Post on 01-Apr-2015
P-wave
S-wave
Particles oscillate back and forthWave travels down rod, not particlesParticle motion parallel to direction of wave propagation
Particles oscillate back and forthWave travels down rod, not particlesParticle motion perpendicular to direction of wave propagation
Boundary Effects
0 0
0u
Boundary Effects
0 0
0u
Boundary Effects
0
0u
2
Boundary Effects
00
0u
At centerline, displacement is always zeroStress doubles momentarily as waves pass each other
Boundary Effects (Fixed End)
0 0
0u
Boundary Effects (Fixed End)
0 0
0u
Boundary Effects (Fixed End)
0
0u
02
Boundary Effects (Fixed End)
00
0u
Boundary Effects (Fixed End)
00
0u
Response at boundary is exactly the same as for caseof 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
Boundary Effects (Fixed End)
00
0u
Response at boundary is exactly the same as for caseof 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.
Displacement
Boundary Effects
0
0
0u = 0
Boundary Effects
0
0
0u = 0
Boundary Effects
0
0
0u = 0
Boundary Effects
0
0
0u = 0
Boundary Effects
0
0
0u = 0
At centerline, stress is always zeroParticle velocity doubles momentarily as waves pass each other
Boundary Effects (Free End)
0
0u = 0
Boundary Effects (Free End)
0
0u = 0
Boundary Effects (Free End)
= 0
0u
Boundary Effects (Free End)
0u = 0
0
Boundary Effects (Free End)
0u = 0
0
Response at boundary is exactly the same as for caseof 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.
Displacement
Boundary Effects (Material Boundaries)
incidentincident
reflectedreflected
transmittedtransmitted
11111 M A E 22222 M A E
incidentincident
reflectedreflected
transmittedtransmitted
Boundary Effects (Material Boundaries)
At material boundary, displacements must be continuous
Ai + Ar = At
equilibrium must be satisfied
i + r = t
incidentincident
reflectedreflected
transmittedtransmitted
iz
t
iz
zr
AA
AA
1
2
1
1
11
22v
vz
iz
zt
iz
zr
1
2
1
1
z = Impedance ratio
Boundary Effects (Material Boundaries)
Using equilibrium and compatibility,
incidentincident
reflectedreflected
transmittedtransmitted
11
22v
vz
z = 0.5
Boundary Effects (Material Boundaries)
Stiff Soft
2 = 1
v2 = v1/2
incidentincident
reflectedreflected
transmittedtransmitted
Boundary Effects (Material Boundaries)
Stiff Soft
Ar = Ai / 3
At = 4Ai / 3
Displacement amplitude is reduced
Displacement amplitude is increased
incidentincident
reflectedreflected
transmittedtransmitted
Boundary Effects (Material Boundaries)
Stiff Soft
r = - i / 3
t = 2i / 3
Stress amplitude is reduced, reversed
Displacement amplitude is reduced
Boundary Effects (Material Boundaries)
Stiff Soft
Consider limiting condition: vConsider limiting condition: v22 0 0
z = 0
Boundary Effects (Material Boundaries)
Stiff Soft
Consider limiting condition: v2 0
z = 0
Ar = Ai
At = 2Ai
Displacement amplitude is unchanged
Displacement amplitude at end of rod is doubled - free surface effect
Boundary Effects (Material Boundaries)
Stiff Soft
Consider limiting condition: v2 0
z = 0
r = - i
t = 0
Polarity of stress is reversed, amplitude unchanged
Stress is zero - free surface effect
Wave Propagation ExampleDr Layer
Three Dimensional Elastic Solids
x
y
z
xx
yy
zz
xy
yx
zy
xz
zy
zx
zyxt
u xzxyxx
2
2
zyxt
v yxzyxyyx
2
2
zyxt
w zzzyzx
2
2
Displacements on left
Stresses on right
Displacements on left
Stresses on right
xx
t
2
2
2
22
2 )2(
t
Three Dimensional Elastic Solids
uxt
u 22
2
)(
vyt
v 22
2
)(
wzt
w 22
2
)(
xsx v
t
22
2
2
222
2
pvt )2(
pv
sv
or
Using 3-dimensional stress-strain and
strain-displacement relationships
Using 3-dimensional stress-strain and
strain-displacement relationships
uxt
u 22
2
)(
vyt
v 22
2
)(
wzt
w 22
2
)(
xx
t
2
2
2
22
2 )2(
t
Three Dimensional Elastic Solids
xsx v
t
22
2
2
222
2
pvt )2(
pv
sv
or
Two types of waves can exist in
an infinite body• p-waves• s-waves
Waves in a Layered Body
Incident PIncident P
transmitted Ptransmitted P
reflected Preflected P
Waves perpendicular to boundaries
p-waves
Incident SHIncident SH
transmitted SHtransmitted SH
reflected SHreflected SH
Waves perpendicular to boundaries
SH-waves
Waves in a Layered Body
Waves in a Layered Body
Incident PIncident P
Refracted SVRefracted SV
Refracted PRefracted P
reflected SVreflected SV
reflected Preflected P
Inclined Waves
Incident p-wave
Incident SVIncident SV
Refracted SVRefracted SV
Refracted PRefracted P
reflected SVreflected SV
reflected Preflected P
Inclined Waves
Incident SV-wave
Waves in a Layered Body
Incident SHIncident SH
Refracted SHRefracted SH
Reflected SHReflected SH
Inclined Waves
Incident SH-wave
When wave passes fromstiff to softer material, it isrefracted to a path closer to being perpendicular tothe layer boundary
When wave passes fromstiff to softer material, it isrefracted to a path closer to being perpendicular tothe layer boundary
Waves in a Layered Body
Vs=2,500 fps
Vs=2,000 fps
Vs=1,500 fps
Vs=1,000 fps
Vs=500 fps
Waves in a Layered Body
Waves are nearly vertical by the time
they reach the ground surface
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
H1
H2
H3
incidentincident
reflectedreflected
Surface waveSurface waveFree surfaceFree surface
• Rayleigh-waves
• Love-waves
Surface Waves
Rayleigh-waves
Comparison of Rayleigh wave and body wave velocities
Rayleigh waves travel slightly more slowly
than s-waves
Rayleigh waves travel slightly more slowly
than s-waves
Horizontal and vertical motion of Rayleigh waves
Rayleigh-waves
Rayleigh wave amplitude decreases quickly with depth
Rayleigh wave amplitude decreases quickly with depth
Attenuation of Stress WavesAttenuation of Stress Waves
The amplitudes of stress waves in real materials decrease, or attenuate, with distance
Material damping
Radiation damping
Two primary sources:
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
Attenuation of Stress WavesAttenuation 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
Attenuation of Stress WavesAttenuation of Stress Waves
Attenuation of Stress WavesAttenuation of Stress Waves
Both types of damping are important, though one may dominate the other in specific situations
Transfer Function
• A Transfer function may be viewed as a filter that acts upon some input signal to produce an output signal.
• The transfer function determines how each frequency in the bedrock (input) motion is amplified, or deamplified by the soil deposit.
Transfer Function(filter)
input output
Transfer Function
Linear elastic layer on rigid base
u
z
H
u(0,t)
u(H,t)
Aei(t+kz)
Bei(t-kz)
At free surface (z = 0),
u(z, t) = 2Acos kz eit
(0, t) = 0 (0, t) = 0 A = B
Factor of 2 amplification
Linear elastic layer on rigid base
u
z
H
u(0,t)
u(H,t)
VHHkHzu
zuH
s** cos
1
cos
1
)(
)0()(
22cos
1)(
ss VHVHH
Amplification factor
Transfer function relates inputto output
Transfer Function
Zero dampingZero damping
Linear elastic layer on rigid base
u
z
H
u(0,t)
u(H,t)
For undamped systems, infinite amplification can occur
Extremely high amplification occurs over narrow frequency bands
Amplification is sensitive to frequency
Fundamental frequency
Characteristic site period
Ts =4HVs
Transfer Function
Linear elastic layer on rigid base
u
z
H
u(0,t)
u(H,t)
Very high, but not infinite, amplification can occur
Degree of amplification decreases with increasing frequency
Amplification is still sensitive to frequency
1% damping1% damping
Transfer Function
Linear elastic layer on rigid base
u
z
H
u(0,t)
u(H,t)
2% damping2% damping
Transfer Function
Linear elastic layer on rigid base
u
z
H
u(0,t)
u(H,t)
5% damping5% damping
Transfer Function
Linear elastic layer on rigid base
u
z
H
u(0,t)
u(H,t)
10% damping10% damping
Transfer Function
Linear elastic layer on rigid base
u
z
H
u(0,t)
u(H,t)
20% damping20% damping
Maximum level of amplification is low
Amplification sensitive to fundamental frequency
Transfer Function
Linear elastic layer on rigid base
u
z
H
u(0,t)
u(H,t)
All dampingAll damping
Amplification
De-amplification
Transfer Function
Linear elastic layer on rigid base
u
z
H
u(0,t)
u(H,t)
10% damping10% damping
Stiffer, thinner
Transfer Function
Transfer Function example
How is it used?
Input motion convolved with transfer function – multiplication in freq domain
Steps:1. Express input motion as sum of series of sine waves (Fourier series)2. Multiply each term in series by corresponding term of transfer function3. Sum resulting terms to obtain output motion.
Notes:1. Some terms (frequencies) amplified, some de-amplified2. Amplification/de-amp. behavior depends on position of transfer function
Transfer Function