Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

10.3 Stiffness matrix method in cylindrical coordinates 161
Example 10.6: Elastic half-space subjected to a tangential point load
at its surface (Chao’s problem)
A lower elastic half-space z < 0 is subjected at its surface to a harmonic, horizontal point
load b x = δ(x) δ(y) δ(z), which is equivalent to a surface traction in the horizontal plane
z = 0 in direction x of the form p x = δ(x) δ(y). In cylindrical coordinates, this corresponds
to a surface traction with azimuthal index n = 1 and components
⎧
⎧ ⎫ ⎧
⎫ ⎧ ⎫
⎨
p 1 =
⎩
p r
p θ
⎫
⎬
⎭
p z (1)
⎫
= 1
2π
⎧
⎨ 1⎬
δ(r)
= T 1 1
⎩ ⎭ 2π r
0
⎨ cos θ ⎬
− sin θ
⎩ ⎭
0
δ(r)
r
⎨cos θ 0 0 ⎬ ⎨1⎬
δ(r)
= 0 − sin θ 0 1
⎩
⎭ ⎩ ⎭ 2π r
0 0 cos θ 0
(10.74)
This is so because the radial and tangential components of the load, when projected in
the x direction, added together, and integrated over a small circular area of radius R
enclosing the origin, give a unit load in direction x:
∫ R ∫ 2π (
) ∫ δ(r)
R
cos 2 θ + sin 2 θ
0 0
2π r rdrdθ = δ(r) dr = 1 (10.75)
0
To solve this problem using stiffness matrices, we begin by casting the load in the
frequency–wavenumber domain:
⎧ ⎫
∫ ∞ ∫ 2π ⎨ 1 ⎬
δ(r)
˜p n = a n r C n T n T 1 1 dθ dr,
0 0 ⎩ ⎭ 2π r a n = 1 (
1 − 1 )
π 2 δ n0
(10.76)
0
Because of orthogonality conditions satisfied by the integral in the azimuth θ, only the
term n = 1 survives, so
⎧
⎧
⎧ ⎫
⎧ ⎫
⎧ ⎫
⎨ ˜p r
˜p 1 = ˜p
⎩ θ
˜p z
⎫
⎬
⎭
(1)
=
∫ ∞
0
1
⎨ J
1
′ kr J ⎫
1 0 ⎬ ⎨1⎬
∫
1
⎩
kr J 1 J
1 ′ δ(r)
∞ ⎨J 0 ⎬
0 1
⎭ ⎩ ⎭ 2π dr = J
0 ⎩ 0
⎭
0 0 J 1 0
0
δ(r)
2π dr = 1
2π
⎨1⎬
1
⎩ ⎭
0
(10.77)
This implies that of all azimuthal components, only ũ 1 will exist. The displacement components
in the frequency–wavenumber domain are then
ũ 1 = K −1 ˜p 1 (10.78)
in which K is the stiffness matrix of the half-space, which is assembled with the elements
given in Table 10.1 for SV-P waves and for SH waves. Formally, the stiffness matrix will
be of the form
⎧
⎪⎨
K11 SVP 0 K SVP ⎫
12 ⎪⎬
K = 0 K SH 0
(10.79)
⎪⎩
⎪⎭
K21 SVP 0 K22
SVP
but because the SV-P components are uncoupled from the SH components, the two
should be assembled and solved separately – and this is especially true for layered media.