Damage Identification Method of Tied-Arch Bridges Based on the Equivalent Thrust-Influenced Line

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Introduction
As one of the most common bridge types in China [1], the arch bridge presents reasonable structural forces and a beautiful shape [2,3].With the increasingly mature analysis theory of arch bridges [4], tied-arch bridges are widely used due to their characteristics of an explicit force transmission path and large span capacity [5,6].In terms of structural stress, tied-arch bridges have a two-hinged main arch structure, with horizontal thrust at the arch foot withstood by the tie beam and no horizontal external force generated at the arch foot [7].Terefore, uneven settlement of the pier will have less infuence on the internal force distribution of the structure, which has a stronger capacity to ft with the terrain.However, in recent years, accidents of through tied-arch bridges and half-through tied-arch bridges have occurred frequently and commonly and the failure of main components (such as suspenders and arch ribs) will bring about changes in the force transmission paths, thus resulting in structural instability and damage [8].Meanwhile, arch ribs and suspenders, as the main stressed components in the tied-arch bridge structural system, comprise the main paths for bridge decks-transmitting loading and once they are severely damaged, major structural safety accidents will occur [9].
In the face of the increasing number of arch bridge accidents, domestic and international scholars have carried out many useful scientifc studies [10].Civera et al. [11] tested and validated a novel multistage clustering algorithm for SHM applications of automatic OMA (AOMA), especially for damage detection and severity assessment of masonry arch bridges.Ge et al. [12] presented an experimental validation for a high-precision vision-based displacement-infuenced line (DIL) measurement system for the purpose of damage detection on bridges.Lonetti et al. [13] proposed a numerical investigation to identify the instability strength of tied-arch bridges due to vertical loads.Bozyigit et al. [14] proposed a dynamic stifness method-(DSM-) based practical approach is developed to calculate vibration frequencies and mode shapes of masonry arch bridges.Chen et al. [15] proposed a quantifcation method of structural damage by reconstructing the defection-infuenced line matrix using the matrix decomposition method, revealing the relationship between structural damage and change in the defectioninfuenced line.Fan et al. [16] proposed an identifcation method for the damage of displacement diference-infuenced lines of tied-arch bridges, derived the displacementinfuenced lines of tied-arch bridges with the force method equation, and verifed the efectiveness of those displacement diference-infuenced lines derived in identifying the suspender damage of arch bridges through the fnite element model.Breccolotti and Natalicchi [17] proposed a bridge damage identifcation method with bridge defection, cornerinfuenced line, and dynamic weighing system combined.Zhu et al. [18] proposed a damage identifcation method for longspan bridges based on the strain-infuenced line and information fusion.Based on the main analytical parameters of extreme events and limit states of tie beam failure, Fan et al. [19] proposed a robustness evaluation framework for tie beam failure, which can be used to evaluate the safety of in-service bridges.Based on the convolutional neural network, Duan et al. [20] proposed a damage identifcation method for tiedarch bridge suspenders, with its robustness having been proved in various noise environments.Based on the Ritz method, Feng et al. [21] introduced the Hellinger-Reissner variational theory to improve the rapid decision-making approximation solution for parameters of tied-arch bridges and used this theory in the deformation and force analysis of tied-arch bridges for the frst time.Tese existing studies have provided theoretical support and a method reference for the damaged calculation and health assessment of arch bridge structures and further revealed the explicit analytical relationship between damage of tied-arch bridges and important mechanical indices, thus providing an important way for scientifcally evaluating the damage status of tied-arch bridges' components.Tis paper has proposed an accurate method for identifying the damage of the main stressed components of tied-arch bridges, thus providing a solution for this key scientifc problem in structural health assessment of bridges.
Because horizontal thrust at the arch feet of tied-arch bridges is balanced by tie beams [22], arch ribs serve to ofset the horizontal thrust through the tension previously exerted by tie beams, so that supports at arch feet will not produce horizontal external force.Under such a circumstance, the infuence line of tie beams' axial force, which is easy to test in practice, can be viewed as the equivalent thrust-infuenced line (ETIL).In the current research, the infuence line is a perfect parameter for assessing the health status of tied-arch bridges [23].In view of this, with the force method equation, two analytical solutions of the axial force-infuenced line of the two-hinged arch tie beam with the variable section are derived in this paper.Tat is, such parameters as the equivalent thrustinfuenced line, explicit expression of damage, and ETIL of tied-arch bridges are established in this paper, with a new damage identifcation index of equivalent thrust-infuenced line diference curvature (ETILDC) proposed.Furthermore, through numerical simulation, the accuracy of the inferred analytical solution is evaluated, with its application feasibility of damage identifcation studied.In summary, this paper has provided theoretical support and an ideal reference for design calculation and health assessment of tied-arch bridges.

The Establishment of Analytical Solution
2.1.Teoretical Basis.As one of the main stressed components of a tied-arch bridge, the arch rib has a typical twohinged arch structure with variable sections.Figure 1 shows the mechanical calculation the model of the arch rib.F P is the unit moving load.Te arch axis equations for the parabola and catenary variable sections of the two-hinged arch (equations ( 1) and ( 2)) [24] and the calculation formula for the arch-axis geometric parameter k are equations ( 3) and (4) [24] are as follows.
where L represents the half-span of the two-hinged arch, f is the vector height, m is the arch coefcient (g j is constant load strength at the arch foot and g d is constant load strength at the arch vault), and x is the horizontal distance from any section of the arch to the arch vault.Arch sections can be set based on Ritter's formula (equations ( 5) and ( 6)) [24], and the variable cross-section heights of two arch-axis types can be calculated with equations ( 7) and ( 8).

2
Structural Control and Health Monitoring Te height of the parabola axis is as follows: Te height of the catenary axis is as follows: where n is the coefcient of change in the arch rib section, φ is the horizontal angle of the arch section, I 0 and h 0 are the moments of inertia of the arch vault and height of the arch vault section, respectively, and A 0 is the cross-sectional area of the arch rib vault position.

Derivation of Analytical Solution.
Te superfuous constraint along the horizontal direction at the arch foot is replaced by redundant force F H . F H represents both the thrust at the two-hinged arch feet and the tie beam axis force of the two-hinged arch, as shown in Figure 1.
Te upper arch ribs of two-hinged arches with and without tie beam have the same form.However, tie beam deformation needs to be considered when the δ 11 parameter of the two-hinged arch structure with the tie beam is calculated with equations ( 9) and (25).
Te fexibility coefcient is as follows: Te displacement equation is as follows [25]: where ∆ 1p is the displacement of the basic structure (Figure 1) in the direction F H under the action of the load alone, E is the elastic modulus of the arch rib, E * is the elastic modulus of the tie beam, A * is the tie beam section area, and l is the tie beam length (l � 2L).
Te bending moment equation is as follows: Te axial force equation is as follows: Te bending moment equations are as follows.
Te elastic displacement at any point subjected to the load can be simplifed according to the displacement reciprocity theorem and virtual work principle of deformation as follows: where Δ is the deformation displacement; EI, EA, and GA are the beam section's bending, tensile, and shear stifness, respectively; and M, F, and Q represent the bending moment, axial force, and shear force, respectively.Te three integral functions represent bending deformation, axial deformation, and shear deformation, respectively.Structural Control and Health Monitoring Terefore, in order to facilitate the analytical calculation, the axial deformation of arch ribs is not considered in the derivation, with only the axial deformation of tie beams considered (equation ( 16)).
Te fexibility coefcient equation is as follows: Te displacement equation is as follows: where x p is the load action position, y is the mathematical expression of the arch axis, φ is the horizontal angle of the arch section, and x is the horizontal distance from any section of the arch structure to the arch vault.
Te parabola ftting formula is as follows: Te catenary ftting formula is as follows: Te thrust-infuenced line expression formula is as follows: With equations (1)-( 5) and ( 20) and ( 21) substituted into equations ( 17) and (18), the analytical expression of redundancy force for the tie beam with two-hinges arch can be obtained, as shown in Table 1.With equations (1), ( 2), (17), and (18) substituted into equation (22), the analytical solution of the tie beam axial force-infuenced line on the twohinged arch of the tie beam is obtained.Te ETIL analytical solution for the tie beam with two-hinged arch can be obtained, as shown in Table 2.

Accuracy Analysis of Analytical Solution.
With the MIDAS/Civil fnite element software, a model of two-hinged arch ribs with the parabola and catenary tie beam is established in this paper, with the fve rise-span ratios of 1/4, 1/5, 1/6, 1/7, and 1/8 applied.Linear-elastic material behavior in the fnite element model is assumed [26].Te parameters of this arch rib model are as follows.
Te span diameter is 50.934m, the vault section size is 1 m × 1 m, and the tie beam section size is 1 m × 1 m.C40 is used as the material of the arch rib.Te elastic modulus of C40 concrete is 3.25e + 07 kN/m 2 , the Poisson ratio of C40 concrete is 0.2, the volumetric weight is 25 kN/m 3 , and the coefcient of linear expansion of C40 concrete is 1.20e − 05 (1/ °C).C50 is used as the material of the tie beam.Te elastic modulus of C50 concrete is 3.45e + 07 kN/m 2 , the Poisson ratio of C50 concrete is 0.2, the volumetric weight is 25 kN/m 3 , and the coefcient of linear expansion of C50 concrete is 1.0e − 05 (1/ °C).
Also, the arch rib variable sections of this model can be determined using equations (7) and (8).
Tis fnite element model has a total of 158 nodes and 158 elements, with its tie beam structure divided into 108 beam elements and a tie beam element length of 1.01868 m, as shown in Figure 2. In this fnite element model, the arch rib is loaded with a quasistatic-infuenced line, with a total of 109 times of loading with a loading step of 0.5 m.To make the model as close to reality as possible, the software does not ignore the axial deformation and shear deformation of the arch rib.In this paper, the ETIL at the 1# measurement location of the arch rib can be extracted.Figure 3 shows the ETIL curve of the two-hinged arch with the tie beam drawn with the horizontal distance of the unit force application position from the arch vault viewed as an independent variable and the infuence line coefcient viewed as a dependent variable.
From Figure 3, it can be seen that the analytical value of the ETIL with the tie beam is less than that without the tie beam and the parabola with the tie beam is close to the amplitude of the ETIL of the two-hinged arch of the catenary with the tie beam.Based on equations ( 17) and ( 22), it can be seen that when the elastic modulus of the tie beam is greater than that of the arch rib, the analytical solution presents basically equal values on the tie beam, with similar force states obtained.Assume that the numerical solution presents a true value of S and the analytical solution presents an approximate value of S, then the relative error  S can be calculated with (23).With the two curve sets ((a)-(e) and (f )-(j)) presented in Figure 3 substituted into equation ( 23), it can be calculated that the maximum relative errors of parabola and catenary two-hinged arches with tie beam are 10.29% under diferent rise-span ratios (and under a 1/8 rise-4 Structural Control and Health Monitoring  Structural Control and Health Monitoring span ratio of catenary two-hinged arch with the tie beam) and it can also be seen that with the decrease in the rise-span ratio, the diference between the fnite element analysis results of numerical simulation and analytical solution gradually increases.Besides their bending deformation, the axial deformation of fat arches with a rise-span ratio less than 1/5 should not be overlooked.Terefore, the analytical solution of ETIL of two-hinged arches with tie beams can present more accurate results on those steep-arch structures with a rise-span ratio greater than 1/5.As shown in Table 3, the relative errors between numerical simulation and analytical solution of the critical section ETIL of two-hinged arches with tie beams are negatively correlated with their rise-span ratios.
According to Table 3, the relative errors between the ETIL analytical solution and ETIL numerical simulation of the critical section of two-hinged arches with tie beams increase with the decrease in arch rise-span ratio, with a maximum relative error of 9.57% under an arch rise-span ratio of 1/7.Terefore, the ETIL analytical solution of the critical section of two-hinged arches with tie beams is applicable for evaluating parabola and catenary two-hinged arch structures with tie beams under a rise-span ratio greater than 1/7. Figure 4 presents the patterns of these relative errors.It can be seen that in the 4l/8 and l/8 sections of these arch structures, there appear the smallest relative errors between the ETIL analytical solution and ETIL numerical simulation.Among those arches with fve diferent rise-span ratios, there are the largest relative errors between the ETIL analytical solution and ETIL numerical simulation on the critical sections of two-hinged arches with tie beams under a rise-span ratio of 1/8.

Damage Identifcation Method of Planar Tied-Arch
Structures.Concerning the ETIL analytical solutions of the parabola and catenary arch structures with the tie beam in an undamaged state as shown in Table 2, the elastic modulus at the damaged arch rib is represented by E ′ , the elastic modulus at the damaged tie beam is represented by E * ′ , and the ETIL at the damaged state is represented by F * H .In the curvature formula, 1/ρ(x p ) � |F H ″ |/(1 + F ′ 2 H ) 3/2 ; the frst derivative of thrust F H is approximately equal to zero.Terefore, this formula can be simplifed to the following formula: 1/ρ(x p ) � |F H ″ |.So, the formula of ETILDC before and after damage can be written as 1 When the load moves to the damaged region, (F H − F * H ) ″ is not equal to zero, leading to an abrupt change in the curve.It proves that the ETILDC can be used for locating bridge damage.
From Table 4, it can be seen that values of ETILDC in all damaged sections are not equal to zero, which indicates the feasibility of locating the damage of the stressed components of the planar tied-arch structures by ETILDC.Te appearance of the suspenders makes the force transmission path of the planar tied-arch system clear; theoretically, the damage of the suspenders can be refected by ETILDC.In order to perform research on the efectiveness of the ETILDC index in identifying damage to planar tied-arches, a planar tied-arch model with suspenders is established in this paper.In this FE model which is established by MIDAS/Civil software, linear-elastic material behavior is assumed.C50 is used as the material of the arch rib, which has a span of 63.8 m and a vector height of 11.86 m, wherein the elastic modulus of C50 concrete is 3.45e + 07 kN/m 2 , the Poisson ratio of C50 concrete is 0.2, the volumetric weight is 25 kN/m 3 , and the coefcient of linear expansion of C50 concrete is 1.20e − 05 (1/ °C).Te elastic modulus of the suspender is 2.06e + 08 kN/m 2 , the Poisson ratio of the suspender is 0.3, the volumetric weight is 76.98 kN/m 3 , and the coefcient of linear expansion of the suspender is 1.0e − 05 (1/ °C).Quasistatic-infuenced line loading is carried out on the beam of the planar tied-arch with suspender in this FE model, as shown in Figure 5.A total of 29 loading steps are set on the 28 units of the tie beam.Te cross-sections of the tie beam have the structure of single room box beam, and the preset initial tension of the suspenders is shown in Figure 6.Bozyigit et al. ([27], [28]) had modelled damage as stifness reduction considering linear rotational spring.In this paper, the damage is simulated with the reduction of the elastic modulus of damaged elements.As shown in Table 5, the damage extent of the midspan arch beam, midspan arch rib, and midspan arch suspender are set as 10%, 30%, and 50%, respectively.Ten, the efectiveness of ETILD in identifying and locating planar tiedarch structure damage is investigated in this paper.
As shown in Figure 7, with ETILDC, damage to the midspan arch beam, arch rib, and suspender of the planar tied-arch structure can be well located.When there is the same extent of damage to diferent stress components in the midspan arch location, ETILDC is more efective in identifying the damage of the tie beam than in identifying the damage of the suspenders.Meanwhile, it is more efective in locating the damage to the suspenders than in locating the damage to the arch rib.Te ETILDC curves presented in Figures 7(a)-7(c) show that the damage of the tie beam and suspender is more obvious.Under the same damage extent of diferent stressed components, there are diferent ETILDC curve patterns among these stressed components.Terefore, based on the amplitudes and patterns of ETILDC curves, the damage of stressed components of planar tied-arch structure can be roughly judged.Tis result shows that the ETILDC index efectively identifes the damage of the main stressed components of the planar tied-arch structure.

Loading Model for Tied-Arch Bridge Damage Identifcation.
In practical engineering, it is difcult to perform direct loading on arch ribs of through tied-arch bridges and halfthrough tied-arch bridges.In this paper, the axial forceinfuenced line of the tie beam at the arch foot is equivalent to the thrust-infuenced line of the arch foot of the twohinged arch.Terefore, the ETIL can be obtained by extracting the axial force-infuenced line of the unit (unit25#) at the connection between the tie beam and the arch foot.ETIL of the tie beam section (Figure 8, unit 25#) of the tied-arch bridge can be extracted through the loading of moving unit force on the bridge deck, and the efectiveness of ETIL in identifying the damage of the main stressed components of tied-arch bridges are further verifed.
In  6, and the damage position numbers of bridge suspenders are shown in Figure 9.A measuring location is set on unit 25# near the arch foot of the bridge tie beam.From Figures 10(a)-10(c), it can be seen that peaks corresponding to load steps 31#, 61#, and 91# appear on the ETILDC curves, and when load steps 31#, 61#, and 91# are performed on the bridge deck, damage at the 3#, 6#, and 9# suspenders are located, respectively.With comparisons among curves presented in Figures 10(a) and 10(d), curves presented in Figures 10(b) and 10(e), and curves presented in Figures 10(c) and 10(f ), it can be seen that when a pair of suspenders on a same cross beam have the same damage extent, the damaged suspender near the measuring location presents large peak values on ETILDC curves than that damaged suspender far away from the measuring location, showing great efectiveness in Structural Control and Health Monitoring damage identifcation.Tis result shows that the ETILDC index presents high accuracy in locating the damage of the suspenders of arch bridges.
From Figure 11, it can be seen that peak values of ETILDC curves occur between load steps, indicating damaged arch ribs on the corresponding load sections in the FE model.Under work condition 7, the maximum damage peak value among ETILDC curves is close to 2.5E− 5 with a damage condition of 50%, while under work condition 8, the maximum damage peak value is only close to 2.5E− 6. Terefore, it indicates that if arch ribs on both sides of a tied-arch bridge are damaged at the same position to the same extent, ETILDC is more effective in identifying the damage of those arch ribs on the same side of the bridge with the measuring location.A comparison of curves presented in Figures 11(e   measurement.Also, the work condition with noise is set as work condition 13, with the following noise introduction equation: where T N i represents the thrust data containing noise under the i loading step; T i is the thrust data without noise under the i loading step; RAND(− 1, 1) is the standard normal distribution of random numbers; and μ is the noise extent level.
As can be seen from Figure 12, the ETILD curve with noise presents a generally consistent pattern with that pattern of the ETILD curve without noise.In order to further investigate the noise-immunity performance of ETIL, grey relation analysis (GRA) [29] was performed in this research to evaluate the noise immunity of damage identifcation indicators.Based on cybernetics, grey correlation analysis applies a method of multifactor statistical analysis [30].Tis method uses colors to represent how much information is known about a system, with white representing sufcient information, black representing a system whose structure is  that is, only a part of the system is understood [31].In this grey system, it is crucial to determine the parent sequence, conduct dimensionless processing of data through the mean value, and calculate the grey correlation coefcient through the following equation: where ξ i (k) is the correlation coefcient of x i with y(k) at point k; |y(k) − x i (k)| is the absolute diference between y and x i at point k; min i min k |y(k) − x i (k)| is the minimum absolute value of the second-order diference between y series and x i series at k point; max i max k |y(k) − x i (k)| is the maximum absolute value of the second-order diference between y series and x i series at k point; and ρ is the grey resolution coefcient, which ranges from 0 to 1, with a general value of 0.5.Meanwhile, the correlation degree r i between x i and y(k) can be obtained through the Structural Control and Health Monitoring substituting of the correlation coefcient of each character into the following equation: Te grey correlation coefcient is calculated in this research with work condition 13 after the introduction of noise retaken as a sample.Te grey correlation coefcient can refect the correlation degrees of data, and with the size of this coefcient, the correlation degree between noiseless data and noise data can be investigated.Trough the calculation in this research, it can be seen that the grey correlation coefcients with and without noise obtained the values of 0.6365 and 0.7173, respectively.Compared with the error result of the thrust-infuenced line without noise, the error result of the thrust-infuenced line with noise is 11.26%, indicating that the ETIL index has good noise immunity in identifying the damage of tied-arch bridges.

Thrust-Influenced Line Recognition Based on
Variational Mode Decomposition VMD (variational mode decomposition) is a signal-processing method with adaptive model variation [32].Te adaptive function of the VMD method is to determine the number of mode decomposition of time series according to the actual situation.Te optimal solution of variational modes can be searched with the iterative method [33,34], and the timedomain signal can be decomposed into K fnite bandwidths of IMF ranging from high frequency to low frequency.It is assumed that IMF with limited bandwidths is disturbed around their respective central frequencies, and K modes can be iteratively searched.Tis research set a constraint condition that a combination of all modes makes the original signal [35,36], and all the models have a minimized combination of estimated bandwidths.Te numerical model is expressed in the following equation: To solve the optimal problem of constrained variation, the quadratic penalty factor α and the alternate multiplication operator λ(t) are introduced to transform ( 27) into an equation of unconstrained variation.Te unconstrained Lagrange function is presented in the following equation: Te "saddle point" in ( 28) is obtained through the use of the alternating multiplication operator, with u n+1 k , ω n+1 k , and λ n+1 k iteratively updated.Te update formulas are presented in equations ( 29)- (31).
where ∧ represents the Fourier transformation and τ represents the time step.With w l k  , u l k  , λ 1 , and n initialized and u k , ε, and λ iteratively updated until the allowable error ε is satisfed, decomposition is stopped and IMF components are fnally put as the output (the number of IMF is K).Te allowable error criterion is expressed in the following equation: Numerical simulation was performed based on the condition of the original ETIL with measured dynamic disturbance in a 30% damage state in work condition 1. VMD is used to eliminate dynamic components in the numerical simulation.Ten, the quasistatic time-history response of the bridge can be obtained, as shown in Figure 13.
It can be seen from Figure 13(a) that with the VMD process method, dynamic components in the thrust response under the loading of the moving vehicle can be efectively eliminated, with a smooth quasistatic thrust infuence curve obtained.Figure 13(b) shows that VMD processing will reduce the peak value of the thrust infuence curve.Terefore, the quasistatic-infuenced line after VMD processing has a smaller shape than the static thrustinfuenced line of the structure.

Practical Process for Tied-Arch Bridge
Damage Identification Structural Control and Health Monitoring infuence lines of tied-arch bridges, it is difcult to simulate the unit concentrated forces of infuence lines through the application of single-axis-concentrated load.It is an important method to research the damage of bridges by the moving load and dynamic response analysis [37].Terefore, this paper has proposed a three-step loading process based on moving load reduction and this process can be used for the rapid diagnosis of bridge damage under short trafc interruption [38], as shown in Figure 14.
(1) Tree three-axle vehicles with the same wheelbase and diferent front and rear axle weight ratios are selected as loading tools.Te front and rear axles of each vehicle can exert diferent amounts of concentrated forces at corresponding positions.In the process of loading, concentrated forces should be efciently applied to the tied-arch bridge to ensure the stability and efectiveness of the structural response measurement.(2) Te three vehicles are, respectively, idle mobile loading on the tied-arch bridge deck to load the tiedarch bridge.Te virtual loading of the tied-arch bridge is conducted repeatedly three times at the same position, which can be achieved by controlling the speeds of loading vehicles and collecting those three thrust response data at the same time.

Conclusions
(1) In this research, it is observed that with the decrease of rise-span ratio, axial deformation has a gradually increased infuence on the arch structures.Under a rise-span ratio greater than 1/7, the calculation error of the derived practical analytical solution for ETIL is less than 9.57%, indicating that this solution can efectively calculate and analyze the damage of arch structure with a tie beam under a rise-span ratio ranging from 1/4 to 1/5.(2) Te analytical diference curvature of ETIL can be efectively used in the damage identifcation of planar tied-arch structures.Tie beams, suspenders, and arch ribs are the main stressed components of arch bridges suitable for damage identifcation.In actual measuring scenarios of ETIL at the arch feet, measuring locations along infuence lines can be arranged near those tie beams that are close to the arch feet units.(3) It has been proved that the curvature method established based on the diferential efects of ETIL is efective in identifying damage in suspenders and arch ribs at diferent positions of tied-arch bridges.Tis method is particularly helpful in assessing the health status of bridge suspenders.Qualitative analysis shows that the identifcation of damage extents of diferent bridge components relies on the index amplitude with good noise immunity by grey correlation analysis.(4) In this research, thrust-infuenced line recognition based on VMD (variational mode decomposition) is introduced and a practical approach for assessing the health status of arch bridges based on quasistatic impact lines was proposed; this approach is suitable for efciently testing the impact lines of crucial components of arch bridges under the static loading of three-axle vehicles.With a combination of existing bridge monitoring methods, the identifcation method of arch bridge damage proposed in this paper has the prospect of facilitating routine health assessment of in-service arch bridges in the future.(5) Tis research is helpful in developing methods for damage diagnosis and load-bearing capacity assessment of tied-arch bridges.It provides a new idea for structural health monitoring of tied-arch bridges.
Te infuence line has the function of refecting the structural characteristics.Te damage identifcation and diagnosis method of the tied-arch bridge based on ETIL is high efciency, so the health detection and monitoring based on the equivalent thrustinfuenced line has a good development prospect.

Figure 2 :Figure 3 :
Figure 2: Finite element model of two-hinged arch rib with the tie beam.
Numerical simulation with tie beam Analytical solution without tie beam Analytical solution with tie beam 0

Figure 4 :
Figure 4: Comparison of relative errors between analytical solution and numerical simulation of the critical sections of two-hinged arches with diferent arch axis types.(a) Relative errors between analytical solution and numerical simulation of the critical sections of parabola two-hinged arches with the tie beam.(b) Relative errors between analytical solution and numerical simulation of the critical sections of catenary two-hinged arches with the tie beam.

Figure 7 :
Figure 7: ETILDC damage identifcation curves of planar tied-arch structure.(a) ETILDC curve of the midspan arch beam.(b) ETILDC curve of the midspan arch rib.(c) ETILDC curve of the midspan arch suspender.

Table 1 :
Analytical expression of redundancy force.

Table 3 :
Comparison between results of analytical calculation and fnite element numerical simulation.
Note.Te load action position refers to the loading position on the critical section of arch structure.
this FE model which is established by MIDAS/Civil software, a tied-arch bridge model with a span of 60 m, a vector height of 12 m, and a total of 11 pairs of suspenders are established and linear-elastic material behavior is assumed, 3, and the coefcient of linear expansion of wind brace is 1.20e − 05 (1/ °C).With a total of 121 loading steps with a loading length of 0.5 m, a quasistatic moving concentrated force is applied to this loading model, as shown in Figure8.3.2.1.Analysis of Suspenders' Damage Identifcation.Te preset damage conditions (1-6) of bridge suspenders are listed in Table

Table 4 :
Expressions of ETILDC at damage region of arch bridges under two arch-axis types.

Table 5 :
Damage conditions of planar tied-arch structure.
44246E − 5, 1.46345E − 5, and 9.29704E − 6, respectively.Besides the fact that ETILDC is most efective in locating the damage of arch ribs at the midspan arch units, there is a negative correlation between the distance between measuring location and damage location and the efectiveness of ETILDC in locating damage.Terefore, the results of this research can be complemented with the addition of more3.2.3.Indices of Noise-Immunity Verifcation.Test or environmental noise may cause deviations between real data and measured data.Terefore, noise is introduced in this paper to verify the immunity of ETILD.Under work condition 1 and with a 10% damage extent applied to the 3# suspender, noise with a 5% level is introduced to the infuence line to simulate the noise situation of actual