half of it is carried
by the structure while the upper half is entirely independent of it,
and, in making this assumption, he believes he is adding a factor of
safety thereto. The area, then, which is assumed to be carried by an
underground structure the depth of which is sufficient to allow the
lines, _V A_ and _J A_, to intersect below the surface, is the lower
half of _A N_{1} V E J N A_, or its equivalent, _A V E J A_, plus the
area, _V E J_, or _A V J A_, the angle, _A V J_, being

1 [phi]
[alpha] = --- ( 90 deg. - [phi] ) + [phi] = 45 deg. + -------.
2 2

It is not probable that these lines of thrust or pressure transmission,
_A N_, _D K_, etc., will be straight, but, for purposes of calculation,
they will be assumed to be so; also, that they will act along and
parallel to the lines of repose of their natural slope, and that the
thrust of the earth will therefore be measured by the relation between
the radius and the tangent of this angle multiplied by the weight of
material affected. The dead weight on a plane, _V J_, due to the
material above, is, therefore, where

_l_ = span or extreme width of opening = _V J_,
_W_ = weight per cubic foot of material, and
_W_{1}_ = weight per linear foot.

2 x (_l_ / 2) tan. [alpha] x _W_
_W_{1}_ = ---------------------------------- =
2
1 / 1 \
--- _l_ tan. { --- (90 deg. - [phi]) + [phi] } _W_ =
2 \ 2 /

_l_ [phi]
----- tan. ( 45 deg. + ------- ) _W_.
2 2

The application of the above to flat-arched or circular tunnels is very
simple, except that the question of side thrust should be considered
also as a factor. The thrust against the side of a tunnel in dry sand
having a flat angle of repose will necessarily be greater than in very
moist sand or clay, which stands at a much steeper angle, and, for the
same reason, the arch thrust is greater in dryer sand and therefore the
load on a tunnel structure should not be as great, the material being
compact and excluding cohesion as a factor. This can be illustrated by
referring to Fig. 3 in which it is seen that the flatter the position of
the "rakers" keying at _W_{1}_, _W_{2}_, and _W_, the greater will be
the side thrust at _A_, _C_, and _F_. It can also be i