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BANDS 2.0 User Manual This document is CONFIDENTIAL. Neither the whole nor any part of this document may be disclosed to any third party without the prior written consent of Shell International Oil Products B.V., The Hague. The copyright of this document is vested in Shell International Oil Products B.V., The Hague. All rights reserved. Neither the whole nor any part of this document may be reproduced, stored in any retrieval system or transmitted in any form or by any means (electronic, mechanical, reprographic, recording or otherwise) without the prior written consent of the copyright owner. Bitumen Business Group May 1998

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Page 1: BANDS2

BANDS 2.0

User Manual

This document is CONFIDENTIAL. Neither the whole nor any part of this document may bedisclosed to any third party without the prior written consent of Shell International Oil Products B.V.,The Hague.

The copyright of this document is vested in Shell International Oil Products B.V., The Hague. Allrights reserved. Neither the whole nor any part of this document may be reproduced, stored in anyretrieval system or transmitted in any form or by any means (electronic, mechanical, reprographic,recording or otherwise) without the prior written consent of the copyright owner.

Bitumen Business GroupMay 1998

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Contents

1. Introduction 32. The BANDS Package 43. System Requirements 54. Installing BANDS 2.0 6

4.1 Installation from CD-ROM 6

4.2 Installation from diskettes 7

5. Starting BANDS 106. Use of SBIT for Bitumen Stiffness Prediction 117. Use of SMIX for Asphaltic Mix Stiffness Prediction 148. Use of NFAT for Fatigue Life Prediction 159. COMB for combined use of SBIT, SMIX and NFAT 1610. Working with Results Tables 1711. Printing in BANDS 2.0 18

Print Preview 19

Printing to the printer 19

12. Error Messages 20Applicable Error Conditions for SBIT 20

Applicable Error Conditions for SMIX 20

Applicable Error Conditions for COMB 20

Appendix I 21SBIT: Bitumen Stiffness Prediction 21

Appendix II 23SMIX: Asphaltic Mix Stiffness Prediction 23

Appendix III 25NFAT: Asphalt Mix Fatigue Life Prediction 25

References 27

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1. Introduction

BANDS is one of the packages in the series of Shell Pavement Design software packages,together with BISAR and SPDM (the Shell Pavement Design Method).

BANDS contains tools to assist the pavement designer in estimating relevant materialproperties of the bituminous binder and the asphaltic mix for use in thickness design relatedcalculations. It should be used in addition to SPDM, which is dedicated to specific designcalculations. BANDS provides calculated material properties for a large range of conditions andcompositions.

This second version of the BANDS package is designed for use under Windows and replacesthe former release BANDS-PC ME 1.1 (1989) for DOS computers. Output of the program canbe copied to the Windows clipboard for importing the results of the calculation into commercialWindows programs for preparing graphics etc.

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2.The BANDS Package

BANDS contains the following tools:

SBIT: The prediction of the stiffness modulus of a bitumen with aid of the Van der Poelnomograph.

SMIX: The calculation of the stiffness of the asphaltic mix from the stiffness of the bitumenand the volumetric composition of the mix.

NFAT: Estimate the relation between critical strain and number of cycles, according to one ofthe nomographs published in the Shell Pavement Design Manual.

COMB: A combination of above calculations.

The well known Van der Poel nomograph, published in 1954, remains a valuable tool toestimate the stiffness modules of bitumens on the basis of standard tests as penetrationsand/or softening point. The nomograph is suitable for S-type bitumens including MULTIPHALTE.The nomograph cannot be used for very waxy bitumens and for polymer modified binders.

The SMIX-nomograph is sometimes called the ‘Ugé-nomograph’. It is designed to estimate thestiffness of an asphaltic mix in the ‘elastic-region’, so at low temperatures and short loadingtimes. It should not be used for rutting calculations. The nomograph was designed for bitumenswithout polymer modification.

The NFAT-nomograph estimates fatigue characteristics of asphaltic mixes on the basis of mixstiffness and volumetric mix composition. The nomograph is based on fatigue tests carried outat constant strain. The nomograph was designed for standard conventional bitumens andshould not be used for polymer modified binders.

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3. System Requirements

The minimum requirements for the computer system to run BANDS 2.0 are as follows:

Part Description

Computer IBM (or compatible) PC with a 486SX/50 MHz or higher processor

Operating system Windows 3.1, Windows 95, Windows NT 3.51, Windows NT 4.0

Memory 4 MB of memory (8 MB recommended for Windows 95, 16 MBrecommended for Windows NT)

Disk-drive The program is supplied on CD-ROM. On request it is available ontwo high density diskettes of 1.44 MB

Hard disk 4 MB of available disk space

Screen VGA display or better

Mouse Microsoft Mouse or compatible pointing device

Printer Any Windows compatible printer

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4. Installing BANDS 2.0

4.1 Installation from CD-ROM

In Windows 3.x

• Start Windows (or close all running programs if Windows is already started)

• Place the CD-ROM in your CD-ROM drive

• Select File/Run from the Program Manager

• Type: D:\BANDS2\SETUP (where D: is the drive letter assigned to your CD-ROM drive)

• Press the [Enter] key

• Follow the instructions in the section “Installation Screens (CD-ROM)” below

In Windows 95/NT

• Start Windows 95/NT (or close all running programs if Windows 95/NT is already started)

• Place the CD-ROM in your CD-ROM drive

• Select Run from the Start menu

• Type: D:\BANDS2\SETUP (where D: is the drive letter assigned to your CD-ROM drive)

• Press the [Enter] key

• Follow the instructions in the section “Installation Screens” below

Installation Screens (CD-ROM)Once the installation of BANDS 2.0 has started, the following screen will be displayed.

It is recommended that BANDS 2.0 is installed in a directory called BANDS2, however, adifferent directory may be specified. Pressing the OK button starts the process of installing theBANDS 2.0 files into the specified directory.

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As files are installed, a progress meter is updated as shown below:

Installation is complete when the following window is displayed:

4.2 Installation from diskettes

In Windows 3.x

• Start Windows (or close all running programs if Windows is already started)

• Place the disk labelled “BANDS 2.0 Disk 1 of 2” in Drive A:

• Select File/Run from the Program Manager

• Type: A:\SETUP

• Press the [Enter] key

• Follow the instructions in the section “Installation Screens” below

In Windows 95/NT

• Start Windows 95/NT (or close all running programs if Windows 95/NT is already started)

• Place the disk labelled “BANDS 2.0 Disk 1 of 2” in Drive A:

• Select Run from the Start menu

• Type: A:\SETUP

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• Press the [Enter] key

• Follow the instructions in the section “Installation Screens” below

Installation Screens (diskettes)Once the installation of BANDS 2.0 has started, the following screen will be displayed.

It is recommended that BANDS 2.0 is installed in a directory called BANDS2, however, adifferent directory may be specified. Pressing the OK button starts the process of installing theBANDS 2.0 files into the specified directory.

As files are installed, a progress meter is updated as shown below:

When the following screen is displayed, remove DISK 1 and replace it with DISK 2 beforepressing the OK button.

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Installation is complete when the following window is displayed:

Installation TroubleshootingIf installation appears to be extremely slow, or if problems are encountered as installationprogresses, files may be manually copied to your hard disk and installed from there. To do this,follow the instructions below.

• Using File Manager (in Windows 3.x) or Windows Explorer (in Windows 95) create atemporary directory on your hard disk.

• Copy all of the files from both BANDS 2.0 installation disks to this temporary directory.

• Run the BANDS 2.0 SETUP program from this temporary directory.

Once installation has completed, the temporary directory may be removed.

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5. Starting BANDS

The program is started by double-clicking the BANDS icon

and the main window appears with following pull-down menus:

• File• Edit• Nomograph• Window• Help

Click on Nomograph to obtain the pull down menu for selecting one of the tools:

• Bitumen Stiffness (SBIT) • Asphalt Mix Stiffness (SMIX)• Fatigue Life Asphalt Mix (NFAT)• Asphalt Mix Performance (COMB).

The use of these tools is explained in the following sections of this manual. It is possible to openmore nomographs and more windows for the same nomograph at the same time.

File offers the following options:

• Results Table• Results Report• Exit

Exit closes the application.

The options Results Table and Results Report are alternatives for the same functions in theseparate nomograph windows. Results Table refers to output suitable for copying to the clip-board. Results Report prepares printed output to the selected printer.

The Edit pull-down menu has one option:

• Copy Results to Clipboard

Copy Results to Clipboard actually copies the content of the window obtained with ResultsTable (see section 10) to the clipboard for pasting the data to other Windows application (e.g. aspread sheet or a graphical application).

The print option Results Report is described in section 11 of this manual.

The Window pull-down menu in the BANDS opening screen offers the possibility to use thewell-known general Windows options

• Cascade• Windowsfor arranging and selecting the various windows.

The Help menu offers the general Windows Help options

• Contents• Search For Help On• About BANDS 2.0Help is also always available when pressing F1, except within Results Report. For details usingHelp, refer to your Windows Manual on Help.

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6. Use of SBIT for Bitumen Stiffness Prediction

This tool predicts the Stiffness of a bitumen, and is an exact simulation of the Van der Poelnomograph. See Appendix I for details and technical background.

This program predicts the stiffness modulus of a bitumen at a selected temperature and loadingtime. The bitumen properties can be supplied in four ways:

• Softening Point (T800Pen) and Penetration Index• Softening Point (T800Pen) and Penetration at a certain temperature (the default)• Use of two penetrations at different temperatures• Penetration at a certain temperature combined with Penetration Index.

The window obtained after selecting Nomograph, Bitumen Stiffness (SBIT) has the followinglay-out

When changing one of the input values, the output is displayed instantaneously.

The following Time of Loading units are available:

• Seconds• Hertz• Minutes• Hours• Days• Weeks• Months (30 days)• Years (365 days).

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It is possible to select ranges for one or more input parameters by clicking the check boxes, e.g.for Time of Loading and Bitumen Temperature:

The results of this request can be viewed with Results Table and with Results Report.

By clicking Results Table (see section 10) the following window will be displayed:

All the data are ‘selected’ and ready for copying to the clipboard, either by pressing the button inthe window or via the Edit pull-down menu. In this way the output can be pasted into any otherWindows application.

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The option Results Report for printing and browsing is explained in section 11.

A survey of possible error messages is given in section 12.

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7. Use of SMIX for Asphaltic Mix Stiffness Prediction

This tool predicts the stiffness of an asphaltic mix, from the bitumen stiffness and the volumetricmix composition. See Appendix II for details and technical background.

The window obtained after selecting Nomograph, Asphalt Mix Stiffness (SMIX) has thefollowing lay-out

It is possible to select ranges for bitumen stiffness, bitumen content and aggregate content.

The option for Results Table is used to copy results to the clipboard (see section 10)

The option Results Report for printing and browsing is explained in section 11.

A survey of possible error messages is given in section 12.

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8. Use of NFAT for Fatigue Life Prediction

This tool predicts the fatigue characteristics from the stiffness of the asphaltic mix and itsbitumen content by volume. See Appendix II for details and technical background.

The window obtained after selecting Nomograph, Fatigue Life Asphalt Mix (NFAT), has thefollowing lay-out

The tool can be used in two modes:

• to calculate the fatigue life (number of fatigue cycles) from the strain

• to calculate the fatigue strain from the applied number of cycles.

It is possible to select ranges for bitumen content, asphalt mix stiffness, fatigue strain andfatigue life.

The option for Results Table is used to copy results to the clipboard (see section 10)

The option Results Report for printing and browsing is explained in section 11.

A survey of possible error messages is given in section 12.

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9. COMB for combined use of SBIT, SMIX and NFAT

This tool is a combination of the SBIT, SMIX and NFAT nomographs in one window and isevoked via Nomograph, Asphalt Mix Performance (COMB)

The input mode for the first (SBIT) part is limited to the default option (softening point andpenetration at a certain temperature). For the fatigue calculations both NFAT options areavailable here.

The option for Results Table is used to copy results to the clipboard (see section 10)

The option Results Report for printing and browsing is explained in section 11. When usingResults Report in COMB, the output is automatically printed in landscape mode.

A survey of possible error messages is given in section 12.

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10. Working with Results Tables

Whenever the Results Table button (or Results Table option on the File menu) is selectedfrom a nomograph input form, a table of calculated results is displayed. The following exampleshows a table of Bitumen Stiffness prediction results:

The Results Table provides a convenient way to view a large number of results (up to 2000).Horizontal and vertical scroll bars will appear, if necessary, to allow navigation when moreresults have been calculated than can be displayed in the Results Table window.

The Results Table is initially displayed with all results selected (highlighted). Pressing the Copyto Clipboard button (or selecting Copy to Clipboard from the Edit menu) will copy the selectedresults to the clipboard so that they may be pasted into another application. For example,copying the results and pasting them into a spreadsheet application would allow the graphing ofresults etc.

If too many results are selected for copying, the following message will be displayed:

If the above error message appears, it is still possible to copy all results to another applicationby copying the results in a number of smaller selections. To copy a smaller selection, use themouse to point to the first required cell and drag to the last required cell (the cells will appearhighlighted). Now use the Copy to Clipboard option to copy the selected cells. It is possible toselect whole columns of results by clicking on the heading for the desired column.

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11. Printing in BANDS 2.0

Whenever a report is created in BANDS 2.0 (by pressing the Results Report button) a previewwindow similar to the following is displayed:

The arrow buttons at the top of the preview window enable you to move backwards andforwards in your report. The action of those buttons is as follows:

Moves you to the first page of the report

Moves you to the previous page

Moves you to the next page

Moves you to the last page of the report

Cancels page formatting. For long reports, page formatting may take a fewmoments. If you want to stop the page formatting, press this button.

Previews the page to be printed

Sends your report to the printer

NOTE:You can also use the keyboard to move around in the print window. [Ctrl]+[Home] movesyou to the first page, [Pg Up] moves you to the previous page, [Pg Dn] moves you to the nextpage, [Ctrl]+[End] moves you to the last page, and [Esc] closes the print window.

NOTE:You can use the scroll bars to move around individual pages of the report.

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Print PreviewThe magnifying glass button is the print preview button. This button lets you see each page inits entirety, as it will print. When you Click this button, the program displays the page that’scurrently in the preview window, reduced in size so the entire page fits in the window at onetime.

Printing to the printerTo send the displayed report to the printer, click the printer button. A screen similar to thefollowing will be displayed. Note that BANDS 2.0 will print the report on your default printer. Toselect a different printer in Windows 3.x use the printers option in the control panel. To select adifferent printer in Windows 95, use the printers option in My Computer.

The print range option allows you to print all or just part of your report. Select ALL to print theentire report or specify a page range for a partial report.

The default number of copies printed is 1, however this may be changed by specifying adifferent number in the copies box.

The Collate Copies option determines how multiple copies of a report are printed. To printmultiple copies of a multiple page report in the order 1,1,1,2,2,2,3,3,3, etc. leave this optionempty. To print multiple copies of a multiple page report in the order 1,2,3...,1,2,3…, etc. thenselect this option. Note that certain printers do not respond to this collating option and willalways print a report in the order 1,2,3…,1,2,3…, etc.

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12. Error Messages

Applicable Error Conditions for SBIT

Error conditions are displayed in the field for Bitumen Stiffness (in screen and reports).

Applicable Error Conditions for SMIX

Error conditions, indicating values outside the applicable range of the nomograph, are displayedin the field for Mix Stiffness (in screen and reports).

Applicable Error Conditions for NFAT

Error conditions, indicating values outside the applicable range of the nomograph, are displayedin the fields for Fatigue Life or Fatigue Strain (in screen and reports).

Applicable Error Conditions for COMB

The error messages are the same as for the separate parts SBIT, SMIX and NFAT.

Page 20

Volume % Bitumen < 5

Volume % Bitumen > 30

Stiffness < 100 Mpa

Stiffness > 50000 Mpa

Fatigue Strain < 10 µm/m

Fatigue Strain >1000 µm/m

Fatigue Life < 10 (x 1000)

Fatigue Life > 100000 (x 1000)

Bitumen Stiffness < 5 MPa

Bitumen Stiffness > 3 GPa

Volume % Bitumen < 5

Volume % Bitumen > 100

Volume % Aggregate < 0

Volume % Aggregate > 100

Sum of Volumes > 100 (Vbit + Vagg > 100)

PI > 7

PI < -3

pen < 1

Time < 1E-6 seconds

Time > 1 E10 seconds

Temp too high Maximum bitumen temperature 130 °C above Softening Point

Temp too low Minimum bitumen temperature 200 °C below Softening Point

Glassy State Bitumen Stiffness > 2.5x103 MPa

White area Bitumen Stiffness < 10-16 MPa.

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Appendix I

SBIT: Bitumen Stiffness Prediction

Details and Background

This nomograph, developed by Van der Poel1 and published in 1954, has proved to be a usefultool for the prediction of the bitumen stiffness at key loading conditions and circumstances.

For the development of the nomograph, illustrated in Figure 1, following observations andassumptions were made:

• there is a linear relationship between the logarithm of the penetration and the temperature• the penetration at the softening point temperature is equal to 800 dmm.

The computer program SBIT provides a simulation of reading off the original nomograph.

The nomograph allows the prediction of the bitumen stiffness at a specific temperature and timeof loading or frequency. The actual stiffness value differs from the predicted value by not morethan a factor of 2, corresponding to only a few degrees in temperature, which is consideredsufficient for engineering purposes.

Within the original nomograph, the bitumen was characterised by softening point and PI. Lateron a single modification has been made and the softening point was formally replaced byT800pen, obtained from penetration measurements at at least two temperatures.

The nomograph cannot be used for very waxy bitumens, especially soft bitumens, as thesoftening point temperature is largely influenced by the presence of considerable amounts ofwax. For good quality (S-type) paving grade bitumens2 , for which the Van der Poel nomographis valid, the softening point temperature is very close to the T800pen temperature. Within thewindows of BANDS, the softening point and T800pen can be used interchangeably.

The Van der Poel nomograph was not designed for Polymer Modified Bitumens.

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Figure 1:

The V

an der Poel N

omograph

Page 22

–10

+7+6+5+4+3+2+1

0–1–2–3

10–4 10–3 10–2 10–110 102

103 104

105

106107

108109

2 x 109 25 x 109

+42

53

6 2 52 5 2 5

+6

+7

+5+4

+3+2

+10

–1–2

–3

7

Temperature difference, °C

Stiffness modulus, N/m2

Above T800 pen

Pen

etra

tion

Inde

x

Below T800 pen

100 50 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200

10 –5

10 –6

10 –7

10 –8

10 –9

10 –10

10–62 4 6 8

2468 2468 2468 2468 2468

2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8

105 104 103 102 10 1 1" 1" 10" 30"1' 2' 5' 10' 30'1h 2h 5h10h1day2d 7d 30d 1 year 10 y 100 y

10–5 10–4

Frequency, Hz

10–3 10–2 10–1 1021 10

Time of loading s

103 104 105 106 107 108 109 1010

The penetration index (PI) has been defined by:20 – PI log pen at T1 – log pen at T2

= 5010 + PI T1 – T2

The stiffness modulus, defined as the ratio stress/strain, is a function oftime of loading (frequency), temperature difference with T800 pen, and PI.T800 pen is the temperature at which the penetration would be 800. This isobtained by extrapolating the experimental log penetration versustemperature line to the penetration value 800.At low temperatures and/or high frequencies the stiffness modulus of allbitumens asymptotes to a limit of approximately 3 x 109 N/m2.

Units:1 N/m2 = 10 dyn/cm2 =1.02 x 10–5 kgf/cm2 = 1.45 x 10–4 lb/in2

1 N s/m2 = 10 P

KSLA, August 1953, 3rd edition 1972Electronic Reproduction, 1998

ExampleOperating conditionsTemperature 10°CLoading time 0.02 seconds

Characteristics of the bitumen in the mixT800 pen (temperature at which the penetration is 8000.1 mm) is 64°CPI (Penetration Index) is 0Connect .02 seconds on time scale with temperaturedifference 64 – 10°C on temperature scale.Record stiffness on network at PI = 0The stiffness of the bitumen detemined with thisNomograph is Sb = 2.0 x 108 N/m2.

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Appendix II

SMIX: Asphaltic Mix Stiffness Prediction

Details and Background

The SMIX part of BANDS 2.0 is based on a method3 to predict the stiffness of bituminousmixes from the stiffness modulus of the bituminous binder and the volumetric composition of themix. This study resulted in the nomograph, also called Ugé-nomograph, depicted in Figure 2.SMIX is based on the underlying equations.

This work covered the study of twelve asphalt mixes, all vastly different in composition, but yetstandard mixes for road or hydraulic applications in a number of countries. In this study, the mixstiffness was measured in two-point bending tests on trapezoidal specimens. The tests weremade at temperatures between -15 and 30 °C and at loading frequencies between 4 and 50 Hz.The strain level was highest in the tests at 30 °C but did not exceed 20x10-6, to avoid effects ofnon-linear behaviour.

The test results were analysed using the bitumen stiffness as obtained with the SBITnomograph. In this way the bitumen stiffness accounts for the effect of temperature, loadingfrequency and bitumen properties. Mix stiffness and bitumen stiffness were correlated via thevolumetric mix composition in terms of aggregate content and binder or void content.

The method is applicable to short loading time conditions, where elastic behaviour is dominant.Data on bitumens (not polymer modified) reported elsewhere were predicted with deviations notexceeding a factor of 2.

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Figure 2: The SMIX Nomograph

Page 24

95

85

75

65

50

30

10

1003020105

Bituminous binder% volume (Vb)

E.g.: Stiffness modulus of the recoveredbinder 2 x 108 N/m2

Vb: Volume of binder 13.1%Volume of mineral aggregate 80.5%

Stiffnessmodulusof the mix1.1 x 1010 N/m2

108

107

109864

2

2

100100 4040VbVb3030 2020 1010 55

864

2

86

9080

70

60

40

20

0

Mineralaggregate% volume (Vg)

Stiffness modulusof the bituminousbinder (N/m2)

Stiffness modulusof the bituminousmix (N/m2)

1010864

2

109864

2

108864

2

10786

4

2

������������

yyyyyyyyyyyy

mineral

aggregate

mineral

aggregate

Bitumen

Air

Volume %

Va% 100–

Vg

6.4

13.1

80.5

Electronic Reproduction, 1998

Page 25: BANDS2

Appendix III

NFAT: Asphalt Mix Fatigue Life Prediction

Details and Background

The NFAT part of BANDS 2.0 represents a computerised version of the nomograph given in theShell Pavement Design Manual4 and is illustrated in Figure 3.

The nomograph was developed using fatigue data obtained in constant strain tests on thirteendifferent mixes which were representative of base course and wearing course mixes as appliedin various countries. Details on the mix compositions are given in Ref5. Predicted fatigue strainsdid not differ by more than 30 to 40 percent of the measured values.

The program requires the volumetric bitumen content and mix stiffness as input. The fatigue life(the number of cycles at which the original stiffness is halved) is calculated as a function of theapplied strain or, vice versa, the permissible asphalt strain is calculated as a function of theapplied number of load cycles.

It is noted that the NFAT program estimates the number of load cycles to the onset of failureunder continuous sinusoidal loading conditions in a constant strain mode as determined in 2-,3- or 4 point bending tests. The NFAT program does not account for lateral distribution andhealing under practical traffic and climate conditions. These aspects are covered in the ShellPavement Design package SPDM.

The NFAT-nomograph was designed for standard bitumens and will, in general, under-estimatethe benefits of the effect of polymer modification on fatigue properties.

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Figure 3: The NFAT Nomograph.

Page 26

30%

201510

5

54

2

86

4

2

8109

1010

N/m2

Stiffness modulus of the mix Smix

Volume of bitumen

in mix Vb'

108

6

4

2

108

Fatigue life

Nfat

107 106

105

104

10–3864

Asphalt fatiguestrain εfat

210–410–5 8642

Example: If

then

= 2 x 109 N/m2Smix

= 10%Vb

= 106N

= 2.7 x 10–4εfat

Electronic Reproduction, 1998

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References

1. C. van der Poel, A general system describing the visco-elastic properties of bitumens and itsrelation to routine test data, Journal of Applied Chemistry, London (1954), 4, 221.

2. J.C. van der Werff, C.D.M. Beverwijk, J.H. Breuker and T. Harrison, Quality of Paving GradeBitumen. Definition on the Basis of Rheology and Qualogon® Targets, 5th EurobitumeCongress in Stockholm, June 1993.

3. F. Bonnaure, G. Gest, A. Gravois and P. Ugé, A new method of predicting the stiffness ofasphalt paving mixtures, Proceedings of the Association of Asphalt Paving Technologists,Volume 46 (1977).

4. Shell Pavement Design Manual, Shell International Petroleum Company Ltd. London, 1978.

5. W. van Dijk and W. Visser, The energy approach to fatigue for pavement design. Proceedingsof the Association of Asphalt Paving Technologists, Volume 46 (1977).

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