Consistent application of horizontal and vertical earthquake components in analysis of a block sliding down an inclined plane

The seismic displacement of a block sliding down an inclined plane is commonly obtained by a double integration of portions of the acceleration time history exceeding the yield acceleration. Although this simple procedure was originally developed by Newmark for a tangential excitation, it is widely used also for horizontal shaking. Strictly speaking, the sliding displacement in this case needs to be modified by a correction factor, a fact that is not very well known and thus often overlooked. The vertical component is usually either ignored or considered by resorting to arbitrary simplifying assumptions. This paper presents an alternative double integration procedure for the calculation of the sliding displacement under the combined action of the horizontal and vertical components. After transforming the excitation to the tangential and normal directions, the procedure is applied to the relative tangential acceleration. Unlike the traditional Newmark approach, the application of the alternative procedure does not require a prior explicit calculation of the yield acceleration, which would not even be a constant when the vertical component is present. The results of a series of case studies performed employing this procedure indicate that the sliding displacement does not necessarily increase when the vertical component is also considered.