# Investigating the Impact of Random Field Element Size on Soil Slope Reliability Analysis
###### Abstract  
The determination of the optimal random field element (RFE) size is crucial in soil slope reliability analysis as it governs the trade-off between precision in failure probability calculations and computational efficiency. Given the substantial computational burden associated with smaller RFE sizes, studies on their impact on slope failure probability are scarce. This research examines the influence of RFE size on failure probability and safety factor, employing the Karhunen-Loeve expansion to generate random fields and integrating the simplified Bishop method with particle swarm optimization (PSO) to assess slope stability. Through Monte Carlo Simulation (MCS), this study investigates the effects of the ratio of slope height to RFE size (\(H/D_{e}\)) on slope reliability metrics across two illustrative cases. Results reveal a notable influence of \(H/D_{e}\) on the distribution of safety factors (\(F_{s}\)) and failure probability (\(PF\)), with overestimation observed at smaller \(H/D_{e}\) ratios. When \(H/D_{e}\) exceeds 10 for Example 1 and 15 for Example 2, the \(F_{s}\) distribution patterns in both scenarios stabilize significantly, displaying minimal variability. The \(PF\) of Example 1 and Example 2 decreases with the increase of \(H/D_{e}\) and remains basically unchanged when \(H/D_{e}\) exceeds 10 and 15, respectively. Consequently, a recommended \(H/D_{e}\) ratio of 20 is proposed based on the analyzed cases, facilitating accurate calculations while mitigating computational overhead. s _Applied sciences_  
**Citation:** [PERSON]. one of the methodologies for generating random fields. This technique is widely utilized by numerous researchers [14; 15] due to its ability to achieve the same accuracy requirements with a lesser number of expansion terms. Over the past few years, increasing attention has been paid to the reliability analysis of slopes considering the uncertainties of soil parameters [15; 16; 17]. [PERSON] [14] presented the Random Limit Equilibrium Method (RLEM), an innovative approach that harmonizes random field theory with the well-established LEM. This integration results in a robust methodology particularly suited to assessing the reliability of soil slope designs. Then, the RLEM was used by various researchers to study the probability of the saturated slope [18], the slope considering the nonlinear failure criterion [16], the slope reinforced permeable polymer [19], and the three dimensional slope [15]. Taking into account the spatial variability of soil parameters, slope reliability analysis can become exceedingly time-consuming [20]. When employing RLEM for slope reliability analysis, a substantial number of numerical simulations must be conducted, with each simulation requiring the determination of the critical slip surface (CSS). Hence, one of the keys to increasing the efficiency of reliability analysis lies in effectively reducing the time needed to determine the CSS of the slope [21]. The Particle Swarm Optimization (PSO) method has been employed by numerous researchers for identifying the CSS of slope [22] owing to its advantages of simple operation, high computational efficiency, and robust stability [23; 24; 25]. When applying the RLEM for slope reliability analysis, the random field is discretized into elements of specific dimensions to characterize the spatial variability of soil. The random field element (RFE) size plays a crucial role in the slope reliability analysis [26]. A larger RFE size may inadequately describe the spatial variability of soil parameters, negatively impacting slope reliability analysis results. Conversely, a smaller RFE size can better describe this variability but may also introduce significant computational demands [26]. It is very important to determine the appropriate RFE size in slope reliability analysis, yet it remains a topic that has garnered limited attention from scholars. The primary objective of this study is to conduct a thorough analysis of the influence of RFE size on slope reliability.