SEISMIC ANALYSIS
8.1 Introduction
Stability of a slope can be affected by seismicity in two ways: earthquake and blasting. These
seismic motions are capable of inducing large destabilizing inertial forces. In general, three
methods of analysis have been proposed for the evaluation of slopes response under such
conditions.
1. Pseudostatic Method: The earthquake’s inertial forces are simulated by the inclusion of
static horizontal and vertical forces in limit equilibrium analysis.
2. Newmark’s Diaplacement 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).
3. Dynamic Finite Element Analysis: This is a coupled two or three dimensional analyses
using appropriate constitutive material model that will provide details of concerning
stresses, strains, and permanent displacement.
When a seismic motion occurs, different types of seismic waves are produced. The main seismic
wave types are Compression (P), Shear (S), Rayleigh (R) and Love (L) waves. P and S waves are
known as body waves, because they propagate outward in all directions from source (such as an
earthquake) and travel through the interior of the earth. Love and Rayleigh waves are surface
waves and propagate approximately parallel to the earth’s surface.
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Figure 8.1: Typical seismogram ( www.geo.mtu.edu)
Figure 8.2: definition of earthquake terms (www.culcanhammer.net)
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8.2 Types of seismic wave
P wave is primary compression type, characterized by alternative compressions and dilations,
with their direction same as the direction in which the wave is propagating. Its is around 5-7
Km/s. among all the seismic waves, P wave travels fastest in materials, so it obtained first on a
seismogram.
The motion of S (secondary wave) is transverse in nature and is in the direction perpendicular to
the direction of propagation. Typical velocity of S wave in earth crust is 3-4 Km/s. These do not
travel through fluids, so do not exist in earth’s outer core (inferred to be primarily liquid iron) or
in air or water or molten rock (magma). S waves travel slower than P waves.
The motion of L (love) or surface wave is transverse horizontal, perpendicular to the direction of
propagation and generally parallel to the earth’s surface. This wave exists because of the earth’s
surface. The amplitude is largest at the surface and decrease with depth. Love waves are
dispersive, i.e. the wave velocity is dependent on frequency; with low frequencies propagating at
higher velocity. Depth of penetration of the Love waves is also dependent on frequency, with
lower frequencies penetrating to greater depth.
R (Rayleigh) wave is another surface wave which has its motion both in the direction of
propagation and perpendicular and phased, so that the motion is generally elliptical either
prograde or retrograde. Rayleigh waves are also dispersive and its amplitudes generally
decreases with depth. Appearance and particle motion are similar to water waves. Depth of
penetration of the Rayleigh waves is also dependent on frequency, with lower frequencies
penetrating to greater depth.
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The magnitude of an earthquake is related to the amount of energy released during the event.
Seismic energy attenuates as it travels away from the zone of energy release and spreads out over
a greater volume of material. Hence, intensity of the bedrock motion decreases as the distance of
a particular site from the zone of energy release increases. Measurement of mechanical
properties of soil and rock masses is, to a certain extent, possible with seismic methods. If P and
S waves are measured, a so-called ‘seismic E modulus’ and ‘seismic Poisson’s ratio’ can be
determined:
𝜆 + 2𝜇
𝜌
P– wave velocity = Vp = �
𝜇
𝜌
S– wave velocity = Vs = �
Seismic Young ′ s modulus = E =
Seismic Poission′ s ratio = σ =
𝜇(3𝜆 + 2𝜇)
(𝜆 + 𝜇)
𝜆
2(𝜆 + 𝜇)
Where, λ and µ, are lame′ s constant, ρ, the desity of material
The relation between the frequency and the wavelength is:
V=f X λ
Where, V is the velocity of seismic wave, f is its frequency, and λ is the wavelength
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The relation by deere et. al. relates Rock Quality Designation to seismic velocities measured in
the field and in the laboratory:
�
V�ield
VLaboratory
� X 100 % ≈ RQD
V field = seismic velocity measured in the field
V laboratory = seismic velocity measured in the laboratory
Where, RQD is the Rock Quality Designation. The seismic velocity measured in the laboratory is
done on intact rock. The velocity measured in the field is the velocity of a signal passing through
intact rock but also through or around discontinuities and will hence result in a lower velocity.
Inhomogeneities in soil or rock mass, such as boulders in a soil or joints in rock mass will not
individually be detected if the wavelength of seismic wave is low compared to the dimensions of
the inhomogeneities.
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8.3 Dynamic Soil Properties
Important elements in a seismic response analysis are: input motions, site profile, static soil
properties, dynamic soil properties, constitutive models of soil response to loading and methods
of analysis using computer programs. The contents include: earthquake response spectra, site
seismicity, soil response to seismic motion, seismic loads on structures, liquefaction potential,
lateral spread from liquefaction, and foundation base isolation.
The properties that are most important for dynamic analyses are the stiffness, material damping,
and unit weight. These properties enter directly into the computations of dynamic response. In
addition, location of the water table, degree of saturation, and grain size distribution may be
important, especially when liquefaction is a potential problem. One method of direct
determination of dynamic soil properties in the field is to measure the velocity of shear waves in
the soil. The waves are generated by impacts produced by a hammer or by detonating charges of
explosives and travel times are recorded. This is usually done in between bore holes.
Figure 3: Relation between number of blows per foot in standard penetration test and velocity of
shear waves
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8.4 Field Measurements of Dynamic Modulus
Direct measurement of soil or rock stiffness in the field has the advantage of minimal material
disturbance. The measurements are not constrained by the size of a sample. Dissipation of energy
during strain, i.e. material damping, requires significant strains to occur. In situ techniques are
based on measurement of the velocity of propagation of stress waves through the soil. The Pwaves or compression waves are dominated by the response of the pore fluid in the saturated
soils. Consequently, most techniques measure S-waves or shear waves. If the velocity of the
shear wave through a soil deposit is determined to be Vs, the shear modulus G is given as:
Where,
G = pVs2 =
Y 2
V
g s
p = mass density of soil.
Y = unit weight of soil.
g = acceleration due to gravity.
There are three techniques for measuring shear wave velocity in in-situ soil; cross- hole, downhole, and uphole. All the three techniques require boring to be made in the in-situ soil. In the
cross-hole method, sensors are placed at one elevation in one or more bore holes. A source of
energy is triggered in another bore hole at the same elevation. The waves travel horizontally
from the source to the receiving hole. The arrival of the S-waves is noted on the traces of the
response of the sensors. The velocity of S-wave can be calculated by dividing the distance
between bore holes by the time for a wave to travel between them. Thus, the measured travel
time reflects the cumulative travel through layers with different wave velocities. Interpreting the
data requires sorting out the contribution of the layers.
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8.5 Ground Motion Estimates
Ground motion attenuation equations are used to determine the level of acceleration as a function
of distance from the source as well as the magnitude of the earthquake. Correlations have been
made between peak acceleration and other descriptions of ground motion with distance for
various events. These equations allow the engineers to estimate the ground motions at a site from
a specified event. They also allow engineers to find out the uncertainty associated with the
estimate. There are a number of attenuation equations that have been developed by various
researchers. Donovan and Bornstein (1978), developed the following equation for peak
horizontal acceleration.
Y = (a)(exp(bM))(r + 25)d
a = (2,154,000)(r)−2.10
b = (0.046) + (0.445) log(r)
Where,
d = (2.515) + (0.486)log (r)
Y = peak horizontal acceleration (in gal)(1 gal = 1cm/sec 2 )
M = earthquake magnitude and
r = distance to energy center in km, default value 5 km.
Several methods for evaluating the effect of local soil conditions on ground response during
earthquakes are now available. Most of these methods are based on the assumption that the main
responses in a soil deposit are caused by the upward propagation of horizontally polarized shear
waves (SH waves). These waves are propagated from the underlying rock formation. Analytical
procedures based on this concept incorporating linear approximation to nonlinear soil behavior
have been shown to give results in fair agreement with field observations in a number of cases.
The analytical procedure generally involves the following steps:
(a) Determining the characteristics of the motions likely to develop in the rock formation
underlying the site. The maximum acceleration, predominant period, and effective duration are
the most important parameters of an earthquake motion. Empirical relationships between these
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parameters and the distance from the causative fault to the site have been established for
earthquakes of different magnitudes.
(b) Determining the dynamic properties of the soil deposit. Average relationships have been
established to estimate dynamic shear modulus as a function of shear strain and static properties
for various soil types (Seed and Idriss, 1970). Average relation between the damping ratios of
soils as functions of shear strain and static properties have also been established. Thus a testing
program to obtain the static properties for use in these relationships will often serve to establish
the dynamic properties with a sufficient degree of accuracy.
(c) Computing the response of the soil deposit to the base rock motions. A one-dimensional
method of analysis can be used if the soil structure is essentially horizontal.
8.6 Factors Affecting Ground Motion:
Factors that affect strong ground motion include:
(a) Wave types—S and P waves that travel through the earth, as well as the surface waves that
propagate along the surfaces or interfaces.
(b) Earthquake magnitude—There are several magnitude scales. Even a small magnitude event
may produce large accelerations in the near field. Consequently, a wide variety of acceleration
for the same magnitude may be expected.
(c) Distance from epicenter or from center of energy release.
(d) Site conditions.
(e) Fault type, depth, and the recurrence interval.
The selection of a design earthquake may be based on:
(a) Known design-level and maximum credible earthquake magnitudes. These are associated
with a fault whose seismicity has been estimated.
(b) Specification of probability of occurrence of the earthquake for a given life of the structure
(such as having a 10 percent chance of being exceeded in 50 years).
(c) Specification of a required level of ground motion. This specification is available in the code
provision.
(d) Fault length empirical relationships.
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8.7 Simulation of seismic effect
Slope movement is secondary effect of earthquake or blasting. For rock slopes, earthquake
induced slope movement is divided into falls and slides. In general there are two basic
approaches to incorporate the seismic effect on slope stability i.e. Inertia slope stability analysis
and weakening slope stability analysis.
It is preferred if material retains its shear strength during earthquake. Weakening slope stability
analysis is preferred if material experiences significant shear strength reduction during
earthquake. During liquefaction, there are two cases of weakening slope stability analyses. Flow
slide develops when the static driving forces exceed shear strength of soil along failure surface.
In lateral spreading, static driving forces do not exceed shear strength of soil along slip surface.
Instead, driving forces only exceed resisting forces during those portions of earthquake that
impart net inertial forces in the downward direction. This results in progressive and incremental
lateral movement.
Soils which dilate during seismic shaking, or do not exhibit reduction in shear strength with
strain, clay with low sensitivity, soils located above water table and landslides having distinct
rupture surface are examples, where material retain shear strength during earthquake. Inertia
slope analysis is preferred for these cases. Foliated or friable rock, which undergo fracture during
earthquake, sensitive clays, overloaded soft and organic soils as well as loose soils located below
water table and under liquefaction induced excess pore water pressure are examples where
material experiences sufficient shear strength reduction during earthquake. Weakening slope
stability analyses is preferred for them.
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8.8 Pseudo-static approach
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 of
gravity. A force is applied to the soil mass equal to the product of the acceleration and the weight
of the soil mass.
The limit equilibrium method is modified to include horizontal and vertical static seismic forces
that are used to simulate the potential inertial forces due to ground accelerations in an
earthquake. In pseudostatic slope stability analyses, a factor of safety against failure is computed
using a static limit equilibrium stability procedure, in which a pseudostatic, horizontal inertial
force, which represents the destabilizing effects of the earthquake, is applied to the potential
sliding mass.
This method is easy to understand and is applicable for both total and effective stress slope
stability analyses. The method ignores cyclic nature of earthquake. 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 F h is calculated as follows:
Where,
Fh = ma =
Wa Wamax
=
= khW
g
g
F h = horizontal pseudostatic force acting through centroid of sliding mass out of slope direction.
m=total mass of slide material
W=total weight of slide mass
a= acceleration, maximum horizontal acceleration at ground surface due to earthquake (= amax )
a max = peak ground acceleration
a max /g=seismic coefficient
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A considerable limitation of the pseudostatic approach is that the horizontal force representing
the effects of seismicity is constant and acts in only one direction. With dynamically applied
loads, the force may be applied for only a few tenths of a second before the direction of motion is
reversed. The result of these transient forces will be a series of displacement pulses rather than
complete failure of the slope.
Seismicity subjects sliding mass to vertical as well as horizontal pseudostatic forces. However,
the effect of vertical pseudostatic force on sliding mass is ignored as it has very little effect on its
slope stability. Although, the weight of the sliding mass W can be readily estimated, selection of
seismic coefficient requires considerable experience and judgement. Certain guidelines regarding
selection of seismic coefficient are as follows:
•
•
Higher the value of peak ground acceleration, higher is the value of k h .
•
k h is also determined as function of earthquake magnitude.
•
Sometimes local agencies suggest minimum value of seismic coefficient.
•
•
When items 1 and 2 are considered, k h should never be greater than
For small slide mass, k h =
amax
g
amax
g
.
.
amax
•
For intermediate slide mass, k h =0.65
•
k h =0.15 for sites near faults generating 8.5 magnitude earthquake.
g
.
For large slide mass, k h =0.1 for sites never generating 6.5 magnitude earthquake and,
k h =0.1 for severe earthquake, = 0.2 for violent and destructive earthquake and=0.5 for
catastrophic earthquake.
Wedge Method
This is simplest type of slope stability analysis. A wedge failure has planar slip surface, inclined
at an angle α to horizontal. Analysis could be performed for the case of planar slip surface
intersecting the face of slope or passing through toe of slope. Factor of safety for pseudostatic
analysis is obtained as follows:
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For total stress analysis:
FOS =
cL + Ntan∅
cL + (Wcos ∝ −Fh sin ∝)tan∅
resisting force
=
=
Wsin ∝ +Fh cos ∝
driving force
Wsin ∝ +Fh cos ∝
For effective stress analysis:
Where,
FOS =
c ′ L + (W cos ∝ −Fh sin ∝ −uL)tan∅′
c ′ L + N tan∅
=
Wsin ∝ +Fh cos ∝
W sin ∝ +Fh cos ∝
FOS= factor of safety for pseudostatic analysis
u= average pore water pressure along slip surface
c’ and ∅’ are the effective cohesion and friction angle material
For total stress analysis, total stress parameters of soil should be known and is often performed
for cohesive soils. For effective stress analysis, effective stress parameters of soil should be
known and is often performed for cohesionless soils. For this purpose pore water pressure along
slip surface should also be known. For soil layers above water table, pore water pressure is
assumed zero. If the soil is below water table and water table is horizontal, pore water pressure
below water table is hydrostatic. In case of sloping water table, flow net can be used to estimate
pore water pressure below water table.
Method of Slices
In this method, failure mass is subdivided into vertical slices and factor of safety is determined
based on force equilibrium equations. A circular arc slip surface and a rotational failure mode is
often used in this method. The resisting and the driving forces are calculated for each slice and
then summed to obtain factor of safety of the slope. Factor of safety can be calculating the
resisting forces to driving forces for each slice and then summed to obtain factor of safety.
However, this method involves more unknowns than equilibrium equations in the method of
slices. Consequently, an assumption is to be made concerning interslice forces. In ordinary
method of slices, resultant of interslice forces is parallel to average inclination of slice, α. Bishop
simplified, Janbu simplified, Janbu generalized, Spencer method and Morgenstern Price method
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are other methods of slices. Because of the tedious nature of calculations, computer programs are
routinely used to perform the pseudostatic slope stability analysis using the method of slices.
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.
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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)
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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,
log d = 0.90 + log ��1 −
ay 253 ay − 1.09
��
��
amax
amax
d= estimated downslope movement due to earthquake in cm.
ay =yield acceleration and
amax =peak ground acceleration of design earthquake.
Essentially amax must be greater than ay . 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, k h is reduced till
pseudostatic factor becomes equal to 1. This value of k h is used to determine ay using Eq. (9.1).
ay and amax are used to determine slope deformation.
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