1. Introduction

2. Thermodynamic modeling of continuous curved steel box girders under variable temperatures

3. Geometric properties of the cross-section

3.3. cross-sectional aspect ratio

It is feasible to study the influence of the cross-sectional λ on the temperature effect of curved box girders by changing the aspect ratio of the section. To fully demonstrate the impact of λ on the temperature effect of curved box girders, twelve groups of different structural parameters were selected, as shown in Table 4, to investigate the relationship between λ, stress, and deformation. To investigate the influence of the cross-sectional aspect ratio λ on the temperature effects in curved box girders, it is feasible to vary the aspect ratio of the section. To comprehensively demonstrate the impact of λ on the temperature-induced effects in curved box girders, twelve groups of different structural parameters were selected, as outlined in Table 4. These groups were used to explore the relationship between λ, stress, and deformation.

Figure 12. Influence of aspect ratio on stress results.

Figure 12. Influence of aspect ratio on stress results.

Figure 13. Influence of aspect ratio on deformation results.

Figure 13. Influence of aspect ratio on deformation results.

4. Influencing factors on temperature effect of longitudinal bridge

4.1. Longitudinal bridge alignment

The structural parameters and constraint methods of a bridge in the longitudinal direction can significantly influence the temperature effects on curved box girders. This research focuses on key parameters of the longitudinal bridge alignment, including the radius of curvature R, the central angle θ, and the total length L of the curved steel box girder bridge. Following trial calculations and analysis, radii of curvature of 69 m, 86 m, and 103 m were selected for study. Four models of curved steel box girders with varying numbers of spans were developed, as detailed in Table 5.

Temperature loads, dead weight, and a secondary constant load of 25 kN/m are applied to 12 curved steel box girder models. The initial temperature of the temperature field is set at 37 °C. Two temperature change scenarios are considered, both starting from this initial temperature: an overall temperature increase of 9 °C and an overall temperature decrease of 9 °C. Analyses and calculations are performed for each scenario. The results of the maximum equivalent stress and deformation for the curved steel box girders, with varying spans under these temperature conditions, are presented in Figure 15, Figure 16, Figure 17, Figure 18.

Figure 14. Model of four-span curved steel box girder bridge with a radius of 86m.

Figure 14. Model of four-span curved steel box girder bridge with a radius of 86m.

Figure 15. Span number and maximum stress under overall temperature rise.

Figure 15. Span number and maximum stress under overall temperature rise.

Figure 16. Span number and maximum displacement under overall temperature rise.

Figure 16. Span number and maximum displacement under overall temperature rise.

Figure 17. Span number and maximum stress under overall cooling.

Figure 17. Span number and maximum stress under overall cooling.

Figure 18. Span number and maximum displacement under overall cooling.

Figure 18. Span number and maximum displacement under overall cooling.

From Figure 15, it is evident that under overall heating conditions, the maximum equivalent stress in curved steel box girders is slightly lower in odd spans compared to even spans. When comparing bridges in Group A and Group C with identical spans, it is observed that with a constant central angle θ, an increase in the radius of curvature R leads to an increase in the maximum equivalent stress caused by temperature effects and permanent loads, with a maximum increase of up to 28%.

From Figure 17, under overall cooling conditions, the maximum equivalent stress in curved steel box girders decreases as the number of spans increases. However, there is a slight increase in stress in even spans compared to adjacent odd spans.

Figure 16 and Figure 18 illustrate that, under both overall heating and cooling conditions, the radius of curvature is the primary factor influencing the temperature-induced deformation of curved steel box girders. Comparing bridges in Group B and Group C with the same span, it is observed that, for a given total bridge length L, a larger curvature ρ results in greater deformation due to temperature effects.

To further verify the observed patterns, a positive radial lateral temperature gradient and a negative radial lateral temperature gradient were applied to 12 models of curved steel box girder bridges. The initial atmospheric temperature was set at 37 °C, with a lateral temperature difference of 9 °C, resulting in a minimum lateral temperature of 37 °C and a maximum temperature of 46 °C. The calculation results for the maximum equivalent stress and deformation of the curved steel box girders, influenced by the lateral temperature gradient as the number of spans changes, are presented in Figure 19, Figure 20, Figure 21, Figure 22.

Figure 19, Figure 20, Figure 21, Figure 22 illustrate that, under lateral temperature gradients, the stress and displacement variation patterns of bridges with three distinct curvature radii, as the number of spans n changes, resemble those observed during an overall temperature increase. For a constant curvature radius R, as the central angle θ increases with the number of spans n, the maximum stress in even-numbered spans exceeds that in adjacent odd-numbered spans. When the central angles θ are equal, an increase in the curvature radius R results in a rise in the maximum equivalent stress in the curved steel box girder due to temperature effects and permanent loads. For equal span lengths L, the curvature ρ influences the temperature effects on the curved steel box girder. Greater curvature ρ with equal span lengths L leads to increased deformation.

4.2. Wide span ratio

As presented in Table 6, this section selects two sets of planar alignment parameters for curved steel box girder bridges to conduct modeling and simulation. By applying the combined effects of temperature and constant load to the two model sets, this study examines the impact of stress and deformation on the curved steel box girder as the span-to-width ratio δ varies with single-span length. This is analyzed under conditions of a 9 °C overall temperature increase and a 9 °C overall temperature decrease, while maintaining a constant bridge width b. The curved steel box girders utilize a single-box, double-cell section with a width of 10 meters. The span-to-width ratio δ for the two model sets ranges from 0.238 to 0.476. The top and bottom plates, as well as the webs, have a thickness of 16 mm, while the diaphragms have a thickness of 12 mm.

Figure 23. Relationship between stress and deformation of group D with span number. (a) Overall heating condition. (b) Overall cooling condition.

Figure 23. Relationship between stress and deformation of group D with span number. (a) Overall heating condition. (b) Overall cooling condition.

Figure 24. Relationship between stress and deformation of group E with span number. (a) Overall heating condition. (b) Overall cooling condition.

Figure 24. Relationship between stress and deformation of group E with span number. (a) Overall heating condition. (b) Overall cooling condition.

When the total bridge length L, radius of curvature R, central angle θ, and the bridge width b are fixed, increasing the single-span length of the curved steel box girder, which increases the width-to-span ratio δ, results in reduced stress on the curved steel box due to temperature effects. However, when the width-to-span ratio δ is less than 0.36, deformation due to temperature effects increases sharply as δ decreases. For a continuous curved steel box girder with a curvature radius of 69 m, increasing the width-to-span ratio δ from 0.28 to 0.46 results in a maximum stress reduction of 38.6% and a maximum deformation reduction of 64.9% due to temperature effects. For a continuous curved steel box girder with a curvature radius of 60 m, increasing the width-to-span ratio δ from 0.24 to 0.47 results in a maximum stress reduction of 47.5% and a maximum deformation reduction of 81.9% due to temperature effects. Therefore, when the bridge width is fixed, increasing the width-to-span ratio δ and reducing the single-span length of the curved steel box girder bridge can significantly reduce stress and deformation due to temperature effects. The larger the δ and the smaller the single-span length, the greater the reduction in stress and deformation caused by temperature effects. Table 7 and Table 8 present the statistical reductions in internal force and deformation values resulting from changes in δ for two model groups under conditions of overall heating and cooling.

Based on the statistical data presented in Table 7 and Table 8, it is evident that when the temperature rises, the rate at which stress decreases becomes more pronounced as the reduction in span diminishes. Conversely, when the temperature falls, the stress reduction rate diminishes as the reduction in span increases.

4.3. Support layout scheme

For continuous curved steel box girder bridges aligned in the north-south direction, differential solar exposure from the east and west leads to temperature variations between the inner and outer sides of the curved girder, creating lateral temperature gradients over the course of a day. This study examines how modifying the constraints of supports on both the inner and outer sides of the curved steel box girder affects its thermal response to these lateral temperature gradients.

With identical single-span lengths and side-to-mid-span ratios, the local maximum stress in a four-span continuous curved steel box girder is notably high under fluctuating temperature conditions. The study focuses on a four-span continuous curved steel box girder from the D group model, utilizing a dual-support simply supported system. By adjusting the positions of the fixed supports, the overall support configuration of the bridge is modified. Supports on the same side or section as the fixed supports employ unidirectional movable supports, while others utilize bidirectional movable supports. The detailed support configuration is provided in Table 9.

Figure 25. Support number.

Figure 25. Support number.

Figure 26. Maximum equivalent stress of fulcrum section when exposed to sunlight in the east.

Figure 26. Maximum equivalent stress of fulcrum section when exposed to sunlight in the east.

Figure 27. Maximum equivalent stress of fulcrum section during sun exposure.

Figure 27. Maximum equivalent stress of fulcrum section during sun exposure.

Figure 28. Comparison of optimal stress arrangement schemes under partial sun exposure.

Figure 28. Comparison of optimal stress arrangement schemes under partial sun exposure.

Considering the self-weight of the bridge and an additional dead load of 25 kN/m, numerical models representing eight distinct support arrangement schemes were analyzed under temperature fields resulting from east and west solar exposure. The maximum equivalent stress at each support section was determined, as illustrated in Figure 26 and Figure 27.

Figure 26 and Figure 27 illustrate that the local stresses at support points vary significantly across different support arrangements when subjected to the combined effects of solar exposure and constant loads in the same temperature variation environment. A comparison between schemes Z1 and Z8 reveals that the maximum stress at the same support point can differ by over 90 MPa. Schemes Z1 and Z3, which feature a single fixed support on the inner (west) side, exhibit lower equivalent stresses at the support points under both temperature conditions, with similar maximum equivalent stress values at the three intermediate support points. As shown in Figure 27, under east exposure conditions compared to west exposure conditions, the stress at the intermediate support points in schemes Z1 and Z3 increases by 36.8% and 38.1%, respectively. This suggests that temperature effects are more significant on the outer web than on the inner web. Under identical temperature conditions, scheme Z1 exhibits a lower average stress across all support point cross-sections compared to scheme Z3, although the cross-sectional stress at the middle support point is higher. Consequently, in practical applications, scheme Z1 should be locally reinforced at the middle support point cross-section.

From the perspective of overall average stress, scheme Z1, featuring a single inner curved support at the middle support point of a fixed multi-span bent steel box girder, emerges as the most rational support arrangement. This configuration effectively minimizes the overall average stress in the bent steel box girder when subjected to lateral temperature gradients.

Figure 28 presents the relationship between the maximum deformation and maximum equivalent stress of a four-span continuous bent steel box girder bridge in relation to variations in slope, as observed under the three temperature conditions detailed in Table 3. It is evident that as the longitudinal slope increases, there is a decrease in maximum deformation, whereas the maximum equivalent stress experiences an increase. The longitudinal slope plays a crucial role in influencing the temperature effects on the bent steel box girder when subjected to longitudinal bridge temperature differences. Particularly, when the longitudinal temperature gradient is contrary to the slope direction, it effectively reduces both the maximum equivalent stress and maximum deformation of the steel box girder.

5. Analysis of the effect of stiffening ribs on the temperature effect of bent steel box girders

5.1. Longitudinal stiffening rib

While keeping other geometric configurations constant, this study examines the effects of varying the number and spacing of closed-end stiffening ribs on the top surface of the steel box girder. The investigation evaluates how additional longitudinal bridging stiffening ribs on the top plate impact the stresses and displacements experienced by the steel box girder under the three temperature change conditions described in Section 3.1. The detailed results of this analysis are presented in Figure 29.

Figure 29 demonstrates that altering the number and arrangement of longitudinal, closed-end stiffening ribs on the top flange does not significantly reduce the structural sensitivity to thermal loads, although it does affect stiffness to some extent. Increasing the number of ribs resulted in only minor variations in structural deformation—limited to a 1.5 mm range—which is deemed negligible, while the maximum stress experienced a slight increase. Further analysis was conducted using a model with 12 stiffening ribs on the top plate, where the number of ribs on the bottom plate was increased from 6 to 8, evenly distributed. A comparative assessment with the original model revealed effects similar to those observed with additional longitudinal ribs on the top plate: the deformation of the curved steel box girder under the three specified temperature conditions showed minimal variation, with changes remaining under 1.5 mm after the addition of bottom plate ribs; however, the internal forces within the structure increased. This indicates that while the addition of stiffening ribs enhances structural stiffness, it does not significantly mitigate the internal forces under thermal loading but instead intensifies the structural stress.

To explore the effect of varying the number of longitudinal stiffening ribs on the web plate’s temperature response, the study utilized the 12-rib top plate model as a basis. The number of ribs on the outer and central webs was varied independently, and the resulting changes in stress and displacement in the steel box girder, due to the increased longitudinal web ribs under identical temperature conditions, are shown in Figure 30 and Figure 31. Figure 29 and Figure 30 reveal a significant increase in the local maximum stress within the bent steel box girder as the number of longitudinal bridging stiffening ribs on the outer web increases. However, this structural change results in a maximum displacement shift of less than 2 mm. While adding more longitudinal stiffening ribs to the central web had a similar impact on structural stress as observed with the side webs, the effect was comparatively milder and had virtually no influence on structural displacements.

While keeping the longitudinal dimensions of the U-ribs on the bottom plate constant, changes were made to the height h and width b of the rib on the top plate, thereby altering the section’s moments of inertia (I_x and I_y). This modification was designed to explore the impact of the flexural stiffness EI of the longitudinal bridging box stiffening rib on the temperature response of the curved steel box girder during an overall temperature increase of 9 ℃.

As shown in Figure 32, an increase in the bending stiffness EI of the closed-end stiffening rib section leads to a slight reduction in both the maximum stress and deformation of the bent steel box girder under a 9 ℃ overall heating scenario. The maximum stress decreases by no more than 2 MPa, and the maximum deformation reduction is limited to 10 mm. Figure 34 illustrates that reducing the width b of the bottom plate’s closed-end stiffening rib slightly decreases the local maximum stress, but by less than 1 MPa. As the flexural stiffness EI of the rib increases with the section height, the local maximum stresses in the bent steel box girder initially rise slightly before falling. When the section dimension h is less than or equal to b, the local maximum stress may increase; however, when h exceeds b, the reduction in local maximum stress becomes more pronounced, influenced by the stiffening rib’s cross-sectional moment of inertia Ix relative to the bottom plate. Despite these changes, variations in the bending stiffness EI of the bottom plate’s stiffening ribs have a negligible impact on the thermal deformation of the bent steel box girder, with the overall deformation remaining largely unchanged.

Figure 29. Stress and displacement variations due to changes in the number of stiffening ribs at different temperature conditions.

Figure 29. Stress and displacement variations due to changes in the number of stiffening ribs at different temperature conditions.

Figure 30. Relationship between the number of longitudinal stiffening ribs in the outer web and the maximum stress and displacement.

Figure 30. Relationship between the number of longitudinal stiffening ribs in the outer web and the maximum stress and displacement.

Figure 31. Relationship between the number of longitudinal stiffening ribs and maximum stresses and displacements in the midweb plate.

Figure 31. Relationship between the number of longitudinal stiffening ribs and maximum stresses and displacements in the midweb plate.

Figure 32. Relationship between the height of the top plate closed-end stiffening rib section and maximum stress and displacement.

Figure 32. Relationship between the height of the top plate closed-end stiffening rib section and maximum stress and displacement.

Figure 33. Relationship between the height of the closed-end stiffening rib section and the maximum stress and displacement of the base plate.

Figure 33. Relationship between the height of the closed-end stiffening rib section and the maximum stress and displacement of the base plate.

5.2. transverse-stiffening rib

Simulation analyses were conducted on models featuring 1 to 9 transverse stiffening ribs, spaced at 240 mm intervals on either side of the central support diaphragm. The outcomes of these simulations are illustrated in Figure 34. The results show that adding transverse stiffening ribs to the bottom plate near the center support diaphragm—close to the fixed bearing where the curved steel box girder experiences significant stress—is highly effective in reducing stress levels. This effect is particularly evident under the combined impact of temperature fluctuations and static load. Specifically, placing two transverse stiffening ribs on either side of the mid-span bearing diaphragm results in a maximum local stress reduction of up to 12% in a five-span curved bridge model under three different temperature conditions.

To evaluate the impact of transverse stiffening rib and support diaphragm spacing D and transverse stiffening rib spacing d on the temperature effects in a curved steel box girder, a series of numerical simulations were conducted using ANSYS software. The simulations began by varying the longitudinal bridging length L of the bearing from 600 mm to 900 mm in increments of 100 mm.

For each bearing dimension, six spacings were chosen, adhering to the condition D=d, with increments of 50 mm ranging from 250 to 500 mm. These 24 models were then subjected to a comparative analysis under a temperature scenario featuring a 9°C longitudinal gradient. The results are compiled and presented in Figure 35.

Figure 34. Relationship between the number of transverse stiffening ribs and the maximum stresses and displacements in the base plate.

Figure 34. Relationship between the number of transverse stiffening ribs and the maximum stresses and displacements in the base plate.

Figure 35. Local maximum stress versus spacing D.

Figure 35. Local maximum stress versus spacing D.

The analysis of Figure 35 reveals that when transverse stiffening ribs are arranged on the base plate with D = d ≤ 400 mm, the optimal spacing between these ribs and the transverse diaphragm is approximately half of the longitudinal bridging length L at the support. This configuration significantly reduces local stress concentration in curved steel box girders under thermal loading, thereby minimizing temperature-induced stress. However, when the longitudinal length of the bearing exceeds 900 mm, the effectiveness of the base plate’s transverse stiffening ribs in reducing localized stress concentration near the bearing diaphragm due to thermal loads decreases. As a result, the previously observed relationship no longer applies, and the optimal spacing tends to be around 400 mm.

5.3. Support-stiffening rib

The magnitude of the eccentricity B of the support stiffening rib, which depends on the transverse width L of the supporting plate, is typically positioned within the span of the supporting plate. With a stiffening rib thickness of t=10 mm and a width a=120 mm (where a/L=0.12), the relationship between the local maximum stresses at the transverse spacer of the support in the curved steel box girder and the ratio B/L under the combined influence of temperature and permanent load is illustrated in Figure 36. The figure indicates that as the B/L ratio increases, there is initially a decrease followed by an increase in the local maximum stress within the pivot transverse spacer, due to the combined effects of overall warming and permanent load. The optimal reduction in stress within the pivot transverse bulkhead due to thermal effects is observed when B/L is approximately 0.25. Beyond this ratio, the effectiveness of the support stiffening rib in reducing local stress on the diaphragm progressively diminishes with further increases in B/L.

With the parameters B/L = 0.25 and a = 120 mm (a/L = 0.12), the impact of varying the thickness t of the support stiffening ribs on the localized stresses due to thermal effects in the curved steel box girder was further examined, as illustrated in Figure 37. It is discerned that the local maximum stress in the transverse diaphragm of the curved steel box girder decreases consistently as the thickness of the supporting stiffening ribs increases. Beyond a thickness t ≥ 10 mm, a linear correlation emerges between the rib thickness t and the local maximum stress. Empirical equations were derived through data fitting, as follows:

$$\sigma =71.536-0.094t$$

(8)

$$D=\left\{\begin{array}{l}{\displaystyle \frac{L}{2}},L\le 0.8m\\ 0.4,L>0.8m\end{array}\right.$$

(9)

Although increasing the thickness t of the support stiffening ribs can reduce the local stress, the maximum local stress is only reduced by 1.3% when the thickness t is increased from 10 mm to 20 mm, which is not obvious to improve the local stress concentration. In the following, t=16 mm and B/L=0.25 are chosen to study the effect of the ratio of the width of the support stiffening rib a and the width of the support mat L on the local stresses in the girder plate under the coupling of temperature and constant load, as shown in Figure 38.

While augmenting the thickness t of the support stiffening ribs yields a reduction in local stresses, an increase in thickness from 10 mm to 20 mm culminates in a mere 1.3% decrement in the maximum local stress, indicating a marginal impact on ameliorating local stress concentrations. Consequently, a thickness t of 16 mm and a B/L ratio of 0.25 are selected for subsequent analyses examining the effect of the width a of the support stiffening rib relative to the width L of the support mat on the local stresses in the girder plate under the synergistic influence of temperature and permanent load, as delineated in Figure 38. It discloses that with a fixed support plate dimension L and a ratio of a/L≤0.3, the local maximum stress in the fixed-pivot diaphragm subject to thermal effects diminishes as the width a of the support stiffening ribs increases. Conversely, with a constant supported diaphragm dimension L and a ratio a/L > 0.3, local stresses in the curved steel box girder escalate with an increasing a/L ratio. This phenomenon suggests that an excessive width a of the supporting stiffening ribs may precipitate local instability risks, thereby attenuating the efficacy of local reinforcement on the support diaphragm.

Figure 36. B/L vs. local maximum stress in the pivot diaphragm with temperature effect.

Figure 36. B/L vs. local maximum stress in the pivot diaphragm with temperature effect.

Figure 37. Thickness t vs. maximum localized stress in the pivot diaphragm for temperature effects.

Figure 37. Thickness t vs. maximum localized stress in the pivot diaphragm for temperature effects.

Figure 38. a/L versus the maximum localized stress in the pivot diaphragm under temperature effect.

Figure 38. a/L versus the maximum localized stress in the pivot diaphragm under temperature effect.

6. Design Recommendations

Figure 39. Maximum Equivalent Stress at the Support Section for Various Parameter Coupling Schemes under Temperature Change Conditions. (a) Overall Heating Condition. (b) Overall Cooling Condition.

Figure 39. Maximum Equivalent Stress at the Support Section for Various Parameter Coupling Schemes under Temperature Change Conditions. (a) Overall Heating Condition. (b) Overall Cooling Condition.

Figure 40. Maximum Displacement at Various Locations of the Entire Bridge for Different Parameter Coupling Schemes under Temperature Change Conditions. (a) Overall Heating Condition. (b) Overall Cooling Condition.

Figure 40. Maximum Displacement at Various Locations of the Entire Bridge for Different Parameter Coupling Schemes under Temperature Change Conditions. (a) Overall Heating Condition. (b) Overall Cooling Condition.

7. Conclusion

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