Forthcoming articles


International Journal of Reliability and Safety


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International Journal of Reliability and Safety (24 papers in press)


Regular Issues


  • Random finite element method for bearing capacity assessment of a shallow foundation under varied uniaxial loadings   Order a copy of this article
    by Amal Hentati, Mbarka Selmi, Tarek Kormi, Nizar Bel Hadj Ali 
    Abstract: This paper focuses on the application of the random finite element method (RFEM) for the assessment of the uniaxial bearing capacities of a shallow foundation subjected to centred vertical, horizontal and rotational loadings. The analysis combines finite element modelling, spatial variability analysis and Monte Carlo simulation. For this, the soil undrained shear strength is assumed to be variable in both horizontal and vertical directions with spatial dependency expressed via a Markovian autocorrelation function. The application of the proposed methodology to a shallow foundation permitted to highlight the insufficiency of the deterministic approach to predict the uniaxial foundation bearing capacities and led to different failure mechanisms.
    Keywords: random finite element method; random fields; spatial variability; shallow foundation reliability.

  • Bayesian networks and probability boxes to model uncertainty in unavailability assessment   Order a copy of this article
    by Walid Mechri, Wassim Snene, Kamel Ben Othman 
    Abstract: In this article, the problem of uncertainty in assessing the unavailability of Safety Instrumented Systems (SIS) is treated. The characteristic parameters of the SIS, including common cause failure factors, are replaced by probability densities families (p-boxes) allowing reliability experts to express their uncertainty on the statement of value probabilities. We show how the imprecision is modelled and propagated in a Bayesian network, which induces uncertainty about the failure probability on demand of the SIS and its safety integrity level. We will demonstrate how the uncertainty on some characteristic parameters values causes significant variations on the level risk.
    Keywords: safety systems; probability of failure on demand; uncertainty; Bayesian networks; common cause failure; p-boxes.

  • A prediction interval approach to developing life test acceptance criteria for progressively censored data   Order a copy of this article
    by Maram Salem, Zeinab Amin, Moshira Ismail 
    Abstract: In this paper we use the prediction-interval approach to construct acceptance criteria to determine whether or not certain batches of products are acceptable. The procedure is intended to protect us against highly defective lots and demonstrate that a required quality level is met with certain confidence. The prediction interval approach is particularly useful to employ when the lifetime of the product represents the quality characteristic of interest. On the basis of a progressively censored sample from the Weibull lifetime distribution, the problem of constructing acceptance criteria by predicting a future lifetime based on an independent past sample of lifetimes from the same distribution is addressed in a Bayesian setting with a dependent bivariate prior. The Metropolis-within-Gibbs sampler algorithm is used to obtain a sequence of draws from the posterior predictive distribution of future observations. This sequence is used to derive the prediction intervals based on which the lot acceptance criteria are determined. An example using real data is illustrated.
    Keywords: acceptance criteria; Metropolis-within-Gibbs sampler algorithm; prediction interval; progressively censored sample; Weibull lifetime model.

  • Availability optimization of Pre Heat Exchanger system with random repair and failure rates using PSO   Order a copy of this article
    by Ajay Kumar 
    Abstract: The main objective of this study is to carry out performance analysis and availability maximization with randomly selected failure and repair rates (FRR) of a Pre Heat Exchanger (PHE) system of a brewery plant using the Particle Swarm algorithm. The behavioral analysis of each system is carried out by Markovian method and the schematic diagram of Pre Heat Exchanger system represents various components and their connectivity (series/parallel/hybrid) whereas the transition diagram explores various possibilities and combinations of working states of the components i.e. full capacity working (FCW), reduced capacity working (RCW) and failed state (FS). The mathematical equations are formulated using the transition diagrams in order to carry out steady state availability (SSA) and transient state availability (TSA) analysis. The prediction of failure and repair rate to attain maximum availability of a system is done by using Particle Swarm Optimization (PSO) technique. These results are beneficial for plant personnel by steering of failure and repair rates to achieve maximum availability in plant.
    Keywords: Performance Modelling; Repair rates; Availability Optimization; Brewery Plant; Failure rate.

  • Reliability analysis of ((e, f), k, lc)/(m, n):F system under multiple failure using universal generating function   Order a copy of this article
    by Km. Meenakshi, S.B. Singh 
    Abstract: In this paper, the ((e, f),k, lc)/(m, n):F system is taken for study under multiple failures. The target system has m and n rows and columns, respectively. The system fails if any (e, f) submatrices fail or any k components fail, or consecutive lc components within any m rows or n columns fail out of (m, n) matrix. A Markov stochastic process is applied to obtain the probability of components of the system. Reliability indices such as reliability, mean time to failure and sensitivity analysis of the considered system are evaluated with the help of universal generating function. Finally, a numerical example is taken to demonstrate the model.
    Keywords: ((e; f),k; lc)/(m; n):F system; reliability; mean time to failure; sensitivity analysis; universal generating function.

  • Performance assessment of repairable system in series configuration under different types of failure and repair policies using copula linguistics   Order a copy of this article
    by Monika Gahlot, V.V. Singh, Hamisu Ismial Ayagi, C.K. Goel 
    Abstract: This paper deals with the study of reliability measures of a complex system consisting of two subsystems, subsystem-1 and subsystem-2, in a series configuration. The subsystem-1 has three units that work under the policy 2-out-of-3: F; and the subsystem-2 has two units working under 1-out-of-2: G; policy. Failure rates of the units of subsystems are constant and assumed to follow an exponential distribution, but the repair supports two types of distribution, general distribution and Gumbel- Hougaard family copula distribution. Two types of repair (general repair and Copula repair) are employed for partially failed and completed failed states. The system is analysed using the supplementary variable technique. Some important measures of reliability, such as the availability of the system, the reliability of the system, (MTTF) and profit analysis are discussed. Computations have taken a particular case by evaluating availability, reliability, MTTF and profit of operation of the system.
    Keywords: k-out-of-n: F; system; k-out-of-n: G; system; availability; MTTF; cost analysis; Gumbel-Hougaard family copula distribution.

  • Model-checking software product line based on multi-valued logic   Order a copy of this article
    by Shuang Liu, Yufeng Shi, Mingyu Huang 
    Abstract: A software product line (SPL) maximises commonality between software products to reduce cost and improve productivity. In this paper, we consider model-checking partial software product line designs. We first propose bilattice-based feature transitions systems (BFTSs) for modelling partial SPL designs, which support description of uncertainty and preserve features as a first class notion; the partial model and final model of product are defined via projection and simulation. We then express system behavioral properties using ACTL formulas and define its semantics over BFTSs. We investigate model-checking software product lines based on bilattice in two efficient ways: we provide the procedures that translate BFTSs to multi-valued Kripke structure and develop a software model-checker assistant BPMCA to leverage the power of the existing model-checking engine called XChek for verification; we decompose the multivalued BFTS into three-valued BFTS. Finally, we implement our approach on a benchmark from the literature.
    Keywords: model checking; software product line; multi-valued.

Special Issue on: IJRS REC2016 Computing with Polymorphic Uncertain Data

  • Uncertainty assessment in the results of inverse problems: applied to damage detection in masonry dams   Order a copy of this article
    by Long Nguyen-Tuan, Carsten Koenke, Volker Bettzieche, Tom Lahmer 
    Abstract: In this work, we study the uncertainties in the results of inverse problems. The inverse problems solve damage identification problems in multifield-multiphase problems for fluid-flow problems in deforming porous materials under non-isothermal boundary conditions. These analyses are important within the structural health monitoring of masonry dams. Results of the inverse problems show a scatter due to different sources of uncertainties in model parameters, measurement data, field of measurements, and in the solving algorithms of the inverse problem. In order to see and analyse the scatter, the inverse problem is solved repeatedly by a sampling process. The uncertainty in the inverse solutions can be quantified by their probability distributions according to the sampling results.
    Keywords: damage identification; masonry dams; optimisation; uncertainty quantification; random field.
    DOI: 10.1504/IJRS.2018.10013786
  • Numerical simulation of wooden structures with polymorphic uncertainty in material properties   Order a copy of this article
    by Ferenc Leichsenring, Christian Jenkel, Wolfgang Graf, Michael Kaliske 
    Abstract: Uncertainties are inherently present in structural parameters such as loadings, boundary conditions or resistance of structural materials. Especially material properties and parameters of wood are strongly varying in consequence of growth and environmental conditions. To include this variation in structural analysis, available data needs to be modelled appropriately, e.g. by means of probability and, furthermore, fuzzy probability based random variables or fuzzy sets. In order to comprehend uncertainties induced by estimating the distribution parameters, the stochastic approach has been extended by fuzzy distribution parameters to fuzzy probability based random variables according to studies by Möller et al. To cope with epistemic uncertainties for e.g. geometric parameters of knotholes, fuzzy sets are used. The consequence for wooden structures is determined by fuzzy stochastic analysis in combination with a Finite Element (FE) simulation using a model suitable for characteristics of a timber structure by Jenkel and Kaliske.
    Keywords: polymorphic uncertainty; fuzzy randomness; stochastic modelling; wood mechanics; structural analysis.
    DOI: 10.1504/IJRS.2018.10013787
  • Using statistical and interval-based approaches to propagate snow measurement uncertainty to structural reliability   Order a copy of this article
    by Árpád Rózsás, Miroslav Sýkora 
    Abstract: Observations are inevitably contaminated with measurement uncertainty, which is a predominant source of uncertainty in some cases. In present practice probabilistic models are typically fitted to measurements without proper consideration of this uncertainty. Hence, this study explores the effect of this simplification on structural reliability and provides recommendations on its appropriate treatment. Statistical and interval-based approaches are used to quantify and propagate measurement uncertainty in probabilistic reliability analysis. The two approaches are critically compared by analysing ground snow measurements that are often affected by large measurement uncertainty. The results indicate that measurement uncertainty may lead to significant (order of magnitude) underestimation of failure probability and should be taken into account in reliability analysis. Ranges of the key parameters are identified where measurement uncertainty should be considered. For practical applications, the lower interval bound and predictive reliability index are recommended as point estimates using interval and statistical analysis, respectively. The point estimates should be accompanied by uncertainty intervals, which convey valuable information about the credibility of results.
    Keywords: measurement uncertainty; snow; structural reliability; interval arithmetic; maximum likelihood; deconvolution.
    DOI: 10.1504/IJRS.2018.10013792
  • Extrapolation of extreme traffic load effects on a cable-stayed bridge based on weigh-in-motion measurements   Order a copy of this article
    by Naiwei Lu, Yang Liu, Michael Beer 
    Abstract: The steadily growing traffic loading may become a hazard for the bridge safety. Compared to short and medium span bridges, long-span bridges suffer from simultaneous presence of multiple vehicle loads. This study presents an approach for extrapolating probabilistic extreme effects on long-span bridges based on weigh-in-motion (WIM) measurements. Three types of stochastic traffic load models are simulated based on the WIM measurements of a highway in China. The level-crossing rate of each stochastic traffic load is evaluated and integrated for extrapolating extreme traffic load effects. The probability of exceedance of a cable-stayed bridge is evaluated considering a linear traffic growth model. The numerical results show that the superposition of crossing rates is effective and feasible to model the probabilistic extreme effects of long-span bridges under the actual traffic loads. The increase of dense traffic flows is sensitive to the maximum load effect extrapolation. The dense traffic flow governs the limit state of traffic load on long-span bridges.
    Keywords: bridge; traffic load; extreme value; level-crossing theory; weigh-in-motion; probability of exceedance.
    DOI: 10.1504/IJRS.2018.10013793
  • Solving the power allocation problem using methods with result verification   Order a copy of this article
    by Ekaterina Auer, César Benavente-Peces, Andreas Ahrens 
    Abstract: Characterising how different types of uncertainty in the multiple-input multiple-output (MIMO) systems influence their performance is an important research topic. In this paper, we focus on the task of power allocation in fixed rate MIMO systems with singular value decomposition based channel separation. The interval analysis is used to develop a verified solution to the problem taking bounded uncertainty in parameters and rounding errors into account. We demonstrate that power allocation improves the bit error rate (BER) using an exemplary 4 × 4 MIMO channel for two distinct choices of the channel matrix and compute an upper bound on the BER under realistic uncertainty conditions. Besides, we show that a combined analytical/numerical procedure produces better results than the purely numerical one and identify parameters the mathematical model is most sensitive to.
    Keywords: interval analysis; result verification; MIMO systems; power allocation.
    DOI: 10.1504/IJRS.2018.10013795
  • Fatigue reliability evaluation of short-span concrete bridges under dynamic impacts of stochastic truck loading   Order a copy of this article
    by Yuan Luo, Donghaung Yan, Ming Yuan 
    Abstract: This study presents an approach for the fatigue stress spectrum simulation of short-span bridges under the dynamic impacts of stochastic traffic loading. This approach is utilised to evaluate the fatigue reliability of existing bridges. Response functions defined by intervals are used to approximate the equivalent fatigue stress range of the bridge. Probability models of the fatigue stress ranges are evaluated with Gaussian mixture models. The effectiveness of the proposed method is validated via a case study of a simply supported bridge. The numerical results indicate that the impact effect of vehicle loads on short-span bridge leads to an obvious increase in both the stress range and the number of stress cycles. Both the degradation of the road surface roughness condition and the traffic growth lead to a significant decrease in the fatigue reliability.
    Keywords: fatigue reliability; vehicle-bridge coupled vibration; response surface method; stochastic traffic flow; road surface roughness; traffic volume.
    DOI: 10.1504/IJRS.2018.10013798
  • Structural dynamic problems in time domain under uncertainty: an interval finite element approach   Order a copy of this article
    by Naijia Xiao, Francesco Fedele, Rafi L. Muhanna 
    Abstract: An analysis of the structural dynamic response under uncertainty is presented. Uncertainties in load and material are modelled as intervals exploiting the interval finite element method (IFEM). To reduce overestimation and increase the computational efficiency of the solution, we do not solve the dynamic problem by an explicit step-by-step time integration scheme. Instead, our approach solves for the structural variables in the whole time domain simultaneously by an implicit scheme using discrete Fourier transform and its inverse (DFT and IDFT). Non-trivial initial conditions are handled by modifying the right-hand side of the governing equation. To further reduce overestimation, a new decomposition strategy is applied to the IFEM matrices, and both primary and derived quantities are solved simultaneously. The final solution is obtained using an iterative enclosure method, and in our numerical examples the exact solution is enclosed at minimal computational cost.
    Keywords: interval finite element method; dynamic response; discrete Fourier transform; matrix decomposition; iterative enclosure method.
    DOI: 10.1504/IJRS.2018.10013800
  • Solution of uncertain linear systems of equations with probability-box parameters   Order a copy of this article
    by Naijia Xiao, Robert L. Mullen, Rafi L. Muhanna 
    Abstract: The solution of linear systems of equations is often a component of engineering simulation and modelling. Often, the system parameters are uncertain. One representation of this uncertainty is the use of probability-boxes (or p-boxes), which do not require complete information about the probability distribution underlying the random variables. P-boxes are the bounds on allowable continuous distribution function for the random variables. Arithmetic operations on p-boxes yield guaranteed bounds on the probability distribution of the solution, regardless the nature of dependency. The solutions of p-box linear systems of equations are presented in the context of FEA of structural systems. Loading and material uncertainties are described by p-boxes. Earlier Monte-Carlo p-box approach was limited to independent uncertainties. The governing p-box linear equations are solved by an iterative approach using a fixed-point formulation. The resulting formulation gives guaranteed bounds of the probability distribution of the structural responses, at a high computational efficiency and a low overestimation level.
    Keywords: uncertainty; probability-box; matrix decomposition; iterative enclosure method.
    DOI: 10.1504/IJRS.2018.10013799
  • Reliable power flow and short circuit analysis of systems with uncertain data   Order a copy of this article
    by Shashwati Ray, Shimpy Ralhan 
    Abstract: This paper addresses the problem of uncertainties in the input parameters by specifying them as compact intervals, taking into consideration the errors in modelling and measurement of transmission line parameters and also the continuous influence of load measurement errors and fluctuations in the load demand. The power flow equations are modelled as a set of nonlinear algebraic equations which are first linearised using Taylor series expansion and the solution is obtained by the Krawczyk's method of interval arithmetic. For the short circuit analysis the prefault conditions are obtained from power flow analysis and the faulty network is then solved using Thevenin's equivalent network as seen from the fault point. The proposed method is applied to 3 bus, 14 bus and 30 bus IEEE test systems where load currents and fault currents for each relay are obtained in bounded form and thus well-defined relay coordination pairs are available.
    Keywords: interval mathematics; load flow analysis; uncertain data; Newton Raphson method; Intlab toolbox; Krawczyk's method; short circuit studies.
    DOI: 10.1504/IJRS.2018.10013806
  • A fuzzy surrogate modelling approach for real-time predictions in mechanised tunnelling   Order a copy of this article
    by Ba Trung Cao, Steffen Freitag, Günther Meschke 
    Abstract: In mechanised tunnelling, it is important to perform reliability analyses with respect to the tunnel face collapse and the damage risks of the tunnel lining and existing structures on the ground surface due to the tunnelling induced settlements. The reliability assessment requires to deal with limited information describing the local geology and the soil parameters due to the availability of only a small number of borehole data. In this paper, it is focused on real-time reliability analyses in mechanised tunnelling considering different types of uncertain data, i.e. combining epistemic and aleatoric sources of uncertainty within polymorphic uncertainty models. The system output of interest in these analyses is time variant tunnelling induced surface settlement fields, which are computed by a finite element simulation model. However, for real-time predictions with uncertain data, efficient and reliable surrogate models are required. A new surrogate modelling strategy is developed to predict time variant high dimensional fuzzy settlement fields in real-time. The predicted results of the new surrogate model show similar accuracy compared to the results obtained by optimisation based fuzzy analyses. Meanwhile, the computation time is significantly reduced especially in case of high dimensional outputs and in combination with the p-box approach in the case of polymorphic uncertain data.
    Keywords: surrogate models; proper orthogonal decomposition; artificial neural networks; uncertainty; reliability analysis; mechanised tunnelling; real-time predictions; fuzzy data.
    DOI: 10.1504/IJRS.2018.10013808
  • Equilibrium equations in interval models of structures   Order a copy of this article
    by Evgenija D. Popova 
    Abstract: A new interval model of linear equilibrium equations in mechanics is based on the algebraic completion of classical interval arithmetic and provides more realistic and accurate models involving interval uncertainties. It replaces straightforwardly a linear deterministic model by an interval model in terms of proper and improper intervals, fully conforms to the equilibrium principle, and provides sharper enclosure of a resultant force than the methods based on classical interval arithmetic. The present paper presents the interval algebraic approach to linear equilibrium equations and discusses its applications to some interval models of structures.
    Keywords: static equilibrium; structures; interval arithmetic; proper and improper intervals; overestimation.
    DOI: 10.1504/IJRS.2018.10013814

Special Issue on: Engineering Design for Safety and Reliability

  • Topology optimisation design of mechanical tee backsheet   Order a copy of this article
    by Wang Jinlong, Chen Junlong 
    Abstract: A finite element model of ASTM 114.3 mechanical tee backsheet is established by using the finite element software ANSYS. The stress distribution of the backsheet under pretension force load is obtained through the strength analysis, and based on this,a topology optimisation calculation of the backsheet is analysed. According to the topology optimisation results, the backsheet structure is redesigned and finite element analysis showed that, by reducing the volume of the model of the backsheet while improving its stress distribution, the maximum equivalent stress is reduced and the chip reliability is enhanced.
    Keywords: mechanical tee backsheet; finite element analysis; topology optimisation; optimal design.

  • Reliability assessment of pressure vessel design methods   Order a copy of this article
    by Hongjun Li, Peng Yang, Xun Huang, Hui Yang 
    Abstract: Several pressure vessel design methods based on elastic analysis and elastic-plastic analysis are available to designers. This paper proposes a reliability analysis method to assess three pressure vessel design methods: stress categorisation method, limit-load analysis in ASME code, and DBA-L method, which was recently proposed by the present authors. It was concluded that, with the same input variables into the three analyses, the responses of calculated results of the three methods were different, which provided an effective guidance to assess and choose the proper design method in engineering practice.
    Keywords: pressure vessel design; reliability; stress categorisation method; limit-load analysis method.

  • Multi-state system reliability analysis methods based on Bayesian networks, merging dynamic and fuzzy fault information   Order a copy of this article
    by Qin He, Ruijun Zhang, Tianyu Liu, Jie Liu 
    Abstract: Traditional Bayesian networks (BNs) have limited abilities to analyse system reliability with fuzzy and dynamic information. To deal with such information in system reliability analysis, a new multi-state system reliability analysis method based on BNs is proposed. The method expands the traditional BNs and effectively solves the deficiencies of existing reliability analysis methods based on BNs incorporating fuzziness and fault information. In this work, fuzzy set theory and changing failure probability function of components are introduced into BNs, and the concept of dynamic fuzzy subsets is introduced. The curve of the fuzzy dynamic fault probability of the leaf node fault state and fuzzy dynamic importance are developed and calculated with MATLAB software. Finally, a case study of a truck system is employed to demonstrate the performance of the proposed methods in comparison with traditional fault tree analysis methods and T-S fuzzy importance analysis methods. The proposed methods proved to be feasible in capturing the fuzzy and dynamic information in real-world systems.
    Keywords: fuzzy subsets; fuzziness; Bayesian network; travel system of a truck.

  • Reliability allocation technique for complex system of systems   Order a copy of this article
    by Antony Gratas Varuvel 
    Abstract: Reliability allocation is one of the important tasks during the design phase, which is to be executed as part of DfR practices. There are many techniques published in the literature for the stated purpose. Appropriate methodology is chosen based on the data available and the factors of influence required to be considered. Among those, the AGREE method is one of the reliability allocation techniques that is widely adopted during the early stages of the product/project definition. The popularity of this model is attributed to the assumption of standard exponential failure rate distribution, which is the simplest and easiest among the statistical failure rate distributions. Despite this, the AGREE method fails to meet the target reliability, when any/many of the importance index[es] is/are less than unity, resulting in impractical allocation of reliability. In addition, the assumption of exponential probability density function, which is the basis of the AGREE method stands valid only to depict failures arising out of randomness in the physical/environmental behaviour resulting in failures. Hence, validity and appropriateness of the AGREE method for a complex system of systems [SoS], wherein electronics, mechanical, and electromechanical systems are to be allocated, cease to converge. Subjectivity involved with the methodology published recently on reliability allocation procedure in complex redundant systems is greater, which leads to inconsistent results. This paper aims to eliminate the shortcomings of both the methods, which are enumerated. Representing the complex SoS and generalisation of a universal model that could be adapted to any domain, for the purposes of reliability allocation during the initial phases of design are the main objectives that are set forth, while conceptualising the model. Verification of the model under various boundary conditions has been carried out. Although the proposed model is aimed for general usage, it has been validated with the available data in the aerospace domain. Results obtained are found to be achieving the target goal set for the platform, which is a complex SoS.
    Keywords: AGREE; allocation; failure rate; fighter aircraft; maintainability; MTBF; MTTF; reliability.

  • New component-based reliability model to predict the reliability of component-based software   Order a copy of this article
    by Dimpal Tomar, Pradeep Tomar 
    Abstract: Component-based software technology has potential impact on the evolution of software development. One of the dominant questions while designing Component-Based Software (CBS) is to preserve its quality in which reliability has a crucial part. Therefore, prognosis of the reliability of a component-based software system is difficult because mostly components are of black type so the prediction of the emergent properties, such as reliability, is particularly difficult. In this paper, we propose a reliability estimation model noted as Component-Based Reliability Model (CBRM) to assess the reliability of individual components, and after integration of components, i.e. based on two factors: component reliability and average number of interaction failures.
    Keywords: component-based software reliability; CBSS; interaction ratio.

  • The probabilistic analysis of fatigue crack effect based on magnetic flux leakage   Order a copy of this article
    by Meor Qram Meor Ahmad, A. Arifin, S. Abdullah, W.Z.W. Jusoh, S.S.K. Singh 
    Abstract: In this paper, probabilistic analysis on the fatigue crack effect was investigated by applying the Metal Magnetic Memory (MMM) method, based on Self-Magnetic Leakage Field (SMLF) signals on the surface of metal components. The precision of MMM signals is essential in identifying the validity of the proposed method. The tension-tension fatigue test was conducted using the testing frequency of 10 Hz with 4 kN loaded, and the MMM signals were captured using the MMM instrument. As a result, a linear relationship was observed between the magnetic flux leakage and cyclic loading parameter, presenting the R-squared value at 0.72 0.97. The 2P-Weibull distribution function was used as a probabilistic approach to identify the precision of the data analysis from the predicted, and experimental fatigue lives, thereby showing that all points are placed within the range of a factor of 2. Additionally, the characteristics of PDF, CDF, failure rate and failure probability data analysis were plotted and described. Therefore, a 2P-Weibull probability distribution approach is determined to be an appropriate method to determine the accuracy of data analysis for MMM signals in a fatigue test for metal components.
    Keywords: MMM signals; fatigue lives; Weibull distribution.