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SEISMIC RESPONSE
SPECTRUM
By
Dr. Jagadish. G. Kori
Professor & Head Civil Engineering Department
Govt. Engineering College, Haveri-581110
INTRODUCTION
• In order to perform the seismic analysis and design of a structure to be built at a particular location, the actual time history record is required.
• However, it is not possible to have such records at each and every location.
• Further, the seismic analysis of structures cannot be • Further, the seismic analysis of structures cannot be carried out simply based on the peak value of the ground acceleration as the response of the structure depend upon the frequency content of ground motion and its own dynamic properties.
• To overcome the above difficulties, earthquake response spectrum is the most popular tool in the seismic analysis of structures.
INTRODUCTION
• Response spectrum is an important tool in the
seismic analysis and design of structures. It
describes the maximum response of damped
single degree of freedom system to a particular
input motion at different natural periods.input motion at different natural periods.
• Response spectrum method of analysis is
advantageous as it considers the frequency
effects and provides a single suitable horizontal
force for the design of structure.
Methods of Seismic Analysis
• Two basic methods are widely used for dynamic seismic analysis,
namely, Response Spectrum and Time History methods
1. Response Spectrum methods
allows determination of
maximum modal response of a
singly supported structural
system or a multiple supported
system where all supports
receive the same excitation.
Time History of recorded ground acceleration at
Capitola, California in the 1989 Loma Prieta
earthquake 1989
2. Time History method of
analysis permits the
simultaneous application of
different excitations at each
support point of uncoupled
model of the system of
interest .
receive the same excitation.
ORIGIN OF THE RESPONSE SPECTRUM METHOD
• In 1971, with the occurrence of the San Fernando, California, earthquake, the modern era of RSM was launched.
• This earthquake was recorded by 241 • This earthquake was recorded by 241 accelerographs, and by combining these data with all previous strong-motion records it became possible to perform the first comprehensive empirical scaling analyses of response spectral amplitudes.
TIME HISTORY DATA� THE MOST DIRECT DESCRIPTION OF AN
EARTHQUAKE MOTION IN TIME DOMAIN IS
PROVIDED BY ACCELEROGRAMS THAT ARE
RECORDED BY INSTRUMENTS CALLED STRONG
MOTION ACCELEROGRAPHS.
� THE ACCELEROGRAPH RECORDS THREE� THE ACCELEROGRAPH RECORDS THREE
ORTHOGONAL COMPONENTS OF GROUND
ACCELERATION AT A CERTAIN LOCATION.
� THE PEAK GROUND ACCELERATION DURATION,
AND FREQUENCY CONTENT OF EARTHQUAKE
CAN BE OBTAINED FROM AN ACCELEROGRAMS.
AN ACCELEROGRAM CAN BE INTEGRATED TO
OBTAIN THE TIME VARIATIONS OF THE GROUND
VELOCITY AND GROUND DISPLACEMENT.
TIME HISTORY DATATime,
sec Acceleration, g
0.00 0.00630
0.02 0.00364
0.04 0.00099
0.06 0.00428
0.08 0.00758
0.10 0.01087
0.12 0.00682
El Centro ground motion
(N-S Component)
May 18, 1940
0.12 0.00682
0.14 0.00277
0.16 -0.00128
0.18 0.00368
0.20 0.00864
0.22 0.01360
0.24 0.00727
0.26 0.00094
0.28 0.00420
0.30 0.00221http://peer.berkeley.edu/smcat/
http://db.cosmos-eq.org/scripts/default.plx
TIME HISTORY DATA
-0.4
-0.2
0
0.2
0.4
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
ug,g
Time, sec
FOR EARTHQUAKE EXCITATION –
i. ANALYTICAL SOLUTION IS NOT POSSIBLE;
ii. NUMERICAL METHODS ARE EMPLOYED TO FIND OTHER ii. NUMERICAL METHODS ARE EMPLOYED TO FIND OTHER
QUANTITIES LIKE
iii. a. VELOCITY; b. DISPLACEMENT ETC.
DIFFERENT NUMERICAL METHODS ARE:
� CENTRAL DIFFERENCE METHOD
� AVERAGE ACCELERATION METHOD
� NEWMARK’S METHOD ETC.
TIME HISTORY DATA ANALYSIS
DEFORMATION RESPONSE SPECTRUM
Deformation Response SpectrumFor a given EQ excitation calculate |u
max|
from SDOF response with a certain ξ
and within a range of natural periods or
frequencies.
|umax
| for each frequency will be found
from the computed u(t) history at this
frequency.
A plot of |umax
| vs. natural period is
constructed
representing the deformation (or
displacement)
response spectrum (Sd).
From this figure, one can directly read the
maximum relative displacement of any
structure of natural period T (and a
particular value of ξ as damping)
VELOCITY RESPONSE SPECTRUM
Plot of V vs. TN
ACCELERATION RESPONSE SPECTRUM
� Plot of A vs. TN
COMBINED D-V-A SPECTRUM
AA=V =ω D
ωn
n
Tn A =V =
2πD
2π Tn
RESPONSE SPECTRUM
CHARACTERISTICS
Response spectrum ( ζ= 0,2,5,
and 10%) and peak values of
ground acceleration, ground
velocity, and ground
displacement for El Centrodisplacement for El Centro
ground motion.
Response spectrum for El
Centro ground motion plotted
with normalized scale A/ϋgo ,
V/ůgo , and D/ugo ; ζ = 0, 2 , 5
and 10%.
RESPONSE SPECTRUM CHARACTERISTICS
Response spectrum for El
Centro ground motion shown
by a solid line together with an
idealized version shown by a
dashed line; ζ = 5%dashed line; ζ = 5%
Response Spectrum Characteristics
Low Frequency
Out-of-Phase
Response
Mid Frequency
Transition from
Out-of-Phase to
In-Phase Response
High Frequency
In-Phase Rigid Static
Response
Frequency1F
2F
ZPAF
F1
= frequency at which peak spectral acceleration is observed
F2
= frequency above which the SDOF (modal) oscillators are in-phase with the transient
acceleration input used to generate the spectrum and in phase with each other
FZPA
= frequency at which the spectral acceleration returns to the zero period acceleration;
maximum base acceleration of transient acceleration input used to generate the spectrum
ACCELERATION RESPONSE SPECTRUMEL CENTRO EARTHQUAKE 5% DAMPING
� IT IS NOT PRACTICALLY
POSSIBLE TO
CALCULATE EXACT
STRUCTURAL PERIOD .
� SPECTRAL� SPECTRAL
ACCELERATION FOR
SHORT PERIOD IS VERY
IRREGULAR.
� FOR PRACTICAL USE IT
HAS TO MADE ‘SMOOTH’
Elastic Design Spectrum
Use recorded ground motions (available)
Use ground motions recorded at similar sites:
Magnitude of earthquake
Distance of site form earthquake fault
Fault mechanism
Local Soil Conditions Local Soil Conditions
Geology/travel path of seismic waves
Motions recorded at the same location.
For design, we need an envelope. One
way is to take the average (mean) of
these values
DESIGN RESPONSE SPECTRUM
(Design Spectrum may include more than one earthquake scenario)
Factor Influencing Response Spectra The response spectral values depends upon the
following parameters,
•I) Energy release mechanism
•II) Epicentral distance
•III) Focal depth
•IV) Soil condition •IV) Soil condition
•V) Richter magnitude
•VI) Damping in the system
•VII) Time period of the system
RESPONSE SPECTRUM
METHOD OF ANALYSISMETHOD OF ANALYSIS
Introduction
�Response spectrum method is favoured by
earthquake engineering community because of:
� It provides a technique for performing an
equivalent static lateral load analysis.
� It allows a clear understanding of the
contributions of different modes of vibration.contributions of different modes of vibration.
� It offers a simplified method for finding the
design forces for structural members for
earthquake.
� It is also useful for approximate evaluation
of seismic reliability of structures.
Contd…� The concept of equivalent lateral forces for earth-
quake is a unique concept because it converts a
dynamic analysis partly to dynamic & partly to
static analysis for finding maximum stresses.
� For seismic design, these maximum stresses are
of interest, not the time history of stress.
� Equivalent lateral force for an earthquake is
defined as a set of lateral force which will
produce the same peak response as that
obtained by dynamic analysis of structures .
� The equivalence is restricted to a single mode of
vibration.
Contd…
� A modal analysis of the structure is carried out
to obtain mode shapes, frequencies & modal
participation factors.
� Using the acceleration response spectrum, an
� The response spectrum method of analysis is
developed using the following steps.
� Using the acceleration response spectrum, an
equivalent static load is derived which will
provide the same maximum response as that
obtained in each mode of vibration.
� Maximum modal responses are combined to
find total maximum response of the structure.
� The first step is the dynamic analysis while , the
second step is a static analysis.
� The first two steps do not have approximations,
while the third step has some approximations.
� As a result, response spectrum analysis is
called an approximate analysis; but applications
show that it provides mostly a good estimate of
Contd…
show that it provides mostly a good estimate of
peak responses.
� Method is developed for single point, single
component excitation for classically damped
linear systems. However, with additional
approximations it has been extended for multi
point-multi component excitations & for non-
classically damped systems.
Seismic code provisions
� All countries have their own seismic codes.
� For seismic analysis, codes prescribe all three
methods i.e. RSA & seismic coefficient method.
�Codes specify the following important factors for
seismic analysis:seismic analysis:
• Approximate calculation of time period for
seismic coefficient method.
• plot.
• Effect of soil condition on
hC Vs T
a
&h
SAor C
g g
Contd…
• Seismicity of the region by specifying PGA.
• Reduction factor for obtaining design forces
to include ductility in the design.
• Importance factor for structure.
� Provisions of a response spectrum in some country code.
The codes include:
• IBC – 2000
• NBCC – 1995
• EURO CODE – 1995
• NZS 4203 – 1992
• IS 1893 – 2002
Contd…
� IS CODE (1893-2002)
• Time period is calculated by empirical
formula and distribution of force is given by:
∑
2
j j
j b N2
j j
j=1
WhF = V (5.65)
Wh
• are the same; they are given by:a
e
SC vs T & vs T
g
j=1
a
1+15T 0≤T≤0.1sS
= 2.5 0.1≤T≤0.4s for hard soil (5.62)g
10.4≤T≤4.0s
T
Contd…
a
1+15T 0 ≤ T ≤ 0.1sS
= 2.5 0.1≤ T ≤ 0.55s for medium soil (5.63)g
1.360.55 ≤ T ≤ 4.0s
T
1+15T 0 ≤ T ≤ 0.1sS
aS
= 2.5 0.1≤ T ≤ 0.67s for soft soil (5.64)g
1.670.67 ≤ T ≤ 4.0s
T
�For the three types of soil Sa/g are shown in Fig
5.13
�Seismic zone coefficients decide about the PGA
values.
6/6
1.5
2
2.5
3
Hard Soil
Medium Soil
Soft Soil
Sp
ectr
al
acce
lera
tio
n c
oe
ffic
ien
t (S
a/g
)
Contd…
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
Time period (sec)
Sp
ectr
al
acce
lera
tio
n c
oe
ffic
ien
t (S
Variations of (Sa/g) with time period TFig 5.13
Seismic force evaluation
• During base excitation
– Structure is subjected to acceleration
• From Newton’s second law
– Force = mass x acceleration– Force = mass x acceleration
• Hence, seismic force acting on structure
= Mass x acceleration
Seismic force evaluation
• For design, we need maximum seismic force
• Hence, maximum acceleration is required
– This refers to maximum acceleration of structure
– This is different from maximum acceleration of – This is different from maximum acceleration of
ground
– Maximum ground acceleration is termed as peak
ground acceleration, PGA
– Maximum acceleration of rigid structure is same
as PGA.
. . .
Seismic force evaluation
• Seismic force = mass x maximum acceleration– Can be written as:
• Force = (maximum acceleration/g) x (mass x g)
= (maximum acceleration/g) x W– W is weight of the structure– W is weight of the structure
– g is acceleration due to gravity
• Typically, codes express design seismic force as:
V = (Ah) x (W)– V is design seismic force, also called design base shear
– Ah
is base shear coefficient
Seismic force evaluation
• Maximum acceleration of structure depends on
– Severity of ground motion
– Soil conditions– Soil conditions
– Structural characteristics
• These include time period and damping
• More about time period, later
• Obviously, base shear coefficient, Ah, will also
depend on these parameters
Seismic force evaluation
• Seismic design philosophy is such that, design
seismic forces are much lower than actual seismic
forces acting on the structure during severe
ground shakingground shaking
– Base shear coefficient has to ensure this reduction in
forces
• Hence, base shear coefficient would also have a
parameter associated with design philosophy
Seismic force evaluation
• Thus, base shear coefficient depends on:
– Severity of ground motion
– Soil condition
– Structural characteristics– Structural characteristics
– Design philosophy
IS 1893 (Part 1):2002
• Ah
= (Z/2). (I/R). (Sa/g)
– Z is zone factor
– I is importance factor
– R is response reduction factor– R is response reduction factor
– Sa/g is spectral acceleration coefficient
IS 1893 (Part 1):2002
• Zone factor, Z
– Depends on severity of ground motion
– India is divided into four seismic zones (II to V)
– Refer Table 2 of IS 1893(part1):2002– Refer Table 2 of IS 1893(part1):2002
– Z = 0.1 for zone II and Z = 0.36 for zone V
IS 1893 (Part 1):2002
• Importance factor, I
– Ensures higher design seismic force for more
important structures
– Values for buildings are given in Table 6 of – Values for buildings are given in Table 6 of
IS :1893
• Values for other structures will be given in respective
parts
• For tanks, values will be given in Part 2
IS 1893 (Part 1):2002
• Response reduction factor, R
– Earthquake resistant structures are designed for
much smaller seismic forces than actual seismic
forces that may act on them. This depends on forces that may act on them. This depends on
• Ductility
• Redundancy
• Overstrength
– See next slide
MaximumLoad Capacity
To
tal H
oriz
on
tal L
oad
Non linear Response
First
Significant
Yield
Linear Elastic Response
Fy
Fs
Fel
Load at First Yield
Due to
Due to
Redundancy
Due to
Ductility
Maximum force if structure remains elastic
Total Horizontal
Load
ΔIS 1893 (Part 1):2002
Design force
To
tal H
oriz
on
tal L
oad
Roof Displacement (Δ)
Δmax
Fdes
ΔyΔw
Overstrength
0
)(F Force Design
)(F Force Elastic MaximumFactor Reduction Response
des
el
=
IS 1893 (Part 1):2002
• Response reduction factor (contd..)
– A structure with good ductility, redundancy and
over strength is designed for smaller seismic force
and has higher value of Rand has higher value of R
• For example, building with SMRF has good ductility and
has R = 5.0 as against R = 1.5 for unreinforced masonry
building which does not have good ductility
– Table 7 gives R values for buildings
• values will be given in IS:1893 (Part 2)
IS 1893 (Part 1):2002
• Spectral acceleration coefficient, Sa/g
– Depends on structural characteristics and soil
condition
• Structural characteristics include time period and • Structural characteristics include time period and
damping
– Refer Fig. 2 and Table 3 of IS:1893
– See next slide
IS 1893 (Part 1):2002
For 5% damping
IS 1893 (Part 1):2002
• For other damping, Sa/g values are to be multiplied by a factor given in Table 3 of IS:1893– Table 3 is reproduced below
% 0 2 5 7 10 15 20 25 30% damping
0 2 5 7 10 15 20 25 30
Factor 3.20 1.40 1.00 0.90 0.80 0.70 0.60 0.55 0.50
� For higher damping, multiplying factor is less
� Hence, for higher damping, Sa/g is less
