Design of spillway crests ASCE

Prévia do material em texto

DESIGN OF SPILLWAY CRESTS 
By Andrew J. Reese,1 M. ASCE, and Stephen T. Maynord,2 
A. M. ASCE 
ABSTRACT: A design procedure for overflow spillway crests, which is appli­
cable to a wide range of approach depths, upstream slopes, and heads on the 
crest, is developed. An earlier crest design procedure is used to proportion a 
series of models covering a wide range of design conditions. Discharge coef­
ficients, crest pressures, and water surface profiles are determined for each model, 
both with and without piers on the crest. Results are presented in graph or 
tabular form. Using these results a design procedure is suggested and further 
design aids developed. 
INTRODUCTION 
The basic purpose of most spillways is to convey large flows due to 
floods through a project without endangering the safety of the dam and 
without unacceptable flood damage either up - or downstream. The de­
sign engineering team attempts to satisfy the following requirements and 
to maintain cost effectiveness: (1) Sufficient length of crest to pass design 
discharge; (2) acceptable minimum pressures on spillway structure; (3) 
acceptable maximum head over the crest; (4) hydraulic stability and de­
terminate hydraulic characteristics; (5) structural stability; (6) acceptable 
exiting flow conditions to mitigate downstream erosions; and (7) geo­
logic, topographic, environmental, and aesthetic considerations. The ca­
pacity of a spillway is determined by its length, shape, height, gate size, 
approach conditions, and pier or abutment geometries. In preliminary 
investigations, it normally appears that a narrow spillway with high unit 
discharge will be less expensive than a longer spillway with moderate 
unit discharge. Thus, within the Corps of Engineers, e.g., allowable heads 
have increased from 40 to about 60 ft (12.2-18.3 m) with an increase in 
unit discharge from 1,000-2,000 cfs/ft (91.8-183.6 m 3 / s /m) . These higher 
design heads can create excessive abutment and pier contractions, cause 
energy dissipation problems and increase the possibilities of cavitation 
or pulsating nappe over the spillway crest. Because of the growing re­
quirement for increased efficiency in spillway design, a need was felt 
for reliable design data and a simple yet flexible design method that can 
both stand alone and serve as a basis for model studies. 
CREST DESIGN 
Lower Nappe Fitting.—In order to meet the foregoing constraints and 
yet maintain a minimum crest length, it is desirable that the overflow 
'Proj. Mgr., MCI Consulting Engrs., Inc., Nashville, TN 57204. 
2Research Hydr. Engr., U.S. Army Engr. Waterways Experiment Sta., Vicks-
burg, MS 39180. 
Note.—Discussion open until September 1, 1987. To extend the closing date 
one month, a written request must be filed with the ASCE Manager of Journals. 
The manuscript for this paper was submitted for review and possible publication 
on May 20, 1986. This paper is part of the Journal of Hydraulic Engineering, Vol. 
113, No. 4, April, 1987. ©ASCE, ISSN 0733-9429/87/0004-0476/$01.00. Paper No. 
21384. 
476 
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I X 
o 
FIG. 1.—Definition Sketch 
crest be shaped so that a high discharge coefficient results and that pres­
sures are fairly uniform and predictable. It has long been felt that these 
constraints can best be met if the shape of the overflow spillway closely 
approximates the lower nappe surface of a fully ventilated nappe of water 
flowing over a sharp-crested weir. The shape of this nappe is affected 
by the relative head on the weir, the approach velocity, and the up­
stream face slope of the weir. Thus, should experimental data be gath­
ered throughout a suitable range of these variables, a spillway design 
method could possibly be developed. The earliest attempts at the fitting 
of equations to lower nappe surfaces utilized the data of Bazin and are 
recounted elsewhere (Cassidy 1970; Rouse and Reid 1935; Bureau of Rec­
lamation 1948). Perhaps the best data were provided by the United States 
Bureau of Reclamation (TJSBR 1948) and has served as a basis for most 
Corps of Engineers (COE) design procedures. 
Earlier Design Procedures.—The complete shape of the lower nappe 
was earlier described by separating it into the two quadrants, one up­
stream and one downstream from the high point (apex) of the lower 
nappe surface (termed the crest axis). See Fig. 1 for a definition of terms 
used. He = the actual total head on the crest; and ha = the approach 
velocity head. The depth used to compute ha is equal to the pool ele­
vation minus the elevation of the upstream toe of the spillway, and flow 
is assumed to be normal to the spillway face. The spillway is propor­
tioned based on some design total head, Hd. He can be greater than, 
equal to, or less than Hd. As the relative height of the spillway de­
creases, the effect of approach velocity becomes important, and at P/Hd 
£ 1.0, it can no longer be neglected according to present Corps of En­
gineers design procedures (COE 1965, 1977). USBR (1948) data in agree­
ment with Rouse (1938) demonstrate that as h„/He increases or upstream 
slope of the weir face decreases, the nappe contraction (Ye) flattens and 
the crest axis moves slightly upstream. Murphy (1973) explains that for 
a low P/Hd value and a vertical face, flow tends to move down the ver­
tical face, creating an eddy rather than up the face as irrotational flow 
theory would indicate. A sloping upstream face tends to prevent this 
occurrence. On the other hand, a high spillway normally does not re­
quire a sloping face for stability purposes, and at times, savings in con­
crete can result by extending the crest beyond the upstream face, cre-
477 
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ating an overhanging crest. Details are provided in Corps of Engineers 
Hydraulic Design Criteria (HDC) (COE 1977). 
To account for these effects, the COE has divided its design procedure 
into four categories: high and low spillways with either vertical or slop­
ing face. The equation for the downstream quadrant of the crest for all 
spillways can be expressed as 
X" = KHT^Y (1) 
where X = horizontal coordinate positive to the right; Y = vertical co­
ordinate positive downward; K = a constant that varies from 1.85-2.08; 
and n = a constant that varies from 1.75-1.85. 
It has proven more difficult though to fit a single equation to the up­
stream quadrant. Rouse and Reid (1935) and Melsheimer and Murphy 
(1970) have shown that the curvature of the crest immediately upstream 
of the crest axis in large part determines the efficiency of the spillway. 
A sudden change in curvature or a discontinuity not only disrupts the 
boundary layer but also can lead to flow separation and cavitation. Mur­
phy (1973) reports a 3% increase in the discharge coefficient when a small 
discontinuity between the upstream face and upstream quadrant was 
removed. Four separate upstream quadrant curves were compared by 
Melsheimer and Murphy (1970), for energy heads between 0.25 Hd and 
1.5 Hd. Of the four tested, two used compound circular curves, one used 
a curve adjusted from relaxation technique fitting (McNown, et al. 1955), 
and one used a quarter of an ellipse. Of the four curves, the tricom­
pound circle (Abecasis 1944; COE 1977) and the ellipse methods yielded 
the most satisfactory results in terms of flow efficiency and pressures. 
The tricompound circle rnethod has been in general use in the Corps of 
Engineers since that time. Its application is hampered,however, since 
design data have been obtained for only a few representative cases with­
out any generally applicable formulation. 
Elliptical Upstream Quadrant.—In an attempt to provide a general 
and flexible design procedure, Murphy (1973) found that by systemati­
cally varying the axes of an ellipse with depth of approach, he could 
closely approximate the lower nappe surfaces generated by USBR. Fur­
thermore, he found that any sloping upstream face could be used with 
little loss of accuracy if it was attached tangent to the ellipse calculated 
for a vertical upstream face. He also found that the downstream quad­
rant could be closely approximated by setting n in Eq. 1 equal to 1.85 
for all cases and varying K with depth of approach. 
The general ellipse equation is expressed as 
2 2 
x v 
—. + —7 = 1 (2) 
A2 B2 w 
where A = one-half horizontal axis of ellipse; B = one-half vertical axis 
of ellipse; x = horizontal coordinate origin at ellipse center; and y = 
vertical coordinate origin at ellipse center. 
These three parameters {A, B, and K) then fully define crest shape. 
Their variation with relative approach depth is given in Fig. 2. This pro­
cedure was verified for high spillways by Murphy, who recommended 
further model studies for other upstream face and crest height config­
urations. 
478 
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8 . 0 — -
6.0 
13 
X o n 
3N
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A
D
 
—
 
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W " ° 
°06 
n? A 
n « '1 
— f— 
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i 
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j 
/ 
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A 
0.21 0.23 0.26 0.27 0.29 0.12 0.14 0.16 0.181.90 2.10 2.30 
A/H r i B/H, K 
FIG. 2.—Crest Coordinate Equation Parameters (Ref. 9, Plate 2) 
Model Studies.—In order to obtain data to verify this design approach 
over a wide range of approach depths, upstream slopes, and heads on 
the crest, a series of model studies were conducted by Maynord (1985). 
Discharges, crest pressures, and water surface profiles both with and 
without crest piers were measured. The studies covered P/Hd values from 
0.25-2.0 and He/Hd values of 0.4-1.5. Upstream face slopes were varied 
from vertical to 2V:3H. 
A 2.5-ft (0.763-m) wide flume with a horizontal floor and vertical sides 
lined with sheet metal was used in the investigation. Permeable baffles 
were used at the inlet to distribute all flow approaching the crest as 
uniformly as possible. Discharges were measured in 8- by 4- and 20- by 
10-in. venturi meters. Calibrations of the venturi meters were checked 
with a 90° u-notch weir. The calibration curves for the venturi meters 
0.038 0.013 
1.696Y 
FIG. 3.—Example Crest Instrumentation Locations 
479 
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and the curve based on the w-notch weir agreed to within less than 1%. 
Water surface elevations were measured with a point gage mounted on 
rails set to grade along the sides of the flume. 
The crests used in the investigation were constructed of sheet metal 
to conform to the shapes required for the design head of 0.80 ft (0.24 
m). Simple piezometers were installed along the face of the crest to mea­
sure the average pressures acting on the crest. A typical crest with pi­
ezometers is shown in Fig. 3. The piezometers are staggered in two rows 
to increase spacing between piezometers. 
Each test began with determination of the discharge coefficient C. Next, 
static pressures along the face of the crest and water surface profiles 
were measured. Then piers were attached to the crests and the pier con­
traction coefficients were determined. Finally, static pressures and water 
surface profiles were measured at the centerline of the bay and along 
the piers. 
TEST RESULTS 
Discharge Coefficients.—A list of the spillway crests tested to deter­
mine discharge coefficients is shown in Table 1. 
The spillway discharge coefficient was determined so as to fit the stan­
dard weir equation: 
Q = CLeHi5 , (3) 
where Q = discharge; C = discharge coefficient; Le = effective crest length; 
and He = energy head on crest. 
The discharge coefficient, C, was found to vary with approach depth, 
P, upstream face slope, and relative head on the crest, He/Ha. This re­
sult was expected and can be compared with other data (Cassidy 1970; 
TABLE 1.—Crests Tested to Determine Discharge Coefficients 
Crest number 
0) 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
P/Hd 
(2) 
0.25 
0.25 
0.25 
0.25 
0.25 
0.50 
0.50 
0.50 
0.50 
0.50 
1.00 
1.00 
1.00 
1.00 
1.00 
2.00 
2.00 
Upstream face slope 
(3) 
Vertical 
3V:2H 
1V:1H 
2V:3H 
1V:2H 
Vertical 
4V:1H 
3V:2H 
IV: 1H 
2V:3H 
Vertical 
4V:1H 
2V.-1H 
IV: 1H 
2V:3H 
Vertical 
1V:1H 
480 
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3-6 3.8 4.0 
FIG. 4.—Discharge Coefficients FIG. 5.—Discharge Coefficients (1H :1V-
(Steeper than 1V:1H Face) 2V:3H Face) 
USBR 1948). The discharge coefficient can be thought of as a measure 
of the flow efficiency of the system and thus is also affected by some 
site-specific factors. Flow angularity or complex flow geometry, e.g., can 
be significant. Should the combination of flow angularity and approach 
velocity become excessive, model studies or conservative estimates of 
abutment and pier effects should be used. In this study, such effects 
were eliminated except for a subsequent introduction of piers. 
Basic model discharge, static head, and computed energy head data 
are not presented but are available in Maynord (1985). Static head mea­
surements for the P/Hd = 0.25 crests were difficult at heads greater than 
the design head because of significant turbulence in the approach flow. 
The energy correction factor was assumed equal to 1.0 for all compu­
tations. 
Although there was a weak general tendency for flatter upstream slopes 
to result in smaller discharge coefficients (especially at higher P/Hd val­
ues), data appear to be best represented by two sets of curves. Fig. 4 
serves for all upstream faces steeper than IV: 1H, and Fig. 5 for all faces 
from 1V:1H-2V:3H. The variation between the curve and measured C 
values using these two curve families is everywhere less than 2%. 
Because of possible scale effects, discharges were not measured below 
He/Hd = 0.4. However, prototype experience has shown that spillway 
crests at very low heads exhibit the same discharge characteristics as a 
broad-crested weir. Therefore, for extrapolation purposes, the discharge 
coefficient is set equal to 3.08 as He/Hd approaches zero in Figs. 4 and 
5 (dotted lines). The P/Hd = 3.4 curve is taken from Melsheimer and 
Murphy (1970) and agrees with the COE design curve for all high spill­
ways with standard crest shape (COE 1977). Thus this curve may be 
used to approximate C values above P/Hd = 3.4. 
481 
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At the other extreme, as P/Hd values decrease and particularly for higher 
values of He/Hd control of the flow begins to shift upstream, efficiency 
is lost, the discharge coefficient decreases, and C would again approach 
that of a broad-crested weir. Also to be noted is the characteristicin­
crease in discharge coefficient for heads greater than design head. Thus, 
the concept of "underdesigning" the spillway crest is applicable and will 
be discussed in a subsequent section. 
Discharge coefficients were found to be 1-3% higher than for venti­
lated sharp-crested weir data (USBR 1948) as expected (COE 1977). This 
is due to the substitution of a solid boundary for the underside of the 
nappe and the resulting lower pressures; however, the form of the data 
curves followed the flat plate data closely. 
Crest Pressures.—Pressures for crests with and without piers were 
measured for vertical and IV: 1H upstream faces for P/Hd values of 0.25, 
0.5, and 1.0. At P/Hd = 0.25, pressures were measured for HJHd = 0.5 
and 1.0 only. Using an underdesigned crest for this low of a P/Hd value 
does not result in a significant increase in discharge coefficient above 
HJHd = 1.0. 
Two piers were placed on the 2.5-ft (0.763-m) wide crests and located 
as shown in Fig. 6. The pier nose used for all crests was the Type 3 
shown in HDC 111-5 (COE 1977). The pier nose was located in the same 
plane as the upstream face for the vertical spillway. For the IV: 1H up­
stream slope, the pier nose location was determined by maintaining the 
same distance from pier nose to crest axis as used in the vertical up­
stream-faced crest. Piezometer taps along the piers were located 0.02 ft 
(6.1 mm) from the edge of the pier to the centerline of the tap hole. 
Figs. 7 and 8 indicate typical pressure distributions for crests without 
and with piers, respectively. The numbers on the curves indicate HJHd 
values. Both figures are for P/Hd = 1.0. 
These pressures indicate some variations that are caused by the method 
of constructing the model crests. The crests were constructed out of sheet 
metal. Plastic crests, which are the best method of accurately reproduc­
ing the true shape of the crest, were used by Melsheimer and Murphy 
(1970) in their tests. However, because of the large number of crests 
used in this investigation, the cost for the use of plastic crests was pro­
hibitive. Sheet metal crests are much less difficult to construct and give 
a good representation of the crest shape. Discharge coefficients should 
be consistent with the plastic crests since previous studies (Johnson 1944; 
Nagler and Davis 1930) have shown surface roughness effects to be neg­
ligible. However, the fact that the sheet metal may not bend in a true 
arc but in a series of small cords can lead to small local variations in the 
pressures measured on the crest and result in a less smooth pressure 
distribution than would otherwise be the case. Notice that at HJHd = 
1, the pressures are near atmospheric. This is an indication that the crest 
is proportioned to correctly fit the lower surface of a sharp-crested weir 
nappe. 
Since it is maximum negative pressure that is of greatest concern to 
designers, plots of dimensionless head, HJHd, versus maximum neg­
ative static pressure head, Hp/Hd, are presented in Figs, 9 and 10. The 
point clusters represent different P/Hd values for a given HJHd ratio. 
The maximum negative static pressure for each value of He/Hd did not 
482 
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FIG. 6.—Pier Location (1 ft = 0.305 m) FIG. 7.—Crest Pressure Distribution 
without Piers {P/Hd = 1.0) 
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CENTER LINE OF BAY 
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HORIZONTAL DISTANCE , W L , , 
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I V : 1H UPSTREAM FACE 
FIG. 8.—Crest Pressure Distribution with Piers (P/Hd = 1.0) 
483 
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FIG. 9.—Minimum Pressure Design FIG. 10.—Minimum Pressure Design 
Curve (No Piers) Curve (with Piers) 
appear to vary systematically with P/Hd. Note that when piers are pres­
ent, it is the minimum pressure along the pier that limits head on the 
crest and not the centerline value. 
It has been recommended (USBR 1977) that an HJHd ratio of 1.33 be 
used for the underdesign of a spillway crest. However, as pointed out 
by Bauer and Beck (1969) and Abecasis (1970), it is the actual minimum 
pressure fluctuation level in relation to atmospheric pressure for a given 
climate and altitude that leads to cavitation. Vacuum tank observations 
by Abecasis (1961) indicated that cavitation on the crest would be incip­
ient at an average pressure of about -25 ft (7.63 m) of water. Since it is 
not average pressure but fluctuating pressures that cause spillway cav­
itation, it is recommended in the Corps of Engineers (COE 1986) that a 
spillway crest be designed so that the maximum expected head will re­
sult in an average pressure no lower than —15—20 ft (-4.6-6.1 m) of 
water with -15 ft (-4.6 m) used for high head design. 
Using this recommendation, Fig. 11 is a transposition of Figs. 9 and 
10. Also plotted for comparison purposes is the lower portion of Abe­
casis' (1970) envelope, which is equivalent to a pressure of -25 ft (7.63 
m). 
Construction irregularities may also limit the allowable underdesign 
of a spillway crest, particularly for high head design. Incipient cavitation 
for various surface irregularities is given by Ball (1976). 
If a crest with piers is designed for negative pressures, the piers must 
be extended downstream beyond the negative pressure zone in order to 
prevent aeration of the nappe, nappe separation or undulation, and loss 
of the underdesign efficiency advantage. For preliminary design pur­
poses when He/Hd equals 1.17, 1.33, and 1.5, the approximate ranges of 
the dimensionless horizontal distance from the crest axis, X/Hd, wherein 
484 
 J. Hydraul. Eng., 1987, 113(4): 476-490 
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1.4 1.5 1.6 
FIG. 11.—Design Curves for -15 ft (4.6 m) and -20 ft (-6.1 m) PressureHead (1 
ft = 0.305 m) 
pressures were found to return positive, are 0.1-0.4, 0.7-0.9, and 1.1-
1.5, respectively. If an aeration slot is used to prevent cavitation dam­
age, the crest should not be undersigned if the slot is located in the zone 
of negative pressure. 
Pier Contraction Coefficients.—Pier contraction coefficients, Kp, were 
determined for P/Hd of 0.25, 0.5, and 1.0, and are shown in Fig. 12. Kp 
modifies the crest length according to Eq. 4: 
Le = U - 2(NKp + Ka)He (4) 
where U = net uncontracted crest length; N = number of piers; and K„ 
-0.15 -0.1 -0.05 0 0.05 0.1 0.15 
FIG. 12.—Pier Contraction Coefficients (Type III Pier) 
485 
 J. Hydraul. Eng., 1987, 113(4): 476-490 
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 J. Hydraul. Eng., 1987, 113(4): 476-490 
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TABLE 2.—Continued 
(c) — = 0.25 
H,, 
WITHOUT PIERS 
X 
H„ 
H, 
Hj 
(1) 
-1.0 
-0.8 
-0.6 
-0.4 
-0.2 
0.0 
0.2 
0.4 
0.6 
0.8 
1.0 
1.2 
1.4 
r 
H4 
0.5 
(2) 
-0.452 
-0.452 
-0.446 
-0.435 
-0.414 
-0.378 
-0.319 
-0.233 
-0.120 
0.020 
0.188 
0.375 
0.578 
1.0 
(3) 
-0.768 
-0.759 
-0.750 
-0.735 
-0.712 
-0.678 
-0.629 
-0.550 
-0.453 
-0.331 
-0.172 
0.008 
0.212 
CENTERUNE OF GATE BAY 
X 
Hd 
H, 
Hd 
(4) 
-1.0 
-0.8 
-0.6 
-0.4 
-0.2 
0.0 
0.2 
0.4 
0.6 
0.8 
1.0 
1.2 
1.4 
y 
Hd 
0.5 
(5) 
-0.469 
-0.469 
-0.464 
-0.454 
-0.438 
-0.405 
-0.358 
-0.260 
-0.151 
-0.018 
0.135 
0.315 
0.528 
1.0 
(6) 
-0.850 
-0.848 
-0.839 
-0.823 
-0.796 
-0.758 
-0.715 
-0.640 
-0.553 
-0.448-0.303 
-0.135 
+0.045 
ALONG PIERS 
X 
Hd 
Hd 
(7) 
-1.0 
-0.8 
-0.6 
-0.4 
-0.2 
0.0 
0.2 
0.4 
0.6 
0.8 
1.0 
1.2 
1.4 
Y 
Hi 
0.5 
(8) 
-0.469 
-0.469 
-0.466 
-0.469 
-0.488 
-0.414 
-0.286 
-0.175 
-0.066 
+0.061 
+0.209 
+0.378 
+0.577 
1.0 
(9) 
-0.838 
-0.835 
-0.833 
-0.835 
-0.894 
-0.900 
-0.756 
-0.615 
-0.471 
-0.311 
-0.139 
+0.044 
+0.250 
used with caution since differing approach conditions can cause signif­
icant variation. 
Water Surface Profiles.—Water surface profile data were taken for crests 
with P/Hd of 0.25, 0.50, and 1.0 and are shown in Table 2. Water surface 
profile data for high crests can be found in HDC (COE 1977). These 
elevations are used in designing walls, determining clearances, etc. Dif­
ferent upstream face slopes had little measurable effect on the water sur­
face elevations. 
DESIGN APPLICATION 
In order to calculate the coordinates of the elliptical spillway, coordi­
nate equations and tangency points must be located. Remembering that 
the vertical coordinate is positive in the downward direction and is mea­
sured from the crest axis, the upstream quadrant coordinates can be found 
from a transposition of 
X2 (B - Yf 
A^ + B2 
= 1 (5) 
or Y = B 1 - 1 
X2 
A2 
Assuming the upstream face slope to be 
F - - & .. 
s dx 
(6) 
(7) 
487 
 J. Hydraul. Eng., 1987, 113(4): 476-490 
D
ow
nl
oa
de
d 
fr
om
 a
sc
el
ib
ra
ry
.o
rg
 b
y 
U
SP
 -
 U
ni
ve
rs
id
ad
e 
de
 S
ao
 P
au
lo
 o
n 
02
/2
7/
19
. C
op
yr
ig
ht
 A
SC
E
. F
or
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er
so
na
l u
se
 o
nl
y;
 a
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ri
gh
ts
 r
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ve
d.
an equating of the derivative of Eq. 6 to Fs yields 
B2X 
A\Y - B) ^ = 7 1 ^ ^ : (8) 
Solving for X and Y, respectively, and substituting into Eq. 6 yields the 
required upstream tangent points: 
X n r ™ (9) 
<P 
B2 
and YUT = B (10) 
<|> 
where $ = (A2F2 + B2)1/2. Eq. 1 gives the downstream coordinates. The 
downstream tangent can be found in a similar way as 
/ K Y"76 
Mi^J * • w 
and YDT = - (j^j Hd (12) 
0.262K1176 
or Ym = n ^ H , (13) 
a ' 
where a = 1/downstream slope = H/V; and subscripts UT and DT = 
upstream and downstream tangents, respectively. 
For design purposes, Figs. 2, 4, 5, and 9-12 and Table 2 can be used 
directly. Depending on available data and design constraints, Figs. 4 or 
5, 9 or 10, and 11 will be used iteratively to determine crest height and 
amount of underdesign allowable. The dashed curves in Fig. 11, for a 
maximum negative pressure of -15 ft, can be approximated as 
Hd = 0.33HJ22 (14) 
Hd = 0.30Hf
126 (15) 
for the uncontrolled (no piers) and controlled (along piers) cases, re­
spectively. 
Example.—For a controlled crest at —15 ft pressure minimum, with 
Qmax = 410,000 cfs at maximum reservoir elevation at the crest of 1,000-
ft NGVD and an approach channel elevation = 920-ft NGVD, then as­
sume a head, He, of 45 ft at the maximum discharge. Then from Fig. 11 
or Eq. 15, Hd = 36.3 ft and He/Hd = 1.24. Dam height, P, then equals 
35 ft and P/Hd = 0.96. From Fig. 4 (assuming a vertical upstream face), 
C = 4.04 at He/Hd = 1.24. 
From Eq. 3, the effective spillway length must be equal to 336.2 ft. 
Use six gates and five piers with Kp = -0.02 for HJHd = 1.24 (Fig. 12). 
Assuming adjacent embankment sections, K„ = 0.19 (see COE 1977, Chart 
111-3/2). For a pier width of 0.267 Hd (see COE 1977, Chart 111-5), the 
total physical crest length between abutments is about 393 ft. This is 
calculated by solving Eq. 4 for L' and adding pier widths. 
488 
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CONCLUSIONS 
The elliptical upstream quadrant spillway design as proposed by Mur­
phy (1973) has been tested over a wide range of conditions. It has been 
shown to yield reliable hydraulic characteristics and has been demon­
strated to closely fit the lower nappe of a ventilated sharp-crested weir. 
Though prototype data are, as yet, lacking, unpublished model study 
results of a spillway designed using this method compare well with pre­
dicted values. 
The advantage to using this method is that it is supported by system­
atic experimental data, and it is applicable to spillways of any practical 
height or upstream face slope. 
ACKNOWLEDGMENTS 
The tests described and the resulting data presented herein, unless 
otherwise noted, were obtained from research conducted under the 
General Spillway Tests of the United States Army Corps of Engineers 
by the Waterways Experiment Station, at Vicksburg, Mississippi. Per­
mission was granted by the Chief of Engineers to publish this infor­
mation. 
APPENDIX I.—REFERENCES 
Abecasis, F. M. (1944). "Letter no. 3281," Office, Chief of Engineers, U.S. Army 
Corps of Engineers. 
Abecasis, F. M. (1961). "Spillways—some special problems," Memorandum No. 
175, National Laboratory of Civil Engineering, Lisbon, Portugal. 
Abecasis, F. M. (1970). Discussion of "Designing spillway crests for high-head 
operation," by J. J. Cassidy, /. Hydr. Engrg., ASCE, 96(12), 2655-2658. 
Ball, J. W. (1976). "Cavitation from surface irregularities in high velocity flow," 
/. Hydr. Engrg., ASCE, 102(9), 1283-1297. 
Bauer, W. J., and Beck, E. J. (1969). "Spillways and stream-bed protection works," 
Section 20, Handbook of Applied Hydraulics, 3rd ed., C. V. Davis and K. E. So-
rensen, Eds., McGraw-Hill Book Co., New York, N.Y. 
Cassidy, J. J. (1970). "Designing spillway crests for high-head operation," /. Hydr. 
Engrg., ASCE, 96(3), 745-753. 
"Design of small dams" (1977). U.S. Dept. of the Interior, Bureau of Reclamation, 
Washington, D.C. 
"Hydraulic design of spillways" (1965). Engineer manual 1110-2-1603, U.S. Army 
Corps of Engineers, Vicksburg, Miss. 
"Hydraulic design criteria, revision 17" (1977). U.S. Army Corps of Engineers, 
Vicksburg, Miss. 
"Hydraulic design of spillways, draft" (1986). Engineer manual 1110-2-1603, U.S. 
Army Corps of Engineers, Vicksburg, Miss. 
Johnson, J. W. (1944). discussion of "Spillway coefficients," by G. H. Hickox and 
"Meter measurements of discharge, University dam," by E. Soucek. Trans., 
ASCE, 109, 120-123. 
Maynord, S. T. (1985). "General spillway investigation," Technical Report HL 85-
1, U.S. Army Corps of Engineers Waterways Experiment Station, Vicksburg, 
Miss. 
McNown, J. S., Hsi, E.-Y., and Yih, C.-S. (1955). "Applications of the relaxation 
technique in fluid mechanics," Trans., ASCE, 120, 650-686. 
Melsheimer, E. S., and Murphy, T. E. (1970). "Investigations of various shapes 
of the upstream quadrant of the crest of a high spillway," Research Report H-
70-1, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss. 
Murphy, T. E. (1973). "Spillway crest design," Misc. Paper H-73-5, U.S. Army 
Corps of Engineers Waterways Experiment Station, Vicksburg, Miss. 
489 
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Nagler, F. A., and Davis, A. (1930). "Experiments on discharge over spillways 
and models, Keokuk Dam," Trans., ASCE, 94, 777-820. 
Rouse, H., and Reid, L. (1935). "Model research on spillway crests," Civ. Engrg., 
5(1), 11-16. 
Rouse, H. (1938). Fluid mechanics for hydraulic engineers, 1st ed., McGraw-Hill Book 
Co., New York, N.Y. 
"Studies of crests of overfall dams" (1948). Boulder Canyon Final Reports, Part VI, 
Bulletin 3, U.S. Department of the Interior, Bureau of Reclamation, Washing­
ton, D.C. 
APPENDIX II.—NOTATION 
The following symbols are used in this paper: 
A = one-half horizontal axis of ellipse; 
B = one-half vertical axis of ellipse; 
C = discharge coefficient; 
E = elevation, L; 
Fs= upstream crest face slope; 
H = measured head on crest excluding velocity head, L; 
Hd = crest design total head, L; 
He = total head on crest, L; 
Hp = pressure head, L; 
h„ = velocity head on crest, L/T; 
K = downstream coordinate coefficient; 
Ka = abutment contraction coefficient; 
Kv = pier contraction coefficient; 
L' = net uncontracted crest length, L; 
Le = effective crest length, L; 
N = number of piers; 
n = downstream coordinate coefficient; 
P = height of spillway above approach channel, L; 
Q = discharge, (I?/T); 
X = horizontal coordinate relative to crest axis, L; 
XUT ,XDT = upstream and downstream tangent points, respectively, 
with horizontal component, L; 
x = horizontal coordinate in ellipse equation, L; 
Y = vertical coordinate relative to crest axis, L; 
Ye = nappe contraction, L; 
^UT/Y'DT = upstream and downstream tangent points, respectively, 
with vertical component, L; 
y = vertical coordinate in ellipse equation, L; 
a = inverse of downstream face slope; and 
t> = (A2F2
3 + B 2 ) 1 / 2 . 
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