Environmental Engineering Reference
In-Depth Information
limited FEM analysis. It would seem to be limited to some extent if one assumes, as the
authors have suggested, that the dam foundation interface has no tensile strength. In that
case, cracking will be a reality so unless the analysis can cope with a possibly varying nat-
ural period as the crack extends, then it would be better to go immediately to a full FEM
analysis. If cracking at the dam foundation interface is limited so that the dam's natural
period does not change very much, then the use of this method is reasonable.
16.9.3.5
Finite element method (FEM)
FEM has been used on many gravity dams since probably the late 1970s. The analyses have
mostly been 2-dimensional analyses, although the authors know of several done in 3-
dimensions. There have also been a number of arch dams analyses done, which have nec-
essarily been 3-dimensional analyses. The most well known of these would be the USBR
studies into the never-built large Auburn dam in California (USBR (1977 and 1978)).
Well into the 1990s, most of the analyses by practising engineers used linear elastic FEM
programs. Given the low level of stresses in a gravity dam, the assumption of linear elastic
behaviour is reasonable, but cracking is hard to include in a full dynamic analysis. Now with
computer programs available with more complex elements that will allow for crack joint
opening, compressive loading, shear loading and sliding, the difficult question of cracking
can be included in the FEM model.
Most FEM analyses are time-consuming. In the 1980s-early 1990s, it was lack of com-
puter power; now (2003) it is very complex programs. In all cases, there is a cost associated
with any FEM analyses. One almost flippant comment to the fourth author many years ago
had and still has much truth in it; “whatever you estimate, it will cost 3 times as much”.
Ideally, if, as the authors believe is the right approach, ft
0 is assumed at the
dam/foundation interface, cracking must be modelled in this zone. Cracks can be mod-
elled elsewhere, if the computer power is available. Once cracking is included, the engi-
neer will be limited to time-history analyses in a step-by-step numerical integration
process. At each small time interval, say 0.01 s, the dam is fully analysed under both the
applied earthquake loading and the static loading in a number of iterations until equilib-
rium is satisfied. The analysis then moves to the next time step. With the analysis being an
effective stress analysis, as are both the foregoing pseudo-static analyses, the pore pres-
sures must be included. Permanent shear displacement can also be allowed for.
The mechanics of an FEM analysis are now reasonably well known and need not be
repeated here. Nevertheless, there are some issues particularly related to dam analyses
that need highlighting. They include:
(a)
Dam-reservoir interaction effects
The USBR (1977, 1978) Auburn study in the late 1970s allowed for these effects to be
included as “added masses” fixed to the upstream face elements. The USBR used the
Westergaard distribution.
The general opinion is that this approach is probably reasonable (Léger et al
(1991)), although, in the case of a 3-D analysis one would question the relevance of
this added mass to a cross-valley earthquake. In the long run, an FEM analysis that
includes the reservoir directly would be the ideal (see ANCOLD (1998), and Léger et
al (1991) for more discussions).
(b)
Inclusion of the foundation rock
Most analyses included the rock mass with fixities at least one “dam height” away
from the dam-foundation interface. For a static analysis, if one ignores the “rebound
effects” on excavation, the foundation has “no mass”. The same approach is often
taken in dynamic analyses, even though the resulting dam/foundation model is slightly
stiffer than a model with rock with mass. For many dams the critical failure surfaces
will be in the foundation so the foundation must be modelled.
