Elastic and plastic bending of the beams by finite difference method (FDM)

The Euler-Bernoulli beam model has a wide range of applications to the real life; such as nano electro mechanical system switches in small scale up to the Eifel tower in large scale. Advantages of FDM like simpler mathematical concept and easier programming have made scientist to choose this numerical method to solve many state-of-the-art physical problems of partial differential equations (PDE). There are researches done by using this method in solving many problems; while, how the nodes at the boundaries can be treated in the best way is still unclear. Therefore, this study is subjected to obtaining the beam behavior with the material in two ranges of elastic and ideal plasticity. Firstly, different schemes of FDM are applied to the PDE of the beam in the elastic range for six cases. Afterwards, loading increases that the material goes to the ideal plastic range. In both ranges, validity of the results by comparing with the analytical solutions need to be studied. Finally, the best FD scheme to implement the boundary conditions are determined. Effect of the point load on FDM is investigated. Moreover, optimum value of the vital parameters like number of nodes, layers and load increments are extracted. Advantages of FDM like simpler mathematical concept and easier programming have made scientist to choose this numerical method to solve many state-of-the-art physical problems of PDE. There are researches done by using this method in solving many problems; while, how the nodes at the boundaries can be treated in the best way is still unclear

Solución numérica de problemas de elasticidad bidimensional, basados en la formulación directa de navier o en funciones potenciales mediante el método de redes: el programa EPSNET_10

[SPA] La complejidad de resolución de los problemas elastostáticos, enunciados en forma general por la Ecuación de Navier, normalmente requiere el uso de técnicas numéricas tales como Elementos Finitos. El principal objetivo de las formulaciones alternativas en términos de funciones potenciales ha sido la obtención de soluciones analíticas. Sólo algunos casos, en los que son aplicables funciones sencillas tales como Airy y Prandtl, se han resuelto numéricamente en términos de potenciales. En esta tesis se presenta la aplicación del método de simulación por redes a la solución numérica de problemas de elasticidad 2D planteados con la formulación de Navier o con funciones potenciales; centrándose en los potenciales de Papkovich-Neuber y formulaciones derivadas, por eliminación de alguna de las funciones potenciales, para los que no se han encontrado soluciones numéricas hasta la fecha. Tras presentar los fundamentos teóricos de esta memoria (Capítulo 2) y discutir las condiciones de completitud y unicidad de la representación de Papkovich-Neuber, se profundiza en las condiciones adicionales para la unicidad numérica (Capítulo 3), cuestión aún sin resolver en toda su extensión. En este sentido, se proponen nuevas condiciones de unicidad numérica para algunas de las formulaciones en potenciales aplicables a casos bidimensionales. En particular, para los potenciales de Boussinesq, se aportan condiciones más sencillas y alternativas a las propuestas hasta la fecha, únicas según Tran-Cong. El diseño de modelos en red y la implementación de las condiciones físicas de contorno, tanto para la formulación de Navier como para la formulación en potenciales, se explica en el capítulo 4. Se ha elaborado un programa en Matlab con interfaz gráfica, EPSNET_10, para la generación de modelos en red, simulación en PSpice y representación gráfica de resultados. Su funcionamiento y las opciones de usuario que contiene se explican en el capítulo 5. En los capítulos 6 y 7 se presentan las aplicaciones a problemas enunciados bajo los dos tipos de formulaciones, Navier y en potenciales, respectivamente. Para verificar la fiabilidad de los modelos propuestos se comparan sus resultados con las soluciones analíticas, cuando existen, o con las de otros métodos numéricos de uso común en elasticidad. [ENG] The complexity of solving elastostatic problems, defined by the Navier equation, usually requires numerical tools such as Finite Element. The main aim of the alternative formulations in terms of potential functions has been to get analytical solutions. Only certain cases, where straightforward functions as Airy and Prandtl can be applied, have been solved numerically in terms of potential. The network simulation method is applied in this PhD Thesis on the numerical solution of 2D elastostatic problems based either in Navier formulation or in potential formulation, focusing on the Papkovich-Neuber potentials and derived solutions, by deleting some of the potential functions, for which no numerical solutions have been investigated up to day. After exposing the theoretical bases of this memory (Chapter 2) and discussing the completeness and uniqueness conditions of the Papkovich-Neuber solution, the additional conditions required for the numerical solution are studied (Chapter 3). This question is still a matter of active interest in the research literature. In this sense, new additional conditions are proposed for some potential solutions applicable to 2D problems. In the case of the Boussinesq solution, the conditions proposed up to day, unique according to Tran-Cong, can be specified in alternative forms, even more simple. The design of the network models as well as the implementation of the physical boundary conditions, for both Navier and potential formulations, is explained in Chapter 4. Software has been developed in Matlab programming language, with graphical interface, EPSNET_10. This contains the routines for the network design, simulation in PSpice and data treatment for the graphical result representation. Its performance and multiple user options are explained in Chapter 5. Applications to problems defined by Navier and potential formulations are presented in Chapters 5 and 6, respectively. The reliability of the proposed models are verified by comparison between its results and analytical solutions, if they exist, or otherwise with standard numerical methods solutions, currently used in elasticity.[ENG] The complexity of solving elastostatic problems, defined by the Navier equation, usually requires numerical tools such as Finite Element. The main aim of the alternative formulations in terms of potential functions has been to get analytical solutions. Only certain cases, where straightforward functions as Airy and Prandtl can be applied, have been solved numerically in terms of potential. The network simulation method is applied in this PhD Thesis on the numerical solution of 2D elastostatic problems based either in Navier formulation or in potential formulation, focusing on the Papkovich-Neuber potentials and derived solutions, by deleting some of the potential functions, for which no numerical solutions have been investigated up to day. After exposing the theoretical bases of this memory (Chapter 2) and discussing the completeness and uniqueness conditions of the Papkovich-Neuber solution, the additional conditions required for the numerical solution are studied (Chapter 3). This question is still a matter of active interest in the research literature. In this sense, new additional conditions are proposed for some potential solutions applicable to 2D problems. In the case of the Boussinesq solution, the conditions proposed up to day, unique according to Tran-Cong, can be specified in alternative forms, even more simple. The design of the network models as well as the implementation of the physical boundary conditions, for both Navier and potential formulations, is explained in Chapter 4. Software has been developed in Matlab programming language, with graphical interface, EPSNET_10. This contains the routines for the network design, simulation in PSpice and data treatment for the graphical result representation. Its performance and multiple user options are explained in Chapter 5. Applications to problems defined by Navier and potential formulations are presented in Chapters 5 and 6, respectively. The reliability of the proposed models are verified by comparison between its results and analytical solutions, if they exist, or otherwise with standard numerical methods solutions, currently used in elasticity.Universidad Politécnica de CartagenaPrograma de doctorado en Tecnologías Industriale