8.9 Inertia Slope Stability – Newmark Method

The Newmark [1965] analysis assumes that relative slope movements would be initiated when the inertial force on a potential sliding mass is large enough to overcome the yield resistance along the slip surface, and these movements stop when the inertial force decreases below the yield resistance and the velocities of the ground and sliding mass coincides. The yield acceleration is defined as the average acceleration producing a horizontal inertial force on a potential sliding mass which gives a factor of safety of one and can be calculated as the seismic coefficient in pseudostatic slope stability analyses that produces a safety factor of one.

 

Newmark’s method assumes: existence of a well-defined slip surface, a rigid, perfectly plastic slide material, negligible loss of shear strength during shaking, and that permanent strains occur if the dynamic stress exceeds the shear resistance. Also, the slope is only presumed to deform in the downslope direction, thus implying infinite dynamic shear resistance in the upslope direction. The procedure requires that the value of a yield acceleration or critical seismic coefficient, ky, be determined for the potential failure surface using conventional limit equilibrium methods.

 

Purpose of this method is to estimate the slope deformation for those cases where the pseudostatic factor of safety is less than 1.0, which corresponds to failure condition. It is assumed that slope will deform during those portions of earthquake when out of slope earthquake forces make pseudostatic factor of safety below 1.0 and the slope accelerates downwards. Longer the duration for which pseudostatic factor of safety is zero, greater the slope deformation. Figure 3 shows horizontal acceleration of slope during earthquake. Accelerations plotting above zero line are out of slope and accelerations plotting below zero line are into slope accelerations. Only out of slope accelerations cause downslope movement and are used in the analysis. In Figrue , ay is horizontal yield acceleration and corresponds to pseudostatic factor of safety exactly equal to 1. Portion of acceleration pulses above ay (darkened portion in Fig. 9.2(a)), causes lateral movement of slope. Fig. 9.2(b) and (c) represent horizontal velocity and slope displacement due to darkened portion of acceleration pulse. Slope displacement is incremental and occurs only when horizontal acceleration due to earthquake exceeds ay.

 

 

Figure 3: Diagram illustrating Newmark method (a) Accelertion versus time (b) velocity versus time for darkened portion of acceleration pulse (c) corresponding downslope displacement versus time in response to velocity pulse (Day, 2002)

 

Magnitude of slope displacement depends on variety of factors. Higher the ay value, more stable the slope is for a given earthquake. Greater the difference between peak ground acceleration amax due to earthquake and ay, larger the down slope movement. Longer the earthquake acceleration exceeds ay, larger the down slope deformation. Larger the number of acceleration pulses exceeding ay, greater the cumulative down slope movement during earthquake. Most common method used in Newmark method is as follows:

 

Where,

            d= estimated downslope movement due to earthquake in cm.

            yield acceleration and

            peak ground acceleration of design earthquake.

Essentially  must be greater than . While using Eq. (9.3), pseudostatic factor of safety is determined first using the technique described in Fig. 9.2. If it is less than 1, is reduced till pseudostatic factor becomes equal to 1. This value of  is used to determine  using Eq. (9.1).  and  are used to determine slope deformation.