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Abstract

The cable-beam anchorage zone is the vital load-bearing component in suspension bridge. For maintenance of such structure, a method of probabilistic optimization was proposed by combining linear elastic fracture mechanics, structural reliability and life cycle cost analysis. In this method, the optimal maintenance time is obtained by determining the relative proportion between the costs under the condition that the structural reliability is higher than the minimum allowable reliability. While, the minimum total maintenance cost is obtained by determining the maintenance interval. Then, an example is presented to verify this method, with the following conclusions: the reliability index is inversely proportional to the failure probability, the change of maintenance cost and failure cost affects the optimal maintenance time, the optimal maintenance time will be ahead of time when consider the risk cost. And finally, when the maintenance time interval is determined, the optimal maintenance cost is affected by the maintenance probability and the failure probability.

Keywords

Reliability cable-beam anchorage fatigue maintenance optimization maintenance interval

Introduction

Suspension bridges have been recognized as the most appealing structures due to their esthetic appearances as well as the structural advantages of cable, due to their distinguished ability to overcome large spans. Whereas, the anchorages between the suspenders and the stiffening beams (cable-beam anchorage zone), is the vital load-bearing component. Playing the role as transferring forces from cable to stiffening beam, the cable-beam anchorage is among the most fatigue-prone components of suspension bridges, owing to the complex geometrical configuration and loading conditions. The mechanical properties of the anchorage are directly related to the safety of the whole bridge.^1–3 Being welded, fatigue damage will emerge in cable-beam anchorage under cyclic loads induced by wind and vehicle loads. With the rapid development in the transportation industry, the traffic flow continues to increase quickly, so the fatigue damage of cable-beam anchorage is prone to grow.⁴

The anchorages fatigue performances of many long-span suspension bridges have been studied.^1–3 However, most of these researches focus on the stress analysis and fatigue performance evaluation of the cable-beam anchorage, but the maintenance evaluation of cable-beam anchorage is less involved. Zhongxiang Liu etc. present the field inspections on the pattern, location and propagation of cracks of the Runyang Suspension Bridge. A multi-scale finite element (FE) model of the bridge is developed To further investigate the mechanism of cracking. Besides, probabilistic FE analyses are performed to predict fatigue lives of the anchorage zone.⁵

This paper is based on the whole life maintenance method,^6,7 as well as the linear elastic fracture mechanics (LEFM). The bridge maintenance cost is divided into inspection cost, repair cost and failure cost, while the inspection cost and repair cost are actual cost, and the failure cost is risk expense. Furthermore, structural reliability theory and life cycle cost method are proposed to optimize the fatigue maintenance of welded steel components.

Reliability evaluation of fatigue life based on LEFM

The cracks are prone to appear in the steel components under cyclic loads, and the steel components will be fatigued failure when the cracks continue to expand.³ The influence factors of fatigue crack propagation are often random, so the fatigue life of steel components tend to have a greater randomness.^8,9 The method of probability analysis is usually used on reliability evaluation of fatigue life of steel component.⁶

The initial crack defect of welding components exists inevitably (hereinafter referred to as the initial cracks), which making the formation time of fatigue cracks is very short. Generally, only the life of fatigue crack propagation of welding component needs to be considered. Fatigue crack propagation is generally calculated by Paris formula¹⁰:

\frac{da}{dn} = C (Δ K)^{m}

(1)

In which, a is the crack size, n is the number of stress cycles, m, C are the fatigue parameters of steels, $Δ K$ is the stress intensity factor. Equation (1) is only valid for stress intensity factor ranges greater than the threshold stress intensity range $Δ K_{th}$ and smaller than the fracture stress intensity range $Δ K_{f}$ . The model keeps its linearity for $Δ K$ far from $Δ K_{th}$ . This threshold value can be taken into account for discriminating between damaging and nondamaging cycles. In the present study, the stress intensity factor range threshold $Δ K_{th}$ is taken equal to zero (all cycles are damaging).¹¹

The stress intensity factor range can be expressed by

Δ K = k_{e} Δ σ Y (a) M_{k} (a) \sqrt{π a}

(2)

k_{e} = 1 / (1 - σ_{m} / σ_{u})

(3)

Where, $Δ σ$ is the stress range, $Y (a)$ is the geometric correction factor of the stress intensity factor, $M_{k} (a)$ is the amplification factor of stress intensity factor, which reflects the influence to stress intensity factor caused by the crack tip when it pass though the stress concentration zone near weld toe. $k_{e}$ is the correction factor of average stress(explained in equation (3)).^6,12 $σ_{u}$ , $σ_{m}$ in equation (3) are the yield stress of materials and mean stress of steel component respectively.

The increase of traffic volume will lead to an increase in the number of stress cycles, so it is necessary to consider the annual growth rate in the number of stress cycles. It is assumed that the annual growth rate of stress cycles is $r_{c}$ , the M(t) is obtained by integrating equation (1).¹²

M (t) = \int_{a_{0}}^{a_{f}} \frac{da}{{[k_{e} Y (a) M_{k} (a) \sqrt{π a}]}^{m}} - C Δ σ^{m} Nt (1 + r_{c})^{t}

(4)

where, M(t) is safety margin of structure after operation time t, $a_{f}$ is the failure crack size, $a_{0}$ is the initial crack size, N is the annual average stress cycle times.

According to the equation (4), if $M (t) > 0$ , it means that the fatigue failure of component not emerged; if $M (t) = 0$ , it means that the component is in a critical state; if $M (t) < 0$ , it means that the fatigue failure of component emerged. The Fatigue failure probability of the component is:

P_{f} = P (M (t) < 0) \approx Φ (- β)

(5)

Where, $Φ (.)$ is the cumulative distribution function of the standard normal distribution, and $β$ is the fatigue reliability index. The evaluation process of fatigue reliability of welded components is shown in Figure 1.

Figure 1.

Flowchart of reliability evaluation of fatigue life based on LFEM.

Optimization of maintenance strategy for welding component

Crack detection probability

Cracks can be detected by inspecting the components, the proper maintenance strategy should be taken according to the detection results. The detection efficiency is related to many factors, such as detection methods, inspection personnel, inspection quality and so on. These influence factors are complex and uncertain. For fatigue problems, the detection quality depends on the probability of crack detection. In a detection, some factors have a direct influence on the results of cracks detection, such as the accuracy of a detection instrument, the uncertainty of the selected detection location and the judgment ability of the operator. Not all cracks can be detected, some cracks may be missed during the inspection process. The detection quality of cracks is expressed by random variable q. When q is equal to 0, no crack is detected, and when q is equal to 1, it corresponds to the perfect detection. Nondestructive testing is usually used to detect cracks, which includes visual inspection, X-ray detection, magnetic particle testing and other methods.^13,14 The detection accuracy, q, of each detection method is different, so the corresponding detection costs are different.⁷ The crack detection probability $P_{d}$ is:

P_{d} = P (a_{t_{i}} > a_{d})

(6)

Where, $a_{t_{i}}$ is crack size at the time $t_{i}$ , $a_{d}$ is minimum crack size that can be detected.

Different distribution models, such as exponential distribution, Weibull distribution and normal distribution, have a little influence on the final detection results. Therefore, the exponential distribution is generally used to simulate the randomness of $a_{d}$ .

Maintenance willingness of decision-maker

Whether or not to carry out maintenance depends on the maintenance willingness of bridge manager, but the maintenance funds, maintenance priorities and political factor and other factors will affect the decision of bridge manager. The probability variable, $P_{rep}$ , is introduced to describe the probability of decision-maker’s maintenance willingness by using the method proposed by Estes and Frangopol.⁶ The formula for the probability variable $P_{rep}$ is

P_{rep} = (\frac{a_{t_{i}} - a_{d}}{a_{c} - a_{d}})^{r_{p}} a_{d} \leq a_{t_{i}} \leq a_{c}

(7)

P_{rep} = 1 a_{t_{i}} \geq a_{c}

(8)

Where $γ_{P}$ is the impact factor, $a_{c}$ is the critical crack size. When the bridge decision manager decides to carry out maintenance, if the $γ_{P}$ is greater than 1.0, it is means that this maintenance is preventive maintenance; if $γ_{P}$ is less than 1.0, it is means that this maintenance is delayed maintenance; if $γ_{P}$ is equal to 1.0, it is means that this maintenance is timely.

Maintenance measures probability

There are three kinds of result as well as treatments after detecting at time $t_{i}$ . The first is that no cracks are detected out, then, the second is that cracks are detected out but not repaired is needed, and the last is that cracks are detected out and maintenance measures should be taken.^6,12 The first two types of result do not change the performance and reliability of components. Also, the appropriate maintenance measures are adopted according to the crack size. The structure cannot be repaired when the crack size exceeds a certain threshold, so in this situation, the replacement strategy should be adopted.

Under the statistical parameters of the component material are independent, it is assumed that the material of the component has the same fatigue characteristics before and after maintenance. The safety margin, $M (t)$ , of a component after repaired fatigue cracks is

M (t) = \int_{a_{0}}^{a_{f}} \frac{da}{{[k_{e} Y (a) M_{k} (a) \sqrt{π a}]}^{m}} - C Δ σ^{m} N (t - t_{i}) (1 + r_{c})^{t - t_{i}}

(9)

After detecting a crack at the time $t_{i}$ , the maintenance probability $P_{r}^{(k)} (t_{i})$ is:

P_{r}^{(k)} (t_{i}) = P ({M (t_{i}) > 0})

(10)

The detection probability, which depends on the detection interval $Δ t$ , reflects that the component will not emerge failure under the detection interval. The formula of detection probability is:

P_{i} (t_{i}) = 1 - P_{f} (t_{i}) = 1 - P (M (t_{i}) \leq 0) = P (M (t_{i}) > 0)

(11)

The probability that the components or structures will fail to achieve the desired function during the period of inspection of the structures is failure probability, so the failure probability in the time interval is expressed as:

P_{f} (t) = P (M (t) \leq 0), 0 \leq t < t_{i}

(12)

Probabilistic optimization of fatigue maintenance strategy

The total cost of bridge maintenance is divided into inspection costs, repair costs and failure costs. The inspection and repair costs are actual incurred expenses. The failure cost is loss caused by structural fatigue failure and it’s belongs to the risk cost. Maintenance should find a reasonable balance between inspection costs, repair costs and failure costs. In general, the better the detection equipment is, the higher the detection accuracy is, the smaller crack size will be detected, and therefore the higher cost of repair will be. These costs are not realistic currencies, so the expected costs need to consider the impact of the discount rate, r, and also multiply the corresponding probability of occurrence.⁷ The optimal maintenance cost is achieved by optimizing the maintenance interval, and it is used in fatigue maintenance activity of the cable-beam anchorage of suspension bridge.

The expected inspection cost, repair cost and failure cost are, respectively¹¹:

C_{I} = C_{i} (q) P_{i} (t_{i}) \frac{1}{{(1 + r)}^{t_{i}}}

(13)

C_{R} = \sum_{k = 1}^{n_{r}} C_{r} (k) P_{r} (t_{i}) \frac{1}{{(1 + r)}^{t_{i}}}

(14)

C_{F} (t_{i}) = C_{f} (t_{i}) ((P_{f} (t_{i}) - P_{f} (0)) \frac{1}{{(1 + r)}^{t_{i}}} + (P_{f} (t_{s}) - P_{f} (t_{i})) \frac{1}{{(1 + r)}^{t_{s}}})

(15)

where, $C_{I}$ is the expected inspection cost; $C_{R}$ is the expected repair cost; $C_{F} (t_{i})$ is the expected failure cost; $P_{r} t_{i})$ is the maintenance probability at $t_{i}$ moment; $t_{s}$ is the design life of the bridge; $t_{i}$ is the detection time.

The bridge fatigue maintenance optimization is transformed to the problem that the expected total cost of maintenance is minimum under the condition that the fatigue reliability of the component is higher than the minimum allowable reliability at a given time, $t_{s}$ . Generally, $t_{s}$ is the design life of a bridge. The optimization variables include the detection time, $t_{i}$ , detection probability, $P_{d}$ , repair willingness, $γ_{P}$ , and maintenance measures, $H^{(k)}$ , and so on. The expected total cost of the maintenance is as follows:

{\begin{matrix} min_{t_{i}, q, k} C_{T} (t_{i}, q, k) = C_{I} + C_{R} + C_{F} \\ s . t . β (t_{i}) \geq β_{min} \end{matrix}

(16)

In equation (16), $β (t_{i})$ is the reliability index of the time $t_{i}$ , $t_{i} \in (0, t_{s}]$ ; $β_{min}$ is the minimum allowable reliability of a component.

A case study

Taking the cable-beam anchorage of a suspension bridge with a main span 788m as an example, the fatigue performance degradation and the optimal time of the first fatigue detection and maintenance in the design life period are analyzed.

General description

The bridge cited in this paper is a steel box suspension bridge which main span is 788 m. The overall layout of this bridge is shown in Figure 2. The main beam is an orthotropic steel box beam, 3.5 m high, 30.7 m width (including wind fairing). The thickness of top plate is 14 mm, the thickness of U-shaped stiffener is 8 mm, and the thickness of bottom plate is 10mm. The thickness of the transverse diaphragm at ear plate is 12 mm, and the remaining diaphragms thickness are 8 mm. The connection components between the steel box beam and the ear plate are anchor plates. The cable-beam anchorage structure of this bridge consists of the ear plates, the stiffening plates, the longitudinal and transverse diaphragms and so on, as shown in Figure 3. Full weld is applied to weld between adjacent panels. The lower part of the ear plate is wholly inserted into the steel box beam. Anchorages of lower part of cable are connect with the transverse diaphragms of stiffened steel box beam, which can position the ear plates and transmit the cable forces to the steel box beam.

Figure 2.

The overall layout of bridge.

Figure 3.

Local detail of cable-beam anchorage structure.

Fatigue reliability calculation of cable-beam anchorage

The initial crack size, $a_{0}$ , and the parameters, C and m, usually have a large uncertainty, so all are considered as random variables. The failure crack size $a_{f}$ can be determined according to the normal using requirements and the size of the welding details. The variation coefficient $Δ σ$ is related to its acquisition method, and the range of values is generally adopt 0.05∼0.1. The $Δ σ$ in this paper is obtained by using standard fatigue vehicles to take structural analysis. The values of each parameter are shown in Table 1. For limited length, this paper only lists the crack propagation and the corresponding maintenance strategy for detail A in Figure 3

Table 1.

Fatigue-related stochastic variables.

Variable designation	unit	Distribution type	Mean value	COV
$a_{0}$	mm	Lognormal	0.10	0.01
$a_{f}$	mm	Lognormal	11.2	1.12
m	mm	Normal distribution	3.0	0.6
C	$\frac{10^{- 13}}{time {(N / mm 3 / 2)}^{m}}$	Lognormal	2.5	0.923
$Δ σ$	MPa	Normal distribution	15.5	1.55
r		Normal distribution	2%	0.2%
$υ$	$\frac{10^{6} time}{year}$	Normal distribution	1.277	0.1277

According to equation (5), the fatigue failure probability of welded steel bridge is related to the fatigue reliability index. $Φ (.)$ , with the cumulative distribution function of the standard normal distribution, so the failure probability index of welded steel bridges increases with the increase of operation time, while the fatigue reliability decreases with the increase of operation time. Therefore, the larger the failure probability is, the smaller the reliability index is. The Monte Carlo method is used to compute the time-variant reliability of welding detail, as shown in Figure 4.

Figure 4.

Reliability during the life span of a detail.

The fatigue cracking of the welded detail in the cable-beam anchorage zone cannot cause overall catastrophic damage of the bridge structure, but it will aggravate the internal corrosion of steel box beam and cracking of deck pavement.^12,14 If the target reliability index $β_{min}$ is taken as 2.33, the corresponding failure probability is about 1%, and fatigue life is 88 year. If $β_{min}$ is taken as 3.03, the corresponding failure probability is about 0.1%, and fatigue life is 34 year. It can be seen, from the Figure 4, when the values of the target reliability index are different, the corresponding fatigue life will be very different.

The best maintenance time

It can be seen from Figure 4 that when the time is 34 year, the reliability index of the anchorage reaches minimum allowable reliability and the maintenance must be carried out, which is a conventional maintenance method. This method corresponds to the strategy that anchorage is only repaired after damage, the anchorage at this time closed to the failure, the failure risk is great, so maintenance at this time is not the best choice. In the practice of bridge maintenance, the costs of maintenance and management are often limited. How to reasonably allocate the cost for fatigue maintenance is an important issue that the maintenance management department should be considered.¹⁵ Combined with the existing research data, the values of optimization of variables are listed in Table 2.^16–20

Table 2.

Values of optimization of variables.

$C_{i} / C^{(n)}$	$C_{r} / C^{(n)}$	$C_{f} / C^{(n)}$	discount rate r
1	20	100,000	5%

Where meanings of $C_{i}$ , $C_{r}$ , $C_{f}$ , $C^{(n)}$ , r, are same as in equations (13–16).

Take data from Table 2 to (equations 13–16) and calculate the costs, and the total cost is the following:

min_{t_{i}} C_{T} = min_{t_{i}} (C_{I} + C_{R} + C_{F})

(17)

The relative proportional relationship between the expected inspection, maintenance, and failure costs is shown in Figure 5. When time is 18 year, the total cost is minimum and maintenance reaches the best time. The best maintenance time in Figure 5 advance 16 year than in Figure 4.

Figure 5.

Relationship of respected inspection, repair and failure costs.

With the increase of operating time, the failure probability of the structure and the cost of failure are gradually increased. The repair probability and repair costs are proportional to the operating time. The probability of failure is proportional to the operating time. Due to the discount rate and the operation time, the total cost will emerge a minimum value that it is correspond to the optimal maintenance time. Because the inspection cost is relatively small, the key factors that affect the optimal maintenance time are the change of maintenance cost and failure cost with time. The issue that determining the maintenance interval to make the total maintenance cost minimum is mainly depends on the repair probability and failure probability.

Conclusion

The optimization fatigue maintenance probability method is present, to assist the maintenance decision of welded steel bridge components.

The detection probability, the probability of maintenance willingness, the repair probability, and the failure probability, are all taken into account in the new optimal maintenance strategy.

In this strategy, optimization goal is to gain the minimum maintenance cost, based on the precondition that components meet minimum allowable reliability. Also, a balance should be kept between inspection cost, repair cost and failure cost.

The reliability index is inversely proportional to the failure probability, and inversely proportional to the fatigue life. As the serve time goes on, the best maintenance time needs to be advanced.

The repair probability and repair costs are directly proportional to the operation time, as well as the failure probability.

Both repair probability and failure probability, are the main influencing factors to the total maintenance cost. Besides, the optimal maintenance time has strong relation with repair cost and the failure cost.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received financial support for the research, authorship, and/or publication of this article: This research was supported by the State Key Laboratory of Mountain Bridge and Tunnel Engineering (CQSLBF-Y16-10), Mountain Bridge and Materials Engineering Research Center of Ministry of Education (QLGCZX-JJ2017-5), Venture & Innovation Support Program for Chongqing Overseas Returnees (cx2018113) and the National Natural Science Foundation of China (grant no. 51478071).

ORCID iDs

Hongmei Tan

Author biographies

Hongmei Tan is an Associate Professor of Chongqing Jiaotong University. Her research direction is maintenance strategy of bridge structures.

Yong Zeng is a Professor of Chongqing Jiaotong University. His research direction is advanced analysis and maintenance strategy of bridge structures.

Qianping Zhang received her bachelor’s degree from Chongqing Jiaotong University in 2019, and his research direction is maintenance strategy of bridge structures.

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Optimization method of fatigue maintenance for cable-beam anchorage zone of suspension bridge

Abstract

Keywords

Introduction

Reliability evaluation of fatigue life based on LEFM

Optimization of maintenance strategy for welding component

Crack detection probability

Maintenance willingness of decision-maker

Maintenance measures probability

Probabilistic optimization of fatigue maintenance strategy

A case study

General description

Fatigue reliability calculation of cable-beam anchorage

The best maintenance time

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

Author biographies

References