1. Introduction
Suspension bridges are the most common type of bridge used to cross the ocean. The clamps of suspension bridges clamp the main cable by the preload of high-strength bolts, which is the key component that transfers the load of the bridge deck to the main cable. When tightening the cable clamp, due to the limited working space, the clamp bolt group is usually tightened by the batch tightening method. Due to the deformation of the main cable and the clamps during the tightening process, the effect of tightening one bolt in the group has an effect on the preload in other previously tightened bolts in the group. Such an effect is called bolt elastic interaction. This significantly affects the uniform distribution of the preload on the cable clamp bolt.
In the theoretical study of the elastic interaction, Bible et al. [
1,
2] investigated the influence law of bolt preload at different locations and proposed the elastic interaction matrix of the bolt group. Li et al. [
3] established a generalized mathematical model for evaluating bolt preload based on the deformation coordination equation and elastic interaction stiffness theory. Alkelani et al. [
4] proposed an elastic interaction model for bolts with gaskets that was able to predict the change in the preload of the later tightened bolt on the previously tightened bolt. Nassar et al. [
5] proposed a formula for the elastic interaction of bolt groups during fastener tightening in planar gasketed joints, which allows optimization of the tightening strategy. Zuo et al. [
6] established the correlation model of the initial bolt preload and the contact stiffness of bolted flange. This model can be employed to reverse determine the tightening strategy of the bolt group according to working conditions. Jiang et al. [
7], combined with the equivalent form of the bolted joint system, proposed a new stiffness model dependent on the strain energy method. Liu et al. [
8] developed an analytical model of elastic interaction through gasket deformation and explored the effect of multidimensional dimensions such as bolt distance, joint width, and thickness on the loss of preload.
Considering the complexity of theoretical analysis, the finite element model has become a new tool to study the elastic interaction of bolt groups. Takaki et al. [
9] investigated the elastic interaction matrix during the tightening of bolts in flanged joint structures by using a FE model to determine the optimum tightening procedure. Bouzid et al. [
10] modeled the elastic interaction of flexible flange rotation and were able to predict the flange rotation angle and bolt preload during the bolt tightening process. Yuan et al. [
11] studied the elastic interaction on a toothed coupling connecting bolt, and the results showed that the elastic interaction could cause non-uniform deformation of the connected parts. Zhu et al. [
12,
13,
14] proposed a new numerical analysis method considering the nonlinear elastic property condition of the gasket and evaluated the distribution of the preload of the flange bolts. Abasolo et al. [
15,
16], based on the FE model, proposed a method capable of achieving a uniform distribution of preload in a bolt group for one or two tightening rounds. Wang et al. [
17] analyzed the stress field of multiple bolted joints, established an analytical model of elastic interaction stiffness, and investigated the effect of elastic interaction stiffness on the initial preload of bolts. Coria et al. [
18,
19] proposed a four-parameter assembly method for a circular bolted joint, which was able to achieve a uniform load distribution of the bolt group in a single tightening sequence. Huang et al. [
20] used the FE model to explore the influence of initial preload and tightening sequence on the elastic interaction and proposed a method to predict the change in the preload. Ding et al. [
21] investigated the elastic interaction of anchorages such as CFRP plates and improved the pressure diffusion mechanism to have a more uniform contact pressure distribution. Most of the above were to study the basic elastic interaction of the bolt group and did not consider the influence of more external factors on the elastic interaction of the bolt group.
Related to the study of the factors influencing the elastic interaction of bolts using the FE model, some research has found that common influencing factors of the bolt group elastic interaction include tightening sequence, thickness of connected components, bolt spacing, and magnitude of applied fastening force [
22,
23]. Considered the influence of other factors on the bolt group’s elastic interaction, Pan et al. [
24] established a prediction model for the residual preload in long plate bolted joint systems and believed that appropriate preloading sequence, increasing the number of preloading steps and having variable amplitude loading can effectively reduce the effect of elastic interaction. Bouzid et al. [
25] established an FE model to investigate the influence of the operating temperature of the fluid inside the flange connection structure on the elastic interaction of the bolt group. Abid et al. [
26,
27] considered the thermal expansion of the connection components and explored the influence of different tightening sequences on the bolt elastic interaction of flange connection structures by using an FE model. Wang et al. [
28] reduced the effect of bolt elastic interaction by optimizing the tightening sequence and used the kriging model to predict the best tightening sequence under different types and tightening clearance. Liu et al. [
29] predicted the optimal tightening sequence of bolt groups by constructing a neural network agent model to obtain a more uniform preload. Wang et al. [
30] considered the preload loss caused by pressure vessel flange connections under the influence of high temperature and creep to explore the law of bolt group elastic interaction.
The above research focuses on the study of elastic interactions in small bolt sets in mechanical engineering. Regarding the elastic interaction of large bolts, He [
31] studied the bolt installation fastening process for up-and-down and left-and-right types of cable clamps and found that fastening the bolt from both ends to the center can effectively reduce the preload loss. Yang et al. [
32] conducted tightening process fastening tests with one pair of tensioners and three pairs of tensioners to analyze the influence of the number of tensioners on the bolt elastic interaction. Miao et al. [
33] investigated the influence of the length of the cable clamp, the number of bolts, and the fastening process on the elastic interaction through a real bridge experiment, and the results showed that symmetrical fastening from the ends of the cable clamps to the middle was a better fastening sequence.
Suspension bridge cable clamps can be regarded as long plate-shaped connection structures. The current research on elastic interactions mainly focuses on small bolt group structures with flange connections, and there is less research on elastic interactions in long plate-shaped connection structures. Due to the existence of strong nonlinearities in the main cables of suspension bridges, it leads that the elastic interaction of cable clamp bolts and their influencing factors in suspension bridges have not been sufficiently researched. This paper investigated the influence of external factors on the elastic interaction of cable clamp bolts of suspension bridges. Firstly, a theoretical model was derived for the elastic interaction of the cable clamp bolt. Secondly, the full-scale experiment was carried out to collect the bolt preload data, and the experiment results were used to verify the finite element model. Thirdly, based on the FE model, the influence of main cable material properties, cable clamp design parameters, and tightening technology on the elastic interaction of cable clamp bolts was discussed. Finally, an optimization method to reduce the effect of the elastic interaction of cable clamp bolts is proposed. This paper can be a guide for improving the tightening efficiency and preload uniformity of cable clamp bolts in suspension bridges.
2. Theoretical Analysis
The elastic interaction of the bolts is essentially reflected in the transfer of deformation or displacement in the connected structure under the action of the bolt preload, which in turn affects the preload of each bolt through the connected structure. In order to better investigate the mechanical relationships in elastic interaction, a double-bolt simplified model of the cable clamp structure is designed, which consists of bolt 1, bolt 2, top clamp, and bottom clamp. The simplified theoretical model is shown in
Figure 1.
In the derivation of the mechanical model, it is assumed that the bolt elongation is equal to the compression deformation of the clamp in the tightened state and that the clamp does not produce distortion effects such as warping. As depicted in
Figure 1, in the first stage, no preload is applied to bolt 1 when the gap between the top and bottom clamp is
.
Then, bolt 1 is tightened, and preload
is applied to bolt 1, which initiates the second stage. As can be seen in the second stage of
Figure 1, the preload
makes the gap
. The part clamped between the top and bottom clamps is the main cable, which is assumed to have a stiffness
. The relationship between preload
, gap
, and stiffness
is shown in Equation (1).
where
is the initial gap between the top and bottom clamps,
is the preload of bolt 1, and
is the stiffness of the clamped main cable.
Under the action of the preload
, the clamp gap at bolt hole 2 changes from
to
. According to the literature [
3], the transfer of displacement in the connected structure is linear. Therefore, there exists a coefficient
between
and
such that it satisfies the relationship in Equation (2).
where
is the gap between the top and bottom clamps at bolt hole 2 after bolt 1 is tightened, and
is a linear coefficient and a constant.
After the second stage, bolt 2 is tightened, applying a preload to bolt 2, and at this point, the gap between the top and bottom clamps at the bolt 2 position disappears. However, due to elastic interaction, the top and bottom clamps at bolt 1 will deflect slightly upward displacement under the preload of bolt 2, which causes the top and bottom clamps at bolt 1 to recreate a small gap, . After deformation, there is an inclination angle between the top and bottom clamps, and the horizontal distance of the inclined section is .
Similarly, the transfer of displacement in the connected structure is linear, Therefore, there exists a coefficient
between
and
such that it satisfies the relationship in Equation (3).
where
is the gap between the top and bottom clamps at bolt 1 after bolt 2 is tightened, and
is a linear coefficient and a constant.
Plug Equation (2) into Equation (3),
Due to the small deformation, the relationship between gap
, inclination angle
, and distance
is shown in Equation (5).
where
is the angle resulting from the deformation, and
is the horizontal distance of the inclined section.
The gap
causes bolt 1 to produce an elongation of
, and the preload of bolt 1 at this point decreases by
. Based on the relationship between deformation and force, Equation (6) is obtained.
where
is the amount of change in the preload,
is Young’s modulus,
is the bolt area, and
is the length of bolt clamping section.
Plug Equations (1) and (4) into Equation (6),
Plug Equation (5) into Equation (6),
Equations (7) and (8) represent the amount of change in preload caused by the elastic interaction of the bolts. The change in preload and the effect of elastic interaction are positively proportional. From Equation (7), it can be seen that the material and size of the bolts, the level of preload, and the stiffness of the main cable affect the elastic interaction of the cable clamp bolts. All other parameters being the same, larger bolt diameter, smaller bolt clamping section length, lower main cable stiffness, and larger preload result in more obvious elastic interaction of the bolts. From Equation (8), it can be seen that larger distance results in more obvious elastic interaction.
In order to verify the accuracy of the theoretical analysis, a full-scale experiment and a finite element analysis were performed.
6. Discussion
There are fewer studies on the factors affecting the elastic interaction of cable clamp bolts in suspension bridges. The purpose of this study was to investigate the laws governing the elastic interaction of bolts due to factors that may occur in the design and construction of cable clamps. Firstly, a theoretical model was derived for the elastic interaction of the cable clamp bolt. Secondly, an indoor full-scale experiment was conducted to validate the FE model using the experiment data. Thirdly, based on the FE model, the influence of main cable material properties, cable clamp design parameters, and tightening method on the elastic interaction of cable clamp bolts was discussed.
The research in this paper can provide suggestions for reducing the elastic interaction of bolts in the design and construction of suspension bridge cable clamps. In our conclusions, the number of bolts has a small effect on the elastic interaction, but in some literature [
33], it is argued that a higher number of bolts leads to a higher elastic interaction effect. Comparison shows that the number of cable clamp bolts in the literature [
33] is much larger than in this article. Our conclusions may be applicable for small changes in the number of bolts. A deeper study of the impact of the number of bolts is needed in the future. In addition, in our theoretical model, there are two linear coefficients, and the coefficients
and
require a more in-depth study. In the FE model, the last tensioned bolt holds the applied preload, but it also experiences a loss of preload in the experiment, which is transient loss due to the bolt thread gap rather than elastic interaction effects. In our research, much research has been done on eight-bolt clamps, and the applicability of the conclusions to clamps with other numbers of bolts needs to be verified. Most of the conclusions of the article are obtained from the FE model, and more experiments need to be conducted in the future to verify the conclusions.