# Safety Evaluation of Steel Temporary Beam Service Status Based on the Combination Weighting-Fuzzy Model of Game Theory

Authors: Xiaozhong Li; Jianshun Wang; Xingchong chen; Caiyun Sun
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

The safe service state of railway steel girders directly affects the safety of frame bridge jacking and the operation of existing lines. For this reason, this paper first adopts a dynamic and static method to analyze the deflection, acceleration, and stress value of temporary steel beams, personnel factors, management factors, and environmental factors. In terms of factors, a safety evaluation index system for the service status of the beams is established, and then, the subjective and objective weights of the indicators are determined by the analytic hierarchy process (AHP) and the coefficient of variation method. The combined weights of the indicators are calculated by introducing game theory. Finally, the fuzzy comprehensive evaluation model is used to obtain the security level of the beam service status. The weight of the indicators that affects the safety of the service state of the beam and the order of risks can be obtained through the calculation. For the indicators ranked first, measures such as increasing the amount of monitoring can be prioritized to reduce the possibility of safety accidents in frame bridge jacking construction and existing line operation.


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## Body

## 1. Introduction

The number of urban roads and existing railway interchange projects is increasing daily. Large-span jacking frame bridge construction technology is used most often when a road passes under an existing railway. To avoid interrupting railway transportation, the line needs to be reinforced [1, 2]. During a frame bridge jacking construction, to ensure normal operation of the existing line, D-type beams are often used to reinforce the existing line within the construction range [3–7]. The train load transfer system is changed from “track ⟶ sleeper ⟶ ballast ⟶ roadbed ⟶ Earth foundation” to “track ⟶ beam ⟶ support pier ⟶ Earth foundation.” The safe service state of the D-type beam directly affects the safety of the frame bridge jacking construction and the operational safety of the existing line.To ensure the safety of steel beam structures under the combined action of design loads and accidental overloads, the beam safety is evaluated by regular monitoring, comparing the measured maximum stress with the allowable stress of the steel beam calculated by the design code [8]. Various fiber-optic sensing systems based on different sensing mechanisms have been developed to evaluate the safety of steel beam structural members [9–11]. Wardenier et al. [12, 13] proposed a fatigue strength evaluation method for steel beams with different failure modes through fatigue tests. Citarelli and Feldmann [14, 15] carried out a statistical analysis on the failure cases of steel beam connection welds and gave fatigue strength evaluation parameters considering different geometric shapes of welds. Liu et al. [16, 17] conducted experimental and theoretical studies on the temperature distribution of steel beam members under the direct action of solar radiation. Hu [18, 19] monitored the settlement and longitudinal deformation of a temporary beam and the horizontal side shift and deflection of the beam. Zhang [20] calculated the vertical deflection of a temporary beam and the reaction force of the support by numerical simulation. Song et al. [21] found that the fulcrum of a temporary beam is subject to relatively large stress and deflection. Du [22] determined the safety of a temporary beam from the stress value, deflection of the beam, and the difference between the heights of the two adjacent temporary beams. Liu [23] compared the overhead construction technology of D-beams with other reinforcement methods. Zhao [24] proposed the use of bored pile piers combined with large steel temporary beams to support D-type beams for line reinforcement. Scholars have studied the deformation state of temporary beams, but they have not fully identified the factors that affect the service state of D-type temporary beams. This study adopts a combination of the dynamic and static methods to establish an index system for evaluating the safety of the service state of a steel temporary beam and uses a combination of qualitative and quantitative analysis to evaluate the safety of the service state of the beam. The research results provide a theoretical basis for the safety of frame bridge jacking construction and existing line operation.


## 2. Establishment of a Safety Evaluation Index System for Steel Temporary Beams in Service

By referring to the monitoring items provided by the specification, the actual monitoring items are selected as the dynamic evaluation indicators according to the actual project situation. The indicators that affect the service status of the beams from the three aspects of personnel, management, and environment are identified as the static evaluation indicators. Dynamic quantitative indicators, such as the deflection, strain value, and acceleration of temporary steel beams, are selected to analyze the service status of the beam from both qualitative and quantitative aspects and increase the comprehensiveness of safety evaluation information. By screening and classifying risk factors and inviting experts to revise, finally, the service status safety evaluation index of temporary steel beams is established from six aspects: temporary steel beam deflection, stress value, and acceleration; personnel factors; management factors; and environmental factors, as shown in Figure1.Figure 1
The safety evaluation index system of the service state for the temporary beam.


## 3. Determining the Weight of Evaluation Indicators

The level of rigor of weight assignment directly affects the accuracy of the complete evaluation model calculation results. At present, a combination of subjective and objective weighting methods is used for the weighting problem, and the outcome is suitable accurate and has achieved good results in practical applications. Therefore, this paper uses hierarchical analysis and the coefficient of the variation method to determine the subjective and objective weights of indicators, respectively, and then introduces game theory to determine the combination weight value of each evaluation index. The analytic hierarchy process (AHP) is a method for solving complex multiobjective decisions proposed by the famous American operations researcher T.L. Satty in the early 1970s. The AHP algorithm is a combination of the qualitative and quantitative decision analysis methods. It is a process that models and quantifies the decision-making thinking process of decision makers for complex systems. By applying this method, the decision maker can derive the weights of different options by decomposing a complex problem into several levels and several factors and by making simple comparisons and calculations among the factors, which provides a basis for the selection of the best option.


### 3.1. Analytic Hierarchy Process

(1)
Determining the evaluation matrixAccording to the hierarchical structure of the established index system, the 9-scale method [25] is used by experts for scoring, and the indices at the same level are compared in pairs. Assuming that there are m indices at a certain level, the evaluation matrix A is formed as shown in formula (1).(1)A=a11a12⋯a1na21a22⋯a2n⋮⋮⋮⋮am1am2⋯amn.(2)
Using the square root method to find the subjective weightθi(2)θi=∏j=1maij1/m∑i=1m∏j=1maij1/mi,j=1,2,⋯m.(3)
Consistency checkTo check whether the evaluation matrix has satisfactory consistency, the CI is compared with the average random consistency index RI, which is called the consistency ratio of the evaluation matrix and is denoted as CR.(3)CR=CIRI<0.1,whereCI=λmax−mm−1, λmax=1m∑i=1mAWiWi, where λmax is the maximum positive eigenvalue of A, CR is the consistency ratio, CI is the consistency index, and RI is the average stochastic consistency index. The RI values are shown in Table 1.If the consistency test conditions are met, the process proceeds to the next calculation step; otherwise, the experts are invited to score again.Table 1
Average random consistency index.
mRI102030.5240.8951.1261.2671.3681.4191.46101.49111.52121.54131.56141.58


### 3.2. Coefficient of the Variation Method

The coefficient of the variation method [26] is a type of the objective weighting method that uses the information contained in an evaluation index to calculate the weight of multiple indices. In the calculation process, the importance of the indicator can be reflected by the coefficient of variation, and a larger coefficient of variation indicates a greater impact of the indicator, that is, greater importance. The coefficient of the variation method is used to calculate the weight, which avoids the influence of expert preference on the results in the subjective weighting method and makes the evaluation results more objective and accurate.(1)
Build a sample matrix. Withn sets of data and m evaluation indicators, the matrix X is formed as follows:(4)X=xijn×mi=1,2,…n;j=1,2,…,m.(2)
Calculate the averagex¯j and standard deviation Sj of each evaluation index:(5)x¯j=1n∑i=1nxiji=1,2,…n;j=1,2,…,m,Sj=1n−1∑i=1nxij−x¯j2i=1,2,…n;j=1,2,…,m.(3)
Calculate the coefficient of variation of each index [27]as follows:(6)Vj=sjx¯ji=1,2,…n;j=1,2,…,m.(4)
Normalize the coefficient of variation to obtain the objective weightσi:(7)σi=Vj∑j=1mVji=1,2,…n;j=1,2,…,m.


### 3.3. Determination of Combination Weights Based on Game Theory

Game theory is a mathematical theory and method that examines how to make decisions and maximize benefits when there are multiple struggling or competitive individuals in a group. In this paper, game theory is applied to the combination weighting among various weights to find a weight that minimizes the deviation from each basic weight to ensure that the calculated weight fits the actual situation.According to the game theory method [28], the weights of risk factors are calculated by L different weight determination methods, and the basic weight set of risk factors is composed of W=w1,w2,⋯,wL. Then, the combined weight coefficient αi needs to satisfy the following equation[29]:(8)∑i=1Lαi∙wi∙wiT=wi∙wiTi=1,2,…,L.In this paper, the subjective and objective weights are determined; then,L is 2, so we set the subjective weight θ and the objective weight as σ. According to the abovementioned formula, we get the following equation:(9)θ∙θTθ∙σTσ∙θTσ∙σT∙α1α2=θ∙θTσ∙σT.The coefficients of the linear combination are normalized, and the normalization formula used is as follows:(10)αk∗=αk∑k=1Lαkk=1,2,…,L.Then, the combined weight is as follows:(11)wji=∑k=1Lαk∗∙wkTk=1,2;j=1,2,…,m,where 0<wj∗<1,∑j=1nwj∗=1,j=1,2,…,m.


## 3.1. Analytic Hierarchy Process

(1)
Determining the evaluation matrixAccording to the hierarchical structure of the established index system, the 9-scale method [25] is used by experts for scoring, and the indices at the same level are compared in pairs. Assuming that there are m indices at a certain level, the evaluation matrix A is formed as shown in formula (1).(1)A=a11a12⋯a1na21a22⋯a2n⋮⋮⋮⋮am1am2⋯amn.(2)
Using the square root method to find the subjective weightθi(2)θi=∏j=1maij1/m∑i=1m∏j=1maij1/mi,j=1,2,⋯m.(3)
Consistency checkTo check whether the evaluation matrix has satisfactory consistency, the CI is compared with the average random consistency index RI, which is called the consistency ratio of the evaluation matrix and is denoted as CR.(3)CR=CIRI<0.1,whereCI=λmax−mm−1, λmax=1m∑i=1mAWiWi, where λmax is the maximum positive eigenvalue of A, CR is the consistency ratio, CI is the consistency index, and RI is the average stochastic consistency index. The RI values are shown in Table 1.If the consistency test conditions are met, the process proceeds to the next calculation step; otherwise, the experts are invited to score again.Table 1
Average random consistency index.
mRI102030.5240.8951.1261.2671.3681.4191.46101.49111.52121.54131.56141.58


## 3.2. Coefficient of the Variation Method

The coefficient of the variation method [26] is a type of the objective weighting method that uses the information contained in an evaluation index to calculate the weight of multiple indices. In the calculation process, the importance of the indicator can be reflected by the coefficient of variation, and a larger coefficient of variation indicates a greater impact of the indicator, that is, greater importance. The coefficient of the variation method is used to calculate the weight, which avoids the influence of expert preference on the results in the subjective weighting method and makes the evaluation results more objective and accurate.(1)
Build a sample matrix. Withn sets of data and m evaluation indicators, the matrix X is formed as follows:(4)X=xijn×mi=1,2,…n;j=1,2,…,m.(2)
Calculate the averagex¯j and standard deviation Sj of each evaluation index:(5)x¯j=1n∑i=1nxiji=1,2,…n;j=1,2,…,m,Sj=1n−1∑i=1nxij−x¯j2i=1,2,…n;j=1,2,…,m.(3)
Calculate the coefficient of variation of each index [27]as follows:(6)Vj=sjx¯ji=1,2,…n;j=1,2,…,m.(4)
Normalize the coefficient of variation to obtain the objective weightσi:(7)σi=Vj∑j=1mVji=1,2,…n;j=1,2,…,m.


## 3.3. Determination of Combination Weights Based on Game Theory

Game theory is a mathematical theory and method that examines how to make decisions and maximize benefits when there are multiple struggling or competitive individuals in a group. In this paper, game theory is applied to the combination weighting among various weights to find a weight that minimizes the deviation from each basic weight to ensure that the calculated weight fits the actual situation.According to the game theory method [28], the weights of risk factors are calculated by L different weight determination methods, and the basic weight set of risk factors is composed of W=w1,w2,⋯,wL. Then, the combined weight coefficient αi needs to satisfy the following equation[29]:(8)∑i=1Lαi∙wi∙wiT=wi∙wiTi=1,2,…,L.In this paper, the subjective and objective weights are determined; then,L is 2, so we set the subjective weight θ and the objective weight as σ. According to the abovementioned formula, we get the following equation:(9)θ∙θTθ∙σTσ∙θTσ∙σT∙α1α2=θ∙θTσ∙σT.The coefficients of the linear combination are normalized, and the normalization formula used is as follows:(10)αk∗=αk∑k=1Lαkk=1,2,…,L.Then, the combined weight is as follows:(11)wji=∑k=1Lαk∗∙wkTk=1,2;j=1,2,…,m,where 0<wj∗<1,∑j=1nwj∗=1,j=1,2,…,m.


## 4. Establishment of the Safety Evaluation Model for the Service State of Steel Temporary Beams

Most of the risk evaluation indices are characterized by random, fuzzy, and other uncertainties, and for the solution of such problems, the fuzzy comprehensive evaluation method has unique advantages. Fuzzy comprehensive evaluation is a method based on fuzzy mathematics, applying the principle of fuzzy relationship synthesis, quantifying some factors with unclear boundaries that are not easy to quantify, and making a comprehensive evaluation of the affiliation level status of the system of interest from multiple factors. The fuzzy comprehensive evaluation method is based on the affiliation theory of fuzzy mathematics, which transforms qualitative evaluation into quantitative evaluation. Since this process can provide an overall evaluation of things or objects whose evaluation boundaries are not very clear, with a simple calculation process and clear results, this paper adopts the fuzzy comprehensive evaluation method to evaluate the safety of the service status of the beams. The main calculation steps are as follows.


### 4.1. Determining the Index Factor Set and Comment Set

The index factor set of fuzzy comprehensive evaluation is each evaluation index, and these indices constitute a set. If there arem indices, they can be expressed as U = (u1, u2,…, um).The set of rubrics for fuzzy integrated evaluation are the criteria for experts to evaluate each indicator. In this paper, the evaluation interval of the index is divided into five levels, expressed asV = (I, II, III, IV, and V), indicating that the security risk is lowest, low, moderate, high, and highest, respectively, forming a comment set with a fuzzy comprehensive rating.


### 4.2. Constructing the Fuzzy Evaluation Matrix

According to the index system established in this paper, two methods of expert scoring and on-site monitoring are used to collect the initial data of the static index and dynamic index evaluation.Assuming thatn sets of data are collected, an evaluation matrix R is formed as follows:(12)R=r11r12⋯r1nr21r22⋯r2n⋮⋮⋮rm1rm2⋯rmn,where m is the number of indicators and n is the number of the data groups.


### 4.3. Determination of the Index Membership Degree

We apply the indicator affiliation function in Figure2; the horizontal coordinate indicates the relative position of each indicator risk characteristic value xij on the domain of each risk evaluation level, a–e indicates the upper bound of five risk intervals, the intersection of each risk level affiliation function curve with the horizontal coordinate is the midpoint of two adjacent values, and the corresponding y value is the indicator affiliation degree relative to each risk evaluation level.Figure 2
Membership function.


### 4.4. Comprehensive Evaluation

According to formula (13), the target layer index membership degree can be obtained as follows [30]:(13)μh=μhXij∙Wij,where μh (Xij) and μh represent the membership degree of the secondary index and the target layer relative to the h-level risk, respectively. Finally, the risk level is evaluated according to the principle of maximum membership degree, and corresponding control measures are taken.


## 4.1. Determining the Index Factor Set and Comment Set

The index factor set of fuzzy comprehensive evaluation is each evaluation index, and these indices constitute a set. If there arem indices, they can be expressed as U = (u1, u2,…, um).The set of rubrics for fuzzy integrated evaluation are the criteria for experts to evaluate each indicator. In this paper, the evaluation interval of the index is divided into five levels, expressed asV = (I, II, III, IV, and V), indicating that the security risk is lowest, low, moderate, high, and highest, respectively, forming a comment set with a fuzzy comprehensive rating.


## 4.2. Constructing the Fuzzy Evaluation Matrix

According to the index system established in this paper, two methods of expert scoring and on-site monitoring are used to collect the initial data of the static index and dynamic index evaluation.Assuming thatn sets of data are collected, an evaluation matrix R is formed as follows:(12)R=r11r12⋯r1nr21r22⋯r2n⋮⋮⋮rm1rm2⋯rmn,where m is the number of indicators and n is the number of the data groups.


## 4.3. Determination of the Index Membership Degree

We apply the indicator affiliation function in Figure2; the horizontal coordinate indicates the relative position of each indicator risk characteristic value xij on the domain of each risk evaluation level, a–e indicates the upper bound of five risk intervals, the intersection of each risk level affiliation function curve with the horizontal coordinate is the midpoint of two adjacent values, and the corresponding y value is the indicator affiliation degree relative to each risk evaluation level.Figure 2
Membership function.


## 4.4. Comprehensive Evaluation

According to formula (13), the target layer index membership degree can be obtained as follows [30]:(13)μh=μhXij∙Wij,where μh (Xij) and μh represent the membership degree of the secondary index and the target layer relative to the h-level risk, respectively. Finally, the risk level is evaluated according to the principle of maximum membership degree, and corresponding control measures are taken.


## 5. Engineering Examples

### 5.1. Project Overview

A certain frame bridge passes under the existing railway line project, and the new frame adopts a (12 + 12 + 12 + 12)m four-hole separated reinforced concrete structure. The existing railroad line has dimensions of 16 m in the vertical railroad direction and 54.7 m in the direction of the railroad as the operation influence area. Under the condition that the train speed is limited to 45 km/h, the jacking construction distance that the frame passes under the railway is 40 m, and the angle between the jacking direction and the existing railway line is 80.9°. K478 + 394 of the existing line is the center mileage of the new frame bridge. The existing bridge at the jacking position has a frame of 1–12.0 m (as shown in Figure 3). According to the requirements of the specification, the beam is used in the range of K478 + 339∼K478 + 449 reinforcement.Figure 3
Schematic diagram of the new bridge frame.


### 5.2. Determining the Risk Level Evaluation Criteria

According to the index system established in this paper, the initial data of the safety risk evaluation of steel beams in service are collected by means of an expert scoring questionnaire, and the initial data of the safety risk evaluation of steel beam deflection, steel beam stress value, and vibration acceleration are collected by means of monitoring. The range of dynamic indices is limited according to the alarm value in the specification, and the range of the remaining indices is set to [0, 100].(1) Stress Value and Deflection of Steel Beam. The project adopts a D16 construction beam with a total length of 16.40 m. By referring to the relevant specification and the second note of Schedule 3-2 of the Safety Rules for Railway Engineering, the allowable deflection of the steel beam is 41 mm, and the basic allowable stress is 240 MPa.(2) Vibration Acceleration. When a train passes, the “Railway Bridge Inspection Code” [31] stipulates that the lateral vibration acceleration of the bridge span structure in the load plane aLmax ≤ 1.4 m/s2; in the research and design of the Qin-Shen passenger line bridge in China, the maximum vertical vibration acceleration of the bridge span aVmax ≤ 3.5 m/s2.With reference to the “Guidelines for Safety Risk Assessment of Highway Bridge and Tunnel Engineering Construction (Trial)-Ministry of Communications 2011,” the risk is divided into 5 levels: lowest (level I), low (level II), moderate (level III), high (level IV), and highest (level V) [32]. The risk level evaluation criteria are shown in Table 2.Table 2
Risk level evaluation criteria.
IndexRisk levelsIIIIIIIVVA1 (mm)[0.0∼2.6)[2.6∼15.4)[15.4∼33.3)[33.3∼36.0)[36.0∼41.0]A2 (MPa)[0∼15.1)[15.1∼90.0)[90.0∼195.1)[195.1∼210.0)[210.0∼240.0]A31 (m/s2)[0∼0.1)[0.1∼0.5)[0.5∼0.9)[0.9∼1.1)[1.1∼1.4]A32 (m/s2)[0∼0.2)[0.2∼1.3)[1.3∼2.8)[2.8∼3.1)[3.1∼3.5]A33 (m/s2)[0∼0.2)[0.2∼1.3)[1.3∼2.8)[2.8∼3.1)[3.1∼3.5]B11. (B12. B13)LanguageStrongModerateWeakWeakerWeaknessQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B21LanguageVery satisfiedSatisfiedModerately satisfiedDissatisfiedStrongly dissatisfiedQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B22. (B23. B24)LanguageVery complete and regularModerately complete and regularComplete and regularIncomplete or irregularIncomplete and irregularQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B25LanguageVery timely and preciseTimely and precisePrecise but untimelyUntimely, with occasional mistakesUntimely, with frequent mistakesQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B26LanguageFrequently checkedLess frequently checkedOccasionally checkedRarely checkedUncheckedQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B31LanguageWithout rainLess rainfallMore rainfallHeavy rainfallVery heavy rainfallQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B32LanguageSmallest temperature differenceSmall temperature differenceLarge temperature differenceLarger temperature differenceLargest temperature differenceQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B33LanguageUnaffectedSlightly affectedModerately affectedMore influentialStrongly influentialQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]


### 5.3. Data Collection

In this paper, the safety evaluation data of each index are obtained by on-site monitoring and expert scoring. To compensate for the lack of information loss caused by using only expert scoring or using only on-site monitoring to obtain research data, this paper collects research data based on the characteristics of the established index system, combining expert scoring and on-site monitoring to increase the comprehensiveness of risk assessment information. When the former is used for the quantification of the indicator, the quantification interval of the indicator is set to [0–100]. When the latter is adopted, the upper limit of the quantification interval of the indicator is set to the alarm value of the relevant monitoring specification, and then, the quantification interval of the indicator is set according to rule A and is divided into four grades, forming a safety grade evaluation standard for the service state of steel temporary beams.


#### 5.3.1. Monitoring

During the construction process of the frame bridge passing through the existing railway line, traditional manual monitoring often cannot be continuously monitored due to trains passing and other reasons. Therefore, the project adopts a combination of the static and dynamic monitoring methods, which can not only improve the accuracy and timeliness of data but also save manpower, material resources, and other resources. The new frame bridge intersects with the existing railway line at mile K478 + 394. According to the “Technical Specification for Construction Work Pit Engineering Monitoring” stipulation that the construction impact range is 3 times the excavation depth and other relevant requirements, the proposed monitoring range for the existing line in this project is K478 + 352∼K478 + 436.(1) Deflection of the Temporary Steel Beam. The deflection of the temporary steel beam is measured using a static level. According to the characteristics of the project, measuring points A1, A2, and A3 are set to measure the deflection of the temporary steel beam, the static level is installed at both ends of the beam and the mid-span position, and a relatively stable reference point is selected. The point should not be larger than 50 m, as shown in Figure 4. Selected data are shown in Figure 5.Figure 4
Layout of relevant measuring points.Figure 5
The deflection of the steel temporary beam.(2) Temporary Steel Beam Strain. We set measuring points B1 and B2 to measure the strain of the temporary steel beam, use the resistance dynamic strain gauge to measure the strain value on the surface of the temporary steel beam, and arrange the monitoring points on the cross section of the temporary steel beam, as shown in Figure 4. Selected data are shown in Figure 6.Figure 6
Strain value of the temporary steel beam.(3) Vibration Acceleration of the Temporary Steel Beam. Measuring point C is set to measure the vibration acceleration of the temporary steel beam, and a three-way magnetoelectric vibration sensor is used to collect the vibration acceleration along the line operation direction (x-axis), the horizontal and vertical directions (y-axis), and the vertical direction (z-axis) of the line operation. When the train travels into the construction area, the deflection deformation of the beam mid-span is the largest, so the three-way magnetoelectric vibration sensor is installed at the mid-span position of the beam, as shown in Figure 4, and it is fixed by welding. Taking the Y-axis as an example, part of the vibration acceleration acquisition data is shown in Figure 7.Figure 7
Y-axis vibration acceleration.


#### 5.3.2. Expert Scoring

Expert scoring is used to summarize the opinions and views of experts or authorities when there is a lack of objective data and information or when there are many uncertainties involved and to make evaluations, assessments, and predictions on research problems.(1)
The weighting of the indicators is highly specialized, so the questionnaire of the weighting of the indicators for the evaluation of the safety risk of steel beam construction and the construction plan of the project are sent to the authorities with relevant work management experience; the importance of each indicator is scored according to the table of the rules for quantifying the importance of indicators (as shown in Table3), and the initial research data of the subjective weighting of each indicator are obtained.(2)
The experts in various professional directions who are assigned values for each risk index are familiar with and have participated in similar projects. Therefore, the safety risk evaluation questionnaire for the service state of steel beams is issued to the relevant personnel involved in the construction, monitoring, and supervision of the project, and the possibility of the occurrence of the indicator is evaluated according to the actual situation of the project in accordance with the risk evaluation criteria (as shown in Table2). In this way, the initial data of the risk evaluation value are obtained.The safety evaluation data of each index are obtained by means of expert scoring and field monitoring, and the average value of the evaluation value of each index is obtained, as shown in the second column of Table4, where X¯ is the average value of each index evaluation value, qij is the subjective weight value, sij is the objective weight value, and wij is the combined weight value.Table 3
Rules for quantifying the importance of indicators.
ImportanceDefinitions1Equally important3Slightly more important5Important7Strongly important9Extremely important2, 4, 6, 8Intermediate valuesCountdownInverse comparisonTable 4
Calculation of the safety evaluation of the service state of the steel temporary beam.
X¯Risk levelsqijsijwijIIIIIIIVVA123.300.0010.0010.6580.3390.0010.09520.10360.0984A285.330.0010.7210.2760.0010.0010.28570.25210.2731A310.340.2860.7110.0010.0010.0010.03170.03130.0316A320.790.0010.0010.6780.3190.0010.15870.18220.1675A331.160.0010.8410.1560.0010.0010.09520.10270.0980B1125.340.2300.7670.0010.0010.0010.03700.02900.0340B1240.510.0010.4650.5320.0010.0010.06170.05400.0588B1333.670.0010.8180.1790.0010.0010.01230.02010.0153B2125.340.2320.7650.0010.0010.0010.04630.03990.0439B2239.870.0010.5040.4930.0010.0010.02780.02180.0255B2320.570.4660.5310.0010.0010.0010.00930.01010.0096B2445.340.0010.2250.7720.0010.0010.02780.03410.0302B2542.150.0010.3820.6150.0010.0010.04630.04840.0471B2623.540.3140.6830.0010.0010.0010.02780.02510.0268B3115.730.7120.2850.0010.0010.0010.00410.00510.0045B3235.650.0010.7230.2740.0010.0010.01230.00980.0114B3352.310.0010.0010.8820.1150.0010.02060.03050.0243


### 5.4. Calculating the Weight of the Index

#### 5.4.1. Calculating the Subjective Weight of the Index

According to formulas (1)–(3), the AHP is used to determine the weight value assigned by each expert to the index. Using one expert’s scoring of the first-level index as an example, the calculation process is shown in Table 5.Table 5
Subjective weight calculation and consistency test results of the primary indicators.
A1A2A3B1B2B3Zi0ZiA11232332.18220.3168lmax = 6.2611RI = 1.26CR = 0.0414 < 0.1A21/2131331.54310.2240A31/31/311/3210.64810.0941B11/2131221.34800.1957B21/31/31/21/2110.55030.0799B31/31/311/2110.61770.0897Similar to the previous discussion, the scoring data of other experts are organized, and each expert’s weighting value for the indicator is calculated. Combined with the assignment of experts, according to formula (2), the comprehensive subjective weight of the first-level index can be obtained. In the same way, the comprehensive subjective weights of other bottom-level indicators on the upper-layer indicators can be obtained, and then, according to the weights of the upper-layer indicators, the comprehensive subjective weights of the indicators on the first-level indicators and the target layer can be obtained, as shown in Figure 8 and the eighth column of Table 4.Figure 8
Schematic diagram of subjective weights.As shown in Figure8, A1 has the largest weight, accounting for 28.58%, followed by A32, accounting for 15.87%; A1 and A33 have the same weight, both 9.52%; and B31 has the smallest weight, accounting for 0.41%. The largest of the A3 series is A32, accounting for 15.87%; the largest of the B1 series is B12, accounting for 6.17%; the largest of the B2 series are B21 and B25, accounting for 4.63%; and the largest of the B3 series is B33, accounting for 2.06%.


#### 5.4.2. Calculating the Objective Weight of Indicators

The coefficient of the variation method is used to obtain the average value, variance, and coefficient of variation of each indicator. The objective weight value of the underlying indicators is shown in Figure9 and the ninth column of Table 4.Figure 9
Schematic diagram of objective weights.As shown in Figure9, A1 has the largest weight, accounting for 25.22%, followed by A32, accounting for 18.22%; B31 has the smallest weight, accounting for 0.51%. The largest of the A3 series is A32, accounting for 18.22%; the largest of the B1 series is B12, accounting for 5.4%; the largest of the B2 series is B25, accounting for 4.84%; and the largest of the B3 series is B33, accounting for 3.05%.


#### 5.4.3. Calculating Index Combination Weights and Relative Affiliation

Bringing the calculated subjective and objective weights into equations (9) and (10), we obtain the subjective weight combination coefficient a1 = 0.6246 and the objective weight combination coefficient a2 = 0.3754. Entering the subjective and objective weights and their combination coefficients into equation (11) obtains the combined weights, as shown in Figure 10 and the last column of Table 4.Figure 10
Schematic diagram of combined weights.The risk evaluation value of each indicator was obtained by collating the data, and then, the data were substituted into the fuzzy comprehensive evaluation model to calculate the relative affiliation of each indicator for the h-level risk according to equation (13). The calculation results are shown in Table 4 (columns 3–7).As Figure10 shows, A1 has the largest weight, accounting for 27.31%, followed by A32, accounting for 16.75%; B31 has the smallest weight, accounting for 0.45%. The largest A3 series is A32, accounting for 16.75%; the largest B1 series is B12, accounting for 5.88%; the largest B2 series is B25, accounting for 4.71%; and the largest B3 series is B33, accounting for 2.43%.


### 5.5. Assessing Construction Safety Risks

According to the membership degree and combination weight of each index for each risk interval, the membership degree of the target layer to each risk intervalμh = (0.043, 0.474, 0.392, 0.090, and 0.001) is finally obtained. According to the principle of maximum membership degree, this project can be implemented. The construction safety risk level is Class II, indicating that the safety risk of the beam in service is low.


## 5.1. Project Overview

A certain frame bridge passes under the existing railway line project, and the new frame adopts a (12 + 12 + 12 + 12)m four-hole separated reinforced concrete structure. The existing railroad line has dimensions of 16 m in the vertical railroad direction and 54.7 m in the direction of the railroad as the operation influence area. Under the condition that the train speed is limited to 45 km/h, the jacking construction distance that the frame passes under the railway is 40 m, and the angle between the jacking direction and the existing railway line is 80.9°. K478 + 394 of the existing line is the center mileage of the new frame bridge. The existing bridge at the jacking position has a frame of 1–12.0 m (as shown in Figure 3). According to the requirements of the specification, the beam is used in the range of K478 + 339∼K478 + 449 reinforcement.Figure 3
Schematic diagram of the new bridge frame.


## 5.2. Determining the Risk Level Evaluation Criteria

According to the index system established in this paper, the initial data of the safety risk evaluation of steel beams in service are collected by means of an expert scoring questionnaire, and the initial data of the safety risk evaluation of steel beam deflection, steel beam stress value, and vibration acceleration are collected by means of monitoring. The range of dynamic indices is limited according to the alarm value in the specification, and the range of the remaining indices is set to [0, 100].(1) Stress Value and Deflection of Steel Beam. The project adopts a D16 construction beam with a total length of 16.40 m. By referring to the relevant specification and the second note of Schedule 3-2 of the Safety Rules for Railway Engineering, the allowable deflection of the steel beam is 41 mm, and the basic allowable stress is 240 MPa.(2) Vibration Acceleration. When a train passes, the “Railway Bridge Inspection Code” [31] stipulates that the lateral vibration acceleration of the bridge span structure in the load plane aLmax ≤ 1.4 m/s2; in the research and design of the Qin-Shen passenger line bridge in China, the maximum vertical vibration acceleration of the bridge span aVmax ≤ 3.5 m/s2.With reference to the “Guidelines for Safety Risk Assessment of Highway Bridge and Tunnel Engineering Construction (Trial)-Ministry of Communications 2011,” the risk is divided into 5 levels: lowest (level I), low (level II), moderate (level III), high (level IV), and highest (level V) [32]. The risk level evaluation criteria are shown in Table 2.Table 2
Risk level evaluation criteria.
IndexRisk levelsIIIIIIIVVA1 (mm)[0.0∼2.6)[2.6∼15.4)[15.4∼33.3)[33.3∼36.0)[36.0∼41.0]A2 (MPa)[0∼15.1)[15.1∼90.0)[90.0∼195.1)[195.1∼210.0)[210.0∼240.0]A31 (m/s2)[0∼0.1)[0.1∼0.5)[0.5∼0.9)[0.9∼1.1)[1.1∼1.4]A32 (m/s2)[0∼0.2)[0.2∼1.3)[1.3∼2.8)[2.8∼3.1)[3.1∼3.5]A33 (m/s2)[0∼0.2)[0.2∼1.3)[1.3∼2.8)[2.8∼3.1)[3.1∼3.5]B11. (B12. B13)LanguageStrongModerateWeakWeakerWeaknessQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B21LanguageVery satisfiedSatisfiedModerately satisfiedDissatisfiedStrongly dissatisfiedQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B22. (B23. B24)LanguageVery complete and regularModerately complete and regularComplete and regularIncomplete or irregularIncomplete and irregularQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B25LanguageVery timely and preciseTimely and precisePrecise but untimelyUntimely, with occasional mistakesUntimely, with frequent mistakesQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B26LanguageFrequently checkedLess frequently checkedOccasionally checkedRarely checkedUncheckedQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B31LanguageWithout rainLess rainfallMore rainfallHeavy rainfallVery heavy rainfallQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B32LanguageSmallest temperature differenceSmall temperature differenceLarge temperature differenceLarger temperature differenceLargest temperature differenceQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]B33LanguageUnaffectedSlightly affectedModerately affectedMore influentialStrongly influentialQuantized[0.0∼6.3)[6.3∼37.5)[37.5∼71.3)[71.3∼87.5)[87.5∼100.0]


## 5.3. Data Collection

In this paper, the safety evaluation data of each index are obtained by on-site monitoring and expert scoring. To compensate for the lack of information loss caused by using only expert scoring or using only on-site monitoring to obtain research data, this paper collects research data based on the characteristics of the established index system, combining expert scoring and on-site monitoring to increase the comprehensiveness of risk assessment information. When the former is used for the quantification of the indicator, the quantification interval of the indicator is set to [0–100]. When the latter is adopted, the upper limit of the quantification interval of the indicator is set to the alarm value of the relevant monitoring specification, and then, the quantification interval of the indicator is set according to rule A and is divided into four grades, forming a safety grade evaluation standard for the service state of steel temporary beams.


### 5.3.1. Monitoring

During the construction process of the frame bridge passing through the existing railway line, traditional manual monitoring often cannot be continuously monitored due to trains passing and other reasons. Therefore, the project adopts a combination of the static and dynamic monitoring methods, which can not only improve the accuracy and timeliness of data but also save manpower, material resources, and other resources. The new frame bridge intersects with the existing railway line at mile K478 + 394. According to the “Technical Specification for Construction Work Pit Engineering Monitoring” stipulation that the construction impact range is 3 times the excavation depth and other relevant requirements, the proposed monitoring range for the existing line in this project is K478 + 352∼K478 + 436.(1) Deflection of the Temporary Steel Beam. The deflection of the temporary steel beam is measured using a static level. According to the characteristics of the project, measuring points A1, A2, and A3 are set to measure the deflection of the temporary steel beam, the static level is installed at both ends of the beam and the mid-span position, and a relatively stable reference point is selected. The point should not be larger than 50 m, as shown in Figure 4. Selected data are shown in Figure 5.Figure 4
Layout of relevant measuring points.Figure 5
The deflection of the steel temporary beam.(2) Temporary Steel Beam Strain. We set measuring points B1 and B2 to measure the strain of the temporary steel beam, use the resistance dynamic strain gauge to measure the strain value on the surface of the temporary steel beam, and arrange the monitoring points on the cross section of the temporary steel beam, as shown in Figure 4. Selected data are shown in Figure 6.Figure 6
Strain value of the temporary steel beam.(3) Vibration Acceleration of the Temporary Steel Beam. Measuring point C is set to measure the vibration acceleration of the temporary steel beam, and a three-way magnetoelectric vibration sensor is used to collect the vibration acceleration along the line operation direction (x-axis), the horizontal and vertical directions (y-axis), and the vertical direction (z-axis) of the line operation. When the train travels into the construction area, the deflection deformation of the beam mid-span is the largest, so the three-way magnetoelectric vibration sensor is installed at the mid-span position of the beam, as shown in Figure 4, and it is fixed by welding. Taking the Y-axis as an example, part of the vibration acceleration acquisition data is shown in Figure 7.Figure 7
Y-axis vibration acceleration.


### 5.3.2. Expert Scoring

Expert scoring is used to summarize the opinions and views of experts or authorities when there is a lack of objective data and information or when there are many uncertainties involved and to make evaluations, assessments, and predictions on research problems.(1)
The weighting of the indicators is highly specialized, so the questionnaire of the weighting of the indicators for the evaluation of the safety risk of steel beam construction and the construction plan of the project are sent to the authorities with relevant work management experience; the importance of each indicator is scored according to the table of the rules for quantifying the importance of indicators (as shown in Table3), and the initial research data of the subjective weighting of each indicator are obtained.(2)
The experts in various professional directions who are assigned values for each risk index are familiar with and have participated in similar projects. Therefore, the safety risk evaluation questionnaire for the service state of steel beams is issued to the relevant personnel involved in the construction, monitoring, and supervision of the project, and the possibility of the occurrence of the indicator is evaluated according to the actual situation of the project in accordance with the risk evaluation criteria (as shown in Table2). In this way, the initial data of the risk evaluation value are obtained.The safety evaluation data of each index are obtained by means of expert scoring and field monitoring, and the average value of the evaluation value of each index is obtained, as shown in the second column of Table4, where X¯ is the average value of each index evaluation value, qij is the subjective weight value, sij is the objective weight value, and wij is the combined weight value.Table 3
Rules for quantifying the importance of indicators.
ImportanceDefinitions1Equally important3Slightly more important5Important7Strongly important9Extremely important2, 4, 6, 8Intermediate valuesCountdownInverse comparisonTable 4
Calculation of the safety evaluation of the service state of the steel temporary beam.
X¯Risk levelsqijsijwijIIIIIIIVVA123.300.0010.0010.6580.3390.0010.09520.10360.0984A285.330.0010.7210.2760.0010.0010.28570.25210.2731A310.340.2860.7110.0010.0010.0010.03170.03130.0316A320.790.0010.0010.6780.3190.0010.15870.18220.1675A331.160.0010.8410.1560.0010.0010.09520.10270.0980B1125.340.2300.7670.0010.0010.0010.03700.02900.0340B1240.510.0010.4650.5320.0010.0010.06170.05400.0588B1333.670.0010.8180.1790.0010.0010.01230.02010.0153B2125.340.2320.7650.0010.0010.0010.04630.03990.0439B2239.870.0010.5040.4930.0010.0010.02780.02180.0255B2320.570.4660.5310.0010.0010.0010.00930.01010.0096B2445.340.0010.2250.7720.0010.0010.02780.03410.0302B2542.150.0010.3820.6150.0010.0010.04630.04840.0471B2623.540.3140.6830.0010.0010.0010.02780.02510.0268B3115.730.7120.2850.0010.0010.0010.00410.00510.0045B3235.650.0010.7230.2740.0010.0010.01230.00980.0114B3352.310.0010.0010.8820.1150.0010.02060.03050.0243


## 5.3.1. Monitoring

During the construction process of the frame bridge passing through the existing railway line, traditional manual monitoring often cannot be continuously monitored due to trains passing and other reasons. Therefore, the project adopts a combination of the static and dynamic monitoring methods, which can not only improve the accuracy and timeliness of data but also save manpower, material resources, and other resources. The new frame bridge intersects with the existing railway line at mile K478 + 394. According to the “Technical Specification for Construction Work Pit Engineering Monitoring” stipulation that the construction impact range is 3 times the excavation depth and other relevant requirements, the proposed monitoring range for the existing line in this project is K478 + 352∼K478 + 436.(1) Deflection of the Temporary Steel Beam. The deflection of the temporary steel beam is measured using a static level. According to the characteristics of the project, measuring points A1, A2, and A3 are set to measure the deflection of the temporary steel beam, the static level is installed at both ends of the beam and the mid-span position, and a relatively stable reference point is selected. The point should not be larger than 50 m, as shown in Figure 4. Selected data are shown in Figure 5.Figure 4
Layout of relevant measuring points.Figure 5
The deflection of the steel temporary beam.(2) Temporary Steel Beam Strain. We set measuring points B1 and B2 to measure the strain of the temporary steel beam, use the resistance dynamic strain gauge to measure the strain value on the surface of the temporary steel beam, and arrange the monitoring points on the cross section of the temporary steel beam, as shown in Figure 4. Selected data are shown in Figure 6.Figure 6
Strain value of the temporary steel beam.(3) Vibration Acceleration of the Temporary Steel Beam. Measuring point C is set to measure the vibration acceleration of the temporary steel beam, and a three-way magnetoelectric vibration sensor is used to collect the vibration acceleration along the line operation direction (x-axis), the horizontal and vertical directions (y-axis), and the vertical direction (z-axis) of the line operation. When the train travels into the construction area, the deflection deformation of the beam mid-span is the largest, so the three-way magnetoelectric vibration sensor is installed at the mid-span position of the beam, as shown in Figure 4, and it is fixed by welding. Taking the Y-axis as an example, part of the vibration acceleration acquisition data is shown in Figure 7.Figure 7
Y-axis vibration acceleration.


## 5.3.2. Expert Scoring

Expert scoring is used to summarize the opinions and views of experts or authorities when there is a lack of objective data and information or when there are many uncertainties involved and to make evaluations, assessments, and predictions on research problems.(1)
The weighting of the indicators is highly specialized, so the questionnaire of the weighting of the indicators for the evaluation of the safety risk of steel beam construction and the construction plan of the project are sent to the authorities with relevant work management experience; the importance of each indicator is scored according to the table of the rules for quantifying the importance of indicators (as shown in Table3), and the initial research data of the subjective weighting of each indicator are obtained.(2)
The experts in various professional directions who are assigned values for each risk index are familiar with and have participated in similar projects. Therefore, the safety risk evaluation questionnaire for the service state of steel beams is issued to the relevant personnel involved in the construction, monitoring, and supervision of the project, and the possibility of the occurrence of the indicator is evaluated according to the actual situation of the project in accordance with the risk evaluation criteria (as shown in Table2). In this way, the initial data of the risk evaluation value are obtained.The safety evaluation data of each index are obtained by means of expert scoring and field monitoring, and the average value of the evaluation value of each index is obtained, as shown in the second column of Table4, where X¯ is the average value of each index evaluation value, qij is the subjective weight value, sij is the objective weight value, and wij is the combined weight value.Table 3
Rules for quantifying the importance of indicators.
ImportanceDefinitions1Equally important3Slightly more important5Important7Strongly important9Extremely important2, 4, 6, 8Intermediate valuesCountdownInverse comparisonTable 4
Calculation of the safety evaluation of the service state of the steel temporary beam.
X¯Risk levelsqijsijwijIIIIIIIVVA123.300.0010.0010.6580.3390.0010.09520.10360.0984A285.330.0010.7210.2760.0010.0010.28570.25210.2731A310.340.2860.7110.0010.0010.0010.03170.03130.0316A320.790.0010.0010.6780.3190.0010.15870.18220.1675A331.160.0010.8410.1560.0010.0010.09520.10270.0980B1125.340.2300.7670.0010.0010.0010.03700.02900.0340B1240.510.0010.4650.5320.0010.0010.06170.05400.0588B1333.670.0010.8180.1790.0010.0010.01230.02010.0153B2125.340.2320.7650.0010.0010.0010.04630.03990.0439B2239.870.0010.5040.4930.0010.0010.02780.02180.0255B2320.570.4660.5310.0010.0010.0010.00930.01010.0096B2445.340.0010.2250.7720.0010.0010.02780.03410.0302B2542.150.0010.3820.6150.0010.0010.04630.04840.0471B2623.540.3140.6830.0010.0010.0010.02780.02510.0268B3115.730.7120.2850.0010.0010.0010.00410.00510.0045B3235.650.0010.7230.2740.0010.0010.01230.00980.0114B3352.310.0010.0010.8820.1150.0010.02060.03050.0243


## 5.4. Calculating the Weight of the Index

### 5.4.1. Calculating the Subjective Weight of the Index

According to formulas (1)–(3), the AHP is used to determine the weight value assigned by each expert to the index. Using one expert’s scoring of the first-level index as an example, the calculation process is shown in Table 5.Table 5
Subjective weight calculation and consistency test results of the primary indicators.
A1A2A3B1B2B3Zi0ZiA11232332.18220.3168lmax = 6.2611RI = 1.26CR = 0.0414 < 0.1A21/2131331.54310.2240A31/31/311/3210.64810.0941B11/2131221.34800.1957B21/31/31/21/2110.55030.0799B31/31/311/2110.61770.0897Similar to the previous discussion, the scoring data of other experts are organized, and each expert’s weighting value for the indicator is calculated. Combined with the assignment of experts, according to formula (2), the comprehensive subjective weight of the first-level index can be obtained. In the same way, the comprehensive subjective weights of other bottom-level indicators on the upper-layer indicators can be obtained, and then, according to the weights of the upper-layer indicators, the comprehensive subjective weights of the indicators on the first-level indicators and the target layer can be obtained, as shown in Figure 8 and the eighth column of Table 4.Figure 8
Schematic diagram of subjective weights.As shown in Figure8, A1 has the largest weight, accounting for 28.58%, followed by A32, accounting for 15.87%; A1 and A33 have the same weight, both 9.52%; and B31 has the smallest weight, accounting for 0.41%. The largest of the A3 series is A32, accounting for 15.87%; the largest of the B1 series is B12, accounting for 6.17%; the largest of the B2 series are B21 and B25, accounting for 4.63%; and the largest of the B3 series is B33, accounting for 2.06%.


### 5.4.2. Calculating the Objective Weight of Indicators

The coefficient of the variation method is used to obtain the average value, variance, and coefficient of variation of each indicator. The objective weight value of the underlying indicators is shown in Figure9 and the ninth column of Table 4.Figure 9
Schematic diagram of objective weights.As shown in Figure9, A1 has the largest weight, accounting for 25.22%, followed by A32, accounting for 18.22%; B31 has the smallest weight, accounting for 0.51%. The largest of the A3 series is A32, accounting for 18.22%; the largest of the B1 series is B12, accounting for 5.4%; the largest of the B2 series is B25, accounting for 4.84%; and the largest of the B3 series is B33, accounting for 3.05%.


### 5.4.3. Calculating Index Combination Weights and Relative Affiliation

Bringing the calculated subjective and objective weights into equations (9) and (10), we obtain the subjective weight combination coefficient a1 = 0.6246 and the objective weight combination coefficient a2 = 0.3754. Entering the subjective and objective weights and their combination coefficients into equation (11) obtains the combined weights, as shown in Figure 10 and the last column of Table 4.Figure 10
Schematic diagram of combined weights.The risk evaluation value of each indicator was obtained by collating the data, and then, the data were substituted into the fuzzy comprehensive evaluation model to calculate the relative affiliation of each indicator for the h-level risk according to equation (13). The calculation results are shown in Table 4 (columns 3–7).As Figure10 shows, A1 has the largest weight, accounting for 27.31%, followed by A32, accounting for 16.75%; B31 has the smallest weight, accounting for 0.45%. The largest A3 series is A32, accounting for 16.75%; the largest B1 series is B12, accounting for 5.88%; the largest B2 series is B25, accounting for 4.71%; and the largest B3 series is B33, accounting for 2.43%.


## 5.4.1. Calculating the Subjective Weight of the Index

According to formulas (1)–(3), the AHP is used to determine the weight value assigned by each expert to the index. Using one expert’s scoring of the first-level index as an example, the calculation process is shown in Table 5.Table 5
Subjective weight calculation and consistency test results of the primary indicators.
A1A2A3B1B2B3Zi0ZiA11232332.18220.3168lmax = 6.2611RI = 1.26CR = 0.0414 < 0.1A21/2131331.54310.2240A31/31/311/3210.64810.0941B11/2131221.34800.1957B21/31/31/21/2110.55030.0799B31/31/311/2110.61770.0897Similar to the previous discussion, the scoring data of other experts are organized, and each expert’s weighting value for the indicator is calculated. Combined with the assignment of experts, according to formula (2), the comprehensive subjective weight of the first-level index can be obtained. In the same way, the comprehensive subjective weights of other bottom-level indicators on the upper-layer indicators can be obtained, and then, according to the weights of the upper-layer indicators, the comprehensive subjective weights of the indicators on the first-level indicators and the target layer can be obtained, as shown in Figure 8 and the eighth column of Table 4.Figure 8
Schematic diagram of subjective weights.As shown in Figure8, A1 has the largest weight, accounting for 28.58%, followed by A32, accounting for 15.87%; A1 and A33 have the same weight, both 9.52%; and B31 has the smallest weight, accounting for 0.41%. The largest of the A3 series is A32, accounting for 15.87%; the largest of the B1 series is B12, accounting for 6.17%; the largest of the B2 series are B21 and B25, accounting for 4.63%; and the largest of the B3 series is B33, accounting for 2.06%.


## 5.4.2. Calculating the Objective Weight of Indicators

The coefficient of the variation method is used to obtain the average value, variance, and coefficient of variation of each indicator. The objective weight value of the underlying indicators is shown in Figure9 and the ninth column of Table 4.Figure 9
Schematic diagram of objective weights.As shown in Figure9, A1 has the largest weight, accounting for 25.22%, followed by A32, accounting for 18.22%; B31 has the smallest weight, accounting for 0.51%. The largest of the A3 series is A32, accounting for 18.22%; the largest of the B1 series is B12, accounting for 5.4%; the largest of the B2 series is B25, accounting for 4.84%; and the largest of the B3 series is B33, accounting for 3.05%.


## 5.4.3. Calculating Index Combination Weights and Relative Affiliation

Bringing the calculated subjective and objective weights into equations (9) and (10), we obtain the subjective weight combination coefficient a1 = 0.6246 and the objective weight combination coefficient a2 = 0.3754. Entering the subjective and objective weights and their combination coefficients into equation (11) obtains the combined weights, as shown in Figure 10 and the last column of Table 4.Figure 10
Schematic diagram of combined weights.The risk evaluation value of each indicator was obtained by collating the data, and then, the data were substituted into the fuzzy comprehensive evaluation model to calculate the relative affiliation of each indicator for the h-level risk according to equation (13). The calculation results are shown in Table 4 (columns 3–7).As Figure10 shows, A1 has the largest weight, accounting for 27.31%, followed by A32, accounting for 16.75%; B31 has the smallest weight, accounting for 0.45%. The largest A3 series is A32, accounting for 16.75%; the largest B1 series is B12, accounting for 5.88%; the largest B2 series is B25, accounting for 4.71%; and the largest B3 series is B33, accounting for 2.43%.


## 5.5. Assessing Construction Safety Risks

According to the membership degree and combination weight of each index for each risk interval, the membership degree of the target layer to each risk intervalμh = (0.043, 0.474, 0.392, 0.090, and 0.001) is finally obtained. According to the principle of maximum membership degree, this project can be implemented. The construction safety risk level is Class II, indicating that the safety risk of the beam in service is low.


## 6. Conclusions

This paper establishes an evaluation index system for the safety of the service state of a temporary steel beam, uses the AHP and the coefficient of the variation method to determine the subjective and objective weights of the indices, introduces game theory to calculate the combined weight of the indices, and finally establishes the service state safety of the beam. Taking the temporary steel beam of a frame bridge under an existing railway line project as an example, the abovementioned theory is used to carry out associated research, and finally, the fuzzy comprehensive evaluation method is used to evaluate the safety of the service state of a frame bridge under the existing railway line project performance, and the following conclusions are drawn.(1)
Using the dynamic and static analysis method, the safety evaluation index system of the service state of the convenient beam is established from six aspects: the stress, the deflection, and the acceleration of the temporary steel beam, a personnel factor, a management factor, and an environmental factor.(2)
In this paper, the subjective hierarchical analysis method is coupled with the objective coefficient of the variation method, and the combined objective and subjective weights of the evaluation indices are determined by the combined weighting model, which reduces the influence of expert preference and makes the weights of the evaluation indices of the safety of steel beams in service more accurate and truly reflect the evaluation object.(3)
In this project, the first three indicators that have a larger weight on the safety of the service state of the beam are the stress value, the acceleration of the vertical line, and the deflection of the temporary steel beam; this stage addresses the service state of the beam. The first three indicators with greater safety risks are the deflection, the acceleration of the temporary steel beam in the vertical line direction, and the train operation. The construction process should prioritize increasing the monitoring frequency of the stress value, deflection, and vertical acceleration of the temporary steel beam and reducing the impact of trains on the existing line by taking measures such as reducing the train running speed.



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*Source: 6271946-2023-01-18.xml*