appendix a and b in geotechnical earthquake engineering

8/18/2019 Appendix a and B in Geotechnical Earthquake Engineering

1/35

ReviewAppendix A and B in Geotechnical

Earthquake Engineering Book

By:

Yuamar Imarrazan Baarah


2/35

G!A"# o$ thi "eon

%& 'now the type o$ vi(ratory motion)& Review *athematical olution o$ imple vi(ration pro(lem

!utline

%& +ype o$ vi(ratory motion ,eriodic loading : #imple -armonic *otion .onperiodic loading

)& /ourier #erie0& #1!/

2ndamped /ree vi(ration 1amped /ree vi(ration

2ndamped $orced vi(ration 1amped $orced vi(ration

3& Repone #pectrum

Review A,,E.1I4 A5B


3/35

%& +ype o$ 6i(ratory *otion

eriodic Loading:It ha ame ,ERI!1 elaped $or % cycle

Ex: harmonic motion

+

Non-Periodic Loading: Random period 7inconitent interva

Ex: impact loading9 exploion9$alling weight9 Earthquake

+

+ +

Impact loading

Earthquake loading


4/35

-armonic "oading

,eriodic "oading

haracterized (y #I.2#!I1A" motionI*,!R+A.+ propertie:%& Amplitude 7A8

)& ,eriod 7+80& /requency 7$83& .atural $requency 7ω);& ,hae 7Φ)

) type o$ notation:%& +rigonometry)& omplex


5/35

+rigonometry $or #imple-armonic "oading

I*,!R+A.+ propertie:%& Amplitude 7A8)& ,eriod 7+80& /requency 7$8

3& .atural $requency 7ω);& ,hae 7Φ)

A

A

+

+


6/35

#imple -armonic "oading

/or % #inuoidal:

/or ) #inuoidal:

= (


7/35

omplex .otation $or #imple -armonic *otion

.ot all dynamic pro(lem9 olved eaily

uing trigonometric .otationAnother impler notation : omplex

.otation It derived directly $rom trigonometric

notation uing Euler> law:

7imaginary num(er8

Real part :

Imaginary part :

Euler> law:

= (


8/35

!ther *eaure o$ *otion

,arameter to decri(e vi(ratory

motion:

1iplacement

6elocity

Acceleration

+rigonometric omplex


9/35

.on5,eriodic "oading ,eriodic "oading

2ing arti?cial @uiet one C

#uperpoition i ued to um therepone o$ ytem to a erie o$imple harmonic loading

Earthquake loading can alo (ede(ri(ed a ,eriodic loading in term o$$ourier erie


10/35

/ourier #erie

1eveloped (y /rench mathematician9 J.B.J. Fourier

,eriodic $unction D a the #2* o$ a erie o$imple harmonic loading

1ierent amplitude9 $requency9 and phae

"oading +ime -ytory

/ourier repreentationo$ loading time

hytorya a um o$ harmonic

alculation o$ +he repone o$ each load/or a #1!/ ytem

Repone o$ each load/or a #1!/ ytem

#ummation o$ harmonicRepone to create complex time hytory o$ repo


11/35


12/35

Example ,age ;0


13/35

+he $ourier erie o$ the quare wave $unction:

5%&; 5% 5&; &; % %&;

5%&;

5%

5&;

&;

%

%&;

Square wave function

#quare wave

$unction

t(Tf)

x(t)

+he ,ro(lem

+he /ourier #erie


14/35

/ourier #erie :)& Exponential $orm

+he a(ove expreion can (e written a:

J mean complex num(er Becaue ω-n = -Kn9 then the expreion can (e compacted into:


15/35

/ourier #erie :0& 1icrete /ourier +ran$orm 71/+8

Ground motion parameter are decri(ed (y a ?nite num(er o$ data +hu9 $ourier coeFcient are o(tained (y ummation rather than integration

+hat ummation D +he 1icrete /ourier +ran$orm 71/+8

2ing Euler> law9 the expreion can (e written a:

+he 1/+ can (e inverted a Invere 1icrete /ourier +ran$orm 7I1/+8:


16/35

6IBRA+I.G #Y#+E*

1e?nition o$ 1egre o$ /reedom:

The number of INDEPENDENT !"I!#LES needed to de$cribethe motion o$ a ytem

*ode o$ vi(ration $or $oundation:

%8 "ongitudinal 7tranlation a(out

y5axi8

)8 Rocking 7rotation a(out z5axi

08 6ertical 7tranlation a(out z5

axi838 Yawing 7rotation a(out z5axi8

;8 "ateral 7tranlation a(out x5

axi8

L8 ,itching 7rotation a(out x5axi8% D&' tran$*ationa* and rotationa*


17/35

6IBRA+I.G #Y#+E*

*ovement!nly in

6ertical direction7% 1!/ D #1!/8

*ovement in6ertical and rocking

7) 1!/8

m%9 m)9 m0tranlation

70 1!/8

In?nite 1!/ In?nite 1!/

Di$crete $+$tem*a concentrateted at a ?nite.um(er o$ location and have a?nite 1o/

,ontinou$ $+$tem*a i ditri(uted throughout yAnd have a in?nite 1!/


18/35

#1!/ haracteritic o$ #1!/:%8 * connected to a pring 7'8)8 * connected to a dahpot

o$ vicou damping coe&7c8

08 * i u(Mected to externalloading 77t88

$I D inerial $orce$1 D vicou damping$orce/ D elatic pring $orce7t8D external load

Example o$ #1!/

Generalized ytem

/orce acting in a ytem

m D mau>>7t8 D accelerationc D coe$ o$ vicou dampingu>7t8 D velocity' D pring contant

u7t8D diplacement

: ra on o


19/35

: ra on o#upport

/or earthquake engineering9dynamic load reult $rom:

S!.IN/ of /"&0ND not $romE4+ER.A" "!A1 applied at the ma le

#taticcondition

1iplacedcondition

u(7t8 D dipl acement o$ (aeu7t8 D diplacement o$ $rame relative to(aeut7t8 D total diplacement

u(>>=u>>

5

#1!/ 2ndamped /ree 6i(ration 7t8


20/35

#1!/ : 2ndamped /ree 6i(ration N D9 7t8D

olution to thi dierential equation i:

2ndamped natural $req&

% and ) can (e o(tained $rom initial condition o$ diplacement andvelocity 7tD8

!m1*itude remain$ con$tantNo energ+ *o$$ (no dam1ing)

&$ci**ate foreverDon2t exi$t in the rea* eng3 a11*ication

Example 2ndamped /ree 6i(ration 7t8


21/35

Example: 2ndamped /ree 6i(ration N D9 7t8D

#1!/ : 1amped /ree 6i(ration N O 7t8


22/35

#1!/ : 1amped /ree 6i(ration N O9 7t8D

ritical dampingcoeFcient

1amping

ratio

e equation can (e expreed a:

In E engineering9 the tructure are alwvirtually underdamped9 then :

#1!/ : 1amped /ree 6i(ration N O 7t8


23/35

#1!/ : 1amped /ree 6i(ration N O9 7t8Dt ω1 (e the damped natural circular $req:

and ) are o(tained $rom initial cond:

+here$ore9

Example: 1amped /ree 6i(ration N 7t8


24/35

Example: 1amped /ree 6i(ration N D9 7t8D

#1!/ : 2ndamped /orced 6i(ration N D 7t8O


25/35

#1!/ : 2ndamped /orced 6i(ration N D9 7t8O

eral olution i +-E #2* o$ the !*,"E*E.+ARY and ,AR+I2"AR olution

-omogeneou olution:

-armonic "oading

,articular olution:7decri(e the repone caued (y external loading8Aume ha$ the $ame form re$1on$e$ a$ harmonic

+uning Ratio: P

Initial condition



26/35


nally9 the equation can (e written a:

) D uo

e cae9 in which the ytem initially at ret in it equili(rium poition 7uoDuo>D

epone due to applied loading 7at $req 8

Repone due to $ree vi(ration eect 7at $req 8

I$ P D % D Q9 the diplacement9 u S

Then harmonic loading applied at o$ #1!/ytem9 the repone goe in?nity 7S8&& In4nit+indicating "ES&N!N,E of the $+$tem3

In rea*it+5 becau$e rea* $+$tem ha$ dam1ing5In4nit+ i$ never reached5 but it can become



27/35


Example: 2ndamped /orced 6i(ration N D 7t8O


28/35

Example: 2ndamped /orced 6i(ration N D9 7t8O

#ee Ex B&%

2(>>7t8Du(>>inωt

#1!/ : 1amped /orced 6i(ration N O 7t8O


29/35

#1!/ : 1amped /orced 6i(ration N O 9 7t8O

e mot general cae : 1amped ytem u(Mected to harmonic loading

A$ter dividing (y m and uing DcQ7)m8 and DkQm9 the equation (ecome:

+he complementary olution repreent damped $ree vi(ration:

t can (e aumed a harmonic particular olution $or damped #1!/9

+he coreponding velocity and acceleration are

(tituting two previou equation into equation o$ motion U grouping the in and

#1!/ : 1amped /orced 6i(ration N O 7t8O


30/35

#1!/ : 1amped /orced 6i(ration N O 9 7t8O

/or in D $or t D and co D %9 thu

+hen9 0 and 3 i olved a:

/or in D % and co D 9 thu

/inally the general olution i o(tained (y com(inaing thecomplementaru and particular olution:

Example: 1amped /orced 6i(ration N


31/35

Example: 1amped /orced 6i(ration N O 9 7t8O


32/35

Repone #pectraition6!7 re$1on$e (!cc5 e*5 Di$1) of a** 1o$$ib*e *inear SD&'

Ground Acc 7E loading8

+ DV D

+ DV D

+ DV D

#1!/ ytem

1e$ormation Repone

1e$ormation Repone #pectrum


33/35

Repone #pectra +he repone will depend on:

• *a 7m8• #tine 7k8• 1amping ratio 7V8

*ethod to contruct Repone #pectru%8 alculated $rom actual time hitori)8 ,#-A

08 Building code

+ype o$ Repone pectrum:%8 1e$ormation 718)8 ,eudo velocity 76808 ,eudo acceleration 7A8

vert $rom 1e$ormation 718 to 6el 768 and Acceleration 7A8

9 69 and A value are the peak repone o$ all poi(le linear #1/ ytem


34/35

Repone #pectra $or El entro ground motion


35/35

+2GA# %

• +uga 6i(ratory *otion:

BraMa *&1a& ,rinciple o$ #oil 1ynamic9 %WW09

-al 353W .o& )&)9 )&09 )&39 )&9 U )&X

• +uga Tave ,ropagation:

#&"& 'ramer& Geotechnichal Earthquake

Engineering9%WWL9 -al %X%5%X0 .o& ;&)9 ;&39 ;&9U ;&W

appendix a and b in geotechnical earthquake engineering

Documents