SHM-based F-AHP bridge rating system with application to Tsing Ma Bridge

Abstract

This paper aims at developing a structural health monitoring (SHM)-based bridge rating method for bridge inspection of long-span cable-supported bridges. The fuzzy based analytic hierarchy approach is employed, and the hierarchical structure for synthetic rating of each structural component of the bridge is proposed. The criticality and vulnerability analyses are performed largely based on the field measurement data from the SHM system installed in the bridge to offer relatively accurate condition evaluation of the bridge and to reduce uncertainties involved in the existing rating method. The procedures for determining relative weighs and fuzzy synthetic ratings for both criticality and vulnerability are then suggested. The fuzzy synthetic decisions for inspection are made in consideration of the synthetic ratings of all structural components. The SHM-based bridge rating method is finally applied to the Tsing Ma suspension bridge in Hong Kong as a case study. The results show that the proposed method is feasible and it can be used in practice for long-span cable-supported bridges with SHM system.

Keywords

structural health monitoring (SHM) / bridge rating method / fuzzy based analytic hierarchy approach / criticality and vulnerability rating / Tsing Ma Bridge

Introduction

Long-span cable-supported bridges begin to deteriorate once they are built and continuously accumulate damage during their service life due to natural hazards and harsh environments such as typhoons, earthquakes, vehicles, temperature and corrosion. To ensure the serviceability and safety of long-span cable-supported bridges, bridge rating methods are often adopted by bridge management authorities as guidance in determining the time intervals for inspections and the actions to be taken in the event of defects being identified. For instance, the Master Maintenance Manual has been developed by the Hong Kong Highways Department (HyD) for the management, operation and maintenance of the Tsing Ma suspension bridge in Hong Kong [1,2], including the guidance in defining the frequencies and time intervals for inspections [3].

Nevertheless, most of the currently-used bridge rating methods are based partly on engineering analysis and partly on practical experience. Efforts have been made by some scholars to pursue more scientific yet practical bridge rating methods. Kushida et al. [4] proposed a membership function to quantify the subjective uncertainty included in empirical knowledge on bridge rating. Melhem [5] presented a fuzzy inference model based on a priority setting obtained through the solution of an eigenvector problem involving a pair-wise comparison matrix of importance. Aktan et al. [6] integrated analytical and experimental researches to assess global conditions and evaluate the serviceability and safety of bridges. Liang et al. [7] set up an evaluation multiple layer fuzzy method for evaluating the damage stage of existing bridges. Kawamura and Miyamoto [8] proposed a concrete bridge rating expert system for deteriorated concrete bridges, constructed from multi-layer neural networks. Sasmal et al. [9,10] developed a systematic procedure and formulations for rating existing bridges using fuzzy mathematics.

On the other hand, structural health monitoring (SHM) technology gains a rapid development recently. SHM systems have been designed and installed in a number of long-span bridges to monitor their serviceability and safety [11]. For example, the wind and structural health monitoring system (WASHMS) has been designed and installed by HyD in the Tsing Ma Bridge since 1997 to monitor its functionality and safety [12,13]. Based on the measurement data recorded by the WASHMS, some analysis has been performed to find loading properties and structural characteristics [14,15]. Some numerical methods [16–18] have also been developed for the prediction of structural responses and potential damages of the bridge by using the loading models and bridge model updated by the measured data. However, the measurement data as well as the updated models have not been fully utilized for the current bridge rating. There is insufficient link between the bridge rating method and the SHM to fulfill the common goals.

In this regard, this paper presents a SHM-based bridge rating method for bridge inspection of long-span cable-supported bridges. The fuzzy based analytic hierarchy process (F-AHP) is employed, and the hierarchical structure for synthetic rating of each structural component of a bridge is proposed. The procedures of determining relative weighs and fuzzy synthetic ratings for both criticality and vulnerability are then suggested. The fuzzy synthetic decisions for inspection are made in consideration of the synthetic ratings of all structural components. The criticality and vulnerability analyses are performed largely based on field measurement data from the SHM system to offer relatively accurate condition evaluation of the bridge and to reduce uncertainties involved in the existing rating method. The SHM-based F-AHP bridge rating method is finally applied to the Tsing Ma suspension bridge in Hong Kong as a case study.

Decision for bridge inspection through F-AHP

Formation of a hierarchical structure

The analytic hierarchy process (AHP) was developed by Saaty based on an axiomatic foundation [19]. AHP has attracted the interest of many researchers because of its interesting mathematical properties and easy applicability. The main steps in the application of AHP to the current problem are as follows: 1) to decompose a general decision problem into hierarchical sub-problems that can be easily comprehended and evaluated; 2) to determine the priorities of the items at each level of the decision hierarchy; and 3) to synthesize the priorities to determine the overall priorities of the decision alternatives. Since a long suspension bridge is a very complex system and the decision-making takes place in a situation in which the pertinent data and the sequences of possible actions are not precisely known, it is important to adopt fuzzy data to express such situations in decision-making of inspection, leading to the so-called F-AHP bridge rating method used in this study.

In the proposed rating system based on the F-AHP for a long-span cable-supported bridge, the top level can be assigned as an objective level upon which the best decision for inspection could be made for each structural component. The next level of the hierarchical structure can be defined as a criterion level upon which the criticality rating and the vulnerability rating can be respectively determined for each structural component based on the criticality and vulnerability rating criteria in the next level called the index level. After the hierarchical structure is constructed, one can then determine the relative weights (priorities) of the items at each level of the decision hierarchy based on the mathematical properties of AHP. Finally, one can synthesize the relative weights at all the levels to make the best decision for inspection. The hierarchical structure for synthetic rating of each structural component of a bridge is shown in Fig. 1. The criticality rating criteria of each component are composed of five criticality factors from C1 to C5. The vulnerability rating criteria are set up based on three vulnerability factors V1 to V3. Each of the vulnerability factors is rated in three serial effects of VA, VB, and VC.

Relative weights for each level

AHP often uses the eigenvalue solution of comparison matrices to find the best relative weights (relative importance) for different elements in each level. The first step is to carry out pair-wise comparisons of elements in each level of the hierarchical structure.

By assuming that the index level for criticality rating consists of A_{\mathrm{1}},A_{\mathrm{2}},\mathrm{...},A_n items, the comparative matrix \mathit{A} can be constructed by comparing objective i with objective j to obtain the relative weights {\alpha }_{ij}={\omega }_i/{\omega }_j (i, j = 1, 2,…,n), as shown in Table 1.

A decision-maker could provide only the upper triangle of the above comparison matrix. The reciprocals placed in the lower triangle do not need any further judgment because of the following characteristics.

(1)\begin{array}{ll}{\alpha }_{ij}>\mathrm{0},& i,j=\mathrm{1},\mathrm{2},\mathrm{...},n;\\ {\alpha }_{ij}=\mathrm{1}/{\alpha }_{ji},& i,j=\mathrm{1},\mathrm{2},\mathrm{...},n;\\ {\alpha }_{ii}=\mathrm{1},& i=\mathrm{1},\mathrm{2},\mathrm{...},n.\end{array}

Each entry of matrix \mathit{A} represents a pair-wise judgement. In the consistent reciprocal matrix \mathit{A}, it can be proved that the matrix \mathit{A} has rank 1 with nonzero eigenvalue equal to n [19]. In most of the practical problems, the pair-wise comparisons are not perfect, and one must find the principal right-eigenvalue that satisfies

(2)\mathit{A}\{\omega \}={\lambda }_{\max }\{\omega \}

where \{\omega \} is the eigenvector with respect to eigenvalue n; and {\lambda }_{\max }\approx n. The next step is to check the consistency of comparison matrices in terms of the consistency ratio CR. The consistency ratio CR is determined by first estimating {\lambda }_{\max } of matrix \mathit{A}. The consistency index CI of the matrix \mathit{A} is defined as

(3)CI=({\lambda }_{\max }-\mathrm{1})/(n-\mathrm{1})\mathrm{.}

Then, the consistency ratio CR is calculated by dividing CI with the random index RI as shown in Table 2. Each RI is an average random consistency index derived from a sample of size 500 of randomly generated reciprocal matrices. If the previous approach yields a CR greater than 0.10 then a re-examination of the pair-wise judgements is recommended until a CR less than or equal to 0.10 is achieved [19].

If the consistency of the comparison matrix is satisfied, the relative weights are calculated based on the normalized eigenvector corresponding to the maximum eigenvalue.

Fuzzy synthetic ratings

The criticality and vulnerability ratings for each structural component are based on the criticality and vulnerability factors in relation to the criticality and vulnerability criteria. Although the utilization of the SHM system will reduce the uncertainties in the estimation of criticality and vulnerability factors, the accuracy of these factors is still not precisely known. Therefore, it is important to treat these factors as fuzzy data in the decision making for inspection. Furthermore, the numerical numbers for each of the factors range from 0 and 100 in this study to facilitate the decision making using the F-AHP based bridge rating method.

The triangular fuzzy numbers [10] are preferred in this study. A fuzzy number M on U\in (-\infty ,+\infty ) is defined to be a triangular fuzzy number if its membership function u_{\mathrm{m}}(x):U\to [\mathrm{0},\mathrm{1}] is equal to

(4){\mu }_{\mathrm{m}}(x)=\{\begin{array}{l}{\displaystyle \frac{x-r}{m-r}},\begin{array}{ll}& x\in [r,m];\end{array}\hfill \\ {\displaystyle \frac{x-u}{m-u}},\begin{array}{ll}& x\in [m,u];\end{array}\hfill \\ \begin{array}{cc}\mathrm{0},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\end{array}\hfill \end{array}

where r≤m≤u; r and u stand for the lower and upper values of the support for the decision of the fuzzy number M, respectively; m is the modal value. The triangular fuzzy number is denoted as M = (r, m, u). Let us select the five-point fuzzy rating set \{G\} as

(5)\{G\}=\left\{\begin{array}{ll}\mathrm{0}& \begin{array}{ll}\mathrm{25}& \begin{array}{ll}\mathrm{50}& \begin{array}{ll}\mathrm{75}& \mathrm{100}\end{array}\end{array}\end{array}\end{array}\right\}.

The triangular fuzzy numbers (0, 0, 50), (0, 25, 75), (0, 50, 100), (25, 75, 100), (50, 100, 100) can be generated to improve the decision making of criticality and vulnerability ratings. Figure 2 shows the five triangular fuzzy numbers defined with the corresponding membership function.

In the F-AHP based bridge rating method, the criticality rating for each structural component can be determined by the following steps:

(a) Figure out the criticality factors for each structural component using SHM-based computation simulations or engineering judgments (see Section 3).

(b) Work out the membership degrees matrix {\mathit{R}}_{\mathrm{c}} based on the fuzzy membership functions and the criticality factors.

(6){\mathit{R}}_{\mathrm{c}}=\left(\begin{array}{ccc}{r}_{\mathrm{11}}& \cdots & r_{\mathrm{1}m}\\ ⋮& \square & ⋮\\ r_{n\mathrm{1}}& \cdots & r_{nm}\end{array}\right),

where n is the number of items in the criticality index level; m is the number of the fuzzy rating values in the fuzzy rating set \{G_{\mathrm{c}}\}; r_{ij} denotes the membership degree of the ith item to the jth fuzzy membership function; the subscript c means the criticality.

(c) The fuzzy synthetic rating vector \{B_{\mathrm{c}}\} for criticality can then be determined by

(7)\{B_{\mathrm{c}}\}={\{{\omega }_{\mathrm{c}}\}}^{\mathrm{T}}{\mathit{R}}_{\mathrm{c}}.

(d) The fuzzy synthetic rating T_c for the criticality of the concerned structural component can be finally obtained as

(8)T_{\mathrm{c}}=\{G_{\mathrm{c}}\}{\{B_{\mathrm{c}}\}}^{\mathrm{T}}.

In the F-AHP based bridge rating method, the vulnerability rating for each structural component can be determined by the following steps:

(a) Figure out the vulnerability factors for each structural component in the vulnerability index level 2.

(b) Calculate the vulnerability factors for each structural component in the vulnerability level 1 using the weighted product model.

(9)\{\begin{array}{l}V\mathrm{1}=VA{\mathrm{1}}^{\mathrm{1}/\mathrm{3}}\times VB{\mathrm{1}}^{\mathrm{1}/\mathrm{3}}\times VC{\mathrm{1}}^{\mathrm{1}/\mathrm{3}};\hfill \\ V\mathrm{2}=VA{\mathrm{2}}^{\mathrm{1}/\mathrm{3}}\times VB{\mathrm{2}}^{\mathrm{1}/\mathrm{3}}\times VC{\mathrm{2}}^{\mathrm{1}/\mathrm{3}};\hfill \\ V\mathrm{3}=VA{\mathrm{3}}^{\mathrm{1}/\mathrm{3}}\times VB{\mathrm{3}}^{\mathrm{1}/\mathrm{3}}\times VC{\mathrm{3}}^{\mathrm{1}/\mathrm{3}}.\hfill \end{array}

(c) Work out the membership degree matrix {\mathit{R}}_{\mathrm{v}} based on the fuzzy membership functions and the vulnerability factors, where the subscript v means the vulnerability.

(d) The fuzzy synthetic rating vector \{B_{\mathrm{v}}\} for vulnerability can then be determined by

(10)\{B_{\mathrm{v}}\}={\{{\omega }_{\mathrm{v}}\}}^{\mathrm{T}}[R_{\mathrm{v}}].

(e) The fuzzy synthetic rating T_v for the vulnerability of the concerned structural component can be finally obtained as

(11)T_{\mathrm{v}}=\{G_{\mathrm{v}}\}{\{B_{\mathrm{v}}\}}^{\mathrm{T}}.

Fuzzy synthetic decision

The fuzzy synthetic rating R at the objective level can be calculated based on the fuzzy synthetic ratings, T_c and T_v, and the relative weights at the criterion level.

(12)R={\{{\omega }_{\mathrm{c}\mathrm{v}}\}}^{\mathrm{T}}\left\{\begin{array}{l}{T}_{\mathrm{c}}\\ T_{\mathrm{v}}\end{array}\right\}.

After the fuzzy synthetic ratings of all the structural components are obtained, the prioritization or optimum for inspection frequency (fuzzy synthetic decision for inspection) can be determined. The larger the value of R, the smaller is the inspection time interval. One example is shown in Table 3. The mapping relationship between R and inspection interval depends on many factors: some are technical and some are economical. The four time intervals, i.e., 6-month, 1-year, 2-year and 6-year, are actually used in the current routine inspection works of the Tsing Ma Bridge [3]. This study just adopts the four time intervals suggested by experts, and then calculates the priority of inspection for different components in terms of the fuzzy synthetic rating value R.

SHM-based criticality and vulnerability analyses

Criticality factors (CF)

The criticality factors include five items in this study for a long-span cable-supported bridge. Table 4 shows the definitions, range and points for each criticality factor. The numerical values of the five criticality factors range from 0 to 100.

(a) Criticality factor C1——it is understandable that the structural component with an alternative load path or redundancy is robust, that is, no serious failure consequence will be induced by limited damage to this structural component. Therefore, the more the redundancy for a given structural component, the smaller is the numerical value of the criticality factor C1 for this component. This criticality factor is represented by three numerical values of 100, 67 and 0 in this study.

To quantify the redundancy of a category of bridge component (substructure), the definition of redundancy should be determined first. An intuitive definition for redundancy might be the ratio of load that causes the collapse of the system to which causes the failure of the first member of the intact substructure [20]. In this regard, nonlinear pushover analysis [20] or progressive collapse analysis [21] may be used to perform the redundancy analysis based on the finite element model of the bridge which represents real structure members and actual nonlinear behaviors. It is worthy to point out that the ultimate load condition used in the pushover analysis or progressive collapse analysis can be determined by the statistical model of loads measured by the SHM system.

(b) Criticality factor C2——this factor represents the strength reliability of a structural component, which is determined based on the strength utilization factor (SUF). SUF is calculated using Eq. (13). If the SUF of a certain structural component reaches the extreme value, the numerical value of the criticality factor C2 for this component should be 100. Otherwise, the numerical value should be the ratio of the strength utilization factor divided by the extreme value.

(13)\mathrm{S}\mathrm{U}\mathrm{F}={\displaystyle \frac{{\gamma }_{\mathrm{L}}{\sigma }_{\mathrm{L}}(\mathrm{1.0}+I)}{\varphi R_{\mathrm{N}}-{\gamma }_{\mathrm{D}}{\sigma }_{\mathrm{D}}}},

where R_{\mathrm{N}} is the as-built nominal resistance; \varphi is the strength reduction factor; {\mathit{\gamma }}_{\mathrm{L}} is the partial load factor for live loads; {\mathit{\gamma }}_{\mathrm{D}} is the partial load factor for dead loads; {\mathit{\sigma }}_{\mathrm{L}} is the stresses due to live loads; {\mathit{\sigma }}_{\mathrm{D}} is the stresses due to dead loads; and I is the impact factor for dynamic live loads.

To calculate SUF for each component of a bridge, the finite element model of the bridge shall be developed based on the approach of one analytical member representing one real member at a stress level. This bridge model can be further updated and verified by the measurement data recorded by the SHM system, to form the so-called structural health monitoring-orientated finite element model (SHM-FEM). The numerical models for various types of loads can also be simulated based on the measured loads from the SHM system. Therefore, local stress for a component without sensors installed can also be directly computed using the SHM-FEM and simulated loads imposed on local members of the bridge [16,17].

(c) Criticality factor C3——this item represents the fatigue reliability of a structural component. In a similar way with the criticality factor C1, the relative fatigue criticality of different structural components could be represented by three numerical values of 100, 67 and 0 in this study.

To determine the criticality factor C3 for each of the structural components, fatigue life of the structural component shall be estimated. The wind loadings and traffic loadings measured by the SHM system can be used to derive the actual loading spectra for fatigue assessment. The stress analysis can then be performed based on the SHM-FEM of the bridge and the actual load spectra [18]. The fatigue life at the fatigue-critical components can finally be estimated based on the Miner’s law or the continuum damage mechanics.

(d) Criticality factor C4——this item emphasizes the imperfections (deterioration/damage) of structural components detected by the previous inspections. In this regard, the severity of imperfections and the urgency to repair is respectively considered and represented by the three numerical values of 100, 67 and 0 in this study. Besides the visual inspection, the SHM system can now be used as a tool to detect imperfections by analyzing the measurement data directly recorded by strain sensors, fatigue sensors, accelerometers and others.

(e) Criticality factor C5——this item represents the ultimate loading-carrying capacity of structural components under extreme loading event. The structural component of foremost failure will be the most critical component and the corresponding energy demand for failure will be the least. The relative ultimate loading-carrying capacity of different structural components could be represented by three numerical values of 100, 67 and 33 in this study. The nonlinear pushover analysis or the progressive collapse analysis used for redundancy analysis can also be applied to determine the criticality factor C5.

Vulnerability factors (VF)

The vulnerability factors include three items in this study. Table 5 shows the definitions, range and points for each vulnerability factor.

(a) Vulnerability factor V1——this item represents damage due to extra-slowly varying effect, such as carbonation of concrete and corrosion of steel. The item V1 consists of three sub-items, VA1, VB1, and VC1. Each sub-item can be represented by the three numerical values of 100, 50, and 0 according to three different ranges.

(b) Vulnerability factor V2——this item represents damage due to rapidly varying effect, for instance, accidental damage caused by vehicle collision and ship collision. The item V2 consists of three sub-items, VA2, VB2 and VC2, each of which can be represented by the three numerical values of 100, 50 and 0 respectively.

(c) Vulnerability factor V3——this item represents damage due to slowly varying effect, for example, movement of bearings and movement joints because of daily temperature changing. The item V3 has also three sub-items, and each sub-item has three difference ranges represented by three numerical values correspondingly.

The item VA1 can be quantified by corrosion rate which is defined as reciprocal of time demand to reach critical corrosion value since present. The corrosion rate can be calculated based on either the directly measured data by corrosion sensors of the SHM system or the numerical model updated by the SHM system. The numerical values for the item VA2 can be allocated by experience just after any extreme accidental event occurs and damages on some components are detected. The item VA3 can be quantified by either visual inspection or analyzing the measurement data recorded by displacement sensors of the SHM system. The numerical values for the items VB1, VB2 and VB3 can be allocated to various structural components based on the experiences of inspectors. The numerical values for the items VC1, VC2 and VC3 can be the same as those for the criticality factor C1.

Case study

The Tsing Ma Bridge in Hong Kong is taken as an example in this study to demonstrate the feasibility of the SHM-based F-AHP bridge rating method as guidance in determining the time intervals for inspection. The Tsing Ma Bridge is a suspension bridge with an overall length of 2132 m and a main span of 1377 m (see Fig. 3(a)). The height of the two reinforced concrete towers is 206 m. The two main cables of 1.1 m diameter and 36 m apart in the north and south are accommodated by the four saddles located at the top of the tower legs. The bridge deck is a hybrid steel structure consisting of Vierendeel cross frames supported on two longitudinal trusses acting compositely with stiffened steel plates (Fig. 3(b)). The bridge deck carries a dual three-lane highway on the upper level of the deck and two railway tracks and two carriageways on the lower level within the bridge deck as shown in Fig. 3(b).

Classification of structural components

The key structural components of the Tsing Ma Bridge are classified into 15 groups and 55 components for criticality and vulnerability analyses. The 15 groups, which are basically the key components of the Tsing Ma Bridge for direct and indirect load-transfer, are: 1) suspension cables, 2) suspenders; 3) towers, 4) anchorages, 5) piers; 6) outer-longitudinal trusses; 7) inner-longitudinal trusses; 8) main cross frames; 9) intermediate cross-frames; 10) plan bracings; 11) deck; 12) rail way beams, 13) bearings; 14) movement joints; and 15) Tsing Yi approach deck. The details of classification in each group are illustrated in Table 6.

Criticality and vulnerability factors

This section takes the criticality factors C2 and C3 as an example to explain how to use the measurement data recorded by the SHM system installed in the Tsing Ma Bridge to determine these factors.

To determine the criticality factor C2 for each of the structural components of the Tsing Ma Bridge, the criticality analysis of the bridge is performed on strength of the bridge under design normal combined loads in terms of the strength utilization factor. The strength utilization factor is one of the five criticality factors in the criticality rating of the bridge. To fulfill this task, the SHM-oriented finite element model of the Tsing Ma Bridge is established (see Fig. 4) based on the approach of one analytical member representing one real member at a stress level using the ABAQUS software package [16]. Seven types of loads (dead loads, super-imposed dead loads, temperature loads, highway loads, railway loads, wind loads, and seismic loads) and three load combinations have been considered in the stress analysis of the bridge. Except for the dead loads, the super-imposed dead loads and the seismic loads, there are 2, 24, 8 and 3 load cases for the temperature loads, the highway loads, the railways loads and the wind loads, respectively. In the three load combinations, there are also a total of 52 load cases. For each load case, the stresses in the major structural components are determined, and the stress distributions are obtained for each of the major structural components. Based on the obtained stress distribution results, the stresses in the structural components at five key bridge deck sections are provided. The strength utilization factors of the major structural components are calculated, from which the critical locations of each major structural components are identified. For the bridge towers made of reinforced concrete, the strength analysis are carried out using the load-moment strength interaction method, and the strength utilization factors of the two tower legs are determined. With the computed strength utilization factors for the major structural components of the bridge, the point can be assigned according to Table 4.

To determine the criticality factor C3 for each of the structural components of the Tsing Ma Bridge, the criticality analysis of the bridge is performed on fatigue life of the structural components. The railway loading and highway loading are considered to be the major contributors to fatigue damage of the bridge [17,18]. The railway and highway loadings measured by the SHM system are then used to derive the actual train spectrum and the actual road vehicle spectrum for fatigue assessment. A traffic induced stress analysis method is proposed based on the SHM-oriented finite element model of the bridge and the influence line method for the determination of stress time histories. The fatigue-critical locations are identified for different bridge components through the rigorous stress analysis. Finally, the fatigue lives due to both train and road vehicles at the fatigue-critical components are estimated using the vehicle spectrum method recommended in British Standard [22]. With the computed fatigue lives at fatigue-critical locations of each bridge component, the point can be assigned according to Table 4.

The measurement data recorded by the SHM system have a relatively short duration of 13 years, and few imperfections or damages have been detected by visual inspections or the SHM system. Also there is no corrosion sensor installed on the bridge. The complex nonlinear pushover analysis and progress collapse analysis for such a large structure have not been completed yet. As a result, except for criticality factors C2 and C3, other criticality factors and vulnerability factors given in the existing Master Maintenance Manual of the Hong Kong Highway Department for the Tsing Ma Bridge have to be adopted at this stage [3]. The criticality factors of each bridge component used in this study are listed in Table 7.

Relative weights

To apply the proposed SHM-based F-AHP bridge rating method to the Tsing Ma Bridge, the relative weights shall be determined for each level. According to the AHP described in Section 2.2, the comparison matrix and the relative weights for the criticality index level 1 (CR1) are found and listed in Table 8. The counterparts for the vulnerability index level 1 (VR1) are listed in Table 9.

Since the comparison matrix is not unique, its effect on the final results shall be investigated. Tables 10 and 11 tabulate another set of comparison matrix and relative weights for the criticality index level 1 (CR2) and vulnerability index level 1 (VR2) respectively.

If the importance of criticality is regarded to be the same as that of vulnerability, the relative weight vector for the criterion level can be taken as \{{\omega }_{\mathrm{c}\mathrm{v}}\}={\{\mathrm{0.5},\mathrm{0.5}\}}^{\mathrm{T}} for both cases.

Inspection based on fuzzy synthetic decision

Based on the two sets of relative weights and according to the proposed SHM-based F-AHP bridge rating method, the decision on the time intervals for inspection can be determined and the results are listed in Table12 for the two cases. It can be seen that the time intervals for inspection are almost the same for the two cases concerned, which indicates that the effect of the relative weights from different comparison matrices is small. It can also be seen that for the bridge components concerned, the time intervals for inspection are either 1 year or 2 years.

Conclusions

A SHM-based F-AHP bridge rating method for long-span cable-supported bridges has been proposed in this study. The proposed bridge rating method has been applied to the Tsing Ma Bridge in Hong Kong. The effects of different comparison matrices and relative weights on the decision of time intervals for inspection have been investigated. The results show that the effects of relative weights from different comparison matrices are small. For the bridge components concerned, the time intervals for inspection are either one or two years. The results from the case study indicate that the proposed bridge rating method is feasible and can be used in practice for long-span cable-supported bridges with SHM systems.

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