fact of driving portions of the tunnels under
the North River for long distances without opening the doors of the
shield or removing any of the material. The case of filling in bogs or
marshes, causing them to sink at the point of filling and rise
elsewhere, is readily explained by the fact that the water is confined
in the interstices of the material, admitting of displacement but no
compression.

The application of the above to pressures over tunnels in materials of
Class A is that the sand or solid matter is virtually assumed to be a
series of columns with their bases in such intimate contact with the
tunnel roof that water cannot exert pressure on the tunnel or buoyancy
on the sand at the point of contact, and that if these columns are
sufficiently deep to have their upper portions wholly or partly carried
by the arching or wedging action, the pressure of any water on their
surfaces is not transferred to the tunnel, and the only aqueous pressure
is that which acts on the tunnel between the assumed columns or through
the voids.

Let _l_ = exterior width of tunnel,
_d_ = depth of cover, as:

_D_{W}_ = depth, water to roof,
_D_{E}_ = " earth to roof,
_D_{X}_ = " of cover of earth necessary to arching stability,

that is:

_l_ / 90 deg. - [phi] \
_D_{X}_ = ----- ( tan. { ------------- } + [phi] ) =
2 \ 2 /

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

where [phi] = angle of repose,
and _D_{W}_ > _D_{E}_ > _D_{X}_.

Then the pressure on any square foot of roof, as _V_{P}_ as at the base of
any vertical ordinate, as 9 in Fig. 2, = _V_{O}_,

_W_{E}_ = weight per cubic foot of earth (90 lb.),
_W_{W}_ = " " " " " water (621/2 lb.), we have

_V_{P}_ = _V_{O}_ x _W_{E}_ + _D_{W}_ x _W_{W}_ x 0.40 =

1
_V_{O}_ x 90 + _D_{W}_ x 62--- x 0.4 = _V_{O}_ 90 + _D_{W}_ x 25.
2

And for horizontal pressure:

_P_{h}_ = the horizontal pressure at any abscissa (10), Fig. 2, = _A_{10}_
at depth of water _D_{W1}_ is

_A_{10}_ x 90 1
_P_{h}_ = --------------- + _D_{W1}_ x 62--- x 0.4 =
tan. [phi] 2

_A_{10}_ x 90
--------------- + _D_{W1}_ x 25.
tan. [phi]

The only question of serious doubt is at just what de