P-wave S-wave Particles oscillate back and forth Wave travels down rod, not particles Particle...

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

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


0

0u

02


00

0u


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


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

At centerline, stress is always zeroParticle velocity doubles momentarily as waves pass each other

Boundary Effects (Free End)

0

0u = 0


= 0

0u


0u = 0

0


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



At material boundary, displacements must be continuous

Ai + Ar = At

equilibrium must be satisfied

i + r = t

incidentincident

reflectedreflected


iz

t

iz

zr

AA

AA

1

2

1

1

11

22v

vz

iz

zt

iz

zr

1

2

1

1

z = Impedance ratio


Using equilibrium and compatibility,

incidentincident

reflectedreflected


11

22v

vz

z = 0.5


Stiff Soft

2 = 1

v2 = v1/2

incidentincident

reflectedreflected



Stiff Soft

Ar = Ai / 3

At = 4Ai / 3

Displacement amplitude is reduced

Displacement amplitude is increased

incidentincident

reflectedreflected



Stiff Soft

r = - i / 3

t = 2i / 3

Stress amplitude is reduced, reversed

Displacement amplitude is reduced


Stiff Soft

Consider limiting condition: vConsider limiting condition: v22 0 0

z = 0


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


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


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


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



Incident PIncident P

Refracted SVRefracted SV

Refracted PRefracted P

reflected SVreflected SV


Inclined Waves

Incident p-wave

Incident SVIncident SV

Refracted SVRefracted SV

Refracted PRefracted P

reflected SVreflected SV


Inclined Waves

Incident SV-wave


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


Vs=2,500 fps

Vs=2,000 fps

Vs=1,500 fps

Vs=1,000 fps

Vs=500 fps


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


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



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


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


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


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


u

z

H

u(0,t)

u(H,t)


Transfer Function


u

z

H

u(0,t)

u(H,t)


Maximum level of amplification is low

Amplification sensitive to fundamental frequency

Transfer Function


u

z

H

u(0,t)

u(H,t)

All dampingAll damping

Amplification

De-amplification

Transfer Function


u

z

H

u(0,t)

u(H,t)


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

P-wave S-wave Particles oscillate back and forth Wave travels down rod, not particles Particle...

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