Stochastic Finite Element Analysis Research Articles

Accuracy of stochastic finite element analyses for the safety assessment of unreinforced masonry shear walls

ABSTRACT To examine structural safety and reliability, an accurate prediction of the variability of structural resistance must first be determined. This may be achieved through extensive physical testing or, more commonly in modern research, synthetic data generation, such as stochastic finite element analyses (SFEAs). Due to the prevalence and versatility of such techniques, and the need for a high level of confidence when performing a safety assessment, an understanding of the accuracy of data derived from SFEAs is essential. In this paper, SFEA models have been developed to predict the responses of 16 unreinforced masonry walls tested in a laboratory under cyclic in-plane lateral loading. Each SFEA has been developed to reflect the as-built conditions of these experimental specimens and focus on predicting the shear capacity of the failure mechanisms that unreinforced masonry shear walls are susceptible to. From the results of these SFEAs, the accuracy (quantified as model error) of this modelling strategy has been estimated by comparing the peak in-plane shear resistances of the laboratory and numerical models.

Robust topology and discrete fiber orientation optimization under principal material uncertainty

This paper introduces a formulation of the robust topology optimization problem that is tailored for designing fiber-reinforced composite structures with spatially varying principal mechanical properties. Specifically, a methodology is developed that incorporates the spatial variability in the engineering constants of the composite lamina into the concurrent topology (i.e., material distribution) and morphology (i.e., fiber orientation distribution) optimization problem for the minimization of the robust compliance function. The spatial variability in the mechanical properties of the lamina is modeled as a homogeneous random field within the design domain by means of the Karhunen-Loe´ve series expansion, and is thereafter intrusively propagated into the stochastic finite element analysis of the composite structure. To carry out the stochastic finite element analysis per iteration of the optimization cycle, the first-order perturbation method is utilized for approximating the current state variables of the physical system. The resulting robust topology and fiber orientation optimization problem is formulated step-by-step for the minimization of the robust compliance function. With the view of solving the optimization problem at hand by means of gradient-based solution algorithms, the first-order derivatives of the involved design functions w.r.t. the associated design variables are analytically derived. The present work concludes with a series of numerical examples, focusing on the benchmark academic case studies of the 2D cantilever and the half part of the Messerschmitt-Bölkow-Blohm beam, aiming to demonstrate the developed methodology as well as to explore the effect that different parameterization instances of the random field bear on the predicted topology and morphology of the beams.

Stochastic finite element analysis using polynomial chaos on a flexible rotor with contact nonlinearity

Stochastic finite element analysis using polynomial chaos on a flexible rotor with contact nonlinearity

Uncovering the dual role of dimensionless radius in buckling of spherical shells with random geometric imperfections

Localized deformation and randomly shaped imperfections are salient features of buckling-type instabilities in thin-walled load-bearing structures. However, it is generally agreed that their complex interactions in response to mechanical loading are not yet sufficiently understood, as evidenced by buckling-induced catastrophic failures which continue to today. This study investigates how the intimate coupling between localization mechanisms and geometric imperfections combine to determine the statistics of the pressure required to buckle (the illustrative example of) a hemispherical shell. The geometric imperfections, in the form of a surface, are defined by a random field generated over the nominally hemispherical shell geometry, and the probability distribution of the buckling pressure is computed via stochastic finite element analysis. Monte-Carlo simulations are performed for a wide range of the shell's radius to thickness ratio, as well as the correlation length of the spatial distribution of the imperfection. The results show that over this range, the buckling pressure is captured by the Weibull distribution. In addition, the analyses of the deformation patterns observed during the simulations provide insights into the effects of certain characteristic lengths on the local buckling that triggers global instability. In light of the simulation results, a probabilistic model is developed for the statistics of the buckling load that reveals how the dimensionless radius plays a dual role which remained hidden in previous deterministic analyses. The implications of the present model for reliability-based design of shell structures are discussed.

Mechanistic mapping of random fields for stochastic finite element simulations of quasibrittle fracture

Spurious mesh sensitivity is a major challenge in continuum finite element (FE) simulations of damage and fracture of quasibrittle structures. It has been shown that the existing localization limiters, which largely focus on energy regularization, are insufficient for addressing the issue of mesh sensitivity in stochastic analysis. In this study, we investigate the mathematical algorithm for mapping the continuous random fields of material properties onto the FE meshes. This is a fundamental problem in stochastic FE analysis, which has profound implications for the mesh sensitivity in the prediction of the statistics of failure behavior. We adopt a continuum damage constitutive model, and develop a mechanistic mapping method. The projection of the random fields of material properties onto the FE mesh is governed by the prevailing damage pattern of the element. The damage pattern of each finite element is determined based on the spatial distribution of the damage of its surrounding elements. Meanwhile, the prevailing damage pattern also dictates the energy regularization of the constitutive response of the finite element. The combination of energy regularization and mechanistic mapping method ensures that, for each time increment, the random tangential stiffness tensor of each finite element is calculated in accordance with the ongoing damage pattern. A direct consequence of the model is that, depending on the damage pattern, the statistics of the tangential stiffness tensor could vary with the mesh size. The model is applied to stochastic FE analysis of both notched and unnotched flexural specimens under different loading configurations, which exhibit different failure behaviors. The numerical analysis also considers different correlation lengths of the random fields of material properties. The simulation shows that, with the energy regularization scheme, the commonly used local mapping and local averaging methods could yield considerable mesh dependence of the statistics of the peak load capacity. The result also reveals the effect of correlation length on the spurious mesh dependence. By relating the mapping algorithm to the underlying damage pattern, the present model is able to mitigate the mesh sensitivity for different specimen geometries, loading configurations, and correlation lengths.

A stochastic numerical model for wave-induced dynamic response in a spatially heterogeneous seabed around structures

The interaction between waves, seabed, and structures is a complex phenomenon that involves multiple physical domains and processes. The spatial variability of properties in marine sediment can significantly affect the dynamic response of seabed. A stochastic numerical model is developed to investigate the response in a spatially heterogeneous poro-elastic seabed around a submerged breakwater. The model is implemented using the COMSOL platform, which allows for solving the wave and seabed sub-models within the same framework. The spatial variability of soil properties is simulated using the random field method employing the Karhunen-Loève expansion. To couple the generated random fields in MATLAB with the integrated model in COMSOL, a LiveLink program is developed, which incorporates a cycle algorithm for stochastic finite element analysis, enabling hundreds or thousands of runs. The model is validated by comparing its results with analytical solutions and experimental data from literature. Probabilistic analysis is performed using the stochastic numerical model and Monte Carlo simulations. The effects of coefficient of variation and correlation lengths in random field on the probabilistic analysis are investigated. The present study provides a valuable tool and insights for the probabilistic assessment of soil response in a spatially heterogeneous seabed around structures.

A study of the natural frequencies of the functionally graded curved beams including porosities: Deterministic and stochastic modeling

In this research article, an investigation into the natural frequencies of functionally graded porous curved beams is presented, covering both deterministic and stochastic domains. The stochastic finite element analysis is based on a three-node, C0 continuous element, and higher-order shear deformation theory, utilizing the first-order perturbation technique (FOPT). The study includes a comparative analysis of three porosity distribution models, including a recently proposed sinusoidal model. Additionally, the study examines the influence of various factors, such as porosity distributions, boundary conditions, volume fraction indices, and geometric configurations, on the natural frequencies. The analysis encompasses uncertain natural frequencies in functionally graded curved beams, accounting for material stochasticity through FOPT. Monte Carlo simulation (MCS) is used to validate the accuracy of the FOPT model, increasing the approach’s reliability. Potential future research in this area could investigate the effects of stochastic variations in geometric parameters, including dimensions and boundary conditions.

Numerical investigation on influence of dispersed rock particles on bimslope stability

Analysing the stability of bimslopes (i.e. slopes made of bimrock (block-in-matrix rock)) is challenging because bimrock has a complex internal structure. In the study reported here, a stochastic finite-element analysis with the shear-strength-reduction method was conducted to elaborate how dispersed oversized rock particles influence the stability of a vertical bimslope. Numerical models with different rock compositions were generated and analysed to characterise the definiteness and randomness. The factor of safety and the failure zone were used to establish the relationship between slope stability and fractal dimension (FD) of rock-size distribution and rock spatial variability. The numerical results show that the FD and spatial locations of the rock particles influence bimslope stability indirectly, with the key indicator for the skid resistance of a bimslope being the compactness of the rock composition crossing the sliding surface with given volumetric block proportion. This finding also highlights the validity and feasibility of stochastic analysis in outperforming deterministic analysis. Overall, the present findings offer a reference for further understanding of the collapse mechanism and reinforcement measures of existing bimslopes.

Stochastic Finite Element Analysis of Root-Reinforcement Effects in Long and Steep Slopes

This article introduces a novel numerical scheme within the finite element method (FEM) to study soil heterogeneity, specifically focusing on the root–soil matrix in fracture treatments. Material properties, such as Young’s modulus of elasticity, cohesion, and the friction angle, are considered as randomly distributed variables. To address the inherent uncertainty associated with these distributions, a Monte Carlo simulation is employed. By incorporating the uncertainties related to material properties, particularly the root component that contributes to soil heterogeneity, this article provides a reliable estimation of the factor of safety, failure surface, and slope deformation, all of which demonstrate a progressive behavior. The probability distribution curve for the factor of safety (FOS) reveals that an increase in the root area ratio (RAR) results in a narrower range and greater certainty in the population mean, indicating reduced material variation. Moreover, as the slope angle increases, the sample mean falls within a wider range of the probability density curve, indicating an enhanced level of material heterogeneity. This heterogeneity amplifies the level of uncertainty when predicting the factor of safety, highlighting the crucial importance of accurate information regarding heterogeneity to enhancing prediction accuracy.

Magnetic Filler Polymer Composites-Morphology Characterization and Experimental and Stochastic Finite Element Analyses of Mechanical Properties.

Polymer composites containing magnetic fillers are promising materials for a variety of applications, such as in energy storage and medical fields. To facilitate the engineering design of respective components, a comprehensive understanding of the mechanical behavior of such inhomogeneous and potentially highly anisotropic materials is important. Therefore, the authors created magnetic composites by compression molding. The epoxy polymer matrix was modified with a commercial-grade thickening agent. Isotropic magnetic particles were added as the functional filler. The microstructural morphology, especially the filler distribution, dispersion, and alignment, was characterized using microscopy techniques. The mechanical properties of the composites were experimentally characterized and studied by stochastic finite element analysis (SFEA). Modeling was conducted employing four cases to predict the elastic modulus: fully random distribution, randomly aligned distribution, a so-called "rough" interface contact, and a bonded interface contact. Results from experiments and SFEA modeling were compared and discussed.

Research on fatigue reliability assessment of engine cylinder head based on neural network

Fatigue reliability (FR) evaluation is crucial for extending the service life and improving the reliability of automotive engines. This study proposed a general framework for assessing high cycle FR based on the back propagation neural network (BPNN). Using a diesel engine cylinder head (CH) as an example, the stochastic finite element analysis was implemented by considering the uncertainties of materials and loads. Subsequently, the stochastic finite element calculation process was replaced with the BPNN model to expedite fatigue life prediction. Ultimately, the reliability assessment of the CH was conducted, taking into account the influence of the load sequence. The research findings indicated that the reliable life t0.95 of the CH considering the load sequence was approximately 75 h, representing a 7.1% increase compared to the linear cumulative damage. During the service life of the CH, the failure probability sensitivity factor for the gas force load exceeded 0.9, offering an important reference for optimizing FR.

Failure analysis and structural reliability of unreinforced masonry veneer walls: Influence of wall tie corrosion

Corrosion appears to be a very common problem in historical and aged structures with steel ties and anchorages. A potential rapid deterioration of the wall tie due to the corrosion leading to a premature structural failure poses a serious threat to the built heritage infrastructures’ safety. This paper describes an overview of the probabilistic failure analysis of the unreinforced masonry (URM) veneer wall system with flexible backup under uniformly distributed out-of-plane loadings. Moreover, a framework is proposed to consider the wall tie corrosion in the stochastic finite element analysis (FEA) while estimating the structural reliability. A probabilistic experimental study where 18 full-scale URM veneer wall systems with theoretically identical geometries and properties were tested under inward and outward lateral loading. Wall failure statistics along with the probabilistic characterisation of veneer wall constituent materials including mortar, wall tie and timber were accomplished. A stochastic computational model was then developed which combines the FEA and Monte Carlo simulation to evaluate the failure progression and system peak load (veneer capacity) while considering the spatial variability of veneer wall material properties. Two scenarios, with and without wall tie corrosion, are considered, and the probabilistic characterisation of the veneer capacities for these scenarios under inward and outward out-of-plane loading is also reported. The model error statistics were combined with the probabilistic load models to determine the reliability index corresponding to the Australian Standard for Masonry Structures AS 3700. Annual reliabilities are compared to target reliabilities recommended by ISO 2394. Capacity reduction factor and design minimum tie strength consideration for corroded wall tie is also discussed.

Probabilistic analysis of derrick frame in a formwork support system

The fastener is the core component of the formwork support system, and its working characteristics have a significant influence on the bearing capacity of the system. The non-intrusive stochastic finite element analysis of the formwork support system was conducted using the Python script process driving the ABAQUS main progress based on the Monte Carlo simulation. The influence of random occurrence of fastener failure with different numbers at any position on the reliability of the system was obtained. The analysis results showed that the increase of the failure probability of the fastener would lead to a significant increase in the magnitude and probability of the decrease of the bearing capacity. The above results also illustrated that the deterministic analysis of the formwork support system based on the integrity of all fasteners is not practical and unsafe.

Structural reliability and partial safety factor assessment of unreinforced masonry in vertical bending

ABSTRACT This paper focuses on a structural reliability-based assessment of clay brick unreinforced masonry (URM) walls subjected to uniformly distributed out-of-plane loads in one-way vertical bending. Stochastic models combining finite element analysis (FEA) and Monte Carlo simulations (MCS) are used to account for spatial variability of the flexural tensile bond strength when estimating the wall failure loads. The strength of URM walls is known to be influenced by the flexural tensile bond strength, which is subject to high spatial variability as batching, workmanship, and environmental exposure alter the strength of this bond. For this assessment, single skin walls have been considered with bond strength statistics seen in typical construction. The model error statistics available for similar walls are combined with the results of the spatial stochastic FEA and probabilistic load models to determine the reliability index corresponding to the Australian Standard for Masonry Structures AS 3700 design of members in vertical bending. It was found that existing levels of reliability exceed target reliabilities, and the capacity reduction factor can be increased from 0.60 to 0.65 for URM walls in one-way vertical bending while still providing an acceptable level of reliability. A sensitivity analysis showed this finding to be robust.

Axial compression capacity of rectangular CFST columns under random pitting corrosion

To study the effect of random pitting corrosion, this paper presents the change law of the bearing capacity of the rectangular concrete-filled steel tube (CFST) column under the action of axial force with the mass loss ratio of the outer steel tube. Existing experimental data, including the axial ultimate bearing capacity and load displacement curves of the uncorroded and corroded rectangular CFST columns, are collected to verify the rationality of the finite element model. ANSYS software was used to establish a model that can simulate the pitting corrosion, and the results of the finite element analysis were compared with the experiment, which illustrates that the established model can accurately predict the actual load changes. For further research, the corroded thickness tc/t, which include constant tc/t and random tc/t, and other influencing factors such as corrosion pit diameter Dc, steel tube wall thickness t, material strength, shape and section size of specimens are discussed, the results reveal the effect of these parameters on the decline of axial compressive load capacity. Meanwhile, to visually reflect this effect, this study introduces the concept of bearing capacity reduction coefficient Rcs and Rc to express the ratio of bearing capacity of specimens before and after corrosion. The extensive stochastic finite element analysis was carried out, and on this basis, a suitable calculation formula for the reduction factor is proposed to predict the axial compression bearing capacity of the corroded rectangular CFST column by measuring the value of local mass loss ratio of the outer steel.

Fuzzy finite element analysis based on the transformation between fuzzy and random variables

In this paper, a fuzzy finite element method (FFEM) for determining the responses of structures is proposed by using the transformation between fuzzy and random variables. Firstly, the formulae for establishing normal random variables equivalent to symmetric triangular fuzzy numbers are presented based on the combination of the principle of insufficient reason and that of maximum specificity. As a result, fuzzy finite element analysis is transferred into stochastic finite element analysis. To solve this problem, the response surface method with the aid of standard normal random variables is utilized to approximate the real responses of structures. Then, the errors between the training and test sets are estimated to select a suitable response surface model amongst the regression models. Lastly, the formulae for determining the mean and the standard deviation values of the responses of structures are established. The accuracy and effectiveness of the proposed method are verified via an illustrative example.

Strain energy density as a Gaussian process and its utilization in stochastic finite element analysis: Application to planar soft tissues

Data-based approaches are promising alternatives to the traditional analytical constitutive models for solid mechanics. Herein, we propose a Gaussian process (GP) based constitutive modeling framework, focusing on planar, hyperelastic and incompressible soft tissues. Specifically, the strain energy density of soft tissues is modeled as a GP, which can be regressed to experimental stress–strain data obtained from biaxial stretching experiments. Moreover, the GP model can be weakly constrained to be convex. A key advantage of a GP-based model is that, in addition to the mean value, it provides a probability density (i.e. associated uncertainty) for the strain energy density. To simulate the effect of this uncertainty, a non-intrusive stochastic finite element analysis (SFEA) framework is proposed. The proposed framework is verified against an artificial dataset based on the Gasser–Ogden–Holzapfel model and applied to a real experimental dataset of a porcine aortic valve leaflet tissue. The results show that the proposed framework can be trained with limited experimental data and fits the data better than several existing models. The SFEA framework provides a straightforward way of using the experimental data and quantifying the resulting uncertainty in simulation-based predictions.

A Modeling Method for Finite Element Analysis of Corroded Steel Structures with Random Pitting Damage

Aiming at simulating the surface morphology of corroded steel and providing a modeling method with higher accuracy, the accelerated corrosion test was used to obtain six groups of corroded specimens, and then applied to stochastic finite element analysis (FEA) for studying the mechanical behavior of corroded steel. The pitting parameters (the depth, width, and diameter–depth ratio) of all specimens were investigated and statistically analyzed. Considering the irregularity of corroded surface, the random pitting model (RPM) was established based on the secondary development of ABAQUS. Moreover, the rough surface meshing method (RSMM) was subsequently proposed to optimize the element quality of the FEA model. At last, the modeling method was applied to investigate the bearing capacity of corroded steel beams. The results indicate that, firstly, the pitting parameters of all specimens obeyed log-normal distribution, and their logarithmic mean values grew with increase in corrosion time. The corroded surface the RPM can reproduce the evolution behaviors of a corroded surface with higher accuracy. In addition, the FEA model of corroded steel structures can be meshed easily into hexahedron elements by using the RSMM and effectively optimizing the number and quality of elements. By comparing with other test results, the calculation results of the FEA model of steel beams established by using the modeling method proposed in this study demonstrate a good accuracy in mechanical behavior analysis. The modeling method provides further support for the study of mechanical properties of corroded steel structures.

Evaluation of Loess Collapsibility Based on Random Field Theory in Xi’an, China

The engineering properties of collapsible loess have significant uncertainty. Accurate prediction of collapsible deformation is crucial for the safety of engineering construction in loess areas. Taking the typical collapsible loess stratum as the research object in Xi’an, based on the random field theory, combined with the Monte Carlo strategy and modulus reduction method, the stochastic finite element analysis of loess self-weight collapsibility is carried out to study the influence of the spatial variability of compression modulus on the self-weight collapsibility of loess. The results show that the loess tends to be stratified and average along the depth direction with the increase of transverse correlation distance. The random field result of self-weight collapsibility considering the spatial variability of compression modulus is significantly greater than the deterministic result of layered average and the calculated value of loess code. Considering the low compression modulus dominance effect of compression modulus with positive skewed distribution of random field, the equivalent characteristic value of the compression modulus calculated by the layered average modeling for the collapsibility evaluation of typical loess strata in Xi'an area is proposed.

Investigation of the Mechanical Properties for Polymer Reinforced by Nanoparticles with Considering Viscoelasticity of the Matrix

Investigation of the Mechanical Properties for Polymer Reinforced by Nanoparticles with Considering Viscoelasticity of the Matrix