Variational mechanics and numerical methods for structural analysis

Abstract

This work focuses on the particular application of the variational principles of Lagrange and Hamilton for structural analysis. Different numerical methods are compared in their computation of the elastic energy through time. According to variational mechanics, the difference between the stored elastic energy and the applied work should be null on each time step, so by computing this difference we can account for the level of accuracy of each combination of numerical methods. Moreover, in some situations when numerical instabilities are difficult to perceive due to high complexities, this procedure allows for the control and straightforward visualization of them, being an excellent source of hindsight on the behaviour of the analysed system. The purpose of this dissertation is to present a scheme where the current numerical methods can be benchmarked in a qualitative as well as in a quantitative manner. It is shown how different combinations of methods, even for a simple model, can give very different results, particularly in the field of dynamics, where often also instabilites arise. The first half of the thesis is a thorough explanation of these concepts and their application in terms of structural analysis. In the second part, a review on the numerical methods in general and of those implemented for our experiments is provided, followed by the experimental results and their interpretation. The model of choice, for simplicity and availability of analytical results is one cantilever column. Bending elastic energy of the column is monitored under transient regimes of different shapes, computing the total action of the system as its integral through time.