Stochastic Finite Element Analysis Research Articles

Spatial variability and stochastic finite element model of unreinforced masonry veneer wall system under Out-of-plane loading

Inconsistency in the quality of workmanship, the weather during construction, and materials may result in the high unit-to-unit spatial variability of mechanical properties in the same masonry structure. This paper develops a numerical modelling strategy and focuses on the stochastic assessment of unreinforced masonry (URM) veneer walls with spatially variable wall constituent material properties subjected to out-of-plane loading. 3-D finite element modelling of experimentally tested veneer walls was conducted using a micro-modelling approach combined with the Monte Carlo Simulation technique. Spatial stochastic finite element analysis considered the spatial variability of the properties of the wall components (mortar flexural tensile strength) and compared them with non-spatial analysis. The non-spatial analysis overestimates the wall system failure compared to spatial analysis, and the spatial analysis is considered to more realistically represent the variabilities of the URM veneer wall system. Sensitivity analysis is conducted to check the sensitivity of the veneer system behaviour to variability in the various input parameters. The variability of experimental results obtained in a parallel study are quantified and compared with stochastic finite element analysis (SFEA) results. The SFEA model developed in this study can estimate the behaviour and system peak load reasonably and are considered to be from the similar population as test results. Model errors are also included to assess the efficiency of the SFEA model.

The exact element stiffness matrices of stochastically parametered beams

Stiffness matrices of beams with stochastic distributed parameters modelled by random fields are considered. In stochastic finite element analysis, deterministic shape functions are traditionally employed to derive stiffness matrices using the variational principle. Such matrices are not exact because the deterministic shape functions are not derived from the exact solution of the governing stochastic partial differential equation with the relevant boundary conditions. This paper proposes an analytical method based on Castigliano’s approach for a beam element with general spatially varying parameters. This gives the exact and a simple closed-form expression of the stiffness matrix in terms of certain integrals of the spatially varying function. The expressions are valid for any integrable random fields. It is shown that the exact element stiffness matrix of a stochastically parametered beam can be expressed by three basic random variables. Analytical expressions of the random variables and their associated coefficient matrices are derived for two cases: when the bending rigidity is a random field and when the bending flexibility is a random field. It is theoretically proved that the conventional stochastic element stiffness matrix is a first-order perturbation approximation to the exact expression. A sampling method to obtain the basic random variables using the Karhunen–Loève expansion is proposed. Results from the exact stiffness matrices are compared with the approximate conventional stiffness matrix. Gaussian and uniform random fields with different correlation lengths are used to illustrate the numerical results. The exact closed-form analytical expression of the element stiffness matrix derived here can be used for benchmarking future numerical methods.

Probabilistic failure analysis of quasi-isotropic CFRP structures utilizing the stochastic finite element and the Karhunen–Loève expansion methods

The accuracy of structural analysis in composite structures depends on the proper estimation of the uncertainties mainly related to the mechanical properties of the constituent materials. On this basis, a sophisticated numerical tool is proposed, able to perform stochastic finite element analysis on composite structures with material uncertainties by distributing stochastic mechanical properties along the domain of a composite structure. The output of the analysis is a probability density function for the deformation, strain, stress and failure fields. The proposed tool exploits the Karhunen–Loève expansion and the Latin Hypercube Sampling methods for the stochastic distribution of the mechanical properties, the well-established First-Order Shear Deformation theory in conjunction with a random variable approach for the calculation of stochastic stiffness matrices, and the Puck’s failure criterion for the conduction of probabilistic analysis of different failure modes in composite structures. A quasi-static tensile testing campaign was conducted with quasi-isotropic coupons in order to assess the fidelity of the method and the efficiency of the stochastic distribution algorithm is compared with the full field data acquired by the digital image correlation approach. The current paper provides a thorough presentation of the development of the proposed stochastic finite element method and validation results which ensure the efficiency of the proposed stochastic numerical tool. • The Karhunen–Loève expansion is utilized for efficient random distribution of mechanical properties. • The Latin Hypercube Sampling is utilized for randomness generation and it outperforms the Monte Carlo Simulation. • Stochastic finite elements are developed for probabilistic failure analysis using Puck’s failure criterion. • Digital image correlation data are obtained from tensile experiments to assess the inherent randomness of composites.

Stochastic finite element analysis of composite cycloidal propeller blade during crash-stop ship maneuver

This article quantifies uncertainty in the dynamic responses of marine cycloidal propeller (MCP) blade by Stochastic Finite Element Method (SFEM) during crash-stop ship maneuver. Stopping a ship in an emergency is known as crash-stop astern maneuver. A MCP contains a horizontal circular metallic disc on which six aerofoil blades are hung vertically. All blades are made up of carbon fiber-reinforced composite laminates. Uncertainty in ship speed and composite material properties are propagated to find out the statistical response of natural frequencies, displacement, and stresses on the blade. SFEM is a combination of Stochastic Response Surface Method (SRSM) and Finite Element Method (FEM). SRSM is a collocation-based non-intrusive polynomial chaos expansion to transform any random distribution into a stochastic response surface equation (surrogate model). During ship maneuvering motion, the whole unit rotates about a vertical axis at the centre. For each rotation of the disc, all blades undergo periodic oscillations. The SRSM for uncertainty propagation is computationally more efficient than the standard Monte Carlo Simulation (MCS) technique without compromising the accuracy of the results. It is observed that uncertainty in vibration responses of MCP blade is significant and worth study.

Effective excitation conditions for the intense motion of the ginkgo seed-stem system during mechanical vibration harvesting

The degree of the motion response of the ginkgo seed-stem system during mechanical vibration harvesting determines the separation force obtained, thereby affecting harvesting efficiency. A response surface stochastic finite element analysis was carried out to obtain the modal shapes and the ranges of natural frequencies of the seed-stem systems. It was initially based on model updating and considering the inherent variability of the ginkgo seed-stem system, Then, the dynamic response of the ginkgo seed-stem system under different vibration parameters was investigated. The results showed that at low frequencies, only the first-order natural frequency could lead to resonance and cause the seed-stem system to swing violently. Under horizontal excitation, the main resonance of the system was at an excitation frequency close to the natural frequency. Under vertical excitation, the system had a sub-harmonic resonance at an excitation frequency close to twice the natural frequency. When the excitation acceleration was much larger than the gravitational acceleration, due to the increase in the excitation frequency, the seed-stem bounced violently, significantly increasing the inertial forces on the ginkgo seeds. Thus, vertical excitation is more likely to cause the intense motion response of the seed-stem system than horizontal excitation. This study not only proposes a correction method for the finite element model of the seed-stem system but it also clarifies the effective excitation conditions for the ginkgo seed-stem system. It provides theoretical guidance for the efficient mechanical vibration harvesting of ginkgo seeds.

Monte-Carlo laboratory testing of unreinforced masonry veneer wall system under out-of-plane loading

This paper presents the results of a probabilistic experimental study into the behaviour of full-scale unreinforced masonry (URM) veneer walls with flexible backup subjected to out-of-plane loading. The actual safety and reliability of the contemporary Australian URM structures are unknown due to the absence of information regarding the probabilistic behaviour of the whole veneer wall system and material characterisation of the wall constituent materials. The study focused on masonry typologies representative of modern URM buildings in the Australian context. In this study, 18 full-scale URM veneer wall systems with theoretically identical geometries and properties were tested under inward and outward out-of-plane loading. For each loading type, one specimen was tested under semi-cyclic loading to check whether the monotonic loading can capture the overall behaviour of the cyclic response. For each mortar batch mixed, bond wrench testing was conducted at the same age as the test for the associated wall constructed using that mix. Batch to batch variabilities were statistically analysed, and probability distributions for flexural tensile strength were established. Lognormal distributions with aggregated means of 0.40 MPa and 0.42 MPa for inward and outward loading, respectively, were estimated for flexural tensile strengths. After the wall tests, all timber studs used to build the veneer walls were tested to evaluate the probabilistic characterisation of timber stiffness. This probabilistic information is essential for a stochastic finite element analysis (FEA) to conduct the reliability analysis. From the wall tests, veneer wall system behaviour was observed and measured until the collapse or 20% post-peak drop of the peak load. Outward loaded specimens exhibited higher variabilities for masonry cracking and system peak load compared to inward loading due to variabilities from materials, testing arrangements and failure mechanism. The true coefficient of variations of system peak loads were calculated as 0.10 and 0.19 for inward and outward loadings, respectively, considering the effect of testing variability.

Mesh objective stochastic simulations of quasibrittle fracture

Continuum finite element (FE) modeling of damage and failure of quasibrittle structures suffers from the spurious mesh sensitivity due to strain localization. This issue has been addressed for deterministic analysis through the development of localization limiters. This study proposes a mechanism-based model to mitigate the mesh sensitivity in stochastic FE simulations of quasibrittle fracture. The interest is placed on the analysis of large-size structures, where the mesh size is conveniently chosen to be larger than the width of the fracture process zone as well as the correlation length of the random fields of constitutive properties. The present model is formulated within the framework of continuum damage mechanics. Two localization parameters are introduced to describe the evolution of the damage pattern of each finite element. These parameters are used to guide the energy regularization of the constitutive law, as well as to formulate the mesh-dependent probability distributions of constitutive properties. Depending on the prevailing damage pattern, different energy regularization schemes and mesh dependence of the probability distribution functions are used in the constitutive law. The model is applied to simulate the stochastic failure behavior of quasibrittle structures of different geometries featuring different failure processes including damage initiation, localization, and propagation. It is shown that using fixed probability distribution functions of constitutive properties could lead to strong mesh dependence of the prediction of the mean and variance of the structural load capacity. The probability distribution functions of constitutive properties must be linked to the damage pattern, which may evolve during the failure process. Such a mechanism-based modeling of the probability distributions of constitutive properties is essential for mitigating the spurious mesh sensitivity in stochastic FE analysis of quasibrittle fracture.

Study of the Mechanical Properties of Wood under Transverse Compression Using Monto Carlo Simulation-Based Stochastic FE Analysis

Wood is an anisotropic material, the mechanical properties of which are strongly influenced by its microstructure. In wood, grain compression strength and modulus are the weakest perpendicular to the grain compared to other grain directions. FE (finite element) models have been developed to investigate the mechanical properties of wood under transverse compression. However, almost all existing models were deterministic. Thus, the variations of geometry of the cellular structure were not considered, and the statistical characteristic of the mechanical property was not involved. This study aimed to develop an approach to investigate the compression property of wood in a statistical sense by considering the irregular geometry of wood cells. First, the mechanical properties of wood under radial perpendicular to grain compression was experimentally investigated, then the statistical characteristic of cell geometry was extracted from test data. Finally, the mechanical property of wood was investigated using the finite element method in combination with the Monte Carlo Simulation (MCS) techniques using randomly generated FE models. By parameter sensitivity analysis, it was found that the occurrence of the yield points was caused by the bending or buckling of the earlywood axial tracheid cell wall in the tangential direction. The MCS-based stochastic FE analysis was revealed as an interesting approach for assessing the micro-mechanical performance of wood and in assisting in understanding the mechanical behavior of wood based on its hierarchical structure.

Stochastic dynamics of veering modes in a symmetric coupled system

ABSTRACT A conceptual model of a coupled symmetric system consisting of three beams coupled using a common base beam was developed. The purpose was to analyse the behaviour of close modes of the repeated system. A study on variation in the dynamic behaviour due to parametric uncertainty was carried out experimentally and theoretically. Experimentally the mass of the system was perturbed and the frequency response function (FRF) was analysed. Theoretically the mean and standard deviation of the frequency response function was determined using stochastic finite element (SFEM). Stochastic finite element analysis was carried out considering uncertainty in stiffness and the system response. Karhunen–Loeve (KL) expansion and polynomial chaos (PC) were carried out for Young's modulus uncertainty and the response uncertainty respectively. Optimal number of KL terms and PC order were determined. Using the optimal SFEM model, the influence of uncertainty in the individual subsystem on the dynamic response was determined. The results from the study indicated that a relaxed manufacturing tolerance could reduce the possibility of shock amplification. The frequency range over which high system response was observed in the FRF increased with lower tolerance, however, the maximum amplitude in the mean response was reduced.

Efficient stochastic finite element analysis of irregular wall structures with inelastic random field properties over manifold

Efficient stochastic finite element analysis of irregular wall structures with inelastic random field properties over manifold

Random fields representation over manifolds via isometric feature mapping‐based dimension reduction

AbstractGenerating random fields over irregular geometries (e.g., two‐dimensional (2D) manifolds embedded in the three‐dimensional (3D) Euclidean space) is a great challenge because the geometry structure is complex and the correlation function can hardly be derived, thus the traditional methods, for example, spatial discretization methods or series expansion methods, cannot be directly adopted. To solve this issue, the present paper develops a two‐stage strategy to simulate random fields over manifolds. The core idea is to map the manifolds into the 2D Euclidean space through Isometric feature mapping (Isomap), with which the geodesic distance between points in the mapped 2D Euclidean space and the original manifold space is kept as the same. Therefore, the correlation function can be readily derived and the conventional methods can be directly employed to generate random fields. To validate the proposed method, several different types of manifolds are dimensionally reduced to 2D planar domains, and the stochastic harmonic function method is used to generate the random fields. Finally, case studies on the stochastic finite element analysis of two different structures are also performed to demonstrate the applicability and efficiency of the proposed method.

Neumann enriched polynomial chaos approach for stochastic finite element problems

An enrichment scheme based upon the Neumann expansion method is proposed to augment the deterministic coefficient vectors associated with the polynomial chaos expansion method. The proposed approach relies upon a split of the random variables into two statistically independent sets. The principal variability of the system is captured by propagating a limited number of random variables through a low-ordered polynomial chaos expansion method. The remaining random variables are propagated by a Neumann expansion method. In turn, the random variables associated with the Neumann expansion method are utilised to enrich the polynomial chaos approach. The effect of this enrichment is explicitly captured in a new augmented definition of the coefficients of the polynomial chaos expansion. This approach allows one to consider a larger number of random variables within the scope of spectral stochastic finite element analysis in a computationally efficient manner. Closed-form expressions for the first two response moments are provided. The proposed enrichment method is used to analyse two numerical examples: the bending of a cantilever beam and the flow through porous media. Both systems contain distributed stochastic properties. The results are compared with those obtained using direct Monte Carlo simulations and using the classical polynomial chaos expansion approach.

Structural Response Prediction of Thin-Walled Additively Manufactured Parts Considering Orthotropy, Thickness Dependency and Scatter.

Besides the design freedom offered by additive manufacturing, another asset lies within its potential to accelerate product development processes by rapid fabrication of functional prototypes. The premise to fully exploit this benefit for lightweight design is the accurate structural response prediction prior to part production. However, the peculiar material behavior, characterized by anisotropy, thickness dependency and scatter, still constitutes a major challenge. Hence, a modeling approach for finite element analysis that accounts for this inhomogeneous behavior is developed by example of laser-sintered short-fiber-reinforced polyamide 12. Orthotropic and thickness-dependent Young’s moduli and Poisson’s ratios were determined via quasi-static tensile tests. Thereof, material models were generated and implemented in a property mapping routine for finite element models. Additionally, a framework for stochastic finite element analysis was set up for the consideration of scatter in material properties. For validation, thin-walled parts on sub-component level were fabricated and tested in quasi-static three-point bending experiments. Elastic parameters showed considerable anisotropy, thickness dependency and scatter. A comparison of the predicted forces with experimentally evaluated reaction forces disclosed substantially improved accuracy when utilizing the novel inhomogeneous approach instead of conventional homogeneous approaches. Furthermore, the variability observed in the structural response of loaded parts could be reproduced by the stochastic simulations.

Fatigue life estimation of the weld joint in K-node of the offshore jacket structure using stochastic finite element analysis

Several methods for finite element modelling and analysis of tubular welded joints are described in various design codes. These codes provide specific recommendations for modelling of the welded joints, using simple weld geometries. In this paper, experimental strain range results from two full-scale semi-automatically welded K-nodes tests are compared to corresponding finite element models. Three dimensional (3D) scans of the weld surfaces have been developed for the automatically welded K-joints. These 3D scans are included in the finite element (FE) models to determine the accuracy of the FE models. Some discrepancies are observed between finite element predictions and test results. Finite element model updating has been applied to improve the accuracy of the developed finite element models. The results are compared to a FE model with a simple weld geometry based on common offshore design codes and a FE model without any modelled weld. The results show that the FE model with 3D scanned welds are more accurate than the other FE models. The precise location of weld toe in the 3D scanned weld is difficult to locate in the FE model, thus stochastic finite element modelling is performed to analyse the probabilistic hot-spot stresses. The results show large variations in the hot-spot stresses. It can be concluded that it is necessary to evaluate the probabilistic nature of stresses for calculating fatigue lives of welded tubular joints in offshore structures.

Application of stochastic numerical analyses in the assessment of spatially variable unreinforced masonry walls subjected to in-plane shear loading

This paper develops a modelling strategy for the finite element analysis of perforated (arched) unreinforced masonry walls subjected to in-plane shear loading. An experimental baseline was used to facilitate an accurate calibration and assessment of the chosen modelling strategy. This study provides the procedure and the results relevant to a stochastic assessment of unreinforced masonry shear walls. These results may be used in future studies of the reliability of these structures and may be applied in the calibration of reliability-based design practices. Utilising a two-dimensional micro-modelling approach, the capacity of a monotonic loading scheme to capture the envelope of a cyclically applied load was examined. It was found that, while the elastic stiffness of the laboratory specimens was overestimated by the finite element models, the peak load and global response was accurately recreated by the monotonically loaded models. Once the applicability of this procedure had been established, a series of spatially variable stochastic finite element analyses were created by considering the stochastic properties of key material parameters. These analyses were able to estimate the mean load resistance of the experimentally tested walls with a greater accuracy than a deterministic model. Furthermore, these analyses produced an accurate estimate of the variability of shear capacity of and the observed damage to the laboratory specimens. Due to the fact that the tested walls failed almost exclusively in a rocking mode, a failure mechanism highly dependent upon the structures’ geometry, the variability of the peak strength was minimal. However, the observed damage and presence of some sliding and stepped cracking indicates that the proposed methodology is likely to capture more variable and unstable failure modes in shear walls with a smaller height-to-length ratio or those more highly confined.

Dynamic Response of Angle Ply Laminates with Uncertainties Using MARS, ANN-PSO, GPR and ANFIS.

In the present work, for the first time, free vibration response of angle ply laminates with uncertainties is attempted using Multivariate Adaptive Regression Spline (MARS), Artificial Neural Network-Particle Swarm Optimization (ANN-PSO), Gaussian Process Regression (GPR), and Adaptive Network Fuzzy Inference System (ANFIS). The present approach employed 2D C0 stochastic finite element (FE) model based on the Third Order Shear Deformation Theory (TSDT) in conjunction with MARS, ANN-PSO, GPR, and ANFIS. The TSDT model used eliminates the requirement of shear correction factor owing to the consideration of the actual parabolic distribution of transverse shear stress. Zero transverse shear stress at the top and bottom of the plate is enforced to compute higher-order unknowns. C0 FE model makes it commercially viable. Stochastic FE analysis done with Monte Carlo Simulation (MCS) FORTRAN inhouse code, selection of design points using a random variable framework, and soft computing with MARS, ANN-PSO, GPR, and ANFIS is implemented using MATLAB in-house code. Following the random variable frame, design points were selected from the input data generated through Monte Carlo Simulation. A total of four-mode shapes are analyzed in the present study. The comparison study was done to compare present work with results in the literature and they were found in good agreement. The stochastic parameters are Young’s elastic modulus, shear modulus, and the Poisson ratio. Lognormal distribution of properties is assumed in the present work. The current soft computation models shrink the number of trials and were found computationally efficient as the MCS-based FE modelling. The paper presents a comparison of MARS, ANN-PSO, GPR, and ANFIS algorithm performance with the stochastic FE model based on TSDT.

Stochastic Finite Element Analysis of Shear-Critical Concrete Structures

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Stochastic Finite Element Analysis of Composite Structures with Partially Restrained Connections under Earthquake Loads

The stochastic analysis of composite structures with partially restrained (PR) connections under seismic loads present some interesting and challenging issues to practicing engineers. This paper proposes an efficient, robust, and accurate method for stochastic finite element analysis of concrete–steel composite structures allowing for PR connections. These are followed by suitable numerical example which indicates that employment of such a stochastic finite element analysis. The Kocaeli earthquake in 1999 is considered as a ground motion. The connections parameters and material properties are random variables. It is essential to properly consider the PR connections in the stochastic dynamic analysis and design of the steel-concrete composite structures since design forces change significantly. The assumption that the connections are rigid, which is routinely used in the application, is not proper. The effect of the variability connection stiffness on the composite structures responses is sufficiently important for consideration in structural safety.

Reliability calibration for the design of multiple-chord CFST trusses by advanced analysis

Concrete-filled steel tubular (CFST) truss with complex configurations are now widely used in practice, especially in large-scale bridge constructions. For such complex composite trusses, traditional structural analysis approaches are readily applied for the safety checks of individual members and connections, whilst inelastic analysis and reliability calibration are very limited. Recently, the development of advanced system-level analysis that takes into account various nonlinearities and uncertainties allows a rational reliability calibration for CFST trusses to assess their structural reliability. This paper aims to present reliability calibrations on three typical CFST trusses (two-chord, three-chord and four-chord) by advanced finite element models implemented with random uncertainties. The established finite element models are validated against experimental results. Afterwards, stochastic finite element analyses (FEA) are conducted taking into account the material and geometric nonlinearities, the random initial imperfections and the potential model errors. Using the obtained statistics of resistance, reliability calibration is then undertaken to calculate the reliability indexes of the three typical CFST trusses in respect of resistance factors under various load conditions, with the target reliability discussed in accordance with AASHTO and ANSI/AISC 360–16. The proposed novel computational approach and the reliability calibration contribute to both the practical design and the standard drafting for CFST trusses.

Stochastic Modeling and Analysis of a Heavy Duty Radiator

Governmental agencies across the globe are constantly evolving with stringent emission laws to tackle the problem of CO2 and NOx/SOx emissions. New emission standards force the Truck OEM’s to redesign the engine. The paper is aimed to measure the header tube joint stress of the radiator subjected to random variations in geometry, shape and material properties. Linear analysis will not consider the uncertainty and randomness due to tolerance, process changes, part-part variation etc. Stochastic finite element analysis (FEA) is carried out to account the uncertainty in the system. The finite element model of radiator system is built and baseline linear simulation is performed to obtain the baseline deformation and baseline stress responses. Then the uncertainty and random variation due to the geometry, material and shape variable is defined by a normal distribution function. Random designs are generated by defining the upper and lower bound limit values for the input design variable. Random designs are populated using Monte-Carlo simulation technique. 250 random design points are created for each design variables. Then stochastic simulation is performed to evaluate the responses at random design points. Statistical and probabilistic tools are used to post process the simulation results. The paper showcases application of stochastic simulation method which aids in indentifying the robust design with minimum variations. This also enables engineers and designers to understand the relationship and significance between different design variables in designing energy efficient systems.