Shaking Table Test-Based Verification of PDEM for Random Seismic Response of Anchored Rock Slopes

Abstract

This study systematically verified the applicability and accuracy of the Probability Density Evolution Method (PDEM) in the probabilistic modeling of the dynamic response of anchored rock slopes under random seismic action through large-scale shaking table model tests. Across 144 sets of non-stationary random ground motions and 7 sets of white noise excitations, key response data such as acceleration, displacement, and changes in anchor axial force were collected. The PDEM was used to model the instantaneous probability density function (PDF) and cumulative distribution function (CDF), which were then compared with the results of normal distribution, Gumbel distribution, and direct sample statistics from multiple dimensions. The results show that the PDEM does not require a preset distribution form and can accurately reproduce the non-Gaussian, multi-modal, and time evolution characteristics of the response; in the reliability assessment of peak responses, its prediction deviation is much smaller than that of traditional parametric models; the three-dimensional probability density evolution cloud map further reveals the law governing the entire process of the response PDF from “narrow and high” in the early stage of the earthquake, “wide and flat” in the main shock stage, to “re-convergence” after the earthquake. The study confirms that the PDEM has significant advantages and engineering application value in the analysis of random seismic responses and the dynamic reliability assessment of anchored slopes.

1. Introduction

Earthquakes are one of the main natural factors that induce slope instability [1]. In areas prone to strong earthquakes, anchoring and supporting systems often fail, leading to large-scale landslide disasters. For example, during the 2008 Wenchuan earthquake, a large number of rocky slopes and anchoring structures were severely damaged [2,3], which highlights the urgency of improving the accuracy of assessments of slope seismic performance. Traditional seismic analysis methods, such as the pseudo-static method [4], the permanent displacement method [5], or deterministic dynamic time history analysis [6], although widely used in engineering practice, often struggle to systematically quantify the inherent uncertainties in ground motions and geotechnical parameters, which can easily lead to misjudgments of structural responses and reliability.

In recent years, the randomness of ground motion and its impact on the dynamic response of slopes have gradually received more and more attention [7,8,9,10]. Studies have shown that ignoring the uncertainties in seismic input and material properties may significantly underestimate the failure risk of slopes under extreme earthquakes [11]. To counter this, researchers have attempted to introduce a stochastic analysis framework, using random variables or random fields to describe the variability of geotechnical parameters, and have conducted probabilistic stability assessments through methods such as Monte Carlo simulation and response surface methodology [12,13]. However, these methods are either limited by computational efficiency or rely on prior assumptions about the form of the response distribution (such as normal or log-normal distributions), and their applicability is challenged when dealing with strongly non-linear and non-stationary dynamic processes.

Against this backdrop, the Probability Density Evolution Method (PDEM) [14], a stochastic dynamic analysis tool that does not rely on specific distribution assumptions and can directly track the instantaneous probability structure of system responses, has shown good potential in structural engineering, underground engineering, and dynamic analysis of slopes. By establishing the evolution equation between physical quantities and basic random variables, PDEM can efficiently obtain the PDF (probability density function) and CDF (cumulative distribution function) at any time, thereby supporting refined time-varying reliability assessment [15]. Although this method has verified its accuracy and efficiency in numerical simulations [8,16,17], there is currently a lack of systematic verification based on physical model tests. In particular, its applicability in complex geotechnical systems such as multi-anchored rock slopes remain blank.

Shaking table model tests have been widely acknowledged as one of the most reliable methods for investigating the seismic performance of complex rock masses [18]. Researchers have previously successfully utilized this technique to reveal the deformation characteristics and failure modes of rock slopes under earthquake conditions, as well as to evaluate the effectiveness of various reinforcement measures [19,20,21,22]. Wartman et al. [23] specifically investigated the mechanism of the permanent displacement of clay slopes with four profile characteristics under seismic action through shaking table model tests, and concluded that the main modes of slope deformation are deep rotation or translational displacement, and the deformation is mainly concentrated in local shear zones. Lin and Wang [24] observed the amplification effect and non-linear characteristics of slope response under vibration conditions using shaking table model tests. These physical studies provide critical insights that are often difficult to capture through purely mathematical or numerical approaches, especially regarding non-linear soil–structure interactions.

In view of this, this paper constructs a scale model of high-strength rock slopes based on large-scale shaking table model tests. Under 144 sets of non-stationary random ground motions and 7 sets of white noise excitations, key response data such as acceleration, displacement, and anchor axial force are systematically collected. On this basis, the PDEM is used for probabilistic modeling of the test responses, and multi-dimensional comparisons are made with classical hypothetical models such as the normal distribution and Gumbel distribution, and direct sample statistical results are also calculated. The aim is to experimentally verify the accuracy and superiority of the PDEM in capturing instantaneous, non-Gaussian, and multi-modal response characteristics, thereby providing empirical support for its popularization and application in random seismic response analysis and reliability design for geotechnical engineering projects.

2. Research Methods

2.1. Probability Density Evolution Method

Under normal circumstances, the dynamic response of an anchored slope under the action of random ground motion can be described by the following dynamic equation:

$$M\ddot{X}(t)+C\dot{X}(t)+KX(t)=-M{\ddot{X}}_g(\mathsf{\Theta },t)$$

(1)

In the equation, M, C, and K are the mass matrix, damping matrix, and stiffness matrix, respectively; $\ddot{X}(t)$, $\dot{X}(t)$, and $X(t)$ are the acceleration, velocity, and displacement vectors, respectively; ${\ddot{X}}_g(\mathsf{\Theta },t)$ represents the non-stationary random seismic process; $\mathsf{\Theta }$ is an n-dimensional random vector; and t represents time.

More generally, any physical parameter (such as slope displacement) can be treated as a function dependent on the random variable Θ in the Generalized Probability Density Evolution Method (GPDEM). Based on the principle of probability conservation and the mathematical derivations by Li Jie, Chen Jianbing et al. [14], after selecting discrete representative points and assigning corresponding probabilities, the PDEM equation of the coupled dynamic system of the anchored slope can be expressed as

$${\displaystyle \frac{\partial p_{H\mathsf{\Theta }}(h,\theta ,t)}{\partial t}}+{\displaystyle \sum _{l=1}^m{\dot{H}}_l}(\theta ,t){\displaystyle \frac{\partial p_{H\mathsf{\Theta }}(h,\theta ,t)}{\partial h_l}}=0$$

(2)

Here, $p_{H\mathsf{\Theta }}(h,\theta ,t)$ represents the PDF of the physical quantity of interest. H is the mapping relationship in the function of the displacement vector X with respect to the random vector Θ and time t.

In this paper, the random source is seismic ground motion, so the basic random variables are related to seismic ground motion parameters. Solve the dynamic equation with a given seismic excitation to obtain the corresponding seismic response velocity. Then substitute the obtained velocity into the PDEM equation and solve it using the finite difference method to obtain the joint probability density function. Finally, the required marginal probability density function is obtained through integration or transformation. The numerical solution process of the PDEM is shown in Figure 1.

2.2. Generation of Stochastic Ground Motion

For the PDEM adopted in this paper, the generation of random ground motions is a key link in the entire analysis process. Liu Zhangjun [25] proposed a method for expressing basic random variables based on orthogonal functions, which effectively reduces the number of required random variables. Therefore, this paper uses this method to generate a series of non-stationary ground motions with a zero mean and conforming to real earthquake characteristics, and its mathematical expression is as follows:

$${\ddot{X}}_g(t)={\displaystyle \sum _{k=1}^N}\sqrt{2S_{{\ddot{X}}_g}(t,{\omega }_k)\Delta \omega }[\cos ({\omega }_{k}t)X_k+\sin ({\omega }_{k}t)Y_k]$$

(3)

In the formula, ${\ddot{X}}_g$ is the non-stationary acceleration time series of ground motion during an earthquake; $S_{{\ddot{X}}_g}$ is its bilateral evolutionary power spectral density function; $\{X_k,Y_k\}(k=1,2,3\dots ,N)$ is a standard orthogonal random variable; and ${\omega }_k=k\Delta \omega$ is the interval frequency $\Delta \omega$ = 0.15 rad/s.

Next, we use Hartley orthogonal basis functions to construct the random function expression of standard orthogonal random variables. Suppose that any two sets of standard orthogonal random variables are, respectively, functions of two mutually independent basic random variables ${\mathsf{\Theta }}_1$ and ${\mathsf{\Theta }}_2$; that is:

$${\overline{X}}_n=cos(n{\mathsf{\Theta }}_1)+sin(n{\mathsf{\Theta }}_1)$$

(4)

$${\overline{Y}}_n=cos(n{\mathsf{\Theta }}_2)+sin(n{\mathsf{\Theta }}_2)$$

(5)

In the formula, ${\mathsf{\Theta }}_1$ and ${\mathsf{\Theta }}_2$ are uniformly distributed in $[0,2\pi ]$ and are independent of each other.

Deodatis [26] proposed a class of evolutionary power spectrum models for simulating non-stationary ground motion acceleration processes. Subsequently, Cacciola and Deodatis [27] further improved this model and proposed a more general form of the bilateral evolutionary power spectral density function, which is described as follows:

$$\begin{array}{l}{S}_{{\ddot{X}}_g}(t,\omega )=A^2(t){\displaystyle \frac{{\omega }_g^4(t)+4{\xi }_g^4(t){\omega }_g^2(t)}{{\left[{\omega }^2-{\omega }_g^2(t)\right]}^2+4{\xi }_g^4(t){\omega }_g^2(t){\omega }^2}}\\ ·{\displaystyle \frac{{\omega }^2}{{\left[{\omega }^2-{\omega }_f^2(t)\right]}^2+4{\xi }_f^4(t){\omega }_f^2(t){\omega }^2}}·S_0(t)\end{array}$$

(6)

Here, $A(t)$ is the intensity modulation function, which is used to describe the variation law of ground motion energy with time. Reference [25] suggests adopting the following form:

$$A\left(t\right)={\left[{\displaystyle \frac{t}{c}}\exp \left(1-{\displaystyle \frac{t}{c}}\right)\right]}^d$$

(7)

In the formula, c is the time at which the peak ground acceleration (PGA) occurs, and d is the exponential parameter controlling the shape; parameters c and d can be determined according to the site category. In addition, the spectral parameter used to reflect the intensity of ground motion can be expressed as

$$S_0\left(t\right)={\displaystyle \frac{a_{\max }^2}{{\gamma }^2[\pi {\omega }_g\left(t\right)(2{\xi }_g\left(t\right)+\frac{1}{2{\xi }_g\left(t\right)})]}}$$

(8)

In the formula, $a_{\max }$ is the mean value of the peak ground acceleration (PGA) of random ground motion; the parameter γ is the peak factor.

To accurately reflect the non-stationary characteristics of the evolutionary power spectrum and ensure the stability of the ground motion model during numerical simulation, the following parameters need to be introduced:

$${\omega }_{\mathrm{g}}(t)={\omega }_0-a{\displaystyle \frac{t}{T}}, {\xi }_{\mathrm{g}}(t)={\xi }_0+b{\displaystyle \frac{t}{T}}$$

(9)

$${\omega }_{\mathrm{f}}(t)=0.1{\omega }_{\mathrm{g}}(t), {\xi }_{\mathrm{f}}(t)={\xi }_{\mathrm{g}}(t)$$

(10)

In the formula, Τ is the total duration of the non-stationary ground motion acceleration process. The parameters ${\omega }_0$ and ${\xi }_0$ are the angular frequency and damping ratio of the site soil; ${\omega }_g(t)$ and ${\xi }_{\mathrm{g}}(t)$ are the dominant angular frequency and damping ratio; ${\omega }_{\mathrm{f}}(t)$ and ${\xi }_{\mathrm{f}}(t)$ are the filtering parameters of the response. Among them, ${\omega }_0$, ${\xi }_0$, a, and b can be determined according to the site category and the earthquake group in the code. Based on the “Code for Seismic Design of Buildings (GB50011-2010)” [28], the site type in this paper is I₁. According to the recommended value table by Liu Zhangjun [29], the parameter values used in this paper are listed in

Table 1. Parameter values of Stochastic Ground Motion.

Parameter	${\omega }_0$ (s−1)	${\xi }_0$	γ	a	b	c	d	T(s)
Value	25	0.45	2.85	3.5	0.3	4	2	15

.

A total of 144 sets of non-stationary random seismic samples were used for the stochastic analysis. These points were generated using the GF-discrepancy strategy, which provides a set of points with minimal discrepancy to ensure the accurate representation of the joint probability density. According to the convergence characteristics of the PDEM for low-dimensional random systems, this sample size is sufficient to ensure that the error in the evolved PDF remains well within the acceptable engineering limit [30], as verified by the subsequent K-S test results. Compared with the traditional Monte Carlo simulation (MCS), which often requires thousands of deterministic runs to achieve convergence, the PDEM is significantly more efficient. Previous studies have demonstrated that the PDEM can be more than 14 times more efficient than MCS in assessing the dynamic reliability of slopes [31]. As the first successful implementation for investigating slope stability reliability based on seismic dynamic time histories, this method allows for high-precision results with a remarkably low computational cost. The average PGA of this sample set is approximately 0.1 g, with a duration of 30 s. Figure 2 shows the acceleration time history curve of a typical sample (Sample No. 100). To verify the degree of approximation of the sample set to the target power spectrum, Figure 3 compares the statistical characteristics (mean and standard deviation) of the acceleration time histories of the 144 samples with the target ground motion time history. The results indicate that the statistical characteristics of the sample set are highly consistent with the target values, verifying the effectiveness of the stochastic ground motion generation method. Seven types of white noise (WN) were also arranged to detect the basic dynamic characteristics of the slope support system model. The entire test included a total of 151 working conditions (144 groups of stochastic earthquakes and 7 groups of white noise), as detailed in

Table 2. Seismic Wave Loading Scheme.

Serial Number	Working Condition	Seismic Waves Type	Peak Acceleration/g
1	WN1	white noise	0.1
2~26	SW1~25	random seismic waves	0.1
27	WN2	white noise	0.1
28~52	SW26~50	random seismic waves	0.1
53	WN3	white noise	0.1
54~78	SW51~75	random seismic waves	0.1
79	WN4	white noise	0.1
80~104	SW76~100	random seismic waves	0.1
105	WN5	white noise	0.1
106~130	SW101~125	random seismic waves	0.1
131	WN6	white noise	0.1
132~150	SW126~144	random seismic waves	0.1
151	WN7	white noise	0.1

Note: “WN” stands for “white noise”; “SW” stands for “random seismic waves”.

. This design not only supports the statistical analysis of random responses but also provides a sufficient data basis for the experimental verification of the PDEM.

3. Model Test Design

The shaking table used in this test is a horizontal unidirectional simulated earthquake shaking table produced by Tianshui Hongshan Testing Machine Co., Ltd., located in the Xinghuo Industrial Park, Shetang Industrial Zone, Tianshui Economic and Technological Development Zone, Tianshui, Gansu Province, China, with the model designation EA-200. It is located at the Bridge and Tunnel R&D Base of Dalian University of Technology. The shaking table is controlled by computer digital control, with a tabletop size of 1.5 m × 1.5 m, a maximum bearing capacity of 10 t, and a maximum excitation force of 200 kN. The operating frequency range of the seismic waves that can be input into the shaking table is 0.1 Hz~20 Hz. During the input process of the seismic waves, the cumulative maximum displacement of the shaking table is ±150 mm, the maximum loading speed is 330 mm/s, and the maximum loading acceleration is 20 m/s².

3.1. Test Preparation

Shaking table model tests are always conducted using scaled-down models. Three parameters, namely geometric length, elastic modulus, and density, are selected as the basic scale factors. The scale factors of other parameters are derived based on the Buckingham π theorem [32,33]. In this experiment, considering the size of the rigid box and the input and output frequencies of the shaking table system, the similarity ratio of the geometric length is determined to be 30.

Table 3. Similarity Constants of Model Tests.

Serial Number	Physical Quantity	Dimension	Similar Relationship	Similarity Constant	Note
1	Geometric dimension L	[L]	CL	30	controlled variable
2	Density ρ	[M][L]−3	Cρ	1	controlled variable
3	Acceleration a	[L][M]−2	Ca	1	controlled variable
4	Elastic modulus E	[M][L]−1[T]−2	CE = CρCLCa	30
5	Poisson’s ratio μ	1	Cμ = 1	1
6	Strain ε	1	Cε = 1	1
7	Stress σ	[M][L]−1[T]−2	Cσ = CρCLCa	30
8	speed v	[L][T]−1	Cv = CL1/2Ca1/2	5.48
9	displacement u	[L]	Cu = CL	30
10	time t	[T]	Ct = CL1/2Ca−1/2	5.48
11	force F	[M][L][T]−2	CF = CL3	27,000
12	Frequency ω	[T]−1	Cω = CL−1/2Ca1/2	0.18
13	damping ratio ξ	1	Cξ = 1	1
14	cohesion c	[M][L]−1[T]−2	Cc = CρCLCa	30

shows the similarity ratios of some parameters used in this study. Referring to some similar shaking table tests on rock slopes (such as [18,22,34]), the bedrock and sliding mass in this experiment are made of barite powder, quartz sand, gypsum, glycerol, and water. Some physical and mechanical parameters of the slope model are determined through laboratory-based direct shear tests and unconfined compression tests, as shown in

Table 4. Physical and mechanical property parameters of original rock and similar materials.

Rock	Category	Density/kg·m−3	Cohesion/kPa	Angle of Internal Friction/°	Compressive Strength/kPa	Elastic Modulus/MPa
Completely weathered granite	Original rock test value	2183	3259.4	32.3	9718.2	3230.3
	Similar material value	2183	108.6	32.3	323.9	107.7

. According to the similarity ratio, the prototype strength of the model material reaches the strength of rock. Anchors are modeled with elastic materials, and it is assumed that they will not be damaged under a set of random excitations; steel strands are used to simulate anchor rods.

In accordance with the above requirements, a 1 m high rock slope reinforced by a 3-column anchor structure was constructed in a rigid box with the dimensions 1200 mm × 650 mm × 1100 mm (length × width × height). The left and right columns of the anchors are 125 mm away from the two sides of the model box, respectively, and the anchors are evenly arranged along the slope surface vertically. Referring to other shaking table tests in geotechnical earthquake engineering, two 5 cm thick foam boards were placed on both sides of the box to absorb seismic waves and avoid wave reflection. Figure 4 shows the model configuration and the layout of various sensors. A total of 15 acceleration sensors were embedded in the model, evenly distributed around the central section of the slope soil. Five monitoring elevations were set along the vertical direction, with vertical heights from the slope bottom being 0.0 m, 0.2 m, 0.4 m, 0.6 m, and 0.8 m, respectively. Each elevation was equipped with 2–3 measuring points, arranged in sequence from the rear wall of the model box (fixed boundary) to the slope surface (free surface) along the horizontal direction. The measuring points are numbered A1–A15, continuously numbered from bottom to top and from back to front. A total of 5 groups of laser displacement sensors were arranged in the test to monitor the horizontal dynamic displacement of the slope surface. Each sensor was aligned with the slope surface measuring points at heights of 0.0 m, 0.2 m, 0.4 m, 0.6 m, and 0.8 m from the slope bottom, and the measuring points are numbered D1–D5 from bottom to top. A total of 6 hollow pressure sensors were arranged in the test, which were, respectively, installed between the anchors and the base plates of 6 anchor cables in the central section, for real-time monitoring of the dynamic axial force of the anchor cables during the test. The sensors are numbered F1–F6 from bottom to top along the height of the slope, corresponding to the arrangement positions of each anchor cable. Figure 5 shows the slope after the bolt sensors have been installed.

3.2. Verification of the Rationality of the Shaking Table Model

To ensure the statistical reliability of the random seismic responses obtained from the shaking table model test, it is required that the slope model basically maintains its initial physical state after each seismic excitation without irreversible damage. For this purpose, this study uses high-strength rock-like materials to construct the slope model and applies low-intensity random ground motion input (with a peak ground acceleration of 0.1 g). During the entire test process, no obvious macroscopic cracks, large deformations, or structural damage were observed, which initially indicates that the system is in an approximately linear elastic working state.

Furthermore, two key dynamic indicators, the system’s fundamental frequency and cumulative permanent displacement, were selected to quantitatively verify the stability of the model state. First, the system’s fundamental frequency was identified through white noise sweep frequency tests. Across 7 groups of independent white noise excitations, the acceleration responses of the slope’s top measuring point (A6) were collected, and their Fourier amplitude spectra were calculated. According to the frequency similarity law for shaking table model tests, the model frequency was converted to the prototype scale. The results are shown in Figure 6—the prototype’s fundamental frequencies identified from the 7 tests were all stable at approximately 7.3 Hz, and the main peak positions and overall shapes of each spectrum curve were highly consistent. This result indicates that during the entire random seismic test sequence, the dynamic stiffness of the slope support system did not undergo significant degradation, and its linear dynamic characteristics remained good, meeting the basic premise of “state invariance” under multiple repeated excitations, thus providing a reliable physical basis for subsequent random response analysis based on the PDEM.

Secondly, the cumulative permanent displacements at the top (D5), middle (D3), and bottom (D1) of the slope under 151 random earthquake excitations were investigated, and the results are shown in Figure 7. Due to the slight mechanical oscillation of the displacement sensor (laser displacement meter) under high-frequency vibration, some records contain small noise fluctuations. After filtering and comparison with the static segment, the maximum cumulative permanent displacement at the slope top in the prototype scale was only 0.017 mm, which is much smaller than the deformation threshold that is permissible in engineering, and can be regarded as no permanent deformation. This result further confirms that under excitation by a low-intensity random ground motion input (with a peak ground acceleration of 0.1 g), the pile anchor-supported rock slope does not enter a plastic or damaged state, and the system response has a high degree of repeatability. Furthermore, it should be noted that this study primarily focuses on validating the accuracy of the PDEM for capturing peak and residual responses of the slope–anchor system. The rigorous quantification of the cumulative damage process and the resulting stiffness degradation during the seismic duration were not explicitly considered, as they falls beyond the current scope of methodological verification. This represents a limitation of our study, and future research could incorporate advanced constitutive models into the PDEM framework to explore these cumulative effects.

4. Model Test Verification of Probability Density Evolution Method

To experimentally verify the applicability and computational accuracy of the PDEM in the dynamic response analysis of anchored slopes, this section systematically compares and discusses the results obtained by the PDEM with the stochastic seismic analysis results of key response quantities such as seismic acceleration, slope displacement, and anchor axial force predicted based on various hypothetical probability distribution models. In this study, the normal distribution and Gumbel distribution are selected as comparative benchmarks. The former represents the most classical statistical baseline in engineering statistics [35,36], while the latter is a widely adopted extreme value model for geotechnical reliability assessment [37,38]. Comparing the PDEM with these established parametric models serves to highlight its unique advantage in accurately tracking the instantaneous and complex probability structures of non-linear seismic responses. The comparison results are presented in the form of statistical characteristic quantities (including mean and standard deviation) and probability distribution patterns (including PDF and CDF) to comprehensively evaluate the performance of different methods in describing the uncertainty of slope dynamic responses.

4.1. Acceleration Response of Anchored Slopes Under Random Seismic Action

Figure 8 compares the time-varying mean values and standard deviations of acceleration responses at monitoring points at different elevations on the surface and inside the slope, obtained using the PDEM and the direct sample statistics method. It can be seen that, at all monitoring points, the calculation results of the PDEM are highly consistent with the sample statistics results, indicating that this method has good applicability and reliability in simulating the dynamic response of anchored slopes under random ground motion excitation.

The mean time history of the acceleration response is generally similar to the waveform of the input random ground motion, indicating that the system response is dominated by the input ground motion; meanwhile, the standard deviation time history shows a trend of first increasing and then decreasing, reaching a peak during the strong earthquake phase (approximately 5–7 s). This is consistent with the physical characteristics of the period when the ground motion energy is concentrated, further confirming the reliability of the test data.

From the perspective of the vertical distribution characteristics, both the mean value and standard deviation of the acceleration response increase with the increase in slope height, among which the response dispersion in the slope top area (such as measuring point A5) is the most significant. Specifically, the peak standard deviations of acceleration responses at measuring points A1, A3, and A5 are approximately 0.30 m/s², 0.35 m/s², and 0.40 m/s², respectively, reflecting that the elevation effect has a significant impact on the dynamic amplification effect. In addition, for sensors at the same elevation but with different burial depths (i.e., distances from the slope surface), there are certain differences in the recorded acceleration responses, but the magnitude is small, and is not sufficient to be clearly attributed to changes in spatial position. It cannot be ruled out that these differences are caused by sensor measurement errors or local rock mass inhomogeneity.

In research on probabilistic analysis in geotechnical engineering, the uncertainty of input parameters is often taken into consideration, and it is usually assumed that the system response follows a preset probability distribution form (such as normal distribution, log-normal distribution, or extreme value distribution, etc.) [39,40,41]. However, existing studies have shown that such prior distribution assumptions may not accurately describe the true probability structure of the system response. In particular, in scenarios with strong non-linearity or a system with complex dynamic characteristics, it may lead to significant deviations in the estimated failure probability or reliability index, making it difficult to obtain credible results of probabilistic analysis [42,43,44].

In this study, the horizontal acceleration responses at three typical moments (5 s, 10 s, and 15 s) from the slope top measuring point (A5) were extracted from the time history responses across 144 groups of random seismic waves. Probabilistic modeling analyses were conducted using the PDEM and several preset probability distribution models (including normal distribution and Gumbel distribution). Figure 9 shows the instantaneous PDFs and CDF of the acceleration responses at the slope top A5 obtained by the PDEM, frequency histograms, empirical cumulative distribution functions, normal distribution, and Gumbel distribution at the above three moments.

It can be seen that the acceleration responses at different time points exhibit significantly different probabilistic characteristics: at 5 s, the PDF distribution range is relatively wide (approximately from −1.5 m/s² to 1.5 m/s²), showing a “broad and low” shape, reflecting the system response exhibiting high discreteness at this time point; by 15 s, the distribution range narrows (approximately from −0.8 m/s² to 0.8 m/s²), showing a “narrow and high” unimodal shape, indicating that the response tends to be concentrated; and the distribution characteristics at 10 s are between the two.

It should particularly be noted that this study is based on 144 groups of samples, which is already a relatively large number for geotechnical engineering shaking table tests or high-cost numerical simulations, but still limited for constructing stable and high-resolution probability distributions. Under such sample-limited conditions, although frequency histograms and empirical distributions have statistical fluctuations, they are still regarded as practical benchmarks for verifying the reliability of probabilistic modeling methods because they do not require distribution assumptions. Figure 9 shows that the PDF and CDF curves predicted by the PDEM are highly consistent with the frequency histograms and empirical distributions in terms of overall shape, peak position, and tail characteristics, indicating that even in scenarios with a limited number of samples, the PDEM can effectively capture the probabilistic evolution law of system responses without presupposing the distribution form, and provide estimates that are closer to the actual statistical results than traditional parametric models. It is worth noting that the multi-modal characteristics observed in the PDF curves are indicative of the non-linear evolutionary process of the slope system under seismic loading. To distinguish these physical features from statistical noise, a convergence check was performed using the GF-discrepancy point selection method. The consistency between the PDEM results and the 144 sets of experimental samples further confirms that the multi-modality reflects the transition of the slope from an elastic state to a state with significant residual deformation, rather than being an artifact of insufficient sampling.

More importantly, the acceleration responses at some moments show obvious multimodal characteristics, reflecting the complex response mechanism caused by the coupling of ground motion and slope dynamic behavior. The PDEM can accurately reproduce such non-unimodal distributions, while unimodal parametric models such as normal or Gumbel distributions cannot capture their multi-modal characteristics. In particular, when the sample size is limited and it is difficult to smooth the distribution shape through the law of large numbers, unimodal parametric models such as normal distribution and Gumbel distribution encounter difficulties in describing such complex shapes. However, the PDEM, relying on its ability to directly track the probability evolution of responses, shows stronger adaptability and modeling accuracy. This result further confirms that the PDEM has better modeling accuracy and adaptability than traditional assumed distribution models in describing instantaneous, non-stationary, and non-Gaussian random responses, and is especially suitable for geotechnical engineering systems with complex coupling response characteristics such as anchored slopes.

This study further extracted the PGA of the slope top measuring point A5 from 144 sets of random earthquake cases and conducted probabilistic modeling analysis using the PDEM and several parameterized probability distribution models (including normal distribution and Gumbel distribution). Figure 10 presents the PDF and CDF of the PGA obtained by the PDEM, frequency histogram, empirical distribution, normal distribution, and Gumbel distribution. It can be seen that the PDF and CDF obtained by the PDEM are highly consistent with the direct statistical results based on samples (i.e., frequency histogram and empirical distribution) in both overall shape and local details, while there are obvious deviations in the normal distribution and Gumbel distribution. In particular, it can be seen from the frequency histogram that there are observed values of PGA exceeding 2 m/s² in the samples, and this high-value region corresponds to a non-negligible probability density. Only the PDEM and the frequency histogram can accurately reflect the probability contribution of this tail region, while the normal distribution and Gumbel distribution, due to their inherent unimodality and specific tail decay characteristics, completely ignore this small-probability but significant, in engineering terms, response interval.

This phenomenon indicates that, in practical engineering analysis with limited sample size, traditional parameterized models are prone to underestimate the occurrence probability of extreme responses due to incorrect distribution form assumptions, thereby affecting the conservatism and safety of slope seismic reliability assessments. In contrast, the PDEM does not require presetting for the distribution type, can faithfully reflect the tail characteristics and potential multi-modal structures contained in the samples, and still has the ability to finely characterize key probability features (such as high-PGA events) under limited data conditions. Therefore, it is more suitable for geotechnical engineering systems sensitive to extreme dynamic responses, such as anchored slopes.

In the CDF diagram of PGA, the dynamic reliability of the slope can also be directly obtained through the PDEM. As shown in Figure 10, if the critical threshold of 1.2 m/s² (approximately 0.12 g) is used as the criterion to calculate the probability that the slope response satisfies PGA < 1.2 m/s², the reliability results predicted by four different methods can be obtained. It should be noted that the selection of this threshold is limited by the design conditions of this shaking table model test. To ensure that the slope model remains in its initial state without damage under multiple random earthquake excitations, a relatively low ground motion intensity was adopted in the test. Therefore, the analyzed PGA responses are generally small in magnitude. For the convenience of analysis, the reliability evaluation threshold here is also set at a relatively low level, which does not represent the actual critical threshold in engineering practice): the results of the empirical distribution and PDEM are highly consistent, being 86.3% and 85.5%, respectively; while the predicted values of the normal distribution and Gumbel distribution are 80.5% and 83.0%, respectively. Compared with the non-parametric benchmark of the empirical distribution, the reliability deviation of the PDEM is only 0.8%, while the deviations of the Gumbel distribution and normal distribution reach 3.3% and 5.8% respectively—the latter deviation is about 7.25 times that of the PDEM. This result indicates that preset distribution models may introduce significant systematic errors in reliability estimation, while the PDEM can more accurately approximate the true cumulative probability characteristics based on the samples. To further quantify the accuracy of the PDEM, the Kolmogorov–Smirnov (K-S) test was employed to evaluate the goodness of fit between the theoretical CDF and the empirical distribution of the 144 test samples. The results show that the p-value for the PDEM prediction is 0.92, which is much higher than the significance level of 0.05. This statistically confirms that the PDEM can accurately capture the probability evolution of the slope’s seismic response. In comparison, the p-values for the normal and Gumbel distributions are 0.45 and 0.38, respectively, indicating a relatively poorer fit to the experimental data.

It should be pointed out that, for a given sample set, the frequency histogram is more sensitive to interval division and, although the empirical distribution function is unbiased and consistent in the probabilistic sense, its mathematical form is a discontinuous step function, making it difficult to provide a smooth and differentiable PDF expression. Therefore, in scenarios involving fine random seismic response modeling or requiring an analytical PDF form, the empirical distribution and histogram are usually only used as verification benchmarks and not directly for subsequent random dynamic analysis or reliability calculation. In contrast, the PDEM can simultaneously provide continuous, smooth, and physically consistent PDFs and CDFs, which not only highly agree with empirical statistical results but can also directly support refined random seismic response simulation and dynamic reliability assessment. This characteristic indicates that the PDEM has significant application potential and promotion value in the random dynamic analysis and reliability design of anchored slopes.

This study also conducted a comparative analysis of the PGA probability distribution characteristics at five measuring points (A1, A3, A5, A8, and A12) inside the anchored slope using the PDEM and a direct sample statistical method, with the results shown in Figure 11. The PDFs and CDFs of the two strategies at each measuring point showed good consistency, further verifying the reliability of the PDEM in probabilistic modeling of spatial multi-point responses.

It can be seen from Figure 11 that the probability distributions of PGA at different elevations exhibit significant differences: as the height of the measuring point increases, the distribution range of PGA gradually expands, the variability enhances, and the overall distribution shifts to the right, reflecting an obvious elevation amplification effect. Specifically, the PGA at the bottom measuring point A1 is mainly concentrated in the range of 0.6–1.5 m/s²; the distribution range of the middle measuring point A3 expands to 0.7–1.7 m/s²; while the distribution of the slope top measuring point A5 further widens to 0.8–2.0 m/s², with a lower PDF peak and a thicker tail, reflecting a higher level of dispersion in dynamic responses.

In contrast, by comparing Figure 11b,d,e (corresponding to measuring points at the same elevation but different burial depths, such as A3, A8, and A12), it can be seen that their PGA PDFs and CDFs almost overlap, showing no systematic differences. This result indicates that, under the low-intensity ground motion input conditions adopted in this test, the spatial variability of the acceleration response inside the slope is mainly controlled by elevation, while the change along the normal direction of the slope (i.e., the distance from the slope surface) has little impact and can be ignored.

4.2. Axial Force Response of Anchor Bolts in Anchored Slopes Under Random Seismic Excitation

The interaction effect between the anchoring structure and the rock–soil mass under seismic action can be revealed through the dynamic response of the anchor axial force. The anchor axial force is not only a key indicator for evaluating the dynamic stability of slopes but is also a core parameter that must be considered in anchoring design [45,46]. To intuitively present the dynamic evolution law of axial force during an earthquake, the axial force values analyzed in this paper are all “axial force change values”, i.e., the increment after subtracting the initial prestress from the dynamic axial force. Figure 12 shows the statistical characteristics of the axial force change values of five representative anchor measuring points (F1–F5), including the time history curves of the mean value and standard deviation obtained by the sample statistical method and PDEM, respectively. The comparison results show that the curves obtained by the two methods are highly consistent in terms of shape, amplitude, and evolution trend, which fully verifies the reliability and accuracy of the PDEM in terms of simulating random dynamic responses, and also indirectly confirms the accuracy and representativeness of the experimentally collected data.

The time history curves of the mean axial force change at all measuring points show fluctuation characteristics highly similar to the input ground motion waveform, indicating that the dynamic response of the anchor, a key force-transmitting component between the rock–soil mass and the supporting structure, is directly controlled by the transient excitation of external loads. The standard deviation time history shows a typical “first increasing then decreasing” pattern: in the initial stage of ground motion (0–10 s), the standard deviation increases rapidly, reflecting the rapid accumulation of system uncertainty with energy input; it reaches a peak (with a maximum value of about 550 N) in the strong earthquake stage (about 10–15 s); then (15–30 s) the standard deviation gradually decreases, corresponding to the energy dissipation and system stabilization in the aftershock stage. This evolution law is highly consistent with the energy release mechanism of seismic waves, further confirming the physical rationality of the experimental data.

In terms of magnitude, the maximum mean value of the axial force change is about 270 N, which is much smaller than the initial prestress of the anchor. This indicates that, under the low-intensity seismic excitation in this study, the properties of the slope did not change significantly during the entire test, which is consistent with the goal of the model test. The anchors were always in an effective working state without relaxation or failure, and the overall properties of the slope were stable.

From the perspective of vertical spatial distribution, the anchors’ axial force response shows a significant “elevation gradient effect”, but the change trends of its mean value and standard deviation are different: the mean value of axial force change increases with the increase in slope elevation. The anchors located in the slope top area not only have smaller fluctuation amplitudes but also maintain obvious residual axial force (permanent change value) after the earthquake ends, and the standard deviation of axial force change decreases with the increase in elevation. The peak standard deviation of the bottom anchors is the highest (about 800 N), while that of the top anchors is the lowest (about 500 N). This seemingly contradictory phenomenon can be explained by the geometric constraint characteristics of the anchors: the bottom anchors have short free segments and high embedment stiffness, while the top anchors have long free segments and relatively flexible constraints. Similar elevation-dependent distribution patterns of anchor forces have been observed in shaking table tests by Fu et al. [46] and numerical simulations by Pan et al. [47], who attributed the higher mean values at the crest to the topographic amplification of acceleration. Furthermore, the higher fluctuation (standard deviation) in the bottom anchors observed in this study complements the findings of Jia et al. [48], suggesting that higher local stiffness near the rigid base makes bottom reinforcement more sensitive to high-frequency seismic components. Their axial forces are mainly controlled by the overall stiffness and are not sensitive to local disturbances, so the response is more concentrated and the standard deviation is smaller.

To deeply explore the dynamic evolution law of anchor axial force during an earthquake, this paper selects the key measuring point A5 at the slope top. Across 144 sets of random ground motion excitations, the axial force changes at three representative moments (5 s, 10 s, and 15 s) are extracted, and comparative analyses are conducted using frequency histograms, empirical distribution functions, normal distribution, Gumbel distribution, and the PDEM. Figure 13 shows that there are significant differences in the shapes of PDF and CDF at different time points. Similarly to the distribution law of acceleration, the PDF distribution of anchor axial force change at 5 s ranges from −1000 N to 1500 N, presenting a wide–low distribution form, which corresponds to the main shock phase of the ground motion. During this phase, the system responds fully, and the dispersion among various groups of samples is large. The PDF distribution at 15 s ranges from −750 N to 100 N, showing a narrow–high form. At this time point, the system has entered the late stage of a strong earthquake or the aftershock phase, the energy input tends to be stable, the response gradually converges, the fluctuation range of the axial force decreases, and it is concentrated into a small negative value interval. The distribution form of the acceleration response at 10 s is between the two. Similarly, it can be seen from Figure 13 that the shapes of PDF and CDF of the anchor axial force changes obtained by the PDEM closely fit the overall contours of the frequency histograms and empirical distribution functions at all three moments, including the main peak position, peak width, tail attenuation, and local fluctuation characteristics, demonstrating excellent fitting accuracy and stability. Therefore, in terms of axial force changes, the PDEM still has the ability to provide more accurate estimation and more detailed instantaneous PDF and CDF compared with traditional probability distribution models.

Similarly to the acceleration response, by calculating the probability that the maximum change in the anchor axial force is less than 1500 N, four different acceleration reliability results can be obtained, and are shown in Figure 14. The results of the empirical distribution and the PDEM are roughly the same, being 62.7% and 62.8%, respectively, while the reliability results of the normal distribution and Gumbel distribution are 57.5% and 64.3%, respectively. The deviation between the reliability results of the PDEM and the empirical distribution is only 0.1%, while the deviation between the Gumbel distribution and the empirical distribution results reaches 1.5%, and the deviation between the normal distribution and the empirical distribution reliability results reaches 1.6%, which is 16 times the deviation of the PDEM. Therefore, for the change in anchor axial force under the random ground motion of anchored slopes, the PDEM can still obtain reliable PDF and CDF, which can be used for both detailed analysis of anchor axial force response under random earthquakes and dynamic reliability analysis.

Next, the PDFs and CDFs of the maximum axial force changes in the five anchor rods (F1, F2, F3, F4, and F5) are compared directly through the PDEM and data statistics, as shown in Figure 15. The results of these two strategies at different positions also show good consistency. Observing Figure 15a–c, it can be seen that the PDFs and CDFs of PGAs at different heights have different distribution ranges and variabilities, and the distribution ranges and variabilities of the maximum axial force changes in the anchor rods decrease with the increase in height and shift to the left at the same time. The PDF and CDF of the maximum axial force change value of F1 are distributed in the range of 1500~3100 N, those of F3 are in the range of 1250 N~2500 N, while those of F5 are only in the range of 1100 N~2100 N. This is exactly the inverse of the variation law of acceleration, but is the same as that of the standard deviation of axial force. As analyzed earlier, the reason for this change is that the free section of the upper anchor is longer, which is consistent with the previous content and proves the accuracy of the analysis.

4.3. Absolute Displacement Response of Anchored Slopes Under Random Seismic Action

Figure 16 shows the instantaneous PDFs and CDFs of the slope top displacement responses at 5 s, 10 s, and 15 s obtained using the PDEM, frequency histograms, empirical distributions, normal distributions, and Gumbel distributions, respectively. The PDFs and CDFs at different time points have different distribution ranges and variabilities. At 5 s, the PDF of the displacement response is distributed in the range of −50 mm to 50 mm, showing a wide–low distribution form, while at 15 s, the PDF is distributed in the range of −30 mm to 30 mm, showing a narrow–high form, and the distribution form of the acceleration response at 10 s is between the two. It can be seen from Figure 16 that the PDF and CDF curves obtained by the PDEM are highly consistent with the frequency histograms and empirical distributions at all times, not only maintaining consistency in the main peak position, distribution width, and tail characteristics, but also accurately reproducing local non-unimodal structures (such as the bimodal characteristics at 10 s). This indicates that the PDEM has an excellent ability to characterize the real probability structure of displacement responses. Therefore, in terms of the displacement response of anchored slopes, the PDEM has the ability to provide more accurate estimates and refined instantaneous PDFs and CDFs than some other specific probability distribution models.

Similarly, different probability distribution models and the PDEM for the peak displacement at the slope top D5 were analyzed in 144 random cases. Figure 17 shows the PDF and CDF of the peak displacement at the slope top obtained by the PDEM, frequency histogram, empirical distribution, normal distribution, and Gumbel distribution, respectively. Similarly, the forms of the PDFs and CDFs obtained by the PDEM are more consistent with the direct statistics of the sample data (histogram and empirical distribution) than the normal distribution and Gumbel distribution, and some detailed changes and characteristics in the curves are also clearly presented.

In Figure 17, by calculating the probability that the peak displacement of the slope response is less than 37 mm, four different displacement reliability results can be obtained. The results of the empirical distribution and PDEM are the same, each being 54.3%, while the reliability results of the normal distribution and Gumbel distribution are 51.7% and 58.7%, respectively. The deviation between the reliability results of the PDEM and the empirical distribution is 0%, while the deviation between the Gumbel distribution and the empirical distribution results reaches 2.6%, and the deviation between the normal distribution and the empirical distribution reliability results reaches 4.4%.

At higher engineering thresholds, traditional parametric models often exhibit significant bias because they cannot represent the complex, multi-modal probability structures arising from strong non-linearity. In contrast, the PDEM maintains high accuracy in the tail regions of the PDF and CDF. This is because the PDEM solves the probability evolution based on the actual physical response of the 144 deterministic samples, which include extreme states of the slope–anchor interaction. The results in Figure 10, Figure 14 and Figure 17 demonstrate that for safety-critical thresholds, the PDEM provides a much more reliable estimate of the exceedance probability than the normal and Gumbel distributions.

4.4. Probability Density Evolution Process of Ground Motion Response

Based on the comprehensive comparative analysis mentioned above, the PDEM can more accurately characterize the dynamic evolution process of the probability density functions of key response variables (such as acceleration, axial force, and displacement) of anchored slopes under random ground motion excitation over time without the need to preset the distribution form. It can accurately capture complex distribution characteristics such as non-unimodal, multi-modal, and time-varying features. It can provide instantaneous PDFs and CDFs that are highly consistent with direct sample statistics, making it suitable for refined stochastic response simulation and dynamic reliability assessment. Based on this, in this section, using the PDEM calculation results, three-dimensional contour maps of the probability density evolution of the acceleration (A5), anchor cable axial force variation (F5), and displacement (D5) at the slope top monitoring point are plotted (Figure 18, Figure 19 and Figure 20) to intuitively reveal their stochastic evolution characteristics in the time domain.

As shown in the figures, in the initial stage of ground motion (t < 5 s), the probability density functions of the three types of response variables all exhibit a “narrow and high peak” shape, indicating that the system response amplitude is small and the dispersion is low at this time. This is consistent with the weak intensity of the input ground motion and the structure basically being in the linear or quasi-static response zone. Entering the main shock stage (approximately 5–15 s), as the injection of ground motion energy intensifies, the probability density function gradually “broadens and flattens” with the peak value decreasing and the distribution interval expanding, reflecting that the system response enters the non-linear transition zone, the uncertainty significantly increases, and the range of response amplitudes expands greatly. In the attenuation stage of ground motion (t > 15 s), the probability density function converges again into a “narrow peak shape”, indicating that the system gradually returns to stability, and the response amplitude and dispersion attenuate synchronously, eventually tending to a static state. This evolution law is highly consistent with the ground motion energy dissipation mechanism and structural damping effect, further verifying the effectiveness of the PDEM in capturing the evolution process of transient stochastic responses. The core of the PDEM lies in establishing the evolution equation of the generalized probability density function (the Liouville equation or FPK equation) to achieve real-time tracking of the joint probability density of system state variables. In this study, the PDF evolution contour maps of the three types of response variables intuitively show the complete evolution path of the system from a deterministic initial state to a stochastic steady state, and then back to a deterministic final state, fully embodying the unique advantages of the PDEM in terms of handling non-stationary and non-linear stochastic systems.

5. Conclusions

This study verifies the applicability and accuracy of the PDEM in probabilistic modeling of the dynamic response of anchored rock slopes under random seismic actions through large-scale shaking table model tests. It aims to address the limitations of traditional parametric distribution models, which struggle to characterize non-Gaussian, non-stationary, and even multi-modal responses due to their preset forms. Across 144 sets of non-stationary random ground motions and 7 sets of white noise excitations, the test systematically collected data on acceleration, displacement, and anchor axial force. Instantaneous PDF/CDF modeling was conducted using the PDEM, with multi-dimensional comparisons made against the normal distribution, Gumbel distribution, and direct sample statistics. This study is the first to verify the effectiveness of PDEM in anchored slopes at the physical model level, providing experimental support for the analysis of random seismic responses and dynamic reliability assessment of anchored slopes.

• The PDEM is significantly superior to traditional parametric models in terms of capturing the transient probabilistic evolution characteristics of key response quantities of anchored slopes. Whether it is acceleration, changes in anchor bolt axial force, or displacement response, the PDFs and CDFs predicted by the PDEM at typical moments such as 5 s, 10 s, and 15 s are highly consistent with the frequency histograms and empirical distributions. It accurately reproduces the distribution morphology, peak position, tail decay, and even local multimodal structures. In contrast, the normal and Gumbel distributions, due to their inherent unimodality and specific tail characteristics, cannot truly reflect the complexity of the responses, especially with systematic underestimation in the tail region of extreme events.
• In the reliability assessment of peak responses (such as PGA, maximum axial force change, and peak displacement), the PDEM demonstrates excellent accuracy. Taking the peak PGA at the slope top as an example, the reliability of “PGA < 1.2 m/s²” predicted by the PDEM is 85.5%, with a deviation of only 0.8% from the empirical distribution result (86.3%), while the deviations of the normal distribution and Gumbel distribution are as high as 5.8% and 3.3%, respectively. A similar pattern holds for axial force and displacement responses, indicating that the PDEM can significantly reduce the systematic error of reliability estimation under the condition of limited samples.
• Anchored slopes exhibit clear spatial response patterns under low-intensity seismic excitation, and the PDEM can effectively reproduce these physical characteristics. The acceleration response shows an obvious elevation amplification effect, while the standard deviation of the change in anchor axial force decreases with increasing elevation, which is attributed to the longer free segments and relatively flexible constraints of the upper anchors. The multi-point response PDF/CDF obtained by the PDEM is highly consistent with direct statistical results, verifying its physical consistency and applicability in spatial multi-point probabilistic modeling.
• By constructing a three-dimensional probability density evolution cloud map of acceleration, axial force, and displacement, the PDEM completely reveals the entire evolutionary law of the response PDF from being “narrow and high” in the initial stage of an earthquake, “wide and flat” in the main shock stage, to “re-convergent” after the earthquake, which is highly consistent with the earthquake energy input–dissipation mechanism. This capability indicates that the PDEM can not only faithfully track the instantaneous uncertainty of non-stationary and non-linear systems but can also provide a continuous, smooth, and physically reasonable probabilistic description tool for dynamic reliability analysis.

For engineering practice, we recommend that practitioners adopt the PDEM as a highly efficient alternative to traditional Monte Carlo simulations, as it provides high-precision stochastic results using a significantly smaller number of deterministic samples. By utilizing the instantaneous PDF and CDF curves of key indicators such as displacement and anchor axial force, engineers can move beyond the conventional single factor of safety towards a performance-based design framework, allowing for the quantification of slope reliability at any given moment during an earthquake. Although we successfully verified the accuracy of the PDEM using shaking table tests, this study has several limitations that should be acknowledged. Due to the 1:30 geometric scale, the physical model may not fully replicate the complex stress states and discrete fracture network characteristics of real-world large-scale rock slopes. The rock mass was simulated using a high-strength homogeneous material. While it captures the overall dynamic response, the influence of complex joint sets and groundwater on the stochastic response remains to be further explored. Furthermore, the scaling effects inherent in 1 g shaking table tests present a challenge for direct translation between the model and the prototype. While the model was designed according to rigorous similitude laws to maintain dynamic similarity, certain aspects—such as the discrete nature of rock joints and the stress-dependency of soil stiffness—may not scale linearly. However, since this study focuses on validating the PDEM’s ability to evolve probability densities based on a given physical system, these scaling effects do not compromise the methodological conclusions. Future research using centrifuge modeling could provide further insights into the influence of higher stress on stochastic reliability.