Response Spectrum

Prof. A. Meher Prasad
Department of Civil Engineering
Indian Institute of Technology Madras
email: prasadam@iitm.ac.in

Typical Accelerograms

From Dynamics of Structures by
A K Chopra, Prentice Hall
Time, sec

Response Spectrum

• If the ground moves as per the given accelerogram, what is the
maximum response of a single degree of freedom (SDOF) system
(of given natural period and damping)?

– Response may mean any quantity of interest, e.g., deformation,
acceleration

a(t)/g T=2 sec,
Damping  =2%

Time, sec

Ground motion time history

Response Spectrum (contd…)

• Using a computer, one can calculate the response of SDOF system
with time (time history of response)

• Can pick maximum response of this SDOF system (of given T and
damping) from this response time history
– See next slide


Maximum response = 7.47 in.

T=2 sec,
U(t) Damping  =2%

Time, sec

Time History of Deformation (relative displacement
of mass with respect to base) response

A(t)/g

Time, sec
Ground motion time history


• Repeat this exercise for different values of natural period.

• For design, we usually need only the maximum response.

• Hence, for future use, plot maximum response versus natural
period (for a given value of damping).

• Such a plot of maximum response versus natural period for a given
accelerogram is called response spectrum.


Displacement Response
Spectrum for the above time
A(t)/g
history

Time, sec

T=0.5 sec U(t)
 =2%

T=1.0 sec U(t)
 =2%

Umax
T=2.0 sec U(t)
 =2%

Time, sec T, sec

Figure After Chopra, 2001

RESPONSE SPECTRUM – IS 1893:2002


• Different terms used in the code:

- Design Acceleration Spectrum (clause 3.5)
– Response Spectrum (clause 3.27)
– Acceleration Response Spectrum (used in cl. 3.30)
– Design Spectrum (title of cl. 6.4)
– Structural Response Factor
– Average response acceleration coefficient (see
terminology of Sa/g on p. 11)
– Title of Fig. 2: Response Spectra for ….

Smooth Response Spectrum

• Real spectrum has somewhat irregular shape with local peaks
and valleys

• For design purpose, local peaks and valleys should be ignored

– Since natural period cannot be calculated with that much
accuracy.

• Hence, smooth response spectrum used for design purposes

• For developing design spectra, one also needs to consider other
issues.

Smooth Response Spectrum (contd…)

Period (sec) Period (sec) Period (sec)
Acceleration Spectra Velocity Spectra Displacement Spectra

Shown here are typical smooth spectra used in design for different
values of damping (Fig. from Housner, 1970)

Floor Response Spectrum

• Equipment located on a floor needs to be designed for the motion
experienced by the floor.

• Hence, the procedure for equipment will be:

– Analyze the building for the ground motion.
– Obtain response of the floor.
– Express the floor response in terms of spectrum (termed as
Floor Response Spectrum)
– Design the equipment and its connections with the floor as per
Floor Response Spectrum.

Response Spectrum versus Design Spectrum

• Consider the Acceleration Response Spectrum
• Notice the region of red circle marked: a slight change in natural
period can lead to large variation in maximum acceleration
Spectral Acceleration, g

Undamped Natural Period T (sec)

Response Spectrum versus Design Spectrum (contd…)

• Natural period of a civil engineering structure cannot be calculated
precisely

• Design specification should not very sensitive to a small change in
natural period.

• Hence, design spectrum is a smooth or average shape without local
peaks and valleys you see in the response spectrum

Design Spectrum

• Since some damage is expected and accepted in the structure
during strong shaking, design spectrum is developed considering
the overstrength, redundancy, and ductility in the structure.

• The site may be prone to shaking from large but distant earthquakes
as well as from medium but nearby earthquakes: design spectrum
may account for these as well.
– See Fig. next slide.

Design Spectrum (contd…)

• Design Spectrum must be accompanied by:

– Load factors or permissible stresses that must be used
• Different choice of load factors will give different seismic
safety to the structure

– Damping to be used in design
• Variation in the value of damping used will affect the design
force.

– Method of calculation of natural period
• Depending on modeling assumptions, one can get different
values of natural period.

– Type of detailing for ductility
• Design force can be lowered if structure has higher ductility.

Design Spectrum (contd…)

• 1984 code provided slightly different design spectrum for two
methods
– Seismic Coefficient Method (static method), and
– Response Spectrum Method (dynamic method)

• It was confusing to use two different sets of terminology for two
methods.

• Present code provides same design spectrum irrespective of
whether static or dynamic method is used.

IS:1893-1984

• Design base shear for a building by Seismic Coefficient Method was
calculated as
Vb= oIKCW

C

Natural Period (sec)

• In a way, one could say that the design spectrum for the seismic
coefficient method in the 1984 code was given by oIKC

IS:1893-1984 (contd…)

• In the Response Spectrum Method, the design spectrum was given
by FoIK(Sa/g)
Sa/g = Average Acceleration Coefficient

Natural Period (sec)

Major Changes in Design Spectrum

• Zone Factor (Z) is specified in place of o and Fo
• Importance Factor (I) is same
• Soil Effect is considered by different shapes of response spectrum;
Soil-Foundation Factor () has now been dropped.
• Response Reduction Factor (R) used in denominator; earlier
Performance Factor (K) was used in numerator.
– For more ductile structures, K was lower.
– Now, R will be higher for more ductile structures.

• Structure Flexibility Factor (Sa/g); earlier C or Sa/g

Soil Effect

• Recorded earthquake motions show that response spectrum shape
differs for different type of soil profile at the site

Period (sec)

Fig. from Geotechnical Earthquake
Engineering, by Kramer, 1996

Shape of Design Spectrum

• The three curves in Fig. 2 have been drawn based on general
trends of average response spectra shapes.

• In recent years, the US codes (UBC, NEHRP and IBC) have
provided more sophistication wherein the shape of design spectrum
varies from area to area depending on the ground motion
characteristics expected.

IS1893:2002

 Local soil profile reflected through a different design spectrum for Rock , Soil
 Normalized for Peak Ground Acceleration (PGA) of 1.0

Rocky or hard sites,
1 + 15 T 0.00 ≤ T ≤ 0.10
Sa / g = 2.50 0.10 ≤ T ≤ 0.40
1.00 / T 0.40 ≤ T ≤ 4.00

Medium soil sites
1 + 15 T 0.00 ≤ T ≤ 0.10 Damping 5%

Sa / g = 2.50 0.10 ≤ T ≤ 0.55
1.36 / T 0.55 ≤ T ≤ 4.00

Soft soil sites
1 + 15 T 0.00 ≤ T ≤ 0.10
Sa / g = 2.50 0.10 ≤ T ≤ 0.67
1.67 / T 0.67 ≤ T ≤ 4.00

Damping 0 2 5 7 10 15 20 25 30
percent
Factors 3.2 1.4 1.00 0.90 0.80 0.70 0.60 0.55 0.50
(new code)

BACKGROUND

Discussed in SDOF System

Spectral Quantities…

This may also be viewed as the equivalent lateral static force which
produces the same effects as the maximum effects by the ground
shaking.
It is sometimes convenient to express Qmax in the form ,

Qmax  CW (B18)

Where W = mg is the weight of the system. The quantity C is the so
called lateral force coefficient, which represents the number of times
the system must be capable of supporting its weight in the direction of
motion.

From Eqn.B17 and B18 it follows that, C=A/g (B19)

Spectral Quantities…

Another useful measure of the maximum deformation, U, is the pseudo
velocity of the system, defined as, V = p U (B20)

The maximum strain energy stored in the spring can be expressed in terms
of V as follows:
Emax = (1/2) (k U) U = (1/2) m(pU)2 = (1/2)mV2 (B21)

Under certain conditions, that we need not go into here, V is identical to ,or
approximately equal to the maximum values of the relative velocity of the
mass and the bays, U and the two quantities can be used interchangeably.

However this is not true in general, and care should be exercised in
replacing one for the other.

Deformation spectra

1.Obtained from results already presented
2.Presentation of results in alternate forms
(a) In terms of U
(b) In terms of V
(c) In terms of A
3.Tripartite Logarithmic Plot

General form of spectrum

..
It approaches U = y0 at extreme left; a value of A  y0 extreme right;

It exhibits a hump on either side of the nearly horizontal central
portion;and attains maximum values of U, V and .. which may be
.
A
materially greater than the values of y0 , y 0 , and y0

It is assumed that the acceleration trace of the ground motion,and
hence the associated velocity and displacement traces, are smooth
continuous functions.

The high-frequency limit of the response spectrum for discontinuous
acceleration inputs may be significantly higher than the value referred
to above,and the information presented should not be applied to such
inputs.

The effect of discontinuous acceleration inputs is considered later.

Generation of results

• General form of spectrum is as shown in next slide

(a) It approaches V= y0 at the extreme left; value of A  &&0 at the
y

extreme right ; it exhibits a hump on either side of the nearly
horizontal central portion; and attains maximum values of U, V and
y0 , y0 and &&0
& y

A, which may be materially greater than the values of
(a) It is assumed that the acceleration force of the ground motion,
respectively.
and hence the associated velocity and displacement
forces, are smooth continuous functions.

(c) The high frequency limit of the response spectrum for discontinuous
acceleration inputs may be significantly higher than the value referred
to above, and the information presented should not be applied to such
inputs.

Acceleration spectra for elastic
system - El Centro Earthquake

SDF systems with 10%
damping subjected to El
centro record

Base shear coefficient, C

Building
Code

Natural period,secs

Spectral Regions

The characteristics of the ground motion which control the deformation of
SOF systems are different for different systems and excitations. The
characteristics can be defined by reference to the response spectrum for
the particular ground motion under consideration .

Systems the natural frequency of which corresponds to the
Inclined left-hand portion of the spectrum are defined as low-frequency
systems;
Systems with natural frequencies corresponding to the nearly horizontal
control region will be referred to as a medium-frequency systems ; and
Systems with natural frequencies corresponding to the inclined right
handed portion will be referred to as high-frequency systems.

Spectral Regions…

Minor differences in these characteristics may have a significant effect on
the magnitude of the deformation induced.

Low frequency systems are displacement sensitive in the sense that their
maximum deformation is controlled by the characteristics of the
displacement trace of the ground motion and are insensitive to the
characteristics of an associated velocity and displacement trace:

Ground motions with significantly different acceleration and velocity traces
out comparable displacement traces induce comparable maximum
deformations in such systems.

Spectral Regions…

The boundaries of the various frequency regions are different
for different excitations and, for an excitation of a particular
form, they are a function of the duration of the motion.

It follows that a system of a given natural frequency may be
displacement sensitive, velocity sensitive or acceleration
sensitive depending on the characteristics of the excitation to
which it is subjected .

Logarithmic plot of Deformation Spectra

It is convenient to display the spectra or a log-log paper, with the
abscissa representing the natural frequency of the system,f, (or some
dimensionless measure of it) and the ordinate representing the pseudo
velocity ,V (in a dimensional or dimensionless form).

On such a plot ,diagonal lines extending upward from left to right
represent constant values of U, and diagonal lines extending downward
from left to right represent constant values of A. From a single plot of
this type it is thus possible to read the values of all three quantities.

Advantages:
• The response spectrum can be approximated more readily and
accurately in terms of all three quantities rather than in terms of a
single quantity and an arithmetic plot.
• In certain regions of the spectrum the spectral deformations can more
conveniently be expressed indirectly in terms of V or A rather than
directly in terms of U. All these values can be read off directly from the
logarithmic plot.

Logarithmic plot of Deformation Spectra

Velocity
sensitive
Displacement
sensitive V0
D0 y0
&
V y0 y
&&0 Acceleration
sensitive
Log
scale A0

U
A

Natural Frequency, f (Log scale)

General form of spectrum

Deformation Spectra for Half-cycle Acceleration pulse:

This class of excitation is associated with a finite terminal velocity
and with a displacement that increases linearly after the end of the
pulse.

Although it is of no interest in study of ground shock and earthquakes
,being the simplest form of acceleration diagram possible ,it is
desirable to investigate its effect.

When plotted on a logarithmic paper, the spectrum for the half sine
acceleration pulse approaches asymptotically on the left the value.
V  yo&

This result follows from the following expression presented earlier for
fixed base systems subjected to an impulsive force,
I
X max 
mp
t1

where I   P (t ) dt
0

t1

Letting P(t )   m &&(t ) and
y X max  U and noting that
 &&(t ) dt  y
0
y &
o

y
&
we obtain, U  o or V  y o
&
p
( This result can also be determined by considering the effect of an
instantaneous velocity change, yo ,i.e. an acceleration pulse of finite
&
magnitude but zero duration. The response of the system in this case
is given by, uo
&
u(t )  uo cos pt  sin pt
p
Considering that the system is initially at rest, we conclude that,
uo  0 and uo   yo
& &
yo
&
where, u(t )   sin pt
p
The maximum value of u(t), without regards to signs, is
yo
&
U  or V  y o )
&
p

Spectra for maximum and minimum accelerations of the mass
(undamped elastic systems subjected to a Half cycle
Acceleration pulse)

Spectra for maximum and minimum acceleration of the mass
(undamped Elastic systems subjected to a versed-sine velocity
pulse)

Deformation spectra for undamped elastic systems
subjected to a versed-sine velocity pulse

‘B’ Level Earthquake (=10% ; μ=1.0)

Deformation spectrum for undamped Elastic systems
subjected to a half-sine acceleration pulse

Example:

For a SDF undamped system with a natural frequency,f=2cps,evaluate
the maximum value of the deformation,U when subjected to the half
sine acceleration pulse. Assume that &&0  0.5 g ,t1=0.1sec. Evaluate
y
also the equivalent lateral force coefficient C, and the maximum spring
force,Q0
ft1= 2 x 0.1 = 0.2
From the spectrum, .
V ; y0

Therefore
2 .. 2 1
2p fU ﾻ f1 y 0  ﾴ 0.1ﾴﾴ 9.81
p p 2
1 0.1 1
U ﾻ 2 ﾴﾴﾴ 9.81  0.024
p 2 2
2 ..
4p ﾴﾴ t 1ﾴ y0
.
A 2p fV 2p ﾴ 2 ﾴ y0 p 8t1 ﾴ 0.5 g
C      8 ﾴ 0.1ﾴ 0.5  0.4
g g g g g
Q0  CW  0.4W

A
Alternatively,one can start reading the value .. from the spectrum
y0
proceeding this may, we find that
A
..
 0.5
y0

A 0.8  1 2  g
Accordingly, C    0.4
g g
Q0  0.4W
..
A 0.8 y0 0.8  0.5  9.81
and U     0.024 m
p 2
p 2
4p  2
2 2

V A
The value of . and .. as read from the spectrum are
y0 y0 A
approximate. The exact value of .. determined is
y0
0.7. This leads to C  0.385 Q0  0.385W and U  0.025

If the duration of the pulse were t1 = 0.75 sec instead of 0.1 sec , the
results would be as follows

ft1  2 ﾴ 0.75  1.5
therefore, A  1.5
..
y0
A 1.5 ﾴ 0.5 g
C   0.75
g g
Q0  0.75W

A 1.5 ﾴ 0.5 ﾴ 9.81
U   0.047 m
p 2
4p ﾴ 2
2 2

If the duration of the pulse were t1,as in the first case, but the natural
frequency of the system were 15 cps instead of 2 cps, the results would
have been as follows: ft1=15 * 0.1=1.5

A A
 1.5 C  0.5  1.5  0.75
Therefore, y
&& g
Q  0.75W

A 0.75  9.81
and U 2   0.00082m
4p   15
2
p 2

• Plot spectra for inputs considered in the illustrative example and compare

y0 For t1=0.75sec
&

y0 For t1=0.1sec
&

V
..
y0 Same as in
both cases

f

• The spectrum for the longer pulse will be shifted upward and to the left by a
factor of 0.75/0.10 = 7.5

Design Spectrum

xmax A
May be determined from the spectrum by interpreting as &&
 xst  0 y
When displayed on a logarithmic paper with the ordinate representing V and
the abscissa f, this spectrum may be approximated as follows:

(Log scale)

=1.5

(Log scale)

Deformation Spectra for Half-Cycle Velocity Pulses

Refer to spectrum for   0
Note the following
• At extreme right A  &&0 . Explain why?
y
• Frequency value behind which A  &&0 is given by ftr= 1.5
y
• y
The peak value of A=2 x 1.6 &&0 Explain why?
In general for pulses of the same shape and duration with different
n
peak values A   ( &&0 ) 2j
j 1
y

• If duration on materially different

be conservative. Improved estimate may be obtained by considering
relative durations of the individual pulses and superposing the peak
component effects.The peak value of V is about 1.6 yo

It can be shown that the absolute maximum value of the amplification
factor V y0 for a system subjected to a velocity trace of a given shape is
0

approximately the same as the absolute maximum value of A0 &&0 for an
y

acceleration input of the same shape.

This relationship is exact when the maximum response is attained
following application of the pulse. But it is valid approximately even
when the peak responses occur in the forced vibration era.

The maximum value of U is yo and the spectrum is bounded on the left
by the diagonal line U = yo

It should be clear that,

(c) The left-hand, inclined portion of the spectrum to displacement
sensitive.

(e) The middle, nearly horizontal region of the spectrum is governed
by the peak value of the velocity trace. It is insensitive to the shape
of the pulse which can more clearly be seen in the acceleration
trace.
(c) The right hand portion is clearly depended on the detailed
features of the acceleration trace of the ground motion. In all
cases, the limiting value of on the right is equal to t1 / td.These limits
appear different in the figure because of the way in which the
results have been normalized.

Note that the abscissa is non-dimensionalised and the ordinate with
respect to the total duration of the pulse and the ordinate with
respect to the maximum ground velocity. It follows that to smaller
y
values of &&0 corresponds to larger values of peak acceleration

Design Rules

Design spectrum for the absolute maximum deformation of
systems subjected to a half cycle velocity pulse
(--undamped elastic systems;continuous input
acceleration functions)

Deformation spectra for undamped elastic systems
subjected to skewed versed-sine velocity pulses

Deformation Spectra for Half-cycle Displacement Pulse

See spectrum for undamped systems, =0, on the next page

Note that:

(a) The RHS of the spectrum is as would be expected from the remarks

already made.

(h) Peak value of V is approximately 3.2 yo. This would be expected, as
the velocity trace of the ground motion, has two identical pulses.

(c) At the extreme left and of the spectrum, U=y0. The system in this
case is extremely flexible and the ground displacements is literally
absorbed by the spring.

Design Rules

Design spectrum for maximum deformation of systems
subjected to a half cycle displacement pulse

However the spectrum is no longer bounded on the left by the line
U= yo, but exhibits a hump with peak value of U0 = 1.6 y0

It can be shown that the peak value of U / y0 for a system subjected
to a displacement trace is approximately the same as the peak
value of V / y0, induced by a velocity input of the same shape.

Further more the peak value of U occurs at the same value of the
dimensionless frequency parameter, f1 as the peak value of V.

However it is necessary to interpret t1 as the duration of the
displacement pulse, rather than of that of velocity pulse.

Deformation spectra for damped elastic systems subjected
to a half cycle displacement pulse

Deformation spectra for full cycle Displacement pulse

As would be expected ,the maximum value of U in this case is
approximately 3.2 yo .Furthermore, the left hand portion of the
spectrum consists of three rather than two distinct parts:

(a) The part on the extreme left for which U = yo .This corresponds to

the first maximum,which occurs at approximately the instant that
y attains its first maximum.

(f) The smooth transition curve which defines the second
maximum. This maximum occurs approximately at the instant that

y(t) attains its second extremum, and is numerously greater than
the peak value of the second pulse of the contribution of the first
pulse.

Effect of Discontinuous Acceleration Pulses

The high frequency limit of the deformation spectrum is sensitive to
whether the acceleration force of the ground motion is a continuous or
discontinuous diagram.
A y
The limiting value given priority applies only to continuous &&0
functions
The sensitivity of the high-frequency region to the detailed
characteristics of the acceleration input may be appreciated by
reference to the spectra given in the following these pages.

These spectra provide further confirmation to the statement made
previously to the effect that low-frequency and medium-frequency
systems are insensitive to the characteristics of the acceleration force
of the ground motion.

Explain high-frequency response to discontinuous functions.

Deformation spectra for damped elastic systems subjected
to a full cycle displacement pulse

Application to Complex Ground Motions

• Compound Pulses
• Earthquake Records
Eureka record
El-Centro record

Design Spectrum
Minimum number of parameters required to characterize the design
ground motion &&, y and y
y &
Max values of &&, y and y
y &
The predominant frequency (or deviation) of the dominant pulses in

The degree of periodicity for (the number of dominant pulses in) each
diagram.
Dependence of these characteristics on
Local soil conditions
Epicentral distance and
Severity of ground shaking

Effect of damping:

• Effect is different in different frequency ranges
• Effect is negligible in the extremely low frequency regime (U = y0)
..
and extreme high frequency ranges (A = y0).
..
u + p2u = y0(t)
.. ..
low frequency u = y(t) .. 0 = y0
u ..
high frequency p2u = A(t) = y(t) A = y0

Eureka, California earthquake of Dec 21,1954 S 11o E
component.

Elcentro ,California Earthquake of May 18,1940,N-S component

V
= pseudo velocity
Yc Maximum Ground Velocity

Undamped Natural Frequency, f, cps

Further discussion of Design Response Spectra

The specification of the design spectrum by the procedure that has
been described involves the following basic steps:

1. Estimating the maximum values of the ground acceleration,
ground velocity and ground displacement. The relationship
.. .
between y0, y0, y0 is normally based on a statistical study of
existing earthquake records. In the Newmark – Blume – Kapur
paper (“Seismic Design spectra for Nuclear Power Plants”, Jr. of
Power Division, ASCE, Nov 1973, pp 287-303) the following
relationship is used.

0.3 m : 0.7 m/sec : 1g for rock

0.9 m : 1.2 m/sec : 1g for Alluvium

1. Estimating the maximum spectral amplification factors, αD, αV, αA ;
for the various parts of the spectrum.

Again these may be based on statistical studies of the respective
spectra corresponding to existing earthquake records.

The results will be a function not only of the damping forces of the
system but also of the cumulative probability level considered.

Following are the values proposed in a recent unpublished paper
by Newmark & Hall for horizontal motions:

Damping One sigma (84.1%) Median (50%)
%critical αD αV αA αD αV αA
0.5 3.04 3.84 5.10 2.01 2.59 3.65
1 2.73 3.38 4.38 1.82 2.31 3.21
2 2.42 2.92 3.66 1.63 2.03 2.74
3 2.24 2.64 3.24 1.52 1.86 2.46
5 2.01 2.30 2.71 1.39 1.65 2.12
7 1.85 2.08 2.36 1.29 1.51 1.89
10 1.69 1.84 1.99 1.20 1.37 1.64
20 1.38 1.37 1.26 1.01 1.08 1.17

Ground Acceleration

• Number of empirical relations available in literature to correlate
shaking intensity with Peak Ground Acceleration (PGA)

• Table on next slide gives some such values.

• Notice that the table gives

– Average values of PGA; real values may be higher or lower
– There is considerable variation even in the average values
by different empirical relations.

Table

Average horizontal peak ground acceleration as a function of earthquake intensity

Intensity (MM Acceleration (as a fraction of g)
Scale)
Empirical Relations

Gutenberg Newmann, Trifunac and Trifunac and Newmann, Murphy
and 1954 Brady, 1975 Brady, 1977 1977 (revised and
Richter, (revised by by Murphy O’Brien,
1956 Murphy and and O’Brien, 1977
O’Brien, 1977)
1977)

V 0.015 0.032 0.031 0.021 0.022 0.032

VI 0.032 0.064 0.061 0.046 0.053 0.056

VII 0.068 0.13 0.12 0.10 0.13 0.10

VIII 0.146 0.26 0.24 0.23 0.30 0.18

IX 0.314 0.54 0.48 0.52 0.72 0.32

Ground Acceleration

• ZPA stands for Zero Period Acceleration.
– Implies max acceleration experienced by a structure having zero
natural period (T=0).

Zero Period Acceleration

• An infinitely rigid structure
– Has zero natural period (T=0)
– Does not deform:
• No relative motion between its mass and its base
• Mass has same acceleration as of the ground
• Hence, ZPA is same as Peak Ground Acceleration

Example: Determine the response spectrum for a design earthquake
with &&  0.3g ye  0.3 m / sec and y0  0.25 m. Take   0.05 and use the
y &
amplification factors given in the preceding page. Take the knee of
amplified constant acceleration point of this spectrum at 8 cps and the
point beyond which A  &&0 at 25 cps
y

d 0.3 x 2.30 = 0.69
e

25
0.3g x 2.71 =0.813 g
50 2.3
0.
=

f
A = 0.3g
01
2.

& 
y0 =0.3 m/sec 2.71
x

V 2.01
25

C = 0.3
0.

&&0 
y =0.3g 0.3g

y0=0.25 m
Q = 0.3W
Y=0.00127
  0.05

0.22 cps 1.81 cps 8 cps 25 cps
f

Note: In the spectra recommended in the Newmark – Blume -Kapur
paper, the line de slope upward to the left and the line of slopes
further downward to the right

Design Earthquakes
Describing the Earthquake

Ground Motion Time Histories

 Ground motion time histories are numerical descriptions of how a certain
ground motion parameter, such as acceleration, varies with time.

 They provide a full description of the earthquake motion, unlike response spectra,
as they show duration as well as amplitude and frequency content.

 They are usually expressed as plots of the ground motion parameter versus time,
but consist of discrete parameter-time pairs of values.

 Idealized time histories are sometimes represented by simple mathematical
functions such as sine waves, but real earthquake motions are far too complex
to be represented mathematically.

 There are two general types of time histories:
- Recorded (often referred to as historical records)
- Artificial

Statistically Derived Design Spectra

 The general procedure for generating statistically derived spectra is as follows:

 Classes of ground motions are selected (based on soil, magnitude, distance, etc.)

 Response spectra for a large number of corresponding ground motions are
generated and averaged

 Curves are fit to match computed mean spectra

 Resulting equations are used to develop a design response spectrum with desired
probability of exceedence

Effect of various factors on spectral values
Soil Conditions

 For soft soils, ag remains the same or
decreases relative to firm soil,but
vg and dg increase, generally.

 Layers of soft clay, such as the Young
Bay Mud found in the San Francisco
Bay area, can also act as a filter,
and will amplify motion at the period
close to the natural period of the soil
deposit.

 Layers of deep, stiff clay can also have
a large effect on site response.

 For more information on site effects, see
Geotechnical Earthquake Engineering
by Kramer.

Near Fault Motions and Fault Rupture Directivity
For near-fault motions ag increases,
but vg increases more dramatically due to
effect of a long period pulse.

This pulse is generally most severe in the
fault normal direction (as it can cause fling),
but significant displacement also occurs in
the fault parallel direction.

The fault parallel direction usually
has much lower spectral acceleration and
velocity values than the fault normal direction.

Sample waveforms are located in a
previous section of the notes,
Factors Influencing Motion at a Site.

No matter the directivity, however,
the motions very close to the
fault rupture tend to be more severe
than those located at moderate distances.

Near Fault Motions and Fault Rupture Directivity (Cont..)

Somerville et al. have developed a
relationship which converts mean
spectral values generated from
attenuation relationships to either the
fault parallel or fault normal component
of ground motion.

See the shift of the spectrum in the
long period range.

Viscous Damping

Friction between and with structural and non-structural elements

Localized yielding due to stress concentrations and residual stresses
under low loading and gross yielding under higher loads

Energy radiation through foundation

Aeroelastic damping

Viscous damping

Analytical modeling errors

Viscous Damping

 Viscous Damping Values for Design

 Many codes stipulate 5% viscous damping unless a more properly
substantiated value can be used.

 Note that actual damping values for many systems, even at higher
levels of excitation are less than 5%.

Effect of Various Factors on Spectral Values
Modifying the Viscous Damping of Spectra

Newmark and Hall's Method

For each range of the spectrum, the spectral values are multiplied by the ratio
of the response amplification factor for the desired level of damping to the
response amplification factor for the current level of damping.

 Consider if we have a median spectrum
at 5% viscous damping and we would
like it at x%.

 If the 5% Joyner and Boore
Sv value is 60 cm/sec on the descending
branch, an estimate of the 2% Sv value
is 60x(2.03/1.65) = change 60x1.47
= 88 cm/sec

Elasto Plastic Force Elasto Plastic system and its
Deformation relation corresponding linear system

Design Values of normalized yield Strength

Construction of Elastic Design Spectrum

Construction of Inelastic Design Spectrum

Response of Elastoplastic system to Elcentro Ground motion

Empirically Derived Design Spectra
Basic Concepts

 The complexity of the previous methods, and the limited number of
records available two decades ago, led many investigators to develop
empirical methods for developing design spectrum from estimates of
peak or effective ground motion parameters.

 These relationships are based on the
concept that all spectra have a
characteristic shape, which is shown
here.


 N. M. Newmark and W. J. Hall's procedure
for developing elastic design spectra starts
with the peak values of ground acceleration,
velocity, and displacement.

 These values are used to generate a baseline
curve that the spectrum will be generated from.

 The values of peak ground acceleration and
velocity should be obtained from a A typical baseline curve plotted on
deterministic or probabilistic seismic hazard tripartite axes is shown above.
analysis

 The value of peak ground displacement is a
bit more difficult to obtain due to the lack of
reliable attenuation relationships.

 Some empirical functions utilizing the
PGA are available to provide additional
estimates of the peak ground displacement.

Empirically Derived Design Spectra (Cont..)

Structural Response Amplification Factors

Structural response amplification factors are then applied to the different
period-dependent regions of the baseline curve

Structural response amplification factors
Damping
Median + One Sigma
(% critical)
a v d a v d
1 3.21 2.31 1.82 4.38 3.38 2.73
2 2.74 2.03 1.63 3.66 2.92 2.42
3 2.46 1.86 1.52 3.24 2.64 2.24
5 2.12 1.65 1.39 2.71 2.3 2.01
7 1.89 1.51 1.29 2.36 2.08 1.85
10 1.64 1.37 1.2 1.99 1.84 1.69
20 1.17 1.08 1.01 1.26 1.37 1.38


Tripartite Plots:

Newmark and Hall's spectra are plotted on a four-way log plot called a tripartite plot.

This is made possible by the simple relation between spectral acceleration,
velocity, and displacement: Sa/w = Sv = Sdw

A tripartite plot begins as a log-log plot of spectral velocity versus period as shown.


 Then spectral acceleration and spectral displacement axes are superimposed
on the plot at 45 degree angles


 All three types of spectrum (Sa vs. T, Sv vs. T, and Sd vs. T) can be plotted
as a single graph, and three spectral values for a particular period can easily
be determined.

 The Sa, Sv, and Sd values for a period of 1 second are shown below.

Constructing Newmark and Hall Spectra

1. Construct ground motion 'backbone' curve using constant agmax, vgmax,
dgmax lines. Take lower bound on three curves (solid line on figure)

Response Spectrum

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