Stochastic Finite Element Research Articles

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.

Variable mass loading effect on the long-term ambient response of a freeway bridge

This study addresses the quantification and compensation of the so-called mass loading effect, a disturbance that typically occurs in structural health monitoring of bridges. Source of this effect is the time-variant traffic, which means that dynamic performance indicators such as identified resonance frequencies are also time-variant. Basis of this study is the ambient acceleration response of a freeway bridge in the Alps recorded in a long-term monitoring campaign. After statistical evaluation of the traffic over the bridge, stochastic finite element simulations are performed to quantify the effect of time-variant truck traffic on a resonance frequency identified from the recorded accelerations, which serves as performance indicator. The corresponding natural frequency of the numerical bridge-truck model is obtained both by modal analyses and response history analyses, whose results however show considerable differences. In an effort to reduce the temporal scatter of the performance indicator, the corresponding natural frequency of the numerical model is used to compensate for the mass loading effect. It it shown that the mass loading effect plays a significant role for the considered bridge, although temperature effects dominate as a disturbance for the identified resonance frequency.

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.

Robust Design Optimization of Car-Door Structures with Spatially Varied Material Uncertainties

This paper presents an effective approach for robust design optimization of car-door structures with spatially varied material properties. This spatially varied material property causes structural response quantities; for example, the natural frequency and the lateral stiffness coefficient become random variables. In this regard, the Karhunen-Loève expansion is first used to represent the elastic modulus and the mass density random fields as a series of random variables. Then, a stochastic finite-element model is formulated for uncertainty quantification of the car-door structure. Combined with a polynomial-based response surface model to mimic the true performance indicator, this allows one to efficiently evaluate probability constraints for the robust design optimization of the uncertain car-door structure. In numerical simulations, design variables of the uncertain car-door structure are defined as thickness values of the tailor rolled blank structure at various regions, whereas multiple design objectives are formulated via the structural weight, the first-order natural frequency, and the lateral stiffness coefficient. Results have shown that the mean value of performance indicators can be generally improved, whereas the response variance is further minimized to archive the robust design objective. The probability-based constraint is significant to relate the Pareto optimum set to the targeted structural safety level. The proposed approach is simple, suggesting an attractive tool for the robust design optimization of car-door structures with spatially varied material uncertainties.

Three-Dimensional Stochastic Modelling of Wavy Carbon Nanotube Reinforced Epoxy Nanocomposites

Carbon nanotubes (CNTs) are frequently used as the nanoscale filler materials. It has been observed that when they are added to the polymer matrix by a small fraction, they are able to form a strong yet light-weight multifunctional composite materials. Previous studies found these properties to be significantly influenced by the morphology of CNTs embedded in the matrix. In general, long CNTs become wavy and randomly oriented when dispersed in the matrix material. Collectively, they form an interconnected network. Over the past several decades, many theoretical and computational studies have been carried out to capture the mechanics of CNT-reinforced nanocomposites. In most of these models, CNTs are modeled as straight fibers. In addition, a perfect interphase between CNT and epoxy is generally assumed. As such, these models may not capture the realistic morphology of CNT reinforced composites. In the current study, we have developed a method to construct stochastic three-dimensional finite element models of wavy CNT reinforced epoxy nanocomposites. Both aligned type and randomly distributed type are considered. Uniform random distributions are assumed for both waviness and orientation angles of the dispersed CNTs. To study the effect of waviness on the elastic properties, finite element models with maximum CNT waviness angles of 0° (straight), 10° , 20° , and 30° are generated for CNT volume fractions ranging from 0.1 to 1%. A significant decrease in the properties is observed as the maximum allowable waviness angle (θmax) of the embedded CNTs in the epoxy is increased. The obtained values are compared with the theoretical models available in the literature. The three-phase model including CNTs, CNT/epoxy interphase and matrix is also studied to evaluate the effect of ‘soft’ (Ei = 10–75% Em) as well as ‘hard’ interphase (Ei = 110–175% Em) on the composite modulus. These results suggest that due to ineffective load transfer between CNT and epoxy in the presence of CNT waviness, the elastic modulus of the CNT composites is compromised by 4–8%. The modulus is further decreased by nearly 2.5% as the CNT/epoxy interphase modulus decreases by 90%. The ‘hard’ interphase model confirms an efficient load transfer from matrix to CNT fibers, leading to the increase in composite stiffness. It is found that the composite modulus is more sensitive to the ‘soft’ interphase.

Stochastic normal mode frequency analysis of hybrid angle ply laminated composite skew plate with opening using a novel approach

In the present work, for the first time, stochastic normal mode frequency analysis of hybrid angle ply laminated composite skew plate with an opening is attempted by a novel approach. The present novel approach employed 2 D C0 stochastic finite element (FE) model based on higher-order shear deformation theory (HSDT) in conjunction with artificial neural network (ANN) and artificial neural network-particle swarm optimization (ANN-PSO). HSDT model used eliminates the requirement of shear correction factor owing to 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 vibration using FE analysis with Monte Carlo simulation, design of experiment (DOE), and soft computing with ANN, ANN-PSO is implemented. With the DOE method, a total of 128 design points are generated. Since there are no results for the present problem in the literature is available. Hence suitable comparisons have been made and found in the 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 present soft computation models shrink the trial and are found computationally in agreement with the Monte Carlo simulation-based finite element modeling (MCS-FEM). Current work reveals that ANN-PSO is a fast and straightforward method with result accuracy close to MCS-FEM, and the future researcher in this area may use ANN-PSO.

Dynamic behavior of the macauba palm (Acrocomia aculeata) fruit-rachilla system using the stochastic finite element method

The search for alternative energy sources has fomented the study of several crops. The macauba palm crop, for instance, has been highlighted because of its particular relevance in Brazil due to its wide distribution across Brazilian territory and its potential for yielding high amounts of oil per cultivated hectare. However, the species is still most commonly harvested via extractivism, which results in low yields. Therefore, we aimed to analyze the dynamic behavior of the fruit-rachilla system when subjected to mechanical vibration to gather baseline information for the subsequent development of macauba harvesting machines. The fruit-rachilla system of the species was modeled for different fruit maturation stages and plant accessions. Natural frequencies and modes of vibration were determined by the stochastic finite element method (FEM), adopting the specific mass and the modulus of elasticity of the system as random variables, which enabled us to compile a dataset of natural frequencies based on the variability of the system properties. The mean values of the natural frequencies obtained in the vibration assays were 26.02 Hz at the green maturation stage and 21.22 Hz at the ripe maturation stage. The mean values of natural frequencies found in the simulation by stochastic FEM, referring to the third mode of vibration, were 26.05 Hz at the green maturation stage and 21.23 Hz at the ripe maturation stage. We concluded that the natural frequencies of the macauba fruit-rachilla system on the basis of different plant accessions showed a decreasing behavior during fruit maturation. The modes of vibration characterized by pendulum displacement did not differ among plant accessions or between fruit maturation stages.

Research on Reliability for Servo Turret Based on Vibration Transmission Path System with Stiffness Degradation Model

The stiffness degradation of the servo turret will inevitably lead to accuracy reduction of the cutter head and the tool change. Considering the degradation process of servo turret with stiffness, by introducing the stiffness cumulative damage theory into the vibration differential equation, combined with stochastic finite element method and reliability theory, the mathematical model of the reliability and reliability sensitivity for vibration transmission path system with random parameters was established. Taking a typical power servo turret for example, the reliability and the reliability sensitivity to each random parameter at the mean value with the excitation frequency and time were obtained. The results showed that the shift of the reliability and reliability sensitivity to random parameters with time was caused by the stiffness degradation, the peak value of reliability sensitivity fluctuated with time, and the peak value in the frequency domain at the initial time was not necessarily the maximum value in the time domain. The accuracy of the proposed method was further proved by the Monte Carlo method. Optimizing sensitive parameters could enhance the system stability and effectively prevent the resonance failure caused by the change of the resonance region.

Uncertainty propagation in dynamics of composite plates: A semi-analytical non-sampling-based approach

In this study, the influences of spatially varying stochastic properties on free vibration analysis of composite plates were investigated via development of a new approach named the deterministic-stochastic Galerkin-based semi-analytical method. The material properties including tensile modulus, shear modulus, and density of the plate were assumed to be spatially varying and uncertain. Gaussian fields with first-order Markov kernels were utilized to define the aforementioned material properties. The stochastic fields were decomposed via application of the Karhunen-Loeve theorem. A first-order shear deformation theory was assumed, following which the displacement field was defined using admissible trigonometric modes to derive the potential and kinetic energies. The stochastic equations of motion of the plate were obtained using the variational principle. The deterministic-stochastic Galerkin-based method was utilized to find the probability space of natural frequencies, and the corresponding mode shapes of the plate were determined using a polynomial chaos approach. The proposed method significantly reduced the size of the mathematical models of the structure, which is very useful for enhancing the computational efficiency of stochastic simulations. The methodology was verified using a stochastic finite element method and the available results in literature. The sensitivity of natural frequencies and corresponding mode shapes due to the uncertainty of material properties was investigated, and the results indicated that the higher-order modes are more sensitive to uncertainty propagation in spatially varying properties.

An efficient iterative model reduction method for stochastic systems having geometric nonlinearities

The dynamic analysis of systems has several applications in the project or monitoring and controlling machines and equipments. In particular, the analysis of nonlinear systems has academic and industrial appeal for owning many particularities such as structural integrity, models updating, stability and real-time predictability. Those particularities complicate the study of these models. However, the development of more representative mathematical models requires normally costly computations. By considering the uncertainties associated with the fabrication process (geometrical dimensions, physical and material properties), as well as those related to the operational conditions (boundary conditions, external forces, etc.), it complicates further the analysis of those nonlinear systems. Thus, this paper proposes a methodology to analyze nonlinear systems subjected to uncertainties using the stochastic finite element method. In view of the high computational cost needed to construct the confidence region predicted with the stochastic nonlinear computational model, it is proposed herein a new model reduction method based on the construction of an adaptive iterative enriched basis to deal with the stochastic nonlinear system addressed herein composed by a thin rectangular plate used as an application. The results demonstrate clearly the efficiency and accuracy of the proposed method as an efficient tool to approximate the nonlinear responses of more complex nonlinear systems subjected to uncertainties. Also, it demonstrates the relevance of taking into account the uncertainties in the modeling of nonlinear systems to consider more realistic situations.

Simplified spectral stochastic finite element formulations for uncertainty quantification of engineering structures

In general, the structure inputs including loading and physical specifications can be deterministic or stochastic and uncertainties can be modeled using a random variable, random field, or a stochastic process. A well-known method developed to analyze structures with uncertainties is the spectral stochastic finite element method (SSFEM). To improve the efficiency of SSFEM, this paper has developed three simplified formulations to separately solve problems involving stochastic loading and constant material properties, stochastic loading and random material properties and stochastic loading and stochastic material properties the main feature of which is that they need fewer finite element (FE) analyses to assess the statistical specifications of the response. To examine their accuracy and efficiency, three practical engineering problems: 1) Euler-Bernoulli beam, 2) square solid and 3) modeling the corrosion effect in a concrete structure were solved and the results were compared with those of the Monte Carlo simulation (MCS). To see if the proposed formulations can be used in structural reliability analyses, a limit state function (LSF) was defined for each problem and the results were compared with those of the corresponding MCS; their significant efficiency and enough potential was, thus, confirmed.

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.

On the performance evaluation of stochastic finite elements in linear and nonlinear problems

Due to the inherent uncertainties in various systems, deterministic approaches may not be able to satisfactorily characterize their response. In such cases, stochastic approaches that can systematically consider uncertainties have to be employed. In the past, many stochastic finite element analysis (SFEA) methods have been developed for uncertainty quantification (UQ), among which the perturbation methods, intrusive and non-intrusive polynomial chaos expansion (IPCE/NIPCE) methods and stochastic collocation (SC) methods have received considerable attention. However, in mechanics, most of the applications of these methods are confined to relatively simple problems, and the applicability and performance of these methods to complex nonlinear mechanics problems are not clear. To this end, this study carried out an investigation on the performance of different SFEA methods in linear and nonlinear problems. Numerical studies show that the NIPCE and SC methods are superior in terms of accuracy among other methods in the linear elastic case. The NIPCE method is also used for UQ in stochastic models with plasticity and nonlocal elastoplastic damage. The results demonstrate that the stochastic averages can be significantly different from the deterministic results, which indicates the necessity of considering UQ for improving response predictions.

Assessing the effective elastic properties of the tendon-to-bone insertion: a multiscale modeling approach.

The interphase joining tendon to bone plays the crucial role of integrating soft to hard tissues, by effectively transferring stresses across two tissues displaying a mismatch in mechanical properties of nearly two orders of magnitude. The outstanding mechanical properties of this interphase are attributed to its complex hierarchical structure, especially by means of competing gradients in mineral content and collagen fibers organization at different length scales. The goal of this study is to develop a multiscale model to describe how the tendon-to-bone insertion derives its overall mechanical behavior. To this end, the effective anisotropic stiffness tensor of the interphase is predicted by modeling its elastic response at different scales, spanning from the nanostructural to the mesostructural levels, using continuum micromechanics methods. The results obtained at a lower scale serve as inputs for the modeling at a higher scale. The obtained predictions are in good agreement with stochastic finite element simulations and experimental trends reported in literature. Such model has implication for the design of bioinspired bi-materials that display the functionally graded properties of the tendon-to-bone insertion.

Stochastic finite elements analysis of large concrete structures’ serviceability under thermo-hydro-mechanical loads – Case of nuclear containment buildings

This work proposes a global Stochastic Finite Element Method (SFEM) to model the effects of concrete ageing uncertainties on the serviceability and durability of large reinforced and prestressed structures with a containment role. As their modelling requires strongly non-linear, coupled and expensive calculations with a large number of parameters, adapted and efficient probabilistic strategies need to be defined aiming at a stochastic analysis within a reasonable cost and a physically admissible representativeness. In this contribution, this is achieved through four steps: (a) the definition of a well-established physical framework based on a staggered Thermo-Hydro-Mechanical + Leakage (THM-L) model; (b) the limitation of random inputs for uncertainty propagation to the most influential ones using a variance-based Hierarchized and Local Sensitivity Analysis (HLSA); (c) the construction of a THM-L response metamodel using Polynomial Chaos Expansion (PCE); (d) the reliability analysis of serviceability criteria using Crude Monte Carlo Method (CMCM) applied to the developed metamodel. For validation purposes and demonstration of achievability within a complex industrial framework, this global methodology is applied to an experimental 1:3 scaled Containment Building of a nuclear reactor. Eventually, it is shown that a complete probabilistic analysis of a physically admissible total dry air leakage rate (indicative of a nuclear containment structure’s performance) and its evolution in time are obtained within a computational time of tens of days only. Such result can provide insights and help during the decision-making process for the design, maintenance and risk assessment of large structures. For Nuclear Containment Buildings (NCB), a direct application would be the evaluation of lifespan extension based on a leakage-rate-defined criterion under operational loads.

Stochastic stress-based topology optimization of structural frames based upon the second deviatoric stress invariant

This work pertains to the stochastic stress-based topology optimization of frame structures considering uncertainty. Specifically, this paper presents an investigation of the second deviatoric stress invariant, J2, as the measure of stress in the domain where the objective is to minimize the maximum of the expected values of the stress invariant. Analytical expressions for the expected value of the J2 invariant are developed employing the perturbation approach. The premise being that the J2 invariant eliminates the square root operator that would otherwise be present if using the von Mises stress, and by eliminating this operator, the nonlinearity in the functional mapping between the random input (magnitude and direction of external forces) and uncertain output (expected values of stress) is reduced. Hence, an improved accuracy can be achieved with an analytical approximation of a given order. The analytical expressions of the expected value of the J2 invariant to the second order are obtained based on a Taylor series expansion along with the associated sensitivities for gradient-based optimization. The analytical expressions are implemented in an optimization scheme and applied for the design of three ground structures considering different variability in the input random variables. For each example, the relative error between the maximum of the expected values of the J2 invariant obtained using the analytical expressions and corresponding values evaluated by performing stochastic finite element analysis, whereby the input distribution is sampled using Monte Carlo methods, finite element simulation performed for each sample realization, then computing the expected value of the stress measure using straightforward statistical expressions, and finally estimation of the maximum via the p-norm, were observed to be less than 2%, even for a coefficient of variation of 0.4. The optimized designs were again analyzed using the von Mises stress and the relative error again computed with respect to those obtained from Monte Carlo sampling and finite element simulation. In general, the relative error using the analytical expressions of the von Mises stress was greater than, except in a couple instances, those obtained using the J2 invariant.

Development of hybrid dimension adaptive sparse HDMR for stochastic finite element analysis of composite plate

A novel adaptive high dimensional model representation (HDMR) is developed for stochastic finite element analysis of composite plate. Uncertainty propagating through the physical system is quantified using this technique where only the significant dimensions are retained. Influence of these dimensions is modeled by different orders which are preferred to be low for computational tractability. These dimensions are identified by a sensitivity analysis using support points in the sparse grid which are generated using multiple levels of hierarchy, joint or marginal probability distributions and dimension adaptiveness. This multi level hierarchical generation of support points fill the domain of the input variables with major emphasis on the regions of interest which may be the entire space, limit state, maxima/minima or tail ends of the probability distribution function for rare events depending upon the nature of the problem. The hybrid characteristics of the proposed dimension decomposition is introduced in this paper where the component functions are expressed using orthogonal bases and the error term which is further modeled using Gaussian random process (i.e. Kriging). The performance of the proposed Hybrid dimension adaptive HDMR (hdA-HDMR) is illustrated using stochastic finite element analysis of laminated composite plate whose material properties are considered as homogeneous non-normal random fields.

Stochastic Finite Element Analysis Framework for Modelling Electrical Properties of Particle-Modified Polymer Composites.

Properties such as low specific gravity and cost make polymers attractive for many engineering applications, yet their mechanical, thermal, and electrical properties are typically inferior compared to other engineering materials. Material designers have been seeking to improve polymer properties, which may be achieved by adding suitable particulate fillers. However, the design process is challenging due to countless permutations of available filler materials, different morphologies, filler loadings and fabrication routes. Designing materials solely through experimentation is ineffective given the considerable time and cost associated with such campaigns. Analytical models, on the other hand, typically lack detail, accuracy and versatility. Increasingly powerful numerical techniques are a promising route to alleviate these shortcomings. A stochastic finite element analysis method for predicting the properties of filler-modified polymers is herein presented with a focus on electrical properties, i.e., conductivity, percolation, and piezoresistivity behavior of composites with randomly distributed and dispersed filler particles. The effect of temperature was also explored. While the modeling framework enables prediction of the properties for a variety of filler morphologies, the present study considers spherical particles for the case of nano-silver modified epoxy polymer. Predicted properties were contrasted with data available in the technical literature to demonstrate the viability of the developed modeling approach.

Stochastic Perturbation-Based Finite Element for Free Vibration of Functionally Graded Beams with an Uncertain Elastic Modulus

The stochastic eigenvalue problem for free vibrations of functionally graded beams with a random elastic modulus is investigated. The effective material properties and beam cross section are assumed to vary continuously in different directions according to the exponential law. The governing equations for the natural frequency of the functionally graded beams are derived from Hamilton’s principle. In the stochastic finite-element method, the random process was discretized by averaging random variables in each element to increase the accuracy of the natural frequency found. A solution of the stochastic eigenvalue problem formulated was obtained using the perturbation technique in conjunction with the finite-element method. The spectral representation was used to generate a random process to employ the Monte Carlo simulation. A good agreement was obtained between the results of the first-order perturbation technique and the Monte Carlo simulation.