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 depth the sand is incapable of arching itself, but, for purposes of safety, the writer has put this at the point, F, as noted above, = D{X}_, although he believes that experiments on a large scale would show it to be nearer 0.67._D_{X}_, above which the placing of additional back-fill will lighten the load on the structure.

We have, then, for D{E}_ < D{X}_, the weight of the total prism of the earth plus the water in the voids, plus the added pressure of the water above the earth prism, that is: 

The pressure per square foot at the base of any vertical ordinate = V{P}_

1
V{P}_ = D{E}_ x 90 + D{E}_ x 62—–­ x 0.40 +
2

1
( D{W}_ — D{E}_ ) x 62—–.
2

To those who may contend that water acting through so shallow a prism of earth would exert full pressure over the full area of the tunnel, it may be stated that the water cannot maintain pressure over the whole area without likewise giving buoyancy to the sand previously assumed to be in columns, in which case there is the total weight of the water plus the weight of the prism of earth, less its buoyancy in water, that is

1 1
V{P}_ = D{W}_ x 62—–­ + D{E}_ x ( 90 — 62—–­ ),
2 2

which, by comparison with the former method, would appear to be less safe in its reasoning.

[Illustration:  COMBINED EARTH AND WATER PRESSURES.  FIG. 12.]

Next is the question of pressure against a wall or braced trench for materials under Class A. The pressure of sand is first calculated independently, as shown in Fig. 6.  Reducing this to a basis of 100 lb. for each division of the scale measured horizontally, as shown, gives the line, B O, Fig. 12, measuring the outside limit of pressure due to the earth, the horizontal distance at any point between this line and the vertical face equalling the pressure against that face divided by the tangent of the angle of repose, which in this case is assumed to be 45 deg., equalling unity.  If the water pressure line, C F, is drawn, it shows the relative pressure of the water.  In order to reduce this to the scale of 100 lb. horizontal measurement, the line, C E, is drawn, representing the water pressure to scale, that is, so that each horizontal measurement of the scale gives the pressure on the face at that point; and, allowing 50% for voids, halving this area gives the line, C D, between which and the vertical face any horizontal line measures the water pressure.  Extending these pressure areas where they overlap gives the line, B D, which represents the total pressure against the face, measured horizontally.