This thesis considers the theoretical determination of the elasto-plastic response of structures to earthquake loading. The particular structures studied are a railway bridge and four multi-storey buildings: one with a tall flexible steel frame, two others with open reinforced concrete frames and the other a reinforced concrete spandrel beam frame. Calculations were made to determine the magnitudes of the plastic deformations which occur in these New Zealand structures under a strong motion earthquake, the N-S component of that recorded at El Centro in 1940. The development of computer programs to calculate the elastic and elasto-plastic response of multi-mass structures to recorded earthquakes and the numerical integration methods on which they are based are outlined. Efficient static analysis programs which may be used to calculate the individual member actions in a multi- storey frame, caused by the earthquake induced lateral forces are also developed. It was found that the lateral deflections assuming elasto-plastic behaviour were of the same order as those calculated assuming yielding did not occur. The plastic action tended to reduce the top storey deflection and increase the lower storey deflections. The member ductilities required by these frames were of reasonable magnitude and could probably be provided with adequate detailing, but it is clear that typical structures must be capable of providing member ductilities greater than four. This research indicated that it is possible to design multi-storey frames so that the columns remain elastic while the beams yield and absorb energy plastically without forming a collapse mechanism.