**1. Introduction**

Stay cables, as significant components of cable-stayed bridges, carry and transmit loads, and their service conditions directly impact the safety of bridge operation. The combined impact of load and environment can accelerate the corrosion degradation of cable systems, resulting in failure, jeopardizing the safety of bridge operation, and incurring considerable economic losses and safety mishaps [1–3].

Scholars have conducted relevant experimental research to uncover the corrosion mechanism of cables in corrosive settings, delay the corrosion rate of cables, and increase the durability of cables. In terms of cable damage and failure, the cable tension plays an increasingly important role in the cable-stayed bridge system and affects the feasibility of optimizing and updating this type of bridge [4]. Greco et al. [5] conducted a nonlinear analysis of bridge cables based on the characteristic parameters of the bridge structure, taking into account the dynamic amplification effect and failure mechanism of the cables under dynamic loads. Ammendolea et al. [6] reproduced the damage law of cables under the coupling effect of the bridge and dynamic load based on the theory of continuous damage mechanics. Mozos [7,8] analyzed 10 cable-stayed bridges and investigated the effects

**Citation:** Li, S.; Yao, G.; Wang, W.; Yu, X.; He, X.; Ran, C.; Long, H. Research on the Diffusion Model of Cable Corrosion Factors Based on Optimized BP Neural Network Algorithm. *Buildings* **2023**, *13*, 1485. https://doi.org/10.3390/ buildings13061485

Academic Editor: Fabrizio Greco

Received: 8 May 2023 Revised: 29 May 2023 Accepted: 7 June 2023 Published: 8 June 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

of cable layout, single and double cable planes, and main beam cross-section on the ultimate failure state of the cables, laying the foundation for the evaluation of the safe service state and long-term performance of the cables. In terms of corrosion fatigue, Yan et al. [9] examined the corrosion of steel wires in service. They discovered that lower anchoring section steel wires were more susceptible to corrosion than higher ones. Stewart et al. [10] conducted tests to investigate the corrosion process of steel wires, and the results revealed that the corrosion form of steel wires is connected to their environment. Through accelerated corrosion testing on parallel steel wires, Rou [11] presented a positive and negative electrode chemical reaction formula for hydrogen evolution and oxygen absorption corrosion. Changqing [12] conducted a corresponding exploration of the law of waste steel wire corrosion based on actual engineering. Betti [13] investigated the corrosion damage mechanism under different environmental corrosion by simulating salt spray corrosion experiments of high-strength galvanized steel wires in acid rain. Furuya et al. [14] conducted atmospheric exposure studies on cable segments in natural environments. The study revealed that humidity and temperature are the primary causes of the deterioration of the cable interior environment. Suzumura [15,16] conducted accelerated corrosion tests on high-strength galvanized steel wires in different environments and comparatively analyzed the effects of temperature, relative humidity, and NaCl solution concentration on corrosion rate. Furthermore, the galvanized layer has variable degrees of effect on corrosion parameters such as corrosion potential and polarization resistance of steel wire [17–20]. Heying [21] discovered that corroded steel wires' elongation and fatigue strength dramatically decrease after corrosion, with the decline of elongation occurring primarily in the latter stages of corrosion [22]. Hamilton [23,24] used seawater as the corrosion media in accelerated corrosion studies on defective cable-stayed cables under static tension. On this basis, researchers investigated the damaging effect of corrosion on steel wires under environmental load coupling using experiments and numerical simulations, as well as analyzed the law of mechanical properties degradation and damage evolution of corroded steel wires [25–29]. Similarly, Rosso et al. [30] compared the model with the half-joint in the actual project and analyzed the degradation mechanism of the mechanical and physical properties of the half-joint of the bridge under different corrosion levels by simulating the corrosion development process.

With the development of computer technology, the machine learning (ML) method, as a relatively advanced data processing approach, has become widely applied in the practical engineering of related bridges. Xin et al. [31–33] used the machine learning method to identify and process the deformation monitoring data of the bridge more efficiently, and reliably laid the foundation for the early warning of bridge deformation. Kim et al. [34] investigated the diffusion of the Cl− in concrete structures using a neural network model and examined the time-varying law of the Cl− diffusion coefficient utilizing measured data from 30 concrete specimens as input to the database. Gupta [35] used adaptive artificial neural network (ANN), and ANN approaches to model the permeation law of Cl− in concrete structures, considering the effect of environmental temperature on the Cl− permeability coefficient. Yong et al. [36] investigated the shear strength of recycled concrete beams (RAC) based on the ANN and random forest (RF) models. Bukhsh [37] and Pengyong [38] designed a bridge state-level prediction approach based on the ML model to solve bridge degradation influenced by various unpredictable factors, boosting forecast accuracy. Boyu [39] promoted the idea of constructing a decision tree (DT) based on density, which lowered the size of the DT and, to some extent, prevented overfitting. The intelligent assessment model of bridge safety risk created using the ML algorithm solved the problem of insufficient use of previous assessment data [40–42]. Shuheng [43] utilized the sparrow search algorithm (SSA) to optimize and alter the initial weights and thresholds of the back propagation (BP) neural network, demonstrating that the SSA has excellent accuracy and can optimize the BP neural network.

The current study mainly focuses on the overall distribution, macroscopic performance, and mechanical changes of cable corrosion. There has been little research on the nonstationary dispersion of internal corrosion factors in cables. The diffusion of corrosion factors and external elements have a complicated nonlinear relationship that is difficult to obtain using one-time conversion or simple linear regression (LR) approaches. Therefore, this article establishes a data-driven model based on the ML method to reveal the effects of cable inclination angle, environmental temperature, humidity, and cable defect size on the concentration and diffusion coefficient of corrosion factors on the cable's surface. Based on multiple nonlinear regression analysis methods, appropriate empirical formulas are provided and validated, providing a reference for theoretical analysis of the long-term performance of stay cables in practical engineering.

## **2. Diffusion Mechanism and the Test Method of Corrosion Factors**
