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cumulative distribution function
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Published: 01 January 2000
Image
Published: 01 January 1997
Fig. 5 Cumulative distribution function for fatigue data from Table 1 based on assumed normal distribution
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Image
Published: 01 January 1997
Fig. 6 Cumulative distribution function for fatigue data from Table 1 based on an assumed log-normal distribution.
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Image
Published: 01 January 1997
Fig. 8 Approximate confidence limits on a cumulative distribution function showing uncertainty in the mean and standard deviation
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Series: ASM Handbook
Volume: 8
Publisher: ASM International
Published: 01 January 2000
DOI: 10.31399/asm.hb.v08.a0009213
EISBN: 978-1-62708-176-4
... on the probability density function, cumulative distribution function, population mean and variance, and parameter and percentile estimation. binomial distribution cumulative distribution function exponential distribution log normal distribution normal distribution Poisson distribution population mean...
Abstract
The six types of statistical distributions are normal distribution, log normal distribution, Weibull distribution, exponential distribution, binomial distribution, and Poisson distribution. This article discusses the applicability of each distribution, providing information on the probability density function, cumulative distribution function, population mean and variance, and parameter and percentile estimation.
Image
in Analysis Methods for Probabilistic Life Assessment
> Analysis and Prevention of Component and Equipment Failures
Published: 30 August 2021
Fig. 9 Illustration of the discrete probability distribution method. CDF, cumulative distribution function
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in Analysis Methods for Probabilistic Life Assessment
> Analysis and Prevention of Component and Equipment Failures
Published: 30 August 2021
Fig. 7 Illustration of sampling from a distribution using a random number generator. CDF, cumulative distribution function
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Image
in Analysis Methods for Probabilistic Life Assessment
> Analysis and Prevention of Component and Equipment Failures
Published: 30 August 2021
Fig. 8 Illustration of the Latin hypercube sampling method. CDF, cumulative distribution function
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Series: ASM Handbook
Volume: 24
Publisher: ASM International
Published: 15 June 2020
DOI: 10.31399/asm.hb.v24.a0006567
EISBN: 978-1-62708-290-7
... of the PSD results in a cumulative distribution function (CDF) representation of the distribution. As an example, Table 1 presents data for a mock PSD. Example data for both a count- and volume-based particle size distribution Table 1 Example data for both a count- and volume-based particle size...
Abstract
This article provides an overview of the general methods of metal powder production. It details the primary methods for particle sizing used in additive manufacturing: sieving, laser diffraction and scattering, and digital image analysis. Methods of interpreting and understanding particle size distribution (PSD) data are presented, with an emphasis on the differences between count- and volume-based PSDs. The article then outlines practices for both qualitative and quantitative assessment of particle morphology.
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002431
EISBN: 978-1-62708-194-8
..., is shown for this case in Fig. 4 . Fig. 4 Histogram of failure modes with their approximate discrete density function Cumulative Distribution Functions Plots of experimental data as density functions, as shown in Fig. 2 and 3 , provide some useful statistical information. Inferences...
Abstract
This article discusses some of the statistical aspects of design from an engineer's perspective. It reviews the commonly used statistical terms and distributions for providing some guidance on the practical engineering applications of these distributions. The article describes the basic statistical procedures that can be used to address variability and uncertainty in an engineering analysis. It contains a table that lists the relevant statistics standards published by the American Society of Testing and Materials.
Series: ASM Handbook
Volume: 13A
Publisher: ASM International
Published: 01 January 2003
DOI: 10.31399/asm.hb.v13a.a0003641
EISBN: 978-1-62708-182-5
... of experimental design and data analysis. A sample of test results of mass loss, mass gain, thickness loss, corrosion potential, corrosion rate, and pitting area may have a normal distribution. The following equations describe the normal probability distribution function, f ( x ), and the cumulative normal...
Abstract
This article details factors that have been used for evaluating the susceptibility of alloys to stress-corrosion cracking. Many considerations impacting the validity and accuracy of information gathered from laboratory testing programs are reviewed. The article highlights the main characteristics of probability distributions, such as normal distribution, log-normal distribution, exponential distribution, Poisson distribution, and extreme-value distribution. It also provides information on the statistical concepts to produce effective test programs.
Series: ASM Handbook
Volume: 11A
Publisher: ASM International
Published: 30 August 2021
DOI: 10.31399/asm.hb.v11A.a0006803
EISBN: 978-1-62708-329-4
... to use 1 M + 1 as a weight instead of 1 M for the empirical distribution, to acknowledge the fact that the extrema (minimum and maximum) are not reached within a sample of value. The empirical (discrete) cumulative distribution function is then: (Eq 1) F ( x J ) = J...
Abstract
This article provides an outline of the issues to consider in performing a probabilistic life assessment. It begins with an historical background and introduces the most common methods. The article then describes those methods covering subjects such as the required random variable definitions, how uncertainty is quantified, and input for the associated random variables, as well as the characterization of the response uncertainty. Next, it focuses on specific and generic uncertainty propagation techniques: first- and second-order reliability methods, the response surface method, and the most frequently used simulation methods, standard Monte Carlo sampling, Latin hypercube sampling, and discrete probability distribution sampling. Further, the article discusses methods developed to analyze the results of probabilistic methods and covers the use of epistemic and aleatory sampling as well as several statistical techniques. Finally, it illustrates some of the techniques with application problems for which probabilistic analysis is an essential element.
Book Chapter
Series: ASM Desk Editions
Publisher: ASM International
Published: 01 December 1998
DOI: 10.31399/asm.hb.mhde2.a0003088
EISBN: 978-1-62708-199-3
...Abstract Abstract This article describes design factors for products used in engineering applications. The article groups these factors into three categories: functional requirements, analysis of total life cycle, and other major factors. These categories intersect and overlap, constituting...
Abstract
This article describes design factors for products used in engineering applications. The article groups these factors into three categories: functional requirements, analysis of total life cycle, and other major factors. These categories intersect and overlap, constituting a major challenge in engineering design. Performance specifications, risk and hazard analysis, design process, design for manufacture and assembly, design for quality, reliability in design, and redesign are considered for functional requirements. Life-cycle analysis considers raw-material extraction from the earth and product manufacture, use, recycling (including design for recycling), and disposal. The other major factors considered include evaluation of the current state of the art for a given design, designing to codes and standards, and human factors/ergonomics.
Book: Fatigue and Fracture
Series: ASM Handbook
Volume: 19
Publisher: ASM International
Published: 01 January 1996
DOI: 10.31399/asm.hb.v19.a0002369
EISBN: 978-1-62708-193-1
... of a histogram where the area under each segment of the curve represents the percentage of all cycle amplitudes within that range. The integral of the pdf is the cumulative distribution function (cdf), which starts at zero and goes to one with increasing amplitude. The complementary cdf, one minus the cdf...
Abstract
There are two parts to deal with uncertainty in fatigue design: determining the distributions of possible values for all uncertain inputs and calculating the probability of failure due to all the uncertain inputs. This article discusses the sources of uncertainty in a fatigue analysis, such as the material properties, distribution of applied stress levels within a given environment, environments or loading intensities, and modeling or prediction. It presents a probabilistic approach for analyzing the uncertainties and determining the level of reliability (probability of failure).
Series: ASM Handbook Archive
Volume: 11
Publisher: ASM International
Published: 01 January 2002
DOI: 10.31399/asm.hb.v11.a0003514
EISBN: 978-1-62708-180-1
... of a component during the intended service life is sought. To help answer these questions, define the probability density function (PDF), f X ( x ), and the cumulative distribution function (CDF), F X ( x ) (see Fig. 2 ). Areas under the PDF define the probabilities as shown. The CDF defines...
Abstract
This article describes the historical background, uncertainties in structural parameters, classifications, and application areas of probabilistic analysis. It provides a discussion on the basic definition of random variables, some common distribution functions used in engineering, selection of a probability distribution, the failure model definition, and a definition of the probability of failure. The article also explains the solution techniques for special cases and general solution techniques, such as first-second-order reliability methods, the advanced mean value method, the response surface method, and Monte Carlo sampling. A brief introduction to importance sampling, time-variant reliability, system reliability, and risk analysis and target reliabilities is also provided. The article examines the various application problems for which probabilistic analysis is an essential element. Examples of the use of probabilistic analysis are presented. The article concludes with an overview of some of the commercially available software programs for performing probabilistic analysis.
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002476
EISBN: 978-1-62708-194-8
.... This second probability is given by the cumulative distribution function (CDF) for the resistance random variable ( F R ) evaluated at x , that is: (Eq 5) P 2 = F R ( x ) With the probability of failure defined as the product of these two probabilities, summed over all possible...
Abstract
Brittle materials, such as ceramics, intermetallics, and graphites, are increasingly being used in the fabrication of lightweight components. This article focuses on the design methodologies and characterization of certain material properties. It describes the fundamental concepts and models associated with performing time-independent and time-dependent reliability analyses for brittle materials exhibiting scatter in ultimate strength. The article discusses the two-parameter and three-parameter Weibull distribution for representing the underlying probability density function for tensile strength. It reviews life prediction reliability models used for predicting the life of a component with complex geometry and loading. The article outlines reliability algorithms and presents several applications to illustrate the utilization of these reliability algorithms in structural applications.
Book: Composites
Series: ASM Handbook
Volume: 21
Publisher: ASM International
Published: 01 January 2001
DOI: 10.31399/asm.hb.v21.a0003382
EISBN: 978-1-62708-195-5
... for fatigue life. For comparison, shape parameter for commonly used metals is approximately 7.0, indicating a greater degree of scatter for composites. The same information can also be cast into another form, called a cumulative Weibull distribution function. This function is a cumulative sum of the density...
Abstract
In the design of composite structures for durability and damage tolerance, the primary concerns are out-of-plane failures, such as delamination, material degradation associated with environment, stability under compression loading, large degree of scatter in fatigue life, and bearing failure of joints. This article presents an introductory discussion on the fatigue damage process, methodologies assessing fatigue behavior, and life prediction models. It describes the damage mechanisms introduced for a quasi-isotropic laminate under tension-compression fatigue loading. Delamination is a critical issue in fatigue and generally results from high interlaminar normal and shear stresses. The article schematically illustrates the structural elements in which high interlaminar stresses are common. It concludes with a discussion on the classification of fatigue models such as mechanistic or phenomenological, for composite materials under cyclic loading.
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002432
EISBN: 978-1-62708-194-8
... ) = e − ∫ 0 t λ ( t ) d t = exp [ − ∫ 0 t λ ( t ) d t ] This function is independent of the specific failure distribution involved. The case of constant failure rate, that is, λ( t ) independent of time, has special interest: (Eq 6a) R ( t...
Abstract
Reliability is a measure of the capacity of equipment or systems to operate without failure in the service environment. This article focuses on reliability in design and presents equations governing the instantaneous failure rate, general reliability function, mean time to failure, mean time between failures, and useful life period. The article describes the calculation of reliabilities for series and parallel arrangements of a complex system. It provides a comparison of probabilistic and deterministic design and concludes with a discussion on reliability growth.
Series: ASM Handbook
Volume: 8
Publisher: ASM International
Published: 01 January 2000
DOI: 10.31399/asm.hb.v08.a0009212
EISBN: 978-1-62708-176-4
... of the random variable at which the cumulative distribution function, F ( x ), is 0.50. In other words, it is the value that divides the population into equal parts. For symmetrical distributions such as the normal distribution, the median and the mean are equal. For nonsymmetrical distributions...
Abstract
This article discusses statistical concepts that form the basis of most of the following articles on specific areas of applied statistics and data analysis. It reviews some of the basic concepts that must be understood to successfully apply the statistical procedures, including probability, random variables, degrees of freedom, confidence limits and intervals, and reliability. Descriptive statistics, measures of central tendency, confidence limits and intervals, and degrees of freedom are also discussed.
Series: ASM Handbook
Volume: 17
Publisher: ASM International
Published: 01 August 2018
DOI: 10.31399/asm.hb.v17.a0006446
EISBN: 978-1-62708-190-0
... Acoustics Consider a longitudinal plane wave of particle displacement u ( x,t ) propagating in the x -direction in a nonlinear isotropic elastic solid. The ultrasonic wave is generated by a transmitter at x = 0, producing a time harmonic displacement function of t ; that is, u 0 ( t ) = U sin...
Abstract
Nonlinear ultrasonic nondestructive examination (NDE) techniques are based on nonlinear interaction of ultrasonic waves with the material to be characterized and defects to be detected. This article introduces the basic principles of nonlinear material-wave interaction, the origin of intrinsic nonlinearity in intact solids, and the main mechanisms of excess nonlinearity in damaged metals. It describes the measurement methods for nonlinear ultrasonic materials characterization and flaw-detection. The article schematically illustrates the instrumentation used for measurements of longitudinal wave and Rayleigh surface acoustic waves. It concludes with information on the applications of nonlinear ultrasonics.