Metodología para la determinación del factor de seguridad probabilístico basado en la validación del modelo teórico

Los modelos ingenieriles están fundamentados en simplificaciones e idealizaciones de la situación real, que afectan a su exactitud. Son modelos cuyo grado de veracidad depende de cuan semejantes sean las condiciones de contorno ideales de las reales. Por lo tanto, antes de ser utilizados en el cálculo de la tensión de diseño para determinar el factor de seguridad del mismo, debe comprobarse que las estimaciones realizadas con el modelo empleado se corresponden con la situación analizada. La validación experimental permite cuantificar el grado de distorsión del modelo simplificado con respecto a la realidad. Para realizar esta cuantificación, ha de considerarse que las magnitudes del modelo teórico y del ensayo experimental, no son únicas, sino que son la mejor estimación de todos los posibles valores atribuibles a ambas; es decir, implica la identificación y cuantificación de todas las potenciales fuentes de incertidumbre. Si las incertidumbres no son tomadas en cuenta, el criterio de aceptación del modelo predictivo, depende de la experiencia de quien diseña, siendo esto una valoración subjetiva que puede resultar errónea. La valoración subjetiva del ingeniero basada en la experiencia es necesaria, pero no puede ser la única herramienta empleada en la comparación de resultados. Es por ello que, en la actualidad, numerosos investigadores, así como comités internacionales, han estado trabajando en el desarrollo de los conceptos fundamentales y la terminología necesaria para la validación de modelos. Sin embargo, aún no existe un enfoque unificado de cuáles son los parámetros estadísticos que se deben usar, que en todo caso, deberán considerar que las predicciones y las observaciones experimentales, deben expresarse en términos probabilísticos. Actualmente, el denominado factor de seguridad de uso común a nivel ingenieril, también se ha sumado a la filosofía empleada en la validación experimental, ya que considera que los parámetros involucrados en su estimación, poseen cierto grado de incertidumbre. A este nuevo enfoque de diseño seguro, se le conoce como factor de seguridad estadístico o probabilístico. En la presente tesis doctoral, se propone una metodología que pretende innovar el concepto actual del factor de seguridad, con miras a mejorar el nivel de confianza de la carga o la tensión de diseño que asegura la obtención del factor de seguridad mínimo, de acuerdo a la probabilidad de fallo seleccionada. La metodología planteada es un procedimiento novedoso que vincula el resultado de la validación experimental del modelo predictivo, con la expresión del factor de seguridad probabilístico. Para ello, se ha introducido un estadístico de validación, de uso común en la comparación entre laboratorios, que permite validar el modelo teórico y que, además, ayuda a definir si el modelo sobre-estima o sub-estima el caso real. Para el cálculo del estadístico de validación, primero se ha elaborado un modelo de incertidumbre, que toma en cuenta tanto la aportación de las variables del modelo, como la debida al incumplimiento de las condiciones de contorno ideales. Esta metodología se aplica a un caso real, en el que 5 secciones de tubo circular, con diferentes diámetros son comprimidos por cargas axiales diametralmente opuestas. Se aplican los conceptos metrológicos expuestos en la guía para la expresión de la incertidumbre de medida, internacionalmente aceptados y se utilizan simulaciones tipo Monte Carlo, tanto para el cálculo de incertidumbre, como para la generación de información adicional. Se comprueba que el modelo cumple con los requisitos de aceptación planteados en este trabajo de investigación. Al mismo tiempo, se verifica la tendencia de dicho modelo y se le relaciona con el factor de seguridad mínimo, de manera que la carga de diseño calculada garantiza que se cumplen con las condiciones de seguridad impuestas, de acuerdo con la probabilidad de fallo asignada. Finalmente, se demuestra que el resultado de la validación afecta al cálculo del factor de seguridad cuando las predicciones tienden a sub-estimar la situación real, ya que aumenta la probabilidad de fallo. Además, se verifica cómo la introducción del estadístico de validación en la expresión del factor de seguridad probabilístico, obliga a que las predicciones en conjunto con la incertidumbre global, nunca superen el valor de la tensión máxima admisible.The engineering models are based on simplifications and idealizations of the real situation that have an effect on their accuracy. The degree of veracity of these models depends on how similar are the ideal boundary conditions to the real ones. Therefore, before the model being used in the calculation of the design stress to determine its safety factor, it must be verified that its predictions correspond with the analyzed situation. The experimental validation allows quantifying the degree of distortion of the simplified model with respect to the reality. For this purpose, it must be considered that the magnitudes of the theoretical model and the experimental test are not unique, but that they are the best estimates of all the possible values; i.e. it implies the identification and quantification of all the potential sources of uncertainty. If the uncertainties are not taken into account, the acceptance of the predictive model depends exclusively on the designer´s experience being thus a subjective validation that could be erroneous. The subjective assessment of the engineer, based on experience is necessary, but may not be the only tool used in the validation of results. This is the reason why many researchers and international committees have been working on the development of the fundamental concepts and the necessary terminology for model validation and verification. Nevertheless, there is still no unified approach of the statistical parameters to be used, but in any case, they have to consider predictions and experimental observations in probabilistic terms. Nowadays, the so-called safety factor commonly used at an engineering level, has also joined the philosophy of considering that the parameters involved in its calculus, have some degree of uncertainty. In this new approach to safe design, it is known as statistical or probabilistic safety factor. In the present thesis, a methodology that innovates the current concept of safety factor is proposed, as it optimizes the accuracy of the load or the design stress that ensure the minimum magnitude of the safety factor, according to the probability of failure selected. The proposed methodology is a novel procedure that links the result of the experimental validation of the predictive model, with the expression of the probabilistic safety factor. To do so, a statistical magnitude commonly used in interlaboratory comparisons is used, that allows to validate the theoretical model and verify if the model overestimates or underestimates the real case. For the calculus of the statistical magnitude of validation, an uncertainty model has been generated that takes into account the uncertainty of the magnitudes in the theoretical model as well as its differences with the real case under study. This methodology is applied to a real case, in which 5 sections of circular tube, with different diameters are compressed by diametrically opposed axial loads. The internationally accepted concepts of the guide for the expression of the uncertainty of measurement as well as Monte Carlo simulations are used, both for the calculus of uncertainty and the generation of additional information. The theoretical two-dimensional model is verified together with its underestimating tendency and the minimum safety factor is determined with the imposed safety conditions, according to the probability of failure assigned. Finally, it is shown that the validation result influences the calculation of the safety factor, when predictions tend to underestimates the real situation, since it increases the probability of failure. Furthermore, it is verified that the introduction of statistical validation into the expression of the probabilistic safety factor, enables that the predictions together with the global uncertainty, never exceed the value of the maximum allowable stress.Programa de Doctorado en Ingeniería Mecánica y de Organización IndustrialPresidente: José Luis San Román García.- Secretario: Francisco Javier González Fernández.- Vocal: Emilio Velasco Sánche

A novel analytical solution for the brazilian test with loading arcs

This research study presents a new theoretical model to calculate the indirect tensile strength for the Brazilian disk with loading arcs, based on numerical simulations, two-dimensional elasticity theory, and Griffith failure criterion. The new expression incorporates a no uniform contact pressure distribution determined by the results of the simulations with the finite element method. A computational experiment design has been developed to test the accuracy of the predictions made with the proposed model. This study demonstrates that the stresses predicted with the new model are closer to those determined by the finite element models than other theoretical solutions available in the literature. Additionally, a comparative analysis with experimental results obtained by other authors also indicates that the new model provides a more accurate magnitude of the indirect tensile strength.The authors of this paper acknowledge the financial support of the PhD fellowship of the Severo Ochoa Program of the Government of the Principality of Asturias (PA-14-PF-BP14-067)

Experimental verification of the boundary conditions in the success of the Brazilian test with loading arcs. An uncertainty approach using concrete disks

The present work analyses the reliability of the Brazilian test with loading arcs. A new testing set up has allowed to determine in an effective way the real load of the failure initiation as this moment was not always or correctly detected by the universal testing machine. The instrumentation used is a simple and low-cost method that allows to know the possible pressure distribution in the contact zone as well as the final contact angle. It has been observed that the success of the test depends mainly on the surface finish of the parts involved, their geometric tolerances and the symmetry of the applied load. These boundary conditions have a direct effect in the contact pressure distribution. The possible failure modes observed experimentally have been simulated with the finite element methods. For this, the contact boundary condition has been changed and the possible stress distribution in term of Griffith equivalent stress has been obtained. The numerical analysis allows to study the influence of the initial contact condition on the success of the test and agrees with the experimental results. Furthermore, an uncertainty analysis in the expression of the tensile strength confirms that, when the test is valid, a crack appears suddenly in the central area of the disk, as observed experimentally, so there is no need to determine if the starting point is in the centre. Additionally, it has been observed that the initial crack length depends on the type of pressure distribution in the contact zone. Finally, a series of recommendations are given in order to minimize both the variability of the final contact angle and the risk of premature failure of the Brazilian disk

An uncertainty model of approximating the analytical solution to the real case in the field of stress prediction

Deterministic mechanics has been extensively used by engineers as they needed models that could predict the behavior of designed structures and components. However, modern engineering is now shifting to a new approach where the uncertainty analysis of the model inputs enables to obtain more accurate results. This paper presents an application of this new approach in the field of the stress analysis. In this case, a two-dimensional stress elasticity model is compared with the experimental stress results of five different size tubes measured with resistive strain gages. Theoretical and experimental uncertainties have been calculated by means of the Monte Carlo method and a weighted least square algorithm, respectively. The paper proposes that the analytical engineering models have to integrate an uncertainty component considering the uncertainties of the input data and phenomena observed during the test, that are difficult to adapt in the analytical model. The prediction will be thus improved, the theoretical result being much closer to the real case

Practical Case Application for Stress Model Validation and Enhancement by Means of Metrological Tools

Models consider ideal and simplified situations that will never be met in the real case. The process of comparing model predictions and experimental observation is in the basis of scientific research. This comparison is however complicated because of the uncertainties of the model input data and the difficulty to control the accuracy of the tests and to obtain a significant statistical sampling. Moreover, there isn't yet a consensus on a validation parameter. This paper presents a three-step validation procedure that allows quantifying the application limits of a two-dimensional stress model in a three-dimensional situation. A global uncertainty model is calculated comprising the uncertainty of the model and also the uncertainty coming from the experimental results. The EN number, a statistical magnitude for interlaboratory comparisons, is used to analyse the compatibility between the experimental and theoretical results. Finally, a bootstrapping method is proposed to calculate the coverage interval of the sampling and determine if new experiments should be carried out. Numerical results of this new validation procedure are provided for the case under study. It is also demonstrated that the computed uncertainty budget can be a useful tool to enhance the two-dimensional model by enlarging its uncertainty limits

Validation and improvement of a bicycle crank arm based in numerical simulation and uncertainty quantification

In this study, a finite element model of a bicycle crank arm are compared to experimental results. The structural integrity of the crank arm was analyzed in a universal dynamic test bench. The instrumentation used has allowed us to know the fatigue behavior of the component tested. For this, the prototype was instrumented with three rectangular strain gauge rosettes bonded in areas where failure was expected. With the measurements made by strain gauges and the forces registers from the load cell used, it has been possible to determine the state of the stresses for different loads and boundary conditions, which has subsequently been compared with a finite element model. The simulations show a good agreement with the experimental results, when the potential sources of uncertainties are considered in the validation process. This analysis allowed us to improve the original design, reducing its weight by 15%. The study allows us to identify the manufacturing process that requires the best metrological control to avoid premature crank failure. Finally, the numerical fatigue analysis carried out allows us to conclude that the new crank arm can satisfy the structural performance demanded by the international bicycle standard. Additionally, it can be suggested to the standard to include the verification that no permanent deformations have occurred in the crank arm during the fatigue test. It has been observed that, in some cases this bicycle component fulfils the minimum safety requirements, but presents areas with plastic strains, which if not taken into account can increase the risk of injury for the cyclist due to unexpected failure of the component

Novel Bayesian Inference-Based Approach for the Uncertainty Characterization of Zhang's Camera Calibration Method

Camera calibration is necessary for many machine vision applications. The calibration methods are based on linear or non-linear optimization techniques that aim to find the best estimate of the camera parameters. One of the most commonly used methods in computer vision for the calibration of intrinsic camera parameters and lens distortion (interior orientation) is Zhang¿s method. Additionally, the uncertainty of the camera parameters is normally estimated by assuming that their variability can be explained by the images of the different poses of a checkerboard. However, the degree of reliability for both the best parameter values and their associated uncertainties has not yet been verified. Inaccurate estimates of intrinsic and extrinsic parameters during camera calibration may introduce additional biases in post-processing. This is why we propose a novel Bayesian inference-based approach that has allowed us to evaluate the degree of certainty of Zhang¿s camera calibration procedure. For this purpose, the a prioriprobability was assumed to be the one estimated by Zhang, and the intrinsic parameters were recalibrated by Bayesian inversion. The uncertainty of the intrinsic parameters was found to differ from the ones estimated with Zhang¿s method. However, the major source of inaccuracy is caused by the procedure for calculating the extrinsic parameters. The procedure used in the novel Bayesian inference-based approach significantly improves the reliability of the predictions of the image points, as it optimizes the extrinsic parameters.This work was supported by the Madrid Government (Comunidad de Madrid Spain) under the Multiannual Agreement with UC3M ("Fostering Young Doctors Research", APBI-CM-UC3M), and in the context of the VPRICIT (Research and Technological Innovation Regional Programme and by the FEDER/Ministry of Science and Innovation -Agencia Estatal de Investigacion (AEI) of the Government of Spain through the projects PID2022-136468OB-I00 and PID2022-142015OB-I00.Publicad

New Procedure for the Kinematic and Power Analysis of Cyclists in Indoor Training

This article belongs to the Special Issue Sensor Technology for Sports Science.In this research, the performance and movements of amateur and professional cyclists were analyzed. For this, reflective markers have been used on different parts of the body of the participants in conjunction with sports cameras and a mobile power meter. The trajectories of the markers have been obtained with the software Kinovea and subsequently analyzed using error ellipses. It is demonstrated that the error ellipses help determine movement patterns in the knees, back, and hip. The covariance of the error ellipses can be indicative of the alignment and symmetry of the frontal movement of the knees. In addition, it allows verifying the alignment of the spine and the symmetry of the hip. Finally, it is shown that it is necessary to consider the uncertainty of the power devices since it considerably affects the evaluation of the cyclists’ performance. Devices with high uncertainty will demand a greater effort from the cyclist to meet the power required in the endurance test developed. The statistical magnitudes considered help to analyze power and evaluate the cyclists’ performance

Influence of anodized depth on fatigue life for bicycle cranks

Bicycle manufacturers are using new materials in order to improve the bicycle performance. In high-level cycling, magnesium, aluminum, titanium and carbon fiber, have replaced steel with the purpose of improving the weight/rigidity ratio. Due to cost factors and ease of machining, aluminum is widely used in bicycle cranks manufacturing. In order to improve the external aspect of the final product and protect the external surface of the component, some surface treatments must be applied. In the case of aluminum, anodizing is the most extended treatment, due to several factors as low cost, visual aspect, variety of colors and finishing. However, this treatment may reduce the fatigue resistance of the component. In this work, the best compromise between anodizing depth and fatigue resistance performance of a bicycle crank has been analysed in order to provide an optimum solution to improve the performance of the component

New methodology for estimating the shear strength of layering in slate by using the Brazilian test

A new method is proposed in order to estimate the shear strength of schistosity planes in slate in terms of Mohr-Coulomb cohesion and internal friction angle. The procedure consists in carrying out the Brazilian method under different loading-foliation angles, for which experimental tests were achieved in slates from the northwest of the Iberian Peninsula (Spain). The experimental fracture patterns were analytically studied and justified by simulating the stress field in the discontinuity planes contained in the whole sample, taking into account the first failure registered in the tests. By combining experimental and analytical studies and a procedure based on the representation of the threshold state of stressesin the elastic regimein the failure plane, it is possible to estimate the foliation's strength envelope through a lineal adjustment according to the Mohr-Coulomb criterion and, thus, to characterize the layering. Finally, the proposed procedure was validated by the direct shear test. The cohesion and the internal friction angle obtained with this convenctional test were very close to that calculated by the proposed method, verifying the methodology developed by the authors. This procedure may be interesting in various engineering applications, either in the study of the properties of cleavage in slate, which is commonly used as an industrial rock, or in dam foundations, underground excavations and slope engineering, since one of the main failures in civil engineering is due to sliding along weak planes.The authors of this paper would like to acknowledge the financial support of the PhD fellowship Severo Ochoa Program of the Government of the Principality of Asturias (PA-14-PF-BP14-067). Also, the authors are grateful to editors and reviewers for their suggestions and help us to improve this manuscript