Interfacial fatigue reliability of CRTS III slab ballastless track under material uncertainty: a combined experimental and numerical study

Reliability assessment of civil structures with incomplete probability distribution information

This chapter introduces probability theory as a system of models, based on measure theory, of some real-world phenomena. The models are measure spaces of total measure 1 and usually have certain distinguished measurable functions defined on them. Section 1 begins by establishing the measure-theoretic framework and a short dictionary for passing back and forth between terminology in measure theory and terminology in probability theory. The latter terminology includes events, random variables, mean, probability distribution of a random variable, and joint probability distribution of several random variables. An important feature of probability is that it is possible to work with random variables without any explicit knowledge of the underlying measure space, the joint probability distributions of random variables being the objects of importance. Section 2 introduces conditional probability and uses that to motivate the mathematical definition of independence of events. In turn, independence of events leads naturally to a definition of independent random variables. Independent random variables are of great importance in the subject and play a much larger role than their counterparts in abstract measure theory. Examples at the end of the section indicate the extent to which functions of independent random variables can remain independent. The techniques in the examples are of use in the subject of statistical inference, which is introduced in Section 10. Section 3 states and proves the Kolmogorov Extension Theorem, a foundational result allowing one to create stochastic processes involving infinite sets of times out of data corresponding to finite subsets of those times. A special case of the theorem provides the existence of infinite sets of independent random variables with specified probability distributions. Section 4 establishes the celebrated Strong Law of Large Numbers, which says that the Cesaro sums of a sequence of identically distributed independent random variables with finite mean converge almost everywhere to a constant random variable, the constant being the mean. This is a theorem that is vaguely known to the general public and is widely misunderstood. The proof is based on Kolmogorov’s inequality. Sections 5–8 provide background for the Central Limit Theorem, whose statement and proof are in Section 9. Section 5 discusses three successively weaker kinds of convergence for random variables—almost sure convergence, convergence in probability, and convergence in distribution. Convergence in distribution will be the appropriate kind for the Central Limit Theorem. Section 6 contains the Portmanteau Lemma, which gives some equivalent formulations of convergence in distribution, Section 7 introduces characteristic functions as Fourier transforms of probability distributions, and Section 8 proves the Levy Continuity Theorem, which formulates convergence in distribution in terms of characteristic functions. Section 9 contains the statement and proof of the Central Limit Theorem, followed by some simple examples. This theorem is the most celebrated result in probability theory and has many applications in mathematics and other fields. Section 10 is a brief introduction to the subject of statistical inference, showing how the Central Limit theorem plays a role in practice through the $t$ test of W. S. Gosset.

Complete monotonic expression of the fourth-moment normal transformation for structural reliability

General regularities related toLagrangian andHamiltonian equations are revealed. Probability distributions for functions ofHamiltonian random variables are considered. It is shown that all probability distributions of this kind are fully determined by the probability distributions for the random variables satisfying the corresponding Lagrangian equations. Some formulas related tocanonically conjugate operators are given. The similarity of these formulas to those related to Hamiltonian random variables is demonstrated. The “quantum approach” to the treatment of Hamiltonian random variables is discussed, and the origin of some peculiarities related to this approach is elucidated; it is explained, in particular, why it is impossible to form the joint probability density for canonically conjugate random variables when using this approach. The peculiarities revealed prove to be common for any objects possessing Hamiltonian random variables, irrespective of the nature of the objects, and coincide, therefore, with those in quantum mechanics. The existence of joint probability distributions for canonically conjugate random variables in the general case is demonstrated through the calculation of the corresponding joint mathematical expectations in an illustrative example. This proves, in particular, that joint probability distributions for canonically conjugate coordinates and momenta exist indeed in the case of mechanical microsystems. The results obtained prove once again that the pecularities of quantum mechanics are not related to the specificity of the measurements of physical quantities for microsystems.

In view of the probabilistic quantizer–dequantizer operators introduced, the qubit states (spin-1/2 particle states, two-level atom states) realizing the irreducible representation of the S U ( 2 ) symmetry group are identified with probability distributions (including the conditional ones) of classical-like dichotomic random variables. The dichotomic random variables are spin-1/2 particle projections m = ± 1 / 2 onto three perpendicular directions in the space. The invertible maps of qubit density operators onto fair probability distributions are constructed. In the suggested probability representation of quantum states, the Schrödinger and von Neumann equations for the state vectors and density operators are presented in explicit forms of the linear classical-like kinetic equations for the probability distributions of random variables. The star-product and quantizer–dequantizer formalisms are used to study the qubit properties; such formalisms are discussed for photon tomographic probability distribution and its correspondence to the Heisenberg–Weyl symmetry properties.

Worst-case conditional value-at-risk based bidding strategy for wind-hydro hybrid systems under probability distribution uncertainties

Introduction. The development of probabilistic approaches to the assessment of mechanical safety of bearing structural elements is one of the most relevant areas of research in the construction industry. In this research, probabilistic methods are developed to perform the reliability analysis of steel truss elements using the p-box (probability box) approach. This approach ensures a more conservative (interval-based) reliability assessment made within the framework of attaining practical objectives of the reliability analysis of planar trusses and their elements. The truss is analyzed as a provisional sequential mechanical system (in the language of the theory of reliability) consisting of elements that represent reliability values for each individual bar and truss node in terms of all criteria of limit states. Materials and methods. The co-authors suggest using p-blocks consisting of two boundary distribution functions designated for modeling random variables in the mathematical models of limit states performed within the framework of the truss reliability analysis instead of independent true functions of the probability distribution of random variables. Boundary distribution functions produce a probability distribution domain in which a true distribution function of a random variable is located. However this function is unknown in advance due to the aleatory and epistemic uncertainty. The choice of a p-block for modeling a random variable will depend on the type and amount of statistical information about the random variable. Results. The probabilistic snow load model and the numerical simulation of tests of steel samples of truss rods are employed to show that p-box models are optimal for modeling random variables to solve numerous practical problems of the probabilistic assessment of reliability of structural elements. The proposed p-box snow load model is based on the Gumbel distribution. The mathematical model used to perform the reliability analysis of planar steel truss elements is proposed. The co-authors provide calculation formulas to assess the reliability of a truss element for different types of p-blocks used to describe random variables depending on the amount of statistical data available. Conclusions. The application of statistically unsubstantiated hypotheses for choosing the probability distribution law or assessing the parameters of the probability distribution of a random variable leads to erroneous assessments of the reliability of structural elements, including trusses. P-boxes ensure a more careful reliability assessment of a structural element, but at the same time this assessment is less informative, as it is presented in the form of an interval. A more accurate reliability interval requires interval-based assessments of distribution parameters or types of p-boxes applied to mathematical models of the limit state, which entails an increase in the economic and labor costs of the statistical data.

The risk of damage during the construction phase of a coastal structure was investigated by utilizing Level II and III reliability methods. Structural risks were incorporated in the planning phase of coastal projects by utilizing a practical risk management approach implemented in the Mezitli (Icel) Yacht Harbor project in Turkey. By specifying the exceedance probability of damage level at the design stage, the delay times of construction activities due to damage were obtained at the project-planning phase, which enabled the estimation of the project completion time. It is found that structural risk of damage in the construction period can be an important factor to be considered in the reliability-based project management, because coastal projects involve considerable damage risk during the construction phase. The application of Monte Carlo simulations (Level III), which is generally performed within a few minutes of CPU time in fast computers, has the advantage of robustness when compared with the second-order methods such as the Hasofer-Lind method, provided that the probability distribution of random variables and their correlation are described with an acceptable accuracy, which should be obtained from data accumulated over a sufficient number of years. In this study, by specifying the exceedance probability of “no damage level” at the design stage, the probability of construction completion times were estimated at the planning phase for the case study of Mezitli Yacht Harbor, where the probability distributions of random variables were obtained from the governmental archives.

The five basic axioms of Kolmogorov define the probability in the real set R and do not take into cons ideration the imaginary part which takes place in the complex set C, a problem that we are facing in applied mathematics. Whatever the probability distribution of the random variable in R is, the corresponding probability in the whole set C equals always to one , so the outcome of the random experiment in C can be predicted totally. This is the consequence of the f act that the probability in C is got by subtracting the chaotic factor from the degree of our knowledge of the syst em. In this study, I will evaluate the complex rand om vectors and their resultant that represents the who le distribution and system in the complex space C. I will also define imaginary and complex expectations and variances and I will prove the law of large numbers usin g the concept of the resultant complex vector. In fact, a fter extending Kolmogorov’s system of axioms, the new axioms encompass the imaginary set of numbers and this by adding to the original five axioms of Kolmog orov an additional three axioms. Hence, the concept of c omplex random vector becomes clear, evident and it follows directly from the new axioms added. This re sult will be elaborated throughout this study using discrete probability distributions. Moreover, any experiment executed in the complex set C is the sum of the re al set R and the imaginary set M. Therefore, the whole probability distribution of random variables can be repr esented totally by the resultant complex random vector Z th at is used subsequently to prove the very well know n law of large numbers. In addition to my previous first paper, this second one elaborates the new field of “Complex Statistics” that considers random variables in the complex set C. Thus, the law of large numbers proves that this complex extension is successful and fruitful.

A review of the tomographic-probability representation of classical and quantum states is presented. The tomographic entropies and entropic uncertainty relations are discussed in connection with ambiguities in the interpretation of the state tomograms which are considered either as a set of the probability distributions of random variables depending on extra parameters or as a single joint probability distribution of these random variables and random parameters with specific properties of the marginals. Examples of optical tomograms of photon states, symplectic tomograms, and unitary spin tomograms of qudits are given. A new universal integral inequality for generic wave function is obtained on the base of tomographic entropic uncertainty relations.

A compatible probabilistic framework for quantification of simultaneous aleatory and epistemic uncertainty of basic parameters of structures by synthesizing the change of measure and change of random variables

In engineering structural systems, two types of uncertainty exist in systems widely. Epistemic uncertainty comes from incomplete information or ignorance while aleatory uncertainty derives from inherent variations. Due to the influence of many uncertainties and vagueness in the available information, all probabilities or probability distributions of random variables are precise known or perfect determination is impossible. For many structural reliability problems lacking information of the uncertain parameters, interval variable is a convenient and effective selection for the uncertainty description. According to this method, this paper suggests a new nonprobabilistic set model of structural reliability based on interval analysis and the satisfaction degree of the interval. The nonprobabilistic reliability of a structure is defined as the satisfaction degree between the stress-interval and the strength-interval. With the nonprobabilistic reliability model presented in this paper, a practical engineering example of the contact fatigue reliability analysis for the gear transmission is calculated and the result is reasonable and reliable.http://dx.doi.org/10.5755/j01.mech.17.1.208

Fatigue reliability analysis of wind turbine tower under random wind load

A sensitivity analysis represents crucial part of uncertainty quantification. The paper is focused on global sensitivity analysis, specifically moment-independent importance measure based on Cramér-von Mises distance. This type of sensitivity analysis takes whole probability distribution of random variables into account in contrast to commonly used Sobol' indices. It leads to more precise sensitivity analysis. Nevertheless, such a method is highly computationally demanding. Therefore, novel idea of utilization the polynomial chaos expansion for the estimation of conditional distributions is presented herein. The paper represents a pilot study of performance of such method using simple example.

Modeling and material uncertainty quantification of RC structural components