Comprehensive Evaluation of Crack Safety of Hydraulic Concrete Based on Improved Combination Weighted-Extension Cloud Theory

Abstract

When multiple elements come together, hydraulic concrete develops cracks of varying widths, which huts the dependability of buildings. Therefore, with pertinent tools or procedures, swiftly ascertaining the safety status of hydraulic concrete cracks under diverse service conditions is required by conducting a quantitative and qualitative analysis of the elements influencing the onset of cracks. This paper took the safety status of hydraulic concrete cracks as the main body of research; every step of hydraulic conservation infrastructure from the ground up—design stage, construction process, operation environment, and impoundment operation—was thoroughly examined. After establishing a multi-dimensional and multi-level system for the safety status evaluation of hydraulic concrete cracks, the subjective exponential AHP and objective CRITIC method were employed to determine the weight of each factor. Then, the two weights were processed using an enhanced combination assignment method to produce a more scientifically developed combination weight. Furthermore, fuzziness and randomness were considered in the quantitative analysis thanks to integrating cloud theory and extension matter elements. In order to determine the safety evaluation findings for hydraulic concrete fractures, the maximum membership principle and the cloud picture were employed. The conclusion reached after using this method to evaluate Dianzhan Dam was that the crack had a safety grade of III, meaning that it greatly impacted the reliability of the dam, and called for prompt acceptance or repair measures to improve building efficiency and safety.

1. Introduction

Concrete’s strong compressive strength, outstanding solidity, and water resistance make it a popular material for dams, reservoirs, embankments, canals, and other hydraulic engineering. Nevertheless, concrete contains discontinuities by nature as a composite building material, and fractures are one of its unavoidable fundamental characteristics [1]. Concrete crack development in hydraulic structures is a complicated process made up of numerous interconnected elements. It is challenging to identify the information that contributes to the failure of concrete safety because each aspect is unpredictable [2]. However, if the fractures are allowed to spread unchecked, they will unavoidably have a negative effect on the building from the ground up, raising the risk of catastrophes like leaks, carbonization, and steel corrosion, as well as significantly lowering the structure’s integrity and quality [3]. Thus, fast management over the stage of crack growth is necessary to improve building service effectiveness and guarantee safety.

In the routine examination of concrete cracks, it is critical to use a variety of tools for evaluating crack width, length, and other indications. This demands workers who are skilled at running testing devices and conducting regular maintenance [4]. However, manual testing under varying working circumstances may cause damage to the testing instrument, affecting test accuracy and risking detecting workers safety, as well as increasing the time necessary to achieve the final test results. Traditional concrete fracture detection technologies are not only labor-intensive, but they also fail to assure efficient and precise crack treatment, especially in complicated engineering circumstances [5]. Mathematical methods have been recognized by many scholars in evaluating complex and disordered research objects due to their advantages of wide applicability, fast arithmetic speed, and precise results. Wang et al. in a mixed concrete laboratory test, on the basis of quantitative analysis, used rough sets theory analysis for the characteristics and pore structure of concrete permeability and salt resistance [6]. Tan et al. took the control parameters of petroleum hydrocarbon pollution in groundwater as the research parameters, carried out the experiment through the soft computing model of an adaptive neural fuzzy reasoning system, and used the hierarchical ladder method of analytic hierarchy process to evaluate [7]. Wang Ming proposed a cloud model membership evaluation method based on conventional cloud theory and a scale-based analytic hierarchy process, applied it to a rock slope project, and the feasibility and practicability of the evaluation results were verified [8].

In conclusion, on the basis of the existing data of hydraulic concrete crack detection, the analysis and arrangement ability of mathematical methods can be used to better control the development state of cracks. However, most of the researchers employed one mathematical technique to analyze or evaluate the research object. This might result in subjective or objective results that failed to capture the simultaneity of the ambiguity and randomness of the influencing factors, biasing the accuracy of the evaluation that followed. In order to fully combine the benefits and drawbacks of different evaluation techniques, take into account the myriad factors that affect the safety state of hydraulic concrete cracks and their influence modes, and ensure that the uncertain and coupled fuzziness of the factors when they are randomly combined can be taken into account, more thorough and scientific methods of evaluating the safety state of hydraulic concrete cracks must be used in order to identify appropriate countermeasures for various cracks.

Therefore, this paper set the safety status of hydraulic concrete cracks as the research object, and conducted a multilevel decomposition of the entire construction and operation process of hydraulic engineering from four aspects: design stage, construction process, operation environment, and impoundment operation. The subjective and objective weights of each factor in the system were solved using the exponential AHP (EAHP) and CRITIC methods, respectively, based on the available data and the constructed multilevel system. The two solutions were then aggregated using the improved combination assignment method to obtain the comprehensive weights that simultaneously characterize uncertainty and ambiguity. In addition, this paper employed the cloud model with extension theory to maintain the uncertainty and randomness of the assessment procedure while conducting a quantitative analysis of the incompatibility issue. In order to provide a referable method for the safety assessment of the crack state, the affiliation and cloud parameters obtained by multilevel feedback were then used to understand the actual situation of each index. The safety evaluation results of hydraulic concrete cracks were obtained through the application of the principle of maximum affiliation and the cloud model cloud diagram.

2. Improved Combination Weighting Method Based on EAHP-CRITIC

2.1. Subjective Weighting Analysis

Analytic hierarchy process (AHP) is a methodical approach to analysis used in making guided decisions. The systematization, modeling, and mathematical representation of decision makers’ thought processes about intricate items are among its attributes, along with simplicity, adaptability, and a pragmatic nature [9]. AHP is centered on two primary facets. The first method involves breaking down a complex issue into its constituent elements and organizing them into dominance-ordered groups to create an ordered recursive hierarchy. The second technique involves conducting pairwise comparisons to ascertain the relative significance of different components within the hierarchy. The methods that are now in use are 1~9 scale method, −1~1 scale method, and e^0/5~e^8/5 scale method.

The n-scaled approach has marginally lower accuracy when it comes to measuring people’s opinions. Nonetheless, the application of exponential scaling could accurately represent the level of consistency in thought, lower the average relative error, and guarantee that the judgment matrix’s consistency index is zero due to the method’s transferability.

Consequently, this research applied the e^0/5~e^8/5 exponential scaling method and created the exponential AHP (EAHP), which could increase the accuracy of the results and the efficiency of the weight calculation procedure. Table 1 displayed the exponential scaling method’s value rules [10,11].

Following the processes listed below, the weights of different components in the hierarchy could be determined and verified when the aforementioned crucial challenges have been resolved [12,13].

• (1). Creation of the matrix of comparisons

The comparison results a_ij of all factors were determined with reference to the mathematical significance of each index form in Table 1, and the comparison matrix A = (a_ij)_n×n was produced.

• (2). Determined the eigenvector and eigenvalue maxima.

The greatest eigenroot λ_max of matrix A, and its related eigenvector W, could be found after the comparison matrix had been established.

• (3). Verification of rationality

The reasonableness of the weight vector λ needed to be verified using the consistency index CR of A.

(1)$CR={\displaystyle \frac{CI}{RI}}$

where CI was the matrix deviation consistency indicator.

(2)$CI={\displaystyle \frac{{\lambda }_{\max }-n}{n-1}}$

RI was the average random consistency index and took the values shown in Table 2.

When CR ≤ 0.1, it meant that the comparison matrix A satisfies the consistency check. If the calibration failed, the values of the elements within A need to be re-evaluated until the consistency calibration was passed.

2.2. Objective Weighting Analysis

Criteria importance through intercriteria correlation (CRITIC) is an objective weight assignment method, whose basic idea is to characterize the variability and conflict between the evaluation indicators using the standard deviation and correlation coefficient. Then, based on the analysis of the relationship between them, the weight of each indicator is determined [14].

Variability, which is represented by the index’s standard deviation, can be used to illustrate how the value size of the same index varies for several samples. The difference in each sample’s value range is larger the higher the standard deviation. On the other hand, correlations between indicators—which are defined by their correlation coefficient—are associated with conflicts [15]. Indicators are given less weight the more repeated information there is between them; that is, the better the correlation between indicators, the lower the conflict. This is how the operation flow was explained [16,17]:

• (1). Construction of the initial indicator data matrix X = (x_ij)_n×n.

• (2). Standardization of raw data.

According to the different manifestations of indicators, different standardized processing methods were chosen, and the normalized data matrix Y = (y_ij)_n×n was obtained.

In actual application, the weight of each variable’s function would vary in the comprehensive evaluation due to the huge variations in the order of magnitude caused by the scale disparities among the variables. As a result, the data must be processed dimensionlessly and converted into pure dimensionless numbers for analysis and comparison. Methods like the polar deviation standardization, Z-score standardization, and linear scale standardization are commonly used to handle data with many dimensions.

• (3). The index comparison intensity was determined.

  (3)$S_j=\sqrt{{\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m{(y_{ij}-{\overline{y}}_j)}^2}}$

• (4). The conflict correlation coefficient was determined.

  (4)$s_{ij}=\mathrm{cov}(Y_k,Y_j)/(S_k,S_j)$

  where cov (D_k, D_j) was the covariance of the evaluation data between the kth and jth indicators, k ≠ j; S_k and S_j were the comparative strengths of the kth and jth indicators, respectively.

• (5). The amount of information synthesized between indicators was determined.

  (5)$G_{ij}=S_j{\displaystyle \sum _{j=1}^n(1-s_{kj})}$

• (6). Determination of weighting factors for indicators was as follows.

  (6)$M_j=G_j/{\displaystyle \sum _{j=1}^n{G}_j}$

2.3. Improved Combinatorial Weighting Analysis

Multiplicative normalizing and linear weighting are the two most common weighing methods used in combined weighting, which is the process of combining two or more weighting methods to create optimal weights. Nevertheless, these techniques are not helpful in identifying accurate and inaccurate information when conducting weight analysis, and the rationale of the weight combination is not adequately described [18].

Therefore, in the process of combination assignment, game theory was introduced in this work. When determining the maximum interest point between the indicators, the weights of the indicators—which were derived from various methods—were coordinated and consistent. In the end, the system’s ideal combination weights for every component are found. The following describes the specific steps [19,20]:

• (1). The weight vector ω_i (i = 1, 2, …, n) was obtained by using N weighting methods, respectively, thus forming the initial weight vector set W_n = {ω₁, ω₂, …, ω_n}, i = 1, 2, …, n. Given a set of combination coefficients (α₁, α₂, …, α_n), the comprehensive weight of every component could be expressed as:

  (7)$W_i={\alpha }_1{w}_1^T+{\alpha }_2{w}_2^T+\cdots +{\alpha }_n{w}_n^T$

• (2). To define the objective function, a Matlab 2023a calculation (α₁, α₂, …, α_N) was used to find the value that minimized the deficiencies of the combination coefficients. This would yield the weight W at its optimal value.

  (8)$\min {\Vert {\displaystyle \sum _{i=1}^N{\alpha }_i{\mathit{W}}_i^T-{\mathit{W}}_i}\Vert }_2$

• (3). The optimal linear equations that are equal to Equation (8) could be found as follows, based on the differential properties of the matrix:

  (9)$\left[\begin{array}{ccc}{W}_1{W_1}^T& \cdots & W_1{W_n}^T\\ ⋮& \ddots & ⋮\\ W_n{W_1}^T& \cdots & W_1{W_n}^T\end{array}\right]\left[\begin{array}{c}{\alpha }_1\\ ⋮\\ {\alpha }_n\end{array}\right]=\left[\begin{array}{c}{W}_1{W_1}^{T,*}\\ ⋮\\ W_n{W_n}^{T,*}\end{array}\right]$

• (4). After the computation was finished and the α_i was normalized to α_i*, the comprehensive weight W based on the improved combinational game theory may be ultimately determined as follows:

  (10)$\mathit{W}={\displaystyle \sum _{i=1}^n{{\alpha }_i}^{*}{\mathit{W}}^T}$

3. Extension Cloud Theory Method

3.1. Basic Concepts of Extension Cloud

Extension theory is a theoretical approach that uses formal models to address the issue of incompatible fusion and investigate the idea of things expanding. The fundamental matter-elements representing things, represented as R = (M, C, V), are formed by combining ordered triples (object M, sample feature C, and eigenvalue V) in matter-element analysis theory [21]. If the object M has n characteristics C₁, C₂, …, Cn, its corresponding quantity value is V₁, V₂, …, V_n, then the R is called n-dimensional matter element, its matrix form is:

(11)$\mathit{R}=(M,C,V)=\left(\begin{array}{ccc}M& C_1& V_1\\ & C_2& V_2\\ & ⋮& ⋮\\ & C_n& V_n\end{array}\right)$

While the influence elements of the safety state of concrete cracks in hydraulic engineering have many characteristics, the characteristic value V of an item in the material element expansion method cannot effectively characterize the assessment object with uncertain variables. It can be challenging to keep the fuzziness and randomness of the elements in traditional evaluation methods when certain evaluation indications are hard to quantify [22,23]. The assessment’s findings are a little strict. For an acceptable transformation between qualitative and quantitative evaluation, the fixed characteristic value of the object in the classical material element model must be changed.

This limitation can be overcome by substituting the fixed numerical characteristic quantity value V in the material element model with the relevant concepts and characteristic parameters (E_x, E_n, H_e) of the cloud model. This allows for the use of dual uncertain evaluation indicator intervals to perform the qualitative-to-quantitative evaluation transformation, retaining the fuzziness and randomness of each evaluation factor, and improving the evaluation results’ proximity to reality.

E_x represents the membership cloud’s center of distribution, a scalar value that best captures the safety performance assessment of hydraulic concrete cracking. E_n represents the element concept’s fuzziness; it can be used to convey the uncertainty of the questionnaire data sample that was acquired throughout the evaluation process as well as the expert’s acceptance range of that data sample. H_e describes the relationship between the randomness and fuzziness of various evaluation process components and quantitatively expresses the uncertainty of entropy with reference to the data dispersion in the hydraulic concrete cracking safety performance evaluation questionnaire [24].

Such ordered triples of “objects, features, and cloud values” are called extension cloud matter elements. Then n features common to m standard things C₁, C₂, …, Cn and its corresponding magnitude μ₁ (x_1i), μ₁ (x_2i),…, μ₁ (x_mi) constitutes the N-dimensional complex cloud element of m things, which is recorded as [25]:

(12)${\mathit{R}}_{mn}=\left(\begin{array}{ccccc} & M_1& M_2& \cdots & M_m\\ C_1& {\mu }_1(x_{1i})& {\mu }_2(x_{2i})& \cdots & {\mu }_m(x_{mi})\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ C_n& {\mu }_1(x_{1n})& {\mu }_2(x_{2n})& \cdots & {\mu }_m(x_{mn})\end{array}\right)$

3.2. Evaluation Steps of Extension Cloud Theory

• (1). Determine the object-element and evaluation criteria cloud of the sample.

A topological cloud theoretical model of Equation (15) was constructed for the object element objects. Usually, m standard intervals were established with the help of existing specifications and expert opinions, and the boundaries of each level of classification were used as a bilateral constraint interval [C_min, C_max]. These three numerical features of the cloud model were calculated by using the transformation relation equation between interval number and cloud model [26]. The calculation formula was as follows.

(13)$\{\begin{array}{l}{E}_x={\displaystyle \frac{C_{\min }+C_{\max }}{2}}\hfill \\ E_n={\displaystyle \frac{C_{\max }-C_{\min }}{2.355}}\hfill \\ H_e=p{E}_n\hfill \end{array}$

where p was a constant determined according to the degree of ambiguity, generally 0.1.

• (2). The correlation degree between the things to be evaluated and each evaluation grade index was determined.

The correlation degree between the deterministic value and the matter element of the cloud model was expressed by the certainty degree of the value relative to the cloud model. Matlab 2023a was used to calculate the correlation degree in this paper:

• Matlab was used to generate a normal random number E_n′ with E_n as the mean, and H_e as the standard deviation.

• Let the deterministic value of the thing to be evaluated be x_i. Then, the calculation formula of cloud correlation degree μ_i was as follows:

(14)${\mu }_i=\exp \left[-{\displaystyle \frac{{(x_i-E_x)}^2}{2{E_n^{\prime }}^2}}\right]$

The (x_i, μ_i) was called a cloud droplet, and the more cloud droplets there were, the more it reflected the overall characteristics of the sample.

It should be noted that when applying Equation (13) to calculate the cloud affiliation of a factor, the existence of randomness would make the result fluctuate, which required multiple operations to improve the calculation accuracy. The rank eigenvalue u_i (x) obtained from the ith calculation was homogenized accordingly [27]. Considering the computational cost and time, this paper set the number of operations as 800 times, and the specific calculation formula was shown in Equation (14).

(15)$\{\begin{array}{l}{E}_x^u={\displaystyle \frac{u_1(x)+u_2(x)+\dots +u_{800}(x)}{800}}\hfill \\ E_n^u=\sqrt{{\displaystyle \sum _{i=1}^{800}{(u_i(x)-E_x^i)}^2}/800}\hfill \end{array}$

• c.. In order to evaluate the reliability of evaluation results more intuitively, the confidence factor θ was introduced [28]. Its value was inversely proportional to confidence, and the formula was as follows:

  (16)$\theta =E_n^u/E_x^u$

• d.. When the value of the confidence factor was reasonable, the cloud affiliation obtained from Equation (16) can be utilized to form the extension cloud matrix Q. It represented the cloud affiliation between each metric in the sample and the standard cloud of the first to mth level.

  (17)$\mathit{Q}=\left(\begin{array}{cccc}{\mu }_{11}& {\mu }_{12}& \cdots & {\mu }_{1m}\\ {\mu }_{21}& {\mu }_{22}& \cdots & {\mu }_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ {\mu }_{n1}& {\mu }_{n2}& \cdots & {\mu }_{nm}\end{array}\right)$

• (3). The combined weight was calculated.

In this step, the subjective weighting method (EAHP) and objective weighting method (CRITIC) were first used to determine the weights of each factor in the sample system, and the weight vectors were w^u and w^v, respectively.

Then, game theory was introduced in the process of combinatorial weighting, and the two weight results were integrated and coordinated to obtain the final weight vector W = (w₁, w₂,…, w_n)^T.

• (4). A comprehensive certainty matrix was established.

  By combining the results (2) and (3), the matrix K of sample certainty was obtained.

  (18)$\mathit{K}=\mathit{W}\mathit{Q}={\left(w_1,w_2,\dots ,w_n\right)}^T\left(\begin{array}{cccc}{\mu }_{11}& {\mu }_{12}& \cdots & {\mu }_{1m}\\ {\mu }_{21}& {\mu }_{22}& \cdots & {\mu }_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ {\mu }_{n1}& {\mu }_{n2}& \cdots & {\mu }_{nm}\end{array}\right)=\left(\begin{array}{cccc}{k}_{11}& k_{12}& \cdots & k_{1m}\\ k_{21}& k_{22}& \cdots & k_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ k_{n1}& k_{n2}& \cdots & k_{nm}\end{array}\right)$

• (5). Each of the corresponding levels had a corresponding degree of certainty, and the same level of certainty was added, according to the principle of maximum membership; the comprehensive evaluation level could be determined [29]. That was, k_j = maxk_j (p), p belongs to grade j.

  In summary, Figure 1 displayed the technology roadmap for this paper.

4. Examples of Engineering Applications

4.1. Establishment of a Hierarchy

Many factors would lead to the production of cracks in dams, embankments, channels, and other hydraulic engineering. Depending on the type of crack, these cracks are typically divided into structural and non-structural categories. The factors that affect the state of hydraulic concrete cracks were systematized at multiple levels in order to establish the hierarchical system structure of Figure 2, which would allow for a thorough analysis of the causes of hydraulic building concrete crack formation and countermeasures, in addition to the crack detection results of various forms of water conservancy facilities [30,31,32,33].

The safety status of hydraulic concrete fractures was the target layer in the hierarchical structure used in this paper. Four key factors—design stage (A), construction procedure (B), operation environment (C), and water storage operation (D)—were considered as the dimension layer while analyzing the entire process of building and running water conservation systems from the ground up. Subsequently, each dimension was then refined to further obtain numerous metrics for the performance layer based on the known timing and causes of hydraulic concrete cracks. For example, in the design stage (A), hydraulic buildings would have different durability performance due to the structural form (A₁), design standards (A₂), computational method (A₃), reinforcement layout (A₄), load combination (A₅), and the form and damage degree of concrete cracks were not the same.

4.2. Determination of Standard Cloud Parameters

This paper synthesized all the factor expressions of the dimension layer and the performance layer in Figure 1 based on Chinese standards [34,35], to categorize the comprehensive safety state of hydraulic concrete cracks. Table 3 displayed the precise values and meanings for the safety status of hydraulic concrete cracks, which created a judgment standard and provides the relevant overhaul procedures.

To acquire the cloud parameters corresponding to different classes, the upper and lower limit values in Table 3 were then substituted into Equation (16). This process produced the standard evaluation cloud map of hydraulic concrete cracks, as shown in Figure 3.

4.3. Determination of Comprehensive Weight

Specialists with specialized understanding of hydraulic concrete structures were hired based on the current design and testing criteria and after analyzing the monitoring data pertaining to concrete fractures. The objective and subjective weights of each indicator were then determined using the CITIC and EAHP methods, respectively, after the experts were asked to assess each indication’s significance using the index scaling criteria in Table 1.

Next, using Equations (7)–(9), the combination coefficients of the dimension layer weights were respectively determined to be 0.462 and 0.538, while the performance layer’s combination coefficients were 0.589 and 0.411, respectively. Ultimately, as indicated in Table 4, the total weights of the indicators were determined.

4.3.1. Comprehensive Weight of Dimension Layer

Table 4 illustrated that, of the dimensional layer parameters, improper handling of the design stage (A) and construction procedure (B) increased the likelihood of fractures developing in hydraulic concrete. Consequently, the building’s operating environment (C) had a greater influence on the concrete cracks than the conditions the building faces during the impoundment operation (D).

Generally speaking, the hydraulic building’s fundamental form and structure were decided upon during the design stage (A), which served as the foundation and assurance for all subsequent work. Unlike other elements, the quality of the design would not have a noticeable impact on the formation of hydraulic concrete fractures. and it usually happened only after a somewhat lengthy use phase, in conjunction with detecting tools to identify parts shortages during the design phase. A structural form that could withstand reasonable working conditions was designed only after considering the topography and other constraints. This would help to improve the stability, durability, and safety of the building by giving each concrete piece a more stable form, reducing or postponing the likelihood and time of cracks.

The performance of construction procedure (B) had a major influence weight on hydraulic concrete fractures as well. Structure building in hydraulic conservation facilities typically required a lot of labor and time, which significantly increased the unpredictable construction process. Construction quality control was a challenging procedure; if a step was missed and not remedied, it would result in incalculable quality issues with the finished structure. As a result, the benefits of concrete were not fully realized and its flaws were exposed, reducing the building’s dependability and service efficiency.

Impoundment operation (D) described the use of hydraulic buildings, which was essential for confirming the structure’s dependability and for displaying the pattern of concrete cracks the easiest. Additionally, the structure was subjected to a range of working condition loads, which caused the concrete members to sustain variable degrees of fatigue damage. Concrete cracks and spalling were more prone to form, particularly in regions with long-term water washout and critical pressures.

Operating environment (C) was one of the external influence factors. Its performance was highly varied, particularly in the service building’s outermost areas where the local environment interferes quite strongly. The operating environment’s impact on hydraulic concrete cracks was less significant than the other components when considering the whole picture, though, as this research aggregates inspection data from water conservation facilities across several locations for assessment.

4.3.2. Comprehensive Weight of Presentation Layer

As can be observed from the previous study, Figure 4 and Figure 5 displayed the comprehensive weight of the presentation layer in the multilevel indicator system.

As can be seen from Figure 4a, for the design stage (A), structural form (A₁), and reinforcement layout (A₄) had the most obvious influence weight on it. This was because in the preliminary design of hydraulic conservancy facilities, program selection needed to be carried out, such as dam type and channel cross-section, only to determine the most cost-effective structural form, in order to enhance the reliability of the building and attenuate the adverse effects of cracks. In addition, the reinforcement layout (A₄), that was, the number of reinforcements and the way, determined the overall structural load-bearing capacity and damage ductility. The great tensile strength of steel may be used to postpone the formation of concrete structure cracks and provide crucial reserve time for engineering maintenance, but only the right reinforcing arrangement can cooperate with concrete to satisfy diverse service requirements.

The structural form (A₁) and reinforcement layout (A₄) for the design stage (A) had the most evident influence weight on it. This was due to the requirement for program selection, including dam type and channel cross-section, in the initial design of hydraulic conservation facilities in order to choose the most economical structural design that would lessen the negative effects of cracks and increase the building’s reliability. Furthermore, the total structural load-bearing capacity and damage ductility were determined by the reinforcement layout (A₄), which basically referred to the quantity and arrangement of reinforcement. Just the right amount of reinforcement was needed to handle a range of service requirements in addition to the concrete. The high tensile strength of the steel bar also helped to postpone the emergence of cracks in the concrete structure, giving project maintenance a valuable buffer period.

The influence weights of design standards (A₂), load combination (A₅), and computational method (A₃) were drastically lowered. The following three indications were less optional throughout the design stage after A₁ and A₄ were decided. Though the load circumstances experienced by individual buildings may vary, it was imperative to adhere to current design requirements when calculating and verifying results. Consequently, the impact of these three indications on the ultimate development of cracks was negligible.

Figure 4b illustrated that, for the construction procedure (B), the mix proportion (B₁), curing results (B₂), raw material selection (B₃), internal and external temperature difference (B₄), restrained shrinkage (B₅), and local climate (B₆) diminished in sequence. Since they were the primary determinants of concrete materials, the importance of the B₁, B₂, and B₃ variables stood out among the rest. The B₁ factor was a crucial metric for evaluating the concrete’s performance, as it indicated the material’s capacity to adjust to varying conditions and requirements. For instance, you could increase the density of concrete and postpone the emergence of surface cracks by lowering the water-to-cement ratio or adding mineral admixtures.

The human control factors were B₂ and B₃. Water moved from the inside to the surface of the concrete during the hardening process and evaporated, leaving the surface of the concrete with some irregular shrinkage fractures. The emergence of micro-cracks could be effectively prevented if the construction site had a good maintenance environment and raw material quality. Temperature difference and water evaporation could also be managed. However, plastic deformation and dry shrinkage fractures would appear in the concrete if the maintenance was not conducted on time or the raw materials were of subpar quality.

B₄, B₅, and B₆ were beyond our control. The temperature differential between the inside and exterior of the concrete would alter, especially if there was a dramatic shift in temperature during the mass concrete production process. In any case, as temperature stress or deformation gradually manifested, surface crack damage became worse. Furthermore, concrete would also show up during the curing process as a result of boundary conditions, cement hardening, and the volume shrinkage phenomena. Because of this shrinkage, the tensile tension during hydration heat release was larger than the pressure stress, and some non-structural cracks would develop. During the building process, the concrete’s ability to maintain itself would be impacted by heavy rain or wind, which may eventually cause cracks in the material to widen.

It could be seen from Figure 5a that among the operating environment (C), the influence weights of earthquake action (C₄), erosion impact (C₃), ambient temperature (C₁), and ambient humidity (C₂) all dropped in turn. Although the design took seismic level influence into account, in the event of a real earthquake, the building’s overall safety could be gravely jeopardized by the occurrence of structural cracks, even though the hydraulic structure would not collapse due to the influence of seismic waves. C₃ described the chemical erosion of concrete, which occurred when certain elements of the material reacted with corrosive media in the surrounding environment to produce new chemicals and harm the structure of the concrete.

There were various processes that cause erosion, such as the alkali aggregate reaction during the initial mixing of the concrete, the carbonization phenomena, and salt ion erosion in later stages. All of these processes would cause irreversible crack damage to the concrete. The building’s service function, the hydraulic structure, and the parts that came into touch with the outside environment were all influenced by the complementary C₁ and C₂ factors. These parts were susceptible to harm from either the freeze–thaw cycles or the dry–wet cycles, and surface cracks in concrete would unavoidably develop. Additionally, the coupling effect of chemical erosion would cause some of the major surface cracks to develop into structural cracks, transforming the concrete from its initial solid and hard state to a loose state that breaks easily.

In the impoundment operation (D) phase, foundation settlement (D₃), bedrock constraints (D₄), impounding height (D₁), hydraulic action (D₅), and wind wave action (D₂) are reduced in order, as shown in Figure 5b. Large-scale hydraulic engineering often had sections of their base built on hard bedrock and others on hard soil layers with varying water contents. This was because the bedrock composition of these projects is not consistent. Long-term pressure would cause the foundation to eventually experience uneven settling, which would lead to uneven deformation and structural cracking of the concrete. On hydraulic buildings, the hydrostatic pressure was determined by D₁, and it was proportional. Furthermore, D₁ was not fixed; rather, it was akin to cyclic variations, which would cause simultaneous damage to the surface concrete due to freeze–thaw cycles or dry–wet cycles and further encourage the formation of concrete cracks when loads were applied.

D₄ was for the impact of water scour. For instance, if there was a hollow area in the flood discharge dam, the water flow would create a suction and direct the direction of the water discharge, which would unavoidably have a significant influence on the concrete. The likelihood of concrete spalling and cracking increased significantly following many flood discharges. D₂ stood for wind, wave, and current loads. These dynamic loads were incredibly unpredictable, and they would cause localized-only stresses in the beams and columns, which would cause the concrete to deform and crack.

In real projects, it was challenging to discern the fissures caused by the various factors mentioned above, and they were interdependent. Under the coupling effect of several causes, even modest non-structural fractures that do not compromise safety would eventually evolve into structural cracks. The likelihood of steel reinforcement corroding increases with time, and concrete spalling, bulging, and other ring-breaking phenomena progressively manifest. These factors significantly compromised the building’s appropriateness, durability, and safety metrics.

4.4. Evaluation Results and Validation

The final step was to calculate the software programming of Equation (14) in order to determine each factor’s cloud affiliation and the overall judgment matrix. Here, the Dianzhan Dam in reference [36] was used as an example to examine, using mathematical statistics, the safety status of its concrete fractures.

The final affiliation results of the crack safety state (W) of the power station dam were arranged, as indicated in Table 5, using the performance layer factor and dimension layer factor of the construction procedure (B) as examples to show their affiliation values in the cloud affiliation matrix construction.

Based on Table 5 and the application of the principle of maximum affiliation in conjunction with mathematics, the cracks in Dianzhan Dam of ref. [18] were classified as III, or medium safety status. Still, the dam’s cracking hazard was comparatively modest, and the affiliation metrics showed that the cracks in the dam were moving from a safer to an intermediate stage. According to Table 2, it was deemed possible to start strengthening and maintaining the dam at this time, and there was enough time to complete the strengthening procedure.

Simultaneously, the confidence factor θ for each component was less than 0.01; this suggested that there was a reasonable amount of random volatility in MATLAB when generating the rank eigenvalues numerous times, meaning that the affiliation and evaluation outcomes that follow were more reliable. As a result, the extension cloud model might guarantee and raise the evaluation results’ confidence in addition to revealing the extent to which each component influences fracture safety at the same assessment level.

Next, as Figure 6 illustrated, MATLAB was used to simulate the comprehensive assessment value of the safety performance level of the power station dam cracks once more.

According to Figure 6, the majority of the cloud dropped the safety status of cracks to fall between “Relatively Safe” and “Moderate” grade. This was consistent with the findings derived from the maximum degree of affiliation principle, meaning that the safety status of cracks in the Dianzhan dam’s concrete fell into III grade. This demonstrated the viability and practicability of using extension cloud theory to assess the safety status of hydraulic concrete fractures.

As a result, more analysis may be performed on the information in Table 4 and Figure 6. U (74.31, 5.28, 1.06) was the evaluation cloud parameter for dam cracks. The cloud droplets’ fuzziness and randomness were represented by the E_n value, which was tiny. This meant that when experts graded the entire system, the differences were smaller, the scores were more concentrated, and the data adhesion was good. Since the cloud droplet condensation degree, or H_e, had a value of about 1, the factor values of each layer in the system had a small deviation from the central value, the cloud droplet dispersion degree was low, the entropy uncertainty degree was low, and the reliability of the results was good.

It is recommended that the existing buffer time be used promptly and completely based on the results of the maximum membership degree and cloud model.

Surface sealing or chemical grouting treatment was chosen based on the various crack manifestations. In order to prevent the repaired cracked elements from becoming delicate or perishable, normative acceptance of crack treatment measures must also be carried out. This would increase the building’s longevity and efficiency.

5. Discussion and Conclusions

(1) The weight of each component in the multi-level system of hydraulic concrete fracture safety conditions was first determined in this paper using the subjective EAHP and the objective CRITIC technique. Next, a more thorough and empirical combinatorial weighing was achieved by applying game theory to enhance the combinatorial weighting technique. Moreover, this idea of weight analysis could also be applied to other research objects.

(2) In order to preserve the fuzziness and randomness of the entire evaluation process, this paper replaced the fixed eigenvalue in the comprehensive evaluation of the multi-level system with the digital eigenvalue of the cloud model. This improved upon the traditional extension theory. Consequently, the accuracy and logic of the evaluation findings were increased when the extension cloud model was utilized to assess the fracture safety status of hydraulic concrete since it can be more appropriate for the real-world scenario. This optimization method could also be applied to other objects to be evaluated, such as bridges, hydropower stations, gates, power grids, river and lake health, etc.

(3) The dam fracture level is III, and the safety state was medium after a thorough assessment of the safety state was conducted using the maximum membership principle and the cloud map data performance. This indicated that the reliability of the dam was directly impacted by the safety state of the cracks. To guarantee the building’s safe operation and dependable service status in the future, it was imperative to utilize the buffer time that was currently in place, apply the surface sealing and chemical grouting treatment on time, and closely inspect maintained components.

(4) It is noteworthy that this research does not take other widely used approaches into consideration while choosing EAHP and CRITIC for performing early weight analysis of different elements in the multi-level system. Obtaining a complete weight of elements still has limits. In addition, although the extended matter aspect is integrated with the cloud model, this paper’s analysis and assessment are predicated on the standard cloud model, which could not be broadly relevant to various research systems. In order to fully assess the fracture status of hydraulic concrete and provide more accurate and sensible findings, it is therefore imperative that future research use several weight calculation techniques and make use of diverse kinds of cloud models.

Author Contributions

Conceptualization, Y.C.; methodology, M.Z. and H.W.; software, Y.D.; validation, G.L.; formal analysis, L.L.; investigation, M.Z.; resources, H.W.; data curation, Y.D. and G.L.; writing—original draft preparation, L.L. and C.X.; writing—review and editing, Y.C.; visualization, Y.C.; supervision, C.X.; project administration, L.L. and C.X.; funding acquisition, C.X. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data presented in this study are available on request from the corresponding author. The data are not publicly available due to restrictions on privacy.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Technology roadmap.

Figure 2. Safety status evaluation of hydraulic concrete cracks.

Figure 3. Cloud diagram of safety criteria evaluation of hydraulic concrete cracks.

Figure 4. Performance layer weights for A and B dimensional factor. (a) Design stage (A); (b) construction procedure (B).

Figure 5. Performance layer weights for B and C dimensional factor. (a) Operating environment (C); (b) impoundment operation (D).

Figure 6. Safety level of concrete cracks in Dianzhan dam.

Table 1

Meaning of the values of the index scale.

Element Values	Judging Standard
e0/5	Xi is as equally important as Xj
e2/5	Xi is slightly more important than Xj
e4/5	Xi is significantly more important than Xj
e6/5	Xi is strongly more important than Xj
e8/5	Xi is absolutely more important than Xj
e1/5, e3/5, e5/5, e7/5	Between the above two neighboring judgments
multiplicative inverse	Importance of Xj compared to Xi

Table 2

RI values.

n	1	2	3	4	5	6	7	8	9	10
RI	0	0	0.52	0.89	1.12	1.26	1.36	1.41	1.46	1.49

Table 3

Safety evaluation grade of hydraulic concrete cracks and cloud model parameter values.

Evaluation Level	Score	Significance	Ex	En	He
Safe (I)	95~100	Essentially harmless	97.5	2.12	0.21
Relatively Safe (II)	80~95	Slightly affects structural performance	87.5	6.37	0.64
Moderate (III)	60~80	Significantly affects structural performance	70.0	8.49	0.85
Relatively Hazardous (IV)	30~60	Seriously affects structural performance	47.5	10.62	1.06
Hazardous (V)	0~30	Fatally affects structural performance	15.0	12.74	1.27

Table 4

Combined weights of factors in the hierarchy.

Dimension Level	Weight			Presentation Layer	Weight
	Subjective	Objective	Combined		Subjective	Objective	Combined
A	0.261	0.279	0.273	A 1	0.266	0.234	0.253
				A 2	0.195	0.201	0.197
				A 3	0.102	0.163	0.127
				A 4	0.279	0.193	0.244
				A 5	0.158	0.209	0.179
B	0.258	0.267	0.264	B 1	0.277	0.213	0.251
				B 2	0.235	0.215	0.227
				B 3	0.176	0.236	0.201
				B 4	0.122	0.139	0.129
				B 5	0.103	0.104	0.103
				B 6	0.087	0.093	0.089
C	0.248	0.228	0.225	C 1	0.232	0.257	0.242
				C 2	0.208	0.235	0.219
				C 3	0.258	0.231	0.247
				C 4	0.302	0.277	0.292
D	0.233	0.226	0.238	D 1	0.248	0.096	0.186
				D 2	0.078	0.154	0.109
				D 3	0.282	0.268	0.276
				D 4	0.261	0.257	0.259
				D 5	0.131	0.226	0.170

Table 5

Affiliation of selected factors.

Factors	I	II	III	IV	V	Results	θ
B 1	0.235	0.344	0.126	0.189	0.106	II	0.0063
B 2	0.212	0.266	0.287	0.114	0.121	III	0.0055
B 3	0.295	0.337	0.178	0.136	0.054	II	0.0032
B 4	0.206	0.243	0.241	0.124	0.186	II	0.0028
B 5	0.204	0.172	0.268	0.253	0.103	III	0.0069
B 6	0.251	0.205	0.183	0.218	0.143	I	0.0074
A	0.256	0.279	0.247	0.102	0.116	II	0.0066
B	0.236	0.282	0.208	0.162	0.112	II	0.0059
C	0.114	0.253	0.333	0.172	0.128	III	0.0042
D	0.006	0.264	0.287	0.242	0.201	III	0.0061
W	0.151	0.270	0.274	0.167	0.138	III	0.0054

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