P-wave

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P-wave
Particles oscillate back and forth Wave travels
down rod, not particles Particle motion parallel
to direction of wave propagation
S-wave
Particles oscillate back and forth Wave travels
down rod, not particles Particle motion
perpendicular to direction of wave propagation
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Boundary Effects
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Boundary Effects
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Boundary Effects
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Boundary Effects
At centerline, displacement is always zero Stress
doubles momentarily as waves pass each other
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Boundary Effects (Fixed End)
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Boundary Effects (Fixed End)
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Boundary Effects (Fixed End)
2
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Boundary Effects (Fixed End)
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Boundary Effects (Fixed End)
Response at boundary is exactly the same as for
case of two waves of same polarity traveling
toward each other At fixed end, displacement is
zero and stress is momentarily doubled.
Polarity of reflected wave is same as that of
incident wave
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Boundary Effects (Fixed End)
Displacement
Response at boundary is exactly the same as for
case of two waves of same polarity traveling
toward each other At fixed end, displacement is
zero and stress is momentarily doubled.
Polarity of reflected stress wave is same as that
of incident wave. Polarity of reflected
displacement is reversed.
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Boundary Effects
s 0
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Boundary Effects
s 0
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Boundary Effects
s 0
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Boundary Effects
s 0
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Boundary Effects
s 0
At centerline, stress is always zero Particle
velocity doubles momentarily as waves pass each
other
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Boundary Effects (Free End)
s 0
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Boundary Effects (Free End)
s 0
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Boundary Effects (Free End)
s 0
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Boundary Effects (Free End)
s 0
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Boundary Effects (Free End)
Displacement
s 0
Response at boundary is exactly the same as for
case of two waves of opposite polarity traveling
toward each other At free end, stress is zero
and displacement is momentarily doubled.
Polarity of reflected stress wave is opposite
that of incident wave. Polarity of reflected
displacement wave is unchanged.
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Boundary Effects (Material Boundaries)
transmitted
incident
reflected
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Boundary Effects (Material Boundaries)
At material boundary, displacements must be
continuous Ai Ar At equilibrium must be
satisfied si sr st
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Boundary Effects (Material Boundaries)
Using equilibrium and compatibility,
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Boundary Effects (Material Boundaries)
Soft
Stiff
r2 r1 v2 v1/2
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Boundary Effects (Material Boundaries)
Soft
Stiff
Ar Ai / 3 At 4Ai / 3
Displacement amplitude is reduced
Displacement amplitude is increased
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Boundary Effects (Material Boundaries)
Soft
Stiff
sr - si / 3 st 2si / 3
Stress amplitude is reduced, reversed
Displacement amplitude is reduced
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Boundary Effects (Material Boundaries)
Consider limiting condition v2 ? 0
az 0
Soft
Stiff
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Boundary Effects (Material Boundaries)
Consider limiting condition v2 ? 0
az 0
Soft
Stiff
Ar Ai At 2Ai
Displacement amplitude is unchanged
Displacement amplitude at end of rod is doubled
- free surface effect
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Boundary Effects (Material Boundaries)
Consider limiting condition v2 ? 0
az 0
Soft
Stiff
sr - si st 0
Polarity of stress is reversed, amplitude
unchanged
Stress is zero - free surface effect
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Wave Propagation ExampleDr Layer
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Three Dimensional Elastic Solids
Displacements on left Stresses on right
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Three Dimensional Elastic Solids
Using 3-dimensional stress-strain and
strain-displacement relationships
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Three Dimensional Elastic Solids

• Two types of waves can exist in
• an infinite body
• p-waves
• s-waves

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Waves in a Layered Body
Waves perpendicular to boundaries
p-waves
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Waves in a Layered Body
Waves perpendicular to boundaries
SH-waves
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Waves in a Layered Body
Inclined Waves
Incident P
Incident p-wave
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Waves in a Layered Body
Inclined Waves
Incident SV
Incident SV-wave
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Waves in a Layered Body
Inclined Waves
When wave passes from stiff to softer material,
it is refracted to a path closer to being
perpendicular to the layer boundary
Incident SH-wave
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Waves in a Layered Body
Waves are nearly vertical by the time they reach
the ground surface
Vs500 fps
Vs1,000 fps
Vs1,500 fps
Vs2,000 fps
Vs2,500 fps
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Waves in a Semi-infinite Body

• The earth is obviously not an infinite body.

• For near-surface earthquake engineering problems
• the earth is idealized as a semi-infinite body
  with
• a planar free surface

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Surface Waves

• Rayleigh-waves
• Love-waves

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Rayleigh-waves
Comparison of Rayleigh wave and body wave
velocities
Rayleigh waves travel slightly more slowly than
s-waves
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Rayleigh-waves
Horizontal and vertical motion of Rayleigh waves
Rayleigh wave amplitude decreases quickly with
depth
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Attenuation of Stress Waves
The amplitudes of stress waves in real
materials decrease, or attenuate, with distance
Two primary sources
Material damping
Radiation damping
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Attenuation of Stress Waves
Material damping
A portion of the elastic energy of stress
waves is lost due to heat generation
Specific energy decreases as the waves travel
through the material
Consequently, the amplitude of the stress
waves decreases with distance
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Attenuation of Stress Waves
Radiation damping
The specific energy can also decrease due to
geometric spreading
Consequently, the amplitude of the stress
waves decreases with distance even though the
total energy remains constant
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Attenuation of Stress Waves
Both types of damping are important, though one
may dominate the other in specific situations
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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.

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Transfer Function
Linear elastic layer on rigid base
u
z
H
At free surface (z 0),
Aei(wtkz)
Bei(wt-kz)
u(z, t) 2Acos kz eiwt
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Transfer Function
Linear elastic layer on rigid base
u
z
Transfer function relates input
H
to output
Amplification factor
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Transfer Function
Linear elastic layer on rigid base
Amplification is sensitive to frequency
For undamped systems, infinite amplification can
occur
Extremely high amplification occurs over narrow
frequency bands
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Transfer Function
Linear elastic layer on rigid base
1 damping
Amplification is still sensitive to frequency
Very high, but not infinite, amplification can
occur
Degree of amplification decreases with increasing
frequency
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Transfer Function
Linear elastic layer on rigid base
2 damping
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Transfer Function
Linear elastic layer on rigid base
5 damping
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Transfer Function
Linear elastic layer on rigid base
10 damping
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Transfer Function
Linear elastic layer on rigid base
20 damping
Amplification sensitive to fundamental frequency
Maximum level of amplification is low
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Transfer Function
Linear elastic layer on rigid base
All damping
Amplification
De-amplification
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Transfer Function
Linear elastic layer on rigid base
10 damping
Stiffer, thinner
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Transfer Function example
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Transfer Function

• How is it used?
• Input motion convolved with transfer function
  multiplication in freq domain
• Steps
• Express input motion as sum of series of sine
  waves (Fourier series)
• Multiply each term in series by corresponding
  term of transfer function
• Sum resulting terms to obtain output motion.
• Notes
• Some terms (frequencies) amplified, some
  de-amplified
• Amplification/de-amp. behavior depends on
  position of transfer function