**1. Introduction**

In suspension bridges, cable clamps are key nodes connecting the main cables and hangers. With the progress and development of suspension bridges, cable clamps have gradually evolved from eyebars and pin-type hinge structures to riding straddle and pintype structures. The choice is closely related to the type of hanger used: riding straddle cable clamps are commonly used for wire rope hangers, whereas pin-type cable clamps are used for parallel wire hangers. In the U.S. and Japan, preference is given to riding straddle cable clamps, whereas pin-type cable clamps are more common in China and Europe [1]. Figure 1 shows the different types of cable clips.

During the construction and operation, the cable clamp will no longer be firmly anchored because of the loosening of the fastening bolts, the main cable has a thinner section under long-term loading, and the linear expansion coefficient is inconsistent under the influence of temperature. These factors are important inducements that cause slippage of the suspension bridge cable clamps. Therefore, the cable clamp slippage is a common phenomenon that is detrimental to suspension bridges. In the inspection manual for suspension bridges developed by the Honshu–Shikoku Bridge Construction Corporation, Japan, cable clamp slippage is listed as a key inspection item [2]. In the United States, the cable inspection and strength assessment guidelines provide detailed inspection requirements for hangers and cable clamps [3]. There are strict regulations on cable clamp slippage in the Chinese Highway Bridge Technical Condition Assessment Standard [4].

**Citation:** Li, H.; Liu, Y.; Li, C.; Hu, H.; Sun, Q. Force Analysis of Self-Anchored Suspension Bridges after Cable Clamp Slippage. *Symmetry* **2021**, *13*, 1514. https:// doi.org/10.3390/sym13081514

Academic Editors: Yang Yang, Ying Lei, Xiaolin Meng and Jun Li

Received: 22 July 2021 Accepted: 16 August 2021 Published: 18 August 2021

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**Figure 1.** Different types of cable clips: (**a**) abridged general view of a riding straddle cable clip; (**b**) abridged general view of a pin-type cable clamp.

Cable clamps with poor anti-sliding performance tend to slide along the main cable, as shown in Figure 2, which directly causes a loss of stress in the entire structure and major safety hazards in the structural system. Insufficient anti-skid performance of the cable clamp will cause the cable clamp to slide down the main cable, which will lead to the redistribution of the internal force of the principal force-bearing components such as the main cable, hanger, and main beam. The internal force deviates from the design value and the safety reserve of the structure is cut. In practice, engineering problems due to the poor anti-sliding performance of cable clamps may force a project to be terminated or even rebuilt. Hence, scholars have extensively studied the problem of cable clamp slippage.

**Figure 2.** Actual images of a cable clip after slipping.

In 2000, Ji et al. analyzed the additional force due to the hanger pull acting on the screws in the hanger, with an objective to improve the cable clamp design of the Kurima Bridge, Japan. The Liaison Bridge Corporation proposed the application of pin-type cable clamps for the first time [5]. Since 2009, scholars have started paying attention to cable clamp slippage in suspension bridges. Ren et al. analyzed the anti-skid friction coefficient of a cable clamp and the main cable void ratio by conducting an anti-sliding test on the main cable and cable clamp [6]. Huang et al. studied the variations in the lateral displacement of the main cable and the lateral deflection angle of the cable clamp under the effect of hanger tension in the main cable [7]. In 2013, Li et al. determined the upper limit of the transverse elastic modulus based on the macroscopic force of an ideally arranged main cable [8]. In 2014, Ruan et al. studied and analyzed the spatial effect and stress distribution characteristics of a cable clamp [9]. Ma et al. conducted jack push tests on the main cable of a suspension bridge to analyze the friction coefficient of the cable clamp against sliding and the internal and external void ratios of the cable clamp [10]. Zhuge et al. studied the friction factor of a carbon fiber reinforced plastic (CFRP) cable–cable clamp interface [11]. Li et al. proposed the Tsai–Hill failure criterion for composite materials [12]. In 2015, Zhou et al. calculated the anti-sliding friction coefficient and the internal and external void ratios of a steel wire main cable clamp coated with a zinc–aluminum alloy [13]. In 2016, Sun et al. derived a simplified calculation formula to determine the increase in the elevation of the main cable mid-span control point and the average increment in the hanger cable force [14]. In 2018, Shen et al. analyzed the change laws of the ultimate anti-sliding friction resistance of a cable clamp under the action of the cable force, and the contact force between the main cable and the cable clamp surface [15,16]. In 2019, Ruan et al. proposed a theoretical model considering transversely isotropic materials based on the generalized Hooke's law. Through anti-slipping performance tests, they determined the actual ultimate sliding resistance and comprehensive friction coefficient of a cable clamp [17]. In 2020, Miao et al. proposed a new slippage-based failure criterion based on an analytical model [18] and briefly revised the original slippage resistance formula using the Coulomb friction law.

In summary, most studies have been based on the interface performance of the main cable and the cable clamp, focusing on the analysis of the effects of the clamping force of the cable clamp and the friction coefficient on the anti-slipping performance. Studies on the total force acting on suspension bridges are lacking. The integrity of the overall structure of a full bridge following cable clamp slippage has rarely been studied. In practice, although cable clamp slippage is minor, it is a key factor causing a change in the forces acting on the entire bridge. Therefore, this study mainly analyzed the overall force acting on suspension bridges following cable clamp slippage.

In this paper, the influence of pin-type cable clamp slippage on a self-anchored suspension bridge is reported. Considering the geometric nonlinear influence of the bridge, a finite element simulation was conducted. First, we selected a large-span self-anchored bridge as an example to describe the geometry of self-anchored suspension bridges and the magnitude of the cable clamp slippage in detail. Subsequently, a method for simulating the cable clamp slippage was proposed and verified through examples. Additionally, the total force acting on the suspension bridge before and after cable clamp slippage was analyzed, and recommendations for operation and maintenance was provided.
