UNITED ST ATES DEPARTMENT OF THE INTERIOR ... 416 .B87 HYD-325 1951 UNITED ST ATES DEPARTMENT OF THE...
Transcript of UNITED ST ATES DEPARTMENT OF THE INTERIOR ... 416 .B87 HYD-325 1951 UNITED ST ATES DEPARTMENT OF THE...
..
- ... TA 416 .B87 HYD-325 1951
UNITED ST ATES DEPARTMENT OF THE INTERIOR
BUREAU OF RECLAMATION
STABLE CHANNEL PROFILES
Hydraulic Laboratory Report No. Hyd-325
ENGINEERING LABORATORIES BRANCH
COMMISSIONER'S OFFICE DENVER, COLORADO
September 27, 1951
..
_,
\
DATE DUE
t..o I Uol o 3
-GAYLORD
PRINTED IN U.S.A.
The information contained in this re -
port may not be used in any publica
tion, advertising, or other prom otion
in such a manner as to constitute an
endorsement by the Government or the
Bureau of Reclamation, either e xpl ic i t
or implicit, of any material, p roduct,
device, or process that may be re
ferred to in the report.
-- -._ .. -
Introduction ••••• Basis ot Computations
CONTENTS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotation • • • . . . • . . • . . . • • • • • . • • . • Stable Channel Shape with Drag Proportional to Depth. Stable Channel with a Drag Force Proportional to
the Square ot the Velocity and no Lateral · Transter ot Drag • • • • • • • • • • • • • • • •
Stable Channel Shape with Lateral Drag Transter Accounted tor ••••••••••••••••••
Discussion ot Compilation of Data and Curves ••••• Example,· • • • • • • • • • • • • • • • • • • • • • • • Acknowledgments • • • • . • • • • • • • • • • • • • •
Computation tor S • 0~3 Computation tor S • 0.5 Computation tor S • o.6 Computation tor S • o.8 Computation tor S • 1.0
LIST OF TABLES
. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIST OF FIGURES
Channel cross section • • • • • • • • • • • • • • • • Element of a wide channel ••••••••••••••
. 2 Comparison of values of (J-) for the drag relation
0
and the modified Chezy relation ••••••••• Properties of stable channels •••••••••••• Stable channel profiles found as a solution tor the
example • • • • • • • • • • • • • • • • tit • • • •
1 1 2 4
5
7 13 15 19
Table
1 and 2 3 and 4 5 and 6 7 and 8 9 and 10
Figure
1 2
3 through 7 8 through 12
13
UNITED STATES DEPARTMENT OF THE INTERIOR
BUREAU OF RECLAMATION
Design and Construction Division Engineering Laboratories Branch Denver, Colorado
Laboratory- Report No. Hyd-.325 Hydraulic Laboratory-Compiled by: R. E. Glover and
Q. L. Florey Reviewed by: E. W. Lane
September 27, 1951
Subject: Stable channel profiles
INTRODUCTION·
In order to deal intelligently with certain problems or maintenance of channels in erodible materials, it is important to know what shape and dimensions a channel should have in order to avoid changes of cross section due to scour or sediment deposition. A knowledge of stable shapes finds application to the problem or designing canals in earth and to the problems or river regimen. The computations of stable shape patterns described herein have been made at the request of Mr. E.W. Lane who has also laid down the basis for the computations.
BASIS OF COMPUTATIONS
The problem of finding a stable channel shape is here based on the following assumptions:
a. The particle is held against the bed by the canponent or the submerged weight or the particle acting normal to the bed.
b. At the edge of the stream the particles are at the angle of repose under the action or gravity.
c. At the· center or the stream the drag force of the flowing water holds the particles at the point or incipient instability.
d. At points between the edge and the center the particles are held at the point or incipient instability by the resultant of the gravity component or the particle's submerged.weight and the drag force or the fiowing water.·
In dealing with the drag force of th~ flowing water, two assumptions have been used as follows: ·· ·
e. The drag fore~ ·acting ··on ·an ar~a of the bed of the stream is produced by the component of the weight of the water directly above the.area, which acts in the di,r~ction of flow. This component is equf:1,1 to :the above weignt multiplied by the .. stream gradient. · · · ·
f. Tqe drag force acting· on a particle is proportional to the square of the mean velocity in the stream at the point where the particle is located.
Use of assumption (e) permits a simpler treatment than is possible with assumption (f) but does not permit the lateral transfer of drag due to velocity gradients in_the .horizontal direction.to be taken into account. This. factor can b,e included,. in the computat_ions when assumption Ji) fs used. ' · · ·
NOTATION ,·· .
' ' . '
A _ r,epresents the _ cross sectional area of the channel
a=; or ~alf the top width
c, Chezy's coefficient
D, depth of_ an in_finitely wide channel
f, longitudinal slope or friction slope
m, _ factor of, ~roportionality
n, coe_f_f'icient of .. roµghness
P, perimeter
Q; quanti,ty . of fl.ow ( cu:t,ic ft./ se_c)
q, drag force on a surface of a lamina of uriit len~h A , . . :
.r • P' tb,e hydr,ulic radi~s
s = tan g, the transver·~.e slope at the outer edge of the channel · · ·
2
T, top width
u. v2
v, mean velocity of the lamina at a distance x from the center of a channel
Ve, mean velocity of the flow in an infinitely wide channel of depth D = Yo
Vm, mean velocity of the flow in a channel of finite width and of depth Yo
V0 , mean velocity of the lamina at the center of a channel
W, submerged weight of a particle
x, distance measured horizontally from the center of a channel
y, depth of a channel at the distance x from the center; or, in the evaluation of m, y represents the distance from the top of the channel of depth D to a horizontal lamina of thickness dy
y0 , depth of a channel at the center
CZ, a variable
J3, a variable
n=L Yo
Q,. angle of repose for the material
l = ~ Yo
.p, density of water
-r.,, an average drag force
i, maximum allowable drag force for soil in lb/ft2
~, angle the tangent at x makes with the horizontal
3
STABLE CHANNEL SHAPE WITH DRAG PROPORTIONAL TO DEPTH
The type of stable channel studied herein has the property that all the particles of its bed are in a state of incipient instability under tpe action of the forces acting.on them. At the edge the drag force is zero; and incipient instability is due to the force of gravity acting on particles resting on the slope S, which is at the angle of repose. At this point the_~atioof the componen,t of force down the slope to the component normal to the slope is· s. At the middle of the channel, where the slope gz is zero, the particles are held in
. . dx the state of incipient instability by the drag force of the flowing water. In this case, also, the ratio of the force along the;bed to the force acting on the particle in the direction nonnal to the bed is s. At all other points the resultant of the drag force and the gravitational component along the bed act on the particle, and the ratio of this resultant to the gravitational force component normal to the bed must also bes. These r~lations may be expressed mathematically in the following way. If W represents the submerged weight of the particle, then, with the notation of Figure 1, the force component normal to the bed is W cos¢. The gravitational component along the bed is W sin ¢ and the drag force of the flowing water per unit of area of the bed is
•L- ¢ .WS Yo cos
Then the requirement that the condition of incipient "instability shall prevail everywhere is
Since
- gz • tan¢ dx
This relation can be put in the form 2 · 2 <f )_ . + s2 i"2 . s2
0
4
• • •
• • •
(1)
(2)
A solution of this differential equation which satisfies the condition that y • y0 when x • 0 is
• • •
Then it mq be concluded that a simple cosine curve represents the shape of a stable channel under the conditions specified in assumption (e).
STABLE CBADEL WITH A DRAG FQRCE PROPORTIORAL TO THE SQUARE OF THE TELOCITY ilD BO LATERAL TRANSFER or DRAG
In this case the drag force tel'lll is assumed t.o have the fol'lll:
wsL .y 2
0
(3)
where V represents the mean velocity in the stream at the distance x from the center1 and V represents the mean velocity at the center. Then the condition ~hat the particles everywhere be on the verge of motion is:
. . . (4)
But since tan ¢ =--! - (dy')
dx and sin¢= , cos¢ • 1 . . . (5)
V 1 + 2 V1 + (~)2 (~)
dx dx
5
\'he above expression ~an. be put in the form! . ,, .· .. . { .. . .
. . . (6)
In order to obtain an expression for the velocities,we may apply the Chezy f'ol'lll\lla
• • • . (7)
to compute the flow through an element such as the shaded element of Figure 1.. Since the as.sumption of .no ;i.ate;-al. tran~fer of drag is equivalent to an assumption of no drag fo~es·on the sides of the element, the hydraulic radius r·inay be computed by dividing the area of the element yd,x by the length of. bed _in contact _with its base. This length is
Then y
(...X.)' . r =v d 2 · 1 +. dx ·
.. ( 8)
and from (7)
(9)
Since
then, by division
(10)
6
If this expression is substituted into Equation (6), it becomes:
This is identical with Equation (2) and, of course, has the same solution, namely:
y • y cosL o Yo
It may be concluded therefore, that if the lateral transfer of drag is neglected, both assumptions (e) and (f) lead to the same shape for the stable channel. If this seems surprising, it should be recalled that the Chezy formula itself is derived by equating a frictional drag force proportional to the square of the velocity to the component of the weight of the water acting in the direction of flow. Therefore, if the Chezy relation is used with the above restrictions to determine the relation between velocity and drag, assumptions (e) and (f) become identical.
STABLE CHANNEL SHAPE WITH LATERAL DRAG TRANSFER ACCOUNTED FOR
Before the effect of lateral drag transfer can be introduced into the equations, it will be necessary to develop some means of computing it. In the present case it will be assumed that the shear force acting on any section through the stream is proportional to the gradient of the square of the velocity. The factor of pr~portionality, here called m, will be determined by computing the mean velocity in an infinitely wide channel according to these relations and then determining the factor m by comparison with the Chezy formula. The modifications needed to introduce the effect of lateral drag transfer can be made as soon as the factor m is evaluated. The detail of the evaluation of m is as follows:
In a channel of depth D and having a friction slope f, such as is shown in Figure 2, the shear on a unit area of the plane at a distance y below the top is
7
- . . (11)
),,
UBRr\RY.
SEP 2 O tUUI
-Bureau of Reclamallon Reclamation Service Cente,
where U = v2 and V repre~ents. the stream velocity at . y. The gravitational component acting on the volume included between unit areas of the planes y and y + dy is
ffdy
where -p represents the weight of a unit volume of water. Then if this gravitational component is to be held in balance by the difference between the drag forces on the top and bottom of the element:
d dU - - (m -) = .of dy dy '
(12)
In setting up,this expression;·rorces in the downstream direction have been considered positive •. By integration
and
dU - m - = .nfy + c1 dy r
where c1 and c2 are constants.
If .
then
Hence
or
dU - m dy = ,prn at y = D
U = 0 at y = D
mU == fr:2 -·4 U = ~ (D2 - y2) (13)
8
and
The mean velocity is D
Vmean .1: -Jvdy = 1_ ~[z . D . D la 2 .o
or
vmean = ~ 1'D T
• • •
D
Vn2 - I-+ B:sin-1 z] · . 2 D
0
-(14)
(15)
A rearrangement which will permit a comparison :wtth the Chezy formula is
For this case Chezy's formula has the form:
Vmean = C '{'Dr'
Then by comparison
C= ,p v~ and :J.t follows that:
~D7T2 m = 32c2
(16)
(17)
. . . {18)
{19)
It will now be possible to modify Equation (9) ,by adding to the original ·gravitation driving force the difference of the frictional drag forces on the two sides of the shaded element of Figure 1. The drag force acting on the side of the shaded element · nearest to the center of the channel is, for a unit length of channel,
q = y'( (20)
9
and acts in the downstream direction if the velocity gradient dV is dx .
negative. Under the same conditions the force on the side away from the origin is given by a similar expression, but here the·force acts upstream. The difference between the two forces is
dq = ~ (y7) dx dx ·
. . . (21)
Modification of' Eq~ation(9)will:be most easily made if this difference is expressed in tenns of an equivalent channel gradient. To do this ae.t
Then
f 1fydX = ~ (yT) dx dx
1 d -r. fl = -· - (yJ)
-pY dx • . • (22)
and the equation as modified to account for the lateral transfer of' drag becomes
Since
v2 -- y Vc2 Yo vl + {~)2
dx .
• • • (23)
(24)
. . . (25)
[ l + m d (Y.~).l pt- Y dX UA ~
10
or
• • (26)
To put this expression into dimensionless form let
Yod7) = dy • • • (27)
!= !_ Yo
and since
then
Yods = dx
2 m. fYo7f
32c2
• • • (28)
v22 - TJ ~ i. + ;~ ::: :1r [ 11:. c:/J} . . . <29J
Ye Vi+ c:J>2 i s s
The qomplicated nature of this expression would make it difficult to obtain a direct solution by integrating the differential equation which.might be obtained by eliminating the ratio of' L between this
Vo equation and Equation ( 6) • A trial method of solution will therefore be adopted inste,ui. By this process a trial profile wi.11 be chosen and the ratio(L(\iill be computed for it from both the Drag Relation
. Vo or· Equation (6) and the Modified Chezy Relation of' Equation (29). It will be considered that a satisfactory solution has been obtained when the two curves agree closely. An interesting feature of' Equation (29) is that the quantity m does not occur in it. This indicates that the stable shape will f'ix a value for the ratio ~} This shape will also
be independent of' size. ·
11
For these purposes, if' the relation between n and .$ can be assumed to be:
or
If
then
1'J = i + al 2 + /3 s4 q- = 2 a§+ 4 _/3 !3
2 4 L.1+a(~) +/3(.!...) Yo Yo Yo
! = 2 a(.!...) + 4/3 ,~,3 Yo Yo
L = O and ~ • - S when x • a Yo dx
· a 2 /3 a 4 0 = 1 +ct(-) + (-) Yo Yo
- s = 2 ac!....> + 4/3c!....>3 . Yo . Yo
from which
c§ !.... - 2> a = _2 _Yi_o --
c!.... ,2 Yo
( S a . ) - -- + 1 /3 = 2 Yo
. c;o>4
12
. . .
. . .
. . .
• • •
. . .
. . .
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
Discussio• or coMPILATI011 oF DATA m·cuana
The computations in tlle tables at the end ot this report are those used to obtain data tor the curves shown on Figures 3 through 7. They consist ot the development of the ratios of the •an velocity of the lamina at the distance x from the center of the channel to the mean velocity at the center tor various top widths and transverse slopes. The procedure followed in making these calculations was to assume a value ot· . ; tor each value of s and to
0
determine the ratio of ~ from the Drag !{elation and from the 0
Modified Chezy Relation. When close agreement. between the two curves was obtained, a satisfactory solution of Equation (29) was assumed to have been attained. In computing the values at t:' from the Modified
.. 0
Chezy Relation, it was necessary to choose a value for the constant y 2m . o which would make the.curve agree with the one obtained from the n-
Drag Relation. rt agreement was not obtained, it was necessary to
choos~ a new value of and make a new set of calculations.
It was found that the curves could be made to agree quite
cloself although !_ was allowed to vary. a great deal. Consequently, Yo .
the greatest source of error was in choo.sing the value o't r· It is . . - 0
· I ·. a . believed that by careful adjustment of Y
O the curves., could be made
to coincide still more closely; however, the agreement shown in this set of curves is as close as is warranted by the approximations used in the development of these equations.
It may be noted that the solutions obtained by the modification for lateral transfer of drag agree very closely with tbe·cosine solution. Since this is t~e case, thE! use of the cosine solution may give equally ·1sattsfactory results· with considerably less work. ·
The range of transverse slope covered by this set of curves is from S • 0.3 to S • 1.0.
13
Ym Graphical integration was used to evaluate -
Vo' p -,
and Q as follows: Yo
a
. . .
0
a
r 2 p
2 l + (dr\.) ~! - . . . Yo df
0
a
A 2 f~ Y\ d_J y 2 - • • •
0
0
a -· 2 /Yo V.
Q - 2V y Y\. d ! . . . 0 0 Yo
0
14
(38)
(39)
(40)
(41)
It will be. not.ed that the curve fot p . is . 11ot . used for solution .· . -Yo .. · ··. ··<· .· ....
ot a problem by this method·. '1'b1.i is because the concept of the wetted perimeter and the h74i'aulicradiua is not in agreement w:l,tli the drag distribution a11umed in thia, solution. However·, the -c,ur:v:e is inserted to permit a comparison of results obtained by this method with similar results obtained from the ordinary formulas ot hydraulics.
EXAMPLE
The procedure involved in the design of a stable channel can best be illustrated by the solution of an example.
For the particular type of soil through which the channel will run, the engineer will know or will be able to determine certain physical characteristics on which to base his design. It will be necessary to know the angle of repose of the soil and the maximum allowable drag force. An appropriate coef'.ficient of roughness can be found in hydraulics tables, and the longitudinal slope on which the channel will run will be a predetermined factor. With these values specified, the quantity of wate.r. which the stable channel will carry will be a fixed value. Since. it .is· desirable to specify the amount of water to be carried by the channel, it is necessary to have some method of modifyingthe stab:te channel shape to increase or decrease the flow. If the desired flow is greater than the ,stable channel shape·would carry, it is proposed to insert a rectangular section at the center; and if the flow is to be les_s than the stable channel shape would carry, then a portion would be removed from the center. The portion to be removed in the latter case is easily determined by-use of Figure 12.
The conditions assumed for the design of this particular channel are:
a. The tangent of. the angle of r.epose of the soil is o.6 (S • 0.6).
b. The material through which the channel is to run will stand a drag force of 0.1 pound per square· foot ( ~ • 0.1) 0
c. The longitudinal slope of the channel will be 0.0004 (f • 0.0004).
d. A reasonable value for the coefficient of roughness would be n • 0.020.
15
In a wide channel of uniform depth the drag force per unit of area ot the bed would be 7' •.,Om and the velocity would be 'c· At the center of a stable channel, ot equal depth, the lateral trauater ot drag reduces the velocity from Ve to Y and the drag relation there becomes 0
An expression for the center depth can then be obtained from the above relation in the form
,,-y O - --.,--2-
.,0f( ~) V
C . V
For the prese1,1t case the upper curve of Figure 9 gives Vo• 0.983. Then c
Y • O.l • 4.146 feet 0 (62.4)(0.0004)(0.983)2
For an infinitely wide channel the following relation holds
I • t to,b l tA \~ (., ~- - \lo.>)
. ~
:, o-:.. D and for the conditions~assumed above the value of C • 94.4 is obtained from hydraulic tables.
Then
'c • 94.4 vc4.146)(o.ooo4)
• 94.4 (0.04072)
• 3.844 tt/sec
16
•
From Figure 9
Then
From Figure 8
Or
Vm • (0.869)(0.983)(3.844)
V • 3.28 ft/sec m
_ .... · '
A • (3.42) (~ •. 146) 2
A• 58.79 ft2
Then since Q • ·Av - ; , . m
Q • (58.79)(3.28)
• Q • i92.83 r.t3 /sec
From Figure 8 the value of !_ is found to be 5.38, so that the Yo
top width of the st_abl~ channel is 5.38 Yo or (5.38)(4 .• 11!,6) .• 22.30. Having now obtained the top width and the center depth of' the stable channel, it remains only to obtain the . x and y coordinates of· the channel bed. This is readily done by making use of Figure 11. For a transverse slope of S • Q.6 the following values of !...c
' . ' . ~ and ;
0 are obta,ined '711d the values of x and y are calculated.
17
X L. Yo X Yo y
0 0 1.000 4.146 0.5 a.073 0.958 3.972 1.0 4.146 0.839 3.478 1.5 6.219 o.649 2.691 2.0 8.292 o.4oo 1.658 2.5 10.365 0.117 o.485
11.15 0
If the channel is to carry leBS th~ the sta"t?le channel will carry, it is necessary to remove a portion of the channel. Suppose the channel is to carry 75 rt3/sec. Then the ratio of Q to Q0 is 75 or 0.389. From Figure 12 the value of !_
192.83 · a is found to be 0.369, that is, 36.9 percent of the top width is to be removed from the center of the channel. The modified section with a top width of 14.07 feet is shown on Figure 13.
If the channel is to carry more than the stable channel will carry, it is necessary to add a rectangular section at the center. Assume that it is desired for the channel to carry 300 tt3/sec, then the amount to be carried by the center section is 300 - 192.83 or 107.17 tt3/sec. The mean velocity of the flow in an infinitely wide channel of depth y0 is Ve, which was determined to be 3.844 ft3/sec in this case. For a center depth of 4.146 fe~t each foot of width would carry (4.146)(3.844) ft3/sec or 15.94 ftj/sec. The necessary width of the center section woµld then be l~t ~z or 6. 72 feet.
It should be noted that there is a discrepancy between the velocity of the center section and the maximum velocity of the stable section. This discrepancy is relatively unimportant, however, since the difference is less than 2 percent of the maximum velocity tor the center section; and, the lateral transfer of drag being small, the velocity at the edge of the center section would soon reach its maximum value at points nearer the center. The modified section with a top width of 29.02 feet is shown on Figure 13.
18
In conclusion, it may be noted that if water flowing in these channels carries sediment of the type coming from soil through which the channels run, it could be expected that the channels as designed would neither scour nor silt up. However, if clear water is to be carried, these channels would be stable to~ any slope less than 0.0004.
ACKNOWLEDGMENTS
The fundamental principles on which this work is based were laid ~own by Mr. E.W. Lane. This analysis has profited by the previous work of Mr. R. G. Conard who made many computations of the type shown in the tables and who succeeded in obtaining a close agreement between the Modified Chezy and Drag relations before he left the Bureau. The tables shown herein and the J:igures 3 to 7, inclusive, represent a further refinement of Mr. Conard's work by Mr. Florey •
19
Tab
le
l
Slo
pe=
0.3
a
= -
.0388
11-3
c;o
> =
5.
5 .f
l =
.0
0019
124
(12)
I+
{II
J
(1)
: (2
) :
(3)
: (4
) :
(5)
: (6
) .. :
(7
) :
(8)
(9)
: (1
0)
: ( 1
1)
: (1
2)
: (1
3)
. .
. .
. .
. .
. .
. .
(!..
..);
(!..
..)2
; (!
....)
3 ;
(.!...
.)4
Yo
: Y
o :
Yo
: Y
o !a
(.!.
...)2
!Jf
(.!....)q
.! (L
) !2
a(.!
....)
!4P
C.!...>
3! (d
y)
!cay/
!1 --
*-<d
y)2
~ cL
)4
• Y
o •
Yo
• Y
o •
Yo
• Y
o •
. d
x
• d
x
• s2
dx
•
V0
. . .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
0:
0.:
0
: 0
: 0
: 0
:1.0
0000
: 0
: .4
: .1
6:
.064
: .0
256:
-.
0062
1: .0
0000
5:
-993
79:
-.03
107:
.8
: .6
4:
.512
: .4
096:
-.
0248
6: .0
0008
:
.975
22:
-.06
215:
1.
2: 1
.44:
1.
728:
2.
0736
: -.
0559
3:.0
0040
: .9
4447
: -.
0932
2:
1.6:
2.5
6:
4.09
6:
6.55
36:
--09
944:
.001
25
: .9
0181
: ·-.
1242
9:
2.0:
4.0
0:
8.00
0:
16.0
000:
-.
1553
7:.0
0306
: .8
4769
: -.
1553
7:
2.4:
5.
76:
13.8
24:
33.1
776:
-.
2237
4:.0
0634
: .7
8260
: -.
1864
4:
2.8:
7.8
4:
21.9
52:
61.4
656:
-.
3045
3:.0
1175
: .7
0722
: -.
2175
1:
3.2:
10.2
4: 3
2.76
8:10
4.85
76:
--39
775:
.020
05:
.622
30:
-.24
858:
3.
6:12
.96:
46.
656:
167.
9616
: -.
5034
0:.0
3212
: .5
2872
: -.
2796
6:
4.0:
16.0
0: 6
4.00
0:25
6.00
00:
-.62
149:
.048
96:
.427
47:
-.31
073:
4.
4:19
.36:
85
.184
:374
.809
6:
-.75
200:
.071
68:
.319
68:
-.34
18o:
4.
8:23
.o4:
110.
592:
530.
8416
: -.
8949
4: .1
0152
:
.206
58:
-.37
288:
5.
2:27
.o4:
140.
608:
731.
1616
:-1.
0503
0:.l
3983
: .0
8953
: -.
4039
5:
2.5:
30.2
5:16
6.3w
:91.
5.~
25:-
1.17
500:
.175
00 :
_
o ~ ~
~427
26:
0 :
.000
05
.000
39:
.001
32:
.003
13:
.006
12:
.010
57:
.016
79:
.025
07:
.035
69:
.048
96:
.065
16:
.084
60:
.107
56:
.127
21:
0 :
0 :
-• 03
102:
• 00
096:
-.
0617
6: .0
0381
: -.
0919
0: .0
0844
: -.
1211
6: •
0146
8:
-.14
925:
.022
28:
-.17
587:
.030
93:
-.20
072:
.o40
29:
-.22
351:
.049
96:
-.24
397:
.059
52:
-.26
177:
.068
52:
-.27
664:
.076
53:
-.28
828:
.083
10:
-.29
639:
.087
85:
-• 29
999:
~ 0_
8999
:
1.0
00
00
:1
.000
00
.989
33
: .9
8838
.9
5766
:
.954
02
.9()
621
: .8
9862
.8
3687
:
~824
76
.752
42
: .7
3602
.6
5630
:
.636
61
-552
29
: .5
3090
."1
1-83
:
.423
66
.338
60
: .3
1958
.2
3859
:
.223
29
.149
58
: .1
3895
.0
7658
:
~070
70
.023
79
: .0
2187
0
. 0
.
Slo
pe=
0.3
(!
.,_)
= 5
.5
Yo
Tab
le 2
2
Vo
m
<,-t
) •
· 75
c;0
:m, (
17)(
18)
1+(1
9)
c!n
(20)
x(22
) 1.
0576
3 (2
3)
om
rm
t'ioJ
(17)
(1
8)
: (1
9}
• V
2
: {2
a)
rm
rm
• 2
• {2
21
• o
m
Yo
:(-)(-).
. •
1° t
Y
·1 +
la
st=
,----=
;~ ~
L)~
(!..)
2]~
te
rm
:\)-
+ (~
)2:
.dx
~
dx
V
• d
x
• .
0 0
: •
• .
. .
. .
.
; (!
..)
; V
2
: V
e :
<-v)
•M
od
ifie
d.
C
• •
: C
hezy
(!..
/ !~
(!..
/;(L
)~(!
..,2
;~~
L)~
(!..
)2J
! (Y
o)
V0
.dx
V0
• y 0
d
x V
0 .d
x
y0
dx
V0
• y
. .
. .
. .
. .
. -.0
7265
1.
0000
0:
.014
58:
.014
53
; -.0
7092
.9
9417
: .0
4358
: .o
li-29
o •
-.065
40
.976
74:
.071
95=
.06
9r
: -.0
5698
.9
4796
: .0
9950
: .0
9l 5
;
-.045
05
.908
16:
.125
60·
.109
87
: -.0
3120
.8
5792
: .1
5010
: .122
J~ :
-.016
52
.797
88:
.173
12•
.12
• -.0
0060
.7
2863
: .1
9435
: .1
2920
:
.015
18
.650
89:
.213
95=
.123
M
: .0
3062
.~
6531
: .2
3192
: .l
lO
: .0
4422
.4
7254
: .2
4945
: ,0
931 9
:
0572
2 • 3
1216
: .2
611a
= .o
1o3o
:
:066
55
.265
89:
.295
02:
.043
68
: .0
7203
.1
4788
: .4
9293
: .0
2 201
.
0 . .
. .
: :
: 1.
0000
0:
-.054
49
: .9
4551
: 1.
0000
0 :1
.000
00:
: 1.
0062
5:
-.053
52
: .9
4648
: 1.
0004
8 :
.993
31:
: 1.
0254
1:
-.050
30
: .9
4970
: 1.
0019
0 :
,973
37:
: 1.
0587
9:
-.045
25
: .9
5475
: 1.
0042
1 :
.940
51:
: 1.
1088
8:
-.037
47
: .9
6253
: 1.
0073
1 :
.895
26:
: 1.
1796
8:
-.027
61
: ,9
7239
: l.
Oll
08
: .8
384-
o:
: 1.
2777
9:
-.015
83
: .9
84-1
7:
1.01
535
: .7
7077
: :
1.41
399:
-.0
0064
:
.999
36:
1.01
995
: .6
9339
: :
1.60
694:
.0
1829
:
1.01
829:
1.
0246
8 :
.607
31:
: 1.
81)1
36:
.043
43
: 1.
0434
3:
1.02
933
: .5
1365
1 :
2.33
934:
.0
7758
:
1.07
758:
1.
0336
9 :
.413
54:
: 3.
1281
3:
.134
24
: 1.
1342
4:
1.03
756
: ,3
08ll
: :
4.84
-074
: .2
4161
:
1.24
161:
1.
0407
2 :
.198
50:
:ll.
l.69
44:
.603
40
: 1.
6034
0:
1.04
300
: .0
8584
-: 0
0
: 0
0
: 0
0
: 1
.04
40
2
: 0
:
.945
51
.940
15
.924
41
.897
95
.861
71
.815
25
.758
57
.692
95
.618
4-2
.535
96
.445
62
.349
47
.246
46
.137
63
0
{21r
J
v2 ~
0
1.00
000
.994
33
.977
68
,949
70
,9ll
37
.862
23
.802
29
.732
88
.654
06
.566
85
.471
30
.369
61
.260
66
.145
56
0
: (2
5)
: · (
26)
. :
. . . V
:c
!..)
(L)
Vo
• V
o Y
o :
. :1
.000
00:1
.000
00
: .9
9716
: .9
9097
.9
8878
: .9
6428
.9
7452
: .9
2040
:
.954
66:
.860
92
: .9
2856
: .7
8713
:
.895
71:
.700
98
: .8
5608
: .6
0544
.8
0874
: .5
0328
• 7
5289
: .3
9807
.6
8651
: .2
9346
.6
0796
: .1
9435
.5
1055
: .1
0547
.3
8152
: .0
3416
0
: 0
Slo
pe=
0.5
T
able
3
a =
-.1
1718
75
c;o>
=
3.2
19
=
• 001
9073
5
iPch>
:
(9)
: (1
0)
: (1
1)
: (1
2)
= rm
(1
) :
(2)
: (3
) :
{4)
: (5
) :
(6)
: (7
) :
{8)
(.!.
.) ;(
.!..)
2;
(.!.
.)3
~ (.!
..) 4
~
ac.!
../
~pc.
!..>
4;
(L)
~2ac
.!..>
. .
. .
. Y
o •
Yo
• Y
o •
Yo
• Y
o
• Y
o •
Yo
•
Yo
~4,·
(.!..)
3~
(~)
! (dY)2
~
l _
1 (d
Y)2
~ (L
) 4
• Yo
•
dx
•
dx
• ~ d
x
• V 0
:
: . .
. .
. .
. .
. .
. .
O:
0 :
0 .
0 .
0 .
0 :1
.000
00:
0 .
·O
. 0
. 0
. 1.
0000
0 :1
.000
00
. .
. .
. .
. ~2
: .0
4:
.008
: • 0
016:
-.
0046
9: .0
00
00
:
.995
31:
-.04
688:
.0
0006
:
-.04
682:
.002
19:
.991
24
: .9
8907
.4
: .1
6:
.064
: .0
256:
-.
0187
5:.0
0005
: .9
8130
: --
0937
5:
.000
49:
-.09
326:
.008
70:
.965
20
: .9
5688
.6
: .3
6:
.216
: .1
296:
-.
0421
9: .0
0025
:
.958
06:
-.14
063:
.0
0165
:
-.13
898:
.019
32:
.922
72
: .9
0523
.8
: .6
4:
-512
: .4
096:
--
0750
0: .0
0078
:
.925
78:
-.18
750:
.0
0391
:
-.18
359:
.033
70:
.865
20
: .8
3699
1.
0: 1
.00:
1
.00
0:
1.00
00:
-.11
719:
.001
91 :
.88
11-7
2: -.
2343
8:
.007
63 :
-.
2267
5: .0
5142
: • 7
9432
:
• 755
47
1.2:
1.4
4:
1.72
8:
2.07
36:
-.16
875:
.003
96:
.835
21:
-.28
125:
.0
1318
: -.
2680
7:.0
7186
: .7
1256
:
.664
79
1.4:
1.9
6:
2.74
4:
3.84
16:
-.22
969:
.007
33:
.777
64:
-.32
813:
.0
2()9
4:
-.30
719:
.094
36:
.. 622
56
: .5
6888
1.
6: 2
.56:
4.
096:
6.
5536
: -.
3000
0: .0
1250
:
.712
50:
--37
500:
.0
3125
:
-.34
375:
.118
16:
.527
36
: .4
7163
1.
8: 3
.24:
5.
832:
10
.497
6:
--37
969:
.020
02:
.640
33:
-.42
188:
.0
4449
: --
3773
9:.1
4242
: .4
3032
:
.376
67
2.0:
4.0
0:
8.00
0: 1
6.00
00:
-.46
875:
.030
52:
.561
77:
-.46
875:
.0
6104
: -.
4077
1:.1
6623
: .3
3508
:
.287
32
2.2:
4.8
4: 1
0.64
8:
23.4
256:
-.
5671
9:.0
4468
: .4
7749
: -.
5156
3:
.081
24:
-.43
439:
.188
69:
.245
24
: .2
0631
2.
4:
5.76
: 13
.821
n .3
3-17
76:
-.67
500:
.063
28 :
.3
8828
: -.
5625
0:
.105
47 :
-.
4570
3: .2
0888
: .1
6448
:
.136
06
2.6:
6.7
6: 1
7.57
6:
45.6
976:
-.
7921
9:.0
8716
: .2
9497
: ~.
6093
8:
.134
09:
-.47
529:
.225
90:
.0964
<>
: .0
7864
2.
8: 7
.84:
21
.952
: 61
.465
6:
-.91
875:
.117
24:
.198
49:
-.65
625:
.1
6748
: -.
4887
7:.2
3890
: .0
4440
:
.035
84
3.0:
9.0
0:
27.0
00:
81.0
000:
-1.0
5469
:.154
50:
.099
81:
-.70
313:
.2
0599
: -.
4971
4:.2
4715
: .0
1140
:
.009
14
3.2:
10.2
4: 3
2.76
8:10
4.85
76:-l
.gO
OO
Q: .
gooo
o :_
Q____
___r
.._._7
5000
: .2
~00
0:
-.50
000:
.250
00:
0 .
0 .
Slop
e a
0.5
(!
-) •
3
.2
Yo
'!'a
ble
4 v
2
( o
m
rt >
•
0.06
0
vo2m
(j
i"T
) (1
7)(1
8)
1+(1
9)
_il
l T
m
( 20
)x(2
2}
l. 01
667
( 23)
(1
4)
: (1
5)
: {I
o}
(17)
(1
6)
: (1
9)
: (2
0)
: V
o2m
Yo
rm
{2
2J
ro1
. .
• 2
• :
: I
: :(~
)(-) •
=-
1 +
la
st=
_
__
: : c
: ) :
(L
)2
Vo
:!..c
!..)
2!(L
)!..(
!..)
2!!.
. fcL
)!..
(!..
)j:
;dx V 0
;
Yo d
x V
o ;d
x [
Yo d
x V
0 ;
{o
) :!.
.. ~
L !
!..
21~··
term
:V1
+ (~
)2:
Y
·c1x
<:.v: >
c1x<
v >
•
• d
x
: :·
0 0
:
: :
V
2 :
C
: (V
) :M
ocl
ifie
d:
c :
Che
zy
: 1.
0000
0:
.994
52:
.978
20:
• 95i
44:
.914
87:
.869
18:
.815
35:
.754
24:
.686
75:
.613
73:
.536
02:
.454
21:
.368
86:
.280
43:
.189
31:
.095
60:
0 :
.027
Ji.o=
.0
2733
.0
8160
= .0
8064
.1
3380
: .1
2974
.1
8285
: .1
7223
.2
2845
: .2
0680
.2
6915
= .2
3146
.3
0555
: .2
4640
.3
37Ji
.5:
.251
42
.365
10=
.246
96
.388
55:
.233
54
.409
()5
1 .2
1255
.4
2675
: .1
8473
.4
Ji.21
5= ·
.15
105
.455
60:
.112
41
.468
55=
.069
88
.478
00:
.023
85
!
-.27
330
-.26
655
-.24
550
-.21
245
-.17
285
-.12
330
-.07
470
-.02
510
.022
30
. 067
10
.104
95
.139
10
.168
40
.193
20
.212
65
.230
15 .
. .
. .
. .
. .
. :
l.00
000:
-.
0164
0 I
.983
60:
1.00
000
:l.0
00
00
: :
l. 00
4 71:
-
• 01.
607
: • 9
8393
: l.
0010
9 :
• 994
23:
: 1.
0190
6:
-.01
501
: .9
8499
: l.
0043
4 :
.977
06:
: 1.
o437
8:
-.01
330
: .9
8670
: 1.
0096
1 :
.948
94:
: 1.
0801
7:
-.01
120
: .9
8880
: 1.
0167
1 :
.910
56:
1.13
030:
-.
0083
6 :
.991
6Ji.:
1.
0253
9 :
.862
81:
1.19
730:
-.
0053
7 :
,994
63:
1.03
531
: .8
0672
: 1.
2859
4:
-.00
194
: .9
9806
: l.
0461
2 :
. 743
36:
1.40
351:
.0
0188
:
1.00
188:
1.
0574
3 :
.673
80:
1.56
169:
.0
0629
I
1.00
629:
l.
0688
4 :
,599
09:
: 1
. 780
09:
.011
21
: 1.
0112
1:
1.07
992
: . 5
2020
: :
2.09
428:
.0
17Ji
.8
: 1.
0174
8:
1.09
027
: .4
3796
: :
2.57
546:
.0
2602
:
1.02
602:
1.
0994
9 :
· .35
314:
:
3.39
018:
.0
3930
:
1.03
930:
1.
1072
0 :
.266
41:
: 5.
0380
4:
.o6J
i.28
: 1.
0642
8:
1.11
306
: .1
7833
: ·:
10.0
1904
: .1
3835
:
1.13
835:
1.
1167
6 :
.089
37:
00
:
oO
:
00
:
1.11
803
: 0
•
• 983
60
.978
25
.962
39
.936
32
.900
36
.855
60
.802
39
.741
92
.675
07
.602
86
.526
03
.445
62
.362
33
.276
88
.189
79
.101
73
0
{2fi1
v2 vc2
1.00
000
.994
56
.978
43
.951
93
.915
37
.869
86
.815
76
.754
29
.686
32
.612
91
.534
80
.453
05
.368
37
.281
50
.192
95
.103
4-2
0
: (2
5)
: (2
6)
V Vo
:(L
)(L
) :
V0
Yo
. :l
.000
00:1
.000
00
.997
28:
.992
60
.989
16:
.970
66
.975
67:
.934
75
.956
75:
.885
74
.932
66:
.825
14
.903
19:
. 754
35
.868
50:
.675
38
. 828
44:
• 590
26
.782
88:
.501
30
.731
30:
.410
82
.673
09:
.321
39
.606
93:
.235
66
.530
56:
.156
50
.Ji.3
926:
.0
8719
.3
2159
: · .0
3210
0
: 0
Tab
le 5
S
lop
e=
o.6
a
=
-.1
6323
73
(!..
.)
= 2.
70
~-.0
0357
518
{121
Y
o 1
+ (
11
)
(1)
: (2
) :
(3)
. (4
) .
(5)
• {6
) .
(7)
. (8
) .
(9}
. (1
0)
: ( 1
1)
: (1
2)
: (1
3)
. .
. .
. .
. .
. •
. .
. •
. •
• .
. .
. .
. .
.. .
. .
. •
. (.
!...)
:(.!
...)2
: (.
!...
)3:
(.!...
)4
:a(.
!...
)2 :
_p(!
...)4
: (L
) :2
ac!.
..)
:4p
(!...
)3:
(~)
:c~
>2
:i _
!...cdy
>2:
!... 4
Y
o •
Yo
• Y
o •
Yo
• Y
o •
Yo
• Y
o •
Yo
• Y
o •
dx
·d
x
• s2
c1x
-<v>
.
. .
. .
. .
. .
• •
• 0
. .
. .
. .
. .
. .
. .
O:.
0
: 0
. 0
: 0
. 0
:1.0
0000
: 0
. 0
. 0
• 0
. 1
.00
00
0
:1.0
00
00
.
. .
. .
. .·.
2:
.o4:
.0
08:
.001
6:
-.00
653:
.000
01
: .9
9348
: -.
0652
9:
.000
11 :
-.
0651
8: .0
0425
: .9
8819
:
.984
01
.'4:
.16:
.0
64:
.025
6:
-.02
612:
.000
09
: -9
7397
: -.
1305
9:
.000
92
: -.
1296
7: ~
0168
1:
.953
30
: .9
3754
.6
: .3
6:
.216
: .1
296:
-.
05
87
6:.
00
04
6:
.941
70:
-.19
588:
.0
03
09
: -.
1927
9:.Q
3717
: .8
9675
:
.864
61
.a:
.64:
.5
12:
.409
6:
-.1
04
47
:.0
01
46
: .8
9699
: -.
2611
8:
.00
73
2:
-.25
386:
.064
44:
.82
10
0
: .7
7130
1
.0:
1.00
: 1.
000:
1.
0000
: -.
1632
4: .0
0358
:
.840
34:
-.32
647:
.0
1430
:
-.31
217:
.097
45:
.729
30
: .6
6454
1
.2:
1.44
: 1.
728:
2.
0736
: -.
23
50
6:.
00
74
1:
-172
35:
--39
177:
.0
24
71
: -.
3670
6:.1
3473
: .6
2575
:
.551
45
1.4
: 1.
96:
2.74
4:
3.84
16:
-.3
19
94
:.0
13
73
: .6
9379
: -.
4570
6:
.03
92
4:
-.41
782:
.174
57:
.515
08
: .4
3853
l.
.6:
2.56
: 4.
096:
6.
5536
: -.
41
78
9:.
02
34
3:
.605
54:
-.52
236:
.0
58
58
: -.
4637
8:.2
1509
: .J&
-025
3 :
.331
28
1.8
: 3
.24:
. 5
.832
: 10
.497
6:
-.52
889:
.037
53::
.5
0864
: -.5
e"76
5:
.083
40 :
-.
5042
5: .2
5427
: .2
9369
:
.234
15
2.0
: 4.
00:
8.00
0:
16.0
000:
-.
6529
5: .0
5720
:
.. 404
25:
-.65
295:
.1
1440
:
-.53
855:
.290
04:
.194
33
: .1
5064
2
.2:
4.84
: 10
.648
: 23
.425
6:
-.7
90
07
:.0
83
75
: .2
9368
: -.
7182
4:
.15
22
7:
-.56
597:
.320
32:
.110
22
: .0
8348
2
.4:
5.76
: 13
.824
: 33
.177
6:
-.9
40
25
:.1
18
62
: .1
7837
: -.
7835
4:
.19
76
9:
-.58
585:
.343
22:
.046
61
: .0
3470
2
.6:
6._7
6:
17.5
76:
45.6
976:
-1.1
0348
: .1
6338
:
.059
90:
-.84
883:
.2
5135
:
-.59
748:
.356
98.;
.008
39
: .0
0618
2
. 7:
7 .2
9:
19.6
83:_
53.1
441:
-1.1
9000
: .19
000
-=~ _
0 :
-.881
48:_
__._
g_81
48
: -.
6000
0: ._
3600
0:
0 .
0 .
Slo
pe.
o.6
(!..
.) •
2.
7 Y
o T
able
6
2 V
0 m
( r
r ) =
0.0
75
2 ril
l ~
m
. 7
('i"
'T)
(17)
(1&
) 1+
(19)
21
(2
0)x(
22)
1.03
095
(23)
(2
5)
: (2
6)
(14)
:
(15)
:
rm
rm
(18)
:
(19)
:
(20)
:
(21)
:
(22)
:
(23)
:
(24)
•
V 2
•
. •
• •
• •
o m
Y
o •
• •
• •
(L)2
!d
cL/!(
L)~(
L/!!
.. [L
)~(L
)j ~
(Yo
) V
0 .i
ii V
o
• Y
o d
x V
0 .d
x
Yo
dx V
0 •
y :
: ,
: :
•(-)(
) •
• •
,__
__
• •
V
2 •
• .o
f -
• l
+ l
ast·
·
• (-)
• •
r-
y •
• d
y 2
• V
2
• V
•
• 2
• te
nn
·
+ (
-)
• (-)
• c
• : ~
~L
)~( !
... )
]= : V
dx
: V
c =
Mo
dif
ied
:
:dx
Y
o dx
Vo
: .
: :
: C
hezy
:
1.00
000:
. 9
9197
: .9
6827
: .9
2984
: .8
7824
: .8
1519
: .7
4260
: .6
6222
: -5
7557
: .4
8389
: .3
8812
: .2
8893
: .1
8628
: .0
7861
: 0
:
.040
15:
.118
50 •
• 192
15:
.25a
oo:
.315
25:
.362
95:
.4o1
90:
.433
25:
.458
40:
.478
85:
.495
95:
.513
25:
.538
35:
.786
10;
.040
02
.116
57
.182
36
.237
19
.273
84
.292
66
.294
62
.281
46
.255
37
.218
57
.173
07
.121
14
.064
14
.023
54
-.40
020
-.382
75
-.328
95
-.274
15
-.183
25
. -.
0941
0 -.
0098
o .0
658o
.1
3045
.1
8400
.2
2750
.2
5965
.2
8500
.4
0600
. .
. .
. ·~
--~
--
~·~---
. .
~
---
· 1.
0000
0:
-.03
002
: .9
6998
: 1.
0000
0 :1
.000
00:
1.00
656:
- .
0288
9 :
.971
11:
1.00
212
: .9
9138
: 1.
0267
2:
-.025
33
: .9
7467
: 1.
0083
7 :
.965
88:
1.06
191:
-.0
2183
:
.978
17:
1.01
842
: .9
2467
: 1.
1148
4:
-.015
32
: .9
8468
: 1.
0317
2 :
.869
41:
1.18
999:
-.
0084
0 :
.991
60:
1.04
759
: .8
o216
: 1.
2947
5:
-.00
095
: .9
9905
: 1.
0652
4 :
.725
05:
1.44
136:
.0
0711
:
1.00
711:
1.
0837
8 :
.640
16:
1.65
142:
.0
1616
:
1.01
616:
1.
1023
1 :
.549
34:
: 1.
9660
3:
.027
13
: 1.
0271
3:
1.11
994
: .4
5417
: :
2.47
372:
.0
4221
:
1.04
221:
1.
1358
o :
.355
92:
: 3
.405
07:
.066
31
: 1.
0663
1:
1.14
905
: .2
5558
: :
5.60
632:
.1
1984
:
1.11
958:
1.
1589
7 :
.153
90:
:16.
6944
9:
.508
35
: 1.
5083
5:
1.16
489
: .0
5142
: 0
0
: -
: C
>Q
:
1.16
619
: 0
:
.969
98
.962
74
.941
41
.904
48
.856
09
.795
42
.724
36
.644
71
.558
22,
.466
49
.370
94
.272
53
.172
30
.077
56
0
v2
2 V
o
l.0
00
00
.9
9254
.9
7055
.9
3247
.8
8258
.8
2004
.7
4678
.6
6466
.5
7550
.4
8o<J
3 .3
8242
.2
8096
.1
7763
.0
7996
0
V
Vo
:(
!...
)(L
) :
V0
Yo
. .
:l.0
0000
:1.0
0000
. 9
9626
: . 9
8976
.9
8516
: .9
5952
.9
6564
: .9
0934
.9
3946
: .8
4269
.9
0556
: • 7
6098
.8
6416
: .6
6743
.8
1527
: .5
6563
• 7
5862
: .4
5937
.6
9349
: . 3
5274
.6
1840
: .2
4999
.5
3006
: .1
5567
.4
2146
: .0
7518
.2
8277
: .0
1694
0
: 0
Slo
pe=
o.8
o T
ab
le 7
er
= -
.300
00
(!.-
) =
2.0
.P
= .0
125
Yo
(12}
1
+ (
11)
(1)
: (2
) :
(3)
. (4
) .
(5)
: (6
) .
(7)
. (B
J .
(9)
. (1
0)
: ( 1
1)
: (1
2)
· :
(13)
.
. .
. .
. .
. .
. .
. :
: .
. .
. (~
)~{~
)2~
(~
)3
~ (~
)4
: X
2
: ,
X
4:
y .
; (dY
)2
;1 -
!_C
dy,2
; (L
) 4
:a(-
) :
(-)
: (-
) :2
a(~
) :4
,PC
~)3
: (d
Y)
Yo
· Yo
•
Yo
· Yo
•
Yo
• Yo
•
Yo
· Yo
·
Yo
• d
x
: d
x
: 82
dx
:
V 0
. .
. .
: :
: :
: :
. .
. .
o:
o· :
0 :
0 :
0 :
0 :1
.000
00:
0 .
0 :
0 .
0 :1
.00
00
0
-:1.
0000
0 .
. .1
: .0
1:
.001
: .0
001:
-.
003
. 0
: .9
9700
: -.
06
: .0
0005
:
-.05
995:
.003
59:
.994
39
: .9
9083
.
.2:
• 04:
.0
08:
.001
6:
-.01
2 : .
0000
2 :
• 988
o2:
-.12
:
.000
40:
-.11
960:
.014
30:
.977
66
: .9
6388
.3
: .0
9:
.027
: .0
081:
-.
027
:.0
00
10
: .9
7310
: -.
18
:
.001
35
: -.
1786
5: .0
3192
: .9
5012
:
.920
73
.4:
.16:
.0
64:
.025
6:
-.04
8 :.
00
03
2:
.952
32:
-.24
:
.003
20
: -.
2368
o: .0
5607
: .9
1239
. :
.8
6395
.5
: .2
5:
.125
: .0
625:
-.
075
:.0
00
78
: .9
2578
: -.
30
:
.oo6
25:
-.29
375:
.086
29:
.865
17
: • 7
9644
.6
: .3
6:
.216
: .1
296:
-.
108
:.0
01
62
: .8
9362
: -.
36
: .0
108o
: -.
3492
0:.1
2194
: .8
0947
:
.721
49
. 7:
.49:
.3
43:
.240
1:
-.14
7 :.
00
30
0:
.856
00:
-.42
:
.017
15:
-.40
285:
.162
29:
.746
42
: .6
4220
.8
: .6
4:
.512
: .4
099:
-.
192
: .00
512
: .8
1312
: -.
48
: ~0
2560
: -.
4544
0:.2
0648
: .6
7738
:
.561
45
.9:
.81:
.7
29:
.656
1:
-.24
3 :.
00
82
0:
.765
20:
-.54
:
.036
45
: -.
5035
5: .2
5356
: .6
0381
:
.481
68
1.0:
1.0
0:
1.0
00
: 1.
0000
: -.
30
:.
01
25
0:
.712
50:
-.60
:
.050
00:
-.55
000:
.302
50:
.527
34
: .4
0487
1.
1: 1
.21:
1.
331:
1.
4641
: -.
363
: .01
830
: .6
5530
: -.
66
: .0
6655
: -.
5934
5:.3
5218
: .4
4972
:
.332
59
1.2:
1.4
4:
1.72
8:
2.07
36:
-.43
2 :.
02
59
2:
.593
92:
-.72
:
.086
40:
-.63
360:
.401
45:
.372
73
: .2
6596
1.
3: 1
.69:
2.
197:
2.
8561
: -.
507
:.0
35
70
: .5
2870
: -.
78
:
.109
85
: -.
6701
5:.4
4910
: .2
9828
:
.205
84
1.4:
1.9
6:
2.74
4:
3 .8
416:
-.
588
:.0
48
o2
: .4
6002
: -.
84
: .1
3720
:
-• 70
28o:
• 49
393:
. 2
2823
:
.152
77
1.5:
2.2
5:
3.37
5:
5.06
25:
-.67
5 : .
0632
8 :
.388
28:
-.90
:
.168
75
: -.
7312
5: .5
3473
: .1
6448
:
.107
17
1.6:
2.5
6:
4.09
6:
6.55
36:
-.76
8 : .
0819
2 :
.313
92:
-.96
:
.204
80:
-.75
520:
.570
33:
.108
86
: .0
6932
1.
7: 2
.89:
4.
913:
8.
3521
: -.
867
: .10
440
: .2
3740
: -1.
.02
: .2
4565
: -.
7743
5:.5
9962
: .0
6309
:
.039
44
1.8,
: 3.
24:
5.83
2:
10.4
976:
-.
972
:.1
31
22
: .1
5922
:-1.
08
: .2
9160
: -.
7884
0:.6
2157
: .0
288o
:
.017
76
1.9:
3 .6
1:
6.85
9:
13.0
321:
-1.0
83
: .16
290
: .0
7990
: '."'
l.14
: .3
4295
:
--79
705:
.635
29:
.007
36.
. :
.004
50
2.0~
4.
00:
8.00
0:
16.0
000:
-1.2
00
: .20
000
: 0
:-1
.20
:
.400
00:
-.8o
ooo:
.64o
oo:
0 .
0 .
Slo
pe=
0.8
0 2
'fab
le 8
(V
A,t
) a
0.0
20
(~
) =
2.0
"'
. Y
o
V 2m
rtB
<,o
°r .)
(17)
(18)
1+
(19)
(2
0)x(
22)
1.01
871
(23)
1
(14)
:
(15)
:
(i6
) .
(17)
.
(18)
:
{19)
:
{20)
:
{21)
:
{22)
:
(23)
C
(2
4)
: (2
5}
. (2
6)
... .
. •v
:2m
Yi
• •
• •
• •
0 0
• •
• •
2 •
: !!
..(!..
.)2; (
L}!..
(!...
)2!!
... f ~}!.
.( !..
.) •J •(-)(-) •
•
• •
• (V
)
• .
• "I
'~
y ·1
+ last·~
·
• -
• .
c!..>
2 (Y
o)
: 2
: te
rm
: +
~ 2
: V
2
: V
e :
. v2
-:.
V
:cL
)(L
) V
o .d
x V
0 •
Yo
dx V
0 .d
x
Yo
dx
V0
• y
,!... [L~
!...) l
' ("") :
<v 0l •M
od
itie
d
: v
2
: V
o ;
Vo
Y
o .
. .
:dx
Yo
d
x Vo
:
: :
Che
zy
: :
. .
. .
. 0
. .
. .
. .
. .
. :
. .
. .
. . .
.
. .
. 1.
0000
0:
()1i.6
:
: -.9
1860
:
1.00
000:
-.0
1837
.
.98i
63:
1.00
000
: 1.0
0000
: • 9
8163
.
1.00
000
:l.0
0000
:1.0
0000
.0
4593
.
• .
• 995
40:
• 00
: .
-.893
50
: 1.
0030
1:
-.017
92
: .9
8208
: 1.
0017
9 :
.995
22:
.·977
38
. .9
9567
:
.997
83:
-~99
484
.981
77: ·==
• 135
28
. .
: -.8
2600
:
1.01
212:
-.0
1672
:
.983
28:
1.00
712
: • 9
8104
: • 9
6464
.
.982
69
·: .9
9131
: .9
7943
.2
1788
.
-959
55:
:300
60:
: -.
7151
0 :
1.02
764:
-.0
1470
.
.985
30:
1.01
583
: .9
5794
: .9
4386
.
.961
52
: .9
8057
: .• 9
5419
• 2
8939
.
. • 9
2949
. 37
o6o.
.
-.586
30
: 1.
0500
7:
-.01
231
. .9
8769
: 1.
0276
5 :
.926
70:
.915
29
. .9
3242
:
.965
62:
.919
58
.348
01
. .
. -8
9243
: 0
4303
0=
: -.
4343
0 :
1.08
0.l.7
: -.
0093
8 .
.990
62:
1.04
225
: • 8
8825
: • 8
7992
.
.896
38
: .9
4677
: -8
7~50
• 3
9144
.
. 84
940·
•
• .
-.287
30
: l.
ll9
o4
: .-.
0064
3 .
.993
57:
1.05
922
: .8
4366
: .8
3824
.
.853
92
: .9
2408
: '.8
2578
·8
0137
: .4
8030
: • 4
2017
.
. .
. -.1
4390
:
1.16
822:
-.
0033
6 .
.996
64:
1.07
810
: .7
9399
: .7
9132
.
.806
12
: .8
9784
: -·. 7
6855
•
• 52
070·
.4
3456
.
. .
• 749
30:
:552
70:
: -.
0161
0 :
1.22
983:
-.
00
04
0
. .9
9960
: 1.
0983
9 :
.740
28:
.739
98 .
.753
82
: • 8
6823
: · ~ 7
0598
.4
3617
.
. .0
9560
:
1.30
685:
.0
0250
-
: 1.
0025
0:
1.11
962
: .6
8345
: .6
8516
.
.697
98
: -8
3545
: .6
3929
•6
9403
• 57
740·
42
661
-:
. 63
629·
•
• :4
0747
:
.191
40
: 1.
4035
1:
.005
37
: 1.
0053
7:
1.14
127
: .6
2430
: .6
2765
.
.639
39
: • 7
9962
: • 5
6973
·5
1611
: •5
9580
: .
.264
60
: 1.
5260
2:
.008
08
: 1.
0080
8:
1.16
283
: .5
6354
: .5
6809
.
.578
72
: • 7
6o74
: -•
4985
1 •
• 61
000·
.3
8101
.
. .5
1571
: ·6
2010
= .
.329
40
: 1.
6837
3:
__ .O
ll0
9
: 1.
0110
9:
1.18
383
: • 5
0169
: • 5
0725
.
.516
74
: • 7
1885
:, • 4
2694
.3
4807
.
. .4
5370
: • 6
2840
: .
.374
10
: 1.
8914
3:
.014
15
: 1.
0141
5:
1.20
378
: .4
3920
: .4
4541
.
.453
74
: .6
7360
: .3
5613
.3
1066
.
. 39
086•
•
. •
.413
70 _
:
2.17
382:
.0
17~
:
1.01
799:
1.
2222
6 :
.376
37:
.383
14
. .3
9031
:
.624
75:
_.28
740
~269
29
· !
. :3
2737
; -6
3490
; .4
4310
:
2.57
546:
.0
22
·=
1.02
282:
1.
2388
4 .
: .3
1342
: .3
2057
-:
-3
~5
7
: . -
5'{1
~:
.• 221
89
2632
9.
.640
80.
.224
98
:· .4
6660
:
3.18
552:
.0
2973
:
1.02
973:
1.
2531
3 :
.250
51:
.257
96
. .2
6279
:
.512
63:
.;t.6
092
:198
60;
.646
90;
.178
32
: .
.4.8
760
: 4.
2123
0:
.041
08
: 1.
0410
8:
1.26
476
: .1
8770
: .1
9541
.
.199
07
: .4
4617
: .1
0592
• 1
3327
: .6
5330
: • 1
2956
:
. .5
0420
:
6.28
062:
.0
6333
:
1.06
333:
1.
2734
1 :
.125
03:
.132
95
. .1
3544
:
.368
02:
.058
60
• 079
14
; .
• 067
08:
.661
90:
.523
40
:12.
5156
4:
.131
01
: 1.
1310
1:
1.27
878
: .0
6248
: .0
7066
.
.071
98
: .2
6829
: .0
2144
o
: .6
7080
: .0
2680
.
. -
. 0
0
. 0
0
.. .
1.28
062
: 0
: 0
. 0
. 0
. 0
. .
. .
-0
.
. .
.
1)
: (2
J
: 3
Slop
e =
1.0
(!
...)
=
1.6
Yo
:
l ~ J
:
Tab
le 9
5 7
:
a =
-.4
6875
j9
=
• 030
5175
8 (1
2)
... -
» -
-i
9 10
:
lll)
:
(12
: ll
3
. .
. .
. .
. .
. .
. .
(!...)~
C!...>2
! c!..
.>3 !
(!...)
4 !ac
!...>2
! sc!
...>4!
cL>
!2ac!.
..> !
4SC-
~>3!
cdy>
!cay>
2 !1
-!...
~ 2~
L
4
Yo
· Yo
·
Y 0
• Yo
•
Yo
.J-
Yo
• Yo
·
. Yo
· J-
Yo
• dx
•
dx
•
82<a
.x> ·
<v)
•
• •
• •
• •
• •
• •
• 0
. .
. .
. .
. .
. .
. .
O:
O :
.1
: .0
1:
.2:
.o4:
.3
: .0
9:
.4:
.16:
.5
: .2
5:
.6:
.36:
. 7
: .4
9:
.8:
.64:
.9
: .8
1:
1.0:
1.0
0:
1.1:
1.2
1:
1.2:
1.4
4:
1.3:
1.6
9:
1.4:
1.9
6:
1.5:
2.2
5:
1.6:
2.~
6:
0 :
.001
: .0
08:
.027
: .0
64:
.125
: .2
16:
.343
: .5
12:
.729
: 1.
000:
1.
331:
1.
728:
2.
197:
2.
744:
3-
375:
4.
096:
0 :
0 :
0 :1
.00
00
0:
0 :
.000
1:
-.00
469:
O
:
-995
31:·
--09
375:
.0
016:
-.
0187
5:.0
0005
: .9
8130
: -.
1875
0:
.008
1:
-.04
219:
.000
25:
.958
06:
-.28
125:
.0
256:
-.
0750
0:.0
0078
: .9
2578
: --
3750
0:
.062
5:
-.11
719:
.001
91:
.884
72:
-.46
875:
.1
296:
-.
1687
5:.0
0396
: .8
3521
: -.
5625
0:
.240
1:
-.22
969:
.007
33
: • 7
7764
: -.
6562
5.:
.409
6:
-.30
000:
.012
50:
.712
50:
--75
000:
.6
561:
-.
3796
9:.0
2002
: .6
4033
: -.
8437
5:
1.00
00:
-.46
875:
.030
52:
.561
77:
--93
750:
1.
4641
: -.
5671
9:.0
4468
: .4
7749
:-1.
0312
5:
2.07
36:
-.67
500:
.063
28:
.388
28:-
1.12
500:
2.
8561
: -.
7921
9:.0
8716
: .2
9497
:-1.
2187
5:
3.84
16:
-.91
875:
.117
24:
.198
49:-
1.31
250:
5.
o625
:-1.
0546
9:.1
5450
: .0
9981
:-1.
4062
5:
6.~5
3e5:
--!_.
_200
00:._
~000
() :
O
_:-:J
...20
000:
0 :
0 :
0 :
.000
12:
-.09
363:
.008
77:
.000
98 :
-.
1865
2: .0
3.47
9:
.003
30:
-.27
795:
.077
26:
.007
81:
-.36
719:
.134
83:
.015
26:
-.45
349:
.205
65:
.026
37:
-.53
613:
.287
44:
.041
87:
-.61
438:
.377
46:
.062
50:
-.68
750:
.472
66:
.088
99:
-.75
476:
.569
66:
.122
07:
-.81
543:
.664
93:
.162
48:
-.86
877:
.754
76:
.210
94:
-.91
406:
.835
50:
.268
19:
-.95
056:
.903
56:
.334
96:
-.97
754:
.955
58:
.4ll
99
: -.
9942
6:.9
8855
: .5
0000
:-1
.000
00:1
..000
00:
1.00
000
.991
23 .
• 965
21
.922
75
.865
17
.794
35
.712
56
.622
54
.527
34
.430
34
.335
07
.245
24
.164
50
.096
44
.044
42
.0ll
45
0
:1.0
00
00
:
.982
61
: .9
3276
:
.856
57
: • 7
6238
:
.658
86
.553
47
: .4
5195
:
.358
09
: .2
7416
:
~201
25
: .1
3976
:
.089
62
: .0
5066
:
.022
71
: .0
0576
0
Slo
pe=
1.0
(!
-..)
=
1.6
Yo
(14)
:
(15)
:
rm
(17)
Tab
le 1
0
2 V
0 m
<
,of)
(17
)(18
) 1+
(19)
(1
8)
: (1
9)
: (2
0)
: V
2m
~
:
(~~
(20)
x(22
) (2
1)
(~)
: (2
3}
: :
:
('1-)2
Vo
~d
(V
)2~(
Y )
d (V
)2~
d tL
)d (
V )
2J~
.dx
V0
• Yo
. dx
V0
.dx
Yo
dx
V0
• .
. .
. .
. .
.
:c.;f
></>
•
=1 +
last=
~:
(~
0)
:~
~L)~
(Y..)
j: te
rm
: 1
+
<!!>
~ :d
x
Yo d
x V
o :
: :
=
cY..>2
=
V
2
: Ve
:
(y)
:Mod
ifie
d:
C
• •
• C
hez
y •
. .
. .
. .
. :
. .
r:000
00·
.991
27;
.087
30:
.087
10
: -1
.742
00
.965
80:
-254
10:
.251
72
: -1
.646
20
.925
51:
-402
90:
.39(
)68
: -1
.389
60
.873
14:
.523
10:
.493
28
: -1
.026
00
.811
70:
.614
40:
.556
18
: -.
6290
0 .7
4396
: -6
7740
: .5
8254
:
-.26
360
.672
27:
.7l6
90:
.578
12
: .0
4420
.5
9841
: .7
3860
: .5
5031
:
.278
10
.523
60:
.748
io:
.506
03
: .4
4280
.4
4861
: .7
4990
: .4
5073
:
.553
00
.373
84:
.74 7
10:
.388
53
: .6
2200
.2
9936
: .7
4480
: .3
2241
:
.661
20
.225
08:
-742
80:
.253
76
: .6
8650
.1
5070
: .7
4380
: .1
8352
:
.702
40
.075
89:
.74 8
10:
.111
58
: .7
1940
o
: .7
5890
; .0
3787
:
.737
10
LOO
OO
O:
-·:or
568
: .9
8432
: 1.
0000
0 : 1
.000
00:
: 1.
0047
1:
-.01
488
: .9
8512
: 1.
0043
8 :
-990
97:
: 1.
0190
6:
-.012
74
: .9
8726
: 1.
0172
5 :
.964
66:
: 1.
0437
8:
-.009
64
: .9
9036
: 1.
0379
1 :
.923
07:
1.08
017:
-.
0061
1 :
.993
89:
1.06
528
: .8
6905
: 1.
1303
0:
-.00
268
: -9
9732
: 1.
0980
2 :
.805
74:
1~19
730:
.0
0048
:
1.00
048:
1.
1346
5 :
.736
09:
1.28
594:
.0
0322
:
1.00
322:
1.
1736
5 :
.662
58:
1.40
351:
.0
0559
:
1.00
559:
1.
2135
3 :
-587
13:
1.56
169:
.0
0777
:
1.00
777:
1.
2528
6 :
.511
09:
: l. 7
8009
: .0
0996
:
1.00
996:
1.
2903
2 :
.435
37:
· ~
2.09
428:
.0
1246
:
1.01
246:
1.
3246
7 :
.360
46:
: 2.
5754
6:
.015
91
: 1.
0159
1:
1.35
481
: .2
8659
: :
3.39
018:
.0
2143
:
1.02
143:
1.
3797
0 :
.213
79:
: 5.
0380
4:
.032
62
: 1.
0326
2:
1.39
842
: .1
4194
:' :1
0.01
904:
.0
6646
:
1.06
646:
1.
4101
6 :
.070
78:
00
:
00
:
00
:
1.41
421
: 0
:
.984
32
: .9
7622
:
.952
37
: .9
1417
:
.863
74
: .8
0358
:
.736
44
: .6
6471
:
.590
41
: .5
1506
:
.439
71
: .3
6495
:
.291
15·
: .
.218
37
: .1
4657
:
.075
48
: 0
:
2 V 0
m
()
"' f
) =
-0
09
1.11
752
(23)
(2
4)
: (2
5)
: (2
6)
: :
: v2
:
V
: V
y
2 :
Vo
.(-){-)
Vo
: ;_
Vo
Yo
: :
1.00
000
:1.0
00
00
:1.0
00
00
.9
9177
:
.995
88:
.991
21
-967
54
: .9
8364
: .9
6524
.9
2873
:
.963
71:
.923
29
.877
50
: .9
3675
: .8
6722
.8
1638
:
.903
54:
.799
38
.748
17
: • 8
6497
: .·7
224 3
.6
7530
.8
2177
: .6
3904
.5
9982
:
.774
48:
.551
82
.523
26
: .7
2337
: .4
6320
.4
4671
: ,
.668
36:
.375
46
.370
76
: .6
0890
: .2
9074
.2
9579
:
.543
86:
.2ll
l7
.221
85
: .4
7101
:-.1
3893
.1
4890
:
.385
88:
.076
59
.076
68
: .2
7691
: .0
2764
0
. 0
: 0
.
D
I
~ I
~---------------0
-->I -- X----- I ------ ->1
-->I -------- : I I I
, , ,. , I ,.t8
y
Yo
FIGURE
I ,. S, '"-- -
.... 1i, .• --: d ,... , "' av
'L ..::.L.. ----Ton•= - dx I\ I \
*"" •. I .a.\ w COS , ,, --' . I \ w. ~ : .. s· • ~ .. ~--w m
r----------------ran-• f
FIGURE 2
I
~
1.0 .a .6
~~J2 .4
.2
00
r--...
. ~
Dra
g R
ela
tio
n -
Slo
pe
= 0.
3 a Yo
=
5.5
·2
W=
o.7
5
~.,
,--
Mo
dif
ied
C
hezy
,_,f:
~
~
,,
'~
2
~ ~ '\ "' '\
Mo
dif
ied
Ch
ezy
-
3 X
Yo
4
FIG
UR
E
3
~ ~
-Dra
g
(..'
----
~
Rei
o ti
on
,
\ ; ,
\ 5
6
CO
MP
AR
ISO
N
OF
T
HE
V
AL
UE
S
OF
{~
0)2 F
OR
T
HE
D
RA
G
RE
LA
TIO
N
AN
D
TH
E
MO
DIF
IED
C
HE
ZY
R
EL
AT
ION
7
1.0
.8
.6
(Y.)2
Vo
.4
.2
00
i---
. ~ '
"-
"' ~- "' ' ~
' ~ '
Slo
pe
=0
.5
~
,-M
od
ifie
d C
hezy
I
a I\.
Y
o= 3
.2
~ ' '
2 '
~; =
0.06
\
I,,.
Dra
g
-~
Re
lati
on
-"
I'\
.5
1.0
1;5
X
i.o
2.5
3.
0
Yo
FIG
UR
E
4
CO
.MP
AR
ISO
N
OF
T
HE
V
AL
UE
S
OF
(~
0)2
FO
R
TH
E
DR
AG
RE
LA
TIO
N
AN
D
TH
E
MO
DIF
IED
C
HE
ZY
R
EL
AT
ION
3.5
1.0
8 -
.6
(0J2 .4
.2
00
r---
r------.
.. r----...
... ....
...... ~~ ~ "' "
"" " ~ '~
, .
. Dra
g
'~
S
lop
e=
0.6
I
Re
lat i
on
I Q
=
2 7
Yo
·
2 ~~ =
0.07
5
.4
.a
1.2
X Yo
1.6
FIG
UR
E
5
"~
I I
'(
,~ ~
Mo
dif
ied
C
hezy
-- --
2.0
2
.4
CO
MP
AR
ISO
N
OF
T
HE
V
AL
UE
S
OF
(~J2
FOR
T
HE
D
RA
G
RE
LA
TIO
N
AN
D
TH
E
MO
DIF
IED
C
HE
ZY
R
EL
AT
ION
" I\ 2
.8
1.0
--.8
.6
(~J2 .4
.2
00
"-~ ~
Dra
g
Re
lati
on
Slo
pe
= o
.e
~o
= 2
.0
1~ =0.
020
.4
rv---
Mo
dif
ied
C
hezy
.... ,~
\. '
.8
' '\, ~ -... \
"' \
1.2
.X.
I.&
Yo
FIG
UR
E
6
~ \ \ 2.
0 2.
4
' 2
CO
MP
AR
ISO
N
OF
T
HE
V
AL
UE
S O
F(~
.)
FOR
T
HE
D
RA
G
RE
LA
TIO
N
AN
D T
HE
M
OD
IFIE
D C
HE
ZY
RE
LA
TIO
N
'
2.8
1.0
.8
.6
(~J2 .4
.2
0 0
----....
"'
-M
od
ifie
d
Che
zy
~,,
Dra
g
?- "~
, ,
Re
lati
on
-,,
"-"' "
~ "' "" " "'
~ ~
, D
rag
S
lope
= 1
.0
,, R
ela
tio
n
, . ~
o =
1.6
~ ,,
v;~ ;0
.00
9
~
,
~
,, , M
od
ifie
d
Che
zy .
., ~
-
' ~
I
.2
.4
.6
.8
1.0
1.2
1.4
X Yo
.FIG
UR
E
7
CO
MP
AR
ISO
N
OF
T
HE
V
AL
UE
S
OF
(~
.)2
FOR
T
HE
D
RA
G
RE
LA
TIO
N A
ND
T
HE
M
OD
IFIE
D C
HE
ZY
RE
LA
TIO
N
1.6
10
.0
8.0
6.0
4.0
2.0
0
\ \ '
~. '~
\ ,T
Yo
' "' ~
Yo =
Dep
th a
t ce
nte
r
P
= P
eri
me
ter
T
= T
op w
idth
A
=
Are
a
0.3
0
.4
"" -~ ~
......
" --..
......
0.5
r---.... ~ ~
~
A
r yJ
:t___
----
0.6
SL
OP
E
(Sl
FIG
UR
E
8
~
I~
---.
.. ~
~
----
0.7
r---
-r---
......
r---
----- ~ ---
0.8
PR
OP
ER
TIE
S
OF
S
TA
BL
E
CH
AN
NE
LS
-· p ,-
-'
Yo
I
~ -
------
---r--
--
0.9
1.
0 .
1.0
0.8
0.6
0.4
0.2
0 J
-I ,
I •
I I
I I
I V
o
I I
I I
~
I I
I ,,
--V
e I
I I
V
I -
' '
-··:
r-~
=f-
fttt
+
P
---t--~
I
~.-
~
l -
t--t--
J -
-=--
t---
~ r-~
.!
-
, I
- --+
--t-
,---
,-V
m
,'
, V
o
' _
-,-
-.-
I -I---,-
-.--
--
. --
i---
-f--
t--r
--
-+
--t-T
--
. I
I I
I I
-I
. I
I -
,.._
__
__
J
I _
.._
, __
_
,--.
J -
I
-.-
-r---
-L
,--.
·=tP
-t-+H
H-=
tp
1_
1
: ==[=j~=
f=~t=j~=
J~=~~~~
~~L_J,~
~~~~:--J
_ '-o~·~::
~ I
I J
I J
' I
' J
J J
I l
1 0
.3
0.4
0,6
S
LO
PE
(S
) 0
.5
0.7
0
,8
0.9
,.o
FIG
L!R
E
9
PR
OP
ER
TIE
S
OF
S
TA
BL
E
CH
AN
NE
LS
1.0 _,,, ·-
/ \
0.9 ___,,,,. V' \
\ \
0.8
/ V \
0.7 V
V \ --\
--(/) -w a. 0.6 0 _J (/)
\ ./ V v\
0.5
V ~
_L----'" V \ ./
V ~\
0.4
~ .....
I"' -------------- '"" ~ ............... 0.3
__.-,-------~ _j,.
~ .....
--~ 1.0 2..0 3.0 4.0 5.0
X Yo
FIGURE 10
PROPERTIES OF STABLE CHANNELS
1.0 .8 9 ---
- (/) w
a..
0 _J .6
5
(1)_
4 ' .2 r-
--
r---
- ~
0
..........
..... ~ r---
--....._
i---r-
--t-
--..
---r-- -
--r--
- ~
i--.
....
._
----r-
--~
-3
,5
4.0
-5
.0
4.5
0.1
0.2
" \
"' I\
-........
.. ~ ~
'K
\ ·'O
Yo
"l.
5
cJ'
,-.....:.,
__ .....
~ ~
"' \
t-,.....
__
~
i---~
~~
"' ~
"' t-
-...
\
----r--
-...
~
' \
i---
~
"' '"'-
----
~
r---
i---
-.... ~
I"--
'\
\ t-
--r-
--r--
~
"' ...
t--.
_
~
r---
....
r-
--r-
------
i---
--.......
'-
\ ~ \
~
-i--
-...
t-
--
~
r---
-r-
---
I---
r---
. r---
,.....__
""-
0.3
0.4
0.5
0.6
0.
7 0
.8
0.9
1.
0 1..
Yo
FIG
UR
E
II
PR
OP
ER
TIE
S
OF
S
TA
BL
E
CH
AN
NE
LS
X
a
1.0
0.8
0.6
0.4
0.2
0 0 \ \ \ ~
~ '
0.2
~ ~
·"-.
. ' 0
.4
- "' '
0.6
Q
ao
" ~~ "' "
-0
.8.
1.0
FI
G'U
RE
12
.
t< -
----
-a-
----
-->
i I
I
I-<-X
-.>:
I I
I I
I ri
~ Is
tl1
e f
racti
on
of
the
ch
an
ne
l to
be
rem
ove
d w
he
n t
he
de
sig
n
Q i
s l
ess t
ha
n t
he
sta
ble
ch
an
ne
l w
ou
ld c
arr
y.
PR
OP
ER
TIE
S
OF
S
TA
BL
E
CH
AN
NE
LS
GI 11
0 .. .. f:; ..
Mo
dif
ied
Sta
b I e
Cho
n n
e I -
_ ..
Q1=
75
c.f
.s.
''"
Sta
b le
Ch
an
ne
l P
rofi
le
Fo
r so
il p
rop
ert
ies a
nd
lon
git
ud
ina
l sl
op
e s
ho
wn
-... ,
Q
= 19
2 . 8
3 C
. f. S
. '
Mo
dif
ied
Sta
b le
Ch
an
ne
l .....
, Q
2 =
30
0 c
.f .s
. '
FIG
UR
E
13
DE
SIG
N
CR
ITE
RIA
T
ran
sve
rse
slo
pe
= 0
.6
Allo
wa
ble
dra
g_ f
orc
e=
0.1
lb
.lft
.2
Lo
ng
itu
din
al
slo
pe
= 0
.00
04
C
oe
ffic
ien
t o
f ro
ug
hn
ess=
0.0
20
o,
= 7
5
c.f.
s.
Q2
= 3
00
C. f
.S.
ST
AB
LE
C
HA
NN
EL
P
RO
FIL
ES