EP 1110-2-12
30 Sep 95
5-4.
Dynamic Analysis by Modal
particular ground motion. One advantage of a
Superposition
smooth design response spectrum is that it is a statis-
tical representation, or an envelope, of the many
possible ground motions that could occur at the site
a. Frequencies and mode shapes. The linear-
rather than only a single ground motion. The super-
elastic response spectrum method utilizes modal
position of the maximum modal responses is accom-
superposition dynamic analysis to determine the struc-
plished by use of one of several statistical methods
tural response.
described in Chapter 7.
b. Time-history analysis. Once the modes are
derived, the response of the complex multiple degree-
5-5.
Types of Design Response Spectra
of-freedom system is reduced to the solution of the
simple, single basic equation of motion for a single
degree-of-freedom (SDOF) system. For time-history
are usually based statistically either on the mean,
analysis, the response is easily obtained using step-
median (50th percentile probability level), or the
by-step integration of the equation of motion for the
median plus one standard deviation (84th percentile
SDOF system for each significant mode based on the
probability level), of the ground motion parameters
frequency (eigenvalue) of the mode. In essence the
for the records chosen. Design response spectra used
response contribution of each mode is determined for
for design of new RCC dams or for evaluation of the
a series of time steps using a prescribed time-step
safety and serviceability of existing dams shall be
interval, and the response at each time step is simply
based on the mean level of the ground motion
the superposition, or addition, of characteristic mode
parameters.
shapes adjusted by coefficients obtained from the
integration procedure. Normally, only a few mode
b. Type of spectrum required. Either a "site-
shapes are found to contribute significantly to the
specific" or a "standard" design response spectra shall
response, so that the modal superposition method
be used to describe the design earthquakes. The type
produces a precise response with minimum computa-
required shall be based on the seismic zone, the prox-
tional effort.
imity of the seismic source, and the maximum height
of the dam.
c. Response spectrum analysis. In a response
spectrum analysis, the step-by-step integration part of
c. Site-specific design response spectra. The
the dynamic analysis, described above for time-
site-specific design response spectra should be
history analysis, is performed in the process of devel-
developed based on earthquake source conditions,
oping the response spectrum. The response spectrum
propagation path properties, and local foundation
may be envisioned as a display of the results of this
characteristics associated with the specific site. This
part of the modal analysis, and it is presented in the
type of design spectra may be established by anchor-
form of "maximum" response versus frequency (or
ing a selected response spectral shape for the site to
period). In the response spectrum modal analysis,
the estimated peak ground acceleration, or by estimat-
eigenvalues, eigenvectors, and modal participation
ing the design spectra directly using response spectral
factors are computed and used in the analysis proce-
attenuation relationships, performing statistical analy-
dure just as they are in a time-history modal analysis.
sis of strong-motion records, or applying theoretical
Precise "maximum" modal responses are easily calcu-
(numerical) ground motion modeling. In the require-
lated from a simple equation that relates these param-
ments that follow, a site is classified as a "high seis-
eters and the appropriate spectral value that
mic risk site" when it is located within 20 kilometers
corresponds to the modal frequency.
of an active fault or area source in the western United
States (WUS), or within a tectonic province in the
d. Combining modal responses. The final step
eastern United States (EUS) where the source or
in a response spectrum analysis consists of correct
province has a maximum local magnitude of 6.0 or
superpositioning of the "maximum" modal responses;
greater. The boundary between the WUS and the
however, there is not a unique solution to this final
EUS is defined as the eastern boundary of the Rocky
step in the response spectrum method. This is
Mountains. Site-specific design response spectra are
because the exact mode contributions at the critical
required for:
point in time when the response peaks are not avail-
able from a response spectrum representation of a
5-2
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