Civil Engineering Reference
In-Depth Information
it represents. On the other hand, a model built by an architect or a railway enthusiast
should look like the real thing.
The rules that govern the conditions for similarity between models and prototypes
are well known and the simplest method for establishing scaling laws is by dimensional
analysis. The basic principle is that any particular phenomenon can be described by a
dimensionless group of the principal variables. Models are said to be similar when the
dimensionless group has the same value and then the particular phenomenon will be
correctly scaled. Often these dimensionless groups have names and the most familiar
of these are for modelling fluid flow (e.g. the Reynolds number).
27.3 Scaling geotechnical models
In constructing a geotechnical model the objectives might be to study collapse, ground
movements, loads on buried structures, consolidation or some other phenomenon
during a construction or loading sequence. In earlier chapters of this topic I showed that
soil behaviour is governed to a very major extent by the current effective stresses (this
is a consequence of the fundamental frictional nature of soil behaviour). Consequently,
the stresses at a point in a model should be the same as the stresses at the corresponding
point in the prototype.
Figure 27.1(a) shows the vertical total stress at a depth
z
p
in a prototype construction
in the ground and Fig. 27.1(b) shows a similar point at a depth
z
m
in a model with a
scale factor
n
(i.e. all the linear dimensions in the model have been reduced
n
times).
In the prototype the vertical stress is
σ
p
=
g
ρ
z
p
(27.1)
9.81 m/s
2
is the accleration due to Earth's
gravity. If the model is placed in a centrifuge and accelerated to
n
times
g
the stress at
a depth in the model
z
m
=
where
ρ
is the density of the soil and
g
=
z
p
/
n
is
ng
ρ
z
p
σ
m
=
ng
ρ
z
m
=
(27.2)
n
and
p
. Since the stresses at equivalent depths are the same the soil properties
will also be the same (provided that the stress history in the model and prototype are
the same) and the behaviour of the soil in the model will represent the behaviour of the
soil in the prototype. Notice that you cannot reproduce the correct prototype stresses
by applying a uniform surcharge to the surface of the model as, in this case, the stresses
σ
=
σ
m
Figure 27.1
Stresses in the ground and in a centrifuge model.
