3. harmonic ((, , y gp, s, rayleigh & love)...
TRANSCRIPT
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3. HARMONIC
(P, S, Rayleigh & LOVE)( , , y g )
WAVESWAVES
George BouckovalasP f f N T U AProfessor of N.T.U.A.
October, 2016
The Seismic motion is due to …….
Pressure (P), Shear (S), Rayleigh (R) and other wave typespropagating through the ground (soil & bedrock) ...GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.1
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3.1 ELASTIC WAVESELASTIC WAVES ((P & S)P & S)in UNIFORM MEDIAin UNIFORM MEDIA
P WAVE S (SH SV) WAVE
in UNIFORM MEDIAin UNIFORM MEDIA
P- WAVEwith planar front
S (SH, SV) - WAVEwith planar front
ww
SV
SH
v v
u u
Compression - extension
P-waves
Wave length
S-wavesshearing
S waves
Wave lengthGEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.2
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Wave equation dx
Wave equation . . .A
x dxx
2
x dxxx
u¶2
2
tu
x ux x
uΌμως σ Dε D
x
¶= =-
¶x, u
2 2x
2 2
σ u uκαι D ρ
x x t
¶ ¶ ¶=- =-¶ ¶ ¶x x t¶ ¶ ¶
DC
General solutionGeneral solution...
( )( )u f Ct x=
why;
Y Ct x=
( ) ( ) ( )22
2 2f Yu ¶¶
( ) ( ) ( )2 2
2 2
f YuC C f ''
t Y
¶¶ = =¶ ¶
¶ ¶2 22u u
άρα πράγματι
2 2
2 2
u f (Y )ενώ f ''
¶ ¶= =
¶ ¶=¶ ¶
22 2
u uC
t x
2 2f
x Y¶ ¶GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.3
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Physical meaning...Physical meaning...
The problem variables are NOT two independent The problem variables are NOT two independent ones (‘x’ and ‘t’) but a combined one: Y= x±Ct
If Y=Ct-x If Y=Ct+x
then, t=0 Y= -xo
t≠0 Y=Ct-xt=-xo
then, t=0 Y= xo
t≠0 Y=Ct+xt=xot 0 Y Ct xt xo& xt = xo + Ct
f(Ct-x) f(Ct+x)
t 0 Y Ct xt xo& xt = xo - Ct
tt=0
f(Ct x)
o)
t t=0
( )
f(-x
o
xo+Ctxo x xo-Ct xo x
P & S – wave propagation velocities& p p gin soil & rock formations
EGS- waves: 12
EG,GVs
P- waves:
21
1G2
211
1ED,DVp
21211
( / )VS (m/s) VP (m/s)dry saturated
loose-recent SOIL deposits <400 <1000 ≈1500
stiff SOILS – soft ROCKS 400-800 800-1600 1500-2000
Rocks >800 >1600 >2000GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.4
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Vibration velocity (at a point) VVibration velocity (at a point) V
f ( Ct )u f ( x Ct )
* *u u X u x XV u
* *V u
t t x tX X
xV C x
xά V C CD
D
Vibration velocity (at a point) V
f ( Ct )
Vibration velocity (at a point) V
u f ( x Ct ) ATTENTION:
* *u u X u x XV u
The velocity of VIBRATION is different
(2 to 3 orders of magnitude lower!)* *
V ut t x tX X
than the velocity of PROPAGATION
xV C x
xά V C CD
DGEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.5
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Special Case: Harmonic wavesp m
( )( ) ( )
ωi Ct x i ωt kxCf Ct x Ae Ae
+ ++ = =( )af Ct x Ae Ae+ = =
( ) ( ) ( )ω
i Ct x i ωt kxCbf Ct x Ae Ae
- -- = =( )bf Ct x Ae Ae
where
k /C 2 /(Τ C) 2 /λ bk =ω/C = 2π/(Τ•C) = 2π/λ = wave number
Ground VIBRATION at a given POINT (i e x=x )
titiikxf * Harmonic vibration
(i.e. x=xο)
tio
tiikxba exueAef o *, with frequency
ω=2π/Τ
fa,b
T=2π/ω
tku*(xo) =Ae+ikxo
fa,b u*(xo) =Ae+ikxo
xxo
GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.6
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Ground DISPLACEMENT at a given INSTANT OF TIME (i e t=t )at a given INSTANT OF TIME (i.e. t=tο)
ikxikxti etueAef o * oba etueAef o ,
Όπως προηγουμένως αλλά τώρα τον ρόλο As before, but now the role of t ς ρ γ μ ς ρ ρτης συχνότητας τον παίζει ο κυματικός
αριθμός k
,frequency is played by
the wave number kΤ=2π/ωx=xkλ=2π => λ=2π/k
Τ=2π/ω
to
x=xo
[k= ω/C λ = 2π/ωC = T·C]u(xo,t)
xo
u(x) t=to
λx
EXAMPLE: Two rods in contact
x k kxti
oeuu 1
xkti2 eu
kxtiB kxtiBeu 1
12ρ C
ρ1,C1
ρ2,C2
i t kx i t kx i t kxou e u u e Be 2 1
x xu uM M M Mx x
2 12 1
i t kx i t kx i t kxoM ke Mu ke MBke GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.7
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EXAMPLE: Two rods in contact
Boundary conditions:
x=0 u1=u2 d l & x=0 u1=u2
σ1=σ2
displacement & stress compatibility
tititioo MBkekeMuekM o21 MBkekeMuekM
22
o C
C
M
M
k
kBu
h k /C M C2 d kΜ C
11CMk
where k=ω/C, M=C2ρ and kΜ=ωCρ
EXAMPLE: Two rods in contact
Buouou 22C1
o21
C
Buouou o
22
11 u
C
C1
C1
B
11
22o C
CBu 11C
S i l
o22
u
C
C1
2
Special cases:
F d C 0 B & Γ 2
11C
Free end: ρ2C2=0 B=uo & Γ=2uo
Fixed end: ρ2C2=∞ B=-u & Γ=0Fixed end: ρ2C2 B uo & Γ 0
GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.8
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3.2 ELASTIC WAVES IN nonELASTIC WAVES IN non--UNIFORM UNIFORM MEDIA ( ith i t f ) MEDIA ( ith i t f ) MEDIA (with interfaces) MEDIA (with interfaces)
SNELL’s LAW:SNELL’s LAW: 2121
C
sin
C
sin
C
sin
C
sin
2121 sspp CCCC
C1 > C2
VIBRATION AMPLITUDE of reflected & transmitted refracted wavesof reflected & transmitted- refracted waves
I th l d k im l f 1 D ti :
medium 1: ρ1 C1 medium 2: ρ2 C2
In the already known, simple case of 1-D wave propagation:
transmitted incident wave
medium 1: ρ1, C1 medium 2: ρ2, C2
wave
reflected wave Cwave 2 2
1 1. .
2 2
1
1refl inc
CC
u uC
Vibration amplitude u:
2 2
1 1
.
1
1 refl
C
uu u
. .
.
1
2
ftransm inc
inc
u uu
.2 2
1 11
incuCC
GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.9
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VIBRATION AMPLITUDE of reflected & transmitted refracted waves
The amplitude of vibration of the reflected and refracted of reflected & transmitted- refracted waves
pwaves in 2-D problems, is computed based on stress equilibrium and strain compatibility (continuity) at the interface. In this case, the amplitude is also a function of the angle of wave incidence.
EXAMPLE: SH waves (Richter 1958)
2 2
. 1 1
cos1
cosrefl
Cu C
θδ
2 2.
1 1
cos1
cosinc
CuC
2
1 1
. . 21refr reflu u
C
θπρ θαν
1
2 2. .
1 1
cos1
cosinc inc
Cu uC
C1>C2SHinc
Criticalrefraction angle: 121 C
sinsinsinsin
refraction angle:1
221
21
1C
i1i
Csinsin
CC
2
2
121
Cii
1C
sin1sin
21
1
2cr2
C
Csinsin
1
21cr2 C
Csin
C1>C2What happens for θ2 > θcr ?
Application:Application:
Seismic refraction method for geophysical exploration geophysical exploration ……….
GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.10
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3.3 SURFACE WAVESSURFACE WAVESWave length
αδιατάραχτο μέσοαδιατάραχτο μέσο
P-wave+
SV wave RayleighRayleighSV-wave+
FREE C/CS
Rayleighwave
SURFACECR=0.94CS
Ground displacements due to R-waves
( ) ( )RR Rz z2 5.8 i ωt k xλ λu u z x t iB 0 55 e e e
- - -ì üï ïï ï= = -í ý( )
( ) ( )RR Rz z2 5.8 i ωt k xλ λ
u u z ,x ,t iB 0.55 e e e
w w z x t B 1 66e 0 92e e- - -
= = í ýï ïï ïî þì üï ïï ïí ý( ) ( )RR Rw w z ,x ,t B 1.66e 0.92e e= = -í ýï ïï ïî þ
k / b w/wo & u/uo
/
kR=ω/CR wave numberλR=CRT wave lengthΒ = constant u/uoΒ = constant
λR z/λR
w/wo
R
u
w|w(0)||w(0)|
|u(0)|=1.6
GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.11
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LOVE WAVES (in layered soil profile)( y p )
They are SH waves “trapped” within the upper soft layer
of a non-uniform soil profile.of a non uniform soil profile.
CS11 S11
CS22
LOVE WAVES (in layered soil profile)( y p )
They are SH waves “trapped” within the upper soft layer
of a non-uniform soil profile.of a non uniform soil profile.
Ground displacement variation with depth
CS11 S11
CS22
GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.12
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( )1 L 1 L Lir k z ir k z i( ωt k x )1u Ae Be e- -= +
C
L
( )2 L Lir k z i( ωt k x )
2u Γe e- -=CS1
C
L Lk ω / C=
CS2
2 2L L
1 22 2
C Cr 1, r 1
C C= - = -
2 2s1 s2C C
( )1 L 1 L Lir k z ir k z i( ωt k x )1u Ae Be e- -= +
C( )2 L Lir k z i( ωt k x )
2u Γe e- -=CS1
C
1( ) : 0z H ύ ά
CS2
1 2 1 20 ( ) : ,z ά u u
1 1 0
0
ir kH ir kHAe Be substituting for
A B
1 1 1 1 2 2
0 , ,
0 :
A B
A G r B G r G r gives
2LC1 «Dispersion»
relationship
L22S 2L 2
2 2S1 1
1CC GωH
tan 1C C G C
-é ùê ú- =-ê úê úë û relationshipL S1 1 L
2S1
C C G C1
Cê úë û -
GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.13
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Observe that, since tan(*) must be a real number, the following condition must be satisfied for the previous following condition must be satisfied for the previous equation to have a real solution:
C C CCS1 < CLOVE < CS2
CCs2
2LC1
C
L22S 2L 2
2 2
1CC GωH
tan 1C C G C
-é ùê ú- =-ê úê úë ûCs1
0
L S1 1 L2S1
C C G C1
Cê úë û -
0 ΩΗ/CL
In addition, LOVE waves are not possible unless the surface layer is softer than the underlying mediumy y g(i.e. CS1 < CS2).
Dispersion: Code name for LOVE (and other) waves h h h b f d d l which exhibit a frequency dependent propagation velocity
(even for a uniform medium).
Implications of dispersion:
The propagation velocity CLOVE decreases with increasing frequency ω (decreasing period Τ).q y g pAs a result….. the low-frequency components of the excitation get separated from the high-frequency excitation get separated from the high-frequency components, as they propagate faster. Thus:
Th d ti f h ki i ith di t f The duration of shaking increases with distance from the source
Th f t t f th ti ( th l ti The frequency content of the motion (e.g. the elastic response spectrum) changes with distance from the sourcesource
GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.14
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Special Case:Soft soil layer upon BEDROCK (i.e. CS2=∞)
CLOVE
CSH2122
LOVE
H
HC
bedrock
2
sC
bedrock
Fi ll LOVE
CLOVE
Finally, LOVE waves cannot exist for f i l th th frequencies lower than the fundamental vibration frequency of the soft soil
CS
frequency of the soft soil layer, i.e. when
Critical frequency
π/2 ωΗ/CS
ω < ωC= (π/2) Cs/HCritical frequencyωc= (π/2) CS/H
GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.15
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PROBLEM SOLVING FOR CHAPTER 3PROBLEM SOLVING FOR CHAPTER 3PROBLEM SOLVING FOR CHAPTER 3PROBLEM SOLVING FOR CHAPTER 3
HWK 3.1:HWK 3.1:The free end of an infinite rod is displaced according to the The free end of an infinite rod is displaced according to the following relationship:
U=0 for t<0U=t (cm) for 0<t<0.1sU=0 2 t for 0 1s<t<0 2sU=0.2-t for 0.1s<t<0.2sU=0 for 0.2s<t
Find the displacement variation along the rod at t=0.3s. Assume that the wave propagation velocity is 300m/s.
(aπό «Σημειώσεις Εδαφοδυναμικής» Γ. Γκαζέτα)
GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.16
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HWK 3 2:HWK 3 2:HWK 3.2:HWK 3.2:
To isolate a precision measurement laboratory (e.g. a geotech lab.) from traffic vibrations it is proposed to Α traffic vibrations, it is proposed to construct the trench ΑΑ’. Which of the following two fill materials is Lab
Α
gpreferable :(α) concrete, with ρ=2.5 Mgr/m3 &
Lab
C=2000 m/s, or(b) pumice, with ρ=0.8 Μgr/m3
groundρ=2.0 Mgr/m3
C=100 m/s ?ρ gC=500 m/s
Α’
HWK 3.3:HWK 3.3:D h H h h h h l f l Draw the SH wave propagation path through the soil profile shown below, and compute the horizontal acceleration applied t h il lto each soil layer.
± 0mCs=300m/s ρ=1.6Mgr/m3
- 20mCs=600m/s ρ=1.8Mgr/m3
- 40mCs=1200m/s ρ=2.2Mgr/m3
- 60m
Cs=2400m/s ρ=2.4Mgr/m3
SH50o
SH
(uo=1.6cm, T=0.40s)GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.17
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HWK 3.4:HWK 3.4:
Draw the reflected and refracted waves
(type and propagation di ti ) direction)
in the special cases h n in th fi shown in the figure .
.
HWKHWK 3 53 5HWKHWK 3.53.5::To get a feeling of the depth (from the free ground surface) which is affected by R-waves,
1) Compute the normalized depth z/λR where ground displacements are reduced to 10% of the value at the free ground surface.
2) What is the value of the above critical depth (range of variation) in the case of soft soil, stiff soil-soft rock and rock;rock;
(Aπό «Σημειώσεις Εδαφοδυναμικής» Γ. Γκαζέτα)
GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.18