Showing papers in "Journal of Engineering Mechanics-asce in 2022"

Analytical Approach to Estimating the Influence of Shotcrete Hardening Property on Tunnel Response

Abstract: Ignoring the shotcrete hardening property in tunnel design may lead to an overestimation of the bearing capacity of shotcrete liners and underestimation of tunnel convergence. However, acco...

Micromechanical Investigation of Particle-Size Effect of Granular Materials in Biaxial Test with the Role of Particle Breakage

Abstract: Understanding the effect of particle size on the shear strength of granular materials is important for geotechnical design and construction. However, previous studies show contradicting results on the relationship between particle size and shear strength. Additionally, the effect of particle breakage on this relationship has not been fully revealed. In this study, a series of biaxial tests have been simulated with the discrete element method to explore the particle-size effect of sand considering the role of particle breakage. The sand specimens have parallel particle-size distributions. The sequential breakage model has been used to simulate particle breakage, which is a combination of replacement and cluster methods. The main conclusions of this study are: (1) the relationship of peak shear strength and particle size depends on the crushability of particles and relative density of specimens; (2) the particle size and crushability have a very slight effect on the residual shear strength; and (3) at the microscale, the relationship between shear strength and particle size is positively related to the friction utilization ratio.

Micromechanical Investigation of Particle-Size Effect of Granular Materials in Biaxial Test with the Role of Particle Breakage

Abstract: Understanding the effect of particle size on the shear strength of granular materials is important for geotechnical design and construction. However, previous studies show contradicting res...

Physics-Informed Multifidelity Residual Neural Networks for Hydromechanical Modeling of Granular Soils and Foundation Considering Internal Erosion

Abstract: Coupled hydromechanical finite-element modeling of granular soils, taking into account internal erosion, is computationally prohibitive. Alternative data-driven approaches require large data sets for training and often provide poor generalization ability. To overcome these issues, this study proposed a physics-informed multifidelity residual neural network (PI-MR-NN) modeling strategy. The model was first trained using low-fidelity data to focus on capturing the main underpinning physical laws. Subsequent training on sparser high-fidelity data was then used to calibrate and refine the model. Physical constraints, e.g., boundary conditions, were incorporated through modifications to the loss functions. Feedforward and long short-term memory neural networks were considered as the baseline algorithms for training models. The PI-MR-NN was first used to reproduce synthetic results generated by the soil constitutive model SIMSAND and a published internal erosion model. The developed data-driven model was then applied to simulate the breach of a practical dike-on-foundation case and to predict its temporal responses. All results indicated that the hydromechanical response of porous media can be accurately captured using the proposed PI-MR-NN model. The novel training strategy mitigates the dependency of model performance on the training data set and architecture of the neural network, and the use of physical constraints improves training efficiency and enhances the model’s predictive robustness.

Probing Fabric Evolution and Reliquefaction Resistance of Sands Using Discrete-Element Modeling

Abstract: Recent case histories have demonstrated that soil liquefaction can occur repeatedly at a site during a sequence of earthquake events. Field observations and laboratory tests imply that reliquefaction resistance can be markedly different, depending on the strain histories and induced fabric change. However, direct observation of fabric evolution during the entire process remains limited. In this study, we perform simulation using a three-dimensional (3D) discrete-element method (DEM) to quantify fabric evolution in granular soils during liquefaction, reconsolidation, and reliquefaction processes, with the goal of investigating the effects of fabrics on reliquefaction resistance. Clumped particles are used to construct realistic particle shapes of Toyoura sand in the DEM, and soil fabric is characterized by a coordination number Z and a degree of anisotropy ac. By reconsolidating samples at different states after the first liquefaction, we describe the relationships between the maximum preshear strains versus volumetric compression and the resulting soil fabrics (Z, ac) after reconsolidation. Finally, we set up correlations between the reliquefaction resistance and soil fabrics (Z, ac). This study shows that the effects of strain histories on reliquefaction resistance are intrinsically attributed to changes in soil fabrics before and after reconsolidation. The DEM simulation also generates data that are consistent with laboratory tests and provides micromechanical insights into the reliquefaction phenomenon.

Statistical Assessment of In-Plane Masonry Panels Using Limit Analysis with Sliding Mechanism

Abstract: Historical masonry structures have a great interest in civil engineering because they constitute a large part of the world’s building heritage. In this paper, the effects that different geometrical (panel ratio, block ratio, and bond type) and mechanical (friction ratio) parameters have on the in-plane structural response of brick masonry panels are investigated. A discrete modeling approach, based on a limit analysis and capable of reproducing sliding mechanisms, formulation by one of the authors has been adopted, enhanced, and implemented. Results, in terms of collapse multipliers and collapse mechanisms, are presented and analyzed following a systematic statistical approach. Statistically significant effects have been found for each factor considered. Furthermore, the statistical model adopted included nonlinear terms that allowed the identification of whether the effect of one parameter on the response depends on the level of any other parameters. Thus, it was observed that two-way factor interactions played an important role in the in-plane response of masonry panels. The panel ratio-friction ratio two-way factor interaction was the one with a more significant effect.

3D Multiscale Modeling of Asphalt Pavement Responses under Coupled Temperature–Stress Fields

Abstract: This study proposed a three-dimensional (3D) multiscale modeling method to investigate the responses of asphalt pavement subjected to coupled temperature-stress fields. In this method, finite element models of asphalt pavement at two different scales, i.e., the macroscale (pavement level) and mesoscale (mixture level), were developed separately and connected through a two-way coupled approach, including a homogenization (upscaling) procedure and a mapping (downscaling) procedure. X-ray computed tomography (CT) scanning technology was adopted to acquire realistic mesostructure images of asphalt concrete, and a digital image processing technology was employed to reconstruct its 3D mesoscale representative volume element model from these CT images. Both thermal and mechanical properties of asphalt concrete at the two scales were considered in the multiscale simulation. Also, actual climatic data sets, including air temperature history, solar radiation history, and mean wind speeds, were incorporated into the computation. The results showed that the developed multiscale method furnishes an in-depth insight into the thermomechanical behaviors of asphalt pavement at different length scales under both tire loading and realistic environmental factors. The consideration of coupled temperature-stress fields varying with time has a significant impact on the accurate determination of the critical responses within asphalt pavement. Because the developed method is capable of simultaneously taking into account multiple factors, including mixture component properties and mesostructures, pavement structures, tire loads, and climatic information, it can be expected to serve as a mechanistic tool for facilitating and enhancing the analysis and design of asphalt pavement.

A Hierarchical Bayesian Similarity Measure for Geotechnical Site Retrieval

Abstract: Geotechnical site retrieval refers to the quantitative identification and extraction of sites similar to a given target site from predocumented generic sites in a database. This is known as the “site recognition challenge.” Recently, the second and third authors of this paper proposed a Bayesian similarity measure between a target site and generic records in the database. However, the proposed method can only retrieve “similar database records” but not “similar database sites”; that is, records are not grouped according to their test locations within a site boundary. The purpose of the current paper was to propose a novel Bayesian similarity measure between the target site and a database site to extract similar sites from a database. This “site retrieval” approach is more “explainable” to a geotechnical engineer because an engineer has an opportunity to accept or reject the identified “similar” sites based on his or her experiences and judgment. The human engineer can engage an explainable algorithm in a decision loop in a more meaningful way. Based on the hierarchical Bayesian model (HBM) previously developed by the second and third authors, this study further proposed a Bayesian method of measuring similarity between the target site and database sites for the purpose of geotechnical site retrieval. This hierarchical Bayesian measure elegantly reduces to the classical Kullback–Leibler divergence for complete multivariate data. The HBM was used to simultaneously model intrasite and intersite variability and construct the site-specific multivariate distribution for the database sites. Site retrieval was performed by measuring the similarity between the target and database sites in the form of a multivariate likelihood. It is shown that the proposed hierarchical Bayesian method can yield a meaningful interpretation of intersite similarity and can successfully be used for site retrieval. The proposed approach can also quantify the statistical uncertainty due to sparse (limited) and incomplete (missing) data.

A Modified Fractional-Order Derivative Zener Model for Rubber-Like Devices for Structural Control

Abstract: Fractional-order derivative (FD) models are widely used to describe the dynamic behavior of rubber-like materials. However, experiments have shown that FD models cannot reproduce the amplit...

Mechanical Behavior of Shape-Memory Alloy Triply Periodic Minimal Surface Foam Based on Schwarz Primitive

Abstract: This work investigated the effective structural and functional behavior of shape-memory alloy (SMA)-based triply periodic minimal surface foams based on the Schwarz primitive (P-foams) using finite-element analysis (FEA) and numerical homogenization methods. The effect of relative density and applied temperature on the homogenized mechanical behavior of the SMA foam, including its superelasticity, and the evolution of the effective martensite volume fraction with the applied load was investigated considering axial and shear loading cases. In contrast to dense SMA, the effective martensite volume fraction in the considered foam was found to vary exponentially with the strain in the case of monotonic loading, asymptotically approaching 1 as the strain increased indefinitely. Moreover, the effective superelasticity of the SMA P-foam was found to be facilitated by decreased temperature and relative density. The onset of phase transformation for the P-foam under various loading scenarios was shown to be well approximated using an extended Hill loading surface.

Modeling Combined Fabric Evolution in an Anisometric Granular Material Driven by Particle-Scale X-Ray Measurements

Abstract: A combined fabric evolution (CFE) model is used to predict real-world fabric evolution of a strongly anisometric granular material under triaxial loading, connecting advances in theoretical...

Physics-Informed Neural Network Solution of Thermo–Hydro–Mechanical Processes in Porous Media

Abstract: Physics-Informed Neural Networks (PINNs) have received increased interest for forward, inverse, and surrogate modeling of problems described by partial differential equations (PDE). However, their application to multiphysics problem, governed by several coupled PDEs, present unique challenges that have hindered the robustness and widespread applicability of this approach. Here we investigate the application of PINNs to the forward solution of problems involving thermo-hydro-mechanical (THM) processes in porous media, which exhibit disparate spatial and temporal scales in thermal conductivity, hydraulic permeability, and elasticity. In addition, PINNs are faced with the challenges of the multi-objective and non-convex nature of the optimization problem. To address these fundamental issues, we: (1)~rewrite the THM governing equations in dimensionless form that is best suited for deep-learning algorithms; (2)~propose a sequential training strategy that circumvents the need for a simultaneous solution of the multiphysics problem and facilitates the task of optimizers in the solution search; and (3)~leverage adaptive weight strategies to overcome the stiffness in the gradient flow of the multi-objective optimization problem. Finally, we apply this framework to the solution of several synthetic problems in 1D and~2D.

Soft Missing Rib Structures with Controllable Negative Poisson’s Ratios over Large Strains via Isogeometric Design Optimization

Abstract: Soft network materials with biomimetic mechanical properties such as a negative Poisson’s ratio have important applications in tissue engineering, biomedical devices, and soft robotics. Several finite-element (FE)-based design strategies have been developed to produce network materials with prescribed mechanical properties. However, obtaining network designs with a prescribed negative Poisson’s ratio over large strain remains a challenge. Here, an optimization framework was developed using isogeometric analysis and a genetic algorithm for the design of soft missing rib structures with controllable negative Poisson’s ratios over large strains. The missing rib structures with six ligaments were optimized to achieve constant negative Poisson’s ratios ranging from −0.1 to −0.6 up to 70% tensile strain under plane stress condition. The optimization framework was employed to obtain a missing rib network design with deformation behavior closely matching that of cat’s skin up to 90% tensile strain. This optimized design was fabricated using a biocompatible material via liquid additive manufacturing and validated experimentally, demonstrating the potential of the soft missing rib designs for biomedical applications.

Two-Scale Discrete Element Modeling of Gyratory Compaction of Hot Asphalt

Abstract: This paper presents a discrete element model for simulations of the compaction process of hot mixed asphalt (HMA). The model is anchored by the concept of a fine aggregate matrix (FAM), whi...

Two-Scale Discrete Element Modeling of Gyratory Compaction of Hot Asphalt

Abstract: This paper presents a discrete element model for simulations of the compaction process of hot mixed asphalt (HMA). The model is anchored by the concept of a fine aggregate matrix (FAM), which consists of the binder and fine aggregates. In the simulation, the coarse aggregates are explicitly modeled as composite particles. Meanwhile, the FAM is considered as the thick coating of the coarse aggregates with complex constitutive laws. Interparticle interactions include influences of (1) particle properties via Hertz–Mindlin relations; and (2) FAM properties via lubrication relationships. The lubrication relationships include a variable for viscosity for which we derive normal and tangential rate-dependent forms using rheology theory of dense granular-fluid systems, verified reasonable for our systems with the discrete element simulations and experiments with FAM. We assimilate these elements into gyratory compaction simulations of HMA of different aggregate size distributions. We compare these with experiments and find that this model is capable of capturing the measured effects of grain size distribution on the overall compaction behavior of HMA. We conclude by highlighting the advantages of this discrete element model for HMA compaction problems.

Experimental and Numerical Study on Mixed Mode I-II Fatigue Crack Propagation in Concrete

Abstract: To ensure the safety of concrete structures under fatigue loading, the fatigue crack propagation in concrete needs to be evaluated accurately. In this paper, a numerical method for mixed mode I-II fatigue crack propagation in concrete is proposed, in which the stress intensity factor (SIF)-based crack propagation criterion is employed, and the degradation of the cohesive force under fatigue loading is considered quantitatively. To validate the applicability of the numerical method, the mixed mode I-II fatigue fracture test of the three-point bending (TPB) beam is conducted. The fatigue crack propagation length is measured with the digital image correlation (DIC) method. Eventually, the applicability of the numerical method is validated by a reasonable agreement between the numerically derived crack propagation path, crack mouth opening displacement (CMOD), crack mouth sliding displacement (CMSD), crack propagation length, and mode I SIF and the experimental results. It is concluded that the proposed numerical method can be used to evaluate the mixed mode I-II fatigue crack propagation process of concrete when the initial fracture toughness, Poisson’s ratio, and Young’s modulus under static loading and the tension-softening constitutive relation under fatigue loading are given. In addition, the experimental results indicate that the mixed mode I-II fatigue failure of concrete occurs when the mode I SIF reaches a critical value, regardless of the fatigue load level and the fatigue life. The numerical results show that the mixed mode I-II fatigue crack propagation path is independent of the fatigue load level and approximately identical to that under static loading.

Bending Fatigue Life Evaluation of Bridge Stay Cables

Abstract: Large-amplitude cable vibrations are remarkably common on cable-stayed bridges due to various aerodynamic loading mechanisms and/or motion of the cable ends. Geometric nonlinearity can be important in the dynamic behavior, and significant local bending stresses can arise at the anchorages, where rotation is restrained. This leads to a concern about the fatigue of these cables from the cyclic stress variations. This paper presents an analytical bending fatigue model for estimating the fatigue life of low-sag cables subjected to harmonic loading. Using this framework, the fatigue life of cables at the anchorage zone and guide deviator (if present) under external loading can be predicted. The results show that the use of a guide deviator can significantly extend the cable’s fatigue life at the anchorage. For cables with a guide deviator subject to severe loading conditions, the fatigue life is limited by the behavior at the guide rather than the anchorage, which is consistent with previous observations. The fatigue life is greatly reduced if the cable jumps to a multimodal dynamic response due to the cable nonlinearity. The single-mode zone where the dynamic response of the cable is always stable in a single mode, leading to a relatively long fatigue life, has been identified. Finally, the effect of cable inclination angle and ratio of cable weight to tension on the fatigue life has been analyzed.

Nonlinear Partial Differential Equation for Unsteady Vertical Distribution of Suspended Sediments in Open Channel Flows: Effects of Hindered Settling and Concentration-Dependent Mixing Length

Abstract: A model on one-dimensional unsteady suspended sediment transport has been developed in this study by including the effect of hindered settling and from mixing length point of view. The sedi...

Near-Real-Time Identification of Seismic Damage Using Unsupervised Deep Neural Network

Abstract: Prompt identification of structural damage is essential for effective postdisaster responses. To this end, this paper proposes a deep neural network (DNN)–based framework to identify seismic damage based on structural response data recorded during an earthquake event. The DNN in the proposed framework is constructed by Variational Autoencoder, which is one of the self-supervised DNNs that can construct the continuous latent space of the input data by learning probabilistic characteristics. The DNN is trained using the flexibility matrices obtained by operational modal analysis (OMA) of simulated structural responses of the target structure under the undamaged state. To consider the load-dependency of OMA results, the undamaged state of the structure is represented by the flexibility matrix, which is closest to that obtained from the measured seismic response in the latent space. The seismic damage of each member is then estimated based on the difference between the two matrices using the flexibility disassembly method. As a numerical example, the proposed method is applied to a 5-story, 5-bay steel frame structure for which structural analyses are first performed under artificial ground motions to create train and test datasets. The proposed framework is verified with the near-real-time simulation using ground motions of El Centro and Kobe earthquakes. The example demonstrates that the proposed DNN-based method can identify seismic damage accurately in near-real-time.

Physics-informed neural network solution of thermo-hydro-mechanical (THM) processes in porous media

Abstract: Physics-Informed Neural Networks (PINNs) have received increased interest for forward, inverse, and surrogate modeling of problems described by partial differential equations (PDE). However, their application to multiphysics problem, governed by several coupled PDEs, present unique challenges that have hindered the robustness and widespread applicability of this approach. Here we investigate the application of PINNs to the forward solution of problems involving thermo-hydro-mechanical (THM) processes in porous media, which exhibit disparate spatial and temporal scales in thermal conductivity, hydraulic permeability, and elasticity. In addition, PINNs are faced with the challenges of the multi-objective and non-convex nature of the optimization problem. To address these fundamental issues, we: (1)~rewrite the THM governing equations in dimensionless form that is best suited for deep-learning algorithms; (2)~propose a sequential training strategy that circumvents the need for a simultaneous solution of the multiphysics problem and facilitates the task of optimizers in the solution search; and (3)~leverage adaptive weight strategies to overcome the stiffness in the gradient flow of the multi-objective optimization problem. Finally, we apply this framework to the solution of several synthetic problems in 1D and~2D.

Experimental Investigation and Multiscale Modeling of VE Damper Considering Chain Network and Ambient Temperature Influence

Abstract: Viscoelastic (VE) dampers are one of the most promising techniques for reducing vibration in engineering structures caused by earthquakes and wind. This work aims to develop a kind of high-dissipation VE damper for civil structures at low frequency and large amplitude in shear mode. First, nitrile rubber (NBR)/organic small-molecule composite VE materials are optimized and then made into VE damper. In order to test the mechanical performance and energy dissipation performance of the VE damper, the dynamic mechanical performance experiments at different temperatures, frequencies, and amplitudes were implemented. The experimental results show that the VE damper exhibits great stiffness and excellent energy dissipation capacity under different loading conditions. Second, a fractional derivative model based on Gauss microchain, Williams–Landel–Ferry (WLF) equation, and internal variable theory is proposed to accurately describe the effects of temperature, frequency, and amplitude on the dynamic mechanical properties of VE dampers. Finally, the accuracy of the mathematical model of VE damper is verified by comparing the calculated results with the experimental results. The study provides a theoretical basis for effective vibration reduction of civil structures with VE dampers at low frequency.

Noncircular Cavity Expansion in Undrained Soil: Semi-Analytical Solution

Abstract: The cavity expansion approach has been a popular tool to interpret a wide range of geotechnical problems over the last several decades. Most previous research focused on the expansion of cylindrical and/or spherical cavities, whereas nonstandard cavities have received much less attention. To address this shortcoming, this paper presents a general theoretical framework for two-dimensional (2D) displacement-controlled undrained noncircular cavity expansion (N-CCE) in undrained soil. The new approach combines strain path method (SPM) concepts and conformal mapping to determine the soil velocity and strain rate fields analytically. The soil displacement and strain subsequently are determined by integrating the soil velocities and strain rates along the strain path using a series of transformed ordinary differential equations. In this study, the modified Cam Clay (MCC) effective stress constitutive model was used to determine the soil stress–strain relationship, and consolidation effects were captured using finite-difference calculations. The proposed methodology was validated by comparing the reduced solution for a circular cavity with traditional circular cavity expansion theory. A parametric analysis subsequently was undertaken to explore the influence of three noncircular cavity shapes on expansion-induced soil deformation mechanisms, shear strains, effective stresses, and pore-water pressure development and consolidation. The proposed solution can be implemented with any critical state–based soil model and can be applied to arbitrary noncircular cavity problems.

Nonlinear Partial Differential Equation for Unsteady Vertical Distribution of Suspended Sediments in Open Channel Flows: Effects of Hindered Settling and Concentration-Dependent Mixing Length

Abstract: A model on one-dimensional unsteady suspended sediment transport has been developed in this study by including the effect of hindered settling and from mixing length point of view. The sediment diffusion term has been related to mixing length, which has been taken as a function of concentration. The mixing length and settling velocity are reduced due to the presence of particles in the flow. By considering these effects in the governing equation, the resulting partial differential equation (PDE), which becomes highly nonlinear, has been solved numerically using the most generalized boundary conditions available in the literature. For the purpose of validation, the derived model is compared with similar existing works under certain specified conditions. Apart from that, the obtained solution has also been compared with available laboratory data for steady and uniform flow because over a large span of time, the model behaves like a steady one. Furthermore, effects of damping function and hindered settling are explained both graphically and physically.

Energy-Based Fracture Criterion of Rock–Concrete Interface Considering Viscoelastic Characteristics

Abstract: To investigate the fracture property of the rock–concrete interface after sustained loading, creep tests were conducted on composite rock–concrete beams with two mechanical grooving interfaces, 4×4 and 7×7 interfaces, under two sustained load levels, 50% and 75% of the maximum load. After 90-day sustained loading, the creep specimens were unloaded and reloaded to failure in the subsequent three-point bending tests. The viscoelastic characteristic of the rock–concrete interface was numerically simulated by employing the Bailey–Norton creep law to describe the constitutive relationships of bilateral materials. Based on the numerical results, the elastic and creep strain energies within the whole specimen and the core region near the crack tip were calculated during the loading process. The results indicated that during the creep tests, the local elastic strain energy decreased due to the stress relaxation near the crack tip, and the local creep strain energy increased due to the accumulated creep strain. It was found that, at crack initiation status, the average elastic strain energy density in the core region after sustained loading was identical to that under static loading, which indicates that the elastic energy in the core region dominated the fracture behavior. Then, an energy-based fracture criterion was proposed to judge the crack initiation of the rock–concrete interface considering viscoelastic characteristics. It was found that the average elastic strain energy density of the rock–concrete interface was not affected by the viscoelasticity, which can be regarded as a material parameter.

One-Dimensional Nonlocal Model for Gyratory Compaction of Hot Asphalt Mixtures

Abstract: Gyratory compaction has widely been used to evaluate the compactability of hot asphalt mixtures. Existing efforts on modeling of gyratory compaction have largely been devoted to sophisticated high-fidelity numerical simulations. This paper presents a one-dimensional nonlocal model for gyratory compaction. The model is anchored by the principle of mass conversation, in which the local densification rate is formulated as a function of the nonlocal packing fraction. The nonlocal model involves a material characteristic length scale, which is independent of the specimen size. The nonlocality gives rise to strong effects of the specimen height on the overall compaction curve as well as on the profile of the local packing fraction. A set of gyratory compaction experiments is performed on specimens of different heights. It is shown that the model is able to capture the measured size effect on the compaction curves. A parametric study is carried out to investigate the effects of nonlocality and model parameters on the predictions of compaction curve and profile of packing fraction.

An Explicit Solution of Modal-Damping Ratios for Higher Modes of a Cable with an External Damper

Abstract: Design of external dampers for stay cables requires repetitive evaluation of damping ratios for target modes of cables under various damper parameters and installation positions, and explic...

A Dynamic Constitutive Model for Rock Materials Subjected to Medium- and Low-Strain-Rate Dynamic Cyclic Loading

Abstract: Rock materials are often subjected to medium- and low-strain-rate dynamic cyclic loading such as earthquake action; however, few dynamic constitutive models for rock materials under dynamic...

Efficient Geotechnical Reliability Analysis Using Weighted Uniform Simulation Method Involving Correlated Nonnormal Random Variables

Abstract: In the context of probabilistic analysis involving uncertain factors, efficient reliability methods play an important role for promoting a wider application in engineering practice. Although the ordinary Monte Carlo simulation (MCS) can simulate the probabilistic performance of a complex engineering system well and has been widely-employed in reliability analysis because of its simplicity and accuracy, the unavoidable computational burden to ensure sufficient accuracy often limits its use as a reference tool only for academic purposes. This paper proposes a modified weighted uniform simulation (WUS) method for reliability analysis involving nonnormal random variables, in which the Nataf transformation is adopted to effectively transform the correlated nonnormal variables into independent standard normal variables. This method takes into account the correlations between random variables, while the sample size is greatly reduced under the same accuracy requirements. Four examples of reliability analysis are presented to demonstrate the feasibility of the WUS method. It is shown that the proposed method can yield sufficiently accurate reliability analysis results with a reasonably small sample size compared to ordinary MCS. In particular, the most probable failure point (MPP), which is the basis for reliability-based design works, can also be directly obtained during the simulation process.