## 1 Introduction
### Point Estimation Method for the First Four Moments of a Random Variable  
This study utilizes the estimation points obtained by [PERSON] and [PERSON] [37] in standard normal space. This method easily transforms any random variable into a standard normal random variable through Rosenblatt transformation. For random variables \(X_{i}\) (\(i=1\), \(2\), \(\ldots\), \(m\)), using Rosenblatt transformation, Equations (2)-(4) can be written as follows [37,38]:  
\[\mu_{X}=\sum_{i=1}^{m}P_{i}T^{-1}(u_{i}) \tag{5}\]  
\[\sigma_{X}^{2}=\sum_{i=1}^{m}P_{i}\Big{[}T^{-1}(u_{i})-\mu_{X}\Big{]}^{2} \tag{6}\]  
\[\sigma_{X}^{k}\alpha_{kX}=\sum_{i=1}^{m}P_{i}\Big{[}T^{-1}(u_{i})-\mu_{X} \Big{]}^{k},k=3,4 \tag{7}\]  
where \(T^{-1}(u)\) is the Rosenblatt transformation; \(u_{i}\) is the \(i\) point estimate of the standard normal space random variable, and \(P_{i}\) is its corresponding weight. The estimated point values and weights of the 7 point estimates are presented in Table 2.  
Figure 3: Overall X-direction axial stress diagram.  
Estimation of Statistical Moments of Random Variables Based on the Point Estimation-Finite Element Method  
To calculate the stress of a reinforced concrete aqueduct, a three-dimensional finite element model of the aqueduct is employed. Considering the high accuracy of the point estimation method for the probability integral of the estimated variable function, a method that combines point estimation with the finite element method to compute the first four moments of random variables is proposed. The steps of the proposed algorithm are detailed below, and a corresponding flowchart is illustrated in Figure 4.  
1. Establish a three-dimensional finite element numerical model of the reinforced concrete aqueduct.
2. Determine the distribution types of random variables and statistical parameters that affect the carbonation depth model of the concrete in the aqueduct structure.
3. Specify the number of point estimators m, the estimated points in the standard space, and their corresponding weights.
4. Substitute the estimated points of the load parameters into the finite element model of the aqueduct, conduct a stress analysis, and obtain the stress response values of the m aqueducs under the given load conditions.
5. Substitute the calculated stress response values of each group and their corresponding weights into Equations (5)-(7) to compute the first four moments of the random variables in the aqueduct concrete carbonation depth model.  
In this study, the structural assumptions for the aqueduct model include the consideration of three primary loads: self-weight, wind load, and crowd load. To account for the  
\begin{table}
\begin{tabular}{c c c} \hline  & **Estimated Point \(u_{i}\)** & **Weight \(p_{i}\)** \\ \hline \multirow{3}{*}{7 point estimates} & \(u_{0}=0\) & \(p_{0}=16/35\) \\  & \(u_{1+}=-u_{1-}=1.1544054\) & \(p_{1}=0.2401233\) \\  & \(u_{2+}=-u_{2-}=2.3667594\) & \(p_{2}=0.0307571\times 10^{-2}\) \\  & \(u_{3+}=-u_{3-}=3.7504397\) & \(p_{3}=5.48269\times 10^{-4}\) \\ \hline \end{tabular}
\end{table}
Table 2: Seven point estimates’ values and weights.  
Figure 4: Flowchart for calculating the first four moments of the random variable.  
uncertainty of these loads, it is assumed that these loads are random variables, each following a specific probability distribution. Furthermore, structural internal forces (e.g., stresses) are considered as random variables and the first four moments of the maximum structural internal forces, which are obtained through the finite element analysis model and point estimation method, reflecting the random response of the aqueduct under different loading conditions. Through these assumptions, the model is able to simulate the randomness of the loads and provide critical support for the reliability assessment of the aqueduct under carbonation effects.