bridge as a case study, finite element analysis (FEA) is conducted to compare the mechanical responses of the composite system under various loading conditions obtained from the proposed method and the conventional direct method. Furthermore, a parameter analysis is performed to investigate the influence of variations in beam height and width on the consistency between the two methods. The results demonstrate that the straight-line substitution method yields computational outcomes highly consistent with those of the direct method across different beam heights and widths. Moreover, the proposed method exhibits superior modeling efficiency compared to the direct method.

1. Introduction

Bridge incremental launching construction technology originated in the 1950s, with its first successful application at Austria’s Ager Bridge in 1959. Introduced to China in 1977, the technique was first implemented on the Dijiaghe Bridge of the Xi’an-Yan’an Railway and the Wanjang Bridge in Guangdong Province [1,2]. Initially applied to prestressed concrete box girders, the method has gained widespread adoption in steel box girder and steel truss bridges with the development and maturation of finite element software [3,4,5,6]. For instance, Henrique M. Martins employed finite element analysis (FEA) software to investigate the global and local mechanical behavior of modular joints and members during incremental launching construction [7]. Owing to its minimal traffic disruption and precise synchronized control, walking incremental launching construction has become a mainstream bridge construction method [8,9,10,11].

During the launching process, stress concentration and localized deformation may occur at the connection between the steel box girder and the guide beam [12,13,14]. As the weight and stiffness of the main girder segments increase, the induced bending moment also rises, particularly in the negative moment zone near the pier tops. Research indicates that while longitudinal stiffeners can effectively enhance web stability, stress concentration often develops at the web-to-bottom plate connection under conventional fabrication processes. Furthermore, some girder segments exhibit low buckling eigenvalues, indicating a potential risk of instability [15,16,17,18]. Geometric control methods, combined with phased finite element analysis, are also employed for alignment control and elevation limit setting [19,20,21]. Additionally, the finite element analysis method has been widely used to study the mechanical properties of steel box girders during incremental launching due to its convenience, efficiency, low cost, and result accuracy [22,23,24].

Li [25] utilized Midas FEA NX 2021 software to establish a hybrid finite element model under maximum cantilever conditions and analyzed the corresponding local stress state. Cao [26] applied finite element analysis (FEA) to investigate the influence of parameters such as support pad thickness, base plate thickness, longitudinal support length, and stiffener configuration on the local stresses in steel box girders. Lou [27] developed a finite element model of a steel truss girder using MIDAS Civil to analyze the residual bearing capacity of the lower chord, the jacking thrust, and support reactions under both maximum cantilever and non-cantilever conditions. R. Chacón [28] employed FEA to examine the stresses, strains, and displacements during the incremental launching process of steel bridges. Wang [29] adopted the “direct method” to create a finite element model of a curved steel box girder in MIDAS Civil, systematically analyzing its stress state and deflection throughout the entire launching process. Li [30] also used the “direct method” to establish a finite element model for verifying the load-bearing capacity and support reactions of steel truss girders. Yang [31] employed the “direct method” in MIDAS Civil to develop finite element models for multiple critical construction phases, analyzing structural responses under seismic loading and examining the influence of the main girder’s curvature radius on the seismic response during construction. Hai [32] applied the “direct method” in MIDAS Civil to establish a curved model, analyzing the stress state and deflection of a curved steel box girder bridge under various operational conditions and comparing the results with field monitoring data. Herein, the “direct method” refers to a finite element modeling technique that discretizes the actual curved geometry using linear beam elements. In contrast, the “straight-line substitution method” is an approach that equivalently transforms a curved alignment model into a straight-line model for analysis. Currently, the direct method is the mainstream modeling technique for the finite element analysis of curved girders during incremental launching construction. However, if the straight-line substitution method can significantly improve computational efficiency while maintaining high consistency with the direct method, its practical value merits in-depth investigation. Presently, this alternative method has not been widely adopted, and its applicability and reliability require systematic validation. Therefore, this study conducts a systematic comparative analysis between the straight-line substitution method and the conventional direct method.

Based on a large-scale curved highway bridge project spanning a river, this study employs both the straight-line substitution method and the direct method for finite element modeling. The stress distribution, bearing reactions, and deflection variations in the composite system—comprising a curved steel box girder and a steel guide girder—are systematically investigated under various working conditions. The computational results obtained from the straight-line substitution method are comprehensively compared with those from the direct method. This research advances the finite element methodology for analyzing mechanical behaviors during the incremental launching of curved girders, effectively improving computational efficiency while ensuring result reliability. The outcomes not only provide more robust theoretical support for construction control in the current project but also offer a convenient and reliable reference for the analysis and design of similar curved girder bridges in the future.

2. Project Profile

The bridge is a large-scale curved highway bridge crossing a river, with a horizontal curve radius of 1000 m. It features a span arrangement of (4 × 30) + (4 × 58) + (3 × 30) m, a total length of 449 m, and an overall width of 37 m. The bridge is located in Changsha City, Hunan Province, China. The superstructure adopts a continuous steel–concrete composite girder system. Each single span has a deck width of 17.5 m and a girder height of 2.9 m. In each transverse half-section, three steel girders are arranged with a spacing of 5.85 m between main girders. The pile layout and cross-sectional details of the curved girder are illustrated in Figure 1 and Figure 2, respectively.

The construction of the continuous steel–concrete composite girder is carried out in two main phases according to its structural components:

Phase One: The lower steel box girders are prefabricated in the factory, transported to the site, and then lifted segmentally into position using the full-span incremental launching method.

Phase Two: After the steel box girders are launched into their final position, the superstructure concrete slab is cast in place using C55 polypropylene fiber-reinforced shrinkage-compensating concrete.

The steel box girder is constructed using the walking incremental launching method. It is divided into 10 segments, each 22 m long, plus an additional 12 m segment, resulting in a total launching length of 232 m. All steel guide girders employ a double-girder configuration with a length of 36 m. The guide girder has an I-shaped cross-section with a variable longitudinal profile, featuring beam heights of 2.52 m, 2.1 m, and 1 m, respectively. The web thickness is 14 mm. The flange at the end is 1000 mm wide, connected via a 1 m-long transition section, beyond which the flange width reduces to 600 mm with a thickness of 16 mm. The stiffener thickness is 12 mm. The main chord member is a steel pipe of 219 mm × 8 mm, while the longitudinal, transverse, and diagonal bracings between the main chords are composed of 159 mm × 6 mm steel pipes. Each set of the walking launching device primarily includes two vertical jacks (400 t capacity each), one horizontal launching jack (100 t capacity), two correction jacks (65 t capacity each), along with sliding bearings, sliding plates, and slideways. The specific arrangement of the launching device is shown in Figure 3.

3. Theory of Finite Element

Throughout launching, the composite system comprising the curved steel box girder and the steel guide girder may experience considerable displacements, yet the resulting internal strains remain relatively small. As the steel material remains in an elastic state, its stress–strain relationship can be considered linear. Therefore, the geometric nonlinearity pertinent to curved girder launching is classified as the first type—characterized by large deformations, large rotations, but small strains [33,34].

In the geometric nonlinear analysis, the element stiffness matrix and the structural stiffness matrix established by the complete Lagrangian formulation method are often asymmetric, which is not conducive to the solution. Therefore, when analyzing geometric nonlinear problems in curved beams during incremental launching construction, the fully Lagrangian formulation with an incremental column matrix is generally adopted [35].

4. Finite Element Numerical Simulation

4.1. Parameter Setting

4.1.1. Basic Model Parameters

In this study, the main curved steel box girder and the I-section steel guide girder are made of Q345 steel, while the circular tubular transverse bracing of the guide girder is made of Q235 steel. The unit weight of steel is taken as 78.5 kN/m³. The design stress criteria for the two steel grades are as follows: for Q235 steel, the design values for bending and axial stress are 180 MPa, and for shear stress is 105 MPa; for Q345 steel, the corresponding design values are 270 MPa for bending and axial stress, and 155 MPa for shear stress.

This model is established based on the design drawings. The nodes are divided following a control unit length of 1 m, meaning that both the “straight-line substitution method” and the “direct method” discretize the structure using a unit length of 1 m. All boundary conditions are set as rigid connections, and the connection between the superstructure and the bearings is also modeled as rigid.

The load parameters are determined according to the actual engineering conditions during the incremental launching of the curved girder. Specific values are presented in

Table 1. Load parameter values.

Serial Number	Member	Load Type	Loading Value
1	steel box beam	dead weight	According to the actual weight, the program is automatically loaded. In order to reserve enough safety reserve, the weight coefficient of steel box beam is 1.2.
2	steel guide girder	dead weight	According to the actual weight, the program is automatically loaded. In order to reserve enough safety reserve, the weight coefficient of steel box beam is 1.2.

.

4.1.2. Selection Analysis of Pushing Condition

During the walking incremental launching process, the curved girder undergoes a continuous cycle of “jacking up, translating, lowering, and retracting”. Consequently, its stress state and boundary conditions are constantly changing. To achieve accurate modeling, it is essential to select representative working conditions. The selected conditions must not only fulfill the verification requirements specified in the “Technical Specifications for Construction of Highway Bridges and Culverts” [36] but also, by simulating the most unfavorable scenarios, provide a quantitative basis for construction safety control. Based on actual on-site construction sequences and adhering to the principle of considering the most critical conditions, the entire curved girder launching process is divided into six distinct working scenarios, as summarized in

Table 2. Selection of jacking conditions.

The Type of Working Conditions	Incremental Launching Situation
Condition TQ1	The steel box girder is advanced by a total of 152 m. Within this distance, the combined cantilever length of the steel box girder and the steel guide girder is 50 m.
Condition TQ2	The steel box girder was advanced by 162 m, resulting in a combined cantilever length of the steel box girder and the steel guide girder reaching 60 m, which represents the maximum cantilever condition.
Condition TQ3	The steel box girder was advanced by 182 m, with the steel guide girder cantilevering 20 m beyond the last support.
Condition TQ4	The steel box girder was advanced by 187 m, with the steel guide girder cantilevering 25 m.
Condition TQ5	The steel box girder was advanced by 192 m, with the steel guide girder cantilevering 30 m.
Condition TQ6	The steel box girder was advanced by 198 m, with the steel guide girder cantilevering 36 m.

Note: Bridge length is measured along the bridge axis (the span centerline).

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4.2. Modeling Scheme

4.2.1. Scheme 1 Modeling with “Straight-Line Substitution Method”

For curved bridges, the finite element models are established using the “straight-line substitution method”. The core of this approach lies in discretizing the curve into multiple short, straight beam elements, thereby equivalently transforming the curved geometric model into a straightened configuration. This method significantly improves modeling efficiency and computational speed while preserving the accuracy required for engineering calculations. It directly utilizes the well-established and computationally efficient theory of straight beam elements, circumventing the complex mathematical derivations and high computational costs associated with sophisticated curved elements. Through appropriate mesh refinement, the mechanical behavior of the actual curved girder can be approximated with minimal error, satisfying the demands for global bridge stress analysis and design. This represents an optimized engineering practice that effectively balances precision and efficiency. The detailed modeling procedure for the straight-line substitution method is illustrated in Figure 4.

In the first scheme, the finite element software MIDAS Civil 2023 is employed. Based on the straight-line substitution method and according to the actual parameters and boundary conditions of the six construction stages, finite element models for each working condition are established to simulate the entire incremental launching process, as shown in Figure 5.

4.2.2. Scheme 2 “Direct Method” Modeling

The “direct method” is a finite element modeling technique that approximates the actual curved geometry by discretizing it into a series of linear beam elements. Using shorter elements with a greater number of segments produces a model that more closely resembles the true curvature, though this also increases the computational burden. The core advantage of this method is its ability to capture complex geometric boundaries more accurately, thereby enhancing the precision of the numerical solution. However, this increase in geometric accuracy also introduces corresponding challenges: when modeling structures such as curved beams, the coordinate calculation during preprocessing becomes tedious, often requiring more complex mathematical derivations and numerical computations, thereby reducing computational efficiency and increasing costs. The detailed modeling procedure for the direct method is illustrated in Figure 6.

The second scheme also utilizes MIDAS Civil 2023, applying the direct method. Following the actual parameters and boundary conditions of the six construction stages, finite element models were established for each scenario to simulate the complete incremental launching process of the curved girder, as shown in Figure 7.

4.2.3. Efficiency Comparison

The fundamental distinction between the “straight-line substitution method” and the “direct method” lies in their treatment of curved geometry. The former simplifies the model by equivalently developing the curved alignment into a straight-line configuration, whereas the latter approximates the actual curved shape using discretized straight-line elements. Because the straight-line substitution method employs linear geometry, node coordinates can be generated directly through iterative calculations from the starting point using the linear equation and element length, resulting in a straightforward computational procedure. In contrast, the direct method relies on the actual curve equation and must determine subsequent nodal coordinates indirectly via geometric relationships (e.g., solving trigonometric functions), which entails more complex computational steps. For the critical stage of node coordinate generation, the straight-line substitution method achieves higher average efficiency than the direct method. The specific calculation procedure is shown in Figure 8. Given that the remaining modeling procedures require comparable time for both approaches, the straight-line substitution method demonstrates superior overall computational efficiency in establishing finite element models.

5. Results and Analysis

The risks associated with the composite system of a curved steel box girder and a steel guide girder during walking incremental launching construction are primarily reflected in the potential loss of control over three critical mechanical indicators. First, excessive stress may induce plastic deformation or brittle failure of the primary structural members. Second, an abnormal distribution of bearing reactions directly jeopardizes the safety of temporary piers and launching equipment, potentially leading to girder instability or overturning. Furthermore, excessive or irregular deflections can cause the structure to deviate from its intended alignment, triggering collisions or overall instability. Therefore, this study focuses on these three key mechanical indicators—stress, bearing reaction forces, and deflection—to systematically compare and analyze the computational results obtained from the straight-line substitution method and the direct method.

5.1. Stress Analysis

In the first scheme, finite element models for each working condition were established in MIDAS Civil based on the straight-line substitution method. The resulting stress distributions are presented in Figure 9 and Figure 10. The results indicate that during the incremental launching of the curved girder, the maximum bending stress of 161.4 MPa occurs in working condition TQ2. At this stage, the steel box girder has been advanced by 162 m, and the composite cantilever of the steel box girder and the steel guide girder reaches its maximum length of 60 m. The maximum shear stress of 42.1 MPa occurs in working condition TQ5, corresponding to a launching distance of 192 m for the steel box girder and a cantilever length of 30 m for the steel guide girder.

In the second scheme, finite element models for each working condition were established in MIDAS Civil using the direct method. The resulting stress distributions are presented in Figure 11 and Figure 12. The analysis indicates that during the incremental launching of the curved girder, the maximum bending stress of 156.85 MPa occurs under working condition TQ2. At this stage, the steel box girder has been advanced by 162 m, and the composite cantilever of the steel box girder and the steel guide girder reaches its maximum length of 60 m. The maximum shear stress of 30.2 MPa occurs in working condition TQ5, corresponding to a launching distance of 192 m for the steel box girder and a cantilever length of 30 m for the steel guide girder.

The maximum stress values in both Schemes 1 and 2 occur under working conditions TQ2 and TQ5. These peak stresses remain below the bending stress design value of 345 MPa specified for Q420 steel in the *Specifications for Design of Highway Steel Bridges* (JTG D64-2015) [37], confirming that the project’s incremental launching construction scheme complies with the code requirements.

A comparative analysis shows that the maximum bending stress decreased from 161.4 MPa in Scheme 1 to 156.9 MPa in Scheme 2, a reduction of 2.8%. Similarly, the maximum shear stress decreased from 32.89 MPa to 30.2 MPa, a reduction of 2.69 MPa. This suggests that the straight-line substitution method used in Scheme 1—which approximates the curve using discrete straight beam segments—may introduce localized stress concentrations. In contrast, the direct method (Scheme 2) more accurately represents the continuous curvature, thereby mitigating this effect. Consequently, the results from the straight-line substitution method are slightly more conservative. Following the principle of maximum safety, the outcomes from Scheme 1 therefore provide a greater safety margin.

Beyond these peak values, comparative diagrams (Figure 13 and Figure 14) and detailed stress contours indicate that under all other working conditions, the stress results from both methods exhibit a high degree of consistency, with the locations of maximum stress in close agreement. In summary, the stress analyses performed using the straight-line substitution method and the direct method for the composite curved steel box girder and guide girder system yield highly consistent results.

5.2. Analysis of Support Reaction Force

As shown in the stress contour plots (Figure 15 and Figure 16), the bearing reaction forces increase progressively with the launching length due to the growing combined weight of the steel box girder and the steel guide girder within the composite system. For both Scheme 1 and Scheme 2, the maximum bearing reaction force occurs under working condition TQ5—when the steel box girder has been advanced by 192 m and the steel guide girder has a 30 m cantilever—and is located at the bearing on the inner side of Pier 6. Specifically, the maximum reaction force is 3368.1 kN for Scheme 1 and 3361.6 kN for Scheme 2. Both values are below the maximum capacity of the vertical jack (3922.7 kN), confirming the feasibility of the selected launching scheme.

Analysis of

Table 3. Variation in Maximum Bearing Reaction Forces under Different Operating Conditions.

The Type of Working Conditions	Scheme 1 Maximum Support Reaction Force (KN)	Scheme 2 Maximum Support Reaction Force (KN)	Maximum Bearing Reaction Force Position
Condition TQ1	2731.6	2630.2	Pier No. 5
Condition TQ2	3293.6	3226.5	Pier No. 6
Condition TQ3	3231.8	3239.9	Pier No. 6
Condition TQ4	3341.3	3332.6	Pier No. 6
Condition TQ5	3368.1	3361.6	Pier No. 6
Condition TQ6	3251.1	3229.6	Pier No. 6

and Figure 17 indicates that the bearing reaction forces calculated by the two schemes are highly consistent. Under TQ5 at Pier 6, the difference between Scheme 1 (3368.1 kN) and Scheme 2 (3361.6 kN) is negligible. Moreover, the reaction forces and their locations match perfectly across all other working conditions. In summary, the bearing reaction results obtained from the straight-line substitution method and the direct method for the composite system of curved steel box girder and steel guide girder show a high degree of consistency in both overall trends and numerical values.

5.3. Deflection Analysis

The deflection results are presented in contour plots (Figure 18 and Figure 19). As the composite system of the steel box girder and steel guide girder advances, the cantilever length varies, leading to corresponding changes in system deflection, as detailed in

Table 4. The change in the maximum deflection of the structure under each working condition.

The Type of Working Conditions	Scheme 1 Maximum Deflection (mm)	Scheme 2 Maximum Deflection (mm)	Maximum Deflection Position
Condition TQ1	261.15	268.56	Guide beam end
Condition TQ2	431.3	427	Guide beam end
Condition TQ3	31.02	29.82	Guide beam end
Condition TQ4	60.1	60.39	Guide beam end
Condition TQ5	103.6	112.3	Guide beam end
Condition TQ6	167.49	176.2	Guide beam end

. The maximum displacement of the composite system is smallest when the cantilever is 20 m. As the cantilever length increases, the maximum displacement also rises. Under working condition TQ2, where the composite cantilever reaches its maximum length of 60 m, the deflection attains its peak value. Therefore, during the incremental launching of the curved steel box girder, structural deflection exhibits a clear increasing trend with cantilever length, and the maximum displacement occurs when the composite system is at its maximum cantilever.

The maximum deflection calculated in Scheme 1 was 427.0 mm, which is 1.0% lower than the 431.3 mm result from Scheme 2. This difference was not statistically significant (p = 0.12 > 0.05). As shown in Figure 20, the deflection curves for Scheme 1 and Scheme 2 are nearly identical, demonstrating a high degree of consistency in the computed structural deflections. This strongly confirms that the deflection results obtained from the straight-line substitution method and the direct method for the composite curved steel box girder and guide girder system are in close agreement.

5.4. Parametric Analysis

Based on an actual engineering project involving the incremental launching of a large-radius curved steel box girder with a radius of R = 1000 m, this study further examines the consistency between the straight-line substitution method and the direct method in analyzing the composite system of curved steel box girders and steel guide beams. To assess the broader applicability and robustness of the findings, a parametric analysis is conducted to investigate how variations in beam height and width influence the agreement between the results obtained from the two methods [38].

5.4.1. Beam Height Analysis

To investigate the influence of beam height variation on the consistency between the straight-line substitution method and the direct method for the composite curved steel box girder and guide beam system, a parametric analysis is conducted. Three beam heights—2530 mm, 2630 mm, and 2730 mm—are selected as variables. Finite element models corresponding to each height are developed, and the results are compared. Owing to space limitations, only a selection of representative contour plots of the results is presented herein, as shown in Figure 21.

Based on the stress calculation results presented in Figure 22 and Figure 23, a comparative analysis of stresses between the two schemes at different beam heights yields the following outcomes.

Group 1 (beam height 2530 mm): The maximum bending stress in Scheme 1 is 161.4 MPa, which is 2.8% higher than the 156.9 MPa obtained in Scheme 2. Across all working conditions, the maximum difference in bending stress between the two schemes is only 2.9%, and the location of maximum stress remains the same.

Group 2 (beam height 2630 mm): The maximum bending stress in Scheme 1 is 155.5 MPa, approximately 2.7% higher than the 150.0 MPa in Scheme 2. The maximum difference in bending stress is 3.1%, with identical locations of peak stress.

Group 3 (beam height 2730 mm): The maximum bending stress in Scheme 1 is 150.0 MPa, about 2.1% lower than the 153.2 MPa in Scheme 2. The maximum difference between the two schemes is 3.1%, and the critical stress location is consistent across all conditions.

Overall, the maximum bending stress in Scheme 1 generally exceeds that in Scheme 2, suggesting that its modeling approach may introduce slight stress concentrations. The consistent locations of maximum bending stress under all loading conditions indicate that the choice of modeling method does not alter the overall stress distribution pattern or the identification of critical zones in the structure. Furthermore, the maximum difference in bending stress between the two methods never exceeds 3.2%, demonstrat- ing that the stress results obtained from the straight-line substitution method and the direct method for the composite curved steel box girder and guide beam system are highly consistent.

Based on the bearing reaction force results presented in Figure 24, a comparative analysis between the two schemes under different beam heights yields the following findings:

Group 1 (beam height 2530 mm): The maximum bearing reaction force in Scheme 1 is 3368.1 kN, differing by only 0.19% from the 3361.6 kN in Scheme 2.

Group 2 (beam height 2630 mm): The maximum reaction force in Scheme 1 is 3375.7 kN, with a difference of only 0.09% compared to 3378.9 kN in Scheme 2.

Group 3 (beam height 2730 mm): The maximum reaction force in Scheme 1 is 3364.6 kN, which differs by only 0.94% from the 3396.2 kN obtained in Scheme 2.

These three sets of results demonstrate that the difference in maximum bearing reaction force between the two schemes remains below 1.0% across all beam height variations, with numerically close values. Moreover, the locations of the maximum bearing reactions are identical in both schemes.

Based on the structural deflection results presented in Figure 25, a comparative analysis between the two schemes under different beam heights yields the following outcomes:

Group 1 (beam height 2530 mm): The maximum deflection in Scheme 1 is 431.3 mm, differing by only 1.0% from the 427.0 mm in Scheme 2.

Group 2 (beam height 2630 mm): The maximum deflection in Scheme 1 is 412.0 mm, with a difference of only 2.7% compared to 423.4 mm in Scheme 2.

Group 3 (beam height 2730 mm): The maximum deflection in Scheme 1 is 413.4 mm, which differs by only 1.4% from the 419.1 mm obtained in Scheme 2.

The above analysis shows that the difference in maximum deflection between the two schemes remains below 2.0% across all beam height variations, with numerically close values. Furthermore, the locations of maximum deflection are identical for both schemes.

The analysis above indicates that across different beam heights, the differences between the two methods in terms of maximum stress, maximum bearing reaction, and maximum deflection are less than 3.2%, 1.0%, and 2.7%, respectively. The numerical results from both methods are in close agreement, and the locations of these critical responses are fully consistent. Therefore, it can be concluded that under varying beam heights, the results obtained from the straight-line substitution method and the direct method for analyzing the composite system of curved steel box girders and steel guide beams exhibit a high degree of consistency.

5.4.2. Beam Width Analysis

To investigate the effect of beam width variation on the consistency between the straight-line substitution method and the direct method for the composite curved steel box girder and steel guide girder system, three beam widths—3500 mm, 3700 mm, and 3900 mm—were selected as parameters. Finite element models corresponding to each width were developed, and the results were compared. For the sake of brevity, only a selection of representative result contours is presented in this section, as shown in Figure 26.

Based on the stress results presented in Figure 27 and Figure 28, a comparative analysis between the two schemes under different beam widths yields the following outcomes:

Group 1 (beam width 3500 mm): The maximum bending stress in Scheme 1 (straight-line substitution method) is 150.0 MPa, approximately 2.1% lower than the 153.2 MPa obtained in Scheme 2 (direct method). The maximum difference across all working conditions is 3.1%.

Group 2 (beam width 3700 mm): The maximum bending stress in Scheme 1 is 143.3 MPa, about 3.4% higher than the 138.5 MPa in Scheme 2. This represents the maximum stress difference observed across all conditions, with identical locations of peak stress.

Group 3 (beam width 3900 mm): The maximum bending stress in Scheme 1 is 144.5 MPa, roughly 3.2% higher than the 139.9 MPa in Scheme 2. The maximum difference across all conditions is 4.2%, and the critical stress locations remain consistent.

Under varying beam widths, the maximum stress difference between the two schemes is less than 4.2% in all cases, with numerically close values and identical locations of maximum stress. In most scenarios, the stress calculated by the straight-line substitution method is slightly higher, suggesting its discretized modeling approach may introduce localized stress concentrations.

Based on the bearing reaction results presented in Figure 29, a comparative analysis between the two schemes under different beam widths yields the following findings:

Group 1 (beam width 3500 mm): The maximum bearing reaction force in Scheme 1 is 3364.6 kN, differing by only 0.94% from the 3396.2 kN calculated in Scheme 2.

Group 2 (beam width 3700 mm): The maximum reaction force in Scheme 1 is 3595.3 kN, representing a 1.5% difference compared to the 3542.9 kN in Scheme 2.

Group 3 (beam width 3900 mm): The maximum reaction force in Scheme 1 is 3679.2 kN, which differs by 1.5% from the 3625.7 kN obtained in Scheme 2.

Across all beam widths, the difference in maximum bearing reaction force between the two schemes remains below 1.5%, with numerically close values. Furthermore, the locations of these maximum reactions are identical for both schemes.

Based on the structural deflection results presented in Figure 30, a comparative analysis between the two schemes under different beam widths yields the following outcomes:

Group 1 (beam width 3500 mm): The maximum deflection in Scheme 1 is 413.4 mm, differing by only 1.4% from the 419.1 mm in Scheme 2.

Group 2 (beam width 3700 mm): The maximum deflection in Scheme 1 is 385.0 mm, with a difference of only 0.3% compared to 383.3 mm in Scheme 2.

Group 3 (beam width 3900 mm): The maximum deflection in Scheme 1 is 376.7 mm, which differs by merely 0.2% from the 376.1 mm obtained in Scheme 2.

Across all beam widths, the difference in maximum deflection between the two schemes remains below 1.4%, with numerically close values. Furthermore, the locations of maximum deflection are identical for both schemes.

In summary, under varying beam width conditions, the differences between the two methods are less than 3.2% for maximum stress, 1.5% for maximum bearing reaction, and 1.4% for maximum deflection. The numerical results show a high level of agreement, and the corresponding locations are fully identical. This demonstrates that, across different beam widths, the analysis results obtained using the straight-line substitution method and the direct method for the composite curved steel box girder and guide beam system exhibit a high degree of consistency.

6. Conclusions

Based on a practical engineering project involving the incremental launching of a large-radius curved steel box girder with a radius of R = 1000 m, this study employs two finite element modeling approaches in MIDAS Civil software: the “straight-line substitution method” and the “direct method.” The former discretizes the curved alignment into multiple short, straight beam elements, whereas the latter uses higher-order parametric elements to directly model the curved geometry. A comparative analysis is conducted on the bearing reactions, stresses, and deflections of the steel box girder and steel guide girder composite system under various working conditions using both methods. To further verify the robustness of the findings, a parametric analysis is performed to examine the influence of variations in beam height and width on the consistency of the results. The main conclusions are summarized as follows:

(1)

The finite element models established using the straight-line substitution method and the direct method produce highly consistent results in terms of calculated stress, bearing reactions, and deflection. This high level of consistency is maintained across variations in beam height and width, confirming the applicability of the straight-line substitution method for analyzing the mechanical behavior of the steel box girder–steel guide girder composite system during the incremental launching construction of curved girders. Furthermore, the straight-line substitution method avoids complex mathematical derivations and substantial computational effort, offering significantly higher modeling efficiency compared to the direct method.

(2)

Based on the finite element models established using the straight-line substitution method, the calculated maximum bending and shear stresses in the steel box girder and steel guide girder composite system exceed those obtained from the direct method. In most cases, the stresses predicted by the straight-line substitution method are approximately 3.2% greater. Consequently, in accordance with the principle of conservative design, the results derived from the straight-line substitution method offer a safer and more conservative estimation.

(3)

Finite element models were established using both the straight-line substitution method and the direct method. The results from both approaches indicate that during the incremental launching of steel–concrete composite curved girders, the deflection of the steel box girder and steel guide girder composite system shows a strong positive correlation with the increasing cantilever length of the system (from 20 m to 60 m). When the cantilever reached its maximum length of 60 m, the system’s peak displacement approached a critical state, confirming that the cantilever effect is the primary factor controlling deflection development. Consequently, during the incremental launching of curved girders, the maximum cantilever condition represents a phase of reduced structural stiffness, necessitating focused monitoring throughout this construction stage.

This study established comparative models based on the straight-line substitution method and the direct method, systematically analyzing their differences in predicting stress, bearing reactions, and deflection responses. Through parameter analysis, the influence of beam height and width on the consistency between the two modeling approaches was examined. However, the effects of two other key parameters curvature radius and material properties on the consistency of the two methods remain unexplored and represent an important direction for future research.