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resists both the thrust, H, and movement of the weight, G.

For the purpose in question, it is necessary to determine the mini-
mum value of II. In the general case, with an irregular surface
bomidary between C and E, it is necessary to determine such a mini-
mum value by trial. Ordinarily, this can most easily be accomplished
by plotting a curve with values of 6 as abscissas, and corresponding
computed values of H as ordinates. The lowest value of II shown by
such a curve is, of course, the desired value.



RECONSTKUCTION OF THE STONY EIVER DAM 967

However, in the special ease in which the upper bounding surface
is horizontal — as represented on Plate XII and Fig. 20 by C-D — the
minimum value of H may be stated mathematically by expressing G
in terms of 6 and then placing the first differential of the resulting
equation equal to zero. Accordingly, it is found that ^ is a minimum
when

a

= 45° + -.

Evidently, in this special case the critical value of 6 (giving the
minimum value of H) cannot be less than 45 degrees.

Condition II. — The frictional resistance in the vertical plane, B-G,
equals /. (See Fig. 21.)

In the general case, under this condition,

cos. (9 — a)

H — G COS. a— ^ ~

siu. (9 — 2cr)

It follows that, since H becomes infinitely great when sin. (6 — 2a)
= 0, must be greater than 2a:. Again, under ordinary physical con-
ditions, G and, hence also, H, become infinitely great for ^ = 90 degrees.
Therefore, excepting for failure in compression, failure cannot occur
unless a is less than 45° — in other words, unless /, the tangent of a, is
less than 1.0.

In the special case where the upper surface of the toe material lies
in the plane, C-D, the minimum value of H may be determined mathe-
matically, as under Condition I, it being found that // is a minimum
when



= tan.



(2 tau. a + sec. a V 2^
1 — tan.2 a J



Actual Toe Resistance. — Under many, if indeed not under most,
foundation conditions, the toe resistance is limited practically, not
by the theoretical minimum values of H as expressed in the foregoing
paragraphs, but by a third condition, namely, the bearing value of the
foundation material. This is the case at the Stony Kiver site. For
design purposes, the maximum load in the plane, B-C, under the con-
ditions illustrated on Plate XII, was limited to the safe bearing value,
in lateral compression, of the soil at the site. This value was assumed
as 4 000 lb. per sq. ft. In other words, the average resistance to slid-
ing in the vertical plane, B-C, was limited to 2 000 lb. per sq. ft.



968 KECONSTRUCTION OF THE STONY EIVER DAM

To illustrate the possibilities of toe resistance, as well as the prob-
able actual values, reference is again made to the typical section of
the strengthened structure at Bay 35, as shown on Plate XII, all cal-
culations being based on normal maximum load conditions, with /
assumed = 0.33 :

Assuming first that the main "plane of least resistance" to sliding
is horizontal (see A-B, Plate XII), and assuming that there is no fric-
tional resistance in the vertical plane, B-C, the inclined "plane of
least resistance" at the toe is found to make an angle, 6, of 44° with
the vertical for the minimum value of H (toe resistance to sliding).
The corresponding total resistance to sliding is approximately 140 080
lb. per lin. ft. If, instead, the frictional resistance in the vertical
plane, B-C, is assumed to be equal to that in the inclined "plane of
least resistance", the latter is found to make an angle of 67° with the
vertical, with a corresponding total resistance of about 183 530 lb.
per lin. ft. However, if limited by the assumed safe bearing value
of the soil in lateral compression, that is 4 000 lb. per sq. ft., the total
resistance to sliding is only 115 080 lb. per lin. ft.

Assuming now at random a different location of the main "plane
of least resistance", viz., A-I (Plate XII), making an angle, ^, of 6°
with the horizontal, it is found that, without frictional resistance in
the vertical plane, I-C, the angle, 9, is 39°, and the total resistance
to sliding is about 131 200 lb. per lin. ft. ; that with full frictional
resistance (/ = 0.33) in the plane, I-C, the angle, 6, is 65°, and the
corresponding total resistance is 159 300 lb. per lin. ft. ; but that, if
limited by the safe bearing value of the soil, the total resistance is
only 121 650 lb. per lin. ft. In order to determine the minimum
values of the total resistance to sliding for each of the three condi-
tions just mentioned, enough different locations of the main "plane
of least resistance" were assumed, and corresponding calculations
were made, to warrant the plotting of a diagram in which the values
of ^ (the angle which the main "plane of least resistance" makes with
the horizontal) and of total resistance to sliding, respectively, were
plotted as rectangular co-ordinates.

It was evident from inspection of the results that, with the resist-
ance at the toe limited by the safe bearing value of the soil, the mini-
mum total resistance to sliding is afforded when the main "plane of
least resistance" is in the horizontal position, A-B. However, from



EECOiSrSTRUCTION OF THE STONY RIVER DAM 969

the diagram which was based on the assumption that there is no
frictional resistance in the vertical plane, jS was found to be approx-
imately 8° for the minimum value of total resistance to sliding, equal-
ing about 128 500 lb. per lin. ft. Correspondingly, a value of 8° 15'
was found for /? under the assumption that the coefficient of frictional
resistance in the vertical plane too is 0.33, the minimum total resist-
ance to sliding being approximately 152 700 lb. per lin. ft. Both
these values of total resistance to sliding are greater than that ob-
tained when the resistance at the toe is limited by the safe bearing
value of the soil, viz., 115 080 lb. per lin. ft. Consequently, for
design purposes, the "planes of least resistance" under "normal max-
imum load" for the section shown on Plate XII were assumed to be
represented by A-B and B-C.

It is interesting to note to what extent the total resistance to slid-
ing at Bay 35 was increased in the reconstruction. Retaining still the
"normal maximum load" conditions, and with resistance at the toe
limited by the safe bearing value of the soil, the total resistance to
sliding was increased from approximately 41 310 lb. per lin. ft., in the
case of the original structure, to approximately 115 080 lb. per lin.
ft., in the case of the strengthened structure. In other words, the
total resistance to sliding was increased by about 178 per cent. Cor-
respondingly, on the basis of the assumption that the resistance at
the toe is not limited by the safe bearing value of the soil, but that no
frictional resistance exists in the vertical plane at the toe, the total
resistance to sliding would be increased by about 211%; and if, in
addition, the coefficient of frictional resistance in the vertical plane
is assumed to be 0.33, the increase would have been about 270 per
cent.

Of course, there is undoubtedly a considerable difference between
theoretical and actual conditions. Thus, referring to the foregoing
assumptions on which the theoretical analysis has been based, earth
is rarely, if ever, strictly homogeneous. The coefficient of frictional
resistance varies, and the "planes of least resistance" are probably
never true planes. Crude experiments made by the writer indicate
that, even in the case of such granular material as sand, the surface
along which failure occurs is by no means a true plane. Under the
Stony River conditions, the "planes of least resistance" are affected
also by boulders. Again, any cohesion existing within the founda-



970 RECONSTRUCTION OF THE STONY RIVER DAM

tion soil causes tlie resistance to sliding to be greater than according
to the preceding analysis. It is evident, however, that procedure
along the lines indicated is amply conservative and safe.

It may not be amiss to point out that anchoring walls such as those
described may be applied properly and economically in the case of
rock foundations, particularly when the foundations consist of lami-
nated rock with relatively low frictional resistance. By this means a
considerable quantity of the foundation rock can be made to afford
resistance to sliding just as reliably as though the corresponding
weight consisted of concrete in the body of the dam. Generally
speaking, anchoring walls, even if not used for the purpose of reducing
the quantity of material in the body or superstructure of a dam, will,
at least, increase the margin of safety.

Stability of Strengthened Structure Against Sliding. — The design
of the features entering into the strengthening of those portions of
the dam which remained intact conformed as closely as practicable to
the general criterion that the ratio of the probable forces resisting
sliding should be twice the probable forces tending to cause sliding.

It will be noted from Plate XII that an earth fill has been placed
down stream from the typical section there shown. Fig. 22 shows
more clearly the limits and height of this fill, which extends along
the entire length of the higher sections of the dam, viz., between the
old and the new spillways. Referring again to Plate XII, it is evident
that the presence of this earth fill increases the quantity of material
lying within the angle, 0. Moreover, the greater the coefficient of fric-
tional resistance of the foundation soil, the greater the angle, 6, and
hence the greater the effective resisting weight due to the earth fill.
The material for the fill was obtained largely from the excavation for
the new spillway channel.

As in the case of the original structure, the margin of safety
resulting from the work of strengthening will be illustrated by several
examples under different sets of conditions. Thus, for the strength-
ened sections of maximum height, the results are as follows:

I. — Under the assumed "normal maximum load" conditions, as
set out in Column (6), Table 3,

Forces resisting slidina: ^ „„

:= 2.03.



Forces tendinsr to cause slidins:




Fig. 22. — Ston"^' River Dam as RKCONSTRncTED. View from West Bank.



-


w




4^


^s


^^fli




gti


-r. 111 IWIIH iiiiHilllWIIBiilillIbi


••.^•a«fe,is»:'i:




Wm



Fig. 23. — C^i'^siox Undee Footing at Heel of Dam.



EECONSTRUCTION OF THE STONY RIVEE DAM 973

Tot the original — that is, the unstrengihened — structure, the corre-
sponding ratio, as previously explained, was 0.61. By comparison of
the conditions set out in Columns (2) and (6), Table 3, it will be
noted that the only difference in the assumed conditions is that in the
case of the strengthened structure. Column (C), factor (c), the as-
sumed elevation of the lower limit of equivalent full hydrostatic
pressure against the dam was raised somewhat. This was warranted
by the fact that considerable pervious material was taken out of the
river-bed, immediately up stream from the dam, and replaced by a
back-fill of impervious clay, the back-fill in turn being covered in
part with a reinforced concrete protecting mat, especially near the
sluice-gates (see Plate XI).

II. — Under the ''most severe conditions within the limits of
reason", as applied to the strengthened structure (see Column (8),
Table 3),

Forces resistiuo: slidins:



Forces tendins: to cause slidins;



= 1.07,



This ratio is to be compared with the ratio of 0.22 for the sections
of maximum height, of the structure as it stood originally. The
comparison is the more striking when it is noted that, inasmuch as
3-ft. flash-boards can be used safely on both spillways of the recon-
structed dam, the assumed elevation of head-water is 3 ft. greater in
the case of Column (8) than in the case of Column (4), which
correspondingly sets out the "most severe conditions within the limits
of reason" for the sections of maximum height of the original
structure.

As to the typical strengthened section at Bay 35, shown on Plate
XII, the margin of safety against sliding is represented as follows :

III. — Under the "normal maximum load" conditions set out in
Column (7), Table 3, the ratio of safety is 2.14.

IV. — Under the "most severe conditions within the limits of
reason", as set out in Column (9), Table 3, the ratio of safety is 1.12.

These ratios, of course, would be bettered by any increase in the
respective coefficients of frictional resistance assumed. There are,
however, certain additional factors, tending toward greater safety
against sliding, which apply in the case of the strengthened portions



9J^4 RECONSTRUCTION OF THE STONY RIVER DAM

of the dam founded on the clayey over-burden, but in none of which
was any specific reliance placed, namely :

(a) In the structure as strengthened there is no tendency to break
at the top of the cut-off wall. The resistance of the old cut-ofF
wall to shear or rupture at the bottom of the new "heel" is,
therefore, a considerable factor in resisting sliding.

(h) Again, if such shear or rupture in the cut-off wall should
occur below the bottom of the "heel", or should extend
below the bottom of the "heel" in a direction such as A-H,
Plate XII, the effect would be to lower still further the "planes
of least resistance" and, consequently, to increase the re-
sistance to sliding.

(c) The existence of cohesion or resistance to shear in the clayey
foundation soil is by no means speculative even though the
amount of such resistance may not be readily and reliably
determinable.

Tests of Shearing Value of Clay. — The writer has found no pre-
viously published data on this subject, and therefore believes that
certain tests made during the Stony Eiver reconstruction by Mr.
D. N. Showalter, Resident Engineer, will be of general interest. The
apparatus contrived by Mr. Showalter for making the test was simple
and yet quite effective.

Specimens of clay were dug out of the foundation over-burden
and cut to such shape as to fit the apparatus. The shearing was
then accomplished by slicing off the clay, as it were, with a 2 by G-iu.
piece of wood which was forced across the specimen by a small set of
blocks and tackle. The tension in the rope was measured by a spring
balance. The specimens tested were fairly moist. The results of these
tests of the shearing values of several different clayey soils are given
in Table 4.

From inspection of the data in Table 4, it will be noted that, as
was to be expected, the presence of sand in the clay very materially
lessens the shearing value. However, even among Specimens 1 to 8,
inclusive, there is considerable variation among the results. The
effect of the length of time of application of a given shearing load
was not investigated, though time is undoubtedly a very important



EECONSTRUCTION OF THE STONY RIVER DAM



975



factor. The results clearly indicate that here is a fruitful field for
further experimentation.

TABLE 4. — Results of Tests of Shearing Yalue of Foundation
Soils at Stony River Dam Site.







Initial area


Load* at

failure, in

pounds.


Shearing value.


Experiment


Character of soil


under shearing


in pounds per


No.


under test.


stress, in


square inch of






square inches.


initial areas.


1


White clay.


9.0


109


12.1


2




4.8


128


26.6


3


»■ •'


6.1


126


20.7


4


"


H.7


226


26.0


5


" "


7.7


176


22.8


6


Black gunnbo.


19.2


184


9.6


7


" "


l'.t.2


309


16.1


8


Black loam.


13.7


219


16.0


9


Sandy yellow clay.


19.2


89


4.6


10




14.6


89


2.7



Average 1;"



* Including proper allowance for weight of apparatus.
Note. — All soil was in natural state of moistness.

To the writer, it appears that it is hardly proper, in the light of
present information, that the shearing value of clay should be taken
into account over and above the resistance to sliding developed under
such conditions as obtained in the tests of frictional resistance re-
ported in Table 2. The resistance of the clay to shear, however,
affords an added margin of safety, indefinite though the extent of
the added margin may be. Accordingly, it is interesting to consider
what might be the effect of such resistance to shear in a typical case,
for example, the section at Bay 35 under "normal maximum load"
conditions.

Referring to Plate XII, it will be remembered that, in case of failure
of the dam by sliding, it was assumed that failure along A-B would
occur by sliding of clay on clay, and that failure along B-G would '
occur by the crushing of the clay or other foundation material. Hence
It is only along A-B that the resistance of the clay to shear could
increase the total resistance to sliding. Such resistance to shear,
however, would act instead of, and not in addition to, the frictional
resistance along A-B, which, under the assumed conditions, is about
75 000 lb. per lin. ft. of dam. The average shearing value shown by
the tests reported in Table 4 is 15.7 lb. per sq. in. If, then, a
resistance to shear of 15 lb. per sq. in., for instance, existed in the



976 RECONSTRUCTION OF THE STONY RIVER DAM

53.5 lin. ft. of clay along A-B, the total resistance along A-B (exclu-
sive of resistance to rupture on the part of the old cut-off wall) would
be about 115 000 lb. per lin. ft. of dam. Thus the total resistance to
sliding would be increased by about 40 000 lb. per lin. ft.

However, for the reasons stated previously, and because of the
varying characteristics of the foundation soil, resistance of the clayey
soil to shear was not relied on to furnish resistance to sliding.

Resistance to Sliding at New Spillway. — Turning now to the new
spillway section, as shown on Plate X, it will be noted that, by rea-
son of the use of the combined anchoring wall and cut-oif at the heel
of the structure, the principal "plane of least resistance" (at the
elevation of the bottom of the anchoring wall) lies in relatively hard
shale. Free from the hampering conditions imposed by a structure
already existing on the site, it was possible at this particular place to
secure more readily and economically given ratios of safety than
obtain in those portions of the dam which were left intact after the
failure. Thus the new spillway section has generally a greater margin
of safety.

I. — Referring to the 15-ft. wide section centering on Buttress 17
(Plate IX), the ratio of safety against sliding under the "normal
maximum load conditions" is 3.69. The conditions assumed in arriv-
ing at this ratio were essentially like those of Column (6), Table 3,
except that, in accordance with Table 1, the coefficient of frictional
resistance was assumed to be 0.5. ^'

II. — For the same location in the new spillway, but imder the
"most severe conditions within the limits of reason," the ratio of
safety against sliding is 2.03. In this case the assumed conditions
were essentially like those of Column (8), Table 3, except that

(a) Head-water was assumed at Elevation 142.25, with no ice
pressure; and

(&) The minimum coefficient of frictional resistance, in accord-
ance with Table 1, was assumed as 0.40.

It is pertinent at this point to note, by reference to Plate XII, that
apparently no harm would result from uplift pressure exerted along
the planes of contact between the footings of the strengthened struc-
ture and the foundation soil. That is, there would be practically no
change in the vertical load above the probable "plane of least resist-



RECONSTKUCTION OF THE STONY RIVER DAM 977

a nee", such as A-B. Likewise, referring now to Plate X, it would
appear that uplift pressure could exist in the laminations of the
foundation shale rock immediately under the buttress footings (but
above the elevation of the bottom of the anchoring wall) without
affecting materially the stability of the new spillway section as regards
sliding. Such uplift, however, would have a material effect on
stability as regards overturning.

Eesistance to Overturning.

In the case of a hollow dam with a deck slope of approximately
45° from the vertical, it is hardly necessary to inquire deeply into the
question of stability as regards overturning. This will be apparent
from the following illustrations:

I. — In the case of the typical section at Bay 35, in its original,
unstrengthened condition (see unshaded portion of Section
B-B, Plate XII), the ratio of safety against overturning under
the assumed "normal maximum load" conditions of Column
(3), Table 3, was 3.65.

II. — For the same section under the "most severe conditions
within the limits of reason", as set out in Column (5),
Table 3, the corresponding ratio of safety was 1.14.

In deriving these ratios, moments were taken about the point, 0,
at the toe of the original structure, as indicated in Section B-B of
Plate XII, and it was assumed that, should the dam begin to overturn,
the original cut-off wall would fail at the construction joint at the
top of the wall. Furthermore, the horizontal thrust was considered to
be taken up entirely by frictional resistance along the base of the foot-
ings; that is, no resistance to horizontal thrust was attributed to the
body of the cut-off wall. In the second of the foregoing sets of
assumptions cognizance was taken of the fact that, in severely cold
weather, the weep-holes in the footings of the original structure would
have frozen up solidly and become ineffective. Consequently, uplift
pressure was considered as affecting a portion, assumed to be 50%,
of the total area of the base.

Referring now to the typical strengthened structure at Bay 35, as
shown on Plate XII :



978 RECONSTRUCTION OF THE STONY RIVER DAM

III. — The ratio of safety against overturning under the assumed
"normal maximum load" conditions of Column (7), Table 3,
is 4.57.

IV. — Under the "most severe conditions within the limits of rea-
son", Column (9), Table 3, the corresponding ratio becomes
1.68.

In developing the last two ratios it was assumed, as before, that
in case of actual overturning, the original cut-oS wall would fail at
the construction joint at the top of the wall. However, as regards the
effect of frictional resistance along the base of the main footings of
the dam, in taking up horizontal thrust, such resistance was assumed
to be equal to only 0.2 of the total concrete and water load, the
remainder of the horizontal thrust causing a uniformly distributed
load on the anchoring wall at the heel. The anchoring wall at the
toe was assumed to carry no load. Such assumed distribution of the
resistance to the horizontal thrust is obviously arbitrary, but affords
the most unfavorable conditions as regards stability against overturn-
ing. The resultant of the assumed load on the "heel" would neces-
sarily act at half the depth of the "heel".

Needless to say, the increased ratios of safety against overturning

in the case of the strengthened structure were not sought after per se.

They resulted from the provision made to increase the margin of

safety against sliding.

Footings.

Bearing Value of Foundation Soil. — In the case of clayey soil,
compressive loading causes appreciable yielding; hence the safe load-
ing or bearing value is measured by the maximum allowable yielding.
As the result of observations made at the dam site, under various con-
ditions during the investigation period, and also of various tests of
the soil at the site, it was concluded that the yielding due to a com-
pressive loading of about 5 000 lb. per sq. ft. would not exceed 0.1 in.
provided the soil is confined against "flow".

It was considered preferable not to exceed a maximum loading of
4 000 lb. per sq. ft., but somewhat greater loads were allowed for the
horizontal bearing of the anchoring walls at heel and toe, where the
soil is well confined, and also for the footings at the eastern portion
of the dam, where the foundation material is of manifestly better
character.



EECONSTEUCTION OF THE STONY EIVER DAM 979

The results of the foregoing tests of the safe bearing value of
the soils at the site are given in Table 5. The following comments
apply to the table:

1. — The variations among the results are characteristic of the
foundation soils in question.

2. — In view of the fact that the areas under load in the several
tests were very small, it is reasonable to assume that under
the actual conditions of loading the foundation soil would
show less yielding for a given load. Generally speaking, the



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