'.

,

(

II"I~~

l

t

!

1.1

i

, I

~ww ~

W ~ IW

... Ii£

122.

..

'J

;. ... ~ 11 2.0 .......

~



~w W

w

...

W

wW w

I&i

~

.........

:~

II~:!!

1 I I

·1I11'~"11I1.4 ~1111.6

i !

I

~III~ 111111.25 111111.4 111111.6

;

'. ~

MICROCOPY RESOLUTION TEST CHART

MICROCOPY RESOLUTION TEST CHART

NATIONAL BUREAU OF STANDARDS·I963-A

NATIONAL BUREAU Of STANDARDS·I963-A

..

,"

..;

.

TECHNICAL BULLETIN

No. 442

N OVEMBl1.1: 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. 01BUCH 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.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

..

....... ..0

c: ~

Ci

e

;;

g

Cl

::s ::s .r:

'" " -

0

~

::s ;;1~ d ~~aJ CIlVlO"lU ~.e ~-; c ~.!::!

~\

CC~

0

,

-:;

\~ 1\ ~,

.13' q'o

~

.2

.".,

4­

..!

cS

,~

\\ ~ '\ '\

.~

~

~

o
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





Q _

Q=

- 23.3 twin

!

1.6

1.2 f-·

,;

I ,

~o~ o:~~~ f

o

ni 0

.:

!

1.81-- I

[r

\

fr

1

-I·--



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(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

r-t( !

a=2~3

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

)~

--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