
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12507-8
10.1016/j.heliyon.2024.e36476
e36476
Research Article
Identification of hanger damage based on deflection influence matrix
Wang Weiwei ab
Chen Weili ab
Xu Hongbin xuhongbin@semi.ac.cn
cd⁎
a Department of Road and Bridge Engineering, Hebei Jiaotong Vocational and Technical College, Shijiazhuang, 050091, Hebei, China
b Hebei Provincial Seasonal Frozen Area Highway Service Safety and Early Warning Technology Innovation Center, Shijiazhuang, 050091, Hebei, China
c Key Laboratory of Structural Health Monitoring and Control, Hebei Province, Shijiazhuang Tiedao University, Shijiazhuang, 050043, Hebei, China
d National Key Laboratory of Green and Long-Life Road Engineering in Extreme Environment, Shenzhen University, Shenzhen, 518060, Guangdong, China
⁎ Corresponding author. Key Laboratory of Structural Health Monitoring and Control, Hebei Province, Shijiazhuang Tiedao University, Shijiazhuang, 050043, Hebei, China. xuhongbin@semi.ac.cn
17 8 2024
15 9 2024
17 8 2024
10 17 e364761 5 2023
11 8 2024
16 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Tied-arch bridges, a vital component of modern infrastructure, are susceptible to various forms of damage, particularly hangers. The detection and identification of such damages are crucial for maintaining structural integrity and safety. However, traditional methods face challenges in terms of accuracy and efficiency. This study aims to develop a refined method for hanger damage identification in tied-arch bridges, to address the limitations of existing techniques. By focusing on deflection changes at the anchoring points between the hangers and tie beams, we sought to enhance the precision of damage detection. We propose an innovative approach based on the concept of influence lines, introducing the ‘generalized deflection difference influence line'and the ‘deflection difference influence matrix’. Then proposed a new identification index for identifying the damaged hanger after matrix. An actual tied-arch bridge was used to validate the proposed approach. A detailed three-dimensional finite element model of the bridge was developed and calibrated using dynamic and static response data. Thirty different hanger-damage conditions were simulated to evaluate the effectiveness of the proposed method. Our findings reveal that the deflection difference influence matrix offers more detailed and comprehensive information on bridge distribution points than traditional methods. Our method proved effective in identifying hanger damage, irrespective of its location on the bridge. In additionally, the identification efficiency of the method can be improved by adjusting the magnitude of the applied load, with larger loads amplifying the detectability of damage. This study highlights the potential of the deflection difference influence matrix to revolutionize hanger damage identification for tied-arch bridges. Its adaptability, accuracy, and efficiency are significant advancements over existing methods. This study successfully demonstrates an innovative and reliable method for hanger damage identification.

Keywords

Influence matrix
Hanger damage
Damage identification
Tied arch bridge
==== Body
pmc List of symbols

Pji	the axle loads acting on the cross-section i of the bridge	
Fi	the concentrated load acting on the north tie beam after the equivalent of section i	
Fi′	the concentrated load acting on the south side of the tie-beam of the section i after the equivalent	
DILk	the deflection influence line at anchorage point k under unit load	
DILk′	the deflection influence line of anchorage point k on the south side tie beam under unit load	
flk	the deflection value of the deflection influence line of k section at the anchorage point l	
wk	the column vector of the deflection influence line under the given load when the bridge undamaged, that is, when the anchor point k is loaded, the column vector is formed by the displacement of each hanger and the anchor point of the beam.	
B	the deflection influence matrix when the structure is undamaged.	
wk′	the column vector of the deflection influence line under the given load when the bridge damaged, that is, when the anchor point k is loaded, the column vector is formed by the displacement of each hanger and the anchor point of the beam.	
C	the deflection influence matrix when the structure is damaged.	
Δwk	the vector of deflection difference influence line (deflection change influence line DCILi)	
D	the damage identification index.	
Δfkl	the deflection difference value at the anchorage point l of the deflection influence line of anchorage point k after hanger damage.	
β	the damage degree of the hanger.	
ΔA	the reduction of the hanger cross-sectional area caused by damage	
A	cross-sectional area without damage.	

1 Introduction

The development of tied-arch bridges has led to significant achievements in the industry. Competitiveness has become stronger. As an essential force transfer and bearing component of this type of bridge, the hanger (cable) bears the cyclic action of service and environmental loads year-round; therefore, it is easily damaged [[1], [2], [3], [4]]. In addition, there is no standard code for replacing the hanger (cable), thus there is no basis for replacing the hanger. Generally, when cables are damaged, the operator replaces all hangers on the bridge to ensure the safety and reliability of the structure during service [5]. The cost of replacing the hanger (cable) is very expensive. When building a new bridge, the construction cost of the hangers (cables) accounts for approximately 25 % of the total construction cost, and the cost of replacing hangers (cables) is 3–4 times that of the new cables [[6], [7], [8], [9], [10]]. Considering the importance of the hanger structure, the cost of replacement, and the impact on bridge safety, the detection and damage identification of hangers are particularly important [10].

Bridge damage identification methods can be classified into local and overall damage identifications. According to different data acquisition methods, general damage identification can be divided into damage identification based on static performance tests and damage identification based on dynamic characteristic tests [[11], [12], [13], [14]]. Local damage detection is often performed by visual inspection, ultrasonic inspection, magnetic flux leakage detection, and other methods, which are either time-consuming, laborious, or costly [15]. Dynamic damage identification based on dynamic performance testing is performed by collecting the vibration response signals of bridges and obtaining modal data, such as natural frequencies or vibration modes, through data processing [[16], [17], [18]]. The frequency of the structure is easily obtained; however, it is not sensitive to local damage. In addition, different types of damage may cause the same frequency change, making it difficult to locate the damage. Although higher-order frequencies more sensitive than lower-order frequencies, obtaining higher-order frequencies for large bridge structures is difficult [19,20]. Damage diagnosis based on the change or curvature of the mode shape requires more sensors to be arranged in the structure to obtain more complete response data of the structure, which is difficult to achieve in practical measurement [21]. In addition, dynamic testing is greatly affected by environmental noise and operating conditions, so its application in practical engineering is still not satisfactory [22].

The overall damage identification of bridge structures based on static tests involves identifying the damage by obtaining static parameters through a bridge static load test during a traffic interruption. Statics-based methods offer the advantages of accurate test data, lower noise interference, and simplicity [23]. Many researchers have been conducted damage identification based on static parameters. Chen et al. [11] proposed the use of a symmetrical deflection difference influence line to diagnose the damage to a prefabricated hollow slab bridge. Sun et al. [21] considered a non-ideal support condition using the displacement influence line data and the quadratic difference of the control points of a continuous curved beam bridge with a small radius to obtain the curvature curve for damage identification. Du et al. [24] proposed using the curvature of the deflection difference influence line to identify damage of a simply supported beam. The deflection difference curvature increase when the moving load is in the damaged area. Wang et al. [25] established a primary beam model considering the uncertainty of the bending stiffness and use the difference index of the mid-span displacement influence line to identify damage. Xie et al. [26] proposed using the deflection difference influence line to identify damage in simply supported beams. The deflection difference influence line is better for single-point damage identification, but it can not locate all damages in the case of multi-point damage. Damage identification based on deflection or deflection difference influence lines can identify and quantify damage and exhibit anti-noise performance [27]. In addition, performing damage identification research on actual structures is generally not realistic; therefore, many researchers have studied structural characteristics and damage by developing a benchmark finite element model of structures [[28], [29], [30]].

In summary, the deflection difference influence line can provide complete information about the structure, and most previous studies have used the deflection difference influence line of a single point to identify structural damage. In this study, the deflection difference influence matrix was used as the damage identification index to study the hanger damage identification of through-tied arch bridges.

2 Theory

The change in the deflection of the tie-beam caused by hanger damage is generally small; therefore, it is reasonable to believe that the research scope of this study is linear elastic analysis. Therefore, it is assumed that the structure is in the range of linear elasticity in accordance with Hooke's law and the superposition principle.

Following the concept of the influence line in structural mechanics, the definitions of the deflection difference influence line and deflection difference influence matrix are provided in this study. In structural mechanics, when a concentrated load pointing to an invariant unit (usually vertically downward) moves along a structure, the figure indicating the change law of a specified value is called the influence line of the value. Therefore, the difference in the deflection influence line of the tied beam under a given load before and after the hanger damage to the arch bridge was defined as the deflection difference influence line. The deflection difference influence matrix refers to the matrix formed by the deflection difference vector at the anchoring point of each hanger and tie beam.

2.1 Definition and acquisition of deflection difference influence line

For a through tied-arch bridge, the deflection of the tie beam (or anchorage point of the hanger and tie beam) caused by hanger damage is small, and the measured deflection difference is likely to be affected by the operating conditions and environment [31,32]. To amplify the deflection change of the tie-beam caused by hanger damage, the maximum load within the normal bearing capacity of the bridge can be selected as the given load.

For the selected real bridge, six trucks were chosen as the given loads and arranged in two rows and three columns. The vehicles were loaded symmetrically on both sides of the target hanger. The specific loading mode is illustrated in Fig. 1. The size of the test load can be adjusted according to the stiffness of the tie beam., if the stiffness of the tie beam is small, a car with a common axle load of 30 tons can be used.; if the stiffness of the tie beam is high, it is necessary to increase the load, so as to increase the beam deflection deformation caused by hanger damage. However, the load must be less than the maximum load value of the completion test.Fig. 1 Loading diagram for a given load.

Fig. 1

It is shown in Fig. 1, six trucks acting on the bridge deck are equivalent to 36 applied concentrated loads. Therefore, based on the lever principle, the six concentrated loads in each row are equivalent to the two concentrated loads acting on the tie-beams on both sides. For example, the six concentrated loads at section A-A ′ are equivalent, and the intersection between the section and tie beams on both sides is regarded as a support. The reaction influence line for the northern support is shown in Fig. 2.Fig. 2 The reaction influence line of the north tie beam (support) at section A–A.

Fig. 2

It can be seen from Fig. 2 and the lever principle that simplifying the six concentrated loads applied to the north tie beam gives(1) Fi=Σi=16pji2yji=p1i2y1+p2i2y2+p3i2y3+p4i2y4+p5i2y5+p6i2y6

where Fi (i=1⋯6) is the concentrated load acting on the north tie beam after the equivalent, pji (j=1⋯6) are the axle loads acting on the cross-section i of the bridge, and yji (i=1⋯6) is the corresponding vertical coordinate on the influence line of the reaction force. The concentrated load acting on the south side of the tie-beam of the same section is:(2) Fi′=Wi−Fi

where Wi is the sum of axial forces acting on the cross section i. After the equivalent, each side tie-beam was equivalent to loaded six concentrated loads. To obtain the deflection influence line of each anchorage point under a given load, it was necessary to acquire the deflection influence line of the beam at each anchorage point under a unit load. The north tie beam is considered an example to study hanger damage identification. When calculating the influence of the deflection difference at each anchorage point of the north tie beam, the load state of the tie beams on both sides should be considered simultaneously. Therefore, two loading conditions should be considered when calculating the deflection influence line of the northern tie-beam under a unit load. A unit load 1 kN passes through the bridge span according to the positions shown in Fig. 3(a) and (b). The deflection influence line of each anchorage point on the north tie-beam can be obtained by recording the deflection curve of the beam when the unit load stops at each loading point. The deflection influence lines at each anchorage point of the north tie beam obtained from Fig. 3 (a), and (b) are denoted as DILi and DILi′, respectively.Fig. 3 Unit load positioning stations.

Fig. 3

When the unit force (generalized force 1) moves along the bridge, the deflection at section k of the bridge is called the deflection influence line of the section. Therefore, DILk (deflection influence line k) represents the deflection influence line at anchorage point k under a unit load, as shown in Fig. 4.Fig. 4 Deflection influence line under a unit load at anchorage point k.

Fig. 4

The deflection of any anchorage point on the tie beam under a given load (equivalent to multiple concentrated loads) can be obtained as the product of each concentrated load and the longitudinal coordinate of the deflection influence line under a unit load at its position.(3) flk=Σi=1mFiDILk(xj)+Σi=1mFi′DILk(xj)=F1DILk(x1)+F2DILk(x2)+F3DILk(x3)+F4DILk(x4)+F5DILk(x5)+F6DILk(x6)+F1′DILk′(x1)+F2′DILk′(x2)+F3′DILk′(x3)+F4′DILk′(x4)+F5′DILk′(x5)+F6′DILk′(x6)

Where, l,k = 1 … …n, n is the number of anchorage points, flk is the deflection value of the deflection influence line of k section at the anchorage point l. In Eq. (3), DILk represents the deflection influence line of anchorage point k on the north side tie beam under unit load; DILk′ represents the deflection influence line of anchorage point k on the south side tie beam under unit load; l represents each anchorage point, m represents the number of loads acting on each tie beam after equivalence; and n represents the number of anchorage points. The deflection influence line vector can be obtained at the anchorage point k according to the steps above. The column vector is then defined as the deflection influence line under the given load.(4) wk=[f1kf2k⋯fnk]T

2.2 Definition and acquisition of deflection difference influence matrix

When the column vector of the deflection influence line corresponding to each anchorage point is placed in the matrix, the deflection influence matrix B can be obtained.(5) B=[f11f12⋯f1nf21f22⋯f2n⋮⋮⋮fn1fn2⋯fnn]

where, flk represents the deflection value at anchorage point l of the deflection influence line of anchorage point k before hanger damage.

The deflection influence line of the corresponding section after hanger damage was obtained using above steps.(6) wk′=[f′1kf′2k⋯f′nk]T

Thus, the deflection influence matrix C after damage can also be obtained,(7) C=[f11′f12′⋯f1n′f21′f22′⋯f2n′⋮⋮⋮fn1′fn2′⋯fnn′]

where flk′ represents the deflection value at anchorage point l of the deflection influence line of anchorage point k after hanger damage.

From the deflection influence line vectors before and after hanger damage, the vector of the deflection difference influence line (deflection change influence line DCILi) can be obtained as(8) Δwk=[Δf1kΔf2k⋯Δfnk]T

Subsequently, the deflection difference influence matrix D can be directly obtained from the deflection difference influence line (or the difference can be obtained from influence matrices C and B).(9) D=C−B=[Δf11Δf12⋯Δf1nΔf21Δf22⋯Δf2n⋮⋮⋮Δfn1Δfn2⋯Δfnn]

where Δflk represents the deflection difference value at anchorage point k of the deflection influence line of anchorage point l after hanger damage. The deflection difference influence matrix D is defined as the damage identification index used to identify the deterioration of the hanger. A real bridge was used to verify the proposed method.

3 Establishment and modification of finite element model

The proposed method was verified using a highway concrete-filled steel tube simply supported tied-arch bridge. The bridge has an east-west orientation. The bridge spans a lake, and the superstructure measures 130 m. The arch axle is a quadratic parabola, the rise-span ratio is 1:4.5, and the bridge deck width is 12 m, as shown in Fig. 5. The arch rib section is dumbbell-shaped. The upper and lower circles are made of steel pipes with a diameter of 1200 mm and a wall thickness of 20 mm; the panel width is 700 mm, the height is 1025 mm, and the wall thickness is 20 mm; and the filling is C50 self-compacting shrinkage compensating concrete. The tie beam is a reinforced concrete beam with a height of 2200 mm and a wide of 1800 mm. Four wind braces are arranged on the entire bridge, and the upper and lower chords of the wind brace are hollow tubes with a diameter of 600 mm and wall thickness of 12 mm; the web consists of hollow steel tubes with a diameter 300 mm and wall thickness of 10 mm. The hanger is an epoxy, unbonded, steel strand cable. The two pairs of short hangers at both ends of the arch foot are of type J15-12, and the rest of the hangers adopt model GJ15-15 with a tensile strength of 1860 MPa.Fig. 5 Actual through tied arch bridge.

Fig. 5

3.1 Finite element modeling

3.1.1 Finite element model

The finite element model of the bridge was established by Midas/civil, as shown in Fig. 6. The tie beam, arch rib, transverse brace, and crossbeam were simulated using beam elements; the bridge deck was simulated using a four-node plate element; and the hanger was simulated using a tension-only truss element. The whole model include 866 nodes and 1322 elements. There were four wind braces between the two arch ribs. The arch rib was a dumbbell-shaped concrete filled steel tube; the diameter of the single pipe was 120 cm; the wall thickness was 2 cm; the steel pipe was filled with C50 self-compacting concrete; and the steel pipe was Q345 steel, which was established by constructing a joint section in the software. The bridge deck system is composed of a longitudinal beam, an end beam, a middle beam and a bridge deck. The constraint at the arch foot was consolidated, and the whole superstructure was simply supported on the lower structure.Fig. 6 Finite element model of the tied arch bridge and arch rib section.

Fig. 6

3.1.2 The influence of meshing on the deflection of beam

To analyze the influence of the meshing accuracy on the analysis results, three types of grids were discussed in the model. The total numbers of grids were 881, 1322, and 3721, respectively. Refer to Table 1 for more details. The calculated results of the anchorage points of No. 10 hanger and beam are compared with the measured values, and the measured deflection value is −10.88 mm.Table 1 Gridding conditions and influence results.

Table 1Gridding condition	Grid size (mm)	Displacement of No. 10 anchorage point (mm)	Number of elements	Error (%)	
1	1000–4000	−9.9114	881	−8.228	
2	500–1000	−10.8459	1322	0.425	
3	50–120	−10.8461	3721	0.427	

It can be observed from Table 1 that the meshing accuracy affects the calculation results for beam deflection. The finer the meshing, the closer the deflection was to the measured value. However, the finer the grid, the longer the calculation time; therefore, considering the calculation time and accuracy, we chose the second meshing method in Table 1.

3.2 Field test

3.2.1 Modal parameter test (frequencies and mode shapes)

Numerous field tests were conducted on an actual bridge to obtain its static and dynamic characteristics, including the natural frequencies, mode shapes, and deflection influence lines at the anchorage point. The specific method of mode shape testing was as follows: The mode shape test sensor (accelerometer) was arranged in the middle of the span and at 1/4 L and 3/4 L of the span to pick up the vibration response of the bridge structure under the action of the environment with a sampling frequency of 50 Hz. Then the first and second-order vibration modes of the structure were obtained using the signal acquisition and analysis system. Subsequently, the finite element model was modified based on the experimental results. Fig. 7 shows the data acquisition for the vibration mode test. An eigenvalue modal analysis was performed using the calibrated finite element model. The first two natural frequencies and corresponding vibration modes were compared with the modal parameters obtained from the field test, as shown in Fig. 8.Fig. 7 Acquisition of bridge vibration modes.

Fig. 7

Fig. 8 Comparison of natural frequencies and vibration modes obtained by numerical analysis with field test results.

Fig. 8

The predicted values were in good agreement with the measured values. In Fig. 8, it can be seen that the frequencies of the first two vertical vibration modes predicted using the FEM were 1.866 and 2.911 Hz, and the corresponding measurement results were 1.855 and 2.895 Hz, respectively, show in Fig. 8. The errors between the predicted and measured values were 0.59 % and 0.55 %, respectively. Thus, the established finite element model could fully represent the overall stiffness of the structure.

3.2.2 Deflection change influence line (DCIL)

In addition to testing the fundamental frequency and vibration mode, the deflection difference influence line caused by loading at the mid-span anchorage point was measured using a precision level. However, it is difficult to measure the deflection of the beam accurately. The change in the deflection of the tie beam caused by hanger damage was not very large; therefor it was more difficult to accurately measure the deflection change caused by hanger damage. To increase the difference in the tie beam before and after the hanger damage, a group of loads was applied to the bridge deck to assess the hanger state. Thus, the vertical deflection of the tie beam increased, and the difference in the vertical deflection caused by the damage to the hanger could be easily measured. For the selected through-tied arch bridge, the loading test vehicle was six triaxial trucks, which were partially loaded along the north tie beam, and the loading vehicles are shown in Fig. 9. The axle loads of the test vehicles are listed in Table 2.Fig. 9 Loading vehicles.

Fig. 9

Table 2 Test vehicle parameter.

Table 2vehicle No.	axle load (kN)	
gross weight	front axle	rear axle	
1	389.6	64.6	162.5 × 2	
2	385.6	64.4	160.6 × 2	
3	388.8	65.2	161.8 × 2	
4	387.6	68.6	159.5 × 2	
5	387.6	63.1	162.1 × 2	
6	391	67.8	161.6 × 2	

The deformation measurement points of the tie beam were located at the anchorage points. The arrangement of the measurement points is shown in Fig. 10.Fig. 10 Layout of measuring points for deflection of tie beam.

Fig. 10

One of the purposes of the field test of the influence line was to compare it with the finite element model and provide a basis for modifying the finite element model. The second purpose was to test whether the given load could cause sufficient deflection of the tie beam. First, the elevation h1 of each measurement point of the bridge deck was precisely measured with a precision level. The elevation h2 of each measurement point was measured after loading at the midspan hanger. The deflection difference of each point was then calculated using Eq. (10). The measured deflection difference was compared with the deflection difference calculated by the finite element method, as shown in Fig. 11.(10) Δw=h2−h1

Fig. 11 Comparison of measured and calculated values of deflection change at anchorage point of hanger No. 10.

Fig. 11

As shown in Fig. 11, the deflection influence line calculated using the finite element method is in good agreement with the measured results. A comparative analysis of the dynamic and static experimental data and finite element calculation data showed that the modified finite element model can simulate the real bridge.

4 Numerical investigation

4.1 Design of hanger damage cases

In this study, only damage to the hanger was studied, and it was assumed that there was no damage to the other bridge components. Thirty damage cases were investigated in the calibrated finite element model. The damage degree of each hanger was set to 12.5 %, 25 %, and 37.5 %. All the damaged hangers were placed on the north side of the bridge. Hanger damage was simulated by reducing the cross-sectional area of the hanger. Table 3 summarizes the damage cases investigated in the study.(11) β=ΔAA×100%

where ΔA and A represent the reduction in the cross-sectional area of the damaged hanger and the cross-sectional area of the un-damaged hanger, respectively.Table 3 Damage cases.

Table 3damage cases no.	damage hanger no.	damage degree (%)	
1–3	1	12.5, 25, 37.5	
3–6	2	12.5, 25, 37.5	
7–9	3	12.5, 25, 37.5	
10–12	4	12.5, 25, 37.5	
13–15	5	12.5, 25, 37.5	
16–18	6	12.5, 25, 37.5	
19–21	7	12.5, 25, 37.5	
22–24	8	12.5, 25, 37.5	
25–27	9	12.5, 25, 37.5	
28–30	10	12.5, 25, 37.5	

4.2 Damage identification results and discussion

The hanger on the north side of the real tied-arch bridge was selected as the research object. The north hangers are numbered from 1 to 19 from east to west. The anchor point of each hanger and tie beam was used as the control point of the influence line to measure the difference in deflection, as shown in Fig. 12. Under each damage condition, the deflection influence line of each anchorage point before and after damage was evaluated. The deflection difference influence line was obtained by determining the difference in the deflection influence line before and after hanger damage. Finally, the deflection difference influence matrix was created from the 19 deflection difference influence lines.Fig. 12 Locations of the 19 control points.

Fig. 12

The obtained deflection change influence matrix was used as the damage identification index to identify and analyze the simulated 30 damage cases. The identification results of damage cases 1–9 can only be used to determine the damage range. The damaged hanger could not be accurately located; therefore, the results of damage cases 1–3 and 6–9 were plotted only. The identification results of damage cases 10–30 were consistent; therefore, the results of damage cases 12–15, 19–21, and 25–27 were plotted and discussed.

Fig. 13 Shows that when hanger No.1 was damaged, the lowest point of the deflection difference influence matrix appeared at hanger No.2, regardless of the degree of damage. This may be due to the hangers at the arch foot having a larger cross-section than the other hangers. In addition, the stiffness of the bridge was higher at the arch foot, therefore, the loading had little effect on the deflection of hanger No. 1. From the plane diagram of damage cases 1–3, it can be seen that when hanger No.1 was damaged, the peak of all the deflection change influence lines appeared near the damaged hanger. A larger degree of damage, corresponded with a larger peak value of the deflection difference influence line under the given load. Therefore, the proposed method can provide the region where a damaged hanger is located when hanger No. 1 is damaged; however, the individual hanger cannot be accurately identified.Fig. 13 Identification results of damage cases 1-3.

Fig. 13

Fig. 14 shows the identification results for hanger No. 3 with 12.5 %, 25 %, and 37.5 % damage, respectively. When hanger No. 3 was damaged, the damaged hanger could be located directly from the deflection influence matrix. It can be seen that the figure of the deflection difference influence matrix has a peak at hanger No. 3. The side elevation of the deflection influence matrix shows that the lowest point appears at hanger No. 3. However, the values for hangers No. 4 and 3 were almost equal. It can be seen from the plane diagram that when hanger No.3 was damaged, the peak of all the deflection difference influence lines appeared at the damaged hanger. In the mid-span direction, several hangers near the damaged hanger changed obviously. When hanger No. 3 was damaged, the line of influence of the deflection difference between hangers No. 4 and 5 was sensitive. In addition, a greater degree of damage, corresponded with a greater peak value of the deflection difference influence line under the given load.Fig. 14 Identification results of damage cases 7-9.

Fig. 14

Fig. 15, Fig. 16, Fig. 17 show the identification results of hangers No.5, 7, and 9 with 12.5 %, 25 %, and 37.5 % damage, respectively. The damage identification results of hangers No. 4 to 10 were consistent; therefore, only the identification results of these three hangers are discussed. The three-dimensional graph shows that the peak value of the influence matrix appears at the positions of hangers No.5, 7, and 9, and the sharp points on the side surface map also correspond to these hangers. Hence, it is easy to locate damaged hangers. In addition, it can be seen from the plane figures that when hanger No. 5 was damaged, the influence line of the deflection difference of hangers No. 4 to7 was more sensitive. The influence line of the deflection difference of hangers No.6 to 9 is more sensitive to the damage of hanger No.7. The influence line of the deflection difference of hangers No. 7 to 11 was sensitive to the damage of hanger No. 9.Fig. 15 Identification results of damage cases 13-15.

Fig. 15

Fig. 16 Identification results of damage cases 19-21.

Fig. 16

Fig. 17 Identification results of damage cases 25-27.

Fig. 17

As seen in Fig. 15, Fig. 16, Fig. 17, when the damaged hanger is near the midspan, the proposed method can accurately detect the damage. Moreover, the closer the damaged hanger was to the middle of the span, the greater the deflection difference of the anchorage point under the given load. This is because when the position of the load changes, the effect on the displacement of the affected section differs. According to the principle of structural mechanics, the displacement of the response section is largest when acting in the middle of the span.

The deflection difference caused by hanger damage can be adjusted by changing the load. The change in deflection can be accurately obtained from actual measurements. Therefore, the proposed method has good prospects for practical engineering applications.

The reference state was not necessarily obtained based on the finite element model for the real bridge hanger damage identification. The reference state of this method can be any working state of the bridge; that is, the deflection influence matrix obtained by testing under the normal working state of the bridge can be used as a reference. The reference deflection influence matrix can be obtained using the same method as that used for damage identification. The deflection difference influence matrix can be obtained by determining the difference between the two states, which is used to identify the damage to the hanger. In addition, the recognition effect of this method can be adjusted by varying the loading amount and mode within the bearing capacity of the bridge to obtain a more accurate deflection difference. In the normal bearing capacity range of the structure, a larger loading value allows for a more reasonable loading method, amplifying the deflection change better, resulting in a more accurate recognition of hanger damage.

In the actual identification of damage, we can reduce the number of sensor placements, not every hanger and tie-beam anchorage point should be included. We found that the three hangers near the arch foot can determine only the damaged area using this method; one hanger anchorage point from hangers 1–3 is sufficient for inclusion in the identification process. The identification results from Nos. 4 to 10 were entirely consistent. The identification index changed the most for the damaged hanger, and the influence line of the deflection difference at the anchorage points of several nearby hangers was also sensitive. Therefore, it is not necessary to place sensors at each anchorage point. For example, for the selected arch bridge, the influence matrix can be constructed using the deflection difference influence lines at the anchorage points of hangers No. 2,5, 8, 11, 14, and 17 and the tie beams. This will result in a non-square influence matrix. Therefore, for a specific real bridge, the sensors can be arranged at intervals.

5 Conclusion

In this study, the influence matrix of the deflection difference between the hanger and tie-beam anchorage points was used as an index to identify damage to hangers. The proposed method is based on the deflection difference influence lines at the anchorage points of each hanger and tie beam. The influence matrix is composed of these influence lines to create the identification index. This was verified using a tied arch bridge. The results show that:(1) In the past, when hanger damage was identified based on static deflection, the deflection change caused by the hanger damage was not significant, and difficult to measure accurately. Therefore, this study proposes the application of a load to a structure to amplify the deflection change in the beam caused by hanger damage. The results showed that the identification effect of this method can be adjusted by changing the loading amount and loading mode. In the normal bearing capacity range of the structure, a larger loading value allows for a more reasonable loading method, amplifying the deflection change better, resulting in a more accurate recognition of hanger damage.

(2) If the damaged hanger is a short hanger located near the arch foot, the proposed method identify the area in which the damaged hanger is located. When a hanger at another locations is damaged, it can be accurately located. The damage index increases with the degree of damage; therefore, this method can also quantify the damage.

(3) The sensitivity of the index differs with the position of the hanger, and when the hanger is closer to the middle of the span, the index is more sensitive to damage to the hanger. (4) In practical applications, sensors need not be placed at the anchoring points of each hanger and beam, and one sensor can be used for the three hangers in the arch foot area. Hangers in other positions can be equipped with a sensor at interval, which can not only obtain better identification results but also be economical.

(5) The deflection difference influence matrix provides more accurate structural stiffness information than the single-point deflection and deflection difference influence lines. The field applications were also found to be more robust.

(6) The identification method proposed in this study is most suitable for long-span rigid arch-rib flexible-beam arch bridges. In future studies, this method can be applied to long-span cable-stayed bridges and suspension bridges.

Additional information

No additional information is available for this paper.

Data and code availability statement

Data related to this study was not deposited in a public repository. The relevant data involved in this paper is part of the ongoing research, and in order to prevent illegal elements from using it, it cannot be shared for the time being, but those who need the data can contact us and provide data under their real names. So the data will be available on request.

CRediT authorship contribution statement

Weiwei Wang: Writing – review & editing, Writing – original draft, Validation, Software, Resources, Methodology. Weili Chen: Methodology, Data curation. Hongbin Xu: Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization.

Declaration of competing interest

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Acknowledgment

This paper is supported by the 10.13039/501100012166 National Key Research and Development Program of China (2022YFB2603300 ), 10.13039/501100001809 National Natural Science Foundation of China (NSFC) under grant No. 51278315 , 2019 graduate innovation funding project of Shijiazhuang Railway University, No.YC2018005 , Shenzhen Science and Technology Program(JCYJ20220818095608018 , KQTD20180412181337494 ), China Postdoctoral Science Foundation(2022M722188 ). The all authors of this paper expresses there sincere gratitude to the government for its financial support.
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