Seismic Analysis of Structures - I

T.K. Datta
Department Of Civil Engineering, IIT
Chapters – 1 & 2
Chapter -1
SEISMOLOGY

T.K. Datta
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

T.K. Datta
Interiors of earth
1
84 −
− 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

T.K. Datta
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

T.K. Datta
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

T.K. Datta
Contd...
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Fig 1.3
Major tectonic plates
Seismology

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 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 1.4
Types of interplate boundaries
Seismology

T.K. Datta
 Faults at the plate boundaries are the likely
locations for earthquakes - inter plate earth-
quake.
 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...
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Seismology

T.K. Datta
Contd...
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Types of fault
Fig 1.5
Seismology

T.K. Datta
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 boundar-
ies & 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

T.K. Datta
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

T.K. Datta
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 1.6
Seismology

T.K. Datta
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

T.K. Datta
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

T.K. Datta
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

T.K. Datta
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 homogeno-
us 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

T.K. Datta
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 particl-
es either in vertical(SV) or in horizontal(SH) plane.
( )( )
( )
)2.1(
12
1
)1.1(
211
1
2/12/1
2/1






+
=





=






−+
−
=
νρρ
ν
νν
ν
ρ
ν
EG
E
s
p
)3.1(
11








−∆=
ps
pT
νν
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Seismology

T.K. Datta
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

T.K. Datta
Contd...
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Body & Surface waves
Fig 1.8
Seismology

T.K. Datta
 P& S waves change phases as PPP, PS, PPS
etc. after reflection & refraction at the surface.
Contd...
PS
P
S
S
SP
P
SS
PP
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Reflection at the earth surface
Fig 1.9
Seismology

T.K. Datta
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 record
Fig 1.10
Seismology

T.K. Datta
 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..
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 Most ground motions are intermediate between
those described before (mixed).
 Amongst them, nearly white noise type earth-
quake records ( having a variety of frequency
compositions are more frequent on firm ground ).
Seismology

T.K. Datta
0.1
0.05
0.0
0.05
0.1
WEST
EAST
Acceleration(g)
Time (sec)
0.5 1.0 1.5 2
(a)
Acceleration
Contd...
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Single Shock
1
0.0
1
WEST
EAST
Displacement(cm)
Time (sec)
0.5 1.0 1.5 2
displacement
Fig 1.11
Seismology

T.K. Datta
Contd..
0 5 10 15 20 25 30
-0.4
-0.2
0
0.2
0.4
Time (sec)
Acceleration(g)
Acceleration
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Mixed frequency
0 5 10 15 20 25 30
-10
-5
0
5
10
15
Time (sec)
Displacement(cm)
Displacement
Fig 1.12
Seismology

T.K. Datta
Contd..
0 1 2 3 4 5 6 7 8
-6
-4
-2
0
2
4
Time (sec)
Displacement(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
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Acceleration(g)
Acceleration
Predominant frequency
Fig 1.13
Seismology

T.K. Datta
 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 parameters
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Seismology

T.K. Datta
 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 definitions
Fig 1.14
Seismology

T.K. Datta
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

T.K. Datta
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 ∆+−= AML
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Seismology

T.K. Datta
 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 ∆+





= hQ
T
A
Mb
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Seismology

T.K. Datta
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
A
Ms
∆
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Seismology

T.K. Datta
 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

T.K. Datta
Contd...
2 3 4 5 6 7 8 9 10
2
3
4
5
6
7
8
9
M
L
M
s
Ms
MJMA
MB
ML
Mb
M
~
M
W
Moment Magnitude Mw
Magnitude
)11.1(0.6log
3
2
10 −= ow MM
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Fig 1.15
Seismology

T.K. Datta
 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...
sM
E 158.4
10 +
=
)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
=−=
=−=
=−=
+=
+=
σ
σ
σ
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Seismology

T.K. Datta
 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...
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Seismology

T.K. Datta
Contd...
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Intensity Evaluation Description
Magnitude
(Richter Scale)
I Insignificant Only detected by instruments 1- – 1.9
II Very Light
Only felt by sensitive persons; oscillation of
hanging objects
2 – 2.9
III Light Small vibratory motion 3 – 3.9
IV Moderate
Felt inside building; noise produced by
moving objects
4 – 4.9
V Slightly Strong
Felt by most persons; some panic; minor
damages
VI Strong
Damage to non-seismic resistance
structures
5 – 5.9
VII Very Strong
People running; some damages in seismic
resistant structures and serious damage to
un-reinforced masonry structures
VIII Destructive Serious damage to structures in general
IX Ruinous
Serious damage to well built structures;
almost total destruction of non-seismic
resistant structures
6 – 6.9
X Disastrous
Only seismic resistant structures remain
standing
7 – 7.9
XI
Disastrous in
Extreme
General panic; almost total destruction; the
ground cracks and opens
XII Catastrophic Total destruction 8 – 8.9
Seismology

T.K. Datta
 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

T.K. Datta
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 / electro-
magnetic means
Damp : electromagnetic/
fluid dampers
Fig 1.16
Seismology

T.K. Datta
u
Horizontal pendulum
Vertical pendulum
u
Contd...
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Fig 1.17
Seismology

T.K. Datta
 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...
2
2 (1.18)x kx w x uν+ + =−&& && &&
)19.1(uxorux ∝−= ν
)20.1(2
uxoruxw &&&& ∝−= ν
(1.21)x u or x uν=− µ& && &
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Seismology

T.K. Datta
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

T.K. Datta
 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

T.K. Datta
 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 motion
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Seismology

T.K. Datta
 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

T.K. Datta
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

T.K. Datta
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

T.K. Datta
 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…
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Seismology

T.K. Datta
Contd…
)25ln(80.1859.074.6PGA(gals)ln +−+= rm
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 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

T.K. Datta
 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

T.K. Datta
 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…
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PSHA is carried out in 4 steps.
Seismology

T.K. Datta
 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…
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 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 ambma
m βαλ −== −
Seismology

T.K. Datta
 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
mm
mfM
−−−
−−
=
β
ββ
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 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

T.K. Datta
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
i
iy
S
γλ
yvsyλ
y
 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
λ−
> = −
Seismology

T.K. Datta
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 −=4logλ
mm 2.151.4log −=λ
mm 8.03log −=λ
Lec-5/5
Fig 1.20
Seismology

T.K. Datta
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

T.K. Datta
 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
105.52
117.09
128.66
140.23
P[R=r]
0.0 10
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

T.K. Datta
Size Uncertainty :
631.010
501.010
110
48.03
3
42.15.4
2
414
1
==
==
==
×−
×−
×−
γ
γ
γ
Contd…
)29.1()(
2
)(][
12
21
2
1
21
amm
mm
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

T.K. Datta
 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.
81
0.0
0.8
4.05
4.15
4.25
4.35
4.45
4.55
4.65
4.75
4.85
4.95
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]
4.8
3
7.14
4.17
4.50
5.16
5.49
5.82
6.15
6.48
6.81
Lec-5/9
Fig 1.24
Fig 1.25
Fig 1.26
Seismology

T.K. Datta
 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
γλ
λ
Lec-5/10
[ ]
[ ] 336.004.27
551.019.4
==
==
rP
mP
Seismology

T.K. Datta
 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 λλλλ ++=
Lec-5/11
Seismology

T.K. Datta
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 tλ
λ
[ ] [ ] 0021.0
50
1.01ln)1(1ln
=
−
=
≥−
=
t
NP
λ
λ
Seismology

T.K. Datta
 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 site
Lec-5/13
Seismology

T.K. Datta
 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

T.K. Datta
Contd...
[ ]
( )
( )
( )
1
1
1
1 1
1
1.54
1
( )
1 1 1 ( )
1 1
( )
1
( ) ( / ) (1.30)
exp exp ( ) (1.32 )
ln (1.32 )
47 (1.33)
1
( ) | (1.37)
1
1 ( ) (1.38)
1 (1.39)
o
s u o
s
o
p
s s
o
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
e
F m P M m M m M
e
P M m F m
P M m e
β
β
β
α β
α α
−
−
− −
− −
− −
=
= − −
=
≥ =
−
 = ≤ ≤ ≤ =  −
≥ = −
≥ = −
Lec-5/15
Seismology

T.K. Datta
Microzonation using hazard analysis
Lec-5/16
Seismology

T.K. Datta
Contd...
Lec-5/17
Seismology

T.K. Datta
Contd...
Lec-5/18
0.35 g
0.1 g
0.25 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 Microzonation
Fig 1.27
Seismology

T.K. Datta
Lec-1/74

T.K. Datta
Chapter -2
SEISMIC INPUTS

T.K. Datta
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

T.K. Datta
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

T.K. Datta
 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
Acceleration(g)
Time (sec)
Fig 2.1(a)
1/3
Seismic Input

T.K. Datta
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)
A
c
c
e
l
e
r
a
t
I
o
n
(g
)
Minor (vertical)
Fig 2.1(b)
Fig 2.1(c)
Seismic Input

T.K. Datta
 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 dv
φ( t ) = + ( 2.1)
dy dx
dw
θ( t ) = ( 2.2)
dx
Seismic Input

T.K. Datta
 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
α =Angle of incidence
Fig 2.2
Seismic Input

T.K. Datta
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 n
n=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)
1
a = x( t ) dt ( 2.4)
T
2
a = x( t ) cosω t dt ( 2.5)
T
2
b = x( t ) sinω t dt ( 2.6)
T
ω = 2πn/T ( 2.7)
1/7
Seismic Input

T.K. Datta
 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
2
A = a + b = x( t ) cosω tdt
T
2
+ x( t ) sinω tdt
T
Contd..
nω
1/8
 
 ÷
 
n n
-1 n
n
n
c = A
b
φ = tan ( 2.10)
a
 Equation 2.3 can also be represented in the form
∑
α
0 n n n
n=1
x( t ) = c + c sin(ω t + φ ) ( 2.9)
Seismic Input

T.K. Datta
 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..
nω
∫
∫
α
-iω t
-α
α
iω t
-α
1
x( iω ) = x( t ) e dt ( 2.11)
2π
x( t ) = x( iω) e dω ( 2.12)
1/9
Seismic Input

T.K. Datta
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
2πj
ω =
T
ω = Nπ/T
Contd..
( )
 
 ÷ ÷
 
1/22 2
j j j
j-1
j
j
N
A = a + b j = 0,....., ( 2.13)
2
b
φ = tan ( 2.14)
a
1/10
Seismic Input

T.K. Datta
1/11
Contd..
0 2 4 6 8 10 12 14 16 18 20
0
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

T.K. Datta
 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
OrdinateFourier
amplitude(logscale)
Fig 2.4
Seismic Input

T.K. Datta
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
nπ
ω = = 157.07 rad/s
T
2π
dω = = rad/s
T
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)
GroundAcceleration(g)
Fig 2.5
Seismic Input

T.K. Datta
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)
Realpart
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

T.K. Datta
2 2 1/2
i i i
-1 i
i i n
i
A =( a + b ) i = 0....N/2
b
j = tanω =( 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
iω
iω
Amplitude spectrum
0 20 40 60 80 100 120 140 1600
0.005
0.01
0.015
0.02
Frequency (rad/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

T.K. Datta
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

T.K. Datta
 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
2
n
0
1
λ = c ( 2.16)
2
∑∫
nω N/2
n
n=00
λ = S( ω ) dω = g( ω ) ( 2.17)
2
n
n
c
S(ω ) = & g( ω ) = S( ω ) dω ( 2.18 )
2dω
2/5
Seismic Input

T.K. Datta
 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 70
0
0.5
1
1.5
2
Frequency (rad/sec)
NormalizedPSDFordinate
Fig 2.8
Seismic Input

T.K. Datta
 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.
ω∫
nω
n
n
0
2
0
λ = ω S( ω ) d ( 2.19a)
λ
Ω = ( 2.19b)
λ
 
 ÷
 
&& d
gmax 0
2.8ΩT
u = 2λ ln ( 2.19c)
2π
Contd.. 2/7
Seismic Input

T.K. Datta
 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 1
2 2
x x x x 1 2
1 1
2 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

T.K. Datta
 Records of actual strong
motion records show that
mean square value of the
process is not stationary
but evolutionary.
Contd..
2
S(ω,t ) = q( t ) S( ω ) ( 2.22)
2/9
Time(sec)
σacc
(m/sec2
)
 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

T.K. Datta
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 80 100 120 140 160
0
1
2
3
4
x 10
-6
Frequency (rad/sec)
PSDF(g
2
sec/rad)
Fig 2.10
2/10
Seismic Input

T.K. Datta
 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 50 100 150
0
0.5
1
1.5
2
2.5
3
x 10
-6
Frequency (rad/sec)
PSDF(g2
sec/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

T.K. Datta
 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

T.K. Datta
 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 d
n 0
v
m d
n
t
-ξω( t-τ)
v g d
0 max
1
x( t ) = - x(τ ) e sinω( t - τ ) dτ ( 2.23)
ω
S
x = S = ( 2.24a)
ω
S = x(τ ) e sinω( t - τ ) dτ ( 2.24b)
2
d
1
E= kS ( 2.25a)
2
&
&
2 2
eq d
eq n d
1 1
mx = kS ( 2.25b)
2 2
x =ω S ( 2.25c)
3/2
Seismic Input

T.K. Datta
 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..
2
a n dS =ω S ( 2.26)
( ) 2
s d n d amax
f = kS = mω S = mS ( 2.27)
3/3
Seismic Input

T.K. Datta
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

T.K. Datta
 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..
 
    
 
&
1
2 2 2
n
2E( t )
= x( t ) + (ω x( t ) ) ( 2.29)
m
       
      
       
∫ ∫&& &&
1
2 2 2t t
g n g d
0 0
2E( t )
= x(τ ) cosω τ dτ + x( τ ) sinω τ dτ ( 2.30)
m
3/5
Seismic Input

T.K. Datta
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 3
0
0.8
1.6
2.4
3.2
4
Time period (sec)
Energyspectrum(g-sec)
Fig 2.13
Seismic Input

T.K. Datta
3/7
Contd..
0 20 40 60 80 100 120 140 1600
0.005
0.01
0.015
0.02
Frequency (rad/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)
Accelerationresponsespectrum(g)
Fig 2.15
Fig 2.14
Seismic Input

T.K. Datta
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 gmax
T→∞
a gmax
T→0
limS =u ( 3.33)
limS =u ( 3.34)
3/8
Seismic Input

T.K. Datta
Fig 2.16
3/9
Seismic Input

T.K. Datta
Fig 2.17
3/10
Seismic Input

T.K. Datta
 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

T.K. Datta
 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=ξ
3/12
Seismic Input

T.K. Datta
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 f
T T−
a
Tb
Tc
Td
Te
Tf
T
Seismic Input

T.K. Datta
 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 RSP
3/14
Seismic Input

T.K. Datta
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
&& &
&
&&
2
gmax gmax
gmax 1 gmax 2
gmax
u u
u = c ; u = c
g u
&& &a gmax d gmax v gmaxα u ;α u ;α u
a d vα , α & α
Seismic Input

T.K. Datta
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 70 100
0.001
0.002
0.003
0.004
0.005
0.007
0.01
0.02
0.03
0.04
0.05
0.07
0.1
0.2
0.3
0.4
0.5
0.7
1
2
3
4
5
7
10
aT
bT cT dT eT fT
Disp.(m
)
Pseudovelocity(m/sec)
2
Acc.(m
/sec
)
& mv guα
& mgu
m
D
gu
α
mgu
&&
m
A
g
u
α
&& mgu
Peak ground acceleration,
velocity and displacement
Elastic design spectra
Time period (sec)
Seismic Input

T.K. Datta
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 small
epicentral distance
Large size earthquake at large epicentral distance
Fig 2.21
Seismic Input

T.K. Datta
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..
& && -1
g g
2
g
1.22
u = u = 0.732 ms
g
( 0.732)
u = = 0.546m
0.6g
&&gu = 0.6g
3/18
Seismic Input

T.K. Datta
3/19
Contd..
50th
84th
Fig 2.22
Seismic Input

T.K. Datta
 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

T.K. Datta
 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 spectra
4/1
Seismic Input

T.K. Datta
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
profile
B
bedrock motion
A
D
Bedrock motion
same as point B
Seismic Input

T.K. Datta
 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 spectra
4/3
Seismic Input

T.K. Datta
 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 accelerograms
4/4
Seismic Input

T.K. Datta
 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 ii
a( t ) = A sin(ω t + φ ) ( 2.39)
Seismic Input

T.K. Datta
 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

T.K. Datta
 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 - lnY
p = ( 2.42)
σ
4/7
Seismic Input

T.K. Datta
 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).
4/8
Seismic Input

T.K. Datta
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 & PHV
calculated by different empirical equations
for M=7; r=75 & 120 km .Note that PHA denotes generally
peak ground acceleration and PHV refers to peak ground
Velocity.
4/9
Seismic Input

T.K. Datta
4/10
Contd..
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

T.K. Datta
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
7=wM
is calculated
using Eqn 1.11 as
35.4 is selected
so that it matches El
Centro earthquake. 10
-2
10
-1
10
0
10
1
10
-1
10
0
10
1
10
2
10
3
Frequency (Hz)
Fourieramplitude(cm/sec)
Elcentro
Equation (2.60)
Fig 2.45
Seismic Input

T.K. Datta
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
4/12
Contd..
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

T.K. Datta
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.
4/13
0 10 20 30 40 50 60 700
0.5
1
1.5
2
Frequency (rad/sec)
NormalizedPSDFofacceleration
Housner and Jennings
Newmark and Rousenblueth
Kanai Tazimi
Clough and Penzien
Fig 2.47
Seismic Input

T.K. Datta

Seismic Analysis of Structures - I

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