Auswahl der wissenschaftlichen Literatur zum Thema „Reliability (Engineering) Mathematical models“

The finite mixed Weibull distribution is an appropriate distribution in modeling the lifetime of the units having more than one possible failure cause. Due to the lack of a systematic statistical procedure of fitting the distribution to a data set, it has not been widely used in lifetime data analyses. Many areas on this subject have been studied in this research. The following are the findings and contributions. Through a change of variable, 5 parameters in a two Weibull mixture can be reduced to 3. A parameter'vector (p₁, η, β) defines a family of two-Weibull mixtures which have common characteristics. Numerous probability plots are investigated on Weibull probability paper (WPP). For a given p₁ the η-β plane is partitioned into seven regions which are labeled by A through F and S. The Region S represents the two Weibull mixtures whose cdf curves are very close to a straight line. The Regions A through F represent six typical shapes of the cdf curves on WPP, respectively. The two-Weibull mixtures in one region have similar characteristics. Three important features of the two-Weibull mixture with well separated subpopulations are proved. Two existing methods for the graphical estimation of the parameters are discussed, and one is recommended over the other. The EM algorithm is successfully applied to solve the MLE for mixed Weibull distributions when m, the number of subpopulations in a mixture is known. The algorithms for complete, censored, grouped and suspended samples with non-postmortem and postmortem failures are developed accordingly. The developed algorithms are powerful, efficient and they are insensitive to the initial guesses. Extensive Monte Carlo simulations are performed. The distributions of the MLE of the parameters and of the reliability of a two Weibull mixture are studied. The MLEs of the parameters are sensitive to the degree of separation of the two subpopulation pdfs, but the MLE of the reliability is not. The generalized likelihood ratio (GLR) test is used to determine m. Under H₀: m=1 and H₁: m=m₁>1, ζ, the GLR is independent of the parameters in the distribution of H₀. The distributions of ζ or -21n(ζ) with n=50, 100 and 150 are obtained through Monte Carlo simulations. Compared with the chi-square distribution, they fall in between x²(4) and x²(6), and they are very close to x²(5). A FORTRAN computer program is developed to conduct simulation of the GLR test for 1 ≤ m₀ < m₁ ≤ 5.

Thesis (MScEng)-- Stellenbosch University, 2013.<br>ENGLISH ABSTRACT: Most of the parameters involved in the design and analysis of structures are of stochastic nature. This is, therefore, of paramount importance to be able to perform a fully stochastic analysis of structures both in component and system level to take into account the uncertainties involved in structural analysis and design. To the contrary, in practice, the (computerised) analysis of structures is based on a deterministic analysis which fails to address the randomness of design and analysis parameters. This means that an investigation on the algorithmic methodologies for a component and system reliability analysis can help pave the way towards the implementation of fully stochastic analysis of structures in a computer environment. This study is focused on algorithm development for component and system reliability analysis based on the various proposed methodologies. Truss structures were selected for this purpose due to their simplicity as well as their wide use in the industry. Nevertheless, the algorithms developed in this study can be used for other types of structures such as moment-resisting frames with some simple modi cations. For a component level reliability analysis of structures different methods such as First Order Reliability Methods (FORM) and simulation methods are proposed. However, implementation of these methods for the statistically indeterminate structures is complex due to the implicit relation between the response of the structural system and the load effect. As a result, the algorithm developed for the purpose of component reliability analysis should be based on the concepts of Stochastic Finite Element Methods (SFEM) where a proper link between the finite element analysis of the structure and the reliability analysis methodology is ensured. In this study various algorithms are developed based on the FORM method, Monte Carlo simulation, and the Response Surface Method (RSM). Using the FORM method, two methodologies are considered: one is based on the development of a finite element code where required alterations are made to the FEM code and the other is based on the usage of a commercial FEM package. Different simulation methods are also implemented: Direct Monte Carlo Simulation (DMCS), Latin Hypercube Sampling Monte Carlo (LHCSMC), and Updated Latin Hypercube Sampling Monte Carlo (ULHCSMC). Moreover, RSM is used together with simulation methods. Throughout the thesis, the effciency of these methods was investigated. A Fully Stochastic Finite Element Method (FSFEM) with alterations to the finite element code seems the fastest approach since the linking between the FEM package and reliability analysis is avoided. Simulation methods can also be effectively used for the reliability evaluation where ULHCSMC seemed to be the most efficient method followed by LHCSMC and DMCS. The response surface method is the least straight forward method for an algorithmic component reliability analysis; however, it is useful for the system reliability evaluation. For a system level reliability analysis two methods were considered: the ß-unzipping method and the branch and bound method. The ß-unzipping method is based on a level-wise system reliability evaluation where the structure is modelled at different damaged levels according to its degree of redundancy. In each level, the so-called unzipping intervals are defined for the identification of the critical elements. The branch and bound method is based on the identification of different failure paths of the structure by the expansion of the structural failure tree. The evaluation of the damaged states for both of the methods is the same. Furthermore, both of the methods lead to the development of a parallel-series model for the structural system. The only difference between the two methods is in the search approach used for the failure sequence identification. It was shown that the ß-unzipping method provides a better algorithmic approach for evaluating the system reliability compared to the branch and bound method. Nevertheless, the branch and bound method is a more robust method in the identification of structural failure sequences. One possible way to increase the efficiency of the ß-unzipping method is to define bigger unzipping intervals in each level which can be possible through a computerised analysis. For such an analysis four major modules are required: a general intact structure module, a damaged structure module, a reliability analysis module, and a system reliability module. In this thesis different computer programs were developed for both system and component reliability analysis based on the developed algorithms. The computer programs are presented in the appendices of the thesis.<br>AFRIKAANSE OPSOMMING: Meeste van die veranderlikes betrokke by die ontwerp en analise van strukture is stogasties in hul aard. Om die onsekerhede betrokke in ontwerp en analise in ag te neem is dit dus van groot belang om 'n ten volle stogastiese analise te kan uitvoer op beide komponent asook stelsel vlak. In teenstelling hiermee is die gerekenariseerde analise van strukture in praktyk gebaseer op deterministiese analise wat nie suksesvol is om die stogastiese aard van ontwerp veranderlikes in ag te neem nie. Dit beteken dat die ondersoek na die algoritmiese metodiek vir komponent en stelsel betroubaarheid analise kan help om die weg te baan na die implementering van ten volle rekenaarmatige stogastiese analise van strukture. Di e studie se fokus is op die ontwikkeling van algoritmes vir komponent en stelsel betroubaarheid analise soos gegrond op verskeie voorgestelde metodes. Vakwerk strukture is gekies vir die doeleinde as gevolg van hulle eenvoud asook hulle wydverspreide gebruik in industrie. Die algoritmes wat in die studie ontwikkel is kan nietemin ook vir ander tipes strukture soos moment-vaste raamwerke gebruik word, gegewe eenvoudige aanpassings. Vir 'n komponent vlak betroubaarheid analise van strukture word verskeie metodes soos die "First Order Reliability Methods" (FORM) en simulasie metodes voorgestel. Die implementering van die metodes vir staties onbepaalbare strukture is ingewikkeld as gevolg van die implisiete verband tussen die gedrag van die struktuur stelsel en die las effek. As 'n gevolg, moet die algoritme wat ontwikkel word vir die doel van komponent betroubaarheid analise gebaseer word op die konsepte van stogastiese eindige element metodes ("SFEM") waar 'n duidelike verband tussen die eindige element analise van die struktuur en die betroubaarheid analise verseker is. In hierdie studie word verskeie algoritmes ontwikkel wat gebaseer is op die FORM metode, Monte Carlo simulasie, en die sogenaamde "Response Surface Method" (RSM). Vir die gebruik van die FORM metode word twee verdere metodologieë ondersoek: een gebaseer op die ontwikkeling van 'n eindige element kode waar nodige verandering aan die eindige element kode self gemaak word en die ander waar 'n kommersiële eindige element pakket gebruik word. Verskillende simulasie metodes word ook geïmplimenteer naamlik Direkte Monte Carlo Simulasie (DMCS), "Latin Hypercube Sampling Monte Carlo" (LHCSMC) en sogenaamde "Updated Latin Hypercube Sampling Monte Carlo" (ULHCSMC). Verder, word RSM tesame met die simulasie metodes gebruik. In die tesis word die doeltreffendheid van die bostaande metodes deurgaans ondersoek. 'n Ten volle stogastiese eindige element metode ("FSFEM") met verandering aan die eindige element kode blyk die vinnigste benadering te wees omdat die koppeling tussen die eindige element metode pakket en die betroubaarheid analise verhoed word. Simulasie metodes kan ook effektief aangewend word vir die betroubaarheid evaluasie waar ULHCSMC as die mees doeltre end voorgekom het, gevolg deur LHCSMC en DMCS. The RSM metode is die mees komplekse metode vir algoritmiese komponent betroubaarheid analise. Die metode is egter nuttig vir sisteem betroubaarheid analise. Vir sisteem-vlak betroubaarheid analise is twee metodes oorweeg naamlik die "ß-unzipping" metode and die "branch-and-bound" metode. Die "ß-unzipping" metode is gebaseer op 'n sisteem-vlak betroubaarheid ontleding waar die struktuur op verskillende skade vlakke gemodelleer word soos toepaslik vir die hoeveelheid addisionele las paaie. In elke vlak word die sogenaamde "unzipping" intervalle gedefinieer vir die identifikasie van die kritiese elemente. Die "branch-and-bound" metode is gebaseer op die identifikasie van verskillende faling roetes van die struktuur deur uitbreiding van die falingsboom. The ondersoek van die skade toestande vir beide metodes is dieselfde. Verder kan beide metodes lei tot die ontwikkeling van 'n parallelserie model van die strukturele stelsel. Die enigste verskil tussen die twee metodes is in die soek-benadering vir die uitkenning van falingsmodus volgorde. Dit word getoon dat die "ß-unzipping" metode 'n beter algoritmiese benadering is vir die ontleding van sisteem betroubaarheid vergeleke met die "branch-and-bound" metode. Die "branch-and- bound" metode word nietemin as 'n meer robuuste metode vir die uitkenning van die falings volgorde beskou. Een moontlike manier om die doeltre endheid van die "ß-unzipping" metode te verhoog is om groter "unzipping" intervalle te gebruik, wat moontlik is vir rekenaarmatige analise. Vir so 'n analise word vier hoof modules benodig naamlik 'n algemene heel-struktuur module, 'n beskadigde-struktuur module, 'n betroubaarheid analise module en 'n sisteem betroubaarheid analise module. In die tesis word verskillende rekenaar programme ontwikkel vir beide sisteem en komponent betroubaarheid analise. Die rekenaar programme word in die aanhangsels van die tesis aangebied.

Structural reliability technology provides analytical tools for management of uncertainty in all relevant design factors in structural and mechanical systems. Generally, the goal of analysis is to compute probabilities of failure in structural components or system having single or multiple failure mode. Alternately, modern optimization methods provide efficient numerical algorithms for locating optima, particularly in large-scale systems having prescribed deterministic constraints. Optimization procedure can accommodate random variables either directly in its objective function or as one of the primary constraints. The combination of elementary optimization and probabilistic design techniques is the subject of this study. Presented herein is a general strategy for optimization when the design factors are random variables and some or all of the constraints are probability statements. A literature review has indicated that optimization technology in a reliability context has not been fully explored for the general case of nonlinear performance functions and nonnormal variates associated multiple failure modes. This research focuses upon development of the theory to address this general problem. Because analysis algorithms are complicated, a computer code, program RELOPT, is constructed to automate the analysis. The objective function to be minimized is arbitrary, but would generally be the total expected lifetime costs including all initial costs as well as all costs associated with failure. Uncertainty is assumed to be possible in all design factors (including the factors to be determined), and they are modeled as random variables. In general, all of the constraints can be probability statements. The generalized reduce gradient (GRG) method was used for optimization calculations. Options for point probability calculations are first order reliability analysis using the Rackwitz-Fiessler (R-F) or advanced reliability analysis using Wu/FPI. For system reliability analysis either the first order Cornell's bounds or the second order Ditlevsen's bounds can be specified. Several examples are presented to illustrate the full range of capabilities of RELOPT. The program is validated by checking with independent and exact solutions. An example is provided which demonstrates that the cost of running RELOPT can be substantial as the size of the problem increases.

Thesis (Ph. D.)--Mechanical Engineering, Georgia Institute of Technology, 2008.<br>Committee Co-Chair: Peak, Russell; Committee Co-Chair: Sitaraman, Suresh; Committee Member: Paredis, Christiaan; Committee Member: Pucha, Raghuram; Committee Member: Wong, C.

"October 2001." Bibliography: leaves 99-103. Establishes several sets of conditioning relating to the variant optimal deign of linear consecutive-k-out-of-n systems and includes a review of existing research in the theory of variant optimal design of linear consecutive-k-out-of-n systems.

Metallic structures dominated by tensile loads are vulnerable to fatigue and fracture. Fatigue is produced by oscillatory loads. Quasi-static brittle or ductile fracture can result from a "large" load in the random sequence. Moreover, a fatigue or fracture failure in a member of a redundant structure produces impulsive redistributed loads to the intact members. These transient loads could produce a sequence of failures resulting in progressive collapse of the system. Fatigue and fracture design factors are subject to considerable uncertainty. Therefore, a probabilistic approach, which includes a system reliability assessment, is appropriate for design purposes. But system reliability can be improved by a maintenance program of periodic inspection with repair and/or replacement of damaged members. However, a maintenance program can be expensive. The ultimate goal of the engineer is to specify a design, inspection, and repair strategy to minimize life cycle costs. The fatigue/fracture reliability and maintainability (FRM) process for redundant structure can be a complicated random process. The structural model considered series, parallel, and parallel/series systems of elements. Applied to the system are fatigue loads including mean stress, an extreme load, as well as impulsive loads in parallel member systems. The failure modes are fatigue, brittle and ductile fracture. A refined fatigue model is employed which includes both the crack initiation and propagation phases. The FRM process cannot be solved easily using recently developed advanced structural reliability techniques. A "hybrid" simulation method which combines modified importance sampling (MIS) with inflated stress extrapolation (ISE) is proposed. MIS and ISE methods are developed and demonstrated using numerous examples which include series, parallel and series/parallel systems. Not only reasonable estimates of the probability of system failure but also an estimate of the distribution of time to system failure can be obtained. The time to failure distribution can be used to estimate the reliability function, hazard function, conditional reliability given survival at any time, etc. The demonstration cases illustrate how reliability of a system having given material properties is influenced by the number of series and parallel elements, stress level, mean stress, and various inspection/repair policies.