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Geotechnical Earthquake

Engineering

by

Dr. Deepankar Choudhury Humboldt Fellow, JSPS Fellow, BOYSCAST Fellow

Professor

Department of Civil Engineering

IIT Bombay, Powai, Mumbai 400 076, India.

Email: [email protected]

URL: http://www.civil.iitb.ac.in/~dc/

Lecture – 20

IIT Bombay, DC 2

Module – 5

Wave Propagation

6

For isotropic material, the coefficients must be independent of direction

Hooke’s law for an isotropic, linear, elastic material allows all

components of stress and strain to be expressed in terms of two Lame’s

constants. λ and µ.

12 21 13 31 23 32

44 55 66

11 22 33 2

c c c c c c

c c c

c c c

2

2

2

xx xx

yy yy

zz zz

xy xy

yz yz

zx zx

xx yy zz

The volumetric strain:

IIT Bombay, DC

Reference : Kramer (1996)

IIT Bombay, DC 7

Common Expressions for

Various Modulus

3 2

2

3

2

E

K

G

All components of stress and strain for an isotropic,

linear, elastic material follows Hooke’s law and can be

expressed in terms of two Lame’s constants, λ and µ.

Young’s modulus,

Bulk modulus,

Shear modulus,

Poisson’s ratio,

8

The equation of motion in x-direction in terms of strains :

Using strain displacement relationships

Equation of motion reduced to

where the Laplacian operator:

Repeating the operation In y and z directions

the obtained equation of motion

2

22 xx xy xz

u

t x y z

2

2

2

uu

t x

2 2 22

2 2 2x y z

, ,xx xy xz

u v u w u

x x y x z

22

2

22

2

vv

t y

ww

t x

Equation of Motion for a 3D Elastic Solid

IIT Bombay, DC 9 Reference : Kramer (1996)

IIT Bombay, DC 10

Solution of 3D Equation of Motion

The solution for the first type of wave can be calculated by differentiating each equations w.r.t. x, y and z and adding them together,

Rearranging, the wave equation is given by,

2

2 2

2t

2

2

2 2

t2

pv

2 2

1 2p

Gv

2 22 22 22 2 2

2 2 2 2 2 2 2 2 2

yy yyxx xxzz zz

t t t x y z x y z


11

Solution of 3D Equation of Motion

The solution of second type of wave can be written in the 2 forms,

2

2

2

xx

t

2 2

1 2

p

s

v

v

A distortional (s) wave propagates through the solid at velocity

Comparing the velocities vp and vs,

s

Gv

2

2

w v w v

t y z y z


IIT Bombay, DC 13


19

Ratio of amplitudes of (a) reflected p-wave, (b) reflectd SV-wave, (c) refracted

p-wave, and (d) refracted SV-wave to amplitude of incident p-wave versus

angle incidence.


20

Ratio of amplitudes of (a) reflected p-wave, (b) reflected SV-wave, (c)

refracted p-wave, and (d) refracted SV-wave to amplitude of incident SV-wave

versus angle of incident.


IIT Bombay, DC 21

22

Waves in Semi-Infinite Body

Motion induced by a typical plane wave that propagates in x-direction.

Wave motion does not vary in the y-direction.

Rayleigh wave


23

Waves in a semi-infinite body

Two potential functions Φ and Ψ can be defined to describe the displacements

in the x and z directions:

The volumetric strain or dilation of the wave is given by

The rotation in x-z plane is given by

............... 5.1

............... 5.1

u ax z

w bz x

xx zz

2 22

2 2

u w

x z x x z z z x x z

2 22

2 22 y

u w

z x z x z z z x z x

Rayleigh waves


24

Substituting the expressions for u and w into the equations of motions gives.

Solving the above equations simultaneously for shows.

2 22 2

2 2

2 22 2

2 2

2

2

x t z t x z

z t x t z x

2 2

2 2and

t t

22 2 2

2

22 2 2

2

2p

p

vt

vt



25

For harmonic wave with frequency ω and wave number kR, Rayleigh

wave velocity vR = ω/ kR, the potential function is expressed as

substituting these in the above equations.

........................... 5.2

.......................... 5.2

R

R

i t k x

i t k x

F z e a

G z e b

222

2 2

222

2 2

R

p

R

s

d F zF z k F z

v dz

d G zG z k G z

v dz

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26

The equations can be rearranged to give the second-order differential equations

The general solution to these equations can be written in the form

where

2 22

2 2

2 22

2 2

0

0

R

p

R

s

d Fk F

dz v

d Gk G

dz v

1 1

2 2

qz qz

sz sz

F z A e B e

G z A e B e

22 2

2

22 2

2

R

p

R

s

q kv

s kv

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27

The potential function can be written as

Since neither shear nor normal stresses can exist at the free surface of the half-

space , when z = 0, therefore

1

2

R

R

qz i t k x

sz i t k x

A e

A e

0xz zz

2 2 0

0

zz zz

xz xz

dw

dz

dw du

dx dz

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28

The free surface boundary condition can be written as

Which can be rearranged to yield

2 2

1 2

2 2

1 2

0 2 2 0

0 2 0

zz R R

zz R R

z A q k iA k s

z iA k q A s k

2 2

1

2

1

2 2

2

21 0

2

21 0

R

R

R

R

q kA

A i k s

A iqk

A s k

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29

Rayleigh Wave Velocity

Adding the above two equations and cross-multiplying gives.

Which upon introducing the definitions of q and s and factoring out a G2 k8R

term, yields.

KRs is the ratio of the Rayleigh wave velocity to the s-wave velocity,

Then

2 2 2 2 24 2R R Rq sk s k q k

2 22 2 2 2

2 2 2 2 2 2 2 2

216 1 1 2 2 ..................... 5.3

p R s R p R s Rv k v k v k v k

RRs

s s R

vK

v v k

2 /

RRs

p p R s R

vK

v v k v k


30

/ 2 1 2 / 2 2

Hence previous equation can be rewritten as

2

22 2 2 2 2 2

2

116 1 1 2 2Rs Rs Rs RsK K K K

Which can be rearranged to the equation

6 4 2 2 28 24 16 16 1 0Rs Rs RsK K K

IIT Bombay, DC Reference : Kramer (1996)

Rayleigh Wave Velocity

31

Variations of Rayleigh wave and P-wave with shear wave for

various values of Poisson’s ratio.

IIT Bombay, DC


IIT Bombay, DC 33

Love Wave

Typical condition for generation of Love Wave

for softer surficial layer (G1/1 < G2/2)

overlaying elastic half-space


34

Love wave traveling in x direction will involve in y direction, and can be

presented by the equation

where v is the particle displacement in y direction, V(z) describes the

variation of v with depth, and kL is the wave number of the Love wave.

Wave equation.

, , .................... 5.4Li k x tv x z t V z e

2

2

v

t

2 21

2 21

2 22

2 22

...............0

.................

G v vz H

x z

G v vz H

x z

IIT Bombay, DC Reference : Kramer (1996)

Love Wave

35

The amplitude can be shown vary with depth

where A and B coefficients describe the amplitude of down going and up

going waves, respectively, and

Since the half space extends to infinite depth, B2 = 0, the requirement that all

stresses vanish at the ground surface is satisfied if

V z

1 1

2 2

1 1

2 2

......................0

.....................

v z v z

v z v z

A e B e z H

A e B e z H

2 2

11 1/

Lkv

G

2 2

22 2/

Lkv

G

1 1 1 1

1 1 1 1 1 1 1 0Li k x t v z v z v z v zV zve A v e v B e A B v e e

z z


Love Wave

36

The love wave velocity can be

obtained by:

1/ 2 2 222

2 211

2 21

1 1

1 1tan

1 1

L s

s L

s L

v vGH

Gv v

v v

Variation of Love wave velocity with frequency.

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Love Wave

IIT Bombay, DC 37

Variation of particle displacement amplitude with depth for

Love wave


38

Three-Dimensional Case: Inclined Wave

Ray path, ray, and wavefront for (a) plane wave and (b) curved wavefront.

sinconsant

i

v

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39

Reflected and refracted rays resulting from incident (a) p-wave,

(b) SV- wave, and (c) SH-wave.

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40

Refraction of an SH-wave ray path through series of successively softer

(lower vs) layers. Note that orientation f ray path becomes closer to

vertical as ground surface is approached. Reflected rays are not shown.

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IIT Bombay, DC 44

Attenuation of Stress Waves

Both types of damping are important,

though one may dominate the other in

specific situations

IIT Bombay, DC 45

Case Studies

IIT Bombay, DC 46

Case Study # 1

• Wave propagation in stratified media has wide applications in petroleum

exploration, geophysical inversion, nondestructive evaluation of highway and airport

pavement structures, countermine technology, structural health monitoring, and

vehicle weigh-in-motion systems.

• Sun et al. (2013) developed a high order thin layer method for analyzing

viscoelastic wave propagation in stratified media.

• It approximates the stiffness matrix involving transcendental functions by truncating

the Taylor series of the stiffness matrix to the fourth order term. A generalized

eigenvalue problem is then formulated, which allows an efficient numerical

algorithm to be designed in a computer program Dyna Thin Layer.

• The new method is most applicable to situations where a large number of layers is

involved or to situations where some natural layers have large thickness.

IIT Bombay, DC 47

Case Study # 1 (contd.)

A multilayered soil strata resting on half

space or bed rock.

The motion of the multilayered viscoelastic

solid is governed by Navier’s equation:

where, F is the displacement vector and f is

the body force

Here, u = u(x, y, z, t), v = v(x, y, z, t) and w =

w(x, y, z, t) are the displacements of the ith layer

along x, y and z directions, respectively.

48


• The vector of internal stresses in any horizontal plane can be written as:

• The present method can be effectively and efficiently used to compute the Green’s

function (fundamental solution) of the stratified media, which is of paramount

importance to many applications having an arbitrary loading condition.

• It can also be embedded into algorithms dealing with inverse problems involved in

nondestructive evaluation of highway and airport pavement structures, petroleum

exploration, countermine technology, geophysical inversion, structural health

monitoring, and vehicle weigh-in-motion systems.

Reference: Sun, L., Pan, Y. and Gu, W.(2013) “High-order thin layer method for viscoelastic

wave propagation in stratified media”, Comput. Methods Appl. Mech. Engrg. , 257 , 65–76

IIT Bombay, DC 49

Case Study # 2

• Zhu and Zhao (2013) studied propagation of obliquely incident waves across

joints with Virtual Wave Source Method (VWSM). The superposition of P wave and

S wave was for the first time mathematically expressed and studied.

• Complete theoretical reflection and transmission coefficients across single joint

described by displacement discontinuity model were derived through plane wave

analysis.

• With increasing joint stiffness, the transmission coefficients across single joint

increased except those whose wave type was different from the incident wave.

• The amplitude of superposed transmitted wave for P wave incidence increases

with incident angle, which is coincident with field observations.

• Both joint spacing and number of joints have significant effects on transmission

coefficients.

IIT Bombay, DC 50


The stresses obtained were given as :

Coordinate system and incident, reflected and transmitted waves for (a) P wave

incidence and (b) S wave incidence.

IIT Bombay, DC 51


• Since P wave and S wave have different velocities, the change of non-dimensional

joint spacing (ξ) resulted in different phase changes of the transmitted waves.

• Complete accurate theoretical reflection and transmission coefficients for obliquely

incident wave upon single joint are derived in matrix form through plane wave

analysis.

• The transmitted wave energy was mainly constrained in the transmitted wave of the

same type as the incident wave for wave propagation across single and multiple

joints.

• Both joint spacing and number of joints have significant effects on transmission

coefficients.

Reference: Zhu, J.B. and Zhao, J.(2013) “Obliquely incident wave propagation across rock

joints with virtual wave source method”, Journal of Applied Geophysics , 88, 23–30

IIT Bombay, DC 52

End of

Module – 5

geotechnical earthquake engineering - nptelnptel.ac.in/courses/105101134/downloads/lec-20.pdf6 for...

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