One-dimensional Random Field Research Articles

Stochastic modeling of periodic beams under uncertain boundary conditions and environmental fluctuations

Periodic structures have been attracting a great deal of academic and industrial interest lately, due to their distinctive vibration and wave propagation behavior, which can be explored for the development of innovative solutions to structural dynamics and vibroacoustic problems. Although such a potential has been demonstrated in a large number of studies, the investigation of detrimental effects, which can be present in practical applications, is still necessary. This paper reports investigations on the combined influence of uncertainties affecting ambient temperature — which alters material properties and induces stress-stiffening due to constrained thermal dilatation — and boundary conditions (BCs) on the bandgap characteristics of periodic beams. The space-dependent temperature fluctuations are represented as a one-dimensional stationary Gaussian random field, discretized using the Karhunen-Loève expansion, while non-ideal BCs, represented as springs, are modeled as discrete random variables. Sampling-based stochastic analyses of the central frequency and bandwidth of the beam’s attenuation bands are performed using Monte Carlo simulations. The results demonstrate that the variability in the attenuation band features is influenced not only by the coefficients of variation (CVs) of the input random quantities, but also by the correlation length of the random temperature fluctuations. Numerical simulations reveal that the bandgap central frequency is primarily affected by the temperature random field, while the BCs govern the bandwidth. Although low CV and standard deviation values are obtained for the dispersion of the bandgap features, reliability analyses indicate that some designs exhibit low reliability. Increased variability in both the bandgap central frequency and bandwidth is observed for greater temperature correlation lengths and CVs. The contributions of the study include the proposal of a comprehensive stochastic modeling procedure duly accounting for relevant random influences, and evidencing that those influences can be significant, requiring consideration in the design of robust periodic structures.

Influence of spatial variability of soil shear-wave velocity considering intralayer correlation on seismic response of engineering site

Shear-wave velocity uncertainty significantly impacts seismic response analysis at engineering sites. Previous studies focused on the correlations of shear-wave velocities within soil layers. However, the influence of the vertical spatial variability of shear-wave velocity within individual soil layers was not considered. This study selected a typical Site Class II engineering site to investigate this influence based on the equivalent linear seismic response analysis method of layered soil deposits. Existing field test data on shear-wave velocity were used to discretize the shear-wave velocity within soil layers into a one-dimensional (1D) random field through covariance matrix decomposition and local average process theory. Monte Carlo simulations generated random shear-wave velocity profiles with different vertical correlation distances. Three artificial records with different earthquake return periods were used as input motions at the engineering bedrock for the 1D free-field model. Considering soil shear-wave velocity uncertainty, the peak shear strain increased by 19–27% with the input ground motion amplitude. Moreover, the site peak shear strain variability increased by 25–34% with the vertical correlation distance. The proposed 1D random field model effectively simulated the interlayer correlation and spatial variability of shear-wave velocity within soil layers, demonstrating its significance in seismic response analysis.

A thermodynamic model for predicting the stress–strain relation of stochastic heterogeneous materials with experimental verification

Materials are intrinsic stochastic heterogeneity regardless of their natural or artificial origin. Traditional models for predicting mechanical properties of materials typically involve homogenization, which exhibits certain limitations in practical applications, such as investigating the size effect in stress–strain relationships. In this study, a random field thermodynamic theory for predicting the mechanical properties of stochastic heterogeneous materials was proposed, and its applicability and reliability were verified by testing hard-drawn 304 stainless-steel wires with gauge lengths of 10–500 mm. In these wires composed of the lamellar martensite with a thickness of ∼47.1 nm, it is found that failure strains decreased significantly with increasing gauge length, whereas fracture stresses show little dependence on the gauge length leading to a great increase of the effective modulus. Digital image correlation investigations revealed a significant strain heterogeneity during the tensile process, even at the elastic deformation stage. Nanoindentation tests show the non-uniformly distributed reduced modulus and hardness, which can be attributed to the intrinsic stochastic heterogeneity of materials. Taking Young's modulus as a one-dimensional stationary random field along the longitudinal direction, our random field model can well demonstrate the variation of failure strain and effective modulus with changing gauge lengths. The size effect on the macroscopic mechanical properties mainly originated from the microscopic mechanical property correlation of different points within the materials. These findings could provide new insights into the size effect on the mechanical properties of materials.

Stochastic free vibration analysis of FG-CNTRC plates based on a new stochastic computational scheme

Stochastic analysis can provide more accurate results compared to deterministic analysis. In this study, the spatial variability of material parameters is introduced into the free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates to achieve a higher degree of accuracy, and a new stochastic computational scheme is proposed to handle the low-level uncertainties. First-order shear deformation theory (FSDT) is employed to establish the displacement field of the plates. The Young's moduli of carbon nanotube (CNT) and matrix are regarded as one-dimensional (1D) and two-dimensional (2D) random fields, respectively, which is discretized by Karhunen-Loève (K-L) expansion. The obtained random variables are substituted into modified perturbation stochastic method (MPSM) and radial point interpolation method (RPIM) to calculate the first two order estimates of the stochastic non-dimensional natural frequencies ω¯. The sensitivity of the first to fourth ω¯ to the random fields is analyzed, and the corresponding stochastic bands are plotted. The results indicate that the presented approach can accurately solve the generalized eigenvalue problem in stochastic free vibration; ω¯ is sensitive to random fields E11CNT and Em, and insensitive to random field E22CNT; the CNT distribution mode also affects the sensitivity.

Investigation of the Eigenvector of Stochastic Finite Element Methods of Functionally Graded Beams with Random Elastic Modulus

This paper presents a stochastic finite element method to calculate the variation of eigenvalues and eigenvectors of functionally graded beams. The modulus of functionally graded material is assumed to have spatial uncertainty as a one-dimensional random field. The formulation of the stochastic finite element method for the functionally graded beam due to the randomness of the elastic modulus of the beam is given using the first-order perturbation approach. This approach was validated with Monte Carlo simulation in previous studies using spectral representation to generate the random field. The statistics of the beam responses were investigated using the first-order perturbation method for different fluctuations of the elastic modulus. A comparison of the results of the stochastic finite element method with the first-order perturbation approach and the Monte Carlo simulation showed a minimal difference.

Experimental Test of Sinai's Model in DNA Unzipping.

The experimental measurement of correlation functions and critical exponents in disordered systems is key to testing renormalization group (RG) predictions. We mechanically unzip single DNA hairpins with optical tweezers, an experimental realization of the diffusive motion of a particle in a one-dimensional random force field, known as the Sinai model. We measure the unzipping forces F_{w} as a function of the trap position w in equilibrium and calculate the force-force correlator Δ_{m}(w), its amplitude, and correlation length, finding agreement with theoretical predictions. We study the universal scaling properties since the effective trap stiffness m^{2} decreases upon unzipping. Fluctuations of the position of the base pair at the unzipping junction u scales as u∼m^{-ζ}, with a roughness exponent ζ=1.34±0.06, in agreement with the analytical prediction ζ=4/3. Our study provides a single-molecule test of the functional RG approach for disordered elastic systems in equilibrium.

A mesostructure-informed cohesion-based numerical method for fracture behavior of slate with foliation structure

The foliation structure and spatial variability are important factors affecting the fracture behavior of slate. According to the statistical analysis of mesostructural information, random slate specimens with the foliation structure are physically reconstructed by coupling a one-dimensional random field with a two-dimensional isotropic random field. A cohesion-based numerical method incorporating the foliation structure is developed to simulate the fracture behavior of slate-like materials. The elastic and fracture parameters are evaluated by direct tensile tests, and the numerical model is validated to be self-consistency. It has been confirmed by numerical simulations and tests that the elastic modulus and tensile fracture increase with the foliation angle, and there is a critical foliation angle from the tensile fracture to the shear fracture for the slate under uniaxial tension. Affected by the spatial variability of the foliation structure, slate fractures in the dense and thick area of the foliation plane, and the mechanical properties follow lognormal distribution.

Experimental Realization of the 1D Random Field Ising Model.

We have measured magnetic-field-induced avalanches in a square artificial spin ice array of interacting nanomagnets. Starting from the ground state ordered configuration, we imaged the individual nanomagnet moments after each successive application of an incrementally increasing field. The statistics of the evolution of the moment configuration show good agreement with the canonical one-dimensional random field Ising model. We extract information about the microscopic structure of the arrays from our macroscopic measurements of their collective behavior, demonstrating a process that could be applied to other systems exhibiting avalanches.

Modeling and analysis of high aspect ratio wing considering random structural parameters

With the application of advanced composite materials in High-Aspect-Ratio wings (HARW), the randomness of structural parameters, such as elastic modulus and Poisson's ratio, is enhanced. Hence, in order to explore the whole picture of aeroelastic problems, it is of great significance to study the role of random structural parameters in aeroelastic problems. In this paper, the dynamic response of flexible HARW considering random structural parameters is analyzed. An aeroelastic model of a one-dimensional cantilevered Euler–Bernoulli beam considering aerodynamic forces acting on the wing is established based on Hamilton's principle. Adopted the idea of simplifying calculation, the effect of random structural parameters is analyzed. Then, considering the elastic modulus and torsional stiffness as continuously one-dimensional random field functions, and discretized by local method. The first and second order recursive stochastic nonlinear finite element equations of wing are derived by using perturbation method. Based on it, statistical expression of aeroelastic effects of the wing is derived. Monte Carlo method is adopted to verify the effectiveness of the method. Numerical simulations indicate that the method proposed can well mirror the statistical characteristics of aeroelastic response.

Reliability analysis of horizontally loaded pile considering spatial variability of soil parameters

In the design and analysis of pile foundations, the horizontal load is an important factor that needs to be considered. The reliability analysis can take the uncertainty of various parameters into consideration and thus reduce the design deviations, especially the uncertainty of soil parameters. This paper proposes a reliability analysis method considering the spatial variability of soil for the horizontally loaded pile. Firstly, a parameter sensitivity analysis was performed by the Sobol sensitivity method, and the critical variables, including the undrained shear strength Cu, the soil strain ε50, which represents the soil strain when principal stress difference reaches half (50%) of the maximum value in triaxial test, and the horizontal load and applied moment, are identified and taken as the random variables. Moreover, the spatial variability of soil parameters (Cu and ε50) are simulated as one-dimensional random field through the orthogonal series expansion method. The failure probabilities of pile foundation under the failure modes of deflection and strength are evaluated by Monte Carlo simulation. Finally, the proposed method is verified through case study in which the influences of the spatial variability of soil parameters, the uncertainty of the horizontal load on the failure probability were investigated. The results show that the uncertainty of horizontal loads and Cu make significant effects on the reliability of pile foundations. The increase of correlation length of Cu and ε50 will lead to the increase of the mean value of pile response, which has different effects on the failure probability under different failure modes.

Dynamics analysis of slender vehicle influenced by random factors

In this paper, the aeroelastic problems of slender vehicles under the influence of random factors and thrust are studied. An aeroelastic dynamic model of a free-free Euler–Bernoulli beam considering thrust and aerodynamic forces is established based on Hamilton’s principle of nonconservative systems. On this basis, considering the influence of random factors, the elastic modulus and viscous drag are regarded as one-dimensional continuous stationary random fields and discretized. The stochastic finite element method is used to solve the dynamic model, and the results are compared with the Monte Carlo simulation results. Then, the influence of the correlation of the random field on the elastic displacement is further analyzed. The following simulation results are obtained: (1) the stochastic factor analysis model established in this paper can reflect the statistical characteristics of aeroelastic response well; (2) the stronger the correlation of the random field is, the greater the expectation of elastic displacement, but as the correlation increases, the expectation tends to be constant; and (3) it is necessary to choose the discrete length of the random field reasonably, and the discrete length depends on the correlation characteristics of the random field studied.

A static analysis of nonuniform column by stochastic finite element method using weighted integration approach

In general, the fluctuation of the elastic modulus of materials is crucial in structural analysis. This paper develops a stochastic finite element method (SFEM) for analyzing a nonuniform column considering the random process in elastic modulus. This random process of elastic modulus is assumed as a one-dimensional Gaussian random field. The weighted integration method is used to discretize the random field and establish the stochastic finite element formulation to compute the first and second moments of displacement fields. The results of the proposed approach are validated with those of the previous study. The response variability of displacement of column and effect of the parameter of the random field is investigated in detail.

A Comparative Assessment of the Methods-of-Moments for Estimating the Correlation Length of One-Dimensional Random Fields

Due to geological processes, soil properties vary in vertical and horizontal directions which is defined as inherent uncertainty of soil. This type of uncertainty may seriously affect the reliability of, among others, deep and shallow foundations (and in turn, of structures, such as, buildings and bridges) and earth retention and shoring systems. The inherent uncertainty of soil properties is modelled as a random field, which is described by the mean, standard deviation and scale of fluctuation (also known as correlation length, θ) of soil properties. In this paper, the effectiveness of eight methods-of-moments for estimating the correlation length θ is investigated. This is done by generating samples of one-dimensional random fields for pre-defined values of the correlation length, which is then estimated by the different methods. For each method, the influence of the sampling domain length D and sampling interval dx on the estimation of θ were investigated, and the results are quantified in the form of errors over the parameter space, defined by the dimensionless ratios D/θ and θ/dx. Through the present analysis, one is able to assess the reliability of θ estimations obtained in practice, by mapping the conditions of any given experiment i.e., sampling domain, interval and estimated correlation length, onto the parameter space. The expected error associated with each method used is also quantified. Through this analysis a comparative assessment of the methods is also obtained.

On the need for dependence characterization in random fields: Findings from cone penetration test (CPT) data

Random field theory is widely used to model spatial variability of soil properties. However, random field modeling focuses mainly on the estimate of spatial correlation structure. The dependence structure that is necessary to construct the joint probability distribution over a random field is usually not characterized. The aim of this research is twofold. First, this paper focuses on characterizing the dependence structure underlying a random field based on cone penetration test (CPT) data. The copula approach is adopted to represent dependencies and the best-fit dependence (copulas) are identified from the CPT data. It is found that the nonGaussian dependencies can be a real phenomenon in spatial fluctuation of the soil shear strength parameter. Second, this paper provides formulations for generating random fields with Gaussian or nonGaussian dependencies, and investigates whether the improper use of the dependence structure could lead to significant bias in failure probability. The generated one-dimensional (1-D) and two-dimensional (2-D) random fields of a cohesive slope under different dependencies are compared. Large deviation in probabilistic results implies that the effect of dependencies on failure probability can be nontrivial. Therefore, the complete random field characterization should involve the estimate of both correlation structure and dependence structure.

Random Field Parameter Estimation of Service Bridge Component and Comparative Analysis of Estimation Methods

Three 36-year-old beams are taken as research objects to estimate the fluctuation scale θ of structural dimensions and material property parameters of reinforced concrete structure in service. First, the protective layer thickness Cd and the concrete compressive strength fc of service bridge component are measured, and the ordinary kriging method is used to interpolate the measured data and gather effective data. Second, the θ values of Cd and fc are estimated by two methods, namely autocorrelation function (ACF) and semivariogram function (SVF). Thereafter, the spatial distribution of R of each corroded steel bar is studied on the basis of measured values of the pitting factor R at different positions of 246 corroded steel bars. On the basis of one-dimensional random field theory, each beam is discretized into several small elements. According to the fluctuation scale obtained by different estimation methods, the random variables are transformed into spatial correlation variables, and the parameters of each discrete element are obtained. Then the bending capacity of each discrete beam element and the corresponding whole beam are obtained on the basis of the value of parameters of each discrete element. Finally, the two estimation methods are compared and analyzed by comparing the theoretical moment value and the bearing capacity test results of the corroded beams. Results show that the mean values of θ of fc and Cd obtained by the SVF method are 2.242 and 2.467, respectively, which are larger than that obtained by the ACF method. Moreover, the fluctuation intensity of concrete compressive strength is smaller with an increase of θ value, and no firm correlation is identified between the spatial distributions of each pit on the surface of a corroded steel reinforcement. The bending moments of three beams from the SVF method are in better agreement with the experimental values than those predicted from the ACF method. In addition, the average relative error of the theoretical bending moment value of all beams based on the SVF estimation is 6.52%, which is 33.19% lower than that obtained by ACF method. Therefore, the SVF method is more effective in describing the spatial properties of material and geometrical parameters compared with the ACF method.

Failure Mechanism and Factor of Safety for Spatially Variable Undrained Soil Slope

This paper aims to investigate the differences in factor of safety (FS) and failure mechanism (FM) for spatially variable undrained soil slope between using finite element method (FEM) , finite difference method (FDM), and limit equilibrium method (LEM). The undrained shear strength of cohesive soil slope is modeled by a one‐dimensional random field in the vertical direction. The FS and FM for a specific realization of random field are determined by SRT embedded in FEM‐ and FDM‐based software (e.g., Phase2 6.0 and FLAC) and LEM, respectively. The comparative study has demonstrated that the bishop method (with circular failure surface) exhibits performance as fairly good as that of SRT both in FS and FM for the undrained slope cases where no preferable controlling surfaces such as hydraulic tension crack and inclined weak seams dominate the failure mechanism. It is, however, worthwhile to point out that unconservative FM is provided by the Bishop method from the aspect of failure consequence (i.e., the failure consequence indicated by the FM from the Bishop method is smaller than that from SRT). The rigorous LEM (e.g., M‐P and Spencer method with noncircular failure surface) is not recommended in the stability analysis of spatially variable soil slopes before the local minima and failure to converge issues are fully addressed. The SRT in combination with FEM and/or FDM provides a rigorous and powerful tool and is highly preferable for slope reliability of spatially variable undrained slope.

Multi-scale stochastic damage model for concrete and its application to RC shear wall structure

PurposeThis paper aims to present a multi-scale stochastic damage model (SDM) for concrete and apply it to the stochastic response analysis of reinforced concrete shear wall structures.Design/methodology/approachThe proposed SDM is constructed at two scales, i.e. the macro-scale and the micro-scale. The general framework of the SDM is established on the basis of the continuum damage mechanics (CDM) at the macro-scale, whereas the detailed damage evolution is determined through a parallel fiber buddle model at the micro-scale. The parallel buddle model is made up of micro-elements with stochastic fracture strains, and a one-dimensional random field is assumed for the fracture strain distribution. To represent the random field, a random functional method is adopted to quantify the stochastic damage evolution process with only two variables; thus, the numerical efficiency is greatly enhanced. Meanwhile, the probability density evolution method (PDEM) is introduced for the structural stochastic response analysis.FindingsBy combing the SDM and PDEM, the probabilistic analysis of a shear wall structure is performed. The mean value, standard deviation and the probability density function of the shear wall responses, e.g., shear capacity, accumulated energy consumption and damage evolution, are obtained.Originality/valueIt is noted that the proposed method can reflect the influences of randomness from material level to structural level, and is efficient for stochastic response determination of shear wall structures.

Probabilistic failure analysis of infinite slopes under random rainfall processes and spatially variable soil

ABSTRACTRainfall-induced landslides occur during or immediately after rainfall events in which the pore water pressure builds up, leading to shallow slope failure. Thereby, low permeability layers result in high gradients in pore water pressure. The spatial variability of the soil permeability influences the probability such low permeability layers, and hence the probability of slope failure. In this paper, we investigate the influence of the vertical variability of soil permeability on the slope reliability, accounting for the randomness of rainfall processes. We model the saturated hydraulic conductivity of the soil with a one-dimensional random field. The random rainfall events are characterised by their duration and intensity and are modelled through self-similar random processes. The transient infiltration process is represented by Richards equation, which is evaluated numerically. The reliability analysis of the infinite slope is based on the factor of safety concept for evaluating slope stability. To cope with the large number of random variables arising from the discretization of the random field and the rainfall process, we evaluate the slope reliability through Subset Simulation, which is an adaptive Monte Carlo method known to be especially efficient for reliability analysis of such high-dimensional problems. Numerical investigations show higher probability of slope failure with increased spatial variability of the saturated hydraulic conductivity and with more uniform rainfall patterns.

Constrained correlation functions from the Millennium Simulation

Context. In previous work, we developed a quasi-Gaussian approximation for the likelihood of correlation functions, which, in contrast to the usual Gaussian approach, incorporates fundamental mathematical constraints on correlation functions. The analytical computation of these constraints is only feasible in the case of correlation functions of one-dimensional random fields. Aims. In this work, we aim to obtain corresponding constraints in the case of higher-dimensional random fields and test them in a more realistic context. Methods. We develop numerical methods to compute the constraints on correlation functions which are also applicable for two- and three-dimensional fields. In order to test the accuracy of the numerically obtained constraints, we compare them to the analytical results for the one-dimensional case. Finally, we compute correlation functions from the halo catalog of the Millennium Simulation, check whether they obey the constraints, and examine the performance of the transformation used in the construction of the quasi-Gaussian likelihood. Results. We find that our numerical methods of computing the constraints are robust and that the correlation functions measured from the Millennium Simulation obey them. Despite the fact that the measured correlation functions lie well inside the allowed region of parameter space, i.e. far away from the boundaries of the allowed volume defined by the constraints, we find strong indications that the quasi-Gaussian likelihood yields a substantially more accurate description than the Gaussian one.

확률론적 침투해석을 통한 무한사면 파괴의 특성 연구

세계의 많은 지역이 강우에 의한 사면파괴가 취약하다. 사면파괴의 발생 메커니즘을 파악하기 위해 지금까지 다양한 방법들이 제안되어져 왔으나 현재 사용되는 방법들은 비균질한 지반분포와 수리학적 거동이 강우로 인한 사면파괴에 미치는 효과를 고려하지 못한다. 본 연구에서는 강우 시 토층의 두께에 따른 사면파괴의 발생 메커니즘을 연구하기 위하여 불투수 기반암 위에 존재하는 풍화 잔류토 사면에 대한 확률론적 사면안정 해석을 수행하였다. 불균질한 투수계수의 공간적 분포로 인한 불확실성이 강우침투에 의한 불포화 사면의 파괴에 미치는 영향을 고려하기 위하여 일차원 랜덤필드에 기초한 일련의 침투해석과 사면 안정해석을 수행하였다. 해석결과에 의하면 확률론적 해석법은 사면에 대한 강우의 침투 평가 시 투수계수의 공간적인 변동에 의하여 발생하는 다양한 파괴 패턴을 효과적으로 고려할 수 있음을 보여준다. Many regions around the world are vulnerable to rainfall-induced slope failures. A variety of methods have been proposed for revealing the mechanism of slope failure initiation. Current analysis methods, however, do not consider the effects of non-homogeneous soil profiles and variable hydraulic responses on rainfall-induced slope failures. In this study, probabilistic stability analyses were conducted for weathered residual soil slopes with different soil thickness overlying impermeable bedrock to study the rainfall-induced failure mechanisms depending on the soil thickness. A series of seepage and stability analyses of an infinite slope based on one-dimensional random fields were performed to consider the effects of uncertainty due to the spatial heterogeneity of hydraulic conductivity on the failure of unsaturated slopes due to rainfall infiltration. The results showed that a probabilistic framework can be used to efficiently consider various failure patterns caused by spatial variability of hydraulic conductivity in rainfall infiltration assessment for a infinite slope.