Chapter I: Part 1

AMERICAN SOCIETY OF CIVIL ENGINEERS INSTITUTED 1852

TRANSACTIONS

Paper No. 1174

PRESSURE, RESISTANCE, AND STABILITY OF EARTH.[A]

BY J.C. MEEM, M. AM. SOC. C. E.

WITH DISCUSSION BY MESSRS. T. KENNARD THOMSON, CHARLES E. GREGORY, FRANCIS W. PERRY, E.P. GOODRICH, FRANCIS L. PRUYN, FRANK H. CARTER, AND J.C. MEEM.

In the final discussion of the writer's paper, "The Bracing of Trenches and Tunnels, With Practical Formulas for Earth Pressures,"[B] certain minor experiments were noted in connection with the arching properties of sand. In the present paper it is proposed to take up again the question of earth pressures, but in more detail, and to note some further experiments and deductions therefrom, and also to consider the resistance and stability of earth as applied to piling and foundations, and the pressure on and buoyancy of subaqueous structures in soft ground.

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.

_Experiment No. 1._--As the sand-box experiments described in the former paper were on a small scale, exception might be taken to them, and therefore the writer has made this experiment on a scale sufficiently large to be much more conclusive. As shown in Fig. 1, wooden abutments, 3 ft. wide, 3 ft. apart, and about 1 ft. high, were built and filled solidly with sand. Wooden walls, 3 ft. apart and 4 ft. high, were then built crossing the abutments, and solidly cleated and braced frames were placed across their ends about 2 ft. back of each abutment. A false bottom, made to slide freely up and down between the abutments, and projecting slightly beyond the walls on each side, was then blocked up snugly to the bottom edges of the sides, thus obtaining a box 3 by 4 by 7 ft., the last dimension not being important. Bolts, 44 in. long, with long threads, were run up through the false bottom and through 6 by 15 by 2-in. pine washers to nuts on the top. The box was filled with ordinary coarse sand from the trench, the sand being compacted as thoroughly as possible. The ends were tightened down on the washers, which in turn bore on the compacted sand. The blocking was then knocked out from under the false bottom, and the following was noted:

As soon as the blocking was removed the bottom settled nearly 2 in., as noted in Fig. 1, Plate XXIV, due to the initial compacting of the sand under the arching stresses. A measurement was taken from the bottom of the washers to the top of the false bottom, and it was noted as 41 in. (Fig. 1). After some three or four hours, as the arch had not been broken, it was decided to test it under greater loading, and four men were placed on it, four others standing on the haunches, as shown in Fig. 2, Plate XXIV. Under this additional loading of about 600 lb. the bottom settled 2 in. more, or nearly 4 in. in all, due to the further compression of the sand arch. About an hour after the superimposed load had been removed, the writer jostled the box with his foot sufficiently to dislodge some of the exposed sand, when the arch at once collapsed and the bottom fell to the ground.

Referring to Fig. 2, if, instead of being ordinary sand, the block comprised within the area, _A U J V X_, had been frozen sand, there can be no reason to suppose that it would not have sustained itself, forming a perfect arch, with all material removed below the line, _V E J_, in fact, the freezing process of tunneling in soft ground is based on this well-known principle.

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.

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_.

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 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.

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 the 14½-ft. circular sewer and 19½ ft. for the 15-ft. sewer. The arch timber segments in the cross-section were 10 by 12-in. North Carolina pine of good grade, with 2 in. off the butt for a bearing to take up the thrust. They were set 5 ft. apart on centers, and rested on 6 by 12-in. wall-plates of the same material as noted above. The ultimate strength of this material, across the grain, when dry and in good condition, as given by the United States Forestry Department tests is about 1,000 lb. in compression. Some tests[C] made in 1907 by Mr. E.F. Sherman for the Charles River Dam in Boston, Mass., show that in yellow pine, which had been water-soaked for two years, checks began to open at from 388 to 581 lb. per sq. in., and that yields of ¼ in. were noted at from 600 to 1,000 lb. As the tunnel wall-plates described in this paper were subject to occasional saturation, and always to a moist atmosphere, they could never have been considered as equal to dry material. Had the full loading shown by the foregoing come on these wall-plates, they would have been subjected to a stress of about 25 tons each, or nearly one-half of their ultimate strength. In only one or two instances, covering stretches of 100 ft. in one case and 200 ft. in another, where there were large areas of quicksand sufficient to cause semi-aqueous pressure, or pockets of the same material causing eccentric loading, did these wall-plates show any signs of heavy pressure, and in many instances they were in such good condition that they could be taken out and used a second and a third time. Two especially interesting instances came under the writer's observation: In one case, due to a collapse of the internal bracing, the load of an entire section, 25 ft. long and 19 ft. wide, was carried for several hours on ribs spaced 5 ft. apart. The minimum cross-section of these ribs was 73 sq. in., and they were under a stress, as noted above, of 50,000 lb., or nearly up to the actual limit of strength of the wall-plate where the rib bore on it. When these wall-plates were examined, after replacing the internal bracing, they did not appear to have been under any unusual stress.

In another instance, for a distance of more than 700 ft., the sub-grade of the sewer was 4 ft. below the level of the water in sharp sand. In excavating for "bottoms" the water had to be pumped at the rate of more than 300 gal. per min., and it was necessary to close-sheet a trench between the wall-plates in which to place a section of "bottom." In spite of the utmost care, some ground was necessarily lost, and this was shown by the slight subsidence of the wall-plates and a loosening up of the wedges in the supports bearing on the arch timbers. During this operation of "bottoming," two men on each side were constantly employed in tightening up wedges and shims above the arch timbers. It is impossible to explain the fact that these timbers slackened (without proportionate roof settlement) by any other theory than that the arching was so nearly perfect that it relieved the bracing of a large part of the load, the ordinary loose material being held in place by the arching or wedging together of the 2-in. by 3-ft. sheeting boards in the roof, arranged in the form of a segmental arch. The material above this roof was coarse, sharp sand, through which it had been difficult to tunnel without losing ground, and it had admitted water freely after each rain until the drainage of a neighboring pond had been completed, the men never being willing to resume work until the influx of water had stopped.

The foregoing applies only to material ordinarily found under ground not subaqueous, or which cannot be classed as aqueous or semi-aqueous material. These conditions will be noted later.

The writer will take up next the question of pressures against the faces of sheeted trenches or retaining walls, in material of the same character as noted above. Referring to Fig. 2, it is not reasonable to suppose that having passed the line, _R F J_, the character of the stresses due to the thrust of the material will change, if bracing should be substituted for the material in the area, _W V J R_, or if, as in Fig. 3, canvas is rolled down along the lines, _E G_ and _A O_, and if, as this section is excavated between the canvas faces, temporary struts are erected, there is no reason to believe that with properly adjusted weights at _W_ or _W_{2}_, an exact equilibrium of forces and conditions cannot be obtained. Or, again, if, as in Fig. 5, the face, _P Q_, is sheeted and rodded back to the surface, keying the rods taut, there is undoubtedly a stable condition and one which could not fail in theory or practice, nor can anyone, looking at Fig. 5, doubt that the top timbers are stressed more heavily than those at the bottom. The assumption is that the tendency of the material to slide toward the toe causes it to wedge itself between the face of the sheeting on the one hand and some plane between the sheeting and the plane of repose on the other, and that the resistance to this tendency will cause an arching thrust to be developed along or parallel to the lines, _A N_, _B M_, etc., Fig. 2, which are assumed to be the lines of repose, or curves approximating thereto. As the thrust is greatest in that material directly at the face, _A O_, Fig. 6, and is nothing at the plane of repose, _C O_, it may be assumed arbitrarily that the line, _B O_, bisecting this angle divides this area into two, in one of which the weight resolves itself wholly into thrust, the other being an area of no thrust, or wholly of weight bearing on the plane of repose. Calling this line, _B O_, the haunch line, the thrust in the area, _A O B_, is measured by its weight divided by the tangent of the angle, _P Q R_ = [phi], which is the angle of repose; that is, the thrust at any given point, _R_ = _R Q_ ÷ tan. [phi].

The writer suggests that, in those materials which have steeper angles of repose than 45°, the area of pressure may be calculated as above, the thrust being computed, however, as for an angle of 45 degrees.

In calculating the bending moment against a wall or bracing, there is the weight of the mass multiplied by the distance of its center of gravity vertically above the toe, or, approximately:

2
Area, _A O B_ × weight per unit × --- height,
3

where _h_ = height,

_W_ = weight per cubic foot of material = 90 lb.,

90° - [phi]
and [beta] = -------------
2

_P_ = pressure per linear foot (vertically),

_h_ 2
then _P_ = _h_ × ----- (tan. [beta]) × _W_ × --- _h_ =
2 3
1
--- _h^{3}_ _W_ tan. [beta].
3

When the angle of repose, [phi], is less than 45°, this result must be reduced by dividing by tan. [phi]; that is,

1
_h_ = --- _h^{3}_ tan. [beta] ÷ tan. [phi].
3

Figs. 1 and 2, Plate XXV, show recently excavated banks of gravel and sand, which, standing at a general angle of 45°, were in process of "working," that is, there was continual slipping down of particles of the sand, and it may be well to note that in time, under exposure to weather conditions, these banks would finally assume a slope of about 33 degrees. They are typical, however, as showing the normal slope of freshly excavated sandy material, and a slope which may be used in ordinary calculations. The steps seen in Plate XXV show the different characteristics of ground in close proximity. In Fig. 2, Plate XXVI,[D] may be seen a typical bank of gravel and sand; it shows the well-defined slope of sand adjacent to and in connection with the cohesive properties of gravel.

The next points to be considered are the more difficult problems concerning subaqueous or saturated earths. The writer has made some experiments which appear to be conclusive, showing that, except in pure quicksand or wholly aqueous material, as described later, the earth and water pressures act independently of each other.

For a better understanding of the scope and purpose of this paper, the writer divides supersaturated or subaqueous materials into three classes:

_Class A._--Firm materials, such as coarse and fine gravels, gravel and sands mixed, coarse sands, and fine sands in which there is not a large proportion of fine material, such as loam, clay, or pure quicksand.

_Class B._--Semi-aqueous materials, such as fine sands in which there is a large proportion of clay, etc., pure clays, silts, peats, etc.

_Class C._--Aqueous materials, such as pure quicksands, in which the solid matter is so finely divided that it is amorphous and virtually held in suspension, oils, quicksilver, etc.

Here it may be stated that the term, "quicksand," is so illusive that a true definition of it is badly needed. Many engineers call quicksand any sand which flows under the influence of water in motion. The writer believes the term should be applied only to material so "soupy" that its properties are practically the same as water under static conditions, it being understood that any material may be unstable under the influence of water at sufficiently high velocities, and that it is with a static condition, or one approximately so, that this paper deals.

A clear understanding of the firm materials noted in Class A will lead to a better solution of problems dealing with those under Class B, as it is to this Class A that the experiments largely relate.

The experiments noted below were made with varying material, though the principal type used was a fine sand, under the conditions in which it is ordinarily found in excavations, with less than 40% voids and less than 10% of very fine material.

_Experiment No. 2._--The first of these experiments, which in this series will be called No. 2, was simple, and was made in order to show that this material does not flow readily under ordinary conditions, when not coupled with the discharge of water under high velocity. A bucket 12 in. in diameter, containing another bucket 9 in. in diameter, was used. A 6 by 6-in. hole was cut in the bottom of the inner bucket. About 3 in. of sand was first placed in the bottom of the larger bucket and it was partly filled with water. The inside bucket was then given a false bottom and partly filled with wet sand, resting on the sand in the larger bucket. Both were filled with water, and the weight, _W_, Fig. 7, on the arm was shifted until it balanced the weight of the inside bucket in the water, the distance of the weight, _W_, from the pivot being noted. The false bottom was then removed and the inside bucket, resting on the sand in the larger one, was partly filled with sand and both were filled with water, the conditions at the point of weighing being exactly the same, except that the false bottom was removed, leaving the sand in contact through the 6 by 6-in. opening. It is readily seen that, if the sand had possessed the aqueous properties sometimes attributed to sand under water, that in the inside bucket would have flowed out through the square hole in the bottom, allowing it to be lifted by any weight in excess of the actual weight of the bucket, less its buoyancy, as would be the case if it contained only water instead of sand and water. It was found, however, that the weight, resting at a distance of more than nine-tenths of the original distance from the pivot, would not raise the inside bucket. On lifting this inside bucket bodily, however, the water at once forced the sand out through the bottom, leaving a hole almost exactly the shape and size of the bottom orifice, as shown in Fig. 1, Plate XXVII. It should be stated that, in each case, the sand was put in in small handfuls and thoroughly mixed with water, but not packed, and allowed to stand for some time before the experiments were tried, to insure the compactness of ordinary conditions. It is seen from Fig. 1, Plate XXVII, that the sand was stable enough to allow the bucket to be put on its side for the moment of being photographed, although it had been pulled out of the water a little less than 3 min.

_Experiment No. 3._--In order to show that the arching properties of sand are not destroyed under subaqueous conditions, a small sand-box, having a capacity of about 1 cu. ft., and similar to that described in Experiment No. 1, was made. The bottom was cut out, with the exception of a ¾-in. projection on two sides, and a false bottom was placed below and outside of the original bottom, with bolts running through it, keying to washers on top of the sand, with which the box was partly filled. One side of the box contained a glass front, in order that conditions of saturation could be observed. The box of sand was then filled with water and, after saturation had been completed and the nuts and washers had been tightened down, the box was lifted off the floor. There was found to be no tendency whatever for the bottom to fall away, showing conclusively that the arching properties had not been destroyed by the saturation of the sand.

The next three experiments were intended to show the relative pressure over any given area in contact with the water in the one case or sand and water in the other.

_Experiment No. 4._--The apparatus for this experiment consisted of a 3-in. pipe about 4-in. long and connected with a ¾-in. goose-neck pipe 17 in. high above the top of the bowl shown in Fig. 8 and in Fig. 2, Plate XXVII. A loose rubber valve was intended to be seated on the upper face of the machined edge of the bowl and weighted down sufficiently to balance it against a head of water corresponding to the 17-in. head in the goose-neck. The bowl was then to be filled with sand and the difference, if any, noted between the weight required to hold the flap-valve down under the same head of water flowing through the sand. The results of this experiment were not conclusive, owing to the difficulty of making contact over the whole area of the sand and the rim of the bowl at the same time. At times, for instance, less than 1 lb. would hold back the water indefinitely, while, again, 2 or 3 lb. would be required as opposed to the 4½ lb. approximate pressure required to hold down the clear water. Again, at times the water would not flow through the neck at all, even after several hours, and after increasing the head by attaching a longer rubber tube thereto. In view of these conditions, this experiment would not be noted here, except that it unexpectedly developed one interesting fact. In order to insure against a stoppage of water, as above referred to, gravel was first put into the bottom of the bowl and the flap-valve was then rubbed down and held tightly while the pipe was filled. On being released, the pressure of water invariably forced out the whole body of sand, as shown in Fig. 2, Plate XXVII. Care was taken to see that the sand was saturated in each case, and the experiment was repeated numberless times, and invariably with the same result. The sand contained about 40% of voids. The deduction from this experiment is that the pressure of water is against rather than through sand and that any excess of voids occurring adjacent to a face against which there is pressure of water will be filled with sand, excepting in so far, of course, as the normal existing voids allow the pressure of the water to be transmitted through them.

If, then, the covering of sand over a structure is sufficiently heavy to allow arching action to be set up, the structure against which the pressure is applied must be relieved of much of the pressure of water against the area of sand not constituted as voids acting outside of the arching area. This is confirmed by the two following experiments:

_Experiment No. 5._--The same apparatus was used here as in Experiment No. 2, Fig. 7, except that the inside bucket had a solid bottom. The inside and outside buckets were filled with water and the point was noted at which the weight would balance the inside bucket at a point some 3 in. off the bottom of the outside bucket. This point was measured, and the bottom of the larger bucket was covered over with sand so that in setting solidly in the sand the inside bucket would occupy the same relative position as it did in the water. The same weight was then applied and would not begin to lift the inner bucket. For instance, in the first part of the experiment the weight stood at 12 in. from the pivot, while in the next step the weight, standing at the end of the bar, had no effect, and considerable external pressure had to be exerted before the bucket could be lifted. Immediately after it was relieved, however, the weight at 12 in. would hold it clear of the sand. No attempt was made to work the bucket into the sand; the sand was leveled up and the bucket was seated on it, turned once or twice to insure contact, and then allowed to stand for some time before making the experiment. No attempt was made to establish the relationship between sands of varying voids, the general fact only being established, by a sufficient number of experiments, that the weight required to lift the bucket was more than double in sand having 40% of voids than that required to lift the bucket in water only.

_Experiment No. 6._--The apparatus for this experiment consisted essentially of a hydraulic chamber about 8 in. in diameter and 1 ft. high, the top being removable and containing a collar with suitable packing, through which a 2½-in. piston moved freely up and down, the whole being similar to the cylinder and piston of a large hydraulic jack, as shown in Fig. 1, Plate XXVIII. Just below the collar and above the chamber there was a ½-in. inlet leading to a copper pipe and thence to a high-pressure pump. Attached to this there was a gauge to show the pressure obtained in the chamber, all as shown in Fig. 9. The purpose of the apparatus was to test the difference in pressure on any object submerged in clear water and on the same object buried in the sand under water. It is readily seen that, if pressure be applied to the water in this chamber, the amount of pressure (as measured by the gauge) necessary to lift the piston will be that due to the weight of the piston, less its displacement, plus the friction of the piston in the collar.

Now, if for any reason the bottom area of the piston against which the water pressure acts be reduced, it will necessarily require a proportionate amount of increase in the pressure to lift this piston. If, therefore, it is found that 10 lb., for illustration, be required to lift the piston when plunged in clear water, and 20 lb. be required to lift it when buried in sand, it can be assumed at once that the area of the piston has been reduced 50% by being buried in the sand, eliminating the question of the friction of the sand itself around the piston. In order to determine what this friction might be, the writer arranged a table standing on legs above the bottom of the chamber, allowing the piston to move freely through a hole in its center. Through this table pipes were entered (as shown in part of Fig. 9). The whole was then placed in the chamber with the piston in place, and the area above was filled with sand and water. It is thus seen that, the end of the piston being free and in clear water, the difference, if any, between the pressure required to lift the piston when in clear water alone and in the case thus noted, where it was surrounded by sand, would measure the friction of the sand on the piston. After several trials of this, however, it was clearly seen that the friction was too slight to be noted accurately by a gauge registering single pounds, that is, with a piston in contact with 6 in. of sand vertically, a friction of 25 lb. per sq. ft. would only require an increase of 1.8 lb. on the gauge. It is therefore assumed that the friction on so small a piston in sand need not be considered as a material factor in the experiments made.

The piston was plunged into clear water, and it was found that the pressure required to lift it was about 4 lb. The cap was then taken off, a depth of about 2 in. of sand was placed in the bottom of the chamber, and then the piston was set in place and surrounded by sand to a depth of some 6 in., water being added so that the sand was completely saturated. This was allowed to stand until it had regained the stability of ordinary sand in place, whereupon the cap with the collar bearing was set in place over the piston, the machine was coupled up, and the pump was started. A series of four experiments, extending over a period of two or three days, gave the following results:

_Test 1._--The piston began to move at a pressure of 25 lb. The pressure gradually dropped to 7½ lb., at which point, apparently, it came out of the sand, and continued at 7½ lb. during the remainder of the test.

_Test 2._--The piston was plunged back into the sand, without removing the cap, and allowed to stand for about 2 hours. No attempt was made to pack the sand or to see its condition around the piston, it being presumed, however, that it had reasonable time to get a fair amount of set. At slightly above 20 lb. the piston began to move, and as soon as a pocket of water accumulated behind the piston the pressure immediately dropped to 9 lb. and continued at this point until it came out of the sand.

_Test 3._--The piston was plunged into the sand and hammered down without waiting for the sand to come to a definite set. In this case the initial pressure shown by the gauge was 17½ lb., which immediately dropped to 8 lb. as soon as the piston had moved sufficiently far to allow water to accumulate below it.

_Test 4._--The cap was again removed, the piston set up in place, the sand compacted around it in approximately the same condition it would have had if the sand had been in place underground; the cap was then set in place and, after an hour, the pump was started. The pressure registered was 25 lb. and extended over a period of several seconds before there was any movement in the piston. The piston responded finally without any increase of pressure, and, after lifting an inch or two, the pressure gradually dropped to 10 lb., where it remained until the piston came out of the sand.

The sum and average of these tests shows a relation of 22 lb. for the piston in sand to about 8½ lb. as soon as the volume of water had accumulated below it, which would correspond very closely to a sand containing 40% of voids, which was the characteristic of the sand used in this experiment.

The conclusions from this experiment appear to be absolutely final in illustrating the pressure due to water on a tunnel buried in sand, either on the arch above or on the sides or bottom, as well as the buoyant effect upon the tunnel bottom under the same conditions.

While the apparatus would have to be designed and built on a much larger scale in order to measure accurately the pressures due to sands and earths of varying characteristics, it appears to be conclusive in showing the principle, and near enough to the theoretical value to be taken for practical purposes in designing structures against water pressures when buried in sand or earth.

It should be carefully noted that the friction of the water through sand, which is always a large factor in subaqueous construction, is virtually eliminated here, as the water pressure has to be transmitted only some 6 or 8 in. to actuate the base of the piston, whereas in a tunnel only half submerged this distance might be as many feet, and would be a considerable factor.

It should be noted also that although the area subject to pressure is diminished, the pressure on the area remaining corresponds to the full hydrostatic head, as would be shown by the pressure on an air gauge required to hold back the water, except, of course, as it may be diminished more or less by friction.

The writer understands that experiments of a similar nature and with similar apparatus have been tried on clays and peats with results considerably higher; that is, in one case, there was a pressure of 40 lb. before the piston started to move.

The following is given, in part, as an analysis and explanation of the above experiments and notes:

It is well known that if lead be placed in a hydraulic press and subjected to a sufficient pressure it will exhibit properties somewhat similar to soft clay or quicksand under pressure. It will flow out of an orifice or more than one orifice at the same pressure. This is due to the fact that practically voids do not exist and that the pressure is so great, compared with the molecular cohesion, that the latter is virtually nullified. It is also theoretically true that solid stone under infinitely high pressure may be liquefied. If in the cylinder of a hydraulic press there be put a certain quantity of cobblestones, leaving a clearance between the top of the stone and the piston, and if this space, together with the voids, be filled with water and subjected to a great pressure, the sides or the walls of the cylinder are acted on by two pressures, one almost negligible, where they are in contact with the stone, restraining the tendency of the stone to roll or slide outward, and the other due to the pressure of the water over the area against which there is no contact of stone. That this area of contact should be deducted from the pressure area can be clearly shown by assuming another cylinder with cross-sticks jammed into it, as shown in Fig. 10. A glance at this figure will show that there is no aqueous pressure on the walls of the cylinder with which the ends of the sticks come in contact and the loss of the pressure against the walls due to this is equal to the least sectional area of the stick or tube either at the point of contact or intermediate thereto.

Following this reasoning, in Fig. 11 it is found that an equivalent area may be deducted covering the least area of continuous contact of the cobblestones, as shown along the dotted lines in the right half of the figure. Returning, if, when the pressure is applied, an orifice be made in the cylinder, the water will at once flow out under pressure, allowing the piston to come in contact with the cobblestones. If the flow of the water were controlled, so as to stop it at the point where the stone and water are both under direct pressure, it would be found that the pressures were totally independent of each other. The aqueous pressure, for instance, would be equal at every point, while the pressure on the stone would be through and along the lines of contact. If this contact was reasonably well made and covered 40% of the area, one would expect the stone, independently of the water, to stand 40% of the pressure which a full area of solid stone would stand. If this pressure should be enormously increased after excluding the water, it would finally result in crushing the stone into a solid mass; and if the pressure should be increased indefinitely, some theoretical point would be reached, as above noted, where the stone would eventually be liquefied and would assume liquid properties.

The same general reasoning applies to pure sand, sand being in effect cobblestones in miniature. In pressing the piston down on dry sand it will be displaced into every existing abnormal void, but will be displaced into these voids rather than pressed into them, in the true definition of the word, and while it would flow out of an orifice in the sides or bottom, allowing the piston to be forced down as in a sand-jack, it would not flow out of an orifice in the top of the piston, except under pressures so abnormally high as to make the mass theoretically aqueous. If the positions of cylinder and piston be reversed, the piston pointing vertically upward and the sand "bled" into an orifice in or through it, the void caused by the outflow of this sand would be filled by sand displaced by the piston pressing upward rather than by sand from above.

It was the knowledge of this principle which enabled the contractors to jack up successfully the roof of a long section of the cast-iron lined tubes under Joralemon Street in Brooklyn, in connection with the reconstruction of the Battery tubes at that point, the method of operation, as partly shown in Fig. 2, Plate XXVIII, being to cut through a section of the roof, 4 by 10 ft. in area, through which holes were drilled and through which again the sand was "bled," heavy pressure being applied from below through the medium of hydraulic jacks. By a careful manipulation of both these operations, sections of the roof of the above dimensions were eventually raised the required height of 30 in. and permanently braced there in a single shift.

If water in excess be put into a cylinder containing sand, and pressure be applied thereto, the water, if allowed to flow out of an orifice, will carry with it a certain quantity of sand, according to the velocity, and the observation of this might easily give rise to the erroneous impression that the sand, as well as the water, was flowing out under pressure, and, as heretofore stated, has caused many engineers and contractors to apply the term "quicksand" to any sand flowing through an orifice with water.

Sand in its natural bed always contains some fine material, and where this is largely less than the percentage of voids, it has no material effect on the pressure exerted by the sand with or without water, as above noted. If, however, this fine material be largely in excess of the voids, it allows greater initial compression to take place when dry, and allows to be set up a certain amount of hydraulic action when saturated. If the base of the material be sand and the fill be so-called quicksand in excess of the voids, pressure will cause the quicksand to set up hydraulic action, and the action of the piston will appear to be similar to that of a piston acting on purely aqueous material.

Just here the writer desires to protest against considering semi-aqueous masses, such as soupy sands, soft concrete, etc., as exerting hydrostatic pressure due to their weight in bulk, instead of to the specific gravity of the basic liquid. For instance, resorting again to the illustration of cubes and spheres, it may be assumed that a cubical receptacle has been partly filled with small cubes of polished marble, piled vertically in columns. When this receptacle is filled with liquid around the piles of cubes there will be no pressure on the sides except that due to the hydrostatic pressure of the water at 62½ lb. The bottom, however, will resist a combined pressure due to the water and the weight of the cubes. Again, assume that the receptacle is filled with small spheres, such as marbles, and that water is then poured in. The pressure due to the weight of the solids on the bottom is relieved by the loss in weight of the marbles due to the water, and also to the tendency of the marbles to arch over the bottom, and while the pressure on the sides is increased by this amount of thrust, the aqueous pressure is still that of a liquid at 62½ lb., and it is inconceivable that some engineers, in calculating the thrust of aqueous masses, speak of it as a liquid weighing, say, 120 or 150 lb. per cu. ft.; as well might they expect to anchor spherical copper floats in front of a bulkhead and expect the hydrostatic pressure against this bulkhead to be diminished because the actual volume and weight of the water directly in front of the bulkhead has been diminished. Those who have had experience in tying narrow deep forms for concrete with small wires or bolts and quickly filling them with liquid concrete, must realize that no such pressures are ever developed as would correspond to liquids of 150 lb. per cu. ft. If the solid material in any liquid is agitated, so that it is virtually in suspension, it cannot add to the pressure, and if allowed to subside it acts as a solid, independently of the water contained with it, although the water may change somewhat the properties of the material, by increasing or changing its cohesion, angle of repose, etc. That is, in substance, those particles which rest solidly on the bottom and are in contact to the top of the solid material, do not derive any buoyancy from the water, while those particles not in contact with the bottom directly or through other particles, lose just so much weight through buoyancy. If, then, the vertical depth of the earthy particles or sand above the bottom is so small that the arching effect against the sides is negligible, the full weight of the particles in contact, directly or vicariously, with the bottom acts as pressure on the bottom, while the full pressure of the water acts through the voids or on them, or is transmitted through material in contact with the bottom.

Referring now to materials such as clays, peats, and other soft or plastic materials, it is idle to assume that these do not possess pressure-resisting and arching properties. For instance, a soft clay arch of larger dimensions, under the condition described early in this paper, would undoubtedly stand if the rods supporting the intrados of the arch were keyed back to washers covering a sufficiently large area.

The fact that compressed air can be used at all in tunnel work is evidence that semi-aqueous materials have arching properties, and the fact that "blows" usually occur in light cover is further evidence of it.

When air pressure is used to hold back the water in faces of large area, bracing has to be resorted to. This again shows that while full hydrostatic pressure is required to hold back the water, the pressure of the earth is in a measure independent of it.

In a peaty or boggy material there is a condition somewhat different, but sufficiently allied to the soft clayey or soupy sands to place it under the same head in ordinary practice. It is undoubtedly true that piles can be driven to an indefinite depth in this material, and it is also true that the action of the pile is to displace rather than compress, as shown by the fact of driving portions of the tunnels under the North River for long distances without opening the doors of the shield or removing any of the material. The case of filling in bogs or marshes, causing them to sink at the point of filling and rise elsewhere, is readily explained by the fact that the water is confined in the interstices of the material, admitting of displacement but no compression.

The application of the above to pressures over tunnels in materials of Class A is that the sand or solid matter is virtually assumed to be a series of columns with their bases in such intimate contact with the tunnel roof that water cannot exert pressure on the tunnel or buoyancy on the sand at the point of contact, and that if these columns are sufficiently deep to have their upper portions wholly or partly carried by the arching or wedging action, the pressure of any water on their surfaces is not transferred to the tunnel, and the only aqueous pressure is that which acts on the tunnel between the assumed columns or through the voids.

Let _l_ = exterior width of tunnel,
_d_ = depth of cover, as:

_D_{W}_ = depth, water to roof,
_D_{E}_ = " earth to roof,
_D_{X}_ = " of cover of earth necessary to arching stability,

that is:

_l_ / 90° - [phi] \
_D_{X}_ = ----- ( tan. { ------------- } + [phi] ) =
2 \ 2 /

_l_ [phi]
----- tan. (45° + ------- ),
2 2

where [phi] = angle of repose,
and _D_{W}_ > _D_{E}_ > _D_{X}_.

Then the pressure on any square foot of roof, as _V_{P}_ as at the base of any vertical ordinate, as 9 in Fig. 2, = _V_{O}_,

_W_{E}_ = weight per cubic foot of earth (90 lb.),
_W_{W}_ = " " " " " water (62½ lb.), we have

_V_{P}_ = _V_{O}_ × _W_{E}_ + _D_{W}_ × _W_{W}_ × 0.40 =

1
_V_{O}_ × 90 + _D_{W}_ × 62--- × 0.4 = _V_{O}_ 90 + _D_{W}_ × 25.
2

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}_ × 90 1
_P_{h}_ = --------------- + _D_{W1}_ × 62--- × 0.4 =
tan. [phi] 2

_A_{10}_ × 90
--------------- + _D_{W1}_ × 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}_ × 90 + _D_{E}_ × 62--- × 0.40 +
2

1
( _D_{W}_ - _D_{E}_ ) × 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}_ × 62--- + _D_{E}_ × ( 90 - 62--- ),
2 2