Study of the earth pressure in rock around or on the boundary of an elliptic tunnel is one of the important problems in mining and civil engineering as well as a circular tunnel. So far as the author is aware, however, this problem has not yet been solved theoretically, therefore he has dealt with the present problem, assuming the earth crust is a semi-infinite elastic solid body and an elliptic tunnel whose major and minor axes are 2
aand 2
blong respectively and depthξ, is excavated horizontally.
The results obtained are as follows;
1. Elliptic boundary of the tunnel is given byα=α
0of the series of curves, which are introduced by the relations
x=
cchαCOSβ,
y=
cShαsinβ,
a=
cchα
0,
b=
cshα
0,
xand
ybeing vertical upward and horizontal axes.
Stress at the apex of the tunnel, i.e., at the upper end of major axis, is given by
ββ= {-σ/1-σ}ρ
yξ+ {1/1-σ(1+a/b)-1/2 (1-σ)}ρ
ga
at the side wall, i.e., at the end of minor axis,
ββ= {σ/1-σ(1+2b/a)}ρ
gξ
at the bottom,
ββ{-σ/1-σ(1+2a/b) +1}ρ
gξ-{1/1-σ(1+a/b)-1/2 (1-σ)}ρ
ga;whereββis stress tangential to the boundary and positive sign represents tension, negative sign compression, ρis densityσPoisson's ratio of the earth crust, g gravity.
2. Whereξ>>
a, andσ=1/4, as in the ordinary case, stress on the, boundary is of negative sign everywhere, i. e., compression, and its maximum value exists at the end of minor axis provided
a<3
b.
3. Where
a=
b, this is the case of circular tunnel.
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