Seismic Analysis of Structures - I

T.K. Datta Department Of Civil Engineering, IIT Delhi

Chapters – 1 & 2

Chapter -1

SEISMOLOGY


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, earthquake waves, measurement of earthquake, effect of soil condition on earthquake, earthquake pre-diction and earthquake hazard analysis

Understanding of these topics help earthquake engineers in dealing seismic effects on structures in a better way.

Further knowledge of seismology is helpful in describing earthquake inputs for structures where enough recorded data is not available.

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Seismology


Interiors of earth

184 kmstoVp

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Before earthquake is looked as a geological process, some knowledge about the structure of earth is in order.

In-side the earth

Crust: 5-40 km; M discontinuity; floating Mantle: lithosphere (120 km); asthenosphere-plastic molten rock (200 km); bottom- homogenous; variation of v is less (1000 km - 2900 km)Core: discovered by Wichert & Oldham; only P waves can pass through inner core (1290 km); very dense; nickel & iron; outer core (2200 km), same density; 25000 C; 4x106 atm;14 g/cm3

Lithosphere floats as a cluster of plates with different movements in different directions.

Fig 1.1

Seismology


Plate tectonics

At mid oceanic ridges, two continents which were joined together drifted apart due to flow of hot mantle upward.

Flow takes place because of convective circulation of earth's 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

Concept of plate tectonics evolved from continental drift.

Fig 1.2

Seismology


Contd...

New crust sinks beneath sea surface; spreading continues until lithosphere reaches deep sea trenches where subduction takes place.

Continental motions are associated with a variety of circulation patterns.

As 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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Seismology


Contd...Lec-1/5

Fig 1.3

Major tectonic plates

Seismology


Three types of Inter plate interactions exist giving three types of boundaries.

Contd...

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Tectonic plates pass each other at the transform faults.

Fig 1.4Types of interplate boundaries

Seismology


Faults at the plate boundaries are the likely locations for earthquakes - inter plate earthquake. Earthquakes occurring within the plate are caused due to mutual slip of rock bed releasing energy- intra plate earthquake.

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

At the faults two different types of slipage are observed- Dip slip; Strike slip.

In reality combination of the types of slipage is observed at the fault line.

Contd...Lec-1/7

Seismology


Contd...

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Types of fault Fig 1.5

Seismology


Causes of earthquake There are many theories to explain causes of earthquake. Out of them, tectonic theory of earthquake is popular.

The tectonic theory stipulates that movements of tectonic plates relative to each other lead to accumulation of stresses at the plate boundaries & inside the plate.

This accumulation of stresses finally results in inter plate or intra plate earthquakes.

In interplate earthquake the existing fault lines are affected while intra-plate earthquake new faults are created.

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Seismology


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 moves along the fault line.

Elastic rebound theory, put forward by Reid, gives credence to earthquake caused by slip along faults.

Large amplitude shearing displacement that took place over a large length along the San andreas fault led to elastic rebound theory.

Modelling of earthquake based on elastic rebound theory is of two types:

Kinematic-time history of slip Dynamic-shear crack and its growth

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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

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Fig 1.6Seismology


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 long time.

Large field of strain reaches limiting value at some point of time.

Slip occurs due to crushing of rock& masses; the strain is released, releasing vast energy equivalent to blasting of several atom bombs.

Strained layers of rock masses bounces back to its unstrained condition.

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Seismology


Contd...

Fault

Before slip Rebound due to slip

Push and pull force Double couple

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Fig 1.7

Slip could be of any type- dip, strike or mixed giving rise to a push & pull forces acting at the fault; slip velocity at an active fault-10 to 100mm/year.

This situation is equivalent to two pairs of coupled forces suddenly acting and thus, moving masses of rock leading to radial waves propagating in all directions.

Seismology


Contd… Propagating wave is complex& is responsible for creating displacement and acceleration of soil/rock 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.

Other theory of tectonic earthquake stipulates that the earthquake occurs due to phase changes of rock mass, accompanied by volume changes in small volume of crust.

Those who favour this theory argues that earthquakes do occur at greater depths where faults do not exist.

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Seismology


Seismic waves Large strain energy released during earthquake propagates in all directions within earth as elastic medium.

These waves, called seismic waves, transmit energy from one point to the other & finally carry it to the surface.

Within earth, waves travel in almost homogenous elastic unbounded medium as body waves.

On the surface, they move as surface waves.

Reflection & refraction of waves take place near the surface at every layer; as a result waves get modified.

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Seismology


Contd... Body waves are of two types- P & S waves; S waves are also called transverse waves.

Waves propagation velocities are given by:

P waves arrive ahead of S waves at a point; time interval is given by:

Polarized transverse waves are polarization of particles 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

pspT

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Seismology


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

L waves: particles move in horizontal plane perpendicular to the direction of wave

propagation.

R waves:- particles move in vertical plane; they trace a retrogate elliptical path; for oceanic waves water particles undergo similar elliptical motion in ellipsoid surface as waves pass by.

L waves move faster than R waves on the surface (R wave velocity ~0.9 )

Contd...

SV

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Seismology


Contd...Lec-2/10

Body & Surface wavesFig 1.8

Seismology


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

Contd...

PS

PS

S

SPP SS

PP

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Reflection at the earth surfaceFig 1.9

Seismology


Records of surface waves

Strong earthquake waves recorded on the surface are irregular in nature.

P PP S SS L

They can generally be classified in four groups:

Practically Single Shock: near source; on firm ground; shallow earthquake.

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

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Typical strong motion recordFig 1.10

Seismology


A long ground motion with prevailing period: filtered ground motion through soft soil,

medium- Loma Prieta earthquake.

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

Contd..Lec-2/13

Most ground motions are intermediate between those described before (mixed).

Amongst them, nearly white noise type earthquake records ( having a variety of frequency compositions are more frequent on firm ground ).

Seismology


0.1

0.05

0.0

0.05

0.1

WEST

EAST

Accele

rati

on

(g

)

Time (sec)0.5 1.0 1.5 2

(a)

Acceleration

Contd...Lec-2/14

Single Shock

1

0.0

1

WEST

EASTDis

pla

cem

en

t (c

m)

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)

Accele

rati

on

(g

)

Acceleration

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Mixed frequency

0 5 10 15 20 25 30-10

-5

0

5

10

15

Time (sec)

Dis

pla

cem

en

t (c

m)

Displacement

Fig 1.12

Seismology


Contd..

0 1 2 3 4 5 6 7 8-6

-4

-2

0

2

4

Time (sec)

Dis

pla

cem

ent

(cm

)

Displacement

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

-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.50.60.70.80.9

1

Acc

eler

atio

n (g

)

Acceleration

Predominant frequency

Fig 1.13

Seismology


They refer to quantities by which size & 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.

Many of those empirical relationships and the parameters are used as inputs for seismic analysis of structures; so they are described along with the seismic inputs.

Earthquake measurement parametersLec-3/1

Seismology


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 earth influenced by the focus is called focal region ; greater the size of earthquake, greater is the focal region.

shallow focal depth <70 km; depths of foci >70 km are intermediate/deep.

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Earthquake definitionsFig 1.14

Seismology


Contd...

Force shocks are defined as those which occur before the main shock.

After shocks are those which occur after the main shock.

Magnitude 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 maximum displace- ments of ground at specified locations.

first developed by Waditi & Richter in 1935.

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Seismology


Contd...

magnitude (M) scale is open ended.

M > 8.5 is rare; M < 2.5 is not perceptible.

there are many varieties of magnitude of earthquake depending upon waves and quantities being measured.

Local magnitude ( ), originally proposed by Richter, is defined as log a (maximum amplitude in microns); Wood Anderson seismograph: R=100 km; magnification: 2800:

LM

pT = 0.8s : ξ = 0.8

)6.1(log7.248.2log AM L

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Seismology


Since Wood Anderson seismograph is no more in use, coda length ( T ), defined as total signal duration, 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 with periods in the range of 1s; first few cycles are used;

Contd...

)7.1(logTbaM L

bM

LM

)8.1(,log

hQT

AM b

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Seismology


Occasionally, long period instruments are used for periods 5s-15s.Surface magnitude ( ) was again proposed by Gutenberg & Richter mainly for large epicentral distance.However, it may be used for any epicentral distance & any seismograph can be used.Praga formulation is used with surface wave period of the order of 20s

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

sM

Contd...

)9.1(0.2log66.1log

T

AM s

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Seismology


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

A- area (m²) ; U- longitudinal displacement(m); G(3x10¹ºN/m²).

Seismic Moment ( ) is measured from seismographs using long period waves and describes strain energy released from the entire rupture surface.

wM

Contd...

( 1.10)oM GUA

oM

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Kanamori designed a scale which relates to .

wM

oM

Seismology


Contd...

2 3 4 5 6 7 8 9 102

3

4

5

6

7

8

9

M L

M s

MsMJMAMB

ML

Mb

M~M W

Moment Magnitude Mw

Mag

nit

ud

e

)11.1(0.6log3

210 ow MM

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Fig 1.15

Seismology


Energy Release, E ( Joules ) is given by :

M(7.3) ~ 50 megaton nuclear explosion M(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

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Seismology


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

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

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

Mercalli – Cancani – Sieberg scale is still in use in

western Europe.

Modified Mercalli Scale (12 grade) is widely used

now.

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 objects

2 – 2.9

III Light Small vibratory motion 3 – 3.9

IV ModerateFelt inside building; noise produced by moving objects

4 – 4.9

V Slightly StrongFelt by most persons; some panic; minor damages

VI StrongDamage to non-seismic resistance structures

5 – 5.9

VII Very StrongPeople running; some damages in seismic resistant structures and serious damage to un-reinforced masonry structures

VIII Destructive Serious damage to structures in general

IX RuinousSerious damage to well built structures; almost total destruction of non-seismic resistant structures

6 – 6.9

X DisastrousOnly seismic resistant structures remain standing

7 – 7.9

XIDisastrous in Extreme

General panic; almost total destruction; the ground cracks and opens

XII Catastrophic Total destruction 8 – 8.9

Seismology


There have been attempts to relate subjective intensity with the measured magnitude resulting in several empirical equations:

Other important earthquake measurement parameters are PGA, PGV, PGD.

PGA is more common & is related to magnitude by various attenuation laws (described in seismic inputs).

Contd...

max1.3 0.6 (1.15)

8.16 1.45 2.46ln (1.16)

1.44 ( ) (1.17)

sM I

I M r

I M f r

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Seismology


Measurement of earthquake

Principle of operation is based on the oscillation of a pendulum.

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Sensor : mass; string; magnet & support

Recorder : drum; pen; chart paper

Amp : optical / electromagnetic means

Damp : electromagnetic/ fluid dampers

Fig 1.16

Seismology


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 kx w x u

)19.1(uxorux

)20.1(2 uxoruxw

(1.21)x u or x u

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Seismology


Contd..

N

S Horseshoe Magnet

Suspension

Copper Mass

Mirror

Light Beam

copper cylinder 2mm / 25mm / 0.7g

taut wire 0.02 mm

reflection of beam magnified by 2800

electro - magnetic damping 0.8

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Wood Anderson Seismograph

Fig 1.18

Seismology


Commonly used seismograph measures earthquake within 0.5-30 seconds.

Strong motion seismograph has the following characteristics:

Contd..

• period & damping of the pick up of 0.06- 25cps ; • preset acceleration 0.005g;

• sensitivity 0.001-1.0g;

• average starting time 0.05-0.1s.

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Seismology


Local Soil condition may have significant influence on ground motions.

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

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

The extent depends upon geological, geographical and geotechnical conditions.

Most influencing factors are properties of the soil and topography.

Modification of ground motionLec-4/6

Seismology


Analysis of collected data revealed interesting features of soil modification:

Contd...

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

For very soft soil, predominant period of ground motion changes to soil period; for rock bed PGA 0.03g (AF=5).

Duration increases also for soft soil.

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

The shape of the response spectrum becomes narrow banded for soft soil.

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Seismology


Contd...

As PGA at the rock bed increases, AF decreases.

For strong ground shaking, PGA amplification is low because of hysteretic behaviour of soil.

At the crest of narrow rocky ridge, increased amplification occurs; AF ≈ 2π/ǿ ( theoretical analysis ).

At the central region of basin, ID wave propagation analysis is valid; near the sides of the valley, 2D analysis is to be carried out.

1D, 2D or 3D wave propagation analysis is carried out to find PGA amplification theoretically.

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Seismology


Seismic hazard analysis It is a quantitative estimation of most possible

ground shaking at a site.The estimate can be made using deterministic

or probabilistic approaches; they require some/all of the following: Knowledge of earthquake sources, fault activity, fault rupture length.

Past earthquake data giving the relationship between rupture length & magnitude.

Historical & Instrumentally recorded ground motion.

Possible ground shaking may be represented by PGA, PGV, PGD or response spectrum ordinates.

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Seismology


Deterministic Hazard Analysis (DSHA):

A simple procedure to compute ground motion to be used for safe design of speciality structures.

Restricted only when sufficient data is not available to carry out PSHA.

It is conservative and does not provide likely hood of failure.

It can be used for deterministic design of structures.

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

Contd…Lec-4/10

Seismology


Contd…

)25ln(80.1859.074.6PGA(gals)ln rm

Lec-4/11

It consists of following 5 steps:

Identification of sources including their geometry.

Evaluation of shortest epicentral distance / hypo central distance.

Identification of maximum likely magnitude at each source.

Selection of the predictive relationship valid for the region.

Seismology


Example 1.1 :

Maximum magnitudes for sources 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 earthquake near 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

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Fig 1.19

Seismology


Probabilistic seismic hazard analysis (PSHA).

It predicts the probability of occurrence of a certain level of ground shaking at a site by considering uncertainties of:

Size of earthquake

Location

Rate of occurrence of earthquake

Predictive relationship

Contd…Lec-5/1

PSHA is carried out in 4 steps.

Seismology


Step 1 consists of following: Identification & characterization of

source probabilistically.

Assumes uniform distribution of point of earthquake in the source zone .

Computation of distribution of r considering all points of earthquake as potential source.

Contd…Lec-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 recurrence law.

)23.1()exp(10 ambmam

Seismology


Using the above recurrence law & specifying

maximum & minimum values of M, following pdf of M can be derived (ref. book)

3rd step consists of the following:

A predictive relationship is used to obtain seismic 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 considered by assuming PGA to be log normally distributed;

the relationship provides the mean value; a standard deviation is specified.

Seismology


Contd…Lec-5/4

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

& predictive relationship by

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

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

drdmrfmfrmyYP RiMi

N

iiy

S

yvsyy

By including temporal uncertainty of earthquake (uncertainty of time) in PSHA & assuming it to be a Poisson process, probability of exceedance of the value of , of the seismic parameter in T years is given by (ref. book)

y

[ ] 1 (1.28 )y T

tP y y e d

Seismology


Example 1.2 :

For the site shown in Fig 1.20, show a typical calculation for PSHA ( use Equation 1.22 with σ = 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 4logmm 2.151.4log

mm 8.03log

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Fig 1.20

Seismology


Solution:Location Uncertainty

1st source

Line is divided in 1000 segments

2nd source

Area is divided 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

Contd…Lec-5/6

Seismology


3rd source :min maxr r r

Contd…

0.0

0.4

27

.04

33

.68

40

.32

49

.96

53

.60

60

.24

66

.88

73

.52

80

.16

86

.80

P[R

=r]

Epicentral distance, r (km)

0.0

0.2

36

.10

47

.67

59

.24

70

.81

82

.38

93

.95

10

5.5

2

11

7.0

9

12

8.6

6

14

0.2

3

P[R

=r]


0.01

0

20

30

40

50

60

70

80

90

10

0

P[R

=r]


1.0

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Fig 1.21

Fig 1.23

Fig 1.22

Seismology


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 10 divisions.

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 7.1

4

4.1

7

4.5

0

5.1

6

5.4

9

5.8

2

6.1

5

6.4

8

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

7.1

4

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


Say, Probability of exceedance of 0.01g is desired for 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

176.004.2719.4

04.27,19.4|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 of can be obtained.

Plot of vs. PGA gives the seismic hazard curve.

λ

λ

Contd...

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

0.01gλ

301.0201.0101.001.0 ||| sourgsourgsourgg

Lec-5/11

Seismology


Contd…Lec-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 to earthquakes (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 seismic hazard determined for a site.

It is defined as:

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

PDF. Attenuation Relationships.

s ix x≥

ix

Seismic risk at a siteLec-5/13

Seismology


Using the above Information, seismic risk can be calculated with the help of either Cornell's approach or Milne & Davenport approach.

Using the concept, many empirical equations are obtained with the help of data / information for regions.

For a particular region, these empirical equations are developed; for other regions, they may 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 )

l n (1.32 )

47 (1.33)

1( ) | (1.37)

1

1 ( ) (1.38)

1 (1.39)

o

s u o

s

o

ps s

o

is

m M

M s o u m M

s M

m Ms

N Y Y c

p m a

T b

P I i e

eF m P M m M m M

e

P M m F m

P M m e

Lec-5/15

Seismology


Microzonation using hazard analysisLec-5/16

Seismology


Contd...Lec-5/17

Seismology


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


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 depends upon the type of analysis at hand.

In addition, some earthquake parameters such as magnitude, PGA, duration, predominant frequency etc. may be required.

The input data may be provided in time domain or in frequency domain or in both.

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

Predictive relationships for different earthquake parameters are also required in seismic risk analysis.

1/1

Seismic Input


Time history records

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

The records may be for displacement, velocity and acceleration; acceleration is generally directly measured; others are derived quantities.

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

Removal of noises by filters

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 accordingly defined.

Stochastically, ground motions in principal directions 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

Acc

eler

atio

n (g

)

Time (sec)

Fig 2.1(a)

1/3

Seismic Input


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)

Acc

eler

atio

n (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)

AcceleratIon

(g)

Minor (vertical)

Fig 2.1(b)

Fig 2.1(c)

Seismic Input


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

For homogeneous field of ground motion, rms / peak values remain the same at two stations but there is

a time lag between the two records.

For nonhomogeneous field, both time lag & difference in rms exist.

Because of the spatial variation of ground motion, both rotational & torsional components of ground motions are generated.

Contd..1/5

du dvφ( t) = + ( 2.1)

dy dx

dwθ( t) = ( 2.2)

dxSeismic Input


In addition, an angle of incidence of ground motion may 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 provides frequency

contents of ground motion.

It provides useful information about the ground motion & also forms the input for frequency domain analysis of structure.

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 cosω t + b sinω t ( 2.3)

1a = x( t) dt ( 2.4)

T

2a = x( t)cosω tdt ( 2.5)

T

2b = x( t) sinω tdt ( 2.6)

T

ω = 2πn/T ( 2.7)

1/7

Seismic Input


The amplitude of the harmonic at is given by

(2.8)

2T/22 2 2n n n n

-T/2

2T/2

n-T/2

2A = a + b = x( t) cosω tdt

T

2+ x( t) sinω tdt

T

Contd..

nω

1/8

n n

-1 nn

n

c = A

bφ = tan ( 2.10)

a

Equation 2.3 can also be represented in the form

α

0 n n nn=1

x( t) = c + c sin( ω t + φ ) ( 2.9)

Seismic Input


Plot of cn with is called Fourier Amplitude Spectrum.

The integration in Eq. 2.8 is now efficiently performed by FFT algorithm which treats fourier synthesis problem as a pair of fourier integrals in complex domain.

Standard input for FFT is N sampled ordinates of time history at an interval of ∆t.

Output is N complex numbers; first N/2+1 complex quantities provide frequency contents of time history other half is complex conjugate of the first half.

Contd..

n

α-iω t

-α

αiω t

-α

1x( iω) = x( t) e dt ( 2.11)

2π

x( t) = x( iω) e dω ( 2.12)

1/9

Seismic Input


is called Nyquest Frequency.

Fourier amplitude spectrum provides a good understanding of the characteristics of ground motion. Spectrums are shown in Fig 2.3.

For under standing general nature of spectra, like those shown in Fig 2.3, spectra of ground accelerations of many earthquakes are averaged & smoothed for a particular site.

j

n

2πjω =

Tω = Nπ/T

Contd..

1/22 2j j j

j-1j

j

NA = a + b j = 0,....., ( 2.13)

2

bφ = tan ( 2.14)

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)

Fou

rier

am

plitu

de (

g-s

ec)

1.4

Narrow band

0 20 40 60 80 100 120 140 1600

1

2

3

4

5

6x 10-3

Frequency (rad/sec)

Ord

inat

e F

ouri

er a

mp

litu

de

(g-s

ec)

Broad band

Fig 2.3(a)

Fig 2.3(b)

Seismic Input


The resulting spectrum plotted on log scale shows:

Amplitudes tend to be largest at an intermediate range of frequency.

Bounding frequencies are fc & fmax.

fc is inversely proportional to duration.

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

Contd.. 1/12

Frequency (log scale)fc fmax

Ord

inat

e F

ouri

er

amp

litu

de

(log

sca

le)

Fig 2.4

Seismic Input


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

YY = 1/16 fft(y,32)

9.81

n

nπω = = 157.07 rad/s

T2π

dω = = rad/sT

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

0.01

0.02

0.03

0.04

Time (sec)

Gro

un

d A

ccel

erat

ion

(g)

Fig 2.5

Seismic Input


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)

Rea

l par

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)

Imag

inar

y p

art

A

Imaginary part

2/2

Fig 2.6a

Fig 2.6b

Seismic Input


2 2 1/2i i i

-1 ii i n

i

A =( a + b ) i =0....N/2

bj = tan ω =( 0..dw...w )

a

Fourier amplitude spectrum is Ai Vs plot & phase

spectrum is Φi Vs plot as shown in Fig 2.7

Contd.. 2/3

ii

Amplitude spectrum

0 20 40 60 80 100 120 140 1600

0.005

0.01

0.015

0.02

Frequency (rad/sec)

Fou

rier

am

pli

tud

e (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)

Ph

ase

(rad

)

Fig 2.7b

Seismic Input


Power spectral density function Power spectral density function (PSDF) of ground

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

It is defined as the distribution of the expected mean square value of the ground motion with frequency.

Expected value is a common way of describing probabilistically a ground motion parameter & is connected to a stochastic process.

The characteristics of a stochastic process is described later in chapter 4; one type of stochastic process is called ergodic process.

For an ergodic process, a single time history of the ensemble 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 ergodic process, then PSDF of future ground motion (say acceleration) may be derived using the concept of fourier synthesis.

Meansquare value of an acceleration time history a(t)using Parsaval’s theorem.

PSDF of a(t) is defined as

Hence,

Contd..

N/2

2n

0

1λ = c ( 2.16)

2

nω N/2

nn=00

λ = S( ω) dω = g( ω) ( 2.17)

2n

n

cS( ω) = & g( ω) = S( ω) dω ( 2.18)

2dω

2/5

Seismic Input


A close relationship between PSDF & Fourier amplitude spectrum is evident from Eqn. 2.18.

A typical PSDF of ground acceleration is shown in Fig 2.8.

Contd.. 2/6

0 10 20 30 40 50 60 700

0.5

1

1.5

2

Frequency (rad/sec)

Nor

mal

ized

PS

DF

ord

inat

e

Fig 2.8

Seismic Input


Some of the important ground motion parameters are described using the moments of PSDF.

Ω is called central frequency denoting concentration of frequencies of the PSDF.

The mean peak accln.(PGA) is defined using Ω, λ, Td.

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

nω

nn

0

2

0

λ = ω S( ω) d ( 2.19a)

λΩ = ( 2.19b)

λ

dgmax 0

2.8ΩTu = 2 λ ln ( 2.19c)

2π

Contd.. 2/7

Seismic Input


An additional input is needed for probabilistic dynamic analysis of spatially long structures that have multi support excitations.

The time lag or lack of correlation between excitations at different supports is represented by a coherence function & a cross PSDF.

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

Contd..

1 2 1 2

1 2 1 2

1 12 2

x x x x 1 2

1 12 2

x x x x 1 2 x 1 2

S = S S coh( x ,x ,ω) ( 2.20)

S = S S coh( x ,x ,ω) = S coh( x ,x ,ω) ( 2.21)

2/8

More discussions on cross PSDF is given later in chapter 4.

Seismic Input


Records of actual strong motion records show that mean square value of the process is not stationary but evolutionary.

Contd..

2S( ω,t) = q( t) S( ω) ( 2.22)

2/9

Time(sec)

ac

c (m

/sec

2 )

The earthquake process is better modeled as uniformly modulated stationary process in which PSDF varies with time as:

From the collection of records ,various predictive relationships for cross PSDF, Fourier spectrum, modulating functions have been derived; they are given later.

Fig 2.9

Seismic Input


Example2.2: For the time history of Example 2.1, find PSDF.

Solution: Using Eqns 2.9, 2.16, 2.18 ordinates of PSDF are obtained. Raw and smoothed PSDFs are shown in Figs 2.10 & 2.11

Contd..

0 20 40 60 80 100 120 140 1600

1

2

3

4x 10

-6

Frequency (rad/sec)

PSD

F (g

2 sec/

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)

PS

DF

(g2 s

ec/r

ad)

Three point averaging(curve fit)

Three point averaging

Five point averaging

Five point averaging(curve fit)

Contd..

Fig 2.11

2/11

Seismic Input


Response spectrum of earthquake is the most favored seismic input for earthquake engineers.

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 not directly as Fourier spectrum does.

Displacement spectrum forms the basis for deriving other spectra.

It is defined as the plot of maximum displacement of an SDOF system to a particular earthquake as a function of & ξ.

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

Response spectrum

nω

gx( t)

3/1

Seismic Input


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

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

Contd..

n

n

t-ξω( t-τ)

g dn 0

vm d

n

t-ξω( t-τ)

v g d0 max

1x( t) = - x( τ) e sinω( t-τ) dτ ( 2.23)

ω

Sx = S = ( 2.24a)

ω

S = x( τ) e sinω( t-τ) dτ ( 2.24b)

2d

1E= kS ( 2.25a)

2

2 2eq d

eq n d

1 1mx = kS ( 2.25b)

2 2x =ω S ( 2.25c)

3/2

Seismic Input


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

Plots of Sd & Sv over the full range of frequency & a damping ratio are displacement & pseudo velocity response spectrums.

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

Maximum force developed in the spring of the SDOF is

Thus, spectral acceleration multiplied by the mass provides the maximum spring force.

Contd..

2a n dS = ω S ( 2.26)

2s d n d amaxf = kS = mω S = mS ( 2.27)

3/3

Seismic Input


Contd.. This observation shows importance of the spectral

acceleration.

While displacement response spectrum is the plot of maximum displacement, plots of pseudo velocity and acceleration are not so.

These three response spectra provide directly some physically meaningful quantities:

Displacement – Maximum deformation Pseudo velocity – Peak SE Pseudo acceleration – Peak force

Energy response spectrum is the plot of against a full range of frequency for a specified damping ratio; it shows the energy cotents of the ground motion at different frequencies.

max

2E( t)m

3/4

Seismic Input


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

For ξ = 0, it may further easily be shown that

Comparing Eqns.(2.8) & (2.30), it is seen that Fourier spectrum & energy spectrum have similar forms.

Fourier amplitude spectrum may be viewed as a measure of the total energy at the end (t = T) of an undamped SDOF.

Contd..

12 2 2

n

2E( t)= x( t) +( ω x( t) ) ( 2.29)

m

12 2 2t t

g n g d0 0

2E( t)= x( τ) cosω τ dτ + x( τ) sinω τ dτ ( 2.30)

m

3/5

Seismic Input


Example2.3: Draw the spectrums for El Centro acceleration for ξ = 0.05

Solution: Using Eqns 2.23 - 2.30, the spectrums are drawn & 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)

En

erg

y s

pectr

um

(g

-sec)

Fig 2.13

Seismic Input


3/7Contd..

0 20 40 60 80 100 120 140 1600

0.005

0.01

0.015

0.02

Frequency (rad/sec)

Fou

rier

am

pli

tud

e (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)

Acc

eler

atio

n r

esp

onse

sp

ectr

um

(g)

Fig 2.15

Fig 2.14

Seismic Input


D-V-A Spectrum All three response spectra are useful in defining the

design response spectrum discussed later.

A combined plot of the three spectra is thus desirable & can be constructed because of the relationship that exists between them

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

The following conditions (physical) help in plotting the spectrum.

d v n

a v n

logS = logS - logω ( 2.31)

logS = logS + logω ( 2.32)

d gmaxT→∞

a gmaxT→0

limS =u ( 3.33)

limS =u ( 3.34)

3/8

Seismic Input


Fig 2.16

3/9

Seismic Input


Fig 2.17

3/10

Seismic Input


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

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

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

Below ’a’ shows constant ; below ‘f’ shows constant .

Between b & c constant ; between d & e constant .

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 maximum velocity of ground motion & most sensitive to damping ratio.

Contd.. 3/11

a gS = u

d gS = u

a a gmaxS =α u

d d gmaxS =α u

Seismic Input


Response spectrum of many earthquakes show similar trend when idealised.

This observation led to the construction of design response spectrum using straight lines which is of greater importance than response spectrum of an earthquake.

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

Solution: Using Eqns. 2.23-2.26, the spectra are obtained & drawn in tripartite plot; it is idealized by straight lines; Fig 2.18 shows Parkfields & El Centro RSPs. Comparison of Ta to Tf between the two is shown in the book.

Contd..

%5

3/12

Seismic Input


Fig 2.18

Table 2.1 Comparison of periods between Parkfield and 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

aTbTcTdTeTfT

Seismic Input


Design response spectrum should satisfy some requirements since it is intended to be used for safe design of structures (book-2.5.4)

Spectrum should be as smooth as possible.

Design spectrum should be representative of spectra of past ground motions.

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

It should be normalized with respect to PGA.

Cunstruction of Design Spectrum

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

Peak values of ground velocity & displacement are obtained as:

Design RSP3/14

Seismic Input


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; Te≈10 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

2gmax gmax

gmax 1 gmax 2gmax

u uu =c ; u =c

g u

a gmax d gmax v gmaxα u ;α u ;α u

a d vα , α & α

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

0.0030.0040.0050.007

0.01

0.02

0.030.040.050.07

0.1

0.2

0.30.40.50.7

1

2

3457

10

aTbT cT dT eT fT

Disp.(m)

Pseu

do ve

locit

y(m

/sec)

2

Acc.(m

/sec)

mv gu

mg

u

mD

gu

mg

u

m

Agu

mgu

Peak ground acceleration,velocityand displacement

Elastic design spectra

Time period (sec)

Seismic Input


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)

Pseu

do-ac

c eler

a tio n

(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 50th percentile & 84.1 percentile in Tripartite plot.

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

αA, = 2.17(2.71) ; αV = 1.65(2.30)

αD =1.39(2.01)

For 5 % damping;

Values within bracket are for 84.1 percentile spectrum.

Plots are shown in Fig 2.22.

Contd..

-1g g

2

g

1.22u = u = 0.732 ms

g

( 0.732)u = = 0.546m

0.6g

gu = 0.6g

3/18

Seismic Input


3/19Contd..

50th

84th

Fig 2.22

Seismic Input


Design Earthquake; many different descriptions of the level of severity of ground motions are available.

Contd..

MCE – Largest earthquake from a source

SSE – Used for NP design

Other terms denoting similar levels of earthquake are, credible, safety level maximum etc & are upper limits for two level concept.

Lower level is called as OBE; other terminologies are operating level,

probable design & strength level.

OBE ≈ ½ SSE

3/20

Seismic Input


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

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

If needed, earthquake data is augmented by earthquake records of similar geological & geographical regions.

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

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

The effect of appropriate soil condition may have to be incorporated by de-convolution and convolution as shown in Fig 2.23.

Site specific spectra4/1

Seismic Input


Contd.. 4/2

Fig 2.23

Rock outcroping motion

CC

Soil profile atsite of interest

convolution

E

Surface motion atsite of interestSurface motion

DeconvolutionGiven soilprofile

B

bedrock motion

A

D

Bedrock motionsame as point B

Seismic Input


Statiscal analysis of available spectrum is performed to find distributions of PGA & spectral ordinate at each period.

From these distributions, values of spectral ordinates with specified probability of exceedance are used to construct the uniform hazard spectra.

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

From these hazard curves, uniform hazard spectrum for a given probability of exceedance can be constructed. An example problem is solved in the book in order to illustrate the concept. These curves are used for probabilistic design of structures (book - Example 2.7).

Uniform hazard spectra4/3

Seismic Input


For many cases, response spectrum or PSDF compatible time history records are required as inputs for analysis.

One such case is nonlinear analysis of structures for future earthquakes.

Response spectrum compatible ground motion is generated by iteration to match a specified spectrum; iteration starts by generating a set of Gaussian random numbers.

Many standard programs are now available to obtain response spectrum compatible time histories; brief steps are given in the book (2.6.1).

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

Synthetic accelerograms4/4

Seismic Input


By considering the time history as a summation of sinusoids having random phase differences, the time history is generated.

Relationship between discussed before is used to find amplitudes of the sinusoids (book – 2.6.2).

Random phase angle, uniformly distributed between , is used to find

Generation of partially correlated ground motions at a number of points having the same PSDF is somewhat involved & is given in ref(6).

Contd..

nc &Sdω

4/5

0 -2π iφ

i i iia( t) = A sin( ωt + φ ) ( 2.39)

Seismic Input


Many seismic input parameters & ground motion parameters are directly available from recorded data; many are obtained using empirical relationships.

These empirical relationships are not only used for predicting future earthquake parameters but also are extensively used where scanty data are available.

Predictive relationships generally express the seismic parameters as a function of M, R, Si ( or any other parameter).

They are developed based on certain considerations.

Prediction of seismic input parameters

iY = f M, R, S ( 2.40)

The parameters are approximately log normally distributed.

4/6

Seismic Input


Decrease in wave amplitude with distance bears an inverse relationship.

Energy absorption due to material damping causes amplitudes to decrease exponentially.

Effective epicentral distance is greater than R.

The mean value of the parameter is obtained from the predictive relationship; a standard deviation is specified.

Probability of exceedance is given by:

p is defined by

Contd..

1P Y ≥ Y =1 - F p ( 2.41)

1

lnY

lnY - lnYp = ( 2.42)

σ

4/7

Seismic Input


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

Many predictive relationships, laws & empirical equations exist; most widely used ones are given in the book.

Predictive relationships for different seismic parameters given in the book include.

Contd..

Predictive relationships for PGA , PHA & PHV. (Eqns: 2.43 – 2.57).

Predictive relationships for duration (Eqn 2.58).

Predictive relationships for arms(Eqns2.59 – 2.62)

Predictive relationship for Fourier & response spectra (Eqns 2.63 – 2.68).

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Seismic Input


Contd..

Predictive relationships for PSDF (Eqns: 2.69 – 2.80).

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

Predictive relationships for coherence function (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


Example 2.9: Compare between the smoothed normalized 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 kmRMmsV ws 100;7;1500 1

0M

7wM

is calculated using Eqn 1.11 as 35.4 is selected so that it matches El Centro earthquake. 10-2 10-1 100 101

10-1

100

101

102

103

Frequency (Hz)

Fo

uri

er

am

pli

tud

e (

cm

/se

c)

Elcentro

Equation (2.60)

Fig 2.45

Seismic Input


Example 2.10: Compare between normalized spectrums obtained by IBC, Euro-8, IS 1893 and that given 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 from Table3.9(book); Gc = 0; PGA=0.35g (obtained)Comparison is shown in Fig 2.46

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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


Contd..

Example 2.11: Compare between the shapes of PSDFs of ground acceleration given by Housner & Jennings (Eqn. 2.70); Newmark & Resenbleuth (Eqn 2.71); Kanai and Tazimi(Eqns 2.72-2.73) & Clough & Penziene (Eqns 2.74-2.75) Solution: All constant multipliers are removed from the equations to compare the shapes; comparison is 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)

Nor

mal

ized

PS

DF

of

acce

lera

tion

Housner and JenningsNewmark and Rousenblueth

Kanai TazimiClough and Penzien

Fig 2.47Seismic Input


Education

Seismic Analysis of Structures - I