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 those above will form themselves into an arch over the “centering” made up of those supported directly by the roof of the structure, thus relieving the structure of any load except that due to this “centering.”

If, now, the line, A B, in Fig. 4, be drawn so as to form with A C the angle, [beta], to be noted later, and it be assumed that it measures the area of pressure against A C, and if the line, C F, be drawn, forming with C G, the angle, [alpha], noted above, then G F can be reduced in some measure by reason of the increase of G C to C B, because the side thrust above the line, B C, has slightly diminished the loading above.  The writer makes the arbitrary assumption that this decrease in G F should equal 20% of B C = F D{1}_.  If, then, the line, B D{1}_ be drawn, it is conceded that all the material within the area, A B D{1} G C A_, causes direct pressure against or upon the structure, G C A, the vertical lines being the ordinates of pressure due to weight, and the horizontal lines (qualified by certain ratios) being the abscissas of pressure due to thrust.  An extreme measurement of this area of pressure is doubtless approximately more nearly a curve than the straight lines given, and the curve, A R T I D{II}_, is therefore drawn in to give graphically and approximately the safe area of which any vertical ordinate, multiplied by the weight, gives the pressure on the roof at that point, and any horizontal line, or abscissa, divided by the tangent of the angle of repose and multiplied by the weight per foot, gives the pressure on the side at that point.

[Illustration:  FIG. 4.]