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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.

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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 , a_{y} 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 a_{y}.

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 a_{max }due to earthquake and a_{y},
larger the down slope movement. Longer the earthquake acceleration exceeds a_{y},
larger the down slope deformation. Larger the number of acceleration pulses
exceeding a_{y}, 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.