The science of engineering is soiled by uncertainties. The experimental data cost. The design stumbles at random. The objective of the design is to maximize the chances of success of a dimensioning. The objective of this work is to allow the users to understand the main methods. This volume is centered on a vast range of statistical distributions met in reliability. The aim is to run statistical measures, to present a report of enhanced measures in mechanical reliability and to evaluate the reliability of the repairable or not repairable systems. To reach these purposes, we present an approach theory /practice based on these themes: Criteria of failures; Bayesian Applied Probability; Chains of Markov; Simulation of Monte Carlo and finally many solved cases of studies.

Ammar Grous is Teacher of Mechanical Engineering at CéGEP de l'Outaouais (Academic College), Gatineau, Quebec, Canada.

Preface

Glossary

Chapter 1 Fracture Mechanisms by Fatigue

Introduction

Principal physical mechanisms of cracking by fatigue

Fracture mechanics

Criteria of fracture (plasticity) in mechanics

Modes of fracture

Directed works

Fatigue of metals: analytical expressions used in reliability

Wöhler’s law

Basquin’s law

Stromayer’s law

Palmgren’s law

Corson’s law

Bastenaire’s law

Weibull’s law

Henry’s law

Corten and Dolen’s law

Manson–Coffin’s law

Reliability models commonly used in fracture mechanics by fatigue

Coffin–Manson’s model for the analysis of crack propagation

Neuber’s relation

Arrhenius’ model

Miner’s law

Main common laws retained by fracture mechanics

Fost and Dugdale’s law

McEvily’s law

Paris’s law

GR Sih’s law

Stress intensity factors in fracture mechanics

Maddox’s model

Gross and Srawley’s model

Lawrence’s model

Martin and Bousseau’s model

Gurney’s model

Engesvik’s model

Yamada and Albrecht’s model

Tomkins and Scott’s model

Harrison’s model

Intrinsic parameters of the material (C and m)

Fracture mechanics elements used in reliability

Crack rate (life expectancy) and sif (Kσ)

Simplified version of Taylor’s law for machining

Elements of stress (S) and resistance theory (R)

Case study, part – suspension bridge (Cirta)

Case study: failure surface of geotechnical materials

Conclusion

Bibliography

Chapter 2 Analysis Elements for Determining the Probability of Rupture by Simple Bounds

Introduction

First-order bounds or simple bounds: systems in series

First-order bounds or simple bounds: systems in parallel

Second-order bounds or Ditlevsen’s bounds

Evaluating the probability of the intersection of two events

Estimating multinomial distribution–normal distribution

Binomial distribution

Hohenbichler’s method

Hypothesis test, through the example of a normal average with unknown variance

Development and calculations

Confidence interval for estimating a normal mean: unknown variance
Conclusion

Bibliography

Chapter 3 Analysis of the Reliability of Materials and Structures by the Bayesian Approach

Introduction to the Bayesian method used to evaluate reliability

Posterior distribution and conjugate models

Independent events

Counting diagram

Conditional probability or Bayes’ law

Anterior and posterior distributions

Reliability analysis by moments methods, FORM/SORM

Control margins from the results of fracture mechanics

Bayesian model by exponential gamma distribution

Homogeneous Poisson process and rate of occurrence of failure

Estimating the maximum likelihood

Type I censored exponential model

Estimating the MTBF (or rate of repair/rate of failure)

MTBF and confidence interval

Repair rate or ROCOF

Power law: non-homogeneous Poisson process

Distribution law – gamma (reminder)

Bayesian model of a priori gamma distribution

Distribution tests for exponential life (or HPP model)

Bayesian procedure for the exponential system model

Bayesian case study applied in fracture mechanics

Conclusion

Bibliography

Chapter 4 Elements of Analysis for the Reliability of Components by Markov Chains

Introduction

Applying Markov chains to a fatigue model

Case study with the help of Markov chains for a fatigue model

Position of the problem

Discussion

Explanatory information

Directed works

Approach for solving the problem

Which solution should we choose?

Conclusion

Bibliography

Chapter 5 Reliability Indices

Introduction

Design of material and structure reliability

Reliability of materials and structures

First-order reliability method

Second-order reliability method

Cornell’s reliability index

Hasofer–Lind’s reliability index

Reliability of material and structure components

Reliability of systems in parallels and series

Parallel system

Parallel system (m/n)

Serial assembly system

Conclusion

Bibliography

Chapter 6 Fracture Criteria Reliability Methods through an Integral Damage Indicator

Introduction

Literature review of the integral damage indicator method

Brief recap of the FORM/SORM method

Recap of the Hasofer–Lind index method

Literature review of the probabilistic approach of cracking law

parameters in region II of the Paris law

Crack spreading by a classical fatigue model

Reliability calculations using the integral damage indicator method
Conclusion

Bibliography

Chapter 7 Monte Carlo Simulation

Introduction

From the origin of the Monte Carlo method!

The terminology

Simulation of a singular variable of a Gaussian

Simulation of non-Gaussian variable

Simulation of correlated variables

Simulation of correlated Gaussian variables

Simulation of correlated non-Gaussian variables

Determining safety indices using Monte Carlo simulation

General tools and problem outline

Presentation and discussion of our experimental results

Use of the randomly selected numbers table

Applied mathematical techniques to generate random numbers

by MC simulation on four principle statistical laws

Uniform law

Laplace–Gauss (normal) law

Exponential law

Initial value control

Conclusion

Bibliography

Chapter 8 Case Studies

Introduction

Reliability indicators (λ) and MTBF

Model of parallel assembly

Model of serial assembly

Parallel or redundant model

Reliability and structural redundancy: systems without distribution
Serial model

Rate of constant failure

Reliability of systems without repairing: parallel model

Reliability applications in cases of redundant systems

Total active redundancy

Partial active redundancy

Reliability and availability of repairable systems

Quality assurance in reliability

Projected analysis of reliability

Birnbaum–Saunders distribution in crack spreading

Probability density and distribution function

(Birnbaum–Saunders cumulative distribution through cracking)

Graph plots for the four probability density functions and

distribution functions

Reliability calculation for ages (τ) in hours of service, Ri(τ) = ?

Simulation methods in mechanical reliability of structures and materials: the Monte Carlo simulation method

Weibull law

Log-normal Law (of Galton)

Exponential law

Generation of random numbers

Elements of safety via the couple: resistance and stress (R, S)

Reliability trials

x Fracture Mechanics

Controlling risks and efficiency in mechanical reliability

Truncated trials

Censored trials

Trial plan

Coefficients for the trial’s acceptance plan

Trial’s rejection plan (in the same conditions)

Trial plan in reliability and K Pearson test χ

Reliability application on speed reducers (gears)

Applied example on hydraulic motors

Reliability case study in columns under stress of buckling

RDM solution

Problem outline and probabilistic solution

(reliability and error)

Adjustment of least squared for nonlinear functions

Specific case study : a Weibull law with two parameters
Conclusion

Bibliography

Appendix

Index

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