is used in the model, model laws of soil mechan-

ics, together with requirements for geometric and

kinematic similarity, lead to scale factors for cen-

trifuge modeling. Several are shown in Table 1 as

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/*N*2

Area

1

type. Following Croce et al. (1985), a number of

1/*N*3

Volume

1

these are derived in Appendix A.

Mass density

1

1

Many of the scale factors listed in Table 1 are

1/*N*3

Mass

1

basic structural modeling scale factors and can

Acceleration

1

be derived from the mechanical considerations

Stress

1

1

1/*N*2

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/*N*3

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/*N*2

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-

nificant engineering variable of the model is re-

Ω=

(1)

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

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

(2)

the forces in the prototype, and so that the model's

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-

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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