Finite element analysis of vibrating elastic structures

Abstract

In this thesis mathematical models for vibrating elastic structures are derived and analysed using finite element approximations where necessary.

The most important contribution in this thesis is the development of the Local Linear Timoshenko (LLT) model and its applications. Using the well-known equations of motion for a one-dimensional solid or rod, these equations are rigorously simplified for planar motion. To complete the model, the constitutive equations for shear and bending is adapted from the linear Timoshenko theory.

A significant property of the model is that existing linear and nonlinear models can be derived from it. This promotes insight into the LLT model itself as well as existing models. In particular, by making the appropriate assumptions for small vibrations, a number of models published by other authors, were derived. Of importance is an adapted version of the linear Timoshenko model which allows for longitudinal vibration and a special case for transverse vibration of a Timoshenko beam with an axial force.

The variational equations of motion for the LLT model was easy to derive but the constitutive equations could not simply be substituted into them. Nevertheless, in the thesis a well defined variational form for the Local Linear Timoshenko model is derived. Using the variational form, finite element approximations of problems can be formulated. A rigorously defined algorithm was developed which is a substantial contribution. Through numerical experiments, convergence was demonstrated. While solutions of LLT and linear models compared well for small vibrations, it was shown that the LLT model can be applied to cases where the solutions of linear beam models are not realistic.

A model for earthquake induced oscillations in vertical structures, based on the Timoshenko model, was derived. The model was transformed to that for a cantilever beam with homogeneous boundary conditions. This made it possible to compare beam models using modal analysis. This adapted Timoshenko model was compared to the Twin-beam model of E Miranda. The models compared poorly and both predicted the measured fundamental period completely wrong. This is due to the lack of reliable information on additional mass not contributing to stiffness.

As an alternative, a building was modelled as a series of beams connected by rigid bodies to represent floors. Correct modelling of interface conditions made it possible to derive the variational form, which is a significant contribution. An adapted Mixed Finite Element approximation was thus possible and a system of ordinary differential equations was derived which can be used for simulations.

Finally, new interface and boundary conditions for a hybrid Timoshenko beam model with a tip body were derived. This model is an improvement on previous versions since elasticity at the interfaces is taken into account. The derivation of the estimates required to apply the general theory for existence needed to be done with care and the proofs were by no means trivial. The new model can also be used to evaluate cases where ``rigid'' boundary and interface conditions may not be realistic.

The numerical experiments in this thesis had limited scope. It was mainly used to complement the theory, for convergence experiments (e.g. LLT model) or to examine the feasibility of a model (e.g. vertical structure and Hybrid model).