Civil Engineering Reference
In-Depth Information
and the circular ring - with the help of the unity deformation condition
from the equilibrium conditions along the tunnel contour. The consideration
of the equilibrium conditions for the individual components of the Fourier
series leads to an equilibrium system, which allows for a more or less easy
calculation of the unknown deformations, resulting in the internal forces.
For the case of a tunnel support with infinite axial stiffness, the equations
of the systems can be decoupled so that explicit equations can be given for
the calculation of the deformations.
The following are the equations, using first order theory, for the case of
a rigid bond between the tunnel support and the ground assuming infinite
axial stiffness. If using segmental lining, the pre-deformations of the
segments as a result of the installation can be considered using second order
theory (Ahrens
et al.
1982). It is assumed that the tunnel lining has an
infinite axial stiffness.
The following equations are from Ahrens
et al.
(1982) and further
explanations can be found in this reference.
Earth pressure:
p
v
= -
h
(B.1a)
p
h
= -K
0
(h + r)
(B.1b)
Transformation into polar coordinates:
p
r
=
-
r0
+
-
r2
cos 2
(B.2a)
p
t
=
-
r2
sin 2
(B.2b)
With
-
r0
= 0.5
[h + (h + r)
K
0
]
(B.3a)
-
r2
=
-
tr2
= 0.5
[h - (h + r)
K
0
]
(B.3b)
The load and deformation variables of the plate are denoted with a super-
script 'D', while the load and deformation variables associated with the
circular ring frame are denoted with a superscript 'R'.
Deformations:
) =
-
0
+
-
2
· cos 2
w(
(B.4)
) = +
-
2
· sin 2
v(
(B.5)
