backwater and discharge at highway crossings with …
TRANSCRIPT
BACKWATER AND DISCHARGE AT HIGHWAY CROSSINGS WITH
MULTIPLE BRIDGES IN LOUISIANA AND MISSISSIPPI
By B. E. Colson and Verne R. Schnexder
U.S. GEOLOGICAL SURVEY
Water-Resources Investigations Report 83-4065
Prepared o.n cooperation with
MISSISSIPPI STATE HIGHWAY DEPARTMENT
Jackson, Mississippi
1983
UNITED STATES DEPARTMENT OF THE INTERIOR
JAMES G. WATT, Secretary
GEOLOGICAL SURVEY
Dallas L. Peck/ Director
For additional information write to:
U.S. Geological Survey Suite 710, Federal Building 100 West Capitol Street Jackson, Mississippi 39269
Copies of this report can be purchased from:
Open-File Services Section Western Distribution Branch U.S. Geological Survey Box 25425, Federal Center Denver, Colorado 80225 (Telephone: (303) 234-5888)
ILLUSTRATIONS
Page
Figure 1. Definition sketch showing the variables used incomputing backwater and discharge 3
2. Definition sketch showing the variables used incomputing backwater and discharge at a multipleopening constriction- 7
Figures 3-10. Graphs showing:
3. The comparison of measured and computed flowdivision points (stagnation points on the upstreamside of the embankment) 28
4. The comparison of the measured and computed discharge 29
5. The comparison of the computed and measured Ah formethod I 30
6. The comparison of the computed and measured ah formethod II 31
7. The comparison of the computed and measured backwater,, for method I 33
8» The comparison of the computed and measured backwater,h- , for method II 34
9. The comparison of the computed and measured backwater,h., , for method I 35
10. The comparison of the computed and measured backwater,h , for method II 36
TABLES
Table 1. Site location and flood date 122. Summary of site data 143. Summary of data for computing backwater and discharge 184. Summary of measured and computed backwater and
discharge 215. Comparison of measured and computed backwater 27
IV
CONTENTS
Page
Abstract---- - - . -- . - ------ .- .- -------- . iIntroduction-- -- -- - ------ *.. . ---------- ------ 2Purpose and scope-- -- -- --t- . ------ . .- 2Methods for computing discharge-- . -- --- 2Methods for computing backwater --- - -- ----- --- - - . . - 5Data collection--- --- -- - - .---»----- . . . . . n
Peak discharge measurement-------- >------ -- . .- ^5Valley cross sections -4 ----- - -- 15Water-surface elevation \ 16Bridge geometry ------ - - 4 -- . - .-- . .- .-- ^Manning's roughness coefficient-- --J- - 15
Analysis of data------ ----- -- - - * .--- --------- . -^Computation of natural profile -- -* ---------- - . -^7Measurements of h.. - - .-- - . . . .- 24Measurements of h3 - - ----- - . .- 24Stagnation points- -- --i-- . ------ . 24Computation of discharge -- ---- -------- - . .- 25Computation of backwater-- - - - - 25Errors xn computed backwater and discharge- - 26
Discussion of results ----- - - ------ .- . . - .--- . . 26Summary and conclusion 32References----- - ----- . - ------ .->.- .-- .- .- . .-- 33
111
SYMBOLS AND UNITS Continued
Symbol Definition Units
K
av
m m n n Q
q*TT 1
n
Controlling conveyance used for computing flowthrough the constriction
Conveyance of the spur dike cross section Portion of the approach conveyance, K-j,
corresponding to the bridge width, b
Distance between cross sections i and jAverage streamline length in the approach reachLength of bridge abutment in direction of flowLength of spur dikesLength of interior embankment left of stagnation point
Distance from approach section to upsteam side of constriction or the toe of the spur dikes when spur dikes are included on the bridge
Channel-constriction ratio, 1 - KU/K-jGeometric channel constriction ratio, 1 - b/BSubscript denoting flow under natural conditionsManning's roughness coefficientTotal dischargeDischarge through the ith opening in a muiti- opening crossing
Channel resistance ratio for opening iMean velocity at cross section i during constricted
flow conditions, Q/A^Horizontal distance from the intersection of abutment and embankment slopes to the location on upstream embankment having the same elevation as the water surface at section 1
Energy coefficient at cross section i, under con stricted and natural flow conditions, respectively
Momentum coefficient at cross-section iAngle of skew; acute angle between the plane of the constriction and a line normal to the thread of the stream
Acute angle between a wing wall and the plane of constriction
ft3/s ft3/s
ft3/s
ft ft ft ft
ft
ftV& ft3/s
ft3/s
ft/s
ft
VI
SYMBOLS AND UNITS
Definition units
2a. Area of subsection i of a specified cross ftsection
oA. Area of fiow at cross section i ft Ain Area of fiow beiow the natrual Water surface
elevation ft2TA. Submerged cross-sectional area <&f piers or pij.es ftb Width of bridge opening ftbt Width of bridge opening at the water surface ftC Coefficient of dischargeC. Backwater ratio/ h 1 /AhGI Coefficient of discharge for equivalent base
type opening (method I) d Subscript .denoting a variable measured at the
cross section across the upstream toe of thespur dikes
e Eccentricity ratio based on conveyance distributionin approach reach
E Slope of the embankments, horizontal distance per unitvertical distance
g Acceleration of gravity ft/s^ h^ Water-surface elevation at cross section i during
constricted flow conditions ft h^n Water-surface elevation at cross section i during
natural flow conditions he Head loss due to flow expansion between sections
3 and 4 ft hf(i--j) Head loss due to friction between upstream and
downstream cross sections i and j duringconstricted flow conditions ft
hf(i--i)n Head loss due to friction between upstreamand downstream cross sections i and j duringnatural fiow conditions ft
h * Total backwater or rise above the naturae water i
surface caused by the constriction at a crosssection ft
hg Water-surface elevation at the stagnation pointson the interior embankment ft
Ah Fall between section 1 and 3, h-j-h^ i Subscript denoting cross-section number (i-j) Subscript denoting a variable measured between an
upstream (subscript i) and downstream (subscript j),cross section
ok. Conveyance of subsection i of a specified cross ft /s
section ^in Conveyance of subsection i during natural flow
conditions ft^/s K. Conveyance of cross section i ft /s
v
FACTORS FOR CONVERTING INCH-POUND UNITS TO
INTERNATIONAL SYSTEM (SI) UNITS
Multiply By To Obtain
inch (in) 25.4foot (ft) 0.3048foot per second (ft/s) 0.3048cubic foot per second (ft3/s) 0.02832mile (nu.) 1.609
mj.li.uneter (nun)meter (m)meter per second (rn/s)cubic meter per second (m-^/s)kilometer (km)
VI1
BACKWATER AND DISCHARGE AT HIGHWAY CROSSINGS WITH
MULTIPLE BRIDGES IN LOUISIANA AND MISSISSIPPI
by
B. E. Colson and Verne R. Schneider
ABSTRACT
Data were collected for nine floods In Mississippi and Louisiana at eight stream crossings having two to Six separate bridge openings. Discharge through each bridge, water-surface profiles/ valley cross sections/ and bridge geometry were measured. The multiple openings were divided into equivalent Single-opening cases by apportioning interior embankments in direct proportion to the area of openings on either Side. Using existing procedures for computing discharge/ the bias in computed discharge was 2 percent With a root mean square error of 18 percent.
Backwater was computed by two current U.S. Geological Survey methods that use the average flow path in the friction loss term for the approach. One method gave a root mean square error of 0.34 feet with a bias of -0.25 feet, suggesting that the method underestimates backwater. The other method gave a root mean square error of 0.39 feet with a bias of -0.03 feet. The results indicate that the method developed for single-opening highway crossings can be applied to the multiple bridge crossings.
INTRODUCTION
The U.S. Geological Survey, in cooperation with the Highway Departments of Mississippi, Alabama, and Louisiana, completed in 1976 a study of backwater computations ftt bridges in the wide, densely vegetated flood plains that are common in the Gulf Coastal Plain areas of the three States. The results of studies for the single-opening bridge case (only one bridge in a highway embankment crossing a stream) have been published by Schneider and others(1976). At the conclusion of that study it was apparent thatadditional field data were needed to extend the results to the multiple-bridge system (more than one bridge in a highway embankment crossing a stream). In 1977, the Surve|y in cooperation With the Mississippi State Highway Department bejgan a study to coilect the necessary fieid data to test the method at muitipie-opening bridges.
Knowledge of the backwater caused by the constriction formed by multiple bridges of highway stream-drossings is essential in the design of bridge openings. Peak discharge of floods is computed from elevation change across the embankment and the geometry of the channels and bridges.
PURPOSE AND SCOPE
The principle objective of this project was to measure the backwater and discharge distribution for multiple bridges. These data have been used to determine if the methods developed by Schneider and others (1976) for singles-opening highway crossings could be applied to multiple bridges. In addition, the method developed by Tracy and Carter (1955) ajnd Cragwall (1958) was modified to use the procedure proposed by Schneiider and others (1976) to calculate friction losses in the approach reach. This modified procedure was then also tested.
METHODS FOR COMPUTI^ DISCHARGE
Kindsvater and Carter (1955) conducted the analytical and experimental work leading to the development of the Survey method of computing discharge through width donstnctions. The discharge relationship is derived from the energfy and continuity baiance between an approach section and the moist contracted section designated sections 1 and 3, respectively, in figure 1,
= CA3 'V' 2g2g
BACKWATER PROFILE CONSTRICTION
NATURAL PROFILE
(a) ELEVATION
SECTION^ B NUMBERS®
(b) PLAN
Figure 1. Definition sketch of the variables used in computing backwater and discharge by the proposed method.
whereQ is the total discharge in cubic feet per second.A is the flow area at section 3 below the measured water-
surface elevation in square feet 4Ah is the difference in water-surface elevation between
sections 1 and 3 in feet.ot is the energy coefficient at cro^s section 1.hf is the head loss due to friction in feet.C is the discharge coefficient.V is the mean velocity at cross section 1 in feet per second.
Laboratory investigations were conducted to define the discharge coefficients for four typical abutment geometries. The coefficient C represents a combination of (1) a coefficient of contraction, (2) a coefficient which takes into account the eddy losses/ and (3) the velocity head coefficient/ 013 / for the contracted section (Kindsvater and others/ 1953). The procedures for selecting the discharge coefficients are discussed by Matthai (1967).
The energy loss (Matthai/ 1967) due to fraction is the product of the geometric mean of the energy slopes at the end cross sections of the reach times the distance between the sections. The energy loss due to friction is obtained from the equation
L Q2 LQ2 wx *hf(1-3) = + ~ (2)
IN. * -K.O ^O
where1^ is the length of the approach reach. L is the length of the bridge opening. K-j is the total conveyances of section 1. K>j is the total conveyances of section 3.
When the approach reach has dense brush and the reach under the bridge is relatively clear, the weighteld conveyance computed from the conveyances of sections 1 and 3 will be too high. A more accurate approximation of the friction loss may be obtained if L (Q2/K.K ) is substituted for the firat term m equation 2.W I QKq is that part of the approach section conveyance corresponding to the projected bridge opening b.
SECTION 0
"Tl<5'c(D
10 *l
0> O ** (D
li
2. 3"
CO §.O O O £
(D «-
" i; ! * ** 3 ~
O O313 Cr+ 3*
(Qo-09 O
09r+
$
09
a a55' o09 ^(Q (D
iio
CD O
SECTION 1SECTION 1
(SECTION 1'
^ J (JJ1L3 gj," ^ _-U2U__
" SECT.ON4 SECT(ON4
SECTION 4
SECTION 5
embankments) are located in direct proportion to the gross fiow area of the opening on either side/ the larger length of embankment being assigned to the larger opening. From the fiow division points, lines are projected paraiiei to the fiow from the embankment upstream to the approach section to form flow boundaries for each opening. This procedure defines a separate approach for each mdividuai bridge. A channei resistance ratio (q*^ based on the approach geometry is computed by iterative proce4ures from the equation:
ii q =1 + 0.46 iQ^f (8)
The discharge q through each bridge is computed by the formula:
Ci A3i* qi = qi
where i = i. . .n. where
q-j^ is the discharge for the individual bridge.q.* is the channel resistance tatio for the individual
bridge opening. k^ is the approach channel conveyance for the individual
bridge opening.A^^ is the area for the individual bridge opening. a^ is the approach area for the individual bridge opening. K-| is the total approach conveyance. A-| is the total approach area. C^ is the coefficient of discharge for the individual
bridge opening. £(CA3) 1 is the sum of the live flow areas for a particular site.
Each bridge opening with the associated fiow boundaries can be treated as an equivalent single-opening crossing for computation of backwater.
Laboratory investigations by David^ian and others (1962) have shown that discharge coefficients developed for bridges at smgie- opening constrictions are valid for multiple-bridge constrictions. These coefficients have been well defined for the four most common types of bridge openings.
Lee (1976) tested the method of Davidian and others (1962) with data from three sites in Louisiana. Two of the sites had two bridges and one site had three bridges. Peak discharge was measured at all bridges so that the actual flow distribution was known. At the three-bridge site and at one two-bridge site, the discharge distributed using equations 8 and 9 agreed within 6 percent of the measured discharge. The main channel bridge at one two-bridge site was in error by 20 percent which induced an error of 100 percent at the small relief bridge.
The most recent method developed by the Survey for computing backwater at single-opening constriction is described by Schneider and
When spur dikes are included on the bridge, the energy ioss due to friction is
2
hf(1-3) = + + O)
whereI^j is the length of the spur dike in the direction of fiow in
feet. K^ is the conveyance of the cross section at the toe of
the spur dikes.
Schneider and others ( 1976) modified the friction loss term forL Q * (equation 2) to more accurately estimate the friction losses
in the approach reach. In the modified term , Lav is the average
flow length in the approach reach and Kc , the controlling conveyance at the downstream end of the approach reach, is the smaller of the conveyances K3 and K^,.
Matthai (1967) recommends that the conveyance that best represents the conditions at the end of the reach be used to compute friction losses. An examination of the data reported by Schneider and others (1976) indicates that Kc was the representative conveyance. Studies indicate that care in selecting the representative conveyance is important because it significantly affects the magnitude of the friction loss and therefore the computation of discharge and backwater.
METHODS FOR COMPUTING BACKWATER
Tracy and Carter (1955) defined backwater, h 1 , as one component of the fail, Ah, between sections 1 and 3 as illustrated in figure 1. The fall was resolved into three components.
Ah = h/ - h3 * +hf( _ (4)where
h< is the increase in the water-surface at section 1 in feet. 1*h- is the backwater at section 3 in feet.hf is the head loss due to friction in feet.n is a subscript denoting flow under natural conditions.
* Tracy and Carter (1955) defined h^ as positive when the contrictedwater-surface elevation at section 3 was below the natural water- surface elevation. After finding that the constricted water-surface elevation at section 3 could be above the natural elevation, Schneider and others (1976) adopted the convention of positive (hi* = h3 - h_ ) when the constricted water-surface elevation is above natural. This convention is followed in this report.
Equation 4 is divided by Ah/
h 1* h3* hf(1-3)n = 1+ - i (5)ah ah ah
The ratio, , is defined as the backwater ratio, C^. Equation 1Ah
is solved for Ah and V3 = Q/A3 is substituted from the continuity equation,
The backwater ratio is a function of the channel-constriction ratio, Manning's n value, and the constriction geometry. Equation 6 is sub stituted into equation 4 and is solved fbr h- .
h*- d )(!L +h a -3 - ' v '~hD' \ , T nf(1-3) ~ »1 T nf(1-3)n \ 2gC2 2g
The procedures for computing backwater by this method are reported by Cragwali (1958).
A laboratory study on backwater at |multipie bridge systems was made by Davidian and others (1962). The( objective was to develop methods for the computation of discharge through multiple-opening constrictions, prediction of maximum bacikwater, and prediction of division of flow through the several openings.
Current one-dimensional methods of computing backwater at multiple-opening highway crossings ail rely on reducing the bridge openings to equivalent single openings. This is accomplished by establishing pseudo-fixed boundaries between separate openings.
The method of apportioning flow through multiple bridges, as given by Davidian and others (1962), requires three items of data: (1) stage-discharge relation at the site, (2) a valley cross-section, and, (3) locations and geometry of all bridges. A definition sketch for flow through a typical multiple-opening constriction is shown in figure 2. The flow division points (along the interior
others ( 1976) . The naturai profiie is computed using a standard step- backwater procedure (Chow, 1959), where frxctxon losses are
hf(i-j)n =KinKjn
The constricted profile is also computed using a standard step-backwater procedure. The approach section is located one bridge- opening length upstream (fig. 2). The friction losses are computed using the average flow length in the approach. Section 4 is located one bridge-opening length downstream from the highway crossing. The water-surface elevation at section 4 is assumed to be at the natural elevation. ftn expansion loss term is applied between sections 3 and 4. The constricted water-surface profile is computed by itera tion, until successive estimates agree within a preselected tolerance. An example of this method is given by Schneider and others (1976).
For computation of friction losses, Schneider and others(1976) divided the approach reach into as many as three separate subreaches. The constriction subreach is considered to be the length, L/2-3) of the abutment in the direction of flow. The friction loss is computed as:
hf(2-3) = L (2-3) (1D\ K3/
For a bridge without spur dikes,
Q2
hf(1-2) = Lav (12)
In this case (no spur dikes)
hf(1-3) = hf(1-2) + hf(2-3) d
When spur dikes are present, the approach reach is further divided into two subreaches.
LavQ2hf (1-d) =
*1*c
and L (d-2)22
So that , when spur dikes are present
hf(1-3) = hf(1-d) + hf(d-2) + hf(2-3)
In the flow expansion reach, the flow is assumed to be at naturax elevation one-bridge-width downstream from section 3. Therefore, the area and conveyance of section 4 are computed at the naturae exevation
"Hie friction losses are estimated from equation 17 using the straight- line distance between sections,
bQ2 hf<3-4) -
where the controlling conveyance, venyances, K, K3 , or K^.
is the smallest of the con-
Schneider and others ( 1976) , following a suggestion by Bender son (1966, page 277, problems 7.1 and 7.2), (present an approximate solution of the momentum, energy, and continuity equation for expansion losses of an ideal abrupt expansion in open-channel flow:
Where
"4 + "3
4(18)
o^ JLS the energy coefficient tf^ is the momentum coefficient
It can be shown that alpha and beta at section 3 are related tothe bridge geometry and can be estimated from the bridge coefficient
1013= (19)
Bi- (20)
Alpha and beta at section 4 can be computed from the cross section properties as
a4 =
and
where
34 = (22)K4/A4
K4 A4
AS the subsection conveyance.AS the subsection area.AS the cross section conveyance.AS the cross sectAon area of Section 4.
Schneider and others ( 1976) teste<^ the method developed by Bradley (1970). They found that for the wide flood plains backwater was undercomputed significantly. The friction losses appear to be underestimated by the method. Hence this procedure was not tested in this report.
10
Several other research attempts have been made to develop flow models. The latter efforts have been primarily in the application of two-dimensional finite-element flow models. Lee (1980) Lee and Bennett, (1981), and Lee and others (1982). These are promising in that they allow much more flexible application of hydraulic theory. Present two-dimensional models require relatively large amounts of manpower and computer time. Rbwever, the probable successful efforts to automate the data handling process, will make the 2D model more accessible.
DATA COLLECTION
Field data were collected usxng the procedures outixned by Benson and Dairympxe (1967) and Matthao. (1967). In generax, the data were collected and reported u.n the same way as outlined by Schneider and others (1976). Data include peak discharge, valley cross sections, water-surface elevations, bridge geometry, and Manning's roughness coefficient, n. ttlgh-water-mark elevations, valley cross-section ground elevations, highway profile, and bridge geometry were surveyed using standard leveling techniques. Highwater- mark elevations were measured with a resolution of 0.01 ft and ground-surface elevations and highway profile to 0.1 ft. The site location and flood date are contained in table 1. A summary of site data is in table 2.
The total data set as reduced and assembled generally includes the following:
1. SummaryA. Location of siteB. Description of siteC. Description of floodD. Description of discharge measurementE. Field surveyF. ComputationsG. Results of computationsH. Datum
2. Topographic map3. Aeri.ai photographs4. Highway plans5. Flood-frequency curve6. Stage-discharge relation7. Discharge measurement notes8. Velocity distribution and measuring section diagram9. Plan of roadway crossing and location of high-water marks10. Bridge geometry11. List of high-water marks12. Water-surface profile along highway embankments13. Valley cross sections14. Flood profiles15. Field notes16. Computer printouts17. Stereoscopic slides documenting flood-plain roughness
11
Table 1. Site location
Flood Station name and location Date of No. flood peak
1 Thompson Creek at Strengthford, Miss., ][at 31°36 I 59", long 03-03-71 88°52'53" in sec. 34, T. 8 N. , R. 9 W., St. Stephens meridian, on county highway 0.3 mile east of Strengthford, Wayne County, Miss.
2 Sipsey Creek near Forest, Miss., lat 32«32 I 33 11 , long 10-17-75 89°21 I 32" in sec. 15, T. 8 N., R. 9 E., Choctaw meridian, on State Highway 21, 15 miles northeast of Forest, Scott County, Miss.
3 Big Black River near Winona, Miss., lat 33 0 22'58", long 03-04-77 89 0 36'52", in sec. 36, T. 18 N., R. 6 E., Choctaw meridian, on State Highway 407, 9 miles southeast of Winona, Montgomery County, Miss.
4 Big Black River near Canton, Miss., lat 32 0 42'26", long 03-08-77 90°05'39", in sec. 16, T. 10 N., R. 5 E., Choctaw meridian, on State Highway 16, 6.8 miles northwest of Canton, Madison County, Miss.
5 Big Black River near Canton, Miss., lat 32 0 42'26", long 04-14-79 90 0 05'39", in sec. 16, T. 10 N., R. 5 E., Choctaw meridian, on State Highway 16, 6.8 miles northwest of Canton, Madison County, Miss.
6 East Fork Amite River near Peoria, Miss., lat 31°05'54 11 , 04-22-77 long 90°43 I 00", in sec. 32, T. 2 N., R. 5 E., Washington meridian, on State Highway 584, 4 miles southwest of Peoria, Amite County, Miss.
7 Castor Creek near Grayson, La., lat 32 a 04'55", long 12-07-71 92°12 I 24", in sec. 30, T. 13 N. , R. 3 fi. , Louisiana meridian, on Louisiana Highway 126, 6.5 miles west of Grayson, Caldwell Parish, La.
8 Bayou de Loutre near Farmerville, La., |lat 32 0 52'25", 03-15-73 long 92 0 22'40", in sec. 20, T. 22 N., i. IE., Louisiana meridian on Louisiana Highway 549, 7 miles north of Farmerville, Union Parish, La.
9 Sixmile Creek near Sugartown, La., lat 30 0 48'52", long 03-25-73 92 0 55'34", in sec. 12, T. 3 S., R. 6 W;, Louisiana meridian on Louisiana Highway 112, 6.5 miles east of Sugartown, Alien Parish, La.
a Elliptical
12
and flood date
Number of Total peak Recurrence Average Channel Dike Manning's bridge discharge interval flood slope type roughness
openings (ft3/s) (years) plain (ft/mi) coefficient (rounded) width
(ft)
2170 2 2000 5.5 0.20
7510 7 2500 4.2 0.08-0.16
6 23300 5 9000 2.9 0.06-0.15
4 30300 7 10000 1.5 a 0.06-0.15
4 85800 100 10000 1.5 a 0.06-0.15
3 27000 100 5000 6.1 a 0.08-0.15
5850 2 2100 1.8 0.14
4900 2 2300 4.5 0.11
2 12900 12 3300 5.0 0.13-0.18
13
Table 2. Summary
Site
1
2
3
4
5
6
7
8
9
MC ROa
Br idge
MCRO-1MCRO-1
MCRO-1RO-2RO-3RO-4RO-5
MCRO-1RO-2RO-3
MCRO-1RO-2RO-3
MCRO-1RO-2
MCRO-1
MCRO-1RO-2
MCRO-1
Abut.type
4444444444333333333334444433
E
4:14:12:12:12:12:12:12:12:12:12:12:12:12:12:12:12:12:12:12:12:12:12:12:12:12:13:13:1
m
0.77.94.70.96.50.66.68.66.68.80.21.83.84.83.35.84.86.85.58.83.88.32.83.29.84.64.25.79
m 1
0.57.96.92.97.82.86.77.68.72.89.72.79.91.88.70.83.89.89.86.81.88.60.85.72.87.93.64.90
b
ft
192569639
37414918537930311464918822034465120622337734316917647413295
11376
606162
bt
ft
192569639
37414918537930311465319122236866922523539635517817847413295
11376
606162
Main channel Relief openingElliptical
14
of site data
L w
et
192569639
374149185379 303 11465319122236866922523539635517817847413295
11376
606162
L av
ft
213145192115527231241447 376 194805252411568788297379641532233290531198118181161697284
Diketype
-- ---
-
aaaaaaaaaaa------
Laft
.------
-
150100120120150100120120175170150-------
L
ft
3737282927222222 22 22404040404040404027272731312727273434
X
ft
_ -_
-
181311128312222___
1111
e
.41
.32
.64
.10
.55
.901.00.35 .57 .75
0.03.63.74
0.15.54.79.67.76.58.64.42.04.48.30.66
1.00
0
53°53°45°45°35°35°35°35° 35° 35°
_ ___ _ _45°45°45°45°45°
-
C
0.74.85.73.79.69.68.68.65 .65 .68.89.90.87.83.81.89.88.83.88.90.91.79.72.85.73.78.79.70
ci
0.64.70.677-3
/ -J
.67
.65
.65
.65
.64
.64
.73
.67
.6564 \J**
.71
.68
.67
.67
.66
.66
.66
.70
.63
.79
.64
.69
.74
.66
15
Peak Discharge Measurement
Peak discharge was measured by current meter at the flood peak or was obtained from stage-discharge relations. The stage-discharge relations were extrapolated several feet at some Sites. Avaxj.ab.Le data on the volume of runoff and the duration of the peak indicated that steady flow existed throughout the r|each during the peak at most sites. When necessary, flow over the highway embankment was computed using the procedure described by Huismg (1967). At these sites the amount of fiow over the highway embankmertt was small compared to the total discharge.
Valley Cross Sections
At least four valley cross sections were selected. Each cross section was approximately one valley width apart. At each site at least two valley cross sections were located upstream and two valley cross sections down-stream of the highway embankment. In addition, an approach cross section was surveyed approximately one-bridgeopening width upstream from the constriction. Additional cross sections were surveyed as required to define road fills, pipeline crossings, and other features affecting the flood profile.
Locations for the valley cross sections were selected using a plot of the flood profiles obtained along each edge of the flood plain and by inspection of topographic maps. The cross sections were drawn on the map at approximately valley-width intervals and were alined perpendicular to the assumed direction of flow. Identifiable landmarks were used to locate the cross sections in the field, where they were oriented to the correct azimuth by compass. The survey datum was established at the bridge. A base line was surveyed from the highway to esitablish horizontal and vertical control for the cross section.
Water-Surface Elevation
Water-surface elevations were determined by high-water marks recovered along the cross sections and base lines. Water surfaces also were marked along the upstream and downstream sides of the embankment during the peak discharge measurement. Additional high- water marks were selected at random locations upstream and downstream of the bridge to describe the lines of constant watersurface elevation in the approach and fiow-expartsion reaches.
Bridge Geometij-y
Bridge geometry data, collected according to the procedures discussed by Matthai (1967), included abutment slope, bridge cross section, and pier and spur dike geometry and location.
Manning's Roughness Coefficient
An attempt was made to field-select Manning's roughness coefficient, n. Selection is usually based on experience obtained by computing water-surface profiles in channels where peak discharge
16
and water-surface elevations are known (n-verification studies) and by studying stereoscopic slides that document features affecting the magnitude of n. Although n was selected by experienced personnel and, at most sites, by the same individual for consistency, neither published n-verification studies nor stereoscopic slides were available for comparative purposes. Therefore, the field-selected n's were adjusted using the measured discharge and the measured water-surface profile downstream of the bridge. The n-values were adjusted so that the water-surface profile computed using a step- backwater procedure (Shearman, 1976) agreed with the measured profile downstream of the bridge. Gross sections were subdivided for major changes in geometry and roughness which persisted throughout the reach and n selected for each subdivision. When the reach included an open field which extended approximately one-half the distance upstream and downstream to the next cross sections, the reach was subdivided and n selected for the open-field condition. Composite n-values were used where frequent roughness changes occurred that did not affect the entire reach.
ANALYSIS OF DATA
Backwater is defined as the difference between the natural and the constricted water-surface elevation. The natural (unconstricted) profile prior to construction of the highway was not available for any of the sites, and was therefore, computed using standard step- backwater techniques (Chow, 1959).
The constricted water-surface elevations were obtained at the approach for each bridge opening by interpolation of the profiles defined by high water marks surveyed aj.ong each edge of the vaney. These elevations were compared with the stagnation elevations observed at each edge of the valley and on the interior embankments.
Backwater was computed by two methods named method I and method II. Method I was the technique developed by Tracy and Carter (1955) and reported by Cragwall (1958). The method was modified in this study so that the average flow distance; Lav , is used in equation 2 and 3 in place of 1^. Because method I, as developed, applies only to sites without spur dikes, it was not applied to sites 4, 5, and 6.
Method II is the procedure developed by Schneider and others (1976). In both methods, the approach is divided into equivalent single openings using the methods developed by Davidian and others (1962). Data for computing backwater and discharge are summarized in table 3. The results for discharge and backwater computation are summarized in table 4.
Computation of Natural Profile
In the step-backwater procedure, peak discharge, cross-section geometry, and n-vaiues were used to compute the natural profile. The water surface profile was defined by highwater marks surveyed along each edge of the valley sufficiently far in each direction to
17
Table
3. Summary of da
ta for
computing backwater an
d discharge
oo
Site
Br
idge
1 MC RO-1
2 MC RO-1
3 MC RO-1
RO-2
RO-3
RO-4
RO-5
4 MC RO-1
RO-2
RO-3
5 MC RO-1
RO-2
RO-3
6 V.
CRO-1
RO-2
7 MC RO-1
8 MC
RO-1
RO-2
9 MC RO-1
Q
ft/s
1440
727
5870
1640
9700
2510
2120
4350
3340
1190
16000
2130
6130
6050
35200
10500
18100
22000
14200
7470
5340
4890
964
2420
1600
878
11100
1850
h. in ft
215.83
215.60
371.63
371.58
302.05
301.76
301.58
301.36
301.12
300.91
189.48
188.87
188.96
18B.52
193.16
193.02
193.04
193.12
267.56
266.93
266.70
13.06
13.17
44.76
44.80
44.82
16.34
16.04
Aln
ft2
2410
2510
6060
4360
9840
5200
3360
6260
4460
1070
13000
1980
9400
7670
20800
7580
17000
22600
13200
5570
5760
5800
2700
1440
3910
2500
8580
7890
K-, In
ft3/s
36300
30500
232000
890 'J
O429000
143000
85900
185000
112000
11000
854000
33200
231000
159000
1453000
255000
678000
796000
551000
184000
150000
183000
61000
50900
141000
58600
475000
183000
aln 1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
4.90
1.00
1.00
1.00
4.15
1.00
1.00
1.00
2.64
1.00
1.00
1.00
1.00
1.00
1.00
1.00
4.03
1.00
hr» 3n
ft
215.63
215.49
371.55
371.55
301.90
301.70
301.50
301.20
301.00
300.86
189.25
188.80
188.80
188.
40-
192.61
192.94
192.96
192.98
267.30
266.80
266.56
12.68
12.80
44.61
44.60
44.62
15.88
15.88
A3n
ft2
860
246
1270
295
3050
1120
763
2080
1770
567
5960
494
1390
1480-
8170
1380
2290
3240
3120
1300
1020
2570
1280
815
659
325
3990
724
K-> 3n
ftVs
29500
7910
55900
9620
209000
66900
36000
185000
97300
29300
559000
20800
943000
sooo
a844000
91400
199000
253000
188000
59300
42500
103000
65900
62300
43200
18700
282000
44100
MC
Main channel
RO
Relief opening
Tabl
e 3, Summary of
data for
computing ba
ckwa
ter
and discharge Continued
Site 1 2 3 4 5 6 7 8 9
MC
RO n.a.
Bridge
MC RO-1
MC RO-1
MC RO-1
RO-2
RO-3
RO-4
RO-5
MC RO-1
RO-2
RO-3
MC RO-1
RO-2
RO-3
MC RO-1
RO-2
MC
RO-1
MC
RO-1
RO-2
MCRO-1
hl ft
216.67
216.68
373.45
373.45
302.75
302.50
302.30
302.12
301.85
301.65
190.10
190.05
189.80
189.50
194.65
194.65
194.64
194.64
269.30
269.30
269.30
13.13
13.17
45.12
45.16
45.18
16.65
16.50
Al
ft2
3120
4400
8160
6870
11200
5980
3930
7170
5260
1850
14400
3050
11400
10400
24100
9540
20400
27800
17400
7700
9350
5880
2700
1560
4230
2890
9100
8590
Kl
ft3/s
55200
68300
362000
187000
504000
187000
112000
231000
147000
26400
941000
67700
319000
237000
1746000
373000
914000
1110000
792000
315000
325000
187000
61000
58200
161000
74900
511000
214000
al
1.00
1.00
1.97
1.00
4.76
1.00
1.00
1.00
1.00
1.00
4.90
1.00
1.00
1.00
3.84
1.00
1.00
1.00
2.39
1.00
1.00
1.00
1.00
1.00
1.00
1.00
4.00
1.00
Kq
12900
3650
109000
7890
256000
71000
37200
81500
47300
5450
757000
11300
53200
45900
1164000
60000
132000
168000
335000
54900
39200
138000
10500
48800
25200
18800
382000
46600
h s
ft
n.a
n.a.
373.16
373.18
302.65
302.45
302.25
302.00
301.76
301.58
190.09
190.04
189.84
189.56
194.32
194.46
194.54
194.56
268.96
268.48
268.04
13.10
13.18
45.08
45.12
45.14
n.a.
n.a.
fla ft2
_ - - - - - - - - -5240
508
3110
2330
8140
2070
4970
4550
3430
2120
1680
- - - - - -
Ka ft3/s
- - - - - - - - -548000
19300
301000
172000
884000
173000
614000
494000
292000
151000
110000
- - - - - -
Main c
hann
el
Relief opening
not av
aila
ble
Table
3. Summary of
da
ta for
computing
backwater
and discharge Continued
O
Site 1 2 n 4 5 6 7 8 9
MC
RO
Bridge
MC RO-1
MC RO-1
MC RO-1
RO-2
RO-3
RO-4
RO-5
MCRO-1
RO-2
RO-3
MC RO-1
RO-2
RO-3
MCRO-1
RO-2
MC
RO-1
MC
RO-1
RO-2
MCRO-1
h3
ft
215.80
215.66
371.80
371.77
302.00
301.95
301.60
301.34
301.09
300.89
189.48
188.90
188.89
188.47
192.87
192 . 66
192.67
192.75
267.34
266.88
266.76
12.72
12.80
44.75
44.69
44.60
15.85
15.85
A3
ft2
892
255
1300
303
3090
1150
781
2L30
1800
570
6110
513
1410
1510
8350
1310
2220
3150
3130
1310
1050
2590
1280
828
669
323
3980
677
K3
ft3/s
31100
8360
56700
9950
212000
69900
37200
114000
99400
29600
578000
21900
96300
82000
8680
0085400
1890
0024
2000
189000
6020
044
600
104000
69000
6360
044200
1850
028
0000
4280
0
A. D ft2
35.5
11.8
115 26.9
118 57.9
38.4
108
87.0
24.4
325 12.8
34.0
37.7
413 31.6
52.9
76.2
77.8
61.8
49.1
129 58.8
38.5
31.4
13.9
227 43.0
h4
ft
213.87
213.
88371.21
371.
28301.47
301.39
301.18
300.
9630
0.84
300.72
188.97
188.
70188.57
188.33
192.65
192.
67192.69
192.
7526
7.04
266.
71266.33
12.60
12.60
44.4
244
.42
44.42
15.44
15.74
A4
ft2
2110
3230
4390
2860
9620
3890
3790
5410
7100
4270
11900
4720
11600
22500
23500
1100
019300
37600
17600
5810
5130
5390
1720
1660
3360
1980
8820
3920
K4
ft3/s
2570
039800
1290
0055
200
358000
9610
0100000
1530
0022
9000
1200
00740000
1300
0033
2000
8030
0013&703&
4510
00830000
1861
000
666000
2040
00125000
1640
0030100
4350
079600
33500
2260
0058
800
«4
1.00
1.00
2.62
1.00
3.64
1.00
1.00
1.00
1.00
1.00
5.35
1.00
1.00
1.00
5,aa
1.00
1.00
1.00
2.22
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
*4 1.00
1.00
1.40
I. 00
1.56
1.00
1.00
1.00
1. 00
1.00
2.11
1.00
1.00
1.00
2.01
1.00
1.00
1.00
1.28
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Main channel
Relief opening
Table 4. Summary of measured and computed backwater and discharge
Site Bridge
1 MCRO-1
2 MCRO-1
3 MCRO-1RO-2RO-3RO-4RO-5
4 MCRO-1RO-2RO-3
5 MCRO-1RO-2RO-3
6 MCRO-1RO-2
7 MCRO-1
8 MCRO-1RO-2
9 MCRO-1
MEASURED BACKWATER
Ah
ft
0.871.021.651.68.75.55.70.78.76.76.62
1.15.91
1.031.781.991.971.891.962.422.54.41.37.37.47.58.80.65
hl*
ft
0.841.081.821.87.70.74.72.76.73.74.62
1.181.12.98
1.491.631.601.521.742.372.60.07.00.36.36.36.31.61
vft
0.17.17.25.22.10.25.10.14.09.03.23.10.09.07.26
-.28-.29-.23.04.08.20.04.00.14.09
-.02-.02-.03
Q meas.
ft 3/s
1440727
58701640970026102120435033401190
160002130613060503520010500181002200014200747053404890964
24201600878
111001850
DISCHARGE
^comp.
ft 3/s
1530841
6880162094703430234048403370719
16400140063005200
4230084101690019200155007590606035001060225019601150
114002340
Diff.
percent
6.315.717.2
- 1.2- 2.431.4L0.411.3
.9-39.6
2.5-34.3
2.8-14.020.2
-19.9- 6.6-12.7
9.21.613.5
-28.410.0
- 7.022.531.02.7
26.5
MC Main channel RO Relief opening
21
Table 4. Summary of measured and computed backwater and discharge Continued
Site Bridge
METHOD I COMPUTED BACKWATER
Diff. ah from meas. h.,* fr
ft ft ft
Diff. Diff. om meas. 113* from meas.
ft ft ft
1
2
3
4
5
6
7
8
9
MC RO
MC 0.RO-1
MC 1.RO-1 1 .MCRO-1RO-2RO-3RO-4RO-5
MCRO-1RO-2RO-3
MCRO-1RO-2RO-3
MCRO-1RO-2
MCRO-1
MCRO-1RO-2
MCRO-1
Main channel Relief opening
53693483843665554862-----------72064026337536
-0.34- .33- .31
.15
.09- .19- .05- .23- .28- .14
-----------.31
- .31.03
- .21- .25- .05- .29
0.41.63.98
1.73.53.26.48.42.37.50-----------.28.05.15.22.22.31.30
-0.43- .45- .84- .14- .17- .48- .24- .34- .36- .24
-----------.21.05
- .21- .14- .14
.00- .31
0.08- .05
.28
.07
.16
.04
.08- .03- .01- .07
-----------.06
- .36.10
- .16- .09- .02- .09
-0.09- .22
.03- .15
.06- .21- .02- .17- .10- .10
-----------.02
- .36- .04- .25- .07
.00- .06
22
Table 4. Summary of measured and computed backwater and discharge Continued
Site Bridge
METHOD II COMPUTED BACKWATER
Diff . Ah from meas.
ft ft
Diff. h * from meas.
ft ft
h 3*
ft
Diff. from meas.
ft
1 MCRO-1
2 MCRO-1
3 MCRO-1RO-2RO-3RO-4RO-5
4 MCRO-1RO-2RO-3
5 MCRO-1RO-2RO-3
6 MCRO-1RO-2
7 MCRO-1
8 MCRO-1RO-2
9 MCRO-1
1.071.961.301.91
.81
.38
.68
.62
.741.20
.621.23
.831.211.252.341.952.111.621.931.60
.59
.28
.45
.35
.39
.62
.40
0.20.94
- .35.23.06
- .17- .02- .16- .02
.44
.00
.08- .08
.18- .53
.35- .02
.22- .34- .49- .94
.18- .09
.08- .12- .19- .18- .25
0.50.48
1.151.74
.62
.12
.46
.75
.731.17
.531.50
.831.291.362.482.002.131.542.372.23
.70
.08
.28
.09
.06
.70
.27
-0.34- .60- .67- .13- .08- .62- .26- .01
.00
.43- .09
.32- .29
.31- .13
.85
.40
.61- .20
.00- .37
.63
.08- .08- .27- .30
.39- .34
-0.38-1.37- .06- .14- .05- .20- .13
.28
.12
.03
.14
.33
.17
.20
.65
.22
.14
.16
.18
.57
.77
.49
.16- .02- .06- .13
.54
.03
-0.55-1.54- .31- .36- .15- .45- .23
.14
.03
.00- .09
.23
.08
.13
.39
.50
.43
.39
.14
.49
.57
.45
.16- .16- .15- .11
.56
.06
MC Main channel RO Relief opening
23
extend beyond the effects of the highway construction. The water surface at the fartherest downstream section (section 5) was used as the starting eievation. Gross sections usuaiiy were divided into three subsections with the main channei separating the f lood piain. Cross sections were divided into oniy two subsections in those sections where the main channei was at the edge of the vaiiey, The fieid-seiected n-vaiues were adjusted where necessary untii the computed water-surface eievation matched the observed watersurface elevation at the most upstream section. The computed profile was examined to ensure that it reflected the known physical features of the flood plain.
*Measurement of h 1
* Backwater h^ at the approach section was measured one
bridge-opening width upstream (fig. 1 ). The observed water surface at the approach section (section 1) was determined from the water- surface elevations surveyed along each edge of the valley. Where a sloping water surface extended across the approach section, the water-surface elevation was determined by interpolation at the boundary between the equivalent single channels as described by Davidian and others (1962) for each bridge-opening. The average of the elevations of the boundaries appropriate for each opening was used for the observed elevation at section 1. The computed natural water-surface eievation was subtracted from the observed water- surface elevation at the approach and is shown at "h.. measured" in table 4.
*Measurement of h
The difference between the contracted water-surface eievation and the natural profile at the downstream side of the contraction is defined as h.. . The contracted water surface was measured as the average of the level determined at the downstream end of the abutments of each opening. The natural profile was determined from the profile computed through step-backwater procedures. Negative values of h_ represent contracted water-f surface elevations that are below the natural profile and positive values of h-j represent those that are above the natural profile (table 4).
Stagnation Poinjs
On the upstream side of the constriction embankments, flow stagnation occurs in the corners formed by the embankment and each edge of the valley. Flow stagnation occurs also at each of the interior embankments between bridge-openings. The location of the point of stagnation is a function of the location and geometry of the constriction and of the hydraulic characteristics of the approach channel. The flow divides at the stagnation point and passes through the openings on each side* This point was readily observable in the field and its elevation is reported as hg in table 3.
An attempt was made to find the analogous stagnation point on the downstream side of the embankment. This point may be visualized
24
as the point where the flow from adjacent bridges converges and turns downstream.
When the stage was just above the low-water channel, very little if any flow appeared to be parallel to the downstream embankment. Parallel flow was observable along the downstream embankment as the stage rose with increasing discharge. However, the point downstream analogous to the upstream stagnation point could not be located.
Computation of Discharge
For computation of discharge and backwater, lines are projected parallel to the flow from the fiow do.vo.sion points to the approach and exit sections (sections 1 and 4). These iines are treated as fixed boundaries of an equivalent single opening construction.
Discharge for each bridge was computed using the recovered highwater marks. The cross section properties were calculated. Total fall, Ah was calculated as the difference between the measured values of h-j and 113. Discharge was then calculated from equation 1 with the energy losses computed from equation 13 for sites without spur dikes and from equation 16 for sites with spur dikes. Since water-surface elevation cannot usually be directly measured at the upstream end of the spur dikes, this elevation was estimated to be 1/2 (h-j + 113) . The dike area, A^ and conveyance K^, in table 3 were calculated for this elevation. The contracted water-surface elevation at section 3 is obtained by extrapolating the measured water-surface profile along the downstream side of the embankment to the intersection of the abutment and embankment for each side and averaging the values obtained. The computed and measured discharges are compared in table 4.
Computation of Backwater
Backwater is the difference between the water-surface profiles for the natural and constructed conditions. The natural, profile is computed using a standard step-backwater procedure (Chow, 1959), where the fraction losses are computed from equation 10. The constricted profile is aj.so computed using a standard step-backwater procedure where the friction o.osses are computed from equation 13 and 16. Both profiles use section 4 as a common starting point. The average flow path needed in equation 13 and 16 is obtained from Schneider and others (1976).
The constricted water-surface profile is computed by iteration because the controlling conveyances are not known. The controlling conveyance, KC, is computed at the natural water-surface elevation and used as the first estimate. Revised estimates of the controlling conveyances are determined at the computed constricted elevations and compared to the previous estimates. Successive estimates of the constricted profile are continued until the controlling conveyances agree within a preselected tolerance. With a tolerance criterion of
25
k-1 k Kc 2 0.95KC (23)
convergance can be achieved in two or three iterations. The super script is the iteration number.
Errors in Computed Backwater and Discharge
Error is calculated as the differenc^ between the computed and measured quantity. Two measures of error are used to evaluate each computation method (table 5). The bias, defined as the algebraic mean error, indicates whether or not the brror magnitudes tend to be evenly distributed above and below zero. The foot fltean square (rais) error is defined as the square root of the mean of the sum
of the squares of the errors. The rms error expresses the magnitude of error likely to occur in any computation using the method in question.
DISCUSSION OF RESULTS
Computations of backwater or discharge at multiple bridge sites depend on the distribution of flow through the several openings and the division of flow boundaries in the vicinity of the constriction. The distribution of flow in direct proportion to the gross area (Davidian and others, 1962) of the bridge opening gave consistent results. The ratio of the interior embankment length of the stagnation point to the total interior embankment length between each pair of openings was computed and these ratios are compared with the observed results in figure 3. There is considerable scatter about the line of equal value but on the average the answer may be a reasonable estimate of the stagnation points.
The description of flow through multiple openings in a highway is complex. Plow through each opening i^ affected by the hydraulic characteristics of the approach channels as weli as the configuration and geometry of adjacent bridges. The methods described in this report to calculate discharge (figure 4) give resuits that are within +15 percent of the measured discharge in 17 of 28 cases, and are relatively simple to apply. Overall the bias was +2 percent with a rms error 4-18 percent. Schneiderjand others (1976) obtained about the same results for the bias (3 percent) but less scatter in the error (rms error about 9 percent). The need to divide the flood plain into a single equivalent channel for each bridge introduces additional error and also affects the backwater results.
The bias and rms error for the total fall. Ah, the backwater (h^) at section 1 and the backwater (h-) at section 3, computed by method I and method II are summarized in table 5. The errors are expressed as the difference in feet between the computed and measured value.
Comparison of the computed total fall with the measured total fall are shown in figures 5 and 6. Figure 5 shows that method I underestimates the total fall for 13 of the 17 bridges to which it
26
Table 5. Comparison of measured and computed backwater based on 17 bridge openings for method I and 28 bridge openings for method II.
Backwater BackwaterSection 1, h* at Section 3,
method I method II method I method II method I method II
Total Fall, Ah at Section I, h* at Section 3, h*
RMSE (ft) 0.23 0.35 0.34 0.39 0.15 0.43 Bias (ft) -.14 -.03 -.25 -.03 -.10 .02
27
IVJ
00
1.0
0.9
0.8
c
o " « « o _J I
°-6
O
0.5
,0 LL
0.4
T3 I
0.3
E
o 0
0.2
0.1
oo
oo
o
oLi
ne o
f eq
ual
valu
e
O
O
0
0.1
0.2
0
.3
0.4
0.5
0
.6
0.7
0
.8
0.9
1
.0
Obse
rved F
low
D
ivis
ion
Lo
ca
tio
n
Fig
ure
3.
Co
mp
ariso
n o
f m
ea
sure
d a
nd c
om
pu
ted
flo
w d
ivis
ion p
oin
ts
(sta
gnatio
n p
oin
ts o
n th
e u
pst
ream
sid
e o
f th
e em
bank
men
t).
50,000
Measured discharge, in cubic feet per second
Figure 4. Comparison of the measured and computed discharge.
29
0) 0)
2.0
1.8
1.6
1.4
1.2
0) g 1.0 a
8 0-8
0.6
0.4
0.2
I I I I I I I
D
Line of equal value
D
D
l _L0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Ah measured, in feet
Figure 5. Comparison of the computed and measured Ah for method I.
30
2.8
2.6
2.4
2.2
2.0
1.80) 0)
.£ 1-6« o
03 1.4 a
o 1.2
1.0
0.8
0.6
0.4
0.2
0
1 T 1 I"I I
D
Line of equal value,
D
D
D
Explanation
S With Spur Dikes D Without Spur Dikes
1 J___L0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2,
Ah measured, in feet
Figure 6. Comparison of the computed and measured Ah for method II.
31
applies. Figure 6 shows that for method II the results for 28 bridges are equally distributed about the .Line of equa... va^ue.
Comparison of the computed backwater with the measured backwater at section 1 are shown in figures 7 and 8. Figure 7 shows that method I underestimates the backwater, h^i for 13 of the 17 bridges. Figure 8 shows that for method II the resuits are equaiiy divided about the line of equal value.
Schneider and others (1976) calculated backwater by the method developed by Tracy and Carter (1955) and found it underestimated backwater. Method I is an improvement of the method to the extent that 1^ was replaced by Lav for computing friction loss. Hence, backwater calculated by method I is more accurate than the method developed by Tracy and Carter (1955) but 1£he bias of -0.25 ft for h- indicates it still underestimates backwater. The bias of -0.03 ft for the backwater, h^, calculated by method II is considered negligible. The slight increase in rms error from 0.34 ft for method I to 0.39 ft for method II probably is not significant. Schneider and others (1976) reported for method II bias of -0.04 ft with a rms error of 0.24 ft for single opening systems. The results from computation of discharge (table 4) indicate similar error in evaluating losses between section 1 and 3 for these sites. Therefore, these resuits indicate that the method developed for single-opening highway crossings can be applied to multiple bridges with comparable results.
The backwater, h3 ,at section 3, (fig. 9) computed by method I was less than that observed for 13 of the 17 bridge openings (tabie 4) and averaged 0.10 ft less than measured. Backwater, h., (fig- 10) computed by method II was equal or greater than measured at 17 of the 28 bridge openings and averaged 0.02 ft more than the measured value. Backwater at section 3 is difficult to measure accurately because of the turbulent flow condition and the large spatial changes in water surface elevation. In general h., is a relatively small value being less than 0.3 ft at all sites.
SUMMARY AND CONCLUSION
Data were collected for nine flood events to supplement laboratory studies of backwater at multiple bridge systems. These data consisted of measured discharge and water surface profiles through 28 bridge openings.
The multiple openings were divided i^nto equivalent single opening cases by apportioning the interior embankments in direct proportion to the area of the openings ori either side (Davidian and others,1962). The discharge was computed using procedures described by Matthai (1967) and by Schneider and otihers (1976). The best results were obtained by using the average flow path (Schneider and others, 1976) for approach friction losses in the method given by Mattahi (1967). This gave computed discharges within 15 percent of the measured values for 17 of the 28 openings. The mean error for
32
O (1)
2.0
1.8
1.6
1.4
1.2
1.0T3 0)
IE o
0.6
I I I I I I I I I
Line of equal value
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Measured h-*, in feet
1.6 1.8 2.0
Figure 7. Comparison of computed and measured backwater, h^* for method I.
33
E o u*T-
Explanation With spur dike
D Without spur dike
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.<
hj* measured, in feet1.6 2.0 2.2 2.4 2.6 2.8 3.0 3.2
Figure 8. Computed and measured backwater, h-j*, for method I.
34
0.4
0.3
2 0.2
oQ) *-*
IO O*« -
-0.2
-0.3
-0.4
1 T I I I T
Line of equal value
D
D
D
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2
(13* measured, in feet
0.3 0.4
Figure 9. Comparison of the computed and measured backwater, hg*, for method I
35
0>0 «*
C.
T3 "3
ao o
1.0
0.8
0.6
0.4
0.2
0
-0.6
-0.8
-1.0
-1.2
-1.4
Line of equal value
DD DD D D
D
Explanation
S With Spur Dikes O Without Spur Dikes
I Q j I I I0 0.2 0.4 0.6 0.8 1.0
f measured, in feet-0.4 -0.2
Figure 10. Comparison bf the computed and measured backwater, h3* f for method H.
36
ail openings was 2 percent with a root mean square error of 18 percent.
Backwater was measured by comparing the computed natural profile with the water-surface elevations obtained from high water marks. Backwater was computed by method I (Tracy and Carter, 1955) in which the friction loss term in the approach reach was modified to account for the average flow path, and by method II (Schneider and others, 1976). The bias and rms error for backwater at section 1 computed by method I are -0.25 ft and ^0.34 ft, respectively. Method I underestimates backwater for 14 of 17 sites. The bias and rms error for backwater at section 1 computed by method II are -0.03 ft and +0.39 ft, respectively, with results about evenly divided by the line of equal value. These results indicate that the method developed for single-opening highway crossings can be applied to the multiple-bridge opening.
37
REFERENCES
Benson, M. A., and Dairympie, T., 1967, General field and officeprocedures for indirect discharge measurements: U.S. Geoiogicai Survey Techniques of Water-Resources Investigations, Book 3, Chap. A1, 30 p.
Bradley, J. N., 1970, Hydraulics of bridge waterways: Federal Highway Administration, Hydraulic Design Series No. 1, 111 p.
Chow, V. T., 1959, Open-channel hydraulics: New York, McGraw-Hill, Inc., 680 p.
Cragwall, J. S., Jr., 1958, Computation of backwater at open-channel constrictions: U.S. Geological Survey Open-Pile Report, 23 p.
Davidian, Jacob, Carrigan, P. H., and Shen, John, 1962, Flow through openings in width constrictions: U*S. Geological Survey Water- supply Paper 1369-D, p. 108-112.
Hulsing, Harry, 1967, Measurement of peak discharge at dams byindirect methods: U.S. Geological Survey Techniques of Water- Resources Investigations, Book 3, Cfiap. AS, 29 p.
Kindsvater, C. E., and Carter, R. W., 1955, Tranquil flow throughopen-channel constrictions: American Society of Civil Engineers Transaction, v. 120, p. 955-980.
Kindsvater, C. E., Carter, R. W., and Triacy, H. J., 1953, Computation of peak discharge at contractions: U.S. Geoiogicai Survey Circular 284.
Lee, F. N., 1976, Field verification of method for distributingflow through multiple-bridge openings: Journal of Research of the U.S. Geological Survey, Vol. 4, No. 5, September-October, 1976, p. 539-543.
Lee, J. K., 1980, Two-dimensional finite element analysis of the hydraulic effect of highway bridge fills in a complex floodplain, in Wang, S. Y., and others, eds., Finite elements inwater resources: International Conference on Finite Elements in Water Resources, 3rd, University, Mississippi, 1980, Proceedings: University, Mississippi, University of Mississippi, School of Engineering, p. 6.3-6.23.
Lee, J. K., and Bennett, C. S., Ill, 198 study of the impact of the proposed
1, A finite-element model 1-326 crossing on flood
stages of the Congaree River near Ctoiumbia, South Carolina: U.S.Geological Survey Open-Fiie Report 81-1194, 56 p.
Lee, J. K., Froehlich, D. C., Gilbert, J. J., and Wiche, G. J.,1982, Two-dimensionai analysis of bridge backwater, in American Society of Civil Engineers, Applying research to hydraunc
38
practice: Hydraulics Division Conference, Jackson, Mississippi, 1982, Proceedings: New York, American Society of Civii Engineers.
Matthai, H. P., 1967, Measurement of peak discharge at widthcontractions by indirect methods: U.S. Geological Survey Techniques of Water-Resources Investigations, Book 3, Chap. A4, 44 p.
Schneider, V. R., Board, J. W., Cblson, B. E., Lee, F. N., andDruffel, L., 1976, Computation of backwater and discharge at width constrictions of heavily vegetated flood plains: U.S. Geological Survey Water-Resources Investigations 76-129, 64 p.
Shearman, J. O., 1976, Computer applications for step-backwater and floodway analyses: U.S. Geological Survey Water-Resources Investigations 76-499, 119 p.
Tracy, H. J., and Carter, R. W., 1955, Backwater effects ofopenchannel constrictions: American Society of Civil Engineers Transaction, v. 120, p. 993-1006.
39