EP 1110-2-12
30 Sep 95
7-5.
Hydrodynamic Effect
where
Ωr = water compressibility significance
a. Dynamic characteristics affected. Hydrody-
parameter
namic load results from the interaction of the reser-
voir and the structural mass of the dam in response to
ground motion. The dam-reservoir interaction
water idealized by a fluid domain of con-
changes the water pressure acting on the face of the
stant depth and infinite length
dam, and directly affects the following dynamic char-
acteristics of the system:
(1) Modal frequencies. As the depth of the
and when
reservoir increases beyond a depth equal to about one
half of the height of the dam, there begins to be a
Ωr ≤ 0.5, compressibility of water is significant
noted decrease in the modal frequencies.
and should be accounted for in determining the
hydrodynamic effect
(2) Mode shapes. The equivalent added mass to
account for reservoir effects, as discussed in para-
d. Standard pressure function curves. In
graph 7-5c, changes the relative distribution of mass
Chopra's system, the hydrodynamic pressure distribu-
in the system. Thus, the normalized mode shapes
tion and equivalent mass system are derived using a
will be affected to some degree.
set of standard hydrodynamic pressure function
curves. The equivalent mass system for the compos-
(3) Effective damping ratio. As the depth of the
ite finite element method may be developed using the
reservoir increases, dam-reservoir interaction tends to
same principles as those for the Simplified Procedure.
increase the effective damping ratio.
The added mass is determined by using the appropri-
ate pressure function curve, certain equations from
b. Added mass based on Westergaard's formula.
Chopra's Simplified Procedure, and the fundamental
Accounting for hydrodynamic effects when using a
mode shape and frequency obtained from the finite
composite finite element model (refer to para-
element analysis of the dam-foundation model. Some
graph 8-1d(3)(a)) requires developing an equivalent
additional requirements applying to added mass are
mass system which strategically adds mass to the
discussed in paragraph 7-8c, and complete details for
dam-foundation model. The amount and location of
deriving the equivalent mass system for the composite
the added lumped masses must be such that they cor-
finite element method are provided in Appendix D of
rectly alter the dynamic properties described above in
this EP.
a manner which will also produce the desired pres-
sure changes. Often the added mass is calculated
e. Hydrodynamic pressure distribution.
based on Westergaard's pressure diagram divided by
Figure 7-3 shows the hydrodynamic pressure distribu-
the acceleration due to gravity to convert it from a
tion associated with the fundamental mode for a
distributed load to a distributed mass.
typical dam with a high reservoir condition. Plot 1
shows the distribution calculated by Chopra's Simpli-
c. Added mass based on Chopra's method.
fied Procedure, where Plot 2 and Plot 3 were
A. K. Chopra's Simplified Analysis Procedure
obtained using the composite finite element method
(Chopra 1978) uses an equivalent mass system to
with equivalent mass systems as discussed above.
consider compressibility of water and the dynamic
The added mass for Plot 2 was based on Wester-
properties of the dam and reservoir bottom. Chopra
gaard's formula, and the added mass for Plot 3 was
suggests that the key parameter that determines the
based on the standard pressure function curves and
significance of water compressibility is
ω1 r
the method described in Appendix D. To extract the
Ωr
hydrodynamic pressure distribution using the compos-
ω1
ite finite element method, the dynamic analysis was
7-5
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