J. C. Meem - Pressure, Resistance, and Stability of Earth
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J. C. Meem >> Pressure, Resistance, and Stability of Earth
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7 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.
[Illustration: FIG. 1. SECTIONS OF BOX-FRAME FOR SAND-ARCH
EXPERIMENT]
_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.
[Illustration: FIG. 2.]
[Illustration: FIG. 3.]
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_.
[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 deg. - [phi] ) + [phi] = 45 deg. + -------.
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 x (_l_ / 2) tan. [alpha] x _W_
_W_{1}_ = ---------------------------------- =
2
1 / 1 \
--- _l_ tan. { --- (90 deg. - [phi]) + [phi] } _W_ =
2 \ 2 /
_l_ [phi]
----- tan. ( 45 deg. + ------- ) _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.
[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 the 141/2-ft. circular sewer and 191/2
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 1/4 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.
[Illustration: PLATE XXV, FIG. 1.--NORMAL SLOPES AND STRATA
OF NEWLY EXCAVATED BANKS.]
[Illustration: PLATE XXV, FIG. 2.--NORMAL SLOPES AND STRATA
OF NEWLY EXCAVATED BANKS.]
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.
[Illustration: FIG. 5.]
[Illustration: FIG. 6.]
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 deg., 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_ x weight per unit x --- height,
3
where _h_ = height,
_W_ = weight per cubic foot of material = 90 lb.,
90 deg. - [phi]
and [beta] = -------------
2
_P_ = pressure per linear foot (vertically),
_h_ 2
then _P_ = _h_ x ----- (tan. [beta]) x _W_ x --- _h_ =
2 3
1
--- _h^{3}_ _W_ tan. [beta].
3
When the angle of repose, [phi], is less than 45 deg., 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 deg., 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.
[Illustration: FIG. 7.]
_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.
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