Energy absorption in structural frames

Based on the theory of fractional calculus and the spectral theory of vibration, a new spectrally-formulated finite-element method of analysis is developed which is capable of making accurate predictions of the dynamic response of structures with added dampers. The frequency-dependent and temperature- dependent damping characteristics of structural materials can be modelled accurately using the fractional derivative model. It is shown that the proposed method can be extended to develop a non-linear damping element which can be used to model structural dampers. The approach has an advantage over the usual viscous treatment, which appears to lack a physical basis. The main features of the complex-spectral finite-element method of analysis have been presented in this thesis. This method is capable of making accurate predictions of the dynamic response of stractural systems. Most structural systems can be analysed and designed by using the conventional finite element method. However, in order to guarantee stability and accuracy of the solution, the number of elements used to model the structure may be very large. Hence, it appears that, for large structures, it may be more effective to use the spectral approach presented in this thesis. A set of the fractional derivative damping models, capable of representing different damping mechanisms, have been derived for solving the dissipation problem in damped systems. The modelling of elastomeric and viscoelastic components in damped structures often requires complex viscoelastic representations. While traditional differential operators are typically employed in such a formulation, fractional operators give rise to a richer variety of functional families, and hence lead to an improved integro-differential type curve fitting of constitutive representations. The fractional derivative viscoelastic damping model enables a single formulation of the complex dynamic stiffness matrix of the damped system to be developed. This leads to the successful formation of the frequency domain equations of motion for a structure containing both elastic and viscoelastic components. However, although the development of the fractional-spectral method focuses primarily on damped structural frames, the results can readily be extended to damped mechanical systems, hi addition, although the analysis of damped systems is the primary object of this study, the spectral finite element method presented in Chapter Four can be used in the case of undamped dynamical systems