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

Abstract

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.