EP 1110-2-12
30 Sep 95
7-4.
Dam-Foundation Interaction, Founda-
fundamental mode of vibration, and has been shown
tion Modulus Effect
to be reasonably close for the significant higher
vibration modes (Fenves and Chopra 1986). In Fig-
ure D-6, damping for the foundation rock is expres-
a. Modulus of deformation. The flexibility of
sed by the constant hysteretic damping factor.
the jointed rock foundation is characterized by the
modulus of deformation which represents the relation-
ship between applied load and the resulting elastic
1
ξ1
ξ1
ξf
plus inelastic deformation. It is best determined by
(Rf)3
in-situ testing, but may be estimated from the elastic
modulus of the rock by applying an appropriate
where
reduction factor. In a linear-elastic analysis, the
modulus of deformation is synonymous with Young's
ξ1 = the effective viscous damping ratio for the
empty reservoir condition
b. Dynamic characteristics affected. The elastic
ξ1 = the viscous damping ratio for the RCC dam
modulus of the foundation influences the response
structure only
because it directly affects the following dynamic
characteristics of the dam-foundation system:
ξ1 = 5.0 percent for the OBE
ξ1 = 7.0 percent for the MCE
(1) Modal frequencies. As the modulus of defor-
mation decreases, the modal frequencies of the com-
Rf = ratio of the fundamental period of the dam on a
posite dam/foundation system also decrease.
rigid foundation to the fundamental period of
the dam on a foundation with a deformation
(2) Mode shapes. As the modulus of deforma-
modulus = Ef
tion decreases, the mode shapes are affected by
increased rigid body translations and rotation of the
ξf = added damping ratio due to dam-foundation
dam on the elastic foundation.
rock interaction taken from Figure D-6
(3) Effective damping ratio. As the modulus of
c. Effect of damping on response. To determine
deformation decreases, the effective damping ratio of
the effect that the damping ratio has on the response
the dam/foundation system increases.
of a dam, the fundamental frequency of the composite
finite element dam-foundation model must be deter-
c. Effect of foundation modulus on response.
mined. It is noted that for the response spectrum
To determine the effect of the foundation modulus on
method, the effects of damping are contained only in
dynamic response, a typical dam model was analyzed
the response spectrum itself. Thus, the ratio of the
on foundations that bracket a wide range of founda-
response of a dam/foundation system responding at
tion stiffness from infinitely stiff (Es/Ef = 0.0), to
one damping factor to the same system responding at
relatively flexible (Es/Ef = 2.5). The response was
a second damping factor is equal to the ratio of the
expressed as the distributed lateral inertia loading
spectral ordinates taken from the two spectra eval-
acting over the full height of the dam. Figure 7-2
uated at the fundamental frequency of the system.
shows the response graphically for three different
values of Es/Ef. It is noted that the total inertia load,
d. Conclusion. The damping characteristics of
or base shear, only varied by 15 percent, but a con-
the foundation can have a great influence on the
siderable variation occurred in the load pattern. As
dynamic response. This indicates the need to care-
the foundation becomes more flexible, the greatest
fully determine the value of the constant hysteretic
inertia load shifts from the upper portion of the dam
damping factor for the foundation rock. This can be
to the lower portion. This would be accompanied by
determined from experimental tests of appropriate
a considerable change in the concrete stresses.
rock samples subject to harmonically varying stress
and strain. From such tests, the inelastic energy lost
and the strain energy stored per cycle are determined
and the hysteretic damping factor is calculated.
7-3
Previous Page
