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Q
0
=
(
EA
0
/h
)
w
0
= kw
0
or
q
0
= Q
0
/A
0
=
(
E/h
)
w
0
= kw
0
/A
0
(12.43)
This is the known as Winkler's model. The soil at the surface reacts at the
surface as a simple spring, and the spring constant is
k = EA
0
/h
. The choice of
h
is
related to the choice of the representative cone, within which the soil stresses are
absorbed. Adopting Rankine stress states, i.e. active or passive, the value of
h
is
given by
r
0
/
tan(¼
+ ½
)
< h < r
0
/
tan(¼
½
)
(12.44)
Here,
r
0
is the (apparent) radius of the foundation and
the internal soil friction
angle. For cohesive soil (
=
0) one finds
h = r
0
corresponding to a cone apex of
=
90
o
.
The previous theory applies for axi-symmetric systems. The same holds for
plane two-dimensional systems, if one adopts the soil elasticity increasing
proportionally to
z
. Thus,
E
is proportional to
z
and for a two-dimensional cone
A
is
also proportional to
z
, so that
EA
is proportional to
z
2
, and the previous elaboration
applies. This model is known as a Gibson soil. Gibson showed that in this case at
the surface Winkler's model is satisfied. It seems realistic, since experiments show
that elasticity increases with depth. For a one-dimensional case, i.e.
A
is constant,
the same solutions may hold, if
E
is proportional to
z
2
.
Soil response to a dynamic load moving along a 'beam'
The response of subsoil to dynamic loading
Q
0
can be approximated by a
bending beam on springs (Fig 12.9). The equilibrium is described by
EIw
0
,xxxx
+
Aw
0
,tt
+
(
k/B
)
w
0
= Q
0
/
B
(12.45)
Q
0
B
v
A
k
Figure 12.9 Schematization of a HSL railway embankment
A
the mass/m of the beam, and
k
is the spring constant of the supporting soil. The
material damping, which affects the amplitude of the induced vibrations during
resonance, has been disregarded. When the load is moving at a constant velocity
v
,
equation (12.45) can be reformulated with a convective coordinate
Here,
w
0
is the surface displacement under the load,
EI
the bending stiffness,
;
= x - vt
,
according to
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