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

_l_ [phi]
----- tan. ( 45° + ------- ) _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 illustrated by
assuming that the arching material is composed of cubes of polished
marble set one vertically above the other in close columns. There would
then be absolutely no side thrust, but, likewise, no arching properties
would be developed, and an indefinite height would probably be reached
above the tunnel roof before friction enough would be developed to cause
it to relieve the structure of any part of its load. Conversely, if it
be assumed that the superadjacent material is composed of large bowling
balls, interlocking with some degree of regularity, it can be seen that