Elastic response spectra are used to estimate the response of linear structures to earthquakes. They plot the peak response of oscillators with varying natural frequencies forced by the same ground motion. Site-specific spectra are developed using recorded ground motions for a site and account for local soil conditions, while code-based spectra provide a standardized approximation. Response spectra are useful for elastic design but have limitations for nonlinear or multi-mode systems.

1. CE-658
EARTHQUAKE ENGINEERING
ELASTIC RESPONSE SPECTRA
Submitted by Alban KURIQI

2. ELASTIC RESPONSE SPECTRA
GENERAL INFORMATION
USES OF ELASTIC RESPONSE SPECTRA
SITE-SPECIFIC ELASTIC DESIGN SPECTRA
STATISTICALLY DERIVED RESPONSE SPECTRA
EMPIRICALLY DERIVED RESPONSE SPECTRA

3. GENERAL INFORMATION
• A response spectrum is simply a plot of the peak or steady-state
response (displacement, velocity or acceleration) of a series of
oscillators of varying natural frequency, that are forced into motion
by the same base vibration or shock. The resulting plot can then be
used to pick off the response of any linear system, given its natural
frequency of oscillation. One such use is in assessing the peak
response of buildings to earthquakes.

4. GENERAL INFORMATION
• Response spectra can also be used in assessing the response of
linear systems with multiple modes of oscillation (multi-degree of
freedom systems), although they are only accurate for low levels of
damping. Modal analysis is performed to identify the modes, and
the response in that mode can be picked from the response
spectrum.
• The science of strong ground motion may use some values from
the ground response spectrum (calculated from recordings of
surface ground motion from seismographs) for correlation with
seismic damage.
• If the input used in calculating a response spectrum is steady-state
periodic, then the steady-state result is recorded. Damping must be
present, or else the response will be infinite. For transient input
(such as seismic ground motion), the peak response is reported.
Some level of damping is generally assumed, but a value will be
obtained even with no damping.

5. GENERAL INFORMATION
. This peak response is then combined to estimate a total response. A
typical combination method is the square root of the sum of the
squares (SRSS) if the modal frequencies are not close.
The result is typically different from that which would be calculated
directly from an input, since phase information is lost in the process
of generating the response spectrum.
The main limitation of response spectra is that they are only universally
applicable for linear systems. Response spectra can be generated
for non-linear systems, but are only applicable to systems with the
same non-linearity, although attempts have been made to develop
non-linear seismic design spectra with wider structural application.
The results of this cannot be directly combined for multi-mode
response.

6. Uses of Elastic Response Spectra
• Elastic design response spectra are extremely useful to structural
engineers. These spectra are the basis for:
• Computing design displacements and forces in systems expected
to remain elastic
• Developing design forces and displacement systems that respond
in elastically by:
- Modifying elastic spectrum
- Evaluating response of equivalent elastic structure
• These elastic spectra can be obtained by several methods, which
are:
• Processing of site specific ground motion time histories
• Statistical relationships
• Empirical relationships
• Code stipulations

7. Uses of Elastic Response Spectra
• Elastic spectra can be presented in several formats, depending on
the needs of the engineer and what information is being presented.
Some of the most common formats are:
• Spectral acceleration vs. period
• Spectral velocity vs. period
• Spectral displacement vs. period
• Spectral acceleration vs. spectral displacement (capacity design
spectrum)
• Tripartite plots (Sa, Sv, and Sd vs. period)
• Also, any of the above (except the capacity design spectrum) can
be plotted versus frequency rather than period.
• Factors which effect elastic spectra include the damping ratio, site
conditions, and near fault ground motion effects such as rupture
directivity.

8. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA
Response spectra for actual ground motions are quite irregular, as
shown below. Do not use them for design — they can be used for
analysis to assess the response to a particular earthquake.
Where site specific ground motions have been compiled, the response
spectra for each record can be averaged. The resulting "mean“
spectrum will be smooth. The COE can be used to establish a
spectrum with a desired probability of exceedance.

9. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA
• TYPES OF DESIGN RESPONSE SPECTRA
a. Probability level. Design response spectra
• are usually based statistically either on the mean, median (50th
percentile probability level), or the median plus one standard
deviation (84th percentile probability level), of the ground motion
parameters or the records chosen.
• Design response spectra used for design of new RCC dams or for
evaluation of the safety and serviceability of existing dams shall be
based on the mean level of the ground motion parameters

10. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA
• b. Type of spectrum required. Either a “site-specific”
• or a “standard” design response spectra shall be used to describe
the design earthquakes.
• The type required shall be based on the seismic zone, the proximity
of the seismic source, and the maximum height of the dam.
• c. Site-specific design response spectra.
• The site-specific design response spectra should be developed
based on earthquake source conditions, propagation path
properties, and local foundation characteristics associated with the
specific site.
• This type of design spectra may be established by anchoring a
selected response spectral shape for the site to the estimated peak
ground acceleration, or by estimating

11. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA
• HORIZONTAL AND VERTICAL DESIGN
• a. Site-specific design response spectra.
• When site-specific design response spectra are required in
accordance with paragraph 5-5c, two independent design response
spectra shall be developed, one to define the horizontal component
of ground motion, and the second to define the vertical component.
• The vertical component of ground motion usually contains much
higher frequency content than the horizontal component, therefore
the spectral shape is quite different than that of the horizontal
component.

12. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA
• B. STANDARD DESIGN RESPONSE SPECTRA.
• When it is acceptable to use standard design response spectra to
define the design earthquakes, the horizontal component of ground
motion shall be defined by anchoring the standard design response
spectra for the appropriate damping factor with the scaling factor
• The vertical component of ground motion shall utilize the same
standard design response spectrum used for the horizontal
component, but it shall be scaled using the appropriate ratio of the
PGA for the vertical component to the PGA for the horizontal
component

13. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA
AMPLIFICATION CALCULATIONS
• To compute the amplification functions, three different computer
programs were used:
• - ProShake [7] and CyberQuake [8] for 1D and
• - Aki-Larner SH according to Bard and Gabriel [9] for 2D
calculations. As input motion on the bedrock, time histories
following as closely as possible the shape of the elastic
• response spectra of the national application document of Euro code
8 [10] (version ENV-1998-1-1) for
• rock and zone 1 were taken. Zone 1 of the Swiss design code
represents the largest part of the highly
• populated area in Switzerland. The selected time histories had to
fulfill the following criteria:

14. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA
• - Occurrence in similar tectonic conditions as Switzerland and
• - Covering the target response spectra (split into the period ranges
of 0.02-0.2s and 0.2-4s).
• Table 1 contains the selected earthquakes. Figures 3 and 4 show
the response spectra of the selected time
• histories in comparison to the target spectra.

15. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA

16. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA

17. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA
• The multi-step code approach for calculating the Seismic Response
Coefficient (Cs in NEHRP), is essentially a way of constructing a
smoothed average response spectrum that accounts for the
damping and ductility characteristics of the building, as well as the
regional seismicity and underlying soil of the site.
• Compare an elastic response spectrum for a Northridge 1994
earthquake motion, with a code design response spectrum
developed with the NEHRP provisions.
• The code spectrum is an approximation of an elastic response
spectrum, scaled down by two factors:
• It is reduced by the factor of safety used in allowable stress design
to account for the fact to achieve the given yield strength, allowable
stress design must aim at a lower strength. (for this case, Fs = 1.5)
• It is reduced by the R factor to account for damping and ductility.
This reduction creates an inelastic spectrum which accounts for the
effect of ductility in limiting force levels. (for this case, R=6.5)

18. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA

19. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA

20. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA
REDISTRIBUTION FOR HEIGHT
• The response spectrum concept is based on the notion that the
structure is a single degree of freedom system, but real structures
are not.
• In particular, the levels of acceleration are not constant throughout
the structure.

21. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA

22. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA
• Effects of local soil conditions on response spectra

23. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA
• TYPES OF SOIL

24. SITE-SPECIFIC ELASTIC DESIGN
SPECTRA
• At periods above about 0.5 s, spectral amplifications for soil sites
are much higher for soil sites than for rock sites. Deep and soft soil
deposits produce greater proportions of long period motion. Use of
single response spectrum shape for all site conditions is not
appropriate

25. STATISTICALLY DERIVED RESPONSE
SPECTRA
• Elastic design response spectra can be predicted in the same
statistical manner as ground motion parameters such as peak
ground acceleration or velocity. Numerous researchers have
developed attenuation relationships for elastic spectra, which are
listed in the references.
• 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 exceedance

26. STATISTICALLY DERIVED RESPONSE
SPECTRA
• Attenuation relationships are developed by statistical analyses
performed on a large number of records which were obtained in
compatible geomorphic regions. Most of these relationships are
updated as new strong ground motion data becomes available and
many now include additional parameters such as fault type and site
soil conditions.
• The relationships are grouped by region, with note to those which
are applicable only to a particular area, such as the Cascadian
seduction zone. It is important to realize that the relationships are
only as good as the data they were generated from, and therefore
the relations for Western North America are the most reliable
because the database for that area is quite rich. The relationships
for Eastern and Central North America should be used with caution
because the relationships have been calibrated with only a few
events. In any event, more than one relation should be used to
predict the motions at a site.

27. STATISTICALLY DERIVED RESPONSE
SPECTRA

28. STATISTICALLY DERIVED RESPONSE
SPECTRA
• where PHA = peak horizontal ground acceleration, PVA = peak
vertical ground acceleration, PHV = peak horizontal ground
velocity, PVV = peak vertical ground velocity, Sah = horizontal
spectral acceleration, and Sav = vertical spectral acceleration
Distance definitions: rrup = closest distance to the rupture surface, rjb
= closest horizontal distance to the vertical projection of the
rupture, rhypo = hypo central distance, rseis = closest distance to the
seismogenic rupture zone

29. EMPIRICALLY DERIVED RESPONSE
SPECTRA
N. M. Newark and W. J. Hall's procedure for developing elastic design
spectra starts with the peak values of ground
acceleration, velocity, and displacement. The values of peak
ground acceleration and velocity should be obtained from a
deterministic or probabilistic seismic hazard analysis.
The value of peak ground displacement is a bit more difficult to obtain
due to the lack of reliable attenuation relationships. A typical
baseline curve plotted on tripartite axes is shown below.

30. EMPIRICALLY DERIVED RESPONSE
SPECTRA
• Use response amplification factors (listed in table on previous page
to determine spectral values in the following period ranges:
• Short period (Tn < 0.03 sec) Sa = ag
• Transition
• Constant amplified acceleration range (Tn > 0.13 sec) Sa = aaag
• Intermediate period range Sv = avvg
• Long period range Sd = addg
• Very long period range Sd = dg (transition unclear)

31. EMPIRICALLY DERIVED RESPONSE
SPECTRA
Connect the lower bound response lines.
If desired, plot the spectrum in a different format, such as the one
shown here.
STRUCTURAL RESPONSE AMPLIFICATION FACTORS
• Structural response amplification factors are then applied to the
different period-dependent regions of the baseline curve. These
factors differ for acceleration, velocity, and displacement, especially
at low values of damping. The factors decrease rapidly with
increasing damping, especially at small damping values. These
factors are shown in the table below.
Newmark and Hall's structural response amplification factors can also
be used to change the damping value of other spectra, such as
those generated using attenuation relationships. This modification
technique is presented in the viscous damping section of the notes.

32. EMPIRICALLY DERIVED RESPONSE
SPECTRA
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.

33. EMPIRICALLY DERIVED RESPONSE
SPECTRA
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.

34. EMPIRICALLY DERIVED RESPONSE
SPECTRA
EFFECT OF VARIOUS FACTORS ON SPECTRAL VALUES
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.

35. EMPIRICALLY DERIVED RESPONSE
SPECTRA
REFERENCES:
[1]-Wikipedia (Response spectrum, general information)
[2]-Design of earthquake resistant building Using site-specific response
spectra”,(Site-specific elastic design spectra)
[3]http://peer.Berkeley.Edu/course_modules/eqrd/index.Htm?C227top.Htm&2
27cont.Htm&eqdef/eqdef4.Htm(eqrd,interactive), Site-specific elastic design
spectra.
[4]http://peer.Berkeley.Edu/course_modules/eqrd/index.Htm?C227top.Htm&2
27cont.Htm&eqdef/eqdef4.Htm(eqrd,interactive), Empirically derived
response spectra.
[5] Bakir P. G., Roeck G. D., Degrade G. and Wong K. K. F., “Site dependent
response spectra and analysis of the characteristics of the strong ground
motion,(Empirically derived response spectra)