The application of the game theory combination weighting-normal cloud model to the quality evaluation of surrounding rocks

The status of surrounding rocks dramatically influences the safety of construction workers, so the quality assessment of surrounding rocks has great significance. The uniaxial saturated compressive strength of rock (X₁), the quality index of surrounding rock (X₂), the frictional coefficient of the structural surface (X₃), the joint spacing (X₄), the state of groundwater(X₅), and the integrity coefficient (X₆) are selected as the initial evaluation index. Then, the game theory combination weighting-normal cloud model is introduced. Second, the certainty degree matrix of each index is established, and the weight coefficients of assessment indexes are determined based on the game theory combination weighting method. Finally, the quality level of surrounding rocks is judged. Compared with the traditional methods, the proposed model solves the fuzziness and randomness of different indexes, improves the reliability of the assessment process, and enhances the predictive accuracy of assessment results. In addition, it can provide a solution scheme for the evaluation indicators, which are difficult to quantify, and reduce the influence of human factors. The results obtained from the suggested model are consistent with the current specification. Its accuracy approaches 100%, and the method is feasible for the quality level assessment of surrounding rocks, providing a new technique and approach to assessing the risk level of surrounding rocks.

1 Introduction

With the development of the economy in China, more extensive infrastructure is being constructed more quickly (Chen et al., 2022). Many large-scale underground projects are used in water conservancy, hydropower, transportation, mining, and other projects (Zhou et al., 2012). At the same time, the stability of underground engineering due to the excavation of a large amount of rock and soil has become a critical problem faced by engineers (Chen and Zhou, 2019). An underground tunnel is often in a complex geological environment. Where the geological conditions and stability of surrounding rock vary, the evaluation of surrounding rock quality is essential to understanding the engineering characteristics (Zhou et al., 2015a). An accurate assessment of the surrounding rock is significant in ensuring a reasonable survey design and smooth construction on site.

Researchers and scholars have performed many investigations on methods of assessing surrounding rocks in recent years (Gu et al., 2021). TAN et al. (2022) established the fuzzy assessment model based on the hierarchy analytic method to predict four indexes of rock mass in each section of the tunnel. WANG and CAO (2013) used a matter-element extension evaluation model and concluded that uniaxial compressive strength and groundwater seepage volume have the most significant effect on the stability of surrounding rocks by WANG and CAO (2013) using the matter-element extension evaluation model. WEI et al. (2016) introduced a cloud model that transforms qualitative concepts and quantitative data to determine the quality method of the rock mass. Qiu (2008) established a quality assessment model of surrounding rocks in the tunnel by using the result of reduction as the input samples of an artificial neural network. In addition, QIN et al. (2016) classified the various factors that affected the stability of the surrounding rock of a deep mine roadway and established the three types of hazard impact factor model using the fuzzy comprehensive evaluation method in combination with the three types of hazard classification method. The ideal point method is applied to calculate the proximity of the surrounding rock by HUANG et al. (2014), and the weight of the corresponding index in the evaluation system is determined using the entropy weight theory.

The above methods have prompted the substantial development of the assessment theory of surrounding rock quality. However, it still has some shortcomings (Gu and Wu, 2016). For example, the evaluation of surrounding rock quality is a nonlinear and complex problem (Gu et al., 2022a). In addition, the fuzziness and randomness of surrounding rock quality evaluation are neglected, and the relative importance of the evaluation index and the calculation of weight distribution must be optimized (Gu et al., 2022b).

The game theory combination weighting-normal cloud model is introduced to overcome the shortcomings of the above methods and to assess the quality level of the surrounding rock in the Pingzitou Tunnel. First, the game theory combination weighting method is introduced to determine the weights of the assessment index. Relative to traditional subjective or objective weighting methods, combination weighting theory not only considers the subjective factors of expert assessments but also involves some objective assessments. This has improved the accuracy of assessment prediction. When the game theory is combined with the combination weighting theory, the contradiction between subjective and objective weights can be dealt with effectively, the advantages of subjective and objective weights are integrated, the agreement and compromise in the conflict of both are searched, and the difference between subjective and objective weights is reduced. Finally, the deviation between the basic weight and the ideal weight is minimized. Therefore, the uncertainty of the evaluation results is lowered (Chen and Zhou, 2024). Third, a new assessment model is formed when the normal cloud model is introduced. It has many virtues, such as the preciseness of algorithms and operability in practice. Compared to traditional cloud theory (Alison et al., 2022), the suggested method needs no significant amount of data, and its operation is easy. In addition, it can provide a solution scheme for the evaluation indicators, which are difficult to quantify, and reduces the influence of human factors (Li and Wu, 2023; Zhou et al., 2015b). The method dramatically improves the traditional cloud model (Zhou et al., 2016).

The paper is organized as follows: in Section 2, the engineering overview is introduced. In Section 3, theory and methodology based on the game theory combination weighting-normal cloud model are presented. In Section 4, the assessment model of the surrounding rock quality is established, and the assessment results of the proposed model are compared. Conclusions are drawn in Section 5.

2 Engineering overview

The Pingzitou Tunnel is located in the Daping village, Pingzi town, Guizhou province, and plotted in Figure 1. The railway tunnel goes under the tunnel’s entrance; the export end is parallel to the railway. In all, the pile number of the left tunnel entrance is Zk2 + 880, the pile number of the tunnel exit is Zk4 + 972; the elevation of the tunnel bottom is 1,473.36–1,686.11 m, the total length is 2092 m, the vertical slope gradient is −2.8%, the plane is located on the curve and straight line section of R = 735 m, and the maximum superelevation of the tunnel pavement is 4%. The karst in the entrance section of the tunnel is well-developed, and the karst is strong along the karst fissure. Most of the surface forms are karst gullies and sinkholes, which are mostly developed vertically. The covering layer in the tunnel section is mainly gravel soil, and lightly weathered dolomite limestone is located at the lower part. The surface strata at the exploration area are mainly Quaternary Holocene (Q_h) strata, mostly composed of macadam soil with small thicknesses and uneven distribution. The underlying bedrock is a Carboniferous dolomite limestone formation. The surface water system in the tunnel area has not been developed. There are three kinds of groundwater in the site: bedrock fissure water, karst groundwater, and structure fissure water. A specific picture of the tunnel is shown in Figure 2.

Figure 1

Figure 1. Geographical location of the survey area.

Figure 2

Figure 2. Picture of the tunnel.

3 Assessment process

3.1 The combination weighting method

The common weight calculation methods are divided into subjective, objective, and combination weights. Combination weighting is a common method; two or three kinds of subjective and objective weights are combined to obtain the comprehensive weight, which can reduce the error caused by a single method to a certain extent (Ding et al., 2022a; Ding et al., 2022b; Ding et al., 2023). Based on the discussion in the introduction, the entropy weight and criteria importance through inter-criteria correlation (CRITIC) methods are applied to represent the subjective and objective factors, and the combination weights are obtained using game theory (Zhou and Yang, 2007).

(1) The entropy method

The entropy weight method is an objective weighting method to determine the weight coefficient according to the degree of information utility value of each evaluation index. The entropy weight method can reflect the degree of discreteness among the index data (Zhao et al., 2021).

Its calculative process is listed as follows:

① Constructing the original matrix of assessment index $X$

Assuming that there are $m$ evaluation indexes and $n$ evaluation objects, $x_{ij}$ is the corresponding value of the $ith$ assessment index at the $jth$ assessment object; then, its origin assessment matrix can be expressed as follows:

$X={\left(x_{ij}\right)}_{m\times n}\left(i=1,2,...,m;j=1,2,...,n\right)(1)$

② Normalization and forward processing

To eliminate the impact of the different types of indicators and dimensional differences, dimensionless processing needs to be performed for each index; the indexes are expressed as follows:

$\left.Y=\left({\mathrm{y}}_{ij}\right)\left(i=\right)1,2,...,m,j=1,2,....,n\right).(2)$

The positive indicator is expressed as follows:

$y_n=\frac{x_{ij}-\min \left(x_{ij}\right)}{\max \left(x_{ij}\right)-\min \left(x_{ij}\right)}.(3)$

The negative indicator is expressed as follows:

$y_n=\frac{\max \left(x_{ij}\right)-x_{ij}}{\max \left(x_{ij}\right)-\min \left(x_{ij}\right)},(4)$

where $y_{ij}$ is the standard value of $ith$ assessment index at the $jth$ assessment object.

③ Calculation of the information entropy of the $ith$ assessment index

$h_i=\frac{1}{\ln n}{\displaystyle \sum _{j=1}^n}{e}_{ij}\ln e_{ij},(5)$

$e_{ij}=\frac{y_{ij}}{{\displaystyle \sum _{j=1}^n}{y}_{ij}}.(6)$

④ Calculation of weights ${\omega }_{1\mathrm{i}}$:

${\omega }_{1\mathrm{i}}=\frac{1-h_i}{m-{\displaystyle \sum _{i=1}^m}{h}_i},(7)$

where $0<{\omega }_{i1i}\le 1,{\displaystyle \sum _{i=1}^m}{\omega }_{1i}=1,i=1,2,...,m.$

(2) The CRITIC method

Criteria importance through inter-criteria correlation (CRITIC) is an objective weighting method proposed by Diakoulaki that synthetically measures the index weight by calculating the variability and conflict of the index. Its calculative procedure follows (Zhou et al., 2014):

① Assuming that there are $\mathrm{m}$ estimated objects and $\mathrm{n}$ assessment indexes, construct a matrix $A={\left(a_{ij}\right)}_{m\times n}$,where $i=1,2,...,m;j=1,2,...,n$.

② Matrix $A$is standardized based on the Z-score method. Its expression is shown as follows:

$a_{ij}^{*}=\frac{a_{ij}-\overline{a_j}}{s_j}\left(i=1,2,...,m;j=1,2,...,b\right),(8)$

where $\overline{a_j}=\frac{1}{a}{\displaystyle \sum _{i=1}^m}{a}_{ij}$; $s_j=\sqrt{\raisebox{1ex}{${\displaystyle \sum _{i=1}^m}\left(a_{ij}-\overline{a_j}\right)$}\!\left/ \!\raisebox{-1ex}{$a-1$}\right.}$; and $\overline{a_j}$ and $s_j$ are, respectively, the mean value and standard deviation of the $jth$ assessment index.

③ Calculate the coefficient of variation of different indexes as follows:

$B{Y}_{\mathrm{j}}=\raisebox{1ex}{$s_j$}\!\left/ \!\raisebox{-1ex}{$\overline{a_j}$}\right.\left(j=1,2,...n\right),(9)$

where $B{Y}_{\mathrm{j}}$ is the variation coefficient of the $jth$ index.

④ The coefficients of correlation are calculated based on the standardization matrix $A^{*}$. Its expression is listed as follows: $X={\left(r_{kl}\right)}_{n\times n}\left(k=1,2,...,n,l=1,2,...,b\right)$, $r_{kl}$ is the coefficient of correlation between the $kth$ and $lth$ index, and

$r_{kl}=\frac{{\displaystyle \sum _{i=1}^m}\left(a_{ik}-\overline{a_k}\right)\left(a_{il}-\overline{a_l}\right)}{\sqrt{{\displaystyle \sum _{i=1}^m}{\left(a_{ik}-\overline{a_k}\right)}^2}\sqrt{{\displaystyle \sum _{l=1}^m}{\left(a_l-\overline{a_l}\right)}^2}}\left(r_{kl}=r{}_{lk};k=1,2,...,m,l=1,2,...,m\right)(10)$

where $a_{ik}$ and $a_{il}$ are, respectively, the standard value of measured values at the $kth$ and $lth$ index for the $ith$ assessment object in the standardization matrix $A^{*}$; $\overline{a_k}$ and $\overline{a_l}$ are, respectively, the mean of standard value of measured values at the $kth$ and $lth$ index in the standardization matrix $A^{*}$.

⑤ Calculate the quantitative coefficient about the degree of independence for different assessment indexes.

Its expression is shown as follows:

${\eta }_j={\displaystyle \sum _{k=1}^n}\left(1-\left|r_{kj}\right|\right)\left(j=1,2,...,n\right).(11)$

⑥ The quantitative coefficients of the comprehensive information and the degree of independence of each index are solved as follows:

$C_{\mathrm{j}}=B{Y}_j{\displaystyle \sum _{k=1}^n}\left(1-r_{kj}\right)\left(j=1,2,...,n\right).(12)$

⑦ The determination of the weight of each evaluation index can be expressed as follows:

${\omega }_{\mathrm{j}}=\frac{C_j}{{\displaystyle \sum _{j=1}^n}{C}_j}\left(j=1,2,...,n\right).(13)$

(3) The combination weighting method of the game theory

Based on game theory, the combination weight $\omega$ is obtained by combining the entropy weight method with the CRITIC method. Its procedure is correlated as follows:

① The weight sets ${\omega }_1$ and ${\omega }_2$ were obtained by the entropy weight method and the CRITIC method. It is assumed that $a_1$ and $a_2$ are, respectively, the linear combination coefficient determined by each method, then weight sets ${\omega }_1$ and ${\omega }_2$ can be linearized as (Zhou et al., 2021):

$\omega =a_1{\omega }_1^T+a_2{\omega }_2^T(14)$

② According to game theory, the linear combination coefficients $a_1$ and $a_2$ in Eq. (10) are optimized and are expressed as follows:

$\min {‖a_k\omega {}_k{}^T-{\omega }_k‖}^2\left(k=1,2\right).(15)$

③ According to the differential properties of the matrix, the linear differential equation group for optimizing the first derivative condition of Eq. (15) is determined as follows:

$\left[\begin{array}{cc}{\omega }_1{\omega }_1^T& {\omega }_1{\omega }_2^T\\ {\omega }_2{\omega }_1^T& {\omega }_2{\omega }_2^T\end{array}\right]=\left[\begin{array}{c}{\omega }_1{\omega }_1^T\\ {\omega }_2{\omega }_2^T\end{array}\right].(16)$

④ The optimal combination coefficients $a_1$ and $a_2$ were obtained by Eqn. (16). The normalization process is obtained as $a_1^{*}=\raisebox{1ex}{$a_1$}\!\left/ \!\raisebox{-1ex}{$\left(a_1+a_2\right)$}\right.$, $a_2^{*}=\raisebox{1ex}{$a_2$}\!\left/ \!\raisebox{-1ex}{$\left(a_1+a_2\right)$}\right.$, then based on the game theory, the comprehensive weight $\omega$ can be obtained as follows:

$\omega =a{}_1{}^{*}{\omega }_1^T+a{}_2{}^{*}{\omega }_2^T.(17)$

3.2 The normal cloud model

The normal cloud model is applied to determine the membership degree of different indicators. It is defined as follows: $x,E,D$ is assumed as a common quantitative set, and $E$ is called the domain, where $x\in E$, $D$ is the qualitative conception in domain $E$. For the random research object $x$ in the domain $E$, there still exists a random number with the stable tendency $u\left(x\right)\in \left[0,1\right]$; then, $u\left(x\right)$ is called either the membership degree of $x$ corresponding to $D$ or the definitive degree. The distribution of definitive degrees in the domain $E$ is called the membership cloud. If $x$ meets with $x\sim N\left(Ex,E{n}^{,2}\right)$, and $E{n}^{,}\sim N\left(En,H{e}^2\right)$, and then, $u\left(x\right)$ can be expressed as follows (Zhou et al., 2008):

$u\left(x\right)=\exp \left[-\frac{{\left(x-Ex\right)}^2}{2E{n}^{,2}}\right],(18)$

where the distribution definitive degree $u\left(x\right)$ in the domain $E$ is also called a normal cloud or Gauss cloud. The expectation $Ex$, the entropy $En$, and the hyperentropy H_e are, respectively, applied to represent the digital features in the cloud model. $Ex$ can represent the point of certain conception in the domain; $En$ reflects the accepting range of conception; H_e demonstrates the uncertainty of entropy, and its magnitude reflects the thickness of the cloud drop. They can, respectively, be expressed as follows:

$Ex=\frac{c^{+}+c^{-}}{2},(19)$

$En=\frac{c^{+}-c^{-}}{6},(20)$

$H_{\mathrm{e}}=k_1,(21)$

where $c^{+}$ and $c^{-}$ are, respectively, the upper and lower bounds corresponding to the grade standard of the specific index. The hyperentropy H_e can be selected as a proper constant $k$, which is set as 0.01 in the investigation.

3.3 The determination of the evaluation index

The quality assessment process of surrounding rocks is very complex, and many influencing factors affect the final evaluation results. The evaluation index of a model is often selected based on the actual case in the engineering site. Otherwise, a more significant deviation will occur (WANG et al., 2010). According to the actual investigation data, six assessment factors are considered the quality assessment index of surrounding rocks. These indexes are the uniaxial saturated compressive strength of rock $R$ (X₁), the quality index of surrounding rock $RQD$ (X₂), the frictional coefficient of structural surface $J_{\mathrm{f}}$ (X₃), the joint spacing $J_{\mathrm{d}}$ (X₄), the state of groundwater $W$ (X₅), and the integrity coefficient $K_{\mathrm{v}}$ (X₆).

According to the relevant specifications, the six evaluation indexes can be classified into five levels in Table 1: risk level I (extremely stable), risk level II (stable), risk level III (common), risk level IV (unstable), and risk level V (extremely unstable). The monitoring values of six assessment indexes of the surrounding rocks determined via site inspections and indoor experiments are shown in Table 2.

Table 1

Table 1. Risk level classification of the surrounding rocks.

Table 2

Table 2. Monitoring value.

3.4 The construction of the evaluation frame

The quality of surrounding rocks dramatically influences the safety of construction workers. So, assessing the risk level of surrounding rocks has great significance.

A new evaluation method of surrounding rocks based on the game theory combination weighting-normal cloud model is provided in this article. The process is outlined in Figure 3. First, to evaluate the risk level of surrounding rocks, a complete assessment index system is established. Second, the weight of each assessment index is determined according to the game theory combination weighting theory. Third, certain degrees are determined using the normal cloud theory. Then, the magnitudes of synthetic certainty degree $M$ (shown in Eq. 22) are determined; finally, the risk level of surrounding rocks is determined.

$M={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{u}_i{\omega }_i.(22)$

Figure 3

Figure 3. Process of evaluating the risk level of the surrounding rocks.

4 Results and discussion

4.1 The determination of index weight coefficients

(1) Calculation of the weight coefficient ${\omega }_1$ based on the entropy method

According to Eqs 1–7, and in combination with Table 1, the corresponding weight coefficient can be calculated as follows:

${\omega }_1=\left[\begin{array}{cccccc}0.1197& 0.1242& 0.2688& 0.1641& 0.2603& 0.0628\end{array}\right].$

(2) Calculation of the weight coefficient ${\omega }_2$ based on the CRITIC method

Based on Eqs 8–10, and in combination with Table 1, the correlation coefficients can be obtained as follows:

$r=\left[\begin{array}{cccccc}1& 1& 0.906& 0.9604& 0.9103& 0.9949\\ 1& 1& 0.909& 0.9584& 0.9131& 0.9942\\ 0.906& 0.909& 1& 0.7522& 0.9999& 0.8588\\ 0.9604& 0.9584& 0.7522& 1& 0.7589& 0.9836\\ 0.9103& 0.9131& 0.9999& 0.7589& 1& 0.864\\ 0.9949& 0.9942& 0.8588& 0.9836& 0.864& 1\end{array}\right].$

According to Eq. 11, the standard deviation of different columns is obtained as follows:

$\eta =\left[\begin{array}{cccccc}0.5085& 0.5092& 0.5493& 0.5025& 0.5519& 0.5017\end{array}\right].$

Similarly, according to Eqs 12, 13, the weight of each evaluation index can be calculated as follows:

${\omega }_2=\left(\begin{array}{cccccc}0.0894& 0.0883& 0.2427& 0.2268& 0.2352& 0.1176\end{array}\right).$

(3) The calculation of the combination weight

Based on Eqs 14–17, and in combination with weight sets ${\omega }_1$ and ${\omega }_2$, the combination weight $\omega$ can be obtained as follows:

$\omega =\left[\begin{array}{cccccc}0.1125& 0.1157& 0.2626& 0.1789& 0.2544& 0.0755\end{array}\right].$

4.2 The determination of digital features in the normal cloud model

Based on Table 2, and in combination with Eqs 19–22, the classification standard of normal cloud is depicted in Table 3.

Table 3

Table 3. Digital features of the cloud model.

According to Table 3, the characters of the cloud model corresponding to different indexes are calculated using the forward cloud generator, which is plotted in Figure 4.

Figure 4

Figure 4. Cloud of each assessment index.

Its horizontal coordinates present the magnitude of different variables; the vertical coordinates present the magnitude of certainty degree. A sub-figure in Figure 4 includes five grades: I (very good), II (good), III (common), IV (unstable), and V (extremely unstable). This is the assessment result for the suggested model. When a certain variable is fixed, the certainty degree of the specific point at the state grade can be obtained.

According to Tables 2 and 3, and with Eqs (17)–(18), a comprehensive membership degree is obtained. Its results are listed in Table 4, and the results compared with the actual investigation are plotted in Figure 5.

Table 4

Table 4. Comprehensive membership degree.

Figure 5

Figure 5. Comparison results of the three methods.

The suggested model is applied to assess the surrounding rocks. The complete results are shown in Table 4. Table 4 shows that the quality levels of three different types of surrounding rocks differ. Based on the maximum membership degree criterion, the quality level of N₁ intermediate weathered dolomite limestone is III; one of the N₂ karst development zones is IV; one of the N₃ structural belts is V. It means that the risk level of intermediate weathered dolomite limestone is common; one of the N₂ karst development zones is unstable, and one of the structural belt N₃ is very unstable. The qualified rate of the quality level of surrounding rock quality is 33.3%. Because the quality level of N₁ intermediate weathered dolomite limestone is common, no measures need to be performed. Necessary consolidation measures must be adopted for the N₂ karst development zone and the N₃ structural belt. For example, rock bolts should be fixed in the surrounding rocks (Shao et al., 2022).

Based on the comparative results of the assessment model in Figure 5, the results assessed by the suggested method are consistent with the actual investigation. Its accuracy rate arrives at 100% in the text method, which is higher than the results from the basic quality indicators (BQ) method (67%) (HUANG et al., 2012). Compared to the BQ method, the suggested model improves the reliability of the assessment process and enhances the predicative accuracy of assessment results. Therefore, it is feasible to estimate the quality level of surrounding rocks using the suggested model. The method not only provides accurate results but also adds detail. For example, $RQD$ for the N₁ intermediate weathered dolomite limestone is 75.4, which should belong to level II according to Table 1. In addition, according to Table 1, the quality level of the other indicators obtained by the suggested model belongs to level III, so the quality level probability at the N1 intermediate weathered dolomite limestone at level III is more significant than that of levels I, IV, V, and II. Therefore, its quality level belongs to level III, and it is very unlikely that it would be assigned to levels I, IV, V, or II. The results obtained using the suggested model can accurately demonstrate the quality level of surrounding rocks.

5 Conclusion

A new assessment method is established in this article based on the game theory combination weighting-normal cloud model, and considering the uniaxial saturated compressive strength of rock (X₁), the quality index of surrounding rock (X₂), the frictional coefficient of structural surface (X₃), the joint spacing (X₄), the state of groundwater(X₅), and the integrity coefficient (X₆).

The proposed method is applied to assess the quality level of surrounding rocks. The result is compared with the current specifications and the BQ method; the results obtained based on the suggested method are consistent with the actual investigation. Its accuracy arrives at 100%, which is higher than the results from the BQ method (67%). The results give various quality grades of surrounding rocks from nos N₁-N₃ samples. The quality level of N₁ intermediate weathered dolomite limestone is III; one of the N₂ karst development zones is IV; one of the N₃ structural belts is V. This means that the risk level of intermediate weathered dolomite limestone is common; one of the karst development zones is unstable, and one of the structural belts is very unstable. The qualified rate of the quality level of the surrounding rock is 33.3%. Necessary consolidation measures must be adopted for the karst development zone and the structural belt. In addition, the quality levels of the other indexes obtained for the N₁ sample by the suggested model belong to level III, so its quality level probability at level III is more significant than that of levels I, IV, V, and II.

In total, the results from the proposed model accurately predict the quality levels of surrounding rocks and further determine the quality grade ranking for different samples at the same level. The suggested method provides a new approach to evaluating the quality grade assessment of surrounding rocks.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

BZ: writing–original draft, investigation, and data curation. CY-Bing Shao: writing–original draft, methodology, and funding acquisition. CY: writing–review and editing, validation, and supervision. CZ: writing–review and editing, resources, and conceptualization.

Funding

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This work is supported by the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (2022QYJ02, 2023QYJ02, and 2022QYY02) and the key scientific research project of colleges and universities in Henan province (23B560019).

Conflict of interest

Author Y-BS was employed by Henan Kaiyang Architectural Design Co., Ltd.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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