Pressure, Resistance, and Stability of Earth | Page 3

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.]
The practical conclusion of this whole assumption is that the material in the area, _F E C B B_{1}_, forms with the equivalent opposite area an arch reacting against the face, _C B B_{1}_ and that, as heretofore noted, the lower half (or its equivalent, _B D_{1} G B_) of the weight of this is assumed to be carried by the structure, the upper half being self-sustaining, as shown by the line, _B_{III} D_{IV}_ (or, for absolute safety, the curved line), and therefore, if rods could be run from sheeting inside the tunnel area to a point outside the line, _F B_{1}_, as indicated by the lines, 5, 6, 7, 8, 11, 12, 13, etc., that the internal bracing of this tunnel could be omitted, or that the tunnel itself would be relieved of all loading, whereas these rods would be carrying some large portion at least of the weight within the area circumscribed by the curve, _D_{II} I T G_, and further, that a tunnel structure of the approximate dimensions shown would carry its maximum load with the surface of the ground between _D_{IV}_ and F, beyond which point the pressure would remain the same for all depths.
In calculating pressures on circular arches, the arched area should first be graphically resolved into a rectangular equivalent, as in the right half of Fig. 4, proceeding subsequently as noted.
The following instances are given as partial evidence that in ordinary ground, not submerged, the pressures do not exceed in any instance those found by the above methods, and it is very probable that similar instances or experiences have been met by every engineer engaged in soft-ground tunneling:
In building the Bay Ridge tunnel sewer, in 62d and 64th Streets, Brooklyn, the arch timber bracing shown in Fig. 1, Plate XXVI, was used for more than 4,000 ft., or for two-thirds of the whole 5,800 ft. called for in the contract. The external width of opening, measured at the wall-plate, averaged about 19 ft. for