is used in the model, model laws of soil mechan-
Table 1. Scale factors for geotechnical centrifuge mod-
ics, together with requirements for geometric and
eling. (After Scott and Morgan 1977, Croce et al. 1985,
kinematic similarity, lead to scale factors for cen-
Savidou 1988.)
trifuge modeling. Several are shown in Table 1 as
Quantity
Prototype
Model
ratios of the model quantity to the prototype quan-
tity for a model that is 1/N the size of the proto-
Linear dimension and displacement
1
1/N
1/N2
Area
1
type. Following Croce et al. (1985), a number of
1/N3
Volume
1
these are derived in Appendix A.
Mass density
1
1
Many of the scale factors listed in Table 1 are
1/N3
Mass
1
basic structural modeling scale factors and can
Acceleration
1
N
be derived from the mechanical considerations
Stress
1
1
1/N2
Force
1
(Langhaar 1951). The scale factor for acceleration,
Strain
1
1
which is N, can be derived from the consider-
Energy density
1
1
ation that weight forces should be scaled as other
1/N3
Energy
1
forces, and it implies the need for an increased
Temperature
1
1
gravity field (Croce et al. 1985). The centrifuge is
Time (for viscous force similarity)
1
1
Time (for inertial force similarity)
1
1/N
used to approximate this condition by subject-
1/N2
Time (for seepage force similarity)
1
ing a model to the constant angular velocity Ω
according to
said to be similar to the prototype when each sig-
Ng
nificant engineering variable of the model is re-
Ω=
(1)
r
lated by a proportionality or scale factor to the
corresponding variable of the prototype. Scale
where g = acceleration due to Earth's gravity
factors are governed by the physics and model
r = representative radius of the model
laws of the problem. They are used to design the
from the axis of rotation
model and to interpret the measured model re-
sponse as the prototype response.
radius.
Physical models of civil engineering structures
In this context, N can be considered as a nominal
are often designed so that the model will be geo-
gravity level:
metrically similar to the prototype, so that sig-
nificant forces in the model are proportional to
r Ω2
N=
(2)
the forces in the prototype, and so that the model's
g
response to loads will be kinematically similar
to that of the prototype. Geometric similarity
and the model can be considered to be subjected
means that all parts of the model have the same
to an inertial field equivalent to N gravities, or N
shapes as the corresponding parts of the proto-
g. As described by Langhaar (1951), if self-weight
type, and kinematic similarity means that the
stresses have a negligible influence on the proto-
motions of the model are similar to the motions
type response, or if partial dynamic similarity is
of the prototype at corresponding times. When all
acceptable to the modeler, the principles discussed
net forces are proportional, dynamic similarity is
here for centrifuge modeling and the scaling re-
said to exist (Langhaar 1951). In general, the me-
lations of Table 1 can be applied to models tested
chanical response of a structural model to forces
at 1 g. However, because self-weight stresses typi-
is observed and interpreted as the prototype re-
cally cannot be neglected for geotechnical proto-
sponse.
types, centrifuge modeling has become a conven-
Centrifuge modeling is a physical modeling
tional technique.
technique in which the weight stresses of a struc-
As indicated in Table 1, there are different time
ture are simulated by the placement of a small-
scales for the forces of viscous, inertial and seep-
scale model in a centrifugal field. This technique
age phenomena. As a result, time scale conflicts
can occur for certain modeling problems, and dy-
is of proven benefit for modeling soil structures,
namic similarity can be impossible to achieve.
because the form and magnitude of the soil re-
Thus the experimenter must consider the limita-
sponse are often greatly dependent on weight-
tions imposed by the model laws and scale fac-
generated effective stresses and because the load-
tors when designing a model experiment. Appen-
ing can be dominated by weight loads. As de-
dix A presents derivations of these time scales.
scribed in Appendix A, when the prototype soil
2
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