Chute Guidelines 100

Guidelines for the Design of Rock Chutes using CHUTE Prepared by Associa Associ at e Pr Pr of ofes essor sor R. J. Kell Keller er CRC for Catchment Hydrology Version 1.0.0 September 2003 www.toolkit.net.au/chute

 © CRC for Catchment Hydrol Hydr ology ogy 2003. Except as permit t ed under the t he Austr alilia an Copyright Act 1968, t hes hese e Design Guidelines may not be copied or distributed by itself except with express written permission of the Director of the CRC for Catchment Hydrology

 G  U I   D

E  L  I   N E   S 

Document History  

Aug 2003 01 Sep 2003

R.J. Keller Nick Murray

1.0.0b 1.0.0

Creation Conversion 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 responsibility and excludes all liability whatsoever in respect of any person's use or reliance on this publication or any part of it.

Document History  

Aug 2003 01 Sep 2003

R.J. Keller Nick Murray

1.0.0b 1.0.0

Creation Conversion 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 responsibility and excludes all liability whatsoever in respect of any person's use or reliance on this publication or any part of it.

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

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% .................................................................................10

4.1.4

  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 ......................................................14

5.2

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

TABLE OF FIGURES

Figure 1: Schematic of a Typical Chute with Explanation of Terms ...............................................5 Figure 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

1 Introduction  A rock chute is a relatively short and steep section of the bed of a channel that has been armoured 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 or  reinforced concrete weirs.  Although the concept of a rock chute is simple, proper hydraulic design is very important to ensure that the chute geometry and rock size are matched with the expected flow conditions such that the rock remains stable under all expected flow conditions. In addition, appropriate rock chute design requires that a number of other issues are adequately addressed. In particular: •

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

•

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 voids and minimise the area of individual rocks exposed to forces from the flow

•

Where the underlying material is largely non-cohesive or where high ground-water  levels 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 rock chutes 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 of  these structures. Indeed many chutes fail for reasons other than inadequacies in their specific hydraulic design. Such failures are often related to poor understanding of the prevailing site conditions such as hydrology, overall stream morphology, floodplain and channel hydraulics, and foundation conditions. CHUTE is written as an EXCEL spreadsheet-based program. The internal calculations are run by macros. The macro security level on your computer must be set to Medium or Low in order for CHUTE to run properly.

1

1.1

The User Guide 1.1.1

Purpose

These guidelines first address the program CHUTE and its proper use to design and analyse the performance of a chute under a range of flow conditions. It is clearly not possible to provide a complete treatise on the other issues of importance. However, design notes on these other issues are provided for guidance only and not as a set of prescribed rules. In all cases it is important to access local knowledge and experience with other chutes on the same stream 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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2 Installation 2.1

Technical Specifications CHUTE requires the following system configuration.

Minimum 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, or  Windows XP + Service Pack 1, or  Windows NT4 + Service Pack 6a (Note that CHUTE has not been extensively tested on Windows NT4) Windows 95, 98 and ME are not supported.

Other software

2.2

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

Installation To install CHUTE: Double-click the setup.exe file on the CD. The CHUTE Setup Wizard appears. 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 licence conditions, you cannot install CHUTE. Click Next to continue the installation. The Select Installation Folder window appears. Leave the installation folder at the default setting. Click Next. The Confirm Installation window appears. Click Next. CHUTE is installed onto your computer, and the Installation Complete window appears. Click Close to exit the installer.

3

2.3

Uninstalling CHUTE To uninstall CHUTE, from Windows: Select Start|Settings|Control Panel|Add/Remove Programs. The Add or Remove programs dialog appears. Find CHUTE in the list of installed programs Click on the name “CHUTE”, then click the Remove button Windows asks you to confirm that you want to remove CHUTE. To remove CHUTE, click Yes. Otherwise, click No. Close the Add/Remove programs dialog.

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3 Hydraulic Design Elements  A typical rock chute is shown schematically in Figure 1 together with an illustration of the various 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 the bed from the erosive forces of the water. Jump location x=0 q

Chute drop

Downstream depth chute slope Downstream bed

1

Chute length

apron rise

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 the rock size is dependent on the bed shear stress, which, in turn, is dependent on the flow profile over the chute, a key element in the design process is the determination of the water  surface profile. Three types of profile may occur and are depicted in Figure 2. It is shown in Appendix A that the location of the point of maximum bed shear stress coincides with the location of the minimum depth. Clearly this location is dependent on the type of profile over the chute.

5

The profile shown in Figure 2 (c) is most likely to be associated with the maximum design flow 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 will be 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 required rock size is a maximum – will be lower than the channel design flow rate – that for which the required channel capacity is a maximum. This is a most important distinction in the design of  a rock chute. A particular feature of the program CHUTE is that it examines the entire range of flows and explicitly determines the flow within that range for which the required rock size for 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 the determination of the water surface profile throughout the chute for a given flow rate and the consequent determination of the minimum value of depth. The normal design situation is shown 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 depth at the chute crest and a known downstream subcritical depth. The downstream depth is dependent on controls further downstream and would normally be established from a onedimensional 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) (Henderson 1966). The location of the hydraulic jump is then determined as the point where the

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upstream and downstream depths form a conjugate pair – ie the Momentum Function is the same. For the case shown in Figure 2 (c), the subcritical profile, controlled by the downstream depth, is everywhere higher than the conjugate profile, computed from the assumed upstream supercritical profile. In this case, the chute is said to be drowned.

q

E n e r g  y g r a  d e l i n  C o n  e   j ug    a  yc t e d  S2 e p    t h  l i ne   

S1 A1 or M1

ydownstream

So 1 Figure 3: Water Surface Profile with Hydraulic Jump on Chute The discussion presented in this section is expanded with the full mathematical equations and solution procedure in Appendix A.

7

4 Design Program The design program is set up as a Microsoft Excel workbook, with separate sheets devoted to site inputs and the downstream rating table and a number of other sheets for presenting aspects of the results. Figure 4 shows the “Input” worksheet and the various inputs in the top table are summarised in the following sub-section.

4.1

Inputs . With reference to the sketch in Figure 1, the following variables describing the dimensions of the proposed structure are entered into the input table: •

The vertical and horizontal distance respectively between the upstream lip of the chute and the lowest point

•

. The vertical rise and horizontal length respectively of the downstream apron section

•

This is used by the program, in conjunction with the flow data, to determine the range of unit flow rates on the chute.

The nominal minimum and the maximum design flow rate specifying the range over which the program will calculate the D50 required. The angle of repose and the specific gravity of the proposed rock that will be used in the chute. Typical values are presented in Section 4.3. This is a critical input that reflects the degree of conservatism built into the design. Further comments on this factor are presented subsequently.

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Figure 4: CHUTE Input Table In the bottom table, the user is required to nominate the computation procedure for the downstream 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 backwater  program, 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 table should be used as the final design. If this option is chosen, pairs of flow rate-depth values are entered into the worksheet marked “Rating Table”. These pairs may be entered manually or inserted using “Copy” a nd “Paste” commands. 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 flow rate-depth values. An interpolated rating table is then automatically calculated matching the range of flow rates that will be tested. Ten flow steps are used. However, a smaller or greater  number of steps can be accommodated by changing the layout of the table. The program counts 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 (or  uniform) depth which is calculated from values of downstream channel depth, slope and channel 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.

9

This option normally produces a more conservative chute design than the “Rating Table” option and should be used where a downstream backwater profile is not available or cannot be computed and where the downstream channel can be approximated as a rectangular  wide channel with a known slope and Manning’s roughness.

4.1.3

ycrit+10%

With this option, the program assumes the downstream boundary depth is 10% greater than critical 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 is completely lacking.

4.1.4

 Average Normal, ycrit

With this option, the program calculates the average of normal depth and critical depth. It is intended only as an additional option available to the designer for investigating design sensitivity to downstream conditions. When all required inputs have been satisfactorily entered into the input sheet, the user simply clicks the “Run” button. This action initiates the running of a macro that takes the required data from the input sheet and the rating table sheet and performs the necessary computations, producing a range of outputs.

4.2

Outputs On completing the calculations, the program automatically switches to the “Results” spreadsheet. This sheet contains two tables – the first and part of the second are presented in Figure 5. The first table contains summary information for each of the flow rates within the specified flow range. The five blocks of information contain: Flow Rate. Required rock size and side slope, representing the required D50 assuming normal (uniform) flow, calculated D50 with full profile calculations, and required chute bank angle for stability. Downstream boundary depths, representing that used by the program and that specified by the rating table. The critical depth is also calculated a nd reported. Specific energy results, representing the specific energy at the spillway crest (“u/s”) and at the toe of the apron (“d/s”). The column labelled “extra” has meaning only for a Scenario 3 profile which is the case where the hydraulic jump is swept downstream with supercritical flow throughout the structure. In this case, the number reported in this column is the additional specific energy required at the toe of the apron for a hydraulic jump to form. Jump conditions. In the first column of this block is the scenario number which represents 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) of Figure 2.

•

Scenario 3:

Jump swept downstream.

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

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depth immediately downstream of the jump respectively are listed. The final two columns list the energy loss across the hydraulic jump and the friction loss down the chute. The sum of  the 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 presented in the second table. Figure 5 lists only the first few lines of this table. Listed in this table are the variations with distance of bed elevation, bed slope, water surface elevation, total energy line, friction slope, velocity, and Froude Number.

Figure 5: Spreadsheet “Results” Output Two final spreadsheets, labelled “Profile” and “Rock Size”, provide graphical output. In “Profile”, the total energy line, water surface, velocity, bed elevation, and friction slope are plotted as a function of distance from the chute crest. The “Rock Size” chart provides the variation of required rock size with flow rate. The downstream boundary depth is also graphed. These graphs are illustrated in Figure 6.

11

(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. This covers the vast majority of cases. For rounded and other types of rock a good summary of  angles 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 1 gives a guide to typical rock densities.

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Sandstone

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 be considered 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 much less than the return period of the flood

•

The likelihood of changes in tailwater conditions due, for e xample, to siltation from other downstream chutes or vegetation

The most important of these factors is the consequence of failure in terms of subsequent erosion, 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 a structure 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 is general 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 is minimal or where active maintenance can be assured. These low factors of safety reflect the generally conservative assumptions built into the rock sizing procedures. The recommended values should lead to design conditions in which there is no significant rock movement under the critical flow condition.

13

5 Other Design Details Typical 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 least 25Mpa. The rock should be free of defined cleavage planes and should not be adversely affected 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 designers discretion, 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 rock quarry by drilling and blasting. A hydraulic rock breaker mounted on a hydraulic excavator  provides 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 to ensure 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 through which underlying material can be washed out. Additionally, it helps to create an interlocking mass of rock which is highly stable.

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3000

3000

min

min

3000

Rock beaching on batter 

min 1000 Typ. Crest R.L.

Rock chute

Minimum 300mm above design water line Bed

Bed 1500 Typ.

Filter layer  1000 Typ.

ELEVATION Excavate and place rock to form a key in bank

Fold of geotextile to form a barrier to fine material

A

B

Transition the cross section of the creek upstream and downstream to match the cross section of the structure as shown

C Where required place and compact excavated material to form an alignment bank.

Extend rock beaching on bank upstream and downstream of  chute by three metres minimum PLAN

Rock armour 

SECTION 'A' Filter layer  Additional rock and geotextile to form key at abutment SECTION 'B'

1000 Typ.

SECTION 'C' Trench continues into bank

Figure 7: Typical Rock Chute Plan and Sections Experience suggests a rock gradation such as that summarised in Table 2. When specifying rock gradation to field staff and contractors, it is helpful to transform this grading by weight into an equivalent grading by number.

15

1.5 – 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 the crest, whether by flow forces, debris impact, or some other means, results in a concentration of flow at that point which can destabilise more rock. The subsequent rill can lead to failure of the chute.  An effective means of guarding against such failure is to construct a solid wall through the chute along the line of the crest. Options for such a wall include reinforced concrete, steel sheet piling, and a wall constructed from piles and timber boards. Inclusion of a fixed crest is recommended for major structures and for structures requiring a high 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 the permeability of the parent material, decreasing cohesion of the parent material, and the height and steepness of the chute.

•

The cutoff may be an impermeable barrier, such as concrete, sheet piling, timber, or  membrane. Alternatively, geotextile can be considered to prevent the passage of  material.

•

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

•

Normally, a filter layer is only necessary where the underlying material is largely noncohesive such as uniform sand or silt, high groundwater levels create large pore pressures, or an unusually high factor of safety is required. If one or more of these conditions prevails, the need for a filter layer can be further assessed from the following required criteria: For stability:

 D15 (chute rock) D 85 ( bed material)

≤5

and

 D50 (chute rock) D 50 ( bed material)

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Issued: 8-Sep-03

For permeability:

 D15 (chute rock) D15 ( bed material)

≤5

If the relationship between bed material grading and chute rock grading is outside these 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 layer  and the chute rock.

•

If geotextile is to be used as a filter layer, special rules apply and specialist texts on geotextile 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 of  geotextile or membrane be brought up through the rock along the line of the crest to the rock surface. This provides a barrier to the passage of fine material through the rock and increases upstream siltation rates.

5.4

Treatment of Abutments  A high proportion of chute failures occur at the abutments. Where a fixed crest is provided by a pile and board wall, or a concrete cutoff, it must be excavated into the abutment and backfilled and compacted with selected clay fill. Where no fixed crest is provided, geotextile should extend into a key excavated in the bank and backfilled with rock, as shown in Figure 8. Rock riprap bank protection should extend upstream and downstream of the chute to provide direct abutment protection.

Rock

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

17

Rock

Geotextile (bi) Rock

Extent of geotextile beneath chute depends on site conditions

Geotextile

(bii)

Rock

Geotextile

Extent of geotextile beneath chute depends on site conditions (c)

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

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 Appendix A  Theory for the Hydraulic Design of  Rock Chutes  Appendix A.1 Theory  The determination of the critical shear stress at which bed particles will start to move on a flat bed is given by:

τ c γ  D50 (S  s − 1)

= constant

(1)

where

τ c

is the critical shear stress

γ 

is the specific weight of water 

D50

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

S s

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 this value, 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 other  studies have yielded lower values. For the present study, a more conservative value of 0.047 is chosen, consistent with the work of (Meyer-Peter and Muller 1948) and (Yalin and Karahan 1979).

19

Noting that, on the bed of a trapezoidal channel, the shear stress may be expressed as τ  = 0.97γ   y S  (Chow 1959), where  y is the depth and S is the local energy slope, and introducing a factor of safety, FS, Equation (1) may be transposed to:

 D50 =

0.97 F S  yS 

(2)

0.047(S  s − 1)

Now, flow over a chute may be considered as being hydraulically wide, leading to the expression of Manning’s Equation as:

q = vy = where n q

1

n

5

 y 3 S 

1

2

(3)

is Manning’s roughness parameter  is the flow rate per unit width

Combining Equations (2) and (3) yields:

 D50 =

0.97 F S  q 2 n 2 0.047(S  s − 1) y

7

(4) 3

Equation (4) demonstrates that, for a given flow rate per unit width, particle density, and Manning’s roughness parameter, the minimum rock size required for stability is strongly inversely proportional to the local depth,  y. It is evident that the critical design condition, ie the condition requiring the largest stone size for stability, will be that for which the depth is a minimum. There are three possible types of flow profile that may occur on a rock chute. These are illustrated 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 by an 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 and type c) is associated with large flows. It is evident that solution of Equation (4) requires the determination of the water surface profile 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 largest stone size for stability, is shown in Figure 10 and corresponds to the case shown in Figure 9(a).

21

q

E n e r g  y g r a  d e l i n  C o n  e   j  u   g   yc a t e  d e  S2  p t h  l i ne   

S1 A1 or M1

ydownstream

So 1 Figure 10: Water Surface Profile with Hydraulic Jump on Chute For the case shown in Figure 10, two depths control the combined flow profile – critical depth 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 a rectangular channel (Henderson 1966):

 y c = 3

q2

(5)

 g 

The downstream depth is dependent on controls further downstream. Program CHUTE currently incorporates four options for determining this depth and these are discussed in the Guidelines. The most reliable and exact is the use of a rating table which 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) (Henderson 1966). The location of the hydraulic jump is then determined as the point where the upstream and downstream depths form a conjugate pair – ie the Momentum Function is the same. 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 upstream supercritical 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 be recognised that the computed profiles are dependent on Manning’s roughness coefficient, which, in turn, is linked to the unknown rock size through the Strickler Equation (Henderson 1966): 1

n = 0.041D506

(6)

 Accordingly, an iterative solution procedure is required. The final stage of the hydraulic design of the chute is the determination of the chute bank angle. The classic equation linking the shear stress on a particle, at the inception of motion, on the channel bank ( τ 0 ) to the critical shear stress for the same particle on the bed of the channel ( τ c ) is (Carter 1953, cited in Henderson 1966):

τ 0 τ c where θ 

22

= cos θ  1 −

tan 2 θ  tan 2 φ 

(7)

is the bank angle

Issued: 8-Sep-03

φ 

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 of  safety, FS, defined in this context as the ratio of restoring forces to disturbing forces. This modification alters Equation (7) to:

τ 0 τ c

= cosθ 

1

F S 2

−

tan 2 θ 

(8)

tan 2 φ 

Noting that, on the bank of a trapezoidal channel, the shear stress may b e expressed as τ  = 0.75γ   y S  (Chow 1959), and incorporating Equation (1) for  τ c yields:

0.75 yS  θ c D50 (S  s − 1)

= cos θ 

1

F S 2

−

tan 2 θ  tan 2 φ 

(9)

 Algebraic manipulation of Equation (9) yields:

 D50 =

0.75 yS  θ c (Ss − 1) cos θ 

1

 F S 2

2

−

(10)

tan θ  tan 2 φ 

 A flat bed is equivalent to θ  = 0 and it is readily shown that, with the multiplying factor  0.75 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, θ  , such that the value of D50, required for stability on the side slope, is no greater than that calculated by Equation (4). In this way, rock stability is assured on both the 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 simply clicks the “Run” button. This action initiates the running of a macro that takes the required data from the input sheet and the rating table sheet and performs the necessary computations. 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 program then goes on to calculate a value for the required rock size at each of the flowrates in the table of flowrates that was generated in the “Rating Table” input sheet. The sequence of steps that 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 is required. This is provided by a simple two-stage process. Firstly, the rock size is taken as 1 metre. With this roughness size on the chute face, it is assumed that the flow reaches uniform depth. Given this calculated depth, the shear stress is calculated and a new rock size determined. This value is passed to the iterative procedure shown in Figure 11, as the starting 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 

23

downstream boundary depth computed earlier. If the flow is indeed sub-critical throughout the chute (drowned) then the program uses the minimum depth value, at the upstream boundary, 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 is achieved, defined as less than 1% difference between the newly calculated value of D50 and the old one. This fully-drowned, subcritical scenario is du bbed “Scenario 1” in the program. The subroutine contains a special routine to detect the presence of a hydraulic jump. If one is found, program flow proceeds to the supercritical section, where the S2 curve of the supercritical flow region is computed. This section begins computations at the upstream boundary, with the assumption that the depth there is 99% of critical depth. Once the water  surface has been computed to the hydraulic jump, the minimum depth is taken as the depth just before the jump, unless the jump is on the apron, in which case the minimum depth is taken as the depth at the beginning of the apron. The required rock size is then calculated using this depth. Again, convergence is checked before exiting the subroutine. This situation is referred to as “Scenario 2” in the program. If convergence has not yet b een achieved, the whole 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 is calculated using the depth at the start of the apron, as before. However, a warning is given that the hydraulic jump exists past the end of the apron, ie, in the bed of the downstream river, as this is an undesirable situation. This situation is termed “Scenario 3”.

24

Issued: 8-Sep-03

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” are handled by separate subroutines. These make use of the generalised form of the non-uniform resistance equation:

dy dx

=

S o − S f  

(11)

1 − Fr 2

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

 Fr  = 2

S  f   =

q2

(12)

 gy 3

q 2n 2  y 10 / 3

(Manning)

(13)

25

Hence,

 q 2 n2   S o − 10 / 3   y dy  =  f  ( y ) =  dx  q2  1 − 3    gy  Note that:

(14)

dx is positive moving downstream, So is positive if the slope is downstream, Sf  is always positive, (1-Fr 2) is positive for subcritical flow a nd 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 4 th order Runge-Kutta numerical technique to forward predict the next y value based on a known y. With reference to Figure 12, at the point 1, y 1is known, and hence

dy

=  f  ( y ) is known from Equation (14).

dx

The Runge-Kutta method estimates y2 as follows

k 1 =  f  ( y1 )

 

k 1dx 

 

k 2 dx 

k 2 =  f   y1 + k 3 =  f   y1 +

2 

2 

k 4 =  f  [ y1 + k 3 dx ]  y 2 =  y1 +

26

dx 6

(k 1 + 2k 2 + 2k 3 + k 4 )

Issued: 8-Sep-03