Description About Seismic Analysis Method includes Pseudostatic method, Newmark's Displacement method and Dynamic Finite element analysis

1. DYNAMIC
ANALYSIS OF
SLOPE
STABILITY
ROSHAN KUMAR PATEL
M.Tech, Roll No. 16152015
BY
INDIAN INSTITUTE OF TECHNOLOGY VARANASI

2. Seismic Analysis
 Pseudostatic Method: The earthquake’s inertial forces are simulated by the inclusion of
static horizontal and vertical forces in limit equilibrium analysis.
 Newmark’s Displacement Method: This method is based on the concept that the actual
slope accelerations may exceed the static yield acceleration at the expense of generating
permanent displacements (Newmark, 1965).
 Dynamic Finite Element Analysis: This is a coupled two or three dimensional analyses
using appropriate constitutive material model that will provide details of concerning stress
, strains and permanent displacement.

3. Important elements in a seismic response analysis are:
• Input motions
• Site profile
• Static soil properties
 Degree of weathering of clay minerals
 Depth
 Texture
 Rock content
• Dynamic soil properties
 Stiffness
 Material Damping
 Unit Weight
 Location of water, degree of saturation and grain size distribution
• Constitutive models of soil response to loading
• Methods of analysis using computer programs

4. Pseudostatic Method
In pseudo-static methods, the cyclic earthquake motion is replaced with a constant horizontal
acceleration equal to kc (g), where kc is the seismic coefficient, and g is the acceleration due
to gravity. A force is applied to the soil mass equal to the product of the acceleration and the
weight of the soil mass.
It assumes that additional static force is applied on the slope due to earthquake. In actual
analysis, a lateral force acting through centroid of sliding mass, is applied which acts out of
slope direction. This pseudostatic lateral force Fh is calculated as follows:

5. Newmark Sliding Block Analysis
The serviceability of a slope after an earthquake is controlled by deformations, analyses
that predict slope displacement provide a more useful indication of seismic slope stability.
Since earthquake induced accelerations vary with time, the pseudostatic factor of safety
will very throughout an earthquake.
If the inertial forces acting on a potential failure mass become large enough that the total
(static plus dynamic) driving forces exceed the available resisting forces, the factor of
safety will drop below 1.0.
Newmark (1965) considered the behaviour of a slope under such conditions. When the
factor of safety is less than 1.0, the potential failure mass is no longer in equilibrium
consequently, it will be accelerated by the unbalanced force. The situation is analogous to
that of a block resting on an inclined plane.

6. Under static conditions, equilibrium of the block (in the direction parallel to the plane)
requires that the available static resting force exceed the static driving force, Assuming
that the block’s resistance to sliding is purely frictional.

7. Now consider the effect of inertial forces transmitted to the block by horizontal
vibration of the inclined plane with acceleration ah(t) = kh(t)g, (the effect of vertical
accelerations will be neglected for simplicity). At a particular instant of time,
horizontal acceleration of the block will induce a horizontal inertial force, khW. When
the inertial force acts in the downslope direction, resolving forces perpendicular to the
inclined plane gives:

8. Obviously the dynamic factor of safety decrease as kh increases and there will be (for a
statically stable block) some positive value of kh that will produce a factor of safety of
1.0 . This coefficient, termed the yield coefficient ky, corresponds to the yield
acceleration, ay = kyg. The yield acceleration is the minimum pseudostatic acceleration
required to produce instability of the block which is given by:
For sliding in the down slope direction. For sliding in the uphill direction (which can
occur when both angles are small),

9. Dynamic Finite Element Analysis and
Monte Carlo Technique
The analysis consists of two parts:
• Finding the stresses at required points using the finite element method.
• Locating the critical slip surface and finding the factor of safety using these stresses.

10. For dynamic analysis, the stiffness matrix [k] and consistent mass matrix [m] are
obtained for each element and the stiffness matrix and mass matrix of each element
are added by direct stiffness method to obtain the overall stiffness matrix [K] and
mass matrix [M]. The damping of the slope is assumed as Rayleigh type and the
damping matrix [C] is obtained using the equation
[C] = α[M] + β[K]
where α and β are the Rayleigh constants. The dynamic equation for the entire slope c
an then be expressed in matrix form as

11. A Monte Carlo technique proposed by Venanzio (1996) as explained below is used to
obtain the factor of safety of the slope at each time interval.
Trial slip circle ABCDE

12. The trial slip surface is divided into ‘n’ number of segments each of length dL. The
overall factor of safety for a particular slip surface is obtained using the equation:

13. References
 R.C.H. Koo, J.W. Pappin, D.C. Rule, M.I. Wallace & I.P.H. Yim; Dynamic Analyses of
Slopes in Hong Kong
 Agrahara Krishnamoorthy; Factor of Safety of a Slope Subjected to Seismic Load
 Sahar Ismail, Fadi Hage Chehade and Riad Al Wardany; SLOPE STABILITY ANALY
SIS UNDER SEISMIC LOADING
 Andrea Lisjak & Giovanni Grasselli ; Combined finite-discrete element analysis of roc
k slope stability under dynamic loading
 Sergei Vadimovich Tsirel, Boris Yurievich Zuev, Anton Anatolevich Pavlovich; The In
fluence of Earthquakes on Open-Pit Slope Stability ; International Journal of Geosciences
, 2012, 3, 799-808