Technical information

5-4
izx
)(
ν
,
ixy
)(
ν
and
ixz
G )(
. Layers are numbered serially, the layer at the top being layer
1 and the half-space, layer
n . Similar to the isotropic case (see Subsection 4.2.1), the
origin of the cylindrical coordinate system is placed at the surface of the first layer with
the
z -axis pointing into the medium and the
r
-axis parallel to the layers. As before, 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 ).
Inspired by Huang (2004) and Lekhnitskii (1963), a stress function that complies
with all of the above requirements is offered:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅+⋅+
⋅+⋅
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅⋅
=
−
−
−⋅⋅−−⋅⋅−
−⋅⋅−−⋅⋅−
)()(
)()(
2
0
3
1
1
)(
),(
iiii
iiii
m
i
m
i
m
i
m
i
i
eDeC
eBeA
m
mJH
λλβλλβ
λλαλλα
ρ
λρφ
................. (5.1.6)
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 5.1.3a-f yields the responses of interest in a given layer
i
due to a
vertical non-dimensional surface load of the form
)(
0
ρ
⋅
⋅
mJm
and not due to a
uniform load distributed over a circular area (this fact is indicated by an asterisk). The
resulting expressions are presented in what follows for completeness of the derivation.
()
()
()
()
ρ
ρ
β
α
ρ
β
α
σ
λλβλλβ
λλαλλα
λλβλλβ
β
λλαλλα
α
)()1(
)()(
1
)()(
)()(
0
)()(
)()(
*
1
1
1
1
⋅⋅−
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅−⋅⋅+
⋅−⋅⋅
+
⋅⋅⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅−⋅⋅⋅+
⋅−⋅⋅⋅
=
−
−
−
−
−⋅⋅−−⋅⋅−
−⋅⋅−−⋅⋅−
−⋅⋅−−⋅⋅−
−⋅⋅−−⋅⋅−
mJb
eDeC
eBeA
mJm
eDeCL
eBeAL
i
m
i
m
ii
m
i
m
ii
m
i
m
iii
m
i
m
iii
ir
iiii
iiii
iiii
iiii
............ (5.1.7a)
()
()
()
()
ρ
ρ
ρσ
λλβλλβ
α
λλαλλα
α
λλβλλβ
β
λλαλλα
α
θ
)(
)()(
1
)()(
)()(
0
)()(
)()(
*
1
1
1
1
⋅
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅−⋅⋅+
⋅−⋅⋅
+
⋅⋅⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅−⋅⋅+
⋅−⋅⋅
=
−
−
−
−
−⋅⋅−−⋅⋅−
−⋅⋅−−⋅⋅−
−⋅⋅−−⋅⋅−
−⋅⋅−−⋅⋅−
mJ
eDeCS
eBeAS
mJm
eDeCQ
eBeAQ
iiii
iiii
iiii
iiii
m
i
m
ii
m
i
m
ii
m
i
m
ii
m
i
m
ii
i
................... (5.1.7b)