Seismicanalysisofstructuresi 130620030503 Phpapp02 [Compatibility Mode]

8/10/2019 Seismicanalysisofstructuresi 130620030503 Phpapp02 [Compatibility Mode]

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T.K. DattaDepartment Of Civil Engineering, IIT Delhi

Chapters 1 & 2

Chapter -1

SEISMOLOGY


Introduction

It is a big subject and mainly deals withearthquake as a geolog ical process.

However, some portions of seismology areof great interest to earthquake engineers.

They include causes of earthquake, earthquakewaves, measurement of earthquake, effect ofsoil condition on earthquake, earthquake pre-diction and earthquake hazard analysis

Understanding of these topics help earthquakeengineers in dealing seismic effects on structuresin a better way.

Further knowledge of seismology is helpful indescribing earthquake inputs for structures whereenough recorded data is not available.

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Seismology


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Interiors of earth

184 kmstoVp

Lec-1/2

Before earthquake is looked as a geologicalprocess, some knowledge about the structure of

ear th is in order.

In-side the earth

Crust: 5-40 km;M discontinuity; floating

Mantle: lithosphere (120 km);asthenosphere-plasticmolten rock (200 km);bottom-homogenous;var iat ion of v is less

(1000 km - 2900 km)

Core: discovered by Wichert &Oldham; only P waves canpass through inner core(1290 km); very dense;nickel & iron; outer core

(2200 km),same density;25000 C; 4x106 atm;14 g/cm3

Lithosphere f loats as a cluster of plates withdifferent movements in different directions.

Fig 1.1

Seismology


Plate tectonics

At mid oceanic ridges, twocontinents which were joinedtogether dri fted apart due toflow of hot mantle upward.

Flow takes place because ofconvective circulation of earth's mantle; energy comesfrom radioactivity inside theearth.

Hot mater ial cools as i t comesup; addi tional crust is formedwhich moves outward.

Lec-1/3

Convective currents

Concept of plate tectonics evolved fromcontinental dri ft .

Fig 1.2

Seismology


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Contd...

New crust sinks beneath sea surface; spreadingcontinues until l i thosphere reaches deep sea

trenches where subduction takes place.

Continental motions are associated with a varietyof circulation patterns.

As a resul t, motions take place through sl iding oflithosphere in pieces- called tectonic plates.

There are seven such major tectonic plates andmany smaller ones.

They move in different directions at differentspeeds.

Lec-1/4

Seismology


Contd...Lec-1/5

Fig 1.3

Major tectonic plates

Seismology


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Three types of Inter plate interactions exist givingthree types of boundaries.

Contd...Lec-1/6

Tectonic plates pass each other at the transformfaults.

Fig 1.4Types of interplate boundaries

Seismology


Faults at the plate boundaries are the likelylocations for earthquakes - inter plate earth-quake.

Earthquakes occurring within the plate arecaused due to mutual slip of rock bedreleasing energy- intra plate earthquake.

Slip creates new faults, but faults are mainlythe causes rather than results of earthquake.

At the fau lts two d if ferent types of sl ipageare observed- Dip slip; Strike slip.

In reality combination of the types of slipageis observed at the fault line.

Contd...Lec-1/7

Seismology


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Contd...

Lec-1/8

Types of faultFig 1.5

Seismology


Causes of earthquake

There are many theories to explain causes ofearthquake.

Out of them, tectonic theory of earthquake ispopular.

The tectonic theory stipulates that movementsof tectonic plates relat ive 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 exist ing faultlines are affected while intra-plate earthquakenew faults are created.

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Seismology


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Contd...

During 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.

Elastic rebound theory, put forward by Reid, givescredence to earthquake caused by slip alongfaults.

Large ampli tude shearing d isplacement that tookplace over a large length along the San andreasfault led to elastic rebound theory.

Modelling of earthquake based on elastic reboundtheory is of two types:

Kinematic-time history of sli p Dynamic-shear crack and its growth

Lec-2/2

Seismology


Contd...

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

Lec-2/3

Fig 1.6

Seismology


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Contd

An earthquake caused by slip at the fault proceeds in

the following way:

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

Large f ield of st rain reaches l imi ting value atsome point of t ime.

Sl ip occurs due to crushing of rock& masses;the st rain is released, releas ing vast energyequivalent to blasting of several atom bombs.

Strained layers of rock masses bounces backto its unstrained condition.

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Seismology


Contd...

Fault

Before slip Rebound due to slip

Push and pull force Double couple

Lec-2/5

Fig 1.7

Sl ip could be of any type-dip, str ike or mixed givingrise to a push & pull forcesacting at the fault ; slipvelocity at an active fault-10to 100mm/year.

This situation is equivalent

to two pairs of coupledforces suddenly acting andthus, moving masses ofrock leading to radialwaves propagating in alldirections.

Seismology


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Contd

Propagating wave is complex& is responsiblefor creating displacement and acceleration of

soil/rock particle in the ground.

The majority of the waves travels through therocks within the crust and then passes throughthe soi l to the top surface.

Other theory of tectonic earthquake stipulatesthat the earthquake occurs due to phasechanges of rock mass, accompanied by volumechanges in small volume of crust.

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

Lec-2/6

Seismology


Seismic waves

Large strain energy released during earthquakepropagates in all directions within earth as elasticmedium.

These waves, cal led seismic waves, transmitenergy from one point to the other & f inal ly carryi t to the surface.

Within earth, waves travel in almost homogeno-us elastic unbounded medium as body waves.

On the surface, they move as surface waves.

Reflection & refraction of waves take place nearthe surface at every layer; as a resul t waves getmodified.

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Seismology


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Contd...

Body waves are of two types- P & S waves;S waves are also cal led transverse waves.

Waves propagat ion velocit ies are given by:

P waves arrive ahead of S waves at a point ; t imeinterval is given by:

Polarized transverse waves are polarization of particl-es either in vertical(SV) or in horizontal(SH) plane.

)2.1(

12

1

)1.1(211

1

2/12/1

2/1

EG

E

s

p

)3.1(11

ps

pT

Lec-2/8

Seismology


Surface waves are of two types - L wavesand R waves.

L waves: particles move in horizontal planeperpendicular to the direction of wavepropagation.

R waves:- particles move in vertical plane;they trace a retrogate elliptical path; foroceanic waves water particles undergo

similar elliptical motion in ellipsoid surfaceas waves pass by.

L waves move faster than R waves onthe sur face (R wave veloci ty ~0.9 )

Contd...

SV

Lec-2/9

Seismology


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Contd...Lec-2/10

Body & Surface wavesFig 1.8

Seismology


P& S waves change phases as PPP, PS, PPSetc. after reflection & refraction at the surface.

Contd...

PS

PS

S

SPP

SS

PP

Lec-2/11

Reflection at the earth surfaceFig 1.9

Seismology


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Records of surface waves

Strong earthquake waves recorded on the surfaceare irregular in nature.

P PP S SS L

They can generally be classified in four groups:

Practically Single Shock: near source; on firmground; shallow earthquake.

Moderately long irregular: moderate distancefrom source; on firm ground-elcentro earthquake.

Lec-2/12

Typical strong motion recordFig 1.10

Seismology


A long ground motion wi th prevailing period:filtered ground motion through soft soil,medium- Loma Prieta earthquake.

Ground motion involving large Scale groundDeformation: land slides, soil liquefaction-Chilean & Alaska earthquakes.

Contd..Lec-2/13

Most ground motions are intermediate betweenthose described before (mixed).

Amongst them, nearly white noise type earth-quake records ( having a variety of frequencycompositions are more frequent on firm ground ).

Seismology


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0.1

0.05

0.0

0.05

0.1

WEST

EASTAcceleration

(g)

Time (sec)0.5 1.0 1.5 2

(a)

Acceleration

Contd...Lec-2/14

Single Shock

1

0.0

1

WEST

EAST

Displacement(cm)

Time (sec)

0.5 1.0 1.5 2

displacement

Fig 1.11

Seismology


Contd..

0 5 10 15 20 25 30-0.4

-0.2

0

0.2

0.4

Time (sec)

Acceleration(g)

Acceleration

Lec-2/15

Mixed frequency

0 5 10 15 20 25 30-10

-5

0

5

10

15

Time (sec)

Displacement(cm)

Displacement

Fig 1.12

Seismology


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Contd..

0 1 2 3 4 5 6 7 8-6

-4

-2

0

2

4

Time (sec)

Displacement(cm

)

Displacement

Lec-2/16

Time(sec)0 2 4 6 8 10 12 14 16 18

-0.5

-0.4-0.3

-0.2-0.1

00.1

0.20.3

0.40.50.6

0.70.8

0.91

Acceleration(g)

Acceleration

Predominant frequency

Fig 1.13

Seismology


They refer to quantities by which size & energyof earthquakes are described.

There are many measurement parameters; someof them are directly measured; some areindirectly derived from the measured ones.

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

Many of those empirical relationships and theparameters are used as inputs for seismicanalysis of structures; so they are described

along with the seismic inputs.

Earthquake measurement parametersLec-3/1

Seismology


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Here, mainly two most important parameters,magnitude & intensity of earthquake are described

along with some terminologies.

Contd...

Most of the damaging earthquakes have

Epicentre Epicentral Distance

Hypocentral DistanceFocal Depth

Focus/Hypocentre

Site

Limited region of earthinfluenced by the focusis called focal region ;greater the size ofearthquake, greater isthe focal region.

shallow focal depth 70 km are intermediate/deep.

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Earthquake definitionsFig 1.14

Seismology


Contd...

Force shocks are defined as those which occurbefore the main shock.

After shocks are those which occur after the mainshock.

Magnitude of earthquake is a measure of energyreleased by the earthquake and has the followingattributes:

is independent of place of observation.

is a funct ion of measured maximum disp lace-ments of ground at specified locations.

firs t developed by Waditi & Richter in 1935.

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Seismology


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Contd...

magnitude (M) scale is open ended.

M > 8.5 is rare; M < 2.5 is not percept ible.

there are many varieties of magnitude ofearthquake depending upon waves andquantities being measured.

Local magnitude ( ), orig inally proposed byRichter, is defined as log a (maximum amplitudein microns); Wood Anderson seismograph:R=100 km; magnif ication: 2800:

LM

pT = 0.8s : = 0.8

)6.1(log7.248.2log AML

Lec-3/4

Seismology


Since Wood Anderson seismograph is no more inuse, coda length ( T ), defined as total signalduration, is used these days:

Body magnitude ( ) is proposed by Gutenberg& Richter because of limitations of instrument &distance problems associated with .

It is obtained from compression P waves withperiods in the range of 1s; firs t few cycles are

used;

Contd...

)7.1(logTbaML

bM

LM

)8.1(,log

hQT

AMb

Lec-3/5

Seismology


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Occasionally, long period instruments are usedfor periods 5s-15s.

Surface magnitude ( ) was again proposed byGutenberg & Richter mainly for largeepicentral distance.

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

Praga formulation is used with surface waveperiod of the order of 20s

A is amp of Rayleigh wave (20s); is in km.

sM

Contd...

)9.1(0.2log66.1log

T

AMs

Lec-3/6

Seismology


Seismic moment magnitude ( ) is a better measure of large size earthquake with the helpof seismic moment.

A- area (m) ; U- longitudinal displacement(m);G(3x10N/m).

Seismic Moment ( ) is measured from

seismographs using long period waves anddescribes strain energy released from the entirerupture surface.

wM

Contd...

( 1.10)oM GUA

oM

Lec-3/7

Kanamori designed a scale which relates to.

wM

oM

Seismology


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Contd...

2 3 4 5 6 7 8 9 102

3

4

5

6

7

8

9

MsMJMA

MB

ML

Mb

Moment Magnitude Mw

Magnitude

)11.1(0.6log3

210 ow MM

Lec-3/8

Fig 1.15

Seismology


Energy Release, E ( Joules ) is given by :

M(7.3) ~ 50 megaton nuclear explosionM(7.2) releases 32 times more energy than M(6.2)M(8) releases 1000 times more energy than M(6)

Some Empirical formulae [L (km); D/U(m);A(km2)]

Contd...

sME 158.410

)14.1()42.0(46.582.0

)14.1()24.0(49.391.0

)14.1()22.0(22.369.0

)14.1(27.4)log32.1(

)13.1(65.5)log98.0(

dMLogD

cMLogA

bMLogL

aUM

LM

LogDw

LogAw

LogLw

Lec-3/9

Seismology


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Intensity is a subjective measure of earthquake;human feeling; effects on structures; damages.

Many Intensity scales exist in different parts of theworld; some old ones:

Gastaldi Scale (1564) Pignafaro Scale(1783) Rossi- forel Scale(1883)

Mercalli Cancani Sieberg scale is still in use inwestern Europe.

Modif ied Mercalli Scale (12 grade) is widely usednow.

Contd...Lec-3/10

Seismology


Contd...Lec-3/11

Intensity Evaluation DescriptionMagnitude

(Richter Scale)

I Insignificant Only detected by instruments 1 1.9

II Very LightOnly felt by sensitive persons; oscillation of

hanging objects2 2.9

III Light Small vibratory motion 3 3.9

IV ModerateFelt inside building; noise produced by

moving objects4 4.9

V Slightly StrongFelt by most persons; some panic; minor

damages

VI StrongDamage to non-seismic resistance

structures5 5.9

VII Very Strong

People running ; some damages in seismic

resistant structu res and serious damage to

un-reinforced masonry structures

VIII Dest ru ct iv e Serio us d am age to st ru ct ures in g en er al

IX Ruinous

Serious damage to well built struct ures;

almost total destruction of non-seismic

resistant structures

6 6.9

X DisastrousOnly seismic resistant struct ures remain

standing7 7.9

XIDisastrous in

Extreme

General panic; almost to tal destruction; t he

ground cracks and opens

XII Catastrophic Total destruction 8 8.9

Seismology


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There have been attempts to relate subjectiveintensity with the measured magnitude resultingin several empirical equations:

Other important earthquake measurementparameters are PGA, PGV, PGD.

PGA is more common & is related to magnitude

by various attenuation laws (described in seismicinputs).

Contd...

max1.3 0.6 (1.15)

8.16 1.45 2.46 ln (1.16)

1.44 ( ) (1.17)

sM I

I M r

I M f r

Lec-3/12

Seismology


Measurement of earthquake

Principle of operation is based on the oscil lation of apendulum.

Lec-4/1

Sensor : mass; string;magnet &support

Recorder : drum; pen;chart paper

Amp : optical / electro-magnetic means

Damp : electromagnetic/fluid dampers

Fig 1.16

Seismology


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u

Horizontal pendulum

Vertical pendulum

u

Contd...Lec-4/2

Fig 1.17

Seismology


Equation of motion of the bob is

If T very large (Long period seismograph)

If T very small (short period seismograph)

If T very close to & 2k very LargegT

Contd...

22 (1 .18)x k x w x u

)19.1(uxorux

)20.1(2 uxoruxw

(1.21)x u o r x u

Lec-4/3

Seismology


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Contd..

N

S Horseshoe Magnet

Suspension

Copper Mass

Mirror

Light Beam

copper cylinder2mm / 25mm /0.7g

taut wire 0.02mm

reflection of beammagnified by 2800

electro - magnetic

damping 0.8

Lec-4/4

Wood Anderson Seismograph

Fig 1.18

Seismology


Commonly used seismograph measuresearthquake wi thin 0.5-30 seconds.

Strong motion seismograph has the followingcharacteristics:

Contd..

period & damping of the pickup of 0.06- 25cps ;

preset acceleration 0.005g;

sensitivity 0.001-1.0g;

average starting t ime 0.05-0.1s.

Lec-4/5

Seismology


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Local Soil condition may have signi ficant influenceon ground motions.

Most of seismic energy at a site travels upwardthrough soil from the crust/rock bed below in theform of S/P waves.

In the process, amplitude, frequency contents &duration of earthquake get changed.

The extent depends upon geological, geographicaland geotechnical conditions.

Most influencing factors are properties of thesoil and topography.

Modification of ground motionLec-4/6

Seismology


Analysis of co llected data revealed interest ingfeatures of soil modification:

Contd...

Attenuation of ground motion through rockbed is significant 0.03g-350km (M=8.1).

For very soft soi l, predominant period ofground motion changes to soi l period; forrock bed PGA 0.03g (AF=5).

Duration increases also for soft soil.

Over a loose sandy soil underlying bymud, AF=3 for 0.035g-0.05g (at rock bed).

The shape of the response spectrumbecomes narrow banded for sof t soi l.

Lec-4/7

Seismology


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Contd...

As PGA at the rock bed increases, AFdecreases.

For s trong ground shaking, PGA amplification islow because of hysteret ic behaviour of soil .

At the crest of narrow rocky ridge, increasedampl if icat ion occurs; AF 2/ ( theoreticalanalysis ).

At the central region of bas in, ID wave propagationanalysis is valid; near the sides of the valley, 2Danalysis is to be carried out.

1D, 2D or 3D wave propagat ion analysis is carriedout to find PGA amplification theoretically.

Lec-4/8

Seismology


Seismic hazard analysis

It is a quantitati ve estimation of most possibleground shaking at a site.

The estimate can be made us ing deterministi cor probabilistic approaches; they requiresome/all of the fol lowing:

Knowledge of earthquake sources, fault activity,fault rupture length.

Past earthquake data g iving the relat ionshipbetween rupture length & magnitude.

Historical & Instrumentally recorded groundmotion.

Possible ground shaking may be representedby PGA, PGV, PGD or response spectrumordinates.

Lec-4/9

Seismology


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Determinist ic Hazard Analysis (DSHA):

A simple procedure to compute groundmotion to be used for safe design o fspeciality structures.

Restricted only when sufficient data isnot available to carry out PSHA.

It is conservative and does not providelikely hood of failure.

It can be used for deterministic design of

structures.

It is quiet often used for microzonation oflarge cities for seismic disaster mitigation.

ContdLec-4/10

Seismology


Contd

)25ln(80.1859.074.6PGA(gals)ln rm

Lec-4/11

It consists of following 5 steps:

Identif ication of sources including their geometry.

Evaluation of shortest epicentral distance / hypocentral distance.

Identif ication of maximum likely magnitude ateach source.

Selection of the predictive relationship valid forthe region.

Seismology


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Example 1.1 :

Maximum magnitudes forsources 1, 2 and 3 are 7.5,6.8 and 5 respectively.

Contd

(-50, 75)

Source 1

(-15, -30)

(-10, 78)

(30, 52)

(0, 0)

Source 3

Source 2

Site

Sources of earthquakenear the site (Examp. 1.1)

Source m r(km) PGA

1 7.5 23.70 0.490 g

2 6.8 60.04 0.10 g

3 5.0 78.63 0.015 g

Hazard level is 0.49g for the site

Lec-4/12

Fig 1.19

Seismology


Probabilistic seismic hazard analysis (PSHA).

It predicts the probabil ity of occurrence of acertain level of ground shaking at a site byconsidering uncertainties of:

Size of earthquake

Location

Rate of occurrence of earthquake

Predict ive relationship

ContdLec-5/1

PSHA is carried out in 4 steps.

Seismology


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Step 1 consists of following: Identification & characterization of

source probabilistically.

Assumes uniform dist ribution of po intof earthquake in the source zone .

Computation of d istribution ofrconsidering all points of earthquake aspotential source.

ContdLec-5/2

2 step consists of following:Determination of the average rate at

which an earthquake of a particular size

will be exceeded using G-R recurrencelaw.

)23.1()exp(10 ambmam

Seismology


Using the above recurrence law & specifyingmaximum & minimum values of M, followingpdf of M can be derived (ref. book)

3rd step consists of the following:

A predict ive relationship is used to obtainseismic parameter of interest (say PGA) for given

values of m , r .

Contd

)26.1()]([exp1

)]([exp)(

0max

0

mm

mmmfM

Lec-5/3

Uncertainty of the relationship is consideredby assuming PGA to be log normally distributed;the relationship provides the mean value; astandard deviation is specified.

Seismology


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ContdLec-5/4

4th step consists of the following:Combines uncertainties of location, size

& predictive relationship by

A seismic hazard curve is plot ted as(say is PGA level ).

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

drdmrfmfrmyYP RiMi

N

i

iy

S

yvsy

y

By including temporal uncertainty of earthquake(uncertainty of time) in PSHA & assuming it to be aPoisson process, probability o f exceedance of thevalue of , of the seismic parameter in T years

is g iven by (ref. book)y

[ ] 1 (1 .2 8 )y T

tP y y e d

Seismology


Example 1.2 :

For the site shown in Fig 1.20,show a typical calculation forPSHA ( use Equation 1.22with = 0.57)

Contd

(-50,75)

Source 1

(-15,-30)

(0,0)

Source 3

Source 2

Site

(5,80)(25,75) (125,75)

(125,15)(25,15)

Source Recurrence Law Mo Mu

Source 1 4 7.7

Source 2 4 5

Source 3 4 7.3

mm 4log

mm

2.151.4log

mm 8.03log

Lec-5/5

Fig 1.20

Seismology


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Solution:Location Uncertainty

1st source

Line is divided in 1000 segments

2nd source

Area is d ivided in 2500 parts (2x 1.2)

min

min

90.12

23.72( interval ( ) 10)

r km

r divide n

)10(32.30

98.145

min

max

nr

kmr

ContdLec-5/6

Seismology


3rd source :min maxr r r

Contd

0.0

0.4

27.0

4

33.6

8

40.3

2

49.9

6

53.6

0

60.2

4

66.8

8

73.5

2

80.1

6

86.8

0

P[R=r]

Epicentral distance, r (km)

0.0

0.2

36.1

0

47.6

7

59.2

4

70.8

1

82.3

8

93.9

5

105.5

2

117.0

9

128.6

6

140.2

3

P[R=r]


0.010

20

30

40

50

60

70

80

90

100

P[R=r]


1.0

Lec-5/7

Fig 1.21

Fig 1.23

Fig 1.22

Seismology


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Size Uncertainty :

631.010

501.010

110

48.033

42.15.42

4141

Contd

)29.1()(2

)(][

1221

2

1

21

ammmm

f

dmmfmmmP

m

m

m

M

Lec-5/8

For each source zone

For source zone 1, mu and m0 are divided in 10divisions.

Seismology


Histogram of M for each source zone are shown

Contd

0.0

0.8

Magnitude, m

0.7

0.5

0.4

0.3

0.2

0.1

0.6

P[M=m]

4.8

3 7

.14

4

.17

4

.50

5

.16

5

.49

5

.82

6

.15

6

.48

6

.

8

1

0.0

0.8

4.0

5

4.1

5

4.2

5

4.3

5

4.4

5

4.5

5

4.6

5

4.7

5

4.8

5

4.9

5

Magnitude, m

0.7

0.6

0.5

0.4

0.3

0.2

0.1

P[M=m]

0.0

0.8

Magnitude, m

0.7

0.5

0.4

0.3

0.2

0.1

0.6

P[M=

m]

4.83 7

.14

4.1

7

4.5

0

5.1

6

5.4

9

5.8

2

6.1

5

6.4

8

6.8

1

Lec-5/9

Fig 1.24

Fig 1.25

Fig 1.26

Seismology


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Say, Probabil ity of exceedance of 0.01g is desiredfor m = 4.19, r = 27.04 km for source zone1

The above probability is given as

Contd..

951.0)(1

65.1

)(104.27,19.4|01.0

ZF

z

ZFrmgPGAP

z

z

176.004.2719.404.27,19.4|01.0

04.27&19.4

101.0

01.0

rPmP

rmgPGAP

isrmfor

g

g

Lec-5/10

336.004.27

551.019.4

rP

mP

Seismology


For different levels of PGA, similar values ofcan be obtained.

Plot of vs. PGA gives the seismic hazardcurve.

Contd...

for other 99 combinations of m & r canobtained & summed up; for source zones 2 & 3,similar exercise can be done; finally,

0 .0 1 g

301.0201.0101.001.0 ||| sourgsourgsourgg

Lec-5/11

Seismology


31/63

11/22


ContdLec-5/12

Example-1.3:

The seismic hazard curve for a region shows that the annual

rate of exceedance of an acceleration 0.25g due toearthquakes (event ) is 0.02.What is the prob. that exactly

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

place in 30 years? Also, find that has a 10% prob. of

exceedance

in 50 yrs.

Solution:

Equation 1.28c (book) can be written as

%2.451)1()(

%333002.0)1()(3002.0

3002.0

eNPii

eteNPi t

0021.0

50

1.01ln)1(1ln

t

NP

Seismology


Seismic risk at a site is similar to that of seismichazard determined for a site.

It is defined as:

P( ) during a certain period (usually 1year).

Inverse of risk becomes return period for .

The study of seismic risk requires:

Source mechanism parameters focal depth;orientation of faults etc.

Recurrence relationship which is used to findPDF.

Attenuation Relat ionships.

s ix x

ix

Seismic risk at a siteLec-5/13

Seismology


32/63

11/22


Using the above Information, seismic risk can

be calculated with the help of either Cornel l'sapproach or Milne & Davenport approach.

Using the concept, many empirical equations areobtained with the help of data / informationfor regions.

For a particular region, these empir icalequations are developed; for other regions, theymay be use by choosing appropriate values for the parameters.

Some equations are given in the following

Many others are given in the book.

Contd..Lec-5/14

Seismology


Contd...

1

1

1

1 1

1

1.541

( )

1 1 1 ( )

1 1

( )1

( ) ( / ) (1.30)

exp exp ( ) (1.32 )

ln (1.32 )

47 (1.33)

1( ) | (1.37)

11 ( ) (1.38)

1 (1.39)

o

s u o

s

o

p

s s

o

i

s

m M

M s o u m M

s M

m M

s

N Y Y c

p m a

T b

P I i e

eF m P M m M m M

eP M m F m

P M m e

Lec-5/15

Seismology


33/63

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Microzonation is delineation of a region of a bigcity into different parts w ith respect to seismic

hazard potential.

Various Parameters indicating hazard potential areused to microzone the area like, local soilcharacteristics, earthquake source properties,epicentral distance, topography, populationdensity, type of Construction etc.

With respect to each parameter, a map may beprepared.

They are then combined ( by giving weightages toeach parameter ) to arrive at a hazard index.

Microzonation using hazard analysisLec-5/16

Seismology


Although each parameter has its importance,soil amplification, earthquake source properties,epicentral distance are considered very importantparameters to denote seismic risk or hazard of aregion.

Thus, DSHA/ PSHA combined with soil amplif icationare quite often used to prepare a microzonationmap. The steps include:

Divide the region into a number of grids consi-dering variation of soil properties.

At the centre of the si te of each gr id find PGAeither by DSHA/ PSHA (giving prob. exceed)

For each si te f ind PGA ampli ficat ion by 1D,2D or 3D wave propagation analysis.

Contd...Lec-5/17

Seismology


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Multiply PGAs obtained from DSHA/ PSHA toobtain f ree field PGAs.

With these PGAs, a microzonation map isprepared.

Contd...Lec-5/18

0.35 g

0.1 g0.25 g

0.4 g

Deterministic Microzonation

Probability of exceedance = 0.1

0.15 g

0.4 g

0.25 g

0.2 g

0.1 g

0.3 g

Probabilistic MicrozonationFig 1.27

Seismology


Lec-1/74


35/63

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Chapter -2

SEISMIC INPUTS


Seismic inputs

Various forms of Seismic inputs are used for earthquake analysis of structures.

The the form in which the input is provided dependsupon the type of analysis at hand.

In addition, some earthquake parameters suchas magnitude, PGA, duration, predominantfrequency etc. may be required.

The input data may be provided in time domainor in f requency domain or in both.

Further,the input data may be required indeterministic or in probabilistic form.

Predictive relationships for different earthquakeparameters are also required in seismic r iskanalysis.

1/1

Seismic Input


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Time history records

The most common way to describe ground mot ion isby way of time history records.

The records may be for displacement, veloc ityand acceleration; acceleration is generally directlymeasured; others are derived quantit ies.

Raw measured data is not used as inputs; dataprocessing is needed. It includes

Removal of noises by fil ters

Baseline correction

Removal of instrumental error

Conversion from A to D At any measuring station, ground motions are

recorded in 3 orthogonal directions; one is vertical.

1/2

Seismic Input


They can be transformed to principal directions;major direction is the direction of wave propagation;the other two are according ly defined.

Stochastically, ground motions in principaldirections are uncorrelated.

Contd..

(a) major (horizontal)

Major (horizontal)

0 5 10 15 20 25 30 35 40-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Acceleration(g)

Time (sec)

Fig 2.1(a)

1/3

Seismic Input


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Contd..1/4

0 5 10 15 20 25 30 35 40-0.3

-0.2

-0.1

0

0.1

0.2

Time (sec)

Acceleration(g)

Minor (horizontal)

0 5 10 15 20 25 30 35 40-0.3

-0.2

-0.1

0

0.1

0.2

Time (sec)

AcceleratIo

n

(g)

Minor (vertical)

Fig 2.1(b)

Fig 2.1(c)

Seismic Input


Because of the complex phenomena involved in thegeneration of ground motion, trains of groundmotion recorded at different stations vary spatially.

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

For nonhomogeneous field, both time lag & differencein rms exist.

Because of the spatial variation of ground motion,

both rotational & torsional components of groundmotions are generated.

Contd..1/5

=dy dx

dw=

Seismic Input


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In addit ion, an angle of inc idence of ground motionmay also be defined for the time history record.

Contd.. 1/6

Major direction

x

y

=Angle of incidence

Fig 2.2

Seismic Input


Frequency contents of time history

Fourier synthesis of time history record providesfrequency contents of ground motion.

It provides useful information about the ground motion& also forms the input fo r frequency domain analysis ofstructure.

Fourier series expansion of x(t) can be given as

a

0 n n n nn=1

T /2

0

-T/2

T /2

n n

-T/2

T /2

n n

-T/2

n

x( t )= a + a cos +

1= ( (

T

2a = x( t )co s

T

2b = x( t ) sin

T

=

1/7

Seismic Input


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The amplitude of the harmonic at is given by

(2.8)

2T/2

2 2 2

n n n n

-T/2

2T/2

n

-T/2

2A = a + b = x( t)cos tdt

T

2+ x( t)sin tdt

T

Contd.. 1/8

n n

-1 nn

n

=

b=

a

Equation 2.3 can also be represented in the form

0 n n nn=1

x( t) = c + c sin( +

Seismic Input


Plot of cn with is called Fourier Ampli tude Spectrum.

The integration in Eq. 2.8 is now efficiently performed byFFT algorithm which t reats four ier synthesis problem asa pair of fourier integrals in complex domain.

Standard input for FFT is N sampled ord inates of timehistory at an interval of t.

Output is N complex numbers; firs t N/2+1 complexquantities provide frequency contents of time historyother half is complex conjugate of the first half.

Contd..

n

-i t

-

i t

-

1x( i =

2

x( t)= x( i

1/9

Seismic Input


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is called Nyquest Frequency.

Fourier amplitude spectrum provides agood understanding of the characteristics ofground motion. Spectrums are shown in Fig 2.3.

For under standing general nature of spectra, like

those shown in Fig 2.3, spectra of groundaccelerations of many earthquakes areaveraged & smoothed for a particular site.

j

n

=T

=

Contd..

1/22 2

j j j

j-1

j

j

= =2

b=

a

1/10

Seismic Input


1/11Contd..

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

frequency (rad/sec)

Fourieramplitude(g-sec)1.4

Narrow band

0 20 40 60 80 100 120 140 1600

1

2

3

4

5

6x 10-3

Frequency (rad/sec)

OrdinateFourieramplitude

(g-sec)

Broad band

Fig 2.3(a)

Fig 2.3(b)

Seismic Input


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The resulting spectrum plotted on log scale shows:

Ampl itudes tend to be largest at an intermediaterange of frequency.

Bounding frequencies are fc & fmax.

fc is inversely proportional to duration.

For frequency domain analysis, frequency contentsgiven by FFT provide a better input.

Contd.. 1/12

Frequency (log scale)fc fmax

Ordina

teFourier

amplitude(logscale)

Fig 2.4

Seismic Input


Example2.1: 32 sampled values at t = 0.02s aregiven as input to FFT as shown in Fig 2.5

YY = 1/16 fft(y,32)

9.81

n = =T

2d = =

Contd.. 2/1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.03

-0.02

-0.01

0

0.010.02

0.03

0.04

Time (sec)

GroundAcceleration(g)

Fig 2.5

Seismic Input


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Contd..

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300-0.2

-0.1

0

0.1

0.2

0.3

Frequnecy (rad/sec)

Realpar

t

A

Real part

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300-0.2

-0.15-0.1

-0.05

0

0.05

0.1

0.15

Frequency (rad/sec)

Imaginarypart

A

Imaginary part

2/2

Fig 2.6a

Fig 2.6b

Seismic Input


2 2 1/2i i i

-1 i

i

= =

bj = tan =

a

Fourier amplitude spectrum is Ai Vs plot & phasespectrum is i Vs plot as shown in Fig 2.7

Contd.. 2/3

i

i

Amplitude spectrum

0 20 40 60 80 100 120 140 1600

0.005

0.01

0.015

0.02

Frequency (rad/sec)

Fourieramplitude(g-sec)

Fig 2.7a

Phase spectrum

0 20 40 60 80 100 120 140 160-1.5

-1

-0.5

0

0.5

1

1.5

Frequency (rad/sec)

Phase(rad)

Fig 2.7b

Seismic Input


43/63

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Power spectral density function

Power spectral density funct ion (PSDF) of groundmotion is a popular seismic input fo r probabilistic

seismic analysis of structures. It is defined as the dist ribut ion of the expected mean

square value of the ground motion wi th frequency.

Expected 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 4; one type of stochastic process iscalled ergodic process.

For 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.

2/4

Seismic Input


If future earthquake is assumed as an ergodicprocess, then PSDF of future ground motion (sayacceleration) may be derived using the concept of fourier synthesis.

Meansquare value of an accelerat ion t ime history a(t)using Parsavals theorem.

PSDF of a(t) is defined as

Hence,

Contd..

N/2

2n

0

=2

n N/2

nn=00

= =

2

nn

cS( = =

2/5

Seismic Input


44/63

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A close relat ionship between PSDF & Fourieramplitude spectrum is evident from Eqn. 2.18.

A typical PSDF of ground acceleration is shownin Fig 2.8.

Contd.. 2/6

0 10 20 30 40 50 60 700

0.5

1

1.5

2

Frequency (rad/sec)

NormalizedPSDFordinate

Fig 2.8

Seismic Input


Some of the important ground mot ion parametersare described using the moments of PSDF.

is called central frequency denoting concentrationof f requencies of the PSDF.

The mean peak accln.(PGA) is def ined using , , Td.

Predominant frequency / period is where PSDF /Fourier spectrum peaks.

n

n

n

0

2

0

=

=

dgmax 0u = 2

2

Contd.. 2/7

Seismic Input


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An addit ional input is needed for probabi list ic dynamicanalysis o f spatially long structures that have multi

support excitations. The time lag or lack of correlation between excitations at

different supports is represented by a coherencefunction & a cross PSDF.

The cross PSDF between two excitations which isneeded for the analysis of such structures is given by

Contd..

1 2 1 2

1 2 1 2

1 1

2 2xx x x 1 2

1 1

2 2xx x x 1 2 x 1 2

S =S S coh(x,x ,

S =S S coh(x,x , =

2/8

More discussions on cross PSDF is given laterin chapter 4.

Seismic Input


Records of actual s trongmotion records show thatmean square value of theprocess is not stationarybut evolutionary.

Contd..

2

S( =

2/9

Time(sec)

acc

(m/sec

2)

The earthquake process is better modeled asuniformly modulated stationary process in whichPSDF varies with time as:

From the collection of records ,various predict iverelation- ships for cross PSDF, Fourier spectrum,modulating funct ions have been derived; they aregiven later.

Fig 2.9

Seismic Input


46/63

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Example2.2: For the time history o f Example 2.1, findPSDF.

Solut ion: Using Eqns 2.9, 2.16, 2.18 ordinates of PSDFare obtained. Raw and smoothed PSDFs are shown inFigs 2.10 & 2.11

Contd..

0 20 40 60 80 100 120 140 1600

1

2

3

4x 10

-6

Frequency (rad/sec)

PSDF(g

2sec/rad)

Fig 2.10

2/10

Seismic Input


Sum of areas of bar = 0.011 (m/s2)2

Area under smoothed PSDF = 0.0113 (m/s2)2

Meansquare value of time history = 0.0112 (m/s2)2

0 50 100 1500

0.5

1

1.5

2

2.5

3x 10

-6

Frequency (rad/sec)

PSDF(g

2sec/rad)

Three point averaging(curve fit)

Three point averaging

Five point averaging

Five point averaging(curve fit)

Contd..

Fig 2.11

2/11

Seismic Input


47/63

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Response spectrum of earthquake is the mostfavored seismic input for earthquake engineers.

There are a number of response spectra used todefine ground motion; displacement, pseudovelocity, absolute acceleration & energy.

The spectra show the frequency contents of groundmotion but not directly as Fourier spectrum does.

Displacement spectrum forms the basis forderiving other spectra.

It is defined as the plot of maximum displacement ofan SDOF system to a particular earthquake as afunction of & .

Relative displacement of an SDOF for a given isgiven by (3rd chapter):

Response spectrum

g

3/1

Seismic Input


At the maximum value of displacement, KE = 0 &hence,

If this energy were expressed as KE, then anequivalent velocity of the system would be

Contd..

n

n

t-(t-)

g d

n0

vm d

n

t-(t-)

v g d

0 max

1x( t) =- x( -

S= =

S = x( -

2

d

2 2

eq d

eq n d

=2 2

x =

3/2

Seismic Input


48/63

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Thus, xeq = Sv; this velocity is called pseudo velocity &is dif ferent from the actual maximum velocity.

Plots of Sd & Sv over the full range of frequency & adamping ratio are disp lacement & pseudo veloci tyresponse spectrums.

A closely related spectrum called pseudo accelerationspectrum (spectral acceleration) is defined as:

Maximum force developed in the spring of the SDOF is

Thus, spectral acceleration multiplied by the massprovides the maximum spring force.

Contd..

2

a n dS =

2s d n d amaxf =kS =m =

3/3

Seismic Input


Contd..

This observation shows importance of the spectralacceleration.

While disp lacement response spectrum is the plot ofmaximum displacement, plots of pseudo veloci ty andacceleration are not so.

These three response spectra provide directlysome physically meaningful quantities:

Displacement Maximum deformation Pseudo velocity Peak SE

Pseudo acceleration Peak force

Energy response spectrum is the plot ofagainst a full range of frequency for a specif ieddamping ratio; i t shows the energy cotents of theground motion at different frequencies.

m a x

( )

m

3/4

Seismic Input


49/63

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At any instant of time t, it may be shown that

For = 0, it may fur ther easily be shown that

Comparing Eqns.(2.8) & (2.30), it is seen that Fourierspectrum & energy spectrum have simi lar forms.

Fourier amplitude spectrum may be viewed as ameasure of the total energy at the end (t = T) of anundamped SDOF.

Contd..

1

2 2 2n

2E(t)= x(t) +(

m

12 2 2t t

g n g d

0 0

2E(t)= x(

m

3/5

Seismic Input


Example2.3: Draw the spectrums for El Centroacceleration for = 0.05

Solution: Using Eqns 2.23 - 2.30, the spectrums aredrawn & are shown in Figs. 2.13 2.15

Tp(Energy) = 0.55 s

Tp(Fourier) = 0.58 s

Tp(Acceleration) = 0.51s

Contd.. 3/6

0 0.5 1 1.5 2 2.5 30

0.8

1.6

2.4

3.2

4

Time period (sec)

Energyspectrum(

g-sec)

Fig 2.13

Seismic Input


50/63

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3/7Contd..

0 20 40 60 80 100 120 140 1600

0.005

0.01

0.015

0.02

Frequency (rad/sec)

Fourieramplitude

(g-sec)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

Time period (Sec)Acceleration

responsespectrum

(g)

Fig 2.15

Fig 2.14

Seismic Input


D-V-A SpectrumAll three response spectra are useful in def ining the

design response spectrum discussed later.

A combined plot of the three spectra is thusdesirable & can be constructed because of therelationship that exists between them

Some limiting conditions should be realised as T 0 & T .

The following conditions (physical) help in plottingthe spectrum.

d v n

a v n

= -

logS = logS + log

d gmaxT

a gmaxT0

=

=

3/8

Seismic Input


51/63

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Fig 2.16

3/9

Seismic Input


Fig 2.17

3/10

Seismic Input


52/63

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The response spectrum of El Centro earthquake isidealised by a series of straight lines.

Straight lines below a & between points b & c areparallel to Sd axis.

Those below f & between d & e are parallel to Sa axis.

Below a shows constant ; below f showsconstant .

Between b & c constant ; between d & econstant .

Left of c is directly related to maximum acceleration;right of d is directly related to maximum displacement.

Intermediate portion cd is directly related to maximumvelocity of ground motion & most sensitive to dampingratio.

Contd.. 3/11

=

=

==

Seismic Input


Response spectrum of many earthquakes showsimilar trend when idealised.

This observation led to the construction ofdesign response spectrum using straight lineswhich is of greater importance than responsespectrum of an earthquake.

Example2.4: Draw the RSP for Park field earthquakefor & compare it with El Centro earthquake

Solut ion: Using Eqns. 2.23-2.26, the spectra areobtained & drawn in tripartite plot ; it is idealized bystraight lines; Fig 2.18 shows Parkfields & El CentroRSPs. Comparison of Ta to Tf between the two is shownin the book.

Contd..

%5

3/12

Seismic Input


53/63

11/22


Fig 2.18

Table 2.1 Comparison of periods between Parkfieldand El Centro earthquakes

3/13

(s) (s) (s) (s) (s) (s)

Park field 0.041 0.134 0.436 4.120 12.0 32.0

El Centro 0.030 0.125 0.349 3.135 10.0 33.0

a fT T

aT

bT

cT

dT

eT

fT

Seismic Input


Design response spectrum should satisfy somerequirements since it is intended to be used for safedesign of st ructures (book-2.5.4)

Spectrum should be as smooth as possib le.

Design spectrum should be representative ofspectra of past ground motions.

Two response spectra should be considered tocater to variations & design philosophy.

It should be normalized with respect to PGA.

Cunstruction of Design Spectrum

Expected PGA values for design & maximumprobable earthquakes are derived for the region.

Peak values of ground velocity & disp lacementare obtained as:

Design RSP3/14

Seismic Input


54/63

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c1 = 1.22 to 0.92 m/s c2 = 6

Plot baseline in four way log paper.

Obtain bc, de & cd by using

c & d points are fixed; so Tc is known.

Tb Tc/4 ; Ta Tc/10; Te10 to 15 s; Tf 30 to 35 s

Take from ref(4) given in the book.

Sa/g may be plotted in ordinary paper.

Contd.. 3/15

2

gmax 1 gmax 2

u u

u =c ; u =cg u

Seismic Input


Fig 2.19

3/16

0.01 0.02 0.05 0.1 0.2 0.3 0.5 0.7 1 2 3 4 5 6 7 10 20 30 50 70 1000.001

0.002

0.0030.0040.005

0.0070.01

0.02

0.03

0.040.05

0.07

0.1

0.2

0.30.40.5

0.7

1

2

3

45

7

10

aTbT cT dT eT fT

Disp

.(m

Pseudovelocity(m/sec)

2

Acc.(

m

v gu

m

gu

m

D

g

u

mg

u

m

A

gu

mgu

Peakgroundacceleration,velocityand displacement

Elastic design spectra

T im e per iod ( se c )

Seismic Input


55/63

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Fig 2.20

3/17

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

Time period (sec)

Sa/g

Hard soil

Medium soil

Soft soil

Time Period (sec)

Pseudo-acceleration(g)

Design spectrum for site

Medium-sized earthquake at smallepicentral distance

Large size earthquake at large epicentral distance

Fig 2.21

Seismic Input


Example2.5: Construct design spectra for the 50thpercentile & 84.1 percentile in Triparti te plot.

Solution: Ta = 1/33s; Tb = 1/8s; Te = 10s; Tf= 33s

A, = 2.17(2.71) ; V = 1.65(2.30)

D =1.39(2.01)

For 5 % damping;

Values within bracket are for 84.1 percentile spectrum.

Plots are shown in Fig 2.22.

Contd..

-g g

2

g

= =

g( 0.732)= =

0.6g

g =

3/18

Seismic Input


56/63

11/22


3/19Contd..

50th

84th

Fig 2.22

Seismic Input


Design Earthquake; many di fferent descriptions ofthe level of severity of ground mot ions are available.

Contd..

MCE Largest earthquake from a source

SSE Used for NP design

Other terms denoting simi lar levels ofearthquake are, credible, safety levelmaximum etc & are upper limits for twolevel concept.

Lower level is called as OBE; otherterminologies are operating level,probable design & st rength level.

OBE SSE

3/20

Seismic Input


57/63

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Site specific spectra are exclusively used for thedesign of structures for the site.

It is const ructed using recorded earthquake data in& around the site.

If needed, earthquake data is augmented byearthquake records of similar geolog ical &geographical regions.

Earthquake records are scaled for uniformity &then modified for local soil condition.

Averaged & smoothed response spectra obtainedfrom the records are used as site specific spectra.(book 2.5.7.1 & Example 2.6).

The effect of appropriate soil condit ion may have tobe incorporated by de-convolution and convolutionas shown in Fig 2.23.

Site specific spectra4/1

Seismic Input


Contd.. 4/2

Fig 2.23

Rock outcroping motion

CC

Soil profile at

site of interest

convolution

E

Surface motion at

site of interestSurface motion

DeconvolutionGiven soil

profileB

bedrock motion

A

D

Bedrock motion

same as point B

Seismic Input


58/63

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Statiscal analysis of available spectrum is performedto find distributions of PGA & spectral ordinate at

each period.

From these distributions, values of spectralordinates with specified probability of exceedanceare used to construct the uniform hazard spectra.

Alternatively, seismic hazard analysis is carriedout with spectral ordinate (at each period for a given) as parameter (not PGA).

From these hazard curves, uniform hazard spectrumfor a given probabil ity of exceedance can beconstructed. An example problem is solved in the

book in order to i llust rate the concept. These curvesare used for probabilistic design of structures (book- Example 2.7).

Uniform hazard spectra4/3

Seismic Input


For many cases, response spect rum or PSDFcompatible time history records are required asinputs for analysis.

One such case is nonlinear analysis of structuresfor future earthquakes.

Response spectrum compatible ground motionis generated by iteration to match a specifiedspectrum; iteration starts by generating a setof Gaussian random numbers.

Many standard programs are now available toobtain response spectrum compatible time histories;brief s teps are given in the book (2.6.1).

Generation of time history for a given PSDFessentially follows Monte Carlo simulation.

Synthetic accelerograms4/4

Seismic Input


59/63

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By considering the time history as a summationof sinusoids having random phase dif ferences,

the time history is generated.

Relationship between discussedbefore is used to find amplitudes of thesinusoids (book 2.6.2).

Random phase angle, uniformly d istr ibutedbetween , is used to find

Generation of partially correlated groundmotions at a number of po ints having the same

PSDF is somewhat involved & is g iven in ref(6).

Contd..

n

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-

i i iia( t)= A sin(

Seismic Input


Many seismic input parameters & ground motionparameters are directly available from recordeddata; many are obtained using empiricalrelationships.

These empirical relationships are not only usedfor predicting future earthquake parameters but alsoare extensively used where scanty data areavailable.

Predict ive relationships generally express theseismic parameters as a function of M, R, Si ( orany other parameter).

They are developed based on certain considerations.

Prediction of seismic input parameters

i

The parameters are approximately lognormally d istributed.

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Seismic Input


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Decrease in wave amplitude with d istance bearsan inverse relationship.

Energy absorpt ion due to material dampingcauses ampli tudes to decrease exponentially.

Effective epicentral distance is greater than R.

The mean value of the parameter is obtained fromthe predictive relationship; a standard deviation isspecified.

Probability o f exceedance is given by:

p is defined by

Contd..

1P Y = -

1

lnY

lnY - lnY=

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Seismic Input


lnY is the mean value ( in ln ) o f the parameter.

Many predict ive relationships, laws &empirical equations exist; most widely used onesare given in the book.

Predict ive relationships for dif ferent seismicparameters given in the book include.

Contd..

Predict ive relationships for PGA , PHA & PHV.(Eqns: 2.43 2.57).

Predict ive relationships for duration (Eqn 2.58). Predict ive relationships for arms(Eqns2.59

2.62)

Predict ive relationship for Fourier & responsespect ra (Eqns 2.63 2.68).

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Seismic Input


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Contd..

Predict ive relationships for PSDF (Eqns: 2.69

2.80).

Predictive relationships for modulatingfunction (Eqn 2.22) given in Eqns 2.81 2.89and Figs. 2.47 2.50

Predict ive relationships for coherencefunction (Eqns 2.90 2.99).

Example 2.8:Compare between the values of PHA & PHVcalculated by different empirical equations

for M=7; r=75 & 120 km .Note that PHA denotes generallypeak ground acceleration and PHV refers to peak groundVelocity.

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Seismic Input


4/10Contd..

Empirical Relationship PHA(g)

75 km 120 km

Esteva (Equation 2.43) 0.034 0.015

Cambell (Equation 2.44) 0.056 0.035

Bozorgina(Equation 2.45) 0.030 0.015

Toro(Equation 2.46) 0.072 0.037

Trifunac(Equation 2.54) 0.198 0.088

Empirical Relationship

PHV(cm/s)

75 km 120 km

Esteva (Equation 2.49) 8.535 4.161

Joyner (Equation 2.56) 4.785 2.285

Rosenblueth (Equation 2.50) 2.021 1.715

Table 2.3: Comparison of PHAs obtained by different empirical equations for M=7

Table 2.4: Comparison of PHVs obtained by different empirical equations for M=7

Seismic Input


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Example 2.9: Compare between the smoothednormalized Fourier spectrum obtained from El Centro

earthquake & that given by McGuire et al. (Eqn 2.68)

Solution: Assume and; comparison

is shown in Fig 2.45.

Contd.. 4/11

HzfHzfc 10;2.0 max

kmRMmsVws

100;7;1500 1

0M

7wM

is calculatedusing Eqn 1.11 as35.4 is selected

so that it matches ElCentro earthquake. 10-2 10-1 100 10110

-1

100

101

102

103

Frequency (Hz)Fourie

ramplitude(cm/sec)

Elcentro

Equation (2.60)

Fig 2.45

Seismic Input


Example 2.10: Compare between normalizedspectrums obtained by IBC, Euro-8, IS 1893 and thatgiven by Boore et al. (Eq.2.66) for M=7; R=50 km &Vs = 400 m/s.

Solution: Values of b1, to b6 are taken fromTable3.9(book); Gc = 0; PGA=0.35g (obtained)Comparison is shown in Fig 2.46

4/12Contd..

Fig 2.46

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

1

1.5

2

2.5

Time period (sec)

Sa/g

Boore

IS Code

Euro Code

IBC Code

Seismic Input


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Contd..

Example 2.11: Compare between the shapes of PSDFs ofground acceleration given by Housner & Jennings (Eqn.

2.70); Newmark & Resenbleuth (Eqn 2.71); Kanai andTazimi(Eqns 2.72-2.73) & Clough & Penziene (Eqns 2.74-2.75)Solution: All constant multipliers are removed from theequations to compare the shapes; comparisonis shown in Fig 2.47.

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0 10 20 30 40 50 60 700

0.5

1

1.5

2

Frequency (rad/sec)Normalize

dPSDFofacceleration

Housner and Jennings

Newmark and Rousenblueth

Kanai Tazimi

Clough and Penzien

Fig 2.47

Seismic Input

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