**3. Simulation of Cable Clamp Slippage**

*3.1. Finite Element Modelingsubsection*

MIDAS Civil was used to establish a finite element model for the theoretical calculations. The bridge is first assembled from a free cables state to a finished bridge state and then changed from a finished bridge state to a slip state. In a structural analysis, the equilibrium equation of the force should be established based on the geometric position of the structure after deformation, and the relationship between the force and the deformation is nonlinear. Therefore, the model cannot be established directly from the slip state. When building the model, the finished bridge state model must be established first, and then the slip state should be established based on the finished bridge state. The models currently employed can mainly be divided into nondamaged bridge completion stage models and single-hanger slip whole bridge models for calculation and comparison. The main girder and the main tower of the whole bridge model are simulated by beam elements, and the main cables and hangers are mainly stimulated by cable elements. Table 2 lists the main modeling material parameters.


**Table 2.** Specifications of the main components.

To ensure that the force condition of the bridge is simulated correctly and is consistent with reality, when the model is established, the boundary conditions are as follows: consolidate the bottom nodes of the main tower; rigidly connect the top of the main cable to the main tower and then release the full fixation in the rest of the direction of the bridge displacement; rigidly connect the bottom of the main cable to the main girder and constrain all direction; elastic support simulation is adopted at each support of the main beam. The entire bridge is divided into 4706 elements and 5409 nodes. Among them, the main cable has 106 elements, and only 98 elements of the hanger are used as tensile members. Figure 6 shows the model diagram.

**Figure 6.** Integral finite element model of a bridge.

### *3.2. Simulation Steps for Cable Clamp Slippage*

For cable clamp slippage, in this study, the method of double-cable element replacement with common nodes is adopted. A new construction stage after the last construction stage of the original bridge type is established and defined as the cable clamp slip stage. We compared the two stages and observed the variations in the structural parameters after cable clamp slippage. In the slipping stage, the original main cable element is passivated, and the adjusted main bridge state cable element is activated to ensure that the model follows the original design. In the sliding stage of the cable clamp, the original main cable element, without slipping of the cable clamp, is passivated, and the main cable element after the cable clamp slips is activated correspondingly. The model takes the original mechanical state when the bridge is erected under the original design without the stress cable length as the initial state, the adjusted cable element is used to re-calculate iteratively, and finally the model converges to the slip state to complete the cable clamp slip. In the simulation of the moved finite element model, the cable element of the aforementioned main cable is adjusted by modifying the unstressed length parameters of the main cable element. The original main cable element is controlled by the unstressed cable length given by the design, and the bridge is in the final state of the bridge with the original design cable length under the working force. The replaced cable element changes its unstressed length relative to the original cable element.

The modifications are as follows: Using the measured or known slippage of the cable clamp in the final slipping state, the stress-free length change in the main cable section before and after the slipping of the cable clamp node is calculated, and the stress-free length of the main cable element, before and after the slipping of the cable clamp node, is manually adjusted. The total unstressed length of the main cable element before and after the adjustment should be kept constant.

### *3.3. Verification of Measured Cable Force Against Model Cable Force*

The hanger is the force transmission member that transfers the stiffening girder's own weight and external load to the main cable. It is the link between the stiffening girder and the main cable that bears the axial tension. The magnitude of the constant load axial force in the hanger determines the behavior of the main cable in the suspension bridge. The overall linearity in the completed state of the bridge also determines the magnitude and distribution of the dead load bending moment of the stiffening girder; therefore, the hanger is key to studying the finished state of the bridge after completion of the suspension bridge. The cable force is the most intuitive measurement standard that reflects the overall force of the hanger [19]. The force distribution, between the main cable and the main girder of the actual spatial self-anchored suspension bridge, is mainly determined by the sling force, and the load on the main girder is transmitted by the hanger. Therefore, the load on the main beam is directly related to the weight of the main beam and the cable force of the sling.

To ensure that the finite element simulation is consistent with the actual force trend in the actual completed state, the hanger cable force was used to verify the model. The verification method was used to measure the hanger frequency using the frequency method when the bridge was completed in 2011, and the result was then compared with the cable force obtained from the finite element method. The suspension bridge structure is a symmetrical structure, so only the cable forces of the model and actual bridge on one side of the structure are compared. As shown in Figure 7, the cable force distribution in the model is consistent with the actual cable force distribution trend of the bridge in 2011. The maximum cable force in the finite element model is 2590 kN, and the maximum cable force measured in the bridge in 2011 was 2629.8 kN. The difference is 39.8 kN, and the positions are all close to the hanger of the main tower.

**Figure 7.** Verification of measured and simulated cable forces.

As shown in Figure 8, the difference between the cable force provided by the finite element model and the measured cable force of the bridge in 2011 is within the range of 60.3 to 86.5 kN, with the difference being <5%. The maximum rate of change on both sides of the hanger is 4.26%. This is because the length of the hanger on both sides is relatively short. In the actual test, the error of the cable force value calculated by the frequency method will be relatively high; nevertheless, the test result and overall trend in the model are in line with those observed in 2011 when the bridge was completed. Therefore, from the above comparison, it can be seen that the proposed model yields the same force state as that of the bridge in 2011. The model can be used to simulate the single point slip of the suspension bridge for a detailed analysis and calculation owing to the reliable accuracy.

**Figure 8.** Variations in the measured and simulated values of the hanger tensions.

To further verify the accuracy of the clamp slipping model simulation, the hanger cable force after clamp slippage in the model is compared with the measured cable force of the suspension bridge. As shown in Figure 9, the trend in the simulated hanger tension after single-clamp slippage and the measured cable force is the same. The greater difference in the individual cable force can be attributed to the reallocation of the hanger tension after repeated cable clamp slippage during actual measurement. Nevertheless, the simulated hanger tension after the most severe clamp slippage coincides with the trend in the measured data. Hence, the simulation of the single-clamp slippage by the finite element model is accurate.

**Figure 9.** Cable forces after simulated slip of a single cable clip and measured hanger tension in 2021.

As shown in Figure 10, the overall cable force after the measured slip in 2021 is greater than the finite element model simulation result. The overall difference between the measured cable force and the simulated cable force is <10%, where the maximum difference in the measured and simulated cable forces after clamp slippage is 137.1 kN, and the simulated cable force is 7.26% lower than the measured cable force. The above comparison shows that the proposed finite element model yields the same result as the force state measured in 2021, indicating that the accuracy of the proposed suspension bridge model is reliable and can be used for further analysis and calculation.

**Figure 10.** Variations in the simulated and measured hanger tensions after the simulated slip of a single cable clamp in 2021.

### **4. Results of Finite Element Calculation**

In the analysis to clearly illustrate the impact of cable clamp slippage on the overall structure of the self-anchored suspension bridge, this study conducted a single-point slip analysis to simulate and analyze the impact on the structure. The most serious clamp slipping location on this bridge is selected for the slippage simulation of the single cable clamp.

### *4.1. Hanger Cable Force before and after Cable Clamp Slippage*

In the study of cable clamp slippage, it is found that the most direct result of slippage is the change in the magnitude of the cable force. As shown in Figure 11, the cable force before and after the slip of a single cable clamp is simulated using the finite element model, and it is found that the slippage decreases the cable force of this hanger, while the cable force of the two adjacent hangers relatively increases.

**Figure 11.** Diagram of hanger tension before and after the slip of a simulated single cable clip.

In combination with the cable force of the entire bridge hanger, the single-cable clamp slippage affects about two hanger positions at the front and rear, and the cable force of the hanger at the farthest distance is <1%. From the overall position, because of the stiffness of the main tower, the slippage in the middle span affects the cable force of the side span hanger by less than 1%.

Figure 12 focuses on the impact of single-cable clamp slippage on the cable force of adjacent hangers on the same side. As observed, the cable force at the slipping position is reduced by 392.6 kN, the hanger cable force drops by 19.2% compared with that before the slip, and the cable force of the four hangers before and after the slip is increased by 62.8, 123.3, 147.7, and 26.2 kN. Compared with pre-slip, the cable force growth rates are 2.45%, 6.19%, 7.25%, and 1.29%. The impact of the remaining hanger cable force is <1%. The single-clamp slippage affects other hanger cable forces. The impact is only in the range of the two hangers before and after.

**Figure 12.** Variation in the hanger tension before and after cable clip slippage.

### *4.2. Force Acting on the Main Girder before and after Cable Clamp Slippage*

The suspension bridge is a flexible structure, wherein the stiffening girder mainly provides the torsional stiffness and load acting surface and transmits the load to the hanger. The hanger connects the main cable and the stiffening girder and distributes the load from the stiffening girder to the main cable. Therefore, the change in the hanger tension directly affects the force state of the main girder, and the effect of single-clamp slippage on the main girder is discussed here.

As shown in Figure 13, the maximum variation in the bending moment at the singleclamp slipping position is 118 kN, and the variation rate is 1.28%. This figure also shows that the magnitude of the effect decreases with the increase in the slipping cable clamp position distance, and the effect of mid-span slippage on the side-span main girder bending moment is <1%.

**Figure 13.** Variation in the bending moment of the main girder.

After cable clamp slippage, because of the change in the cable force of the horizontal symmetric hanger, the force of the main girder is no longer on the same horizontal plane. As a result, the change in the cable force causes a local torsion effect on the stiffened girder. To clearly analyze the influence range of the torque change in the main girder, the main girder at the cable clamp slipping position is taken as the 0 point, and the distance is used to represent the influence range. The change in the internal force of the main girder after slipping was carefully studied, and it was found that the influence of the slippage in the middle span was only within the middle span, and the influence on the side span was low.

From Figure 14, the biggest change in the torque of the main girder after the single cable clamp slips is the main girder between the W1 tower and W2 tower, while the main girder torque change of the side span is smaller. From a local analysis, the slippage of the cable clamp only has an impact on the torque near the hanger, and the range of the four hangers is approximately its range of influence. Through distance analysis, the influence range of the single cable clamp on the main girder torque after slipping is 24 m in total. Among them, the slip of the single cable clamp has a greater impact on the torque of the main beam close to the W1 main tower, with an impact range of 16 m, and an impact range of 8 m in the mid-span direction.

**Figure 14.** Torque of the main girder.

As shown in Figure 15, cable clamp slippage has a significant impact on the local torque of the main girder. The torque after slippage is 3.5 times that before slippage, and the maximum torque change after slippage is 1785 kN. In terms of the long-term torque effect, this is unfavorable to the main girder in terms of the force.

**Figure 15.** Variations in the main girder torque before and after cable clamp slippage.

### *4.3. Forces Acting on the Main Cable before and after Cable Clamp Slippage*

The main cable is the main load-bearing member of the structural system and is a geometrically varying body, mainly subjected to tension. The main cable has a high initial tensile force under constant loading, providing "geometric stiffness" for the subsequent structure, not only through its own elastic deformation, but also through the geometry to affect the system equilibrium. Hence, cable clamp slippage will have a corresponding effect on the tensile force.

As shown in Figure 16, the maximum effect of single-clamp slippage on the main cable force is at the slippage location, while the maximum main girder force is 527.6 kN, a change of 1.21% before and after slippage. The maximum change is 1.21%. The remaining positions of the main cable force change is <1%, and the middle-span clamp slippage has a certain effect on it. The main tower stiffness is higher on the mid-span side of the main cable, and the force change on the side span side of the main cable tension is <1%.

**Figure 16.** Rate of change of main cable tension before and after cable clip slippage.

### *4.4. Forces in the Main Tower before and after Cable Clamp Slippage*

In a suspension bridge, the main tower is a compression-bending member. In the vertical direction, the main tower bears the vertical force of the main cable mainly through the constant load. In the longitudinal direction, the horizontal force of the main cable is balanced on both sides of the tower by the effect of the constant load, and no bending moment is generated on the tower. After cable clamp slippage, the unbalanced state of the load causes an unbalanced tension on both sides of the main tower, resulting in a horizontal displacement of the tower top. The single-clamp slippage at the mid span causes the main tower to produce a longitudinal top displacement of 0.242 mm.

After the horizontal displacement of the top of the tower, the horizontal force of the main cable on both sides of the main tower is rebalanced, the main tower is subjected to an unbalanced main cable tension, and the bending moment is changed. As shown in Figure 17, the trend in the main tower bending moment after slippage is consistent with that before slippage. Due to the single-clamp slippage, the maximum change value of the main tower is 0.0558 kN·m.

**Figure 17.** Bending moment diagram of main tower before and after cable clip slippage.

### *4.5. Suggestions for Handling Cable Clamp Slippage*

A force analysis of self-anchored suspension bridges after single-clamp slippage revealed that cable clamp slippage, which typically brings about a minor change, directly affects the force of each main component. Practically, cable clamp slippage is not a fixed state, but a state that continues to deteriorate with the operational phase of the bridge. As more clamps slip, the slipping distance gradually increases, and the structural forces of the bridge tend to develop unfavorably. If this problem is ignored, the safety of the bridge will be compromised. Therefore, if slipping of cable clamps in suspension bridges is detected during structural inspection, a detailed investigation should be conducted immediately, and for cable clamps with slippage <5 mm, measures, such as re-torqueing of the bolts, should be taken to restore the slip resistance of the clamps to prevent further slippage. For cable clamps with a slippage >5 mm, immediate restoration and adjustment should be performed to ensure structural safety [20].
