Keywords

1 Introduction

As a crucial connecting component between the hanger cable and the main cable, the cable clamp mainly improves its anti-slip ability by increasing the friction force with the main cable through bolt fastening [1, 2]. Once the cable clamp slips, it will cause cracking and damage to the coating near the clamp and straight seams, redistribute the internal force of the main cable, change the line shape of the main beam, affect the safety of the bridge structure, and the process of change has a continuous divergent effect, which is irreversible [3].

In order to investigate the influencing factors of cable clamp slippage, Miao et al. [4, 5] proposed a new criterion for cable clamp sliding, which corrected the traditional Coulomb friction formula and analyzed the anti-slippage performance of cable clamps under different bolt tightening schemes through finite element modeling. Liu [6] and Ruan [7] conducted small-scale model tests on cable clamp anti-slippage, collected a large amount of test data, and obtained the influence laws of cable clamp tightening force, friction coefficient, and creep under different factors. Tang [8] and Zhang [9] mainly studied the reasons for the decay of cable clamp pre-tightening force. Shen [10, 11] studied the nonlinear relationship between the steel wires of the main cable at the cable clamp site and researched the anti-slippage friction resistance of the cable clamp based on a multi-scale model. Zhou [12], Zhao [13], and Zhao [14] based their research on actual engineering projects and produced large-scale models of main cable-cable clamp tests. The test data showed that the measured anti-slippage friction coefficient was greater than the specification requirement of 0.15, and the contact surface of the cable clamp was subject to nonlinear changes in force and had a complex stress composition. In addition to using traditional materials for the main cable in the above studies, Li [15], Zhuge [16], and Hou [17] also conducted related research on the sliding relationship between CFRP materials for the main cable and cable clamps, making it possible to further apply new material structures to suspension bridges.

The above research findings indicate that current studies have mostly focused on the influence of factors such as bolt pre-tensioning force, cable tension, and main cable creep. Suspension bridges are typical cable-supported structures, with the main load-bearing structure being made of steel. They are sensitive to temperature changes, and this effect is more pronounced in regions with large temperature differences between day and night [18]. Therefore, further research is needed to investigate the anti-slippage performance of cable clamps on suspension bridges under the influence of temperature.

2 Calculation of Temperature Effects on Cable Clamp Friction Resistance

For the main cable, temperature changes can cause variations in the cross-sectional area of the cable. This leads to the concept of the coefficient of thermal expansion for the main cable, which is expressed as follows:

$$ \alpha_s^{\,} { = }\frac{\pi (r_T^2 - r^2 )}{{\pi r^2 \Delta t}} $$
(1)

In the equation, \(\alpha_s^{\,}\) represents the coefficient of expansion of the main cable surface; \(r_T\) represents the radius after temperature change; \(r\) represents the initial radius; \(\Delta t\) represents the temperature difference.

Then the change amount of the main cable radius before and after the temperature change is:

$$ \Delta r = r(\sqrt {\alpha_s \cdot \Delta t + 1} - 1) $$
(2)

According to the coordination relationship between the main cable, cable clamp and screw deformation, it can be known:

$$ \varepsilon_L = 2\Delta r/L $$
(3)

In the equation， \(\varepsilon_L\) represents the helical strain caused by the deformation of the main cable section； \(L\) represents the initial length of the helix.

Therefore, the influence relationship of the cable clamp tightening force under the temperature change of the main cable can be expressed by the following formula:

$$ P_{mc} = \frac{{2 \cdot r(\sqrt {\alpha_s \cdot \Delta t + 1} - 1)}}{L} \cdot E_L A_L $$
(4)

In the formula, \(E_L\), \(A_L\) are the elastic modulus and cross-sectional area of the screw respectively.

According to the principles of material mechanics, the tightening force changes under the temperature change of the screw itself are as follows:

$$ P_{sc} = \alpha_L \Delta t \cdot E_L A_L $$
(5)

\(\alpha_L\) is the screw linear expansion coefficient.

Therefore, the total influence expression of screw tightening force under the influence of temperature is:

$$ P = \left[ {{\text{2r}}(\sqrt {\alpha_s \cdot \Delta t + 1} - 1)/L + \alpha_L \cdot \Delta t} \right] \cdot E_L A_L $$
(6)

According to the classic Coulomb friction law, the ultimate anti-slip friction resistance between the cable clamp and the main cable can be expressed as \(F_{fr}\).

$$ F_{fr} = 2N_r = \frac{4P}{{\mu_\theta }}(1 - e^{ - \mu_\theta \pi /2} ) $$
(7)

Then the friction formula after the influence of temperature is:

$$ F_{fr} { = 4}\frac{{\left[ {{\text{2r}}(\sqrt {\alpha_s \cdot \Delta t + 1} - 1)/L + \alpha_L \cdot \Delta t} \right] \cdot E_L A_L }}{\mu_\theta }(1 - e^{ - \mu_\theta \pi /2} ) $$
(8)

3 Solar Radiation Data

The Xi He Big Data platform [19] introduces various meteorological data sources, and based on artificial intelligence and machine learning algorithms, it performs downscaling calculations on existing meteorological elements, and optimizes fusion and calibration of meteorological data based on its own data grid. In order to verify the reliability of the data results, the platform's internal meteorological data and on-site radiation measurement data were compared for inspection.

The data collected by a solar radiation sensor installed at the construction site of the Qingshui River Bridge project is used as a comparison source for verification. The time span selected is August, when the sunlight is relatively strong, and the data collection time is from 8:00 to 17:00 (August), with a time interval of 1 h. Rainy days in each month are excluded. The data is shown below in Fig. 1.

Fig. 1.
figure 1

Comparison between radiation prediction and actual measurement in August

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Based on the data curve in the graph, it can be observed that the measured values and predicted values are well aligned. There is a relatively large deviation in the high radiation value area, with a maximum deviation of 15.2% in July and 17.6% in August. Overall, the measured values tend to be slightly higher. According to the calculation formula of the MAPE function, the M value for July is 9.95%, and for August it is 11.08%. These values meet the requirements for accurate estimation and can be used in practical engineering projects.

4 Project Overview and Model Construction

4.1 Project Overview

The Changshou Economic Development Zone Bridge is a single-span simply supported steel box girder suspension bridge with a main span of 739 m and a vertical span ratio of 1:9.11. The longitudinal spacing of the suspenders is 12 m, and the suspenders near the tower are 15.5 m away from the tower centerline. The bridge cable clamps mainly consist of upper and lower half clamps and M42 high-strength bolts. The design clearance between the two half clamps is set at 4 cm, and the effective length of the bolts is greater than 0.7 times the diameter of the main cable inside the clamps in Fig. 2.

Fig. 2.
figure 2

Cable clamp design dimensions. (unit: mm)

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4.2 Model Parameter Selection

Main Cable Properties.

In reference [10], a large amount of actual field data on the tension forces of the cable clamps and the dimensions of the main cable were analyzed. The three-dimensional (axial, tangential, and radial) anisotropic equivalent material property relationship of the main cable was proposed, and its material adaptability was confirmed through experimental data. The fitting curve and relationship equation are shown in Table 1.

Table 1. Equivalent material parameter table
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The author compared the main cable dimensions, construction methods, and design aspects of the experimental bridge in reference [10] with those of the Changshou Economic Development Zone Bridge, and found that except for slight differences in the quantity of galvanized steel wires used for the main cable, all other indicators are completely consistent. Therefore, disregarding factors such as construction errors, it can be considered that the three-dimensional anisotropic material for the main cable in reference [10] is equally applicable to the main cable model in this paper.

Thermophysical Parameters.

Zhang [20] summarized the research results on the thermal properties of the main cable in the past. By fitting the corresponding thermal property parameter model of the main cable through laboratory and on-site main cable model test data, the correctness of the parameter model has been verified in the paper, and can be used for temperature field calculation and analysis of the main cable of suspension bridges. As shown in Table 2.

Table 2. Thermophysical parameters.
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4.3 Finite Element Model Establishment

A model was created using the ABAQUS software, a large-scale general-purpose finite element software. The main cable and cable clamps were simulated using C3D8T thermal-coupled hexahedral elements, while the high-strength bolts were simulated using spatial beam elements. The contact surfaces between the ends and the nuts on the cable clamps were coupled to simulate the transmission of bolt forces, with pre-tension applied through bolt loading. The axial friction coefficient was set to 0.15, the circumferential friction coefficient was set to 0.2, and the friction coefficient at the cable clamp stopper was set to 0.15. The ends of the main cable were fixed constraints, and a vertical displacement was applied at one end of the cable clamp. According to the principle of action and reaction, the frictional resistance of the cable clamps could be obtained by extracting the boundary reaction forces of one end of the main cable. The computational model is shown in Fig. 3.

Fig. 3.
figure 3

Finite element model diagram

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5 Result Analysis

5.1 Analysis of the Influence of Screw Temperature

Based on the equation for calculating the frictional resistance of the cable clamps and the finite element model, different values of bolt temperature were determined to analyze the differences between the finite element solution and the analytical solution. The results are shown in the table below. For different bolt temperatures, the maximum difference ratio between the finite element solution and the analytical solution is -4.2%, and the minimum is -0.3%. The comparison of the calculation results in these two cases shows small differences, indicating that the finite element calculation results are close to the analytical solution, further confirming the accuracy of the established finite element model (Table 3).

Table 3. Comparison of friction resistance at different screw temperaturesa
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5.2 Force Analysis of Cable Clamp Under Steady Temperature Field

During daytime solar radiation, factors such as radiation intensity, radiation angle, and wind speed at different times can all affect the temperature field results. Therefore, to simplify the calculation, we take the example of the day with the highest radiation at the bridge site in mid-July 2023, with time set at 7 AM and 12 PM. The solar radiation is assumed to be horizontal at 7 AM and vertically from top to bottom at 12 PM. The temperature field results are shown in Fig. 4.

Fig. 4.
figure 4

Temperature field results. (unit:℃)

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The temperature results mentioned above were loaded into the model in the form of a mapping field, with the ambient temperature based on the data provided by the platform, and the bolt pre-tightening force set at 500 kN without applying temperature load to the bolt.

The tangential friction force at the contact point of the cable clamp was extracted, as shown in Fig. 5. According to the results in the figure, compared with the original model, at 7 AM, the numerical range of the inner friction force distribution of the upper cable clamp is basically consistent, and the distribution area of higher friction force is expanding. Overall, the frictional resistance of the cable clamp is increasing at this time. However, at 12 PM, although the maximum frictional force has increased, it is mainly concentrated in the corner area of the cable clamp and local positions near the screw hole, and the frictional force in a larger area at the top is decreasing. Combined with the temperature field results, it can be inferred that during the transition from low temperature to high temperature, the frictional resistance of the cable clamp will increase to a certain extent. However, after exceeding a certain temperature, the deformation of the cable clamp relative to the radial deformation of the main cable increases, causing some deformation inconsistency and a decrease in contact density, resulting in a decreasing trend in frictional resistance. Comparing with the results of cable clamp frictional force, it is basically consistent with the description above. The cable clamp frictional force at the side is greatly increased under high temperature, while most areas show a decreasing trend.

Fig. 5.
figure 5

Cable clamp friction force results. (unit: N)

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5.3 Screw Force Analysis Under Steady Temperature Field

Observing the bolt axial stress results shown in Fig. 6, it can be found that the bolt axial stress increases correspondingly with the increase of temperature. At 7 AM, the increase in bolt axial stress on the left and right sides is not consistent. Considering the direction of radiation, the temperature on one side is higher than the other, and the axial stress on the side with higher temperature will be higher than that on the other side with lower temperature. In addition, the increase in bolt axial force caused by the uneven temperature difference on both sides of the cable clamp is greater than the effect of uniform temperature difference. Therefore, in actual bolt axial force testing, if the site cannot reach a constant temperature state, testing can also be performed under uniform temperature difference.

Fig. 6.
figure 6

Screw shaft stress results. (unit: Pa)

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6 Conclusion

Based on the validated platform radiation data, this article establishes a thermal-mechanical coupling model of the cable clamp-main cable using ABAQUS software, and compares the finite element solution with the analytical solution to obtain the effect of bolt temperature. In addition, the corresponding stress situation under steady-state temperature field is obtained for the frictional force of the cable clamp and the axial stress of the bolt. The following conclusions are drawn:

With the increase of temperature, the bolt pre-tightening force decreases, and the analytical solution is close to the finite element solution, with a maximum error of 4.2%, which can be used for actual on-site engineering calculations.

The increase of temperature causes structural deformation of the main cable and cable clamp under thermal variation, which increases the frictional force of the cable clamp to a certain extent, which is beneficial to reducing the probability of slippage during use.

Based on the above two analysis results, in actual projects, the cable clamp pre-tightening should be performed at low temperature to avoid pre-tightening force loss caused by cooling after high-temperature pre-tightening. In addition, thermal insulation treatment can be applied to the surface of the bolt to maximize the increase in frictional force caused by structural warming and improve the anti-slippage safety factor.