Environmental Engineering Reference
In-Depth Information
determined from harmonic ground motion, are included in the analysis by attaching a
“block of water” to the face of the dam. This block or mass of water, perhaps the first time
that the term “added mass” was used, has a shape for a vertical faced dam of:
b
7
8
h
y
(16.25)
where b
width of the added mass measured normal to the face horizontal at depth y.
h
height of the dam at the section concerned (generally the height above the reservoir
floor, not necessarily the structural height of the dam).
The total inertia force is found by applying the design peak ground acceleration (pga)
to the structure. The movements and shear forces at the base and at other sections follow
in the normal way for a conventional static load analysis.
FERC (2000) advises that this method should not be used and the authors would agree.
It may be reasonable for low pgas but once these pgas are 0.2 g and beyond, the analysis
simply proves that the dam would be unsafe, particularly if the engineer has assumed an
f
t
0 foundation. This outcome is probably not helped by designers treating the analysis
as a conventional static load analysis, with the uplift at any section and at the base being
redistributed once cracking has occurred.
16.9.3.3
The Fenves-Chopra refined pseudo-static method
The Fenves and Chopra (1987) method is a significant improvement on the Westergaard
method. It uses data from more complex finite element analyses to allow the engineer to
do a far more realistic analysis. In it:
-
Dam response is included by taking from a suitable response spectrum the spectral
accelerations at the first and selected higher modes for a nominated damping factor.
For dams up to, say, 20 m in height, the dam's response may not be too critical;
-
The water interaction model is more refined than the one often used (see next section);
-
The flexibility of the foundations can be included;
-
The effect of displacement of the crest upstream can be assessed.
The method gives an equivalent static loading applied to the upstream face of the dam
that may be used to complete an analysis of the dam with the inclusion of the static loads
(concrete weight, reservoir pressure, internal pore pressures and so on).
This method clearly has advantages over the Westergaard pseudo-static methods, but it
does not allow for cracking of the base or at any other section that could change the dam's
dynamic loading characteristics. For very stiff dams, cracking could see a substantial
increase in the earthquake inertia loading, whereas in a large dam with a natural period of
0.3 seconds, the reverse might be the case.
One real advantage in this method is that if the stresses in the top of the dam are of con-
cern, as was the case at Pine Flat and Koyna dams, the stresses there can be quickly
assessed.
FERC (2000) suggests that this method is a reasonable one to use and the authors
would agree, subject to some careful review of the results.
16.9.3.4
The US Corps of engineers method
This method is presented in Guthrie (1986) and an excellent discussion on it is given in
ANCOLD (1998).
This method goes somewhat further than the Fenves and Chopra (1987) method in that
depending on the level of tensile stress developed, the engineer may have to undertake a
