Research on the effects of curvature and shear keys on torsional stiffness and interfacial stress of segmented U-shaped curved bridge

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The effects of curvature radius ( R ) of the bridge and shear keys (number, arrangement, and size) at inter-segment bonding joints on the bridge's torsional stiffness and interfacial stress are analyzed. The results demonstrate a nonlinear relationship between bridge curvature and torsional stiffness. As the curvature radius increases, torsional stiffness enhances while the growth rate declines gradually. Increasing the quantity of shear keys in the web and floor regions of U-shaped bridge segments, along with extending their width, effectively enhances the bridge's torsional stiffness and global deformation resistance while ameliorating the interfacial stress state between segments. These findings offer both theoretical and practical guidance for the design of segmental U-shaped curved bridges and shear key systems. Physical sciences/Engineering Physical sciences/Materials science Segmental U-shaped curved bridges Curvature radius Shear keys Torsional stiffness Interfacial stress Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 1 Introduction In recent years, Rail transit industry has developed rapidly, and various forms of beams have been widely used in this field, such as U-shaped bridges, T-shaped beams, box beams, and so on 1 , 2 . Compared to other beams, U-shaped bridges have many advantages, such as better noise reduction effects due to their lower construction height, aesthetic appearance, and high economic benefits. Therefore, U-shaped bridges are widely used in the rail transit field 3 – 7 . In 1962, a bridge crossing over the Seine River in France was constructed—regarded as the inaugural precast segmental concrete bridge on a global scale. Segmental bridges have garnered significant attention due to their numerous advantages, including fast construction speed, environmental friendliness, and superior quality control. By adopting this technology, construction time and traffic disruptions can be significantly reduced, thereby greatly enhancing economic benefits 8 – 10 . Against the backdrop of increasing global demand for bridges, the appeal of segmental bridges is growing 11 , 12 . For segmental bridges, joints are the weak points in their structure, thus they require special attention and handling 13 . This is because the strength, maintainability, and structural behavior of precast concrete segmental bridges largely depend on the performance of the joints between the segments. There are various existing forms of joints, including dry joints, wet joints, and epoxy resin joints. Compared to other types of joints, epoxy resin joints are considered to perform better in terms of durability and ultimate shear load capacity. The use of epoxy resin can in-crease the shear load capacity of the joint 14 , 15 . Additionally, uncured epoxy resin can act as a lubricant for the joint, compensating for irregularities between the joint surfaces 16 . In bridge engineering, curved bridges are widely used to adapt to terrain and traffic requirements. However, in practical use, curved bridges face serious torsion problems, which can affect the stability and safety of the bridge, and in severe cases may lead to structural overturning and damage 17 . The case of overturning involving the Yangmingtan Bridge exemplifies a critical vulnerability inherent in bridges with curvature. Observable is that, when subjected to eccentric loading conditions, such curved structures are notably susceptible to experiencing torsional instability 18 . Therefore, significant to in-depth research are the torsional stiffness attributes inherent within curved bridges, and from enhancements of anti-overturning capabilities can overall safety appreciably benefit. Many scholars have conducted research on the bending and shear behavior of bridges, but studies on the torsional behavior are relatively fewer in number. Meyyada et al. 19 conducted a review on the torsional strengthening of reinforced concrete (RC) beams using externally bonded (EB) composite materials and established a database for RC beams retrofitted with EB fiber-reinforced polymer (FRP) and fiber-reinforced cementitious matrix (FRCM). Zhao et al. 20 discussed the torsional behavior of a novel reinforced concrete U-shaped steel-concrete composite beam (RCUCB) system through a series of torsion tests. Mane et al. 21 proposed a high-yield-strength welded wire mesh that can be wrapped around concrete members to improve composite performance and avoid sudden cracking, while also increasing the torsional stiffness of reinforced concrete beam specimens. Numerous scholars have conducted research on the shear keys of segmental bridge joints. Jiang et al. 22 tested full-scale specimens with shear keys under different levels of confining stress and proposed a shear failure mechanism for the continuous failure of multiple shear keys in dry joints based on the experimental results. Celia et al. 23 designed a new push-off dry joint test under combined forces, testing 24 push-off dry joint specimens under axial force, bending force, and shear force, and concluded the effects of parameters such as the number of shear keys, confining stress level, and eccentricity on the shear resistance of segmental bridges. Hua et al. 24 designed and developed 3D-printed concrete shear keys with epoxy mortar infill and studied the various designs of 3D-printed concrete shear keys with different shear key angles and shear joint inclinations through push-off and slant shear tests. Compared with straight bridges, curved bridges exhibit more complex stress patterns, and the problem of eccentric loads and torsion is more prominent. At present, the research on segmental U-shaped curved bridges is not sufficient, especially the effects of the arrangement, number and size parameters of shear keys and different curvature radii ( R ) of bridge on the torsional stiffness and interfacial stress of the bridge are not systematically studied in combination with the construction characteristics of segmental assembly and the section characteristics of U-shaped bridges. The above effect mechanism can provide theoretical support for the design optimization and engineering application of segmented U-shaped curved bridge, and promote the technical development and promotion of this kind of bridge structure. 2 Research background This study relies on the Bogotá Metro Line 1 Metro bridge project, which is designed with a total length of 23.86 kilometers and a total of 742 span bridges. The segmental U-beam assembly process is adopted, of which the curved bridge has 180 spans. Overview of the bridge and details is shown in Fig. 1 , the standard span of the bridge in the project is mainly displayed, which is 30 \(\:\text{m}\) long and 10 \(\:\text{m}\) wide, and consists of 11 segments. The R of the bridge in Fig. 1 is 300m. The effects of various factors, including the arrangement, number, size of shear keys and curvature of the bridge, on the torsional stiffness and interfacial stress of segmented U-shaped curved bridges are investigated using numerical methods in this paper. 3 Finite element analysis model Due to the symmetry of the structure, the equivalent simplification principle in structural mechanics is adopted to select the half-bridge (15 m) of the full-bridge (30 m) as the research object, and the half model is used to simplify the calculation process, as shown in Fig. 2 (a) and (b). The finite element analysis is performed using ANSYS. The boundary conditions and load of the half-bridge are shown in Fig. 2 (b). The symmetric boundary conditions are applied to the symmetric face, and the bearings at the end of the bridge are simply supported. The structure is subjected to uniform load of 48 k N/m, two concentrated loads of 150 k N and standard gravity 25 . The single-span bridge with 11 segments of U-shaped bridge girder contains 10 joints in total. as shown in Fig. 1 . Joints between bridge segments are bonded with epoxy resin adhesive. Tie constraint (Pink section) is used to connect the key block and keyway of the shear key, while friction contact(Green section) is used to simulate the flat part of the joint, and 0.45 is set as the friction coefficient 26 , as shown in Fig. 3 . The shear key of the finite element model are set to the same specification in this paper, which is more convenient to discuss the effect of the shear key on the torsional stiffness and interfacial stress of the bridge under the control of different parameters 11 . The parameters of the shear keys are marked in Fig. 3 . The elastic constitutive relation of concrete and prestressed tendons is used in finite element simulation. The concrete strength class is C50. The prestress applied to the prestressed tendons is 1395MPa, the cross-sectional area of curved prestressed tendons is 1540 mm², and that of straight prestressed tendons is 1400 mm². All data refer to the engineering drawings of the Bogotá project. Material parameters detailed in Table 1 . The layout of the prestressed tendons is shown in Fig. 4 . Table 1 Material Parameters. Material Elastic Modulus (GPa) Poisson's Ratio Density(kg/m 3 ) Concrete 35.04 0.2 2500 Prestressed tendons 206.0 0.3 7850 To ensure the accuracy of the finite element simulation in this study, a mesh convergence analysis is carried out based on a model with R = 200 m, l = 300 mm, w = 200 mm, h = 50mm, α = 45°, and 24 shear keys. The shear keys are uniformly distributed on the web and floor of the bridge, as shown in Fig. 3 . The maximum vertical deformation (deformation along the Y axis) of the bridge is calculated for multiple sets of models with varying numbers of elements. As shown in Fig. 5 , the results indicate that as the number of elements in the model increases, the maximum vertical deformation of the bridge gradually stabilizes, thereby confirming the validity of the finite element simulation. Considering the calculation speed and accuracy, 1.87×10 5 elements are finally selected as the number of calculation elements. 4 The effect of the curvature of curved bridges on torsional stiffness and interfacial stress To investigate the effect of bridge curvature radius on the torsional stiffness and interfacial stress, three different finite element calculation models are established by changing R . The parameters corresponding to models with different curvature radii are shown in Table 2 . The arrangement of shear keys for the models in Table 2 is shown in Fig. 3 . The shear keys are uniformly distributed on the web and floor of the bridge. Table 2 Parameters corresponding to models with different curvature radii. Name R (m) Number l (mm) w (mm) h (mm) α (°) R = 200 200 24 300 200 50 45 R = 300 300 24 300 200 50 45 R = 400 400 24 300 200 50 45 To facilitate result analysis, three paths (Path-1, Path-2, Path-3) and two cross - sections (C1 and C2) are set on the curved bridge deck, as shown in Fig. 6 (a). Path-1, Path-2 and Path-3 are used to obtain the inner, central and outer deformation of the bridge from the inside to the outside. C2 is in the midspan and C1 is in the quarter span of the bridge. The results of three different curvature radii models are shown in Fig. 7 and Fig. 8 . The overall vertical deformation nephogram of R = 200 model is shown in Fig. 6 (b), and the corresponding nephogram of other models are similar. The overall deformation of the structure is asymmetric. The bridge deforms upwards along Path-1 and deflects downwards along Path-3, indicating a clear trend of torsional deformation, which is likely due to the structural characteristics of curved bridges. The outer side of a curved bridge is longer, and its relative weight has a greater impact, with a higher stiffness on the outer side. And the inner side is short, with less impact on relative weight and relatively low stiffness. When subjected to the same prestress, the effect of the inner prestress is more pronounced, resulting in upward deformation on the inner side. On Path-2, the downward vertical deformation of the bridge from the simply supported end to the midspan gradually increases, and reaches the maximum at the midspan. As shown in Fig. 7 , under the same load conditions, the vertical deformation of different curvature radius models along Path-2 is basically the same. However, as the curvature radius increases, the vertical deformation of both the inner and outer sides of the bridge decreases significantly, and the difference in deformation between the inner and outer sides also decreases. When the curvature radius increases from 200m to 300m and 400m, the difference in deformation between the inner and outer sides at the quarter span decreases by 34.04% and 50.64%, respectively, while that at the mid-span decreases by 34.38% and 49.84%. Sudden changes may occur on the curve due to the structural characteristics of different beam joints. The stress conditions of C1 and C2 in curved bridges with different curvature radii are showed in Fig. 8 . For C1, the change of the interfacial stress of the bridge with the curvature radius is not obvious, and the overall performance is that the interfacial stress increases with the decrease of the curvature radius. For C2, as the curvature radius increases, the tangential stress along the x-axis (S11) shows an upward trend, while the absolute value of the normal stress along the z-axis (S33) gradually decreases. 5 The effect of shear keys on torsional stiffness and interfacial stress The joints between bridge segments are equipped with multiple pyramid shaped shear keys. The number, arrangement, and size of these shear keys can all affect the torsional stiffness and interfacial stress of segmental U-shaped curved bridges. The effects of different shear key configurations on the torsional stiffness and interfacial stress of segmental U-shaped curved bridges are investigated. 5.1 The effect of shear key number Four different finite element calculation models are established by changing the number of shear keys. The parameters corresponding to models with different numbers of shear keys are shown in Table 3 . The four types of interface shear key arrangement between bridge segments are established according to Table 3 , as shown in Fig. 9 . The shear keys are uniformly distributed on the web and floor of the bridge. 12keys, 16keys, 20keys and 24keys models are used for finite element simulation analysis. The results for these four models are presented in Fig. 10 . Table 3 Parameters corresponding to models with different number of shear keys. Name R (m) Number l (mm) w (mm) h (mm) α (°) 12keys 200 12 300 200 50 45 16keys 200 16 300 200 50 45 20keys 200 20 300 200 50 45 24keys 200 24 300 200 50 45 As shown in Fig. 10 (a) and (b), when the number of shear keys increases from 12 to 24, the vertical deformation of the model along the three paths gradually decreases. Taking path-2 as an example, compared with key 12, the vertical deformation of keys 16, 20, and 24 at the quarter span is reduced by 25.07%, 34.73%, and 41.25%, respectively. At mid-span, the reductions are 24.93%, 31.06%, and 36.18% respectively. Results show that increasing shear key number effectively controls vertical deformation of segmental U-shaped curved bridge. Notably, the deformation reduction when the number of shear keys increases from 12 to 16 is significantly greater than the reduction during the stage from 16 to 24 keys. Considering the manufacturing difficulty and cost factors, the lowest reasonable number of shear keys should be matched for different segmented U-shaped curved bridge. As shown in Fig. 10 (b) and (c), with the increase of shear keys, the vertical deformation on the inner or outer side decreases, but the difference in deformation between the inner and outer sides remains basically unchanged. Therefore, changing the number of shear keys will not significantly change the overall torsional stiffness of the bridge. Figure 10 (d) shows the effect of different shear key numbers on the interfacial stress of the bridge. With the increase of shear keys, for C1 and C2, S11 decreases, and the absolute value of S33 increases and becomes negative, indicating that the interface is under overall compression along the bridge direction. Therefore, increasing the number of shear keys can enhance the overall integrity of the bridge and improve the shear bearing capacity at the joints. Increasing the number of shear keys in the design can improve the connection stiffness of the bridge and effectively improve the interfacial stress state of the bridge structure. However, increasing the number of shear keys has little effect on improving the torsional stiffness of the bridge. 5.2 The effect of shear key arrangement Three different finite element calculation models are established by changing the arrangement of shear keys. The parameters corresponding to models with different arrangement of shear keys are shown in Table 4 . The arrangement of the shear keys used in the model, as shown in Fig. 11 , each scheme has 12 shear keys. In Combination 1, all shear keys are on the web. In Combination 2, all are on the floor. In Combination 3, all shear keys are evenly distributed between the floor and web. Table 4 Parameters corresponding to models with different arrangement of shear keys. Name R (m) Number l (mm) w (mm) h (mm) α (°) Combination1 200 12 300 200 50 45 Combination2 200 12 300 200 50 45 Combination3 200 12 300 200 50 45 Combination1, Combination2 and Combination3 models are used for calculation and analysis, and discusses the effect of shear key arrangement on the torsional stiffness and interfacial stress of the bridge. The calculation result is shown in Fig. 12 . As shown in Fig. 12 (a) and (b), on Path-1 andPath-2, the vertical deformation of Combination 2 and 3 is basically the same and less than that of Combination 1. On Path-3, compared with Combination 1, the vertical deformation of Combination 2 and Combination 3 at the quarter span decreased by 17.05% and 35.86% respectively, and that at the mid span decreased by 24.24% and 32.20% respectively. Therefore, the arrangement of shear keys has a significant effect on the control of vertical deformation. Considering all three paths, Combination 3 is the most effective in controlling vertical deformation, followed by Combination 2, and Combination 1 has the worst effect. As shown in Fig. 12 (c), there are significant differences in the vertical deformation difference between the inner and outer sides of the bridge under different shear key arrangement. Combination 2 has the largest vertical deformation difference, followed by Combination 3, and Combination 1 has the smallest vertical deformation difference. The arrangement of shear keys will affect the torsional deformation of bridges, with Combination 1 being the most effective in suppressing torsional deformation, followed by Combination 3, and Combination 2 being relatively ineffective. Therefore, setting more shear keys on the web is more beneficial to improve the torsional stiffness of the bridge. Figure 12 (d) shows the bridge interfacial stress under three different shear key arrangements. For C1 and C2, the order of S11 from maximum to minimum is Combination 2, Combination 3, and Combination 1. The absolute values of S33 are in ascending order as Combination 3, Combination 2, and Combination 1. This shows that Combination 1 is the best in reducing the tangential stress, and has the maximum normal compressive stress. Therefore, the priority of setting shear keys on the web can enhance the integrity of the bridge and improve the shear bearing capacity of the joints. In summary, increasing the shear key on the web can increase the torsional stiffness and improve the interface stress. The addition of shear keys on the floor is beneficial to increase the normal connection stiffness of the bridge interface, which should be comprehensively selected. 5.3 The effect of shear key size Five different finite element calculation models are established by changing the size of shear keys. The parameters corresponding to models with different size of shear keys are shown in Table 5 . The arrangement of shear keys is the same as 24keys, as shown in Fig. 9 . To avoid sharp corners in the shear keys, h has been changed from 50mm to 30mm 27 . Table 5 Parameters corresponding to models with different size of shear keys. Name R (m) Number l (mm) w (mm) h (mm) α (°) S100-100 200 24 100 100 30 45 S200-100 200 24 200 100 30 45 S300-100 200 24 300 100 30 45 S100-200 200 24 100 200 30 45 S100-300 200 24 100 300 30 45 The five models S100-100, S200-100, S300-100, S100-200 and S100-300 are used for the calculation and analysis, and the results are shown in Fig. 13 . As shown in Fig. 13 (a) and (b), as the size of the shear key gradually increases, the vertical deformation of the model along the three paths shows a decreasing trend. Taking path-2 as an example, when the width w of the shear key is kept constant at 100mm, compared with the length l of 100mm, the vertical deformation of the model at the quarter span is reduced by 15.40% and 26.43% respectively when l is 200mm and 300mm, and at the mid span the vertical deformation is reduced by 14.21% and 23.98% respectively. Similarly, when l remains constant, compared to w of 100mm, the vertical deformation of the model at the quarter span is reduced by 29.46% and 41.43% at w of 200mm and 300mm, respectively, while at the mid span the vertical deformation is reduced by 23.76% and 34.95%, respectively. Although S200-100 and S100-200 have the same base area, the latter exhibits superior performance in controlling vertical deformation, with effects comparable to or even slightly better than S300-100. In addition, compared with S300-100, S100-300 exhibits more outstanding ability in controlling vertical deformation. Therefore, increasing the size of shear keys can effectively reduce the vertical deformation of bridges. Under the same foundation area, increasing the width of shear keys is more effective in controlling vertical deformation than increasing their length. Figure 13 (c) indicates that the variation in shear key size has little effect on the difference in vertical deformation between the inner and outer sides of the bridge. However, increasing the size of the shear keys slightly helps to improve the torsional stiffness of the bridge by carefully comparing the subtle differences in the curves. Under the same basic area, longer shear keys provide slightly stronger torsional stiffness. Observing Fig. 13 (d), different shear key size configurations will significantly affect the stress at the bridge interface. When w is constant and l increases from 100 mm to 200 mm and then to 300 mm, both S11 and S33 at C1 and C2 decrease. Similarly, when l is fixed and w increases from 100 mm to 200 mm, the stresses reduce significantly. However, increasing w from 200 mm to 300 mm results in a smaller decrease in S11 and even a slight increase in S33 at C1, while S33 at C2 continues to decrease significantly. This also confirms that the sensitivity of different interfaces of bridges to changes in shear keys varies. Under the same substrate area, widening the shear key can more effectively reduce interfacial stress than lengthening the shear key. This indicates that when reducing the stress at the bridge interface, increasing the width of the shear keys can be prioritized to achieve lower stress levels. Increasing the size of the shear key can effectively reduce the vertical deformation and interfacial stress of the bridge. Under the same base area, increasing the width of the shear key can more effectively control the vertical deformation and reduce the interfacial stress than increasing length. Although longer shear keys provide slightly stronger torsional stiffness under the same base area, overall, the contribution of increased shear key size to improving bridge torsional stiffness is still not significant. 6 Conclusions This study systematically analyzes the effects of curvature and shear key design of segmental U-shaped curved bridge on torsional deformation and interface stress of the bridge. The main conclusions are as follows: 1. The curvature radius has little effect on the vertical deformation, but has significant effect on the torsional deformation and interfacial stress. With the increase of curvature radius, the torsional deformation of the bridge decreases gradually, and the interfacial stress at the midspan increases gradually. 2. Increasing the number of shear keys helps control vertical deformation and improves the interfacial stress state but has limited effect on enhancing torsional stiffness. Increasing appropriately the number of shear keys in design is recommended. 3. Shear keys are evenly distributed between the floor and web or concentrated on the floor, effectively controlling the vertical deformation. Setting all shear keys on the web can maximize the torsional stiffness and reduce the tangential stress most effectively. The shear key should be set on the web first, and then set on the floor according to the project needs 4. Increasing the shear key size can significantly reduce the vertical deformation and interfacial stress, but has little effect on the torsional stiffness. Under the same base area, increasing the width of the shear key can improve the deformation resistance of the bridge and reduce the interfacial stress more than increasing the length. Declarations Competing interests The author(s) declare no competing interests. Funding The research was funded by the Technology Research and Development Program of China Harbour Engineering Co., Ltd. (METRO1-CS-E-230812). Author Contribution Conceptualization, S.Z. and C.W.; methodology, S.Z.; software, C.W. and W.X.; validation, S.Z., C.W., W.X., H.Q., and L.X.; formal analysis, S.Z. and Q.F.; investigation, C.W. and H.Q.; resources, C.W. and L.X.; data curation, S.Z. and W.X.; writing—original draft preparation, S.Z., C.W., and Q.F.; writing—review and editing, S.Z., C.W., W.X., H.Q., L.X., Q.F., C.B.W., and Y.H.; visualization, S.Z., W.X., and C.B.W.; supervision, W.X. and Q.F.; project administration, L.X., H.Q., S.Z., C.W., and Y.H; funding acquisition, Q.F. and Y.H. All authors have read and agreed to the published version of the manuscript. Acknowledgement The authors want to thank the editor and anonymous reviewers for their valuable suggestions for improving this paper. Data Availability The authors confirm that the data supporting the findings of this study are available within the article. References Li, X. & Zheng, F. 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Cite Share Download PDF Status: Published Journal Publication published 09 Oct, 2025 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 29 Jul, 2025 Reviews received at journal 29 Jul, 2025 Reviews received at journal 21 Jul, 2025 Reviewers agreed at journal 21 Jul, 2025 Reviewers agreed at journal 17 Jul, 2025 Reviewers invited by journal 14 Jul, 2025 Editor assigned by journal 14 Jul, 2025 Editor invited by journal 09 Jul, 2025 Submission checks completed at journal 09 Jul, 2025 First submitted to journal 07 Jul, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7063152","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":486076202,"identity":"770b91c8-92a8-4501-8aa6-2998c38ee4a4","order_by":0,"name":"Shuai Zhang","email":"","orcid":"","institution":"Central South University","correspondingAuthor":false,"prefix":"","firstName":"Shuai","middleName":"","lastName":"Zhang","suffix":""},{"id":486076203,"identity":"f6c869ad-d5c8-42ac-914c-3ea0e378bca2","order_by":1,"name":"Chuang Wang","email":"","orcid":"","institution":"China Communications Construction Company, Second Harbor Engineering Co., Ltd","correspondingAuthor":false,"prefix":"","firstName":"Chuang","middleName":"","lastName":"Wang","suffix":""},{"id":486076204,"identity":"2d511688-b9dd-46d5-bfdd-5671f05b7278","order_by":2,"name":"Hongchun Qu","email":"","orcid":"","institution":"China Communications Construction Company, Second Harbor Engineering Co., Ltd","correspondingAuthor":false,"prefix":"","firstName":"Hongchun","middleName":"","lastName":"Qu","suffix":""},{"id":486076206,"identity":"75e2e13d-aab9-466f-9b9d-b57a2460fd75","order_by":3,"name":"Wenjie Xu","email":"","orcid":"","institution":"China Communications Construction Company, Second Harbor Engineering Co., Ltd","correspondingAuthor":false,"prefix":"","firstName":"Wenjie","middleName":"","lastName":"Xu","suffix":""},{"id":486076207,"identity":"64c93a76-53f4-4c2b-8da1-53ac9e56d22f","order_by":4,"name":"Lin Xiao","email":"","orcid":"","institution":"China Communications Construction Company, Second Harbor Engineering Co., Ltd","correspondingAuthor":false,"prefix":"","firstName":"Lin","middleName":"","lastName":"Xiao","suffix":""},{"id":486076209,"identity":"b7859202-47db-4ce2-8630-28c93a2fc71e","order_by":5,"name":"Qingsong Fan","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAxElEQVRIiWNgGAWjYBACefbGxgcSFTU8/OwNRGox7Dl82MDizDE5yZ4DxFpzwy1NorKN2djgRgKROhhn8BhI3GxjS9xw8/HGGww1NtEEtbBL9xgYzjgnkzjzdlqxBcOxtNwGgrbMOWOQLFHGlth3O8dMgrHhMGEtDDdyDA7/YWNObLh5hmgtaYkNEkDvC9zgIVILKJAZJMCBDPRLAjF+AUZl+w9IVB7eeONDjQ0RDkMCBhIJpCiHaCFVxygYBaNgFIwMAAA+0ERGL1EV8wAAAABJRU5ErkJggg==","orcid":"","institution":"Central South University","correspondingAuthor":true,"prefix":"","firstName":"Qingsong","middleName":"","lastName":"Fan","suffix":""},{"id":486076210,"identity":"03abc417-9afd-4ef8-b725-d75444390735","order_by":6,"name":"Chuanbiao Wang","email":"","orcid":"","institution":"Chongqing Transportation Planning and Technology Development Cente","correspondingAuthor":false,"prefix":"","firstName":"Chuanbiao","middleName":"","lastName":"Wang","suffix":""},{"id":486076212,"identity":"8b2218c4-106a-4293-b89b-790a39806651","order_by":7,"name":"Yanqun Han","email":"","orcid":"","institution":"Central South University","correspondingAuthor":false,"prefix":"","firstName":"Yanqun","middleName":"","lastName":"Han","suffix":""}],"badges":[],"createdAt":"2025-07-07 08:38:32","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7063152/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7063152/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-025-19337-4","type":"published","date":"2025-10-09T15:57:03+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":86956532,"identity":"c201ee3f-9e81-4b92-9776-738f543eedb0","added_by":"auto","created_at":"2025-07-17 15:13:10","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":183172,"visible":true,"origin":"","legend":"\u003cp\u003eOverview of the bridge and details (unit: m): (a) Bridge plan view; (b) 30 m span U-shaped spliced beam (11 segments); (c) Detail drawing of a segment; (d) Shear key details.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/a4dec194fba7ea12ef8bbcf0.png"},{"id":86955230,"identity":"b86ffda6-82a7-4d0a-888f-eff99b054e1c","added_by":"auto","created_at":"2025-07-17 15:05:10","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":131006,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Full-bridge model; (b) Half-bridge model, boundary and load.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/d7470e839e1f54a89d79117d.png"},{"id":86955233,"identity":"3993080e-5739-4a5c-8648-523162eb56b4","added_by":"auto","created_at":"2025-07-17 15:05:10","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":53910,"visible":true,"origin":"","legend":"\u003cp\u003eContact mode and details of shear key parameters.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/a7ce17573326abcf8fd2e3c0.png"},{"id":86955235,"identity":"0014736a-0dc9-46ba-baef-ddd56751bd13","added_by":"auto","created_at":"2025-07-17 15:05:10","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":124890,"visible":true,"origin":"","legend":"\u003cp\u003ePrestressed tendons distribution.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/51f6f53e51c35e7c2412134b.png"},{"id":86955232,"identity":"d5f2a4c6-9c2e-4148-a72c-5ac914bf84c6","added_by":"auto","created_at":"2025-07-17 15:05:10","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":44094,"visible":true,"origin":"","legend":"\u003cp\u003eMesh convergence analysis.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/23b5b8f29c97879940c4d4fe.png"},{"id":86955247,"identity":"4f471fc3-b8db-4578-9507-d463a6e1c760","added_by":"auto","created_at":"2025-07-17 15:05:11","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":136569,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Bridge paths and section settings; (b) Verticaldeformation nephogram.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/866c1591c3a1c0c260e17ba6.png"},{"id":86956794,"identity":"7d8ee82d-fc9b-44c6-8318-0d9bef13922e","added_by":"auto","created_at":"2025-07-17 15:21:10","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":69871,"visible":true,"origin":"","legend":"\u003cp\u003eDeformation of curved bridge models with different curvature radii: (a) Path-1 deformation; (b) Path-1 and Path-3 deformation.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/0a4cbe78f0198f88ba65a5c3.png"},{"id":86955241,"identity":"6b069048-084f-4b39-aeaf-2dc83737bc6b","added_by":"auto","created_at":"2025-07-17 15:05:10","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":15691,"visible":true,"origin":"","legend":"\u003cp\u003eStress Distribution in Curved Bridge Models with Different Curvature radii.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/e30b78cea7b90b0d2f59923b.png"},{"id":86956534,"identity":"9dd5ece5-593a-4d0d-9923-b18692e2f91f","added_by":"auto","created_at":"2025-07-17 15:13:10","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":58616,"visible":true,"origin":"","legend":"\u003cp\u003eLayout of different number of shear keys.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/a3a4bd9f1dc5f3834021e974.png"},{"id":86957812,"identity":"34934b4c-e13c-4326-a72f-6e84fd49b65b","added_by":"auto","created_at":"2025-07-17 15:29:11","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":130731,"visible":true,"origin":"","legend":"\u003cp\u003eEffect of different shear key numbers on bridge model calculation results: (a) Vertical deformation along Path-2; (b) Vertical deformation along Path-1 and Path-3; (c) The difference in vertical deformation along Path-1 and Path-3; (d) Stress at different bridge interfaces.\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/a0c0a336d40ae26c7f610fd0.png"},{"id":86956539,"identity":"10fc7f4a-baf7-4766-9b56-c3df916dc32d","added_by":"auto","created_at":"2025-07-17 15:13:11","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":45767,"visible":true,"origin":"","legend":"\u003cp\u003eshear key arrangement scheme.\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/9e296570733a8aa991c52f88.png"},{"id":86955239,"identity":"6356c45d-ea1b-4bb7-b42d-69c549af2dd6","added_by":"auto","created_at":"2025-07-17 15:05:10","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":124487,"visible":true,"origin":"","legend":"\u003cp\u003eEffect of different shear key arrangements on bridge model calculation results. (a) Vertical deformation along Path-2; (b) Vertical deformation along Path-1 and Path-3; (c) The difference in vertical deformation along Path-1 and Path-3; (d) Stress at different bridge interfaces.\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/a0f8d2145c7ecbdc94d73cd2.png"},{"id":86955248,"identity":"3c9e8834-c51b-48b8-935d-6ffd5c1dc43f","added_by":"auto","created_at":"2025-07-17 15:05:11","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":146734,"visible":true,"origin":"","legend":"\u003cp\u003eEffect of different shear key size on bridge model calculation results: (a) Vertical deformation along Path-2; (b) Vertical deformation along Path-1 and Path-3; (c) The difference in vertical deformation along Path-1 and Path-3; (d) Stress at different bridge interfaces.\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/f29d59df2e9739d3c6bea415.png"},{"id":93419704,"identity":"7f590852-270c-4eb5-b0ff-ff38181e075c","added_by":"auto","created_at":"2025-10-13 16:06:13","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1722656,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7063152/v1/8dd24cd2-152d-4cfa-8fc0-7b916aee0690.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Research on the effects of curvature and shear keys on torsional stiffness and interfacial stress of segmented U-shaped curved bridge","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eIn recent years, Rail transit industry has developed rapidly, and various forms of beams have been widely used in this field, such as U-shaped bridges, T-shaped beams, box beams, and so on\u003csup\u003e\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e,\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e. Compared to other beams, U-shaped bridges have many advantages, such as better noise reduction effects due to their lower construction height, aesthetic appearance, and high economic benefits. Therefore, U-shaped bridges are widely used in the rail transit field\u003csup\u003e\u003cspan additionalcitationids=\"CR4 CR5 CR6\" citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eIn 1962, a bridge crossing over the Seine River in France was constructed\u0026mdash;regarded as the inaugural precast segmental concrete bridge on a global scale. Segmental bridges have garnered significant attention due to their numerous advantages, including fast construction speed, environmental friendliness, and superior quality control. By adopting this technology, construction time and traffic disruptions can be significantly reduced, thereby greatly enhancing economic benefits\u003csup\u003e\u003cspan additionalcitationids=\"CR9\" citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u003c/sup\u003e. Against the backdrop of increasing global demand for bridges, the appeal of segmental bridges is growing\u003csup\u003e\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e,\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/sup\u003e. For segmental bridges, joints are the weak points in their structure, thus they require special attention and handling\u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/sup\u003e. This is because the strength, maintainability, and structural behavior of precast concrete segmental bridges largely depend on the performance of the joints between the segments. There are various existing forms of joints, including dry joints, wet joints, and epoxy resin joints. Compared to other types of joints, epoxy resin joints are considered to perform better in terms of durability and ultimate shear load capacity. The use of epoxy resin can in-crease the shear load capacity of the joint\u003csup\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e,\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u003c/sup\u003e. Additionally, uncured epoxy resin can act as a lubricant for the joint, compensating for irregularities between the joint surfaces\u003csup\u003e\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eIn bridge engineering, curved bridges are widely used to adapt to terrain and traffic requirements. However, in practical use, curved bridges face serious torsion problems, which can affect the stability and safety of the bridge, and in severe cases may lead to structural overturning and damage\u003csup\u003e\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e\u003c/sup\u003e. The case of overturning involving the Yangmingtan Bridge exemplifies a critical vulnerability inherent in bridges with curvature. Observable is that, when subjected to eccentric loading conditions, such curved structures are notably susceptible to experiencing torsional instability\u003csup\u003e\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u003c/sup\u003e. Therefore, significant to in-depth research are the torsional stiffness attributes inherent within curved bridges, and from enhancements of anti-overturning capabilities can overall safety appreciably benefit.\u003c/p\u003e\u003cp\u003eMany scholars have conducted research on the bending and shear behavior of bridges, but studies on the torsional behavior are relatively fewer in number. Meyyada et al.\u003csup\u003e\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u003c/sup\u003e conducted a review on the torsional strengthening of reinforced concrete (RC) beams using externally bonded (EB) composite materials and established a database for RC beams retrofitted with EB fiber-reinforced polymer (FRP) and fiber-reinforced cementitious matrix (FRCM). Zhao et al.\u003csup\u003e\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e\u003c/sup\u003e discussed the torsional behavior of a novel reinforced concrete U-shaped steel-concrete composite beam (RCUCB) system through a series of torsion tests. Mane et al.\u003csup\u003e\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e\u003c/sup\u003e proposed a high-yield-strength welded wire mesh that can be wrapped around concrete members to improve composite performance and avoid sudden cracking, while also increasing the torsional stiffness of reinforced concrete beam specimens.\u003c/p\u003e\u003cp\u003eNumerous scholars have conducted research on the shear keys of segmental bridge joints. Jiang et al.\u003csup\u003e\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u003c/sup\u003e tested full-scale specimens with shear keys under different levels of confining stress and proposed a shear failure mechanism for the continuous failure of multiple shear keys in dry joints based on the experimental results. Celia et al.\u003csup\u003e\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/sup\u003e designed a new push-off dry joint test under combined forces, testing 24 push-off dry joint specimens under axial force, bending force, and shear force, and concluded the effects of parameters such as the number of shear keys, confining stress level, and eccentricity on the shear resistance of segmental bridges. Hua et al.\u003csup\u003e\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e\u003c/sup\u003e designed and developed 3D-printed concrete shear keys with epoxy mortar infill and studied the various designs of 3D-printed concrete shear keys with different shear key angles and shear joint inclinations through push-off and slant shear tests.\u003c/p\u003e\u003cp\u003eCompared with straight bridges, curved bridges exhibit more complex stress patterns, and the problem of eccentric loads and torsion is more prominent. At present, the research on segmental U-shaped curved bridges is not sufficient, especially the effects of the arrangement, number and size parameters of shear keys and different curvature radii (\u003cem\u003eR\u003c/em\u003e) of bridge on the torsional stiffness and interfacial stress of the bridge are not systematically studied in combination with the construction characteristics of segmental assembly and the section characteristics of U-shaped bridges. The above effect mechanism can provide theoretical support for the design optimization and engineering application of segmented U-shaped curved bridge, and promote the technical development and promotion of this kind of bridge structure.\u003c/p\u003e"},{"header":"2 Research background","content":"\u003cp\u003eThis study relies on the Bogot\u0026aacute; Metro Line 1 Metro bridge project, which is designed with a total length of 23.86 kilometers and a total of 742 span bridges. The segmental U-beam assembly process is adopted, of which the curved bridge has 180 spans. Overview of the bridge and details is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the standard span of the bridge in the project is mainly displayed, which is 30 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{m}\\)\u003c/span\u003e\u003c/span\u003e long and 10 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{m}\\)\u003c/span\u003e\u003c/span\u003e wide, and consists of 11 segments. The \u003cem\u003eR\u003c/em\u003e of the bridge in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e is 300m. The effects of various factors, including the arrangement, number, size of shear keys and curvature of the bridge, on the torsional stiffness and interfacial stress of segmented U-shaped curved bridges are investigated using numerical methods in this paper.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"3 Finite element analysis model","content":"\u003cp\u003eDue to the symmetry of the structure, the equivalent simplification principle in structural mechanics is adopted to select the half-bridge (15 m) of the full-bridge (30 m) as the research object, and the half model is used to simplify the calculation process, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e(a) and (b). The finite element analysis is performed using ANSYS.\u003c/p\u003e\u003cp\u003eThe boundary conditions and load of the half-bridge are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e (b). The symmetric boundary conditions are applied to the symmetric face, and the bearings at the end of the bridge are simply supported. The structure is subjected to uniform load of 48 k N/m, two concentrated loads of 150 k N and standard gravity\u003csup\u003e\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe single-span bridge with 11 segments of U-shaped bridge girder contains 10 joints in total. as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Joints between bridge segments are bonded with epoxy resin adhesive. Tie constraint (Pink section) is used to connect the key block and keyway of the shear key, while friction contact(Green section) is used to simulate the flat part of the joint, and 0.45 is set as the friction coefficient\u003csup\u003e\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u003c/sup\u003e, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. The shear key of the finite element model are set to the same specification in this paper, which is more convenient to discuss the effect of the shear key on the torsional stiffness and interfacial stress of the bridge under the control of different parameters\u003csup\u003e\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u003c/sup\u003e. The parameters of the shear keys are marked in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe elastic constitutive relation of concrete and prestressed tendons is used in finite element simulation. The concrete strength class is C50. The prestress applied to the prestressed tendons is 1395MPa, the cross-sectional area of curved prestressed tendons is 1540 mm\u0026sup2;, and that of straight prestressed tendons is 1400 mm\u0026sup2;. All data refer to the engineering drawings of the Bogot\u0026aacute; project. Material parameters detailed in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. The layout of the prestressed tendons is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eMaterial Parameters.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eMaterial\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eElastic Modulus (GPa)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003ePoisson's Ratio\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eDensity(kg/m\u003csup\u003e3\u003c/sup\u003e)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eConcrete\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e35.04\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e2500\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ePrestressed tendons\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e206.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e7850\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eTo ensure the accuracy of the finite element simulation in this study, a mesh convergence analysis is carried out based on a model with \u003cem\u003eR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;200 m, \u003cem\u003el\u003c/em\u003e\u0026thinsp;=\u0026thinsp;300 mm, \u003cem\u003ew\u003c/em\u003e\u0026thinsp;=\u0026thinsp;200 mm, \u003cem\u003eh\u003c/em\u003e\u0026thinsp;=\u0026thinsp;50mm, \u003cem\u003eα\u003c/em\u003e\u0026thinsp;=\u0026thinsp;45\u0026deg;, and 24 shear keys. The shear keys are uniformly distributed on the web and floor of the bridge, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. The maximum vertical deformation (deformation along the Y axis) of the bridge is calculated for multiple sets of models with varying numbers of elements. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, the results indicate that as the number of elements in the model increases, the maximum vertical deformation of the bridge gradually stabilizes, thereby confirming the validity of the finite element simulation. Considering the calculation speed and accuracy, 1.87\u0026times;10\u003csup\u003e5\u003c/sup\u003e elements are finally selected as the number of calculation elements.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"4 The effect of the curvature of curved bridges on torsional stiffness and interfacial stress","content":"\u003cp\u003eTo investigate the effect of bridge curvature radius on the torsional stiffness and interfacial stress, three different finite element calculation models are established by changing \u003cem\u003eR\u003c/em\u003e. The parameters corresponding to models with different curvature radii are shown in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The arrangement of shear keys for the models in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. The shear keys are uniformly distributed on the web and floor of the bridge.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eParameters corresponding to models with different curvature radii.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eName\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003e\u003cem\u003eR\u003c/em\u003e(m)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eNumber\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003e\u003cem\u003el\u003c/em\u003e (mm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003e\u003cem\u003ew\u003c/em\u003e (mm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003e\u003cem\u003eh\u003c/em\u003e (mm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e\u003cp\u003e\u003cem\u003eα\u003c/em\u003e (\u0026deg;)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cem\u003eR\u0026thinsp;=\u003c/em\u003e\u0026thinsp;200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e24\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cem\u003eR\u0026thinsp;=\u003c/em\u003e\u0026thinsp;300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e24\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cem\u003eR\u0026thinsp;=\u003c/em\u003e\u0026thinsp;400\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e400\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e24\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eTo facilitate result analysis, three paths (Path-1, Path-2, Path-3) and two cross - sections (C1 and C2) are set on the curved bridge deck, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e(a). Path-1, Path-2 and Path-3 are used to obtain the inner, central and outer deformation of the bridge from the inside to the outside. C2 is in the midspan and C1 is in the quarter span of the bridge. The results of three different curvature radii models are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eThe overall vertical deformation nephogram of \u003cem\u003eR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;200 model is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e(b), and the corresponding nephogram of other models are similar. The overall deformation of the structure is asymmetric. The bridge deforms upwards along Path-1 and deflects downwards along Path-3, indicating a clear trend of torsional deformation, which is likely due to the structural characteristics of curved bridges. The outer side of a curved bridge is longer, and its relative weight has a greater impact, with a higher stiffness on the outer side. And the inner side is short, with less impact on relative weight and relatively low stiffness. When subjected to the same prestress, the effect of the inner prestress is more pronounced, resulting in upward deformation on the inner side. On Path-2, the downward vertical deformation of the bridge from the simply supported end to the midspan gradually increases, and reaches the maximum at the midspan.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, under the same load conditions, the vertical deformation of different curvature radius models along Path-2 is basically the same. However, as the curvature radius increases, the vertical deformation of both the inner and outer sides of the bridge decreases significantly, and the difference in deformation between the inner and outer sides also decreases. When the curvature radius increases from 200m to 300m and 400m, the difference in deformation between the inner and outer sides at the quarter span decreases by 34.04% and 50.64%, respectively, while that at the mid-span decreases by 34.38% and 49.84%. Sudden changes may occur on the curve due to the structural characteristics of different beam joints.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe stress conditions of C1 and C2 in curved bridges with different curvature radii are showed in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e. For C1, the change of the interfacial stress of the bridge with the curvature radius is not obvious, and the overall performance is that the interfacial stress increases with the decrease of the curvature radius. For C2, as the curvature radius increases, the tangential stress along the x-axis (S11) shows an upward trend, while the absolute value of the normal stress along the z-axis (S33) gradually decreases.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"5 The effect of shear keys on torsional stiffness and interfacial stress","content":"\u003cp\u003eThe joints between bridge segments are equipped with multiple pyramid shaped shear keys. The number, arrangement, and size of these shear keys can all affect the torsional stiffness and interfacial stress of segmental U-shaped curved bridges. The effects of different shear key configurations on the torsional stiffness and interfacial stress of segmental U-shaped curved bridges are investigated.\u003c/p\u003e\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\u003ch2\u003e5.1 The effect of shear key number\u003c/h2\u003e\u003cp\u003eFour different finite element calculation models are established by changing the number of shear keys. The parameters corresponding to models with different numbers of shear keys are shown in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. The four types of interface shear key arrangement between bridge segments are established according to Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e. The shear keys are uniformly distributed on the web and floor of the bridge. 12keys, 16keys, 20keys and 24keys models are used for finite element simulation analysis. The results for these four models are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eParameters corresponding to models with different number of shear keys.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eName\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003e\u003cem\u003eR\u003c/em\u003e(m)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eNumber\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003e\u003cem\u003el\u003c/em\u003e (mm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003e\u003cem\u003ew\u003c/em\u003e (mm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003e\u003cem\u003eh\u003c/em\u003e (mm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e\u003cp\u003e\u003cem\u003eα\u003c/em\u003e (\u0026deg;)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e12keys\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e12\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e16keys\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e16\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e20keys\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e20\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e24keys\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e24\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e (a) and (b), when the number of shear keys increases from 12 to 24, the vertical deformation of the model along the three paths gradually decreases. Taking path-2 as an example, compared with key 12, the vertical deformation of keys 16, 20, and 24 at the quarter span is reduced by 25.07%, 34.73%, and 41.25%, respectively. At mid-span, the reductions are 24.93%, 31.06%, and 36.18% respectively. Results show that increasing shear key number effectively controls vertical deformation of segmental U-shaped curved bridge. Notably, the deformation reduction when the number of shear keys increases from 12 to 16 is significantly greater than the reduction during the stage from 16 to 24 keys. Considering the manufacturing difficulty and cost factors, the lowest reasonable number of shear keys should be matched for different segmented U-shaped curved bridge.\u003c/p\u003e\u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e (b) and (c), with the increase of shear keys, the vertical deformation on the inner or outer side decreases, but the difference in deformation between the inner and outer sides remains basically unchanged. Therefore, changing the number of shear keys will not significantly change the overall torsional stiffness of the bridge.\u003c/p\u003e\u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e (d) shows the effect of different shear key numbers on the interfacial stress of the bridge. With the increase of shear keys, for C1 and C2, S11 decreases, and the absolute value of S33 increases and becomes negative, indicating that the interface is under overall compression along the bridge direction. Therefore, increasing the number of shear keys can enhance the overall integrity of the bridge and improve the shear bearing capacity at the joints.\u003c/p\u003e\u003cp\u003eIncreasing the number of shear keys in the design can improve the connection stiffness of the bridge and effectively improve the interfacial stress state of the bridge structure. However, increasing the number of shear keys has little effect on improving the torsional stiffness of the bridge.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\u003ch2\u003e5.2 The effect of shear key arrangement\u003c/h2\u003e\u003cp\u003eThree different finite element calculation models are established by changing the arrangement of shear keys. The parameters corresponding to models with different arrangement of shear keys are shown in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. The arrangement of the shear keys used in the model, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e, each scheme has 12 shear keys. In Combination 1, all shear keys are on the web. In Combination 2, all are on the floor. In Combination 3, all shear keys are evenly distributed between the floor and web.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eParameters corresponding to models with different arrangement of shear keys.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eName\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003e\u003cem\u003eR\u003c/em\u003e(m)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eNumber\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003e\u003cem\u003el\u003c/em\u003e (mm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003e\u003cem\u003ew\u003c/em\u003e (mm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003e\u003cem\u003eh\u003c/em\u003e (mm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e\u003cp\u003e\u003cem\u003eα\u003c/em\u003e (\u0026deg;)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCombination1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e12\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCombination2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e12\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCombination3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e12\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eCombination1, Combination2 and Combination3 models are used for calculation and analysis, and discusses the effect of shear key arrangement on the torsional stiffness and interfacial stress of the bridge. The calculation result is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e (a) and (b), on Path-1 andPath-2, the vertical deformation of Combination 2 and 3 is basically the same and less than that of Combination 1. On Path-3, compared with Combination 1, the vertical deformation of Combination 2 and Combination 3 at the quarter span decreased by 17.05% and 35.86% respectively, and that at the mid span decreased by 24.24% and 32.20% respectively. Therefore, the arrangement of shear keys has a significant effect on the control of vertical deformation. Considering all three paths, Combination 3 is the most effective in controlling vertical deformation, followed by Combination 2, and Combination 1 has the worst effect.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e (c), there are significant differences in the vertical deformation difference between the inner and outer sides of the bridge under different shear key arrangement. Combination 2 has the largest vertical deformation difference, followed by Combination 3, and Combination 1 has the smallest vertical deformation difference. The arrangement of shear keys will affect the torsional deformation of bridges, with Combination 1 being the most effective in suppressing torsional deformation, followed by Combination 3, and Combination 2 being relatively ineffective. Therefore, setting more shear keys on the web is more beneficial to improve the torsional stiffness of the bridge.\u003c/p\u003e\u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e (d) shows the bridge interfacial stress under three different shear key arrangements. For C1 and C2, the order of S11 from maximum to minimum is Combination 2, Combination 3, and Combination 1. The absolute values of S33 are in ascending order as Combination 3, Combination 2, and Combination 1. This shows that Combination 1 is the best in reducing the tangential stress, and has the maximum normal compressive stress. Therefore, the priority of setting shear keys on the web can enhance the integrity of the bridge and improve the shear bearing capacity of the joints.\u003c/p\u003e\u003cp\u003eIn summary, increasing the shear key on the web can increase the torsional stiffness and improve the interface stress. The addition of shear keys on the floor is beneficial to increase the normal connection stiffness of the bridge interface, which should be comprehensively selected.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\u003ch2\u003e5.3 The effect of shear key size\u003c/h2\u003e\u003cp\u003eFive different finite element calculation models are established by changing the size of shear keys. The parameters corresponding to models with different size of shear keys are shown in Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. The arrangement of shear keys is the same as 24keys, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e. To avoid sharp corners in the shear keys, \u003cem\u003eh\u003c/em\u003e has been changed from 50mm to 30mm \u003csup\u003e\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eParameters corresponding to models with different size of shear keys.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eName\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003e\u003cem\u003eR\u003c/em\u003e(m)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eNumber\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003e\u003cem\u003el\u003c/em\u003e (mm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003e\u003cem\u003ew\u003c/em\u003e (mm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003e\u003cem\u003eh\u003c/em\u003e (mm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e\u003cp\u003e\u003cem\u003eα\u003c/em\u003e (\u0026deg;)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eS100-100\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e24\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e100\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e100\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eS200-100\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e24\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e100\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eS300-100\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e24\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e100\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eS100-200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e24\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e100\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eS100-300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e24\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e100\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e300\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe five models S100-100, S200-100, S300-100, S100-200 and S100-300 are used for the calculation and analysis, and the results are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e (a) and (b), as the size of the shear key gradually increases, the vertical deformation of the model along the three paths shows a decreasing trend. Taking path-2 as an example, when the width \u003cem\u003ew\u003c/em\u003e of the shear key is kept constant at 100mm, compared with the length \u003cem\u003el\u003c/em\u003e of 100mm, the vertical deformation of the model at the quarter span is reduced by 15.40% and 26.43% respectively when \u003cem\u003el\u003c/em\u003e is 200mm and 300mm, and at the mid span the vertical deformation is reduced by 14.21% and 23.98% respectively. Similarly, when \u003cem\u003el\u003c/em\u003e remains constant, compared to \u003cem\u003ew\u003c/em\u003e of 100mm, the vertical deformation of the model at the quarter span is reduced by 29.46% and 41.43% at \u003cem\u003ew\u003c/em\u003e of 200mm and 300mm, respectively, while at the mid span the vertical deformation is reduced by 23.76% and 34.95%, respectively. Although S200-100 and S100-200 have the same base area, the latter exhibits superior performance in controlling vertical deformation, with effects comparable to or even slightly better than S300-100. In addition, compared with S300-100, S100-300 exhibits more outstanding ability in controlling vertical deformation. Therefore, increasing the size of shear keys can effectively reduce the vertical deformation of bridges. Under the same foundation area, increasing the width of shear keys is more effective in controlling vertical deformation than increasing their length.\u003c/p\u003e\u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e (c) indicates that the variation in shear key size has little effect on the difference in vertical deformation between the inner and outer sides of the bridge. However, increasing the size of the shear keys slightly helps to improve the torsional stiffness of the bridge by carefully comparing the subtle differences in the curves. Under the same basic area, longer shear keys provide slightly stronger torsional stiffness.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eObserving Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e (d), different shear key size configurations will significantly affect the stress at the bridge interface. When \u003cem\u003ew\u003c/em\u003e is constant and \u003cem\u003el\u003c/em\u003e increases from 100 mm to 200 mm and then to 300 mm, both S11 and S33 at C1 and C2 decrease. Similarly, when \u003cem\u003el\u003c/em\u003e is fixed and \u003cem\u003ew\u003c/em\u003e increases from 100 mm to 200 mm, the stresses reduce significantly. However, increasing \u003cem\u003ew\u003c/em\u003e from 200 mm to 300 mm results in a smaller decrease in S11 and even a slight increase in S33 at C1, while S33 at C2 continues to decrease significantly. This also confirms that the sensitivity of different interfaces of bridges to changes in shear keys varies. Under the same substrate area, widening the shear key can more effectively reduce interfacial stress than lengthening the shear key. This indicates that when reducing the stress at the bridge interface, increasing the width of the shear keys can be prioritized to achieve lower stress levels.\u003c/p\u003e\u003cp\u003eIncreasing the size of the shear key can effectively reduce the vertical deformation and interfacial stress of the bridge. Under the same base area, increasing the width of the shear key can more effectively control the vertical deformation and reduce the interfacial stress than increasing length. Although longer shear keys provide slightly stronger torsional stiffness under the same base area, overall, the contribution of increased shear key size to improving bridge torsional stiffness is still not significant.\u003c/p\u003e\u003c/div\u003e"},{"header":"6 Conclusions","content":"\u003cp\u003eThis study systematically analyzes the\u0026nbsp;effects\u0026nbsp;of curvature and shear key design of segmental U-shaped curved\u0026nbsp;bridge on torsional deformation and interface stress of the bridge. The main conclusions are as follows:\u003c/p\u003e\n\u003cp\u003e1. The curvature radius has little effect on the vertical deformation, but has significant effect on the torsional deformation and interfacial stress. With the increase of curvature radius, the torsional deformation of the bridge decreases gradually, and the interfacial stress at the midspan increases gradually.\u003c/p\u003e\n\u003cp\u003e2. Increasing the\u0026nbsp;number\u0026nbsp;of shear keys helps control vertical deformation and improves the interfacial stress state but has limited effect on enhancing torsional stiffness. Increasing appropriately the\u0026nbsp;number\u0026nbsp;of shear keys in design is recommended.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e3. Shear keys are evenly distributed between the floor and web or concentrated on the floor, effectively controlling the vertical deformation. Setting all shear keys on the web can maximize the torsional stiffness and reduce the tangential stress most effectively. The shear key should be set on the web first, and then set on the floor according to the project needs\u003c/p\u003e\n\u003cp\u003e4. Increasing the shear key size can significantly reduce the vertical deformation and interfacial stress, but has little effect on the torsional stiffness. Under the same base area, increasing the width of the shear key can improve the deformation resistance of the bridge and reduce the interfacial stress more than increasing the length.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003ch2\u003eCompeting interests\u003c/h2\u003e\u003cp\u003eThe author(s) declare no competing interests.\u003c/p\u003e\u003c/p\u003e\u003ch2\u003eFunding\u003c/h2\u003e\u003cp\u003eThe research was funded by the Technology Research and Development Program of China Harbour Engineering Co., Ltd. (METRO1-CS-E-230812).\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eConceptualization, S.Z. and C.W.; methodology, S.Z.; software, C.W. and W.X.; validation, S.Z., C.W., W.X., H.Q., and L.X.; formal analysis, S.Z. and Q.F.; investigation, C.W. and H.Q.; resources, C.W. and L.X.; data curation, S.Z. and W.X.; writing\u0026mdash;original draft preparation, S.Z., C.W., and Q.F.; writing\u0026mdash;review and editing, S.Z., C.W., W.X., H.Q., L.X., Q.F., C.B.W., and Y.H.; visualization, S.Z., W.X., and C.B.W.; supervision, W.X. and Q.F.; project administration, L.X., H.Q., S.Z., C.W., and Y.H; funding acquisition, Q.F. and Y.H. All authors have read and agreed to the published version of the manuscript.\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eThe authors want to thank the editor and anonymous reviewers for their valuable suggestions for improving this paper.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe authors confirm that the data supporting the findings of this study are available within the article.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eLi, X. \u0026amp; Zheng, F. Experimental and numerical investigation of the water-entry behavior of an inverted T-shaped beam. \u003cem\u003eSci. Rep.\u003c/em\u003e \u003cstrong\u003e14\u003c/strong\u003e, 29367 (2024).\u003c/li\u003e\n\u003cli\u003eOjha, S., Pal, P. \u0026amp; Mehta, P. Computational analysis of curved prestressed concrete box-girder bridges using finite element method. \u003cem\u003eSci. 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Finite element modelling of the shear behavior of joints in precast segmental UHPC bridge girders. \u003cem\u003eEM\u003c/em\u003e \u003cstrong\u003e40\u003c/strong\u003e, 85-98, 256 (2023).\u003c/li\u003e\n\u003cli\u003e黄方林, 孟宪冬, 冯帆, 高英杰 \u0026amp; 温伟斌. 预制桥面板方台形剪力键 湿接缝受力性能分析. \u003cem\u003eJ. Railw. Sci. Eng.\u003c/em\u003e \u003cstrong\u003e20\u003c/strong\u003e (2023).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Segmental U-shaped curved bridges, Curvature radius, Shear keys, Torsional stiffness, Interfacial stress","lastPublishedDoi":"10.21203/rs.3.rs-7063152/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7063152/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper investigates the segmental U-shaped curved bridge of Bogot\u0026aacute; Metro Line 1 project in Colombia. The effects of curvature radius (\u003cem\u003eR\u003c/em\u003e) of the bridge and shear keys (number, arrangement, and size) at inter-segment bonding joints on the bridge's torsional stiffness and interfacial stress are analyzed. The results demonstrate a nonlinear relationship between bridge curvature and torsional stiffness. As the curvature radius increases, torsional stiffness enhances while the growth rate declines gradually. Increasing the quantity of shear keys in the web and floor regions of U-shaped bridge segments, along with extending their width, effectively enhances the bridge's torsional stiffness and global deformation resistance while ameliorating the interfacial stress state between segments. These findings offer both theoretical and practical guidance for the design of segmental U-shaped curved bridges and shear key systems.\u003c/p\u003e","manuscriptTitle":"Research on the effects of curvature and shear keys on torsional stiffness and interfacial stress of segmented U-shaped curved bridge","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-17 15:05:06","doi":"10.21203/rs.3.rs-7063152/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-07-29T14:28:59+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-07-29T08:01:53+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-07-21T07:19:40+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"208275289193455814488179929145876025001","date":"2025-07-21T04:10:42+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"265711487751709918523806800855094359018","date":"2025-07-17T09:08:42+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-07-15T01:21:38+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-07-15T01:11:19+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-07-09T20:20:52+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-07-09T05:13:59+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-07-07T08:30:58+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"d017ee25-0b4f-4201-99c1-2e3e1da21680","owner":[],"postedDate":"July 17th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":51600106,"name":"Physical sciences/Engineering"},{"id":51600107,"name":"Physical sciences/Materials science"}],"tags":[],"updatedAt":"2025-10-13T16:00:45+00:00","versionOfRecord":{"articleIdentity":"rs-7063152","link":"https://doi.org/10.1038/s41598-025-19337-4","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2025-10-09 15:57:03","publishedOnDateReadable":"October 9th, 2025"},"versionCreatedAt":"2025-07-17 15:05:06","video":"","vorDoi":"10.1038/s41598-025-19337-4","vorDoiUrl":"https://doi.org/10.1038/s41598-025-19337-4","workflowStages":[]},"version":"v1","identity":"rs-7063152","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7063152","identity":"rs-7063152","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}