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MICROCOPY RESOLUTION TEST CHART
MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS·I963-A
NATIONAL BUREAU Of STANDARDS·I963-A
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TECHNICAL BULLETIN
No. 442
N OVEMBl<R 1934
. . ~ ·BRIDGE PIERS AS
CHANNEL OBSTRUCTIONS'·
By
DAVID L. YARNELL
Senior Drohill!/.e En~lneer D1Yislon of DlNllna~\! and Soil Erosion Control Bureau of Allrlculr.ural$niUneerinll UNITED STATES DEPARTMENT OF AGRICULTURE, WASHINGTON,
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Technical Bulletin No. 442
November 1934
CNITED STATES DEPARTMENT OF AGRICULTURE WASHINGTON, D.C. BRIDGE PIERS AS CHANNEL OBSTRUCTIONS Senior
draintl\~z
By DAVID L. YARNELL I
engineer, Division of Drainage and Soil Erosion Control, Bureall of Agricultural Engineering The Bureau of Agricultural Engineering in Cooperation with the College of Engineering, University of Iowa CONTENTS
Page
Introd uction_ ______________________ ___ ___ _____ I
Backwater occurrent'C!____________________ 2
Principal earlier bridge-pier studi~~____________ 3
Scope oC the investigations_____________________ 4
Description oC experimental plant_____________ 6
University laboratory_____________________ 6
Measuring weir____________________________ 6
Pier models_______________________________ 6
Piezometer and piezomoter tubes__________ 7
Theory
oC _____________________________________
the obstruction oC bridge piers to /low i
oC water
Test procedure_ __________________________ _____ 10
EtTect oC shape oC pier on coefficient____________ 11
EtTect
length-wldtb
ratio oC pier
coefficient.oC
____
_____________________
__ ____on____
___ _ 15
Page
EITect or channel contraction on coefficient_ ___ 16
J,ITect on coefficient oC setting piers lit an~le
with current______________________________ __ 20
Use oC data illustrated by exnmples_ . __________ ~'O
Example
1_________________________________
2()
Example 2________________ .. ______________ 24
Summary and conclusions____________________ 24
Literature cited _________________________ "' ___ ..
Annotated
reCerences relating to bridge piers 85 26
channelobstructions_________________________
Appendix:
Energy method oC computing" hrading-up"
~
~5
due to piers ______________________________
32
Empirical Cormulas and grapbicsolutlons.. _ 34
Velocity distribution around piers_________ 48
INTRODUCTION
This bulletin presents the results of about 2,600 cxperimcnts on
thc obstructivc effcct of bridge piers t,o flow of water, using larger
piers and u more extensive rtmge of conditions than has hitherto been
attempted. The tests were conducted by the Bureau of Agricultural
Engineering and the University of Iowa during 1927 to 1931, at
the hydraulic laboratory of the university at Iowa Oity, Iowa_
The investigation was undertaken for the purpose of determining
(1) the effect of shape of pier upon the height of backwater caused
by the pier, (2) the effect of length of pier upon the height of back
I The autbor acknowledges his Indebtedness to Sherman M. Wood ward and the late Floyd A. Nagler oC the
University oC Iowa Cor assistance as consltltlng engineers during the conduct oC the experiments and prep­
aration oC the report oC this im·esUgRtion. He Rcknowledges also the assistance and suggestions given
by Martin E. Nelson, engineer, oC the U.S. Engineer Office, and by Ralph W. Powell oC Ohio State Uni­
versity. Paul L. Hopkins, junior civil engineer, oC the Burea!1 oC Agricultural Engineering, assisted in
running tbe tests and making the computations, as did also Jesse .C_ Ducommun, C. L. Barker, Harold
E. Cox, .T. stuart Meyers, R. N. Weldy, Noilln Page, J,. H. Heskltt, H. E. Howe, Montok Tom, R. N.
Brudenell, R. A. Kampmeier, Frnnk W. Edwards, and C. H. Morris, graduate and senior engineering
stndenls in hydraulics at tbe University of Iowa.
68815°-34-1
2
TECHNICA.L. BULLE'l'IN 442, U. S. DEI'T. O.Jt' AGIUCUL'l'URE
water, und (3) the efl'cet of lnngnitude of ehannel ('.ontruction upon
the height of backwater.
The problem of the obstruction of bridge piers to the flow of water
is becoming more important with the passing of time. Population
is increasing. Industrial and manufacturing interests also are
inerensing. The larger streams are '''-itnessing a gr'eat revival of
inland-waterway transportation, and the rniiToads themselves utilize
the ensy grades of the river bottoms for their roadbeds, while power
plants, both steam and hydroclectrie, a.ro generally located nt the
riverside, all resulting in the concentration of industrial do' '''pment
along streams. This in turn means f'n inereasing number 01 :)l'idges,
w'hich results in a borger number of piers. Also, prices of materials
for steel and altel'l1ate construction types, IlS well as other con­
siderlltions, may tend to promote building of short-span bridges
with many piers. Hence, it becomes importul1t to determme what
forms of piers ofTer least obstruction to the flow of water, and
how much clifl'erence thore is in the hydraulic efficiency of differont
shapes.
The amolmt of obstruction which a bridge pier causes depends
upon (1) the shape of the pier nose, (2) the shape of the pier tail,
(3) the percentage of channel rontraction caused by th r pier, (4)
the length of the pier, (5) the angle which the pier makes with the
tlU'ead of the stream, and (G) the quantity of flow.
8ACKW ATER OCCURRENCE
The erection of one 01' more bridge piers in U stream forces the
riYer to flow through a reduced cross section nnd hence in pnssing
through this section, the wnter must acquire a. yelocity greater than
that existing in the unobstrueted cbanne1. 2 The increase in Yelocity
can be produced only by eleya.ting the water SWl'lllce in the reach
upstream frem the piers which produce the ~'o:rl,trnetiQn in area.
Thus, as the stream enters the contrncted area, n drop in t110 water
surftlce is noted accompllnying the increuso in yelocity. Howeyer,
when the stream expands again into the unobstructed channel down­
stream from the pier, the water sllrface fails to rise agnin to the
loyel of the water surface upstrcam from the pier. This permanent
drop in water lcyel is inclicfLtiYe of energy losses which may originate
from three SOUl'COS, (1) friction of the water on the pICr walls, (2)
coutmction of flow caused by the picr nose, 1111d (3) expansion of
the stream as it pusses out from between the piers.
The changes in cross section and velocity in passing the bridge piers
cause much disturbance in the flow, espeeially when the pier does
not conform in shape to the direction of the contracting filaments of
water. The curvature of the stream lines around the upstreum nose
of a pier induces high and erosive veloeities at that point. Eddies
may be formed along the sides and below tho tail of the pier. These
high velocities and resultant eddies often scour out the beels of streams
next to the piers to such 1m extent thllt the foundations may be
endangered and e'~en "undermined. Like Illany other hydraulic
phenomeno" the height of backwater varies with the square of the
veloeity of the water) aud thus with the square of the qnnntity of £low
, It is Ilssumed that the yelocity or the Willer in tho unobstructed channel is less thun critical. H tho
Yelority of the wilter in tho unobst.ructed chllnnel is at critical stagn or ~reater, thon the water will rise
at the point ur Obstruction. 'rbis condition or flow is soldom encountered in actunl prllctice.
BRIDGE PIERS AS CHANNEL OBS'l'RUC'l'IONS
3
if the depth of water remains the same. Thus doubling the discharge
would quadruple the amount of backwater caused by the pier. In
many formulas the height of backwater also varies inversely with the
square of the ()oefficient of contraction.
Lawsuits sometinles occur because of damages caused by backwater
due to bridge piers. If such damages seem probable, the engineer in
designing the bridge for a certain location may calculate by means of
some formula the height of backwater that may result from con­
struction of the bridge. Every backwater formula contains a coeffi­
ciE:nt which varies with the shape of the nose and tail of the pic 1.••
Thus the accuracy of the calculation depends greatly. upon the
correctness of this coefficient, which can be determined only by
experiment.
PRINCIPAL EARLIER BRIDGE··PIER STUDIES
Probably one of the earliest writers 011 the obstruction of bridge
piers to flow of water was Dubuat (5),3 who in 1786 attempted to show
by mathematics that the nose of a pier should have convex curves to
cause the minimum obstruction to flow. Dubuat also derived a
formula for computing the amount of backwater caused by different
velocities, and by various percentages of channel contraction. In'
addition he conducted some experiments in losses of head caused by
bridge piers. About this same time Bossut (2), in a mathematiClll
solution, proved to his own satisfaction that the pier nose should be
triangular, the tip being a right angle. Eytelwein (6) in 1801 pre­
sented !1 bridge-pier formula in his handbook on hydraulics.
These early hydraulicians were followed in turn by Dupuit,
Gauthey, D'Aubuisson (1), Debauve (4·), and Weisbach (16, 17).
The lust about 1862 performed a few experiments on a small round
pier, 0.02 meter in diameter in a channel 0.028 meter wide. This
was a much greater percentage of channel contraction than is usuaUy
found in practice. His channel was so small that the experiments
may be of questionable value. eresy (3) in 1865 reported eight
e~. periments he had made on various shapes of piers 15 centimeters
in thickness tested in a canal in which the depth of the water was
controlled by stop boards.
Investigations were made in 1914 by Nagler (9), who conducted 256
tests on 34 model piers of various types at Ann Arbor, Mich. He
suggested a new formula for computing backwater caused by bridge
piers, and derived coefficients for use in it. His tests were made on
sinr.;le piers 6 inches wide placed in the center of a channel 2.138 feet
wiae, contracting the channel 23.3 percent. In 1915-16 Lane (8)
conducted at Dayton, Ohio, a series of experiments on the :flow of
water through contractions in an open channel 0.3 feet wide. His
tests were made with the following contractieils: A I-foot opening
with rounded edges; a I-foot and a 2-foot opening each with sharp
edge contractions; short flumes with I-foot op{mings and with both
sharp and rounded edges; and an expanding or Venturi flume. The<;le
conditions caused 70.5 and 85.3 percent contraction. Lane computed
coefficients for use in the D'Aubuisson and Weisbach formulas, and
pointed out certain conditions that somewhat limit the use of
D'Al1buisson's formula.
aItalic numbers in parentheses refer to Literature Cited, p. 25.
4
'rECHNICAL BULLETIN 442, U. S. DEPT. OF AGRICUL'l'URE
From 1917 to 1922 Rehbock (10; 11; 12; 13; 14, pp. 106-115)
carried 0-:-;' at the hydraulic laboratory of the technical Hochschule at
Karlsruhe, Germany, an elaborate series of tests on br.idge piers of
various sizes and shapes and developed formulas for computing the
amount of bednvater. As a result of more than 2,000 experiments,
the effort to develop a backwater formula suitable for all possible
flow conditions was abandoned, and flow past piers was divided into
three classes.
While considerable experimental work has been done on piers of
solid construction, no tests pdor to those reported herein for the first
time have ever been made, so far as the writer knows, on piers con­
sisting of two cjrcular pillars connected either by a web of structural
braces or by a solid diaphl'llgm. Fowler states, in his discussion of
N agler's paper on bridge-pier tests (9), that thiR type of pier which is
now used widely offers much greater resistance to flow of water than
the ordinary forms of solid masonry.
The amount of obstruction to flow is increased when the piers are
set at an angle with the thread of the stream, but apparently this is
another problem of installation that never has been investigated
before.
SCOPE OF THE INVESTIGATIONS
In this investigation, tests were l'1~n on piers giving four percentages
of channel contraction, namely, 11.7, 23.3, 35, and 50 percent. Each
pier model was built up of a rectangular barrel 3 feet 6 inches long,
and a nose and a tail of lengths depending upon their shapes and
upon the width of the pier.
The first series of tests was run on a single pier 14 inches wide
placed in the center of the testing canal10 feet wide. Each" square"
or semicircular or 90 0 triangular end added 7 inches to the barrel
length, and other shapes added somewhat more. The twin eylinders
were 3 feet 6 inches apart on centers. The over-all lengths of the
piers thus ranged from 4 feet 8 inches to 6 feet 7 inches. Eleven
combinations of end shapes were tested, as follows:
1. Square nose and square tail.
2. Square nose and semicircular tail.
3. Semicircular nose and semicircular tail.
4. Semicircul.!Lr nose and square tail.
5. Triangular nose of 53° angle and semicircular tail.
6. Triangular ~lose of 60° angle and semicircular tail.
7. Triangular nose of 90° angle and semicircular tail.
8. Convex nose and tail each formed by two curves tangent to sides of pier
and described on an equilateral triangle.
9. Lens-shaped nose and tail each formed by two convex curves tangent
to sides of pier and of radius twice the pier width.
10. Twin-cylinder pier without diaphragm.
11. Twin-cylinder pier with diaphragm.
The second series of tests was run on two piers each 14 inches wide
placed in the 10-foot canal, doubling the percentage of channel con­
traction used in the first series. These tests covered the following 14
combinations of end shapes:
1.
2.
3.
'1.
Square nose and square tail.
Semicircular 110se and semicircular tail.
Convex 1I0se and tail. Lens-shaped nose and tail. Tech. Bul. 442. U.S. Dept. o( Agriculture
PLATE 1
FLOW PAST PIERS OF DIFFERENT SHAPES.
l'il'rs o( slnndnrd Il'n~tlt. (our t illl~:; \ridt It. • t, '('win·cylinder pier wi'lumt dinphra!!lJl; ('honnel ronlrnction
11.; pen.'C'nt. 11, 'I'win~cylindcr pi(lrs wil h dinphra!!m; Chillllwi cOlltrncliull !!3.:i pcrecIlt. G', l'icrs with
convex noses ulltil.'Olln'X LUjl~; ehamwI (.'(;lltraclion ~3.a IJCn'Cl1l.
PLATE 2
Tech. Bul. -442. U.S. Dept. of Agriculture
FLOW PAST PIERS OF DIFFERENT SHAPES.
Piers of stnndnrd length, fllur t inws width. A, Viers with S0mirirc,ulor no~cs on,1 tnils; chonnel rontrnrt ion 2:1.3 perrclll. R. Piers wit h sel1licir~lll:lr noses nnd tnils, nntl hnllcr 1:24
011 nround: channel contrn~tion 2:C\ perecnL Q==S2.2:~ ('uhie feet l)Cr sCl'ond, ]). =!U:O fellt.. C. Pief with sLlInit'irrlllnr nO'sll mHI tail haYing 1:2·} hatter, weh recessed; channel con­
trnction 11.7 percent, (/=82.1)(; ('uhic feet rwrsC!cond, fl,=2.tj4 [c<t. Ih=2.HO fect. J), JJjc""S with sl,!tlIicirculnr noses and tails huying 1:2.1 biltter, web recessed; channel contraction,
23.:1 percent, Q=82.5 cuhic f!!et ]ler second, fJJ=2.il feet, DJ=2.61 fect.
en
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5
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Tech. Bul. 442. U.S. Dept. of Agriculture
PLATE 4
FLOW PAST PIERS WITH SEMICIRCULAR NOSE AND TAIL HAVII\:C
WITH DIFFERENT WEB CONSTRUCTION,
1:24 DATTER. AND
J'ierso(stulltlnrtllcllilih. (ollr t.hllcs width. (,hnll1lL'l ('O:;lrm·tkn II.i P('... ·l'lIl . ..'1. Xo 1:llcts III ('llllSo(
reteSSNl we!>. Il. (~Il~r:or·rolllld fillets in weh n'('l'S~(~ m Inil. C, Qllnr:rr·rotll:() Gllets nt tllillllld tri­
angular fillets 11t
JW~C
in
\'''o~J
recesses. }),
,rch same thic:.. IH;sg lOS noso a~,tl tnil.
Tech. Bul. 442. U.s. Dept. of Agriculture
PLATES
FLOW PAST LONG PIERS.
Chnnnell'nntrnction 2:ta pcrCl'ut. A. Piers with length s('vcnlimcs witlt h, nnd square noses and tnils; (l=7S.0 ('uliit, frel pl'rsCl'ond. /), ;::::2.•;:~ [el't. D1=2.-11 feet. B, Piers with len~t h
sevell I hue...; width, ;nul scmicirculnr noses und tuils; Q=f~ n ('uhie (tlet Jlel' S{I('nJlCl. n,=2.01 feet. Dj=2.00 [l'l'L (', I'iers with length thirteen I illlt.'s witHh. nlHi squure noses IH1d
tnils; Q=75.0 (~Ilhic feet pl'r ~ec(JudJ JJJ= 2.:iO fcet, DJ='!!.:!S feel. j), J'icrs with Icngl h Ihirtl!cn t iml!s witH h, :nul SCIIlil'irclllar IIOSe.::; tlIHI tuils; (1:;;;;7.0 cubit." feet 1,er second. DJ=2.53
feet, lh=2..1U (eel..
ID
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Tech. nul. 442. U.S. Dept. of A~ricultlJrc
PLI\TE 7
FLOW PAST P:ERS SET AT ANGLE WITH CURRENT.
Piprs of len~1I1 four tiull'S \\1<1111. (lud lI'ith sl'Illi('in'ulnr IHlsr nu<I sl'llll<'irc'ulllr Inil. A. Plrr fit JO.de~rco
allgle with lIo\\'; q;,:-:7.i.fi ('uhit· (.'ttl P{\J" :w('ond. J>,··.:!.il!I (,llIt. /)1 ~!.!.Ut (l'l't. 111 Plpr nt 20·dcgrce unglo
,\Ulllh)wj q=U:-i.; ('ubic (('l't pt'r !'IN'olHI t j)1-<t~a (!.'l'll l)J :!.Ii fl'l't.
PLATE 8
Ted•• Bul. 442. U.S. Depl. of t.gricullure
A, nfuldng \"('locUr trn\'l.'l'ses
1I1'011l1d pill!" ill lesling-l'hnnnci.
ca 1111 I. IJ, 'l'win I)jcl'S hl1ing- IO\\'l.'l'l d 11110 testing
l
BRIDGE PIERS AS CHANNEL OBSTRUC'TIONS
5
5.
6.
7.
8.
9.
10.
11.
Twin-cylinder piers without diaphragm.
Twin-cylinder piers with diaphragm.
Tr.iangular nose and tail of 90° angle.
Triangular nose and tail of 90° angle with 1:12 batter.
Semicircular nose and lens-shaped tail.
Lens-shaped nose and semicircular tail.
Semicircular nose and semicircular tail with 1:24 batter on ends and sides;
web one-third thickness of nose and tail at flood-stage level.
12. Semicircular nose and semicircular tail with 1:24 batter OIl ends and sides;
web one-third thicknes6 of nose and tail at flood-stage level; quarter
round fillets in recesses at tail.
13. Semicircular nose and semicircular tail with 1:24 batter on ends and sides;
web one-third thickness of nose and tail at flood-stage level; q uartel'­
round fillets ill recesses at tailalld triang'llul'-shaped fillets ill l'eCeSSeR
at nose.
14. Semicircular nose and semicircular tail; web same thickness as nose and
tail; 1:24 butter ullaround.
The third series of tests was made on a single pier 3.5 feet wide
placed in the center of the 10-foot testing canal. Two shapes of noses
and tails were tested in this series: (1) Square nose and square tail,
and (2) semicircular nose and semicircular tail. The over-all lengths
were 7.0 feet.
The fourth series of tests was made on a single pier 5.0 feet wide
placed in the center of the 10-foot canal. Two shapes of noses and
tails were tested in this series: (1) Square nose and square tail, and
(2) semicircular nose and semicircular tail. The over-all lengths were
8.5 feet.
The effect of the length of the pie.!' upon the coefficient was deter­
mined by expp-:;iments upon piers of 14-inch width with barrels 2 times
and 4 times the length of those of the piers upon which the regular
tests were run, thus giving ratios of over-all length to width of 7 to 1
and 13 to 1. Only two shapes of noses and tails were tested on these:
(1) Square nose and square tail, and (2) semicircular nose and semi­
circular tail.
In addition to the above experiments, two series of tests were made
on a pier placed at angles of 10° and 20° with the current. The pier
was 14 inches wide and had semicircular nose and tail.
The loss of head caused by the various set-ups was determined for
quantities of flow ranging from about 10 to 160 cubic feet per second.
Different depths of flow were obtained for each discharge and for
each type of pier by means of an adjustable weir located in the channel
downstream from the piers. The depths past the piers ranged from
about 0.6 foot to more than 3 feet.
A series of tests were also run in which (1), a constant depth was
maintained upstream from the piers, and (2), a constant velocity was
maintained upstream from the piers.
The combination of these conditions gave a total of about 2,600
experimen ts.
In addition to m.easUling the loss of head for each type of pier for
various quantities and depths of flow, an investigation of velocity
distribution was made for one type of pier, with 1 discharge and 2
depths of flow. The direction of flow of valious filaments of water
as weUas the elevations of the water surface at the pier site were also
obtained for these two tests.
Views of some of the piers tested are shown in plates 1 to 7.
6
TECHNICAIJ BULLETIN 442, U. S. DEPT. OF AGRICULTlTRE
DESCRIPTION OF EXPERIMENTAL PLANT
UNIVERSITY I.ABORATOUY
The hydraulic laboratory of the University of Iowa is located on
the west bank of Iowa River south of the university dam. The
laboratory provided, in addition to other facilities, a gravity water
supply feeding a concrete testing canal 312 feet long, 10 feet wide,
and 10 feet deep. At the upstream end of tbis canal, where it joins
the end of the dam, is an electrically operated gate 10 feet wide by 10
feet deep. The dam is 9 feet high, thus insuring a head of from 9 to
10 feet, depending upon the £t.uge of the river, and an ample quantity
of water.
MEASURING WEIR
For measuring the quantity of water flowing past the piers, a sht1rp­
crested rectangular weir 10 feet long of the suppressed typ('. was built.
The weir was located 61 feet dmvnstream from the head gate. The
quantity of water pu:ssing over the weir was regulate.). by raising or
lowering the head gate.
Since the water entered the canallmder the head ga.te with a high
velocity, a submerged baffle 4 feet high was built on the bottom of
the canal immediately downstream from the head gate. This bailie
consisted of three 2- by 12-inch planks placed on edge 6 inches apart.
Ten feet below the head gate, a baffie of 2- by 4-inch by 10-foot ti.mbers
placed horiz.vntally 1 inch apart was constructed. One foot down­
stream from this batHe, another batHe was built of 2- by 4-inch timbers
})laced vertically 1 inch apart.
To avoid commotion of the water as it approached the pier, three
baffles were installed below the weir. The first of these, located 6
feet from the weir, was built of 2- by 4-inch timb~rs 10 feet long placed
horizontally 1~ inches apart. The next baffie, located 7 feet from
the weir, consisted of 2- by 4-inch timbers 4 feet long placed vertically
1% inches apart. The last baffle, located 10.5 feet from the weir,
consisted of oval-shaped timbers placed vertically, the spaces between
being 1~ inches close to the upstream face enlarging to 4}~ inches wide
at the downstream face. This baffle also was 4 feet high. These
three baffles had the desired efl'ect of stilfulg the turbulence of the
water. Meas1.ll·ements taken ahead of the piers showed uniform
velocity as the water approached the piers.
To determine the head on the weir, a hook gage was placed on the
east wall of the canal. A l}6-inch pipe passillg through the wall of
the canal 15.77 feet upstream from the weu' and 10% inches above the
bottom of the canal was connected to a 15- by 36-inch cylindrical
galvanized tank which served as a stilling well over which was mounted
the hook gage. Bazin's formula was used to detel'mjne the diseharge
over the weir.
PIER MODELS
Each pier model was mounted as shown in plate 1in the bottom of the
Several of the set-ups tested are shown in plates 1 to 7,
incluslve.
The pier models were made of wood. For the regular tests, a cen­
tral section 1.167 feet wide by 3.50 feet long was used for all the piers;
to this were aUached the different forms of noses and tails, in changing
the design. Thus the total lengths of the piers varied somewhat
depending upon the shape of the ends.
testin~ canal.
BRIDGE PIERS AS CHANNEL OBS'l'RUCTIONS
7
The centml sectio11 of Lhe pior model wns secured to the floor in the
canal some 40 feet downstreum from the weir.
l'IEZOME'I'EltS AND l'IEZOME'I'EIt TUDES
The loss of head caused by the piers was measured by means of
piezometers instead of the eustomary stilling wells and hook gages
located upstream ltnd downstream from the pier. Thirty-seven
piezometer openings splteed throughout a distance of 69 :feet were
made through the east wltlI, 4 inches above a level :f:loor built in the
bottom of the testing canal. Ten openings spaced 2}f feet n.part
were placed upstream from the pier, 15 openings spaced 6 inches
apart were placed at the sit0 or the pier models, while 12 openings
spaced 2}~ feet apltrt were rIMed downstream from the pier models.
Connections were mack fn)111 the piezometer openings by means of
rubbt'r tubing to 1-ineh ';lnss tubes 3 feet long attached to white­
enameled gage stufl's secured to the outside wall of the testing eanal.
These gnge stltffs, 3.3 feet long, wore graduated with d~Yisions of
0.02 foot, and the markings were sueh that they could be read to the
nearest 0.01 foot with little ehance of errol'.
'l'he depth of flow and wfltcr surfilce grndient upstream and down­
stream from the piers as well as the depth in the contracted seetion
along the piers were obtained from these piczometers. In addition,
several staff' gages which were. used for cheek readings were set at
various points along both walls of the ennal, the zeros of all staffs
being set even with the levelf:!oor ('onstructed in the canal.
An n,djustftble w(>1.1: 6 f(>et high some 80 feet downstreaJll from the
center of the piers was used to regulate the wlltcr level downstream
from the piers. ffhis weir WitS hung 011 hinges and was adjusted by
menns of fl· block nnd tllckle,
THEORY OF THE OBSTRUCTION OF BRIDGE PIERS TO FLOW OF
WATER
Let .figure 1 represent 11 bridge pier \vith the wat(,l' flowing tlll'ough
Ow contl'lleted a 1'('11. Th(' following symbols are llSNl:
Q=the quantity of Imter flowing in volume per second.
DJ = the melll! depth of the wuter upstream from the nose of the pier at a
distance equul to the len~th of the pier,
/)2 = the lUcan depth of the streum in the lIlost can tracted section of the
c1mnnel.
/)a=the mean depth of the wuter in the ehallllel below the contraction;
that is, the depth ill the unobstructed ('hallne!.
fl'J "., the Ille:ln width of the chllnnelabove the contraction.
W2 the menn width of flow at the most COlltmeted sectioll of the chunnel.
IVa ~. the mean width of the channel iJelow the contraction, ordinarily equal
to WI,
OC""
1'1 = thl' mcan \'cloeity of the wutcr' above the contraction,
= l(~bl'
V z= the mean "elocity of the water ill the most contracted seetion of ihe
Q
chunnel= W2D~'
V3=the mean \'elocity of thr water ill the channel below the contraction
='II~D3' which will ordinarily be equal to 1l~D3'
Hz= the drvp of the W:l tel' surface at the most contracted section = D1 - Dz.
H~= the drop of the wHter stlrfat'c in p:lssing through the COlltr:WtiOJl
=D t -D3.
8
TECHNICAL BULLETIN 442, U. S. DEPT. OF AGRICUI,TURE
g=the
VN2g=the
V 22!2g=the
V32 j2g=the
acceleration of gravity.
velocity head of the water above the channel contraction.
velocity head in the most contracted section of thc channel.
velocity head of the water below the contraction.
cross-sectional area of obstructions
a = channel-contraction ratio = '~c::'r'o=-s::';s:":-sc.:.e:":ct":;i:.;co':"ll-al;;"::"'ar-e';;;'a':"o-"f;':'c='hc.:a'::n:':'n':":e";:'l=
_ velocity head below contraction
VN2g
w- depth of flow below contraction ~
Da
K = a pier-shl1pe coefficient to take account of the losses due to friction,
impact, eddies, etc. The subscripts D' A, W, N, and R designate
D'Aubuisson, Weisbnch, Nagler, Ilnd Rehbock formulus, respec­
tively.
00= pier-shupe coefficient ill Hehbock gcneral bridge-picr formula (see
equation 7).
Pie,.
-
VI
~I\------ ~-------------if3
1DI
I"--r-­
1
z
-'!!-
Bottom of ch.nn.' - '
LONGITUDINAL
I
WI
I
1
t
--­
r
v,
PROFI;"E
"
~
--­ I
!
1
W,
JoWz
2
PLAN
FIGURE I.-Diagram of bridge pier showing symbols used in formulas: "crtical scalc exaggerated.
The real backwater height as shown in figure 1 is lla. The surface
drop H2 in the contracted area is sometimes erroneously called the
backwater height.
D'Aubuisson (3) p!,obably first advanc~d the theory that the drop
H2 was merely the dIfference of the velocIty heads for' points Dl and
D2 • His formula becomes, by substituting DI for (D 2 +H2 ),
Q2(
H2 =,?
-0
1III ~D
T/2
.L1.. - D'.4 ' 2
2'i
-
1)
11'
' 'I 2D I ~
(1 )
in which R D , A is the D'Aubuisson pier-shape coefficient.
The true backwater is not exactly represented by H21 but ordi­
narily in practical field installations there will be little difference
between H2 and Ha, and hence little difference between D2 and Da.
BRIDGE PIERS AS CHANNEL OBSTRUCTIONS
9
Therefore, only the values of the D'Aubuisson coefficient using Ha
and Da are given. Transposing and rearranging the terms ill equation
1, substituting VI for Q/WID" and Ha and Da for H2 and D2 and
solving for Q, equation 1 for practical use becomes
Q=Kn'A W2Da.J2g Ha+ Vl 2
whence
(2)
(3)
Weisbach (17, pp.114--116) hased his formula upon the assumption
tbat the total discharge throu~h the contracted section may be calcu­
lated as the sum of two quantIties, one quantity consisting of the flow
throu~h a submerged orifice of width W 2 and height D2 , and another
quantIty consisting of the flow over a weir with a crest length of WI
and a head of H 2 • The formula then becomes
Q=Kw.J2u[JWI(H2+ VN2g)3/2+ W2D2(H2+V 12/2g)
2.2
2
.8
/
/
1/2]
(3-a)
-
V
J
.6
/
4
If
J
I. 2
olL
I. 0
10
20
30
40
50
60
70
80
9()
I 00
Percentage of :'lannel obstructed by pier
FIGURE 2.-Values oC coefficient {J in Nagler bridge-pier Cormula.
Nagler's (9) formula is
Q=KNWn !2g[Da- 8Vl/2gJ.JH+f3VN2g
(4)
in which the coefficients 8 and {3 depend upon the conditions at the
site of the bridge pier. The coefficient 8 is merely a correction
coefficient, and the factor 8Vi/2g is intended to correct Ds to give a
smaller depth of flow similar to that at the most contracted section.
This coefficient has little effect upon the results obtained when the
depth of the stream is several feet or more. In all computations, t.he
value of 8 was taken as 0.3. The coefficient {3 varies with the per­
centage of channel contraction, the amount of change being greatest
for channel contractions between 5 and 30 percent_ This coefficient
may be obtained from figure 2, prepared by Professor No,gler.
68815°-34-2
10
L'!'URE
TECHN ICAL BULLE TIN 442, U. S. DEP'.I:. OF AGRICU
~
Rehbo ck (11) divides the flow into tlu'ee classes as follows:
obstruc tion with
1. Ordinar y or "steady " flow, in which the water passes the
very slight or no turbulen ce.
displays a
2. Interme diate flow, in which the water passing the obstruc tion
ce.
turbulen
of
degree
modera te
tion becomes
3. "Chang ed" flow, in which the water passing the obstruc
t.
turbulen
etely"
"compl
These t~rree clas~es of flow are defi.ned according to Rehbo ck by
the followmg equatIOns.
1
(5)
a A =0.97+ 21w 0.13
aB=0. 05+ (0.9-2. 5w)2
(6)
the first
According to Rehbo ck, the moving water will he iI\cluned in than
the
less
is
site
pier
the
of
a
class as long as the obstru ction ratio
site
pier
the
of
a
of
value
the
When
5.
on
equat.i
in
limitin g value
5 and aB
under investi gation lies between the vnIues of aA in formulathe
value
when
and
s,
prevnii
flow
of
on
conditi
second
the
6
in formul a
ion
condit
third
the
6
n
equatio
in
of a of the pier site exceeds that given
of flow mdsts.
The Rehbo ck (13) equatio n for compu ting the backw ater height,
H 3 , for all pier shapes in a channel of rectang ular cross section with
.
ordina ry flow! is as follows:
A simple equatio n for \mdge backwat'U' is, according to llehbo ck(11)
(8)
It is probab le that the D'Aub uisson , Weisbach, and Nagler formu­
ck.
las apply only to the first class of flow as defined by Rehboch
formul a
Weisba
the
for
ients
coeffic
pier
bridgeDeterm ination s of
that
were attemp ted, but the extrem ely discord ant results indica ted
ned.
abando
was
effort
the
and
d
unsoun
ically
this formul a is theoret
There are many other backw ater formulas mentio ned in foreign
ing all
publications on hydraulics, of which Tolma n (15), in Teview
as most
formulas and metho ds of which he knew in 1917, mentio ns 'Aubui
s­
promin ent those of Dupui t, Eytelwein, Flame nt, Freyta g-D
nari,
l\/fonta
cke,
l\1ehm
s,
son, Gauthe y, Heinem ann, Hofma nn, Lesbro
s of
Navier, Ruhim ann, Tolkm itt, Turazz a, and Wex. For reason
which
as
formul
these
for
economy, coefficients were not determ ined
are seldom mentio ned in Englis h texts on hydraulics.
TEST PROCE DURE
Tests were run with quanti ties of flow rangin g from 10 to 160 cubic
3.3 feet.
feet per second, find with depths of flow, V 3 , from 0.6 foot to of
water
foot
0.4
about
of
head
The experiments were begun with a
ive
success
with
tests
by
d
followe
weir,
ring
measu
the
discharging over
the
until
weir
ing
measur
the
on
increases of about 0.1 foot in head
at
greate st possible quanti ty was obtaine d. Different stages of flow
BRIDG E PlEHS AS CHANN EL OBS'l'HUC1.'IONS
11
the site of the pier for euch head on the measlll'ing weir were
by means of the adjusta ule weir downs tream from the pier. obtuin ed
In pru·t of the tests the adjusta ble weir was set at a definite
n
while a comple te set of runs for the pier set-up was obtainpositio
variou s heads on the measur ing weir. After the desired headedonwith
the
measm ing weir WitS obtain ed, the observer fh-st read the hook gage
above the weir; t.hen he obtain ed reading s to the neares t 0.01 foot
011
the 37 piezom eters along the wuU of the cunal, and recorde d
depth
of the water in the channe l ns shown by the variou s stnl!' the
gnges set
along the inside wnIl of the cltl1al. A ren.ding wus then taken
the
weir hook gage to see if the water level had yal"ied. This proceson
s
was
repeate d for each head on the ,,'eu·. The adjustl tble weir was then
set Itt ar.othe r position and anothe r series of runs obtnine d.
In the majori ty of the experim ents the pier models were placed in
the dry testing channel and water allowed to f\O\\' throug h the
canal
past the piers. The difference between the wn.ter surfnces immed
i­
ately upstre am and downs tream from the pier was called tbe bnckw ater.
To preclude any criticism of the method used in determ
backw ater, an extensive series of experiments was run inining the
in
which water was permit ted to flow throug h the unobst ructed 1929
testing
channe l and the "Titter-surface profile was obtaine d, nfter
the
pier models were set in the channe l nnd the witter-surface which
elevati
ons
again obtaine d. The difference between the elevlttion of the water
surface upstrea m from the piers and the elevati on of the water surfuce
with no piers in tbe channe l is the bnckw ater caused by
piers.
The results obtaine d from the two method s of testing were the
identic al.
The second metho d of testing was slow nncl expensive, considerable
time being spent in raising the benNY pier models (weighing consid
er­
ably over a ton) from the testing channel and lowering them back into
the channe l to measur e the backw ater. In ordet· to conduc t the tests
by this metho d it wus llecessltry to build :t special device (plate 8,
B)
which would set the piers exactly in place in tbe channel. For these
reason s, in the great majori ty of tests as stated above, the piers were
built in the channe l before the water wns allowed to enter.
Ordina rily SL,{ tests were obtaine d for each head on the weir, two
in each class of flow as defined by Rehbock. Thus it was possibl
obtain a compa rison of Rehbo ck's formullt, considering his classese to
of
flow, with D'Aub uisson 's and Nngler's. For each quanti ty of flow
one test was run in which the adjustu ble weir WfiS lowered to its limit.
For most discharges with this position of the adjusta ble weir, criticnl
velocit y was obtaine d at the site of the pier. This condition, howev
er,
will seldom occur at bridge locatio ns in na.tural stremllS.
Levels were taken on the weir hook gage, and on nIl piezometer
{:;ages during the progress of tbe experiments, to see that these
lllstrum ents did not change in elevation.
EFFECT OF SHAPE OF PIER ON COEFFICIENTr
The relativ e amoun t of obstruc tion a bridge pier offers to the flow
of water may be expressed in the form of n. pier-sh ape coeffic
The coefficient depends upon the backwater' forlllula used. In ient.
D'Aub uissoll , 'Veisbn.ch, aud Nagler formulas the pier-shnpe the
efficient varies directl y with the discharge. FOI' a f?iven nmoun co­
backw ater, depth of flow, and channe l contrac tion, If the pier-sht of
ape
coefficient is increased 10 percen t the discharge is increased 10 percen
t.
12
TECHNICAL BULLETIN 442, U. S. DEPT. OF AGRICULTURE
The coefficient is, in some measure, an index number of the hydraulic
efficiency of the pier.
The effect of the shape of pier upon the coefficient is shown in table
1. Tius table is a summary of the average values of the coefficients
for the different classes oJ flow for the formulas of D'Aubuisson,
Nagler, and Rehbock.
TABLE
l.-Bridge-pier coe.tficients as determined for different shapes of piers
[Stllndnrd plcrs. longth rour times width. in tesUng c\llllnel 10 rect wide)
ON~;
SQUAUE NOSES AND TAILS;
CIIISS I llow
';o'ormula
'I'est-.,
liver·
aged
I,:
AND TWO l'lERS
I
I
(,Inss ,2 lIow
Class 3 llow II A\'cr. A vc·
r
---;----1----;---- age co. nge co·
I
'i'ests ') Coem
Il\·er·
I t'
aged 1 c en
Coem
. t'
clcn
,'--I
c1l1cicm efficient
'I'esls coom·
ror
ror
1I\'or' .
cinsses Cllk"'o5
1 2
aged
Clont ~l nnd 22 and:3 'I
I
I'
N,t/ll· - - ' NIt/ll'
D'Auhuissoll-1(n'.-t ...... ~ ... ~ .. ~.~ ........ ~._
Naglcr-Ks................. _.........
Rehbock-~,..
__ .... _ ............ .
Rehbock-K u..... . _" ............... . SEl\IICIH(,U1~AH
j
n~~g~~t~-,;.~:::::::.:::::,::::.::::1
.D' Aubuigson-l\~o'.4 ~ ~ ~
N1I11I' - - - " , - - - - - ­
ber
ber
iO
O. nlo I
47
70: .858
-Ii
70 I 6.82 i,. __ ....
j~.....~~~J H
I
NOSES AND 'I'AILS; ONE AND
l)'Aubuisson-Kf)',I .......... __ ... __
Nagler-Ks.. · ................. " .. '
CONVJ~X
ber ,
W, 1.018
99:
.8il
(Ill j 4.7S
99
3. 11
al
I. Oill
• !l34
3.3.1
2.08
:11
:n
31
ao
30
ao
II I
1. 001
• \J05
5.50
30, 4.0\)
I
'1'wo
20
20
I..·....·
I
26
0.1152
1.113
(3)
6.08
1
0.988
O. UHf) .866
.\I~~J 5.62 1_...... . 3.92! -1.52.
PIERS
I
0:\41
1.041
I.fI:IS
.020
l. 027
'I. 40 :........
5. 3:1 1 3.07 \ 3. i5
I.
1. 2i8
(3)
NOSES AND 'rAILS; ONE AND 'I'WO PU,RS
~ . . ___ .... ~ .. ~
_..... ii~~~~~~~~;':::: ::::::::::::::::::.::: Rchbock-KIl ..•,•...•.• '_ •.. ____ ..... 31i
1.08!!
.940
1.0.17
36
2. tiS
4,;16
1.48
3.11
ao
:16
.ual
!
ao
aO
I. 042
1. 251
,_....... (3)
5.34
\
30
I. 074
.9,\0
Ug
1.06·\
1.0:1-1
--T26'
J,ENS·SHA1'I,D NOSES AND TAli,S; ONE AND TWO PlEHS
D'AubuissOll-!'f)·.L.................
ii~g~~~~~~;:.·:·:: ::~:::.:: ::::::::::: Rehbock-Ku..................__ ..... a7
:Ii
37
37
I!
2:1
2:1
23
23
1,0.51
.952
3.55
2.05
ao
1.0:13
.932
4.79
:10
3.37
30
~:~~~~,~~;:~:::::: :::::::::::::::::: nehbock-Ku............._••___ .... ..
a7
a7
1.1)65
t
37
.887 \
a.54
37
2.48
45 j 0.985!
451.883:
:~
I. 044
.944
PIERS
58
O.Wi
58
I. 1·15
1. 021
.885
5. i2
3: ~~
~: g} \.... ·58·
(3)
I. 05.1
1.002
U~ "':i~43' '1'wo
!JO. 'l'JUANGULAR NOSES AND 'rAILS Wl'l'llOU'l' BATTEH.;
U·Aubuisson-Kf)·A ............. __ .. .. 1. Oil;
1. :lSi
(3)
5.17
1. 011
.W3
'''4;40'
00· TnIANGULAH NOSES AND TAILS WITll BATTER 1:12; TWO PIERS
D'Aubnisson-K,,' A •• . . ._ •• _ . . . . ___ __ ~:~~~~~~~;.::::::::::::::::::::::::::
nehbock-Ku.._._......__....____._•. See footnotes at end of table.
28
28
28
28
1.109
.006
2.64
1.89
30
30
30
30
I.OOl'i 1
2:1
.894 \
23
4.W ........ .
3.00
23
i
0.960
.986
(3) 5.70
1. 055
. \}()()
1. 028
.924
~: gg '--3~74'
13
BRIDGE PIERS AS CHANNEL OBS'l'RUCTIONS
TABLE
l.-Bridge-pier coefficients as determined for different shapes of piers­
Continued
SEMICIRCULAR NOSES AND LENS-SHAPED 'l'AILS; TWO _PIERS
Class I fiow
Class 2 fiow
Class a Ilo-w
Aver- A ver­
age co- eIDclent
age co­
eIDclent
Tests: CoemCor
cl~es
aver- - ciell' classes
,
agfeo 1_ _'_ Inno 2' a~;/:i I
'----;---1---:----1-----,-
Formula
Ta\~esrts_ CoeID- Tests
Coemaver.
I
_____________. ~~ ~~ ~__ ~:~_
Num-
NumD'Aubllisson-Kn'.< _________________ _
~~~~~~:-t:~==::==:::::::::::=::=:::
Rehbock-KR______ . __________
h
_____ _
ber
32
32
32
32 1.152
.928
1.88
1.41
ber
36
36
a6
a6
NU7ll- :
ber I
1.018
.901
4.67
a.69
28i 0.991
28'
_______
J
28
1. 081
.913
3.36
1. Of18
(3)
1.045
.958
2.62
5.53
a.47
1
------------'-------.--- -------- .------'------'-----'-_.SEMICIRCULAR NOSE AND TAIL: PIER AXIS AT 10° -"_~rGLE WITH CURRENT: ONE PIER (COEFFlOIENT INCLUDES EFFECT OF ANGLE WITH CURRENT) D'Aubuisson-Kn'A _________________ )
Nagler-K"'________________• ___•••• ___
Rehbock-6o____ ._. __ ••_._•• __ ••• __ •• _
Rehbock-KR___ ••____._______________
j
2511.032/
25
.936 ('
251 3.80
21i
2.22
28 1 0.949
28 II .9IU
28
8.21
28 j 1i.49
21
21
21
0.929
.979
(3)
7.84
0.988
O. Uil
.927
.942
6.13 _______ _
a.ali
5.05
SEMICIRCUI,AR NOSE AN}) TAIL: PIER AXIS AT 20° ANGTJE W1T11 CURRENT; ONE
PIER (COEFFICmN'l' INCLUDES EFFEC'I' OF ANGLE WITH ('URRENT)
})'Aubulsson-KD' A __________________
N agler-K.Y___ •• _____ • ________•_____ ••
Rehbock-80__________________________
Reh bock-K R _____ ____________________ 38
38
38
38
0.943
.876
8.22
4.69
:18
38
38
38
0.805
.801
13.29
8.78
34
34
34
0.879
.957
(3)
O. g06
.896
0.904
.869
10.75
6.74
10.28
---7:83­
LENS-SHAPED NOSES AND SEMICIRCULAR TAILS; TWO PIERS
D'Aubuisson-Kn' A _________________ _
Nsgler-KN _______________ -- _________ _
Rehbock-80 _________________________ _
Reh bock-K R ________________________ _
36
36
30
36
1.162
.932
1. 73
1. 31
38
38
38
38
1.042
.912
4.11
:1.26
34
34
1. 007
1. OS1
(3)
34
5.17
1.100 II 1. Oil
.922 _______
.972_
2.95
2.:11
3.21
TWIN-CYLINDER PIERS WITHOUT DIAPHRAGMS: 1 AND 2 PIERS
D'Aubulsson-Kn' A __________________
N agler-KN ___________________________
Rehbock-.lo. ___________ • _____________
Rehbock-K R ____ _____________________ 79
79
79
79
0.991
.892
6.13
3.62
69
69
69
09
0.957
.S94
7.26
5.07
40
40
40
O. 97~
.89:1
0.66
4.30
0.985
1.054
(3)
Ii. 70
0.977
.. 927
---:j:iii"
TWIN-CYLINDER PIERS WITH DIAPR1tAGMS; 1 AND 2 PIERS
U'Aubllisson-Kn'A__________________
_ _ _ =::==::=:::::==::::=::
~~g~~:~,-.-
Rehbock-KR_________________________
65! 0.966
i
~
651
dl8
3.48
61
0.975
gl 4.75
0: g~5
61
44
_____ ~~_
44
0.991
1(3~41
I O..906:
986 I
0, 987
.941
jl 6.41 '_______ _
5.50
4.10
1
4.48
SEMICIRCULAR NOSE AND SQUARE TAIL; 1 PIER
D'Aubulsson-KD' A _________________ _ __
~:~~~~:~o:::=::::=::=:==::=::::::
Rehbock-K R ____
____________________ _ 20
20
20
20
1.014
.941
4.45
2.5:1
4
4
4
4
1. 002 !___ •___ _
.938 ;_______ _
5.12 (
3.03 '_______ _
0.940
.923
8.44
5.52
,
SQUARE NOSE AND SEMICIRCULAR TAIL; 1 PIER
D'Aubulsson-Kn' A__________________
Nagler-KN__________________________ _
Rehbock-.lo____________________ .-'"••
Rehbock-KR ____ • __ __ ••• __ "- _____ -- __ _
See footnotes at end oC table.
19
19
19
19
0.976
.912
6.30
3.55
6
6
o
6
-------- --------)
0.926
.912 ... _---- .. ------,.­
9.20
5.99
...... _.. _-- .. _------
-------- --------1
0.964
.912
i.Ol
4.13
14
TECHNICAL BULLETIN 442, U. S. DEn'. OF AGRICUI,TURE
TABLf'
.~hape8
1.-Bridge-1Jier cpe.Uicients a8 determined Jor different
Continued
5:1° TRIANOUL,\ R NOSE AND
SJ~MICIRCULAH
Class 1 flow
CIIISS 2 flow
oj
1Jier.~­
'l'AIL; 1 pmn
Clnss 3 floll'
Aver.
I.
" '·cr·
nge co- t nge ,co..
emcicnt,cll1cl,~nt
Formuln
r~~
IIged
Num·
D'Aubuisson-Kn' A. __ .. _____ .. ____ ..
Nngler-K.v..__.......__ •• _... __ ._•••• Hehbock-clo__ • _..___...._______ .... __
Reh bock- K n___ ____ ••• _____ .. ________
cocm·
cient
Coem.
cient
Tests
~~:d
(or I (or
cocm· clllsses I clnsses
clent 1 and 21, 1. 2,
I and a I
Tests
~~:d
--- ---,--- --- --- -----Num·
Num.\
ber
ber
19
19
L......
1.012
.947'.. _... ..
71 6.44
4.06
19
19
ber
71 0.978
7: .950
1.024
.945
2,29
7 f
,
~:~?
4.21
,
I::::::::
90° TRIANOULAR NOSE AND SEMICIRCULAR TAIL; 1 PIER
D·Aubuisson-Kn·A
______ .. ________..
Nagler-K........._______________
....__ •
Hehbock-.lo..____________ ...... ___ ...
Rehbock-KIt_________..______________
2020
20
20
II
1•• 027
948
3.90
2.22
1
61
6
6
6
0,957
.933
7.48
4.82
-"--"'1'
..
--- .. - __.. __ ..
._......
_.._______ " ___'
._.. __.. ________
1.011
.1144 4.72
2.82
1
00° THIANOULAR NOSE AND SEMICIRCULAR TAIL; 1 PIER
m
1: 8!1~
if:r,~~:_.J;____::::==:::::=::::::::::::
23
Rebbock-KIl ______ •__ .. __..__ ....__ __
23
D'Aubulsson-Iln·A ...... __ •___ .. __..
4.83
2.69
I
7
j'
,
7
7
7
0'.952
98311-------________
.... _____..
____ •
6.34
4.15
1.001
.937
____.. ______ .___
____.... ___"___
5.18
3.03
SEMICmCULAR NOSES AND TAILS. WEB SAME THICKNESS AS NOSES AND TArI,S; BATTER 1:24 ALL AROUND; 2 PlEHS D'Aubuisson-Kn' .t __________________
N agler-K.v __" ______ •_____________ •__
Rehbock-.lo____ •••____...__ ..________
Rehbock-Kn_______________ •__.._____
21
21
21
21
155
1..930
1.80
1. 36
1
I
22
22
22
22
0.903
.887
22
22
0.972
1.030
(I)
5.83
-....22·
5.29
4. 12
1.072
.908
3.58
2.77
1.038
.949
--Tsi­
SEMICIRCULAR NOSES AND TAILS CONNECTED BY WEB; BATTER 1:24 ON ENDS
AND SIDES OF NOSES AND 'l'AILS; WEB THIOKNESS ONE·THIHD OF PIEH WIDTH AT
FLOOD !,EVEL; 2 PIERS
D'Aubuisson-Kn' A. _________________
if:~be~:__f.:::.:.__~::______:_-__:_-:::::::::
Rebbock-KR______ •____________.. __..
34
34
34
34
1.020
.863
4. 51
3.10
43
43
43
43
0.951
.865
6.36
4.92
36
36
1
-----36·1
0.959
.988
(3)
5.91
0.982
.864
5.54
4.12
0.975
.904
'-Toii'
SEMICIHCULAR NOSES AND TAILS WITH 1:24 BATTEH, CONNECTED BY WEB WITH
QUAHTER·HOUND FILLETS AT TAIL; 2 PIERS
.!
D'Aubuisson-ICf)·.' _. ___ •___________ ..
Nagler-K.v..___________ ••______ • ____
Rehbock-to.•• ______ .. _..____________
Rehbock-KR_....______ ....... _•• __ ..
23
23
23
23
1.087
.897
3.05
2.17
22
22
22
22
0.989
.880
5.35
4.15
22
22
·----22·
0.002
.953
(3)
5.56
i
1.038
.392
4.17
~.14
1. 014
.912
--Tii:i' SEMICInCULAR NOSES AND TAILS WITH 1:24 BATTER, CONNECTED BY WEB WITH
FILLETS QUAHTER·HOUND .AT TAIL AND TnIANOULAR AT NOSE; 2 PIEHS
D'Aubulsson-Kn' A_ •• __ • _______. .___
N agler-K.v..___ .._____________.. _____
Rebbock-60.•• _. __ ._...._____.._____ _
Rebbock-KR_..____ .._____ ..__ ..__ .. _
24
24
24
24
1.082
.895
3.15
2:24
26
26
26
26
I__ _
I
0. 977 '
5: rt1
4.45
22
:~.
22
0.954
.922
(')
5.55
1.027
.888
:Jg
1. 005
.898
"'4:05'
I Cbsnnel contractions for all set-ups in tbis table were 11.7 percent (or 1 pier and 23.3 percent (or 2 piers.
'Average (or 811 tests In the 2 or 3 classes. not 8\erage o( the determinations (or separate classes shown In
preceding columns.
J ao was not computed (or class 3 How.
a.
11.7
L=4W
RR
55
.941
92~)
.923
.864
(.861)
.954
.943
.943
P90 ' P90 ,
.887
0
.698
0
0
0
.883
P90'B
0
.907
P9O' B
.906
SR
0
.941
912
(.914)
(.866)
LR
NC
Pw R
P6o·R
.945
~
.932
Ne l
NC z
~
Pgo·R
0
.948.~
bh
.904
RL
.928
.932
C90 5)
I I I ! X~
.863
.897
.895
oN)
.930
.986
D0
.B88
1.108
1
10
11.7
L- 4W
.936
11.7 L- 4W
201}
.B76
23.3 L=7W
~
0
0
RS
0 0
.867
50
L=I.7W
T-T
o0~ ~ ~ ~ ~ ~
I
23.3
35
L-2W
TT
LL
0 ~ ,J 0000~ ~ ~ ~
t.
0
.908
23.3
L= 4W
CC
~ ~ ~ ~
. B55
.916
.843
,901
Con frae fion of ehanne I in percent._________________ r::t.
Widfh 0 f pier____________ _____________________________ .W
To fa I leng fh 0 f' pier._•• _______ . _______________________ L
Raiio of lengfh fo widfh applies only fo 55 and RR
S qu are___________ •_______ •.• 5
Round (semicircular}••_•• _R
Con vex_._.__________________.C
Cylindrical. ___• __• ____ • __ ....T
Lells- shaped....._. _____.....L Poin fe ri __......_...... _____..cP
Recessed...._._•• _._, ____ ._.N. C.
23.3
L=13W
Ptl:I'ItE
3.-DlslITammatic summarr of Iowa test. results showing Nagler brldg~llier coemclent for dlfterent pier shapes. ('-ahles In JlIIrenthese:!
were det~rmlned h)' Nagler In his Mlchlltan e",leriments.)
MltS
0
u.s. ;OW:llllllm PRINTING OfFICE : Illi
15
TIRIDGE PllmS AS CHANN I..:IJ OBS'I'RUC'L'IONS
In 111nking a study of the effect of slutpe of pier upon the coefficient,
and hence UpOIl the discharge, it is desil'fible to confine the prrlimillrrry
investigation to the values obtained in fL single forlllula. 11'01' con­
ycnience in JUaking comparisons the coeHieicllts for the N nglcr for­
mulu,lllwe been selected becttUse N ngler (D), some 20 yeurs ngo, 11111(le
u greut muny experiments on vllrious shrrpes of piers. It should be
remembered, howeyer, that these studies nre indicl1tive only of the
comprrrutive hydraulic. efficiency of ynrious slmpes of piers, since the
sume number of tests V;1)1'O not run on every shape of pier, nor did
these tests covel' eXllctJ3' the same runges of '1U1mtity nne! velocity.
For example (table 1), 00 expl'l'iments were run on the piers with
squnre noses nud sq un,re tnils (two channel contrnctions), whilc only 20
('xperiments were run on pit'rs with semicircular nose nnd squure tniI.
A dingmmnmtic summnry of the N nglcr coefficient for dnss 1 flow
for the ynrious picr shapes tested is given in figure 3. The vnlut's
ohtained by Professor N ngler for similnr shup!'s of picrs hn,ve bt'(\11
im;(,j·t('d ill this dingrnm.
~~FFECT OF
LENGTH-WIDTH RATIO OF PIER ON COEFFICIENT
TIll' ('omputed bridge-pier coefficients, grouped to show the elred
of It'ugth of pier, hnvo beon plotted in figUl'o 4.
A study of table 2 shows thn,t, in the D' Aubuisson and N ngler
formulns, nil iucl'enso in the length of the pier usually menns n, slight
decrease in the pier-shape coefficient. A plotting of KD',l ngninst
Lin' gives 11 sOlllcwhnt ClIITed lino, the decrense ill coeHicient per unit
incrense in LI II' not beinb constant. For the rnngo considered (LIlY
\"lUTing from 4 to 13), the nvernge change per unit is nbout 0.5 percent
of the vnlue of the coefficient when LIII'=4. '1'he following equations
nppt'al' to fit the datu, fnirly w('11:
'd··1
fI(v =0.873-0.0023L/lV
] i' .
01 squaro nOSe.111 squnre till _.. - - - - - -[KD',l = 1.05 -O.0052LI1V
.,
I ur nose n.n(Iseml('Il'(,U
"
1nrtu
t 'I{J(N
=0.04--0.003L/W
].i~ OJ' semwu'cu
KD,.l = 1.17 -0.006L/1V
TAII(,F.
2.-Bridl/c-pier cor.f/icielll.s (/s determined for diffare/li lengths of piers
crwo piers, each 1.li fceL wide; chllnnel contruction 23.3 percent)
PlEHS WI'I'1l SQt:.\RE NOSES AND ']'AILS
; (,Ius.~ 1 !low
Picr length
(fcel)
Formnln
i__.. ,~._,..,
I
I 'rests
,
Il\'er.
! _.J
ll~cd
I
I
lil'
MI
~I
.
30
yO
30
30
A \'cr. AYcr·
n~e CQ- ul10 .co- .
.etlicient clllC1ent
'I'csls I
for
for
u\·cr.: C'~ctli. clusses classes
ngcd j ClcnL :1\ nud 21 1, 23, ~ud
_
I C'9 c!fi·
tl~ed! Clcnt
-.
I.OW
,MI
-I.:Ji
I
I
i
I
1.003
.855
4.02'
64
.08.1
yO
.843 ,
f
j
t Average ror 1111 tests in the 2 or 3 classes, not fivefhge of
in precedIng' columns.
'0. wus nOL computed ror class 3 !low.
n
.l7
!
O.Of,2 \ 0.994
1.113:
0.084
.850
.921
(H
6.46 •..••.
(') 1 5. ~'9 ....... . tl-J, 4.58
4i· O. tl8 : 3.81
4.51
~'9 l .935
23' .026·, . 90S
.956
2'J
.S5G:
ZJ
.9il
.855
.S89
2\l
2\l
i
I
lSI/mba
0.950
.8M
64
3.39'
?52
.1. 70
__
,--.--------
:,Y/l1IIbcr'l
Hl~. 9i
28
2S
28
28
Clnss 3 tlow
,____
i
'1'('51S I
CICUt.
Xllmber l
,{D'AUhUiS50n-IlIl'A ..•\
! r-inglcr-/{.,'••....•... ,
• 'f •. , .. ·····1 H~hbo~~-Ou; •......:
Hehhock-llll.........
1 J)'Anbuisson-KIl'''':
01Nn~lcr-Il,,·...
•. "
~. , ........,{ Hehbock-oo__ ....... '.
Rchbock-j(II........
D',\ubliisson-Illl'.• ,.:
1'1- ...... 1 Ntlgll'r-~(.\"_'_ . "'j
H. , ,
{ Hchbock-c5o .. __ ~~ H~
Rehbock-KII .••., •. /
-¥-_~
(9 c1h • i nycr·
1_ _ _ _ _ 1
40-
Clnss 2 !low
30
:~O
.10
th~
I
0.S3
I 5.20
.Ill~
I
I'"
2:l
.S·ll \
7.~2;.
5.05
('l
0.02
:IO,.?~
i
30
_
I.
5.89
4.34
,,),_,
(2~
;m· o. SO
950
....... .
5.00
. ~4~
.845.
,58,
l 6.3~ L~ .. _..;.:..
; 4. Ub
5.31
i
determinations for separ,lle classes shown
16
TECHNICAL BULLETIN 442, U. S. DEPT. OF AGRICULTURE
TABLE
2.-Bridgc-pier coe.fficients as determined for different lengths of 1Jiers-Con.
PIERS WITH SEMICIRCULAR NOSES AND 'l'AII,S
----,'
Glass 1 flow
Pier length
(feet)
Formula
Tests
aver·
aged
Coello
clent
-].r~1f.ber
{D'Aubulsson-Kn' A_.
4.67••••.••••• Nagler-KN.••••••••••
Rehbock-.lo•••••••••••
Rehbock-KR•••••••••
{D'Aubulsson-Kn' A ••
Nagler-KN•••••••
••••
8.17•••.•••••. Rehbock-.lo•••••••••••
Rehbock-KR._•• _••••
{D'Aubulsson-KD' A __
Nagler-KN••• _•• __ •••
15.17••••••• _. Rehbock-.lo •••••• __ ••_
Rehbock-K R••• ••• _.
100
16
16
16
16
26
26
26
26
34
34
34
34
Tests
aver·
aged
Cacm·
clent
-I--
Class 3 flow
Tests
aver.
aged
I
Aver· Aver·
age co. age .co·
ellclent(llC,ent
for
Cor
coem· classes 1 classes
clent 1 and 2 11, 23 nnd
-----,
Number
1.150
.927
2.03
1.62
1.128
.916
2.28
1.67
1.096
.901
2.90
2.07
-~----------
Class 2 flow
23
23
23
23
29
29
29
29
37
37
37
37
Number
1.014
26
.8H6
26
4.84
3.81 ·····20·
1.021 I
23
.904
23
4.58
3.62 '·'·'23·
.996
31
.890
31
5.19 -------4.07
31
1.034
1.278
(2)
6.33
.980
1.011
(2)
5.H
.975
1.036
(I)
5.74
1.008
.908
3.73
2.91
1.072
.910
3.49
2.69
1.044
.895
4.09
3.11
1.054
1.1158
···3~89·
1.045
.939
··-3~50·
1.023
.0'38
3.91
was not computed Cor class 3 flow.
With the Rehbock formula, not only is the variation in 00 larger
lor a given change in LIW, hut it is in the opposite direction, increas­
ing for an increase in length, as is of course to be expected since 00
increases with the height of backwater. It should be noted, however,
that Rehbock's experiments showed 00 to have a minimum when
LIW equalled 4.5, and then to increase again for lengths shorter than
that, which could not be verified from these tests since no ratios less
than 4 were used.
When 00 was plotted against LIPl and a strl\ight line drawn as
nearly as possible through the points, the resulting equations were:
For square nose and square tail, 00 =3.94+0.12 L/W.
.
For semicircular nose and semicircular tail, 00 =1.48+0.11 LIW.
The corresponding equations given by Rehbock are:
For square n05'e and square tail, 50 =3.10+0.12 LIW.
For semicircular nose and semicircular tail, 50 =1.27+0.12 LIW.
rhe agreement as to the effect of increase in length of pier is very
satisfactory, but there is a surprising difference in the coefficient L
for the square-ended pier.
A plotting of [(R against L/W gives unsatisfactory results, the
curve being convex downward in the case of the square ends; and
convex upward, with a minimum value of KR when L/W=7, in the
case of the round ends. Therefore, no further attempt is made to
discuss the effect of variations in length on the Rehbock simplified
formula. Rehbock does not seem to have treated it either.
All of the above is based on values for class 1 flow only.
For most practical cases, considering the ratios of length to width
ordinarily used in bridge piers, the results of these experiments seem
to indicate that this variation with length can be neglected.
EFFECT OF CHANNEL CONTRACTION ON COEFFICIENT
In figure 5 the computed bridge-pier coefficients have been plotted
in a manner to show the effect of the degree of channel contraction.
A study of the mathematical structure of the four backwater for­
P\4UltE....
~ou~'t>
~~1)
q
t\,
OF
BULLE;n~
•
P\~Uf!.E.. 5 PoUt-lD
A-,.. 'EN D OF &utJ__eTl ~.
17
BRIDGE PIERS AS CHANNEL OBSTRUCTIONS
mulas will show that an attempt has been made in each one of the
formulas to take care of channel contraction. The success which has
been attained in tIns respect is shown in table 3. This table is a
summary of the coefficients determined for the four backwater for­
mulas for four channel contractions; namely, 11.7, 23.3, 35.0, and
50 percent. In an ideal formula K would be constant for piers of a
certain shape regardless of the amount of channel contraction.
TABLE
3.-Bridge-pier coefficients as determined for different channel-contraction
ratios
PIERS WITH SQUARE NOSES AND SQUARE TAILS
Class 1 flow
Class 2 flow
Class 3 flow
,
Channel contrac­
tion ratio
A ver- _-\ ver­
age
age
coeffi- coeffi­
cient cient
I ~\~~ IC~effi- cl~,es
.
I
I
I--------~I-,-~·-u-m-- --I' Num-I-- :;::1-- -~. --..
Formula
~:e~
~~ffit- ~;~~
aged
aged
C'!JCffi­
C1ent
aged
I
C1ent
cl:es
I. 2.
and 3'
J and
2'
{ ber
D'AUbUisson-KD'A_f
ber O·~~~7~~er.ier 1.17 by { i:~~~~~~;====::::1
18
!i
0.909
D'AUbUlsson-KIi"'_
0.233 (2 piers 1.17 by Nugler-:K.,.__________
'1.67Ieet). { Rehbock-clo_________
Rehbock-KR________
D'Aubuisson-KIi' A _
0.35 (l pier 3.5 by 7.0 Nagler-Ks__________
feet). { Rebbock-clo_________
Rehbock-Ku________
81
SI
81
81
28
28
28
28
1.029
.864
4.37
2.97
.009
.867 f
4.79 I
4.26 i
2:1
23
6.79
{~~·il~~~~~~=_!.{!:_A_:
0.5 (1 pier 5.0 by 8,5
feet).
Rehbock-clo_________
Rehbock-KR________
(~!
I
I
~ 5.42
:lli
I
I
ber f
1
610.!103 .,, ____ ,-_ .... _j 0.952 " ..•• _
g11~:[f
~:i I:::::::
::::::{:::::;
64
.050
47! .0.12· .004: .98·1
64.854
47 ' I. 113
.8.19
. O~I
64 0.4fj ------- (;)
5.29_._ ...
IH 4.88
47 6.68 '3.81 I 4.51
29.918
26.9.18.958.
.1I.i7
1
29
.847
20 , I. 002
.8';7
.1121
29 10.36 _______ i ('J ! 5.•i9 1_____ __
~'91' 6.01
20 0.62
5.15: 5.liI
~~ J~g
2717.52
27 9.81
i
~ I l: ~f~
251
i
'. :~ J______
: e~b
_
_______ 1 (') ; 6.55
7.76 1 8. 42
I
8.20
PIEHS WITH SEMICIRCULAR NOSES AND TAILS
n.~J7_(1 pi or 1.l7 by
1.6, feet).
l{
O.~3
(2 piers 1.17 by
1.67 feet). 0.35 (l pier 3.5 hy 7.0
feef,).
D'Aubuisson-KIJ' A _
Nagler-~s----------
, Rellboek 00 __ ...____
{g~'~~~~;;;;fn'::I~;'~._:
{
g~X~~~;;;;~~'::KI/.._:
Nagler-:K,\'____._____
Rehhock-clo_________
Nagler-:[(,v__________
Rehbock-clo__________
I Rehbock-Ku...... __
,{D'AUbUiSSOn-[{IJ'" _
0.50 (J pier 5.0 by 8••1 I' Nagler-K,\'__________
feet).
Hehbock-clo_________
Hehbock-Ku____ .___
I
I
16
16
16
I. 012
.941
4. S7
:~
U'.927'k
1.1
IS
2.m
:lO
30
:lO
24
24
24
24
.986
I. 73
I. 81
I. 312
I. lOS
1.65
2.81
~g
l: ~
~
\ . 9~
!___ .. ..I._____ _ .006
, . 9 3 / ---_---:- .. ---- .040
7.06 1_______ !_____ __ 5..51
7
];
tgi41'----2ii-h~ii34-
~~
t ~!I I
I
2 3 . 8U6
26 1.278
23 4.84 ______ ., ('l
~ I U~fl
32.936
28 ! l. ISS
32 I a.63 ' _______ , ('l
32 3. f>4 I
28 S.39
24
. !/91
24 11.118
2 4 . U30
2·1 , I. 317
24 4.94
(ll
24 6.85
24 6.37
I"-----1
3.35
1.008
. !lOS
3.73
2.91
1.146
.U61
--j:iiS4
1.0.18
a.so
1.112
1.022
2. i~
2. i5
1.152
l. 022
3.30
4.83
3.57
1.140
1.120
--5. :i.'i­
---~--~----~--~--~----~--~--~---
TWIN-CYLINDER PIERS WITHOUT DIAPHRAGM
0.117 (1 pier 1.17 by
4.67 Ceet). {
J).Aubuisson-~~__::__;:I·;;;--- ~1~ 1 0. 929
Nagler-Ks.-________
51
.898
32.915
Rehbock-clo.._______
51 7.44
32 i 9.28
Rehboek-KR________
D'AUbUisson-KIJ' A_
0.233 (2 piers 1.17 by Nagler-K.,·____ ._____
4.67 Ceet).
{ Rehbock-clo_________
Rehbock-Kn________,
51 4.17
28 I. 055
28.883
28 3.74
28 2.61
.
-
.-----
..
'-."
4 0.9431 O. 946
O. \WI
4.963.904
.!lO7
_______ ('J 8.15 __ ----.
32 ! 6.0.1
4 6.47 .4.89
4.96
37 I .982
36 1.060
.9!10 I' I..879
013
1.005
3-, ! 5.. 85715
36
.943
_____._ (')
4.75
37
37 ; 4.22
36 5.67
3.53
4.29
----------~~---------, Average Cor all test& in the 2 or 3 classes. not average oC the determinations Cor separate classes shown
in preceding columns.
'60 was not computed Cor claSs 3 flow.
08815°-34--3
18
LTURE
TECHN ICAL BULLE TIN 442, U. S. DEPT. 01<' AGRICU
TABLE
ion
3.-Brid ge-pier coefficients as determined for different channel-colltract
ratios-C ontinue d
TWIN·C YLINDE R PIERS WITH DIAPHR AGM
'-_
-
.......
.
.
Channel contrac·
tlon ratio
Formula
Class 3 now
Class 2 now
Aver·
age
coem·
cient
Cor
Tests Coem· Tesls Coem. Test.~ Coem· classes
over.. clent aver- cient aver· cionl J and
aged
aged
aged
21
Cln'!S 1 now
.lVurn·
litr
{DOAubu lsson-Kv ' A. 51
51
•
N•••••••••
Nagler-K
by
0.117 (1 pier 1.17
51
Rehbock-olo•.•••••• ,
4.67 reet). 51
Rehbock- KR.__ ••.••
14
{DOAUbUisson-K/I' A.
14
0.233_(2 piers 1.17 by N:/:ler-K N......... .
14
Re bock-60•• •••••••
4.6, reet).
14
Rehbock- KR.•••••_.
!
Aver'
age
coem·
cient
Cor
ch"ses
It 2,
and 3 1
, - - - -- - ­
--1-- -I'tU11lNU7ll'
0.009
.007
6.83
3.86
1.094
.904
2.93
2.11
ber
30
30
30
30
31
31
31
31
ber
0.932
.919
7
i
li.OO
1.017
.893
4.66
3.58
7
37
37
n.lO
------­37
[
0.953
0.935 0.95.'i
.915
.911
.953
(I)
7.07
··.j~iiii·
fA
4.
6.68
1.023
1.000 1.041
.009
.897
1.056
4.12
(Il
4.13
3.12
5.35
PIERS WITH CONVEX NOSES AND TAILS
22
22
22
22
14
14
14
14
D'Aubuis son-Kn' A.
0.117 (1 ),ier 1.17 by Nngler-K.....__ .......
{ Rehbock-.5R __ .....__
Rehbock -K R ••• __ •••
D'Aubuis son-Kv' A.
0.233 (2 piers 1.17 by Nngler-K N......... .
__
4.67 Ceetl.
{ Rehbock-.5o•••____
Rehbock- KR•••____ •
4.67 reet).
1.034
.954
3.48
1.97
1.178
.943
1. 44 1.15
11
11
11
11
22
22
22
22
0.007
.938
7.13
4.60
1. 100.
.920
2.97
2. 36
•••• ___ ••____ • 1.012 •••• __ _
"'"'' ••.•_•• .948 .... __ •
••_••.•.•• • _•• 4.70
'_"'_' _...... 30 1. 042 i:~o --i:iiiiii
1. 078
.934
30 1. 251
(Il
.,._".
·T.jii'
U~
30 5. 34
PIERS WITH LENS·SH APED NOSES AND TAILS
I
D'Aubuis son-Kn' A.'
0.117 (1 pier 1.17 by Nngler-K N........ __ ,
reet).
4.67
{Rehbock-.5o........ .. ,
Rehbock -Kn··· ..···i
D'Aubuis son-Kn' A.I
0.233 (2 piers 1.17 by Nagler-K .....____.... .•.,i
4.67 reet).
{ Rehbock- .lo.....___ .;
Rehbock- KR.......
32
32
32
32
5
5
5
5
1.032
.953
3.84
2.18
1.166
.943
1. 61
1.28
10
10
--·----I.... --·~
__.....
0191......
O. 978 __..... __ ..... : 1..952
__
.947
I:::::::
19 ~J~ :::::::1:::::.: 1
~:;;g
1.100 I 1.085
31l 11.076
13 1.075
30,1.387
.920
13
13 3.42 ....... 1· (Il
30 5.17
13 2.67
.927 11.214
2.92 •• --.,. 4.00
2.2U
160 was not computed for class 3 now.
noses
The coefficients for piers with square and with semicircular
1 flow are
class
for
and
tions
contrac
l
channe
four
the
for
tails
and
shown graphically in figure 6.
rcular
An examination of the coefficients for the piers with semicicoeffi­
the
as
formul
the
all
in
that
noses and semicircular tails shows
The
cients vary considerably for different channel contractions. l con­
channe
in
e
increas
an
with
es
D'Aub uisson coefficient increas
e in
tractio n. The Rehbo ck coefficient 00, decreases with an increas
es
becom
it
t,
percen
23.3
of
ction
contra
a
channol contra ction until at
for
same
the
ally
practic
is
ient
coeffic
Nagler
The
nt.
consta
fairly
channel contra ctions of 11.7 and 23.3 percent, but increases for con­
tractio ns greater than 23.3 percent.
In an examination of the coefficients for the piers with issquare
noses and square tails it appear s that the Nagler coefficient t. prac­
The
tically consta nt for all channe l contra ctions up to 50 percenction
of
contra
l
channe
the
with
t
highes
was
ient
coeffic
uisson
D'Aub
of
ctions
contra
l
chaune
with
alike
ally
23.3 percen t and was practic
ient,
coeffic
ck
Rehbo
the
hand,
other
the
On
t.
percen
50
and
11.7
BRIDGE PIERS AS CHANNEL OBSTRUCTIONS
19
decreased with an increase in channel contraction up to a con­
traction of 23.3 percent and then increased again.
A study of the coefficients for the twin-cylinder piers with and
without diaphragms and for the IJ'iers with convex and lens-s.haped
noses and tails (table 3) shows thll.t for the 1l.7 and 23.3 percent
channel contractions the Rehbock coefficient, 15o, without exception
decreased with an increase in channel contraction whereas the
D'Aubuisson coefficient increased with an incrcase in channel con­
traction. On the other hand, the Nagler coefficient was practically
constant regardless of the shape of tlJC pier.
These same tendencies are noted in a study of the avernge coeffi­
cients for both class 1 and class 2 flows.
15 01
~ i
Piers with semicircular noses and tails•..o
\ Piers with square nOses and tails ____.C
~ ~ 1.3C~----~~~---+-------+------~----~
~
O.BO' - - - - -•....,J~----_.",J,:------~-----....,.J:,.----~ I
0"
10
20
40
50
Channel contraction. a
FIGURE 6.-Variation in pier coefficient with channel contraction (or class 1 flow.
The difficulties which engineers have experienced in the past in
obtaining reasonable results in backwater computations ha,ve been
due largely to the fact that they used a pier-shape coefficient obtained
for channel contractions other than those wluch prevailed in their
problems. These data show the necessity of selecting a coefficient
which is applicable to the problem under considemtion. On the
other hand, the limited amount of data available for large-percentage
channel contractions (35 to 50 percent), involving tests on only two
shapes of single piers, indicates thc need of additional investigation
before the tendencies indicated in figure 6 may be considered fully
established.
20
'l'ECHNICAJ. BULLE'J'IN 442, U. S. DEP'l'. OF AGRWUL'l'URE
Since the majority of bridge openings have channel contractions
less than 23 percent, it would seem that the Nagler formula is best
suited for practical use beC!UlSe coefficients derived for it appear to
show the least variation below that percentage of channel contraction.
EFFECT ON COEFFICIENT OF SETTING PIERS AT ANGLE WITH
CURRENT
It is reasonable to expect more backwater when piers are placed
at an angle with the current thun when placed in line with the current.
No definite information exists, however, on the additional buckwater
caused by such setting. Experiments were run on a pier with a semi­
circular nose and semicircular tail placed at two angles with the cur­
rent, namely, 100 and 20 0. In each case, the percentage of channel
contraction used in the formula was taken the same 11S if the pier
were placed in line with the current, so that the ef}'ect of the angle
was reflected in the coefficient. The individual coefficients for these
set-ups are shown in figme 7, and the average coefficients are given
in table 1.
Since a single pier with semicircular nose and semicircular tail
was used in the experiments on the effect of angularity, any compari­
son of these coefficients should be made with coefIiciellts obtained
from tests on a single pier of the same shape placed in line with the
current. It will be noted that the Nagler coefIicient for the pier set
at an angle of 100 is 0.936, whereas the coefficient for the same pier
placed in line with the current is 0.941 (table 3, channel contraction
11.7 percent), showing practically no difference. The Nagler coeffi­
cient for the pier placed at 11 20 0 angle with the current is 0.876, ab.out
7 percent less than the coefficient for the same pier placed in line with
the current.
It would appear from these studies that piers placed at angles of
100 or less with the current offer little more obstruction to flow than
the same piers placed in line with the current. As the angle increases
the amount of additional obstruction to flow increases until at 20 0
the Nagler and D' Aubuisson coefficients decrease about 7 percent.
That is to say, for a given height of backwater, depth of flow, and
percentage of channel contraction, a stream containing piers placed
at a 20 0 angle with the current will discharge about 7 percent less
water than if the same piers arc placed in line with the current.
USE OF DATA ILLUSTRATED BY EXAMPLES
These experiments have mnde available coefficients for use in
hydraulic formulas for computing either the backwater due to the
obstruction of piers to the flow of water or, knowing the backwater
height, the quantity of water passing through a bridge opening. If
either quantity of flow 01' backwater height is definitely known, it is
possible to compute the other with a reasonable degree of accuracy.
This procedure can best be illustrated by practical examples.
EXAMPLE 1
A stream discharging at flood stage 45,000 cubic feet per second
has 11 cross-sectional area of flow at the bridge site as shown in figure 8.
The piers have semicircular noses and semicircular tails, the ends and
F\C:aUR.E
~utJ1)
of"
Al
1
E-U» DLU.LE:rt N
·
21
BRIDGE PIERS AS CHANNEL OBSTRUCTIONS
sides are built with a batter of one-half inch per foot of height; mtio
of width to length is 1 to 4. The total cross-sectional area of flow is
W
..J
e
S!
e
«
u
1Il
..J
«
<Xl
u
e
I­
ID
a:
We
w-t
>
lJ..
0
I-
a
N
e
a
N
W
Z
«
~
I­
N
a:
X
'0
Q)
.c
"o
I'CJ
a=
I
I
~
~
I
I
I
I
:i
II
I
I
II
~
Z
e
e
\
W
I
~
I I
I
I
II
~
I
I
g
I
I
I
II
Eid
q
.8
a;
~
S
~
;;
Z .8'"
c(~
...J~
D.f
.-J
o
I
II
.=
.~
--'
'"
II
I
~
I
II
/I
II
I I
~
7,560 square feet, of which the piers take up 720 square feet. Thus
the channel contraction is 9.5 percent. It is desired to compute the
l
22
TECHNICAL BULLETIN 442, U. S. DEPT. OF AGRICULTURE
backwater which may be expected to occur at this bridge site for the
flood of 45,000 cubic feet per second.
The problem may be computed either by treating the entire flood
opening as a unit or by treating the channel area proper and the bank
overflow area separately.
In the first method of computation Fa will be 45,000/7,560 or 5.95
feet per second. The total width of flood area, BTl may be taken as
390 feet. The foUl' piers, cuch 10 feet wide, take up 40 fret. Thus
lr2 =390-40=350 feet. The average depth downstream f!"Om the
bridge opening, D a, would be 7,560/390=19.4 feet.
Thus the known factors are as follows:
Q=45,000 cubic feet pCI' second WI =390 feet W2 =350 feet Da= 19.4 feet F a=5.95 feet pl:'r s('('ond - 720
-0 091':?
a7 ,560 - .. ,)..
1 "/? 0 t:. W = 'a~:{! = 19~~ = 0.0284 7
The coefficients for the various formulas have been obtained bv
extending the CUlyeS of figure 6 for piers with semicircular noses ancl
semicircular tllils without batter. These curves were used because,
insufficient dnta were available to prepare similar curves for such
piers with batter fiS used in this example, and because such data us
were aVllilable (for 23.3 pereent channel contraction) showed quite
close comparison between the two types (tables 1 ulld 3). Figure 6
flpplies to cluss 1 flow only, and it is within thnt cluss thnt this example
fnlls, for w hilS been computed as 0.0284 which satisfies equation 5
(p. 10).
The coefficients, then, nrc as follows: Nagler J(N=0.95 D'Aubuisson J(n·A=O.97 'Rehbock 50 =5.50 For the Nagler formula (no. 4) we find from figure 2 that /1=1.24.
Using K,v IlS 0.95 and 0 as 0.30, the formula may be solved for H3 by
nssuming a value to obtain VI, and then recomputing to cbeck the
assumption. The solution gives H3=0.16 feet.
Similarly, using the coefficient.s as found above, the D'Aubuisson
formula ghTes Ha=0.19 foot and the 'Rehbock formula Ha=0.14 foot.
Since velocity of flow depends upon slope, depth of flow, vegetation,
etc., the velocity will not be uniform throughout such a cross~sectional
area of flow, and it may be argued that the backwater sbould be
figured separately for tbe main channel and for the overflow on the
banks.
It would seem reasonable to e)!.llect a somewbat higher velocity in
the main channel of the stream than throu~h the openin~s on the
banks of the stream since the depth of flow III the former IS 28 feet
and the average depth of flow in the latter is about 10 feet. A
velocity downstream from the piers of 6.5 feet per second has been
assumed for the main channel and to make the quantity of flow cbeck,
the velocity on the banks will have to be about 4.46 feet per second.
BRIDGE PIERS AS CHANNEL OBSTRUCTIONS
23
The channel contraction for the main channel is 9.2 percent and for
the banks is about 10.4 percent.
Thus the known factors for the main channel are as follows:
Q=35,940 cubic feet per second
W 1=200 feet
W 2 =180 feet
D3=27.65 feet
1'3=6.5 feet per second a
area occupied by piers = 510 = 0 092 total area
5,530' 2
Cal= V3 /2g = 0.657 = 0 0237 D3
27.65
.
{3 in the Nagler formula= 1.24 Using the same bridge-pier coefficients for the Nagler and D'Aubuis­
son formulas as before but changing Rehhock's to 5.60 (fig. 6) we get
for main stream the following backwater:
By the Nagler formula, K N =0.95, 113=0.14.
By the D'Aubuisson formula, K D ·A=0.97, 113=0.21.
By the Rehbock formula (no. 7), 00=5.60, 113 =0.16.
The known factors for the remainder of the opening (taken as a
whole) are as follow§!:
Q=9,060 cubic feet per second
W 1=190 feet
W 2 =170 feet
D3=1O.68 feet
V3=4.46 feet per second
a=area occupied by piers 210
total area
2030=0.1035.
,
= Fl/2g = 0.309 = 0 02894
Cal
D3
10.68
.
{3 in the Nagler formula=1.27. Using coefficients from figure 6, we get for the area of flow on the
banks the following backwater:
By the Nagler formula, K N =0.93, 113=0.07.
By the D'Aubuisson formula, K D •A=0.99, 113=0.09.
By the Rehbock formula (no. 7), 00=5.0, 113=0.08.
These results .represent an unstable. condition. That is to say,
assuming a level surface on a transverse section upstream from the
piers, the water immediately below the piers i'3 shown to be about
0.10 foot lower in the center of the channel than along the banks.
This may be true just below the piers, but the water will immediately
become level transversely as the water passes downstream. About
four times as much water has been assumed to flow in the main
channel as over the banks and the final average drop-down would
therefore seem to be an average of that in the main channel and that
on the banks, weighted in proportion to the quantity of water flowing
in each section. The final average would about equal the drop-down
computed by treating the entire cross-sectional area as a unit.
24
TECHNICAL BULLETIN 442, U. S. DEPT. OF AGRICULTURE
EXAMPI,E 2
A highway bridge 350 feet long, built across It riYer Yllllcy was
subjected to Il flood. At the crest of the flood, the drop-down in the
water surface at the bridge opening was found to be an Ilyernge of
0.30 foot for the seven openings. The six piers, cnch 8 feet wide with
semicircular nose and semidrculllr tnil, are spnced 50 feet center to
center. The avcl'llge depth of flow immcdin.tely downstream from
the opening WIlS 8 feet. It is desil'('(\ to compu te t he discharge
tlll'ough the opening.
The yulues of the known fnctors tlrc ns follows:
W 1=350 feet
lV2 =302 feeL
l-I3=0.30 fooL
D3=8.0 feet 0:=0.137 The Nagler formula using 0=0.30, /3=1.45, und K N =0.93, giyes u
discharge of 22,300 cubic feet per second.
The D'Aubuisson formula with K O •A =1.05, gives a discharge of
22,800 cubic feet pl:'r second.
Th(' Rehbock formula with 00=:3.20, gives It dischllrge of 2:3.200
('ubie feet pel' s('cond.
SUMMARY AND CONCLUSIONS
The bridge-pier forll1ulns most commonly used in the United States
are D'Aubuisson's, Nagler'S, 'Yeisbach's, nnd Rehbock's. The dis­
cordant l'esults obtained with the 'Yeisbach formula show it to be
theoretically unsound.
None of the aboye formulas give for a certn.in shape of pier a con­
stant coefficient for nll channel contractions. This factor is of
vital importance and is the reason for the inconsistent results obtained
in the pnst by engineers nttempting to solve problems involving back­
water from bridge piers. The majority of such problems concern
cnses having channel contractions of less than 20 percent.
As long as the velocities are low enough to keep within what
Rehbock calls class 1 flow, anyone of the three formulns will ~ive
results close enough for prncticni purposes, if the propel' coeffiCIent
is used. This coefficient varies with the channel contrnction ns well
as the pier shape, as is shown by table 3 and figure 6. Proper values
for channel contractions of less than 11.7 percent were not determined,
and for most of the pier shapes they also n.re not determined for con­
tractions greater than 23.3 percent. However, most bnckwater
problems fall ,dthin tills range, but as the D'Aubuisson and Rehbock
formulas give quite different coefficients at 11.7 percent than they
do at 23.3 percent, and as no points are known between them, tho
shape of the curve remains undetermined. This objection does not
apply to the Nagler formula because thore is 1ittle difference in the
coefficients for 11.7 percent and 23.3 percent, und the tests of tho
squire and semicircular shapes indicate that a constunt uveruge yulue
can be used throughout the range. The Nagler formultt also applies
through Rehbock's class 2 and into the beginning of class 3. The
other two formulas do not apply at these higher velocities (except
BRIDGE PIERS AS CHANNEl, OBS'fRlJC'1'IONS
2.5
with continunlly vnrying coefficients), nnd thus fnil in th(' most serious
cases of "hending-up" du(' to extreme floods.
The conclusions to be drawn f!'Olll tIl(' ('xperiments on bridge piNs
of various shnpes nne! siz('s mny be sumlllflriz('cL ns follows:
The height of the bnckwnter due to bridge piNs vuri('s diI·('ctly ns
the depth of the unobstructed chnl1nel.
Certnin forlllulns hel'etoforc proposed give npPI'Q;\,;mn.t('ly correct
results fOI' onlillnry nlocities when the propl'I' eoeffiei(,llts nr('· us('d,
but they do not hold for extremely high nlodties. TIl(' ('oeffieients
for these formulns ns determined by these tests n re listed in tnbles 1
und 3.
For the lower velocities (class 1 flow) th(' mOI'(' ('fIici('nt shnprs nre
lens-shaped nose und tail, lells-slwpecl nose tlnd semieir{'ulur tuil,
semicircular 110se nIHl lens-shaped tnil, COll\'ex nosc nnd tnil, nnd
semieil'culnl' nose and tail. SinC'c the lattel' is the only one test('d
with piers obstl'ucting a lnrge pcr'centage of tir(' cluLlHlel, the t('sts
mnde n t Iowa City were not suffk-iell t to distinguish betwcen these
shnpes.
Twin-cylinder piers either with 01' without {'ol1ll('cting diapltl'ngms,
pier's with QOo tri:ll1gulnl' nOSl'S and tnils, nlld pief's "'itl! l'('cessed webs
nre less erncient hydl'l1ulirnlly tlHln those just llIentioned, nnd piers
with squnr(' noses :lTId titils nre lenst efficient.
The ndditioll of bntter to the ends of piers slig}1 tly increases their
hvdrnulic eHkiency .
• Incrensing the length of fi, pier 1'1'0111 4 times the width to 1:3 times
the width hns compnrnti,'ely little el)'rct on its hydmulic efIiciency.
In some {'nses it incl'ens('s it nncl in some {'nses decl'enses it. The
optirmrm rntio of pier I('ngth to wid th probably varies with the
velocity and is gellernlly between 4 nnd 7.
Plncing the piers Ilt nIl nngle with thL' current hns I1.n insignificnut
efreet on the flmoun t of bnckwn tel'if the nngle is jess thnn 10°. Placing
the piel's at nn nnglc of 20° 01' mOrt' with the current mnterinlly in­
creases the i11110unt of buekwuter, the incrense depending upon the
quantity of flow, the depth, ilnd the chullllel contrnctions~
UTERATURE CITED
(11
AUTIUlRfiOX DE \'0[1'[X8, .J. F. D'
1852. A 'l'IU}ATISB os IIYDIIAL"f,[CS, FOU TilE L'RE OF ENGlXEERS.
TrallsL
from the Frcllch and ndnp(cd to the English units of mcasurc, by ,J.
Bcnnctt. 532 pp., ill us. BORton.
(2) BossL''l', C.
1786-87. TUA[TB THI~OIl[QrB B'l' EXPI~IIDIEXTAL D'HYDI!ODYXA~UQUE.
2 V., iIlus. Paris.
(3) CIlESY, E.
1865. AX EXCYCLOI'AEllIA O~' C[\'U, 1~XG!XEEII1XG. HlSTOIUC'AL, Til EOHBT[CAL, AXU I'HACT[CAL.
1752 pp., ilIus. London.
(4) DBIIAUVB, A. A.
(5)
1878. ~IASL'EI, DE L"xGES1Et:H DEl'. I'OX,!'S E'r CIIAUSS}}ES .•• [1] \'. Paris.
DUIIUAT-NASCAY, L. U.
1786. PIlrnC1PES D'IIYD)(AL'L1Qt:[,;, \'EltlFII::S PAIl UN OlUND NO~IBHE
D' EXPB1U[,;NCER FA1TES I'.~ll OUDUE DU GOYEHNlIIENT.
Nouy. cd.,
rc\'. & consideralJlcmcllt augm.
(6) EYTELWE1X, ,r. A. 1801.
(7) KOCII, A.
H126.
2
Y.,
iIlus.
Paris.
HANDBUCH DElllll[';CIIANIK USD DJo}(t IIYDIlAUL1K. VON D[';H IlEWBGUNG DJo;S WASS};HS UND DEN DAIlEI AUFl'RETEN­
DEN KRXFTEN ••. 228 P11., iIlus.
Bcrlin. 68815°-34-4 26
TECHNICAL BUI~LETIN 442, U. S. DEPT. OF AGRICULTURE
(8) LANE, E. W.
1920. EXPEUlMENTS ON TIlE FI,OW OF WATER THROUGII CONTRACTIONS
IN AN OPgN CHANNEL. Amcr. SOC. Civ. Engin. Trans. (1919­
20) 83: 1149-1219, illus.
(9) NAGLER, F. A.
1918. OBSTRUCTION OF BRIDGg I'ams TO TlIg F[,OW OF WATgR. Amcr.
Soc. Civ. Ellgin. Trans. 82: [334)-395, ill us.
(10) REHBOCK, T.
1917. ngTRACIITUNGEN u~mEH AIH'LUSS, STAU UXO W.\LZgNIlILOUNG . . .
114 pp., illus. Bcrlin.
(11)
1919.
ZUR
FHAG~}
DES IlHiicKgNSTALES.
Zcntbl. BCLUYCI'\\'ttitullg 39:
197-200, illus.
(12l
1921. ImiicKENSTAU UNO WAI,ZgNIIII,IlUXG.
BauiJ.;;cniclll' 2:
3·11-347,
iIIus.
(13)
1()21. YEIU'AHlmN ZUH
nl;STI~IMUNG
Df;S
S'l'ulhJgNDEM WASS~}HIlUHCIII·'LUSS.
illus.
BHi'CK~}NS'I'AUI;S
In} I REIN
BlLuingcnicul' 2: 603-609,
(141
1926. DAS
~'I,USSflAUI,AflOHA1'OItlUM
KARLSUUII~J.
DEH TECIIXISClIgX 1I0CIISCIIULg IN
gU­
VI, lJn; WASSEllllAULAlIOHATOmgX
nOPAs, pp. 106-115, iIIus. Bcrlin.
Ch.
(15) TOL~[AX, 13.
1917. CUlm om Imlo;CIINUNG DgS fllleCK~;XSTAUgS, 120 Jlp. Praguc.
(16) ,,'gISBACII, .T.
1847-48. I'UlXCIP[,m'; OF 1'l/l, ~H;CIIAXICS OF MAClllXI;HY AND gXGlXggR­
ING. 2 Y., iIIus. London.
(17)
1877. 'l'lmOllgTICAI. ~lgCIIANIC::;. Trans. from cd. ,1 by A. J. Dubois.
Ncw YOI·k.
(181 YAIlXI~I,I" D. 1.,., NAGl,gH, F, A., and WOODWAIW, 8. 1'1.
1926. FLO\\' OJ.' WA'I'EH TIIHOUGII CULn;JtTS. lown t:'nh·. Studics Engin.
Bull. I, 128 pp., ill us.
ANNOTATED REFERENCES RELATING TO BRIDGE PIERS AS CHANNEL
OBSTRUCTIONS
AKADE~IIS(,lIgX
\'gHEIN H l'TTg, Bcrlin.
"I1UTTg" DES IXGgXmURS T.U;ClIgXIlCCII. Aufi. 25, neu­
bcarb., 4 \'., ill us. Berlin.
Givcs forllluia for computing backwatcr, formub rcally
being D' Aubuisson's. Mcntions Rchbock's and Krey's
work.
BLANCHARD, A. H., cel.
1919. A~mHlCAl\ H1GIlWA1' fJXGrNEEHS' IIAl\IlIIOOK. 1,658 pp., ill us.
Ncw York.
Quotcs forlllultl gh'cn in McrrimlLn's Hydmnlics.
Statcs cocfficient K lIlay bc takcn as 0.9 foJ' piers with round
cnds, and 0.8 for triangular cutwatcrs.
BLIGH, W. G.
1910. TilE I'ltACTICAL IltJSIGl\ OF I1tItlGATIOX WOUKS. Ed. 2, rev. Itnd
cnl., 449 pp., ill us. Ncw York.
Suggcsts treating obstruction as l\ submergcd ovcrCall using
Castcl cocfficient of 0.66.
Box, T.
1902. PRACTICAL HYDRAULICS. A sgnms OF RUI,ES ANIl TABLES FOR
THE USE OF ENGDlggllS, gTC., ETC. Ed. 13, 80 pp., ill us.
London.
States that" thc hcad lost by a strcam in passing throngh a
bridge is principally that due to vclocity alonc, the length of
the channel bcing in most cases so short as to have little
influence on the discharge."
1925-28.
BRIDGE PIERS AS· CHANNEL OBSTRUCTIONS
27
.BUSQUET, R.
1906.
A MANUAL OF HYDRAULICS. Trans. by A. H. Peake. 312 pp.,
illus. London.
Quotes D' Aubuisson's formula and gives \'alues of K as
0.85 for pier with square nose and 0.95 for pier with nose
tapered to narrow edge forming sharp quoins.
DELACHENAL and LEFonT, R.
1911. OBSEHVATIONSFAITES sun LA SEINE A PARIS PENDAN'l' LA GRANDE
cnm, DE 11110. Ann. Ponts et Chaussees (9) 4: 11-53, ill us,
Sho\\'s diagrams of many bridges o\,er the Seine during flood
of UHO, giving location of eddies, water surface curves, etc.
DUHAND-CI,AYE, A.
1873. IIYDnAULIQUE-EXPERIENCES sun LES AFFOUILLEMENTS. Ann.
Ponts ct Chaussees (5) 5:467-483.
Gives data on rectangular, round, and triangular piling and
makes comparison of the three forlllS.
ENGELS, H. 1894. SCHUTZ VON STnOMP~'EILEHFUNDAI\IENTEN GEGEN UNTEHSPULUNG.
Ztschr. Bauwesen 44: 407-415, iIIus.
1914.
lJANDBUCH DES WASSEHBAUES, FUll DAS STUDlUlII Ul'\D DIE PHAXIS.
2 v., iIIus. Leipzig and Berlin.
Gives formula
and \'nlues of the pier-shape coefficient K as 0.90 for 90-degr.ee
triangUlar nose, 0.95 for 60-degree triangular nose, 0.95 for
semicircular nose, and 0.97 for convex nose.
FLAlIlANT, A. A.
1909. HYDRAULIQUE. Ed. 3, pp. 277-278. Paris.
Givcs four shapes of piers. Quotes writings of A. Durand­
Claye. Shows picr with triangular nose and semicircular tail
as suitable form. FORCHHEIMER, P. 1930. RYDRAULIK. Ed. 3, 595 pp., iIIus. Leipzig and Berlin. Discusses writings of Eytelwein, Gauthey, Navier, Sonne,
Montanari, and Tolman. Goes into experimental work of
Rehbock in considerable detail. Calls attention to Rehbock's
findings that the least backwater is developed when, for the
same shape of pie!). the length of the pier is from three to five
times its width. ~uotes Rehbock's formula and gives some of
.Rehbock's coefficients for various shapes of piers.
FOWLER, C. E.
1920. A l'HACTICAL TREATISE ON ENGINEERING AND BUILDING FOUNDA­
TIONS, INCLUDING SUB-AQUEOUS FOUNDATIONS. Ed. 4, rev.
and enl., 531 pp., iIlus. New York.
Mentions Bossut's mathematical solution which showed that
the nose should be 90-degree triangular, and Dubuat's math­
ematical solution which showed the faces of the pier nose
should be convex. Describes in some detail Cresy's experi­
ments made with models being 15 centimeters in thickness,
and shows forms of piers tested.
FREEMAN, .T. R., ed.
1929. HYDRAULIC LABOnATORY PRACTICE... 868 pp., illus. New
York.
Comprising n tmnslation, revised to 1929, of Die Wnsser­
baulaboratorien Ellropas whieh was published in 1926 by
Vereill Deutscher Ingenieure, and including also descriptions
of other European nnd American laboratories and notes on the
theory of experiments with models.
GIBSON, A. H.
1925. HYDRA ULICB AND ITS APPLICATIONS. Ed. 3, 801 pp., illus. New
York.
28
TECHNICAL BULLETIN 442, U. S. DEPT. OF AGRICULTURE
Gives the following formula for loss of head due to bridge'
piers:
Wt~D12)
The value of K, he says, varies from 0.95 (Eytelwein) for a,
pier with pointed cutwaters to 0.85 for a pier \\' ith square ends
or cutwaters.
GILLMORE, Q. A.
1882. OBSTRUCTION TO RIVER DISCHARGE BY BRIDGE PIERS. Vnn
Nostmnd's Engill. Mag. 26: [441)-452.
Discusses effect on flow of railroad bridge crossing Chemung
River in New York. In formula V=C.,fRS assumes.O equals
100 on bed of main streltlll fLnet 84.3 on shfLllow flfLtS.
HAWES, C. G., and KAHAI, H. S.
1929. REPOUT ON EXPElUMENTS CAURIED OUT AT THE KAHACIII Jl!ODEL,
'rESTING STA'I'ION ON A Jl!ODEL OF A FJ.UlIIBD REGULATOR.
Bombay Pub. Works Dept. Tech. Paper no. 31, 7 pp., iIlus.
Experimcnts werc mnde on fL fiumed rcgulntor having one
and two piers in the thront of thc regulntor. Six different
shnpes of noses and seven different shapes of tfLils were tested
with the single pier, while three noses and seven tails were
tested with the two piers. The channel at the pier site wns
1.61 feet wide fLnd the single piers were 0.18 foot wide, con­
tmcting the channel n little more than 11 perccnt. The bar­
rels of the piers "'ere t1bout 2.5 feet long exclusive of the nose
and tail, lllnking the mtio of L/lV somcwhat over 14. When
the two piers were used, each one was made 0.09 foot wide
thus the two piers caused the SfLlllC channel contraction as the
single pier 0.18 foot wide.
The 1:5 cutwn.ter was found so little better, in regfLrd to loss
of head, thfLn the curved nose ('Yith mdillS double the pier
width) tlmt it would not pfL)' to use 1:5 noses. Thc shfLpe of
the tfLilnppel1red to hfLve little effect on the amount of loss of
hefLd (contmdictory to the findings in TechnicfLl Pa,per No. 29).
Two piers in the channel cnlt~cd yery little more loss than a
single pier with like cut and efLHe water.
HOOL, G. A., and KINNE, \V. S., editors, nssisted by BAKER H. S.
.
1923. FOUNDATIONS, ABUTMENTS, AND FOOTINGS. Compiled by a staff
of specialists. 414 pp., ill us. New York.
HOUK, 1. E.
1918. CALCULATION OF FLOW IN OPEN CHANNELS. Miami Conservancy
Dist.· Tech. Repts., 1)t. 4, 283 pp., iIlus.
Discusses cfLlculation of dischfLrge from measurements at
contracted openings. Uses Bernouilli's theorem to derive a
formula which can be used to compute drop if the other ffLctors
are known. FormulfL derived is same fLS d' Aubuisson's.
StfLtes friction loss at contmcted section may be considerable
and should be determined.
Gives Merrimnll's formula and snys this is an crroneous
drop-off formula. The pnrt to which Houk takes exception is
the first term ill the formula which considers part of the water
as passing over a weir. Houk says, "The objections to this
method fLre tWO: first, the essence of a weir is a crest which con­
tracts the cross-section of the moving stream, and such a con­
traction is the absolutely essential basis of the weir formula.
At the plfLce chosen for the calculation, upstream from the
drop-off, there is no crest and no contrnction. Second, the
water moving in the upper surface layers at points upstream
from the drop-off, passes fLt the point of contraction through
t11e area treated as a submerged orifice. This water is an essen­
tial pfLrt of that flowing through the submerged orifice, and is
necessary to help keep up the supply moving through the orifice
with the increased velocity due to the drop-off."
BRIDGE PIERS AS CHANNEL OBSTRUCTIONS
29
HOWDEN, A. C.
1868. FLOODS IN THE NERBUDDA VALLEY: WITH ltElIfARKS ON MONSOON
FLOODS IN INDIA GENERALLY. Inst. Civ. Engin. [England]
Minutes Proc. (1867-68) 27: 218-273, iIllls.
Gives data on backwater at Towah Viaduct during Hoods of
1865 and 1866 and at Gunjal Viaduct in 1866, as being respec­
tively 10, 15, and 3. 5 feet. Cross sections of stream and profiles of
the water surface are given, but no velocities werc measured.
Unwin used one of these cases as illustration of backwater due
to bridge piers in articles on Hydraulics in ninth and eleventh
editions of Encyclopedia Britannica, but inadvertently took a
velocity measured .in 1~64 as being that for 1865.
HUTTON, W. R.
1882. ON THE DETERMINATION OF THE FLOOD DISCHARGE OF RIVERS AND
OF THE BACKWATER CAUSED BY CONTRACTIONS. Amer. Soc. Civ.
Engin. Trans. 11: 211-241, iIlus.
Discusses case of New York, Lackawanna & Western Railway
v. New York, Lake Erie & Western Railway where Erie Railroad
objected to the proposed bridge piers of the Lackawanna Railroad
in Chemung River. There was a great discrepancy in amount of
backwater which proposed bridge would cause. Mentions
Dupuit's, Debauve's, and Gauthey's formulas.
HINDERKS, A.
1928. GRUNDSTROMUNG UND GESCHIEBEBEWEGUNG AN UMFLOSSENEN
STROMPFELIEItN. Bauteclmik 6: 133-135, iIlu~,.
INGLIS, C. C., and HAWES, C. G., with REID, J. S., JOGLEKAR, D. V., aud KALAK­
KAR, H. V.
1929. NOTE ON EXPERIMENTS CARRIED OU'!' Wl'rH VAItIOUS DESIGNS OF
PIERS AND SILLS IN CONNECTION WITH TIH] LLOYD BAItItAGE AT
SUKKUIt. Bombay Pub. Works Dept. Tech. Paper No. 29,
55 p., iIlus.
Numerous tests made on piers with various shapes of noses
and tails, the pier models being one forty-eighth of the full
size. States that loss of head caused by piers is composed (1)
impact loss, (2) eddy loss, and (3) friction loss. These losses are
affected not only by the shape of the nose and tail but also by
the length of the piers. States that for normal conditions, noses
and tails with equilateral arcs of circles are good and semicir­
cular noses and tails are probably as good.
JACKSON, L. D'A. 1875. HYDRAULIC MANUAL AND STATISTICS. Ed. 3, London. Gives table showing for various stream velocities the amount of
backwater caused by different percentages of channel contrac­
tion. Table computed from Dubuat formula,
H3=(K~~g +8) [(~y -1 ]
in which
A and a are the normal and reduced sectional areas; 8 is the
sine of the hydraulic slope of the river; and K is the experimental
coefficient. In most cases 8 is less than 0.0001 and may be
neglccted; then when K=0.96 the formula becomes
H 3 =0.0169V 2 [(~y-lJ
KEUTNER, C.
1932.
'KING, H.W.
1929.
STROMUNGSVORGANGE AN STROMPFEILERN VON VERSHCIEDENEN
GItUNDRISSFORMEN UND IHRE EINWIRKUNG AUF DIE FLUSSSOHLE.
ERMITTLUNG DER ZWECKMASSIGS~rEN GRUNDRISSFORlI1 UND DEn
WlRKSAMEN KOLKABWEHR. Bautechnik. 10: 161-170, illus.
HANDBOOK OF HYDRAULICS FOR THE SOLUTION OF HYDRAULIC
PROBLEMS. Ed. 2, 523 pp., illus. New York.
Discusses the loss of head caused by piers and abutments.
Mentions Nagler's experiments and shows pier shape found to
be efficient.
30
TECHNICAL BULLETIN 442, U. S. DEPT. OF AGRICULTURE
KING, H. W., and WISLER, C. O.
1922. HYDRAULICS. 237 pp., mus. New York.
KREY, H.
1919. BERECHNUNG DES STAUES INFOLGE VON QUERCHNITT VER ENGUNG­
EN. Zentbl. Bauverwaltung 39: 472-475, illus.
1923.
DER WIDERSTAND VON EINBAUTEN IN FLUSSEN UND ANDEREN
OFFENEN GERINNEN AUF DAS STROlllENDE WASSEH. Baute­
chnik 1: [415]-410, illus.
MCCULLOUGH, C. B.
1929. ECONOMICS OF HIGHWAY BHIDGE TYPES. 24G pp., illus. Chicago.
Discusses minimum spacing of piers for the purpose of (1)
keeping within a predetermined limit of backwater height during
extreme flood flow; (2) keeping flood currents below a certain
critical velocity because of erosive tendencies. '
MERRIMAN, M.
1914. HYDHAULICS. Ed. 9, 5G5 pp., mus. New York.
Discusses formula considering discharge as consisting of two
parts, first that passing over a weir of breadth WI under It head
H 3 , and second that passing through the submerged orifice of
breadth W2 and height D2 under the head H 3•
MOLESWORTH, G. L., and MOLESWOHTH, H. B.
1931. POCKET-BOOK OF USEFUL FORMULAE &; MEMORANDA FOR CIVIL AND
MECHANICAL ENGINEERS. Edited by A. P. Thurston. Eel. 30,
rev. and enl., 935 pp., mus. London and New York.
Gives the following formula for rise of water due to obstruc­
tions in rivers:
( V2
_) [(A)2
a- 1]
R= 58.G +0.01)
NEVILLE, .r.
1875.
in which
A=sectional area of river unobstructed, in feet.
a=sectional area of river at obstruction, in feet.
V = velocity previous to obstruction, in feet per second.
HYDRAULIC TABLES, COEFFICIENTS, AND FORMULAS. Ed. 3, pp.
141-144. London.
Apparently treats the discharge through a contraction as
consisting of two parts. States that quantity of water passing
through the lower depth, D 2, is equal to
KW2 Dn /2gH2
and the quantity of water overflowing throngh H2 equals
hence the total discharge through the contracted section
becomes
When the velocity of approach is considerable, or when
VN2g becomes It large portion of H 2, its effect must not be
neglected. In this case, as before, the discharge through the
depth D2 is equal to
KW2 D2.J2g(H2
+ l'N2g)
BRIDGE PIERS AS CHANNEL OBSTRUCTIONS
31
and the discharge through the depth H2 equal to
~ KWn /2(i[(H + VN2g)3~- (VN2g)"'-j
2
and hence the total discharge becomes
Q=KlI'z.,/2g{ D2 (H2 +
Vl'/2g)M+~ [(H2 +VNg)%- (l'N2g)~,)}
For square ended piers, K equals O.Gj wh('n noses arc obtuse
K =0. 7, and when curved or acute K =0.8.
SCBOKLITSCB, A.
1930. DER WASSERBAUj EIN HANI>BUCH FUR STUDIOM UNI> PRAXIS.
[1) v., illus., Wiell.
Discusses Rehbock's work and givcs Rehbock's formula for
computing backwater.
STRECK, O.
1924. AUFOABEN AUS DEM WASSERBAU, 3G2 pp., iIllls. Bcrlill.
Gives the formula
and works out a problem.
Uses K 1=K2 =0.90 for "dip. stumpfwinkligen Pfeiler Kopfe
setzen wir."
TIMONOFF, V. E.
'927. VERS UCHE UBER DIE ANORDNUNG DER STROMPFEILER BEl NEBENEIN­
ANDER STEHENDEN BRUCKEN Abhandl. St. Petersburger (Lenin­
grader) Pamphlet 5. (1927.)
(Cited by John R. Freeman, cd., in Hydraulic Laboratory
Practice, 1929j p. 3GG.)
TRAUTWINE, .1. C.
1909. TH>;J CIVIL ENGINEER'S POCKET-BOOK. Rev. by J. C. Trautwine,
Jr., and J. C. Trautwine, 3d. Ed. 19, 1257 pp., ill us. New
York.
Gives no formulasj merely quotes table (with many correc­
tions) from Nicholson's Architecture. Knowing the original
stream velocity and the proportion of area of the original
waterway occupied by the obstruction, the head of water pro­
duced at the obstruction is given for piers with upstream ends
rounded or pointed. Nicholson says that if the piers arc square­
ended, head will be increased about 50 percent. Trautwine
states subject is extremely intricate and admits of 110 precise
solution.
WADDELL, J. A. L.
1916. BRIDGE ENGINEERING. 2 v., illus. New York.
"The amount of backing up or increase of head can be
ascertained by considering the discharge between the piers as
composed of two elements, viz., the discharge through a sub­
merged orifice, having a width equal to the distance between
the. piers and a depth equal to that below them, aud a flow over
a weir of length equal to the distance between the piers and a
head equal to the difference in depths above aud below them."
WEYRAUCH, R.
,
1930. HYDRAULISCHES RECHNEN . . . Auf!. 6, neubeart. und verm. VOIl A.
Strobel. 370 pp., iIIus., Stuttgart.
Gives D'Aubuisson's formula and quotes values of coefficients
for different shapes of piers.
WILLL\MS, C. C.
1922. THE DESIGN OF MASONRY STRUCTURES AND FOUNDATIONS. 555 pp.,
illus. New York.
Quotes the Eytelwein .formula and givcs values for coefficient
of contraction.
32
TECHNICAL BULLETIN 442, U. S. DEPT. OF AGRWULTURE
APPENDIX
ENERGY METHOD 0.' COMPUTING" HEADING-UP" DUE TO PIERS
A paper I by Fred C. Scobey, /i;ives a method for 10C!Lting the water surface
through constrictions, based somewhat 011 the work of Koch (7) who developed
a graphical method of determining backwater heights which, however, disre­
garded friction and pier shape and gave results appreciably different from test
measurements. Their method suggests that the backwater resulting from pion;
might be computed without the aid of laboratory tests or pier-shape coefficients.
In order to discllss this question, a brief statement of the method will be nccCs­
sur\,
G.
Using the sallle notlttiun
ilK
~!'
hefore (pp. 7-8) ami '1=
Q
I[
1'2
'12
V= WD=[) ami 2~q='2g[)2
w= 1~2.gg =
1)
~!f_ or .'L2 = /)3w '2gJ)3
(9)
2g .
It is well known that the critical depth ill the ullobstructed channel occurs when
• w= !~, and that the maximum discharge for any given total energy head is qr=·liiJJ5.
But if an obstruetioll is introduced into the channel, reducing the width fro III 11'
to (1 -a) lV, the unit-width discharge in the contracted section will be q2= I ~a
and the maximum q3 for any given total energy head will be (l-a)q2 when '12 i~
critical, 01' (1- a) .JYD3~2. But when t.he flow through the contracted sectioll iil
critical, D2 will be two-thirds of the total onergy heae!. Neglecting for the momcnt
allY energy loss. the total energy head is equal to
1'32
D:t+ 2g =Da+wD;I=D 3 (I+w)
When ('riUrnl
\'(~locil.y
exist:; at D2 •
lLnd the Jl)UXilllllll1 q3 will be given by the formula
Q
q32= (l-a)2q22= (l-a)2gD 23= (l-a)2g27Di(1+w)3
Bill. from (0). qj",.,.,'2gwD 33.
Equating these vullles of
'13 2
and solving we get
WI.
4 (1
)" (I+WLP='27
-a ­
(HI
where: WI. indicates the" limiting yalue" of w, which would cause critical flow be­
tween the piers. Equation (10) is plotted in figure 9, which also gives Rehbock's
two criterions, and shows that this diyides his class 2 just about in the middle.
To avoid confusion, Rehbock's classification is designated 1, 2, and 3, and the
Iowa classification A and B.
If the curve dividing classes A and B in figure 9 were extended beyond w=O.50,
it would have turned upward again and a second value of WL would have been
indicated for each value of a 6 • Class A flow exists when W is less than the lower
WI., and class B flow when w lies between the two values of WI..
I Thesis. Leland Stanford 'C'nh·ersity.
"rhe subject is nlso trented quite fully by Rehbock (11. Ie. IS). The matbemntlenl treatment given in
tbls bulletin, however. is not nn nbstract of Hehboek's work but was worked out independently•
• In fuet there is a third vnlue ul$o. but it is negatl\·c. The three roots of the equation are
~-1
I-a
•
where 6 is defined b~' the equation: CJS 3O=-(J-a). The flrstexpre.o;sion gh'es the higherwL. the second the
lower WL. and the third is ulwnys negative and without meaning. 'I'he values of WL for the ditferent chunnel
contractions used in the te.<ts nrc given in table 4.
33
BRIDGE PIERS AS CHANNEL OBSTRUC'l'IONS
~r---~rr----,.r------r----~------.------.------'
A5~---+~~----~~--~r------r------+-----~------~
.'
35,~-----ft--~--r---~rr----~~--~~-----1----~
30'~-----+~·---4~--~~~----~------~-----+----~
tf
•~
~ ~
~ .25,~------~--~~-r~--~-+----~~+-------4--------+-------4
.0
.::~ ~
.~
~ ~
o
'",
•• '
/
~----+-------~~---1
u .20
~
~
§
~'
~~-#..'?) ~
if \,:i .15/-------j-------t­
.IO/------+---+---~.
,
:'
o
.05
.15
.10
.25
.20
Values of
.35
t&)
FIGURE 9.-LiDlit..~
o( clifiercnt classes o( fio\\' accordinll to Rehhock's and to Iowa classifications. Note
however, that Rchbock's classification is made accordlDg to the fiow in the C1nobstructed channel but the
Iowa classification relates to the fiow through the contracted section.
TABLE
4.--Limiting values of
Cos 8
8
a
w
Sin (8-
Higher
Lower
30°)
"'L
"'L
0.5000
1.152
1.684
2.:147
3.596
0.5000
---- ------ ------ ------- - - ----0.000 .117
.233
.350
.500
°
I
50
46
43
40
41
41
31
60 00
0.5000
.fl336
.6860
.7252
.7660
00
-
0.5000
.3532
.2871
.2337
.1736
.11194
.1235
.0785
.0419
1
34
TECHNICAL BULLE'l'IX 442, L'. S. DEPT. Oi" AGRICULTURE
The distinction between class .A and class 13 is showlI clearly by figures 10, 11,
and 12, which give the water surface profiles for several runs on piers with semi­
circular noses and tails. Figure 12 shows the case of a flow of about 68.8 cubic
feet pcr second past a pier which gave 50 perccnt contraction. The critical
depth (in thc unobstructed channel) for this quantity was 1.15 feet. When the
downstream weir was in its lowest position, class 3 flow resulted with Da=0.70,
while upstream DI=2.71. (Eleven feet downstream from the pier the depth was
only 0.45 foot, probably due to eddies which formed a virtual contraction even
1U0re serious than the pier itself.) "'hen the adjustable weir was raised until
Da \\'as ] .83, DI remained practically unchanged. The same was true as Da was
increased successively to 2.12 and 2.32, D I, being now about 2.74. But when
Da I\'ns made 3.08, DI was increased to 3.16.
The "limiting depth" corresponding to the WI. of equation (10) was in this case
2.60 and cnn be computed from the equation
(11)
which comes from cquation (9) when W=WI.. Values of D I• are plotted against.
I'alues of qa and ex in figure 13. As long as Da is less than D I., Dit would be the
~alJle as DJ- exccpt for the energy loss.
Whenever Da is greater than D I ., DI will
be correspondingly greater by an amount equal to the difference in velocity heads
above amI below the obstruction in addition to the ellergy loss in passing the
pier. Then when
Y'a2 l?
(12)
Da>D L , DI=Da+iri-2g +L.i
and when
(13)
where L.i is the energy head loss and LJI is approximately the pier-nose loss.
It Illay be noted that in both figures 10 and 11 one run was almost exactly at the
limiting depth. It should also be pointed 0111, that the actual stream conditions
are opposite to those shown in figurcs 10 to 12, because as the depth increases q
increases so much more rapidly that DI• incrcases more rapidly than Da, and may
overtake it, especially if the proportion of obstruction is large. If D I• becomes
greater than D., the floll' becomes strangUlated (to lise Rehbock's term) and
serions bnckwater lUay rcsult. This will occur very rarely, however, except in
stcep mountain streams where the backwater is unimportant because it would
"rull out" in a short distance upstrcam. The whole matter is made clearer by
figure ] 4 which shows the values of Chezy's C and slope necessary to cause
"strangulated" flow (class B) for variolls channel-contraction ratios. Unless
the slope exceeds 0.001 or the Chezy C exceeds 90, ex can be 0.23 withont exceeding
the limiting val lies of wand Q2.
Koch's method is therefore valuable because it brings out the different sorts of
backwuter caused by piers. In class 13 flow it will give the approximate back­
water, which can be llIade exact by adding LJI as explained later (p. 40). But
in class A flow, the type usually occurring in practice, it fails entirely, as according
to Koch there would be no backwater at all if no energy were lost and he gives no
wlty of computing the loss.
In class A flow the velocity between the piers will be less than the critical
velocity. In class 13 flow the velocity would be, nccording to Koch, just at the
critical I'ILllle and the water surface would be level. Figure 11 shows that this
is approximately trlle in the case of long piers (length 13 times the width), but
figures 10 and 12 show that it is far from true in the case of short piers (length
4. times the width).
EMPIRICAL FORMULAS AND GRAPHIC SOLUTIONS
FORMULA
FOR
CLASS
A
FLOW
A study was made of the variation of L" in equation 12. The first assumption
was that as the losses probably varied somcwhat as the square of the velocity,
they might be taken as varying with V a2/2g, the velocity head in the natural
stream. This was found to be the case in tests where w was approximately
constant, but different valucs of W gave different values of L" for the same value of
lTa2/2g. An attempt was therefore made to develop un empirical formula for
LA in terms of V:,2/2g and wand a pier-shape cocfficient. Bllt since in class A
flow the velocity head above the piers is only slightly less than that below, the
3 l
-
1&
~
A
~
,,
,
'.J>
~
,,, .......
,,,
~
- -
~'
~
~
I. ;
...o A
1Xf
~
.....
Between classes Aand B,limiting w O.1235, limiting depth.'.85~1
a
.
~.:t-
_
)
A CIa•• I, f.,f 769. Q. 70.36. ",.0.0353
B Closs I ,1~sI77/.a:t'70.62.w.O.0635
,,,
C Class 2 ,Ies; 784. Qfl7Z.66,t.J a O.I042
o Closs 2 I lest 754. Q= 70.65.w~O.1245
E Closo;J,f.sf73J.a·7Z.29.w.O.3Z9
F CloS$ 3 ./~sI689.Q·71.39,w.I.244
D3c'~I)
4
.-
'0
j
.J>.
....&--
E
~F
~
2: 2:
t;;:!
t"
o
~
,,,
,
,,,
§
>'l
...,
o
,,
,"
>0
t;;:!
f;l
I:d
,,
0. ;
~
C":l
:~ -oM-'.....-i t--....
I
t;;:!
Ul
t\
~
8C)
:>
~
:!:
I·
B
~V
~
""o
­
,
)
o
,,
,,
t<>"
it:
i-PIER -,,
,,
,
,,
2. ;
-:;:- 2.
,.
30
40
,50
,60
2:
Ul
Distance along testing channel (feet)
FI (lURE 10.-\\'''ler surClIce profiles with twin slnndllr<l piers hnving semicircular ends. Chnnnel contrnctioll 23.3 pcrt'Cnt, flow 71 cubic Ceel per se('Ond.
c,.:)
~
C/j.
0':1.
-----1------,-----.,-------,---------.--------.
3.0r,
~6A6 ~--~--~L~-A ~ ~
~
C
661
~
b::1
I
I
~
.JO..
1
~
0
4J
01
q
0-0
B
10
0
t"
0
~
~
C
Between classes Aand B,limiting ""O.123s:ii:nitin!f
L.L!!:::::t:=~~
~
Z
dep;';~
::
....
....
I":>
~
c::'
D3C=~
iKO
I
A Clo.. I,I..1906,Q.68.7S. w ,O.0496i
B CIa.. I ,1..1907. Q.68.7Z,w.0.0748 I C Clo.,2,leslS08, Q. 68.7Z, w.O.{NO , D Clws2 ,Iesl881, Q.68.53,w.0.1681
E Clcss3 ./~s;880. Q a68.66.w.O.2048.1
0.5 F C/oss3,lesl87S, Q.68.66,w,O.889
--..,.c"""
10
ZO
F
I
'It _
-.c I
I
I
30
I
~
o
>
G')
::e
....
c
I
40
50
60
Distance along' testing channel (feet)
FIGURE l1.-'Vnter surrnce I,roflles with twin long piers IIIl\'ing semicircular ends,
t;j
t;j
..,
'=::
,
I
I
I
I
I
~
,
" ' 1...............
""'..._­
,
I
I
,
0' ~---
~
Chunnel contruction 2;1.3 pcrt'Cnt; now (is.S cubic reet per second.
70
q
~
~
.....--
3.5i
I
1--
~
3.0l
...
1"-..
I
...
..
.:t
-
I
I
~
~
0,;
':;
~,.s
1,/~sI6-#7.Q,68.B5.w~O.OZOI
B Class I .It!s';' 648. Q;69.05. w .. O.OZ53
C Class 2 ,1t!3f 649. a-G8.59. "".0.0577
o Closs 2 , It!sl 650. a..68.59. w.O.0792
E CfossJ,t.,;65I,Q'68.19,w.O.IZOO
F Closs,J , It!sf65Z • a.68.72. w.I.O;'8
O. i
~
I
I
£
~
V
L..{'~
I
/'
~
V
!
I
I
I
I
I
I
i
I
I
30
~
Ul
>­
Ul
n
i:I:
>­
~
~
Ole'
~
.!.:!1­
o
to
/
-
40
Ul
~
F __
~
........... I
I
I
I
-
E
~
t.".1
t.".1
0
I
t:1
G:l
"'C
....
C
II
I
-­20
Between classes AandB,limiting u a O.0419,limiting degth:l.60,
I
I
I
10
'"
ttl
~
....
I
~
A Class
B
I
I
c:)
1.0
­
A
I
1\
2.0
~
J,...P
I
~
.
I
I
I
I
~
2.5i
-:;::.
--- .... --,
PIER
~
..;;
..:::
o
7!
rii
50
60
Distance along testing channel (feet)
}'IGUln: 12.-Wllter surfut-e I)rotiles with single wide pier hnving semielreulur ends. Channel contraction iiO perrent; now 6S.S euhie feet Iier second.
~
-..l
c...:I
00
/
20 ~
'U
'"
~
=:
l.l
...Z
~IO~====t=~==t=t=tj~~=====t==j==t=t=t~~~~ ~
Class A flow
above upper ahne+4------4---~~~~H+~~L-~~~,~~~~~~~~~~~~
~
~
§
5~-----r--~~~r4-r+14------+---t
n
>
r-<
t:
q
~
t:'"
r;:j
~
is
Z
'"
~
u
....
"'"
'"
Co
l,;)
~
II)
~
Ul
§
.s:
t:;j
t=j
'.t:
8
":i
~
~
~
'=:j
~0.5~----~~~~~~~~~~/~~79~~~~b+~
--I-
.t::
.;:::
>
o
...
~
:~
(")
~
'-I
0.2~~~~~~~~~~~~~~~~L-~~J-~~----~L-~--~~~~LL----~~
0.1 0.5
I
5
10
50
100
250
Discharge per foot of width in unobstructed channel (cubic feet per second)
FIGURE 13.-Limitlng depth DL in unohstructedchannel for various channel contractions and for various discharges per footofwidth in unohstrurtecl chunnel.
"
d~
r;:j
BRIDGE PIEnS AS CHANNEl" ons'rnUCTlONS
39
I.oor--,-----r---,----,----r---,.----,-----,
~
~
.g
P, N
OJ
.r:
0
Q
;J
.5
.,
'0
~
Q
'0
~
::s
<3
..
ti
~
::s
'"0
..
.~
...
~ .50
.g
~
~
ci'
.g
Cl
IS\>
..
....... ..0
c:
~
Ci
e
;;
g
Cl
::s
::s
.r:
'"
"
-
0
~
::s
<3
.....
,3
e~
I.
:!:
[5'"
"
ri;
40
TECHNICAL BULLETIN 442, U. S. DEPT. OF AGRICUVfURE
surface drop down If.1 is generally only about 10 per cent more than LA. There­
fore the possibility of d~veloping a formula for fIa directly was iIwestigated and
found to be equally satisfactory.
After many trials the following formula was developed:
H a=2K(l(
+10w-0.6) (a+ 15a
l)
V·
2;-
(14)
This applies only to class A flow, and for some values of a and w the departure
from the curve which represents the avernge of the data is greater than lhe prob­
able experimental error. But it lies well within the range of the individual tests
and was adopted in preference to any more complicated formula which might
have been developed to fit the data better.
The values of K for the principal pier shapes tested are listed below.
Semicircular nose and semicircular taiL _______ .. _. ____ . _. _ O. 9
Twin-cylinder piers with eOllnecting diaphragm_._ _
.95
Twin-cylinder piers without diaphragm. ____ . _
1. 05
!)OOtriangular nose and 90° triangular taiL.. _
1. 05
Square nose and square taiL _. __ • _________ • _. . _
1. 25
Piers with one or both ends lens-shaped or convex-shaped appear tu have a
coefficient very nearly the same as that for one with semicircular ('nds.
Thc above coefficients are for piers with length four times the width. The
t'ffcet of increase in ratio of length tu width is discussed on page 41i.
}'OnMUI,,\S
In the case of
Cl!lHfl
~'on
CI,ASS R FLOW
H lIow it was natural to fluppose that
1.1/1
might be roughly
proportional to the square of the velocity of approach i that is, to
\,2
--.f.
-y
III fuet,
IT'
it wu.~ found that when LI/ for allY given a and pier shape waR plutted agllillst '7)1'
-g
a straight line could be drawll through the origin and fairly close to the poillts.
That is, we have:
'J
(15)
"'hen the values of (.'/1 for euch pier lihap(' were plotted ngainst a, it \\'as found thnt
they could ))e representC'd by the equation
C/I=0.50+](/I(5.5a 3 +.OS)
(16)
where KI/=5 for piC'rs with square cnds ulld 1\11=1 for piers with scmicirculul'
ends.
The urithmeticul averllge of the discrepallcies betweell the observed vulues of
/-fa and those cOl11puted by tIl(' usc of formulus (15) nnd (l(i) are shown in tablc 5.
TAnL}~
5.-Differences be/ween ob.~crved and cOif/puled backwater heigh/s for cla.~.~
13 flow
,A \'cmge
obsen'cll
: (,hannel· _ Tests
\ contnlCtion n\'cmged
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _1_ mtlo _.:
Shupe of pier
Square en<l5._._ ••••• _••• -....................... •• ..
Semlclrculurends............................
Lens·shaped ends .. _.- _...... _..• -....... -- •
·1I{
""j{
...\
,\'umber
0.1167 .
.2:l3:l
.:~'iO :~
.2:l33
I
2 f
8-1
~I
.wo
• 1I67
.233:l
llJ
51
I
70
45
48
45
Fat
0.18
.U-l
.5;
.79
.13
A\'ern~e
diITerenl'e.
Fut
0.02.1
.(~!O
.4115
.011
.011
.026
.024
.012
.0
.0;2
.010
.O'll
-.,.,
.t_
____l.____._. ___ _l....... _. ___ ... _.._ ._.
IJEVEI.OI'MENT OF' GRAPHS
A series of curves hu.') becn prepared to fncililute the computation of backwater
for actual cases. Figure 13 gives the D/. for allY given fJ and a. If DJ is mol'C
t.hnll D/., the flow is eluss Ai if less, it is class B.
41
BRIDGE PIERS AS CHANNEL OBSTRUC'l'IONS
For class A flow, turn to the upper part of figure 15 and from the appropriate
V3 2
values of a and Va read the value of x= 2g a(l + 15a3 ). Then enter the lower
a.
ItI
+
~
~ .4~--4----+~~4.~~q-~~-~~~~--~~1-------~~~i-~
H
W
y.2
= ..2.zg ..!..D3
•
o
.12 1.0 I.S
Backwater, H3
2.0
2.5
(feet)
FIGURE 15.-Chart for determining backwater in class A
part with this value and the proper values of wand K, and read the backwater,
H 3• If the backwater is given and it is required to find the discharge, it will be
42
'rECHNlCAL BULLE'l'IN 442, U. S. DEI)'l'. OF AGRICUL'l'URE
necessary to assume a value of Va and work by trial error until the solution is
found.
For class B flow, turn to figure 16 and go up from the given a to the curve
representing the given pier shape, then horizontally to
~~2 and
then vertically
to LB. Values of ~~2 can be obtained from figure 13. Ae a first approximation
assume DI equal to D/,. Then add the L/J thus obtained to DL to give a corrected
D, and with this value (and using the same scale as for D L ) read a new value of
i~~ from figure 13.
With this new
~~2 enter figure 16 and get a revised LB.
If it
s appreciably different from the first value, the process may be repeated. The
backwater height H a, is given from the relation lIa= D1 - Da.
Although this method of computing backwater for class B was fairly satis­
factory in the case of piel's with square, semicircular, or lens-shaped noses, it
gave unsatisfactory results in the case of twin-cylinder piers with diaphragm.
An attempt was therefore made to develop an entirely empirical method.
It was assumed that
(17)
From dimensional considerations it is clear tho,t if Ha varies as SOllle power of Da
it must be as the first power. Of course, Ha could have been taken as varying
V a2
Y I2
1'a2
Va2 Y I2
with 2~ or with Da+ 2~' but 2(j=wDa and Da+2(j= 2~ (l+w), so the only
change would be in the form of 12(w).
Values of Hal Da for the same pier shape and channel contraction were therefore
plotted against wand an attempt made to derive the equation of the line. Then
by finding similar equations for different values of a determination of the form
of" (a) was attempted, and then of the form 12 (w). However, it was finally
discovered that the simplest way of including the effect of the channel contraction
was to plot Hal Da against wlwL where WI. is the lower limiting w. It was found that
this automatically took care of the variation in channel contraction as is shown
in figure 17, where all the values for the pier with square nose and square tail fall
on the same curve. To avoid confusion, the individual points for the piers with
semicircular nose and semicircular tail are not shown although they plotted about
as well as those for the square-ended piers. For the other pier shapes no tests
were run with a=0.35 and a=0.50, so that data are lacking for the higher values
of wlw/,. A number of the shapes are shown for a more restricted range, bllt at
at larger scale, in figure 18.
Certain limitations of this mE'thod must be noted, however. The limits are
indicated on figurc 17. Nor does it hold for very small values of a. As a ap­
proaches zero, the maximum value of wlwL approaches one; and the upper limit
of Hal Da might be as great as from 0.9 to 1.5, depending on the pier shape, when
it should be practically zero. The methods given below should therefore not be
expected to give reasonable results for a less than about 0.05.
Considerable time was spent in trying to derive an empirical formula for these
curves in figures 17 and 18, especially one that would hold for different pier shapes
with the change of but a single constant. No such formula could be derived, but
the following gives fairly good results for the shapes listed for values of W/WL
greater than unity:
Ha_ 0.4+ (0.045x2+ l.lx-0.98)x
Da -.liX
x2+3.3x+1'
7;-
wherein x=~
WI.
and K=0.136 for square nose and square tail,
0.096 for 90°-triangular nose and 90°-triangular tail,
0.070 for semicircular nose and semicircular tail, lens-shaped nose and lens-shaped tail, lens-shaped !lose and semicircular tail, and convex nose and convex tail. (18)
43
BRIDGE PIERS AS CHANNEL OBS'l'.RUC'l'lONS
r---~~----~~--'---~--~~------~o
LI)
0
~
.;t
..n
...0
+
0
..n
~
d
,g
P<
«!
0
C'J
" ~
]
c:
..:t
..--..
+Q)
0
".j::
U
III
'+-
LI')
'-" "
«l
L.
Ul
'a"
's....
()
....
.8
NQ)
,g
0
<+­
0
::l
"
."
+­ OJ'"
C '0
...J
::l"<l:
,S"
0­
" C
....
~
c;
c: I
co
...c:: Sr.l
~
(J
0:
P
"
~
o
~
2. I
Maximum values of
,
~L
which 'may be
u~ed
for gi'ven values of Cl
21.4­
-Cl=0.30
I
I
-(1=0.10
I
5.01
~
-~
,.,~v:.
3.06
1.6i-Cl=0.05~
£18
.....Q
., 1.2
OJ
~
V
~
0.8I
/:"
'l/l'I
?
>
t'I
b::!
~
t'I
~
.."
Z
"""
"""
~
~
Cl
(1
(1
Cl
Ul
= 11.7
____________________________ D
=23.3 % _____________________________ A
= 35.0 "'. _____________________________ x
= 50.0 % • _________ •• _________________ .+
~L
~
t.:.1
~
o
1:;1
>
o
...
t:o
c
7
.
2
fo--
fo-
.----"
Semicircular nose and taiL____- ­
Square nose and tail __________- - -
""
W
----­
~
~
~
,,'
,,/
0.4
o
~
9.24­
2.0I-Cl=0.20
~
g
Z
...
c
~
4
.
6
.8
FIGURE 17.-Vurintioll of /h'DJ with
..
10
12
Values of ~L
W,'WL
14-
16
IS
tor piers with square und semicin'ulnr noses Ilnd tails.
20
22
24
r
~
~
~t.:.1
45
BRIDGE PIERS AS CHANNEL OBSTRUCTIONS
However, it generally will be simpler to use the curve itself rather than this
formula. Find W from equation (9) and WL from equation (10), table 4, figure 9;
then with W/WL enter figure 18 or figure 17 and read Hal Da for the given pier shape.
This multiplied by Da gives H a, the backwater height. Or, if D" Da, IV2, lira, and
the pier shape are known and it is required to find the discharge, HalDa may be
calculated from Hal Da= (D 1 - Da)IDa, W/WL be taken from figure 18 or figure 17,
1\1\ ~
i
1\
1\ ',\
\ \ 1\ \
I
1\
\
\
DO oQ •
: : :E
l!: : !i 1
•
t
:~
:.~
l~~l
._"'1]
~
N
d
.:.
U
~\
~l:~
\
~
,
~
:
E
l­
:
•
~ta:- l -
~-.; ,_
3:~ "-"'0
I!:~~u
~ ~~~- I ­
.. -g ~'ii
... .., .-~12""- I ­
;.a. 3~~.~
,...f"-5
-g~
~~c~
i\\ 1\\\ \
fit
""1::;
~i£
; ; t-
1"'1)
: :
\
~
nI
C
~
L-
~
:~.!:
.? 8 'gt~
a
0 ~ ~~~- I - o
E·tf.E~
+ C •u •uu.
~ 'E
!:
'"
C"'ClO >;;1~ d ~~aJ
CIlVlO"lU
~.e ~-;
c
~.!::!
~\
CC~
0
,
-:;
\~
1\ ~,
.13'
q'o
~
.2
.".,
4­
..!
cS
,~
\\ ~
'\ '\
.~
~
~
o
<1
'\. ~
! J
-!~!-
"V~
1--0­
-+-------­
-
~~
~~
------
'\
I
:--i-·
I
I
-
I
I
"
-Iz:
I
..
-0-
I
=
'"
'"
.
­
\
"
a= (Wa- H'2)IWa, andwLobtained from figure 9 or equation (10). The "alue of W
can then be computed, and from it Q by the equations Q5=2gwDg and Q=qa IVa.
Figure 19 is an alignment chart which may be substituted for this method for
piers with square, triangular, or semicircular ends if wlw/. does not excred 1.8.
Difficulty was encountered in trying to develop a single alignment chart that
would hold throughout the entire range of the experiments and all classes of flow.
TECHNICAL BULJ~E'l'IK 442, U. S. DEP'l'. OF AGRIC'UT.TURE
46
Curves were also drawn of Ha/Da against w/w/. for the piers where the ratios of
length to width were 7 and 13 instead of 4. Comparison of these curves with
those in figures 17 and 18 for piers with square and with semicircular ends is
shown in table 6.
1.0
1.5
20
.50
2.0
15 2.0
1.5
1.5
1.0
1.0
.05 200 .20
-----------------..-----..------::---:.. ---____ !:~L.. -5-~--------~::~=:..
~-... -------­
~
---------­
____________ ----- .10
·~.5
....
10 3
.3
.3
6.3
..
-'!
6
.2
.,..
0 . 05
.,..
<::
.1 "
'b
~.2
..
~
:l! .1
<::
;:,
:J
~.1
.02
.50
.2 ~
~
"-
:a....
L'' ;
.20
.03
5
.01
UH3 90· f-iangIJlar
nose and tai 1
.12
.50
.A
4
o
c
B
F
E
FIGURE 19.-Aliglllnent chart for determining backwater. From depth nnd dischnrge (senles A and D)
determine 01 (scale 0); froIII 01 and channel contraction (scnles 0 and D) determine W/Wl, (senle E; from
depth and wlwL (scales A andE)deterrnine backwater IT, (scale F for approprinte piershnpe). Hensonahle
nc'Curacy ennnot be expected froIII extensions of these scnles.
TABLE
6.-Ratios of values of Ha/Da{;or pier length-width ratio.~ of 7 und 18 to value
of Ha! D3 for ength-undth ratio of 4
_.
.--_-
--~
-.---­
Piers with S'IUnnl nose l'iers with semicirculnr
nnd tnil
nose and Inil
011011.
L/II'=7
--"-O••S
1.1
2.0
:1.0
7.0
I
0.82 I
.00
.1l7
.07
.98
L/IV=I:I
L/1I'=7
0.9a
.97
n.8:1
.O:!
1.00
1.00
.98
1.00
,
I L/W=13
j
LIM
0.89
1.00
I. OIl
I.OS
1.02
I.(}!
47
BRIDGE PIERS AS CHANNEL OBS'l'RUCTIONS
As only these three valnes of LinT were investigated, it is uncertain whether 7
is the optimulll length-width ratio for a pier with square nose and tail. It seems
probable that the optimum ratio is different for different values of w/W[,. In the
case of the semicircular nose and tail the optimulll is apparently less than 7, being
definitely so for W/WL greater than 2.
Apparently there arc two opposing effects. Increasing the length of the piers
must increase the friction loss but up to a certain length it decreases the eddy loss,
and the latter is greatcr, especially with the sqllare nose and tail. It has been
pointed out .elsewhere (18) that an abrupt entrance tends to decrease frietion
losses for a short distance downstream because of its effect on "elocity distribu­
tion, the velocity along the walls being reduced. This is probably one reason why
the backwater caused by the long piers was greater when the pier ends were semi­
circular than when they were square.
COMPARISON OF
FOR~IULAS
Figure 20 gives the curve obtained from the Iowa tests for piers with semicir­
cular noses and tails (the same as in figs. 17 and 18), and the curves which would
be given by the D' Aubuisson, Nagler, and Rehbock formulas for scveral values
of a using the average coefficients for classes 1 and 2 as givcn in table 1.
These three formulas may be changed into the following forms:
.35r-----~----,-----,-----_r----_r----_r----_.--,__.----_.----.n
.30 Iowa 'est dlta (semicircular nose and hi! ) ....... - - Nailer formula I KN -.923" /3 from fig.2 ........ - - - ­
O"Aubuiuon formula. KDAI= 1.010 ................ - - - _j_------,;:-l----+-----l-bC--J
Rehbock formul. , 6os3.11 ............................-----Rlhbock formul.. edtnded beyond ct.ss 1....- ­
.2~1-----+-----!-----l-----+-----+-----j.
.201-----+_----+---+----+----_j_---jLj-----:
itO'
't
II
~.151_---_I_----+_---_I_---+----_¥_----_j_.",c.___,04....".!~:.....j.~~_:q----_f
.:l!
.101-----t----+----+-----b'C---~5oo"'7''_:b?;...<.:_t_;<_
.05f----+-----+---=~.:,..,,:II5o.j;.<S-'=F----+"""'-::.+----+---_t_----_l
.B
1.0
1.2
1.6
I.B
2.0
V.luu of -l:l-L
FJOUUE
20.-C'ompnri~on
of bridge picr formulns with results of Town tcsts on !Jnsiso( mlion( If,/JJ,low/WL
,
.
ll" [
4
(1 +w/,)3 (D")~J
D AlIblllsson: -D-" = ?7K2
' -W
_
D A
wI,- - ~D"-I
Nagler: Rehbock:
Tl3=[
4
(I +w/,~ __ {:J(D:I)~JW
Da
271(N2w/.(I-.3w)2 • DI
HI
D~ =[o"-",(oo-1)][O.4",+",2+g,,,4]w(1 +2w)
If the best coefficient for each channel contraction had been used, all of the eurves
would have fitted fairly well up to about the limit of class 1, or about w/wL=O.7.
(Rehbock's formula is not supposed to fit beyond this.) For higher values of W
the D' Aubuisson formula departs from the test curve far beyond the range of
experimontal error, unless the coefficient is made to vary with w as well as with
the pL".r shape. The Nagler formula fits fairly well to a higher limit. If the coeffi­
cients in each of thc formulas arc kept constant for each pier shape, a.~ appears
48
TECHNICAL BULLETIN 442, U. S. DEPT. OF AGRICULTURE
to be intended by their authors, the results depart widely from these tests for
most values of a, as is shown by figure 20.
All the tests were made in channels with rectangular cross section. How
satisfactorily these formulas can be applied to actual river channell" by esti­
mating equivalent rectangular cross sections, is not yet determined.
It is believed that these formulas will apply to larger piers without error due
to the scale r)f the model, because the formulas are dimensionally homogeneous
(except for ,,:2ji, which is constant) and because they check with Kagler"s Mich­
igun tests which ,,'ere on slllaller llIodels.
VBI.OCITY DISTRIBUTION AROUND "IBRS
In conncction with the experiments on the obstructioll of bridge picrs to flow
of watcr, two sets of tests were made in which the \'elocity distribution around
the pier was determined. With a flo\\' of 24.1 cubic feet pel' seeolld, velocities
passing a pier with square Ilose and square tail were measurcd at various points
in 11 cross sections in the channel, and the water-surface contours around this
pier were determined for two depths of tlow.
To locate the position of the point at which each velocity rcading was taken,
it was nccessary to build a device by means of which the coordinates of each
point of observation could be determined. A 4-inch angle iron 12 feet long was
secured horizontally to each wall of the clllial at the site of the pier. Telemeter
rods graduated to lllllldredths of a foot were bolted to these angle irons. Sup­
ported by these bars and spanning the cunal was a main guide frame consisting
of two telemeter rods] 0 feet long placed on 3-ineh T -bars 3 feet apart (pl. 8).
This guide frame was easily 1110ved back and forth along the side rods. On it
were two short cross frallles, each consisting of two telemeter rods 3 feet long spuced
about 3 inches upar·t. Metal supports which could be moved along the cross
frame held either hook gages for determining water surface elevations or Pitot
tubes for making \'elocity traverses.
Thus two observers could take readings simultaneously and obtain the co­
ordinates of the points of observation. The hook gage fmme could be moved
lengthwise of the cllUIInel through a range of 3 feet on the smull cl'oss-fmll1es, and
each cross fmIlle was easily moved transversely of the charlIlel on the main guide
frame. The main guide frame was mo\'ed lengthwise of the channel as necessary.
'With each reading of his hook-gage vernicr, the obsen'er determined the location
of his gage on the short cross fra!l1e, as well as the position of this cross frallle on
the main guide fraIlle and the setting of the main guide fraIlle on the side rods.
For taking the velocity readings, the hooks were reIllo"ed frolll their fmmes LWei
Pitot tubes made especially for these tcsts were inserted and clamped in place.
'1'he pressure and velocity columns were mounted on a wooden frame which was
secured to the hook-gage fraIlle.
In the velocity distribution studies shown in figures 21 Iwd 22, each section
represents a cross section of the channel showing the actual measured velocities
in their respective locations. In addition to measuring the velocities at the sitc
of the pier, the direction of flow at various points in the channel was also deter­
mined. The studies showed that the direction of flow ncar the pier nose is to some
extent, away from the pier, whereas the direction of t10w at points some distance
from the pier is directly downstream. The Pitot tubes wcre faced directly
upstream, so the measurements recorded and platted arc only the longitudinal
components where flow was oblique to the axis of the canal. Figure 21 shows
velocities next to the pier appreciably greater thun velocities some distunce away,
along the upper portion of the pier. Figure 22 shows a gencmlly similar "elocity
distribution, with a greater depth but the same quantity of flow and therefore
lower velocity. Critical velocity existed at certain sections in figure 22, partic­
ularly at stl1tions 4.75, 5.75. and 6.75. This condition, which cllused the un­
usually high velocities arOUIIl. the pier nose, would seldom be met with in actual
practice.
Ele\'ations of the water surface were tuken at various poiuts in the channel.
These elevations were plotted in their respective locations IlS shown in figures 23
and 24, ill which contours make the drop dowlI of the water surface uround the
pier nose readily apparent.
j
49
BRIDGE PIERS AS CHANNEL OBSTRUC'l'IONS
Wah,.
3 ,
3.'
3'J
3.3
3.'
3 ,
3"
3"
3.'
3·,
3.3
3.3
,.
3·1 ......
J.I
3.1
'.9
~:~"~
3.3
3.1
3.2
3.2
3.2
3.3~-.!j
4
3.3
3.3
3..2_~_"l4- __ 3.2
I
,_'_Ji
4'3
ti
~
C
3.'
3.3
'"
..;
,;
.,...
'0;"
ti
..,
'"
'...,;"
II)
..,<>
3.3
3.1./
.... 3.4
3.2 ~.O
3.": _ _ 3.0
2.9
3"
"5
1.8
3.Z
3.2
Z.8
Z.'
2'1
.6
..
.6
.2
0
1.0
.8
.6
.4
.2
0
1.0
.8
...Ii
.4
.2
0
2'9
3.1
~5
3••
3.l....--3. _
Z
.6
3.1
3·1
/~
~
,..
,,..
Z!4
2·5
2.5
Z.8
1.6
.. ,..
8
,~,.,
(,.7
,
..
'.J
,
,
2.4
,·7
2 5
A
1.B
2 5
2.'
2.'
3 0
2.8
~
2.4
4
5
Widlh of channel
~'IGUllE
3.2
....
3.3
<:S
II)
3'3
~·,J.4
4'~}
3.9
..,
~
,~3'3
"z
4.~
4.2 J."O
3.'
3.1
2.9
6
7
8
10
(feet)
21.-Vclority distribution around pier with square noso nnd squurc hlii. .Fiow 2·1.1 cuhic loot per
second; fl, upproximntciy 1.0 loot. Pier 1I0S0 uL stntion 2.0:1 und pier tuil ni statioll 7.30.
30
TBt:HNIt:A1J BULLB'l'IN <U2,
.:::
C)
,;
0;
.8
.6
c. S.
DN}''J'. OF AGltlCUJJl'UHB
4.1
4.3
3 5
.4
4.,L
4.3
.2 _3;3
36
0
~
4.3
=-­
'"...
.;
.;
0;
.'"
...:
.;
Z;
".'".;
..
ti
"
'"vi"
ti
Z;
~
'1
~
..;
.;
3.9
''"r.;"
3'6
3.9
..,
.;
...'"
...
.,~
3.B
37
3 a
3'9
3"S
3.6
®
3.9
'"'"
'.,~"
C)
32
3.3
3 I
30
3.0
3.1
3.1
.1
'"cl
3.3
.;
3.2
~
3.2
2.6
3.1
31
3.2
3.1
3.1
30
3.2
3.1
3.3
3.3
3 0
0
3.1
9
10
channel
FIGURE 22.-Volocity distribution around plor with square nosc nnd squnre tnil. Flow 24.1 cubic feet pcr
second, fl, IIpproxilDnlol~' 0.75 (oot. Pier nose nt stntion 2.63 and plcr tail at station 7.30.
BRIDGE l'lEHS AS CHANN ElL OBS'l'RUC'l'lONS
51
II
10
9
-:;:­.,
~ 8
~
......,
c:
7
§
~ 6
Cl)
c:
.95
~ 5
()
<!>
0 4
§
....
.~ 3
<::l
2
o
2
5
.1
6
7
6
10
Distance across channel (feet)
FIGUItE 23.-Water surfut'C contours (in feet) nbout squnrc-cnded pier with fiow of 24,1 cubic feet per second
nnd D, about 1.0 foot. (See fig. 24.)
II
10 ~
'70...--"
.65
.60 ,,--------.....
-:;:­.,
~8
.,c:.
.....
§
..;:
o
(),
c:
~
()
.,
o
~
cS
3
o
j
4
;,
8
,
9
10
Dist ance across channel (feet)
FIGUItE 24.-Watcr surface contours (in feet) about squnre-cnded pier with fiow of 24.1 cubic feel per second
and D, ahout 0.75 foot. (See fig. 23.)
ORGANIZATIONS OF THE UNlTED STATES I>EPARTMENT OF AGRICULTURE WIlEN TillS PUBLICATION WAS LAST PRINTEI> Secretary of Agricult.ure ___________________
Under Secretary _______ - _____________ • __
Assistant SccretnTY __ . _________ •. ________
Director of Extension Work_______ _____ _
Director of Personnel_ • . - _. _ ._ __ __.
Director of Information ________ __________
Director of Ji''i'lIl1IlCC______
_ ________ .
Solicitor___________________ ___ ... __ •
.Agricultural Adjustmenl. Admitlistralion _ ___
Bureau of Agricultural Economics_____ _ _
Burc01Lof Agricu.ltural E11gincering _____ .
Bureau of Animal Industry_________ _
BlIrenll.of Biological Survell ______________
Bureau of Chcmi.5try anel Soils. _____ •
Office of Cooperative Extension Work____ _ __
Burcau of Dniry Ineluslry_____ __ ___
i3Ul'cnlt of Entomology alld Plnnt Quarantine.
OjficeofExperimelltStations ___ - _____ __ .
Food and Drug Administration ___ . _____ ."
Forest Service ________ •• __ ..... _ ________
Grain Futures Administration_ •• __________
Bureau 0/ /lollle Economics _______________
Librnry __ • ______________________ • _ ____ __
Bureau 0/ Plant Industry _________________
i3urenu of Pllblic Ronds __________________
lVeather Burenu_________________________
H~}NRY
A.
REXf'OHO
.1'.1. L.
W.
C.
,v.
W,\I,LAf'E.
TUGWELL.
O.
WILSON.
'VAHnuHToN.
W.
STOf'Kn~}nGEIt.
1\1. S. EISENHOWEH.
W. A..JU~II·.
SETH THOMAS. Administrator. A. OI,S~}N, Chief· S. H. MCCRORY, Chief·
.JOIIN R. Mom,EIt, Chief·
.J. N. DAm.ING, Chief·
11. O. KNJflJl~', Chief.
C. B. SMJ1'll, Chief·
O. Eo REED, Chief.
LEE A. STHONG, Chief. ,JAMES T. JAUOINl" Chief· WALTEH O. CAMPIIEI,L, Chief· FEHOINAND A. SILCOX, Chief. ,J. W. T. DUV'EI" Chief· LOUISE S'l'ANLEY. Chicf. CI,AItlII1!lL R. BA ItNgT'l', Librarian. l". D. RWHEY, Chief· THOMAS ll. 1\1,H'DoNALD. Chief· WlhLIS R. OHEGG, Chief·
CllESTEH C. DAVIS,
NILS
This bullctiu is a contribution from
BlI-reC/1/. of Agriculllli'al BI/{/inccrinU ________ S. R.
Division 0/ Drnilluoc and Soil Erosion
COlltroL _________________________ L. A.
McCnoltY,
JONES,
Chief·
Chicf.
52
U.S. GOYEfI;" "IttlT PR.UitING OHICt.
uu
2
3
r---',
234
4
,
D:A.U BU ISSON-KD~
I
1.81-
I - f-I
a. ~ 11.7 single
23.3 twin
Types of flow
for this shaped pier
I:St--
1.41-
1.2
I
~
I
o _q~
I
a - 11.7 single
I
I
a. = 11.7 single
- 23.3 twin
i
I
4
I
,','. __ r
3
REHBOCK-KR
I
0..11.7 single
23.3 tWin rFrr
rr
2
4
3
REHBOCK- 6 0
NAGLER-r.",K N
1",
1928
1929 +---1-,--1---+--1
Class r ___ 0 _____ 0
f
Class l I ___ x _____ +
I
Class lIL_.r
.... '1--I
2
23.3 twin
1.2
+
j
II I
I
1.0 .
12~---!---+---+--r-----t----+----+~
)(
)(
x
Xl( X
n
I
j
I"
1.4
r
rp 1.4 1.0 P"-a~~ !a5B~'W~"tL___+__+_-_t_.--F- ~~-.+-
~~~~~rt
~.~
1
.B~~~~~
I
(
)
(
1.2
(
) I1 r+_
rrr
0.=.11.7 single
a = 11.7 single
23.3 twin - 4 - - , " - 23.3 twin L
1.8f--
(L=
r
r
.~
.-+--4--I----!-----!.!r f'r
i
-1-­
,,1 I----t------t----+-+----t--, - 1 - -
+­
11.7 single
23.3 twin
I--!
Q=
11.7 single
23.3 twin .8
-+--+-i,+t
I! ---i-;
-1---+-1I ----t.-t--+- ._+-'-t!
I
j
121--­
1.0 C)
I i !
;
rr
j
)
I
I
1.4
1.2 1.0
1.2
1.0
<11.7 single>
<11.7 single> Q _
Q=
- 23.3 twin
!
1.6
1.2 f-·
,;
I
,
~o~
o:~~~ f
o
ni 0
.:
!
1.81-- I
[r
\
fr
1
-I·--
<11.7 single> 0.-
+__ ~
12 I - - - T
,
.~- L. ...
.;
;
··t-·
1.0
!
1., - ..1._
1
;
23.3 twin ......;.~I-- 23.3 tw,ln,
I
I
1.61--_'t-1__
'
23.3 twin
-+--
--. . -,- -.. t----,...l--I
.........lI
--~.-.F-jr- --i--'
i
I
+ -. _ •
i
-1--+-+--1---'
< 11.7 single>
0.-
rr ~~_
I
. !'r4r
!j!
Q=
23.3 twin
23.3 twin r
I
1.41-----,-l!---i---I
<--->
<----->
11.7 single
11.7 single
Q=
---Lt ~
I
r
c____>
>
< 11.7 single
11.7 single 0.-
- 23.3 twin
Q _
- 23.3 twin I
1
+--~:-+-
~_t-,i
I
1.4~--+------1---!--I---t----;C--
1.2
j
i
rr
I
2
a
1.4 - -
3
o
=
4
0
11.7 sin~le 23.3 tWin
2
3
0
4
o o
0
a.='23.3
1.7 sin~le y+­
tWin rrF
;
I
r
t
I
1.4r- -
()===()
0===0
-t-.- f - - .
ct'" 11.7 single;
23.3 tWin .
a=11.7single r r
23.3 twin
rr
1.2 f----+--o-~-+--___-'-I---4--+-,-I~r
_f5)-,,~ x.~tt':
1.0
ra-...,~e~~I'"r:~,r~·--t-·
...&;::
o 0 ~~~ I~~
0
"i:)
o
og I.e:.. ....oJ"'~
r
"'ji:";rxr
o~.",. x·~,!
C"'__---'
~i~I!1.7
1.2 r--
~
0
60
I _ 0.= 11.7
)
T
I
-<
I
60·
a,=II.7
r
i
I
2
3
4
2
3
4
;2
I
2.0
-
- .
"
~
­
'tl
t::
I
I
tJ 1.6
\.0
:-
I..
3
4
2
3
4
I
I
I
I
TYPES OF FLOW
Class 1 ___ . __ 0 Class2 _______ X Class 3 _______ r Channe I contraction. a = 23.3 %
Width I W = 1.17 feet D:AUBUISSON-KD~
1.2t---
I
I
NAGLER- KN
I--
1
0.-11.7
1.0 .. ".
00
Q
I
a, = 11.7
.... ~,~fil(l<)(
-~~' ~,.)(,,~)(
.8 I
1.8t---
o
fP I
1
I
1.61----
Types of flow for this
shaped twin pier
.
'928
1929
Class 1 __ .o ____
Class 2 --.x ____ .+ Class 3._.r
a
I
I
"1-
--+_..... 1
a. - 23.3 I
I
1.0
-~--.-I--
._
1
Ir-L-:=::J--'
14t--
1.2
I
1_ rr
l !,
I
i
I 1
rl
rFr
f
I
--
0
1.6
[
I
= 23.3 -',I---r'---t--
~---t--
1.8
1
a
'I
r-----t­
=
I
23.3
i
I
1.4
I
a. = 23.3
1,2
1.0
o1----4---1-­
I
I
I
D: D
'r
r r
!
I
r
a. = 35 ~
t:::a
Q)
i
?KX~~xl<lC"ax
.....
u
9o~ ~xf.: ~\l)4 I
0
I
u
,
i
I
1
f
I
~~
~~ ;r~._
,0 o>(J'
0
~+--
­
!
i
I, 1.4
I
a=- 35
--+.---~J
-+j
~
t:::
,
T-
.~
;.
!
I
I
!
i
u
~I.O
Q)
i
,
I
1.2
I
j
0
u
2.2
....
~2.0
~
~1.8
at::: 1.6
C::t
1.4
t:::
0
I
VI
.~ 1.2
:::,
-.Q
:::,
I
o
~1.0
CJ
1.2
,
1.4f--­
0
1.0
I
o o
I
I
o o
D~UBUISSON-KD:'"
REHBOCK - KR
NAGLER - KN
:.;
+":
1.2~ (
'i
I--
)
a
ones, it> 0
1.0
00
11.7
oree
Io';':c
-+--+--­
X
(
_ _--. - I - -
(
)
a
=
o
o
11.7
4
~
0
), 0
q,
0
a -11.7
"'0
o (
;-
,
1.8f-~:--:--If---HI---+--+-.l::..-t-i
If'r
x
-l'x
a = 11.7
Xx
)
I
I--
,,010
)­
I
1
I
c
ca.
)
1
(
I
I
)
;
)
=.
I
)
(
a. = 23.3
23.3
I-­
~
t:: 1.2
.~
u
~1.0
III <:) (,J
2.2-
TYPES OF FLOW
Class 1 __ .... o
CI ass 2 _ __ x
Class 3 _____ .. r
a
c
percent channel contraction
a = 50
t-­
t-­
1.2
~
.- a. '" 11.7
~
1--
1.4
O:=<) +-~-+
'n
?':I. ':I.
I
..
~:;o
1
--
I
10=<)
n
?
~
1.
<....
a
1.8~
=
./
;;>
<....
<....
,;>
a- 11.7 single ~
Q.=
~
:\',
11.7 single
23.3 twin -4--1-- 23.3 tWin .J._ +-­
r
>
<.
a = 11.7 single 11.7 single
23.3 tWin
23.3
tWin
r?
rl
r
,
1.61---+--+---+~---+------4----j-..:~
C',
r
r
h-r
1.41----+---I----I--+--+-----io~--I--I
12
o
To
1---+:~::___Io--=-~-t--_I_--4------1__ 8~~~--~-~~---+---+---+-~
r
1.2
1.0
4
<
90·
)
- i-~-4--~-t
0.=
,
23.3
90'
a=23.3
+
-,...--f--
"-t-----i---+
!
--+---t----l--f:
f
< >
1
+-+,--+---;
l
1.4
1.2
,.. -1-- - - - i - - i
!i
,
!
1.0 C
1.2
1.4f--
<;
90'
-<
:>
90·
-I-
a=2l3
>r-t( !
a=2~3
<;
90'
>
<;
1.0
:>
a=23.3
0.=23.3
I
90·
8
1.2 .
4
B
i- ,-
.
'---r-~t~
,
.Br,·, -
-~"
~_~1
2
-
~l-
cJJ
,.+.••• ,,,,.f.
,,' -, -
~~
"
"lI,.. ..
1
I
4
2
3
c____>
1.4
I
~
~
L2
4
1
J"
a
+
I
i
,t
,
___
' ___
' ~_ _~_~____ -L_
3
1.0
4
_ L
:3
_ _ ._~
4
_,_--L
1 --.1...----'---'
2
3
4
r
Velocity upstream from pier. VI (f!
1·
I
I "tI 1
0
<;
f---
o
w
0
0
~
60
<
)
00.'"
11.7
OU 0,
)
f---
B
<! 6~0
0
0
\1.7
4
_SbOv.
p )(
0
o)()(~
tI
o
no. &.
0
u:.0' I"'"
00
;:- Ia= "T
0
Q,c
Yeo 1o"'lDQO'- ~
o
o
3
2
4
a,=
°
o
)(
-<
0
234
Velocity upstream from pier, VI (feet per second)
Q)
(Q
0.=23.3
;'
2
3
4
from pier. VI (feet per second)
2
3
4
2
3
)
r-­
)
t--­
11.7
0.-
60°
11.7
0..=
o~
0
0
90
~
0_00 n
0
234
0
OOnQ
)- -
i.-J
~
0
\1.7
'~%~
0 0
~
p
I
~Q.900
0p,
0
60·
)(>sc~ l~~
0
oCb)
o 0 ~Sc*
0
.dJJ......CP~
1.0
t---
4
)~
<::
)
a. = 11.7
• a.-1I.7
OOd
0
1.2 I - -
G 90·
)
90°
_0 ) v
lo~
~
~~~
o CO
2
3
4
"". h....
. I,..
8 0 )(~)(
)(
,Irll."",.r"
>lx)(x<:;
TYPES OF FLOW
Class 1 ________ 0
Class 2 ________ x
Class 3 _______ r
Channel contraction. a - 23.3 %
Width,W
1.17 feet
8.
4.
(
(
)
:-. L= 7 W --i
8.
)
l-L=7W-l
o
!
I
1.4f- (
I
i
) ---
j-
(
i r/ jr
f---L=13W---i
1.21---+--+----+----1~- -
o~~~~~
r
1----/---
i'
~ ... -
(
L=13 W---j
)
f-04---
L:o 13 w----l
8.
)
L 13 W~I
rr°rtrrr r I.
1.0 1----'1~Vl,.¥-r-F-F·-F-r-+rr..:......r-rrr---+rrr-.j-~--o~
oQ..nO.:IL~
,,(fJO
)
:0
4.
n
.81----I-----+---l---_t__
DAUBUISSON-Ko'A
4
3
2
NAGLER - KN
234
o.
REHBOCK-oo
REHBOCK - KR
234
2
3
4
Velocity upstream from pier, VI (feet per second)
Fun-Itt: 4.-I':II.cl of Icn~lh of picr "I'on Ihr ,·".meionls in hri"~I'·pier forrnulas. Pio'rS of Icn~lhs 4, i, nnlll:! limes Ihe pier widlh lire shown.
hoth with sQtlUre tillct:: and wifh ~C'rnicirl'lIll\r ends.
61815
0 U.S. GOVERNMENT PRINT'NG OfTICE : I'U
.
>1 ~
r.r
tJ
~
~.B
~I. 8
't)1.6
c:::
t:s
I
r
r
~
~1.8
._- t-­
'r-
r
r
r
'tJ
I::; I.S
CD
50
of"
'r....
r
~
~2.0
r ,.
-D
0.,=
2.2
t..
I
1.4 f - - ­
l
Q)
(J
r
r
\)
1.4
c:::
C)
I/')
.0.0
olb"OU'
o
x..
0.=
x
....... 0
0
Q..
a
c;b'Q
o~
."
o OJV"~
Jh~
rr
,
I-rr
I
0
0
0
0
0
o
-a=23.3
-~
..
T
I
_~
1.2
1.0
J o~~..
)<r:
0
C___j__
r
r
r
A';t.
0
0
1.4
'
o~V)('
DAUBUISSON-Ko'A
3
o
~;rt-
I
x'"
~)('*F~ ~~r~r
2
T
0
I
O.r r:L- +--a. = 23.3 ,..
;PF'I
.8
!
!
'--
1.4 ---
1.0
!
11.7
=
.8
1.2
~
,--I ­
,
r",
:J
~1.0
0
0
a - 11.7
a.&.
::,
-.Q
I
I
btp?i",.,o i..&
I
0
.~ 1.2
-
-~
.8
1.0
50
4
t
NAGLER - KN
234
1.2
0
1.0
I
o
j
.L---L.__ J.
REHBOCK-oo
_-----L .•
REHBOCK - KR
234
2
3
.8
4
Velocity upstream from pier,VI (feet per second) F",t'IIE [•. -Elrc,', III ChMII(l('1 ('onrrnclioll upnll the l'oeUicients In hri<lge'I,lrr lormnI8",
Chnnnel L"ntrRl'lIons 0111.7.23.3
~~
9
.;I
J
If:,t
'~1-
,~
TYPES OF FLOW
Class 1 _______ 0
Class 2 _______ x
Class 3 _______ r
Cl~ percent channel contraction
I
0 0
I
8
I
a. 5 0
~
x ><,«.
x-,;*,
~
~
I
o
0
00;
8
QO~
4
~~=11.7
0
'by
o
Oc!og
a.
=
rr,,?,
r .__
0
I~
I
ex,
o
:°L
234
0
.
!
= 23.3
.
~~, :,~t771-
lC
!
r
("rfArT
~: x~~~_
:
,
~ci~e~~x
a. = 23.3
4
4
~
co
23.3 3
1
00
}~
2
11.7
o~ ~~
riix
I~
I
~+CL =
0
1.4~
,
1
0
n
Qi) ~~yo
"
~
r--- _.-
0=:0
0
,..,..
I
I
0
12
r
r r.
~
I"~
~0'0C'j5'C OJ
a. = 50
~
,()C
>eJI~i", ",..o<IS<
0:;
4
1
-.---'--.- --
REHBOCK- 00
1.-1._
REHBOCK- KR
234
I
2
3
4
Velocity upstream from pier,VI (feet per second)
Channel contractions or 11.7, 23.3, 3.5, IIlId 50 perl'''"I.
ll~lng
Iliers both wilh ''tUIlc(' ends 111111 wilh ~1.'UlI(·irCllhlr l'n!I.<.
51115
0
u.s.
GOVUHNENT ~RIIITIHG OffiCE: 1114
rr
c.. _ __>_r--t---i
1.4
r
a= 23.3
C~_~~>--f-_
B
1.4
a=23.3
r
I'
1.2 t---t::d-~IQch'r-+-f---+---t---i'--l
'"
I.Ot--Ji1~vAfi;;;;;~r*-t--+--=-f---+--j
o
-+ . - + - - - I - - f - - - i - - - - l - - 4 - - l
.Br---+--+---+-,r---+--+2
3
3
2
4
4
2
3
4
234
Velocity upstream from pier. VI I
J
S;;::SIOO
12 t-t-xJ-;~;::===::;=t
I!
x >< 1 x x
Xx x
Xx)(,( .
x"'xx
a. = 11.7
x
8
!-tI
i
v
~Cb
Po"
x~x
00
~200
a = 11.7
I. 2 f--+--
-+-t-J--
.6
DAUBUISSON-Ko'A
2
3
4
5
2
3
4
5
2
3
':
5
2
3
4
5
Velocity upstream from pier, VI (feet per second)
t'JGI:RE i,-EIT,'rt o( slllll'e o( pier upon thO co~mcient in th~ \'OrioliS hridge-pier formuills.
In(\I\'idl1l1l CO'lmcienis (or \'ariolls ShR,"'" of pie
;,
.J
I I
I
I
I
r~
)1
a=23.3-
I
I I
It
rr
I,.JI
~.o
mfrom
IV.
2
3
4
••
NAGLER - KN
2
3
4
«:
I
T
D
0.=23.3 A:~_
~K"~~
X>$(
.'" I'K
<beg, ~:d)..
d ""f""'"
tV
1~~-q
!iDi\UBUISSON-K o'A
JJ
4
00
I"'-r
a."'2 ~.3
~~ ~xJ~)(
\ -
8
I
I
I
r rr
rr
I
Ii
-
REHBOpK.- 00
2
3 T
4
REHBOCK- KR
2
3
4
pier, VI (feet per second)
~IO·
~ 0.= 11.7
I
rr
"x
TYPES OF FLOW
x
Class 1 ________ 0
Class 2 _______ X
CI ass 3 ________ r
Direction of flow in all pier sketches ~>
a=percent channel contraction
.­
5
clents for "arlous ghal ..... of ple/ll plotted a~aillst_ "elocil)' upstrenm from pier.
Stllndnrd pier IIscllln which lenllth of barrel was coosl..nt.
688\5
0 U.S. GOVERNMENT PRINTING OFrleE ,
I!~