Selecting relevant effects in factorial designs

Industrial contexts tend to be as much or more concerned about the probability of ignoring an effect when its influence on the response is relevant (Type II error) than about the probability of considering an effect to be active when in fact, it is not (Type I error). Here, we present a methodology for taking into account both types of error by fixing an effect value that is considered large enough to control the probability of it going unnoticed. In addition, we propose a plot to visualize the results obtained.     

Chain Sampling Scheme under Constant Inspection Errors

The primary aim of this paper is to extend the inspection error consideration to chain sampling schemes, an area that has not been dealt with in the literature. A mathematical model is developed to investigate the performance of chain sampling schemes under constant inspection errors. Expressions of performance measures, such as operating characteristic function, average total inspection and average outgoing quality, are derived to aid the analysis of a general chain sampling scheme, ChSP‐4A ( ) r, developed by Frishman. This study reveals that as Type I inspection error increases the probability of acceptance will decrease and as Type II inspection error increases the acceptance probability will increase. The effect of Type II error on the probability of acceptance is very marginal compared with that of Type I error, especially when the true fraction non‐conforming is small. In addition, the effects of inspection errors can be ‘eliminated’ by transforming to its equivalent perfect inspection counterpart, hence greatly reducing the complexity of the analysis. The effects of other sampling parameters are also studied to serve as a foundation for future plan designing purposes. Copyright © 2006 John Wiley & Sons, Ltd.     

The Effect of Estimation Error on Risk‐Adjusted Survival Time CUSUM Chart Performance

Research on risk‐adjusted control charts has gained great interest in healthcare settings. Based on monitored variables (binary outcome or survival times), risk‐adjusted cumulative sum (CUSUM) charts are divided into Bernoulli and survival time CUSUM charts. The effect of estimation error on control chart performance has been systematically studied for Bernoulli CUSUM but not for survival time CUSUM in continuous time. We investigate the effect of estimation error on the performance of risk‐adjusted survival time CUSUM scheme in continuous time with the cardiac surgery data. The impact is studied with the use of the median run lengths (medRLs) and the standard deviation (SD) of medRLs for different sample sizes, specified in‐control median run length, adverse event rate and patient variability. Results show that estimation error affects the performance of risk‐adjusted survival time CUSUM chart significantly and the performance is more sensitive to the specified in‐control median run length (medRL₀) and adverse event rate. To take the estimation error into account, the practitioners can bootstrap many samples from Phase I data and then determine the threshold that can guarantee at least a medRL₀ with certain probability under which false alarms occur less frequently and meanwhile out‐of‐control alarms don't signal too slow. Moreover, additional event occurrences can be used to update the estimation but should be from in‐control process. Finally, non‐parametric bootstrap can be applied to reduce model misspecification error. Copyright © 2015 John Wiley & Sons, Ltd.     

A model for evaluating the effectiveness of motor vehicle inspection programs

A statistical model for the evaluation of the effectiveness of motor vehicle inspection programs in reducing highway crashes is presented. The model is based on the assumption that the waiting time between highway crashes follows an exponential distribution. Since highway crashes are relatively rare events, it is assumed that the length of the study period is such that censoring occurs. Under these assumptions, maximum likelihood estimates of the mean waiting time θ until a crash for the non-inspected (inspected) vehicles is obtained and the corresponding test statistic is derived. As mechanically-caused accidents are but a small part of the overall accident picture and since inspection should only affect this portion, sample size requirements are investigated for various combinations of θ, Δ (increase in average time until a crash due to the effect of inspection), L (length of study period), and = β (probability of Type I error equalling probability of Type II error). For reasonable Δ, the sample required is indeed sizable.     

Impact of censoring types on the two-stage method for analyzing reliability experiments with random effects

Many reliability experiments are not completely randomized. Instead they involve subsamples, blocks, split-plot structures, etc. A common analysis often uses random effects to account for the impact of the experimental protocol. The two-stage method is an easy way for practitioners to incorporate random effects in the analysis. This article compares performance of the two-stage method under Type I, Type II censored, and uncensored data from a Weibull distribution. We evaluate the effects of censoring type, censoring rate, sample size, and shape parameter on the two-stage method. Then, we apply the two-stage method to a real experiment. Finally, we give practitioners some recommendations for designing and analyzing reliability experiments.     

A reliability allocation method of mechanism considering system performance reliability

It is generally believed that the reliability of a mechanical system is determined by its composition. The system operates properly when all of its components do not fail. Based on this assumption, the reliability of the system can be represented by the reliability of its components. A problem arises when applying this hypothesis to a system containing motion mechanisms. There is a phenomenon in motion mechanism that the components do not happen structural failure (we call it “Type I failure”) and joint failure (we call it “Type II failure”), but the function of the mechanism cannot meet the requirements (we call it “Type III failure”). A reliability allocation method, which synthetically considers the composition and Type III failure modes of the motion mechanism, is proposed to solve this problem. A relative dispersion factor is introduced to describe the failure dependence of components and is proposed to calculate the complexity and criticality. A series system reliability allocation model considering three types of failure modes is established. Finally, using an airplane gear door lock mechanism as an example, a comparative analysis of the system reliability allocation results with and without considering Type III failure modes is made. The allocation results show the component reliability value without considering Type III failure modes is less than that when considering them, which will increase the system hazards. The result considering Type III failure modes is more reasonable than that from the traditional method.     

An enriched meshless method for non‐linear fracture mechanics

This paper presents an enriched meshless method for fracture analysis of cracks in homogeneous, isotropic, non‐linear‐elastic, two‐dimensional solids, subject to mode‐I loading conditions. The method involves an element‐free Galerkin formulation and two new enriched basis functions (Types I and II) to capture the Hutchinson–Rice–Rosengren singularity field in non‐linear fracture mechanics. The Type I enriched basis function can be viewed as a generalized enriched basis function, which degenerates to the linear‐elastic basis function when the material hardening exponent is unity. The Type II enriched basis function entails further improvements of the Type I basis function by adding trigonometric functions. Four numerical examples are presented to illustrate the proposed method. The boundary layer analysis indicates that the crack‐tip field predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered, whether for the near‐tip asymptotic field or for the far‐tip elastic field. Numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the J‐integral for the applied load intensities and material properties considered. Also, the crack‐mouth opening displacement evaluated by the proposed meshless method is in good agreement with finite element results. Furthermore, the meshless results show excellent agreement with the experimental measurements, indicating that the new basis functions are also capable of capturing elastic–plastic deformations at a stress concentration effectively. Copyright © 2003 John Wiley & Sons, Ltd.     

Comparison of Weibull small samples using Monte Carlo simulations

The evaluation of the functional reliability of different designs is a common task and times to failure can be compared using the likelihood ratio test. In the microelectronics industry, as in many others, the high cost of testing places severe restrictions on the sample size. Moreover, the products in these tests are often new and do not have previous reliability histories. These factors make the selection of the Type I and Type II errors in comparison tests very difficult. This paper presents the Monte Carlo simulation results of Type II errors for the likelihood ratio test of comparison as a function of the Type I error and the (small) sample size. Our conclusions are summarized as follows: (1) the common microelectronics industry standard sample size of 32 is often insufficient to reach satisfactory conclusions; (2) small sample tests should only be used for prescreening for significant differences; and (3) when only small samples are available, the Type I and the Type II errors must be selected carefully to prevent misleading conclusions. Copyright © 2006 John Wiley & Sons, Ltd.     

A review of: “Melal Cutting Principles.” By MILTON C. SHAW. (Clarendon Press, 1984.) [Pp. 594.] Price £60.

This paper develops formulas that can be used in the design of multiple criteria sampling plans or charts for fraction nonconforming, with sampling on variables or attributes. Products often have multiple requirements and the usual acceptance tests or charts do not take this into account, and hence the overall quality of the product may be in poor control. We design tests or charts on the basis of probability of Type I and Type II errors (α and β) that refer to acceptable and rejectable levels of the overall fraction of the product that is nonconforming. Further, recognizing that the average proportion of the product that is actually nonconforming on each of the characteristics may vary independently of the other characteristics, our formulas give protection on a ‘worst case’ basis.     

The role of first non‐singular stress terms in mixed mode brittle fracture of V‐notched components: an experimental study

This paper investigates the effects of the first non‐singular stress terms on the fracture assessment of sharp V‐notches under mixed mode loading. First, numerical studies have been performed on a fracture test configuration called single V‐notched ring (SVR) specimen. Then, the notch stress intensity factors as well as the coefficients of the first non‐singular stress terms, which are vital parameters in brittle fracture of V‐notched components, were calculated via a finite element over‐deterministic algorithm for a wide range of loading and geometry conditions. The obtained results demonstrate that the SVR specimen is able to provide a complete range of mode mixities from pure mode I to pure mode II loading conditions. The numerical results, next, have been converted to dimensionless parameters and are illustrated in several graphs. Indeed, these graphs can be easily employed by the engineers for rapid calculation of the corresponding notch stress intensity factors and the coefficients of the first non‐singular stress terms in the SVR specimen. The obtained fracture parameters are then submitted to the maximum tangential stress criterion to assess the effects of the first non‐singular terms on fracture behaviour of the specimen. Finally, an experimental study has been performed on the SVR specimen made of Nayriz Marble rock for two notch angles with a complete range of mode mixities. The obtained experimental data confirm the significant role of the first non‐singular stress terms. In fact, these results show that considering only the singular stress terms may induce an average error of 38% in the predicted fracture loads, which can be decreased to about 12% just by adding the contribution of the first non‐singular terms to the maximum tangential stress criterion.     

Inertia-sensitive impact energy-absorbing structures part II: Effect of strain rate

By employing the elastic-plastic structural model introduced in part I [1], which contains four compressible bars and four elastic-plastic “hinges” of finite length, the entire dynamic deformation history of Type II structures is traced. In contrast to part I, strain-rate effects are incorporated into the analysis throughout the entire response of the structure. The Cowper-Symonds relation is adopted and the yield stress varies with the current strain-rate during the dynamic response of the model. The numerical examples presented show that the strain-rate effect plays an equally important role to that of inertia on the dynamic behaviour of this kind of energy-absorbing structure if the material of the structure is rate-sensitive, e.g. made of mild steel. Compared with the corresponding quantities in the quasi-static case, the combined effects of strain-rate and inertia make the peak load much higher and the final displacement much smaller. It is also found that because the increase of the yield stress due to strain-rate sensitivity expands the range of elastic deformation, the elastic strain energy stored in the structure made of rate-dependent material is notably larger than that in the structure made of rate-independent material. This implies that when strain-rate effects are taken into account in the analysis, elasticity must play a more significant role and should not be neglected.     

Consequences of using estimated response values from negligible interactions in factorial designs

This article analyzes the increase in the probability of committing type I and type II errors in assessing the significance of the effects when some properly selected runs have not been carried out, and their responses have been estimated from the interactions considered null from scratch. This is done by simulating the responses from known models that represent a wide variety of practical situations that the experimenter will encounter; the responses considered to be missing are then estimated, and the significance of the effects is assessed. Through comparison with the parameters of the model, the errors are then identified. To assess the significance of the effects when there are missing values, the Box‐Meyer method has been used. The conclusions are that one missing value in eight run designs, and up to three missing values in 16 run designs experiments can be estimated without hardly any notable increase in the probability of error when assessing the significance of the effects.     

Calculation of strict error bounds for finite element approximations of non‐linear pointwise quantities of interest

This paper deals with the verification of simulations performed using the finite element method. More specifically, it addresses the calculation of strict bounds on the discretization errors affecting pointwise outputs of interest which may be non‐linear with respect to the displacement field. The method is based on classical tools, such as the constitutive relation error and extraction techniques associated with the solution of an adjoint problem. However, it uses two specific and innovative techniques: the enrichment of the adjoint solution using a partition of unity method, which enables one to consider truly pointwise quantities of interest, and the decomposition of the non‐linear quantities of interest by means of projection properties in order to take into account higher‐order terms in establishing the bounds. Thus, no linearization is performed and the property that the local error bounds are guaranteed is preserved. The effectiveness of the approach and the quality of the bounds are illustrated with two‐dimensional applications in the context of elastic fatigue problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Evaluating pavement management treatment selection utilising continuous deflection measurements in flexible pavements

The objective of this study was to develop a new structural index based on rolling wheel deflectometer (RWD) deflection data to describe pavement structural capacity and to improve pavement management treatment selection. To achieve this objective, pavement conditions including surface cracking, rutting, roughness and asphalt layer thicknesses were categorised and sorted according to their AC layer thickness and divided into various subgroups. The cumulative distribution function of the new RWD index in each subgroup was generated so that various percentiles were calculated and used to define the boundary between structural and functional rehabilitation. Results showed that the Louisiana criteria may recommend structural rehabilitation for pavements with sound pavement structure (Type I error) and functional rehabilitation to pavements with weak pavement structure (Type II error). Therefore, the state pavement management system should consider both pavement structural indices and surface distress indices currently available when making recommendations for pavement preservation and rehabilitation. RWD testing technology and indices derived from its data are one of the most promising candidates to fulfil this need.     

Determining sample size for specification limits verification with tolerance intervals

In the traditional industrial verification process, when the aim is the compliance to assigned specifications, it is difficult to find an affordable statistical method for the purpose. Most data tables in industrial procedures and standards deal with tolerance limits neglecting the potential needs to verify assigned specification limits. A two-sided tolerance interval, combined with a bivariate statistical hypothesis test can be used to address this problem. The proposed risk-based approach leads to the determination of the minimum sample size with preestablished probabilities of Type I and Type II errors, that are essential elements for estimating the safety and reliability risk. A novel method is proposed for determination of the tolerance interval testing factors. This approach calculates the testing factors based on the deviation of the mean and the variance from the null hypothesis when a specified value of Type II error is found. The deviations of the mean and variance are determined in such a way that an assigned proportion of the population falls within the specification limits. Additional studies are provided to assess the robustness of the method for nonnormal environments and to compare it with other methods.     

Identification and Avoidance of Systematic Measurement Errors in Lamb Wave Observation With One‐Dimensional Scanning Laser Vibrometry

Abstract: Scanning laser vibrometry is a widely used tool to observe Lamb wave fields for structural health monitoring (SHM) purposes. Lamb waves propagate over long distances in thin‐walled structures and interact with structural inhomogeneities, for example, damages, in spite of wavelengths several times of the damage size. In SHM of sheets and glass‐ or carbon‐fibre‐reinforced plastic plates, this effect is used for determining the position as well as the size of structural faults. With the often employed one‐dimensional vibrometry, a geometrically induced, systematic error occurs when measuring oblique‐angled motion. This error can be, in the specific case of Lamb waves, of a non‐negligible quantity. The nature of this geometrical measurement error in general and concerning Lamb waves in special is discussed analytically for both amplitude and phase data. It is shown that this matter should be taken into account in some applications.     

Static assessment of plain/notched polylactide (PLA) 3D‐printed with different infill levels: Equivalent homogenised material concept and Theory of Critical Distances

A novel approach based on the equivalent homogenised material concept and the theory of critical distances is formulated to perform static assessment of plain/notched objects of polylactide (PLA) when this polymer is additively manufactured with different infill levels. The key idea is that the internal net structure resulting from the 3D‐printing process can be modelled by keeping treating the material as linear elastic, continuum, homogenous, and isotropic, with the effect of the internal voids being taken into account in terms of change in mechanical/strength properties. This idea is initially used to assess the detrimental effect of the manufacturing voids on the static strength of the plain (ie, unnotched) material. This is done by addressing this problem in a Kitagawa‐Takahashi setting via the Theory of Critical Distances. Subsequently, this approach is extended to the static assessment of notched components of 3D‐printed PLA; ie, it is used to take into account simultaneously the effect of both manufacturing voids and macroscopic geometrical features. The accuracy and reliability of this design methodology were checked against a large number of experimental data generated by testing, under axial loading, plain specimens, as well as notched samples (including open notches) of PLA. These specimens were manufactured by making the infill level vary in the rage 10% to 90%. This validation exercise allowed us to demonstrate that the proposed approach is highly accurate, returning estimates falling within an error interval of ±20%. This remarkable level of accuracy strongly supports the idea that static assessment of 3D‐printed materials with complex geometries and manufactured with different infill levels can be performed by simply post‐processing conventional linear elastic finite element (FE) solid models, ie, without the need for modelling explicitly the detrimental effect of the manufacturing voids.     

Structural reliability assessment of cracked pipes: The role of probability of detection data

The structural reliability of industrial pipes, including those in the nuclear, oil, and gas industries, has a significant impact on the safety of people and the environment. This work aims to develop a computational structural reliability model method in conjunction with the failure assessment diagram method and the user‐defined probability of detection curves of non‐destructive testing are used. The concept of “reliability factor of a repair” is proposed. Then, the effects of the pipe inspection considering different user‐defined probability of detection curves and different values of the reliability factor of a repair on probability of failure are discussed. The main results include the identification of cases where performing repairs do not guarantee an improved reliability, as well as the consequences of considering the repair as a “perfect process” which result in non‐conservative assessments.     

Short crack approach for multiaxial fatigue assessment

A significant part of the fatigue life is spent during short crack growth. Therefore, modelling of short fatigue crack growth offers an opportunity to improve the accuracy of numerical life assessment. Besides stating some general remarks on the short crack approach itself and on multiaxial fatigue criteria, a short crack growth based fatigue life prediction approach for multiaxial non‐proportional loading is presented. This approach accounts for the geometrical size effect by considering the geometry correction functions for semi‐elliptical surface cracks in inhomogeneous gradient stress fields. The geometrical size effect is becoming significant for notch radii smaller than four times the defined technical crack size. Additionally, life influencing factors due to the statistical size effect have been taken into account. The comparison of calculated and experimentally observed fatigue lives of shouldered shafts made of S460N with notch radii of 0.2 to 4.0 mm under non‐proportional tension and torsion loading yields a satisfying accuracy.