The aim of the work presented in this thesis is to examine numerical solution schemes for both linear and non-linear dynamic behaviour, to evaluate their accuracy, stability and computational resources. Matrix formulation of a structural system in motion is described and its parameters presented in matrix form, both at the element and at the global level. Four direct integration methods are used for the solution of the resulting equations of motion. The four methods are presented together with the dynamic relaxation method and the modal synthesis method which are used as a means of comparison. Computer programs are written for all methods for the elastic dynamic response of a beam in bending. The results are compared with theory and the methods are compared with each other with respect to accuracy, sensitivity to the chosen time-step, computing time and effect of element size. The most efficient and stable of the implicit direct integration methods is modified to accommodate non-linear material behaviour. The stability accuracy and efficiency of the method is compared to that of the explicit Central Difference direct integration method. Recommendations for further work are made. Some information on the Graphical routines used for the presentation of the results, and flow charts of the programs used are given in Appendices I and II respectively.