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recovery of the pile itself?

With a modulus of elasticity, Ec, taken at 2 000 000, the elastic
recovery of a pile, 14 in. in diameter and 40 ft. long, imder a load of
40 tons, would be 0.01 ft., or J in.
L = 40 ft.;
Diameter = 14 in.;
Area = 154 sq. in., nearly; and
P = 40 tons = 80 000 lb.

Then, if Eg is assumed at the usual value of 2 000 000, there will
result: p 80 000

/ = — r = ^^, = 520 lb. per sq. in.
A 154

Deformation ^ elastic recovery (nearly) ^ e ^ ^ —

• 520 lb. X 40 ft.
= 2 000 000 =0-OWft..oriin.

• New York City.



110 discussion: underpixning teixity vestry building

Mr. The average final rebound, of the four tests in which it was recorded,
^^^^' appears to have been 0.2715 in., or slightly more than i in. The clastic
recovery of the pile itself is shown to be just about one-half of this, if
the modulus of elasticity is taken at 2 000 000. Considering the con-
ditions under which the concrete is placed, it would not be surprising
if the modulus were less than 2 000 000, in which case the elastic
recovery of the pile itself would be a still larger proportion of the
total final "rebound". The remainder of the final rebound — probably
not more than 4 in., and possibly less — might be just as well described
and explained as the "elastic recovery" of the material supporting the
pile, as to call it a "readjustment of the particles". In the last analysis,
there may not be much distinction between the two descriptions or
explanations, but if the phenomena can be explained by the action of
Avell understood properties of matter, there would not seem to be any
good reason for seeking new theories.

If the figures given by the speaker are correct, and the rebound is
due, in about equal parts, to the recovery of the pile and of the material
supporting it, then both would be of about equal importance. The
records of the five tests indicate stress-strain relations not very dif-
ferent from those characteristic of concrete columns. This woidd
naturally be expected, as both the concrete and the supporting material
are imperfectly elastic and, like the concrete, the supporting material
probably has a variable modulus of elasticity.
Ml-. H. DE B. Parsons,* M. Am. See. C. E. (by letter). — The writer will

Parsons, gj^^jg^ygj. iq make clear some of the points raised in the discussion.
Replying to Mr. Harte: The rebound of a concrete pile occurred
as soon as the pressure was removed. In other words, the top of the
..-,'.i pile rose while the hydraulic pressure was being relieved from the
jack. When all the pressure was relieved, the rebound ceased.

Replying to Mr. Connelly: Concrete piles used in subway under-
pinning work, as described in the paper, are designed to carry safe
loads, but are tested to loads in excess of the permanent safe loads.
Thus, a building pier might support a load of 200 tons. If each
pile were intended to carry 40 tons, there would have to be at least
five piles beneath the pier. In such a case, each pile could be safely
tested to a load in excess of 40 tons and be held in place under such
a load. When the last pile was tested, it would temporarily relieve
the excess loads from the others.

Although there is nothing in the paper which refers to a pressure
of 77 tons on an empty pile made up of short steel sections, pressures
as great as this are sometimes developed in sinking the empty steel
casings.

Replying to Mr. Branne: Each jill-frame consisted of three 6 by
12-in. timbers, making the wooden part about 36 in. in width, sup-

* New York City.



discussion: underpinning trinity vestry building 111

ported on an iron frame composed of two 12-in., 40-lb. I-beams, about Mr.
18 in. centers, as shown in Fig. 10. Only one jill-frame was pushed ^^^^o^^^-
forward at a time, and the center frame was kept slightly in advance
of the side frames. The pressure used in moving a frame naturally
varied owing to local resistances. The frames were moved by 3i-in.,
hydraulic rams, and the average pressure to start a frame was about
5 000 lb. per sq. in., but this fell, as the frame moved, to about 3 000 lb.
per sq. in.

Referring in general to the remarks on the rebound of the concrete
piles: It can be stated that the rebound was not attributable in any
way to looseness of the joints of the steel casings. The function of
the steel casings was to hold back the sand and to act as a form for
the concrete. The loading was placed on the concrete and not on '
the casing.

The rebound may be due in part to the elastic recovery of the
concrete, as mentioned by Mr. Buel, although his estimate for the
modulus of elasticity is probably somewhat low. The rebound appa-
rently was caused by the particles of sand under and around the
foot of the pile readjusting themselves as the pressure on the pile was
reduced. This "elastic recovery" of the sand and the "elastic recovery"
of the concrete are probably both intimately connected with the
rebound, although with present knowledge it is not possible to tell
how much is due to the one and how much to the other.



edit



>yfi v ddi bah



AMERICAN SOCIETY OF CIVIL ENCINEERS

INSTITUTED 1852



TRANSACTIONS



This Society Is not responsible for any statement made or opinion expressed
In Its publications.



Paper No. 1381

SURGES IN AN OPEN CANAL

By E. D. Johnson, Esq.



With Discussion by Messrs, Karl R. Kennison and Irving P. Church.



Synopsis.
This paper points out a rational theory on which to base research
into the question of the rise of water in a canal following an inter-
ruption of flow, due, for example, to a shut-down of a water-i)ower
plant; it calls attention to the analogy between this surge and the
phenomenon known as the "hydraulic jump".



This interesting subject always comes up in connection with the
problem of how high to build fore-bay walls to avoid overflow in case
the motion of the water in a canal is suddenly arrested by a short
circuit. Trouble from this source seems to be very infrequent, and yet
a sound theory for the computation of the height of the surge wave
has never come to the writer's attention. The reason that wash-overs
have not been more common seems to be due to the fact that, for
ordinary velocities, the surge is comparatively small, and a good fair
guess usually proves a sufficient safeguard; nevertheless, it may be
interesting to set forth what appears to be the beginning of a sound
theory, applicable to such cases.

Neglecting friction, in a smooth, rectangular flume, the sudden
dropping of a gate would seem to cause a backward rolling wave which
consumes a part of the energy of the oncoming water in eddy losses,
and accounts for the remainder in an increased depth behind the wave,



SURGES IN AN OPEN CANAL 113

the water standing still and level between the wave and the gate.
On this theory, an equation from which the depth of the water may
be determined is expressible through recourse to the well-known law
that force is equal to the rate of change of momentum; for, if the
depth of the water in motion is d and that of the water at rest is D,
the total free force acting (for unit weight of fluid and unit width of

flume) is and, in the time, /, during which Qt cubic feet of

water passes, with a velocity, v, and also Qt cubic feet of water is
projected backward over the top, so to speak, of the oncoming stream,
the quantity of water brought to rest is Dt multiplied by the velocity
of propagation of the wave, or,

the change of momentum is

DQtv



and the rate of change of momentum is

DQk

g(D — dy

whence,

1)2 — d^_ Ddv^

2 ~ (D-d)g ^^

from which D may easily be determined by trial.

It may be observed that as velocity is only relative, the height of
the "jump" which takes place in this case should agree exactly with
the formula for the "hydraulic jump", if the proper corrections are
made in the velocities relative to the earth, in such manner that the
wave would "stand still" in the ordinary acceptation of the term. In
other words, no error in theory is introduced if, while the above
phenomenon is in progress, the whole flume is regarded as moving,
bodily, with a velocity, relative to the earth, equal and opposite to
that of the wave propagation; and such modifications ought to, and
do, reveal the formula for the ordinary "hydraulic jump".

In this case, the absolute velocity of the water approaching the
wave would be

, Q



;"114 SUKGES IN AN OPEN CANAL

and the absolute velocity of the deeper water, at depth, D, would be

O •>

I'j, and Vg — «!= u, as before.



D — d

Q D



The new quantity, Q' =^ D v^ ^



D — d

The formula for the hydraulic jump is,
D^ _ d^ Q'

-^- = T ^"^ - "■>

and as D and d are unchanged, we may substitute for the foregoing
values of v^, v,, and Q', their equivalents in terms of Q and v, as follows:

D^ — d^_ Q Dv _ Ddv'
2 "" g (D — d) ~ (D — d) g'

thus disclosing the identity of the two formulas and justifying, to
some extent, the reasoning outlined in the premises.

To complete the analogy, it may come to mind that, as water
cannot "jump" unless it has a velocity greater than V gd, it would be
well to demonstrate that the sum of the velocities, v and v,, is always
greater than V gd.

Note, from Equation (1), that,

j^ _ (D + d) (D — df
y ~ 2 D d

and we are to show that ••.r.fv? nT.oTt



that is, that



V D /— -

« + ^3 or -^Z~d > ^ ^^^'



i;2 d (D — df

— >



9 ^ D'



or, eliminatinsr — , that

° g



D + d j}^ .^libQ6>'

, ....:;;{.,. 2d ^ D'

which is obvious so long a,s D ^ d.

The surge, S, above the level of the quiet water previous to its accel-

eration into the canal entrance is evident!}' equal to D — d — - — ,

^ 9

and it may be shown by calculus methods that the maximum possible



SURGES IN AN OPEN CANAL 115

value of S is equal to 0.714(i, which occurs for a critical velocity of
V = 7.448 VJ.

Modifications Involving Friction. — It now seems clear that Equation
(1) represents the relation between the depths on each side of the
backward rolling wave when friction is neglected. Without attempt-
ing to go further into the subject at this time, it may be stated,
nevertheless, that the surge probably cannot exceed the value of S
derived from this equation, when friction is taken into account. On
the other hand, there would seem to be little danger of extravagance
if the height of the canal and forebay walls was regvdated by the
foregoing considerations.



116 DISCUSSION ON SURGES IN AN OPEN CANAL

DISCUSSION



Mr. Karl K. Kennison,* Assoc. M. Am. Soc. C. E. (by letter). — The

Kennison. j^^^]^qj,'s determination of the height of the surge in an open canal,
following a sudden interruption of the flow, appears to be based on
sound theory. It is particularly interesting to the writer on account
of the intimate relationship between this surge and the hydraulic
jump, the discussions of which are published with the writer's paper
"The Hydraulic Jump, In Open-Channel Flow at High Yelocity".t
The author's treatment of the hydraulic theories involved is complete,
and requires little to be said in addition. The same conclusions, how-
ever, may be reached in a different way, at the same time bringing
out some interesting characteristics of the hydraulic jump which were
suggested by reading this paper; also, a formula for the canal surge
is submitted herewith which is simpler than that deduced by the
author.

There are points of difference between the ordinary hydraulic jump
or standing wave and the author's receding wave which at first seem
inconsistent, but which are really in agreement. It has already been
shownf that in an open channel, carrying a certain quantity of water
under a certain head, there are only two surface levels at which the
water can flow steadily. If the velocity is less than yjg X depth, it
is flowing at the upper alternative stage, and, if a dam of the proper
height is interposed, it will drop to the lower alternative stage. If
the velocity is greater than \/g X depth, it is already at the lower
stage, and may jump to the upper stage by meeting either an obstruc-
tion, which, if of the right height and smoothness, may avoid all but
incidental eddy losses, or a change in channel conditions sufficient
to cause the normal jump with its eddy losses, as ordinarily observed.
The question may arise: how can the level in a low-velocity canal,
which is already at the upper alternative stage, jump any higher on
the sudden closing of a gate, even higher than the level of quiet water
before its acceleration into the canal entrance, as the author states?

The explanation is that the conclusions previously drawn, with
reference to the hydravdic jump, assumed that the jump was
in every case stationary, not moving up or down stream. Now,
if this standing wave travels along the channel, we may, since
velocity is only relative, correct all the velocities by an amount
equal to the velocity of the wave, and then the conclusions re-
garding the hydraulic jump apply correctly to all such moving waves.
For example, a suddenly interrupted canal flow, though flowing appa-
rently at the upper low-velocity stage, is approaching the (receding)
wave at so high a relative velocity that it is relatively at the lower

* Providence, R. I.

t Transactions, Am. Soc. C. E.. Vol. LXXX, p. 338.



DISCUSSION ON SURGES IN AN OPEN CANAL 117

stage and capable of jumping higher. In fact, when we consider the Mr.
standing wave or jump as movable along the stream, instead of
stationary, there are, instead of two, an indefinite number of possible
water levels. It can even be shown that absolutely still water in an
open channel can theoretically be made to drop to any lower level
or to rise to any higher level by the passage of a standing wave.

This relationship between the hydraulic jump and the surge in an
open canal is already clear to one who has followed the author's
admirable mathematical analysis. At the risk of some uninteresting
repetition, an attempt is made to say the same thing in a different way,
and also to show graphically some peculiarities of the hydraulic jump
and its relation to the canal surge. The writer has found that ele-
mentary diagrams like these are often helpful in getting a clear idea
of the subject.

In Fig. 1, two smooth obstructions or dams are assumed to be kept
a \miform distance apart and moved along the bottom of a rectangular
flume containing still water, with the result that the water level
drops, as shown, and rises again to still water, neglecting, of course,
friction and incidental eddies. The dimensions are chosen so that
direct comparison may be made, if desired, with Figs. 3 to 8 in the
writer's paper, "The Hydraulic Jvmap, In Open-Channel Flow at High
Velocity". Higher dams moved at lower velocity would cause a drop
lower than shown, and lower dams at higher velocity a drop not as
low. The same height of dams, moved much more slowly than shown,
would cause only a local depression over each dam. They could not
be moved faster without raising the level of the still water ahead,
until their velocity is increased to that shown in Fig. 2. Then the
water would rise theoretically as shown and drop again to still water,
neglecting friction and incidental eddies, which, of course, would
actually be considerable at this velocity. In Fig. 3, one of the dams
is replaced by a gate; and, in Fig. 4, the other dam is removed, result-
ing in a case exactly similar to the suddenly interrupted canal flow
described by the author.

All velocity being relative, the absolute velocities in Fig. 4 are
also expressed relative to the velocity of the standing wave, illustrating
the normal hydraulic jump, and also relative to the gate, illustrating
the canal surge. These four figures are not necessary to show this
relation, but they may prove interesting in a study of the hydraulic
principles involved.

It is apparent, therefore, that to find a general expression (neglect-
ing friction) for the resulting depth, D, in a channel of rectangular
cross-section in which the water, flowing with depth, d, and velocity,
V, is suddenly checked, it is merely necessary to take Professor Unwin's
formula for the hydraulic jump,* which is in excellent agreement with

* Transactions, Am. Soc. C. E., Vol. LXXX, p. 410.



118



DISCUSSION ON SURGES IN AN OPEN CANAL



Mr. experiment and is apparently based on sound theory, and substitutes

ennison. ^^^ ^^^ velocity before the jump its value in terms of the difference

in velocities before and after the jump. The result checks exactly

with the equation deduced independently by the author. Since this



RECTANGULAR FLUME



FRICTION NEGLECTED




5.^2 ft. per sec.

EiG: 1;— Dams A and B moved through still water.




Fig . 2,- Same as 'Fig-. 1, except that A aad B are moved about six times as fast.




31.1 ft. per sec.

EiG. 3. -Same as Fig; 2, except that B is replaced by gate moving about half as fast.




17.4 ft. per sec.



32.jt f t. per sec. V.
(or stationary) ^
( or 15.0 ft, per sec.)



(or 15.0 ft. per sec.)



still Water
g (o r 32.1 ft. per sec. 1

I /or 17.1 ft. per sec. \ :^'i



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