Soil dyn __3 corr

Soil Dynamics
3. Wave Propagation
Cristian Soriano Camelo1
1Federal University of Rio de Janeiro
Geotechnical Engineering
August 03rd, 2017
Cristian Soriano Camelo (UFRJ) Soil Dynamics August 03rd, 2017 1 / 47

Outline
1 Introduction
Wave Propagation
2 Waves in unbounded media
One-dimensional, Three-dimensional
3 Waves in a semi-inﬁnite body
Rayleigh waves, Love waves, response
4 Waves in a layered body
One-dimensional, three-dimensional
5 Attenuation of Stress Waves
Material damping, Radiation damping

Outline
1 Introduction
Wave Propagation

Introduction
Geologic materials
Must be treated as continua
Dynamic response in terms
of: wave propagation
While most structures can readily be idealized as assemblages of discrete
masses with discrete sources of stiﬀness, geologic materials cannot.

Outline
1 Introduction
Wave Propagation

Waves in unbounded media
One dimensional wave propagation
-Longitudinal vibration: e.g a thin rod extents and contracts.
-Torsional vibration: e.g a thin rod rotates about its axis.
-Flexural vibration: e.g the axis of a thin rod moves laterally.

Longitudinal waves
∂σx
∂x = ρ∂2x
∂t2 - In terms of the inertial force induced by acceleration of
the mass of the element;
In this form, the equation of motion is valid for any stress-strain
behaviour but cannot be solved directly because it mixes stresses with
displacements.

Longitudinal waves
To simplify the equation of motion, the left side of the previous
equation can be expressed in terms of displacement by using the
stress-strain relationship, σx = Mεx, where the constrained modulus
M = (1 − ν)/[(1 + ν)(1 − 2ν)]E, and the strain-displacement
relationship, εx = ∂u/∂x.
Therefore,
∂2u
∂t2 = M
ρ
∂2u
∂x2 - Longitudinal wave equation, M=constrained modulus,
ρ=density;
∂2u
∂t2 = v2
p
∂2u
∂x2 - In terms of wave propagation velocity, vp = M/ρ.

Longitudinal waves
It can be seen that propagation velocity depends only in the properties
of the material (stiffness and density). The wave propagation velocity
increases with stiffness and with decreasing density
Example 1: Wave propagation velocity for different materials.
vp = M
ρ = M
(SG)ρw
= Mg
(SG)γw
g = 9.81m/s2 and γw = unit weight of water.
Material Specific Gravity (SG) M (KPa) vs (m/s)
Steel 7.85 2.79x108 5956
Rubber 1.2 1.15x109 30976
Water 1.0 2.34x106 1531

Torsional waves
∂2θ
∂t2 = G
ρ
∂2θ
∂x2 = v2
s
∂2θ
∂x2
Where, vs = G/ρ is the velocity of propagation of the torsional wave
and G the shear modulus.

Example 2: Shear wave propagation velocity for diﬀerent materials.
vp = G
ρ = G
(SG)ρw
= Gg
(SG)γw
g = 9.81m/s2 and γw = unit weight of water.
Material Speciﬁc Gravity (SG) G (KPa) vs (m/s)
Steel 7.85 7.93x107 3178
Rubber 1.2 1.15x106 979
Water 1.0 0 0
Water can produce no resistance to shear stresses and consequently
cannot transmit torsional waves.

Solution of the one-dimensional equation of motion The general
equation is on the form: ∂2u
∂t2 = v2 ∂2u
∂x2
And the solution can be written as: u(x, t) = f(vt − x) + g(vt + x)
If the rod is subjected to some steady-state harmonic stress
σ(t) = σ0cos(ωt)
Where ω is the stress wave circular frequency, and deﬁning the wave
number, k = ω/v, the solution can be expressed as:
u(x, t) = Acos(ωt − kx) + Bcos(ωt − kx)

The wave number is related to the wavelength, λ, of the motion by:
λ = 2π
k , where k is the wave number
Example 3: Wavelength of longitudinal and torsional waves along
steel and rubber, harmonic frequency of 10 Hz.
Steel λ =
vp
f
5956m/s
10sec−1
= 595.6m
Rubber λ =
vp
f
30976m/s
10sec−1
= 3097.6m
Steel λ = vs
f
3178m/s
10sec−1
= 317.8m
Rubber λ = vs
f
979.55m/s
10sec−1
= 97.9m

Since earth is three-dimensional and sources of seismic energy are
three-dimensional, seismic waves must be described in terms of
three-dimensional wave propagation.
Drrivation of 3-D wave equations follow the same steps as those used
for 1-D propagation, the equations are formulated from:
-Equilibrium consideration
-Stress-strain relationships
-Strain-displacement relationships

Review of stress notation
Six independent components of stress are required to deﬁne the stress
state of the element.

Review of strain notation

Review of strain notation
εxx = du
dx εyy = dv
dy εzz = dw
dz
εxy = dv
dx + du
dy εyz = dw
dy + dv
dz εzx = du
dz + dw
dx
-Three components of normal strain
-Three components of shear strain

Stress-strain relationship Hooke’s law for an isotropic, linear elastic
material expressed in terms of Lame Constants, λ and µ








σxx
σyy
σzz
σxy
σxz
σyz








=








λ + 2µ λ λ 0 0 0
λ λ + 2µ λ 0 0 0
λ λ λ + 2µ 0 0 0
0 0 0 µ 0 0
0 0 0 0 µ 0
0 0 0 0 0 µ








=








εxx
εyy
εzz
εxy
εxz
εyz








Where,
λ =
Eν
(1 + ν)(1 − 2ν)
; µ =
E
2(1 + ν)

Equation of motion for a three-dimensional elastic solid
ρ∂2u
∂t2 = (λ + µ) ▽ ∆ + µ ▽2 u
Where,
∆ = ∂u
∂x + ∂v
∂y + ∂w
∂z
▽2
= ∂2
u
∂x2 + ∂2
v
∂y2 + ∂2
w
∂z2
This is the equation of wave propagation in homogeneous, isotropic
and elastic solids.

Compressional Waves (p-waves):Particle displacement is parallel
to the direction of wave propagation, also known as primary waves.
vp = λ+2µ
ρ = G(2−2ν)
ρ(1−2ν)

Shear Waves (s-waves): The particle motion is constrained to a
plane perpendicular to the direction of wave propagation. The close
relationship between s-wave velocity and shear modulus is used to
advantage in many ﬁeld and laboratory tests.
vs = µ
ρ = G
ρ

Outline
1 Introduction
Wave Propagation

Waves in a semi-inﬁnite body
The earth is not an inﬁnite body, it is a very large sphere with an outer
surface on which stresses cannot exist. For near surface earthquake
engineering problems two types of surface waves are of primary
importance:
Rayleigh waves: Motion is both in the direction of propagation and
perpendicular (in a vertical plane), and phased so that the motion is
generally elliptical either prograde or retrograde.
Love waves:They move only on the horizontal plane, transversally
with respect to the direction of propagation.

Rayleigh waves:

Rayleigh waves: Amplitude of elliptical motion exponentially
decreases with depth.

Love waves:

Love waves: This wave occurs when an elastic half space is overlain
by a softer surface layer. Generally, Love wave velocities are greater
than Rayleigh waves, so Love waves arrive before Rayleigh waves on
seismograph.

Seismograms recorded by a 3-component seismograph at Nana, Peru for an
earthquake located near the coast of central Chile on September 3, 1998

Outline
1 Introduction
Wave Propagation

Waves in a layered body
In the earth, conditions are much more complicated with many diﬀerent materials of
variable thickness.

Kobe, Japan - Earthquake 1995: Large sections of the main Hanshin Expressway
toppled over. This was particularly likely where the road crossed areas of softer,
wetter ground, where the shaking was stronger and lasted longer.

One dimensional case
Ai + Ar = At ; σi + σr = σt
Continuity of displacements and equilibrium

Using equilibrium and compatibility
Ar = 1−αz
1+αz
Ai σr = αz−1
1+αz
σi
At = 2
1+αz
Ai σt = 2αz
1+αz
σi
→ αz = ρ2v2
ρ1v1
Is deﬁned as the impedance ratio

ρ2 = ρ1 v2 = v1
2
αz = 0.5

Ar = Ai
3 Displacement amplitude is reduced
At = 4Ai
3 Displacement amplitude is increased
σr = −σi
3 Stress amplitude is reduced
σt = 2σi
3 Stress amplitude is reduced

Seismic wave paths: reﬂection and refraction of seismic waves from
the source (focus) of an earthquake by the diﬀerent layers of the earth.

Three-dimensional case: inclined waves
In general waves will not approach interfaces at 90 degrees. The
orientation of an inclined body wave can strongly inﬂuence the manner
in which energy is reﬂected and transmitted across an interface.

Three-dimensional case: inclined waves Since incident p- and
SV-waves involve particle motion perpendicular to the plane of the
interface; they will each produce both reﬂected and refracted p- and
SV- waves.

Three-dimensional case: An incident SH wave does not involve
particle motion perpendicular to the interface; consequently, only SH
waves are reﬂected and refracted.

Seismic waves refraction: when a fault ruptures below the earth
surface, body waves travel away from the source in all directions. By
the time the waves reach the ground surface, multiple refractions have
often bent them to a nearly vertical direction.

The angle of refraction is uniquely related to the angle of incidence by
the ratio of the wave velocities of the materials on each side of the
interface.
Example of refraction of an SH wave ray through a series of
successively softer layers.

Outline
1 Introduction
Wave Propagation

Attenuation of stress waves
The amplitude of stress waves in real materials decrease or attenuate
with distance.
Two sources:
Material damping
Radiation damping

Material damping
-A part of the elastic energy of a traveling wave is always converted to
heat.
-Speciﬁc energy decreases as the waves travel through the material.
-Consequently, the amplitude of stress waves decrease exponentially
with distance.
-Soils dissipate their energy by grain slippage and their damping is
somewhat insensitive to frequency.

Material damping

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.

For Further Reading I
Steven L. Kramer.
Geotechnical Earthquake Engineering.
Prentice Hall, 1996.
Towhata, I.
Geotechnical Earthquake Engineering.
Springer, 2008.
http://www.strongmotioncenter.org/
http://www.civil.utah.edu/ bartlett/CVEEN6330/
http://web.ics.purdue.edu/ braile/edumod/waves/WaveDemo.htm

Soil dyn __3 corr

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