dynamic soil structure interaction_02_chapter2_nagano

8/2/2019 Dynamic soil structure interaction_02_Chapter2_nagano

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Dynamic Soil-Structure-

InteractionMasayuki Nagano, Dr. Eng.

Professor

Department of Architecture

Faculty of Science & Technology

Tokyo University of Science


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Amplification of ground motion in surface soil

Structural response

Dynamic soil spring

Foundation input motion

Ground motion on surface

Amplification of ground motion in surface soil is important in that it is directly

reflected into earthquake input motion to the structure and it controls wholestructural response.

Ground motion on

outcropped

engineering bedrock


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CHAPTER 2 :AMPLIFICATION OF

GROUND MOTION INSURFACE SOIL


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Chapter 2, amplification of ground motion in surfacesoil, consists of following three sections;

2.1 One-Dimensional Shear Wave Propagation Theory

2.2 S-Wave Propagation in Multi-Layered Strata

2.3 Response Analysis of Multi-Layered StrataConsidering Soil Nonlinearity


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2.1 One-Dimensional Shear Wave Propagation Theory

Consider a shear column with unit cross-section area,and it has Gof shear modulus and of mass density.

z

G.L

G Soil

Shear column

The positive z is taken as downward direction.

z

dzz

+

2

2

t

udz

u dz


6/42

(1) Equation of Motion

Shear stress is expressedby

G: Shear modulus

z

u

: Shear strain

where,

)1(0dzt

udz

z 2

2

=

+

)2(zt

u2

2

=

)3(z

uG

=

Consider the dynamic equilibrium of a small element,

z

dzz

+

2

2

t

udz

u dz


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Putting Eq.(3) into Eq.(2)

leads to

Eq.(4) is an equation of motion for one-dimensionalshear wave propagation.

)4(z

u

Gt

u2

2

2

2

=

z

dzz

+

2

2

t

u

dz

u dz


8/42

(2) General solution in time domain

The velocity VS of shear wave (S-wave) is expressed by:

Using VS, Eq.(4) can be transferred to:

The general solution of Eq.(6) is given by:

where, E(t+z/VS) and F(t-z/VS) are arbitrary functions in

terms of t and z.

)5(/GVS =

)6(z

u

Vt

u2

22

S2

2

=

)7(VztF

VztE)z,t(u

SS

+

+=


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E(t+z/VS) indicates the

displacement due to abackward propagatingwave which propagates inthe negative z direction.

It is called "upward wave".

while F(t-z/VS) due to aforward propagating wavepropagating in the

positive zdirection.

It is called "downwardwave".

z

xE

F

Characters E & F areconventionally used toexpress the "upward &downward" waves in the 1-

D wave propagation."2E" is used as a jargon toexpress motions on

outcropped bedrock, aswill be discussed later.

z=0


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(3) Free surface condition

The simplest, but most important, boundary conditionis free surface condition of the ground.

Let z=0 ground surface, free surface condition is givenby,

)8(0)0z(z

uG)0z( ==

==

From Eq.(7), shear stress is expressed by,

)9(VztF

VztE

V1

z)z,t(u

SSS

+=

&&

where dot express first derivative with respect to time.


11/42

From Eqs.(8) and (9), assuming initial condition is zero,

( ) ( ) ( ) ( ) )10(tFtEtFtE == &&

Then, Eq.(7) yields

)11(V

ztE

V

ztE)z,t(u

SS

+

+=

On ground surface, putting z=0 into Eq.(11), one canobtain,

( ) ( ))12()t(E2tEtE)0,t(u =+=

Ground motion on surface is twice as verticallyincident wave.


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This is a case where S-wave vertically impinges at thebottom of homogeneous half-space. Maximum amplitudeof upward wave is 0.5. Maximum amplitude of the groundmotion on surface becomes 1.0, as we learned.


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Consider the wavepropagation in twosemi-infinite media

as shown in thefigure.

When the wave E2(t+z/VS2) propagates upward and

impinges on the interface (z=0) between two media, theincident wave is transformed into the transmission waveE1(t+z/VS1) in medium 1 and the reflection wave F2(t-z/VS2)

in medium 2. Notice that downward wave does not exist inmedium 1.

(4) Transmission and Reflection at Interface

z

: Transmission Wave

: Incident Wave : Reflection wave

G1 1

G2 2

Medium 1

Medium 2z=0

)V/zt(E 1S1 +

)V/zt(E 2S2 + )V/zt(F 2S2


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)13(VztE)z,t(u

1S

11

+=

The displacement and shear stress in medium 1:

z 2,VS2

1(z,t) u1(z,t)E1(t+z/VS1)

1,VS1Medium 1

Medium 2

)14(V

ztE

V

G

V

ztE

z

G

z

)z,t(uG)z,t(

1S

1

1S

1

1S

11

111

+=

+

=

=

&

1S1

1S

2

1S1

1S

1 V

V

V

V

G=

=

Considering that

Eq.(14) yields

)15(V

ztEV)z,t(

1S

11S11

+= &


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Similarly, for medium 2:

z

2,VS2

2(z,t) u2(z,t)

E2(t+z/VS2) F2(t-z/VS2)

1,VS1Medium 1

Medium 2

)16(VztF

VztE)z,t(u

2S

2

2S

22

+

+=

)17(V

z

tFVV

z

tEV)z,t( 2S22S2

2S22S22

+=&&


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At the interface(z=0), displacement and stress must beconsistent, resulting in following boundary consition;

z

2,VS2

2(z=0,t) u2(z,t)

E2(t+z/VS2) F2(t-z/VS2)

1,VS1E1(t+z/VS1)

1(z=0,t) u1(z,t)

Medium 1

Medium 2

)18()0,t(u)0,t(u 21 =

)19()0,t()0,t( 21 =

( ) ( ) ( ) )20(tEtFtE 122 =+

Substituting Eqs.(13), (15) ,(16) and (17) into Eqs.(18) and(19), following relationships

can be obtained;

( ) ( ) ( )tEtFtE 122&&&

=( ) ( ) ( ) )21(tEtFtE 122 =

2S2

1S1

V

V

=where is impedance ratio


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Amplitudes of transmission and reflection waves can beexpressed in terms of incident wave.

( ) ( ) )22(tE1

2tE 21 +

=

( ) ( ) )23(tE1

1tF 22 +

=

from (20)+(21)

from(20)-(21)

Define the transmission Tand reflection coefficient Ras:

Transmission coefficient :

Reflection coefficient )25(1

1R

)24(1

2T

+

=

+=


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Using Tand R, the transmission wave E1(t+z/VS1) andthe reflection wave F2(t-z/VS2) are given by:

)27()V

zt(E)(R)

V

zt(F

)26()Vzt(E)(T)

Vzt(E

2S

2

2S

2

2S

2

1S

1

+=

+=+

z

2,VS2

E2(t+z/VS2)

F2(t-z/VS2) =R() E2(t+z/VS2)

1,VS1E1(t+z/VS1)=T() E2(t+z/VS2)

=1VS1/2VS2

Medium 1

Medium 2

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 0.5 1 1.5 2 2.5 3

T R

stiffsoft softstiff


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This is a case where S-wave vertically impinges at thebottom of a layer built on its half-space. Impedance ratio

is 0.5. Maximum amplitude of upward S wave is 0.5.


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QUESTIONS

Q1. Find max. amplitude of transmitted wave in surface layer.Q2. Find max. amplitude of reflected wave in bottom layer.

Q3. Find max. amplitude of ground motion on surface.

Q4. Find max. amplitude of transmitted wave to the lowerhalf-space when the reflected wave from ground surfaceimpinges on boundary.

Q5. Find max. amplitude of reflected wave to the surfacelayer when the reflected wave from ground surface impingeson boundary.


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QUESTIONS

Q1. Find max. amplitude of transmitted wave in surface layer.Q2. Find max. amplitude of reflected wave in bottom layer.

Q3. Find max. amplitude of ground motion on surface.

Q4. Find max. amplitude of transmitted wave to the lowerhalf-space when the reflected wave from ground surfaceimpinges on boundary.

Q5. Find max. amplitude of reflected wave to the surfacelayer when the reflected wave from ground surface impingeson boundary.

3

1

5.01

5.01

1

1R

3

4

5.01

2

1

2T

1

1

=+

=+

=

=+

=+

=

3

42T5.0:3Q

6

1R5.0:2Q

3

2T5.0:1Q

1

1

1

=

=

=

3

1

21

21

1

1R

3

2

21

2

1

2T

2

2

=+

=+

=

=+

=+

=

9

2

)3

1

(3

2

RT5.0:5Q

9

4

3

2

3

2

TT5.0:4Q

21

21

==

==


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(5) General solution in Frequency Domain

The equation of the motion is given by Eq.(6):

Putting:

in which denotes circular frequency(rad./sec).

)6(z

uV

t

u2

22

S2

2

=

)28(e)z(U)t,z(u ti=

Substituting Eq.(28) into Eq.(6) and deleting exponentialterm, following Helmholtz equation can be obtained;

)29(0z

)z(UV)z(U2

22

S

2 =

)30(0)z(UVz

)z(U

2S

2

2

2

=

+


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where,

SV

k

= is wave number(rad.s/m) and transformed as,

SSSS

2

TV

2

V

f2

Vk

=

=

=

=

where, f is frequency(Hz), S is wave length(m).

The solution of Eq.(30) :

is given by:

where Eand Fare arbitrary constants.

)30(0)z(UVz

)z(U2

S

2

2

2

=+

)31(FeEe)z(Uz

Viz

Vi

SS

+=


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Substituting Eq.(31) into Eq.(28):

the displacement u(z,t) can be expressed by:

The first term indicates the wave propagating in thenegative zdirection, while the second term in the positivez direction. Eand Fexpress the amplitude of the waves.

Hereafter, the time term eit is not written.

)32(FeEe)t,z(u SSV

zti

V

zti

+

+=

The shear stress T(z) in frequency domain is expressed by:

)33(FeEeVi

FeEeV

i

dz

)z(dU

)z(T

zV

izV

i

S

zV

izV

i

S

SS

SS

=

==

)28(e)z(U)t,z(u ti=


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z

x

E

F

As we already discussed,

characters E & F are usedto express the "upward &downward" waves in the 1-D wave propagation.

It is easy to understandconsidering form of Eq.(32).

z=0

)32(FeEe)t,z(u SSV

ztiV

zti

+ +=

Free surface condition E=F and transmission andreflection at interface between two media are similar tothe time domain case, as previously discussed.


26/42

Substituting free surface condition E=F into Eq.(32)

yields, considering Euler's formula ,

)34(z

2cosE2zV

f2cosE2z

VcosE2

eeEEeEe)z(U

SSS

zV

izV

izV

izV

iSSSS

=

=

=

+=+=

Again, S is S-wave length(m).At z=S/4, amplitude is equal to zero. This is the firstnodal point for harmonic incident S-wave. Nodal pointsunder surface can be also seen at,

)35(),3,2,1,0n()1n2(

4

z S L=+

=

=

sinicosei


27/42


28/42

We consider the SH wave propagation in a surfacestratum on the engineering bedrock, when the SH wavewith amplitude E

O

incidents.

(6) Amplification of a surface stratum

on the engineering bedrock

zEO(t): Incident Wave

G1 1 V1

G2 2 V2

G.L

H

Surface Stratum

Engineering Bedrock

1S11 V,,G

2S22 V,,G


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Consider the wave propagation in two layered strata asshown, when the SH wave with amplitude Eo incidents onthe interface between the surface stratum and theengineering bedrock.

z1

z2

H

E1 exp(i k1t)

F1 exp(-i k1t)

Eo exp(i k2t) : Incident Wave

F2 exp(-i k2t)

G.L

G1

1

1

G2 2 2

Surface Stratum

Engineering Bedrock

1S11 V,,G

2S22 V,,G

)tV

iexp(F1S

1

)tV

iexp(E1S

1

)tV

iexp(E

2S

0

)tV

iexp(F2S

2


30/42

The displacement and shear stress in both strata are givenby:

(1) for the surface stratum:

(2) for the engineering bedrock:

The subscripts 1 and 2 indicate the surface stratum and

the engineering bedrock, respectively. (except E0)

)36(eFeE)z(U1

1S1

1S

zV

i

1

zV

i

111

+=

)37(eFeEVi)z(T1

1S1

1S

zV

i

1

zV

i

11S111

=

)38(eFeE)z(U2

2S2

2S

zV

i

2

zV

i

022

+=

)39(eFeEVi)z(T

22S

22S

zV

i

2

zV

i

02S222

=


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Therefore, the displacement and the

shear stress in the surface stratumare:

The shear stress at the groundsurface (z1=0) becomes zero.

G.L

Surface Stratum

Engineering Bedrock

Z1

Z2

1(z1=0) = 0

( ) 0FEVi)0z(T 111S111 ===

)40(FE11

=

)41(eeEeEeE)z(U1

1S

1

1S

1

1S

1

1S

z

V

iz

V

i

1

z

V

i

1

z

V

i

111

+=+=

)42(eeEVieEeEVi)z(T1

1S1

1S1

1S1

1S

zV

izV

i

11S1

zV

i

1

zV

i

11S111

=

=

G LThe boundary conditions are


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G.L

Surface Stratum

Engineering Bedrock

Z1

Z2

2(z2=0) = 1(z1=H)

u2(z2=0) = u1(z1=H)

H

The boundary conditions aregiven at the interface between

the surface stratum and theengineering bedrock.

Substituting Eq.(41) and Eq.(42)into Eq.(43):

Putting Eq.(41) and Eq.(42) intoEq.(44):

)43()Hz(U)0z(U 1122 ===

)44()Hz(T)0z(T 1122 ===

)45(eeEFE

HV

iHV

i

120

1S1S

+=+

( ) )46(eeEViFEViH

ViH

Vi

11S1202S21S1S

=

)47(eeEFEH

ViH

Vi

1201S1S

=

2S2

1S1V

V

=where is impedance ratio

HiHi


33/42

Eq.(45) :

plus Eq.(47) :

leads to,

)47(eeEFE HViHVi1201S1S

=

++=

H

V

iH

V

i

101S1S e)1(e)1(EE2

)45(eeEFEH

ViH

Vi

1201S1S

+=+

)48(

)ee()ee(

E2

e)1(e)1(

E2E

H

V

iH

V

iH

V

iH

V

i

0

H

V

iH

V

i

01

1S1S1S1S1S1S

++

=

++

=

Similarly, Eq.(45) minus Eq.(47) yields :

)49(E

)ee()ee(

)ee()ee(F 0

HV

iHV

iHV

iHV

i

HViHViHViHVi

2

1S1S1S1S

1S1S1S1S

++

+++=


34/42

The displacement US on ground surface is

G.L

Surface Stratum

Engineering Bedrock

Z1

Z2

US = u1(z1=0)

H

EO

)50(

)ee()ee(

E4E2)0z(UU

HV

iHV

iHV

iHV

i

0

111S

1S1S1S1S

++

====

= sinicose iPutting Euler's formula

into Eq.(50) yields;

)51(

)HV

sin(i)HV

cos(

E2U

1S1S

0S

+

=

Ratio of US to 2E0 is expressed by;


35/42

EO

2EO

Engineering Bedrock

Outcrop

of Engineering Bedrock

2EO is the displacement of the

engineering bedrock, when thesurface stratum is removed and theengineering bedrock is in outcrop.

Absolute value of transfer function US/(2EO) is:

)53(

)HV(sin)HV(cos

1

E2

U

1S

22

1S

20

S

+

=

)52()H

Vsin(i)H

Vcos(

1

E2

U

1S1S

0

S

+=

Ratio of US to 2E0 is expressed by;

This expression is known as

transfer function.


36/42

Similarly, the displacementat the interface is given by:

G.L

Surface Stratum

Engineering Bedrock

Z1

Z2

UB = u2(z2=0)

= u1(z1=H)

H

EO

Absolute value of UB/(2EO) is:

)54(

)HV

tan(i1

E2

)HV

sin(i)HV

cos(

)HV

cos(E2

EE)0z(UU

1S

0

1S1S

1S

0

2022B

+

=

+

=

+===

)55(

)H

V

(tan1

1

E2

U

1S

220

B

+

=

G

US


37/42

G.L

Surface Stratum

Engineering Bedrock

Z1

Z2

UB

H

EO

The ratio of US to UB is :

)56(

)HV

cos(

1

)H

V

sin(i)H

V

cos(

)HV

sin(i)HV

cos(

)H

V

cos(

1

)HV

sin(i)HV

cos(

)HV

tan(i1

U

U

1S

1S1S

1S1S

1S

1S1S

1S

B

S

=

+

+

=

+

+=

Notice that transfer function US/UB is determined only by

VS1,H, and irrelevant with soil properties in engineeringbedrock.


38/42

0

1

2

3

4

5

6

0 1 2 3 4

=0.2

0.4

0.6

0.4

0.2

0.6

H/V1

x(/2)

Dotted Line : Abs.US/(2EO)

Abs.UB/(2EO)

Solid Line : Abs.US/(2EO)

Abs.US/(2EO)

Abs.(US/2EO) and Abs.(UB/2EO)

Th fi l


39/42

0

1

2

3

4

5

6

0 1 2 3 4

=0.2

0.4

0.6

0.4

0.2

0.6

H/V1

x(/2)


Abs.UB/(2EO)


Abs.US/(2EO)

1 2

The second naturalcircular frequency:

2=3 1

The first naturalcircular frequency:

1H/VS1=/2 1=(/2)(VS1/H)

Th fi l i d T


40/42

0

1

2

3

4

5

6

0 1 2 3 4

=0.2

0.4

0.6

0.4

0.2

0.6

H/V1

x(/2)


Abs.UB/(2EO)


Abs.US/(2EO)

1 2The second natural period T2 and frequency f2:

T2= T1/3 f2= 3f1

The first natural period T1and frequency f1:

T1=2/ 1=(4H)/ VS1f1=1/ T1=VS1/(4H)

These formulas arefrequently used and are

good to memorized.

I E (53) Ab l t


41/42

0

1

2

3

4

5

6

0 1 2 3 4

=0.2

0.4

0.6

0.4

0.2

0.6

H/V1

x(/2)


Abs.UB/(2EO)


Abs.US/(2EO)

1 2

max. Us/2Eo

The amplitude ratio atthe resonant frequency

is given by,

1/When the impedance ratio is smaller, theamplification at the natural

frequency becomes larger.

That is, when the surface stratum is much softer than theengineering bedrock, the amplification becomes much

larger.

In Eq.(53), Absolutevalue of transfer

function US/(2EO),


42/42

E N DAll animation files are available at

http://www.rs.noda.tus.ac.jp/nagano-l/ssi.html

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