Backwater of arch bridges under free and submerged conditions.pdf

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Backwater of arch bridges under free and submerged

conditionsJ.P. Martin-Vide

a& J.M. Prio

a

aTechnical University of Catalonia , Jordi Girona 1-3, 08034, Barcelona, Spain

Published online: 02 Feb 2010.

To cite this article: J.P. Martin-Vide & J.M. Prio (2005) Backwater of arch bridges under free and submerged conditions,Journal of Hydraulic Research, 43:5, 515-521, DOI: 10.1080/00221680509500149

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Journal of Hydraulic Research Vol. 43, No. 5 (2005), pp. 515-521 2005 International Association of Hydraulic Engineering and Research

Backwater of arch bridges under free and submerged conditionsRemous de pont a voute dnoy et submergeJ .P . MARTIN-VIDE, Technical University of Catalonia, Jordi Girona 1-3, 08034 Barcelona, Spain. Tel: +34 93 401 64 76;fax: +34 93 401 73 57; e-mail: videgrahi.upc.edu (author for correspondence)J.M. PRIO, Technical U niversity of Catalonia, Jordi Girona 1-3, 08034 Barcelona, SpainABSTRACTExperimental research on backwater effects in semicircular arch bridges is reported. Both pressurized and free-surface flows at the bridge wereinvestigated. Flows on a mobile bed in clear-water conditions were compared to those with a rigid bed. Methodologies for backwater computationby Yarnell, the US Geological Survey, the US Department of Transportation and the HEC-RAS model were compared with the experimental data.A simple expression for the head loss coefficient as a function of the obstructed bridge area is derived.RSUMLa recherche exprimentale sur des effets de remous avec des ponts a voute semi-circulaire est dcrite. Des coulements en charge et a surface libreau droit du pont ont t tudis. Des coulements sur un lit mobile en eau claire ont t compares a ceux sur un lit fixe. Des methodes de calcul deremous par Yarnell, Ie "US Geological Survey", Ie "US Department of Transportation" et Ie modle de HEC-RAS ont t compares aux donnesexprimentales. Une expression sim ple pour le coefficient de perte de charge en fonction du secteur obstru de pont en est d duite.Keywords: Backwater, bridge, afflux, arch, scour.1 Introduction and object ivesThis research was motivated by the numerous examples of o ldbr idges in Europe, dating back to Roman and medieval t imes,many of them being still in use at important river crossings.The objective was to provide data on their backwater effects.Since much has already been done on the subject of bridgebackwater, a second objective was the assessment of differenttechniques for backwater computation with respect to a new setof laboratory data.

2 Descr ipt ion of experimen tal set-upThe flume used for the experiments was horizontal, 6.0 m long,1.5 m wide and 0.5 m high. A recess 0.5 m deep and 2.7 m longwas located halfway across the flume. The bridge was placed inthe recess, supported laterally by a mechanical device able to lif tit and keep it at various elevations (Fig. 1). The recess was filledwith loose material for mobile-bed tests or covered for rigid-bedtests . The complete br idge d imensions, inspired by Europeanexamples of ancient br idges, are presented in Fig . 1 . The br idgevaults were cut f rom PV C pipes , which determ ined the arch d iam eter, so that four equal sections, including one span and two half

piers each, fitted the flume width. The discharge was measuredby m eans of a th in-plate V-notch weir . At the downstream end ofthe flume, the tailwater was controlled with a thin-plate weir.

Table 1 shows the exper imental program in terms of d ischarge(Q), tai lwater depth (y) , bed condit ion , and Froude number (Fr) .The series A test served as a reference in the sense that the otherser ies were "dev iations " to h igher y (ser ies B) , lower y (series C),lower Q and y (D), and finally a movable bed (E), for whicha fairly uniform natural sand of d5a = 0.86 mm and s tandarddeviation {d^/d^Y12 = 1.34 was used. This sand proved tobe on the threshold of movement, so that there was no generalscour but only contraction and local scour in the mobile-bed tests.The f low was always subcr i t ical .

The elevation of the br idge s tructure was reduced by 1 cm ineach test of series A-D, from free surface flow at the beginning,involving only the bridge piers, to submerged flow at the end.In Fig. 2 the first and the last tests are represented. The ratio ofpier widths to the channel width , cal led the obstruction rat io m,amounted to i ts minimum of 0 .324 at the beginning. In the las ttest the elevation of the lowest arch point (abutment) was closeto the flume bottom (s > 0, see Fig. 2). Pressurized flow meansthat both the upstream and downstream faces are submerged.Serie s E containe d 11 tests at eleva tions 2 cm ap art. Watersurface elevations were m easured with a poin t gauge of 0 .1 mmRevision received March 27, 2004/Open for discussion until August 31 , 2006.

515
http://grahi.upc.edu/http://grahi.upc.edu/http://grahi.upc.edu/


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516 Martin-Vide and Pril i f t i n g s u p p o r t

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S5 _ _ ^ _ 2 5 ^ - 5 5

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Figure 1 Bridge dimensions in the experiments.Table l Experimental program

SeriesABCDE

Q (l/s)124.3124.3124.370124.3

v (cm)25.231.519.821.525.2

Be dRigidRigidRigidRigidMobile

F r ( - )0.2090.1490.3000.1490.209

First Test

Cross Sect ion Front View

Last Test"""y

^ ^ ^ ^ ^^ ^ ^ ^< - Q

y |1

X Cross Sect ion F r o n t V ie w 'Figure 2 Sketch of the experimental program (see Table 1).

accuracy along the center l ine of the second arch f rom the r ight( looking downstream). The scour around the second pier f romthe r ight was measured dur ing the exper iments of ser ies E to0.5 cm. Each scour tes t las ted 4 h . Although th is was too shor t

to develop equilibrium scour, this duration was warranted for thecompar ison between tes ts . The f inal bed topography was alsorecorded.

3 Theoret ical framew ork for backw aterThe s im plest theoretical approach to br idge backw ater in subcri t-ical f low is illustrate d in Fig. 3. Impo rtan t sectio ns are: 1 flowaccelerat ion , 2 and 3 r ight at the upstream and downstreambridge faces (both plane and vertical) and 4 tailwater with conditions uninflu enced by the bridg e structure . Sectio ns 1 and 4are chosen at distances equal to one span width (B) and fourspan w idth s, respectively, followin g the traditiona l 1 : 1 contract ion and 4 : 1 expansion rat ios . All method ologies to be revieweduse Bernoull i ' s equation to compute backwater elevation . For a

S i d e V i e wFigure 3 Control volume and notation for the theoretical framework.
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Backwater of arch bridges under free and submerged conditions 517horizontal bottom it can be written as

< 2 V2 + A HMv\ vf V1yi+ai~= yi + ,-^- + Ai_,- + X-,2g (1)2g 2gwhere i 2, 3, 4 is the number of the cross-section downstream. AH denotes the friction loss and the local head lossacross the bridge is expressed as a coefficient X times a kineticenergy head (V 2/2g), which includes the corresponding Corio-lis coefficient a. The backwater elevation difference is definedas Ay = y\ y4, i.e. the difference with and without the presence of the bridge, since the tailwater depth is assumed to holdeverywhere if there isno bridge (dotted line, Fig. 3). For mobile-bed tests with bed scour, the backwater is the difference in thefree-surface elevations.The empirical equation of Yarnell (1934) can be viewed as if allterms inEq. (1), except for the backwater Ay and the local headloss, were dropped and aggregated in the em pirical coefficient X.

The local head loss is expressed with V = V4.lt then readsX Av = 2k\k+ 10v\llg 0.6 (m 15m 4) (2 )v%/2g \ y4

The right-hand side of Eq. (2) depends on the pier shape expressedwith coefficient k, the velocity and flow depth atsection 4 and theobstruction ratio (m) defined as the ratio ofpier width to channelwidth. A value k 1.25 was found for piers with a square nosegeometry and tail features as in the present experiments.The approaches of Kindsvater et al . (1953), Kindsvater andCarter (1955), and Tracy and Carter (1955) consist in applyingEq. (1) up to section 3 and using V = V3 in the local head lossterm. A discharge coefficient c is worked o ut from Eq . (1) to yield (3)A 3 y 2g (y, - y3 - AHx_3 + ax vj/2g)where c = [a3 + X] ~1/2. Because 03 > 1, and X is positive,c < 1. The authors plotted c versus thedegree of obstruction m.In the approach by Bradley (1978), Eq. (1) is applied up tosection 4. However, the local head loss is computed with thevelocity in section 2 as if y2 was equal to V4. Although not true,this assumption avoids the computation of y2. The expressionsfor the backwater and the local head loss coefficient are

Ay = X^-Av-

a i 2gX 0LXA\

2g1 1 (4)

vz2/2g v-4where, by comparison with Eq. (1), friction losses are neglectedand C4 = aj. Charts were provided with values of X/a2 as afunction of the degree of obstruction m.The HEC-RAS model (US Army Corps of Engineers, 1997)uses Eq. (1) between sections 1, 2, 3 and 4, with the local headloss at every step of the calculation expressed as,,2 (5 )V

2X = X2g

2v- vCti - , - + ]2g 2gwhere i 1, 2, 3. The values recommended for X are 0.3 forcontraction and 0.5 for expansion, even though values of 0.6and 0.8 are allowed for abrupt transitions. This equation is used

throughout the computation either in free-surface (called "lowflow") or pressurized flow ("high flow").4 Experim ental resultsThe backwater, measured as the free-surface elevation difference25 cm upstream of the bridge face and 100 cm dow nstream of it,is plotted in Fig. 4 against the bridge elevation 5. The backwater increases as the bridge is lowered except for the mobile bedtests (E). These are based on the final profile after scour, resultingin practically no backwater increase. For each series, free flowsare shown to the left of a vertical line and submerged flows tothe right. These lines are drawn through the first test in pressurized flow in each series. Thevertical lines fall apart becausethe submergence of the bridge is very sensitive to the tailwatercondition.

A dimensionless backwater (y\ yA)l(v\/2g) is plottedagainst the obstruction ratio m in Fig. 5. The ratio m is now thebridge face area over the channel area, both computed with thedownstream free-surface elevation (and its rigid bed elevationin mobile-bed tests). The graph relating m and bridge elevation s is added. As expected, the experimental data collapse intoa narrower band than in Fig. 4, except for the mobile bed tests.The borders b etween the free-surface and pressurized flows comecloser.5 Comparison between experiments and predictionsFigure 6 shows the comparison with Yarnell's equation. The left-hand side in Eq. (2) is divided by the shape and flow factors,leaving the function of m on the right-hand side. In this way, theequation for all tests is a single plot. The experimental data arehandled similarly, byusing the measurements at sections 1 and 4.The change from free-surface to pressurized flows is indicated bya vertical band.

A E C and D

Ao BA Cx D^ E

80 9 v25 21

3

13s [cm]Figure 4 Experimental results of backwater versus abutment elevation.
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10

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A

20 s [cr

PRESSURIZFLOW X yA / O* /

ED ^/ OX

X

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XX

Scour

Ao BA Cx D

- * - E*

0.3 0.4 0.5 0.6Figure 5 Dimensionless experim ental data. On the left-hand cornerm s functions.

Yarnell (1934) considered cylindr ical br idge p iers , with a max imum obstruction rat io m = 0 .50. Figure 6 suppor ts an extensionto higher values of m, to arch bridges and to submerged flow.However , Yarnell ' s equation overpredicts the results , special lyfor low m values. The reason may be the fact that the pointswhere Eq. (2) should be applied are not clear in the original work(1934). The difference in the pier length to width ratio (4 inYarnell, 3 in present work) is considered minor. Finally, Yarnelldid not consider any bed scour, resulting in a larger discrepancyfor the mobile-bed (E) than for the other tests.

The compar ison with the second approach is shown in Fig . 7 .Two c m curves for ratios of bridge length over relative spansL/b equal to 1.0 and 1.5 (close to L jb 1.44 in the exper im ents) ,

4 .5

3 . 5 -

tEi n 2 . 5 -+Eil>> 1.5-

0 . 5 -

- 0 . 5 -

F R E ES U R F A C EF L O W

* ? o ** x o x

A *XX X

Y a rn e l l (1 9 3 4 ) Ao BA Cx Dx E

P R E S S U R I Z E D - ' F L O W . - ' ?.* o A

X Q AAf O * X

X

x .'

X

X

0. 3 0.4 0.5 0. 7.6m[- ]

Figure 6 Comparison between experimental results and Yarnell'sequation (Yarnell, 1934).

are p lo tted (Matthai , 1967) . These curves for ver t ical embankment and abutment, Fr = 0 .5 and no br idge submergence arecorrec ted for the actual Fr (in A series) and for both Fr and bridgesubm ergence by two addit ional curves . The las t correction m akesthe curve bend downwards at the onset of pressurized flow withinthe ver t ical band. The exper im ental data are in troduced in Eq. (3) ,using the mea sure me nts at sections 1 and 3, the friction losses( A # ! _ 3 ) computed fo llowing Chow (1959) and a = 1.082. ThisCor io lis coeff icient was computed f rom velocity measurementstaken with a current me ter 2 .0 m upstream of the br idge.This methodology overpredicts the d ischarge coeff icient c ,i.e. the local head loss coefficient X is underest imated , and sois the to tal backwater . The or ig inal curves were drawn to f i texper imen ta l c values with in the range 0 .70-0 .95 , very closeto the range here, but for greater obstructions. The correctionfor bridge submergence fails to fit the experimental data of thepresent work. The trend of the data does not depart from thecurve for unsubmerged condit ions. Hamill (1999) suggested th issubmergence factor to be one when both br idge faces are submerged. The flow area A3 can include the scoured area in themobile-bed tests. If not, the values of c for the E series turnout to be much larger than 1, which is absurd ("E" pointsin Fig. 7). On the contrary, with the scoured area ("E modified") the experimental results do not behave different from theother tes ts .Figure 8 compares the data with Bradley ' s (1978) approach.H is X m curve is for wingwall abutments at 90 to the bridgespan. Both a 2 and a\ are set to 1. Measurements at sections 1and 4 are used in Eq. (4). A one-quarter section of the bridgeis considered so that no pier effects are taken into account. ThisX m curve covers so me f ield data, g iv ing h igher X than in a previous curve based mainly on laboratory data. The field bridges

Figure 7 Com parison between experimental results and Kindsvateretal. (1953).


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Backwater of arch bridges under free and submerged conditions 5192.0

1.5

1.0

0.5

0.0

X

< AO Xx xo a Ax o o

PRESSURIZEDFLOW

" j

a AO CX

Ao BA Cx Dx EBradleyGraph

FREE'* SURFACEX

x 0 \ X0.3 0.4 0. 5 0.8.6 0.7

1-m [-]Figure 8 Comparison between experimental results and Bradley(1978).had width to depth ratios up to 700 and rough vegetated flood-plains, whe reas th is ratio is close to 1 here. By adding the scouredarea to the flow area A2 in Eq. (4), the points for the E seriesfall far from the rest and close to the curve. Scour probablydeveloped in the prototype bridges used to derive the graph inFig. 8.

Pressurized flow is treated separately by Bradley (1978) byneglecting upstream velocity head and friction losses in Eq. (3).A constant value c = 0 .80 was proposed. Figu re 7 shows thatthis value does not keep track with the decreasing trend of cas m increases . For pressur ized f low, Naudascher and Medlarz(1983) showed that the coefficient X should be propor tional tothe obstruction ratio m. Assuming X = m, then c = X/s/m + a.This is plotted in Fig. 7 with a = 1.082 to result in a reasonablefit with the new data.

6 Compar i son be twe e n e xpe r ime nts and H EC-R ASBy letting X vary in Eq. (5) the exper imental and numericalbackwaters Av are compared based on the least-square best-fitfor A.. Because the exper im ents involved d if ferent obstructions, Xis assumed to depend on the ratio m. This is justified by theresults in Fig. 5 and also is a common feature in the previous methods. Biery and Delleur (1962) found that the actualarch geometry did not influence the backwater significantly, provided that m is the same. Values of X = 0.1 and 0.3, for gentlecontraction and expansion, give a reasonable agreement for thepier case (m = 0.324). Then, the function X(m) chosen forbest-fit is 0.1C for contraction and 0.3C for expansion, whereC = 1 + k(m 0.324) and k = 5.75 is the best-fit constant forall r ig id-bed exper im ents .

Therefore, the total local head loss coefficient (i.e. the sumof the contraction and expansion) is X = 2.30 m 0.345, for0.324 < m < 0.65, i.e. from 0.40 for the pier case up to 1.15

for pressurized flow. This equation should not be extrapolatedto severe contractions (m -* 1) or very gentle contractions(m 0) . Applying the analysis of Nau dasch er and M edlarz(1983) for pressurized flow to the arch bridge geometry, a goodapproximation is X = mC^, where d is the drag coefficient ofthe br idge. Combining the two expressions for X, it follows thatC d = 2 .30 - 0 .34 5 /m .

7 Discussion of the effect of scour on backw aterThe backwater measured in mobile bed was almost constant ,around 0 .6-0 .7 cm, ir respective of br idge elevation (Fig . 4) .Figure 9(a) shows a d irect compar ison between r ig id-bed (A)

0. 3( b )

0.4 0.5m[- ] 0.6Figure 9 Experimen tal results for series A and E (up), (down): plot ofscour versus obstruction.


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520 Martin-Vide and Pri10

ra>CD"Oa >C O =13 cm

s=15 cm< s=19 cm s=21 cmi s=23 cm o s=25cm

Distance [cm]Figure 10 Final bed profiles in the mob ile bed experime nts.

and mobile-bed (E) tes ts for the same hydraulic condit ions. Form > 0.40 the backwa ter in the r ig id-bed tes ts exceeded that of themobile bed. This is the expected ef fect bec ause scour reduce s thevelocity difference between the approaching f low and the flowunder the br idge. Surpr is ingly enough, the opposite occurs w henm < 0.40.

A scoured volume is compu ted by using the scour depths atfour points. It is then d iv ided by the f low volume (computeddowns t r eam) to give a dimensionless scour as the ordinate ofFig . 9(b) , where it is plotted versus m. A 1:1 slope line is added.It can be seen that scour increases quite linearly with m. In otherwords, after 4 h the flow has scoured roughly as much volume asvolume lost by the flow due to the br idge. Therefore, the meanf low velocity should remain near ly constant through the br idgeand the head loss should be prod uce d by the cha nge in the shape ofthe flow area only, which narrows and deepens. The fact that thehead loss is almost constant , i r respective of the br idge elevation ,is thought to be caused m ainly by th is "shap e resis tance" . The bedprofiles through the scoured area are similar in shape , as shownin Fig. 10.

Bradley (1978) recommended the use of a factor to decreasethe backwater when scour is present . This factor is in the range0.50-0.75 for the present scoured area over flow area. Our resultsdo not suppor t th is correction because the decrease showed inFig. 9(a) is much h igher for large m. In addit ion , the agreementfound between the Bradley curve and the mobile-bed exper im ents(Fig. 8) would d isappear if th is correction were applied .

2. Scour reduces the bac kwater for h igh obstruction rat ios . Thescour ad ds as muc h flow area as is lost due to the bridge itself.How ever, du e to the shape of the scour hole, there is an almo stconstant backwater throughout the tests, irrespective of m.This conclusion refers to clear-water scour only .3. Since most actual br idges are neither in rigid bed nor underlong-duration clear -water f low, it is difficult to extend theresults to proto types. The method by Bradley (1978), basedon live-bed field data, results in much larger backwater thanour exper imen ts , but the agreement improves for clear -watermobile-bed tes ts . His scour factor is not suppor ted by thisresearch. Yarnell 's equation, based on laboratory data, givesa larger backwate r too but i t t races well the trend of backwaterin pressurized flow. The method of the US Geological Surveygives a lower backwater , is suitable for both free-surface andpressure flow and the effect of the scour can be includedsuccessfully. His submergence factor is not supported by this

research .

NotationA = flow areab bridge span (also B)c = discharge coefficient in the approach by Kindsvater etal.

Ci = drag coefficientF r = v/(gy)1/2Froude numberg = gravitat ional accelerat ionk = pier shape factor in Yarnell formula; constant in localhead loss coefficientL br idge p ier lengthm = obstruction ratio, i.e. obstructed area/channel areaQ water d ischarge.y = br idge elevation (abutment elevation with respect tochanne l bo t tom)y = water depth; tai lwaterv mean velocity

V = reference mean velocitya Coriolis velocity distribution coefficientA H = friction head loss

A v = backwater elevation equal to y\ y4X = local head loss coefficient

8 Conclusions1. The experiments support that the local head loss coefficient

X of arch br idges depends mainly on the obstruction rat iom, def ined as the obstructed area d iv ided by the dow nstreamflow area. The expression X 2.30 m 0.345 (0 .324

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Backwater of arch bridges under free and submerged conditions.pdf