EP 1110-2-12
30 Sep 95
(c) Reservoir. The impounded water of the
and bottom of the rectangular foundation block rather
reservoir is idealized as a fluid domain of constant
than at the ground surface (dam-foundation interface)
depth and infinite length. This can be interpreted as
where the design earthquake ground motion is speci-
a series of subchannels of infinite length discretized
fied. To account for this, the foundation is assumed
to match the common upstream nodal points of the
massless. Therefore, no wave propagation takes
dam. The reservoir bottom absorption is modeled by
place in the massless foundation so the ground
adjusting the boundary condition at the reservoir
motion is transmitted to the dam-foundation interface
bottom. This substructure representation of the reser-
without modification.
voir produces more accurate hydrodynamic response
to horizontal and vertical ground motion than does an
(c) Flexibility in modeling. The composite finite
equivalent mass system representation as described in
element model may be formulated to represent a
variety of design conditions for both 2-D and 3-D
paragraph 8-1d(3)(a).
models. For example, most any geometric shape may
be accommodated, various zones of superior RCC
(d) Specialized computer program. This type of
mix may be incorporated in the dam model, and
model requires a specialized computer program to
discontinuities such as fault zones or changes of
allow the foundation and the reservoir effects to be
deformation modulus in the foundation may also be
formulated in the manner described above. Also, the
included.
substructure method requires the foundation to be
modeled as a uniform homogeneous material. Pres-
ently, a computer program is available which devel-
8-2.
Comparison of Dynamic Analysis
ops a 2-D finite element-substructure model for
Methods
gravity dams. Refer to paragraph 8-2b.
This section will describe the attributes associated
(3) Composite finite element-equivalent mass
with the most commonly used dynamic analysis
system model. This method models both the dam
methods, and the methods will be evaluated and
and the foundation as an assembly of discrete finite
compared.
elements. Either solid quadrilateral plane stress or
plane strain elements are used for 2-D models or 3-D
a. Chopra's simplified method. This method
isoparametric solid elements are used for 3-D models.
uses the standardized model described in para-
The foundation consists of a rectangular block with a
graph 8-1d(1). Other attributes include 2-D repre-
width in the upstream-downstream direction about
3 times the base width of the dam at the foundation
sentation, linear-elastic material behavior, and
plane, and with a height about 1.5 times the height of
response spectrum definition of the design earth-
the dam.
quake. This method is not flexible because all of
these attributes are fixed.
(a) Reservoir effects. The reservoir effects are
modeled by developing an equivalent mass system
(1) Equivalent lateral force. The simplified
which consists of adding mass to the finite element
method develops the maximum response to the first
model to correctly alter the dynamic properties. The
mode as a set of equivalent lateral forces. It also
added mass is active in the direction normal to the
approximates the equivalent lateral forces associated
vertical upstream face of the dam. This method also
with the higher vibration modes using a "static cor-
allows the reservoir bottom absorption characteristics
rection" method. The two sets of equivalent lateral
to be incorporated into the analysis by using Chopra's
forces are treated as statically applied distributed
standard hydrodynamic pressure function curves to
lateral loads. At present, response to a vertical com-
determine the added mass. Although use of these
ponent of ground motion is not possible with this
curves in developing the equivalent mass system is
type of model. Stresses may be hand calculated by
only approximate, it has been shown to be reasonably
beam theory treating the dam as a simple cantilever
beam, or the static load may be applied to a finite
accurate. Refer to paragraphs 7-5c and 7-5d and
element model of the dam to gain a more realistic
Appendix D for details.
stress distribution pattern.
(b) Boundary conditions. With this type of
(2) Advantages and limitations. The simplified
model, the earthquake ground motion is introduced at
method is easy to use and can be done without a
the rigid boundary. This boundary is along the sides
8-3