Pressure, Resistance, and Stability of Earth / American Society of Civil Engineers: Transactions, Paper No. 1174, Volume LXX, December 1910

In order to make this paper complete in itself, it will be necessary, in some instances, to include in substance some of the matter of the former paper, and indulgence is asked from those readers who may note this fact.
If, then, instead of removing the mass, J E V , it is allowed to remain and is supported from the mass above, one must concede to this mass in its normal state the same arching properties it would have 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.
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.
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.
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.