Translation random field with marginal beta distribution in modeling material properties

doi:10.1016/j.strusafe.2016.04.001

Structural Safety

Volume 61, July 2016, Pages 57-66

https://doi.org/10.1016/j.strusafe.2016.04.001 Get rights and content

Highlights

•
Unbiased and consistent estimators for bounds of a beta distribution are proposed.
•
The proposed estimators are more efficient than the method of moments.
•
Scale of fluctuation in a beta random field can be preserved.
•
Non-iterative algorithm to determine the autocorrelation function of an underlying Gaussian field.

Abstract

For modeling material properties having a bounded range, the beta distribution may be adopted as the marginal distribution of a second-order non-Gaussian random field. Three aspects related to the simulation of such random field are discussed in this study. First, an unbiased and consistent estimator for the lower (and upper) bound of the beta distribution based on sample data is proposed. This estimator is shown to be generally more efficient than that given by the method of moments. Second, a simple explicit function relating the auto-correlation function of the non-Gaussian random field to that of the underlying Gaussian field is proposed. The relationship facilitates control on the scale of fluctuation of the non-Gaussian field. Third, an algorithm is proposed for generating random fields with an approximate marginal beta distribution and a prescribed cross-correlation, where the latter can range from −1 to 1. Numerical examples are given to illustrate the effectiveness and efficiency of each of the three aspects. The estimation of the lower bound of material property is exemplified through field data from a real project.

Introduction

The presence of uncertainties in the material properties has been acknowledged by the engineering community and accounted for in codes of practice [1], [2]. By treating the material properties as random variables or in some cases as random fields, statistics-based reliability analysis and the specification of characteristic values are techniques adopted to handle uncertainties in a consistent manner [3], [4], [5], [6].

The normal and lognormal distributions are widely used to describe random variables in practice [6] due to their simplicity and well-established properties, notwithstanding that often the actual distributions are hard to establish due to the limited amount of data. However, there are engineering variables that have obvious lower and upper limits and there are instances where large volumes of data are available. For the latter, high order moments (e.g. skewness and kurtosis) of a random variable may even be estimated with fairly good accuracy. Some distributions are only uniquely defined by higher order moments, such as the maximum entropy distribution [7], [8], [9], Hermite polynomials of normal variates [10] and Johnson’s system of frequency distributions [11]. Recently, Low [12] developed a four-parameter distribution to reflect the first four moments of sampled data.

The current study adopts the beta distribution as it not only can capture the first four moments of a random variable, but is bounded. In addition, it has well-established properties for computer-aided simulations. Many publications [3], [4], [13], [14] have shown the flexibility of the beta distribution in reflecting the variability of material properties. Harrop-Williams [15] analytically showed that the strength parameters (cohesion and the tangent of friction angle) of uniform soils follow the beta distribution. Therefore, it is of practical interest to establish methods to efficiently and accurately estimate the parameters of beta distribution from sampled data.

Unlike the normal and lognormal distributions which are completely defined by the sample mean and variance, the beta distribution requires additional information, such as the distribution bounds or higher order moments. Cooke [16] proposed a method to estimate the bounds of general random variables. Based on Cooke’s method, He [17] proposed an iterative algorithm to estimate the bounds of beta-distributed random variables. This algorithm was simplified by Liu et al. [18]. However, both the original and simplified iterative algorithms have certain conditions for convergence, and the iterative process may lead to biased estimators. Another commonly used approach [3], [19] is to fix the range of the distribution to six sample standard deviations centered at the sample mean, which may not reflect correctly the skewness of the distribution. In this paper, a non-iterative approach is proposed to estimate the bounds of a beta variable. It will be shown that this leads to unbiased and consistent estimators. The approach will be compared with the method of moments [20], [21], where the bounds are obtained through the first four moments of sampled data.

In the case of a medium with material property that fits the marginal beta distribution, a second-order beta random field [22] may be considered. One way to generate such beta field is to translate from an underlying Gaussian field [23]. However, the translation changes the correlation structure [24]. Yamazaki and Shinozuka [25] proposed a method to rectify the simulated correlation structure by iteratively updating the power spectral density function (PSDF) (which is the Wiener-Khintchine transform of correlation function) of the underlying Gaussian field until a desired PSDF of the translation field is achieved. This updating method has been extensively developed [26], [27], [28]; in particular, Shields et al. [28] proposed an iterative translation approximation method to speed up the calculation. However, the iteration process in simulating random field might be time-consuming [29], [30], where additional effort in generating the underlying Gaussian field is needed to match the PSDF. Matching the PSDF may preserve the form of the correlation function; however, the exact scale of fluctuation (SOF) may not be realized. This paper proposes an alternative method where the SOF can match exactly the prescribed value without iteration, assuming that the exact form of the auto-correlation function is not as critical in stochastic finite element analysis. On the other hand, based on Grigoriu’s [10] work, the lower bound of correlation of a translation random field is generally greater than -1 [31]. Some material properties indeed have negative correlation (close to −1), such as the friction angle and effective cohesion of soils [13], [32]. The lower bound in correlation may be necessary and sufficient to reflect the cross-correlation in such kinds of material properties. To circumvent this difficulty, a method based on Cholesky decomposition [33] is proposed to generate a beta field with a prescribed cross-correlation ranging from −1 to 1.

Section snippets

Probability density function

A random variable X distributed in the interval [a, b] is said to follow the beta distribution, if its probability density function (PDF), f_X(x; α, β), satisfies: $f_{X} (x; α, β) = \{\begin{matrix} \frac{{(x - a)}^{α - 1} {(b - x)}^{β - 1}}{B (α, β) {(b - a)}^{α + β - 1}}, a ⩽ x ⩽ b \\ 0, otherwise \end{matrix}$ in which a and b are the lower and upper bounds of X, respectively; α and β are the shape parameters, which are positive real numbers; B is the beta function [20]. The cumulative distribution function (CDF) of X, F_X(x; α, β), can be obtained from Eq. (1) via integration.

Estimation of bounds

Let X₁,

Beta field

Let $X (u, ϖ)$ be a second-order beta field defined on a probability space (Ω, A, P), where u is the position vector and $ϖ$ is the symbol of elements in the sample space Ω. We omit the symbol $ϖ$ hereafter for simplicity. X(u) can be translated from a Gaussian field via [23]: $X (u) = F^{- 1} \circ Φ (G (u))$ in which $Φ$ and F = F_X(x; α, β) are the CDFs of the standard Gaussian and target beta variables, respectively; G(u) is a zero-mean, unit-variance, stationary Gaussian field with correlation function ρ(|τ|), in which τ

Engineering application

Deep mixing and jet grouting with cement are widely used to improve soft clayey soils as part of stability intervention in excavation support and ground improvement [52]. It is well-known that significant heterogeneity can be induced into the improved ground during the process of chemical improvement [53], [54]. Due to the heterogeneity and spatial variation, the unconfined compressive strength (UCS) of cement-admixed soils adopted in design is often several times lower than the

Conclusions

The simulation of second-order beta field is discussed in this paper for material properties with upper and lower bounds. The beta marginal distribution is chosen due to its flexibility in representing distributions having a large variety of shapes. Three fundamental and practical issues in simulating a beta field are discussed in this study.

An unbiased and consistent estimator for the lower (and upper) bound of the beta distribution based on sample data is proposed and shown to be more

Acknowledgements

This research is supported by the National Research Foundation Singapore under its Competitive Research Programme (CRP award no. NRF-CRP 6-2010-03) and the Key Research & Development Plan Science and Technology Cooperation Programme of Hainan Province, P. R. China (Grant no.: ZDYF2016226).

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View full text

Translation random field with marginal beta distribution in modeling material properties

Highlights

Abstract

Introduction

Section snippets

Probability density function

Estimation of bounds

Beta field

Engineering application

Conclusions

Acknowledgements

Probab Eng Mech

Probab Eng Mech

Probab Eng Mech

Struct Saf

Comput Struct

Probab Eng Mech

Probab Eng Mech

Probab Eng Mech

Comput Struct

Probab Eng Mech

Struct Saf

Probab Eng Mech

Probab Eng Mech

Struct Saf

Probab Eng Mech

EN 1990:2002 – Eurocode

Reliability and statistics in geotechnical engineering

Mechanics of particulate media, a probabilistic approach

Reliability-based design in civil engineering

Probability concepts in engineering: emphasis on applications to civil engineering environment engineering

Risk assessment in geotechnical engineering

Applied non-Gaussian processes: examples, theory, simulation, linear random vibration, and MATLAB solutions

Systems of frequency curves generated by methods of translation

Biometrika

Safety factors and the probability distribution of soil strength

Can Geotech J

Model selection for a class of stochastic processes or random fields with bounded range

Probab Eng Mech

Probability distribution of strength parameters in uniform soils

J Eng Mech

Statistical inference for bounds of random variables

Biometrika

Estimating the distributions of manufactured dimensions with the beta probability density function

Int J Mach Tools Manuf