Intelligent Assessment of Landslide Impact Force Considering the Uncertainty of Strength Parameters

Abstract

Accurately predicting the peak impact force exerted by landslides on bridge piers is crucial for evaluating structural safety. However, the reliability of such predictions is frequently undermined by the spatial variability and uncertainty inherent in soil and rock strength parameters. To quantify the influence of this uncertainty, in this study, a three-dimensional numerical model of a landslide impacting bridge piers was developed using LS-DYNA software (version R11.0.0). A neural network was then trained on the peak impact forces simulated by the numerical model. Based on the neural network predictions, the impact mechanisms were categorized into two distinct modes, namely, a low-impact mode and a high-impact mode, for a comparative analysis. The results revealed statistically significant differences in soil parameters between these modes. Specifically, low-impact forces (F < 467 kN) were found to correlate with higher cohesion (18.5–24.9 kPa) and lower internal friction angles (15–22.4°). Conversely, high-impact forces (F ≥ 467 kN) were associated with lower cohesion (14.0–21.6 kPa) and higher internal friction angles (18.1–25.3°). This negative correlation highlights the decisive role that the combined uncertainty of strength parameters plays in predicting the peak impact force. Moreover, the surrogate model developed in this study effectively addresses the computational inefficiencies commonly associated with Monte Carlo simulations. This methodology provides a valuable tool for evaluating the vulnerability of infrastructure systems exposed to landslide hazards.

1. Introduction

Landslide hazards present a significant threat to the stability of bridge pile foundations. The dynamic forces generated by landslides can cause severe damage to bridge foundations or supporting structures, potentially leading to catastrophic collapse [1]. Additionally, the accumulation of landslide debris often obstructs roadways and bridges, resulting in widespread traffic disruptions. Such events can render highways and bridges impassable, significantly impacting transportation operations and regional connectivity [2]. In March 1989, the B5434 highway from Trevor to Froncysyllte was severed when the underlying slope failed due to prolonged and intense rainfall [3], triggering a landslide. In a separate case, Luis et al. [4] investigated the collapse of the Caracas–La Guaira viaduct, which was also caused by a landslide. In June 2025, prolonged heavy rainfall triggered a landslide that caused the collapse of a critical section of the G76 Xiarong Expressway Gedu Line at K1264 along the route from Guangxi to Duyun, Guizhou [5]. Their study integrated findings from geotechnical investigations to comprehensively analyze both the causes of the viaduct failure and the associated damage mechanisms. These incidents underscore the significant risks posed by landslides—including impacts from debris, subsidence, and structural fractures—to bridge structures, emphasizing the urgent need for advanced and in-depth research in this critical area.

Geotechnical engineering is inherently subject to uncertainties, which primarily comprise three types: parameter uncertainty, model uncertainty, and measurement uncertainty [6]. Parameter uncertainty arises from the inherent spatial and temporal variability of geotechnical materials; model uncertainty stems from the simplification of complex mechanical behaviors in constitutive models; and measurement uncertainty is caused by equipment limitations, sampling disturbances, and operational errors. In slope stability analysis, the results are influenced by multiple factors, including soil stiffness, the internal friction angle, cohesion, and groundwater conditions. Among these factors, soil strength parameters—especially cohesion and the internal friction angle—are widely recognized as the most critical. Extensive research has explored their influence using methods such as limit equilibrium analysis, strength reduction techniques, and numerical simulations [7,8,9]. Additionally, probabilistic and statistical approaches have been employed to analyze their distribution characteristics [10,11,12]. Despite these efforts, a clear research gap persists. Specifically, the influence of uncertainties in these critical strength parameters on the peak impact force generated during landslide–pier interactions has not been explicitly evaluated.

Current approaches for studying parameter uncertainty along engineering routes encompass the Monte Carlo simulation, the response surface method, and the First-Order Reliability Method (FORM). Although the Monte Carlo simulation is highly versatile, it is computationally expensive [13]. The response surface method is efficient for building simplified models, but its accuracy is sensitive to experimental design and suited for relatively simple systems [14]. The FORM is computationally efficient, yet its accuracy depends entirely on the linearity of the limit state function [15]. To address the computational inefficiency of Monte Carlo simulations, many researchers have turned to machine learning techniques. Models such as support vector machines [16] and artificial neural networks [17] have been widely adopted as surrogate models in geotechnical applications. For example, Kang et al. [18] applied machine learning to predict slope stability, while Mahmoodzadeh et al. [19] evaluated the performance of various machine learning models in slope stability analysis. Wang et al. [20] further utilized machine learning to predict the landslide displacement and runout distance. Although extensive research has been conducted on slope stability and landslide risk, few studies have focused on predicting impact forces, especially in scenarios involving large-deformation landslides impacting bridge piers. This process involves a highly complex three-dimensional soil–structure interaction problem, posing significant challenges for numerical simulation. Consequently, understanding the probability distribution of peak impact forces under uncertain strength parameters has emerged as a critical and urgent issue that requires attention.

This study established a three-dimensional numerical model to simulate the landslide impact force on bridge piers using the Smoothed Particle Hydrodynamics (SPH) method in LS-DYNA. While previous studies, such as Mahmoodzadeh et al. [19] and Wang et al. [20], have successfully applied machine learning to both slope stability and landslide runout analyses, they have primarily focused on the intrinsic stability of geomaterials or kinematic end states. In contrast, this work pioneers a mechanism-driven probabilistic framework, shifting the paradigm toward quantifying the dynamic physical processes of landslide–structure interactions. Sampling points were carefully chosen by considering the correlation between cohesion and the internal friction angle [21], enabling precise simulations of the dynamic response of bridge piers under landslide impacts. The subsequent analysis focused on the peak displacement at the pier top. Additionally, a surrogate model based on a neural network was developed to achieve rapid predictions of the peak landslide impact force. This approach facilitates systematic investigations of the probability distribution characteristics of the peak impact force as it affects bridge abutments.

2. Stochastic Analysis Method for Peak Impact Force of Landslide on Piers Based on Neural Network

2.1. Three-Dimensional Numerical Model of Landslide Large-Deformation Sliding Impact Pier

In the Smoothed Particle Hydrodynamics (SPH) method, the problem domain is discretized into a set of independent particles, each of which carries fundamental material properties and tracks key state variables, such as deformation and stress, throughout its motion. Because these particles are not bound by fixed meshes or absolute positional relationships, the SPH method is capable of effectively capturing large deformations and complex flow patterns. The overall mechanical behavior of the system is then determined by integrating the motion and interaction of all individual particles.

In the SPH method, the approximation and derivative of the macroscopic variable for a particle are represented as follows:

$$<f\left(x_i\right)>=\underset{j=1}{\stackrel{N}{\Sigma }}{\displaystyle \frac{m_j}{{\rho }_j}}f\left(x_j\right)W\left(x_i-x_j,h\right)$$

(1)

$${\displaystyle \frac{\partial <f\left(x_i\right)>}{\partial x_i}}=\underset{j=1}{\stackrel{N}{\Sigma }}{\displaystyle \frac{m_j}{{\rho }_j}}f\left(x_j\right){\displaystyle \frac{\partial W\left(x_i-x_j,h\right)}{\partial x_i}}$$

(2)

where W is the smooth kernel function, which represents the weight of the particle; x is the position vector of the particle; i and j are particle indices; ρ represents the density of particles; m represents the mass of particles; h indicates smooth length; and N represents the number of particles associated with the particle to be calculated. Neglecting thermal effects, the motion of a landslide impacting a pier is governed by the conservation of mass and momentum [22,23].

In this numerical simulation, the landslide mass is treated as a fluid, and its continuity equation is derived based on the principle of mass conservation:

$${\displaystyle \frac{d{\rho }_i}{dt}}=\underset{j=1}{\stackrel{N}{\Sigma }}{m}_j\left(v_i^{\beta }-{\nu }_j^{\beta }\right){\displaystyle \frac{\partial W_{ij}}{\partial x_i^{\beta }}}$$

(3)

where v is the velocity of the particle, and W_ij represents the value of the smooth kernel function calculated by particle j at particle i.

By combining the conservation of momentum and Newton’s second law, the momentum equation of a fluid can be derived:

$${\displaystyle \frac{d{\nu }_i^{\alpha }}{dt}}=\underset{j=1}{\stackrel{N}{\Sigma }}{m}_j\left({\displaystyle \frac{{\sigma }_i^{\alpha \beta }+{\sigma }_j^{\alpha \beta }}{{\rho }_i{\rho }_j}}-{\delta }_{\alpha \beta ,\Pi }\right){\displaystyle \frac{\partial W_{ij}}{\partial x_i^{\beta }}}+g_i^{\alpha }$$

(4)

where ${\sigma }_i^{\alpha \beta }$ and ${\sigma }_j^{\alpha \beta }$ are the stress tensors of the particle; α and β represent the coordinate direction, $g_i^{\alpha }$ represents the magnitude of gravitational acceleration, and δ_αβ,Π is the artificial viscosity applied to ensure the stability of the calculation result. The artificial compression ratio is introduced to calculate the derivative of pressure for time, according to the equation of state, as follows:

$$p=p_0\left[{\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)}^{\gamma }-1\right]$$

(5)

where p₀ is the initial pressure; γ is the constant general case, which is taken as γ = 7; and ρ₀ is the reference density.

To enhance computational efficiency, the bridge pier is represented as a rigid body. The soil–structure interaction between the landslide and the bridge pier is simulated using an automatic point-to-surface contact algorithm grounded in the master–slave concept. Within this framework, the pier nodes are treated as the slave surface, whereas the contact surface of the landslide serves as the master surface. The corresponding contact condition can be mathematically expressed as

$$\left\{\begin{array}{l}{g}_N=(x_s-x_m)\cdot n\ge 0\\ p_N\le 0, p_N{g}_N=0\\ p_T=\mu p_N{\displaystyle \frac{v_T}{\left|\right|v_T\left|\right|}}\end{array}\right.$$

(6)

where g_N is the normal clearance distance, x_s is the coordinate of the slave surface node, x_m is the coordinate of the nearest point on the main plane, n is the normal vector of the main surface, P_N is the normal pressure of the normal phase, P_T is the tangential friction vector, and v_T is the relative sliding velocity vector.

The model’s time step is automatically calculated by the LS-DYNA solver based on the Courant stability condition, contingent upon the smallest element size and material sound speed, and it converges to a value of 2.901 × 10⁻⁵ s. The establishment of the model’s boundary conditions comprises three key components: the direct application of gravitational loads, the constraint of rigid body degrees of freedom, and, particularly critical, the computation of SPH–structure contact interactions. Central to this process is the implementation of the CONTACT_AUTOMATIC_NODES_TO_SURFACE algorithm. This method treats SPH particles as discrete material points (nodes) and performs automatic contact detection with adjacent rigid surfaces. This interaction is modeled as a dynamic computational process based on the penalty method. At each simulation timestep, the program performs a real-time calculation of contact forces to prevent interpenetration, thereby accurately simulating the physical interaction between the SPH domain and the structure. Consequently, the defined boundary conditions represent the physical constraints imposed on the system, while the computed contact forces are the resultant dynamic responses automatically determined by the solver in accordance with these constraints.

2.2. Neural Network Proxy Model for Peak Impact Force of Landslide

The neural network is a powerful mathematical model capable of autonomously learning and identifying patterns from data in the machine learning technique. Its fundamental architecture comprises three components, as depicted in Figure 1: an input layer, a hidden layer, and an output layer [24]. The input layer receives the raw data, which are subsequently processed by the hidden layer for feature extraction and pattern recognition. The output layer then generates the final prediction or result based on the transformed representations [25].

When a neural network surrogate model is applied for regression analysis, its performance can be quantitatively evaluated using the Pearson correlation coefficient (P) and the mean squared error (MSE). A declining MSE during training signifies that the model is effectively capturing the underlying patterns in the data. At the same time, a Pearson correlation coefficient approaching 1.0 indicates stronger linear alignment between the predictions and targets, reflecting improved prediction accuracy and reliability.

$$MSE={\displaystyle \frac{1}{n}}{{\displaystyle {\sum }_{i=1}^n(Y_i-\widehat{Y_i})}}^2$$

(7)

$$P={\displaystyle \frac{\Sigma (T_i-\overline{T})(P_i-\overline{P})}{\sqrt{\Sigma {(T_i-\overline{T})}^2}\cdot \sqrt{\Sigma {(P_i-\overline{P})}^2}}}$$

(8)

2.3. Stochastic Analysis Method for the Peak Impact Force of Landslides on Bridge Piers

The process used in the stochastic analysis to determine the peak impact force on landslide-impacted piers is depicted in Figure 2. Initially, soil strength parameter samples were generated, taking into account their inherent uncertainty. Subsequently, the corresponding peak impact forces were obtained via numerical simulation using LS-DYNA. The soil strength parameters—cohesion and the internal friction angle—served as the feature inputs for the neural network model, while the simulated peak impact forces were utilized as label data for supervised learning. The dataset was divided into training, validation, and test subsets in a 7:1.5:1.5 ratio. The training subset was leveraged to develop the neural network model, whereas the validation and test subsets were employed to optimize hyperparameters and assess the accuracy of the trained surrogate model, respectively. Ultimately, the validated model was integrated within a Monte Carlo simulation framework to predict the probability distribution of the peak impact force under varying parameter conditions.

3. Validation of the Analysis Model of Landslide Impacting Rigid Structure

Experiment of Dry Sand Impacting Baffle

Based on the stochastic framework introduced in Section 2, it is essential to initially assess the applicability of the SPH analysis method for simulating large-deformation motion. The flume experiment conducted by Ahmadipur [26] was selected as a benchmark case. As depicted in Figure 3, the experimental setup comprises a flume tank, a crane, a sand container, a swing door, a rigid plate, and a force sensor. The flume dimensions are 320 cm in length, 40.0 cm in width, and 43.2 cm in height, while the sand container measures 43.2 cm × 40.0 cm × 43.2 cm. For each test, 30 kg of dry sand is placed in the container. The inclination angle of the flume can be varied from 0° to 60° using the crane, facilitating an investigation into how impact forces on the rigid plate change with different slope angles. Since the SPH method is employed in this study to simulate soil-structure interaction, validating its ability to capture impact responses against rigid structures is essential. Although real landslides often involve cohesive-frictional soils, publicly available model test data are mostly based on dry sand, which provides a well-defined benchmark for numerical validation. This verification aims to confirm the SPH method’s capability in capturing key mechanical features of the impact process. The SPH constitutive model adopted here is also applicable to the simulation of cohesive soils.

Based on the experimental setup described earlier, a corresponding numerical model was developed using LS-DYNA, as shown in Figure 4. The dry sand was characterized by a density of 1523 kg/m³, an internal friction angle of 31.8°, and zero cohesion. The Mohr–Coulomb model was adopted to represent the constitutive behavior of the dry sand. The simulation utilized a time step of 1 × 10⁻⁷ s, with a total of 30 million steps defined to cover the 3-s duration required for the dry sand flow to stabilize in the tests. To simulate impacts at varying inclination angles, a gravity decomposition approach was employed.

Figure 5 presents a comparison of the experimental and simulated impact forces under various inclination angles. The results indicate that the simulated peak impact forces exhibit a close alignment with the experimental measurements. However, at smaller inclination angles, the simulated residual impact force surpasses the experimental values. This discrepancy can be explained by the increased volume of dry sand reaching the structure at shallow angles, leading to a larger proportion of static pressure contributing to the overall force composition. In the design of rigid structures, the peak impact force is a critical parameter for assessing strength, while the residual impact force typically does not play a decisive role and is often omitted from consideration. Consequently, the SPH simulation method showcased in this study offers a reliable approach for determining the peak impact force in such scenarios.

4. Prediction and Analysis of Peak Impact Force of Landslide Impacting Bridge Piers

4.1. Numerical Simulation of a Conceptual Landslide Impacting Piers

To investigate the influence of soil strength parameters on the peak impact force in landslide–pier interactions, this study accounted for both the randomness and correlation of cohesion and the internal friction angle. The cohesion was characterized by a mean value of 20 kPa and a coefficient of variation (COV) of 0.2, while the internal friction angle had a mean of 20° and a COV of 0.1. Both parameters were assumed to follow a lognormal distribution, with a correlation coefficient of −0.7 between them [27]. Nevertheless, the findings of this study are entirely contingent on the chosen set of initial parameters, which, derived from regional geological data, may not fully account for the local variability of specific sites. Based on this probabilistic framework, 50 sets of strength parameters (Figure 6) were generated and used in subsequent numerical simulations. To effectively mitigate the risk of overfitting during the training phase with only 50 data points, we employed a comprehensive strategy. First, a feedforward neural network from the MATLAB (2023a) Neural Network Toolbox was utilized. Second, the optimal model architecture was determined using a grid search approach combined with cross-validation, with the objective of identifying the model with the lowest complexity and highest performance. This method fundamentally reduces the risk of overfitting by optimizing the model structure.

The landslide–pier impact model was built in LS-PrePost. The material parameters for the landslide mass included a density of 1950 kg/m³, a shear modulus of 1 × 10⁷ Pa, and a Poisson’s ratio of 0.495, along with the cohesion and internal friction angle values. Among these, cohesion and the internal friction angle were treated as random variables in the analysis. This focus on their intrinsic variability is justified as they constitute the most direct and fundamental parameters governing shear strength within the Mohr-Coulomb criterion, allowing us to isolate a primary source of uncertainty in this foundational study. The sliding surface of the landslide was simulated using a rigid boundary condition, and the pier was likewise modeled as a rigid material. This dual rigid-body simplification serves to establish a controlled baseline by eliminating energy dissipation through both bed and structural deformation, thereby allowing a clearer investigation into the flow-impact mechanics and the propagation of strength parameter uncertainty to the peak impact force. The time-history curve of the impact force was obtained by extracting the contact output force between the landslide and the first pier in the path of the sliding mass. We explicitly acknowledge in the manuscript that this rigid-pier assumption likely results in a conservative overestimation of the peak force. However, for the primary objective of this study—which is to assess the probability distribution of landslide peak impact force governed by the uncertainty of soil strength parameters—the clarity and computational efficiency afforded by this simplification are paramount, and the identified competitive mechanism between cohesion and internal friction angle remains robust under this assumption. Future work will incorporate structural flexibility to refine predictions of absolute force values.

The particle size was selected as 0.3 m. This value represents an optimal compromise determined by balancing the requirements of physical fidelity against computational feasibility. This selection was based on the following key considerations. The Resolution of Critical Geometric Features: While the landslide exhibited a longitudinal extent of 7 m, its dominant characteristic dimension (the average thickness) ranged between 1 and 3 m. Employing a particle size of 0.3 m ensured that even the minimum thickness of 1 m was resolved by at least three particle layers in the thickness direction. This satisfied the minimum resolution criterion necessary for the SPH method to accurately capture shear flow behavior and velocity gradients. The Management of Computational Cost: The choice of a 0.3-m particle size effectively constrained the total particle count to a computationally tractable level. Reducing the particle size to 0.1 m would increase the total number of particles by approximately an order of magnitude (a factor of 27), resulting in prohibitively high computational costs. Therefore, the 0.3-m particle size maximized computational efficiency while maintaining sufficient accuracy in simulating the physical processes of the landslide.

By leveraging the 50 parameter sets selected earlier, the corresponding peak impact force for each set was calculated and utilized for subsequent neural network training and validation. The main sliding process of a representative sample is depicted in Figure 7, with different colors illustrating distinct model components.

4.2. Training and Accuracy Verification of Neural Network

In the training of the neural network model, 50 datasets were divided into training, validation, and test sets, with 34, 8, and 8 samples, respectively. The network architecture consisted of an input layer with 2 neurons, corresponding to the soil strength parameters cohesion c and internal friction angle φ; a hidden layer with 15 neurons, determined via grid search with cross-validation; and an output layer with 1 neuron for predicting the peak impact force. The hidden layer employed a hyperbolic tangent sigmoid activation function to introduce nonlinearity, enabling the network to learn complex input–output mapping relationships, while the output layer utilized a linear transfer function, consistent with regression problems, allowing continuous output across the real number range. Training was performed using the Levenberg–Marquardt optimization algorithm, a second-order method that does not employ a conventional fixed learning rate. Instead, it controls the weight update process through an adaptive damping parameter $\mu$. The weight update is computed as $\mathsf{\Delta }\omega =-{\left(J^{T}J+\mu I\right)}^{-1}{J}^{T}e$, where $J$ is the Jacobian matrix of errors with respect to weights. The initial damping parameter was set to ${\mu }_0=0.001$ and is adaptively adjusted during training: μ decreases when the error is reduced (accelerating convergence) and increases when the error rises (enhancing stability). This approach combines the advantages of gradient descent and the Gauss–Newton method, proving particularly suitable for small- to medium-scale datasets and converging rapidly to an optimal solution within a limited number of iterations.

To prevent overfitting and enhance generalization, multiple regularization strategies were implemented. Systematic data partitioning ensured independent model evaluation; early stopping terminated training when the validation error failed to improve for six consecutive epochs; the streamlined network architecture served as an implicit form of structural regularization. Mathematically, the Levenberg–Marquardt algorithm incorporates an L2 regularization (weight decay) mechanism during optimization. The objective function minimized by the optimizer is expressed as:

$$J\left(\omega \right)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\left(y_i-{\hat{y}}_i\right)}^2}+\lambda {\displaystyle \sum _{j=1}^M{\omega }_j^2}$$

(9)

where the first term is the mean squared error and the second term is the L2 penalty on all network weights. This explicit penalty term suppresses excessively large weight values, thereby further improving model smoothness and stability. The mean squared error (MSE) was adopted as the loss function, and model fitting was performed using the least squares regression.

Figure 8 compares the predicted and simulated results across the training, validation, and test sets. The MSE values for all three sets decreased synchronously and eventually stabilized, indicating that the model effectively captured the underlying patterns in the data, with a final MSE of 1846.7. The best performance was achieved by the model from the sixth training epoch. A comparison between the target and predicted peak impact force values showed that predictions for all three datasets aligned closely along the y = x line. The Pearson correlation coefficients for the training, validation, and test sets all approached 1.00, demonstrating the strong predictive capability of the trained neural network. This high performance confirms that the training dataset, generated through systematic variation in the key strength parameters (cohesion and internal friction angle) within a simplified yet controlled numerical framework, is sufficient to capture the dominant physical relationship for accurate prediction. The model can therefore be reliably employed for rapid and accurate predictions of the peak impact force in slope–pier interaction scenarios, even under varying soil strength parameters. Both the validation set and the test set showed results that clustered closely around the y = x line, indicating excellent agreement between the predicted and actual values. Meanwhile, the Pearson correlation coefficients of the training set (P_trian = 0.957), validation set (P_validation = 0.992), and test set (P_trian = 0.957) were all close to one. These results highlight the robust predictive capability of the trained neural network, thereby demonstrating its suitability for quickly predicting peak impact forces in landslide–pier interactions across a range of soil strength parameters.

4.3. Probability Distribution of the Peak Impact Force

In this section, the cohesion and internal friction angle are characterized by mean values of 20 kPa and 20°, respectively, with coefficients of variation of 0.2 and 0.1. Both parameters were assumed to follow a lognormal distribution. Based on the 3σ principle, the parameter ranges were established as 8–32 kPa for cohesion and 8–32° for the internal friction angle while maintaining a correlation coefficient of −0.7 between them. Using this configuration, 10,000 sample sets of soil strength parameters were generated. These samples were subsequently input into the trained neural network model to predict the corresponding peak impact force for each parameter set. The frequency distribution of the predicted peak impact forces is depicted in Figure 9. The bimodal shape of the distribution clearly indicates the existence of two distinct impact mechanisms for landslides impacting piers: one characterized by a low-impact-force mechanism and the other by a high-impact-force mechanism.

The probability density function of peak impact forces, denoted as f(F), can be considered a composite function consisting of two distinct subdistributions: a low-impact-force subdistribution f₁(F) and a high-impact-force subdistribution f₂(F). The mathematical expression for this function is as follows:

$$f(F)={\omega }_1\cdot f_1(F)+{\omega }_2\cdot f_1(F)$$

(10)

where ω₁ and ω₂ are weight coefficients. Then, using the kernel density estimation, a smoothed probability density function can be obtained as follows:

$$\hat{f}(F)={\displaystyle \frac{1}{nh}}{\displaystyle {\sum }_{i=1}^{n}K(\frac{F-F_i}{h})}$$

(11)

where K is the kernel function, and h is the bandwidth. Thereafter, the boundary threshold can be determined by identifying the local maximum values and calculating their arithmetic average, which are shown in Equations (11) and (12). In this study, the critical value obtained for separating the high- and low-impact forces (T = 467 kN) represents the observed statistical boundary within the dataset. However, it should be noted that this critical value is not a universal or absolute benchmark applicable across all datasets.

$$peaks=\left\{F_0 | f^{\prime }(F_0)=0, f^{″}(F_0)<0\right\}$$

(12)

$$T={\displaystyle \frac{F_{peak1}+F_{peak2}}{2}}$$

(13)

According to the obtained demarcation threshold (T = 467 kN), the 10,000 data points were categorized into high- and low-impact groups, as shown in Figure 10. In this figure, the red dots denote the high-impact group, while the green dots represent the low-impact group. The high-impact group comprises 4343 data points, with impact forces ranging from 571.1 kN to 858.1 kN. The corresponding cohesion values vary between 14 kPa and 21.6 kPa, while the internal friction angles range from 18.1° to 25.3°. In contrast, the low-impact group consists of 5657 data points, with impact forces between 126.9 kN and 319.3 kN. The cohesion in this group ranges from 18.5 kPa to 24.9 kPa, and the internal friction angle from 15° to 22.4°. These distributions clearly indicate a distinct trend: as cohesion increases, the peak impact force tends to decrease, whereas high-impact forces are predominantly linked to higher internal friction angles. This behavior can be explained by the distinct mechanical roles played by cohesion and internal friction under high-velocity-impact conditions. Cohesion, which arises from interparticle cementation, is responsible for the static strength of the soil. However, under extreme dynamic loading, stress waves rapidly disrupt these cemented bonds, leading to brittle failure of the cohesive component. Once cohesion is compromised, its contribution to strength is substantially diminished, reducing its influence during both the peak and residual phases of the dynamic response. In contrast, the frictional strength, characterized by the internal friction angle, is highly rate- and stress-dependent. Under high-impact-force conditions, sliding friction at interparticle contacts and dilatancy effects are significantly amplified. Furthermore, the high contact stresses induced by impact result in the frictional strength increasing linearly with normal stress. As a result, under the combined conditions of high stress and high impact forces, the internal friction angle becomes the dominant parameter controlling impact resistance.

5. Conclusions

In this study, we introduce an integrated framework that combines Smoothed Particle Hydrodynamics (SPH) with artificial neural networks to predict the peak impact force exerted by landslides on piers. Utilizing Monte Carlo simulation, we develop a stochastic analysis approach that accounts for uncertainties in material strength parameters. The principal findings are as follows:

• Trained on 50 SPH simulation datasets, the neural network surrogate model achieved strong predictive accuracy, as evidenced by Pearson correlation coefficients of 0.957 for the training set, 0.992 for the validation set, and 0.992 for the test set. The high consistency across all datasets confirms the model’s robustness and reliability for subsequent probabilistic analyses.
• Monte Carlo simulations involving 10,000 samples revealed a bimodal distribution in peak impact forces. Two distinct failure modes were identified: a low-impact mode (F < 467 kN), which dominated when cohesion exceeded 20 kPa, and a high-impact mode (F ≥ 467 kN), primarily governed by internal friction angles above 22°. These results highlight the competitive interplay between strength parameters in determining failure behavior.
• The developed surrogate model enhances risk assessment efficiency by several orders of magnitude, enabling rapid regional-scale screening of infrastructure’s vulnerability to landslides. It offers direct applicability in revising bridge impact design standards and formulating reinforcement strategies for existing structures, thereby providing a theoretical foundation for resilience-based design.
• The surrogate model is trained on a focused dataset of 50 SPH simulations, which may constrain its extrapolation capability to geological conditions and structural configurations beyond those parameterized in this work. Future research will expand the training database to encompass a wider range of scenarios, thereby enhancing the model’s robustness and generalizability for practical engineering applications.