## Finite element formulation for large displacement analysis

##### Abstract

160 h.

##### Abstract

Abstract: In structural engineering, shells are usually curved objects which can support remarkable external forces without fracture or damage. Due to this remarkable stiffness behaviour and the low amount of material used in the fabrication process, nowadays shell structures are getting lots of followers in the engineering world. In consequence, scientists and the industry in general are keen with the idea of developing efficient and accurate models to predict their behaviour. In the manufacturing industry, inspection procedures are becoming more common every day. This task constists on measuring how much difference exists between the original-ideal object specifed by the designer, and the object that just comes out from the manufacturing process. Generally, this goal is achived by placing the manufactured part in the location where it was designed to be, such as the assamblage of a machine. Subsequently, a 3-D scan of the placed part is made. The registered measurments are compared with the CAD geometry of the original-ideal object in order to acept or reject the manufactured part. However, this procedure represents an inefficient and very time consumig process. If instead, the part is scanned when it just comes out from the fabrication process, and the scanned geometry is placed into the assemblage via a Finite Element software; a more efficient procedure could be achieved assuming that the simulation is fast and reliable. The goal of this work is to provide this virtual inspection enviroment. Given the nature of the manufactured parts, the physics of the problem corresponds to shell structures subjected to large displacements. In consequence, the formulation, implementation, and validation of a shell element based on a large displacement hypothesis have to be fully understood. Nevertheless, before arriving to shell elements, it has been considered as convenient to comprehend the Finite Element formulation proposed in the literature for bars, beams, and plane stress elements. Although all the topics analysed in this report have been somehow studied by several authors and published in FEM journals and books, the information recopilated into this text does not represent a mere attempt to make a literature review of the Finite Element Method for large displacements. Instead, every single piece of information included in this text has been considered as very valuable because either it introduces a keen and important FEM topic, or because the mathematical treatment accomplished to be able to get to the _nal results suggested by FEM literature, has not been fully explained or published. In other words, most of the times FEM authors assume that the reader is able to reach the same results they publish without giving many clues about the right way to get to that point. This text gets into the detail of the mathematics required to deal with finite deformations and large displacements applied to the Finite Element Method; for that purpose, it was necessary to understand several author's point of view and sometimes comprehend very different mathematical notations to explain the same idea. The final products obtained (both the written report and the C++ shell program) represent a big shortcut for those who want to gain good knowledge and understanding of the Finite Element Method under the large displacement hypothesis. The chapter 1 of this text contains a brief review of the Total Potential Energy principle explained in terms of a variational formulation. This is done because this principle represents the base of the Finite Element method. In chapter 2 the principle of Virtual Displacements is used to deduce the Finite Element method, arriving to the general 3-D Finite Element equations to be used in a small displacement scenario. Next, in chapter 3 a large displacements hypothesis is used, and the Finite Element method is amended for this situation; furthermore, some new stress and strain measures needed for this formulation are introduced. Bars, beams and plane-stress elements are studied in chapters 4, 5, and 6 in order to introduce some concepts required for the Shell formulation. The shell formulation explained in chapter 7 was implemented as a C++ program and the displacements results obtained are validated in chapter 8. At the end of the text there are included some appendixes with some material which was considered as important for a better understanding of the Finite Element method. The user's manual of the shell software programmed is included in the Appendix D.