Newman's sliding block model is widely used in evaluating earthquake-induced permanent displacements for rock slopes. However, as are generally encountered in other earthquake-related problems, the calculated permanent displacements may be of little value if the uncertainty about the possible future motions is not accounted for properly. Previous attempts (e.g. Sarma, 1975) did not include sufficient ground motion characteristics in assessing the possible earthquake-induced permanent displacements. Therefore, their applications are limited only to the cases with similar data base. In this paper, an alternative method is described. The method utilizes an equivalent stationary model of ground motions, as well as the results from random vibration formulations. It gives not only an improved estimate of the expected value, but also the probability distribution of the permanent displacements. The conditional distribution presented in Figures 5 and 6 is sufficient for most rock mechanics applications. It is derived by using the characteristics of 58 ground accelerations recorded on rock sites from several earthquakes in California. This method presented also permits the incorporation of the uncertainty in the evaluation of the limiting acceleration of rock slopes. Combining with information from conventional earthquake hazard analysis, this method constitutes a reasonable basis for more sophisticated analysis such as cost benefit evaluation.
Newmark has indicated, in 1965, that a soil or a rock slope does not necessarily fall just because the dynamic transient stress, induced by earthquake motions, reaches its strength. This conclusion has an important implication that the stability of a rock slope during an earthquake excitation can not be correctly reflected by the conventional factor of safety through pseudo-static analysis. Because factor of safety drops below one at one time does not pose a serious problem. What really matters is the magnitude of the permanent displacement caused by the earthquake when the factor of safety lies below one. The evaluation of the earthquake-induced permanent displacements for a rock slope can readily be accomplished by using the Newmark's sliding block model. However, this assessment is of lit-tle use if the uncertainty associated with the possible future motions cannot be incoroorated. To account for the uncertainty of the future possible motions, Newmark has scaled several strong motions to give an approximate upper bound solution. Sarma(1975) has improved this estimation by employing several shapes of pulses to represent earthquake motions. But the expected displacement so obtained differs depending on the shapes used. Morever, due to the highly erratic nature of the ground motions the permanent displacements scatter over a significant wide range as illustated in Fig. 1 for normalized motions. Accordingly, even a good estimation of the expected displacement alone is not enough. The range of the possible scatter should also be quantified. To overcome the above problems, a model (Lin, 1982) was derived which gives a better estimate of the expected displacement, as well as its probability distribution. Before the presentation of the method, the ground motion modelling is described.
GROUND MOTION MODELLING
Ground motion modelling may be very different depending on the type of response under study.