Analytical solutions for the sliding displacement of model slopes simulating both the frictional and rotational effects under piecewise linear acceleration pulses

Abstract

Simplified methods predicting earthquake‐induced displacement of slopes are based on the Newmark sliding‐block model. This model does not consider the decrease in inclination of the sliding mass as a result of its downward motion and this effect is usually modeled by improving the above model by assuming that the critical acceleration value for relative motion of the block increases linearly with the distance moved. This work, for the first time, derives analytical solutions predicting the sliding displacement under idealized acceleration pulses of the model described above simulating both the frictional and rotational effects. First, a general solution is derived in dimensionless form for the motion of the sliding mass under a piecewise linearly varying pulse for the cases of both static stability and instability. The solution is obtained in each interval by (a) solving for the displacement and velocity in terms of these quantities at the end of the previous interval, (b) estimating the time corresponding to a sliding velocity equal to zero, and (c) investigating if the estimated time solution is consistent with the time interval and thus if it can be considered as the global solution. Then, based on this solution and the conditions where downward and upward motion starts, a procedure predicting the final displacement of the sliding mass for a piecewise linearly varying pulse is proposed. Finally, analytical solutions are derived, given in graphical form, partly validated, and discussed for the particular cases of half and full cycle pulses of rectangular, triangular, and trapezoidal shapes.

Explicit solutions for the ground displacement of a sliding model simulating both the frictional and rotational effects under idealized acceleration pulses

“…The block friction follows the Mohr-Coulomb strength law and every time where the applied acceleration is larger than the critical acceleration, either downward (a c ) or upward, the block slides. [2][3][4][5][6][7] Yet, the upward movement is in most cases zero or much less than the downward one, 4,7 so, as a first approximation, it can be neglected. The model of Figure 1A is used for the prediction of permanent seismic movement of slopes by replacing the critical and applied accelerations of the block with those of the potential sliding mass under consideration.…”

“…The critical acceleration of the potential sliding mass (a c ) can be obtained by performing stability analyses of the potentially unstable mass using a resistance which corresponds to the representative one during the seismic event. 4,[8][9][10][11] This resistance for the dry case corresponds to the in-situ dry strength. For the saturated case, at the first stages of earthquake loading, the soil strength may decrease, as a result of excess pore pressure, to the undrained soil strength, which may be dramatically different from that for the dry case.…”

“…For the saturated case, at the first stages of earthquake loading, the soil strength may decrease, as a result of excess pore pressure, to the undrained soil strength, which may be dramatically different from that for the dry case. 4,9,10 If this strength value is less than the resistance required to maintain static equilibrium, static instability occurs and a c takes negative values. 4,9,10 In the case of large permanent displacement, the level of accuracy of the conventional sliding block model of Figure 1A is limited primarily because of changes of the geometry of the sliding mass towards a gentler inclination.…”

“…4,9,10 If this strength value is less than the resistance required to maintain static equilibrium, static instability occurs and a c takes negative values. 4,9,10 In the case of large permanent displacement, the level of accuracy of the conventional sliding block model of Figure 1A is limited primarily because of changes of the geometry of the sliding mass towards a gentler inclination. [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] This rotational effect has been modeled with reasonable accuracy by the sliding chain model of Figure 1B, where the critical acceleration of the sliding model (a c ) increases linearly with the downward distance moved by a factor which is inversely proportional to the length of the slip surface.…”

“…Depending on the geometry of the slope and the location of relevant critical structures, tolerable displacement may vary from (a) a few centimeters to (b) meters. 4 It is inferred that for design, prediction of earthquake-induced displacement varying in this broad range may be needed.…”