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A STUDY OF FLUVIAL GEOMORPHOLOGY
ASPECTS OF HYDRAULIC DESIGN
A STUDY OF FLUVIAL GEOMORPHOLOGY
ASPECTS OF HYDRAULIC DESIGN
A. David Parr, Ph.D.and John Shelley
CEAE DepartmentUniversity of Kansas
(Funded by KDOT)
A. David Parr, Ph.D.and John Shelley
CEAE DepartmentUniversity of Kansas
(Funded by KDOT)
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Jim Richardson
Brad Rognlie
Mike Orth
KDOT Bridge Section
Jim RichardsonJim Richardson
Brad RognlieBrad Rognlie
Mike OrthMike Orth
KDOT Bridge SectionKDOT Bridge Section
AcknowledgmentsAcknowledgments
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Stable Channel DesignStable Channel DesignStable Channel DesignKDOT is sometimes required to realign
short reaches of alluvial channels to
facilitate highway improvements or toprovide protection for highway structuresor roadway embankments.
The new stream reaches should be dynamicallystable and should have geomorphic properties thatare characteristic of natural streams in similar
settings.
They should also be hydraulically and ecologicallycompatible with the contiguous upstream anddownstream stream reaches.
KDOT is sometimes required to realignKDOT is sometimes required to realignshort reaches of alluvial channels toshort reaches of alluvial channels to
facilitate highway improvements or tofacilitate highway improvements or toprovide protection for highway structuresprovide protection for highway structuresor roadway embankments.or roadway embankments.
The new stream reaches should be dynamicallyThe new stream reaches should be dynamicallystable and should have geomorphic properties thatstable and should have geomorphic properties thatare characteristic of natural streams in similarare characteristic of natural streams in similarsettings.settings.
They should also be hydraulically and ecologicallyThey should also be hydraulically and ecologicallycompatible with the contiguous upstream andcompatible with the contiguous upstream anddownstream stream reaches.downstream stream reaches.
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Stream Modification - Road ProjectStream ModificationStream Modification -- Road ProjectRoad Project
Old RoadOld Road
Old StreamOld Stream
New StreamNew Stream
New RoadNew Road
(b)(b)
(a)(a)
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Protection - Meanders on the Kansas RiverProtection - Meanders on the Kansas River
(a)
(b)
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Bank-full Discharge ConditionsBankBank--full Discharge Conditionsfull Discharge Conditions
Copeland* states Bank-full discharge is
the maximum discharge that a steam can
convey without overflowing into the
floodplain. The water surface elevation
for this condition is called the bank-fullstage.
Bank-full discharge is also referred to asthe channel-forming discharge.
Copeland* statesCopeland* states BankBank--full discharge isfull discharge is
the maximum discharge that a steam canthe maximum discharge that a steam can
convey without overflowing into theconvey without overflowing into the
floodplainfloodplain.. The water surface elevationThe water surface elevation
for this condition is called the bankfor this condition is called the bank--fullfullstage.stage.
BankBank--full discharge is also referred to asfull discharge is also referred to asthethe channelchannel--forming dischargeforming discharge..
*http://chl.erdc.usace.army.mil/library/publications/chetn/pdf/chetn-viii-5.pdf*http://chl.erdc.usace.army.mil/library/publications/chetn/pdf/chetn-viii-5.pdf
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Upstream Supply ReachUpstream Supply ReachUpstream Supply Reach
Project SiteProject Site
Flow
Riffle on Stable Upstream
Supply ReachRiffle on Stable Upstream
Supply Reach
Supply ReachCross SectionSupply ReachCross Section
Sinuosity = Lstream/Lvalley
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Bankfull Conditions for Supply
Reach Cross Section(1.2 to 2 year recurrence interval)
Bankfull Conditions for SupplyBankfull Conditions for Supply
Reach Cross SectionReach Cross Section((1.2 to 2 year recurrence interval)
Wbf
Abf
dmax
Wbf
= bankfull width
Abf= bankfull area
dbf= Abf/Wbf= bankfull depth
Bankfull
Stage
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Determination of Bank-full Stage(http://www.stockton.edu/~epsteinc/rosgen~1.htm)
Determination of Bank-full Stage(http://www.stockton.edu/~epsteinc/rosgen~1.htm)
Involves assessing the elevation wherethe channel, under bank-full dischargeconditions, ends and the floodplain begins.The indicators used to assess thiselevation are as follows:
Top of the point bar
A change in vegetation
Slope change in channel cross section
Top of the undercut slopeChange in particle size (where soils end andsediments begin),
Drift lines and water marks
Involves assessing the elevation wherethe channel, under bank-full dischargeconditions, ends and the floodplain begins.The indicators used to assess thiselevation are as follows:
Top of the point bar
A change in vegetation
Slope change in channel cross section
Top of the undercut slopeChange in particle size (where soils end andsediments begin),
Drift lines and water marks
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University of Kansas StudiesUniversity of Kansas StudiesUniversity of Kansas Studies
Guidelines for Stream Realignment Design KAMMethod
McEnroe, Young and ShelleyReport No. K-TRAN KU-08-2
Stream Realignment Design using a ReferenceReach ARR Method
McEnroe, Young and ShelleyReport No. K-TRAN KU-09-4
A Study of Fluvial Geomorphology Aspects ofHydraulic Design (HEC-RAS applications)
Parr and Shelley
Report No. K-TRAN: KU-08-5
Guidelines for Stream Realignment DesignGuidelines for Stream Realignment Design KAMKAMMethodMethod
McEnroe, Young and ShelleyMcEnroe, Young and ShelleyReport No. KReport No. K--TRAN KUTRAN KU--0808--22
Stream Realignment Design using a ReferenceStream Realignment Design using a ReferenceReachReach ARR MethodARR Method
McEnroe, Young and ShelleyMcEnroe, Young and ShelleyReport No. KReport No. K--TRAN KUTRAN KU--0909--44
A Study of Fluvial Geomorphology Aspects of
Hydraulic Design (HEC-RAS applications)Parr and ShelleyParr and Shelley
Report No. KReport No. K--TRAN: KUTRAN: KU--0808--55
This StudyThis Study
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KAM and ARR MethodsKAM and ARR MethodsKAM and ARR Methods
Consider alluvial (noncohesive) and threshold(cohesive) channels.
StrengthsInclude planform design for stream meanders and
pool spacingDesigns pool depth
Uses a simple version of the Meyer-Peter Mueller
sediment transport equation for analytical methodsARR incorporates features of both analytical andreference reach methods
Consider alluvial (noncohesive) and thresholdConsider alluvial (noncohesive) and threshold(cohesive) channels.(cohesive) channels.
StrengthsStrengths
Include planform design for stream meanders andInclude planform design for stream meanders and
pool spacingpool spacingDesigns pool depthDesigns pool depth
Uses a simple version of the MeyerUses a simple version of the Meyer--Peter MuellerPeter Mueller
sediment transport equation for analytical methodssediment transport equation for analytical methodsARR incorporates features of both analytical andARR incorporates features of both analytical andreference reach methodsreference reach methods
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KAM and ARR Methods (Cont.)KAM and ARR Methods (Cont.)KAM and ARR Methods (Cont.)
Limitations
Plane bed (no bedforms)
Wide channels (Large width to depthratios)
No consideration of grain sizedistribution other than d50Does not allow for separation of bedand bank hydraulic roughness
LimitationsLimitations
Plane bed (no bedforms)Plane bed (no bedforms)
Wide channels (Large width to depthWide channels (Large width to depthratios)ratios)
No consideration of grain sizeNo consideration of grain sizedistribution other than ddistribution other than d5050Does not allow for separation of bedDoes not allow for separation of bedand bank hydraulic roughnessand bank hydraulic roughness
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Study ObjectivesStudy ObjectivesStudy ObjectivesDevelop procedures to use HEC-RAS 4.0 in the
design of stable channel reaches for alluvial
streams using the Analytical Approach.
Provide examples for streams with
Sand beds
Gravel/cobble beds.
Compare HEC-RAS methods with McEnroesKAM and ARR Methods.
Develop procedures to use HECDevelop procedures to use HEC--RAS 4.0 in theRAS 4.0 in thedesign of stable channel reaches for alluvialdesign of stable channel reaches for alluvial
streams using thestreams using theAnalytical ApproachAnalytical Approach..
Provide examples for streams withProvide examples for streams with
Sand bedsSand beds
Gravel/cobble beds.Gravel/cobble beds.
Compare HECCompare HEC--RAS methods with McEnroeRAS methods with McEnroessKAM and ARR Methods.KAM and ARR Methods.
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Stable Channel Design in HEC-RASStable Channel Design in HECStable Channel Design in HEC--RASRAS
Uses Steady Flow modeling to determine parameters
needed for the sediment transport modeling components.
Velocity
DepthArea
Only Mannings n-values can be used in the HEC-RAS
steady flow model for resistance.
Uses Hydraulic Design Functions to perform uniform flow
and sediment transport capacity calculations. Brownlie,
Strickler, Limerinos and Manning equations can be usedto account for channel resistance. Brownlie and
Limerinos resistance equations account for bed form
resistance as well as resistance due to grains.
UsesUses Steady FlowSteady Flow modeling to determine parametersmodeling to determine parameters
needed for the sediment transport modeling components.needed for the sediment transport modeling components.
VelocityVelocity
DepthDepthAreaArea
Only ManningOnly Mannings ns n--values can be used in the HECvalues can be used in the HEC--RASRAS
steady flow model for resistance.steady flow model for resistance.
UsesUses Hydraulic Design FunctionsHydraulic Design Functions to perform uniform flowto perform uniform flow
and sediment transport capacity calculations. Brownlie,and sediment transport capacity calculations. Brownlie,
Strickler, Limerinos and Manning equations can be usedStrickler, Limerinos and Manning equations can be usedto account for channel resistance. Brownlie andto account for channel resistance. Brownlie and
Limerinos resistance equations account for bed formLimerinos resistance equations account for bed form
resistance as well as resistance due to grains.resistance as well as resistance due to grains.
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HEC-RAS Hydraulic Design
Functions used in Analytical Design
HECHEC--RAS Hydraulic DesignRAS Hydraulic Design
Functions used in Analytical DesignFunctions used in Analytical Design
Stable Channel Design
Uniform Flow
Sediment Transport Capacity
Stable Channel DesignStable Channel Design
Uniform FlowUniform Flow
Sediment Transport CapacitySediment Transport Capacity
Sand BedsSand Beds
Gravel/Cobble BedsGravel/Cobble Beds
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Resistance Formulas Used
in
Hydraulic Design Functions
Resistance Formulas UsedResistance Formulas Used
inin
Hydraulic Design FunctionsHydraulic Design Functions
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HEC-RAS Resistance Formulas
for Alluvial Channels
HECHEC--RAS Resistance FormulasRAS Resistance Formulas
for Alluvial Channelsfor Alluvial Channels
Equation Applicability Strengths Limitations
Manning All natural and artificialstreams.
Easy to use and to understand.
Required for HEC-RAS hydraulic
modeling.
Requires a high level of
engineering judgment to
choose an appropriate
value from a table or from a
book of reference streams.
Stricker Cobble bed streamsdominated by grain sizefriction.
Quantified the hydraulics lossesdue to grain size friction based on
measurable parameters.
Does not include losses dueto bed forms. May be
unrealistically low.
Limerinos Stream beds withsediment sizes from
coarse sand to cobble
under an upper flowregime.
Includes losses due to both grain
roughness and bedforms. Based
on measurable parameters.
Not applicable to other
sediment sizes or to the
lower regime flow.
Brownlie Sand bed streams ofeither an upper or a
lower regime.
Includes losses due to both grain
roughness and bedforms. Based
on measurable parameters. Can be
used for either the upper or the
lower flow regime. Correlates with
the Brownlie sediemnt transport
function.
Not applicable to other
sediment sizes.
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Mannings EquationManningMannings Equations Equation
V= Mean Velocity in ft/sec,
1.49 = coefficient for English Units (1.0 for Metric),n = Mannings n value,
R = Hydraulic radius, ft. = Area/Wetted Perimeter,
S = Slope of the Energy Grade Line.
(Bed slope for uniform flow)
V= Mean Velocity in ft/sec,V= Mean Velocity in ft/sec,
1.49 = coefficient for English Units (1.0 for Metric),1.49 = coefficient for English Units (1.0 for Metric), n = Manningn = Mannings n value,s n value,
R = Hydraulic radius, ft. = Area/Wetted Perimeter,R = Hydraulic radius, ft. = Area/Wetted Perimeter,
S = Slope of the Energy Grade Line.S = Slope of the Energy Grade Line.
(Bed slope for uniform flow)(Bed slope for uniform flow)
2 /3 1/ 21.49n R S
V=
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Strickler EquationStrickler EquationStrickler Equation
1/ 6
s
s
Rn k
k
=
50
90
, ,
.
0.0342
0.0342
s
s
where k Nikuradse equivalet sand roughness ft or m d
for natural channels and d for riprap lined channels
R Strickler function for natrual channelsk
for velocity and stone size calculations in riprap chann
= =
= =
=
0.038
els
for discharge calculations in riprap channels
R hydraulic radius
==
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Limerinos EquationLimerinos EquationLimerinos Equation
d84
= the particle size, ft, for which 84% of thesediment mixture is finer. Data ranged from0.00328 to 0.820 ft (1.5 to 250 mm). BIG STUFF
n = Mannings n value. Data ranged from 0.02 to
0.10.
R = Hydraulic radius, ft. Data ranged from 1 to 6ft (0.35 to 1.83 m).
dd8484 = the particle size, ft, for which 84% of the= the particle size, ft, for which 84% of the
sediment mixture is finer. Data ranged fromsediment mixture is finer. Data ranged from0.00328 to 0.820 ft (1.5 to 250 mm).0.00328 to 0.820 ft (1.5 to 250 mm). BIG STUFFBIG STUFF
n = Manningn = Mannings n value. Data ranged from 0.02 tos n value. Data ranged from 0.02 to
0.10.0.10.
R = Hydraulic radius, ft. Data ranged from 1 to 6R = Hydraulic radius, ft. Data ranged from 1 to 6ft (0.35 to 1.83 m).ft (0.35 to 1.83 m).
1/ 6
10
84
0.0926
1.16 2.0 log
Rn
Rd
=
+
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Brownlie Resistance Equations
Sand Only
Brownlie Resistance EquationsBrownlie Resistance Equations
Sand OnlySand Only
d50 = the particle size, ft, for which 50% of thesediment mixture is finer by weight,
s = the geometric standard deviation of the
sediment mixture.
dd5050 = the particle size, ft, for which 50% of the= the particle size, ft, for which 50% of thesediment mixture is finer by weight,sediment mixture is finer by weight,
ss = the geometric standard deviation of the= the geometric standard deviation of the
sediment mixture.sediment mixture.
( )
( )
0.1374
0.1670.1112 0.160550
50
0.0.6620.1670.0395 0.1282
50
50
1.6940 0.034
1.0213 0.034
Lower Regime
Rn S dd
Upper Regime
Rn S d
d
=
=
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Brownlie Resistance Equations
(Cont.)
Brownlie Resistance EquationsBrownlie Resistance Equations
(Cont.)(Cont.)'
' '
'
'1/ 3
50
0.8 1.25
0.006
1.74
( 1)
( 2.65 )
g g
g g g
g g
g
g
s
s
Lower Regime F F
Transition F F F
Upper Regime S or F F
FS
VF grain Froude number
s gdwhere
s sediment specific gravity for sand
=
= =
=
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HEC-RAS
Sediment Transport
Equations
HECHEC--RASRAS
Sediment TransportSediment Transport
EquationsEquations
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Sand Beds - Brownlie SedimentTransport Equation
Sand BedsSand Beds -- Brownlie SedimentBrownlie Sediment
Transport EquationTransport Equation
Used in the Stable Channel Hydraulic Design
Function for sand bed channels only. Themethod is called the Copeland Method.
Based on dimensional analysis and regressionof a very large body of field and laboratorysediment transport data for sand beds.
Only applied to the movable bed does notconsider sediment transport from the mainchannel banks.
Used in the Stable Channel Hydraulic DesignUsed in the Stable Channel Hydraulic Design
Function forFunction for sand bed channels onlysand bed channels only. The. Themethod is called the Copeland Method.method is called the Copeland Method.
Based on dimensional analysis and regressionBased on dimensional analysis and regressionof a very large body of field and laboratoryof a very large body of field and laboratorysediment transport data for sand beds.sediment transport data for sand beds.
Only applied to the movable bedOnly applied to the movable bed does notdoes notconsider sediment transport from the mainconsider sediment transport from the mainchannel banks.channel banks.
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Brownlie Sediment Transport Equation
(Sand Bed Natural Channels)
Brownlie Sediment Transport EquationBrownlie Sediment Transport Equation(Sand Bed Natural Channels)(Sand Bed Natural Channels)
( ) 0.33011.978 0.6601
509022( ) /g goC F F S r d
=
( )
*
50
50
0.5293 0.14
/ /
4.596o
g s
go
where C bed material concentration in ppm by weight
r representative grain roughness height
d geometric mean grain size of bed material
S slope of energy grade line
F V gd grain Froude number
F S
==
=
=
= =
=
( )( )
*
05 0.1606
7.7
0.6
3
50
0.22 0.06(10)
/
/
o
g
Y
g
g s
g
critical grain Froude number
Y critical shear stress
geometric standard deviation of bed material
Y R
R gd grain Reynolds number
=
= + =
=
=
= =
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Gravel/Cobble Beds Sediment Transport
Potential Functions
Gravel/Cobble BedsGravel/Cobble Beds Sediment TransportSediment Transport
Potential FunctionsPotential Functions
Ackers-White
Engelund-Hansen
Laursen-Copeland
Meyer-Peter Muller
Toffaleti
Yang
AckersAckers--WhiteWhite
EngelundEngelund--HansenHansen
LaursenLaursen--CopelandCopeland
MeyerMeyer--Peter MullerPeter Muller
ToffaletiToffaleti
YangYang
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Ranges for Sediment Transport FunctionsRanges for Sediment Transport FunctionsRanges for Sediment Transport Functions
dran
ge,mm
dran
ge,mm
Mean
d,mm
Mean
d,mm
Veloc
ity,fps
Veloc
ity,fps
Depth
,ft
Depth
,ft
Energ
yGrad
Energ
yGrad
Width
ft
Width
ft
Temp
,oF
Temp
,oF
Spec
Gravity
Spec
Gravity
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Sand Beds
HEC-RAS Stable ChannelDesign
Sand BedsSand Beds
HECHEC--RAS Stable ChannelRAS Stable ChannelDesignDesign
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HR Stable Channel DesignHR Stable Channel DesignHR Stable Channel Design
Copeland Method using Brownlie Resistance
and Sediment Transport Eqs.
Sand Bed Channels Only.Resistance Due to Sidewall Roughness, Grains
of the Bed Material and Bed Forms.
Sediment Transport from Bed Only.
Sidewall Roughness Method Applied.
Does not specify channel plan form geometry or
profile features. (See McEnroe KAM and ARR
methods.)
Copeland Method using Brownlie ResistanceCopeland Method using Brownlie Resistance
and Sediment Transport Eqs.and Sediment Transport Eqs.
Sand Bed Channels Only.Sand Bed Channels Only.Resistance Due to Sidewall Roughness, GrainsResistance Due to Sidewall Roughness, Grains
of the Bed Material and Bed Forms.of the Bed Material and Bed Forms.
Sediment Transport from Bed Only.Sediment Transport from Bed Only.
Sidewall Roughness Method Applied.Sidewall Roughness Method Applied.
Does not specify channel plan form geometry orDoes not specify channel plan form geometry or
profile features. (See McEnroe KAM and ARRprofile features. (See McEnroe KAM and ARR
methods.)methods.)
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HR Stable Channel Design
Requirements
HR Stable Channel DesignHR Stable Channel Design
RequirementsRequirements
Upstream Supply Channel: (Trapezoidal channel
geometry required.) Bottom width, depth,channel slope, side slopes, discharge,
Mannings n for sidewalls, sediment gradation or
sediment conc.Design Channel: Mannings n for sidewalls, side
slopes, sediment gradation ,and either the
bottom width, depth, or channel slope.Both: Need d16, d50 and d84
Upstream Supply Channel:Upstream Supply Channel: (Trapezoidal channel(Trapezoidal channel
geometry required.) Bottom width, depth,geometry required.) Bottom width, depth,channel slope, side slopes, discharge,channel slope, side slopes, discharge,
ManningMannings n for sidewalls, sediment gradation ors n for sidewalls, sediment gradation or
sediment conc.sediment conc.Design Channel:Design Channel: ManningMannings n for sidewalls, sides n for sidewalls, side
slopes, sediment gradation ,and either theslopes, sediment gradation ,and either the
bottom width, depth, or channel slope.bottom width, depth, or channel slope. Both:Both: Need dNeed d1616, d, d5050 and dand d8484
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Procedure for Stable Channel Design of
Sand Bed Channels
Procedure for Stable Channel Design ofProcedure for Stable Channel Design of
Sand Bed ChannelsSand Bed Channels Establish the bank-full properties of an upstream
reference reach riffle cross section. Discharge, cross
section geometry via station-elevation data, stage, bedmaterial (d16, d50 and d84), longitudinal energy grade line
slope.
Open the Uniform Flow function with the Manning for the
bank resistance and Brownlie for the movable bed
resistance. Input the slope and discharge. By iteration,
determine the bank n-values needed to obtain the
desired bank-full water surface elevation (stage) for thegiven bank-full discharge and slope.
Establish the bankEstablish the bank--full properties of an upstreamfull properties of an upstream
reference reach riffle cross section. Discharge, crossreference reach riffle cross section. Discharge, cross
section geometry via stationsection geometry via station--elevation data, stage, bedelevation data, stage, bedmaterialmaterial ((dd1616, d, d5050 and dand d8484)), longitudinal energy grade line, longitudinal energy grade line
slope.slope.
Open the Uniform Flow function with the Manning for theOpen the Uniform Flow function with the Manning for the
bank resistance and Brownlie for the movable bedbank resistance and Brownlie for the movable bed
resistance. Input the slope and discharge. By iteration,resistance. Input the slope and discharge. By iteration,
determine the bank ndetermine the bank n--values needed to obtain thevalues needed to obtain the
desired bankdesired bank--full water surface elevation (stage) for thefull water surface elevation (stage) for thegiven bankgiven bank--full discharge and slope.full discharge and slope.
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Procedure for Stable Channel Design of
Sand Bed Channels (Cont.)
Procedure for Stable Channel Design ofProcedure for Stable Channel Design of
Sand Bed Channels (Cont.)Sand Bed Channels (Cont.)
Using the bank n-values from the previous step,change the resistance formula for the bed to
Manning then by iteration determine theappropriate n-value for the bed to obtain thedesired bankfull stage.
Create an upstream supply reach that has threeof the natural channels using the bank and bedn-values determined above for the bankfullchannel.
Run the HEC-RAS model.
Using the bank nUsing the bank n--values from the previous step,values from the previous step,change the resistance formula for the bed tochange the resistance formula for the bed to
Manning then by iteration determine theManning then by iteration determine theappropriate nappropriate n--value for the bed to obtain thevalue for the bed to obtain thedesired bankfull stage.desired bankfull stage.
Create an upstream supply reach that has threeCreate an upstream supply reach that has threeof the natural channels using the bank and bedof the natural channels using the bank and bednn--values determined above for the bankfullvalues determined above for the bankfullchannel.channel.
Run the HECRun the HEC--RAS model.RAS model.
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Procedure for Stable Channel Design of
Sand Bed Channels (Cont.)
Procedure for Stable Channel Design ofProcedure for Stable Channel Design of
Sand Bed Channels (Cont.)Sand Bed Channels (Cont.)
Determine an equivalent trapezoidal channelthat has the same conveyance as the natural
supply reach. Open the Stable Channel Designfunction.
Input the side slopes, base width, bank n-values
and the energy grade line slope of theequivalent upstream supply channel.
Input the side slopes and bank n-values for thedesign channel.
Run the Stable Channel Design model.
Determine an equivalent trapezoidal channelDetermine an equivalent trapezoidal channelthat has the same conveyance as the naturalthat has the same conveyance as the natural
supply reach. Open the Stable Channel Designsupply reach. Open the Stable Channel Designfunction.function.
Input the side slopes, base width, bank nInput the side slopes, base width, bank n--valuesvalues
and the energy grade line slope of theand the energy grade line slope of theequivalent upstream supply channel.equivalent upstream supply channel.
Input the side slopes and bank nInput the side slopes and bank n--values for thevalues for thedesign channel.design channel.
Run the Stable Channel Design model.Run the Stable Channel Design model.
S d B d E l
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Sand Bed ExampleSand Bed ExampleSand Bed Example
49.021.6
0
5
10
15
20
25
0 10 20 30 40 50 60 70 80
Station (ft)
Ele
vation
(ft)
Sta-Elev Bankfull Elevation Movable Bed
49.021.6
0
5
10
15
20
25
0 10 20 30 40 50 60 70 80
Station (ft)
Elev
ation
(ft)
Sta-Elev Bankfull Elevation Movable Bed
2828 4444
Bank-full Conditions
d16, d50, d84 = 1.33, 2 and 3 mm, respectively
Q = 325 cfsStage = 7 ft
Slope = 0.00157
BankBank--full Conditionsfull Conditions
dd1616, d, d5050, d, d8484 = 1.33, 2 and 3 mm, respectively= 1.33, 2 and 3 mm, respectively
Q = 325 cfsQ = 325 cfs Stage = 7 ftStage = 7 ft
Slope = 0.00157Slope = 0.00157
U if Fl ith Fi l M i lU if Fl ith Fi l M iU if Fl ith Fi l M i ll
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Uniform Flow with Final Mannings n values(Initially Brownlie for movable bed, unknown for banks)
Uniform Flow with Final ManningUniform Flow with Final Mannings n valuess n values(Initially Brownlie for movable bed, unknown for banks)(Initially Brownlie for movable bed, unknown for banks)
N t l S l R hN t l S l R hN t l S l R h
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Natural Supply ReachNatural Supply ReachNatural Supply Reach
E i l t T id l Ch lE i l t T id l Ch lE i l t T id l Ch l
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Equivalent Trapezoidal ChannelEquivalent Trapezoidal ChannelEquivalent Trapezoidal Channel
Abnk Pbnkh
0
5
10
15
20
0 10 20 30 40 50 60 70 80
Station(ft)
E
levation
(ft)
StaElevPoints Bankfull WaterSurface
EquivalentChannel MovableBed
0
5
10
15
20
0 10 20 30 40 50 60 70 80
Station(ft)
Ele
vation
(ft)
StaElevPoints Bankfull WaterSurface
EquivalentChannel MovableBed
Equivalent
Trapezoidal ChannelEquivalent
Trapezoidal Channel
T id l Ch l S l R hT id l Ch l S l R hTrape oidal Channel S ppl Reach
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Trapezoidal Channel Supply ReachTrapezoidal Channel Supply ReachTrapezoidal Channel Supply Reach
St bl Ch l D i F tiStable Channel Design F nctionStable Channel Design Function
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Stable Channel Design FunctionStable Channel Design FunctionStable Channel Design Function
ComputeComputeCompute
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ComputeComputeCompute
Select Design Channel for b = 20 ftSelect Design Channel for b = 20 ftSelect Design Channel for b = 20 ft
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Select Design Channel for b = 20 ftSelect Design Channel for b = 20 ftSelect Design Channel for b = 20 ft
20 FT20 FT
Stability Curve Width vs SlopeStability Curve Width vs SlopeStability Curve Width vs Slope
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Stability Curve, Width vs. SlopeStability Curve, Width vs. SlopeStability Curve, Width vs. Slope
181.42 ppm181.42 ppm
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Gravel/Cobble Beds
HEC-RAS Sediment
Transport Capacity
Function
Gravel/Cobble BedsGravel/Cobble Beds
HECHEC--RAS SedimentRAS Sediment
Transport CapacityTransport Capacity
FunctionFunction
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HR Sediment Transport Capacity (STC)HR Sediment Transport Capacity (STC)HR Sediment Transport Capacity (STC)Grain size classes are input as grain size
and percent finer.
Computes STC for each size class, gsi
Total STC is computed by the equationgs,total = pigsiwhere pi = fraction of size class i in the bed.
Can compute the total STC for all sixSediment Transport Potential functions.
Grain size classes are input as grain sizeGrain size classes are input as grain sizeand percent finer.and percent finer.
Computes STC for each size class, gComputes STC for each size class, gsisi
Total STC is computed by the equationTotal STC is computed by the equationggs,totals,total == ppiiggsisiwhere pwhere pii = fraction of size class i in the bed.= fraction of size class i in the bed.
Can compute the total STC for all sixCan compute the total STC for all sixSediment Transport Potential functions.Sediment Transport Potential functions.
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Gravel/Cobble ExampleGravel/Cobble ExampleGravel/Cobble Example
The stream has the following bank-full
conditions
Water surface elevation = 11.7 feetDischarge = 3,100 cfs
Slope = 0.0015.
The stream has the following bank-full
conditions
Water surface elevation = 11.7 feetDischarge = 3,100 cfs
Slope = 0.0015.
12027.78
0
10
20
30
0 20 40 60 80 100 120 140 160
Station (ft)
E
levation(ft)
Sta-Elev Bankfull Elevation Movable Bed
12027.78
0
10
20
30
0 20 40 60 80 100 120 140 160
Station (ft)
Elevation(ft)
Sta-Elev Bankfull Elevation Movable Bed
5252 9696
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Pebble Count for Gravel/Cobble StreamPebble Count for Gravel/Cobble StreamPebble Count for Gravel/Cobble StreamINCHES PARTICLE MILLIMETER SIZE CLASS COUNT % CUM % Dtop(mm)
Silt/Clay < 0.062 S/C 12 12 12
Very Fine .062 - .125 S 7 7 19 0.125
Fine .125 - .25 A 2 2 21 0.25
Medium .25 - .50 N 2 2 23 0.5
Coarse .50 - 1.0 D 4 4 27 1
.04 - .08 Very Coarse 1.0 - 2 S 3 3 30 2.00
.08 - .16 Very Fine 2 - 4 12 12 42 4.00
.16 - .24 Fine 4 - 5.7 G 2 2 44 5.7
.24 - .31 Fine 5.7 - 8 R 3 3 47 8.00
.31 - .47 Medium 8 - 11.3 A 1 1 48 11.3
.47 - .63 Medium 11.3 - 16 V 0 0 48 16.00
.63 - .94 Coarse 16 - 22.6 E 2 2 50 22.6
.94 - 1.26 Coarse 22.6 - 32 L 5 5 55 32.00
1.26 - 1.9 Very Coarse 32 - 45 S 7 7 62 45.00
1.9 - 2.5 Very Coarse 45 - 64 6 6 68 64.00
2.5 - 3.8 Small 64 - 90 C - 6 6 74 90.00
3.8 - 5.0 Small 90 - 128 O L 6 6 80 128
5.0 - 7.6 Large 128 - 180 B E 6 6 86 180
7.6 - 10 Large 180 - 256 B S 5 5 91 256
10 - 15 Small 256 - 362 B D 1 1 92 362
15 - 20 Small 362 - 512 O E 1 1 93 512
20 - 40 Medium 512 - 1024 U R 0 0 93 1024
40 - 160 Lrg to Very Lrg 1024 - 2048 L S 0 0 93 2048
BEDROCK BDRK 7 7 100
NOTENOTE
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Log-probability plot of Bed MaterialLogLog--probability plot of Bed Materialprobability plot of Bed Material
Sand Gravel Cobble
Make New HEC-RAS Model with oneMake New HECMake New HEC--RAS Model with oneRAS Model with one
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cross section and no dischargecross section and no dischargecross section and no discharge
Uniform Flow Bed uses Limerinos banksUniform FlowUniform Flow Bed uses Limerinos banksBed uses Limerinos banks
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Uniform Flow Bed uses Limerinos, banks
use Manning (T&E gives nbank = 0.077)
Uniform FlowUniform Flow Bed uses Limerinos, banksBed uses Limerinos, banks
use Manning (T&E gives nuse Manning (T&E gives nbankbank = 0.077)= 0.077)
Uniform Flow Bed uses Mannings BanksUniform FlowUniform Flow Bed uses Mannings BanksBed uses Mannings, Banks
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Uniform Flow Bed uses Mannings, Banks
use n = 0.077 (T&E gives nbed = 0.0363)
Uniform FlowUniform Flow Bed uses Mannings, BanksBed uses Mannings, Banks
use n = 0.077 (T&E gives nuse n = 0.077 (T&E gives nbedbed = 0.0363)= 0.0363)
Create and Run Natural Supply ReachCreate and Run Natural Supply ReachCreate and Run Natural Supply Reach
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pp ypp ypp y
Sediment Transport Capacity FunctionSediment Transport Capacity FunctionSediment Transport Capacity Function
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Sediment Transport Capacity FunctionInput Grain Sizes (Fake size for banks)
Sediment Transport Capacity FunctionSediment Transport Capacity FunctionInput Grain Sizes (Fake size for banks)Input Grain Sizes (Fake size for banks)
Diam, mm % Finer Diam, mm % Finer Diam, mm % Finer
2000 19 0.125 19 2000 19
2000 21 0.25 21 2000 21
2000 23 0.5 23 2000 23
2000 27 1 27 2000 27
2000 30 2 30 2000 30
2000 42 4 42 2000 42
2000 44 5.7 44 2000 44
2000 47 8 47 2000 47
2000 48 11.3 48 2000 48
2000 48 16 48 2000 48
2000 50 22.6 50 2000 502000 55 32 55 2000 55
2000 62 45 62 2000 62
2000 68 64 68 2000 68
2000 74 90 74 2000 74
2000 80 128 80 2000 80
2000 86 180 86 2000 86
2000 91 256 91 2000 91
2000 92 362 92 2000 92
2000 93 512 93 2000 93
2000 93 1024 93 2000 93
2000 93 2048 93 2000 93
ROBLOB Main
Compute Sediment Rating Curve PlotCompute Sediment Rating Curve PlotCompute Sediment Rating Curve Plot
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Compute, Sediment Rating Curve Plot,
Generate Report
Compute, Sediment Rating Curve Plot,Compute, Sediment Rating Curve Plot,
Generate ReportGenerate Report
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Meyer-Peter Mueller Function ResultsMeyerMeyer--Peter Mueller Function ResultsPeter Mueller Function Results
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1010
77
Design 1 (b = 35 ft m = 3:1 hor: vert)Design 1 (b = 35 ft m = 3:1 hor: vert)Design 1 (b = 35 ft m = 3:1 hor: vert)
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Design 1 (b = 35 ft, m = 3:1 hor: vert)Design 1 (b = 35 ft, m = 3:1 hor: vert)Design 1 (b = 35 ft, m = 3:1 hor: vert)
Assume slope
Create 3 cross section model with
trapezoidal xsecs with same ns and Q assupply reach
Run steady flow model
Run Sediment Transport Capacity function
See if STC equals 1083 tons/day if not back
to the top with a new slope
Assume slopeAssume slope
Create 3 cross section model withCreate 3 cross section model with
trapezoidal xsecs with same ntrapezoidal xsecs with same ns and Q ass and Q assupply reachsupply reach
Run steady flow modelRun steady flow model
Run Sediment Transport Capacity functionRun Sediment Transport Capacity function
See if STC equals 1083 tons/daySee if STC equals 1083 tons/day if not backif not back
to the top with a new slopeto the top with a new slope
Design 1 (b = 35 ft, m = 3:1 hor: vert)Design 1 (b = 35 ft, m = 3:1 hor: vert)Design 1 (b = 35 ft, m = 3:1 hor: vert)
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S = 0.003 S = 0.0017
b = 35 Natural Design b = 35 Natural Design
Function Function
A-W 836000 410900... NA A-W 836000 113600... NAE-H 8217 26640 0.31 E-H 8217 9883 0.83
Laur 645200 212400... NA Laur 645200 750300 0.86
MPM 1083 2427 0.45 MPM 1083 1151 0.94
Toff 680.4 627 1.09 Toff 680.4 586.7 1.16
Yang 27290 88360 0.31 Yang 27290 32300 0.84
S = 0.0016 S = 0.00162
b = 35 Natural Design b = 35 Natural Design
Function Function
A-W 836000 992100 0.84 A-W 836000 102100... NA
E-H 8217 8908 0.92 E-H 8217 9105 0.90
Laur 645200 673200 0.96 Laur 645200 688800 0.94
MPM 1083 1063 1.02 MPM 1083 1081 1.00
Toff 680.4 582.9 1.17 Toff 680.4 583.7 1.17
Yang 27290 29000 0.94 Yang 27290 29670 0.92
Nat/Des
Nat/Destons/day
Nat/Des
tons/day
tons/day
Nat/Destons/day
S = 0.003 S = 0.0017
b = 35 Natural Design b = 35 Natural Design
Function Function
A-W 836000 410900... NA A-W 836000 113600... NA
E-H 8217 26640 0.31 E-H 8217 9883 0.83
Laur 645200 212400... NA Laur 645200 750300 0.86
MPM 1083 2427 0.45 MPM 1083 1151 0.94
Toff 680.4 627 1.09 Toff 680.4 586.7 1.16
Yang 27290 88360 0.31 Yang 27290 32300 0.84
S = 0.0016 S = 0.00162
b = 35 Natural Design b = 35 Natural Design
Function Function
A-W 836000 992100 0.84 A-W 836000 102100... NAE-H 8217 8908 0.92 E-H 8217 9105 0.90
Laur 645200 673200 0.96 Laur 645200 688800 0.94
MPM 1083 1063 1.02 MPM 1083 1081 1.00
Toff 680.4 582.9 1.17 Toff 680.4 583.7 1.17
Yang 27290 29000 0.94 Yang 27290 29670 0.92
Nat/Des
Nat/Destons/day
Nat/Des
tons/day
tons/day
Nat/Destons/day
T & E gives S = 0.00162T & E gives S = 0.00162
Final DesignFinal DesignFinal Design
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Design 2 - Select slope and sideDesign 2Design 2 -- Select slope and sideSelect slope and side
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g
slopes, find bslopes, find bslopes, find b
b = 15
bnk ht = 15.62 S = 0.0018 m = 2.5 Function All Grains
RS 0 RS 100 RS 200 A-W 192600...
Station Elevation Station Elevation Station Elevation E-H 12280-46.550 15.620 -46.550 15.800 -46.550 15.980 Laur 901500
-7.500 0.000 -7.500 0.180 -7.500 0.360 MPM 1091
7.500 0.000 7.500 0.180 7.500 0.360 Toff 438.7
46.550 15.620 46.550 15.800 46.550 15.980 Yang 37320
STC in Tons/Dayb = 15
bnk ht = 15.62 S = 0.0018 m = 2.5 Function All Grains
RS 0 RS 100 RS 200 A-W 192600...
Station Elevation Station Elevation Station Elevation E-H 12280-46.550 15.620 -46.550 15.800 -46.550 15.980 Laur 901500
-7.500 0.000 -7.500 0.180 -7.500 0.360 MPM 1091
7.500 0.000 7.500 0.180 7.500 0.360 Toff 438.7
46.550 15.620 46.550 15.800 46.550 15.980 Yang 37320
STC in Tons/Day
S = 0.0018, m = 2.5:1 hor: vertS = 0.0018, m = 2.5:1 hor: vertS = 0.0018, m = 2.5:1 hor: vert
S = 0.0018, m = 2.5:1 hor: vertS = 0.0018, m = 2.5:1 hor: vertS = 0.0018, m = 2.5:1 hor: vert
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S 0.0018, m 2.5:1 hor: vertS 0.0018, m 2.5:1 hor: vertS 0 00 8, 5 o e t
Design Channel, b = 15', hor:vert = 2.5:1, S = 0.0018
0
4
8
12
16
-60 -40 -20 0 20 40 60
Station (ft)
Elevation
(ft)
Design Channel Bankfull Elevation Movable Bed
Design Channel, b = 15', hor:vert = 2.5:1, S = 0.0018
0
4
8
12
16
-60 -40 -20 0 20 40 60
Station (ft)
Elevation
(ft)
Design Channel Bankfull Elevation Movable Bed
b = 15 ft, bank ht = 15.62 ftb = 15 ft, bank ht = 15.62 ft
Design 3 - S = 0.0013, m = 2.5:1 hor: vertDesign 3Design 3 -- S = 0.0013, m = 2.5:1 hor: vertS = 0.0013, m = 2.5:1 hor: vert
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g ,gg ,
0
5
10
-60 -40 -20 0 20 40 60
Station (ft)
Elevation
(ft)
Design Channel Bankfull Elevation Movable Bed
0
5
10
-60 -40 -20 0 20 40 60
Station (ft)Elevation
(ft)
Design Channel Bankfull Elevation Movable Bed
b = 76 ft, bank ht = 8.73 ftb = 76 ft, bank ht = 8.73 ft
4 b = 76.0 1082bnk ht = 8.73 S = 0.0013 m = 2.5 Tons/day
RS 0 RS 100 RS 200 Function All Grains
Station Elevation Station Elevation Station Elevation A-W 388200
-59.825 8.730 -59.825 8.860 -59.825 8.990 E-H 4981
-38 0.000 -38 0.130 -38 0.260 Laur 545400
38 0.000 38 0.130 38 0.260 MPM 108259.825 8.730 59.825 8.860 59.825 8.990 Toff 1078
Yang 19240
Final Design 2MPM Gs (Tons/day) =4 b = 76.0 1082
bnk ht = 8.73 S = 0.0013 m = 2.5 Tons/day
RS 0 RS 100 RS 200 Function All Grains
Station Elevation Station Elevation Station Elevation A-W 388200
-59.825 8.730 -59.825 8.860 -59.825 8.990 E-H 4981
-38 0.000 -38 0.130 -38 0.260 Laur 545400
38 0.000 38 0.130 38 0.260 MPM 1082
59.825 8.730 59.825 8.860 59.825 8.990 Toff 1078
Yang 19240
Final Design 2MPM Gs (Tons/day) = Final Design 3
Sidewall Correction MethodSidewall Correction MethodSidewall Correction Method
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bd
yd
md
1
md
1 1
md
1
md
nwnw
nb
Wd
} 4 /32 /3 1/ 2 2
2 4 /3
3/ 4 3/ 43/ 42 4 /3 2 2
2 4 /3 2 3/ 2
1 1'
1 1 1
square
co
AMetric version of Manning s Equation V R S V S
n n P
Einstein assumed V and S are constant in bank area and sidewall area
V A V A A V
S n P S n P n P S
= =
= = =
( ) ( ) ( )
3/ 2 3/ 2 3/ 2
3/ 2 3/ 2 3/ 2 3/ 2 3/ 2 3/ 2
2 /3 3/ 2 3/ 23/ 2 3/ 2
3/ 2 3/ 2
1
nstant
Constantw b
w b
w b
w bw b w b w w b b
w w b bw w b b w b
P PPn n n
A A A
P PPA A A n n n n P n P n P
n P n Pn n P n P also A A and A A
P n P n P
= = =
= + = + = +
= + = =
64748
Aw/2Aw/2 AbAbAw/2Aw/2
Sidewall ExampleSidewall ExampleSidewall Example
S 0 0002
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50
5
n = 0.025
n = 0.045n = 0.045
12
12
S = 0.0002
nw = 0.040 nw = 0.040nb = 0.025
( ) ( )
2 2
2 2 2
2 /3 2 /3
3/ 2 3/ 2 3/ 2 3/ 2
50 ; 2 5 10 22.4 72.4
(5)(10)50*5 250 ; 2 50 300
2
1 1(0.040) 22.4 (0.025) 50
72.4
0.0300
1.49 (300
0.0300
b w b w
rect tria rect tria
w w b b
P ft P ft P P P ft
A ft A ft A A A ft
n n P n PP
n
Q
= = + = = + =
= = = = = + =
= + = +
=
=
( ) ( )
5/33
2/3
3/ 2 3/ 22 2
3/ 2 3/ 2
2 2
)0.0002 543 /
(72.4)(0.040) (0.025)22.4 50
300 143 300 157(0.030) 72.4 (0.030) 72.4
50 250
w b
w b
ft s
A ft and A ft
Geometric values A ft and A ft
=
= = = =
= =
ARR Analytical MethodARR Analytical MethodARR Analytical Method
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( )
( )( )
5/3
2/3
3/ 250
3/2
50
50
50
1.49. 4 2
8 0.047 . 4 6
0.047 1. 4 8
0.047 1
, , , , 2.65, , .
dd d
d d
s
m sm
m d d m
d d s
d d d m d s m m
AManning for Design Channel Q S ARR Eq
n P
bMPM B yS d ARR Eq
y S G dB B b b ARR Eq
y S G d
Given Q m n S S G d b and y
=
=
= =
= =
. 4 2 4 8
, .
d dUse iteration to solve Eqs and for b and y by iteration
Subscripts d and m denote design channel and match reach channels respectively
bd
yd
md
1
md
1 1
md
1
md
nwnw
nb
Wd
nd = Manningscomposite n
ARR vs HEC-RASComposite n 0 061 from HEC RAS Supply Reach
ARR vs HECARR vs HEC--RASRASComposite nComposite n 0 061 from HEC0 061 from HEC RAS Supply ReachRAS Supply Reach
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Composite n 0.061 from HEC-RAS Supply ReachComposite nComposite n 0.061 from HEC0.061 from HEC--RAS Supply ReachRAS Supply ReachARR solution for Design 1 ( HR solution bd= 35 ft) dm(mm)
md = 3 Sd = 0.001622 nd = 0.061 22.6
yd bd Ad Pd Wb Qb Q
10.96 42.9 829.9 112.2 108.6 3100 0.000
ARR solution for Design 2 ( HR solution bd= 15 ft) dm(mm)
md = 2.5 Sd = 0.0018 nd = 0.061 22.6
yd bd Ad Pd Wb Qb Q
13.38 22.8 752.1 94.8 89.7 3100 0.000
ARR solution for Design 3 ( HR solution bd = 76 ft) dm (mm)
md = 2.5 Sd = 0.0013 nd = 0.061 22.6
yd bd Ad Pd Wb Qb Q
11.91 58.8 1055.6 123.0 118.4 3897.35 797.350
ARR solution for Design 1 ( HR solution bd= 35 ft) dm(mm)
md = 3 Sd = 0.001622 nd = 0.061 22.6
yd bd Ad Pd Wb Qb Q
10.96 42.9 829.9 112.2 108.6 3100 0.000
ARR solution for Design 2 ( HR solution bd= 15 ft) dm(mm)
md = 2.5 Sd = 0.0018 nd = 0.061 22.6
yd bd Ad Pd Wb Qb Q
13.38 22.8 752.1 94.8 89.7 3100 0.000
ARR solution for Design 3 ( HR solution bd = 76 ft) dm (mm)
md = 2.5 Sd = 0.0013 nd = 0.061 22.6
yd bd Ad Pd Wb Qb Q
11.91 58.8 1055.6 123.0 118.4 3897.35 797.350
ARR solution for Design 1 ( HR solution bd= 35 ft) dm(mm)
md = 3 Sd = 0.001622 nd = 0.061 22.6
yd bd Ad Pd Wb Qb Q
10.96 42.9 829.9 112.2 108.6 3100 0.000
ARR solution for Design 2 ( HR solution bd= 15 ft) dm(mm)
md = 2.5 Sd = 0.0018 nd = 0.061 22.6
yd bd Ad Pd Wb Qb Q
13.38 22.8 752.1 94.8 89.7 3100 0.000
ARR solution for Design 3 ( HR solution bd = 76 ft) dm(mm)
md = 2.5 Sd = 0.0013 nd = 0.061 22.6yd bd Ad Pd Wb Qb Q
11.91 58.8 1055.6 123.0 118.4 3897.35 797.350
ARR solution for Design 1 ( HR solution bd= 35 ft) dm(mm)
md = 3 Sd = 0.001622 nd = 0.061 22.6
yd bd Ad Pd Wb Qb Q
10.96 42.9 829.9 112.2 108.6 3100 0.000
ARR solution for Design 2 ( HR solution bd= 15 ft) dm(mm)
md = 2.5 Sd = 0.0018 nd = 0.061 22.6
yd bd Ad Pd Wb Qb Q
13.38 22.8 752.1 94.8 89.7 3100 0.000
ARR solution for Design 3 ( HR solution bd = 76 ft) dm(mm)
md = 2.5 Sd = 0.0013 nd = 0.061 22.6yd bd Ad Pd Wb Qb Q
11.91 58.8 1055.6 123.0 118.4 3897.35 797.350
bHR = 35 ft
bARR= 42.9 ftbHR/bARR= 0.82
bHR = 35 ft
bARR= 42.9 ftbHR/bARR= 0.82
bHR = 15 ft
bARR= 22.8 ftbHR/bARR= 0.66
bHR = 15 ft
bARR= 22.8 ftbHR/bARR= 0.66
ARR did not converge
bHR = 76 ft
bARR= 58.8 ftbHR/bARR= 1.29
ARR did not convergebHR = 76 ft
bARR= 58.8 ftbHR/bARR= 1.29
ARR vs HEC-RASComposite n values from HR Design Reaches
ARR vs HECARR vs HEC--RASRASComposite n values from HR Design ReachesComposite n values from HR Design Reaches
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Composite n values from HR Design ReachesComposite n values from HR Design ReachesComposite n values from HR Design Reaches(b) ARR solution for Design 1 ( HR solution b
d
= 35 ft)
md = 3 Sd = 0.00162 nd = 0.066 dm (mm)= 22.6
yd bd Ad Pd Wb Qb Q
12.35 33.2 867.7 111.3 107.3 3100 0.000
(c) ARR solution for Design 2 ( HR solution bd= 15 ft)
md = 2.5 Sd = 0.0018 nd = 0.072 dm (mm)= 22.6
yd bd Ad Pd Wb Qb Q
15.19 17.8 847.3 99.6 93.8 3100 0.000
(d) ARR solution for Design 3 ( HR solution bd= 76 ft)
md = 2.5 Sd = 0.0013 nd = 0.054 dm (mm)= 22.6
yd bd Ad Pd Wb Qb Q
11.91 58.8 1055.6 123.0 118.4 4402.562 1302.562
(b) ARR solution for Design 1 ( HR solution bd= 35 ft)md = 3 Sd = 0.00162 nd = 0.066 dm (mm)= 22.6
yd bd Ad Pd Wb Qb Q
12.35 33.2 867.7 111.3 107.3 3100 0.000
(c) ARR solution for Design 2 ( HR solution bd= 15 ft)
md = 2.5 Sd = 0.0018 nd = 0.072 dm (mm)= 22.6
yd bd Ad Pd Wb Qb Q
15.19 17.8 847.3 99.6 93.8 3100 0.000
(d) ARR solution for Design 3 ( HR solution bd= 76 ft)
md = 2.5 Sd = 0.0013 nd = 0.054 dm (mm)= 22.6
yd bd Ad Pd Wb Qb Q
11.91 58.8 1055.6 123.0 118.4 4402.562 1302.562
n = 0.066
bHR = 35 ft
bARR= 33.2 ft
bHR/bARR= 1.05
n = 0.066
bHR = 35 ft
bARR= 33.2 ft
bHR/bARR= 1.05
n = 0.072
bHR = 15 ft
bARR= 17.8 ftbHR/bARR= 0.84
n = 0.072
bHR = 15 ft
bARR= 17.8 ftbHR/bARR= 0.84
ARR did not converge
n = 0.054bHR = 76 ft
bARR= 58.8 ftbHR/bARR= 1.29
ARR did not converge
n = 0.054bHR = 76 ft
bARR= 58.8 ftbHR/bARR= 1.29
ARRs Simplified MPM EquationARRARRs Simplified MPM Equations Simplified MPM Equation
RR simplified MPM equation
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( ) 3/ 2
3/ 2
'
'
( )
8( ) 0.047
b s m
b
b
b
b
eyer Peter Mueller MPM
bB RKR R S d
nRKR Nikuradse roughness ratio
n
n Manning coefficient for grain size
n total Manning coefficient
=
= =
=
=
}
( )}
50
3/ 21
3/ 28 ( ) 0.047
dy
b s m
RR simplified MPM equation
bB RKR R S d
=
64748
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Analysis of Bed Grain SizeAnalysis of Bed Grain SizeAnalysis of Bed Grain Size
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DistributionDistributionDistribution
Sieve AnalysisVisual-Accumulation Tube
Pebble Count
Sieve AnalysisSieve AnalysisVisualVisual--Accumulation TubeAccumulation Tube
Pebble CountPebble Count
Log-normal DistributionLogLog--normal Distributionnormal Distribution
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84.1
log 15.9
50
loglog
log84.1 84
15.9 16
log
log log1 1(log ) exp
22
10 10
(log ) (log ) (log ) /100
d
dd
d
d
g
d
Probability Density Function PDF
d df d
Cumulative Distribution Function CDF
d d
d d
F d f d d d P
Standa
=
= = =
= =
84.1
log 15.9
84.1 84.1log 84.1 15.9
15.9 15.9
log84.1 84
15.9 16
1 1(log log ) log log
2 2
10 10d
d
d
d
g
rdDeviation
d dd d
d d
Geometric standard deveiation
d d
d d
= = =
= = =
Standardized Random VariableMean = 0, standard deviation =1
Standardized Random VariableStandardized Random VariableMean = 0, standard deviation =1Mean = 0, standard deviation =1
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Mean 0, standard deviation 1Mean 0, standard deviation 1,
2
50
log
log log 1( ) exp
22
( ) ( ) ( ) 1 ( )
d
z
d d zz PDF f z
CDF F z f z dz where F z F z
= =
= =
F(z) for Standard Normal Random Variable zF(z) for Standard Normal Random Variable z
z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.53590.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7703 0.7734 0.7764 0.7793 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8906 0.8925 0.8943 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7703 0.7734 0.7764 0.7793 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8906 0.8925 0.8943 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
Example - Sand Bed MaterialExampleExample -- Sand Bed MaterialSand Bed Material
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Log-Probability Plot of Sand Bed DataLogLog--Probability Plot of Sand Bed DataProbability Plot of Sand Bed Data
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d(mm)
Cumulative Distribution Function Expressed as a probability (%)
d84=0.363 mm
d50=0.232 mm
d16=0.158 mm
( ) ( )log 50
84.1log
15.9
0.181
0.386 log 0.386 0.181 log 0.232 .565
65
0.363log log 0.181
0.158
10 1.52
10 10 10 0.272d
d
g
d
d
d
d mm
+ +
= = =
= =
= = = =
Log Probability Plot using NORMSINVLog Probability Plot using NORMSINVLog Probability Plot using NORMSINV
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Function in EXCELFunction in EXCELFunction in EXCEL
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
-4 -3 -2 -1 0 1 2 3
log10
(d)
d15.9
NORMSINV(P/100)
d50 d84.1
log(d84)log(d84))
log(d50)log(d50))
log(d16)log(d16))
NORMSINV and NORMSDIST
EXCEL Functions
NORMSINV and NORMSDISTNORMSINV and NORMSDIST
EXCEL FunctionsEXCEL Functions
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d50= 0.48 mm log10(d50)= -0.3188
g= 1.28 mm log10(g)= 0.1072
P (% finer) F(z) z=NORMSINV(F) di (mm) F(z)=NORMSDIST(z)
1 2 3 4 5 6
20 0.2 -0.8416 d20= 0.390 0.2
40 0.4 -0.2533 d40= 0.451 0.4
60 0.6 0.2533 d60= 0.511 0.6
80 0.8 0.8416 d80= 0.591 0.8
99.99 0.9999 3.7190 d99.99= 1.202 0.9999
50 0.5 0.0000 d50= 0.480 0.5
di (mm) z F(z)=NORMSDIST(z) P (% finer)
1 2 3 4
0.41 -0.6385 0.2616 26.2
0.63 1.1016 0.8647 86.5
50
50
( ) ( )
( ) ( )
( )
10 i g
ii
g
log d z log
i
log d log d z
log
d
+
=
=
50( ) ( )
( )i
i
g
log d log d z
log
=
d50= 0.48 mm log10(d50)= -0.3188
g= 1.28 mm log10(g)= 0.1072
P (% finer) F(z) z=NORMSINV(F) di (mm) F(z)=NORMSDIST(z)
1 2 3 4 5 6
20 0.2 -0.8416 d20= 0.390 0.2
40 0.4 -0.2533 d40= 0.451 0.4
60 0.6 0.2533 d60= 0.511 0.6
80 0.8 0.8416 d80= 0.591 0.8
99.99 0.9999 3.7190 d99.99= 1.202 0.999950 0.5 0.0000 d50= 0.480 0.5
di (mm) z F(z)=NORMSDIST(z) P (% finer)
1 2 3 4
0.41 -0.6385 0.2616 26.2
0.63 1.1016 0.8647 86.5
50
50
( ) ( )
( ) ( )
( )
10 i g
ii
g
log d z log
i
log d log d z
log
d
+
=
=
50( ) ( )( )
ii
g
log d log d zlog
=
Geometric Standard DeviationGeometric Standard DeviationGeometric Standard Deviation
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( ) ( )
( ) ( ) ( ) ( )( )
( ) ( )
( )
84 50 50 16
84 50 50 16 84 50 16 50
84 16 84 16 84 16
84 16
84 50 84 50
2log log log log log
2log log / log / log /
12log log / log log / log /2
/
log log log /
log
top top
d d d d
d d d d d d d d
d d d d d d
d d
Alternative Method in HEC RAS Manuald d d d
= +
= + =
= = =
=
= =
( )
( )
50 16 50 16
84 50
50 16
log log /
0.5 0.5
bot bot
ave top bot
d d d d
d d
d d
= =
= = + = +
Geometric Standard Deviation when
P for smallest sample d is greater than 0.16
Geometric Standard Deviation whenGeometric Standard Deviation when
PP for smallest sample d is greater than 0.16for smallest sample d is greater than 0.16
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Pifor smallest sample d is greater than 0.16PP
iifor smallest sample d is greater than 0.16for smallest sample d is greater than 0.16
( )
( )
1
1 1
1
1
84
84 1
(1 )84 84
1
(1 )
84
(1 )log log log
loglog loglog log
(1 ) (1 )
P
zP P
P
z
P
z d d
d
d d d d
z z d
dd
=
= = =
=
Let Pi = the lowest percent finer from the pebble
count analysis and let zi = the standardized
normal variable that gives F(z) = Pi/100.
Let Pi = the lowest percent finer from the pebble
count analysis and let zi = the standardized
normal variable that gives F(z) = Pi/100.
Example for Smallest d > d16Example for Smallest d > dExample for Smallest d > d1616
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( )
1
84 32
1
11 1
(1 )(1 0.469 ) 1.469
84
8450 84
8450
8 2
32 ( ) 0.32 ( ) 1 ( ) 1 0.32 0.68
0.469 0.469
5.7 5.72.04
2 2
log log log 2.04 log2.04
2.
z
P
Given d mm and d mm
P F z F z F z
z z
d
d
dd d
dd
= =
= = = = =
= =
= = = =
= =
= 5.7 2.7904 2.04
mm = =
Pebble Count for Gravel/Cobble StreamPebble Count for Gravel/Cobble StreamPebble Count for Gravel/Cobble Stream
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Sand Gravel Cobble
Theoretical Justification for Pebble
Count
Theoretical Justification for PebbleTheoretical Justification for Pebble
CountCount
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0
, , ,
/ ; ; ( )
(1 )
/
(1 )
L
pores total pores pores pores
s
i
i i s
i s i i ii
s total s s total s total total
i
porosity V V A nA V A x dx
A n A area of A occupied by soil
A area of A occupied by particles of a specified size range
f A A
W V V V p
W V V n V
p
= = =
= =
=
=
= = = =
( ) [ ]( )
[ ]( )
0 0 0 0
0 0
(1 )
(1 ) (1 ) (1 ) (1 )
(1 ) (1 )(1 )
(1 ) (1 ) (1 )
L L L L
i i i s i
total
L L
i i
i
i
i i
A dx A dx f A dx p n A dx
n V n AL n AL n AL
f n A dx f n Adxf n AL
p n AL n AL n AL
p f
= = = =
= = =
=
L
SOIL PARTICLES
PORES
A
Actual Pebble Count Shielding, settling, etc.Actual Pebble Count Shielding, settling, etc.
Mixed Sand and Gravel BedsWatershed Institute, Inc. Pebble Count Data
Mixed Sand and Gravel BedsMixed Sand and Gravel BedsWatershed Institute, Inc. Pebble Count DataWatershed Institute, Inc. Pebble Count Data
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MT043442RR01
D16 D50 D84
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
normsinv(P/100)
log10(D)
Pebble Count Data D16 D50 D84 Extrapolated
D (mm)
D16 0.0763
D50 0.344
D84 14.8
d (mm)
d16 0.0763
d50 0.344
d84 14.8
d16 d50 d84MT043442RR01
D16 D50 D84
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
normsinv(P/100)
log10(D)
Pebble Count Data D16 D50 D84 Extrapolated
D (mm)
D16 0.0763D50 0.344
D84 14.8
d (mm)
d16 0.0763d50 0.344
d84 14.8
d16 d50 d84
SN321115RR
D16 D50 D84
-3
-2
-1
0
1
2
3
-1.5 -1 -0.5 0 0.5 1 1.5 2
normsinv(P/100)
log10(D)
Pebble Count Data D16 D50 D84 Extrapolated
D (mm)
D16 0.00224
D50 0.500D84 20.6
d (mm)
d16 0.00224
d50 0.5d84 20.6
d16 d50 d84SN321115RR
D16 D50 D84
-3
-2
-1
0
1
2
3
-1.5 -1 -0.5 0 0.5 1 1.5 2
normsinv(P/100)
log10(D)
Pebble Count Data D16 D50 D84 Extrapolated
D (mm)
D16 0.00224
D50 0.500
D84 20.6
d (mm)
d16 0.00224
d50 0.5
d84 20.6
d16 d50 d84
Mixed Sand and Gravel Beds (cont.)Mixed Sand and Gravel Beds (cont.)Mixed Sand and Gravel Beds (cont.)Two Bed Materials
60
Two Bed Materials
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0
10
20
30
40
50
60
-1.5 -1 -0.5 0 0.5 1 1.5 2
log D
Wfiner(gm)
Gravel Sand
Sand Gravel
D50 0.4 10
g 1.5 2
W (gm) 40 50
Sand Gravel
d50 0.4 10
g 1.5 2
W (gm) 40 50
log10(d)
0
10
20
30
40
50
60
-1.5 -1 -0.5 0 0.5 1 1.5 2
log D
Wfiner(gm)
Gravel Sand
Sand Gravel
D50 0.4 10
g 1.5 2
W (gm) 40 50
Sand Gravel
d50 0.4 10
g 1.5 2
W (gm) 40 50
log10(d)
Two Bed Materials
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-10 -5 0 5 10norminv(P/100)
logD
Sand Gravel
Sand Gravel
D50 0.4 10
g 1.5 2
W (gm) 40 50
Sand Gravel
d50 0.4 10
g 1.5 2
W (gm) 40 50
log10
(d)
NORMINV(P/100)
Two Bed Materials
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-10 -5 0 5 10norminv(P/100)
logD
Sand Gravel
Sand Gravel
D50 0.4 10
g 1.5 2
W (gm) 40 50
Sand Gravel
d50 0.4 10
g 1.5 2
W (gm) 40 50
log10
(d)
NORMINV(P/100)
Mixed Sand and Gravel Beds (cont.)Mixed Sand and Gravel Beds (cont.)Mixed Sand and Gravel Beds (cont.)Combined
100
Combined
100
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0
10
20
30
40
50
60
70
80
90
-1.5 -1 -0.5 0 0.5 1 1.5 2
log D
WFiner(gm)
Sand GravelD50 0.4 10
g 1.5 2
W (gm) 40 50
log10(d)
Sand Gravel
d50 0.4 10
g 1.5 2
W (gm) 40 50
0
10
20
30
40
50
60
70
80
90
-1.5 -1 -0.5 0 0.5 1 1.5 2
log D
WFiner(gm)
Sand Gravel
D50 0.4 10
g 1.5 2
W (gm) 40 50
log10(d)
Sand Gravel
d50 0.4 10
g 1.5 2
W (gm) 40 50
Combined
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-8 -6 -4 -2 0 2 4
norminv(P/100)
lo
g
D
log D50 D50(mm)
0.537 3.44
Sand Gravel
D50 0.4 10
g 1.5 2
W (gm) 40 50
Sand Gravel
d50 0.4 10
g 1.5 2
W (gm) 40 50
log
10
(d)
log10(d50) d50 (mm)
0.537 3.44
Combined
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-8 -6 -4 -2 0 2 4
norminv(P/100)
log
D
log D50 D50(mm)
0.537 3.44
Sand Gravel
D50 0.4 10
g 1.5 2
W (gm) 40 50
Sand Gravel
d50 0.4 10
g 1.5 2
W (gm) 40 50
log10
(d)
log10(d50) d50 (mm)
0.537 3.44