Analysis of the Risk Factors for PCCP Damage via Cloud Theory

Abstract

Research on prestressed concrete cylinder pipes (PCCPs) has focused primarily on their failure mechanisms, monitoring methods, and the effectiveness of repairs. However, gaps in the study of damage risks associated with PCCPs remain. Based on existing relevant research, this study focused on analysing the uncertainties in the material production and manufacturing processes of PCCPs to assess their damage risk. The research employs onsite test data about the compressive strength of C55 concrete and the real prestressing force exerted on prestressed steel wires, utilising the measured compressive strength of the concrete core in PCCPs alongside the actual prestressing force applied to the steel wires. An inverse cloud generator was employed to obtain the expected value Ex, entropy En, and hyperentropy He of the characteristic numbers. These values are then combined with the forward cloud model in cloud theory to train random parameters. By combining cloud theory with the Monte Carlo method, a risk analysis model for PCCP pipelines was established. Using internal water pressure monitoring data from the Qiliqiao Reservoir to the Xiayi Water Supply Line in the South-to-North Water Diversion Project, along with relevant PCCP pipeline data, the failure probability of the PCCP pipeline was calculated. The reliability index of this pipeline section under 0.6 MPa loading was found to be 4.49, demonstrating the reliability of the PCCP pipeline in this section of the water supply line.

1. Introduction

As composite structures, prestressed concrete cylinder pipes (PCCPs) exhibit notable benefits over conventional water supply pipes in terms of water resistance, pressure resistance, and corrosion resistance [1]. Since PCCP are strengthened with prestressed steel wires, they have strong compressive and bending strengths, which allow them to tolerate high internal pressures and external loads [2,3,4]. Owing to the increasing use of PCCPs, pipeline breakdowns have become a major problem that can impact urban water supplies and safe use, possibly resulting in fatalities and financial losses. Damage to PCCP presents issues to the structural integrity and longevity of engineering projects, leading to immediate financial losses and increasing long-term expenses. It also induces ecological harm and pollution threats, reducing operational efficiency and elevating safety hazards [5,6,7]. Thus, increasing attention to pipeline safety and dependability, establishing efficient safeguards to stop such failures from recurring, and guaranteeing the security of lives and property are essential. To reduce the risk of pipeline accidents, sophisticated detection systems should be implemented. Routine pipeline inspections, maintenance, and upgrades should be performed. A strong management structure should be set up [8,9,10,11,12].

Scholars worldwide have conducted much research to address problems pertaining to the use of PCCPs in engineering projects. By reducing the likelihood of malfunctions and mishaps during operation, these studies seek to improve the safety and dependability of PCCPs. Fang [13] proposed an innovative reinforcement technology of EPS + CFRP liner, established and assessed by the experimental results, a 3D FE model, analysed the mechanical properties of the PCCP under three conditions. Fu [14] proposed a new method for PCCP in-service repair by external bonding with CFRP layers. A three-dimensional numerical model of the repair of PCCPs with broken wires using externally bonded CFRP layers was built using the finite element software package ABAQUS (Version 6.5). The experimental results agreed well with the simulated results. Hu [15] conducted an internal pressure test of a full-scale PCCP model, measured the strain response of different layer structures of PCCP under internal working pressures with a broken wire percentage of 0% (intact), 5%, and 10% and to obtain the load response of the pipe with broken wires during the pressurisation process. A three-dimensional numerical model of a PCCP with broken wires was established based on the actual structural parameters of the test pipe and accounting for the bond stress of the protective mortar coating.

The circumferential breakdown of PCCP pipes occurs gradually. Monitoring the operational condition of PCCP pipes and swiftly identifying current and potential concerns is essential for guaranteeing their safe operation. Wang [16] presented an experimental study to identify the number of wire breaks in a PCCP using piezoelectric sensing technology. Wire breaks were simulated on a large-diameter embedded DN4000 PCCP (Beijing HanjianHeshan Pipeline Co., Ltd., Beijing, China) and a small-diameter lined DN700 PCCP (Beijing HanjianHeshan Pipeline Co., Ltd., Beijing, China), with a testing device designed to realise the identification method using body stress waves as a medium. Based on the analysis of the output voltage, two types of damage indices were adopted to evaluate the wire break damage. Hassi [17] conducted an experimental study to improve the durability of PCCP mortar coatings exposed to chemical attacks. Electrochemical impedance spectroscopy (EIS) was used to identify the changes in the microstructure of the mortar coatings and the reinforcement bars. Dong’s electrochemical equivalent model was implemented and fitted to the experimental results. Wang [18] developed a Rayleigh wave-based monitoring method for cracks on the mortar coating of PCCP based on the working environment. The optimal arrangement of the piezoelectric lead zirconate titanate patches on the mortar coating was determined by stress analysis under internal pressure, with the feasibility of the technique confirmed by comparing the experimental observations with the results of the voltage output analysis. Goldaran [19] employed the acoustic emission (AE) technique to detect corrosion in PCCP. To achieve this object, three experimental specimens were made in the laboratory of Middle East Technical University. It was observed that significant changes in some captured AE parameters occur as the pipe pressure exceeds the previous level at which the condition assessment is made feasible. Zhao [20] established a high-precision finite element model of buried PCCP with prestressed steel wire corrosion and conducted a bearing test based on this model. The results showed that the corrosion of prestressed steel wires has the greatest impact on the mortar protective layer and outer core concrete, and the corrosion point at the waist of the pipe is the most detrimental to the pipeline. Hu [21] proposed a novel corrosion monitoring method for PCCP spigots that combines the Tafel extrapolation and surface acoustic wave (SAW) methods. Tafel extrapolation and the SAW method were used to determine the self-corrosion rates and output voltage–time curves for a single steel plate and PCCP spigot at different corrosion rates.

To fix damaged PCCPs with broken wires, Zhai [14] suggested a novel retrofitting method that uses externally bonded prestressed carbon fibre reinforced polymer (CFRP). After the experimental measurements are successfully reproduced, the impacts of the wire breakage ratio, number of CFRP layers, and prestress level are simulated using the numerical model. Assuming a linearly distributed pattern of bond stress, Guo [22] proposed a theoretical model for determining the stress distribution of steel wires. By comparing the shrinkage values from an experiment on a PCCP with those of broken wires, the theoretical model is assessed. A PCCP with an inner diameter of 1400 mm and a length of 6000 mm was the subject of a full-scale experiment by Dong [23], who examined the effects of various recovery stresses, Fe-SMA bar diameters, and material types on the strengthening effect. A chloride diffusion model appropriate for PCCPs, Hu [24] developed a chloride diffusion model appropriate for PCCPs. Using the RCM test and the chloride diffusion simulation test, the nonsteady-state chloride diffusion coefficient and chloride concentration of the steam-cured concrete were determined. Ji [25] suggested replacing conventional prestressed steel wires in PCCP manufacturing with prestressed basalt fibre-reinforced polymer (BFRP) rods to overcome the problem of wire breakage caused by corrosion and hydrogen embrittlement. The ultimate bending strength, hoop prestressing, minimal reinforcement area, and cost effectiveness of BFRP reinforcement were examined using theoretical computation techniques. By combining Inception, ResNet, and LSTM networks, Zhang [26] proposed a PCCP-E structural deformation prediction model, or Inception-ResNet-LSTM. The model was then used to predict the structural deformation of each layer of the PCCP-E structure using prototype experiments and comparisons with other DL and ML algorithms on the same dataset. The model’s superiority and efficacy are fully illustrated in real-world applications.

A reasonable understanding of the potential failure hazards of PCCP pipelines is essential when performing a risk analysis. We used pertinent knowledge to perform a thorough analysis of the failure probability of PCCP pipelines based on this understanding. This is crucial for maintaining the regular operation of pipelines and putting into practice efficient hazard mitigation and reinforcing techniques. In this work, we used the prestressing steel wire strength and the concrete strength of the pipe core as the primary indicators for determining the ultimate load-bearing capacity of PCCP pipelines. We conducted a thorough evaluation of the load-bearing capacity of the pipelines by examining how these two strengths were distributed. We integrated cloud theory with more suitable distribution models to perform more accurate risk analysis. Furthermore, we conducted a thorough and methodical investigation of the failure hazards of PCCP pipelines by cutting-edge techniques, including cloud theory and Monte Carlo simulation. By using these techniques, the likelihood of pipeline failure can be more accurately predicted and evaluated, providing a scientific foundation for pipeline operation and maintenance management. With its ability to quantify and integrate uncertainty, dynamically model complex risk factor interdependencies, integrate intelligent algorithmic optimisation, advance standardised software applications, and achieve cross-domain technological convergence, cloud theory exemplifies innovation in PCCP risk analysis. Cloud models are more in line with the fuzziness, randomness, and multi-factor coupling features that are intrinsic to engineering practice than conventional techniques like probabilistic statistics and the Analytic Hierarchy Process. They greatly improve the accuracy and usefulness of risk prediction by offering a dynamic, scientific, and visual decision support tool for the whole lifecycle risk assessment of PCCP pipelines.

2. Methods

2.1. Cloud Theory

Renowned Chinese Professor Li Deyi created a novel artificial intelligence model called the Cloud Model [27] by fusing fuzzy mathematics and probability theory. With its primary goal of accurately converting qualitative data into quantitative data, this approach is praised as a paradigm of quantitative interchange. The Cloud Model’s distinctive characteristic is its combination of fuzzy mathematics’ advantage in handling uncertainty with probability theory’s capacity to handle certainty, creating a more complete and potent instrument for dealing with uncertainty problems. The goal of the Cloud Model is to handle and model this type of uncertainty efficiently. The ability to convert qualitative information into quantitative information is one of its design goals. Artificial intelligence systems can better comprehend and handle ambiguous and unclear information because of this feature, which increases the system’s ability to adapt to the complexity of the real world.

The normal cloud model is typically utilised to address issues in practical applications since it is generally accepted that the normal distribution meets most events. Ex (expected value), En (entropy), and He (hyperentropy) are the three numerical properties of clouds, or D (Ex, En, He). The cloud’s shape is described by hyperentropy, the entropy value represents the uncertainty of the information, and the expectation value represents the centre or typical value of anything. Cloud computing, cloud reasoning, cloud clustering, and other techniques can be used to construct specific cloud models [28]. Finally, the cloud droplet process is generated using cloud digital elements, turning the abstract cloud model into a tangible model. The concrete relationship can be defined as follows using the analysis:

$$\mu \left(x\right)=e^{{\displaystyle \frac{{\left(x-Ex\right)}^2}{{2E}^{`2}}}}$$

(1)

The following are the ways that each component of the cloud model adds to the qualitative ideas:

$$∆C\approx \mu A\left(x\right)\times {∆}_x/\left(\sqrt{2\pi }En\right)$$

(2)

The analysis of the cloud model leads to the following expression for the overall contribution of each component to the qualitative concepts:

$$C={\displaystyle \frac{{\int }_{-\infty }^{+\infty } {\mu }_A\left(x\right)dx}{\sqrt{2\pi }En}}={\displaystyle \frac{{\int }_{-\infty }^{+\infty } e^{-(x-Ex{)}^2/\left(2{Ex}^2\right)}dx}{\sqrt{2\pi }En}}=1$$

(3)

An examination of the cloud model reveals that the following is how all its components contribute to the qualitative concepts:

$$C_{Ex\pm 3En}={\displaystyle \frac{1}{\sqrt{2\pi }En}}{\int }_{Ex-3En}^{Ex+3En} {\mu }_A\left(x\right)dx=99.74\%$$

(4)

Approximately 33.33% of the cloud model sample clouds fall between [Ex − 2En, Ex + 2En] and [Ex − 3En, Ex + 3En]. The cloud model sample points make up approximately 4.30% of the qualitative concepts. The performance metrics are displayed in Figure 1 and

Table 1. Cloud model.

Interval	Contribution	Percentage	Importance
[Ex − 0.67En, Ex + 0.67En]	50%	22.33%	Backbone element
[Ex − En, Ex + En]	68.26%	33.33%	Basic Elements
[Ex − 2En, Ex − En], [Ex + En, Ex + 2En]	27%	33.33%	Peripheral elements
[Ex − 3En, Ex − 2En], [Ex + 2En, Ex + 3En]	4.30%	33.33%	Weak peripheral elements

:

Cloud theory is generally applied using the normal cloud algorithm, which can be divided into forwards clouds and inverse clouds in Figure 2 and Figure 3, and the formula can be expressed as:

$$\mu =e^{-{\displaystyle \frac{(x-Ex{)}^2}{2(En{)}^2}}}$$

(5)

2.2. The Monte Carlo Method

An analytical methodology that blends statistical experimental methods with random sampling techniques is the Monte Carlo method [29]. There are many uses for the Monte Carlo approach, especially in domains such as engineering risk assessment. For example, evaluating the dependability of engineering machinery is frequently required in engineering design to ascertain whether it can function normally for the duration of its design. In these situations, the chance of machinery failure can be ascertained using reliability analysis using the Monte Carlo approach. Complex problems can be handled via the Monte Carlo approach, which yields accurate findings. This is because the Monte Carlo approach can generate enough random samples of random variables to minimise sampling error by performing a high number of random samples.

Assuming that the probability density functions are $f_{x1}, f_{x1}, {\dots , f}_{x1}$… $f_{xn}$, respectively, where X₁, X₂, X₃…, X_n are independent statistical random variables, the function is as follows:

$$\mathrm{Z} = \mathrm{g} \left(x_1, x_2, x_3\dots x_n\right)$$

(6)

By randomly sampling each variable $x_1, x_2, x_3\dots x_n$ the quantile values of each variable are substituted into the functional function $Z_i$ = g $\left(x_1, x_2, x_3\dots x_n\right)$ toobtain the $Z_i$ value. Assume that N samples are taken and that N. values are obtained if the number of values is sufficient. $Z_i$ ≤ 0 is L, and if N is sufficiently large, then the structural failure probability can be calculated using the following formula:

$$P_f={\displaystyle \frac{L}{N}}$$

(7)

If the permissible error is set to $\mathsf{\epsilon }$, the expression at a confidence level of 95% is as follows:

$$\epsilon =\sqrt{{\displaystyle \frac{2\left(1-P_f\right)}{N\times P_f}}}$$

(8)

To ensure the accuracy of the permissible error value, the value of N must satisfy the following formula:

$$N\ge {\displaystyle \frac{100}{P_f}}$$

(9)

We test and demonstrate that random parameters statistically follow a cloud distribution by combining cloud theory with the Monte Carlo method. Based on the parameters, we use a cloud generator to create N cloud droplets, and for computation, we substitute a cloud distribution diagram for the distribution histogram. Cloud distributions provide more accurate and adaptable simulation results than traditional distribution techniques do, especially when data or parameter distributions are complex or ambiguous. They are better able to handle different complexities or uncertainties. When paired with cloud theory, the Monte Carlo method is more accurate and flexible in managing uncertainties, better captures the uncertainty of variables or parameters, and yields better simulation results and decision support.

2.3. Materials and Experiment

2.3.1. Concrete Compressive Strength Test

A YAW-5000 microcomputer-controlled electrohydraulic servo pressure testing machine is used to measure the compressive strength of the concrete. The test is negative for compression and positive for tension. The experiment uses 42.5 grade ordinary silicate cement of the Tianrui brand as the cement, and washed river sand is chosen as the fine sand. The specification stipulates that the maximum coarse aggregate particle size in concrete cannot exceed 31.5 mm and cannot exceed 2/5 of the thickness of the concrete layer. After sieving, 5–20 mm stones were chosen as the coarse aggregate. The test is carried out on both saturated concrete and the mixing process. The tap water from Zhengzhou city is utilised for the saturation and mixing processes.

Table 2. Concrete mix ratio design Unit: KG/m^3.

Cement	Sand	Stone	Water	Water-Cementratio
490	708	1155	147	0.3

displays the C55 concrete mix ratio that was utilised in the test, which was carried out in accordance with the hydraulic concrete test specification SL/T 352-2020 [30].

In accordance with the procedure outlined in the hydraulic concrete test specification SL/T 352-2020, the test samples measured 100 mm × 100 mm × 100 mm. The aggregates larger than 20 mm were removed using the flat sieve method, and the aggregates smaller than 5 mm were screened via the vibrating table screening method. After screening for aggregates with a particle size of less than 5 mm, the materials were combined with a concrete mixer and placed into vibration-moulding moulds. The test blocks were left to stand for two days at a temperature of 20 °C ± 5 °C after production was finished. Once the samples were allowed to stand, they were demoulded, numbered, and placed in the curing room for curing. The equivalent test was conducted following 28 days of curing.

After the samples were removed from the curing area, a damp cloth was placed over them. The test apparatus was cleaned, the sample was wiped, its appearance was examined, and pressure was applied to the top and bottom sides of the sample to guarantee a smooth and level surface. This will lessen the impact of stress concentration in the compressive strength test of the concrete test block. Prior to preloading, the test block was positioned in the middle of the lower pressure plate. Five kN is the preloading load, and 5 mm/min should be the loading rate. Once the preloading is finished, official loading begins. The formal loading rate should be set at 20 MPa/min, and the testing apparatus loads the sample consistently and continuously until it is damaged to provide accurate and trustworthy test findings. To determine the compressive strength of a sample, the load at the moment of destruction must be noted during the loading procedure. The following formula is used to determine the compressive strength of a cubic concrete sample:

$$f_{cc}={\displaystyle \frac{P}{A}}\times 1000$$

(10)

where A represents the specimen’s compressive area (mm²), P represents the destructive load (KN), and $f_{cc}$ represents the concrete’s compressive strength (MPa).

2.3.2. Stress Test of Prestressed Steel Wires

Following the completion of the pipeline manufacturing distribution, the actual prestressing steel wire of the PCCP pipeline is measured using the following test. This is accomplished by adding prestressing steel wire to the pipeline to increase its compressive capacity. Typically, tension is applied to the steel wire to prestress it. The experimental test materials and equipment are presented in

Table 3. Experimental materials and equipment table.

Name	Quantity	Name	Quantity
PCCPDN 3200 × 5000 pipe core	25	DH3816N Static stress Tester (16 channels)	1
Strain gauge	Several	Three-core shielded wire	Several
Soldering iron	1	Solder wire	Several
Silicone	Several	502 glue	Several

.

The monitoring arrangement on the pipe core considers comprehensiveness and symmetry by using the technique of directly pasting strain gauges on steel wires and adding sleeves at both ends of the pigtail of each section of the pipe core for protection. Four monitoring sections are laid out on the pipe core, each 90 degrees apart. Two strain gauges and a fibre grating temperature sensor are placed on each section, and the sensors of each longitudinal section are connected by a single-core armoured fibre optic cable that is led out of the pipe’s end. The longitudinal and circumferential monitoring sections are positioned in the centre of the core and on both sides of the core, considering the variation in stress on the outer surface of the concrete core along the pipe’s length. The DH3816N static stress test system gathers the data, and the strain gauge configuration is numbered in accordance with certain guidelines, as illustrated in Figure 4.

3. Results

3.1. Data Analysis

The 25 sets of data obtained from the test are shown in

Table 4. Data table of reference samples of the ultimate bearing capacity of PCCP pipes.

Serial Number	Compressive Strength of Concrete Core/MPa	Measured Prestressing Force of Prestressing Steel Wire/MPa
1	62.93	1528.12
2	64.19	1547.98
3	63.97	1520.28
4	67.1	1613.07
5	64.66	1611.29
6	65.1	1606.22
7	61.99	1589.83
8	65.67	1718.46
9	59.09	1565.79
10	63.1	1608.26
11	68.93	1526.73
12	62.01	1578.31
13	64.88	1638.87
14	64.52	1399.55
15	60.04	1479.35
16	63.91	1742.88
17	65.69	1640.43
18	65.4	1598.91
19	66.35	1604.49
20	64.64	1865.33
21	64.79	1443.40
22	65.16	1536.30
23	64.02	1531.40
24	64.76	1746.90
25	65.68	1574.50

:

The bar plot is plotted as follows, using the data number from Table 4 as the horizontal coordinate, the measured prestressing force of the prestressing wire as the vertical coordinate, and the compressive strength of the pipe core’s concrete as the horizontal coordinate.

Figure 5 and Figure 6 show that in the 25 test groups, the compressive strength of the concrete ranged from 59.09 MPa to 68.93 MPa, with an average strength of 64.34 MPa and a median of 64.66 MPa. With an average applied prestressing force of 1597.44 MPa and a median of 1589.33 MPa, the actual prestressing force applied by the prestressing steel wires varied from 1399.54 MPa to 1865.33 MPa.

The distribution histogram can be used to provide a more intuitive understanding of the distribution of the data. The distribution histogram of the above experimental data is shown in Figure 7 and Figure 8.

Figure 7 and Figure 8 show that the prestressing steel wire data are concentrated primarily in the interval [1500, 1800], whereas the concrete compressive strength data are concentrated primarily in the interval [62, 66]. The data distribution is more uniform and symmetrical, and the distributions of the core concrete compressive strength and prestressing steel wire measured prestressing data follow a normal distribution. The core concrete compressive strength data are distributed primarily in the middle of the upper number, whereas the measured prestressing data are distributed primarily in the middle of the lower number. The majority of the data in each figure is distributed near the middle, and the two ends contain fewer data.

A study of the histogram suggests that the data adheres to a normal distribution. A P-P plot test was performed on the experimental data to further validate the distribution of the PCCP ultimate bearing capacity index.

The principle of P-P plot analysis states that if the distribution function of the population X is F(x), then X₁, X₂, …, X_n represent simple random samples from population X, where X₁, X₂, …, X_n are independent and identically distributed, sharing the same distribution as population X. Their order statistics are X(1) ≤ X(2) … ≤ X(n), and the empirical distribution function of the population X is defined as

$$F_n(\mathrm{x})=\left\{\begin{array}{c}0,x<x_{\left(1\right)}\\ {\displaystyle \frac{1}{n}}, x_{\left(1\right)}\le x<x\\ \dots \dots \dots \dots \\ {\displaystyle \frac{k}{n}},x(k)<x<x(k+1)\\ \dots \dots \dots \dots \\ 1,x\ge x\left(n\right)\end{array}\right.$$

(11)

$$F_n={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n I\left\{x_i\right\}$$

(12)

$$x=F^{-1}\left(\mathrm{y}\right)=\mathrm{i}\mathrm{n}\mathrm{f}\{x:F(\mathrm{x})>y\}$$

(13)

$$\left\{F_n\left(X_{\left(\mathrm{i}\right)}\right),F\left(X_{\left(\mathrm{i}\right)}\right)\right\}=\left(\mathrm{i}/\mathrm{n},F\left(X_{\left(\mathrm{i}\right)}\right)\right),\mathrm{i}=\mathrm{1,2},\dots n$$

(14)

The test results are shown in the Figure 9 and Figure 10:

The P-P plot test indicates that if the scatter points align with the line y = x in a Cartesian coordinate system, the variables can be deemed to adhere to a normal distribution. The Figure 9 and Figure 10 illustrates that the steel wire prestressing closely aligns with the line y = x, although the concrete compressive strength exhibits a minor deviation from this line; however, the discrepancy is negligible, and the general trend remains linear. The ultimate bearing capacity indicators of PCCP pipes exhibit a regular distribution.

3.2. Cloud Distribution Test

In practical applications, the influence of multiple factors on the results at the same time should be fully considered rather than relying on a single factor of the normal distribution analysis. This occurs because both the concrete compressive strength and the prestressing steel wire applied obey a normal distribution. Thus, cloud dispersal can be examined.

The curve that represents the shape of a normal cloud is called its expectation curve, and it has unique characteristics. The variance of the random variable X made up of cloud droplets is D(X) = En² + He², whereas the expectation is E(X) = Ex based on the mathematical characteristics of the typical cloud.

Table 5. Distribution verification summary.

Indicators of Load-Bearing Capacity	Probability	Expectation	Entropy	Hyperentropy
Concrete strength	Normal cloud	64.3432	1.823	0.9703
Prestressing	Normal cloud	15.8929	0.7264	0.2787

displays the three numerical properties that are obtained from the uncertainty-free inverse cloud generator and correlate to the applied prestress and the compressive strength of the concrete.

As shown in Figure 11 and Figure 12, 1000 cloud droplets were successfully simulated and generated through multiple cycles of the forward and reverse cloud generators, yielding the concrete strength cloud distribution map and the prestressing wire applied prestress cloud distribution map, respectively.

The “3D” concept of cloud theory is satisfied by Figure 11 and Figure 12, which shows that the majority of the cloud droplets produced by the characteristic numbers of the two indicators are dispersed within [62, 66] and [1500, 1700]. In other words, both indicators follow the typical cloud distribution. The cloud theory-based research approach is useful for simulating the distribution of small sample data. Compared with the conventional histogram of the distribution and its fitted curves, the generalised normal distribution of the cloud manages the uncertainty more precisely. The cloud theory-based research approach has several benefits in the uncertainty analysis process and is better able to capture the discrete nature of finite sample data.

Concrete compressive strength tests and prestressing steel wire evaluations were performed to analyse the distribution of the core concrete’s compressive strength within PCCP pipes and the actual prestress exerted on the prestressing steel wires. The analysis utilised graphical approaches, P-P tests, and cloud distribution tests to directly assess the data distributions. Results demonstrate that the compressive strength of PCCP pipe core concrete and the actual prestress exerted on prestressing wires conform to normal and normal cloud distributions, respectively. Analysis of various testing methodologies indicates that cloud distribution diagrams for the ultimate load-bearing capacity indicators of PCCP pipes, grounded in cloud theory, exhibit enhanced efficacy in mitigating uncertainty concerns. The research methodology of cloud theory facilitates more precise estimation of probability distributions from limited sample data, notably excelling in investigations that involve uncertainty. Cloud theory provides more flexibility, effectively addressing the many variables and variances inherent in practical engineering applications.

3.3. PCCP Pipeline Damage Risk Analysis Based on Cloud Theory

3.3.1. Establishment of the PCCP Pipeline Instability Risk Analysis Model

The primary determinants of the stable operation of a PCCP pipeline are its bearing load L and final bearing capacity R. The PCCP pipeline breaks when L > R, which interferes with the regular operation of the pipeline. On this basis, a mathematical model for PCCP pipeline instability risk analysis can be created.

The function will be:

$$g\left(·\right)=g\left(L,R\right)=R-L$$

(15)

$$P_f=P(L>R)={\iint }_{g(·)<0} f_{R,L}(r,l)drdl$$

(16)

where $f_{R,L}(r,l)$ represents the joint probability density distribution function of the PCCP under load L and ultimate bearing capacity R. Due to its complexity, this equation is converted into the following equation throughout the computation process to make it simpler:

$$P_f={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} {\int }_0^l f_R(r)dr{f}_L(l)dl={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} F_R(l)f_{L}dl$$

(17)

It can be easier to calculate when the value is indirectly estimated using a probability combination. The joint probability density function of the PCCP pipeline’s ultimate carrying capacity R is as follows:

$$f_{AB}(\mathrm{a},\mathrm{b})=f(\mathrm{a}/\mathrm{b})$$

(18)

where $f(\mathrm{a}/\mathrm{b}$) represents the conditional probability density function of the pipe ultimate load-carrying capacity for the given conditions.

$${ P}_f={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} {\int }_0^l f_R(r)dr{f}_L(l)dl={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} F_R(l)f_{L}dl$$

(19)

The two parameters are subjected to the Monte Carlo method, in which the cloud distribution graphs of the two parameters are used instead of the distribution histograms in the calculation process of the Monte Carlo method. The random number of the normal cloud distribution can be obtained according to uncertain cloud theory:

$$\stackrel{`}{E}x(l)={\displaystyle \frac{{\sum }_{i=1}^n \stackrel{`}{f}\left(l_i\right)l_i}{{\sum }_{i=1}^n \stackrel{`}{f}\left(l_i\right)}}$$

(20)

$$\stackrel{`}{E}n(l)=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n \stackrel{`}{f}\left(l_i\right)\left(l_i-\stackrel{`}{E}x\right)}{{\sum }_{i=1}^n \stackrel{`}{f}\left(l_i\right)}}}$$

(21)

$$\stackrel{`}{H}e(l)=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n {\left[\stackrel{`}{f}\left(En\left(l_i\right)\right)\left(En\left(l_i\right)-\stackrel{`}{E}n\left(En\left(l_i\right)\right)\right)\right]}^2}{{\sum }_{i=1}^n \stackrel{`}{f}\left(En\left(l_i\right)\right)}}}$$

(22)

3.3.2. Example Analysis of the River Diversion Project and the Huaihuai Project

An essential component of the northwards river water transmission system is the Henan portion of the Diversion River Jihuai Project, which uses the Xi’an and Qingshui rivers to transport water to Henan. A vast water conveyance network is formed by the connection of the Xi’an and Qingshui rivers’ water conveyance channels along the border between the provinces of Henan and Anhui. The Qingshui River is chosen as the primary water transmission channel in the Henan portion of the project to meet the water transmission goal. The Qingshui River is 47.46 km long, whereas the Luxin Canal is 16.26 km long, making the transmission line’s total length in the preliminary design stage 195.14 km. The 61.62 km long Qiliqiao Storage ReservoirXiayi transmission line is part of the Henan section’s transmission line.

Table 6. Specifications of the PCCP pipes.

Pipe Inner Diameter	Pipe Length	Pipe Core Concrete Thickness	Thickness of The Steel Drum	Diameter of Prestressing Steel Wire	Number of Layers	Pitch	Mortar Thickness	Burying Depth
3200 mm	5000 mm	345 mm	1.5 mm	7 mm	1	15.5 mm	25 mm	5000 mm

displays the PCCP pipeline specifications used in this transmission line. With a design flow rate of up to 13.8 m³/s, a significant amount of water resources can be transmitted efficiently. The average annual water allocation for the short-term planning level year of Henan Province in 2030 will reach 500 million m³, and for the long-term planning level year of 2040, it will reach 634 million m³ due to the rise in social development and water consumption. This offers trustworthy assurance for Henan’s water supply and demonstrates the project’s long-term planning and thorough assessment of future water demand. Through a clever water transmission system, the Henan portion of the Diversion River Jihuai Project brings river water northwards into Henan, providing the area with a plentiful supply of water.

Through multiple cycles of the forward and reverse cloud generators, 1000 cloud droplets were successfully simulated to collect the compressive strength of the concrete core and the prestressing force of the steel wire in this section of the PCCP pipeline. The resultant cloud distributions of the concrete strength and the prestressing force applied by the prestressing steel wire are displayed in Figure 13 and Figure 14.

The compressive strength of the concrete in the pipe core ranges from 55.41 MPa to 82.36 MPa, with an average of 64.35 MPa, and it ranges primarily between 60 and 65 MPa. With an average value of 1598.58 MPa, the actual applied prestressing force of the prestressing steel wires varied from 1295.57 MPa to 1934.18 MPa, ranging primarily between 1500 and 1700 MPa.

Table 7. Summary of the distribution test results.

Load-Bearing Capacity Index	Probability	Expectation	Entropy	Hyperentropy
Concrete strength	Normal cloud	64.3432	1.823	0.9703
Applied prestressing	Normal cloud	15.8929	0.7264	0.2787

displays the three numerical features that the inverse cloud generator with uncertainty produces in relation to the applied prestressing force and the compressive strength of the concrete.

Figure 15 displays the results of entering the cloud model characteristic figures, along with two indicators regarding the pipeline’s final carrying capacity into the mathematical model for Monte Carlo technique computations.

The data from Figure 15 are dispersed between the range of 0.73–0.77 MPa, whereas the pipeline’s estimated ultimate bearing capacity ranges from 0.59 to 0.93 MPa, with an average value of 0.74 MPa. After the computation findings were examined, the likelihood that this pipe segment would fail when subjected to a 0.6 MPa force was 0.000004.

Following the Unified Standard for Structural Reliability Design of Water Conservancy and Hydropower Projects (GB50199-2013) [31],

Table 8. Correspondence between failure probability and reliability indices.

Probability of Failure	Reliability Index
1.59 × 10−1	1.00
5.05 × 10−2	1.64
2.27 × 10−2	2.00
1.35 × 10−3	3.00
1.04 × 10−4	3.71
3.17 × 10−5	4.00
3.40 × 10−6	4.50

illustrates the corresponding relationship between the reliability index and the probability of failure. The dependability index for this PCCP pipeline segment is 4.49, according to the comparison table.

Analysis of the computation findings revealed that the compressive strength of the concrete in the core of the pipe is dispersed primarily between 60 and 65 MPa. The range of 1500 MPa to 1700 MPa is where the majority of the actual prestressing force supplied by the prestressing steel wire is distributed. The range of 0.73–0.77 MPa is where the final bearing capacity of the pipe is mostly dispersed. The reliability of the pipe’s ultimate bearing capacity can be guaranteed despite the relatively small distribution interval of the prestressing steel wire data in this calculation because the mathematical model shows that the concrete strength of the pipe core has a greater impact on the pipe’s ultimate bearing capacity than does the prestressing steel wire.

4. Discussion

The study examines the uncertainties associated with the failure process of PCCP pipelines and presents the principles and concepts of cloud theory. It offers a comprehensive elucidation of the foundational principles in risk analysis and presents frequently employed methodologies. This study integrates cloud theory into the analysis of uncertainties in the failure process of PCCP pipelines, providing practical tools and guidance for thorough and dependable risk assessments in PCCP pipeline engineering, thus improving the safety and reliability of pipeline engineering.

Normal distribution tests were performed on the compressive strength of the concrete core of PCCP pipes and the actual prestress exerted by the prestressing wires, utilising both graphical and computational techniques. An inverse cloud technique incorporating uncertainty was employed to ascertain the characteristic values of the compressive strength of the concrete core in PCCP pipes and the actual prestress exerted by the prestressing wires, resulting in the creation of cloud diagrams. The findings demonstrate that the compressive strength of the concrete core in PCCP pipes and the actual prestress exerted by the prestressing wires conform to a normal cloud distribution. The cloud distribution diagram of the ultimate bearing capacity index of PCCP pipes, grounded in cloud theory, exhibits superior efficacy in mitigating uncertainty challenges. The study methodologies of cloud theory can more precisely deduce the probability distribution of constrained sample data, especially in investigations concerning uncertainty.

The present research predominantly utilises cloud theory to assess the damage risk of PCCP pipelines, employing measured prestressing of the core concrete and prestressed steel wires as critical influencing elements, while the internal pressure load of the pipeline acts as the dependability indicator. Nevertheless, due to constraints in knowledge repositories and data investigation conditions, specific deficiencies persist that necessitate additional research and enhancement:

The potential damage risk of PCCP pipes has not accounted for the influence of external loads on the pipeline. To improve the precision of forthcoming studies on PCCP pipeline damage risk, the impact of external pipeline loads must be integrated into the existing framework.

This study exclusively examines the effects of predictable loads on the pipeline. Nevertheless, unforeseen causes like hydrogen embrittlement may still undermine the pipeline’s load-bearing capacity during operation. Future research should examine the impact of these factors on the risk of damage to PCCP pipelines.

5. Conclusions

The strength of the concrete and prestressed steel wires in the PCCP pipeline core is considered a key indicator of the pipeline’s ultimate bearing capacity. Its distribution is also investigated, and a more appropriate distribution model is chosen for risk analysis in conjunction with cloud theory and the Monte Carlo method, which are related to pipeline failure risk. The conclusions are as follows.

The study selects the compressive strength of the concrete pipe core and the measured prestressing force of the prestressing steel wire as the primary factors. The risk analysis model of PCCP pipeline instability is created by combining the Monte Carlo method and analysing the cloud distributions of the two risk factors. The influences of ambiguity and randomness in the risk analysis of the compressive strength of the concrete core of a PCCP pipeline and the actual prestressing force of the prestressing steel wire are successfully resolved by introducing the idea of cloud theory. The cloud distribution is more realistic than the conventional normal distribution. The risk of the PCCP pipeline in the Henan section of the Jiangji-Huai project is assessed using the cloud theory in conjunction with the Monte Carlo method, and the reliability index of this pipeline section under 0.6 MPa loading is found to be 4.49 based on the collection of internal water pressure monitoring data of the pipeline during the operation of the water transmission line from the Qiliqiao Storage Reservoir to Xiayi.

The utilisation of cloud theory in forecasting PCCP risks offers the industry, policymakers, and practitioners accurate risk quantification tools and sophisticated decision support, while also advancing the PCCP sector’s evolution from ‘reactive response’ to ‘proactive prevention’ and transitioning from ‘experience-driven’ to ‘data-driven’ operations. As technology matures and regulatory support persists, Cloud Theory is set to be the primary catalyst for digital transformation in the PCCP sector. It will advance the sector towards safer, more efficient, and environmentally sustainable development, ultimately transitioning from ‘risk prediction’ to ‘value generation’.

The implementation of cloud theory in forecasting PCCP threats will evolve from discrete technological advancements to comprehensive solutions. By utilising intelligent, real-time, and standardised risk management systems, the safety, durability, and sustainability of PCCP pipelines will be improved, offering strong protections for worldwide water conveyance and urban infrastructure development. With continuous technical advancement and consistent legislative support, Cloud Theory is set to be the primary catalyst for digital transformation in the PCCP sector, driving the industry towards improved efficiency, heightened safety, and increased environmental sustainability.