Active Fault-Tolerant Control to Prevent Hanger Bending During Configuration Transformation of 3D Cable System in Suspension Bridges

Abstract

Effectively preventing hanger bending damage during the configuration transformation of the spatial main cables in suspension bridges is a critical challenge, particularly under the influence of construction errors. This study proposes an active fault-tolerant control method that integrates real-time data feedback, tolerance interval inversion technology, and a suspended lateral bracing (SLB) system to mitigate the risk of hanger bending damage in real time. The method establishes a dynamic inversion mechanism, utilizing data feedback, constraint function reconstruction, and secondary optimization to compensate for construction errors. This ensures that hangers remain undamaged throughout the transformation process. Construction errors are quantified as intervals, with the lower bound of the reliability interval used to account for extreme disturbances. This transforms the inversion process into a multi-objective optimization problem constrained by the worst reliability conditions. By integrating finite element analysis (FEA), reliability analysis, surrogate modeling, and interval analysis, the proposed approach establishes a direct relationship between design variables and the lower bound of the reliability interval. Case study results demonstrate that the proposed method not only ensures structural performance and hanger safety, but also significantly enhances the constructability of the configuration transformation. Additionally, it provides a larger fault tolerance margin, thereby improving the overall efficiency and safety of the construction process.

1. Introduction

The rapid urbanization process in China has intensified the environmental and traffic-related repercussions of bridge construction, where associated indirect costs frequently surpass direct construction expenses [1,2]. To address these challenges, the bridge engineering sector has increasingly embraced industrialized approaches centered on factory-based prefabrication and on-site modular assembly [3,4,5]. This methodology effectively shortens on-site construction duration while alleviating socioeconomic disruptions. While these methods offer efficiency, they rely heavily on precise segmental assembly, where any failure in connections leads to costly rework and delays. Among various bridge types, self-anchored suspension (SAS) bridges are increasingly employed as iconic urban structures. This preference arises from their harmonious blend of architectural elegance and material-efficient design, which fulfills both esthetic and functional demands of modern infrastructure [6,7,8]. However, the construction of SAS bridges poses considerable challenges, especially during the spatial transformation of the main cable configuration.

One of the most critical issues in SAS bridge construction is preventing hanger bending damage during the spatial configuration transformation of the main cable [9]. The tensioning process of hangers involves complex interactions between the main cable, tower, and the girder system. Since the hangers are pre-fabricated and installed before the cables reach their final tensioned state, they are highly susceptible to angle deviation and bending damage. Furthermore, discrepancies between the free-hanging cable state (FHC) and fully-loaded cable state (FLC) exacerbate the risks associated with hanger installation. These risks are further compounded by construction errors and environmental factors, which can cause unpredictable disturbances during the process [10,11].

Addressing these challenges requires effective assembly fault tolerance, which is essential for maintaining structural safety. However, practical engineering applications face constraints such as time, technology, and investment. Leveraging the self-shape-finding and rebalancing properties of flexible cable systems can enhance hanger tolerance and improve adaptability to construction errors, reducing scheduling and cost pressures. However, fault tolerance in assembly involves complex, interrelated factors—such as structural reliability, cable shape optimization, and local damage resistance—that are difficult to isolate and address using conventional methods [12].

While existing methods, such as interval optimization based on reliability analysis, offer potential solutions, they have limitations [13,14]. Jiang et al. [15] proposed an interval optimization method considering tolerance design, which can determine the optimal design scheme and the maximum allowable manufacturing error range. Wang et al. [16] focused on the industrialized construction of SAS bridges and proposed an inverse analysis method for assembly fault-tolerant intervals based on the geometric nonlinearity redundancy of cable systems, which guides the fault tolerance range of hanger tensioning. Existing methods typically rely on passive fault-tolerant mechanisms, which address errors after they occur [17]. However, these methods lack adaptability to real-time disturbances and fail to account for unforeseen issues that may arise during construction. To address these limitations, there is a growing need for an active fault-tolerant control method that can dynamically monitor and compensate for construction errors as they occur.

Notably, research in the aviation and mechanical engineering fields provides valuable insights for this approach [18,19,20]. For example, Mao et al. [21] proposed an online active fault-tolerant control method for sensor faults in morphing aircraft, demonstrating the effectiveness of dynamic fault compensation in real-time control processes. Similarly, Amin A. A. et al. [22] developed a robust active fault-tolerant control method for internal combustion engines which employs fault detection and reconstruction techniques to enhance system reliability.

This study proposes an active fault-tolerant control method to mitigate the risk of hanger bending damage during the configuration transformation of SAS bridges. The proposed method integrates real-time data feedback, tolerance interval inversion technology, and an SLB system. By establishing a dynamic inversion mechanism based on data feedback, constraint function reconstruction, and secondary optimization, the method actively compensates for construction errors. Specifically, construction errors are quantified as intervals, and the lower bound of the reliability interval is used to characterize extreme disturbances, thereby establishing a multi-objective optimization model constrained by the worst reliability conditions. The proposed method is validated through a case study on the Baheyuanshuo Bay Bridge, demonstrating its effectiveness in preventing hanger bending damage while improving the overall constructability of the SAS bridge configuration transformation.

2. Hanger Bending Damage and Suspended Lateral Bracing

2.1. Hanger Bending Damage During the Configuration Transformation

During the spatial configuration transformation of the main cable in a self-anchored suspension bridge, there is a significant lateral position difference between the main cable in the FHC and FLC. The tensioning process of hangers involves complex interactions between the main cable, tower, and girder system. As the main cable and hangers are progressively tensioned, the cable and cable clamps undergo longitudinal, vertical, and lateral displacements [8]. However, since the installation angles of the cable clamps and conduits are designed based on the FLC, the tension line between the upper and lower anchor points of the hangers deviates from these angles during construction. This deviation can cause angle deviation between the hanger and the cable conduit, which, when exceeding a certain threshold, results in severe bending damage to the hangers at the cable clamps or the anchorage box conduits. This issue represents a critical challenge during the configuration transformation process of self-anchored suspension bridges, particularly in the early stages of tensioning when the main cable is still adjusting to its final configuration. This section analyzes the causes of hanger bending during the classic “Erect Cable Before Girder (ECBG)” and “Erect Cable After Girder (ECAG)” construction methods.

In the ECAG method, the main girder is typically installed using temporary supports. After the girder is erected, the main cables are installed and the hangers are tensioned, completing the configuration transformation. As shown in Figure 1a, during the configuration transformation, the position of the main girder and its cable conduit remain fixed. The hangers are temporarily pulled using an extension rod during tensioning, and the extension rod is removed once the configuration transformation is completed. However, due to the lateral position difference between the FHC and FLC, significant lateral angle deviation occurs during initial hanger tensioning. This angle deviation may prevent the extension rod from properly inserting the hangers into the conduits, causing hanger bending damage at the cable clamps or anchorage box conduits.

In the ECBG method, the main cables are installed first, followed by the hoisting of the main girder and its connection to the hangers. During this process, the position of the cable conduits moves with the main girder. Since the extension rod cannot be used in this method, the girder must be raised until the lower ends of the hangers can connect to the cable conduits. As shown in Figure 1b, as the main girder is raised, the lateral angle deviation between the upper and lower anchor points of the hangers increases. This can cause severe bending damage, particularly for the shorter hangers located in the mid-span, where the length of the hangers may be insufficient to reach the cable conduits.

In the FLC, d and L represent the vertical and horizontal distances between the hanger’s upper and lower anchorages, respectively. The vertical and horizontal displacements of the upper anchorage in the FHC relative to the FLC are denoted by ΔY and ΔZ, while L_d corresponds to the hanger’s unstressed length. The lateral deflection angle Δθ quantifies the transverse angle deviation during construction.

Furthermore, as the hangers are progressively tensioned, the tower experiences longitudinal displacement, and the cable clamps move longitudinally with the main cable. This movement results in the development of a longitudinal tilt angle between the upper and lower anchor points of the hangers, as shown in Figure 2. The tilt angle increases as the hanger length decreases, and short hangers in both the side span and the main span experience significant longitudinal tilt. This tilt angle may exceed the allowable limits of the cable conduit in the main girder, leading to longitudinal bending damage of the hangers, especially at shorter hangers.

2.2. Suspended Lateral Bracing for Configuration Transformation

To address the issue of hanger bending during the configuration transformation, this paper proposes an SLB system. By applying tension to the SLB, the lateral displacement of the main cable in the FHC can be controlled, simplifying the hanger tensioning process and making it more similar to that of traditional planar cable suspension bridges. As shown in Figure 3, the SLB system consists of several key components: the main trusses, wire ropes, remote-controlled winches, a vertical tensioning system, and a lateral stabilizing system. The main trusses serve as axial elements that counteract the lateral forces exerted by the main cable, while the winches and wire ropes are used to position the main cable according to design specifications. The combined action of the vertical tensioning system and the lateral stabilizing system ensures that the temporary structure remains stable during construction. Once the wire ropes are tensioned and the main cable reaches the desired position, the winches are locked in place, effectively reducing the risk of hanger buckling during installation.

The SLB system functions similarly to a spring, offering a flexible restraint for the main cable. This flexibility allows the cable to undergo controlled lateral displacement under sudden loads. After the load is removed, the system automatically returns to its original position, preventing brittle failure at the connection points between the cable and SLB. This dynamic response mechanism significantly enhances the adaptability and stability of the construction process, providing additional fault tolerance during the configuration transformation of the main cable. By employing the SLB system, the construction process becomes more flexible, and the risk of hanger bending due to angle deviation during the configuration transformation is minimized. The system’s ability to adapt to dynamic load changes ensures the stability and safety of the bridge during this critical stage.

3. Active Fault-Tolerant Control Method for Configuration Transformation

3.1. Framework

This section proposes an active fault-tolerant control framework for dynamically adjusting the tension of hangers and the SLB during the main cable spatial configuration transformation. The framework focuses on dynamically adjusting the fault-tolerant inversion model based on data feedback from each construction stage. This results in a dynamic optimization mechanism that incorporates data feedback, verification, and secondary optimization. The framework is highly effective in addressing assembly failures, improving fault tolerance, and preventing hanger bending damage, ultimately enhancing the constructability of the construction process. The proposed framework is shown in Figure 4, and the key steps in the proposed framework are as follows:

Step 1: Determine the initial design space of optimization parameters based on engineering experience and fault tolerance requirements defined by the construction team. This step sets the boundaries within which the construction parameters can vary.

Step 2: Use Latin Hypercube Sampling (LHS) [23] to extract a set of input samples within the design space. These samples are then used to compute the corresponding structural responses using the FEA method, which provides a comprehensive understanding of how the structure will behave under different conditions.

Step 3: Establish a fast mapping relationship between the optimization variables and multiple performance objectives using an artificial rabbit optimization–backpropagation neural network (ARO-BP) surrogate model [24,25]. The worst reliability prediction method from Section 3.3 is applied to formulate the worst reliability constraint function and the worst hanger angle deviation constraint. This step effectively decouples the reliability calculation from the multi-objective optimization process.

Step 4: Apply the NSGA-II [26] multi-objective optimization algorithm to solve the fault-tolerant inversion model for each construction stage. The Pareto solution set is generated for the current stage, and the solution with the highest fault tolerance is selected for guiding construction. Nominal values are used for subsequent construction parameters until real-time feedback is obtained.

Step 5: Perform construction quality testing for the i-th construction stage and provide feedback based on the measured data. If the measured data d_i^* fall within the fault-tolerant range, the assembly is considered successful, and the process moves to Step 6. If the measured data fall outside the fault-tolerant range, rework is required, and the stage undergoes re-testing.

Step 6: Based on the measured values from the i-th construction stage, reconstruct the worst hanger angle deviation constraint for the next stage (i + 1) and repeat Steps 3–5 to complete fault-tolerant analysis for subsequent construction stages.

3.2. Fault-Tolerant Interval Inversion Method

This section proposes an interval inversion model suitable for active fault-tolerant control during the configuration transformation. The method is based on reliability-based design optimization (RBDO), aiming to enlarge the tolerance range of controllable parameters while ensuring that structural reliability, design optimality, system redundancy, and local damage resistance criteria are all satisfied. To evaluate the tolerance range of parameters, we introduce a dimensionless indicator W. This indicator reflects the constructability of the main cable configuration transformation process. The tolerance range of the controllable parameters is directly proportional to W, meaning that a larger value of W results in a wider tolerance range for the parameters, ultimately enhancing the feasibility of the construction process. However, as the tolerance range increases, the design objective optimality (represented by performance indicators such as hanger tension uniformity and main cable shape accuracy) may slightly decrease. Therefore, a trade-off is necessary between constructability and design optimality. The proposed interval inversion model is shown as follows:

$$\begin{array}{ll}find& D^I={[d_1^I{,d}_2^I\dots {,d}_n^I]}^T,i=1,2,\dots n\\ obj.& \mathit{min}{f}_j\left(D^I\right),j=1,2,\dots ,m\\ & \mathit{min}\left(-W\right)\\ s.t.& \beta (G_k(X,D^I))\ge {\beta }_k^T,k=1,2,\dots ,r\\ & D_l\le D^I\le D_u\end{array}$$

(1)

$$W=\sqrt[n]{{\prod }_{k=1}^n\left(d_k^w/d_k^c\right)}$$

(2)

In the proposed interval inversion model, the assembly controllable parameters are represented as a set of interval variables D^I, with each parameter having a lower bound D_l and an upper bound D_u. These bounds are determined based on the tolerance requirements and the design space. X = [x₁, x₂, …, x_m]^T is the m-dimensional vector of random variables accounting for stochastic uncertainties. Here, G_k is the limit state function; β and β^T are the corresponding reliability indexes and reliability constraint thresholds, respectively. For the k-th tolerance parameter d_kⁱ, the parameters d_k^w and d_k^c refer to the midpoint and radius of the tolerance interval, respectively. The index f_j represents the performance of the inversion model. After introducing interval uncertainty, the reliability index fluctuates within the range [β^L, β^R]. The model is solved iteratively, with each step determining the optimal values for the midpoint and radius of the tolerance intervals of the controllable parameters, ensuring that the system operates within the required reliability and safety margins.

3.3. Worst Reliability Prediction Method

The inversion model described in Section 3.2 is essentially a multi-level nested optimization process, which includes multi-objective optimization, worst reliability assessment, reliability calculation, and structural response analysis. The core challenge in decoupling this nested optimization process is efficiently predicting the worst reliability index, β^L, under probability-interval hybrid uncertainty. This section introduces a method for worst reliability prediction based on the ARO-BP surrogate model, which is designed to provide a fast and accurate estimation of the worst reliability index in the presence of interval uncertainty. The FEA model is used for nonlinear structural response analysis during the construction process, and the ARO-BP surrogate model is employed to express the direct mapping relationship between design variables and the worst reliability index. This surrogate model simplifies the calculation of β^L and helps decouple the complex nested optimization process. The procedure for the proposed worst reliability prediction method is shown in Figure 5. The key steps are as follows:

(1)

Structural response surrogate: Input samples are obtained through the LHS method, and corresponding structural responses are computed using FEA. These samples are then used to establish the ARO-BP surrogate model for the limit state function based on the required input–output sample set.

(2)

Reliability surrogate: Reliability calculations are performed using the First-Order Reliability Method (FORM). A surrogate model is created to represent the relationship between input samples and the reliability index. The explicit function for the reliability index β(U,D) is extracted, allowing for efficient reliability analysis. Here, U = [u₁, u₂, …, u_m]^T is the m-dimensional standard normal vector obtained by normalizing the random variable X.

(3)

Worst reliability calculation: The interval variable D^I is introduced, and the reliability boundaries are calculated using the interval analysis method described in Equation (3) [27], ultimately obtaining the worst reliability index β^L.

$$\left\{\begin{array}{l}{\beta }^L({X,D}^I)=\beta ({X,D}^c)-{\displaystyle \sum _{j=1}^n\left|\frac{\partial \beta ({X,d}^c)}{\partial d_j}\right|}{d}_j^w\\ {\beta }^R({X,D}^I)=\beta ({X,D}^c)+{\displaystyle \sum _{j=1}^n\left|\frac{\partial \beta ({X,d}^c)}{\partial d_j}\right|}{d}_j^w\end{array}\right.$$

(3)

(4)

Worst reliability surrogate: The ARO-BP model is used to build a surrogate model between the interval variable D^I and the worst reliability index β^L. This surrogate model enables rapid prediction of β^L for any given set of input variables, thus decoupling the multi-level nested optimization process.

3.4. Multi-Objective Optimizer: NSGA-II

NSGA-II is considered one of the most efficient methods for multi-objective optimization [26]. Compared to its predecessor, NSGA, NSGA-II offers three key advantages: ① NSGA-II employs a fast non-dominated sorting algorithm based on classification, which reduces computational complexity from O(mN³) to O(mN²). This improvement accelerates the optimization process, making it highly efficient for large-scale problems. ② NSGA-II introduces the concept of crowding distance, which eliminates the need to specify a sharing radius as required in traditional fitness sharing strategies. The crowding distance comparison operator allows for a more effective selection of diverse solutions, improving the algorithm’s robustness. ③ NSGA-II integrates elitist strategies, which ensure that the best individuals from the current generation are always retained in the next generation. This feature helps prevent the loss of high-quality solutions during the optimization process.

These improvements make NSGA-II highly efficient for solving multi-objective optimization problems, especially in complex engineering applications where multiple conflicting objectives must be balanced. By using NSGA-II, the proposed optimization framework can effectively explore the trade-offs between design objectives while maintaining structural integrity and reliability.

4. Case Study

4.1. Engineering Description

The Baheyuanshuo Bay Bridge (Figure 6a), a symmetric double-pylon self-anchored suspension bridge with a span layout of 50 m + 116 m + 300 m + 116 m + 50 m, is located in Xian, China. The bridge is 56 m wide with spindle-shaped pylons and has 49 pairs of hangers arranged throughout its span. The lateral distance between the IP nodes of the main cables is 0.5 m, while that of the anchored nodes of the hangers is 45.5 m.

The detailed finite element model was developed using the analysis software ANSYS (ANSYS 17.0), accounting for nonlinear effects such as large displacement and the sag effect of the structure. In this model, 380 spatial beam elements are used to represent the pylon–girder system, and 226 spatial elastic catenary cable elements are used to model the cable–hanger system. The key modeling details are shown in Figure 6b, with the coordinate system origin placed along the road centerline.

The bridge is constructed using the ECAG method, where the prefabricated girder is first assembled to form a temporary continuous beam system with multiple points of support. After the girder is erected, the main cables are installed, and the configuration transformation begins. The key construction procedures for this process are shown in Figure 7.

4.2. Inversion Problem Description

As shown in Figure 6, hangers H1, H2, H22~H28, H48, and H49 are considered short hangers. These short hangers are primarily located in critical regions of the main cable, where their tension deviations can significantly affect the final shape of the bridge and its structural performance. Short hangers are more sensitive to tension measurement errors due to their relatively large bending stiffness, making it more difficult to guarantee precise control during the construction process compared to longer hangers. As a result, tensioning these short hangers requires careful monitoring and adjustment to ensure the correct final configuration.

Additionally, since the SLB provides a flexible constraint to the main cable, its tension directly affects the lateral position of the main cable during configuration transformation. Consequently, the permissible tension range of the SLB also significantly impacts the constructability and stability of the bridge. Given these factors, the tension range of the short hangers and the SLB is treated as adjustable assembly parameters, with the midpoint and radius of their respective intervals serving as optimization variables. The formulation of the fault-tolerant interval inversion model is described in Equation (4).

$$\begin{array}{ll}find& D^I={[d_1^I{,d}_2^I\dots {,d}_n^I]}^T,d_i^I=[d_i^c-d_i^w,d_i^c+d_i^w],i=1,2,\dots n\\ obj.& \stackrel{\wedge }{f_1}=\mathit{min}\{\Delta Z^c(D^I)\}\\ & f_2=min\left\{{\sum }_{k=1}^n(|d_k^c-d_k^{*}|+\mathit{max}(|d_k^c-d_k^w-d_k^{*}|,|d_k^c+d_k^w-d_k^{*}|))\right\}\\ & f_3=\mathit{min}\{-W\}\\ s.t.& \stackrel{\wedge }{g_1}=\stackrel{\wedge }{{\beta }^L}(G(X{,D}^I))\ge {\beta }^T\\ & \stackrel{\wedge }{g_2}=\mathit{max}\left|\theta (D^I)\right|\le {\theta }^u\\ & \stackrel{\wedge }{g_3}=\mathit{max}\left|v(D^I)\right|\le v^u\\ & d_k^c\in [95\%d_k^{*},105\%d_k^{*}];d_k^w\in [0,15{\%d}_k^{*}]\end{array}$$

(4)

The inversion model includes three objective functions and three constraint functions which aim to balance the trade-offs between constructability, design optimality, and reliability. The objective functions are as follows:

(1)

Objective function f₁: Minimize the ΔZ (namely the deviation between the main cable shape at the completion stage and the design target), thus achieving the optimality of the design target.

(2)

Objective function f₂: Minimize the deviation between the tension of the hangers at the completion stage and the design tensions, thus ensuring tension uniformity. d* is the tension in the hangers given in the original design scheme.

(3)

Objective function f₃: Maximize the constructability of the configuration transformation process, considering the feasibility of achieving the design target with minimal rework.

The optimization process is constrained by three key factors:

(1)

Constraint function g₁: The asymmetric tensioning of the hangers during mid-span and the uncertainties arising from construction errors can lead to an unbalanced force in the main cable. If the unbalanced force exceeds a predefined limit, it may cause local damage to the bridge tower. This constraint is designed to ensure that the longitudinal displacement of the tower top remains within acceptable limits, with the corresponding limit state function described as G(X) = a_max − a(X). And a_max is the allowed maximal longitudinal displacement of the bridge tower.

(2)

Constraint function g₂: During configuration transformation, both the upper and lower anchor points of the hangers experience lateral and longitudinal angular deviations. To prevent bending damage, the maximum angular deviations θ(D^I) of the already tensioned and untensioned hangers at each stage must be controlled within the specified limits θ^u.

(3)

Constraint function g₃: The main girder’s performance is constrained by limiting its maximum vertical displacement to an acceptable value, ensuring that the girder’s movement v(D^I) does not exceed structural safety limits v^u.

In the inversion model described above, the design space of the optimization variables is shown in

Table A1. Design space of optimization variables (unit: kN).

Optimization Variables	Lower Bound	Upper Bound	Optimization Variables	Lower Bound	Upper Bound
d1c (H1 and H49)	1889.8	2834.6	d20c (H20 and H30)	2220.7	2454.4
d2c (H2 and H48)	2035.3	3052.9	d21c (H21 and H29)	2229.0	2463.7
d3c (H3 and H47)	2377.3	2627.5	d22c (H22 and H28)	2232.8	2467.8
d4c (H4 and H46)	2417.1	2671.5	d23c (H23 and H27)	2221.9	2455.8
d5c (H5 and H45)	2439.2	2696.0	d24c (H24 and H26)	2230.2	2465.0
d6c (H6 and H44)	2441.3	2698.3	d25c (H25)	2247.0	2483.5
d7c (H7 and H43)	2386.2	2637.4	d26 (SLB-1 and SLB-5)	213.8	236.3
d8c (H8 and H42)	2215.6	2448.8	d27 (SLB-2 and SLB-4)	266.0	294.0
d9c (H9 and H41)	2215.6	2448.8	d28(SLB-3)	223.3	246.8
d10c (H10 and H40)	2088.5	2308.4	d1w (H1 and H49)	0.0	354.3
d11c (H11 and H39)	2227.3	2461.8	d2w (H2 and H48)	0.0	381.6
d12c (H12 and H38)	2212.1	2445.0	d22w (H22 and H28)	0.0	352.5
d13c (H13 and H37)	2151.8	2378.3	d23w (H23 and H27)	0.0	350.8
d14c (H14 and H36)	2160.9	2388.3	d24w (H24 and H26)	0.0	352.1
d15c (H15 and H35)	2218.6	2452.1	d25w (H25)	0.0	354.8
d16c (H16 and H34)	2211.1	2443.8	d26w (SLB-1 and SLB-5)	0.0	33.8
d17c (H17 and H33)	2211.1	2443.8	d27w (SLB-2 and SLB-4)	0.0	42.0
d18c (H18 and H32)	2214.4	2447.4	d28w (SLB-3)	0.0	35.3
d19c (H19 and H31)	2217.0	2450.4

Remarks: d_k^c is the interval center of the k-th hanger tension, and d_k^w represents the fault-tolerant radius of the k-th short hanger tension.

. In each iteration, the interval midpoint and radius are uniformly sampled within the design space to form a new interval variable. The worst reliability under the influence of the interval variables is calculated using the worst reliability prediction method described in Section 3.3. The probability distribution characteristics of the random variables are shown in

Table A2. Stochastic characteristics of random variables (unit: kN).

Variables	Mean Value	Standard Deviation	Distribution Type	Variables	Mean Value	Standard Deviation	Distribution Type
x1 (H1 and H49)	2362.2	157.5	Normal	x15 (H15 and H35)	2335.3	38.9	Normal
x2 (H2 and H48)	2544.1	169.6	Normal	x16 (H16 and H34)	2327.4	38.8	Normal
x3 (H3 and H47)	2502.4	41.7	Normal	x17 (H17 and H33)	2327.4	38.8	Normal
x4 (H4 and H46)	2544.3	42.4	Normal	x18 (H18 and H32)	2330.9	38.8	Normal
x5 (H5 and H45)	2567.6	42.8	Normal	x19 (H19 and H31)	2333.7	38.9	Normal
x6 (H6 and H44)	2569.8	42.8	Normal	x20 (H20 and H30)	2337.6	39.0	Normal
x7 (H7 and H43)	2511.8	41.9	Normal	x21 (H21 and H29)	2346.4	39.1	Normal
x8 (H8 and H42)	2332.2	38.9	Normal	x22 (H22 and H28)	2350.3	39.2	Normal
x9 (H9 and H41)	2332.2	38.9	Normal	x23 (H23 and H27)	2338.8	155.9	Normal
x10 (H10 and H40)	2198.4	36.6	Normal	x24 (H24 and H26)	2347.6	156.5	Normal
x11 (H11 and H39)	2344.6	39.1	Normal	x25 (H25)	2365.2	157.7	Normal
x12 (H12 and H38)	2328.5	38.8	Normal	X26 (SLB-1 and SLB-5)	225.0	15.0	Normal
x13 (H13 and H37)	2265.0	37.8	Normal	X27 (SLB-2 and SLB-4)	280.0	18.7	Normal
x14 (H14 and H36)	2274.6	37.9	Normal	X28 (SLB-3)	235.0	15.7	Normal

, with all variables following a normal distribution.

4.3. Precision Verification of the Surrogate Model

In the interval inversion model constructed in Section 4.2, the objective function f₁ and three constraint functions are fitted using the ARO-BP surrogate model, where f₁, g_2, and g₃ represent the worst structural responses, and g₁ is the worst reliability constraint. While the surrogate model enables hierarchical decoupling and improves the inversion efficiency, it inevitably introduces error accumulation. To ensure the reliability of the inversion process, the priority is to assess the fitting accuracy of the surrogate model.

To evaluate the model’s precision, the goodness of fit for several key performance indices is calculated. As shown in Figure 8, the goodness of fit for the maximum displacement of the main cable max(ΔZ) is 0.98289, with a maximum error of 0.96%; the goodness of fit for the maximum displacement of the girder max(v) is 0.99377, with a maximum error of 0.44%; the goodness of fit for the worst reliability β₁^L of the main tower is 0.99974, with a maximum error of 0.186%; the goodness of fit for the worst lateral and vertical deflection angles θ^L of the short hangers is 0.99791, with a maximum error of 2.01%. In conclusion, the surrogate models involved in the inversion model exhibit high prediction accuracy, meeting the engineering requirements.

4.4. Active Fault-Tolerant Inversion Result Analysis

With the aid of the surrogate model, the fault-tolerant interval inversion model is solved by NSGA-II, with the following parameter settings: population size = 500, maximum number of generations = 200, crossover probability = 0.8, and mutation probability = 0.01. Testing has demonstrated that these parameter settings provide good robustness in finding the Pareto front.

This section presents the results of the active fault-tolerant inversion analysis for the configuration transformation process. To improve assembly fault tolerance, the Pareto solution with the largest fault tolerance for each construction stage is selected. This solution guides the construction process for that stage, with subsequent construction parameters initially set to nominal values. As real-time data become available from the current stage, these data are incorporated into the fault-tolerant inversion model for the next stage, and the optimization model is updated accordingly. This iterative process continues throughout the construction, ensuring that the construction parameters are continuously adjusted to account for construction errors and environmental disturbances.

The Pareto front of the optimization results for each construction stage is shown in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, with a selection of Pareto solutions presented in

Table A3. Partial pareto solution.

Construction Stage	Solution No.	f1	f2	−W	Construction Stage	Solution No.	f1	f2	−W
Stage 1	1	−10.769	0.000	−0.1326	Stage 5	1	−11.940	1994.280	−0.1324
	2	−10.858	0.000	−0.1109		2	−11.820	1702.010	−0.1056
	3	−10.941	0.000	−0.0844		3	−11.677	1506.725	−0.0802
	4	−11.003	0.000	−0.0580		4	−11.613	1425.754	−0.0550
	5	−11.043	0.000	−0.0293		5	−11.595	1409.779	−0.0250
Stage 2	1	−11.843	352.462	−0.1222	Stage 6	1	−11.947	0.000	−0.1304
	2	−11.708	296.428	−0.1128		2	−11.974	0.000	−0.1188
	3	−11.372	190.597	−0.0974		3	−11.979	0.000	−0.1123
	4	−10.943	59.154	−0.0581		4	−11.981	0.000	−0.1076
	5	−10.692	1.105	−0.0278		5	−11.985	0.000	−0.0967
Stage 3	1	−11.881	1287.806	−0.1386	Stage 7	1	−12.142	2539.347	−0.1403
	2	−11.954	1223.215	−0.1242		2	−12.102	2117.881	−0.1150
	3	−12.040	1279.491	−0.1125		3	−11.980	1820.226	−0.0992
	4	−12.021	1196.422	−0.1006		4	−11.961	1683.205	−0.0694
	5	−11.945	946.366	−0.0888		5	−11.841	1645.455	−0.0294
Stage 4	1	−12.048	1758.241	−0.1279	Stage 8	1	−11.860	2524.266	−0.1462
	2	−12.042	1413.549	−0.1107		2	−11.895	2222.702	−0.1211
	3	−12.024	1257.915	−0.0805		3	−11.988	2098.512	−0.0939
	4	−11.968	1196.214	−0.0531		4	−12.055	2032.339	−0.0686
	5	−11.889	1176.865	−0.0271		5	−12.191	2004.081	−0.0328

. In these stages, stage 1 refers to the construction using only the SLB device for the configuration transformation of the main cable, while stage 6 refers to the construction of the long hangers. These two stages do not involve the inversion of the short hangers, so only two objective functions are considered in these stages.

The partial Pareto solution in Table A3 clearly demonstrates the trade-offs between design objective optimality and tolerance capability. Under the constraint of structural reliability, the value of W is directly proportional to the tolerance range of the assembly controllable parameters. Increasing W helps to improve the feasibility of configuration transformation and speeds up construction. However, as the tolerance range increases, there is a slight sacrifice in the optimality of design objectives, as indicated by the combined effects of f₁ and f₂. From the table, it is evident that increasing W leads to a rise in f₂, suggesting a reduction in hanger tension uniformity. Therefore, decision-makers must carefully balance tolerance capability and design objective optimality based on the specific needs of the project.

For each construction stage, the solution with the maximum W value is selected as the construction plan for that stage. Interval inversion for each stage is performed using the active fault-tolerant control method, resulting in the determination of the active fault-tolerant parameters for the entire configuration transformation process. These parameters are updated and refined in each stage, ensuring that the construction process remains within the necessary safety and performance margins. The final active fault-tolerant results for the entire construction process are shown in Figure 17, which provides a comprehensive overview of the adjustments made to the construction parameters at each stage.

Furthermore, the maximum angular deviation of the hangers, calculated at each stage, is presented in Figure 18. For example, in stage 3, 300 sample points were randomly selected within the allowable tolerance range for the tension parameters, and corresponding structural responses were calculated using FEA. After calculating the angular deviations for hangers H25–H24 in stage 3 and H23 in stage 4, the maximum angular deviation was obtained, as shown in Figure 18c. The results show that the maximum angular deviations in all stages are less than the allowable angular tolerance of 6° for the hanger ducts, demonstrating the effectiveness of the active fault-tolerant control method in preventing hanger bending damage.

Additionally, it is worth noting that the objective function f₂ aims to minimize the deviation between the hanger tensions at the completion stage and the design tensions. However, due to the deviations in hanger tensions obtained from the active fault-tolerant inversion, it is necessary to assess the girder. Based on the final hanger tensions obtained from the active fault-tolerant analysis, random values within a ±5% deviation range of the tension are selected, and the finite element analysis model is applied for solving. The vertical displacement fluctuations at the girder control point and the main cable control point are shown in Figure 19 and Figure 20, respectively. The vertical displacement fluctuation range at the girder control point is [−16.23 mm, −14.01 mm], and at the main cable control point, it is [−12.95 mm, −10.75 mm]. Both ranges meet the engineering practice requirements.

5. Conclusions

This paper addresses the problem of hanger bending damage during the main cable spatial configuration transformation process by proposing an active fault-tolerant control method, which is verified through a case study. The main conclusions are as follows:

(1)

The proposed active fault-tolerant control method effectively addresses construction errors by establishing a dynamic inversion mechanism that combines data feedback, constraint function reconstruction, and secondary optimization. This method actively compensates for construction errors, ensuring that hangers remain undamaged throughout the configuration transformation process. The case study results show that the maximum angle deviation of hangers at each construction stage is less than 6°, effectively preventing hanger bending damage.

(2)

The flexible adjustability of the SLB system aligns well with suspension bridges, enhancing the fault tolerance of the construction process. This provides a solid physical foundation for the active fault-tolerant control method. By providing additional support to the main cable, the SLB system helps reduce the risk of hanger bending, ensuring both the safety and constructability of the project.

(3)

By integrating the FEA, reliability analysis, surrogate models, and interval analysis methods, this paper establishes a multi-objective optimization model based on the worst reliability constraint. This model ensures structural performance and hanger safety, while also achieving the inversion of the maximum fault tolerance interval for controllable parameters at each construction stage, thereby reducing delays caused by rework.

(4)

Author Contributions

Conceptualization, Y.B. and X.W.; methodology, Y.B. and X.W.; software, Y.B.; validation, Y.B.; writing—original draft preparation, Y.B.; writing—review and editing, Y.B., X.W., Z.Z. and H.W.; visualization, Y.B.; supervision, X.W., Z.Z. and H.W.; funding acquisition, X.W. All authors have read and agreed to the published version of the manuscript.