# Cost-Effective Maintenance Policy for Sliding Surfaces of Bridge Bearings Using a Gamma Stochastic Process for Forecasting

Authors: Xiang Xu; Michael C. Forde; Antonio Caballero; Yuan Ren; Qiao Huang
Journal: Structural Control and Health Monitoring
 (2023)

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## Abstract

To determine the optimal alert threshold for sliding surface replacement of bridge bearings, a cost-effective maintenance policy is proposed in this paper using a gamma stochastic process. First, the sliding surface-triggered run-to-failure process of bridge bearings is discussed based on existing field inspection and maintenance records. Then, the wear thickness of sliding materials is estimated step-by-step based on the cumulative travel distance and wear rate. The gamma stochastic process is used to model degradation of sliding surfaces by using the indicator of wear thickness to depict uncertainties during the degrading process. Next, the optimal alert threshold for replacement of sliding surfaces is determined based on the cost-effective maintenance policy by minimizing the long-term expected maintenance cost rate. Finally, bearings of a long-span suspension bridge are employed to demonstrate the potential effectiveness of the proposed methodology. As a result, the wear thickness of sliding materials approximately follows a linear degrading law. Based on the gamma degrading model, the objective function subject to the long-term expected cost rate is formed. After optimization, the optimal alert threshold for replacement of sliding surfaces is 2.016 mm to achieve a minimum long-term expected maintenance cost rate of US$9,427 per day. In addition, the estimated service life subject to the alert threshold obeys a Gaussian distribution with a mean of 1278 days based on the one-year monitored displacement data.


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## Body

## 1. Introduction

Bearings are critical articulation components of bridge structures, supporting the superstructure while accommodating rotational or translational movements [1, 2]. Bridge bearings not only carry static loads induced by permanent weights of superstructures but also variable forces due to traffic, temperatures, winds, and earthquakes. However, owing to the motion of bearings, they are more vulnerable, especially sliding surfaces, when compared to other components. According to existing field engineering records, the service life of sliding surfaces is frequently much lower than expected. Premature failures of sliding surfaces have been increasingly observed in recent years, especially on large-span bridges, due to unexpectedly large amounts of accumulated travel distance of girder ends, significantly weakening the serviceability of bridges [3, 4]. The failure of sliding surfaces will lead to serious defects in a short time such as locking up of bearings. The failure of bearings will result in potential damage to surrounding structural components due to increased transferred forces. For instance, the Birmingham Bridge in Pennsylvania suffered damage induced by the failure of bearings in 2008, which cost an estimated repair fee of $8 million [5]. Furthermore, the failure of bearings will change the dynamic behaviour of bridges since bearings are critical constraints for bridge structures.Given that the traditional visual inspection is incapable of offering a timely alert on potential deterioration or malfunctioning of bearings, the long-term monitoring technology is developed to monitor the wear condition of sliding surfaces [6]. The travel distance of girder ends, i.e., cumulative displacement, is one of the dominant factors contributing to the wear of sliding surfaces. Displacements are mainly caused by temperatures, winds, traffic, and earthquakes, where the linear relationship between displacements and temperatures is well studied [7–10]. The temperature-induced displacement is used to estimate the maximum displacement at the bridge design stage to help choose an appropriate type of bearings or expansion joints. However, vehicle-induced dynamic displacements, rather than static ones due to temperatures, contribute to the major part of cumulative sliding displacements, which dominate the wear condition of sliding materials [4, 6, 11, 12]. Although some methodologies were proposed to estimate cumulative travel distances based on dynamically monitored data or random traffic flows [4, 13], the displacement transducer is still the most straightforward way to obtain cumulative travel distances. Newly built large-span bridges are always equipped with displacement transducers to monitor the displacement behaviour of girder ends.In addition to cumulative travel distances, the wear rate (i.e., wear thickness per sliding distance unit) is the other key factor contributing to the wear condition of sliding surfaces. Many studies have been undertaken to investigate the tribological behaviours (i.e., friction and wear) of sliding materials (e.g., PTFE and its composites) [14–16]. Various factors, including contact pressure, temperature, lubrication, sliding surface material, sliding speed, and mating surface, influence the wear rate of sliding materials [17]. Although fruitful experiments were conducted in the lab, the size of the studied samples was relatively small compared to that of the sliding surfaces used in bridge bearings. In this regard, the size effect is the main limitation when using laboratory conclusions in tribology. Full-size experiments were designed and completed by Ala et al. [18] and Stanton et al. [19] for plain PTFE and its composites to investigate their wear behaviour. They found that the wear rate significantly depended on the contact pressure and travel speed. A model was established to determine the wear rate based on the product of contact pressure and travel speed. A relatively reliable wear rate could be estimated based on these full-size experiment results.Based on the measured displacements and the wear rate of sliding materials, the derived wear thickness is regarded as the indicator to depict the state of sliding surfaces for condition-based maintenance [4, 11]. Although wear thickness is available continuously, an optimal alert threshold subject to maintenance operations requires further work. Existing alert/serviceability thresholds for maintenance are based on engineering experience (Table 1).Table 1
Bearing replacement alert thresholds defined in codes in China, Europe, and the USA.
CodeAlert level 1Alert level 2China [4, 20]1 mm of sliding surfaces remains: recommend replacement0.8 mm of sliding surfaces remains: immediateEurope [21]h ≤ 1: more frequent inspection (2.2 mm ≤ ℎ ≤ tp/2.2, where tp is the thickness of PTFE)0 protrusionℎ: immediateUSA [11]10%–25% of sliding surfaces remainsNaThe objective of the maintenance policy could be divided into economy/availability and safety ones. For damage that has a large probability of causing tragic casualties, a safety objective is prioritized. For damage with a small probability of leading to casualties, an economy/availability objective is applied. In this paper, the topic is bearing failure triggered by the wear of sliding surfaces, which is a long-term effect under normal operational conditions. In addition, the sliding surface-triggered failure of bearings has a very low probability of causing casualties like earthquake or typhoon. Within this context, we set the economy/availability as the objective of the maintenance policy. In practice, one cannot wait until the failure level to schedule maintenance due to the large number of temporary works that may be involved. Thus, an alert threshold should be more conservative than the failure level. The optimal alert threshold is determined by maximizing the availability of the bridge or minimizing the lifetime expected cost rate. If the alert threshold is defined at a high level of wear, the probability of breakdown is large, leading to a significant cost of correction or even impacting structural or operational safety. If the alert threshold is defined at a low level of wear, the material is underutilized, resulting in higher lifetime cost.Many stochastic maintenance models were proposed under the context of continuously monitored deteriorating systems to determine the optimal alert threshold [22, 23]. The maintenance model is first determined based on the given case, including perfect maintenance and imperfect maintenance. A perfect maintenance action restores the device to a “good or new” state [24]. For sliding surfaces of bridge bearings, the only option for maintenance actions is replacement, belonging to perfect maintenance. In addition, the stochastic model should be chosen to capture the uncertainty of the degrading process. The commonly used models contain the gamma process, Wiener process, and inverse Gaussian process. The gamma process is good at describing monotonic degradation paths, such as the degradation in the form of cumulative damage [25]. The Wiener process is more suitable for modelling degradation with nonmonotonic deterioration over time [26]. The inverse Gaussian process is a limiting compound Poisson process that is appropriate for modelling heterogeneous degradation of systems deteriorating in a random environment [27]. In this study, the gamma process is adopted since the degrading process of sliding surfaces is monotonic. Finally, the objective function is established, which aims to achieve maximum availability or the minimum cost rate. Compared to availability, bridge owners prefer to minimize the long-term expected cost rate. Therefore, a stochastic gamma process, together with perfect maintenance actions and cost-based objective functions, is desired to determine the optimal alert threshold under the context of continuous monitoring.To determine the optimal alert threshold for replacement of sliding surfaces of bridge bearings, a cost-effective maintenance policy is proposed in this paper by assuming continuously monitored degrading systems. First, the sliding surface-triggered run-to-failure process of bearings is discussed based on existing field recorded inspection and maintenance information. Subsequently, the degrading process of sliding surfaces is developed according to the wear rate of sliding materials and monitored displacements, which is modelled by the gamma stochastic process. Next, the cost-effective maintenance model is built to consider a delayed maintenance operation, and the optimal alert threshold is proposed after optimization of the maintenance model. Finally, actual spherical bearings of a large-span suspension bridge are used to verify the effectiveness of the proposed cost-effective maintenance policy.


## 2. Methodology

Since the maintenance cost largely depends on the timing of intervention, an optimal alert threshold is developed to lower potential costs [28, 29] (see Figure 1). Monitored girder end displacement data and wear rate models of sliding materials (e.g., PTFE) are used to calculate wear thickness. Based on the wear thickness of sliding surfaces within a relatively long time window, a gamma stochastic model is used to model the thickness reduction of the sliding surface. Considering the run-to-failure process of bearings, a maintenance model is built, including maintenance plans, maintenance actions, and maintenance completions. Finally, the maintenance cost rate is set as the objective function, and the optimization algorithm is used to find the optimal thickness for replacement of the sliding surface by minimizing the cost rate.Figure 1
Flowchart of the workflow subject to the proposed cost-effective maintenance policy.


### 2.1. Sliding Surface-Triggered Run-To-Failure Process of Bearings

There are multiple failure models for bearings in practice, such as excessive pressures and corrosion or cracking of steel components. In this paper, we focus on the sliding surface-triggered run-to-failure process.Spherical bearings, usually used on large-span bridges, are examined. The specific configuration is demonstrated in Figure2, which comprises upper plates, middle plates, lower plates, sliding surfaces, anchor bolts, and anchor sockets.Figure 2
Configuration of spherical bearings.Although the run-to-failure process of bearings is well studied under seismic loads [30], the run-to-failure process under normal operating conditions is rarely investigated due to a lack of valid information. From inspection and maintenance records of some actual large-span bridges, the sliding surface-triggered run-to-failure process is as follows:(1)
First, with the accumulation of travel distances of girder ends, sliding materials are worn gradually, resulting in the reduction of thickness.(2)
Then, if failed sliding surfaces are not replaced timely, direct steel-to-steel contact of plates occurs, which will quickly lead to locking up of bearings.(3)
Finally, due to the requirement for motion of bridges, anchor bolts will fracture by large shear forces, leading to total failure of bearings.The time duration from failure of sliding surfaces to total failure of bearings is quite short in practice. The total failure of bearings impacts the serviceability of bridges, leading to operational and structural safety issues. In this regard, it is assumed that the maintenance cost is equal to the potential expense of sliding surface replacement before the failure of sliding surfaces plus the potential expense of bearing replacement after the failure of sliding surfaces.


### 2.2. Stochastic Degrading Model of Sliding Surfaces

#### 2.2.1. Degrading Process of Sliding Surfaces

PTFE and its composites are always adopted as the material for sliding surfaces of bridge bearings due to their high wear resistance and low friction factor. The tribological behaviour of PTFE and its composites (e.g., friction and wear) has been well studied over the past five decades [17, 31].Various factors, including travel distance, contact pressure, temperature, lubrication, sliding surface materials, sliding speed, and mating surface, influence the wear performance and behaviour of sliding materials [19]. With the accumulation of travel distance, the wear degree of sliding surfaces increases progressively. Composite materials (e.g., glass-filled reinforced PTFE) always perform better in wear performance than the plain ones.In general, the quantitative expression of wear volumes is(1)Vw=∫DWvdD,where Vw is the total volume reduction, D is the travel distance, and Wv is the wear rate in volume, namely, volume reduction per kilometer of travel distance. In practice, the wear degrees over contact areas are assumed to be uniform. Thus, Equation (1) is simplified as(2)Tw=∫DWtdD,where Tw is the total thickness reduction and Wt is the wear rate in thickness, i.e., thickness reduction per kilometer of travel distance.In equation (2), the wear rate Wt is not constant and travel speed is the dominant parameter, along with pressure, temperature, lubrication, sliding surface material, and mating surface [19]. Thus, a general equation is proposed to estimate the wear rate as follows:(3)Wt=αW,where α is a synthetic modification factor considering the impacts of temperature, lubrication, and mating surfaces, and W is a base wear rate subject to a determined sliding surface material, which is a function of the product of the travel speed V and pressure P, termed as WPV.Two distinct base wear rate values were observed with an increase in PV: one is associated with mild-wear regimes (lowW) and the other is subject to severe-wear regimes (high W) [32]. There was an abrupt transition between the two regimes at the so-called PV limit. Figure 3 plots the measured base wear rate values with an increase in PV for plain PTFE tests performed by Ala et al. [18] and NCHRP 432 tests [19], where full-size experiments were designed and carried out. Once the value of PV exceeds the limit, the base wear rate increases quickly. Moreover, linear functions are proper to fit the measurements both in low and high wear regimes. Similar rules were found for composited PTFEs, such as ultrahigh-molecular weight PTFE and glass-filled reinforced PTFE [18]. Thus, the base wear rate is expressed as a piece-wise function to distinguish between low and high wear regimes as follows:(4)Wx=k1x,x≤PVlimit,k2x+b2,x>PVlimit,where x is the product of pressure and travel speed, i.e., PV, k1 is the slope within the low wear regime, and k2 and b2 are slopes, and the Y-intercept corresponds to the high wear regime.Figure 3
Variations of the base wear rate against PV for plain PTFE.To investigate the degrading process, the degrading indicator should be defined in advance. For sliding surfaces of bearings, the wear thickness of sliding materials is regarded as a quantitative indicator to depict the wear condition of sliding surfaces. Therefore, the degrading process of sliding surfaces is defined as(5)Tw=∫αDWxdD.The displacements of girder ends are monitored by using transducers, which could indirectly reflect wear status of sliding surfaces [33]. Thus, the monitored data of displacements should be introduced to express the degrading model of sliding surfaces. Since measurement noise largely affects calculated results, denoising is an essential data preprocessing step. Herein, the Butterworth low-pass filter with a cutoff frequency of 0.5 Hz is employed [6, 34]. According to Equation (5), the degrading process of sliding surfaces using the monitored displacement vector d1,d2,…,dN is(6)Tw=∑i=2Nαdi−di−1Wxi,where(7)Wxi=k1Pifdi−di−1,Pifdi−di−1≤PVlimit,k2Pifdi−di−1+b2,Pifdi−di−1>PVlimit,where Pi is the contact pressure at instant i and f is the sampling frequency of displacement transducers.The variation of pressures is not significant since the permanent load (or dead load), rather than the variable load (or live load), i.e., vehicle-induced loads, contributes to the major part of contact pressures. If the contact pressure is not monitored, the following assumption could be made. Two signature pressures are defined herein, including the permanent load/dead load pressure and all load pressure (i.e., permanent plus variable). The all load pressure is the load taken by bearings under the combined loading condition. During normal operational phases, the contact pressure will fluctuate between the permanent load and all load pressures. Thus, it is acceptable to use a normal distribution (μ, σ) to model the contact pressures based on permanent load and all load pressures, where(8)μ+3σ=PD;μ−3σ=P0,where PD is the all load pressure and P0 is the permanent load pressure.The parameters,k1, k2, and b2, of the linear fit are derived from existing full-scale testing results. Then, according to the product of pressure and travel speed, the base wear rate is determined, as shown in Figure 3.The synthetic modification factorα should be obtained through tests. However, at present, insufficient data are available to work out the factor [11]. Until further tests are conducted, the factor is assumed to be 1 herein.Based on the aforementioned discussions, the steps to generate the degrading process are summarized as follows:(1)
Based on the monitored displacements of girder ends, the travel distanceDi from instant (i − 1) to i is calculated as |di − di−1|.(2)
The average sliding speedVi during the time window (i−1, i) is then computed as fDi, where f is the sampling frequency.(3)
The average contact pressurePi is randomly generated, where Pi∼NPμ,σ.(4)
According to the product value ofPiVi and Figure 3, the base wear rate W(PiVi) is determined.(5)
According to Equation (6), the degrading indicator Tw at instant i is calculated, where α equals to 1.(6)
We repeat steps (1)–(5), fromi = 2 to i = N, where N is the number of measured displacement time series.


#### 2.2.2. Gamma Stochastic Model

The gamma process is a popular stochastic process to model the degrading process due to its three features: (1) independent increments, (2) not decreasing, and (3) homogeneous in time [35]. The gamma process is a Levy process that has explicit marginal probability density functions to facilitate computations. In this paper, the gamma stochastic model is applied to model the degrading process.Replacement is the only optional maintenance action for sliding surfaces in practice. Once replacement is completed, the device is assumed to be “good or new,” and the degrading indicator is set at zero. The system evolution after replacement is independent of the past. Although multiple replacement actions are conducted during the service life of bridges, the maintenance policy only needs to be studied within one replacement cycle since the degrading process of each replacement cycle is independent and identical. Within the replacement cycle of sliding surfaces, the degrading indicatorTw(t) is assumed to evolve following a gamma stochastic process. The gamma process X^t subject to the service age t has the following three attributes:(1)
X^0=0(2)
For all 0 ≤ti < tj, the increment in the degrading indicator X^δt subject to the time window δt=tj−ti follows a gamma probability density:(9)fx;αδt,β=βαδtΓαδtxαδt−1e−βx,whereαδt is the shape parameter and β is the scale parameter. The two parameters of the gamma distribution can be estimated using degrading data by the statistical procedure.(3)
X^t has independent increments.


### 2.3. Cost-Effective Maintenance Policy

#### 2.3.1. Maintenance Model

The wear thickness of sliding surfaces is calculated continuously based on monitored displacement data. According to the continuous degrading indicator, the maintenance policy is shown in Figure4 and specifically described as follows:(1)
When the value of the degrading indicator equals or exceeds the alert thresholdAL, the maintenance plan starts getting scheduled. tA denotes instant when the degrading indicator is greater than AL:(10)tA=inft:Twt≥AL.(2)
When activated, a maintenance action effectively starts after a delayed durationτ, i.e., at instant tB = tA + τ. The needed delayed time before the maintenance operation is for a global maintenance setup.(3)
During the maintenance setup, failure of the whole bearing might occur. IfTwtB≤TwtF (where tF is the instant corresponding to the failure level FL), the whole bearing functions well, and the maintenance duration is subject to the time of replacement of sliding surfaces ρ1. If Tw(tB) > FL, the whole bearing fails, and the maintenance duration is the time duration of replacement of the whole bearing ρ2. In this regard, the expression of the expected duration of maintenance actions is as follows:(11)Eρ=PTwtB≤FLρ1+PTwtB>FLρ2,whereP(Tw(tB) ≤ FL) is the probability that the bearing does not fail, ρ1 is the time duration of replacement of sliding surfaces, P(Tw(tB) > FL) is the probability of failure of the whole bearing, and ρ2 is the time duration of replacement of the whole bearing.(4)
At the end of the maintenance operationtC, the performance of sliding surfaces is as good as new, and the degrading indicator equals to zero. The degrading process of the replaced sliding surfaces is independent of the past.Figure 4
Description of the maintenance policy.Since the optimization objective of the maintenance model is to minimize the long-term expected maintenance cost rate, the cost model needs to be discussed in advance. It is noted that the degrading process shown in Figure4 is a regenerative process with regeneration times being the dates of the end of replacement. Once replacement is completed, the degrading indicator is equal to zero, and the following stochastic evolution process is independent of the past. Therefore, the long-term expected maintenance cost rate could be regarded as the cost of the mean downtime in a cycle. The maintenance cost model consists of two parts, including the replacement cost of sliding surfaces Cs and the replacement cost of the whole bearing Cb based on the failure criterion. In addition, the overall cost of maintenance operations includes the economic loss due to bridge shutdowns. The specific long-term expected cost of the maintenance actions is as follows:(12)EC=PTwtB≤FLρ1C0+Cs+PTwtB>FLρ2C0+Cb,where C0 is the toll loss per time unit.


#### 2.3.2. Model Optimization

The objective of the proposed cost-effective maintenance policy is to minimize the long-term expected cost per time unit by selecting a proper alert threshold valueAL. The objective function U is set as(13)U=ECEtC=PTwtB≤FLρ1C0+Cs+PTwtB>FLρ2C0+CbEtA+τ+ρ.Since the maintenance setup duration is constant, Equation (13) is further expressed as(14)U=PTwtB≤FLρ1C0+Cs+PTwtB>FLρ2C0+CbEtA+τ+Eρ.Since the degrading process is modelled by using the gamma process, the probabilitiesPTwtB≤FL and PTwtB>FL can be calculated as follows:(15)PTwtB≤FL=FFL−AL;ατ,β=∫0FL−ALfx;ατ,βdx,PTwtB>FL=1−FFL−AL;ατ,β,where F is the cumulative density function of the gamma distribution and f is the probability density distribution, whose specific expression is listed in Equation (9).To calculateEtA, the expected wear thickness per time unit should be determined in advance, which is as follows:(16)ETw=∫0∞fx;ατ,βxdx.Then,EtA is derived as(17)EtA=ALETw=AL∫0∞fx;ατ,βxdx.E(ρ) can be calculated by following Equation (11).In this regard, the objective function can be expressed by the object variableAL, gamma degrading process function, and known parameters (i.e., ρ1, C0, Cs, ρ2, Cb, τ, and FL).Once the objective function is obtained, the optimization algorithm is applied and summarized as follows:(1)
Estimate values of the shape and scale parameters of the gamma degrading process by using the obtained degrading data.(2)
Determine the known parameters, namely, the maintenance setup durationτ, the time duration of sliding surface replacement ρ1, the time duration of bearing replacement ρ2, the direct expense of sliding surface replacement Cs, the direct expense of bearing replacement Cb, the expected toll loss per time unit C0, and the failure level FL.(3)
Start with a relatively small value ofAL within the range (0, FL).(4)
Calculate the objective function listed in Equation (14) for selected AL.(5)
AdjustAL by a small increment unless reaching FL, and repeat steps (3) and (4).(6)
Choose the feasible value ofAL to minimize the long-term expected maintenance cost rate.


#### 2.3.3. Service Life Prediction

In addition, the service life corresponding to the alert thresholdAL can be approximately estimated in advance of the degrading indicator reaching the alert threshold in practice. Based on the gamma degradation process, the estimated service life subject to AL is(18)NAL=infN:∑w,iN△Tw,i>AL,where ΔTw is the degrading indicator increment in a given time step and follows the gamma distribution listed in Equation (9), namely, ΔTw∼Γαδt,β. Since the increment ΔTw is a stochastic variable, the estimated service life NAL suffers from uncertainties. After repeating Equation (18) many times, the distribution of the service life is achieved.The estimated service life totally depends on the gamma degrading process. The gamma process is developed based on the available degrading indicator measurements. In this regard, the estimated service life is highly related to the existing monitored data. With the accumulation of newly monitored data, the gamma process and service life will be updated accordingly.


## 2.1. Sliding Surface-Triggered Run-To-Failure Process of Bearings

There are multiple failure models for bearings in practice, such as excessive pressures and corrosion or cracking of steel components. In this paper, we focus on the sliding surface-triggered run-to-failure process.Spherical bearings, usually used on large-span bridges, are examined. The specific configuration is demonstrated in Figure2, which comprises upper plates, middle plates, lower plates, sliding surfaces, anchor bolts, and anchor sockets.Figure 2
Configuration of spherical bearings.Although the run-to-failure process of bearings is well studied under seismic loads [30], the run-to-failure process under normal operating conditions is rarely investigated due to a lack of valid information. From inspection and maintenance records of some actual large-span bridges, the sliding surface-triggered run-to-failure process is as follows:(1)
First, with the accumulation of travel distances of girder ends, sliding materials are worn gradually, resulting in the reduction of thickness.(2)
Then, if failed sliding surfaces are not replaced timely, direct steel-to-steel contact of plates occurs, which will quickly lead to locking up of bearings.(3)
Finally, due to the requirement for motion of bridges, anchor bolts will fracture by large shear forces, leading to total failure of bearings.The time duration from failure of sliding surfaces to total failure of bearings is quite short in practice. The total failure of bearings impacts the serviceability of bridges, leading to operational and structural safety issues. In this regard, it is assumed that the maintenance cost is equal to the potential expense of sliding surface replacement before the failure of sliding surfaces plus the potential expense of bearing replacement after the failure of sliding surfaces.


## 2.2. Stochastic Degrading Model of Sliding Surfaces

### 2.2.1. Degrading Process of Sliding Surfaces

PTFE and its composites are always adopted as the material for sliding surfaces of bridge bearings due to their high wear resistance and low friction factor. The tribological behaviour of PTFE and its composites (e.g., friction and wear) has been well studied over the past five decades [17, 31].Various factors, including travel distance, contact pressure, temperature, lubrication, sliding surface materials, sliding speed, and mating surface, influence the wear performance and behaviour of sliding materials [19]. With the accumulation of travel distance, the wear degree of sliding surfaces increases progressively. Composite materials (e.g., glass-filled reinforced PTFE) always perform better in wear performance than the plain ones.In general, the quantitative expression of wear volumes is(1)Vw=∫DWvdD,where Vw is the total volume reduction, D is the travel distance, and Wv is the wear rate in volume, namely, volume reduction per kilometer of travel distance. In practice, the wear degrees over contact areas are assumed to be uniform. Thus, Equation (1) is simplified as(2)Tw=∫DWtdD,where Tw is the total thickness reduction and Wt is the wear rate in thickness, i.e., thickness reduction per kilometer of travel distance.In equation (2), the wear rate Wt is not constant and travel speed is the dominant parameter, along with pressure, temperature, lubrication, sliding surface material, and mating surface [19]. Thus, a general equation is proposed to estimate the wear rate as follows:(3)Wt=αW,where α is a synthetic modification factor considering the impacts of temperature, lubrication, and mating surfaces, and W is a base wear rate subject to a determined sliding surface material, which is a function of the product of the travel speed V and pressure P, termed as WPV.Two distinct base wear rate values were observed with an increase in PV: one is associated with mild-wear regimes (lowW) and the other is subject to severe-wear regimes (high W) [32]. There was an abrupt transition between the two regimes at the so-called PV limit. Figure 3 plots the measured base wear rate values with an increase in PV for plain PTFE tests performed by Ala et al. [18] and NCHRP 432 tests [19], where full-size experiments were designed and carried out. Once the value of PV exceeds the limit, the base wear rate increases quickly. Moreover, linear functions are proper to fit the measurements both in low and high wear regimes. Similar rules were found for composited PTFEs, such as ultrahigh-molecular weight PTFE and glass-filled reinforced PTFE [18]. Thus, the base wear rate is expressed as a piece-wise function to distinguish between low and high wear regimes as follows:(4)Wx=k1x,x≤PVlimit,k2x+b2,x>PVlimit,where x is the product of pressure and travel speed, i.e., PV, k1 is the slope within the low wear regime, and k2 and b2 are slopes, and the Y-intercept corresponds to the high wear regime.Figure 3
Variations of the base wear rate against PV for plain PTFE.To investigate the degrading process, the degrading indicator should be defined in advance. For sliding surfaces of bearings, the wear thickness of sliding materials is regarded as a quantitative indicator to depict the wear condition of sliding surfaces. Therefore, the degrading process of sliding surfaces is defined as(5)Tw=∫αDWxdD.The displacements of girder ends are monitored by using transducers, which could indirectly reflect wear status of sliding surfaces [33]. Thus, the monitored data of displacements should be introduced to express the degrading model of sliding surfaces. Since measurement noise largely affects calculated results, denoising is an essential data preprocessing step. Herein, the Butterworth low-pass filter with a cutoff frequency of 0.5 Hz is employed [6, 34]. According to Equation (5), the degrading process of sliding surfaces using the monitored displacement vector d1,d2,…,dN is(6)Tw=∑i=2Nαdi−di−1Wxi,where(7)Wxi=k1Pifdi−di−1,Pifdi−di−1≤PVlimit,k2Pifdi−di−1+b2,Pifdi−di−1>PVlimit,where Pi is the contact pressure at instant i and f is the sampling frequency of displacement transducers.The variation of pressures is not significant since the permanent load (or dead load), rather than the variable load (or live load), i.e., vehicle-induced loads, contributes to the major part of contact pressures. If the contact pressure is not monitored, the following assumption could be made. Two signature pressures are defined herein, including the permanent load/dead load pressure and all load pressure (i.e., permanent plus variable). The all load pressure is the load taken by bearings under the combined loading condition. During normal operational phases, the contact pressure will fluctuate between the permanent load and all load pressures. Thus, it is acceptable to use a normal distribution (μ, σ) to model the contact pressures based on permanent load and all load pressures, where(8)μ+3σ=PD;μ−3σ=P0,where PD is the all load pressure and P0 is the permanent load pressure.The parameters,k1, k2, and b2, of the linear fit are derived from existing full-scale testing results. Then, according to the product of pressure and travel speed, the base wear rate is determined, as shown in Figure 3.The synthetic modification factorα should be obtained through tests. However, at present, insufficient data are available to work out the factor [11]. Until further tests are conducted, the factor is assumed to be 1 herein.Based on the aforementioned discussions, the steps to generate the degrading process are summarized as follows:(1)
Based on the monitored displacements of girder ends, the travel distanceDi from instant (i − 1) to i is calculated as |di − di−1|.(2)
The average sliding speedVi during the time window (i−1, i) is then computed as fDi, where f is the sampling frequency.(3)
The average contact pressurePi is randomly generated, where Pi∼NPμ,σ.(4)
According to the product value ofPiVi and Figure 3, the base wear rate W(PiVi) is determined.(5)
According to Equation (6), the degrading indicator Tw at instant i is calculated, where α equals to 1.(6)
We repeat steps (1)–(5), fromi = 2 to i = N, where N is the number of measured displacement time series.


### 2.2.2. Gamma Stochastic Model

The gamma process is a popular stochastic process to model the degrading process due to its three features: (1) independent increments, (2) not decreasing, and (3) homogeneous in time [35]. The gamma process is a Levy process that has explicit marginal probability density functions to facilitate computations. In this paper, the gamma stochastic model is applied to model the degrading process.Replacement is the only optional maintenance action for sliding surfaces in practice. Once replacement is completed, the device is assumed to be “good or new,” and the degrading indicator is set at zero. The system evolution after replacement is independent of the past. Although multiple replacement actions are conducted during the service life of bridges, the maintenance policy only needs to be studied within one replacement cycle since the degrading process of each replacement cycle is independent and identical. Within the replacement cycle of sliding surfaces, the degrading indicatorTw(t) is assumed to evolve following a gamma stochastic process. The gamma process X^t subject to the service age t has the following three attributes:(1)
X^0=0(2)
For all 0 ≤ti < tj, the increment in the degrading indicator X^δt subject to the time window δt=tj−ti follows a gamma probability density:(9)fx;αδt,β=βαδtΓαδtxαδt−1e−βx,whereαδt is the shape parameter and β is the scale parameter. The two parameters of the gamma distribution can be estimated using degrading data by the statistical procedure.(3)
X^t has independent increments.


## 2.2.1. Degrading Process of Sliding Surfaces

PTFE and its composites are always adopted as the material for sliding surfaces of bridge bearings due to their high wear resistance and low friction factor. The tribological behaviour of PTFE and its composites (e.g., friction and wear) has been well studied over the past five decades [17, 31].Various factors, including travel distance, contact pressure, temperature, lubrication, sliding surface materials, sliding speed, and mating surface, influence the wear performance and behaviour of sliding materials [19]. With the accumulation of travel distance, the wear degree of sliding surfaces increases progressively. Composite materials (e.g., glass-filled reinforced PTFE) always perform better in wear performance than the plain ones.In general, the quantitative expression of wear volumes is(1)Vw=∫DWvdD,where Vw is the total volume reduction, D is the travel distance, and Wv is the wear rate in volume, namely, volume reduction per kilometer of travel distance. In practice, the wear degrees over contact areas are assumed to be uniform. Thus, Equation (1) is simplified as(2)Tw=∫DWtdD,where Tw is the total thickness reduction and Wt is the wear rate in thickness, i.e., thickness reduction per kilometer of travel distance.In equation (2), the wear rate Wt is not constant and travel speed is the dominant parameter, along with pressure, temperature, lubrication, sliding surface material, and mating surface [19]. Thus, a general equation is proposed to estimate the wear rate as follows:(3)Wt=αW,where α is a synthetic modification factor considering the impacts of temperature, lubrication, and mating surfaces, and W is a base wear rate subject to a determined sliding surface material, which is a function of the product of the travel speed V and pressure P, termed as WPV.Two distinct base wear rate values were observed with an increase in PV: one is associated with mild-wear regimes (lowW) and the other is subject to severe-wear regimes (high W) [32]. There was an abrupt transition between the two regimes at the so-called PV limit. Figure 3 plots the measured base wear rate values with an increase in PV for plain PTFE tests performed by Ala et al. [18] and NCHRP 432 tests [19], where full-size experiments were designed and carried out. Once the value of PV exceeds the limit, the base wear rate increases quickly. Moreover, linear functions are proper to fit the measurements both in low and high wear regimes. Similar rules were found for composited PTFEs, such as ultrahigh-molecular weight PTFE and glass-filled reinforced PTFE [18]. Thus, the base wear rate is expressed as a piece-wise function to distinguish between low and high wear regimes as follows:(4)Wx=k1x,x≤PVlimit,k2x+b2,x>PVlimit,where x is the product of pressure and travel speed, i.e., PV, k1 is the slope within the low wear regime, and k2 and b2 are slopes, and the Y-intercept corresponds to the high wear regime.Figure 3
Variations of the base wear rate against PV for plain PTFE.To investigate the degrading process, the degrading indicator should be defined in advance. For sliding surfaces of bearings, the wear thickness of sliding materials is regarded as a quantitative indicator to depict the wear condition of sliding surfaces. Therefore, the degrading process of sliding surfaces is defined as(5)Tw=∫αDWxdD.The displacements of girder ends are monitored by using transducers, which could indirectly reflect wear status of sliding surfaces [33]. Thus, the monitored data of displacements should be introduced to express the degrading model of sliding surfaces. Since measurement noise largely affects calculated results, denoising is an essential data preprocessing step. Herein, the Butterworth low-pass filter with a cutoff frequency of 0.5 Hz is employed [6, 34]. According to Equation (5), the degrading process of sliding surfaces using the monitored displacement vector d1,d2,…,dN is(6)Tw=∑i=2Nαdi−di−1Wxi,where(7)Wxi=k1Pifdi−di−1,Pifdi−di−1≤PVlimit,k2Pifdi−di−1+b2,Pifdi−di−1>PVlimit,where Pi is the contact pressure at instant i and f is the sampling frequency of displacement transducers.The variation of pressures is not significant since the permanent load (or dead load), rather than the variable load (or live load), i.e., vehicle-induced loads, contributes to the major part of contact pressures. If the contact pressure is not monitored, the following assumption could be made. Two signature pressures are defined herein, including the permanent load/dead load pressure and all load pressure (i.e., permanent plus variable). The all load pressure is the load taken by bearings under the combined loading condition. During normal operational phases, the contact pressure will fluctuate between the permanent load and all load pressures. Thus, it is acceptable to use a normal distribution (μ, σ) to model the contact pressures based on permanent load and all load pressures, where(8)μ+3σ=PD;μ−3σ=P0,where PD is the all load pressure and P0 is the permanent load pressure.The parameters,k1, k2, and b2, of the linear fit are derived from existing full-scale testing results. Then, according to the product of pressure and travel speed, the base wear rate is determined, as shown in Figure 3.The synthetic modification factorα should be obtained through tests. However, at present, insufficient data are available to work out the factor [11]. Until further tests are conducted, the factor is assumed to be 1 herein.Based on the aforementioned discussions, the steps to generate the degrading process are summarized as follows:(1)
Based on the monitored displacements of girder ends, the travel distanceDi from instant (i − 1) to i is calculated as |di − di−1|.(2)
The average sliding speedVi during the time window (i−1, i) is then computed as fDi, where f is the sampling frequency.(3)
The average contact pressurePi is randomly generated, where Pi∼NPμ,σ.(4)
According to the product value ofPiVi and Figure 3, the base wear rate W(PiVi) is determined.(5)
According to Equation (6), the degrading indicator Tw at instant i is calculated, where α equals to 1.(6)
We repeat steps (1)–(5), fromi = 2 to i = N, where N is the number of measured displacement time series.


## 2.2.2. Gamma Stochastic Model

The gamma process is a popular stochastic process to model the degrading process due to its three features: (1) independent increments, (2) not decreasing, and (3) homogeneous in time [35]. The gamma process is a Levy process that has explicit marginal probability density functions to facilitate computations. In this paper, the gamma stochastic model is applied to model the degrading process.Replacement is the only optional maintenance action for sliding surfaces in practice. Once replacement is completed, the device is assumed to be “good or new,” and the degrading indicator is set at zero. The system evolution after replacement is independent of the past. Although multiple replacement actions are conducted during the service life of bridges, the maintenance policy only needs to be studied within one replacement cycle since the degrading process of each replacement cycle is independent and identical. Within the replacement cycle of sliding surfaces, the degrading indicatorTw(t) is assumed to evolve following a gamma stochastic process. The gamma process X^t subject to the service age t has the following three attributes:(1)
X^0=0(2)
For all 0 ≤ti < tj, the increment in the degrading indicator X^δt subject to the time window δt=tj−ti follows a gamma probability density:(9)fx;αδt,β=βαδtΓαδtxαδt−1e−βx,whereαδt is the shape parameter and β is the scale parameter. The two parameters of the gamma distribution can be estimated using degrading data by the statistical procedure.(3)
X^t has independent increments.


## 2.3. Cost-Effective Maintenance Policy

### 2.3.1. Maintenance Model

The wear thickness of sliding surfaces is calculated continuously based on monitored displacement data. According to the continuous degrading indicator, the maintenance policy is shown in Figure4 and specifically described as follows:(1)
When the value of the degrading indicator equals or exceeds the alert thresholdAL, the maintenance plan starts getting scheduled. tA denotes instant when the degrading indicator is greater than AL:(10)tA=inft:Twt≥AL.(2)
When activated, a maintenance action effectively starts after a delayed durationτ, i.e., at instant tB = tA + τ. The needed delayed time before the maintenance operation is for a global maintenance setup.(3)
During the maintenance setup, failure of the whole bearing might occur. IfTwtB≤TwtF (where tF is the instant corresponding to the failure level FL), the whole bearing functions well, and the maintenance duration is subject to the time of replacement of sliding surfaces ρ1. If Tw(tB) > FL, the whole bearing fails, and the maintenance duration is the time duration of replacement of the whole bearing ρ2. In this regard, the expression of the expected duration of maintenance actions is as follows:(11)Eρ=PTwtB≤FLρ1+PTwtB>FLρ2,whereP(Tw(tB) ≤ FL) is the probability that the bearing does not fail, ρ1 is the time duration of replacement of sliding surfaces, P(Tw(tB) > FL) is the probability of failure of the whole bearing, and ρ2 is the time duration of replacement of the whole bearing.(4)
At the end of the maintenance operationtC, the performance of sliding surfaces is as good as new, and the degrading indicator equals to zero. The degrading process of the replaced sliding surfaces is independent of the past.Figure 4
Description of the maintenance policy.Since the optimization objective of the maintenance model is to minimize the long-term expected maintenance cost rate, the cost model needs to be discussed in advance. It is noted that the degrading process shown in Figure4 is a regenerative process with regeneration times being the dates of the end of replacement. Once replacement is completed, the degrading indicator is equal to zero, and the following stochastic evolution process is independent of the past. Therefore, the long-term expected maintenance cost rate could be regarded as the cost of the mean downtime in a cycle. The maintenance cost model consists of two parts, including the replacement cost of sliding surfaces Cs and the replacement cost of the whole bearing Cb based on the failure criterion. In addition, the overall cost of maintenance operations includes the economic loss due to bridge shutdowns. The specific long-term expected cost of the maintenance actions is as follows:(12)EC=PTwtB≤FLρ1C0+Cs+PTwtB>FLρ2C0+Cb,where C0 is the toll loss per time unit.


### 2.3.2. Model Optimization

The objective of the proposed cost-effective maintenance policy is to minimize the long-term expected cost per time unit by selecting a proper alert threshold valueAL. The objective function U is set as(13)U=ECEtC=PTwtB≤FLρ1C0+Cs+PTwtB>FLρ2C0+CbEtA+τ+ρ.Since the maintenance setup duration is constant, Equation (13) is further expressed as(14)U=PTwtB≤FLρ1C0+Cs+PTwtB>FLρ2C0+CbEtA+τ+Eρ.Since the degrading process is modelled by using the gamma process, the probabilitiesPTwtB≤FL and PTwtB>FL can be calculated as follows:(15)PTwtB≤FL=FFL−AL;ατ,β=∫0FL−ALfx;ατ,βdx,PTwtB>FL=1−FFL−AL;ατ,β,where F is the cumulative density function of the gamma distribution and f is the probability density distribution, whose specific expression is listed in Equation (9).To calculateEtA, the expected wear thickness per time unit should be determined in advance, which is as follows:(16)ETw=∫0∞fx;ατ,βxdx.Then,EtA is derived as(17)EtA=ALETw=AL∫0∞fx;ατ,βxdx.E(ρ) can be calculated by following Equation (11).In this regard, the objective function can be expressed by the object variableAL, gamma degrading process function, and known parameters (i.e., ρ1, C0, Cs, ρ2, Cb, τ, and FL).Once the objective function is obtained, the optimization algorithm is applied and summarized as follows:(1)
Estimate values of the shape and scale parameters of the gamma degrading process by using the obtained degrading data.(2)
Determine the known parameters, namely, the maintenance setup durationτ, the time duration of sliding surface replacement ρ1, the time duration of bearing replacement ρ2, the direct expense of sliding surface replacement Cs, the direct expense of bearing replacement Cb, the expected toll loss per time unit C0, and the failure level FL.(3)
Start with a relatively small value ofAL within the range (0, FL).(4)
Calculate the objective function listed in Equation (14) for selected AL.(5)
AdjustAL by a small increment unless reaching FL, and repeat steps (3) and (4).(6)
Choose the feasible value ofAL to minimize the long-term expected maintenance cost rate.


### 2.3.3. Service Life Prediction

In addition, the service life corresponding to the alert thresholdAL can be approximately estimated in advance of the degrading indicator reaching the alert threshold in practice. Based on the gamma degradation process, the estimated service life subject to AL is(18)NAL=infN:∑w,iN△Tw,i>AL,where ΔTw is the degrading indicator increment in a given time step and follows the gamma distribution listed in Equation (9), namely, ΔTw∼Γαδt,β. Since the increment ΔTw is a stochastic variable, the estimated service life NAL suffers from uncertainties. After repeating Equation (18) many times, the distribution of the service life is achieved.The estimated service life totally depends on the gamma degrading process. The gamma process is developed based on the available degrading indicator measurements. In this regard, the estimated service life is highly related to the existing monitored data. With the accumulation of newly monitored data, the gamma process and service life will be updated accordingly.


## 2.3.1. Maintenance Model

The wear thickness of sliding surfaces is calculated continuously based on monitored displacement data. According to the continuous degrading indicator, the maintenance policy is shown in Figure4 and specifically described as follows:(1)
When the value of the degrading indicator equals or exceeds the alert thresholdAL, the maintenance plan starts getting scheduled. tA denotes instant when the degrading indicator is greater than AL:(10)tA=inft:Twt≥AL.(2)
When activated, a maintenance action effectively starts after a delayed durationτ, i.e., at instant tB = tA + τ. The needed delayed time before the maintenance operation is for a global maintenance setup.(3)
During the maintenance setup, failure of the whole bearing might occur. IfTwtB≤TwtF (where tF is the instant corresponding to the failure level FL), the whole bearing functions well, and the maintenance duration is subject to the time of replacement of sliding surfaces ρ1. If Tw(tB) > FL, the whole bearing fails, and the maintenance duration is the time duration of replacement of the whole bearing ρ2. In this regard, the expression of the expected duration of maintenance actions is as follows:(11)Eρ=PTwtB≤FLρ1+PTwtB>FLρ2,whereP(Tw(tB) ≤ FL) is the probability that the bearing does not fail, ρ1 is the time duration of replacement of sliding surfaces, P(Tw(tB) > FL) is the probability of failure of the whole bearing, and ρ2 is the time duration of replacement of the whole bearing.(4)
At the end of the maintenance operationtC, the performance of sliding surfaces is as good as new, and the degrading indicator equals to zero. The degrading process of the replaced sliding surfaces is independent of the past.Figure 4
Description of the maintenance policy.Since the optimization objective of the maintenance model is to minimize the long-term expected maintenance cost rate, the cost model needs to be discussed in advance. It is noted that the degrading process shown in Figure4 is a regenerative process with regeneration times being the dates of the end of replacement. Once replacement is completed, the degrading indicator is equal to zero, and the following stochastic evolution process is independent of the past. Therefore, the long-term expected maintenance cost rate could be regarded as the cost of the mean downtime in a cycle. The maintenance cost model consists of two parts, including the replacement cost of sliding surfaces Cs and the replacement cost of the whole bearing Cb based on the failure criterion. In addition, the overall cost of maintenance operations includes the economic loss due to bridge shutdowns. The specific long-term expected cost of the maintenance actions is as follows:(12)EC=PTwtB≤FLρ1C0+Cs+PTwtB>FLρ2C0+Cb,where C0 is the toll loss per time unit.


## 2.3.2. Model Optimization

The objective of the proposed cost-effective maintenance policy is to minimize the long-term expected cost per time unit by selecting a proper alert threshold valueAL. The objective function U is set as(13)U=ECEtC=PTwtB≤FLρ1C0+Cs+PTwtB>FLρ2C0+CbEtA+τ+ρ.Since the maintenance setup duration is constant, Equation (13) is further expressed as(14)U=PTwtB≤FLρ1C0+Cs+PTwtB>FLρ2C0+CbEtA+τ+Eρ.Since the degrading process is modelled by using the gamma process, the probabilitiesPTwtB≤FL and PTwtB>FL can be calculated as follows:(15)PTwtB≤FL=FFL−AL;ατ,β=∫0FL−ALfx;ατ,βdx,PTwtB>FL=1−FFL−AL;ατ,β,where F is the cumulative density function of the gamma distribution and f is the probability density distribution, whose specific expression is listed in Equation (9).To calculateEtA, the expected wear thickness per time unit should be determined in advance, which is as follows:(16)ETw=∫0∞fx;ατ,βxdx.Then,EtA is derived as(17)EtA=ALETw=AL∫0∞fx;ατ,βxdx.E(ρ) can be calculated by following Equation (11).In this regard, the objective function can be expressed by the object variableAL, gamma degrading process function, and known parameters (i.e., ρ1, C0, Cs, ρ2, Cb, τ, and FL).Once the objective function is obtained, the optimization algorithm is applied and summarized as follows:(1)
Estimate values of the shape and scale parameters of the gamma degrading process by using the obtained degrading data.(2)
Determine the known parameters, namely, the maintenance setup durationτ, the time duration of sliding surface replacement ρ1, the time duration of bearing replacement ρ2, the direct expense of sliding surface replacement Cs, the direct expense of bearing replacement Cb, the expected toll loss per time unit C0, and the failure level FL.(3)
Start with a relatively small value ofAL within the range (0, FL).(4)
Calculate the objective function listed in Equation (14) for selected AL.(5)
AdjustAL by a small increment unless reaching FL, and repeat steps (3) and (4).(6)
Choose the feasible value ofAL to minimize the long-term expected maintenance cost rate.


## 2.3.3. Service Life Prediction

In addition, the service life corresponding to the alert thresholdAL can be approximately estimated in advance of the degrading indicator reaching the alert threshold in practice. Based on the gamma degradation process, the estimated service life subject to AL is(18)NAL=infN:∑w,iN△Tw,i>AL,where ΔTw is the degrading indicator increment in a given time step and follows the gamma distribution listed in Equation (9), namely, ΔTw∼Γαδt,β. Since the increment ΔTw is a stochastic variable, the estimated service life NAL suffers from uncertainties. After repeating Equation (18) many times, the distribution of the service life is achieved.The estimated service life totally depends on the gamma degrading process. The gamma process is developed based on the available degrading indicator measurements. In this regard, the estimated service life is highly related to the existing monitored data. With the accumulation of newly monitored data, the gamma process and service life will be updated accordingly.


## 3. Case Study

### 3.1. The Studied Bridge

A double-tower, single-span steel box girder suspension bridge with a 1200 m main span, a 360 m west approach viaduct, and a 480 m east approach viaduct in China is regarded as a studied object. The 40.5 m width deck accommodates four traffic lanes in each direction, and the designed traffic volume and speed limit are 100,000 vehicles per day and 100 kilometers per hour. A sophisticated structural health monitoring (SHM) system was devised and implemented for the suspension bridge to monitor environmental factors, external actions, and structural responses. The SHM system comprises various types of sensors, including strain gauges, accelerometers, global positioning system (GPS), and temperature sensors. Four displacement transducers, denoted as DSQ-DIS-T02-001, DSQ-DIS-T02-002, DSQ-DIS-T05-001, and DSQ-DIS-T05-002, were installed at the ends of girders to monitor displacements, as shown in Figure5. The sampling frequency of the displacement transducer is 10 Hz, which satisfies the requirement to obtain high-frequency displacement components induced by vehicle loadings [4]. Since tremendous differences in cumulative travel distances between the DSQ-DIS-T02-001 and DSQ-DIS-T02-002 sensors were observed, faults of the DSQ-DIS-T02-001 and DSQ-DIS-T02-002 sensors were identified as the cause of the difference. In contrast, the measurements of the DSQ-DIS-T05-001 and DSQ-DIS-T05-002 sensors vary in a similar way, as shown in Figure 6, indicating the two transducers to be in good working order [36].Figure 5
Layout of displacement transducers installed at the end of girders.Figure 6
Displacement measurements from the two sensors on June 1, 2019.


### 3.2. Discussion of the Recorded Displacements

The displacement transducer, DSQ-DIS-T05-002, was adopted for the following discussion. To study the wear condition of sliding surfaces, the cumulative travel distance is a critical factor. To understand the wear situation of sliding surfaces, the daily cumulative and total cumulative travel distances in June 2019 are calculated and plotted in Figure7, where the Butterworth low-pass filter with a cutoff frequency of 0.5 Hz is employed for denoising. It is observed that the daily cumulative distance is relatively stable, varying between 40 m and 55 m, and the monthly cumulative distance approaches 1.5 kilometers.Figure 7
Daily cumulative travel distance and total cumulative travel distance in June 2019.In addition to the cumulative travel distance, the base wear rate is also important for the wear of sliding surfaces. According to Equation (7), the product of pressure and travel speed determines the base wear rate, where contact pressures slightly fluctuate between the permanent load pressure and all load pressure as assumed. Moreover, the travel speed can be calculated by using the monitored displacements. Time histories of travel speeds on June 1, 2019, are plotted in Figure 8, where travel speeds vary between 0 cm/s and 1.75 cm/s. The distribution of travel speeds is demonstrated in Figure 9, where the majority of the travel speeds are concentrated in the range from 0 cm/s to 0.3 cm/s.Figure 8
Travel speeds on June 1, 2019.Figure 9
Statistical results of travel speeds on June 1, 2019.


### 3.3. Site-Specific Degrading Model of Sliding Surfaces

Prior to construction of the degrading model, variables (i.e., contact pressures and base wear rates) should be determined. According to Equation (8), the distribution of pressures is plotted in Figure 10, with the dead/permanent load contact stress P0 = 10.34 MPa and all load stress PD = 17.24 MPa, according to the AASHTO code [1]. The base wear rate is generated based on Figure 3.Figure 10
Distribution of contact pressures.Based on the monitored displacements from the studied bridge in June 2019, the daily and monthly cumulative wear thicknesses of sliding surfaces are obtained and plotted in Figure11. The daily cumulative wear thickness in June 2019 fluctuates between 0.0004 mm and 0.0015 mm, and the monthly cumulative wear thickness is approximately 0.0225 mm.Figure 11
Daily cumulative wear thickness and total cumulative wear thickness in June 2019.One-year data (from June 2019 to May 2020) obtained from the displacement transducer (i.e., DSQ-DIS-T05-002) are used to describe the degrading process. With the growth of travel distances, the wear thickness of PTFE, regarded as the degrading indicator, is computed, as shown in Figure12. It is observed that the wear thickness of sliding surfaces approximately follows a linear degrading law. The estimated wear thickness in one year is 0.57 mm. Moreover, according to the visual inspection records, the reduction in the thickness of sliding surfaces was approximately 1 mm within 20 months (i.e., the annual mean is equal to 0.6 mm). The estimated annual mean thickness reduction (i.e., 0.57 mm) is in line with the inspection result (i.e., 0.6 mm), which verifies the effectiveness of the proposed degrading process.Figure 12
Daily cumulative wear thickness and total wear thickness within one year.The daily cumulative wear thickness data in one year are used to fit the gamma distribution. The histogram and gamma fit are plotted in Figure13. The fitting results show that the shape parameter is 1.2154 subject to a time interval of one day, and the scale parameter is 0.0013.Figure 13
Histogram and gamma fit of the daily cumulative wear thickness.


### 3.4. Cost-Effective Maintenance Policy

The objective of the cost-effective maintenance policy is to minimize the long-term expected cost rate by selecting an optimal alert threshold. First, costs of field maintenance actions need to be investigated. It is known that the cost varies significantly for different companies. For instance, based on existing records, replacement of PTFE will cost US$111,161 and 13 weeks for a British company, while it cost US$33,585 and 6 weeks for a Chinese one in 2007 [20]. Considering the actual situation of the studied bridge, we prefer to refer to the prices in the Chinese market. Moreover, the total cost of replacement should contain two parts, which are the direct expense for replacement and the indirect cost for bridge shutdowns. Based on statistical analysis, the mean daily toll income for a similar suspension bridge is approximately US$500,000 [37]. In this paper, the influence of bridge shutdowns on the social economy is neglected since the expense is difficult to estimate within existing information.Since no standard guideline regarding sliding surface replacement is published, the costs within the analysis are determined according to the experience and investigation. The cost subject to the replacement of sliding surfaces is estimated as US$33,585 and 6 weeks, where the mean daily loss of the toll income is US$310,980, while the cost corresponding to the replacement of bearings is US$388,726 and 12 weeks. Therefore, the total expenses for the replacement of PTFEs and bearings are US$13,094,785 and US$26,511,125, respectively. It is noticed that the major part of the expense is induced by the loss of toll income.The maintenance setup duration is determined as three months in this case study. During the maintenance setup time, the replacement plan is first made, and then, the preparatory work follows such as material purchase and labour arrangement. The failure criterion of sliding surfaces is defined as the remaining thickness of 0.8 mm [20]. Since the initial thickness of the PTFE pad is 3 mm, the failure level is indicated by a wear thickness of 2.2 mm in this case.The initial alert thresholdAL is set at 1 mm, and the increment is determined as 0.001 mm. The long-term expected maintenance cost rate is calculated and plotted in Figure 14. With the growth of alert thresholds at an earlier stage, the long-term expected cost rate decreases since the failure probability is low and the expected service life gains. When the alert threshold approaches 2.0160 mm, the objective function reaches its lowest point. Once the alert threshold exceeds 2.0160 mm, the potential cost of failure increases to enlarge the long-term expected cost rate. Thus, the optimal alert threshold is determined as 2.0160 mm to achieve a minimum long-term expected maintenance cost rate of US$9,427 per day. The conclusion drawn from the proposed cost-effective maintenance policy is similar to that from engineering experience, which recommends replacing sliding surfaces when the wear thickness reaches 2 mm [20]. Although the conclusions are similar, this study provides a solid theoretical foundation and could be straightforwardly transferred to other case studies.Figure 14
Long-term expected cost rate along with the potential alert threshold.In addition, the service life of PTFE can be estimated based on the gamma degradation process. Due to the uncertainty within the degradation process, the service life subject to the alert threshold follows a Gaussian distribution with a mean of 1278 days and a standard deviation of 32 days (see Figure15). It should be pointed out that the estimated service life totally depends on the one-year monitored displacement of the girder end. With the accumulation of newly monitored data, the gamma degradation process should be updated, and the distribution of estimated service life will be updated accordingly.Figure 15
Distribution of the estimated service life before the alert threshold.


## 3.1. The Studied Bridge

A double-tower, single-span steel box girder suspension bridge with a 1200 m main span, a 360 m west approach viaduct, and a 480 m east approach viaduct in China is regarded as a studied object. The 40.5 m width deck accommodates four traffic lanes in each direction, and the designed traffic volume and speed limit are 100,000 vehicles per day and 100 kilometers per hour. A sophisticated structural health monitoring (SHM) system was devised and implemented for the suspension bridge to monitor environmental factors, external actions, and structural responses. The SHM system comprises various types of sensors, including strain gauges, accelerometers, global positioning system (GPS), and temperature sensors. Four displacement transducers, denoted as DSQ-DIS-T02-001, DSQ-DIS-T02-002, DSQ-DIS-T05-001, and DSQ-DIS-T05-002, were installed at the ends of girders to monitor displacements, as shown in Figure5. The sampling frequency of the displacement transducer is 10 Hz, which satisfies the requirement to obtain high-frequency displacement components induced by vehicle loadings [4]. Since tremendous differences in cumulative travel distances between the DSQ-DIS-T02-001 and DSQ-DIS-T02-002 sensors were observed, faults of the DSQ-DIS-T02-001 and DSQ-DIS-T02-002 sensors were identified as the cause of the difference. In contrast, the measurements of the DSQ-DIS-T05-001 and DSQ-DIS-T05-002 sensors vary in a similar way, as shown in Figure 6, indicating the two transducers to be in good working order [36].Figure 5
Layout of displacement transducers installed at the end of girders.Figure 6
Displacement measurements from the two sensors on June 1, 2019.


## 3.2. Discussion of the Recorded Displacements

The displacement transducer, DSQ-DIS-T05-002, was adopted for the following discussion. To study the wear condition of sliding surfaces, the cumulative travel distance is a critical factor. To understand the wear situation of sliding surfaces, the daily cumulative and total cumulative travel distances in June 2019 are calculated and plotted in Figure7, where the Butterworth low-pass filter with a cutoff frequency of 0.5 Hz is employed for denoising. It is observed that the daily cumulative distance is relatively stable, varying between 40 m and 55 m, and the monthly cumulative distance approaches 1.5 kilometers.Figure 7
Daily cumulative travel distance and total cumulative travel distance in June 2019.In addition to the cumulative travel distance, the base wear rate is also important for the wear of sliding surfaces. According to Equation (7), the product of pressure and travel speed determines the base wear rate, where contact pressures slightly fluctuate between the permanent load pressure and all load pressure as assumed. Moreover, the travel speed can be calculated by using the monitored displacements. Time histories of travel speeds on June 1, 2019, are plotted in Figure 8, where travel speeds vary between 0 cm/s and 1.75 cm/s. The distribution of travel speeds is demonstrated in Figure 9, where the majority of the travel speeds are concentrated in the range from 0 cm/s to 0.3 cm/s.Figure 8
Travel speeds on June 1, 2019.Figure 9
Statistical results of travel speeds on June 1, 2019.


## 3.3. Site-Specific Degrading Model of Sliding Surfaces

Prior to construction of the degrading model, variables (i.e., contact pressures and base wear rates) should be determined. According to Equation (8), the distribution of pressures is plotted in Figure 10, with the dead/permanent load contact stress P0 = 10.34 MPa and all load stress PD = 17.24 MPa, according to the AASHTO code [1]. The base wear rate is generated based on Figure 3.Figure 10
Distribution of contact pressures.Based on the monitored displacements from the studied bridge in June 2019, the daily and monthly cumulative wear thicknesses of sliding surfaces are obtained and plotted in Figure11. The daily cumulative wear thickness in June 2019 fluctuates between 0.0004 mm and 0.0015 mm, and the monthly cumulative wear thickness is approximately 0.0225 mm.Figure 11
Daily cumulative wear thickness and total cumulative wear thickness in June 2019.One-year data (from June 2019 to May 2020) obtained from the displacement transducer (i.e., DSQ-DIS-T05-002) are used to describe the degrading process. With the growth of travel distances, the wear thickness of PTFE, regarded as the degrading indicator, is computed, as shown in Figure12. It is observed that the wear thickness of sliding surfaces approximately follows a linear degrading law. The estimated wear thickness in one year is 0.57 mm. Moreover, according to the visual inspection records, the reduction in the thickness of sliding surfaces was approximately 1 mm within 20 months (i.e., the annual mean is equal to 0.6 mm). The estimated annual mean thickness reduction (i.e., 0.57 mm) is in line with the inspection result (i.e., 0.6 mm), which verifies the effectiveness of the proposed degrading process.Figure 12
Daily cumulative wear thickness and total wear thickness within one year.The daily cumulative wear thickness data in one year are used to fit the gamma distribution. The histogram and gamma fit are plotted in Figure13. The fitting results show that the shape parameter is 1.2154 subject to a time interval of one day, and the scale parameter is 0.0013.Figure 13
Histogram and gamma fit of the daily cumulative wear thickness.


## 3.4. Cost-Effective Maintenance Policy

The objective of the cost-effective maintenance policy is to minimize the long-term expected cost rate by selecting an optimal alert threshold. First, costs of field maintenance actions need to be investigated. It is known that the cost varies significantly for different companies. For instance, based on existing records, replacement of PTFE will cost US$111,161 and 13 weeks for a British company, while it cost US$33,585 and 6 weeks for a Chinese one in 2007 [20]. Considering the actual situation of the studied bridge, we prefer to refer to the prices in the Chinese market. Moreover, the total cost of replacement should contain two parts, which are the direct expense for replacement and the indirect cost for bridge shutdowns. Based on statistical analysis, the mean daily toll income for a similar suspension bridge is approximately US$500,000 [37]. In this paper, the influence of bridge shutdowns on the social economy is neglected since the expense is difficult to estimate within existing information.Since no standard guideline regarding sliding surface replacement is published, the costs within the analysis are determined according to the experience and investigation. The cost subject to the replacement of sliding surfaces is estimated as US$33,585 and 6 weeks, where the mean daily loss of the toll income is US$310,980, while the cost corresponding to the replacement of bearings is US$388,726 and 12 weeks. Therefore, the total expenses for the replacement of PTFEs and bearings are US$13,094,785 and US$26,511,125, respectively. It is noticed that the major part of the expense is induced by the loss of toll income.The maintenance setup duration is determined as three months in this case study. During the maintenance setup time, the replacement plan is first made, and then, the preparatory work follows such as material purchase and labour arrangement. The failure criterion of sliding surfaces is defined as the remaining thickness of 0.8 mm [20]. Since the initial thickness of the PTFE pad is 3 mm, the failure level is indicated by a wear thickness of 2.2 mm in this case.The initial alert thresholdAL is set at 1 mm, and the increment is determined as 0.001 mm. The long-term expected maintenance cost rate is calculated and plotted in Figure 14. With the growth of alert thresholds at an earlier stage, the long-term expected cost rate decreases since the failure probability is low and the expected service life gains. When the alert threshold approaches 2.0160 mm, the objective function reaches its lowest point. Once the alert threshold exceeds 2.0160 mm, the potential cost of failure increases to enlarge the long-term expected cost rate. Thus, the optimal alert threshold is determined as 2.0160 mm to achieve a minimum long-term expected maintenance cost rate of US$9,427 per day. The conclusion drawn from the proposed cost-effective maintenance policy is similar to that from engineering experience, which recommends replacing sliding surfaces when the wear thickness reaches 2 mm [20]. Although the conclusions are similar, this study provides a solid theoretical foundation and could be straightforwardly transferred to other case studies.Figure 14
Long-term expected cost rate along with the potential alert threshold.In addition, the service life of PTFE can be estimated based on the gamma degradation process. Due to the uncertainty within the degradation process, the service life subject to the alert threshold follows a Gaussian distribution with a mean of 1278 days and a standard deviation of 32 days (see Figure15). It should be pointed out that the estimated service life totally depends on the one-year monitored displacement of the girder end. With the accumulation of newly monitored data, the gamma degradation process should be updated, and the distribution of estimated service life will be updated accordingly.Figure 15
Distribution of the estimated service life before the alert threshold.


## 4. Conclusions

To determine the optimal alert threshold for replacement of sliding surfaces of bridge bearings, a cost-effective maintenance policy is developed and proposed in this study to consider maintenance setup time. The following conclusions can be drawn from the research:(1)
The sliding surface-triggered run-to-failure process of bridge bearings is discussed in this study based on existing inspection and maintenance records. Three stages are observed within the failure process, including failure of sliding surfaces, locking up of bearings, and failure of entire bearings.(2)
The gamma process is used to model the degrading process of sliding surfaces by using the indicator of wear thickness, which is derived from the continuously monitored displacements and the wear rate model of sliding materials. The cumulative travel distances and sliding speeds can be derived from the measured displacements, and the base wear rate is determined by the travel speed and contact pressure. The gamma stochastic model is used to model the uncertainty within the degrading process.(3)
The optimal alert threshold is determined through optimization analysis to minimize the long-term expected maintenance cost rate, where the cost includes the direct expense of maintenance operations and the economic loss of bridge shutdowns. The objective function is built based on the potential cost of failures and service ages. The parameters in the objective function are derived from the gamma degrading model and known factors such as maintenance setup time.(4)
Bearings in the case study suspension bridge are used to validate the effectiveness of the proposed methodology. Measured displacement data of girder ends are used to build the gamma degrading model of sliding surfaces. As a result, the wear thickness of sliding materials approximately follows a linear law, and the estimated wear thickness in one year is nearly 0.57 mm based on one-year field monitored data.(5)
Based on the cost-effective maintenance model, the optimal alert threshold is determined as 2.0160 mm subject to a minimum long-term expected cost rate of US$9,427 per day. In addition, the estimated service life of PTFE before the alert threshold follows a Gaussian distribution with a mean of 1278 days and a standard deviation of 38 days.This paper makes a contribution to decision-making of maintenance policies of sliding surfaces. In the future, full-scale experiments regarding the tribology of sliding materials are needed to calibrate the base wear rate since only limited data are available currently. Furthermore, the process from the failure of sliding surfaces to failure of the entire bearings still needs more data support for calibration. Efforts should be made to depict this process through either experimental studies or field observations.



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*Source: 5751636-2023-02-08.xml*