(

\.

w \0

,.

I I ' ! I .. I : I .. .. J " I !

I

...

; w

I -I ...

II

'1-/fl . '1- 1- t-'"- TJ" r= v -# rr ,-

1 r!fiJ 711 7

11 J

r-7iJ: rfrl If/

Ill I -T

7 II

I I ' .

...

I

AvtltAGl YlLOCITY, IPHIIt1CAL OIAM(T!It 0 50, FT

IASIC EQUATIONS WHtM:

r: (,.,_.,_),1'.12( )"' v Cl2g rw 050

0 $0 ( tW )"3

so nl's

V vtLOCITY, FPS Yt II'ICIFIC STOHl WIIGHT, L8/P'TJ Yw SPICifiC W[ICOHT or WAT(ft, 11.5 LIIP'TJ Wto W(IGHT OF STOH[. SU81Cit1PT ONOT[S

PCitCNT M TOTAL WEIGHT M MATllttAL CONTAINING STON( M LIIS W[IGHT.

Dto IPH(ftiCAL OIAWTIIt M STOHl HAv.NG TH( SAW( wttGHT AS Wte

C IS8ASH C:OHSTANT (O.C. rOit MtGH LEvt\. rLOW AHO 1.10

,Oit LOW TUitlt\.t.EMCI Ll11. fLOW) t ACClLfltAT10N OF' GltAYITY, "IKC:1

FIGURE 3-8 (Sheet l of 2)

STONE STABILITY VELOCITY VS STONE DIAMETER

HSIGM C:HAitT 111 -I (SHUT I

wue-P

Isbach- Velocity Versus Stone Diameter (from Hydraulic Design Criteria (14))

" -

! f I I i c .. I ~ ..

I -, J .II

I I

+::-0 '

10,000

-- 10.000

~ao AA.~~~~-~~~~~~-r~-f

v

t- 1-

III Ill

1.11 ~ ~

::~ l ~ LOW l~. f-1 ~ 1 U~~r r~n:~fJ~+I-+H-H--4-1-4-4-++H

... . ,.

l--+--+-+-4-41f-+l I II 1/ll Ill II J

J/1 11/

TABLE 3-6

BOTTOM R!PRAP SIZ~ FOR SAFE DESIGN BY ISBASH METhOD \

DISCHARGE BOTTOM BOTTOM SIDE DSO DEPTH DSO/D F SLOPE WIDTH SLOPE;

CFS FT/FT FT FT FT 13382 000415 100 4 060 100 0060 0533 21853 o. 01107 100 ''" 1~60 100 0"160 0870 .. 27857 001798 1 00 4 2 "60 1 o.-o o.-260 1" 11 0 31&4 11. 000207 100 4 060 20"0 o ~ o3o 0377 56192 000553 1 oo. " 1"60 2o-.o 0080 o;.615 71632 o. oo899 too. ~a . 2.; 60 20"0 o..-130 0"785 53528" 000207 2oo. ~a.- o.- 60 20 ... 0 o.-o3o o.-377 87410 o. oo553 2oo. I& 160 200 o-.o8o o.-615

111427" o . oo899 200" " .. 2 "60 200 o.- 130 0785 91762. o .-oot38 2oo. ~a ; o.--6o 30 .. 0 o.o2o 0308 149846 o. 00369 2oo.- " ... 1 .. 60 30"0 o.-o53 0503 t91017 .. o . oos99 2oo. 4 ... 2 "60 3o.-o o.-o87 "" 0. 641 120437 o.-oo 138 3oo.- ~~ -.- o-.6o 3o.o 0"020 0"308 196673 000369 3oo.- I&.- 60 3o.-o o.os3 0503 25071 o.-- o.-oos99 3oo. " ... 2 .. 60 30" 0 o.o87 0641

1131&3" 000346 too-. 3 ... o-.so 1 o. o o-.o5o 0487 19647 0"01037 1 oo. 3 ... 50 t o-.o 0"150 0843 25365 o-.ot729 1 oo.- 3 ... 2-.so to-.o o.2so 1088 27922" o.-oot73 aoo 3 ... o.so 2o.o oo2s o. 344 48363" o-.oos19 100 3-. a.-so 2o.o oo1s o-.s96 62436 o.-oos6s too-. 3 ... 2-. ~o 2o.-o o.12s Oe769 45374 ... o.oo173 2oo.- 3-. o-.so 20 0 o.-o2s 0"344 78589 o.-oost9 2oo.- 3 1"50 2o.o o-.o7s 0596

101459" o.oo865 2oo-. 3 ... 2 "50 2oo o.-12s 0769 75914 o.-oo1ts 2oo. 3 .... o-.so 3o.-o o.-o17 0281

131486 ... o-.oo346 200-. 3 1.so 30"0 ooso 0"487 169748" o.-oo576 200 3 2"50 300 o.o83 0628 102091 000115 3oo. 3 " o.-so 3o.-o o. 011 0"281 176826 o.- 00346 3oo.- 3 ... 1"50 3o.o o-.o5o 0487 228282 ... 0"00576 3oo.- 3 2 .-so 30 .. 0 o.o83 0"628

9365" o-.oo277 1 oo.- 2 " o.- 40 1 o.-o o-.o.t&o 0 "435 17521 000968 1 oo. 2 1"40 10 .. 0 o.-140 08tl& 22940" o .ot660 too. 2 " 2 "40 10 .. 0 0"240 1066 21853 o..-oo138 too. 2 ... o.-4o 20 ... 0 o .- o2o 0"308 40882 .. .o.- oo.ta84 1 oo.- 2 ... 1-.40 ~o-. o o.-o7o 0576 53528 ... o .- oo830 1 oo-. 2 " 2 "40 20"0 o.-t2o 0"754 37462 ... o.- oo 138 2oo.- 2 o .t&o 20" 0 0020 o-.3o8 7 0084 o. oo484 200 ... 2 " , . 40 2o.-o o.o1o 0576 91762" o. 00830 2oo. 2 ... 2 ... 40 20 .... 0 o.-t2o 0 754 60875" 000092 2oo.- 2 ... o.-4o 30 . .'0 o.-o13 o.-251

113887 ... o. oo323 2oo.- 2 ... , ... 40 3o.-o o. 0111 0"470 149113 ... 000553 2oo.- 2 ... 2 ".-40 30 .... 0 o...-o8o o.-615 84289" o. ooo92 3oo.- 2 ... o.4o 30"0 o..-ot3 o.-251

157690 ... o. oo323 3oo-. 2 ... 1 ... 40 30 .... 0 o.- 047 0"470 206464 o.-ooss3 3oo.- 2-. 2 ... 40 3o-.o o.o8o o.-6t s

41 1

J

Dso DEPTH

,.

EE ~~ ~~

!::E: ~ ;E: ~E F=E:

~~ ~ EE EE f::E

O.l

~ ~ ~ E:: t:

0.01 0.1

I+

~ :m

m ~

Iff ~ Mit

IW I*

II

ffi11

'

FIGURE 3- 9

- m

w I

I

I m if!:

= -~ -H - -

1 I ~ K ~

~ I I I I I II I

1.0

n50/Depth Versus F - Bottom Riprap, Isbach , Safe Design

42

~

motion as determined from the model test of riprap stability in decel-

erating flow.

D50 = 3 ~ 0.22F depth ( 3 bis)

Values of n50/depth and Froude numbers for safe design computed by the

Isbash approach fell on th~ safe side of tQe incipient motion curve for

Froude numbers less than 1.9. A least-squ~es fit of these values

results in

D I 50, = o. 21F2 0 dept~

43

(28)

I

I ,

l

IV. DEVELOPMENT OF SIDE SLOPE CRITERIA

The coefficient C in Equation l will be determined for riprap

-on channel side slopes. Results from the model tests will be used to

determine C and this value will be compared -to existing criteria ...

4-l Model Tests

Tests of the lV:4H and lV:3H channels.showed that failure occurred

on both the channel bottom and the channel side slopes. Therefore the

equation for channel bottom riprap at incipient motion

D50 depth

- 3 = 0.22F (3 bis)

is applicable to channel side slope riprap at incipient motion for side

slopes of lV:3H or flatter.

The results of the model tests of the 1V:2H channel are shown in

Table 2-1. In every test, the 1V:2H channel failed on the side slope

only. A plot of D50

/depth versus Froude number for these tests is

shown in Figure 4-l. The relationship for incipient motion for riprap

on a lV:2H side slope as shown in Figure 4-l is

4-2 Existing Criteria

D50 depth

3 = 0.25F (29)

Anderson's criteria for sizing riprap on channel side slopes are

also based on the tractive force or shear stress method. The critical

shear stress that is required to initiate motion is reduced by the

44

Dso DEPTH 0.1

II

~01 uu~~~llll~~~~~~~~ww~~~~~~~~~~~~

0.1 F=

v 1.0

JtOErrH

FIGURE 4-1 D5o/Depth Versus F - 1V:2H Side

Slope Riprap, Model Tests, Incipient Motion

I __ ./

'

factor K which is a function of the angle of the side slope e and

the angle of repose of the material ~

K = 1 -. 2 e

s~n

. 2 "' s~n 'f' (30)

The critical shear stress for incipient motion for channel side

slopes is

T =5D K c 50

(31)

The critical shear stress for safe design for channel side slopes

~s

The tractive force exerted by the flowing water on the channel

side slope is

T = CyRS s

where C is a function of the aspect ratio and is determined from

Figure 4-2.

(32)

(33)

Solution of Anderson's approach to side slope riprap is the same

as Anderson's approach to channel bottom riprap.

Rock sizes for typical channel disch~ges, bottom widths, and

channel bottom slopes are determined for 1V:2H side slopes using

Anderson's approach and o50/depth and Froude numbers are computed for

46

J

-)( 0 E '* -~ 0:: ~

II .. a:

0

2.0

1.8

1.6 SII:J' .Si. Op~ z,.

6

1.4

1.2

1.0 1.$

o.8 ~----_.------~------~------~----~----~~------~----~ 0 1 2 3 4

ely

FIGURE 4-.2

5

Maximum Boundary Shear Stress on

6

Sides of Trapezoidal Channels (After Anderson (2))

47

7 8

I

each condition. Incipient motion conditions are shown in Table 4-1.

By comparing incipient motion conditions for 1V:2H side slopes from

Table 4-1 to incipient motion conditions for channel bottoms from

Table 3-1, a relation between the two conditions can be determined for

the value C to be used in Equation 1.

Cl:2SS = 1 135CBOTTOM (34)

Equation 1 for incipient motion for 1V:2H side slopes becomes

D50 3 depth = 0.25F (35)

based on the ratio of Anderson's bottom criteria to side slope

criteria.

The riprap stability criteria presented by Ramette was based on ' ..

tests of channel side slopes of 1V:2H and 1V:3H. Incipient motion rock

sizes are determined by solving Equations 18, 19, and 20. The veloc-

ity used in Equation 20 is taken 0.8 depth above the toe of the slope.

Based on the velocity profiles in Figures 2-3 to 2-9, this value can

be estimated by

V(0.8 depth above toe) - 1.2V(average channel velocity) (36)

Incipient motion rock sizes for typical channel discharges, bot-

tom widths, and channel bottom slopes are determined for 1V:2H side

slopes using Ramette's approach and n50

/depth and Froude numbers are

computed for each condition as shown in Table 4-2. By comparing rock

48

TAN.g 4-1

SIDE SLOPE RIPRAP SIZES FOR DiCIPIENT MOTION BY ANDERSON METHOD

DISCHARGE BOTTOM BOTTOM SIDE DSO DEPTH D50/D F SLOPE WIDTH SLOPE

CFS FT/FT FT FT FT 11873 o.oo304 100 2 040 100 0040 0552 18027 "~ o. o 1 063 100 2 " 1 "~40 1 o-.o o.-140 0"838 21575" .. o-.o1822 100 2 ... 2"~40 1 o.-o o.-240 . ... 003 28399" 000148 180 2 " o-.-4o 20 -~0 o.-o2o 0 "400

I 43119 "~ o ~o0517 1 oo. 2 " 1"40 2o.-o o-.-o7o 0""607 51606" o . oo886 1 oo . 2 ".- 2 ""40 2o.o o.-120 o.-121 53309 000152 200 ~ 2 o.4o 20 0 o . 020 0"438 80939 000531 2oo. 2 " 1.-40 20"0 o.o7o o66S 96869 000911 2oo. 2 " 2 "~40 20"0 0"120 o.-796 87335 000098 2oo. 2 0"40 3o.-o o~o 13 0"360

132602 000342 200 2 " 1 "~40 3o-.-o o.o47 o ~547 158701 000587 2oo. 2 ".- 2".-40 30"i0 o~ o8o o.-655 128330 o.oo1o1 3oo-. 2 " o-.4o 3o-.o o. ol3 0"383 194844 000354 3oo.- 2 " 1"40 3o.-o o. 047 o-.581 233193 o. oo607 300 2"~ 2-."40 3o-.-o o.-o8o o-.695

TlBieE 4-2

SIDE SLOPE RIPRAP SIZES POR INCIPIENT MOTION BY RAMETIE METdOD

DISCHARGE BOTTOM BOTTOM SIDE 050 DEPTH D50/D F SLOPE WIDTH SLOPE

CFS FT/FT FT FT FT 10048 o.oo337 100 2 040 100 0040 0467 15575 001181 1 oo. 2 140 1 o. o 0140 0724 18577 o. o2o24 100 2 240 100 0"240 0863 25668 000169 1 oo-. 2. 040 200 0020 0361 40502 000590 100 2 140 200 a. 010 0570 48794 001012 1 oo. 2 2"40 200 0"120 0687 44003 000169 2oo. 2" 040 2o. o 0020 0361 69432 000590 200 2" 140 200 0010 0"570 83646 001012 200 2 2 "40 2o. o 0120 0687 75128 000112 200 2 0"40 300 o.-o 13 o-.31 o

119606 o. o0394 2oo. 2-.- 1.-40 3o.-o o. 047 0494 144801 000675 200 2. 2 "40 3o.-o o. o8o 0598 1 04023 ... o.oo112 3oo. 2 0"40 3o.-o 0013 0"31 0 165608 000394 300 2 140 3o-.-o o.o47 o ~494 200494 000675 300 2 240 3o.-o 0080 o598

50

sizes for 1V:2H side slopes from Table 4-2 with rock sizes for channel

bottom riprap from Table 3-4, a relation between the two conditions

can be determined for the value C -ta be used in Equation 1. , . ,,

Cl:2SS = 1 29CBOTTOM

Equation 1 for incipient motion for 1V:2H side slopes becomes

,. . "' :050 = a.284F3 depth

(37)

(38)

based on the ratio of Ramette's bottom criteria to side slope criteria. r

Side slope criteria used by the Corps of Engineers in EM 1110-2-J

1601 (4') is similar to that used by Ramette. The critical shear stress

determined from Equation 23 is reduced by given in Equation 30.

(39)

The velocity u:sed in Equation 25 is the average V'elocity in the

vertical at the toe of the slope. Based on the velocity profiles in

Figures 2-3 to 2-9

V ( 0. 6 depth at toe) = V ( average channel velocity) (40)

Rock sizes for typical channel discharges, bottom widths, and

channel bottom slopes are determined for 1V:2H side slopes using the

Corps approach and n50

/depth and Froude numbers are computed for each

condition as shown in Table 4-3. By comparing rock sizes for 1V:2H

side slope from Table 4-3 with rock sizes for channel bottom riprap

51 (

, I

TABLE 4-3

SIDE SLOPE RIPRAP SIZES FOR SAFE DESIGN BY C .0 .E. METhOD

DISCHARGE BOTTOM BOTTOM SIDE D50 DEPTH D50/D F SLOPE WIDTH SLOPE

CFS FT/FT FT FT FT 10897 000198 100 2 040 100 0040 0506 15921. 000695 100 2 140 10 .. 0 o-.140 0740 18331 001191 100 2 2"40 100 0240 0852 28507 000099 100 2 040 2o.o o.-o2o 0401 42914 000347 100 2" 1"40 2o.o 0010 0604 50319 o-.oo595 100 2 .. 2"40 2o-.o 0"120 0"709 48869 o. ooo99 200 2 ... 0"40 200 0020 o-.401 73567 000347 200 2 140 200 o.-o7o 0"604 86261 000595 200 2' 2"40 20' 0 0'120 0"709 84432 000066 200 2 .. o.-4o 3o.-o 0013 0"348

128938 000232 200 2 1 .. 40 30" 0 0047 0532 152472 000397 2oo. 2 2"40 3o.-o o.-o8o 0629

116906" o. ooo66 300 2" 040 3o.-o o..-o13 0348 178530 o .- oo232 3oo. 2 " .. , . 40 3o.o 0"047 0"532 2111.15 000397 300 2 2" 40 3o.-o o.oso o-.629

. .

,

f

52

from Table 3-5, a relation between the two conditions can be determined

for the value C to be used in Equation 1.

cl.2SS = 1 236CBOTTOM

Equation 1 for incipient motion for 1V:2H side slopes becomes

D50 3 ----::~ = 0. 272F depth

(41)

(42)

based on the ratio of EM 1110-2-1601 (4) bottom criteria to side slope

criteria.

4-3 Design Curves

A summary of the values of C for incipient motion on 1V:2H side

slopes is as follows:

Method

Model tests

Anderson

Ramette

EM 1110-2-1601

c

0.25

0.25

0.284

0.272

For this investigation a C value of 0.26 will be used. Equa-

tion 1 for incipient motion on 1V:2H side slopes as shown in Figure 4-3

is

D50 _ 3 ----::- - 0. 26F depth

(43)

The curve for safe design with a factor of 1.5 x incipient motion

for 1V:2H side slopes based on the average stone weight is

53

Dso DEPTH

:

~01 uu~~~llll~~~~~~~~~~~uwww~ww~~~~~w

~ v ~ u F = I tOEf'rH

. '

FIGURE 4-3 n50/Depth Versus F - 1V:2H Side Slope Riprap, Incipient Motion

54

050 3 depth= 0.30F (44)

and a factor of 2.0 x incipient motion for 1V:2H side slopes based on

the average stone weight is

D50 _ 3 depth - 0 3~F (45)

I '

'

' . .. .

55

V. DEVELOPMENT OF BEND CRITERIA

Information on sizing riprap in channel bends is relatively

scarce. In Figpre ~-1 the shear distribution in a channel bend is

shown as presented in EM 1110-2-1601 (4). Tbe maximum shear in a chan-

nel bend as a function of bend radius and water surface width is shown

in Figure 5-2. This figure was taken. from EM . lll0-2~1601 and is a good

summary of the work previously conducted in the field of shear distri-

bution in channel bends. Additional research is needed to determine

the effects of total bend angle and side slope angle on the shear dis-

tribution in a channel bend. Figure 5-2 was based on a channel with

1V:2H side slopes and a 60 bend angle. Figure 5-2 was used to deter-

mine tentative values of C in Equation 1 for sizing riprap in channel

bends.

The equation for rough channel conditions as shown in Figure 5-2

is

where

T 0

( )-0.5

3.2 !. w

Tb = maximum boundary shear as affected by bend

T - average boundary shear in approach channel 0

r = center line radius of bend

w = water surface width at upstream end of bend

The Shields' equation for the critical shear stress is

T = o.o4(y - y )D 0 s w 50APPROACH

in the approach channel and

(46)

(47)

C=

Dso DEPTH

FJ

0.1 1.0

I

BEND RADIUS 10.0

WATER SURFACE WIDTH

FIGURE 5-3 C Versus Bend Radius/Water Surface Width, Incipient Motion

59

20.0

Tb = o.o4(y - y )n50 s w bend (48)

in the channel bend. Substituting Equations 47 and 48 into equation 46

From Equation 1, let

and

D50 -0.5 __ ....;;b;..;;e;;.;.;n;.;.;d~ = 3 2 !:.. D w 50 APPROACH

D50 bend

3 = Cbend depth F

(49)

(50)

D = 0.22 depth F3 50 APPROACH (3 bis)

Substituting

-0.5 c = 0.70!:. bend w (51)

as shown in Figure 5-3. This curve represents incipient motion for

only the point on the curve where the shear stress is the highest.

Based on Figure 5-l the point of maximum shear is located on the side

slope of the outside bank at the downstream end of the bend.

Additional work is needed to determine the coefficients that

should be used for safe design for the entire length of the curve and

the area downstream that is affected by the curve.

60

VI. SUMMARY AND SAMPLE PROBLEM

A summary of the coefficients determined in this investigation for

the equation for riprap stability

D50 3 _.:;..___ = CF depth

is as follows:

Condition

Straight channel, bottom riprap, incipient motion Straight channel, bottom riprap, F.S. - 1.5 Straight channel, bottom riprap, F.S. = 2.0

' Straight channel, 1V:3HSS or flatter, incipient motion Straight channel, 1V.:3HSS or flatter, F.S. - 1.5 Straight channel, 1V:3HSS or flatter, F.S. = 2.0

Straight channel, 1V:2HSS, incipient motion Straight channel, lV:2HSS, F.S. - 1.5 Straight channel, 1V:2HSS, F.S. = 2.0

Curved channel, incipient motion* C = 0.70(r/w)-0.50

(1)

Coefficient C

0.22 0.25 0.28

0.22 0.25 0.28

0.26 0.30 0.33

* Incipient motion for only the point on the curve where the shear is

highest.

A sample problem to illustrate the use of the Froude number ap-

proach is as follows:

Design Data--Straight channel 100-ft bottom width 1V:3H side slopes 0 . 004 ft/ft bottom slope Design discharge = 30,000 cfs

Determine the required rock size to provide a safety factor of 1.5 and

the depth of flow at the design discharge.

I

I

Solution: Assume n50

= 1.0 ft

n = 0.0395D~~6

n = 0.0395

From Manning's equation

Normal depth = 16.2 ft

Velocity = 12.5 ft/sec

F = 0.55

From Froude's number concept

050 3

depth = 0.25F

n50 = 0.67 ft This 0

50 not close enough to the assumed lS

Ass tune 050 - 0.75 ft

n = 0.0395D~~6

n = 0.038 From Manning's equation

Normal depth = 15.9 ft

Velocity = 12.8 ft/sec

F = 0.57

From Froude's number concept

050 3

depth = 0.25F

n50

:; 0.72 ft

62

(9 bis)

: . :

(4 bis)

0 50

..

The assumed n50

of 0.75 ft is close enough to the computed

n50

= 0.72 ft. The channel requires a ri~rap blanket with a 9-in.

n50

on both the channel bottom and side slope.

.. ' I '

f ' .

63

. . . \ . ' . VII. CONCLUSIONS .

The results of this investigation show that riprap stability can

be described by parameters that are known or easily computed. Froude

number and depth of flow are used to determine stable riprap size.

Comparison of the Froude number approach with existing shear stress

design methods shows that Froude number and depth of flow properly de-

scribe riprap stability.

The model tests show that riprap on channel side slopes of 1V:3H

or flatter require no increase in rock size to maintain stability.

Appropriate relations for determining stable rock sizes on 1V:2H side

slopes were developed from the model test and existing design concepts.

Additional research is needed so that stable rock sizes in channel

bends can be determined.

64

VIII. BIBLIOGRAPHY

1. Corps of Engineers, "Additional Guidance for Riprap Protection," ETL 1110-2-120 (1971).

2. Anderson, A. G., "Tentative Design Procedure for Riprap Lined Channels," National Cooperative Highway Research Program,, St. Anthony Falls Hydraulic Laboratory Report No. 96 (1968).

3. Li, R., Simons, D. B., Blinco, P. H., and Sa.ma.d, M. A., "Probabi-listic Approach to Design of Riprap for River Bank Protec-tion," Symposium on Inland Waterways for Navigation, Flood Control, and Water Diversions, vol. II, CSU (1976).

4. Corps of Engineers, "Hydraulic Design of Flood Control Channels," EM 1110-2-1601 (1970).

5. Ramette, M., "Riprap Protection of River and Canal Banks," U. s. Army Engineer Waterways Experiment Station Translation No. 63-7, Vicksburg Miss. (1963).

6. Lane, E. W., "Progress Report on Results of Studies on Design of Stable Channels," USBR, HYD 352 (June 1952).

7. Stevens, M.A., Simons, D. B., and Lewis, G. L., "Safety Factor for Riprap Protection," Journal of the Hydraulics Division, ASCE, vol. 102, No. HY5, pp. 637-655 (May 1976).

8. Grace, J. L., Jr., Calhoun, C. C., Jr., and Brown, D. N., "Drain-age and Erosion Control Facilities, Field Performance Inves-tigation," U. S. Army Engineer Waterways Experiment Station, CE, MP H-73-6 (June 1973).

9. Rothwell, E. D. and Bohan, J.P., "Investigation of Scour and Pro-tection Around Bridge Piers," U. S. Army Engineer Waterways Experiment Station, CE, Memorandum Report (November 1974).

10. Shields, A., Anwendung der Achnlichkeitsmechanik und der Turbu-lenzforschung auf die Geschiebebewegung, Mitteilungen der Pruess Versuchsanstalt fur Wasserbau m1d Schiffbau Heft 26, Berlin (1936).

11. Gessler, J., "The Beginning of Bed Load Movement of Mixtures Investigated as Natural Armoring in Channels," W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, Calif. (1965).

65 >

J

12. Keulegan, G. H., "Laws of Turbulent Flow in Open Channels," Journal of Research, vol. 21, National Bureau of Standards, Washington, D. C., pp 701-741 (1938).

13. Isbash, S. V., "Construction of Dams by Dumping Stones in Flowing Water," Translated by A. Dorijikov, Eastport, Maine, CE (1935).

14. Corps of Engineers, Hydraulic Design Criteria, Sheet 712-1, Stone Stability-Velocity Versus Stone Diameter (1970).

66