seismicanalysisofstructuresi-130620030503-phpapp02

8/17/2019 seismicanalysisofstructuresi-130620030503-phpapp02

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Chapters – 1 & 2

Chapter -1

SEISMOO!"


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Introduction It is a big subject and mainly deals with

earthquake as a geological process. However, some portions of seismology are of

great interest to earthquake engineers.

They include causes of earthquake, earthquakewaves, measurement of earthquake, effect of soilcondition on earthquake, earthquake pre- dictionand earthquake haard analysis

!nderstanding of these topics help earthquakeengineers in dealing seismic effects on structures in

a better way."urther knowledge of seismology is helpful in

describing earthquake inputs for structures whereenough recorded data is not available.

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Interiors of earth

184 −− kmstoV p

Lec-1/2

#efore earthquake is looked as a geological

process, some knowledge about the structure of earth is in order.

In-side the earth

Cr%st $-%& km' ( discontinuity' floating

Mantle lithosphere )*+& km' asthenosphere-plastic

molten rock )+&& km' bottom- homogenous'variation of v is less

)*&&& km - +&& km

C#re discovered by ichert /0ldham' only 1 waves can

pass through inner core

)*+& km' very dense' nickel / iron' outer core

)++&& km, same density'+$&&& 2' %3*&4 atm'*% g5cm6

7ithosphere floats as a cluster of plates withdifferent movements in different directions.

"ig *.*

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1late tectonics

8t mid oceanic ridges, two continents which were joined together drifted apart due to

flow of hot mantle upward.

"low takes place because ofconvective circulation ofearth9s mantle' energy comes

from radioactivity inside the earth.

Hot material cools as it comes up' additional crust is formed

which moves outward.

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Convective currents

2oncept of plate tectonics evolved from

continental drift.

"ig *.+

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2ontd...

:ew crust sinks beneath sea surface' spreading

continues until lithosphere reaches deep sea trenches where subduction takes place.

2ontinental motions are associated with a variety of circulation patterns.

8s a result, motions take place through sliding of lithosphere in pieces- called tectonic plates.

There are seven such major tectonic plates and many smaller ones.

They move in different directions at different speeds.

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2ontd...Lec-1/5

"ig *.6

(ajor tectonic plates

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Three types of Inter plate interactions e3ist giving

three types of boundaries.

2ontd...Lec-1/6

Tectonic plates pass each other at the transform faults.

"ig *.%Types of interplate boundaries

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"aults at the plate boundaries are the likely

locations for earthquakes - inter plate earthquake.

;arthquakes occurring within the plate are caused due to mutual slip of rock bed

releasing energy- intra plate earthquake.


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2ontd...Lec-1/8

Types of fault "ig *.$

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2auses of earthquake There are many theories to e3plain causes of

earthquake. 0ut of them, tectonic theory of earthquake is popular.

The tectonic theory stipulates that movements

of tectonic plates relative to each other lead toaccumulation of stresses at the plate boundar-ies / inside the plate.

This accumulation of stresses finally results ininter plate or intra plate earthquakes.

In interplate earthquake the e3isting faultlines are affected while intra-plate earthquake

new faults are created.

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2ontd...

=uring earthquake, slip takes place at the fault'

length over which slip takes place could be several kilometres' earthquake origin is a point that movesalong the fault line.

;lastic rebound theory, put forward by >eid, givescredence to earthquake caused by slip alongfaults.

7arge amplitude shearing displacement that took place over a large length along the


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2ontd...

Fault Line

After earthquake

Direction of motion

Direction of motion

Road

Fault Line

Before Straining

Direction of motion

Direction of motion

Fault Line

Strained (Before earthquake)

Direction of motion

Direction of motion

Road

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"ig *.4

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2ontdA

8n earthquake caused by slip at the fault proceeds in

the following way?

0wing to various slow tectonic activities,strains accumulate at the fault over a longtime.

7arge field of strain reaches limiting value atsome point of time.


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2ontdA

1ropagating wave is comple3/ is responsible

for creating displacement and acceleration ofsoil5rock particle in the ground.

The majority of the waves travels through the rocks within the crust and then passes through the soil to the top surface.

0ther theory of tectonic earthquake stipulates

that the earthquake occurs due to phasechanges of rock mass, accompanied by volume

changes in small volume of crust.

Those who favour this theory argues thatearthquakes do occur at greater depthswhere faults do not e3ist.

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L 2/7


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eflection / refraction of waves take place near the surface at every layer' as a result waves get

modified.

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2ontd... #ody waves are of two types- 1 / < waves'

< waves are also called transverse waves.

aves propagation velocities are given by?

1 waves arrive ahead of < waves at a point' time interval is given by?

1olaried transverse waves are polariation of particl-es either in vertical)


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waves.

7 waves? particles move in horiontal planeperpendicular to the direction of wavepropagation.

> waves?- particles move in vertical plane'they trace a retrogate elliptical path' foroceanic waves water particles undergosimilar elliptical motion in ellipsoid surface

as waves pass by. 7 waves move faster than > waves on

the surface )> wave velocity D&.

2ontd...

SV

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2ontd...Lec-2/10

#ody /


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1/ < waves change phases as 111, 1eflection at the earth surface"ig *.

Seism#l#g$

Lec-2/12


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>ecords of surface waves


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8 long ground motion with prevailing period?

filtered ground motion through soft soil, medium- 7oma 1rieta earthquake.

Fround motion involving large


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2ontd..

0 5 10 15 0 5 #0-0.$

-0.

0

0.

0.$

Time (sec)

A c c e l e r a t i o n ( g )

Acceleration

(i3ed frequency

0 5 10 15 20 25 30

-10

-5

0

5

10

15

Time (sec)

! i s "

l a c e m e n t

( c m )

!is"lacement

"ig *.*+

Seism#l#g$

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2ontd..

! " # $ % & ' &

$

"

"

$

*ime (sec)

D i s p l a c e m e n t ( c m )

!is"lacement

*ime(sec) " $ & ! !" !$ !& !

+%

+$

+#

+"

+!

+!

+"+#

+$+%

+&

+'+

+,!

A c

c e l e r a t i o n ( g )

Acceleration

1redominant frequency

"ig *.*6

Seism#l#g$

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They refer to quantities by which sie / energy

of earthquakes are described.

There are many measurement parameters' some of them are directly measured' some are indirectly derived from the measured ones.

There are many empirical relationships that are developed to relate one parameter to the other.

(any of those empirical relationships and theparameters are used as inputs for seismic

analysis of structures' so they are describedalong with the seismic inputs.

;arthquake measurement parameters

Seism#l#g$

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Here, mainly two most important parameters, magnitude / intensity of earthquake are described along with some terminologies.

2ontd...

(ost of the damaging earthquakes have

E"icentre E"icentral !istance

%&"ocentral !istance'ocal !e"th

'ocus%&"ocentre

Site

7imited region of earth influenced by the focus

is called focal region 'greater the sie ofearthquake, greater isthe focal region.

shallow focal depth GB& km' depths of foci B& km are intermediate5deep.

;arthquake definitions"ig *.*%

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2ontd...

"orce shocks are defined as those which occur

before the main shock.

8fter shocks are those which occur after the main shock.

(agnitude of earthquake is a measure of energy released by the earthquake and has the following

attributes? is independent of place of observation.

is a function of measured ma3imum displace- ments of ground at specified locations.

first developed by aditi / >ichter in *6$.

Seism#l#g$

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2ontd...

magnitude )( scale is open ended.

( E.$ is rare' ( G +.$ is not perceptible.

there are many varieties of magnitude ofearthquake depending upon waves and

quantities being measured.

7ocal magnitude ) , originally proposed by>ichter, is defined as l#g a )ma3imum amplitude

in microns' ood 8nderson seismograph?

>*&& km' magnification? +E&&?

L M

pT = 0.8s :ξ = 0.8

)6.1(log7.248.2log ∆+−= A M L

Seism#l#g$

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ichter because of limitations of instrument /distance problems associated with .

It is obtained from compression 1 waves withperiods in the range of *s' first few cycles are

used'

2ontd...

)7.1(log T ba M L +=

b M

L M

( ) )8.1(,log ∆+

= hQ

T

A M b

Seism#l#g$

2ontdLec-3/6


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0ccasionally, long period instruments are used for periods $s-*$s.

ichter mainly for largeepicentral distance.

However, it may be used for any epicentraldistance / any seismograph can be used.

1raga formulation is used with surface waveperiod of the order of +&s

8 is amp of >ayleigh wave )+&s' is in km.

s M

2ontd...

)9.1(0.2log66.1log +∆+

= T A

M s

∆

Seism#l#g$

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2ontd...

2 3 4 5 6 7 8 9 102

3

4

5

6

7

8

9

M L M s

Ms

MJMAMB

ML

Mb

M ~ M W

Moment Magnitude Mw

M a g n i t u d e

)11.1(0.6log3

210 −= ow M M

"ig *.*$

Seism#l#g$

2ontdLec-3/9


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;nergy >elease, ; ) Moules is given by ?

()B.6 D $& megaton nuclear e3plosion()B.+ releases 6+ times more energy than

()4.+

()E releases *&&& times more energy than()4


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Intensity is a subjective measure of earthquake'

human feeling' effects on structures' damages.

(any Intensity scales e3ist in different parts of the world' some old ones?

Fastaldi


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2ontd...Intensity ;valuation =escription

(agnitude)>ichter


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There have been attempts to relate subjective

intensity with the measured magnitude resulting in several empirical equations?

0ther important earthquake measurementparameters are 1F8, 1FC, 1F=.

1F8 is more common / is related to magnitude by various attenuation laws )described in seismic

inputs.

2ontd...

max1.3 0.6 (1.15)

8.16 1.45 2.46 ln (1.16)

1.44 ( ) (1.17)

s M I

I M r

I M f r

= += + −

= +

Seism#l#g$

(easurement of earthquakeLec-4/1


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(easurement of earthquake

1rinciple of operation is based on the oscillation of a

pendulum.

Sensor : mass; string;magnet &support

Recorder : rum; pen;c!art paper

Amp : optica" / e"ectro-magnetic means

Damp : e"ectromagnetic/

#"ui ampers

"ig *.*4

Seism#l#g$

2ontdLec-4/2


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u

%oriontal "endulum

*ertical "endulum

u

2ontd...

"ig *.*B

Seism#l#g$

2ontd...Lec-4/3


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;quation of motion of the bob is

If T very large )7ong period seismograph

If T very small )short period seismograph

If T very close to / +' very 7arge g T

2ontd...

22 (1.18) x kx w x uν + + = −&& && &&

)19.1(u xor u x ∝−= ν

)20.1(2 u xor u xw ∝−= ν

(1.21) x u or x uν = − µ& && &

Seism#l#g$

2ontd..Lec-4/4


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2ontd..

! %orseshoe +agnet

Sus"ension

Co""er +ass

+irror

,ight eam

copper cylinder+mm 5 +$mm 5

&.Bg

taut wire &.&+ mm

reflection of beam magnified by +E&&

electro - magneticdamping &.E

ood 8nderson


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2ommonly used seismograph measures

earthquake within &.$-6& seconds.


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7ocal


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8nalysis of collected data revealed interesting features of soil modification?

8ttenuation of ground motion through rock bed is significant &.&6g-6$&km )(E.*.

"or very soft soil, predominant period ofground motion changes to soil period' for

rock bed 1F8 &.&6g )8"$.

=uration increases also for soft soil.

0ver a loose sandy soil underlying by

mud, 8"6 for &.&6$g-&.&$g )at rock bed.

The shape of the response spectrumbecomes narrow banded for soft soil.

Seism#l#g$

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8s 1F8 at the rock bed increases, 8" decreases.

"or strong ground shaking, 1F8 amplification is low because of hysteretic behaviour of soil.

8t the crest of narrow rocky ridge, increased

amplification occurs' 8" R +S5 ) theoreticalanalysis .

8t the central region of basin, I= wave propagation analysis is valid' near the sides of the valley, +=

analysis is to be carried out.

*=, += or 6= wave propagation analysis is carried out to find 1F8 amplification theoretically.

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=eterministic Haard 8nalysis )=estricted only when sufficient data is

not available to carry out 1


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2ontdA

)25ln(80.1859.074.6PGA(gals)ln +−+= r m

It consists of following $ steps?

Identification of sources including their geometry.

;valuation of shortest epicentral distance 5 hypocentral distance.

Identification of ma3imum likely magnitude ateach source.


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;3ample *.* ?

(a3imum magnitudes for sources *, + and 6 are B.$, 4.E and $ respectively.

(-50 /5)

Source 1

(-15 -#0)

(-10 /)

(#0 5)

(0 0)

Source #

Source

Site

%ources o# eart!ua'enear t!e site ()*amp. 1.1+

Source m r(km) P-A

! '+% "#+' +$, g

" &+ &+$ +! g

# %+ '+ +!% g

,aar "ee" is 0.49g #or t!e site"ig *.*

Seism#l#g$

2ontdALec-5/1


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1robabilistic seismic haard analysis )1


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recurrencelaw.

)23.1()exp(10 ambmam β α λ −== −

Seism#l#g$

2ontdALec-5/3


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!sing the above recurrence law / specifying

ma3imum / minimum values of M, followingpdf of ( can be derived )ref. book

6rd step consists of the following?

8 predictive relationship is used to obtainseismic parameter of interest )say 1F8 forgiven values of m , r .

)26.1()]([exp1

)]([exp)(

0max

0

mm

mmm f M −−−

−−=β

β β

!ncertainty of the relationship is consideredby assuming 1F8 to be log normally distributed'

the relationship provides the mean value' astandard deviation is specified.

Seism#l#g$

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%th step consists of the following?2ombines uncertainties of location, sie

/ predictive relationship by

8 seismic haard curve is plotted as )say is 1F8 level .

)27.1()()(],|[1

∫∫ ∑ >==

dr dmr f m f r m yY P ! M!

"

!

! y

#

γ λ

y$s yλ y #y including temporal uncertainty of earthquake

)uncertainty of time in 1 = −

Seism#l#g$

2ontdA

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;3ample *.+ ?

"or the site shown in "ig *.+&,show a typical calculation for1


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6rd source ?mn maxr r r = =

0.0

0.$

/ . 0 $

# # .

$ 0 . #

$ 2 . 2

5 # . 0

0 . $

.

/ # . 5

0 . 1

. 0

3 4 6 r 7

E"icentral distance r (8m)

0.0

0.

# . 1 0

$ / . /

5 2 . $

/ 0 . 1

. #

2 # . 2 5

1 0 5 . 5

1 1 / . 0 2

1 .

1 $ 0 . #

3 4 6 r 7


0.0 1 0 0 # 0 $ 0 5 0 0 / 0 0 2 0 1 0 0

3 4 6 r 7


1.0

"ig *.+*

"ig *.+6

"ig *.++

Seism#l#g$

2ontdALec-5/8


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Histogram of ( for each source one are shown

0.0

0.

+agnitude m

0./

0.5

0.$

0.#

0.

0.1

0.

3 4 + 6 m 7

$ . # /

. 1 $

$ . 1 /

$ . 5 0

5 . 1

5 . $ 2

5 .

. 1 5

. $

. 1

0.0

0.

$

. 0 5

$

. 1 5

$

. 5

$

. # 5

$

. $ 5

$

. 5 5

$

. 5

$

. / 5

$

. 5

$

. 2 5

+agnitude m

0./

0.

0.5

0.$

0.#

0.

0.1

3 4 + 6 m 7

0.0

0.

+agnitude m

0./

0.5

0.$

0.#

0.

0.1

0.

3 4 + 6 m 7

$ . #

/ . 1 $

$ . 1 /

$ . 5 0

5 . 1

5 . $ 2

5 .

. 1 5

. $

. 1

"ig *.+%

"ig *.+$

"ig *.+4

Seism#l#g$

2ontd..Lec-5/10


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"or different levels of 1F8, similar values ofcan be obtained.

1lot of vs. 1F8 gives the seismic haard curve.

λ

λ

for other combinations of m / r can obtained / summed up' for source ones + / 6,

similar e3ercise can be done' finally,

0.01g λ

301.0201.0101.001.0

||| sour g sour g sour g g

λ λ λ λ ++=

Seism#l#g$

2ontdALec-5/12

;3ample * 6?


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;3ample-*.6?

The seismic haard curve for a region shows that the annual

rate of e3ceedance of an acceleration &.+$g due toearthquakes )event is &.&+.hat is the prob. that e3actly

one one such event and at least one such event will take

place in 6& yearsU 8lso, find that has a *&V prob. of

e3ceedance

in $& yrs.


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!sing the above Information, seismic risk can

be calculated with the help of either 2ornell9s approach or (ilne / =avenport approach.

!sing the concept, many empirical equations are obtained with the help of data 5 information

for regions.

"or a particular region, these empiricalequations are developed' for other regions, they

may be use by choosing appropriate values for the parameters.


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[ ]

( )

( )( )

1

1

1

1 1

1

1.54

1

( )

1 1 1 ( )

1 1

( )

1

( ) ( / ) (1.30)

exp exp ( ) (1.32 )

l n (1.32 )

47 (1.33)

1( ) | (1.37)

1

1 ( ) (1.38)1 (1.39)

o

s u o

s

o

p

s s

o

!

s

m M

M s o u m M

s M

m M

s

" Y Y c

p m a

T b

P I ! %

% ( m P M m M m M

%

P M m ( m P M m %

β

β

β

α β

α α

−

−

− −

− −

− −

=

= − −=

≥ =

− = ≤ ≤ ≤ = −

≥ = −

≥ = −

Seism#l#g$

(icroonation using haard analysisLec-5/16


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Seism#l#g$

2ontd...Lec-5/17


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Seism#l#g$

2ontd...Lec-5/18


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0.#5 g

0.1 g0.5 g

0.$ g

!eterministic +icroonation

3ro9a9ilit& o: e;ceedance 6 0.1

0.15 g

0.$ g

0.5 g

0. g

0.1 g

0.# g

3ro9a9ilistic +icroonation"ig *.+B

Seism#l#g$

Lec-1/74


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Chapter -2


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Carious forms of


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The most common way to describe ground motion is byway of time history records.

The records may be for displacement, velocity andacceleration' acceleration is generally directlymeasured' others are derived quantities.

>aw measured data is not used as inputs' dataprocessing is needed. It includes

>emoval of noises by filters

#aseline correction

>emoval of instrumental error

2onversion from 8 to = 8t any measuring station, ground motions are

recorded in 6 orthogonal directions' one is vertical.

Seismi( Inp%t

They can be transformed to principal directions' major

2ontd..*56


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They can be transformed to principal directions' majordirection is the direction of wave propagation' the other

two are accordingly defined.


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% ! !% " "% # #% $+#

+"

+!

+!

+"

*ime (sec)

A c c e l e r a t i o n ( g )

+inor (horiontal)

0 5 10 15 0 5 #0 #5 $0-0.#

-0.

-0.1

0

0.1

0.

Time (sec)

AcceleratI

on

(g)

+inor (vertical)

"ig +.*)b

"ig +.*)c

Seismi( Inp%t

#ecause of the comple3 phenomena involved in

2ontd..*5$


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p pthe generation of ground motion, trains of groundmotion recorded at different stations vary spatially.

"or homogeneous field of ground motion, rms 5 peakvalues remain the same at two stations but there isa time lag between the two records.

"or nonhomogeneous field, both time lag / difference in rms e3ist.

#ecause of the spatial variation of ground motion, both rotational / torsional components of ground

motions are generated.

du dvφ( t ) = + ( 2.1)dy dx

dwθ( t ) = ( 2.2)

dxSeismi( Inp%t

2ontd..*54


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In addition, an angle of incidence of ground motion

may also be defined for the time history record.

Ma%o& di&e'tion

1

2

3Angle of incidence

"ig +.+

Seismi( Inp%t

"requency contents of time history "ourier synthesis of time history record provides frequency

*5B


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contents of ground motion.

It provides useful information about the ground motion / alsoforms the input for frequency domain analysis of structure.

"ourier series e3pansion of 3)t can be given as

∑∫

∫

∫

a

0 n n n nn=1

T2

0

!T2

T2

n n

!T2

T2

n n

!T2

n

x( t) = a + a "o#$ t + % #in$ t ( 2.)

1a = x( t ) dt ( 2.')

T

2a = x( t ) "o#$ tdt ( 2.)

T2

% = x( t ) #in$ tdt ( 2.) T

$ = 2*nT ( 2.)

Seismi( Inp%t

Th lit d f th h i t i i b

2ontd..

$

*5E


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The amplitude of the harmonic at is given by

(2.8)

∫

∫

2 T2

2 2 2

n n n n

!T2

2 T2

n

!T2

2, = a + % = x( t ) "o#$ tdt T

2+ x( t ) #in$ tdt

T

n$

÷

n n

!1 nn

n

" = ,

%φ = tan ( 2.10)

a

;quation +.6 can also be represented in the form

∑-

0 n n nn=1

x( t ) = " + " #in($ t + φ ) ( 2.)

Seismi( Inp%t

1lot of cn with is called "ourier 8mplitude


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The integration in ;q. +.E is now efficiently performed by

""T algorithm which treats fourier synthesis problem as apair of fourier integrals in comple3 domain.


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is called :yquest "requency. "ourier amplitude spectrum provides a

good understanding of the characteristics ofground motion.


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The resulting spectrum plotted on log scale shows?

2ontd..*5*+


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8mplitudes tend to be largest at an intermediate

range of frequency.

#ounding frequencies are f c / f ma3.

f c is inversely proportional to duration.

"or frequency domain analysis, frequency contentsgiven by ""T provide a better input.

Frequenc2 (log scale)f c f ma1

5 r d i n a t e

F o u r i e r

a m p l i t u d

e ( l o g s c a l e )

"ig +.%

Seismi( Inp%t

;3ample+.*? 6+ sampled values at Wt &.&+s arei i t t ""T h i "i + $

2ontd..2/1


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given as input to ""T as shown in "ig +.$

YY *5*4 fft)y,6+

9.81

n n*$ = = 1.0 rad# T

2*d$ = = rad#

T

+! +" +# +$ +% +& +'+#

+"

+!

+!

+"

+#

+$

*ime (sec)

- r o u n d A c c e l e r a t i o n ( g )

"ig +.$

Seismi( Inp%t

2ontd..

+#

2/2


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" $ & ! !" !$ !& ! " "" "$ "& " #+"

+!

+!

+"

Frequnec2 (rad4sec)

R e a l p

a r t

A

eal "art

" $ & ! !" !$ !& ! " "" "$ "& " #+"

+!%

+!

+%

+%

+!

+!%

Frequenc2 (rad4sec)

6

m a g i n a r 2 p a r t

A

Imaginar& "art

"ig +.4a

"ig +.4b

Seismi( Inp%t

"ourier amplitude spectrum is 8i Cs plot / phase

t i Z C l t h i "i + B

2ontd..2/3

!ω ω


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2 2 12i i i

!1 ii i n

i

, =( a + % ) i =0....2%

j = tan$ =( 0..dw...w )a

spectrum is Zi Cs plot as shown in "ig +.B!ω

Am"litude s"ectrum

" $ & ! !" !$ !&

+%

+!

+!%

+"

Frequenc2 (rad4sec)

F o u r i e r a m p l i t u d e (

g s e c )

"ig +.Ba

3hase s"ectrum

" $ & ! !" !$ !&!+%

!

+%

+%!

!+%

Frequenc2 (rad4sec)

P h a s e ( r a d )

"ig +.Bb

Seismi( Inp%t

1ower spectral density function 1ower spectral density function )1


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motion is a popular seismic input for probabilisticseismic analysis of structures.

It is defined as the distribution of the e3pected meansquare value of the ground motion with frequency.

;3pected value is a common way of describingprobabilistically a ground motion parameter / isconnected to a stochastic process.

The characteristics of a stochastic process is describedlater in chapter %' one type of stochastic process iscalled ergodic process.

"or an ergodic process, a single time history of theensemble represents the ensemble characteristics 'ensemble r.m.s is equal to that of the time history.

Seismi( Inp%t


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8 close relationship between 1


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amplitude spectrum is evident from ;qn. +.*E.

8 typical 1


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p g pare described using the moments of 1


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The time lag or lack of correlation between e3citations at different

supports is represented by a coherence function / a cross 1ecords of actual strong motion records showthat mean square value of the process is notstationary but evolutionary.

2ontd..2/9

4 s e c " )


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23($t ) = 7( t ) 3( $ ) ( 2.22)

Time)sec

a c c ( m 4

The earthquake process is better modeled asuniformly modulated stationary process in which1


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g q , ,smoothed 1


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8rea under smoothed 1


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There are a number of response spectra used to define ground

motion' displacement, pseudo velocity, absolute acceleration /energy.

The spectra show the frequency contents of ground motion but notdirectly as "ourier spectrum does.

=isplacement spectrum forms the basis for deriving other spectra.

It is defined as the plot of ma3imum displacement of an


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8t the ma3imum value of displacement, @; & / hence,

If this energy were e3pressed as @;, then an equivalentvelocity of the system would be

∫ && n

n 0

vm d

n

t

!8$( t!9)

v 4 d

0 max

$

3x = 3 = ( 2.2'a)$

3 = x(9) / #in$( t! 9) d9 ( 2.2'%)

2

d

1:= ;3 ( 2.2a)

2

&

&

2 2

/7 d

/7 n d

1 1mx = ;3 ( 2.2%)

2 2

x =$ 3 ( 2.2")

Seismi( Inp%t


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2ontd..

This observation shows importance of the spectralacceleration.

3/4


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hile displacement response spectrum is the plot ofma3imum displacement, plots of pseudo velocity andacceleration are not so.

These three response spectra provide directlysome physically meaningful quantities?

=isplacement P (a3imum deformation 1seudo velocity P 1eak


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"or ^ &, it may further easily be shown that

2omparing ;qns.)+.E / )+.6&, it is seen that "ourier spectrum /energy spectrum have similar forms.

"ourier amplitude spectrum may be viewed as a measure of the totalenergy at the end )t T of an undamped


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" $ & ! !" !$ !&

+%

+!

+!%

Frequenc2 (rad4sec)

F o u r i e r a m p l i t u d e

+% ! !+% " "+% # #+% $

+"

+$

+&

+

!

*ime period (Sec) A c c e l e r a t i o n r

e s p o n s e s p e c t r u m ( g )

"ig +.*$

"ig +.*%

Seismi( Inp%t

=-C-8


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8 combined plot of the three spectra is thus desirable / can

be constructed because of the relationship that e3istsbetween them

a 4max T0

lim3 =u ( .)

lim3 =u ( .')

Seismi( Inp%t

3/9


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"ig +.*4

Seismi( Inp%t

3/10


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"ig +.*B

Seismi( Inp%t


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>esponse spectrum of many earthquakes show similartrend when idealised.

2ontd.. 3/12


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This observation led to the construction of designresponse spectrum using straight lines which is of greaterimportance than response spectrum of an earthquake.

;3ample+.%? =raw the >


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=esign response spectrum should satisfy somerequirements since it is intended to be used for safe designof structures )book-+.$.%

=esign >


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c* *.++ to &.+ m5s c+ 4

1lot baseline in four way log paper.

0btain bc, de / cd by using

c / d points are fi3ed' so Tc is known.

Tb R Tc 5% ' TaR Tc 5*&' TeR*& to *$ s' Tf R 6& to 6$ s

Take from ref)% given in the book.


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"ig +.*

&.&* &.&+ &.&$ &.* &.+ &.6 &.$ &.B * + 6 % $ 4 B *& +& 6& $& B& *&&&.&&*

&.&&+

&.&&6&.&&%&.&&$

&.&&B

&.&*

&.&+

&.&6&.&%&.&$

&.&B

&.*

&.+

&.6

&.%&.$

&.B

*

+

a + - e

P s e * - o # e l o + ! , ( m / s e + )

2

m / s

e + ) 6

m g u

m g

m g

u

6 6 m

A g u

α

6 m g u

Pea3g"o*n-a++ele"a!on,#elo+!an- -spla+emen!

mepe"o- (se+)

Seismi( Inp%t

3/17

( g )

0ediumsi8ed earthquake at smallepicentral distance


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+% ! !+% " "+% # #+% $

+%

!

!+%

"

"+%

#

*ime period (sec)

S a 4 g

:ard soil

0edium soil

Soft soil

*ime Period (sec)

P s e u d o a c c e l e r a t i o n (

Design spectrum for site

Large si8e earthquake at large epicentral distance

"ig +.+*

Seismi( Inp%t


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3/192ontd..

84t!


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50t!

84

"ig +.++

Seismi( Inp%t

=esign ;arthquake' many different descriptions of the level of severity of ground motions areavailable.

2ontd..

3/20


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(2; P 7argest earthquake from a source


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site. If needed, earthquake data is augmented by earthquake records of

similar geological / geographical regions.

;arthquake records are scaled for uniformity / then modified forlocal soil condition.

8veraged / smoothed response spectra obtained from the recordsare used as site specific spectra.) book P +.$.B.* / ;3ample +.4.

The effect of appropriate soil condition may have to be incorporatedby de-convolution and convolution as shown in "ig +.+6.

Seismi( Inp%t


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probability of e3ceedance are used to construct the uniform haardspectra.

8lternatively, seismic haard analysis is carried out with spectralordinate )at each period for a given ^ as parameter )not 1F8.

"rom these haard curves, uniform haard spectrum for a givenprobability of e3ceedance can be constructed. 8n e3ample problem is

solved in the book in order to illustrate the concept. These curves areused for probabilistic design of structures )book - ;3ample +.B.

Seismi( Inp%t

"or many cases, response spectrum or 1


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q

>esponse spectrum compatible ground motion is generated byiteration to match a specified spectrum' iteration starts bygenerating a set of Faussian random numbers.

(any standard programs are now available to obtain response spectrumcompatible time histories' brief steps are given in the book )+.4.*.

Feneration of time history for a given 1


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>elationship between discussed before is used to findamplitudes of the sinusoids )book P +.4.+.

>andom phase angle, uniformly distributed between , is used to find

Feneration of partially correlated ground motions at a number of points having thesame 1


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available. 1redictive relationships generally e3press the seismic parameters as a function

of (, >,


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The mean value of the parameter is obtained from the predictive relationship' astandard deviation is specified.

1robability of e3ceedance is given by?

p is defined by

[ ] ( )1C @≥ @ = 1 ! D E ( 2.'1)

( )1ln@

ln@ ! ln@E = ( 2.'2)

F

Seismi( Inp%t

lnY is the mean value ) in ln of the parameter.

(any predictive relationships, laws / empirical equations e3ist' most widely usedones are given in the book.

1redictive relationships for different seismic parameters given in the book include.

2ontd.. 4/8


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1redictive relationships for 1F8 , 1H8 / 1HC. );qns? +.%6 P +.$B.

1redictive relationships for duration );qn +.$E.

1redictive relationships for arms);qns+.$ P

+.4+ 1redictive relationship for "ourier / response

spectra );qns +.46 P +.4E.

Seismi( Inp%t

2ontd..

1redictive relationships for 1


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1redictive relationships for modulatingfunction );qn +.++ given in ;qns +.E* P +.Eand "igs. +.%B P +.$&

1redictive relationships for coherencefunction );qns +.&P +..

;3ample +.E?2ompare between the values of 1H8 / 1HCcalculated by different empirical equations

for (B' rB$ / *+& km .:ote that 1H8 denotes generallypeak ground acceleration and 1HC refers to peak groundCelocity.

Seismi( Inp%t

4/102ontd..

mp"+al ela!onsp PA(g)

75 m 120 m

Table +.6? omparison o# ,s otaine i##erent empirica" euations #or M 7


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s!e#a (*a!on 2.43) 0.034 0.015

amell (*a!on 2.44) 0.056 0.035

o:o"gna(*a!on 2.45) 0.030 0.015

o"o(*a!on 2.46) 0.072 0.037

"*na+(*a!on 2.54) 0.198 0.088

mp"+al ela!onsp

P;(+m/s)

75 m 120 m

s!e#a (*a!on 2.49) 8.535 4.161


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s 400 m/s.


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