Chute Guidelines 100

8/3/2019 Chute Guidelines 100

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chuteGuidelines for the Design

of Rock Chutes using

CHUTE

Prepared by

Associate Professor R. J. KellerCRC for Catchment Hydrology

Version 1.0.0

September 2003

GUID

ELINES

CRC for Catchment Hydrology 2003. Except as permit ted under the Australian Copyright Act 1968, these

Design Guidelines may not be copied or distributed by itself except with express written permission of the

Director of the CRC for Catchment Hydrology

www.toolkit.net.au/chute


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Document History

Date Author Revision Description of Change Aug 2003 R.J. Keller 1.0.0b Creation

01 Sep 2003 Nick Murray 1.0.0Conversion to Toolkit Template. This manual

applies to CHUTE version 1.0.0

Copyright notice

CSIRO Australia 2003

Legal Information

To the extend permitted by law, CSIRO (including its employees and consultants) accepts no responsibilityand excludes all liability whatsoever in respect of any person's use or reliance on this publication or anypart of it.


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TOC

Acknowledgements

These Design Guidelines, in part, reproduce material originally prepared by Ian Drummondand Associates (now Earth Tech Pty Ltd). The permission of Dr. John Tilleard to freely use thismaterial is appreciated.


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TOC

CHUTECONTENTS

1 Introduction ..................................................................... 1

1.1 The User Guide........................................................................................2

1.1.1 Purpose.......................................................................................2

1.1.2 Structure .....................................................................................2

2 Installation....................................................................... 3

2.1 Technical Specifications ............................................................................3

2.2 Installation...............................................................................................3

2.3 Uninstalling CHUTE..................................................................................4

3 Hydraulic Design Elements ............................................... 5

3.1 Explanation of Terms................................................................................5

4 Design Program ............................................................... 8

4.1 Inputs ......................................................................................................8

4.1.1 Rating Table................................................................................9

4.1.2 Normal Depth. ............................................................................9

4.1.3 ycrit+10% .................................................................................104.1.4 Average Normal, ycrit................................................................10

4.2 Outputs .................................................................................................10

4.3 Rock Details and Factor of Safety ............................................................12

4.3.1 Angle of Repose ........................................................................12

4.3.2 Specific Gravity of Rock..............................................................12

4.3.3 Factor of Safety .........................................................................13

5 Other Design Details...................................................... 14

5.1 Rock quality, Grading, and Thickness ......................................................145.2 Incorporation of a Fixed Crest.................................................................16

5.3 Filters and Hydraulic Cutoffs ...................................................................16

5.4 Treatment of Abutments .........................................................................17

Appendix A Theory for the Hydraulic Design of Rock Chutes.... 19

Appendix A.1 Theory ........................................................................................19

Appendix A.2 Program Flow Details ..................................................................23

Appendix A.3 References ..................................................................................27


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TOC

ii

TABLE OF FIGURES

Figure 1: Schematic of a Typical Chute with Explanation of Terms ...............................................5Figure 2: Alternate Flow Profiles over Chute and Location of Minimum Depth...............................6

Figure 3: Water Surface Profile with Hydraulic Jump on Chute ....................................................7

Figure 4: CHUTE Input Table...................................................................................................9

Figure 5: Spreadsheet Results Output................................................................................... 11

Figure 6 : Graphical Output from CHUTE...............................................................................12

Figure 7: Typical Rock Chute Plan and Sections .......................................................................15

Figure 8: Alternative Arrangements for Filters and Cutoffs .........................................................17

Figure 9: Alternate Flow Profiles on Rock Chute and Location of Minimum Depth .......................21

Figure 10: Water Surface Profile with Hydraulic Jump on Chute ................................................22

Figure 11: Flowchart for section of program that calculates rock size .........................................25

Figure 12: Adjacent Sections in Computation of Water Surface Profile .......................................27


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Introduction

1

1 IntroductionA rock chute is a relatively short and steep section of the bed of a channel that has beenarmoured with rock. Its normal function is to either:

Stabilise an erosion head and prevent it from moving upstream in the channel.

Reduce the overall grade of a channel by providing a weir within the channel bed.

A rock chute offers an alternative to other forms of drop structure such as sheet piling orreinforced concrete weirs.

Although the concept of a rock chute is simple, proper hydraulic design is very important toensure that the chute geometry and rock size are matched with the expected flow conditionssuch that the rock remains stable under all expected flow conditions. In addition, appropriaterock chute design requires that a number of other issues are adequately addressed. Inparticular:

Chutes should be located where they can serve their function most efficiently andeffectively

The abutments must be treated to prevent failure by outflanking of the crest

The grading of rock sizes within the rock mixture must minimise the presence of voidsand minimise the area of individual rocks exposed to forces from the flow

Where the underlying material is largely non-cohesive or where high ground-waterlevels or seepage occur, consideration should be given to the use of filter layers.

The program CHUTE has been developed as a design tool in the hydraulic design of rockchutes and these design guidelines are primarily concerned with this aspect. It is emphasised

very strongly, however, that this is only one aspect of the correct application and design ofthese structures. Indeed many chutes fail for reasons other than inadequacies in their specifichydraulic design. Such failures are often related to poor understanding of the prevailing siteconditions such as hydrology, overall stream morphology, floodplain and channel hydraulics,and foundation conditions.

Note CHUTE is written as an EXCEL spreadsheet-based program. Theinternal calculations are run by macros. The macro security level onyour computer must be set to Medium or Low in order for CHUTE torun properly.


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CHUTE Design Guidelines

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1.1 The User Guide

1.1.1 Purpose

These guidelines first address the program CHUTE and its proper use to design and analysethe performance of a chute under a range of flow conditions. It is clearly not possible toprovide a complete treatise on the other issues of importance. However, design notes onthese other issues are provided for guidance only and not as a set of prescribed rules. In allcases it is important to access local knowledge and experience with other chutes on the samestream or under similar conditions.

1.1.2 Structure

These guidelines describe:

hydraulic design elements

the design program CHUTE

other relevant design details

In the appendix, the hydraulic theory underpinning the program CHUTE is presented.


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Installation

3

2 Installation

2.1 Technical Specifications

CHUTE requires the following system configuration.

Type of machine Intel based PC with CD-ROM driveMinimum CPU Pentium III, 400 MHz. Realistically, you should use a Pentium IV 1.6 GHz or

faster for production-level modelling

Minimum memory 128 Mb

Minimum available

disk space

10 MB for the spreadsheet, data may require more space

Operating system Windows 2000 + Service Pack 4, orWindows XP + Service Pack 1, orWindows NT4 + Service Pack 6a (Note that CHUTE has not been extensivelytested on Windows NT4)

Windows 95, 98 and ME are not supported.

Other software Microsoft Excel 2000 or 2002/XP, with all service packs applied

2.2 Installation

To install CHUTE:

1 Double-click the setup.exe file on the CD. The CHUTE Setup Wizard appears.2 Click Next. The Licence Agreement window appears. To confirm that you agree with the

CHUTE licence conditions, click the I agree button. If you do not agree to the licenceconditions, you cannot install CHUTE.

Click Next to continue the installation. The Select Installation Folder window appears.

3 Leave the installation folder at the default setting.4 Click Next. The Confirm Installation window appears.5 Click Next. CHUTE is installed onto your computer, and the Installation Complete

window appears.6 Click Close to exit the installer.


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2.3 Uninstalling CHUTE

To uninstall CHUTE, from Windows:

1 Select Start|Settings|Control Panel|Add/Remove Programs. The Add or Removeprograms dialog appears.

2 Find CHUTE in the list of installed programs3 Click on the name CHUTE, then click the Remove button4 Windows asks you to confirm that you want to remove CHUTE.5 To remove CHUTE, click Yes. Otherwise, click No.6 Close the Add/Remove programs dialog.


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Hydraulic Design Elements

5

3 Hydraulic Design

ElementsA typical rock chute is shown schematically in Figure 1 together with an illustration of thevarious terms used in describing and designing a rock chute. From a hydraulic point of view,the primary elements are the chute face and the apron, since these provide protection to thebed from the erosive forces of the water.

Jump location

x=0

q

Chute

drop Downstream depth

chute slope

Downstream bed

1 apron rise

Chute length Apron length

Figure 1: Schematic of a Typical Chute with Explanation of Terms

3.1 Explanation of Terms

The primary design output is the rock size required to ensure a stable structure. Because therock size is dependent on the bed shear stress, which, in turn, is dependent on the flowprofile over the chute, a key element in the design process is the determination of the watersurface profile.

Three types of profile may occur and are depicted in Figure 2.

It is shown inAppendix Athat the location of the point of maximum bed shear stress

coincides with the location of the minimum depth. Clearly this location is dependent on thetype of profile over the chute.


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The profile shown in Figure 2 (c) is most likely to be associated with the maximum designflow in the river. With lower controlling tail water levels, the profiles shown in Figure 2 (a)and Figure 2(b) are associated with flow rates less than the maximum. Because of the lower

minimum depths associated with the latter profiles, it is likely that the bed shear stresses willbe larger with a consequent increase in the rock size required for stability.

This discussion demonstrates that the chute design flow rate that for which the requiredrock size is a maximum will be lower than the channel design flow rate that for which therequired channel capacity is a maximum. This is a most important distinction in the design ofa rock chute. A particular feature of the program CHUTE is that it examines the entire rangeof flows and explicitly determines the flow within that range for which the required rock sizefor stability is a maximum.

Figure 2: Alternate Flow Profiles over Chute and Location of Minimum Depth

It is clear also from this discussion that determination of the critical rock size requires thedetermination of the water surface profile throughout the chute for a given flow rate and theconsequent determination of the minimum value of depth. The normal design situation isshown in Figure 3 and corresponds to the case shown in Figure 2 (a).

For the case shown in Figure 3, two depths control the combined flow profile critical depthat the chute crest and a known downstream subcritical depth. The downstream depth isdependent on controls further downstream and would normally be established from a one-dimensional program such as HEC-RAS.

The S2 profile downstream of the chute crest and the A1 or M1 profile over and upstream of

the chute apron are computed using the standard step method, described in (eg) (Henderson1966). The location of the hydraulic jump is then determined as the point where the


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Hydraulic Design Elements

7

upstream and downstream depths form a conjugate pair ie the Momentum Function is thesame.

For the case shown in Figure 2 (c), the subcritical profile, controlled by the downstreamdepth, is everywhere higher than the conjugate profile, computed from the assumedupstream supercritical profile. In this case, the chute is said to be drowned.

1

y

So

S1

S2

downstreamA or M1 1

yc

EnergygradelineConjugatedepthline

q

Figure 3: Water Surface Profile with Hydraulic Jump on Chute

The discussion presented in this section is expanded with the full mathematical equations andsolution procedure inAppendix A.


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4 Design ProgramThe design program is set up as a Microsoft Excel workbook, with separate sheets devoted tosite inputs and the downstream rating table and a number of other sheets for presentingaspects of the results. Figure 4 shows the Input worksheet and the various inputs in the toptable are summarised in the following sub-section.

4.1 Inputs

1 Structure Variables. With reference to the sketch in Figure 1, the following variablesdescribing the dimensions of the proposed structure are entered into the input table:

Chute Drop, Chute Length. The vertical and horizontal distance respectivelybetween the upstream lip of the chute and the lowest point

Apron Rise, Apron Length. The vertical rise and horizontal length respectivelyof the downstream apron section

Chute Width. This is used by the program, in conjunction with the flow data, todetermine the range of unit flow rates on the chute.

2 Flow Rate. The nominal minimum and the maximum design flow rate specifying therange over which the program will calculate the D50 required.

3 Rock Variables. The angle of repose and the specific gravity of the proposed rock thatwill be used in the chute. Typical values are presented in Section 4.3.

4 Factor of Safety. This is a critical input that reflects the degree of conservatism built intothe design. Further comments on this factor are presented subsequently.


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Design Program

9

Figure 4: CHUTE Input Table

In the bottom table, the user is required to nominate the computation procedure for thedownstream boundary condition. The options currently available and comments on each are

presented in the following.

4.1.1 Rating Table

This option is the most reliable and exact, but requires the use of an external backwaterprogram, such as HEC-RAS, to determine the tail water levels over a range of flow rates.

Although the other options may be used for initial trials, wherever possible, a rating tableshould be used as the final design.

If this option is chosen, pairs of flow rate-depth values are entered into the worksheet markedRating Table. These pairs may be entered manually or inserted using Copy and Pastecommands. The zero datum for depth (h) is the bed level at the downstream end of the chute

apron.Following entry of these data, the program automatically converts them to a table of unit flowrate-depth values. An interpolated rating table is then automatically calculated matching therange of flow rates that will be tested. Ten flow steps are used. However, a smaller or greaternumber of steps can be accommodated by changing the layout of the table. The programcounts the number of occupied cells in this table before starting.

4.1.2 Normal Depth

With this option, the downstream boundary condition for chute design is the normal (oruniform) depth which is calculated from values of downstream channel depth, slope andchannel width keyed into the Downstream Channel Input Table. Note that these values are

not required for the Rating Table option or for the ycrit+10% option.


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This option normally produces a more conservative chute design than the Rating Tableoption and should be used where a downstream backwater profile is not available or cannotbe computed and where the downstream channel can be approximated as a rectangular

wide channel with a known slope and Mannings roughness.

4.1.3 ycrit+10%

With this option, the program assumes the downstream boundary depth is 10% greater thancritical depth. This is the most conservative of the options. It is valid for preliminary design,but should only be used for detailed design where downstream channel information iscompletely lacking.

4.1.4 Average Normal, ycrit

With this option, the program calculates the average of normal depth and critical depth. It isintended only as an additional option available to the designer for investigating designsensitivity to downstream conditions.

When all required inputs have been satisfactorily entered into the input sheet, the user simplyclicks the Run button. This action initiates the running of a macro that takes the requireddata from the input sheet and the rating table sheet and performs the necessarycomputations, producing a range of outputs.

4.2 Outputs

On completing the calculations, the program automatically switches to the Resultsspreadsheet. This sheet contains two tables the first and part of the second are presented inFigure 5.

The first table contains summary information for each of the flow rates within the specifiedflow range. The five blocks of information contain:

1 Flow Rate.2 Required rock size and side slope, representing the required D50 assuming normal

(uniform) flow, calculated D50 with full profile calculations, and required chute bankangle for stability.

3 Downstream boundary depths, representing that used by the program and thatspecified by the rating table. The critical depth is also calculated and reported.

4 Specific energy results, representing the specific energy at the spillway crest (u/s) andat the toe of the apron (d/s). The column labelled extra has meaning only for aScenario 3 profile which is the case where the hydraulic jump is swept downstream withsupercritical flow throughout the structure. In this case, the number reported in thiscolumn is the additional specific energy required at the toe of the apron for a hydraulicjump to form.

5 Jump conditions. In the first column of this block is the scenario number whichrepresents the type of flow profile as follows:

Scenario 1: Jump contained within the structure - (a) and (b) of Figure 2.

Scenario 2: Chute drowned out (c) ofFigure 2.

Scenario 3: Jump swept downstream.

The second column is a scenario description. In the next three columns, the location of thehydraulic jump from the chute crest, the depth immediately upstream of the jump, and the


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Design Program

11

depth immediately downstream of the jump respectively are listed. The final two columns listthe energy loss across the hydraulic jump and the friction loss down the chute. The sum ofthe last two columns represents the total energy loss within the chute.

Details of the flow profile at the flow rate for the maximum required rock size are presentedin the second table. Figure 5 lists only the first few lines of this table. Listed in this table arethe variations with distance of bed elevation, bed slope, water surface elevation, total energyline, friction slope, velocity, and Froude Number.

Figure 5: Spreadsheet Results Output

Two final spreadsheets, labelled Profile and Rock Size, provide graphical output. InProfile, the total energy line, water surface, velocity, bed elevation, and friction slope areplotted as a function of distance from the chute crest.

The Rock Size chart provides the variation of required rock size with flow rate. Thedownstream boundary depth is also graphed. These graphs are illustrated in Figure 6.


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(a) Profile Output

(b) Rock Size Output

Figure 6 : Graphical Output from CHUTE

4.3 Rock Details and Factor of Safety

4.3.1 Angle of Repose

For angular rock larger than 100mm, the natural angle of repose of the rock is 41-420. Thiscovers the vast majority of cases. For rounded and other types of rock a good summary ofangles of repose is presented by Simons and Senturk (1977).

4.3.2 Specific Gravity of Rock

The specific gravity is defined as the density of rock relative to the density of water. Table 1gives a guide to typical rock densities.


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Design Program

13

Rock Type Specific GravitySandstone 2.1 2.4

Granite 2.5 3.1 (typically 2.65)

Limestone 2.6

Basalt 2.7 3.2

Table 1: Typical Specific Gravity of Rock Types

4.3.3 Factor of Safety

This parameter relies on the judgement and experience of the designer. Factors to beconsidered include the following:

The consequences of failure

The reliability of estimates of input parameters

The quality, consistency, and grading of available rock

The likely standard of construction

The construction details proposed for the chute (see Section 5)

The return period and likely duration of the design flood

The likelihood of the chute being stabilised by vegetation within a time period muchless than the return period of the flood

The likelihood of changes in tailwater conditions due, for example, to siltation fromother downstream chutes or vegetation

The most important of these factors is the consequence of failure in terms of subsequenterosion, loss of habitat, and potential threat to other assets such as bridges.

As a guide, a factor of safety of 1.3 or greater is appropriate for any major structure or to astructure where failure would threaten an asset or cause major loss.

Lower factors of safety of 1.0 to 1.3 may be applicable to chutes where the objective isgeneral erosion control and where channel stability is to be supplemented by vegetation.Lower factors of safety may also be considered where the risk of catastrophic failure isminimal or where active maintenance can be assured.

These low factors of safety reflect the generally conservative assumptions built into the rocksizing procedures. The recommended values should lead to design conditions in which thereis no significant rock movement under the critical flow condition.


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5 Other Design DetailsTypical plan and sections of a rock chute are presented in Figure 7.

Design considerations include a number of factors such as:

Specification for rock quality and grading and thickness of layer

Possible incorporation of a fixed crest within the rock structure

Details of filters and hydraulic cutoffs

Treatment of abutments

Each of these is considered in the following sections.

5.1 Rock quality, Grading, and Thickness

Rock should be hard, tough, and durable. It should have a crushing strength of at least25Mpa. The rock should be free of defined cleavage planes and should not be adverselyaffected by repeated wetting and drying.

The rock should be predominantly angular in shape with not more than 25% of rocks,distributed through the gradation, having a length more than twice the breadth or thickness.No rock should have a length exceeding 2.5 times its breadth or thickness.

Where rock fails to meet this specification, it may still be considered at the designersdiscretion, provided allowance is made in the design for its shortcomings.

Rock to meet the necessary size and strength criteria will normally be won from a hard rockquarry by drilling and blasting. A hydraulic rock breaker mounted on a hydraulic excavatorprovides an excellent means of producing rock to design size specifications.

Rock should not be single sized, but, instead, should be a well-graded mixture designed toensure that all interstices between large rocks are filled with rock of progressively smaller size.This has the effect of ensuring that no significant voids occur in the rock blanket throughwhich underlying material can be washed out. Additionally, it helps to create an interlockingmass of rock which is highly stable.


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Other Design Details

15

Rock chute

Filter layer

Bed

Rock beaching on batter

ELEVATION

3000

min

3000

min 1000 Typ.

Crest R.L.

1500 Typ.

1000 Typ.

Bed

Minimum 300mmabove design

water line

3000

min

Excavate and place rock to

form a key in bank

Fold of geotextile to form

a barrier to fine material

Transition the cross section of the

creek upstream and downstream

to match the cross section of the

structure as shown

Extend rock beaching on bank

upstream and downstream of

Where required place and compact

chute by three metres minimum

excavated material to form analignment bank.

PLAN

A B C

Additional rock and

SECTION 'A'

1000 Typ.

Rock armour

Filter layer

geotextile to formkey at abutment

SECTION 'B'

Trench continues into bank

SECTION 'C'

Figure 7: Typical Rock Chute Plan and Sections

Experience suggests a rock gradation such as that summarised in Table 2. When specifyingrock gradation to field staff and contractors, it is helpful to transform this grading by weight

into an equivalent grading by number.


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Equivalent Spherical Diameter Percent by Weight of Rock of Smaller Size1.5 2.0 D50 100%

D50 50%

0.3 0.4 D50 10 20%

Table 2: Suggested Rock Gradation

5.2 Incorporation of a Fixed Crest

One of the most vulnerable parts of a rock chute is its crest. Dislodging of a rock from thecrest, whether by flow forces, debris impact, or some other means, results in a concentrationof flow at that point which can destabilise more rock. The subsequent rill can lead to failureof the chute.

An effective means of guarding against such failure is to construct a solid wall through thechute along the line of the crest. Options for such a wall include reinforced concrete, steelsheet piling, and a wall constructed from piles and timber boards.

Inclusion of a fixed crest is recommended for major structures and for structures requiring ahigh factor of safety.

5.3 Filters and Hydraulic Cutoffs

General guidelines for filters and cutoffs for rock chutes, based on experience, are presented

in the following. Some form of cutoff is essential to reduce the risk of piping failure beneath the

structure or through the abutments. The importance of the cutoff increases with thepermeability of the parent material, decreasing cohesion of the parent material, andthe height and steepness of the chute.

The cutoff may be an impermeable barrier, such as concrete, sheet piling, timber, ormembrane. Alternatively, geotextile can be considered to prevent the passage ofmaterial.

The need for a filter layer between the rock forming the chute and the parent materialis also influenced by the above factors. The filter layer prevents bed material beingwashed through residual interstices in the rock layer.

Normally, a filter layer is only necessary where the underlying material is largely non-cohesive such as uniform sand or silt, high groundwater levels create large porepressures, or an unusually high factor of safety is required. If one or more of theseconditions prevails, the need for a filter layer can be further assessed from thefollowing required criteria:

For stability:

5material)bed(D

rock)chute(

85

15 D

and

25material)bed(D

rock)chute(

50

50

D


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17

For permeability:

5material)bed(D

rock)chute(

15

15 D

If the relationship between bed material grading and chute rock grading is outsidethese limits, the need for a filter layer becomes more paramount.

A granular filter layer can be designed by applying the above relationships twice once between the bed material and the filter layer and once between the filter layerand the chute rock.

If geotextile is to be used as a filter layer, special rules apply and specialist texts ongeotextile design should be consulted.

Various arrangements for providing filters and hydraulic cutoffs are illustrated in Figure 8.Selection of an appropriate arrangement depends on site conditions and the size of the

structure.For minor structures, where no fixed crest is incorporated, it is recommended that a fold ofgeotextile or membrane be brought up through the rock along the line of the crest to therock surface. This provides a barrier to the passage of fine material through the rock andincreases upstream siltation rates.

5.4 Treatment of Abutments

A high proportion of chute failures occur at the abutments. Where a fixed crest is provided bya pile and board wall, or a concrete cutoff, it must be excavated into the abutment andbackfilled and compacted with selected clay fill.

Where no fixed crest is provided, geotextile should extend into a key excavated in the bankand backfilled with rock, as shown in Figure 8.

Rock riprap bank protection should extend upstream and downstream of the chute to providedirect abutment protection.

(a)

Rock

Figure 8: Alternative Arrangements for Filters and Cutoffs


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(bi)

Rock

Geotextile

(bii)

Rock

GeotextileExtent of geotextile beneathchute depends on site conditions

(c)

Rock

GeotextileExtent of geotextile beneathchute depends on site conditions

Figure 8 (cont): Alternative Arrangements for Filters and Cutoffs


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19

Appendix A

Theory for theHydraulic Design ofRock Chutes

Appendix A.1 Theory

The determination of the critical shear stress at which bed particles will start to move on a flatbed is given by:

( )constant

150=

s

c

SD

(1)

where

c is the critical shear stress

is the specific weight of water

D50 is the particle size for which 50% of the sample is finer

Ss is the relative density of the bed material

Equation (1) is correct provided the particle Reynolds Number is above a certain value,typically taken to be about 400. This corresponds to a particle size of about 6mm(Henderson 1966). In all practical chute designs, the rock size will always be larger than thisvalue, so the use of the equation is justified.

The magnitude of the constant is the subject of some difference of opinion in the literature.The classic work of Shields (Henderson 1966) produced a value of 0.056, although otherstudies have yielded lower values. For the present study, a more conservative value of 0.047is chosen, consistent with the work of (Meyer-Peter and Muller 1948) and (Yalin and Karahan1979).


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Noting that, on the bed of a trapezoidal channel, the shear stress may be expressed as

Sy97.0 = (Chow 1959), wherey is the depth and S is the local energy slope, andintroducing a factor of safety, FS,Equation (1) may be transposed to:

( )1047.097.0

50

=s

S

S

ySFD (2)

Now, flow over a chute may be considered as being hydraulically wide, leading to theexpression of Mannings Equation as:

21

351Sy

nvyq == (3)

where n is Mannings roughness parameter

q is the flow rate per unit width

Combining Equations (2) and (3) yields:

( ) 37

22

50

1047.0

97.0

yS

nqFD

s

S

= (4)

Equation (4) demonstrates that, for a given flow rate per unit width, particle density, andMannings roughness parameter, the minimum rock size required for stability is stronglyinversely proportional to the local depth,y. It is evident that the critical design condition, iethe condition requiring the largest stone size for stability, will be that for which the depth is aminimum.

There are three possible types of flow profile that may occur on a rock chute. These areillustrated in Figure 9.


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Figure 9: Alternate Flow Profiles on Rock Chute and Location of Minimum Depth

In Figure 9, for each type of profile, the location of the point of minimum depth is shown byan asterisk.

The circumstances under which each profile will occur will vary according to site conditions.However, in general, profile types a) and b) are associated with low and mid-range flows andtype c) is associated with large flows.

It is evident that solution of Equation (4) requires the determination of the water surfaceprofile throughout the chute for a given flow rate and the consequent determination of the

minimum value of depth. In most cases, the critical design situation, requiring the largeststone size for stability, is shown in Figure 10 and corresponds to the case shown in Figure9(a).


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1

y

So

S1

S2

downstreamA or M1 1

yc

EnergygradelineConjugatedepthline

q

Figure 10: Water Surface Profile with Hydraulic Jump on Chute

For the case shown in Figure 10, two depths control the combined flow profile criticaldepth at the chute crest and a known downstream subcritical depth. Determination of critical

depth at the chute crest is developed from the standard equation for critical depth in arectangular channel (Henderson 1966):

3

2

g

qyc = (5)

The downstream depth is dependent on controls further downstream. Program CHUTEcurrently incorporates four options for determining this depth and these are discussed in theGuidelines. The most reliable and exact is the use of a rating table which would normally beestablished from a one-dimensional program such as HEC-RAS.

The S2 profile downstream of the chute crest and the A1 or M1 profile over and upstream ofthe chute apron are computed using the standard step method, described in (eg) (Henderson

1966). The location of the hydraulic jump is then determined as the point where theupstream and downstream depths form a conjugate pair ie the Momentum Function is thesame.

For the case shown in Figure 9(c), the subcritical profile, controlled by the downstream depth,is everywhere higher than the conjugate profile, computed from the assumed upstreamsupercritical profile. In this case, the chute is said to be drowned.

Although the computational details for the water surface profile are standard, it needs to berecognised that the computed profiles are dependent on Mannings roughness coefficient,which, in turn, is linked to the unknown rock size through the Strickler Equation (Henderson1966):

61

50041.0 Dn = (6)

Accordingly, an iterative solution procedure is required.

The final stage of the hydraulic design of the chute is the determination of the chute bankangle.

The classic equation linking the shear stress on a particle, at the inception of motion, on the

channel bank ( 0 ) to the critical shear stress for the same particle on the bed of the channel

( c ) is (Carter 1953, cited in Henderson 1966):

2

2

0

tan

tan1cos =

c

(7)

where is the bank angle


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23

is the natural angle of repose of the bank material

Equation (7) was obtained by equating the disturbing forces on the particle to the restoring

forces. In the present study, this approach was modified by the introduction of the factor ofsafety, FS, defined in this context as the ratio of restoring forces to disturbing forces. Thismodification alters Equation (7) to:

2

2

2

0

tan

tan1cos =

Sc F(8)

Noting that, on the bank of a trapezoidal channel, the shear stress may be expressed as

Sy75.0 = (Chow 1959), and incorporating Equation (1) for c yields:

( )

2

2

2

50tan

tan1cos

1

75.0=

Ssc

FSD

yS(9)

Algebraic manipulation of Equation (9) yields:

( )

2

2

2

50

tan

tan1cos1

75.0

=

S

cF

Ss

ySD (10)

A flat bed is equivalent to = 0 and it is readily shown that, with the multiplying factor0.75 replaced by the value 0.97, Equation (10) devolves to Equation (2).

Equation (10) is used by the design program to determine the maximum allowable value of

the side slope angle, , such that the value ofD50, required for stability on the side slope, isno greater than that calculated by Equation (4). In this way, rock stability is assured on boththe bed and side slopes of the chute.

Appendix A.2 Program Flow Details

When all required inputs have been satisfactorily entered into the input sheet, the user simplyclicks the Run button. This action initiates the running of a macro that takes the requireddata from the input sheet and the rating table sheet and performs the necessarycomputations. The process is briefly described in the following paragraphs.

The first part of the process is simply a matter of retrieving the various geometry and flow

inputs from the input sheets and checking them for correctness and consistency. The programthen goes on to calculate a value for the required rock size at each of the flowrates in thetable of flowrates that was generated in the Rating Table input sheet. The sequence of stepsthat the program follows for each flowrate are summarised schematically in Figure 11.

The subroutine alternates between the calculation of water surface profile and rock size.Since these two calculation processes are mutually dependent, a starting value of rock size isrequired. This is provided by a simple two-stage process. Firstly, the rock size is taken as 1metre. With this roughness size on the chute face, it is assumed that the flow reaches uniformdepth. Given this calculated depth, the shear stress is calculated and a new rock sizedetermined. This value is passed to the iterative procedure shown in Figure 11, as thestarting value of D50_trial.

The subroutine is a repetitive loop containing a sub-critical loop (at the top), and a

supercritical section at the bottom. The initial assumption is that the flow will be sub-critical;hence the program begins the computations at the downstream boundary, using the value of


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24 Issued: 8-Sep-03

downstream boundary depth computed earlier. If the flow is indeed sub-critical throughoutthe chute (drowned) then the program uses the minimum depth value, at the upstreamboundary, to compute a value of D50. As this will normally differ from the initial trial value,

the water surface profile must be recalculated. This process is repeated until convergence isachieved, defined as less than 1% difference between the newly calculated value of D50 andthe old one. This fully-drowned, subcritical scenario is dubbed Scenario 1 in the program.

The subroutine contains a special routine to detect the presence of a hydraulic jump. If one isfound, program flow proceeds to the supercritical section, where the S2 curve of thesupercritical flow region is computed. This section begins computations at the upstreamboundary, with the assumption that the depth there is 99% of critical depth. Once the watersurface has been computed to the hydraulic jump, the minimum depth is taken as the depthjust before the jump, unless the jump is on the apron, in which case the minimum depth istaken as the depth at the beginning of the apron. The required rock size is then calculatedusing this depth. Again, convergence is checked before exiting the subroutine. This situationis referred to as Scenario 2 in the program. If convergence has not yet been achieved, thewhole process is repeated using the average of the new value of D50 and the old value,starting with the subcritical flow at the downstream boundary.

In the event that supercritical flow is found throughout the chute and the apron, the D50 iscalculated using the depth at the start of the apron, as before. However, a warning is giventhat the hydraulic jump exists past the end of the apron, ie, in the bed of the downstreamriver, as this is an undesirable situation. This situation is termed Scenario 3.


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Figure 11: Flowchart for section of program that calculates rock size

The sections in the flow chart labelled compute S1 curve and compute S2 curve arehandled by separate subroutines. These make use of the generalised form of the non-uniformresistance equation:

2

1 Fr

SS

dx

dy fo

= (11)

In the case of wide flow with R = y, we have

3

22

gy

qFr = (12)

3/10

22

y

nqSf = (Manning) (13)


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Hence,

( )

==

3

2

3/10

22

1gy

q

y

nq

Syf

dx

dy o(14)

Note that: dx is positive moving downstream,

So is positive if the slope is downstream,

Sf is always positive,

(1-Fr2) is positive for subcritical flow and negative for supercritical flow.

Equation (14) will be undefined at critical depth, (Fr = 1). The use of the equation should be

restricted to depths that are at least 5% smaller or greater than critical.The water surface profile is evaluated using the 4th order Runge-Kutta numerical technique toforward predict the next y value based on a known y. With reference to

Figure 12, at the point 1, y1is known, and hence )(yfdx

dy= is known from Equation (14).

The Runge-Kutta method estimates y2 as follows

)( 11 yfk =

+=

2

112

dxkyfk

+=

2

213

dxkyfk

[ ]dxkyfk 314 +=

( )432112 226

kkkkdx

yy ++++=


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y1

dx

y2

12

1

So

Figure 12: Adjacent Sections in Computation of Water Surface Profile

The above method assumes that Mannings n can be reasonably predicted from Equation(6). There is considerable uncertainty about the true value of n for rock chutes, and thismethod may require revision as more data becomes available.

Appendix A.3 References

Carter, A. C. (1953). Critical Tractive Forces on Channel Side Slopes. U. S. Bureau ofReclamation, Hydraulic Laboratory Report Hyd-366, February

Chow, V. T. (1959). Open Channel Hydraulics, McGraw-Hill, New York

Hemphill, R. W., and Bramley, M. E. (1989). Protection of River and Canal Banks,Butterworths, London

Henderson, F. M. (1966). Open Channel Flow, Macmillan, New York

Meyer-Peter, E., and Muller, R. (1948): Formulas for Bed-Load Transport. Procedings ofthe 2nd Congress of the International Association for Hydraulics Research, IAHR, Stockholm,

Sweden, JuneSimons, D. B. and Senturk, F. (1977): Sediment Transport Technology, Water ResourcesPublications, Fort Collins, Colorado

Yalim, M. Selim, and Karahan, E. (1979): Inception of Sediment Transport.Journal of theHydraulics Division,ASCE, Vol. 105, No. HY11, Proc. Paper 14975, November, pp 1433-1443.

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