Threshold Channel Design - Part 654 Stream Restoration Design National Engineering Handbook - Directives System
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United States
Department of Part 654 Stream Restoration Design
Agriculture
National Engineering Handbook
Natural
Resources
Conservation
Service
Chapter 8 Threshold Channel DesignChapter 8 Threshold Channel Design Part 654
National Engineering Handbook
Issued August 2007
Cover photo: Threshold channels have erosion-resistant boundaries.
Advisory Note
Techniques and approaches contained in this handbook are not all-inclusive, nor universally applicable. Designing
stream restorations requires appropriate training and experience, especially to identify conditions where various
approaches, tools, and techniques are most applicable, as well as their limitations for design. Note also that prod-
uct names are included only to show type and availability and do not constitute endorsement for their specific use.
The U.S. Department of Agriculture (USDA) prohibits discrimination in all its programs and activities on the basis
of race, color, national origin, age, disability, and where applicable, sex, marital status, familial status, parental
status, religion, sexual orientation, genetic information, political beliefs, reprisal, or because all or a part of an
individual’s income is derived from any public assistance program. (Not all prohibited bases apply to all programs.)
Persons with disabilities who require alternative means for communication of program information (Braille, large
print, audiotape, etc.) should contact USDA’s TARGET Center at (202) 720–2600 (voice and TDD). To file a com-
plaint of discrimination, write to USDA, Director, Office of Civil Rights, 1400 Independence Avenue, SW., Washing-
ton, DC 20250–9410, or call (800) 795–3272 (voice) or (202) 720–6382 (TDD). USDA is an equal opportunity pro-
vider and employer.
(210–VI–NEH, August 2007)Chapter 8 Threshold Channel Design
Contents 654.0800 Purpose 8–1
654.0801 Introduction 8–1
654.0802 Design discharges 8–2
654.0803 Allowable velocity method 8–3
(a) Calculate average velocity.............................................................................. 8–3
(b) Determine allowable velocity......................................................................... 8–6
(c) Soil Conservation Service allowable velocity approach............................. 8–8
654.0804 Allowable shear stress approach 8–10
(a) Calculate applied shear stress...................................................................... 8–10
(b) Calculate allowable shear stress.................................................................. 8–14
(c) Procedure for application of allowable shear stress method................... 8–23
(d) Limitations and cautions............................................................................... 8–24
654.0805 Tractive power method 8–26
654.0806 Grass-lined channels 8–27
(a) Allowable velocity.......................................................................................... 8–27
(b) Allowable shear stress................................................................................... 8–28
(c) Species selection, establishment, and maintenance of grass-lined......... 8–30
channels
(d) Determination of channel design parameters............................................ 8–30
(e) General design procedure............................................................................. 8–32
654.0807 Allowable velocity and shear stress for channel lining materials 8–37
654.0808 Basic steps for threshold channel design in stream restoration 8–38
projects
654.0809 Conclusion 8–43
(210–VI–NEH, August 2007) 8–iChapter 8 Threshold Channel Design Part 654
National Engineering Handbook
Tables Table 8–1 General guidance for selecting the most appropriate 8–2
channel design technique
Table 8–2 Suggested minimum radius of curvature in stable soils 8–4
without bank protection
Table 8–3 Maximum permissible canal velocities 8–6
Table 8–4 Allowable velocities 8–7
Table 8–5 Characteristics of methods to determine allowable 8–23
shear stress
Table 8–6 Allowable velocities for channels lined with grass 8–27
Table 8–7 Classification of degree of retardance for various 8–29
kinds of grasses
Table 8–8 Characteristics of selected grass species for use in 8–31
channels and waterways
Table 8–9 Retardance curve index by SCS retardance class 8–31
Table 8–10 Properties of grass channel linings values 8–31
Table 8–11 Allowable velocity and shear stress for selected lining 8–37
materials
Figures Figure 8–1 Design velocities for natural channels 8–5
Figure 8–2 Design velocities for trapezoidal channels 8–5
Figure 8–3 Allowable velocity—depth data for granular materials 8–7
Figure 8–4 Allowable velocities for unprotected earth channels 8–9
Figure 8–5 Applied maximum shear stress, τb, on bed of straight 8–11
trapezoidal channels relative to an infinitely wide channel, τ∞
Figure 8–6 Applied maximum shear stress, τs, on sides of trapezoidal 8–11
zoidal channels relative to an infinitely wide channel, τ∞
Figure 8–7 Lateral distribution of shear stress in a trapezoidal 8–13
channel
8–ii (210–VI–NEH, August 2007)Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
Figure 8–8 Applied maximum shear stress, τbs and τsc on bed and 8–15
sides of trapezoidal channels in a curved reach
Figure 8–9 Applied maximum shear stress, τbt and τst on bed and 8–15
sides of trapezoidal channels in straight reaches
immediately downstream from curved reaches
Figure 8–10 Shields diagram 8–16
Figure 8–11 Gessler’s reformulation of Shields diagram 8–16
Figure 8–12 Variation in Shields parameter with decreasing sediment 8–17
load
Figure 8–13 Probability of grains to stay on the bed 8–18
Figure 8–14 Angle of repose for noncohesive material 8–20
Figure 8–15 K values for allowable stress, sides of trapezoidal 8–20
channels
Figure 8–16 Allowable shear stress for granular material in straight 8–21
trapezoidal channels
Figure 8–17 Allowable shear stress in cohesive material in straight 8–22
trapezoidal channels
Figure 8–18 USDA textural classification chart 8–22
Figure 8–19 Unconfined strength and tractive power as related to 8–27
channel stability
Figure 8–20 Manning’s roughness coefficients for grass-lined 8–28
channels
Figure 8–21 Allowable shear stress for noncohesive soils 8–33
Figure 8–22 Soil grain roughness for noncohesive soils 8–33
Figure 8–23 Basic allowable shear stress for cohesive soils 8–34
Figure 8–24 Void ratio correction factor for cohesive soils 8–34
Figure 8–25 Effect of flow duration on allowable velocities for 8–38
various channel linings
Figure 8–26 Spreadsheet calculations for threshold channel using 8–42
critical shear stress
(210–VI–NEH, August 2007) 8–iiiChapter 8 Threshold Channel Design
654.0800 Purpose 654.0801 Introduction
Threshold channel design techniques are used for rigid A stable threshold channel has essentially rigid bound-
boundary systems. In a threshold channel, movement aries. The streambed is composed of very coarse
of the channel boundary is minimal or nonexistent for material or erosion-resistant bedrock, clay soil, or
stresses at or below the design flow condition. There- grass lining. Streams where the boundary materials are
fore, the design approach for a threshold channel is remnants of processes no longer active in the stream
to select a channel configuration where the stress system may be threshold streams. Examples are
applied during design conditions is below the allow- streambeds formed by high runoff during the reces-
able stress for the channel boundary. Many sources sion of glaciers or dam breaks, streams armored due
and techniques for designing stable threshold channels to degradation, and constructed channels where chan-
are available to the designer. This chapter provides an nel movement is unacceptable for the design flow.
overview and description of some of the most com-
mon threshold channel design techniques. Examples A threshold channel is a channel in which movement
have been provided to illustrate the methods. of the channel boundary material is negligible during
the design flow. The term threshold is used because
the applied forces from the flow are below the thresh-
old for movement of the boundary material. Therefore,
the channel is assumed to be stable if the design stress
is below the critical or recommended stress for the
channel boundary. Design issues include assessing
the limiting force and estimating the applied force. A
requirement for a channel to be considered a threshold
channel is that the sediment transport capacity must
greatly exceed the inflowing sediment load so that
there is no significant exchange of material between
the sediment carried by the stream and the bed. Non-
cohesive material forming the channel boundary must
be larger than what the normal range of flows can
transport. For boundaries of cohesive materials, minor
amounts of detached material can be transported
through the system.
Threshold channels, therefore, transport no significant
bed-material load. Fine sediment may pass through
threshold streams as throughput. In general, this
throughput sediment should not be considered part
of the stream boundary for stability design purposes,
even if there are intermittent small sediment deposits
on the streambed at low flow.
An additional requirement for threshold channel
design is to maintain a minimum velocity that is suffi-
cient to transport the sediment load through the proj-
ect reach. This sediment may consist of clays, silts,
and fine sands. This is necessary to prevent aggrada-
tion in the threshold channel.
(210–VI–NEH, August 2007) 8–1Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
Threshold channels differ from movable bed or alluvial
channels which show interaction between the incom-
ing sediment load, flow, and channel boundary. In an
654.0802 Design discharges
alluvial channel, the bed and banks are formed from
material that is transported by the stream under pres- Threshold channel design methods are appropriate
ent flow conditions. The incoming sediment load and where sediment inflow is negligible and the proposed
bed and bank material of an alluvial channel interact channel boundary is to be immobile, even at high
and exchange under design or normal flow conditions. flows. Threshold channels do not have the freedom to
Essentially, the configuration of a threshold channel adjust their geometry under normal flow conditions.
is fixed under design conditions. An alluvial channel Therefore, channel-forming discharge is not necessar-
is free to change its shape, pattern, and planform in ily a critical factor in determining channel dimensions
response to short- or long-term variations in flow and in a threshold channel. Design flows are traditionally
sediment. The design of alluvial channels is addressed based, at least in part, on programs and policy deci-
in detail in NEH654.09. sions.
Approaches that fall into four general categories for As described in NEH654.07, the classification of a
the design of threshold channels are addressed in this stream as alluvial or threshold may not be clear. One
chapter. These approaches are the permissible velocity reach of the stream may be alluvial, while another
approach, allowable shear stress approach, and allow- may have the characteristics of a threshold channel. A
able tractive power approach. The grass-lined channel threshold stream reach can be changed to an alluvial
design approach, which is a specific case of either the reach by flattening the slope to induce aggradation
permissible velocity or allowable shear stress ap- or increasing the slope so that the boundary material
proach, is also described. Table 8–1 provides general becomes mobile. At flows larger than the design flow
guidance for selecting the most appropriate design or during extreme events, threshold channels may de-
technique. This is a general guide, and there are cer- velop a movable boundary. It is important to evaluate
tainly exceptions. For example, the allowable velocity channels through their entire flow range to determine
technique, being the most historical, has been applied how they will react to natural inflow conditions.
more broadly than indicated in table 8–1. Where there
is uncertainty regarding the appropriate technique, it Design of a stream project may involve a hybrid ap-
is recommended that the designer use several of the proach. For example, project goals may require that
most appropriate techniques and look for agreement the planform is rigid, while the cross section can
on critical design elements. vary. In this situation, a design approach might be to
Table 8–1 General guidance for selecting the most appropriate channel design technique
Significant
Boundary Boundary No baseflow in
sediment load Boundary material
material material channel. Climate can
Technique and movable does not act as
smaller than larger than support permanent
channel discrete particles
sand size sand size vegetation
boundaries
Allowable velocity X
Allowable shear stress X
Tractive power X
Grass lined/tractive stress X
Alluvial channel design X
techniques
8–2 (210–VI–NEH, August 2007)Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
stabilize the grade and toe of a stream in place, and
allow the upper bank to adjust naturally. Threshold 654.0803 Allowable velocity
channel design approaches, such as the use of riprap
(NEH654.14), are also used to size stream features method
such as toe protection, riffles, stream barbs, and de-
flector dikes. The allowable or permissible velocity approach is typi-
cally used with channels that are lined with grass, sand,
or earth. Limiting forces for soil bioengineering and
manufactured protective linings can also be expressed
as permissible velocities.
To design a threshold channel using the allowable
velocity method, average channel velocity is calculated
for the proposed channel and compared to published
allowable velocities for the boundary material. The
average channel velocity in the design channel can be
determined using a normal depth equation or a com-
puter backwater model. Increased velocities at bends
can be accounted for, using applicable charts and equa-
tions. Allowable velocities have been determined for a
large variety of boundary materials and are provided
in many texts and manuals. These tables have primar-
ily been applied to the design of irrigation and drain-
age canals and were developed from data in relatively
straight, uniform channels with depths less than 3
feet. It is common practice to apply allowable velocity
data in meandering, nonuniform channels with depths
greater than 3 feet, but such application should be done
with caution. Allowable velocities can be increased or
decreased to account for such irregularities as mean-
dering alignments and increased sediment concentra-
tions, using applicable charts. Allowable velocities are
somewhat less than critical velocities so that a factor
of safety is included in the values presented.
(a) Calculate average velocity
The first step in applying the allowable velocity design
approach is to calculate the average velocity of the
existing or proposed channel. Computing the average
channel velocity requires a design discharge, cross sec-
tion, planform alignment, average energy slope, and flow
resistance data. If the design channel is a compound
channel, it may be necessary to divide the channel into
panels and calculate velocities for each panel. In chan-
nels with bends, the velocity on the outside of the bend
may be significantly higher than the average velocity. Ve-
locity can be calculated using normal depth assumptions
or by a more rigorous backwater analysis if a gradually
varied flow assumption is more appropriate.
(210–VI–NEH, August 2007) 8–3Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
A normal depth calculation is easier than a backwater or at a constriction. The calculation of hydraulic
analysis and can be accomplished using a flow resis- parameters for both existing and proposed channels is
tance equation such as Manning’s. The normal depth critically important to design. A more complete treat-
assumption is applicable for uniform flow conditions ment of the subject is provided in NEH654.06.
where energy slope, cross-sectional shape, and rough-
ness are relatively constant in the applicable reach. In Minimum radius of curvature
a natural channel, with a nonuniform cross section, Caution is recommended in applying this approach on
reliability of the normal depth calculation is directly channels with sharp bends. Section 16 of the National
related to the reliability of the input data. Sound en- Engineering Handbook (U.S. Department of Agricul-
gineering judgment is required in the selection of a ture (USDA) Soil Conservation Service (SCS) 1971)
representative cross section. The cross section should provides guidance for minimum radius of curvature
be located in a uniform reach where flow is essen- for drainage ditches with very flat topography (slopes
tially parallel to the bank line with no reverse flow less than 0.00114). Table 8–2 provides guidance for
or eddies. This typically occurs at a crossing or riffle. channels in stable soil without bank protection. Con-
Determination of the average energy slope can be dif- ditions outside the range of table 8–2 and in erodible
ficult. If the channel cross section and roughness are soils require use of the more detailed analysis pro-
relatively uniform, water surface slope can be used. vided in this chapter. The curved channel may require
Thalweg slopes and low-flow water surface slopes may bank protection.
not be representative of the energy slope at design
flows. Slope estimates should be made over a signifi- Maximum velocity in bends
cant length of the stream (a meander wavelength or 20 Adjustments to the calculated average channel veloc-
channel widths). ity that account for flow concentration around bends
is provided as part of the USACE riprap design method
A computer program such as the U.S. Army Corps of (USACE 1991b.) The method is based on a large body
Engineers (USACE) HEC–RAS can be used to perform of laboratory data and has been compared to available
these velocity calculations. Such programs allow the prototype data (Maynord 1988). The method is appli-
designer to account for nonuniform sections and for cable to side slopes of 1V:1.5H or flatter. The method
backwater conditions that may occur behind a bridge calculates a characteristic velocity for side slopes,
Table 8–2 Suggested minimum radius of curvature in stable soils without bank protection
Minimum radius of Approximate degree
Type of ditch Slope curvature of curve
(ft) (m) (degrees)
Small ditches with maximumChapter 8 Threshold Channel Design Part 654
National Engineering Handbook
Vss, which is the depth-averaged local velocity over Figure 8–2 Design velocities for trapezoidal channels
the side slope at a point 20 percent of the slope length
from the toe of the slope. This has been determined 1.5
to be the part of the side slope that experiences the 120º Bend angle
1.4
maximum flow velocity. The ratio Vss/Vavg, where Vavg 1.3
Vss/Vavg
is the average channel velocity at the upstream end of 80º
1.2
the bend, has been determined to be a function of the 40º
ratio of the of centerline radius of curvature, R, and 1.1
the water surface width, W. Figure 8–1 illustrates the 1.0
2 3 4 5 6 7 8 10 20 30 40 50
relationship for natural channels. Figure 8–2 illustrates Centerline radius/water surface width
the relationship for trapezoidal channels. The data for Bottom width/depth = 3.3
trapezoidal channels shown in figure 8–2 are based on 1.4
numerical model calculations described in Bernard 120º Bend angle
1.3
(1993). The primary factors affecting velocity distribu- 80º
Vss/Vavg
tion in riprap lined bendways are R/W, bend angle, and 1.2
40º
aspect ratio (bottom width-to-depth). Vavg, R, and W 1.1
should be based on main channel flow only and should 1.0
2 3 4 5 6 7 8 10 20 30 40 50
not include overbank areas. Centerline radius/water surface width
Bottom width/depth = 6.7
1.4
120º Bend angle
1.3
Vss/Vavg
1.2
80º
1.1
40º
1.0
2 3 4 5 6 7 8 10 20 30 40 50
Centerline radius/water surface width
Bottom width/depth ≥10
Notes: Vss is depth-averaged velocity at 20% of slope length up from
toe, maximum value in bend. Curves based on STREMR model (Ber-
nard 1993), Vavg = 6 ft/s, 1V:3H side slopes. n = 0.038, 15 ft depth
Figure 8–1 Design velocities for natural channels. Note: Vss is depth-averaged velocity at 20% of slope length from toe
1.6
Vss R
= 1.74 − 0.52 Log
Vavg W
1.4
Vss/Vavg
1.2
1.0
0.8
2 4 6 8 10 20 40 50
R/W
(210–VI–NEH, August 2007) 8–5Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
(b) Determine allowable velocity Fortier and Scobey (1926) presented a table of maxi-
mum permissible velocities for earthen irrigation ca-
The design velocity of the existing or proposed chan- nals with no vegetation or structural protection. Their
nel must be compared to the allowable velocity for work was compiled based on a questionnaire given to
the channel boundary. The allowable velocity is the a number of experienced irrigation engineers and was
greatest mean velocity that will not cause the chan- recommended for use in 1926 by the Special Commit-
nel boundary to erode. Since the allowable velocity tee on Irrigation Research of the American Society of
is a design parameter that has a factor of safety, it is Civil Engineers. This compilation is presented in table
somewhat less than the critical velocity (the velocity 8–3.
at incipient motion of the boundary material).
USACE (1991b) provides allowable velocity criteria for
The allowable velocity can be approximated from nonscouring flood control channels in table 8–4.
tables that relate boundary material to allowable
velocity, but tabular estimates should be tempered by Theoretical objections to use of average velocity as an
experience and judgment. In general, older channels erosion criterion can be overcome by using depth as a
have higher allowable velocities because the channel second independent variable. An example of a veloc-
boundary typically becomes stabilized with the depo- ity-depth-grain size chart from the USACE (1991b) is
sition of colloidal material in the interstices. Also, a shown in figure 8–3. This particular chart is intended
deeper channel will typically have a higher allowable to correspond to a small degree of bed movement,
velocity than shallow channels because erosion is a rather than no movement. Values given in this chart
function of the bottom velocity. Bottom velocities in are for approximate guidance only.
deep channels are less than bottom velocities in shal-
low channels with the same mean velocity.
Table 8–3 Maximum permissible canal velocities
Mean velocity, for straight canals of small slope,
after aging with flow depths less than 3 ft (0.9 m)
Water
Water transporting
Clear water, no
transporting noncolloidal silts,
detritus
colloidal silts sands, gravels, or
rock fragments
Original material excavated for canals ft/s m/s ft/s m/s ft/s m/s
Fine sand (noncolloidal) 1.5 0.46 2.5 0.76 1.5 0.46
Sandy loam (noncolloidal) 1.75 0.53 2.5 0.76 2.0 0.61
Silt loam (noncolloidal) 2.0 0.61 3.0 0.91 2.0 0.61
Alluvial silt (noncolloidal) 2.0 0.61 3.5 1.07 2.0 0.61
Ordinary firm loam 2.5 0.76 3.5 1.07 2.25 0.69
Volcanic ash 2.5 0.76 3.5 1.07 2.0 0.61
Stiff clay (very colloidal) 3.75 1.14 5.0 1.52 3.0 0.91
Alluvial silt (colloidal) 3.75 1.14 5.0 1.52 3.0 0.91
Shales and hardpans 6.0 1.83 6.0 1.83 5.0 1.52
Fine gravel 2.5 0.76 5.0 1.52 3.75 1.14
Graded, loam to cobbles (when noncolloidal) 3.75 1.14 5.0 1.52 5.0 1.52
Graded silt to cobbles (when colloidal) 4.0 1.22 5.5 1.68 5.0 1.52
Coarse gravel (noncolloidal) 4.0 1.22 6.0 1.83 6.5 1.98
Cobbles and shingles 5.0 1.52 5.5 1.68 6.5 1.98
8–6 (210–VI–NEH, August 2007)Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
Table 8–4 Allowable velocities
Mean channel velocity
Channel material
(ft/s) (m/s)
Fine sand 2.0 0.61
Coarse sand 4.0 1.22
Fine gravel 6.0 1.83
Earth
Sandy silt 2.0 0.61
Silt clay 3.5 1.07
Clay 6.0 1.83
Grass-lined earth (slopesChapter 8 Threshold Channel Design Part 654
National Engineering Handbook
(c) Soil Conservation Service allowable Step 5 Determine the basic average allowable
velocity approach velocities for the channel from one or more of the
available design guidelines (tables 8–3, 8–4, fig.
Basic allowable velocities may be determined from 8–4 (USDA SCS 1977; Federal Interagency Stream
figure 8–4 (USDA SCS 1977). In this figure, allowable Restoration Working Group (FISRWG) 1998)).
velocities are a function of sediment concentration, Step 6 Multiply the basic allowable velocity by
grain diameter for noncohesive boundary material, the appropriate correction factors (fig. 8–4).
and plasticity index and soil characteristics for cohe-
sive boundary material. Adjustments are given in fig- Step 7 Compare the design velocities with the
ure 8–4 to the basic allowable velocity to account for allowable velocities. If the allowable velocities
frequency of design flow, alignment, bank slope, depth are greater than the design velocities, the design
of flow, and sediment concentration for both discrete is satisfactory. Otherwise, three options are avail-
particles and cohesive soils. These design charts were able:
compiled from the data of Fortier and Scobey (1926), • Redesign the channel to reduce velocity.
Lane (1955a), and the Union of Soviet Socialist Repub-
lic (USSR) (1936). Soil materials are classified using • Provide structural measures (riprap, grade
the Unified Soil Classification System. control) to prevent erosion.
• Consider a mobile boundary condition and
Procedure for application of allowable velocity evaluate the channel using appropriate sedi-
method (USDA SCS 1977) ment transport theory and programs.
Step 1 Determine the hydraulics of the system.
This includes hydrologic determinations, as well Design of Open Channels, TR–25 (USDA SCS 1977)
as the stage-discharge relationships for the chan- contains several examples to guide the user through
nel considered. the allowable velocity approach.
Step 2 Determine the soil properties of the bed
and banks of the design reach and of the channel
upstream.
Step 3 Determine the concentration of the
suspended sediment load entering the reach. This
is best accomplished by measurements. Channels
with suspended sediment concentrations less than
1,000 parts per million are considered sediment
free for this analysis, in that the sediment load is
not sufficient to decrease the energy of the stream
flow. Sediment-free flows are, therefore, consid-
ered to have no effect on channel stability. Chan-
nels with suspended sediment concentrations
greater than 20,000 parts per million are consid-
ered to be sediment laden. Sediment-laden flows
are considered to enhance stream stability by
filling boundary interstices with cohesive material.
If a significant portion of the inflowing sediment
load is bed-material load, it is likely that the chan-
nel is alluvial, and threshold design methods are
not applicable.
Step 4 Check to see if the allowable velocity
procedure is applicable using table 8–1.
8–8 (210–VI–NEH, August 2007)Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
Figure 8–4 Allowable velocities for unprotected earth channels
2.0 1.0
Correction factor A
Correction factor F
1.8
0.9
Frequency of design flow
1.6 Alignment
0.8
1.4
0.7
16 14 12 10 8 6 4
1.2 Curve radius ÷ water surface width
1.0 1.0
Correction factor B
Notes:
1 2 3 4 5 6 7 8 9 10 In no case should the
Flood frequency (% chance) 0.8 allowable velocity be
exceeded when the 10%
1.5 0.6 chance discharge occurs,
Bank slope regardless of the design
1.4 0.4 flow frequency.
Correction factor D
1.5 2.0 2.5 3.0
1.3 Cotangent of slope angle, z
1.2 1.2
Correction factor Ce
1.1 1.1 SM
Depth of design flow CH
,S
1.0
C,
1.0 ,M
H
GM
,G
0.9 CL
C
0.9 ,M
2 4 6 8 10 12 14 16 18 20 Density L
Water depth (ft) 0.8
0.2 0.4 0.6 0.8 1.0 1.2 1.4
Void ratio, e
Basic velocities for coherent earth materials, vb
7.0
Basic velocity for discrete particles of earth materials, vb
6.5 Fine S Sand Gravel Cobble
13.0
CH 12.0
Basic velocity (ft/s)
6.0 GC
11.0 Enter chart with D75 particle size
5.5 M 10.0 to determine basic velocity.
CL,G
SC 9.0
Basic velocity (ft/s)
5.0 ,OH
MH 8.0
4.5 7.0
M 6.0
Sediment-laden
4.0 L,S
,O 5.0
ML Sediment-laden flow
3.5 4.0
10 12 14 16 18 20 22 24 3.0
Plasticity index Sediment-free
2.0
5.5 1.0
0.0
1 1 1
5.0 8 4 2 1 2 4 6 8 10 15
Grain size (in)
CH
Basic velocity (ft/s)
4.5 GC Allowable velocities for unprotected earth channels
4.0 Channel boundary materials Allowable velocity
,SC Discrete particles
,CL
3.5 GM ,OH
Sediment-laden flow
MH D75 >0.4mm Basic velocity chart value x D x A x B
3.0 D75 0.2mm Basic velocity chart value x D x A x B
,O
ML Sediment-free flow
D75 10 Basic velocity chart value x D x A x F x Ce
Plasticity index PIChapter 8 Threshold Channel Design Part 654
National Engineering Handbook
depth. Spatial and temporal variation may result in a
654.0804 Allowable shear stress higher or lower point value for shear stress. The equa-
tion approximates average bed shear stress.
approach
The shear stress can also be expressed as a function
The allowable shear approach (sometimes referred to of the velocity and the ratio of hydraulic radius and
as the tractive stress approach) is typically used with boundary roughness. Keulegan (1938) presented such
channels that are lined with rock, gravel, or cobbles. a formula.
Limiting forces for soil bioengineering and manu- ρV 2
τ=
factured protective linings can also be expressed as 1 R
2
(eq. 8–2)
allowable shear, as well. κ ln k + 6.25
s
To design a threshold channel using the allowable where:
shear stress approach, the average applied grain bed V = depth-averaged velocity, ft/s or m/s
shear stress is compared to the allowable shear stress ρ = density of water, lb-s2/ft4(slugs/ft3) or kg/m2
for the boundary material. The applied grain bed shear κ = von Karman’s constant (usually taken to be
stress can be calculated from the hydraulic parameters 0.4)
determined for the design channel and the character- ks = roughness height, ft or m
istics of the channel boundary material. The hydraulic
parameters are calculated using the same methods Actual shear stress values should be calculated for
as in the allowable velocity approach. For noncohe- the banks, as well as for the bed of a trapezoidal earth
sive soils, the average allowable shear stress can be channel. Maximum stresses occur near the center of
calculated using a critical shear stress approach and the bed and at a point on the bank about a third up
then adding a factor of safety or by using an empirical from the bottom. The designer should note that com-
equation with a factor of safety included. For cohesive puter programs such as HEC–RAS may only provide
particles, the electrochemical bonds related primarily average boundary shear stress in the output. For most
to clay mineralogy, are the most significant sediment trapezoidal sections and depths of flow, bed stress val-
properties that determine allowable shear stress. ues are somewhat higher than bank stress. Figures 8–5
Although some empirical data are available, laboratory and 8–6 provide actual shear stress values for the bed
tests to determine allowable shear stress for a specific and sides of straight trapezoidal channels in coarse
cohesive soil are preferred. grained soil materials.
Grain shear stress
(a) Calculate applied shear stress The total applied bed shear stress may be divided
into that acting on the grains and that acting on the
The first step in applying this approach is to calculate bedforms. Entrainment and sediment transport are a
the hydraulics of the study reach. The total average function only of the grain shear stress; therefore, the
shear stress on the boundary can be approximated grain shear stress is the segment of interest for thresh-
from equation 8–1, using any consistent units of mea- old design. Einstein (1950) determined that the grain
surement: shear stress could best be determined by separating
τ o = γRS (eq. 8–1) total bed shear stress into a grain component and a
form component, which are additive. The equation for
where: total bed shear stress is:
τo = total bed shear stress (lb/ft2 or N/m2) τ o = τ ′ + τ ′′ = γRS (eq. 8–3)
γ = specific weight of water (lb/ft3 or N/m3)
R = hydraulic radius (ft or m) where:
S = energy slope, dimensionless τ′ = grain shear stress (shear resulting from size of
the material on the bed)
In wide channels where the width is more than 10 τ″ = form shear stress (shear resulting from bed
times the depth, R is generally taken to be equal to the irregularities due to bedforms)
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National Engineering Handbook
Figure 8–5 Applied maximum shear stress, τb, on bed of Figure 8–6 Applied maximum shear stress, τs, on sides of
straight trapezoidal channels relative to an trapezoidal channels relative to an infinitely
infinitely wide channel, τ∞ wide channel, τ∞
1.0 1.0
z = 1.5 and z = 2
0.9 0.9
0.8 0.8 z=2
z = 1.5
0.7 0.7
0.6 0.6
Values of τb/τ∞
Values of τs/τ∞
z=1 z=0
0.5 0.5
z=0
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
b/d ratio b/d ratio
Note:
b = bottom width
d = depth
z = side slope, zH:1V
τ∞ = shear stress on a straight, infinitely wide channel
τb = applied shear stress on a channel bed
τs = applied shear stress on the side of a channel
(210–VI–NEH, August 2007) 8–11Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
Einstein also suggested that the hydraulic radius could plished using the sidewall correction procedure, which
be divided into grain and form components that are separates total roughness into bed and bank rough-
additive. The equations for grain and form shear stress ness and conceptually divides the cross-sectional area
then become: into additive components. The procedure is based on
(eq. 8–4) the assumption that the average velocity and energy
τ ′ = γR ′S
gradient are the same in all segments of the cross sec-
τ ′′ = γR ′′S (eq. 8–5) tion.
A total = A b + A w (eq. 8–9)
where:
R′ = hydraulic radii associated with the grain rough-
ness A total = PbR b + Pw R w
(eq. 8–10)
R″ = hydraulic radii associated with the form rough- where:
ness A = cross-sectional area (ft2 or m2)
P = perimeter (ft or m)
These hydraulic radii are conceptual parameters, use-
ful for computational purposes and have no tangible Subscripts b and w are associated with the bed and
reality. The total bed shear stress can be expressed as: wall (or banks), respectively. Note that the hydrau-
lic radius is not additive with this formulation, as it
τ 0 = γR ′S + γR ′′S (eq. 8–6) was with R′ and R″. Using Manning’s equation, with
a known average velocity, slope, and roughness coef-
Slope and the specific weight of water are constant so ficient, the hydraulic radius associated with the banks
that the solution is to solve for one of the R compo- can be calculated:
nents. The grain shear stress can be calculated with 2 2
the Limerinos equation, using any consistent units of V R 3 R w3
measurements. 1
= = (eq. 8–11)
n nw
CME S 2
V R′ 3
= 3.28 + 5.66 log10 (eq. 8–7) 2
U *′ D84 V (eq. 8–12)
R w = nw
1
CME S
2
U *′ = gR ′S
(eq. 8–8)
where:
where: CME =1.486 in English units and 1.0 in SI units
V = average velocity (ft/s or m/s)
U *′ = grain shear velocity (ft/s or m/s) Total hydraulic radius and shear stress, considering
D84 = particle size for which 84% of the sediment grain, form, and bank roughness, can be expressed by
mixture is finer (ft or m) equations 8–13 and 8–14:
g = acceleration of gravity (ft/s2 or m/s2)
Pb (R ′ + R ′′ ) + Pw R w (eq. 8–13)
R total =
Limerinos (1970) developed his equation using data Ptotal
from gravel-bed streams. Limerinos’ hydraulic radii
ranged between 1 and 6 feet; D84 ranged between 1.5
P (R ′ + R ′′ ) + Pw R w
and 250 millimeters. This equation was confirmed for τ total = γS b (eq. 8–14)
plane bed sand-bed streams by Burkham and Dawdy Ptotal
(1976). The equation can be solved iteratively for
R′ and τ′, when average velocity, slope, and D84 are Lane’s tractive force method
known. Lane (1952) developed an analytical design approach
for calculation of the applied grain shear stress and
Whenever the streambanks contribute significantly to the shear distribution in trapezoidal channels. The
the total channel roughness, the applied shear stress tractive force, or applied shear force, is the force that
to the banks must be accounted for. This is accom- the water exerts on the wetted perimeter of a channel
8–12 (210–VI–NEH, August 2007)Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
due to the motion of the water. Lane determined that 1
D6
in most irrigation canals, the tractive force near the ns = 75 with D75 expressed in inches (eq. 8–15)
middle of the channel closely approaches 39
γdSo 1
D6
where: ns = 75 with D75 expressed in millimeters (eq. 8–16)
66.9
γ = specific weight of water
d = depth The grain roughness is combined with other roughness
So = bed slope assuming uniform flow elements to determine the total Manning’s roughness
coefficient, n. The friction slope associated with grain
He also determined that the maximum tractive force roughness, St, can then be calculated using equation
on the side slopes was approximately 0.75 γdSo. Lane 8–17:
also found that the side slopes of the channel affected n
2
the maximum allowable shear stress. He developed S t = s Se (eq. 8–17)
n
an adjustment factor, K, to account for the side slope
effects. Detailed information on the tractive force where:
approach is found in Design of Open Channels, TR–25 Se = total friction slope determined from Manning’s
(USDA SCS 1977) and Chow (1959). A summary of the equation
method follows.
The applied shear stress acting on the grains in an
When the boundary of the channel consists of coarse- infinitely wide channel is then calculated from equa-
grained discrete particles, Lane (1952) determined that tion 8–18.
the grain roughness, ns, could be determined as a func- τ ∞ = γdS t (eq. 8–18)
tion of the D75 of the boundary material. Applied grain
shear stress can then be calculated using Manning’s In open channels, the applied shear stresses are not
equation. The D75 range for which Lane found this distributed uniformly along the perimeter as is shown
relationship to be applicable was between 0.25 inches in figure 8–7 (Lane 1952). Laboratory experiments and
(6.35 mm) and 5.0 inches (127 mm). This is similar to field observations have indicated that in trapezoidal
determining the grain shear stress using the Limerinos channels the stresses are very small near the water
equation. surface and corners of the channel. In straight chan-
Figure 8–7 Lateral distribution of shear stress in a trapezoidal channel
1 1
1.5 1.5
y
0.750wyS
4y 0.750wyS
0.970wyS
w=specific weight of water, y=depth, and S=slope
(210–VI–NEH, August 2007) 8–13Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
nels, the maximum shear stress occurs on the bed near initiation of particle motion) and product of the grain
the center of the channel. The maximum shear stress diameter and the submerged specific weight of the
on the banks occurs about a third the way up the particle. The grain Reynolds number is defined as the
bank from the bed. Figures 8–5 and 8–6 can be used to ratio of the product of shear velocity and grain diam-
determine the shear stress distribution in a trapezoidal eter to kinematic viscosity. Shields parameter and
channel, relative to the applied shear stress in an infi- grain Reynolds number are dimensionless and can be
nitely wide channel with the same depth of flow and used with any consistent units of measurement. The
energy slope (USDA SCS 1977). relationship between τ* and R* represents an average
curve drawn through scattered data points that were
The magnitude of applied shear stresses is not uni- determined experimentally from flumes or rivers.
form in turbulent flow. Calculations using traditional Therefore, a wide range in recommended values ex-
equations provide an average value of shear stress. In ists for the Shields parameter, depending on how the
design, therefore, a factor of safety is typically applied experiment was conducted and the nature of the bed
to account for this fluctuation. This fluctuation may material being evaluated.
also be addressed in certain design approaches using
probability methods presented later in this chapter. Once τ* has been assigned, the critical shear stress
for a particle having a diameter, D, is calculated from
Applied shear stress on curved reaches equation 8–19.
Curved channels have higher maximum shear stresses
than straight channels. Maximum stress occurs on the τc = τ * ( γ s − γ ) D (eq. 8–19)
inside bank in the upstream portion of the curve and
where:
on the outer bank in the downstream portion of the
τ* = Shields parameter, dimensionless
curve. The smaller the radius of curvature, the more
R* = grain Reynolds number = u*d/ν, dimensionless
the stress increases along the curved reach. Maximum
τc = critical shear stress (lb/ft2 or N/m2)
applied shear stress in a channel with a single curve
also occurs on the inside bank in the upstream por- γs = specific weight of sediment (lb/ft3 or N/m3)
tion of the curve and near the outer bank downstream γ = specific weight of water (lb/ft3 or N/m3)
from the curve. Compounding of curves in a channel D = particle diameter (ft or m)
complicates the flow pattern and causes a compound- u* = shear velocity = (gRS)1/2 (ft/s or m/s)
ing of the maximum applied shear stress. Figure 8–8 ν = kinematic viscosity of the fluid (ft2/s or m2/s)
gives values of maximum applied shear stress based g = acceleration of gravity (ft/s2 or m/s2)
on judgment coupled with very limited experimental
data (USDA SCS 1977). It does not show the effect of Shields (1936) obtained his critical values for τ* exper-
depth of flow and length of curve, and its use is only imentally using uniform bed material and measuring
justified until more accurate information is obtained. sediment transport at decreasing levels of bed shear
Figure 8–9, with a similar degree of accuracy, gives the stress, and then extrapolating to zero transport. The
maximum applied shear stresses at various distances Shields curve is shown in figure 8–10 (USACE 1995c).
downstream from the curve (USDA SCS 1977). The Shields’ data suggest that τ* varies with R* until the
designer should note that these adjustments are simi- grain Reynolds number exceeds 400. At larger values
lar to rules of thumb. of R*, τ* is independent of R* and is commonly taken
to be 0.06. The Shields curve may be expressed as
an equation, useful for computer programming and
(b) Calculate allowable shear stress spreadsheet analysis.
The applied shear stress must be compared to the τ* = 0.22β + 0.06 × 10 −7.7 β (eq. 8–20)
allowable shear stress. Shear stress at initiation of −0.6
1 γ −γ
motion can be calculated from an empirically derived
β= s
gD3 (eq. 8–21)
ν
γ
relationship between dimensionless shear stress
(Shields parameter), τ*, and grain Reynolds number,
R*. The dimensionless shear stress is defined as the The Shields diagram is the classic method for deter-
ratio of the critical shear stress (shear stress at the mining critical shear stress. However, subsequent
8–14 (210–VI–NEH, August 2007)Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
Figure 8–8 Applied maximum applied shear stress, τbs Figure 8–9 Applied maximum applied shear stress, τbt
and τsc on bed and sides of trapezoidal chan- and τst on bed and sides of trapezoidal chan-
nels in a curved reach nels in straight reaches immediately down-
stream from curved reaches
2.0
1.0
1.9
0.9
1.8 0.8
1.7 0.7
τst–τs
s
τ –τ
Values of τbc/τb or τsc/τs
1.6 0.6
sc
or
τbt–τb
b
τ –τ
1.5 0.5
bc
Values of
1.4 0.4
1.3 0.3
1.2 0.2
0.1
1.1
0
1.0 0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Values of Lc
Rc b
b
Note:
Rc = radius of curvature
b = bottom width
d = channel depth
Lc = length of curve
τb = applied shear stress on a channel bed
τs = applied shear stress on the side of a channel
τbc = applied shear stress on channel bed in a curve
τsc = applied shear stress on channel side in a curve
τbt = applied shear stress on channel bed immediately downstream of a curve
τst = applied shear stress on channel side immediately downstream of a curve
(210–VI–NEH, August 2007) 8–15Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
work identified three significant problems associated shear stress would be too high for a natural bed with
with the curve itself. First, the procedure did not ac- no bedforms. Gessler reanalyzed Shields’ data so that
count for the bedforms that developed with sediment the critical Shields parameter represented only the
transport. Second, the critical dimensionless shear grain shear stress (fig. 8–11). This curve is more ap-
stress is based on the average sediment transport of propriate for determining critical shear stress in plane
numerous particles and does not account for the spo- bed streams with relatively uniform bed gradations.
radic entrainment of individual particles at very low With fully turbulent flow (R* >400), typical of gravel-
shear stresses. Thirdly, critical dimensionless shear bed streams, τ* is commonly taken to be 0.047 using
stress for particles in a sediment mixture may be dif- Gessler’s curve.
ferent from that for the same size particle in a uniform
bed material. In general, for purposes of design of
threshold channels, in which no bed movement is a
requirement, the Shields curve will underestimate the
critical dimensionless shear stress and is not recom- Figure 8–11 Gessler’s reformulation of Shields diagram.
mended unless a factor of safety is added. τ is critical grain shear stress and k is grain
diameter.
Adjustment for bedforms
0.10
Gessler (1971) determined that Shields did not sepa-
0.08
rate grain shear stress from bedform shear stress in his Motion
experimental flume data analysis. Bedforms developed 0.06
with sediment transport for the fine-grained bed mate-
τ
0.04
s
rial in some of Shields flume data. Since a portion of No motion
the total applied shear stress is required to overcome
the bedform roughness, the calculated dimensionless 0.02
10 20 40 60 80 100 200 400 600
uk
R*= *ν
Figure 8–10 Shields curve
γg in gm/cm3
(γ D − γ ) Ds
Amber 1.06
Lignite (Shields) 1.27
τo
Granite 2.7
Fully developed turbulent velocity profile Barite 4.25
* Sand (Casey) 2.65
=
1.0 + Sand (Kramer) 2.65
Dimensionless shear stress, τ *
x Sand (U.S. WES.) 2.65
0.8
Sand (Gilbert) 2.65
0.6 2.61
0.5 Sand (White)
0.4 Turbulent boundary layer Sand in air (White) 2.10
0.3 Steel shot (White) 7.9
Ds γ
0.2 Value of 0.1 s − 1 gDs
ν γ
2 4 6 8 1 2 4 6 100 2 4 6 1,000
0.1
0.08
0.06 *
0.05 x *x
0.04 x +
+x+ * * * *
0.03 * *
Shields curve
0.02
0.2 0.4 0.6 1.0 2 4 6 8 10 20 40 60 100 200 500 1,000
UD
Boundary Reynolds number, R * = * s
ν
8–16 (210–VI–NEH, August 2007)Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
Adjustment for mixtures The minimum value for τ* was found to be 0.020.
Natural streambeds seldom have uniform bed grada- According to Andrews, the critical shear stress for
tions. The critical bed shear stress equation must be individual particles has a very small range; therefore,
modified for mixtures. There are two approaches: the entire bed becomes mobilized at nearly the same
one is to select a τ* that is characteristic of mixtures; shear stress. However, Wilcock (1998) and Wilcock
the other is to select a percent finer grain size that and McArdell (1993) have demonstrated that this
is characteristic of initiation of motion. Meyer-Peter near-equal mobility result applies only to unimodal
and Muller (1948) and Gessler (1971) determined that sediments with a small to modest standard deviation.
when R* >400, the critical Shields parameter for sedi- In coarse beds with a wide range of sizes (especially
ment mixtures was about 0.047 when median grain mixtures of sand and gravel), the fines may begin to
size was used. Neill (1968) determined from his data move at flows much smaller than the coarse grains.
that in gravel mixtures, most particles became mobile
when τ* was 0.030, when median grain size was used Gessler’s concept for particle stability
for D. Andrews (1983) found a slight difference in τ* Critical shear stress is difficult to define because en-
for different grain sizes in a mixture, and presented the trainment is sporadic at low shear stresses caused by
equation 8–22: bursts of turbulence. Due to the difficulty in defining
−0.872 initiation of motion in a flume, the Shields curve was
D
τ *i = 0.0834 i (eq. 8–22) developed by extrapolating measured sediment trans-
D50 port rates back to zero. Unfortunately, the relationship
between the Shields parameter and sediment transport
where:
is not linear at low shear stresses. This phenomenon
subscript, i =Shields parameter and grain size for
was demonstrated by Paintal (1971) (fig. 8–12). Note
size class i
that the extrapolated critical dimensionless shear
D50 =median diameter of the subsurface
stress was about 0.05, but the actual critical dimen-
material
sionless shear stress was 0.03.
Figure 8–12 Variation in Shields parameter with decreasing sediment load
0.07
0.06
τc *
0.05
0.04
τ0*
0.03
D=7.95 mm
0.02
0 10 20 30 40 50 60 70 80
gs (lb/ft/h)
0.07
0.06
τc *
0.05
0.04
τ0*
0.03
D=2.5 mm
0.02
0 10 20 30 40 50 60 70
gs (lb/ft/h)
(210–VI–NEH, August 2007) 8–17Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
Gessler (1971) developed a probability approach to where:
the initiation of motion for sediment mixtures. He P = probability function for the mixture (depends
reasoned that due to the random orientation of grains on the frequency of all grain sizes in the under-
and the random strength of turbulence on the bed, for lying material)
a given set of hydraulic conditions, part of the grains fi = fraction of grain size i
of a given size will move, while others of the same size
may remain in place. Gessler assumed that the critical If the gradation of the channel bed is known, τc for
Shields parameter represents an average condition, each size class is determined from figure 8–11, and P
where about half the grains of a uniform material will for each size class is determined from figure 8–13. P
remain stable and half will move. It follows then that can then be calculated from equation 8–23. Gessler
when the critical shear stress was equal to the bed suggested that when P was less than 0.65, the bed was
shear stress, there was a 50 percent chance for a given
unstable.
particle to move. Using experimental flume data, he
developed a probability function, p, dependent on
The probability concept was presented in an empirical
τc /τ where τc varied with bed size class (fig. 8–13). He
determined that the probability function had a normal fashion by Buffington and Montgomery (1997). They
distribution, and that the standard deviation (slope analyzed critical shear stress data from many inves-
of the probability curve) was a function primarily of tigators and suggested ranges for the critical Shields
turbulence intensity, and equal to 0.057. Gessler found parameter. For visually base data, where initiation of
the effect of grain-size orientation to be negligible. The motion was determined by investigator observation,
standard deviation also accounts for hiding effects; Buffington and Montgomery suggested a range for
that is, no attempt was made to separate hiding from τ* between 0.073 and 0.030 for fully rough, turbulent
the overall process. Gessler’s analysis demonstrates flow (R* >400). They concluded that less emphasis
that there can be entrainment of particles, even when should be placed on choosing a universal value for τ*,
the applied shear stress is less than the critical shear while more emphasis should be placed on choosing
stress; and that not all particles of a given size class on defendable values for particular applications. Buffing-
the bed will necessarily be entrained, until the applied ton and Montgomery also provided the compiled data
shear stress exceeds the critical shear stress by a fac-
tor of 2. The design implications of this work are:
• If near-complete immobility is desired in the Figure 8–13 Probability of grains to stay on the bed
project design, the Shields parameter used to
determine critical shear stress should be on the
0.99
order of half the typically assigned value.
• To assure complete mobility of the bed (fully
0.95
alluvial conditions), the applied grain shear
stress should be twice the critical shear stress. 0.90
The inherent dangers of using 50 percent or 200 per- 0.80
cent of critical shear stress are that the channel could 0.70
Probability, P
aggrade or incise. 0.60
0.50
Gessler used the probability approach to determine if 0.40
the bed surface layer of a channel was stable (immo-
0.30
bile). He suggested that the mean value of the prob-
abilities for the bed surface to stay in place should be 0.20
a good indicator of stability:
0.10
i max
0.05
∫
i min
P 2 fi D i
P= i max (eq. 8–23)
0.01
∫
i min
Pfi Di 0 0.5 1.0 1.5 2.0
8–18 (210–VI–NEH, August 2007)Chapter 8 Threshold Channel Design Part 654
National Engineering Handbook
from many investigators, including data from natural where:
streams. units of γ are in lb/ft3
Lane’s method for coarse grained soils Figure 8–15 (from TR–25) provides adjustment values
Lane (1955a) concentrated on the force exerted over a for allowable bank stress in trapezoidal channels,
given surface area of the channel, rather than the force based on angle of repose and side slope steepness.
exerted on a single particle, as in the Shields parame- The allowable stress for the channel sides is thought
ter and Gessler approaches. He also built in a factor of to be less than that of the same material in the bed
safety to the critical shear stress, so that his equation because the gravity force adds to the stress in moving
more appropriately can be called an allowable shear the materials.
stress equation. This factor of safety accounts for the
shear stress fluctuations in turbulent flow. Lane’s method for fine-grained soils
Allowable shear stress in fine-grained soils (D75You can also read
