EP 1110-2-12
30 Sep 95
responses as random variables. The functions may be
Cr = velocity of pressure waves in the
more complex involving modal frequencies or both
foundation rock = 12 Ef /ρr
modal frequencies and damping factors as is the case
with the complete quadratic combination method
Ef = deformation modulus of the foundation rock
(CQC) and the double sum method (DSM). The
in lb/in.2
more complex methods give additional accounting in
the coefficient calculation when the frequencies of the
two modes under consideration are close. Two
7-7. Method of Combining Modes
closely spaced modes are coupled, and when one of
the modes is excited, it tends to excite the other
a. Maximum modal responses. The maximum
mode. However, the modal frequencies associated
modal response in a response spectrum analysis is the
with gravity dams are normally fairly well separated.
maximum possible contribution that a particular mode
can make to the dynamic response. However, all the
d. Comparing methods. The base shear was the
modes do not arrive at their maximums at the same
response parameter used for comparing the four com-
point of time during the period of ground motion.
bination methods. By using several load cases, foun-
Thus, for a single ground motion record there is one
dation conditions, and damping ratios, eight sets of
point in time when the maximum dynamic response is
maximum modal base shear values were made avail-
reached, and this maximum response is made up of
able to test the combination methods. The more
various fractional parts of the individual maximum
complex methods, CQC and DSM, increased the
modal responses. The "fractional parts" are unique
coupling coefficients for closely spaced mode which
for each ground motion record. If a response spec-
produced greater combined responses than the SRSS
trum analysis is made for a single ground motion
method. The spacing of the modal frequencies for
record, the maximum dynamic response can only be
the TPM was such that no two modes qualified for
approximated because the exact makeup of the "frac-
"additional accounting," so the combined response for
tional parts" of the maximum modal responses cannot
the TPM is the same as SRSS. The two most often
be computed. A time-history analysis is required to
used methods are SRSS and CQC.
determine the exact solution for a given ground
motion record.
e. Conclusion. The mode combination method
does not greatly affect the order of accuracy of the
b. Statistical combination methods. A smooth
dynamic analysis. The factors discussed previously
design response spectrum may be considered as a
have far greater influence on the dynamic response.
convenient representation of many possible ground
The preliminary design of new dams, and the final
motion records that could make up the design earth-
design of dams not considered to be under critical
quake. As discussed in paragraph 5-5a, design
seismic conditions, may use either the SRSS or the
response spectra are often referred to as statistical
CQC method. Final design of dams under critical
representations of the ground motion records used in
seismic conditions and evaluation of existing dams
their development (such as mean, median, 84th per-
shall use the more refined CQC method.
centile). On a similar basis, the maximum modal
responses of a response spectrum analysis are com-
bined by statistical methods to produce a reasonable
7-8.
Vertical Component of Ground Motion
dynamic response to the many possible ground
motions that could make up the design earthquake.
a. Factors that contribute to the response. It is
very difficult to make a general assessment of the
c. Coupling coefficients. Tables 7-1 and 7-2
influence of the vertical component of ground motion
present four commonly used mode combination meth-
on the total dynamic response because of the number
ods. The difference in the methods is in the calcula-
of factors involved. The vertical component of
tion of the coupling coefficient between modes. The
ground motion can be significant under certain condi-
coupling coefficients may be simple discrete functions
tions. The most important factors that affect the
as is the case with the square root of the sum of the
contribution of the vertical component to the response
squares method (SRSS) which treats the modal
are:
7-8
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