Pressure, Resistance, and Stability of Earth | Page 2

had if frozen, excepting, of course, that a greater thickness of key should be allowed, to offset a greater tendency to compression in moist and dry as against frozen sand, where both are measured in a confined area.
If, in Fig. 2, E V J = [phi] = the angle of repose, and it be assumed that A J, the line bisecting the angle between that of repose and the perpendicular, measures at its intersection with the middle vertical (A, Fig. 2) the height which is necessary to give a sufficient thickness of key, it may be concluded that this sand arch will be self-sustaining. That is, it is assumed that the arching effect is taken up virtually within the limits of the area, _A N_{1} V E J N A_, thus relieving the structure below of the stresses due to the weight or thrust of any of the material above; and that the portion of the material below V E J is probably dead weight on any structure underneath, and when sustained from below forms a natural "centering" for the natural arch above. It is also probably true that the material in the areas, _X N_{1} A_ and A N U, does not add to the arching strength, more especially in those materials where cohesion may not be counted on as a factor. This is borne out by the fact that, in the experiment noted, a well-defined crack developed on the surface of the sand at about the point _U_{1}_, and extended apparently a considerable depth, assumed to be at N, where the haunch line is intersected by the slope line from A.
[Illustration: PLATE XXIV, FIG. 1.--INITIAL SETTLEMENT IN 3-FT. SAND ARCH, DUE TO COMPRESSION OF MATERIAL ON REMOVING SUPPORTS FROM BOTTOM.]
[Illustration: PLATE XXIV, FIG. 2.--FINAL SETTLEMENT OF SAND ARCH, DUE TO COMPRESSION IN EXCESS LOADING.]
In this experiment the sand was good and sharp, containing some gravel, and was taken directly from the adjoining excavation. When thrown loosely in a heap, it assumed an angle of repose of about 45 degrees. It should be noted that this material when tested was not compacted as much, nor did it possess the same cohesion, as sand in its normal undisturbed condition in a bank, and for this reason it is believed that the depth of key given here is absolutely safe for all except extraordinary conditions, such as non-homogeneous material and others which may require special consideration.
Referring again to the area, _A N_{1} V J N A_, Fig. 2, it is probable that, while self-sustaining, some at least of the lower portion must derive its initial support from the "centering" below, and the writer has made the arbitrary assumption that the lower half of it is carried by the structure while the upper half is entirely independent of it, and, in making this assumption, he believes he is adding a factor of safety thereto. The area, then, which is assumed to be carried by an underground structure the depth of which is sufficient to allow the lines, V A and J A, to intersect below the surface, is the lower half of _A N_{1} V E J N A_, or its equivalent, A V E J A, plus the area, V E J, or A V J A, the angle, A V J, being
1 [phi] [alpha] = --- ( 90° - [phi] ) + [phi] = 45° + -------. 2 2
It is not probable that these lines of thrust or pressure transmission, A N, D K, etc., will be straight, but, for purposes of calculation, they will be assumed to be so; also, that they will act along and parallel to the lines of repose of their natural slope, and that the thrust of the earth will therefore be measured by the relation between the radius and the tangent of this angle multiplied by the weight of material affected. The dead weight on a plane, V J, due to the material above, is, therefore, where
l = span or extreme width of opening = V J, W = weight per cubic foot of material, and _W_{1}_ = weight per linear foot.
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