Reliability and scale effect in toughness of concrete structures

In the present work, the distribution of the random toughness characteristics (i.e. critical energy release rate, G_1c) has been evaluated on the basis of experimental observations. Fracture test results from three groups of geometrically similar concrete specimens of size (width×total depth×thickness), 420×420×50–1680×1680×200 mm³, made with different maximum aggregate size of 9.5, 19, 38, and 76 mm were analyzed using a recently proposed distribution of extremes. In applications of probability, it is important to use an appropriate distribution type and adequate techniques for estimating the parameters of distribution. In this study, a new type distribution of minima is employed for probability computations. It was noticed that the entropy of distribution increases with the crack length, i.e. the uncertainty of toughness, G_1c, value increases with crack length. A non-linear reduction of the maximum allowable splitting force with the defect size, a, was noticed. For large specimens, the maximum allowable splitting load is more sensitive to the required reliability level than that for small specimens. Reliability increases with aggregate size when all other conditions were constant.     

Efficient Spreadsheet Algorithm for First-Order Reliability Method

A new spreadsheet-cell-object-oriented algorithm for the first-order reliability method is proposed and illustrated for cases with correlated nonnormals and explicit and implicit performance functions. The new approach differs from the writers earlier algorithm by obviating the need for computations of equivalent normal means and equivalent normal standard deviations. It obtains the solution faster and is more efficient, robust, and succinct. Other advantages include ease of initialization prior to constrained optimization, ease of randomization of initial values for checking robustness, and fewer required optimization constraints during spreadsheet-automated search for the design point. Two cases with implicit performance functions, namely an asymmetrically loaded beam on Winkler medium and a strut with complex supports are analyzed using the new approach and discussed. Comparisons are also made between the proposed approach and that based on Rosenblatt transformation.     

A Bayesian approach to the multivariate Behrens-Fisher problem under the assumption of proportional covariance matrices

Summary Two independent random samples of sizesN ₁ andN ₂ from multivariate normal populationsN _p (θ₁,∑₁) andN _p (θ₂,∑₂) are considered. Under the null hypothesisH ₀: θ₁=θ₂, a single θ is generated from aN _p(μ, Σ) prior distribution, while underH ₁: θ₁≠θ₂ two means are generated from the exchangeable priorN _p(μ,σ). In both cases Σ will be assumed to have a vague prior distribution. For a simple covariance structure, the Bayes factorB and minimum Bayes factor in favour of the null hypotheses is derived. The Bayes risk for each hypothesis is derived and a strategy is discussed for using the Bayes factor and Bayes risks to test the hypothesis.     

Rolling and Lifting Probabilities for Sediment Entrainment

This study addresses the rolling and lifting probabilities for sediment entrainment by incorporating the probabilistic features of the turbulent fluctuation and bed grain geometry. The lognormally distributed instantaneous velocity and uniformly distributed initial grain position, along with a relation between lift coefficient and particle Reynolds number, are used to extend the theoretical formulation of the entrainment probabilities in smooth-bed flows. The two threshold conditions identified herein enable us to precisely define the probabilities of entrainment in the rolling and lifting modes. The results obtained in this study coincide well with the published data. The lifting probability increases monotonously with the dimensionless shear stress θ, which is consistent with the earlier results yet displays improved agreement with the experimental data. The maximum value of rolling probability, with a magnitude of 0.25, occurs at θ ≈ 0.15. For θ<0.05 (or θ>0.6), the rolling (or lifting) probability makes up more than 90% of the total entrainment probability and thus can be used as an approximation to the total probability of entrainment. The proposed rolling and lifting probabilities are further linked to the two separate criteria for incipient motion to explore the critical entrainment probabilities. The results reveal that a consistent probability corresponding to the critical state of sediment entrainment cannot be found.     

Extremes of random fields over arbitrary domains with application to concrete rupture stresses

To find the exact probability distribution of the global maximum or minimum of a random field within a bounded domain is a pending problem even for Gaussian fields. Except for very special examples of fields, recourse must be taken to approximate reasoning or asymptotic considerations to be judged with respect to accuracy by simulations. In this paper, the problem is addressed through a functional equation that leads to the definition of a class of distribution functions that depend solely on process or field characteristics and domain quantities that can be calculated explicitly. This distribution function class is studied for Gaussian processes in earlier works by the author and it has been obtained explicitly for Gaussian fields on rectangular domains in the plane. Simulation studies show that rather good predictions are obtained for sufficiently smooth wide band Gaussian processes and fields. In this paper, the distribution function is obtained in general for Gaussian fields over arbitrary bounded domains with piecewise continuous and differentiable boundaries, and as in earlier works the distribution function is tested against empirical distribution functions obtained by simulation of sample functions of a smooth approximately Gaussian field, herein called a broken line Hino field. For completeness this particular field type is defined in appendix a and appendix b. The paper concludes with a statistical application on data for plain concrete tensile strength.     

A method for multi-parameter PDF estimation of random variables

The probability distribution function (PDF) of a random variable Z is approximated with c·e^Q(z), where Q(z) is a polynomial function and c is normalizing constant. Based upon the weighted residual method, general linear algebraic equations have been derived for the evaluation of the unknown parameters in the polynomial. Numerical examples are presented and the results show that the PDFs obtained using the proposed method converge to those obtained from Monte Carlo simulation as the number of parameters in the approximate PDF increases.     

Seismic safety of lifelines—an optimum design method

Lifeline systems are vulnerable to two types of hazards arising from potential earthquake sources. These are the hazard of a fault-rupture strike on elements of a lifeline system and the hazard of overstress induced in different elements of the system because of the ground vibration. An optimum design method is presented in this paper for the design of a lifeline system for a maximum accepted probability of failure because of any of the two modes of failure. The method may be used to determine an optimum path between a number of fixed points which represent supply or demand stations in the system.     

Microstress estimate of stochastically heterogeneous structures by the functional perturbation method: A one dimensional example

A new functional perturbation method (FPM) for calculating the probabilistic response of stochastically heterogeneous, linear elastic structures is developed. The method is based on treating the governing differential operator as well as the unknown displacement function as a functional of material modulus field. By executing a functional perturbation around the homogeneous case, a set of successive differential equations is obtained and solved, from which the average and variance of any local parameter (displacements, stresses, strains) can be found. For a linear problem, the equations to be solved in each approximation order differ from the one for the homogeneous case by a pseudo external loading (right hand side) part only. Thus, only the Green function for the homogeneous case is needed for an analytical solution of the corresponding heterogeneous problem. A one dimensional stochastically heterogeneous rod embedded in a uniform shear resistant elastic medium is solved as an example. The statistical variance of displacements and stresses are found analytically, including the edge regions. Morphological (grain size) and material (modulus) effects on the stochastic response are demonstrated. The above results are essential for estimating the stochastic features of local stress concentrations, which are the source for many strength-related macro properties of materials. Extensive usage of generalized functions (Dirac operator and its derivatives) is needed for the analysis.     

根据概率分布建立有源器件的统计模型

给出一种新的统计模型的建立方法。在有源器件物理模型的基础上，通过优化物理模型参数的均值和均方根，使所建立统计模型的Ｓ参数在各个频率上的概率分布逼近测量数据的概率分布。从建立微波ＭＥＳＦＥＴ统计模型的算例结果看出，本方法不仅能比较好地拟合样本Ｓ参数的概率分布，而且提高了计算效率。