Evaluation of the Effect of Loose Bridge Bearing on Onboard Measured Track Geometry Using Numerical Analysis

Abstract

Detection of loose bridge bearings with an uplift gap in steel bridges requires visual in situ inspection, which is labor-intensive. This study investigated the effect of loose bearings on track geometry using numerical calculation, as part of a fundamental investigation into detecting loose bearings using track geometry. A non-linear spring representing the loose bearing was introduced into the existing calculation tool, identifying the loaded track geometry considering the structural deformation. Results of a simulation using this tool clarified that the displacement of loose bearings appears in track geometry as a local fluctuation with a half wavelength of about 5 m, regardless of the size of the uplift gap at the loose bearing.

1. Introduction

An important inspection item in the maintenance and management of steel railway bridges is the detection of loose bearings [1]. A loose bearing refers to a phenomenon in which gaps that have formed between the sole plate and the lower bearing, or between the lower bearing and the bearing seat―due to damage to the bearing seat mortar, settlement of the bearing, or corrosion and wear―are compressed and uplifted when a train passes over. Loose bearings are a concern since they can increase stress in the bearing parts and surrounding components, which leads to fatigue cracks and loose bolts. Therefore, the occurrence and condition of loose bearings need to be correctly detected and monitored, and measures should be taken as needed [2].

Loose bearings are detected by visual inspection from under the girder, but this requires considerable time and expenditure [3]. In addition, inspection on lines with little traffic is even more time consuming because of the need to wait for a passing train to detect loose bearings based on their movement when a train passes. Therefore, detection methods using sensors installed under girders have been developed to reduce the labor and cost intensity of such inspections [4] [5]. However, there was a period when steel railway bridges on conventional lines were built with standard girders, many such bridges are still standing today [6]. Installing equipment under girders on these bridges still requires massive costs and effort.

Meanwhile, progress has been made in recent years in research that uses data measured by onboard sensors to evaluate bridge performance as they pass over them [7]. An issue in previous research has been the elimination of track geometries measured under unloaded conditions such as rail irregularities and distortions, as well as the extraction of structural deformation components. Some of these issues were solved using Matsuoka et al.’s theory [8] based on the use of multiple data from onboard measurements. This has resulted in the development of a method for estimating bridge resonance conditions and girder deflection from onboard measured track geometries, which has been used in practical applications [9] [10] [11]. These methods were mainly aimed at high-speed railways, but onboard measurement data for conventional railways has also been examined in recent years. Hattori et al. [12] [13] [14] developed a girder deflection-track geometry conversion program that calculates loaded track geometry from bridge deformation in order to establish a girder deflection estimation method using two track geometries obtained by a two-bogie track inspection vehicle used for track inspection on many conventional railways (henceforth, “two-bogie inspection vehicle”). This program also allowed for the evaluation of the influence of adjacent bridges, which was an issue when estimating girder deflection using a two-bogie inspection vehicle.

Improving the above-mentioned method for estimating girder deflection using track geometry and detecting loose bearings on bridges using onboard-measured track geometry can significantly reduce the labor needed for inspections from below the girder. However, the detection of loose bearings using onboard measurement data such as track geometry has hardly been examined to date, and to begin with, the effect of loose bearings on track geometry has not even been clarified. Therefore, the effects and characteristics of loose bearing behavior and track geometry need to be clarified, and a method for detecting loose bearings using those characteristics needs to be developed.

Based on these considerations, we conducted fundamental research to develop a method for detecting loose bearings from track geometry obtained with a two-bogie inspection vehicle. To this end, we first extended the existing girder deflection-track geometry conversion program to consider a nonlinear spring representing loose bearings. Next, we used the extended conversion program in order to conduct a numerical analysis of a steel railway bridge with a span length of 12.3 m. We clarified the loose bearing behavior and its effect on track geometry when a train passes. Specifically, we analyzed the bearing spring displacement, obtained track geometry and its wavelength characteristics.

2. Numerical method

2.1 Existing analysis method

In this section we explain the existing girder deflection-track geometry conversion program [12], which can calculate structure deformation from track geometry.

Figure 1 shows the calculation flow of the girder deflection-track geometry conversion program, and Fig. 2 shows the track geometry measurement using a two-bogie inspection vehicle. This program focuses on the two-track geometries measured onboard the two-bogie inspection vehicle shown in Fig. 2, namely the asymmetrical chord offset track geometry (ACTG) measured at axles 124 (“124 ACTG”) and the ACTG measured at axles 134 (“134 ACTG”), which are measured under different load conditions. The difference between these two track geometries is then taken to estimate the girder deflection [12].

Fig. 1　Calculation flow of girder deflection-track geometry conversion program

Fig. 2　Track geometry measurement using two-bogie inspection vehicle

The program consists of a structural analysis module that uses a two-dimensional finite element method (2D FEM) and a signal processing module that uses the track maintenance management database system LABOCS [15], which is used in the track maintenance field. The entire program is controlled by the numerical analysis software MATLAB.

The structural analysis models the structure using 2D elements such as springs and beams, as in normal FEM, and calculates the structure response when a loading sequence with the same axle arrangement and axle load as the two-bogie inspection vehicle runs through it. Because the effect of the dynamic response of the structure can be neglected, the structural analysis module calculates the static response of the structure repeatedly when a train passes through it using a simulation in which the loading sequence is gradually moved. It should also be noted that the “dynamic” and “static” terms differ from the “loaded” and “unloaded” track geometry terms used later in this text.

As a result, the loaded track geometries 124 and 134 ACTG, are calculated from the four axle position rail displacements obtained from the two-bogie inspection vehicle. There is a phase difference between the 124 ACTG and 134 ACTG, so these values were converted by filter processing into the 10 m symmetrical chord offset track geometry (124) and 10 m SCTG (134), which have no phase difference. Hereafter, the 10 m symmetrical chord offset track geometry (124) and (134) are referred to as 10 m SCTG (124) and 10 m SCTG (134), respectively.

Figure 3 shows the characteristics of the filter for converting from ACTG to 10 m SCTG. Although not subject to calculations in this study, the system has a function for calculating the difference between the 10 m SCTG (124) and 10 m SCTG (134), called the loaded track geometry difference, which has a linear relationship with the girder deflection.

Fig. 3　Characteristics of filter for converting ACTG to 10 m SCTG

Meanwhile, preliminary analysis showed slight differences in loose bearing behavior between each axle pass. Hence, the influence on the difference in the loaded track geometry was also small. Therefore, this paper focuses on the 10 m SCTG. In this case, the unloaded track geometry, which is removed by the difference in the loaded track geometry difference, must be removed separately from the 10 m SCTG. This will be described in detail in Section 3.

2.2 Expansion for loose bearings

The nonlinear spring was introduced into the structural analysis module of the existing girder deflection-track geometry conversion program. This expansion allows for the expression of loose bearings. In this paper, the gaps in the loose bearing are collectively referred to as “uplift gap” and modeled. Cases where the uplift gap is present result in a low bearing stiffness, but when the uplift gap is closed by the train load, the stiffness is thought to return to the same value as in the bearing without the looseness (uplift gap). Therefore, this is modeled as a bilinear nonlinear spring with one break point. We sought to reduce the computational load in the inverse analysis that we plan to develop in the future by using the following method, which does not require convergent computations, in order to solve the stiffness equation, including the nonlinear spring. It should be noted that the load and displacement are considered only in the vertical direction below.

First, we create a stiffness matrix K₁ that considers only the primary stiffness k₁ of a bilinear nonlinear spring defined by primary stiffness k₁ and secondary stiffness k₂. We then calculate the provisional displacement x* of all nodes when the external force vector F acts from the stiffness equation using Eq. (1):

  

$$x^{\text{*}}=K_1^{-1}F$$(1)

The relative displacement of the two ends of the nonlinear spring in x* gives the provisional displacement δ* of the nonlinear spring (displacement when the nonlinear spring is linear with a primary stiffness k₁). The break-point displacement of the nonlinear spring is set as δ₁, and the ratio r of the break-point displacement δ* of the nonlinear spring is defined as Eq. (2):

  

$$r=\frac{{\delta }^{\text{*}}}{{\delta }_1}$$(2)

When r <1, the nonlinear spring does not reach the secondary stiffness region, and the displacement x* obtained by Eq. (1) is the solution. We show the calculation method when r ≥ 1 below.

The calculations up to this point are for linear systems, so the external force vector F₁ required for the nonlinear spring to reach the break point can be calculated as Eq. (3) using the displacement ratio r:

  

$$F_1=\frac{F}{r}$$(3)

When this external force vector F₁ acts, the nonlinear spring has a displacement that just reaches the break point. The displacement vector x₁ of all nodes at this time can be calculated from Eq. (4):

  

$$x_1=K_1^{-1}{F}_1$$(4)

Since r ≥ 1, the load F₁ is a part of F. Therefore, we additionally calculate the deformation corresponding to the remaining load F₂, which is obtained by subtracting F₁ from the external force vector F. The load F₂ is given by Eq. (5):

  

$$F_2=F-F_1=F-\frac{F}{r}=\frac{r-1}{r}F$$(5)

The displacement of the nonlinear spring is the break point displacement δ₁ at the time of loading F₁, so only the secondary stiffness of the nonlinear spring is effective when F₂ acts. Therefore, the displacement x₂ of all nodes after the nonlinear spring reaches δ₁ can be calculated as follows, using the stiffness matrix K₂, in which the primary stiffness k₁ in the stiffness matrix K₁ is changed to the secondary stiffness k₂:

  

$$x_2=K_2^{-1}{F}_2$$(6)

Adding this to x₁ allows for the final displacement x ′ of all nodes to be calculated using Eq. (7):

  

$$x\text{'}=x_1+x_2=K_1^{-1}\frac{F}{r}+K_2^{-1}\frac{r-1}{r}F$$(7)

Figure 4 shows the load-displacement relationship of the bilinear nonlinear spring to be modeled, where P and P₁ in the Fig. 4 represent the load acting on the spring when the loading sequence is F and F₁, respectively, and δ represents the final spring displacement considering the secondary stiffness of the spring.

Fig. 4　Load-displacement relationship of bilinear springs to be modeled

2.3 Analysis target

The span length and bending stiffness can be set as parameters when analyzing the effect of loose bearings on track geometry. However, in this study, we have already assumed an actual bridge section to which the detection method is to be applied, so we decided to conduct a study using a model for that section [6].

Figure 5 shows the FEM model of the bridge section targeted in this study. We assumed an actual section with a continuous simple girder (steel bridge) with a bridge length of 13.1 m and a span length of 12.3 m, and we modeled a seven-girder bridge. From the left, the bridges were called B1, B2, …, B7. The nonlinear spring representing the loose bearing was placed on the train entrance or exit side of B4 at the center. The track inspection vehicle is assumed to be a two-car train running on the relevant section, with the first car being a two-bogie inspection vehicle and the second car being a diesel railcar accompanying the track inspection vehicle in the running direction. Only one side of the rail and bridge was modeled, and the loading sequence was the static wheel load of the track inspection vehicle converted to that of one rail. The bridge and rail were modeled using beam elements, and the track pad and bridge bearings were modeled using spring elements. FEM only modeled the vertical component, so the horizontal displacement of the bearing was not modeled.

Fig. 5　FEM model of bridge section analyzed in this study

Table 1 shows the specifications of the bridge and rail used. The girder and rail specifications were taken from drawings, and the stiffness of the track pad was set from material test results [16].

Table 1　Specifications of bridge and rail used in analysis

Rail bending stiffness	6.47×106 (N·m2)
Track pad stiffness	4.00×107 (N/m)
Girder bending stiffness	1.41×109 (N·m2)
Normal bearing stiffness	1.00×1011 (N/m)

The leading axle's initial position was approximately 25 m from the left end of the leftmost bridge B1. The vehicle moved 0.1 m to the right in the Fig. 5, and the quasi-static calculation was repeated until the rearmost load left the rightmost bridge B7. We recorded the bridge displacement and axle position, rail displacement at each load position, and calculated the 124 and 134 ACTG. We also calculated the 10 m SCTG (124) and 10 m SCTG (134) using a conversion filter from the two ACTGs.

2.4 Analysis case

We sought to understand the loose bearing behavior and its effect on track geometry in the section by analyzing the central bridge B4, changing the uplift gap amount in the bearing expressed by the break point displacement, and the entry and exit sides. Specifically, we considered not only a “no uplift gap” case but also 1, 2, and 5 mm uplift gap cases for the entry and exit sides, for a total of 7 cases.

Figure 6 shows the load-displacement relationship of the nonlinear spring representing the loose bearing. The secondary stiffness was set to 1.0×10¹¹ (N/m), the same stiffness as a bearing without an uplift gap. The actual state of the primary stiffness was unclear, so this was set to 1.0×10⁷ (N/m) in order to ensure that the stiffness was sufficiently smaller than the support stiffness of a normal bearing.

Fig. 6　Load-displacement relationship of nonlinear spring used in analysis

According to the Maintenance Standards for Railway Structures [1], the uplift amount is judged as AA if it impairs running safety, but there is no description of the uplift amount. The Design Standards and Commentary for Railway Structures [17] summarized running safety with respect to bearing displacement in terms of a vertical misalignment on the track surface, and set a design limit value for girder and misalignment under multiple-coupled conditions on conventional lines of 4 mm. It should be noted that the uplift gap amount did not directly become the amount of girder end misalignment. Given the above, we set the uplift gap at 5 mm as the amount that exceeds the design limit value of the girder end misalignment and that needs to be reliably detected, and set 2 mm and 1 mm, which are 1/2 and 1/4 of the design limit value, as realistic values.

3. Result of numerical analysis

3.1 Spring displacement of the bearing part

Figure 7 shows the spring displacement of the B4 entry side bearing with 1 mm, 2 mm, and 5 mm uplift gap cases, as well as a no uplift gap case. The no uplift gap case had only a slight spring displacement of ≤0.1 mm when a train passes, whereas the uplift gap cases exhibited a total of three downward displacement peaks occurring when a train passes. These three displacement peaks correspond to the front bogie of the first car (two-bogie inspection vehicle), rear bogie of the first car (two-bogie inspection vehicle), front bogie of the second car (diesel railcar), and rear bogie of the second car (diesel railcar), which each passed through the loose bearing points. The 1-mm and 2-mm uplift gap cases exhibited a bearing spring displacement that remained almost constant after reaching the set uplift gap. The 5-mm uplift gap case exhibited a peak spring displacement of less than 4 mm at the first peak, and the gap remained in this case. Additionally, at the second peak, the rear bogie of the first car (two-bogie inspection vehicle) and the front bogie of the second car (diesel railcar) were simultaneously positioned on bridge B4, which caused the spring displacement to reach an uplift gap amount of 5 mm, transitioning to secondary stiffness.

Fig. 7　B4 entry side bearing displacement for B4 entry side bearing uplift gap of 1 mm, 2 mm, and 5 mm, and for no uplift gap

Next, we look at the slope of the spring displacement. In Fig. 7 (1) slope, where the spring displacement increases in the direction in which the bearing uplift gap is crushed, the slope was −1.0 × 10⁻³. However, in Fig. 7 (2) slope, where the spring displacement decreases in the direction of the loose bearing, the slope is 4.0 × 10⁻⁴, thereby exhibiting a different slope value. This case has a loose bearing on the entry side of the bridge, and the load acting on the bearing increased sharply when transferring from the exit side of the adjacent bridge to the entry side of the bridge, so the absolute value of the slope when the spring displacement increased was thought to be large.

Figure 8 shows the displacement distribution of the rail, bridge, and loose bearing when the leading axle passes through the loose bearing part in the analysis result with a 2-mm uplift gap. We can confirm from the results that the spring displacement of the bearing increased sharply immediately after the leading axle entered the bridge. Cases where the bearing spring displacement decreased (e.g., Fig. 7 (2) slope) were due to a decrease in the load sharing rate of the entry side bearing as the load moved from the entry side to the exit side of the bridge, and thus had a smaller slope than in cases where the bearing spring displacement increased. The above discussion is also supported by the fact that the axle movement distance at the peak decreases when the uplift gap was 5 mm roughly corresponded to the span length. Given the above, we assumed that different structural elements were involved in the loose bearing behavior when a train passes, depending on whether the spring displacement increased (uplift gap is crushed) or decreased (uplift gap occurs). The increase in spring displacement occurred when the train moved over between bridges, suggesting that the load sharing by the rails between the bridges played an important role. Figure 8 also showed that large local deformation occurred in the rail at the loose bearing location. Meanwhile, the decrease in spring displacement was due to the load sharing of the bearings within the bridge, suggesting that this was mainly dependent on the bridge span length and axle arrangement.

Fig. 8　Loose bearing behavior during bridge transfer (B4 entry side bearing uplift gap 2 mm)

Figure 9 shows the bearing spring displacement when there was a loose bearing on the exit side. Because of the loose bearing on the exit side, the magnitude of the slope when the bearing spring displacement increases (Fig. 9 (3) slope) or decreases (Fig. 9 (4) slope), is roughly the opposite of the entry side case in Fig. 7. Additionally, the 5-mm uplift gap case exhibited a somewhat larger displacement at the axle 1 position of 70 m onward at the peak of the second downward displacement compared to Fig. 6. This was the effect of the second diesel railcar being loaded at the same time.

Fig. 9　B4 exit side bearing displacement for B4 exit side bearing uplift gap of 1 mm, 2 mm, and 5 mm, and for no uplift gap

In this paper, we differentiated the bearings between the entry side and exit side, but if the front and rear of a two-car track inspection vehicle runs in reverse, then the bearing spring displacements for the entry and exit sides will be obtained since this is a static analysis. In other words, what governs the phenomenon is not whether it is on the entry side or exit side, but the vehicle’s axle arrangement and the magnitude of the wheel load. In this paper, we used the terms “entry” and “exit” only for convenience, and it should be noted that the analysis was based on an example in which the two-bogie inspection vehicle is the leading car of a two-car track inspection vehicle.

3.2 Track geometry

Figures 10 and 11 show the 10 m SCTG (124) and 10 m SCTG (134) in the loose bearing on the entry side of B4, respectively. The 10 m SCTG (124) in the no-uplift gap case corresponded to a girder deflection shape, with a maximum at the bearing part, a minimum near the center of the span, and a waveform with a wavelength roughly equal to the span length. Cases where there is a loose bearing exhibit fluctuations in track geometry around the uplift gap location. Specifically, the upward peak position of the loose bearing location moved toward the train entry side, and the maximum value also increased. Furthermore, an increasing uplift gap amount resulted in the downward peak position at the center of the B4 span shifting toward the entry side, and the peak value also increasing.

Fig. 10　10 m SCTG (124) for B4 entry side bearing uplift gap of 1 mm, 2 mm, and 5 mm, and for no uplift gap

Fig. 11　10 m SCTG (134) for B4 entry side bearing uplift gap of 1 mm, 2 mm, and 5 mm, and for no uplift gap

The 10 m SCTG (134) shown in Fig. 11 showed the same tendency as the 10 m SCTG (124) in that the fluctuation in the peak position around the loose bearing location tended to shift to the entry side as the uplift gap amount increased, but unlike the 10 m SCTG (124), the peak value did not change considerably. This is because the 10 m SCTG (134) calculated the displacement of the axle 3 position relative to those of axles 1 and 4, but since axle 2 was still on the bridge when axle 3 passed through the loose bearing, the change due to loose bearing in the axle 3 position relative to the axles 1 and 4 was small.

Figure 12 shows the 10 m SCTG (124) when loose bearings were introduced on the exit side of B4. As with the entry side, fluctuations in peak position and peak value occurred around the loose bearing location. An increasing uplift gap resulted in an upward peak at the loose bearing location moving to the exit side, in the opposite direction to that of the entry side. The maximum value did not change when there was a 1 mm uplift gap, but increased when there was a 2 mm or 5 mm uplift gap. The same trend as on the entry side generally occurred in the direction opposite to the running direction.

Fig. 12　10 m SCTG (124) for B4 exit side bearing uplift gap of 1 mm, 2 mm, and 5 mm, and for no uplift gap

In summary, although tendencies differed somewhat depending on the size of the uplift gap on the loose bearing, both the track geometry corresponding to the girder deflection, and the phase shift and amplitude fluctuations of the loose bearing location, were superimposed at the 10 m SCTG obtained at the loose bearing location. The phase shift and increase in amplitude of this fluctuation differed depending on the size of uplift gap at the loose bearing location. Specifically, in the target range, the phase shift was only approximately 2 m when the uplift gap was 1 mm. However, in addition to the phase shift, there was also an amplitude increase of approximately 0.4 mm when the uplift gap was 2 mm or 5 mm. Therefore, loose bearings could be detected from this feature of the track geometry.

3.3 Analysis of loose bearings and wavelength of track geometry

In this numerical analysis, we evaluated not only the effect of loose bearings but also the track geometry, including the component of girder deflection. Previous research [12] showed that the component due to girder deflection could be evaluated from the loaded track geometry difference. Therefore, being able to separately estimate the component of the track geometry due to girder deflection and subtracting it from the loaded track geometry is thought to enable the extraction of only the loose bearing component with high accuracy. Therefore, further analysis was conducted only on the effects of loose bearings with the girder deflection component removed.

Figure 13 shows the loose bearing component of the track geometry calculated by subtracting the analysis result without a loose bearing from the analysis result of the 10 m SCTG at the loose bearing location. For each case, the fluctuation component at the loose bearing location was extracted, and its half wavelength (distance for one peak) was approximately 5 m regardless of the extent of the uplift gap. The two factors that caused this were thought to be the filter characteristics of the 10 m SCTG and the load dispersion effect of the rail. First, for the filter characteristics, the 10 m SCTG that was used as an index in this study had a large gain near a half wavelength of 5 m, as shown in Fig. 3, and a gain of 0 at half-wavelengths of 2.5 m or less. Therefore, even in the case of a rapid change in rail displacement at the bearing location due to loose bearings, evaluating this as a 10 m SCTG is thought to remove the short-wavelength components with no gain of 2.5 m or less, and the components near a half-wavelength of 5 m could be extracted. In addition to the filter characteristics of the 10 m SCTG, the load dispersion effect of the rail may have also affected the components with a half-wavelength of approximately 5 m. This is presumed to be due to the concentrated load of the train being dispersed by the rail, which makes the displacement change of the bearing spring more gradual, and due to the measured track geometry having a smooth waveform without discontinuities.

Fig 13　Loose bearing component of 10 m SCTG calculated excluding girder deflection component

4. Conclusion

The results of this study are summarized below.

・A bilinear nonlinear spring representing a loose bearing was introduced into the existing loaded track geometry calculation program (girder deflection-track geometry conversion program) in order to expand the program so that the loaded track geometry at loose bearing locations could be calculated.

・We targeted a seven-girder bridge with a span length of 12.3 m, and calculated the 10 m SCTG on loose bearing using the above-mentioned program. Results showed that loose bearings occurred as a half-wavelength of approximately 5 m at the girder ends of the bridge in question and adjacent bridges.

・For the bridge targeted in this study, the fluctuation wavelength of the 10 m SCTG occurring at the loose bearing locations had a half-wavelength of approximately 5 m, regardless of the bearing uplift gap amount. This was thought to be due to the filter characteristics of and the load distribution of the rail.

Additionally, this analysis only covered bridges with a specific span length that are assumed to be the actual target of application, so the findings obtained in this study still need to be generalized. Furthermore, this analysis did not consider unloaded track geometry which is superimposed in actual track geometry under loaded conditions in addition to the deformation components of the structure examined in this analysis[8]. Unloaded track geometry causes an error in loose bearing detection, so it needs to be removed. For example, measuring the unloaded track geometry separately from the track inspection vehicle and subtracting it from the loaded track geometry enables the extraction of only the deformation components of the structure from the loaded track geometry.

For future study, we plan to apply the proposed method in order to track geometries measured on actual tracks to detect loose bearings and verify this by comparing it with the measurement results of the uplift gap of loose bearings under the girder.

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Authors

	Koji HATTORI Researcher, Structural Mechanics Laboratory, Railway Dynamics Division Research Areas: Drive-by Measurement, Structural Health Monitoring, Track Geometry
	Kodai MATSUOKA, Dr.Eng. Senior Researcher, Data Analytics Laboratory, Information and Communication Technology Division Research Areas: Bridge Dynamics, Vehicle and Bridge Interaction, Damage Detection
	Hirofumi TANAKA, Dr.Eng. Manager, Track Geometry & Maintenance Laboratory, Track Technology Division Research Areas: Track Geometry Maintenance, On-board Monitoring, Signal Processing