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Ground Response Analysis
Part - I
Lecture-25
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Source-Path-Site
Source
Path
SiteFault
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Source-Path-Site
A complete ground response analysis should include:
•Rupture mechanism at source of an earthquake (source)
• Propagation of stress waves through the crust to the top of
bedrock beneath the site of interest (path)
• How ground surface motion is influenced by the soils that
lie above the bedrock (site)
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Source-Path-Site
In reality,
•Mechanism of fault rupture is very complicated and
difficult to predict in advance
• Crustal velocity and damping characteristics are generally poorly known
• Nature of energy transmission between the source and site
is uncertain
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Source-Path-Site
In Practice,
•Seismic hazard analyses (probabilistic or deterministic)
are used to predict bedrock motions at the location of the
site.
• Seismic hazard analyses rely on empirical attenuation
relationships to predict bedrock motion parameters.
• Ground response problem becomes one of determining
response of soil deposit to the motion of the underlying bedrock.
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Objectives of ground response analysis
•Predict ground surface motions
Time histories
Response spectra
Scalar parameters
•Evaluate dynamic stresses and strains
Liquefaction hazards
Foundation loading•Evaluate ground failure potential
Instability of earth structures
Response of retaining structures 6
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Rock Outcropping motion
Rock outcropping motion - the motion that would occur
where rock outcrops at a free surface
Rock
Soil
Rock outcroppingmotion
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Bedrock motion
Bedrock motion - the motion that occurs at bedrock overlain by
a soil deposit. Differs from rock outcrop motion due to lack of
free surface effect.
Rock
Soil
Rock outcropping
motion
Bedrock motion 8
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Free surface motion
Free surface motion - the motion that occurs at the surface of a
soil deposit.
Rock
Soil
Rock outcropping
motion
Bedrock motion
Free surface motion
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Common problem 1
Rock outcrop motion is known - usually obtained from
attenuation relationship (based on database of rock outcrop
motions). Free surface motion is to be determined.
Rock
Soil
KnownUnknown
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Common problem 2
Free surface motion is known - usually obtained from
attenuation relationship (based on database of soil outcrop
motions) Free surface motion is to be determined for site with
different soil conditions
Rock
Soil1
Known
Rock
Soil2
unknown
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Ground response analysis
Two Basic Approaches:
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Linear Ground response analysis
•A known time history of bedrock (input) motion is represented
as a Fourier series, usually using the FFT.
•Each term in the Fourier series of the bedrock (input) motion is
then multiplied by the transfer function to produce the Fourier
series of the ground surface (output) motion.
• The ground surface (output) motion can then be expressed in
the time domain using the inverse FFT.
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Transfer function
•The transfer function determines how each frequency in the
bedrock (input) motion is amplified, or deamplified by the soil
deposit.
•A transfer function may be viewed as a filter that acts upon
some input signal to produce an output signal.
Transfer Function
(Filter)
Input Output
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Transfer function Evaluation: cases
•Uniform undamped soil on rigid rock
•Uniform damped soil on rigid rock
•Uniform undamped soil on elastic rock
•Uniform damped soil on elastic rock
•Layered damped soil on elastic rock
Transfer Function
(Filter)
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Uniform undamped soil on rigid rock
Assume harmonic base motion. Then, response should also be harmonic
u(z,t) = A ei(ωt + kz) + B ei(ωt - kz)
Wave traveling
in -z direction
(upward)
Wave traveling
in +z direction
(downward)
H
u
z A ei( t+kz)
B ei( t-kz)
2
2
2
2
z
uG
t
u
Wave Equation :
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Uniform undamped soil on rigid rock
Displacement
u(z,t) = A ei(ωt + kz) + B ei(ωt - kz)
Stress
t(z,t) = Gg(z,t) = GikA ei(ωt + kz) - GikB ei(ωt - kz)
At z = 0 (ground surface)
t(z,t) = 0 = Gik(A - B) eiωt
A = B
H
u
z A ei( t+kz)
B ei( t-kz)
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Uniform undamped soil on rigid rock
Displacement
u(z,t) = A ei(ωt + kz) + B ei(ωt - kz)
Substituting A = B
u(z,t) = 2A [(eikz
+ e-ikz
)/2] eiωt
= 2A cos(kz) eiωt
Defining a transfer function as the ratio of the displacement
at the ground surface to the bedrock displacement
kH ekH A
Ae
t H u
t uF
t i
t i
cos)cos(),(
),()(
1
2
20
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Uniform undamped soil on rigid rock
The modulus of the transfer function is the amplification function(|F()|)
kH ekH A
Ae
t H u
t uF
t i
t i
cos)cos(),(
),()(
1
2
20
sV H kH
F
coscos
)( 11
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Uniform undamped soil on rigid rock
As kH = H/Vs goes to zero, denominator goes to zero and the transfer function
goes to infinity, causing resonance. From this condition, the natural frequency
(n) and the fundamental period (TS)can be estimated as:
Natural Frequencies:
H nV S n 2
Fundamental Period
S s V H T 42 0
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Uniform damped soil on rigid rock
Damping is handled by Complex shear modulus G*
G* = ρ (v* s )2 = ρ (ω/k*)2
k* = [ρω2 /G*] 1/2 Complex wave number
)1(/** ivGv s s )1(*
*
ik v
k
s
H
u z A ei( t+kz)
B ei( t-kz) Damped
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Uniform damped soil on rigid rock
Transfer function for this case is evaluated as it is done
for the case of undamped soil. The equation for
transfer function now becomes:
H
u z A ei( t+kz)
B ei( t-kz) Damped
H k e H k A
Ae
t H u
t uF
t i
t i
**cos)cos(),(
),(),(
1
2
20
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Uniform damped soil on rigid rock
*
*
coscos),(
),(),(
sV
H H k t H u
t uF
110
2222
11
ss V H V H kH kH
F
//cos)()(cos
),(
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Uniform damped soil on rigid rock
Observations:
Natural frequencies still exist
Amplification is strongly dependent
on the frequencies
Low natural frequencies are amplified
High natural frequencies are weaklyamplified
Very high frequencies are de-
amplified
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Kramer (1996) Geotechnical Earthquake Engineering, Prentice Hall.
Villaverde, R. (2009) Fundamental Concepts of Earthquake Engineering , CRC
Press.
Towhata, T. (2008) Geotechnical Earthquake Engineering, Springer.
References