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TECHNICAL BULLETIN No. 442 N OVEMBl

Technical Bulletin No. 442 November 1934

CNITED STATES DEPARTMENT OF AGRICULTURE

WASHINGTON, D.C.

BRIDGE PIERS AS CHANNEL

OBSTRUCTIONS

By DAVID L. YARNELL I Senior draintl\~z 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 PageIntrod uction_ ______________________ ___ ___ _____ I EITect or channel contraction on coefficient_ ___ 16

Backwater occurrent'C!____________________ 2 J,ITect on coefficient oC setting piers lit an~le Principal earlier bridge-pier studi~~____________ 3 with current______________________________ __ 20 Scope oC the investigations_____________________ 4 Use oC data illustrated by exnmples_ . __________ ~'OExample 1_________________________________ 2()Description oC experimental plant_____________ 6

University laboratory_____________________ 6 Example 2________________ .. ______________ 24 Measuring weir____________________________ 6 Summary and conclusions____________________ 24 Pier models_______________________________ 6 Literature cited _________________________ "' ___ .. ~ ~5 Piezometer and piezomoter tubes__________ 7 Annotated reCerences relating to bridge piers 85channelobstructions_________________________ 26Theory oC the obstruction oC bridge piers to /low oC water _____________________________________ i

Appendix:Test procedure_ __________________________ _____ 10 Energy method oC computing" hrading-up"due to piers ______________________________ 32EtTect oC shape oC pier on coefficient____________ 11 EtTect oC length-wldtb ratio oC pier on coeffi- Empirical Cormulas and grapbicsolutlons.. _ 34cient. ____ _____________________ __ ____ ____ ___ _ 15 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 preparation 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 University. 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 considerlltions, 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,

r(3) the percentage of channel rontraction caused by th 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.

8ACKWATER 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 downstream 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.

3 BRIDGE PIERS AS CHANNEL OBS'l'RUC'l'IONS

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 construction of the bridge. Every backwater formula contains a coefficiE:nt which varies with the shape of the nose and tail of the pic1.•• 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 presented !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

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 consisting 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 900 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 contraction used in the first series. These tests covered the following 14 combinations of end shapes:

1. Square nose and square tail. 2. Semicircular 110se and semicircular tail. 3. Convex 1I0se and tail.

'1. 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 2Tech. 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 contrnction 11.7 percent, (/=82.1)(; ('uhic feet rwrsC!cond, fl,=2.tj4 [c

en

5 W

a.

Tech. Bul. 442. U.S. Dept. of Agriculture PLATE 4

FLOW PAST PIERS WITH SEMICIRCULAR NOSE AND TAIL HAVII\:C 1:24 DATTER. AND WITH DIFFERENT WEB CONSTRUCTION,

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 triangular 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 Iilllt.'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=2.53feet, lh=2..1U (eel..

5 ID W

a.

..:

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

Ted•• Bul. 442. U.S. Depl. of t.gricullure PLATE 8

A, nfuldng \"('locUr trn\'l.'l'ses 1I1'011l1d pill!" ill lesling-l'hnnnci. IJ, 'l'win I)jcl'S hl1ing- IO\\'l.'l'll d 11110 testing ca 1111 I.

5 BRIDGE PIERS AS CHANNEL OBSTRUC'TIONS

5. Twin-cylinder piers without diaphragm. 6. Twin-cylinder piers with diaphragm. 7. Tr.iangular nose and tail of 90° angle. 8. Triangular nose and tail of 90° angle with 1:12 batter. 9. Semicircular nose and lens-shaped tail.

10. Lens-shaped nose and semicircular tail. 11. 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 determined 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 semicircular 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 sht1rpcrested 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 downstream 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 testin~ canal. 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 central 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.

7 BRIDGE PIERS AS CHANNEL OBS'l'RUCTIONS

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 whiteenameled 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 downstream 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 OC"" 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,

1'1 = thl' mcan \'cloeity of the wutcr' above the contraction, = l(~bl' Vz = 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

=Dt -D3.

8 TECHNICAL BULLETIN 442, U. S. DEPT. OF AGRICUI,TURE

g=the acceleration of gravity. VN2g=the velocity head of the water above the channel contraction. V 22!2g=the velocity head in the most contracted section of thc channel. V32j2g=the 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, respectively.

00= pier-shupe coefficient ill Hehbock gcneral bridge-picr formula (see equation 7).

Pie,.

~I\------ ~-------------if3

VI v, 1 I"--r- - DI --z

-'!!- 1 r Bottom of ch.nn.' -'

LONGITUDINAL PROFI;"E

I t "

1 WI W,

I ~ IJoWz --2 !1 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 (D2+H2),

Q2( 1 1) (1 )H2 =,? T/2D'.4 III ~D2'i - 11'' 2DI ~-0 .L1.. - '2 'I in which R D , A is the D'Aubuisson pier-shape coefficient.

The true backwater is not exactly represented by H21 but ordinarily in practical field installations there will be little difference between H2 and Ha, and hence little difference between D2 and Da.

9 BRIDGE PIERS AS CHANNEL OBSTRUCTIONS

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 (2)whence

(3)

Weisbach (17, pp.114--116) hased his formula upon the assumption tbat the total discharge throu~h the contracted section may be calculated as the sum of two quantIties, one quantity consisting of the flow throu~h a submerged orifice of width W2 and height D2 , and another quantIty consisting of the flow over a weir with a crest length of WI and a head of H2• The formula then becomes

Q=Kw.J2u[JWI(H2+ VN2g)3/2+ W2D2(H2+V12/2g) 1/2] (3-a) 2.2

2 -

.8 /

/

V

.6 J

4 /If

I.2 J

olL I. 0 10 20 30 40 50 60 70 80 9() I00

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 percentage 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

TECHNICAL BULLETIN 442, U. S. DEP'.I:. OF AGRICUL'!'URE10

~

Rehbock (11) divides the flow into tlu'ee classes as follows:

1. Ordinary or "steady" flow, in which the water passes the obstruction with

very slight or no turbulence.2. Intermediate flow, in which the water passing the obstruction disp

lays a

moderate degree of turbulence.3. "Changed" flow, in which the water passing the obstruction be

comes

"completely" turbulent.

These t~rree clas~es of flow are defi.ned according to Rehbock by

the followmg equatIOns. 1

aA =0.97+21w 0.13 (5)

aB=0.05+ (0.9-2.5w)2 (6)

According to Rehbock, the moving water will he iI\cluned in the first

class as long as the obstruction ratio a of the pier site is less than the

limiting value in equat.ion 5. When the value of a of the pier site

under investigation lies between the vnIues of aA in formula 5 and aB

in formula 6 the second condition of flow prevniis, and when the value

of a of the pier site exceeds that given in equation 6 the third condition

of flow mdsts.The Rehbock (13) equation for computing the backwater height,

H3 , for all pier shapes in a channel of rectangular cross section with

ordinary flow! is as follows: .

A simple equation for \mdge backwat'U' is, according to llehbock(11)

(8)

It is probable that the D'Aubuisson, Weisbach, and Nagler formu

las apply only to the first class of flow as defined by Rehbock. Determinations of bridge-pier coefficients for the Weisbach formula

were attempted, but the extremely discordant results indicated that

this formula is theoretically unsound and the effort was abandoned.

There are many other backwater formulas mentioned in foreign

publications on hydraulics, of which Tolman (15), in Teviewing all

formulas and methods of which he knew in 1917, mentions as most

prominent those of Dupuit, Eytelwein, Flament, Freytag-D 'Aubuis

son, Gauthey, Heinemann, Hofmann, Lesbros, l\1ehmcke, l\/fontanari,

Navier, Ruhimann, Tolkmitt, Turazza, and Wex. For reasons of

economy, coefficients were not determined for these formulas which

are seldom mentioned in English texts on hydraulics.

TEST PROCEDURE

Tests were run with quantities of flow ranging from 10 to 160 cubic

feet per second, find with depths of flow, V 3 , from 0.6 foot to 3.3 feet.

The experiments were begun with a head of about 0.4 foot of water

discharging over the measuring weir, followed by tests with successive

increases of about 0.1 foot in head on the measuring weir until the

greatest possible quantity was obtained. Different stages of flow at

BRIDGE PlEHS AS CHANNEL OBS'l'HUC1.'IONS 11 the site of the pier for euch head on the measlll'ing weir were obtuinedby means of the adjustaule weir downstream from the pier.In pru·t of the tests the adjustable weir was set at a definite positionwhile a complete set of runs for the pier set-up was obtained withvarious heads on the measuring weir. After the desired head on themeasming weir WitS obtained, the observer fh-st read the hook gageabove the weir; t.hen he obtained readings to the nearest 0.01 foot 011the 37 piezometers along the wuU of the cunal, and recorded the depthof the water in the channel ns shown by the various stnl!' gnges setalong the inside wnIl of the cltl1al. A ren.ding wus then taken on theweir hook gage to see if the water level had yal"ied. This process wasrepeated for each head on the ,,'eu·. The adjustltble weir was thenset Itt ar.other position and another series of runs obtnined.In the majority of the experiments the pier models were placed inthe dry testing channel and water allowed to f\O\\' through the canalpast the piers. The difference between the wn.ter surfnces immediately upstream and downstream from the pier was called tbe bnckwater.To preclude any criticism of the method used in determining thebackwater, an extensive series of experiments was run in 1929 inwhich water was permitted to flow through the unobstructed testingchannel and the "Titter-surface profile was obtained, nfter which thepier models were set in the channel nnd the witter-surface elevationsagain obtained. The difference between the elevlttion of the watersurface upstream from the piers and the elevation of the water surfucewith no piers in tbe channel is the bnckwater caused by the piers.The results obtained from the two methods of testing were identical.The second method of testing was slow nncl expensive, considerabletime being spent in raising the benNY pier models (weighing considerably over a ton) from the testing channel and lowering them back intothe channel to measure the backwater. In ordet· to conduct the testsby this method it wus llecessltry to build :t special device (plate 8, B)which would set the piers exactly in place in tbe channel. For thesereasons, in the great majority of tests as stated above, the piers werebuilt in the channel before the water wns allowed to enter.Ordinarily SL,{ tests were obtained for each head on the weir, twoin each class of flow as defined by Rehbock. Thus it was possible toobtain a comparison of Rehbock's formullt, considering his classes offlow, with D'Aubuisson's and Nngler's. For each quantity of flowone test was run in which the adjustuble weir WfiS lowered to its limit.For most discharges with this position of the adjustable weir, criticnlvelocity was obtained at the site of the pier. This condition, however,will seldom occur at bridge locations 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 theselllstruments did not change in elevation.

EFFECT OF SHAPE OF PIER ON COEFFICIENTr

The relative amount of obstruction a bridge pier offers to the flowof water may be expressed in the form of n. pier-shape coefficient.The coefficient depends upon the backwater' forlllula used. In theD'Aubuissoll, 'Veisbn.ch, aud Nagler formulas the pier-shnpe coefficient varies directly with the discharge. FOI' a f?iven nmount ofbackwater, depth of flow, and channel contraction, If the pier-shapecoefficient is increased 10 percent the discharge is increased 10 percent.

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)

SQUAUE NOSES AND TAILS; ON~; AND TWO l'lERS I

IACIIISS I llow I,: (,Inss ,2 lIow Class 3 llow rII A\'cr. vc· ---;----1----;---- age co. nge co·

c1l1cicm efficient';o'ormula 'I'est- 'i'ests ' 'I'esls I ror I ror ., Coem ) Coem coom· Cllk"'o5liver· . t' Il\·er· I t' 1I\'or' . cinsses 1 2

aged clcn aged 1 c en aged Clont ~l nnd 22 and:3 'I I' I

N,t/ll· --' NIt/ll' ,'--I N1I11I' ---",----- ber , ber ber

D'Auhuissoll-1(n'.-t ...... ~ ... ~ .. ~.~ ........ ~._ W, 1.018 iO O. nlo I 47 0.1152 0.988 O. UHf)

Naglcr-Ks................. _......... 99: .8il 70: .858 -Ii 1.113 .866 .\I~~J

Rehbock-~,.. __ .... _ ............ . (Ill j 4.7S 70 I 6.82 i,. __ .... (3) 5.62 1_...... .

Rehbock-Ku..... . _" ............... . 99 I 3. 11 j~.....~~~J H 6.08 1 3.92! -1.52.

SEl\IICIH(,U1~AH NOSES AND 'I'AILS; ONE AND '1'wo PIERS

l)'Aubuisson-Kf)',I .......... __ ... __ j al I. Oill ao I 1. 001 I 20 I. 0:\41 1.041 I I.fI:IS Nagler-Ks.. · ................. " .. ' :11 • !l34 30 • \J05 20 1. 2i8 .020 l. 027 :n 3.3.1 ao I 5.50 I.. · ....· (3) 'I. 40 :........n~~g~~t~-,;.~:::::::.:::::,::::.::::1 31 2.08 30, 4.0\) I 26 5. 3:1 1 3.07 \ 3. i5

CONVJ~X NOSES AND 'rAILS; ONE AND 'I'WO PU,RS

.D' Aubuigson-l\~o'.4 ~ ~ ~ ~ ... ___ .... ~ .. ~ _..... 31i 1.08!! I. 042 I. 074 1.06·\1.0.17 ! ao ao .940 .ual aO 1. 251 .9,\0 1.0:1-1 (3)ii~~~~~~~~;':::: ::: ::::::::: ::::::.::: 36 2. tiS 4,;16 ,_.......

Rchbock-KIl ..•,•...•.• '_ •.. ____ ..... :16 1.48 3.11 \ 30 5.34 Ug --T26' J,ENS·SHA1'I,D NOSES AND TAli,S; ONE AND TWO PlEHS

D'AubuissOll-!'f)·.L................. 2:1 1.0:13 ao 1. Oil; I. 044 I. 05.1 :Ii .952 2:1 .932 :10 1. :lSi .944 1.002a7 I! 1,0.51 ii~g~~~~~~;:.·:·:: ::~:::.:: ::::::::::: 37 3.55 23 4.79 (3)

Rehbock-Ku..................__ ..... 37 2.05 23 3.37 30 5.17 U~ "':i~43'

!JO. 'l'JUANGULAR NOSES AND 'rAILS Wl'l'llOU'l' BATTEH.; '1'wo PIERS

U·Aubuisson-Kf)·A ............. __ .. .. a7 1.1)65 t 45 j 0.985! 58 O.Wi 1. 021 1. 011 a7 I. 1·15 .885 .W3451.883: 58.887 \ (3)

nehbock-Ku............._••___.... .. 37 2.48 :~ ~: g} \.... ·58· 5. i2 3: ~~ '''4;40'~:~~~~,~~;:~:::::::::::::::::::::::: 37 a.54

00· TnIANGULAH NOSES AND TAILS WITll BATTER 1:12; TWO PIERS

D'Aubnisson-K,,'A •• ..._ ••_....___ __ 28 1.109 30 I.OOl'i 1 2:1 0.960 1. 055 1. 028 28 .006 30 .894 \ 23 .986 . \}()() .924~:~~~~~~~;.:::::::::::::::::::::::::: 28 2.64 30 4.W ........ . (3)

nehbock-Ku.._._......__....____._•. 28 1.89 30 3.00 i 23 5.70 ~: gg '--3~74' See footnotes at end of table.

_____

BRIDGE PIERS AS CHANNEL OBS'l'RUCTIONS 13

TABLE l.-Bridge-pier coefficients as determined for different shapes of piersContinued

SEMICIRCULAR NOSES AND LENS-SHAPED 'l'AILS; TWO _PIERS

Class I fiow Class 2 fiow Class a Ilo-w A ver- A ver'----;---1---:----1------ age co- age co,- eIDclent eIDclent

Formula Ta\~esrts_ CoeID- Tests Coem- Tests: Coem- Cor cl~es aver- . aver- - ciell' classes ,I

_____________.~~~~~__~:~_ agfeo 1__'_ Inno 2' a~;/:i I

Num- Num- NU7ll- : ber ber ber ID'Aubllisson-Kn'.< _________________ _ 32 1.152 36 1.018 28i 0.991 1. 081 1.045

32 .928 36 .901 28' 1. Of18 .913 .958 _______ J~~~~~~:-t:~==::==:::::::::::=::=::: 32 1.88 a6 4.67 (3) 3.36Rehbock-KR______ . __________ h _ 32 1.41 a6 a.69 28 5.53 2.62 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 _________________ ) 2511.032/ 28 1 0.949 21 0.929 0.988 O. Uil Nagler-K"'________________• ___•••• ___ 25 .936 (' 28 II .9IU 21 .979 .927 .942jRehbock-6o____ ._. __ ••_._•• __ ••• __ •• _ 251 3.80 28 8.21 (3) 6.13 _______ _ Rehbock-KR___ ••____._______________ 21i 2.22 28 j 1i.49 21 7.84 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 __________________ 38 0.943 :18 0.805 34 0.879 0.904 O. g06N agler-K.Y___ •• _____ • ________•_____ •• 38 .876 38 .801 34 .957 .869 .896Rehbock-80__________________________ 38 8.22 38 13.29 (3) 10.75Rehbock-K R _____ ____________________ ---7:8338 4.69 38 8.78 34 10.28 6.74

LENS-SHAPED NOSES AND SEMICIRCULAR TAILS; TWO PIERS

D'Aubuisson-Kn'A _________________ _ 36 1.162 38 1.042 34 1. 007Nsgler-KN _______________ -- _________ _ 1.100 II 1. Oil Rehbock-80__________________________ 2.95 ________ 36 .932 38 .912 34 1. OS1 .922 .972 Rehbock-K R _________________________ 30 1. 73 38 4.11 (3)

36 1. 31 38 :1.26 34 5.17 2.:11 3.21

TWIN-CYLINDER PIERS WITHOUT DIAPHRAGMS: 1 AND 2 PIERS

D'Aubulsson-Kn'A __________________ 79 0.991 69 0.957 40 0.985 O. 97~ 0.977N agler-KN___________________________ 79 .892 69 .S94 40 1.054 .89:1 .. 927Rehbock-.lo. ___________ • _____________ 79 6.13 69 7.26 (3) 0.66Rehbock-K R ____ _____________________ ---:j:iii"79 3.62 09 5.07 40 Ii. 70 4.30

TWIN-CYLINDER PIERS WITH DIAPR1tAGMS; 1 AND 2 PIERS

U'Aubllisson-Kn'A__________________ 65! 0.966 61 0.975 44 0.991 I O. 986 I 0, 987.906: .9416.41 '________ Rehbock-KR_________________________ 651 3.48 61 4.75 44 5.50 4.10 1 4.48 ~~g~~:~,-.-______=::==::=:::::==::::=:: ~ dl8i gl 0: g~5 _____~~_ 1(3~41 jl

SEMICIRCULAR NOSE AND SQUARE TAIL; 1 PIER

D'Aubulsson-KD' A _________________ _ 1. 002 !___ •___ _20 1.014 4 0.940 .938 ; _______ _20 .941 4 .923~:~~~~:~o:-____::=::::=::=:==::=:::::: 20 4.45 4 8.44 5.12 (Rehbock-K R ____ ____________________ _ 3.03 '_______ _20 2.5:1 4 5.52 , SQUARE NOSE AND SEMICIRCULAR TAIL; 1 PIER

D'Aubulsson-Kn' A __________________ 19 0.976 6 0.926 0.964Nagler-KN__________________________ _ 19 .912 6 .912 -------- --------) .912..._---- .. ------,.Rehbock-.lo____________________.- '"•• 19 6.30 o 9.20 ...... _.. _-- .. _------ i.OlRehbock-KR ____ • __ __ ••• __ "- _____ --__ _ 19 3.55 6 5.99 4.13-------- --------1

See footnotes at end oC table.

http:Rehbock-.lohttp:Rehbock-.lo

14 TECHNICAL BULLETIN 442, U. S. DEn'. OF AGRICUI,TURE

TABLf' 1.-Bridge-1Jier cpe.Uicients a8 determined Jor different .~hape8 oj 1Jier.~Continued

5:1° TRIANOUL,\ R NOSE AND SJ~MICIRCULAH 'l'AIL; 1 pmn

Class 1 flow CIIISS 2 flow Clnss 3 floll' Aver. I . " '·cr· nge co- t nge ,co.. emcicnt,cll1cl,~nt

Formuln (or I (orr~~ cocm· Tests Coem. Tests cocm· clllsses I clnsses IIged cient ~~:d cient ~~:d clent 1 and 21, 1. 2,

I and a I --- ---,---------------

Num· Num.\ Num· ber ber ber

D'Aubuisson-Kn'A. __.._____..____.. 19 1.024 71 0.978 1.012 L......Nngler-K.v..__.......__ ••_... __ ._•••• 19 .945 7: .950 .947'.._... ..Hehbock-clo__ • _..___...._______ .... __ 19 4.06 71 6.44Reh bock-K n___ ____ ••• _____..________ 19 2,29 4.217 f ~:~? I::::::::

90° TRIANOULAR NOSE AND SEMICIRCULAR TAIL; 1 PIER , , D·Aubuisson-Kn·A ______..________.. 2020 1•• 027 61 0,957 ..---.. - -"--"'1' 1.011Nagler-K........._______________ ....__ • 948 1 6 .933 ._...... __.. __ .. .1144 Hehbock-.lo..____________...... ___ ... 20 3.90 6 7.48 _.._______" ___' 4.72 Rehbock-KIt_________..______________ 20 II 2.22 6 4.82 ._..__.. ________ 2.82

1

00° THIANOULAR NOSE AND SEMICIRCULAR TAIL; 1 PIER

D'Aubulsson-Iln·A...... __ •___..__.. 7 I

7 0'98311-------- ________ 1.001m 1: 8!1~ j' .952 .... _____..____• .937 6.34 ____..______ .___ 5.18

Rebbock-KIl______ •__ .. __..__ ....__ __ if:r,~~:_.J;____::::==:::::=:::::::::::: 23 4.83 7

23 2.69 7 4.15 ____.... ___"___ 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__________________ 21 155 22 0.903 22 0.972 1.072 1.038N agler-K.v __" ______ •_____________ •__ .930 22 .887 22 1.030 .908 .949Rehbock-.lo____ •••____...__ ..________ 21 1. 121 1.80 22 5.29 (I) 3.58-....22·Rehbock-Kn_______________ •__.._____ --Tsi21 I 22 4. 12 5.83 2.771. 36

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. _________________ 34 1.020 43 0.951 36 0.959 0.982 0.975 34 .863 43 .865 36 .988 .864 .904if:~be~:__f.:::.:.__~::______:_-__:_-::::::::: 134 4. 51 43 6.36 (3) 5.54Rebbock-KR______ •____________..__.. 34 3.10 43 4.92 -----36·1 5.91 4.12 '-Toii'

SEMICIHCULAR NOSES AND TAILS WITH 1:24 BATTEH, CONNECTED BY WEB WITH QUAHTER·HOUND FILLETS AT TAIL; 2 PIERS

D'Aubuisson-ICf)·.' _. ___ •___________ .. 23 1.087 22 0.989 22 0.002 i 1.038 1. 014

Nagler-K.v..___________ ••______ •____ .! 23 .897 22 .880 22 .953 .392 .912Rehbock-to.•• ______.._..____________ 23 3.05 22 5.35 (3) 4.17·----22·Rehbock-KR_....______......._•• __ .. 23 2.17 22 4.15 5.56 ~.14 --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

I D'Aubulsson-Kn'A_ •• __• _______..___ 24 1.082 26 0. 977 ' 22 0.954 1.027 1. 005N agler-K.v ..___.._____________.._____ 24 .895 26 .922 .888 .898Rebbock-60.•• _. __ ._...._____.._____ _ 24 3.15 26 (')Rebbock-KR_..____.._____..__ ..__ .._

5: rt1 I_____:~. :Jg "'4:05'24 2:24 26 4.45 22 5.55 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.

http:Rehbock-.lohttp:Hehbock-.lo

a. 55 RR CC LL TT T-T RS SR PwR P6o·R Pgo·R 011.7

0 .908 .941 .954 .698 .907 .941 912 .945 .932 .948.~

L=4W 0 0 ~ ,J ~ 0 0 ~ ~ 0 23.3

L= 4W 0 00 0 0 0 ~ ~ ~ ~ 0 0 b h

.943 .943 .883 .904.864 t. 92~)(.861) .923 (.914) (.866) C90 5) P90' P90, P90'B P9O' B RL LR NC Ne l NCz

23.3 oN)I o 0 ~ ~ ~ ~ ~ ~ I I I ! X ~

.887 .906 .928 .932 .863 .897 .895 .930

35 L-2W 0 0

.867 .986

50 L=I.7W D0

.B88 1.108

11.7 101 Con frae fion of ehanne I in percent._________________ r::t.L- 4W Widfh 0 f pier____________ _____________________________.W .936 To fa I leng fh 0 f' pier._••_______ . _______________________ L

Raiio of lengfh fo widfh applies only fo 55 and RR11.7 201} L- 4W S quare___________ •_______ •.•5

.B76 Round (semicircular}••_••_R Con vex_._.__________________.C Cylindrical. ___• __•____ • __ ....T

23.3 Lells- shaped....._. _____.....L

Poin fe ri __......_...... _____..cPL=7W Recessed...._._••_._, ____ ._.N. C . ~ ~ ~ ~

. B55 .916

23.3

L=13W

.843 ,901

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

TIRIDGE PllmS AS CHANN I..:IJ OBS'I'RUC'L'IONS 15

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' conycnience in JUaking comparisons the coeHieicllts for the N nglcr formulu,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 squn,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 Nnglcr 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 Nngler 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 ngninstLin' 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:

] i' . 'd··1 fI(v =0.873-0.0023L/lV01 squaro nOSe.111 squnre till _.. - - - - - -[KD',l = 1.05 -O.0052LI1V

] ~ ., I I" 1 t 'I{J(N =0.04--0.003L/W.i OJ' semwu'cu ur nose n.n( seml('Il'(,U nrtu 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 Clnss 2 !low Clnss 3 tlow A \'cr. AYcr· i__.. ,~._,.., -¥-_~ ,____ _ __ n~e CQ- ul10 .co- .

Picr length Formnln I i .etlicient clllC1ent (fcel) I 'rests , . '1'('51S I 'I'csls I for for

! Il\'er. (9c1h• i nycr· I C'9c!fi· u\·cr.: C'~ctli. clusses classes

ll~cd CICUt. tl~ed! Clcnt ngcd j ClcnL :1\ nud 21 1, 23, ~ud 1_____ 1 _.J

I

-. ,--.--------Xllmber l :,Y/l1IIbcr'l lSI/mba !

,{D'AUhUiS50n-IlIl'A ..•\ lil' I.OW I 64 0.950 I n O.Of,2 \ 0.994 0.084 40- ! r-inglcr-/{.,'••....•... , MI ,MI i 64 .8M .l7 1.113: .850 .921

• 'f •. , .. ·····1 H~hbo~~-Ou; •......: ~I -I.:Ji I (H 6.46 •..••. (') 1 5. ~'9 ....... .

Hehhock-llll......... Hl~. 9i tl-J, 4.58 4i· O. tl8 : 3.81 4.51

1 J)'Anbuisson-KIl'''': 28 1.003 ~'9 l .935 23' .026·, . 90S .956 01- INn~lcr-Il,,·... •. " 2S .855 2'J I .S5G: ZJ .9il .855 .S89 ~. , ........,{ Hehbock-oo__ ....... '.' 28 4.02' 2\l 0.S3 ('l 5.89 ....... .I'"

Rchbock-j(II........ 28 3.39' 2\l I 5.20 2:l 0.02 I 4.34 5.00 D',\ubliisson-Illl'.• ,.: 30 I .08.1 i yO .Ill~ :IO,.?~ .950 . ~4~

1'1- ......1 Ntlgll'r-~(.\"_'_ . "'j yO .843 , 30 I .S·ll \ 30 ,,),_, .845. ,58, H. , , Hchbock-c5o .. __ ~~ H~ 30 ?52 f :~O 7.~2;. _ (2~ l 6.3~ L~ .. _..;.: ..{

Rehbock-KII .••., •. / 30 .1. 70 j .10 5.05 i ;m· o. SO ; 4. Ub i 5.31 t Average ror 1111 tests in the 2 or 3 classes, not fivefhge of th~ determinations for separ,lle classes shown

in precedIng' columns. '0. wus nOL computed ror class 3 !low.

--------

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

----,' -~---------- -I--Glass 1 flow Class 2 flow Class 3 flow Aver· IAver·

age co. age .co· Pier length ellclent(llC,entFormula(feet) Tests Tests Tests for CorCoello Cacm· aver. coem· classes 1 classes

aged clent aged clent aged clent 1 and 2 11, 23nndaver· aver·

-------,].r~1f.ber Number Number

{D'Aubulsson-Kn'A_. 16 1.150 23 1.014 26 1.034 1.008 1.054 Nagler-KN.•••••••••• 16 .927 23 .8H6 26 1.278 .908 1.11584.67••••.••••• Rehbock-.lo••••••••••• 16 2.03 23 4.84 (2) 3.73·····20· ···3~89·Rehbock-KR••••••••• 16 1.62 23 3.81 6.33 2.91

{D'Aubulsson-Kn'A •• 26 1.128 29 1.021 I 23 .980 1.072 1.045 Nagler-KN••••••• •••• 26 .916 29 .904 23 1.011 .910 .9398.17•••.•••••. Rehbock-.lo••••••••••• 26 2.28 29 4.58 (2) 3.49

··-3~50·Rehbock-KR._••_•••• 26 1.67 29 3.62 '·'·'23· 5.H 2.69 {D'Aubulsson-KD'A __ 34 1.096 37 .996 31 .975 1.044 1.023

Nagler-KN••• _•• __ ••• 34 .901 37 .890 31 1.036 .895 .0'3815.17•••••••_. Rehbock-.lo •••••• __ ••_ 34 2.90 37 5.19 (I) 4.09 Rehbock-KR••• •••_. 34 2.07 37 4.07 31 5.74 3.11 3.91

100 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, increasing 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

http:1.27+0.12http:3.10+0.12http:1.48+0.11http:3.94+0.12http:Rehbock-.lohttp:Rehbock-.lohttp:Rehbock-.lo

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BRIDGE PIERS AS CHANNEL OBSTRUCTIONS 17

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 formulas 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 A ver- _-\ verage age

, coeffi- coeffiChannel contrac cient cientFormulation ratio ~:e~ ~~ffit- ~;~~ I ~\~~ IC~effi- cl~,es cl:es

I C'!JCffi

aged aged C1ent aged I C1ent J and I. 2. . I 2' and 3'

I--------~I-,-~·-u-m-- --I'Num-I-- :;::1---~. --.. { ber ber 1 ber f

D'AUbUisson-KD'A_f 18 0.909 610.!103 .,, ____ ,-_ .... _j 0.952 " ..•• _

O·~~~7~~er.ier 1.17 by {i:~~~~~~;====::::1 !i (~! g11~:[f ::::::{:::::; ~:i I::::::: D'AUbUlsson-KIi"'_ 81 1.029 I 64 .050 47! .0.12· .004: .98·1

0.233 (2 piers 1.17 by Nugler-:K.,.__________ SI .864 64.854 47 ' I. 113 .8.19 . O~I {'1.67Ieet). Rehbock-clo_________ 81 4.37 I 64 0.4fj ------- (;) 5.29_._...

Rehbock-KR________ 81 2.97 IH 4.88 47 6.68 '3.81 I 4.51 D'Aubuisson-KIi' A _ 28 .009 29.918 26.9.18.958. .1I.i7

0.35 (l pier 3.5 by 7.0 Nagler-Ks__________ 28 .867 f 29 .847 20 1, I. 002 .8';7 i .1121 feet). Rebbock-clo_________ 28 4.79 I 29 10.36 _______ i ('J ! 5.•i9 1_____ __

Rehbock-Ku________ 28 4.26 i ~'91' 6.01 20 i 0.62 5.15: 5.liI 0.5 (1 pier 5.0 by 8,5

{

~~·il~~~~~~=_!.{!:_A_: ~ :lli I ~~ J~g ~ I l: ~f~ '. :~ J : e~b feet). {Rehbock-clo_________ 2:1 5.42 2717.52 _______ 1 (') ; 6.55 ______ _

Rehbock-KR________ 23 6.79 I 27 9.81 251 7.76 18. 42 I 8.20 PIEHS WITH SEMICIRCULAR NOSES AND TAILS

D'Aubuisson-KIJ' A _ 16 I. 012 ~ \. 9~ !___ ....I. _____ _.006 n.~J7_(1 pi or 1.l7 by Nagler-~s---------- 16 .941 ,.93/ ---_---:- .. ---- .040

1.6, feet). , Rellboek 00 __ ...____ 16 4. S7 7 7.06 1_______ !_____ __ 5..51l{

{3.35g~X~~~;;;;~~'::KI/.._: :~ U''k ]; tgi41'----2ii-h~ii34- 1.008 --j:iiS4

O.~3 (2 piers 1.17 by Nagler-:K,\'____._____ 1.1 .927 23. 8U6 26 I 1.278 . !lOS 1.0.18 1.67 feet). Rehhock-clo_________ IS 2.m 23 4.84 ______ ., ('l 3.73

2.91 a.sog~'~~~~;;;;fn'::I~;'~._: ~g l: ~ ~~ t ~!I I ~ IU~fl 1.146 1.112 0.35 (l pier 3.5 hy 7.0 Nagler-:[(,v__________ :lO .986 32.936 28 ! l. ISS .U61 1.022

feef,). I Rehbock-clo__________ 30 I. 73 32 I a.63 ' _______ , ('l 2. i~{I Rehbock-Ku...... __ :lO I. 81 32 3. f>4 I 28 S.39 2. i5 3.57 ,{D'AUbUiSSOn-[{IJ'" _ 24 I. 312 24 . !/91 I 24 11.118 1.152 1.140

0.50 (J pier 5.0 by 8••1 I' Nagler-K,\'__________ 24 I. lOS 24. U30 2·1 , I. 317 l. 022 1.120 feet). Hehbock-clo_________ 24 1.65 24 4.94 (ll 3.30"-----1IHehbock-Ku____ .___ 24 2.81 24 6.85 24 6.37 4.83 --5. :i.'i

---~--~----~--~--~----~--~--~---

TWIN-CYLINDER PIERS WITHOUT DIAPHRAGM . - .-----.. '-."

{ J).Aubuisson-~~__::__;:I·;;;--- ~1~ 1 0. 929 4 0.9431 O. 946 O. \WI 0.117 (1 pier 1.17 by Nagler-Ks.-________ 51 .898 32.915 4.963.904 .!lO7 4.67 Ceet). Rehbock-clo.._______ 51 7.44 32 i 9.28 _______ ('J 8.15 __ ----. Rehboek-KR________ 51 4.17 32 ! 6.0.1 4 6.47 .4.89 4.96 D'AUbUisson-KIJ'A_ 28 I. 055 37 I .982 36 .9!10 I' I. 013 1.0050.233 (2 piers 1.17 by Nagler-K.,·____ ._____ 28.883 3-, ! 5.. 85715 36 1.060 .879 .943

4.67 Ceet). {Rehbock-clo_________ 28 3.74 37 _____._ (') 4.75 Rehbock-Kn________, 28 2.61 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

TECHNICAL BULLETIN 442, U. S. DEPT. 01

BRIDGE PIERS AS CHANNEL OBSTRUCTIONS 19

1501 decreased with an increase in channel contraction up to a contraction 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 contraction. 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 coefficients for both class 1 and class 2 flows.

i Piers with semicircular noses and tails•..o \ Piers with square nOses and tails ____.C~

~

~ 1.3C~----~~~---+-------+------~----~

~

O.BO' -----•....,J~----_.",J,:------~-----....,.J:,.----~ I0" 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 semicircular nose and semicircular tail placed at two angles with the current, namely, 100 and 200. 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 comparison 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 coefficient for the pier placed at 11 200 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 200 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 200 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

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BRIDGE PIERS AS CHANNEL OBSTRUCTIONS 21

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 « u

e 1Il W ..J

S! II ~ e « ~ II w-t lJ.. 0

a Z N « ~ e ~ II IIIa Z N e :i IIN

e a:

X II II II II

.8 a; ~ ~ S\ II II ~ ;;

Q) '0 II II

Z .8'".c II ~ c(~I'CJ " II IIo ...J~

a= g D.f .-J o .= .~II II '" II II --'

II II II II

Eid ~ q II II /I II II II

~ 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':?a-7, 560 - .. ,).. 17 "/? 0 t:.

W = 'a~:{! = 19~~= 0.0284

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.

http:J(n�A=O.97http:J(N=0.95

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 W1=200 feet W2=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'

Cal= V3

2/2g = 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'Aubuisson formulas as before but changing Rehhock's to 5.60 (fig. 6) we getfor main stream the following backwater:

By the Nagler formula, KN =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 W2=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, KN =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.

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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 the discharge tlll'ough the opening.

The yulues of the known fnctors tlrc ns follows:

W1=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 discordant 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 constant 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 backwater 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 contractions 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

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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 increases 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., illus. 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, TilEO-HBT[CAL, AXU I'HACT[CAL. 1752 pp., ilIus. London.

(4) DBIIAUVB, A. A. 1878. ~IASL'EI, DE L"xGES1Et:H DEl'. I'OX,!'S E'r CIIAUSS}}ES .•• [1] \'. Paris.

(5) 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. 2 Y., iIlus. Paris.

(6) EYTELWE1X, ,r. A.

1801. HANDBUCH DElllll[';CIIANIK USD DJo}(t IIYDIlAUL1K.

(7) KOCII, A. H126. 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. (191920) 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, illus. (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 BHi'CK~}NS'I'AUI;S In} I REIN

S'l'ulhJgNDEM WASS~}HIlUHCIII·'LUSS. BlLuingcnicul' 2: 603-609, illus.

(141 1926. DAS ~'I,USSflAUI,AflOHA1'OItlUM DEH TECIIXISClIgX 1I0CIISCIIULg IN

KARLSUUII~J. Ch. VI, lJn; WASSEllllAULAlIOHATOmgX gUnOPAs, pp. 106-115, iIIus. Bcrlin.

(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., illus.

ANNOTATED REFERENCES RELATING TO BRIDGE PIERS AS CHANNEL OBSTRUCTIONS

AKADE~IIS(,lIgX \'gHEIN H l'TTg, Bcrlin. 1925-28. "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., illus.

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., illus. 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., illus. 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."

27 BRIDGE PIERS AS· CHANNEL OBSTRUCTIONS

.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, illus, 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. DurandClaye. 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 mathematical solution which showed the faces of the pier nose should be convex. Describes in some detail Cresy's experiments 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. HYDRAULICB AND ITS APPLICATIONS. Ed. 3, 801 pp., illus. New

York.

http:90-degr.ee

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, contmcting the channel n little more than 11 perccnt. The barrels 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., illus. New York. HOUK, 1. E.

1918. CALCULATION OF FLOW IN OPEN CHANNELS. Miami ConservancyDist.· 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 contracts the cross-section of the moving stream, and such a contraction 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 essential 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 respectively 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