guidelines for writing good asphalt related research...

highway research journal, july – DeceMBer 2013 17

GuIDELINES FOR WRITING GOOD ASPHALT RELATED RESEARCH PAPERSprithvi s. Kandhal* & raJiB B. malliCK**

ABSTRACT

Review of numerous asphalt related Technical Papers published in India has revealed that their quality is generally not up to international standards. This Paper gives guidelines for improving the quality of research papers on Asphalt Technology.

Comprehensive Guidelines have been given for various typical sections generally contained in Research Papers such as abstract, introduction, scope of study and objectives, review of literature, materials, testing procedures and experiment plan, construction data (if any), test data and statistical analyses, conclusions and recommendations, acknowledgements, references, and appendices. Many of the Guidelines are also applicable and useful for Research Papers on other Areas of Transportation.

1 BACKGROuND

It has been observed from the review of numerous Technical Papers related to Asphalt Technology published in India that their quality is generally not up to international standards. This also applies to Research Papers/Reports produced by premier institutions in India.

The following guidelines have been prepared based on the authors’ experience for improving the quality of papers on asphalt technology. Guidelines are given in order of various typical sections generally contained in a Research Paper such as abstract, introduction, scope of study and objectives, review of literature, materials, testing procedures and experiment plan, construction data (if any), test data and statistical analyses, conclusions and recommendations, acknowledgements, references, and appendices.

2 ABSTRACT

The Abstract (or Synopsis) should be very brief, generally, not exceeding 500 words. The reader should get a general idea about what is contained in the Paper. It needs to have four primary segments: Introduction/Importance of the Subject (2-3 sentences); Objective (one sentence); Scope of the Study or Variables Examined (2-3 sentences); and Main Conclusions (2-3 sentences).

3 INRODuCTION

The Introduction should be brief. It should mention the importance of the specific research topic keeping in perspective the general field and/or needs of India. The Introduction should lead into the next section of objective(s).

4 SCOPE OF STuDy AND OBJECTIVE(S)

Briefly outline the scope of this study so that the reader clearly understands its limitations. Then give the primary and secondary (if any) objectives of this study. It has been observed that sometimes many objectives are given but some are not reported later in the Paper. This usually happens if the Research Paper has been prepared from a large Research Report by “copy and paste”.

5 REVIEW OF LITERATuRE

Before formulating an experimental research plan it is necessary to review the literature on the subject to determine what work has been done in the past by other researchers and what results/conclusions were drawn. This is necessary to avoid repetitions or pitfalls. All Papers which have reported either positive or negative findings on a new technology should be included. Sometimes, when the researchers have obtained positive results from a process or a product (such as any additive), they tend to exclude those Papers which reported negative results on a similar product. It is not unusual and should be accepted that researchers sometimes get different results when similar products or technologies are investigated.

The review of literature can be reported either in a chronological order or can be subdivided into subsections of the larger subject if deemed appropriate. For example, if a product has been evaluated in the laboratory as well as field it can be subdivided into laboratory and field performance.

The researchers must use the internet to access literature on the subject being researched. There are many research engines such as Google and Yahoo. However, the most

* Associate Director Emeritus, National Center for Asphalt Technology, USA ([email protected]) ** Professor, Worcester Polytechnic Institute, Worcester, USA ([email protected])

The views expressed in the Paper are personal views of the author. For any query, the author may be contacted by e-mail.

highway research journal, july – DeceMBer 201318

prithvi s. Kandhal & raJiB B. malliCK on

guidelines For Writing good asphalt related researCh papers

important and comprehensive highway related database is: (a) US Transportation Research Board (TRB)’s Transportation Research Information Services (TRIS) Database and (b) the Organisation for Economic Cooperation and Development (OECD)’s Joint Transport Research Centre’s International Transport Research Documentation (ITRD) Database.

Transportation Research Information Database (TRID) is a newly integrated database that combines the records from TRB’s TRIS Database and the OECD’s ITRD Database. Together, TRID provides access to over 9,00,000 records of Transportation Research. TRID, released in 2011, is the world’s largest and most comprehensive bibliographic resource on transportation research information. It is produced and maintained by the Transportation Research Board of the National Academies with sponsorship by State Departments of Transportation, the various administrations at the U.S. Department of Transportation, and other sponsors of TRB’s core technical activities. ITRD is produced by ITRD member organizations under the sponsorship of the Joint Transport Research Centre (collectively JTRC) of the International Transport Forum and the Organisation for Economic Cooperation and Development (OECD).

The Transportation Research Board’s TRID Website is a leading tool for transportation professionals to stay updated on the status of world-wide transportation research. TRID covers all modes and disciplines of transportation and contains more than 9,00,000 records of references to books, technical reports, conference proceedings, and journal articles in the field of transportation research. Almost 500 serial titles are regularly scanned and indexed for TRID. More than 64,000 records contain links to full-text documents.

TRID can be accessed at the following link:

http://trid.trb.org/search.aspx

Although the TRID search will not give the whole paper, it will give a full record of each Paper/Report containing the Title, Authors’ Name(s), Name of Journal, its Volume Number, and an Abstract. By reading the abstracts, the researchers can determine which papers are worth obtaining from technical library.

Use of proper keywords and their combinations thereof is essential to get the optimum results from a literature search on the internet. For example, a literature search is needed on “Field Performance of Steel Slag in Bituminous Mixtures”. If one key word “bituminous mixture” is used it will produce a large number of references. The literature search can be narrowed down further by including “steel slag” as

additional keyword. That is, use “bituminous mixture” and “steel slag”. If needed, it can be further narrowed down by adding “field performance” keyword.

Quite often, we also have to use “or” when there are other synonyms for a term or keyword. For example, try “bituminous mix” or “asphalt mix” or “hot mix asphalt” to get maximum number of references. For example, in the previous example we can try: “bituminous mix” or “asphalt mix” or “hot mix asphalt” and “steel slag” or “open hearth slag”.

Select the desired Papers from the TRID search and print out full records (including abstract) before beginning the research. If the full Paper is not available in the nearby technical library, send an e-mail to authors. With good luck, they may provide a soft copy via e-mail. Many full Papers on Asphalt Technology published by a specific author are also sometimes available on the internet, for example at the following link:

http://www.scribd.com/doc/47323198/Kandhal-Asphalt-Literature-With-Web-Links

Finally, do search on Google and Yahoo Search Engines. Surprisingly, additional references to Papers/Reports/Trade Journals can be available. 6 MATERIALS

All materials used in laboratory and/or field evaluation should be characterized, documented and reported. References should be made to standard tests used such as ASTM and BIS (Bureau of Indian Standards). If non-standard experimental or research tests are used they should be described briefly and references should be made to published Papers where such tests have been described in detail.

Examples of some materials used in asphalt research are given below:

6.1 Bitumen as a Binder

Bitumen binder should be classified as per prevailing grading system in India. For example, Viscosity Graded (VG) bitumen such as VG-30 and VG-10 should be used rather than old penetration graded bitumen such as 60/70 and 80/100. Sometimes, researchers just reproduce the test data supplied by the refineries. This should be discouraged. Bitumen binder should actually be tested, especially, viscosity at 60C by the researchers’ laboratory. In case of modified bituminous binders, its type such as elastomer and plastomer and the type of polymer should be reported.




In case of Crumb Rubber Modified Bitumen (CRMB), the size and amount of crumb rubber used should be reported together with the details of its manufacturing process such as blending temperature and digestion time.

6.2 Aggregates

The geological type of the aggregate such as limestone, granite and sandstone should always be reported because it can influence mix test results such as moisture susceptibility (stripping). Besides standard tests non-standard tests which are germane to the research should also be conducted to fully characterize the aggregates used. Test values should not be reported to many unnecessary decimal values because that makes them rather incomprehensible to reader.

For example, specific gravity values (bulk, apparent or effective) should be reported to three decimal places (not less or more) and gradation should be rounded off to whole number except the percentage passing 0.075 mm sieve should be reported to one decimal place. Table 1 gives typical aggregate test values rounded off appropriately.

Table 1 Aggregate Test Values (Example)

Test Rounded off value

Bulk specific gravity 2.672

Water absorption 1.2 %

Flakiness and elongation index 24.5 %

Aggregate impact value 17.8 %

Los Angles abrasion value 28.3 %

Sodium or magnesium soundness 8.3 %

6.3 Additive

If an additive is used for the bituminous binder or bituminous mix, it should at least be described generically if it is not advisable to mention its brand name. Some researchers do not do it and just state “certain additive” was used. Such research is hardly useful to other researchers and the technical community at large if they cannot procure that “certain additive” to replicate the research in their laboratory.

Many times there is no harm in mentioning the brand name of the additive if it is acknowledged under the Acknowledgement Section with the following statement: “The research sponsoring organization and the authors do not endorse any proprietary products or technologies mentioned in this paper. These appear herein only because they are

considered essential to the objective of this paper”. This is permitted by the US Transportation Research Board.

6.4 Bituminous Mixes

The procedure used in preparing the bituminous mixes such as mixing temperature and the procedure used in compacting the bituminous mix such as Marshall and Superpave gyratory compaction, should be described. Again, the test values obtained for compacted bitumen mixes should be rounded off appropriately as shown in Table 2. For example, it is of no use to report air voids as 5.173 per cent which rather makes it incomprehensible to reader; it should be reported as 5.2 per cent.

Table 2 Mix Design Test Data

Property Rounded off value

Specific gravity of bitumen 1.028

Bulk specific gravity of aggregate 2.672

Effective specific gravity of aggregate 2.715

Maximum specific gravity of mix 2.529

Bulk specific gravity of compacted specimen

2.409

% Air voids 4.8

% VMA 13.7

% VFA 65.9

Stability, kg 2,210

Flow, unit 4.2

7 LABORATORy AND/OR CONSTRuCTION TEST DATA

It is a normal practice to conduct laboratory or field tests in triplicate and average values reported. It should always be mentioned whether the reported test values are individual test results or averages of duplicate or triplicate specimens.

Construction details should be documented and reported; for example, prevailing ambient temperature, lay down mix temperature, compaction temperature, type and number of rollers used. Construction test data such as core density should be obtained at random locations. Air voids in compacted mat have highly significant effect on pavement performance. For example, if control test section has 10 per cent air voids at the time of construction and the experimental test section has 6 per cent air voids at the time of construction, the field




performance of the former is expected to be worse than the latter. Therefore, just reporting the relative field performance without reporting percent air voids in the mat at the time of construction may be misleading.

Obviously all pertinent test information which has a direct effect on performance should be reported. If some information is not available it should be acknowledged in the Paper.

Any unusual circumstances and/or behaviour of the mix during construction (such as tenderness, harshness and difficulty in placing /handwork) should be documented because these factors also affect the performance of the test sections.

Relative costs of construction for control and experimental sections, if available, should also be reported so that life/cycle costs can be determined later.

8 STATISTICAL ANALySES OF TEST DATA

Statistical analyses of test data cannot be done successfully unless there is an experimental program which lists all independent variables and all dependent variables; tests to be conducted; number of samples to be tested; and replicates of each test. As a minimum such an experimental program would yield means and standard deviations for the test values.

There are two basic necessities of writing a successful research paper: what we say should be “new” and, it should be “convincing”. Statistical analyses help the author in writing the Paper in a convincing manner. Such analysis can be carried out by various software such as SPSS. For example, see the following link:

http://www-142.ibm.com/software/products/in/en/ spss-stats-standard

Commonly used statistical methods are described below, with examples. All quoted equations have been taken from the following textbook: Steel, Robert G. and James H. Torrie, "Principles and Procedures of Statistics", New York: McGraw, 1960.

The reasons for conducting each statistical method are also presented.

8.1 Analysis of Variance (ANOVA)

One very common item in Research Papers is the presentation of a new material or a construction process,

and the discussion on its effect on a measured test property. In such a case, Analysis of Variance (ANOVA) must be carried out before making the conclusion regarding the applicability of the new material/method. One way ANOVA will help to determine what proportion of variance is accounted for by the systematic effect and what proportion is not accounted for; this helps us to determine whether there is a significant effect of x (independent variable) values on the y (dependent) values. An example is as follows:

Suppose researchers use four different treatments of fiber to reduce the draindown of asphalt binder in Stone Matrix Asphalt (SMA) mixes. Draindown tests were carried out using wire baskets and the total draindown at 120 min was recorded for three replicates for each treatment plus one case (termed control) in which no fiber was used. Test results are shown in Fig. 1.

Fig.1 Plot of Additive Type and Amount Versus Draindown

To make convincing conclusions regarding the effect of the fibers, the first question that needs to be answered is the following: Is there at least one treatment mean that is different from the others? ANOVA can be conducted to answer this question, as shown in Table 3.

Table 3 Example of ANOVA

Source SS DF MS F

Treatments 3.124 4 0.781 8.089 > 3.48

Error 0.966 10 0.097

Total 4.090 14

At 95 per cent (Probability, p = 0.05) confidence level Fcritical = 3.48; (In fact, the F value matches Fcritical at a p value of 0.004)




The different statistics are described as follows:

SS Treatments (SST) = ΣK

i=1ri n

Ti2 T

2

Ti=ith Treatment TotalΣk

i=1T = Ti is the grand total

ri = number of replicates for ith treatment

k = number of treatments

SS Error (SSError) =TSS-SST

TSS = S2all * df all

S2all = Σ

k

i=1(Xi-- MeanTotal )

2/ N-1

N = Σk

i=1ri

MeanTotal = TN

MSTreatment = SST

k-1

MSError = SSError

N-K

F = MSTreatment

MSError

Since the F value exceeds the Fcritical value at 95 per cnet confidence level, it can be concluded that there is at least one treatment that differs from the others. In other words, it is confirmed that there is a significant effect from at least one treatment of fibers.

8.2 Separation of Means (Ranking)

The next question that arises is: which treatment(s) are significantly different, and how can they be ranked? This answer can be obtained by conducting a multiple comparison technique, such Tukey’s Honestly Significant Difference (HSD) test, which compares the difference between two means against a standard error.

Let us utilize the last example (effect of fiber on draindown) to conduct Tukey’s HSD test. Recall that in our example, from the ANOVA Table, MSerror = 0.097; DFerror = 10; n (or r, number of replicates) = 3

The decision rule is as follows:

If , (X1-X2)SE

q critical

then the difference between the means X1 and X2 will be declared significant.

where,

Standard Error (SE) = MSError

n√Where, critical is obtained from Tukey’s probability Table for the specific DFTreatment and DFError)

In the given example,

SE = 0.179; DFTreatment = 4; DFError = 10

Table 4 shows the results of Tukey’s HSD analysis, and Table 5 shows the groupings. Note that means with the same groups do not differ significantly. From this analysis it can be concluded that the addition of fiber A at 0.3 per cent or B at 0.1 and 0.3 per cent definitely reduces draindown, as compared to the no additive mix.

Table 4 Tukey’s HSD Analysis

Comparisons Tukey’s parameter

(X1-X2)SE

qcritical Significantly different?

No Additives

Fiber-A-0.1%

2.137 4.33 No

No Additives

Fiber A-0.3%

5.463 Yes

No Additives

Fiber B-0.1%

5.704 Yes

No Additives

Fiber B-0.3%

6.800 Yes

Fiber-A-0.1%

Fiber A-0.3%

3.326 No

Fiber-A-0.1%

Fiber B-0.1%

3.567 No

Fiber-A-0.1%

Fiber B-0.3%

4.664 Yes

Fiber A-0.3%

Fiber B-0.1%

0.242 No

Fiber A-0.3%

Fiber B-0.3%

1.338 No

Fiber B-0.1%

Fiber B-0.3%

1.096 No




By considering the rut depth as a dependent on the level of percentage of sand used, one possible mathematical model is:

yij = μi + εij

yij = μ + (μ-μ) + εij

yij = μ + τi + εij

where, yij = jth response (rut depth) observed for the ith treatment (sand percentage) level, μ = overall mean, μi = mean of all of the responses of the ith treatment, εij = error

The responses yij can be used to test:

H0 = μ1 = μ2 =μ3 = μ4

→ τi = 0 for, i = 1, 2, 3 and 4 HA : at least one treatment mean differs → τi ≠ 0 for, i = 1, 2, 3 and 4using an F ratio test based on the following ANOVA:

Table 7 ANOVA of Rut Depth Data

Source SS DF MS F

Treatments 363.40 3 121.13 15.09

Error 128.40 16 8.03

Total 491.80 19

If the F test fails to reject H0 then the model

yij = μi + εij

is sufficient to explain the yij

When an analysis of variance is performed the variance of the response yij is broken down according to the components in the model being used to explain the responses.

For example, with the model yij = μ + τi + εij

yij - μ + τi + εij

we can describe the variability in yij - μ (Total Sums of Squares about the mean = TSS) using two components; variability attributable to the τi treatments effects (SStreatments)and the remaining unexplainable variability known as pure error ( [ SS ] ↓ Error)

However, if the treatments levels are quantitative (treatment i has xi amount of x which causes response yij) it may be reasonable to assume that the mean response μi depends on the treatment level xi in a linear manner (i.e. μi = μx=0 + βxi). The model to explain treatment responses would then be

yij = μx=0 + βxi + εij

Table 5 Grouping of the Treatments on the Basis of Tukey’s HSD Analysis

Type of treatment Group

No Additives A

Fiber-A-0.1% AB

Fiber A-0.3% BC

Fiber B-0.1% BC

Fiber B-0.3% C

8.3 Simple Linear Regression

Suppose it is necessary to determine whether there is a relationship between the percentage of natural sand (for example river sand with relatively rounded aggregates) in the total fine aggregates of the mix and rut depths observed from wheel tracking tests. An experiment involving four levels of sand percentage: 15, 20, 25 and 30 per cent was conducted. For each percentage of sand, five mixes were prepared and tested with a wheel tracking test. The results are as given in Table 6 and are shown graphically in Fig. 2.

Table 6 Rut Depths in mm from Wheel Tracking Experiments

Percentage of natural sand

Test No. 15 20 25 30

1 7 12 14 19

2 7 17 18 25

3 15 12 18 22

4 11 18 19 19

5 9 18 19 23

The research question is: Can the average rut depth be explained with a straight line relationship between average rut depth and percentage of sand used?

Fig. 2 Plot of Percentage of Natural Sand versus Rut Depth




The variance of the estimator for β is:

The standard error for β , σb can be estimated by replacing σ2 with MSE (see ANOVA, Table 5) in the variance formula and taking the square root:

√σβ = [Σ

ni xi2)- ﴾Σ ni xi 2]

N

MSE

In the example problem, √σ β = [ 625.0]

8.03

= 0.1133

If we want to test the hypotheses:

H0 : β = 0 HA : β ≠ 0

Using t statistic, we calculate:

tcalculted = σ β β - β0

tcritical = 0.1133

= 6.63440.752 - 0

t 0.05 (2), n = 16 = 2.12

Since at

α = 0.05 level, tcalculted >

tcritical

we reject H0 ; we conclude that there is a linear relationship between rut depth and sand percentage.

The variability of by the linear regression needs to be determined to estimate the strength of the regression.

SSregression = β ﴾Σni xi ŷi – ﴾Σ ni xi) ﴾Σ ni ŷi)NIn this example,

SSregression = 0.752 ﴾7,715 – 20 = 353.441,44,900 )

Hence, from Table 7, we can have Table 8 as follows:

Table 8 Partitioning of Sum of Squares

Source SS DF MS FTreatments 363.40 3 121.13 15.09 Linear 353.44 1 353.44 44.02 Remainder 9.96 2 4.98 0.62Error 128.40 16 8.03Total 491.80 19

This defines a simple linear regression model for explaining the responses.

To test whether a simple linear regression effect exists between the response and the treatment levels, the following hypothesis are tested:

H0 : β = 0 HA : β ≠ 0 In order to test β = 0 we need an estimate of β and the standard error for the estimate.

To find a best estimate β for β, consider the estimate α + β xi for the regression μx=0 + βxi to be best if it has the property that the sum of the squared distances from the sample points (xi , yij) to the points on the estimate of the sample regression line (x^i, α + βùx^i) is as small as possible.

For this, the requirement is:

Σ Σ[(﴾yij

[( - ﴾α+ β xi﴿﴿2

needs to be minimized with respect to α and β

The values of α and β which minimize (Σ Σ [(﴾yij

[( - ﴾α + β xi﴿﴿

2 are called the least square estimates. The least square estimates can found as follows:

β =

Σ ni xi ŷi – ﴾Σ ni xi) ﴾Σ ni ŷi)

Σ ni xi2-

﴾Σ ni xi)2N

N

α = [(Σ ni ŷi – β ( Σ ni xi)]

N

where,

ni = the number of responses measured at xi

N = the total number of responses taken in the experimentIn the given example, the least square estimates are found to be:

20β = 7715.0 – ﴾144900.0)

= 0.752625.0

α = [(322.0 – 0.752 (450)]/20= -0.82

Therefore, the estimate of the true average rut depth μi at a level xi is:

ŷ = -0.82+0.752 xi

24




It is also possible to test:

H0 : No linear relation between percentage sand and rut depthHA : There is a linear relation between percentage sand and rut

depth

by using the F statistic,

F = 8.03 = 44.02 >353.44 Fcritical , 0.05 (1) 16

For assessing how good the linear regression model is for reducing errors in predicting rut depth based on percentage sand, the coefficient of determination (R2) is used, where:

R2 = Total sum of squaresSSRegression

In this example,

R2 = 491.8 = 0.7186353.44

This means that 71.86 per cent of the variation in rut depth can be explained by the linear relationship rut depth versus percentage of sand.

We can also conduct a Lack of Fit test, to test whether a linear relationship between the rut depth and the percentage of sand is reasonable to assume by interpreting the remainder of the treatment sum of squares as sum of squares attributable to departure from a linear relationship.

The notation SSLack of Fit is used for the part of the Treatment Sum of Squares that is attributable to deviation from a straight line model.

Partitioning the Treatment Sum of Squares into SSRegression

and SSLack of fit the ANOVA Table can be presented as in Table 9.

Table 9 Further Partitioning of Sum of Squares

Source SS DF ms FTreatments 363.40 3 121.13 15.09 Linear 353.44 1 353.44 44.02 Lack of Fit 9.96 2 4.98 0.62Error 128.40 16 8.03Total 491.80 19

To test whether the assumption of a straight line model is reasonable or not, an F statistic test can be conducted, comparing the MSLack of fit to MSError.

In this example,

H0 : The assumption of a straight line relationship between average rut depth and percentage sand is reasonable

HA :A straight line relationship is unreasonable Fcalculated = MSError

MSLack of Fit = 0.6 > – Fcritical , 0.05 (1), 2, 16

Hence, H0 cannot be rejected.

8.4 Confidence Interval

Sometimes it is good to know what is the interval within which researchers are confident to certain degree that the average value of a dependent variable will lie for a specific value of the independent variable, that is the confidence interval.

If x0 denotes a given value of x, an estimate of μy, xo is

ŷ0 = α + β x0

The standard error for y0 is

S^﴾ŷ0﴿ =√﴾MSE(1/N + (x^0 – x) 2/ (Σ ≡ nî xî 2– (Σ ≡ n̂ i xî N 2/N ))

To form the (1-a) 100% confidence interval use

ŷ0 ± ta(2),v MSE S ŷ0

For the given example, let us find a 95 per cent confidence interval when 20 per cent sand is used. Here,

ŷ = _ 0.82+0.752x and xo = 20

So, ŷO = _ 0.82+0.752 (20) = 14.22

20 20

8.03 1√ = 0.69410750 - 4502

(20-22.5)2+ )(S ŷO =

t 0.05 (2) 16 = 2.1199

Hence, the 95 per cent confidence interval for μy, x20 is

14.22 ± 2.1199( 0.694) 12.75< μy, x20 < 15.69

This means that the researchers are 95 per cent confident that the true average rut depth is between 12.75 and 15.69 mm when the percentage of sand is 20.

8.5 Detection of outlier

Sometimes there are bad data points resulting from poor experimental methods or practices. These data do not follow the trend of the other data, and deviate markedly from the

^ ^^




rest of the observations. Such data need to be identified and deleted from the rest of the data before the set of data is analyzed. There are various standard methods of detecting outliers and different statistical methods employ different procedures.

One simple procedure is through the use of box plots. The method consists of the following steps:

(i) Find the lowest (LO), highest (HI), and the 25th, 50th and the 75th percentile data. Mark these either on a horizontal or a vertical scale.

(ii) Draw a box so that the 25th and the 75th percentiles represent the ends of the box. Also, draw a line parallel to these ends at the 50th percentile.

(iii) Calculate the value C = 33 (x.75- x.25)

(iv) If HI ≥ x.75 +C draw line from x.75 to HI. Otherwise draw line from x.75 to x.75 +c Cand mark of HI with an asterisk (to detect outliers)

(iv) If lO ≥ x.25 – C draw a line from x.25 to LO. Otherwise draw a line from x.25 to x.25 – C and mark LO with an asterisk (de detect outliers)

Consider the following example. In 2004, rut depth data (mm) were obtained from several asphalt paving projects constructed in 1998, year as shown in Table 10. Can the researchers identify the outliers?

Table 10 Rut Depth Data of Projects from 1998, as Obtained in 2004

Project No. Rut depth, mm1 2.542 12.73 1.274 5.085 5.336 6.357 7.628 3.819 2.5410 5.5911 4.8312 11.4313 2.79

14 5.08

LO = 1.27; HI = 12.7; x.25 =3.048; x0.5 =5.08; x.75 =6.159

Fig. 3 Box Plot and Outliers

With the data, Fig. 3 is drawn. It can be seen that the data points 11.43 and 12.7 are identified as outliers.

8.6 Determination of Percent Within Limits (PWL)

In many specifications, the evaluation of a job for acceptance or rejection is based on the Percent Within Limits (PWL) of results for each lot of the pavement. PWL (or percent conforming ) is defined as the percentage of the lot falling above the lower specification limit (LSL), beneath the upper specification limit (USL), or between the USL and LSL. Although tests can be applied to verify it, in general, the population of most test results is assumed to be normally distributed, and the use of this procedure ensures the consideration of both average and variability of the test results for evaluation of the “quality” of the product. The PWL approach is used by many states in the US for acceptance of asphalt paving projects. The PWL concept is based on the use of the area under a standard normal distribution. Consider the following example.

Suppose the bitumen content of five samples have been determined to as shown in Table 11. The permissible range is 5% ± 0.4%. Determine the Percent Within Limits (PWL).

Table 11 Bitumen Content Data

Sample No. Bitumen content1 4.62 5.23 4.94 4.85 4.5




The steps are as follows.

(i) Determine mean value, μ = 4.8; standard deviation, σ = 0.27

(ii) The specification limits are from 4.6 per cent to 5.4 per cent

(iii) Calculate Z statistic for upper limit:

0.27= 2.22x - μZ= =σ

5.4 - 4.8

From Table 12, it can be concluded that percentage above 5.4 is 1 - 0.5 - 0.4868 = 0.0132

(iv) Calculate the Z statistic for lower limit:

0.27= 0.74x - μZ= =σ

4.6 - 4.8

From Table 12, it can be concluded that the percentage below 4.6 is 1 - 0.5 - 0.2704 = 0.2296

(v) Hence, pWl = (1-(0.0132 + 0.2296))*100 = 75.72%

Many state highway agencies in the US have linked PWLs to pay factors, which varies from state to state. For example, one state determines the pay factors as given in Table 13. In such Tables, PWLs are listed in increment of one per cent; Table 12 lists some excerpts only from one such Table.

Table 12 Areas under the Standard Normal Distribution

Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.03590.1 0.0398 0.0138 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.07530.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.00410.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.15170.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.18790.5 0.1915 0.1950 0.1986 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.22240.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2464 0.2486 0.2517 0.25490.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.28520.8 0.2881 0.2910 0.2939 0.2967 0.3925 0.3023 0.3051 0.3078 0.3106 0.31830.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.33891.0 0.3413 0.3438 0.3461 .03485 0.3508 0.3531 0.3554 0.3577 0.3599 0.36211.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.38301.2 0.3849 0.3969 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.40151.3 0.4032 0.4949 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.41771.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.43191.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.44411.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.45451.7 0.4554 0.4564 0.4573 0.4582 0.4581 0.4599 0.4608 0.4616 0.4625 0.46331.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.47061.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.47672.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.48172.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4216 0.4850 0.4854 0.48572.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.48902.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.49162.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.49362.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.49522.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.49642.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.49742.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.49812.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.49863.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.49803.1 0.4990 0.4991 0.4991 0.4991 0.4992 0.4992 0.4992 0.4992 0.4993 0.49933.2 0.4993 0.4993 0.4994 0.4994 0.4994 0.4994 0.4994 0.4995 0.4995 0.49953.3 0.4995 0.4995 0.4995 0.4996 0.4996 0.4996 0.4996 0.4996 0.4996 0.49973.4 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4998




Table 13 Example of PWLs Linked to Pay Factors

Percent Within Limits (PWL)

Pay Factor (Percent of Bid Price)

100 10090 9580 7870 6064 50

Less than 64 Remove and replace

9 SuMMARy, CONCLuSIONS AND RECOMMENDATIONS

Many readers do not have time to read the whole Paper including test data and analyses. They rather prefer to read this section to get an essence of the whole Paper. Therefore, a brief summary of the research (especially, its importance and main objectives) consisting of a short paragraph will be useful to such readers.

Conclusions should be in proper order and should always be supported by reported test data. Many research projects are not comprehensive in scope and, therefore, firm conclusions cannot be drawn. In such cases, it should be acknowledged that the conclusions are based on limited test data.

Recommendations may include (a) need for more research or study and (b) how the research results can be implemented by the highway community.

10 ACKNOWLEDGEMENTS

It is customary to acknowledge the organization which sponsored the research reported in the Paper. Names of key people who assisted in conduction the research should also be acknowledged.

If brand names of some materials are used in the Paper, the following statement can be made: “The research sponsoring organization and the authors do not endorse any proprietary products or technologies mentioned in this Paper. These appear herein only because they are considered essential to the objective of this Paper”.

The following statement can also be made if applicable: “The contents of this Paper/Report reflect the views of the authors who are solely responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views and policies of the research sponsoring organization. This Report does not constitute a standard, specification, or regulation”.

11 REFERENCES

References can be listed in alphabetical order and quoted in the text accordingly. References can also be listed in the order they are quoted in the Paper. Guidelines given by the publishers of the journal need to be followed.

References should be complete in all respects: authors’ names; title of the Paper/Report; and Journal name, volume number and year.

If reference is made to a URL address from an internet website, the date it was accessed by the researchers should also be mentioned.

12 APPENDICES

There may be a need to report voluminous test data, describe a test method in detail or include a tentative specification in the Paper. Appendices are very effective for such purposes. They reduce the size of the main body of the Paper making it more readable. Only those readers who desire more details can refer to the appropriate appendices.


CALIFORNIA BEARING RATIO (CBR) AT DIFFERENT RELIABILITy LEVELS AND COEFFICIENT OF VARIATIONS

swapan Kumar Bagui*

ABSTRACT

This Paper presents the reliability analysis of California Bearing Ratio (CBR). There are several methods for reporting CBR values namely 10th percentile/90th percentile CBR Method, average CBR of same source of material and other methods. There are several factors for performance of CBR. These are: moisture content variation, compacting effort, thickness variation, drying and wetting or freezing and thawing. It is possible to achieve Laboratory CBR in the field. This Paper presents the reliability analysis of the CBR value with a case study using cumulative distribution approach and determine lower and upper limit of CBR value and expected CBR value using Monte Carlo Simulation Method. Finally, this Paper presents CBR at different reliability levels which may be useful in finalization of design CBR value.

1 INTRODuCTION

In India, more than10,000 km roads have been constructed with four lanes configuration in the last ten years. Major roads are found incapable of carrying design traffic within few year/ before design life. Rut and fatigue failure occurred within few years after construction of road. The causes of failure are: Poor subgrade CBR, Poor level of subgrade compaction, laboratory CBR and field CBR were not correlated during execution and inadequate quality control of the work during execution, substandard asphalt concrete work and asphalt uses in the mix, etc. This Paper presents CBR with consideration of different reliability levels which were not considered in design and develop a methodology for determination of CBR at different reliability levels.

2 LITERATuRE REVIEW

Three samples shall be tested for CBR and average values shall be reported if variation of CBR values does not vary as shown in Table 1 [(IRC:37-2001) and (IRC:37-2012)].

Table 1 Permissible Variation of CBR

CBr (%) Maximum Variation (%)5 ±1

5-10 ±210-30 ±3>30 ±5

Where variation is more as reported in Table 1, minimum six samples shall be tested and average value shall be reported (IRC:37-2001).

Subgrades are inherently variable in nature and reflect the changes in topography, soil type, and drainage conditions that generally occur along an existing or proposed road alignment. Hence the selection of a subgrade CBR value requires adequate

consideration of the degree of variability within a particular project section, and the quantity and quality of data on subgrade properties. The investigation methodology and the strength (or stiffness) assessment techniques adopted to determine the design support condition should be consistent with the required level of performance risk for the pavement under consideration. More comprehensive testing programs and/or conservative design values are commonly selected when the consequences of premature pavement distress are highly significant or considered unacceptable (Austroads 2010).

There are primarily two modes of testing available for estimating subgrade support values: field testing and laboratory testing. Field testing is applicable to situations where the support values from the in situ subgrade soil conditions are expected to be similar to those of the proposed pavement. Laboratory testing is applicable both in that situation and also when subgrade support is to be determined from first principles. Consideration should be given to the sample density, moisture, and soaking conditions which simulate the expected pavement support while in-service. To ensure homogeneous sub-sections of subgrade, the CBR values should have a coefficient of variation (i.e. Standard deviation divided by the mean) of 0.25 or less. The ten percentile level (i.e. 90 per cent of results exceed this level) is commonly adopted for the design of highway pavements. For roads in arid climates, or roads of lesser importance, higher percentile values may be appropriate (Vic Roads 2004 and 2005).

Presumptive CBR approach may be used when no other relevant information is available. It is particularly useful for lightly-trafficked roads where extensive investigations are not warranted, and also when conducting preliminary designs for all roads. Typical presumptive values of CBR are given in Table 2. However, such values should only be utilized on the basis that the information will be supplemented by taking account of local experience (Austroads 2008).

* C-78, Shalimar Garden, Flat-F1, Sahibabad, Ghaziabad (UP) ; ( [email protected])

The views expressed in the Paper are personal views of the author. For any query, the author may be contacted by e-mail.


sWapan Kumar Bagui on

CaliFornia Bearing ratio (CBr) at diFFerent reliaBility levels and CoeFFiCient oF variations

Table 2 Typical Presumptive Subgrade Design CBR Values

Soil Type Presumptive CBR (%)

Highly plastic clay 2-5

Silt 2-4

Silty Clay /Silty Clay 3-6Sand 5-10

The design subgrade CBR must not exceed a value calculated using the following expression:

Design CBR = C-KS (RTA Austroads Guide supplements 2012);

where C = Mean of all CBR determinations within a single design unit; S = Standard deviation of all CBR determinations within a single design unit and; K = Reliability Factor depending on climatic and drainage condition.

Percentile CBR value is used by various countries considering reliability of CBR value.

Existing CBR at field density has been carried out during preparation of many Detailed Project Reports (DPR) in India (Widening and strengthening project). It has been found that existing CBR is on the lower side of Design CBR adopted during preparation of DPR / construction of existing road. This means that the laboratory reported CBR has some reliable factors which were not taken attention during preparation of the DPR.

3 OBJECTIVE AND SCOPE

Based on lead from past study and need of present research work, the objective and scope are identified as:

The object is to analyze a road project to report CBR at different reliability levels. This needs to be analyzed properly.

Based on this objective, the scope of the present work is the determination of CBR taking a case study using Cumulative Distribution Approach and Monte Carlo Simulation Techniques for the analysis of CBR at different reliability levels.

4 PROPOSED METHODOLOGy

The proposed methodology to determine CBR at different reliability levels has been described as follows:

Following steps have been considered to determine CBR:

Determine average CBR same source/borrow area. 1. Varying Standard deviation/mean (Coefficient of 2. variation) of CBR values as 0.05, 0.075, 0.1, 0.15 and 0.20.Prepare Cumulative Frequency distribution graph3. Determine CBR at 80, 85, 90, 95, 97.5 and 99.99 4. per cent confidence level both upper and lower side of mean. Calculate lower level, base level and upper level5. Prepare graphs, regression equations, etc.6. Determine expected CBR value using Monte Carlo 7. Simulation Method.

5 CASE STuDy

Project road corridor is selected section of NH-79A, NH-79 and NH-76. Three samples were collected from each borrow area and thirty borrow areas were considered for analysis. Average CBR of 7, 10 and 15 per cent were found for these borrow areas.

6 DISCuSSION

Lower and upper limit of CBR has been calculated by Cumulative Distribution Method as shown in Fig. 1. Analysis has been calculated using Average CBR values of 10 and has been plotted in graph and illustrated in Fig. 2 for CBR 10 per cent. It has been found that R2 value decreases with increasing value of coefficient of variation. Normal distribution graph has been plotted using data as presented in Annexure 1. Table A1 has been developed from Annexure 1. Values of CBR at 85 Percent confidence level for coefficient of variation 0.2 are found 7.92 and 12.08 per cent as shown in Annexure 1. Similarly values of CBR at other coefficient of variations have been calculated and reported in Table A1 and these data have been used to prepare Fig. 2. From Fig. 2, it is revealed that lower and upper limit CBR varies linearly with positive and negative slope for upper limit and lower limit of CBR and zero slopes for base CBR. Non-dimensional regression equations have also been developed. The regression equations are shown in Tables 3 (CBR 10 per cent), and 4 (CBR 15 per cent) for different confidence levels ( 80 to 99.99 per cent) and varying σ/µ from 0.05 to 0.20. From regression equations (reported in Tables 3 and 4), it is found that data have been correlated with good correlation (R2 values vary from 0.99 to 1.00). Similar equations can be developed for other CBR values. From known CBR, CBR at various reliability levels can be determined. Tables 3 and 4 are very useful for constructing good road,




to determine CBR varying σ/µ (Standard Deviation / Mean CBR) and confidence levels / reliability coefficient, K σ non-dimensional regression factor has been prepared for easily calculating CBR at different confidence levels and coefficient of variations. This is shown in Tables 3 and 4. K σ values are shown in Table 5. Expected CBR has been determined using Monte Carlo Simulation Method and presented in Table 6. Roads can be classified as following groups:

Expressway; ●National Highway; ●State Highway; ●Major District Road; ●Ordinary District Road; ●PMGSY / Rural Road; ●

Reliability levels and coefficient of variations may be recommended for Indian condition based on American and Australian practices and presented in Table 7.

Table 3 CBR at Different Confidence Levels for Average CBR 10 %

Cotnfidence Level upper Range Lower Range Reported

80 10+8.4×COV 10 – 8.44×COV CBR×(1±0.842×COV)

85 10+10.4×COV 10-10.4×COV CBR×(1±1.04×COV)

90 10+13.08×COV 10 – 13.1×COV CBR×(1±1.31×COV)

95 10+16.37×COV 10 – 16.44×COV CBR×(1±1.64×COV)

97.5 10+19.56×COV 10 – 19.52×COV CBR×(1±1.954×COV)

99.99 10+37.04×COV 10 – 37.48 ×COV CBR×(1±3.73×COV)

Table 4 CBR at Different Confidence Levels for Average CBR 15 %

Confidence Level upper Range Lower Range Reported

80 1.5(10+8.4×COV) 1.5(10 –8.44×COV) CBR×(1±0.842×COV)

85 1.5(10+10.4×COV) 1.5(10-10.4×COV) CBR×(1±1.04×COV)

90 1.5(10+13.08×COV) 1.5(10 – 13.1×COV) CBR×(1±1.31×COV)

95 1.5(10+16.37×COV) 1.5(10 – 16.44×COV) CBR×(1±1.64×COV)

97.5 1.5(10+19.56×COV) 1.5(10 – 19.52×COV) CBR×(1±1.954×COV)

99.99 1.5(10+37.04×COV) 1.5(10 – 37.48 ×COV) CBR×(1±3.73×COV)

Table 5 Determination of K σ Factor for Determining CBR at Different Reliability Levels

Confidence Level (%)

Coefficient of Variation

0.050 0.100 0.150 0.200

80 0.958 0.916 0.874 0.832

85 0.948 0.896 0.844 0.792

90 0.935 0.869 0.804 0.738

95 0.918 0.836 0.754 0.672

97.5 0.903 0.805 0.708 0.610

99.99 0.814 0.627 0.441 0.254

Table 6 Expected CBR at 95 % Confidence Level for σ/µ=0.1

Assumed Error (%) Actual Error (%) Expected CBR for Base CBR 10

2 1.33 9.94

5 3.86 9.86 7.5 5.33 9.84 10 7.31 9.82

Fig. 2 CBR with Varying Standard Deviation / Mean Ratio at 85 % Confidence Level for CBR 10

Fig.1 Cumulative Probability of Average CBR 10

Coefficient of Variation

Lower Level CBR

upper Level CBR Base CBR

0 10 10 10

0.05 9.48 10.52 10

0.1 8.96 11.04 10

0.15 8.46 11.52 10

0.2 7.92 12.08 10

Table A1 CBR at 85 per cent Confidence Levels




Table 7 Proposed Confidence Level and Coefficient of Variation

Road Classification Confidence Level for CBR

Maximum Coef-ficient of Variation

Expressway 97.5-99.99 0.05

National Highway/State Highway 95.0-97.5 0.10

Ordinary District Road/Major District Road

85-90 0.15

PMGSY/Rural Road 80-85 0.20

7 CONCLuSIONS

Based on present research work, following conclusions may be drawn:

CBR at reliability level should be an important criterion for the evaluation. Cumulative distribution method may be used to determine CBR at different reliability levels. Lower and upper limit of CBR at different reliability levels are to be determined using proposed method. Regression equation may be developed for different confidence limits varying standard deviation / mean ratio to determine CBR at various reliability levels.

Expected CBR may be determined using Monte Carlo Simulation Method. Following equations developed in this present studies may be used directly to determine CBR at different reliability levels from known base CBR value of a project at various values of σ/µ:

CBR at Reliability 1. 99.99=Base CBR×(1-3.73 × σ/µ) (99.99 % confidence level)

CBR at Reliability 2. 97.5=Base CBR×(1-1.95 × σ/µ) (97.5% confidence level

CBR at Reliability3. 95 =Base CBR ×(1-1.64 × σ/µ) (95% confidence level)

CBR at Reliability 4. 90 =Base CBR×(1-1.31 × σ/µ) (90% confidence level)



For other confidence levels, similar equations may be 7. developed.

Expected CBR value can be obtained using Monte 8. Carlo Simulation Technique considering upper, lower and base CBR.

The proposed method may be useful for the 9. Government /Consultants to determine expected CBR at various reliability levels.

REFERENCES

IRC:37-2001, 2012. Guideline for Flexible Pavement Design," 1. Government of India

Austroads 2010, "Guide to Pavement Technology: Part 2: Pavement 2. Structural Design", AGPT02-12, Austroads, Sydney, NSW.

Austroads 2008, "Guide to Pavement Technology: Part 2: Pavement 3. Structural Design", AGPT02-12, Austroads, Sydney, NSW.

National Highways Authority of India (NHAI) "Six Lane of 4. Krishanganj Upiapur Schiemset NH8, 76, 79A and 79 Project", Government in India

Road Traffic Authority, RTA (2012), "Austroads Guide 5. Supplements 2012",

VicRoads 2004, "Assignment of CBR (Strength) and Percent Swell 6. to Earthworks Fill and Pavement Materials", Code of Practice RC/MTD 500.20, VicRoads, Kew, Vic.

VicRoads 2005, "Code of Practice for Selection and Design of 7. Pavements and Surfacing", Code of Practice, RC/MTD 500.22, VicRoads, Kew, Vic.

79




z x f(x) f(x) f(x)1- -4 2 6.69E-05 3.17E-05 0.999968

-3.92 2.16 9.19E-05 4.43E-05 0.999956

-3.84 2.32 0.000125 6.15E-05 0.999938-3.76 2.48 0.00017 8.5E-05 0.999915-3.68 2.64 0.000229 0.000117 0.999883-3.6 2.8 0.000306 0.000159 0.999841

-3.52 2.96 0.000407 0.000216 0.999784-3.44 3.12 0.000537 0.000291 0.999709-3.36 3.28 0.000705 0.00039 0.999610-3.28 3.44 0.00092 0.000519 0.999481-3.2 3.6 0.001192 0.000687 0.999313-3.12 3.76 0.001535 0.000904 0.999096-3.04 3.92 0.001964 0.001183 0.998817-2.96 4.08 0.002496 0.001538 0.998462-2.88 4.24 0.003153 0.001988 0.998012-2.8 4.4 0.003958 0.002555 0.997445

-2.72 4.56 0.004936 0.003264 0.996736-2.64 4.72 0.006116 0.004145 0.995855-2.56 4.88 0.00753 0.005234 0.994766-2.48 5.04 0.009212 0.006569 0.993431-2.4 5.2 0.011197 0.008198 0.991802-2.32 5.36 0.013524 0.01017 0.98983-2.24 5.52 0.01623 0.012545 0.987455-2.16 5.68 0.019353 0.015386 0.984614-2.08 5.84 0.022931 0.018763 0.981237

-2 6 0.026995 0.02275 0.97725




z x f(x) f(x) f(x)1- -1.92 6.16 0.031578 0.027429 0.972571-1.84 6.32 0.036703 0.032884 0.967116-1.76 6.48 0.042388 0.039204 0.960796-1.68 6.64 0.048641 0.046479 0.953521-1.6 6.8 0.05546 0.054799 0.945201

-1.52 6.96 0.062832 0.064255 0.935745-1.44 7.12 0.07073 0.074934 0.925066-1.36 7.28 0.079112 0.086915 0.913085-1.28 7.44 0.087924 0.100273 0.899727-1.2 7.6 0.097093 0.11507 0.88493-1.12 7.76 0.106535 0.131357 0.868643-1.04 7.92 0.116149 0.14917 0.85083-0.96 8.08 0.125822 0.168528 0.831472-0.88 8.24 0.135432 0.18943 0.81057-0.8 8.4 0.144846 0.211855 0.788145-0.72 8.56 0.153926 0.235762 0.764238-0.64 8.72 0.162531 0.261086 0.738914-0.56 8.88 0.170523 0.28774 0.71226-0.48 9.04 0.177766 0.315614 0.684386-0.4 9.2 0.184135 0.344578 0.655422

-0.32 9.36 0.189515 0.374484 0.625516-0.24 9.52 0.193808 0.405165 0.594835-0.16 9.68 0.196934 0.436441 0.563559-0.08 9.84 0.198834 0.468119 0.531881

2.47E-15 10 0.199471 0.5 0.50.08 10.16 0.198834 0.531881 0.4681190.16 10.32 0.196934 0.563559 0.4364410.24 10.48 0.193808 0.594835 0.4051650.32 10.64 0.189515 0.625516 0.3744840.4 10.8 0.184135 0.655422 0.344578

0.48 10.96 0.177766 0.684386 0.3156140.56 11.12 0.170523 0.71226 0.287740.64 11.28 0.162531 0.738914 0.2610860.72 11.44 0.153926 0.764238 0.2357620.8 11.6 0.144846 0.788145 0.2118550.88 11.76 0.135432 0.81057 0.189430.96 11.92 0.125822 0.831472 0.1685281.04 12.08 0.116149 0.85083 0.149171.12 12.24 0.106535 0.868643 0.1313571.2 12.4 0.097093 0.88493 0.11507




z x f(x) f(x) f(x)1-

1.28 12.56 0.087924 0.899727 0.100273

1.36 12.72 0.079112 0.913085 0.0869151.44 12.88 0.07073 0.925066 0.0749341.52 13.04 0.062832 0.935745 0.0642551.6 13.2 0.05546 0.945201 0.054799

1.68 13.36 0.048641 0.953521 0.0464791.76 13.52 0.042388 0.960796 0.0392041.84 13.68 0.036703 0.967116 0.0328841.92 13.84 0.031578 0.972571 0.027429

2 14 0.026995 0.97725 0.022752.08 14.16 0.022931 0.981237 0.0187632.16 14.32 0.019353 0.984614 0.0153862.24 14.48 0.01623 0.987455 0.0125452.32 14.64 0.013524 0.98983 0.010172.4 14.8 0.011197 0.991802 0.0081982.48 14.96 0.009212 0.993431 0.0065692.56 15.12 0.00753 0.994766 0.0052342.64 15.28 0.006116 0.995855 0.0041452.72 15.44 0.004936 0.996736 0.0032642.8 15.6 0.003958 0.997445 0.002555

2.88 15.76 0.003153 0.998012 0.0019882.96 15.92 0.002496 0.998462 0.0015383.04 16.08 0.001964 0.998817 0.0011833.12 16.24 0.001535 0.999096 0.0009043.2 16.4 0.001192 0.999313 0.0006873.28 16.56 0.00092 0.999481 0.0005193.36 16.72 0.000705 0.99961 0.000393.44 16.88 0.000537 0.999709 0.0002913.52 17.04 0.000407 0.999784 0.0002163.6 17.2 0.000306 0.999841 0.000159

3.68 17.36 0.000229 0.999883 0.0001173.76 17.52 0.00017 0.999915 8.5E-053.84 17.68 0.000125 0.999938 6.15E-053.92 17.84 9.19E-05 0.999956 4.43E-05

4 18 6.69E-05 0.999968 3.17E-05

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