Referring again to the area, _A N_{1} V J N A_, Fig. 2, it is probable
that, while self-sustaining, some at least of the lower portion must
derive its initial support from the "centering" below, and the writer
has made the arbitrary assumption that the lower 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° - [phi] ) + [phi] = 45° + -------.
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.