Stiffness Separation Method for Damage Identification of Steel Truss Bridge: Exploring Diverse Separation Interfaces

Abstract

Effective identification and quantification of bridge damage are critical for ensuring infrastructure safety and longevity. This study introduces a damage identification approach for steel truss bridges based on the stiffness separation method. This method simplifies large‐scale problems by partitioning structures into substructures through separation interfaces. To enhance interface adaptability, the method conducts distinct analyses of nodes and members and a combined analysis involving both. A case study of the New Yellow River Bridge validated the effectiveness of the proposed method. Furthermore, a comparative analysis of the Nelder–Mead (NM) simplex and Interior Point (IP) methods was performed across various damage and separation scenarios. The findings confirm the accuracy and efficiency of the proposed method for damage detection, highlighting its importance for maintaining the safety of large bridge structures.

1. Introduction

Civil infrastructure, particularly bridges, constitutes a key component of urban transportation networks and plays a crucial role in the sustainable development of cities. These structures are subjected to environmental factors and material aging, necessitating robust solutions to ensure structural safety. This research contributes to advancing structural damage detection and assessment methods essential for maintaining infrastructure reliability and safety [1–3]. With the growing demand for structural health monitoring technologies, various damage identification techniques have been explored. However, these methods often encounter computational complexities when analyzing large-scale structures, particularly those with high degrees of freedom (DOFs) and numerous unknown parameters, which can impact the accuracy and reliability of damage detection [4–6]. Many scholars have proposed solutions to this issue. Crognale et al. [7] proposed an integrated global-local methodology for fatigue damage identification in steel truss bridges, combining vibration-based analysis and image processing techniques. Applied to the Quisi Bridge, the approach has been successfully validated. Diaferio et al. [8] developed a method for detecting localized damage in beams and trusses using modal frequencies. The approach accurately identified damage location and severity. Gul and Catbas [9] proposed a time series analysis methodology using auto-regressive models with exogenous (ARX) for structural health monitoring, which successfully detected, localized, and quantified damage under noisy conditions. Azim and Gül [10] introduced a sensor clustering-based time-series approach for damage detection in truss bridges. Do and Gül [11] presented an output-only vibration method that uses time-series models to detect and quantify damage in shear-type structures. Mizuno et al. [12] proposed a Haar wavelet-based method for damage detection and data compression in civil structures. Monroig et al. [13] presented a method for modeling the dynamics of systems driven by unknown inputs using multivariate output observations, which can be used for structural damage identification through the detection of deviations from a baseline model. Monroig and Fujino [14] developed a decentralized local damage identification method based on ARX models, successfully detecting damage in a 4-story building.

Furthermore, substructure analysis offers notable benefits in addressing these challenges. By dividing large structures into smaller, more manageable substructures, this method simplifies the analysis and enhances the accuracy of damage localization, enabling more efficient and targeted detection. There is a correlation between the substructure, main structure, and parameters to be identified, and effectively managing separation interfaces is crucial for accurately delineating the substructure model and damage identification. A key challenge is the difficulty in measuring all DOFs at the interfaces, particularly when these interfaces are complex, which limits the practical applicability of this method. Therefore, it is essential to diversify the separation interfaces to enhance the feasibility of practical implementation.

Previous studies introduced a stiffness separation method [15, 16] for the substructure isolation and damage identification. This method extracts the relevant matrix elements from the stiffness matrix of the entire structure. It effectively streamlines the objective function and reduces the number of iterations required. The innovation of this approach is eliminating the requirement to calculate interface forces. Furthermore, computational efforts [17] and partial-model-based damage detection [18] have been studied for the proposed method. By dividing the global stiffness matrix into substructures, this method reduces computational effort. Additionally, the proposed method enables damage identification based on a partial model, which simplifies the modeling process for large-scale structures.

Large structures often experience damage across multiple regions and components. To achieve multiregion inversion, the overall structure must be divided into multiple substructures for analysis. The primary challenge in substructure separation lies in isolating the substructures from the overall structure through separation interfaces. However, large structures are often highly complex, and the structural form may constrain the installation of instruments, making it challenging to measure data at separation points. Therefore, it is necessary to study the separation interfaces further to enhance their operability. To address these challenges, this study introduces diverse separation interfaces. This method bypasses the issues associated with the sensor arrangement and enables accurate identification and localization of damage within the structure, even in large and complex structures.

This study provides a comparative analysis of three separation interfaces: node, member, and hybrid. The distinction between these strategies is crucial, as they offer different benefits depending on the sensor types used and the specific structural aspects of the bridge. The node separation method targets the joints or connections within a structure. The member separation method involves strategically positioning sensors onto structural components such as beams or girders. The combined node-member approach provides a comprehensive monitoring strategy that can adapt to various structural configurations and diagnostic requirements.

The following parts of this article are as follows: Section 2 introduces the theoretical background of the proposed identification method. Section 3 analyzes its application to large steel truss bridges, including the parameter estimation using two optimization algorithms. Finally, Section 4 concludes the paper with a summary of new findings.

2. Damage Identification Method

Nondestructive damage detection is critical for evaluating the stability and functionality of steel structures, making efficient damage identification essential. Structural damage identification plays a vital role by updating analytical models to reflect the actual, or “as-is” structural response. Damage in structures mainly manifests as a reduction in local or global stiffness. Static load-displacement or load-strain relationships offer a direct measure of structural stiffness. Changes in these relationships provide clear indicators of damage existence and offer its approximate location and severity. Through damage estimation, engineers can quantify damage by comparing the estimated response to the “as-is” condition [19, 20]. In this study, the cross-sectional areas of the damaged members were selected as the unknown parameters to be identified.

First, the individual stiffness matrices of the members are combined to construct a global stiffness matrix K for the structure. Employing the stiffness method, the force-displacement relationship in the structure is defined as Q = KD, which is further divided into the following equations for known and unknown displacements: 1 $\begin{array}{}\left[\begin{array}{c}{\mathbf{Q}}_k\\ {\mathbf{Q}}_u\end{array}\right]=\left[\begin{array}{cc}{\mathbf{K}}_{11}& {\mathbf{K}}_{12}\\ {\mathbf{K}}_{21}& {\mathbf{K}}_{22}\end{array}\right]\left[\begin{array}{c}{\mathbf{D}}_u\\ {\mathbf{D}}_k\end{array}\right].\end{array}$

In equation (1), Q_k and Q_u represent the known and unknown loads, respectively, while D_k and D_u denote the corresponding displacements. The formulation for the unknown displacement D_u is 2 $\begin{array}{}{\mathbf{D}}_u={\mathbf{K}}_{11}^{-1}\left({\mathbf{Q}}_k-{\mathbf{K}}_{12}{\mathbf{D}}_k\right).\end{array}$

The methodology for damage detection incorporates a comparative analysis of the measured strains in specific truss members and their calculated analytical counterparts. This comparison forms the basis of the optimization problem. The objective function quantifies discrepancies between the measured and analytical strains and is expressed as follows: 3 $\begin{array}{}f\left({\mathbf{A}}_d\right)={\displaystyle \sum }_{s=1}^M{\left[{{\epsilon }_s}^{\prime }-{\epsilon }_s\left({\mathbf{A}}_d\right)\right]}^2.\end{array}$

Here, ε^′ and ε represent the measured and analytical strains, respectively, M is the total number of strain measurement points, and A_d is the unknown cross-sectional area. The number of unknowns is equal to M. The optimal cross-sectional area ${{\mathbf{A}}_d}^{\mathrm{\ast }}$ of the damaged elements is determined by minimizing the objective function. 4 $\begin{array}{}{{\mathbf{A}}_d}^{\mathrm{\ast }}=\arg \underset{{\mathbf{A}}_d}{\min }\left[f\left({\mathbf{A}}_d\right)\right].\end{array}$

2.1. Stiffness Separation

The stiffness separation method [15–18] addresses the challenges of damage detection in large-scale truss structures, which are characterized by high DOFs and numerous unknown damages. By decomposing the complex structure into smaller and more manageable substructures, this method simplifies damage identification.

This approach isolates the substructure-related matrix from the stiffness matrix of the entire structure. To extract the relevant stiffness matrix, displacements, and loads, the mapping matrices P and R are constructed based on the DOF numbering system. The arrangement of these elements is determined by the numbering of the unknown and known displacements in the substructure. Consequently, for an independent substructure, submatrices K₁₁ and K₁₂ can be derived using the following equations: 5 $\begin{array}{c}{\mathbf{K}}_{11}={\mathbf{P}}^T\mathbf{KP},{\mathbf{K}}_{12}={\mathbf{P}}^T\mathbf{KR},\end{array}$ where T represents the transpose operation of the matrices. Therefore, the unknown displacements of the substructure can be determined using equation (2). The relationship between the known displacements of the substructure and all displacements, including known and unknown, in the overall structure is expressed as D_k = R^TD. The known displacements at the separation interfaces, used in the substructure isolation process, can be obtained from various sources such as direct displacement transducers, computer vision-based measurements, or predefined boundary conditions. The known loads on the substructure are given by Q_k = P^TQ. Subsequently, an independent objective function can be established for the substructural damage analysis.

2.2. Separation Interface on Member

The substructure method isolates a substructure from the overall structure. An interface exists between the substructure and the remaining structure. While previous studies set interface nodes on joints [15–18] this study divides the overall structure by placing interface nodes on members. Figure 1 shows an example of substructure separation. Figure 1(a) illustrates the interface nodes on the joint and member. Figure 1(b) shows the separated substructure. Placing the interface nodes on members creates an interface member with one end fixed and the other pinned. An example of such an interface member is shown in Figure 1(b). In Figure 1, the solid and hollow circle nodes represent the fixed and pinned ends, respectively.

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To study the fixed-pin element, this section shows the force-displacement relationship of the frame element. Figure 2 illustrates the node displacement and force of the frame element in the local coordinate system. q and d represent the load and displacement in the local coordinates, respectively.

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The stiffness equation of the frame element is expressed as follows: 6 $\begin{array}{}\left[\begin{array}{c}{q}_1\\ q_2\\ q_3\\ q_4\\ q_5\\ q_6\end{array}\right]=\left[\begin{array}{cccccc}\frac{AE}{L}& 0& 0& -\frac{AE}{L}& 0& 0\\ & \frac{12EI}{L^3}& \frac{6EI}{L^2}& 0& -\frac{12EI}{L^3}& \frac{6EI}{L^2}\\ & & \frac{4EI}{L}& 0& -\frac{6EI}{L^2}& \frac{2EI}{L}\\ & & & \frac{AE}{L}& 0& 0\\ & & & & \frac{12EI}{L^3}& -\frac{6EI}{L^2}\\ \text{Sym}& & & & & \frac{4EI}{L}\end{array}\right]\left[\begin{array}{c}{d}_1\\ d_2\\ d_3\\ d_4\\ d_5\\ d_6\end{array}\right].\end{array}$

In equation (6), A, E, L, and I denote the cross-sectional area, elastic modulus, member length, and moment of inertia, respectively.

Figure 3 shows an element with rigid-pin ends. q₁ to q₃ and d₁ to d₃ denote the load and displacement at the fixed end, respectively. q₄ to q₅ and d₄ to d₅ are the corresponding values at the pinned end.

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The force-displacement equation of the end-release element can be determined from equation (6). The stiffness equation for the rigid-pin element is as follows: 7 $\begin{array}{}\left[\begin{array}{c}{q}_1\\ q_2\\ q_3\\ q_4\\ q_5\end{array}\right]=\left[\begin{array}{ccccc}\frac{AE}{L}& 0& 0& -\frac{AE}{L}& 0\\ & \frac{3EI}{L^3}& \frac{3EI}{L^2}& 0& -\frac{3EI}{L^3}\\ & & \frac{3EI}{L}& 0& -\frac{3EI}{L^2}\\ & & & \frac{AE}{L}& 0\\ \text{Sym}& & & & \frac{3EI}{L^3}\end{array}\right]\left[\begin{array}{c}{d}_1\\ d_2\\ d_3\\ d_4\\ d_5\end{array}\right].\end{array}$

3. Illustrative Case Studies of Steel Truss Structures

This study used the New Yellow River Bridge, the longest simply supported steel truss railway bridge in China, as a case study to demonstrate the effectiveness of the separation interfaces [21]. The bridge spans 156 m across the Yellow River near Yumenkou Station on the Huang–Han–Hou railway line (Figure 4). This bridge connects Shaanxi Province to the southern Shanxi Province and plays a vital role in energy transportation. Figure 5 shows a two-dimensional model representing the trusses of the bridge. The truss members are numbered in red, and the nodes are labeled in blue. Node 1 is equipped with a hinged support, while Node 23 has a roller support. The “as-built” cross-sectional areas of the truss members are detailed in Table 1.

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Table 1 “As-built” cross-sectional areas.

Member numbering	Cross-sectional area (mm2)
1, 3, 41, 43	107,904
5, 7, 37, 39	89,600
9, 11, 13, 31, 33, 35	71,424
15, 21, 23, 29	54,400
17, 27	64,800
19, 25	46,000
4, 40	113,600
8, 36	136,704
12, 32	183,296
16, 28	213,984
20, 24	238,400
2, 42	80,024
6, 38	114,800
10, 34	161,616
14, 30	216,144
18, 22, 26	240,800

3.1. Left Region Damage Identification

In this study, three different separation interfaces were employed for structural damage identification: node, member, and hybrid. Damage was assumed to occur in the diagonal members on the left side of the bridge. The number of damaged members and their corresponding “as-is” cross-sectional areas are listed in Table 2, with their locations shown in Figure 5. As the damage was concentrated in the left region of the structure, this region was selected as a substructure for damage identification.

Table 2 “As-is” cross-sectional areas in the left region of the structure.

Member numbering	Cross-sectional area (mm2)
1	97,113.6
3	86,323.2
5	62,720
7	57,360
9	42,854.4
11	49,996.8

Figure 6 shows the separation in the left region. Figures 6(a), 6(b), 6(c) depict the overall structure and the three separation strategies employed to isolate the damaged region, including node, member, and hybrid separations. The blue points represent interface nodes, and the red regions indicate the separated substructure portions. A vertical load of 100 kN was applied at node 5 to excite the damaged region. Figures 6(d), 6(e), 6(f) show the isolated substructures based on node, member, and hybrid separation interfaces, respectively, with the corresponding node numbers labeled in these figures. The red members denote the artificially simulated damaged components.

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The damaged cross-sectional areas were determined by incorporating the analytical and measured strain values into an objective function. In this study, strain measurements were set on the damaged members and derived from artificially induced damage cases. Two optimization methods were employed: the Nelder–Mead (NM) simplex and Interior Point (IP) methods.

The NM simplex method is a commonly used nonlinear optimization algorithm [22–24]. It iteratively adjusts the position and shape of the simplex to approximate the minimum point of the objective function, gradually reducing the size of the simplex until the convergence criteria are met. For the optimization problem, the starting points for the NM simplex method were set according to the “as-built” cross-sectional areas. Figures 7(a), 7(b), 7(c) illustrate the changes in optimized values during the iterative process using the NM simplex method for the three separation interfaces. The dashed lines represent the reference values for the actual conditions. These figures show that the optimized values converge toward the “as-is” values of the damaged members. Figure 8 shows the objective function values with iteration steps. The function value gradually approaches zero.

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The IP method is an effective algorithm for solving linear and nonlinear constrained optimization problems. It identifies optimal solutions within constraints, thereby avoiding boundary complexities. This method is particularly suitable for large-scale optimization problems [25, 26]. For the optimization problem, the starting points for the IP method were set as half of the “as-built” cross-sectional areas. The lower and upper bounds for the variables were defined as 0 and the “as-built” condition, respectively. Figure 9 illustrates the variation in the parameter values with the number of iterations based on the IP method. Figures 9(a), 9(b), 9(c) show the results for the node, member, and hybrid separation strategies, respectively. The results indicated that the optimized values successfully converged to the actual values of the damaged components. Figure 10 shows that the objective function values approach zero.

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This study utilizes the mean relative error (MRE), a standard statistical measure, as an accuracy metric. The MRE is calculated using equation (8): 8 $\begin{array}{}\text{MRE}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left(\frac{\left|{A_{d_i}}^{\prime }-{A_{d_i}}^{\mathrm{\ast }}\right|}{{A_{d_i}}^{\prime }}\right).\end{array}$

In equation (8), n represents the total number of damaged components. For each component, ${A_{d_i}}^{\mathrm{\ast }}$ denotes the identified cross-sectional area of the i-th component, and ${A_{d_i}}^{\prime }$ represents its “as-is” cross-sectional area. Figures 11(a) and 11(b) illustrate the objective function and MRE values during the iterations of the NM simplex and IP methods, respectively. Both metrics converged to zero upon completion of the iterations, thereby validating the accuracy of the identified results.

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3.2. Middle Region Damage Identification

In this study, the damage was assumed to be concentrated in the middle region of the structure. The “as-is” cross-sectional areas of the damaged members are detailed in Table 3. Figure 12 shows the separation of the middle region. Figures 12(a), 12(b), 12(c) show the overall structure, with the damaged region separated based on the node, member, and hybrid interfaces. A vertical load of 100 kN was applied at node 11 to excite the structure. Figures 12(d), 12(e), 12(f) illustrate the corresponding localized models used in the analysis.

Table 3 “As-is” cross-sectional area of the middle side of the structure.

Member numbering	Cross-sectional area (mm2)
17	45,360
19	27,600
21	32,640
23	38,080
25	36,800
27	51,840

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Similarly, the NM simplex and IP methods were employed for damage identification in the middle region. Figures 13(a) and 13(b) show the convergence history of the objective function and the variation in the MRE values with the number of iterations. The objective function values and MRE gradually approached zero, demonstrating the success of the optimization process.

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3.3. Right Region Damage Identification

The damage was assumed to occur in the diagonal members located in the right region of the structure. Table 4 lists the “as-is” cross-sectional areas and member numbers of the damaged components. Figure 14 shows the separation of the right region. A vertical load of 100 kN was applied at node 19. The node, member, and hybrid separations are shown in Figures 14(a), 14(b), 14(c), respectively. Figures 14(d), 14(e), 14(f) present the corresponding separated substructures.

Table 4 “As-is” cross-sectional areas in the right region of the structure.

Member numbering	Cross-sectional area (mm2)
33	49,996.8
35	42,854.4
37	53,760
39	62,720
41	86,323.2
43	97,113.6

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Figures 15(a) and 15(b) illustrate the variation in the objective function and MRE values throughout the iterative process using the NM simplex and IP methods, respectively. As shown in the figures, both values gradually converge to zero, indicating that the identified results are accurate.

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Tables 5 and 6 summarize the results of the NM simplex and IP methods, respectively. The iteration steps and final values of the function for the node, member, and hybrid separations are provided. Compared to the NM simplex method, the IP method requires fewer iteration steps in all cases. In this study, the IP method is more efficient in identifying damage.

Table 5 NM simplex method.

Damage region	Separation method	Iteration steps	Final function value
Substructure L	Node	395	4.54 × 10−6
Member	398	3.65 × 10−6
Hybrid	368	4.20 × 10−6
Substructure M	Node	500	6.25 × 10−6
Member	556	6.31 × 10−6
Hybrid	517	5.18 × 10−6
Substructure R	Node	425	7.95 × 10−6
Member	349	3.89 × 10−6
Hybrid	621	2.94 × 10−6

Table 6 IP method.

Damage region	Separation method	Iteration step	Final value of function
Substructure L	Node	169	2.67 × 10−8
Member	174	2.83 × 10−8
Hybrid	178	3.46 × 10−8
Substructure M	Node	176	3.29 × 10−8
Member	164	2.61 × 10−8
Hybrid	167	3.93 × 10−8
Substructure R	Node	190	2.73 × 10−8
Member	188	2.00 × 10−8
Hybrid	193	2.40 × 10−8

4. Conclusion

This study investigates different separation interfaces for truss damage identification using the stiffness separation method. The node, member, and hybrid separation interfaces are studied. By exploring these distinct interfaces, a flexible and efficient approach to separating the substructure is proposed. The node-based method focuses on joints, while the member-based method targets structural components like beams. The hybrid method combines both, offering a comprehensive solution adaptable to various bridge configurations.

The proposed method provides diverse separation strategies and investigates the damage identification for the large-scale structure. A case study involving the New Yellow River Bridge validated the effectiveness of the approach. Different separation scenarios, combined with two optimization methods, demonstrated the accuracy and efficiency of identifying structural damage. The proposed method reduces the overall complexity of damage identification by simplifying large-scale truss structures into manageable substructures. It shows that both optimization algorithms successfully find the optimal solutions efficiently, but the IP method has less iterations than the NM simplex method.

A major advantage of this method is its flexibility in selecting separation interfaces. By reducing the reliance on large-scale structural analyses, the proposed method provides various substructure separation schemes and simplifies the damage identification in large-scale structures. In addition, the separation interfaces can be applied to other structural types, including truss arch bridges and truss roof systems. Besides, the impacts of the measurement errors and environmental factors on the proposed method will be further studied.

Data Availability Statement

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Funding

This study was supported by the Fundamental Research Funds for the Central Universities (grant no. 30925010544) and the Natural Science Foundation of Jiangsu Province, China (grant no. BK20200492).

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