Technical information

4-5
in which the asterisk is used to indicate that the response is due to vertical surface
loading
)(
0
ρ
⋅⋅ mJm
as can also be seen in equations 4.2.4a-b. Equations 4.2.4c-f
express the continuity of stresses and displacements inside the structure at the layer
interfaces; full bonding is suggested by equation 4.2.4f. Equation 4.2.4g means that all
response types (denoted using
R
) must vanish for the n
th
layer and at infinite depth
(i.e.,
0lim =
∞→
R
z
). Finally, the response due to a uniform load
q
distributed over a
circular area of radius
a is obtained by performing the integration:
dmmJ
m
R
qR
m
⋅⋅⋅⋅⋅=
∫
∞
=
)()(
1
0
*
αα
.................................................................... (4.2.5)
in which
Ha /=
α
and
R
is the stress or displacement of interest. Strains are thereafter
obtained using the constitutive relations (i.e., equations 4.2.1a-d).
For the purpose of this study, the entire aforementioned derivation was
programmed into an Excel worksheet (see program ELLEA1 in Appendix B). This was
done for the case of five layers and considering two separate loaded areas. The
combined effect of the two independent loads is calculated using superposition after
converting the axially symmetric results in each case to a Cartesian coordinate system.
The integration in equation 4.2.5 was carried out numerically between the first 200
zeros of the Bessel functions involved. The Gauss integration scheme was used for this
purpose whereby the first interval was integrated using a 30-point Gaussian formula, the
second interval was integrated using a 20-point formula, the third interval was
integrated using a ten-point formula and the remaining intervals were integrated using a
five-point formula. In order to speed the computational time, the number of matrix
inversions required for solving equations 4.2.4 was limited to 96, corresponding to 96
predetermined values of the integration variable
m in the range of 0 to 50,000. A cubic
spline interpolation scheme was used to derive intermediate results within this range.
Furthermore, in order to improve the convergence of the integration, especially for
points residing close to the surface, one step of Richardson extrapolation was employed
(Sugihara, 1987).