APPENDIX A: SOME SCALE FACTORS FOR
CONVENTIONAL CENTRIFUGE MODELING
scalar model laws. Closely following the analy-
PHYSICAL MODELING CONCEPTS
sis of Croce et al. (1985), this approach is presented
Following the presentation of Langhaar (1951),
here. Two assumptions are made: that soil can
be treated as a continuum and that soil proper-
consider a physical variable of a prototype fp(xp,
ties are not affected by a change in acceleration.
yp, zp, tp) and the corresponding model quantity
Dynamic similarity requires that all forces (and
fm(xm, ym, zm, tm), where x, y and z are coordi-
all kinds of forces) have the same scale factor. The
nate measures of a point within the structure, t is
forces of interest include the weight force Fw, the
time, and the subscripts p and m refer to the pro-
external force Fe, the viscous force Fv, the inertia
totype and model, respectively. The function fm
force Fi and the seepage force Fs.
is said to be similar to the function fp if the ratio
Scale factors for length l, mass m and time t
fm/fp is constant for homologous points and ho-
are
mologous times. The constant ratio f = fm/fp is
called the scale factor for the function f. Scale fac-
lm
Kl =
,
(A1)
tors can be derived from model laws, i.e. ratios
lp
of physical laws for the model and prototype.
When similarity is achieved, a model's response
mm
Km =
is interpreted using the scale factors to infer the
,
(A2)
mp
response of the corresponding prototype.
and
The ideal for physical modeling is complete
similarity, which includes geometric, kinematic,
tm
Kt =
.
(A3)
thermal and dynamic similarity. Complete simi-
tp
larity is usually not achieved for a model test.
Scale factor conflicts and scale effects, i.e., dis-
Thus the scale factors for area A and volume V
turbing influences that are associated with the
are
small scale of the model, can limit similarity to a
Am
partial similarity.
KA = Kl2 =
(A4)
Ap
When a model is made of the material of the
prototype, and when there are no scale factor con-
and
flicts or disturbing scale effects, then the model
Vm
KV = Kl3 =
material will have the same constitutive behav-
.
(A5)
Vp
ior as the prototype material, and prototypes with
general constitutive behavior such as nonlinear
Since the same material is used in the model and
elasticity and plasticity can be modeled.
prototype at the same mass density, the scale factor
for mass is equal to the scale factor for volume:
mm
CROCE'S DERIVATION OF
Km = Kl3 =
.
(A6)
mp
SCALE FACTORS FOR CONVENTIONAL
GEOTECHNICAL MODELING USING
For identical effective stresses in the model and
SCALAR MODEL LAWS
prototype,
σ′
m = σ m = um = 1
The objective of conventional small-scale
Kσ ′ = Kσ = Ku =
(A7)
geotechnical modeling is to achieve a model re-
σ′
σp
up
p
sponse that is similar to the mechanical behavior
where σ′, σ and u are, respectively, the effective
of the prototype, and the approach is to use the
prototype soil in a geometrically similar model.
stress, the total stress and the excess pore water
Thus it is important to ensure that the effective
pressure, including the capillary pressure caused
stresses in the model are the same as in the pro-
by surface tension. Considering eq A4 and A7 and
totype at homologous points and times. The scale
the definition of stress, it follows that the scale
factors for these conditions can be derived using
factor for all forces F should be
11
Previous Page
