Technical information
4-4
coordinate system is placed at the surface of the first layer with the
z -axis drawn into
the medium and the
r
-axis parallel to the layers. The depth to the individual interfaces,
measured from the surface, is denoted by
i
z ( 1..,2,1
−
=
ni ). Hence,
1
z
is the thickness
of layer 1,
2
z is the combined thickness of layers 1 and 2, and so on. The combined
thickness of the
1−n
layers is denoted by
H
(i.e.,
1−
=
n
zH ).
Following Huang (2004), a ‘stress function’ that complies with all of the above
requirements is:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅⋅⋅−⋅⋅⋅+
⋅−⋅
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅⋅
=
−
−
−⋅−−⋅−
−⋅−−⋅−
)()(
)()(
2
0
3
1
1
)(
),(
ii
ii
m
i
m
i
m
i
m
i
i
emDemC
eBeA
m
mJH
λλλλ
λλλλ
λλ
ρ
λρφ
.... (4.2.3)
in which
Hr /=
ρ
,
Hz /=
λ
,
Hz
ii
/
=
λ
and m is a unitless parameter;
i
A
,
i
B
,
i
C
and
i
D
are all unitless functions of m ;
k
J
denotes a Bessel function of the first kind of
order
k ; and the subscript
i
refers to the layer number. Substitution of this equation
into equations 4.2.2 yields the response of interest in a given layer
i due to a vertical
non-dimensional surface load of the form
)(
0
ρ
⋅
⋅
mJm
. The value of the functions
)(mA
i
,
)(mB
i
,
)(mC
i
and
)(mD
i
cannot be expressed analytically; they must be
determined, for any given value of
m , by solving a set of linear equations. This set of
equations transpires from the boundary and continuity conditions of the problem as
follows:
0)()(
01
*
=⋅⋅=
λρσ
formJm
z
............................................................ (4.2.4a)
00)(
1
*
==
λτ
for
rz
............................................................................... (4.2.4b)
iiziz
for
λλσσ
==
+1
**
)()(
.................................................................... (4.2.4c)
iirzirz
for
λλττ
==
+1
**
)()(
.................................................................... (4.2.4d)
iii
forww
λλ
==
+1
**
)()(
.................................................................... (4.2.4e)
iii
foruu
λλ
==
+1
**
)()(
...................................................................... (4.2.4f)
∞→=
λ
forR
n
0)(
*
........................................................................... (4.2.4g)