spatial and statistical distribution models using the cbr · pdf filethe california bearing...

Australian Geomechanics Vol 44 No 1 March 2009 37

SPATIAL AND STATISTICAL DISTRIBUTION MODELS USING THE CBR TEST

Burt Look Connell Wagner Pty Ltd, QLD, Australia

ABSTRACT The California Bearing Ratio (CBR) is the most common test used in the design of pavements. This simple test has many pitfalls in its application. This is an empirical test, and one must be aware of its many considerations if the design value is to be appropriately applied.

A design value must consider its spatial variation, level of compaction and its relationship with its surrounding layers. The design risk is used to determine the characteristic value for a given project. Characterisation using the spatial and statistical variation of a CBR is used for a project site in Queensland to illustrate the requirement to use an appropriate prediction model. The results of this curve fitting show the normal distribution is inappropriate due to negative values and the lognormal distribution is an appropriate statistical model to characterize the design CBR.

1 INTRODUCTION The subgrade design modulus is principally derived from the California Bearing Ratio (CBR) Test. This very common and simple test has many pitfalls in its application. This is an empirical test carried out at a high strain level and low strain rates, while subgrades below pavements normally experience relatively low strain levels and higher strain rates. It is widely recognised that the CBR test is outdated, yet its use continues due to its long-standing historical application. Modulus conversions are used to convert the CBR (a strength index) to the stiffness. Look (2007) discussed these considerations and this paper uses that same data. The equilibrium moisture conditions and associated swell behaviour are discussed in Look (2005).

Pavement designs based on the CBR need to consider the following issues

• Spatial variation • Soaked vs unsoaked tests • Modified vs standard compactive effort • Equilibrium moisture content and density • Laboratory compaction vs construction specification • Design risk • Design value based on a characteristic value or lowest value • Swell characteristics • Relationship with moduli • Relationship with the adjacent layers

Characterisation of the subgrade properties of a site located north of Brisbane in south east Queensland is used to show the application of some of the above in the design process.

2 SITE ASSESSMENT AND SPATIAL VARIATION The Caboolture to Beerburrum rail site upgrade is approximately 13 km in length. The results of the initial 4-day soakad CBR tests are tabulated in Figure 1, which also shows the moving average. This moving average assessment provides a visual assessment of the changes occurring across the site for zonation purposes. While additional testing was subsequently carried out, the initial data provided the basis of the subgrade assessment adopted for preliminary costing purposes of the required lengths of capping and ballast layers.

SPATIAL AND STATISTICAL DISTRIBUTION MODELS USING THE CBR TEST B LOOK

38 Australian Geomechanics Vol 44 No 1 March 2009

Figure 1: Variation of soaked CBR values for Caboolture to Beerburrum.

Site variability limited the ability to differentiate small changes in CBR values between material zones, for example separating CBR 4% from CBR 5% was not practical. CBR testing can have a coefficient of variation (COV) of 17% to 58% (Lee et al., 1983) and this was one of the considerations in deriving the simplified zonation shown in Table 1.

Table 1: Simplified zonation areas.

Chainage 51440 54940

54940 56440

56440 59940

59940 64440

Length (km) 3.5 1.5 3.5 4.5

No. Of values 31 6 38 21 Design CBR 4% 15% 4% 7%

Even when longitudinal zones have been classified, the designer must be wary of vertical variation of properties.

Figures 2 and 3 show how these variations occurred during further testing for another stage of the project when materials needed to be won from these sources for embankment fill.

Variation occurs both transversely and vertically. Therefore if one CBR test value is used as representative for these cuttings, then significant errors can result. In addition, the common assumption that CBR value is likely to improve with depth and that this is conservative is not always correct. For Cut 5 that assumption was generally valid, where a CBR < 5% was obtained near the surface. However, while a CBR of 10% to 12% would be appropriate just below the surface, the CBR value decreases to 5% to 8% at the base of the cutting. At Cut 6, CBR values greater than 15% were often obtained at the top of the cutting, yet these dropped to less than 5% at depth.

These cuttings (of the same geological origin) illustrate the need for rigorous testing where significant volumes of materials are to be used. Even in the design stage, one must be aware of the likely quantity of fill to be used from the cuttings as this should be the basis for assessment on the quantity of testing required. Figures 2 and 3 suggest that any testing frequency less than one CBR test per 5,000 cubic metres may lead to errors even at the design stage. This should be factored for the site geology.

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5551440

51940

52440

52940

53440

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54940

55440

55940

56440

56940

57440

57940

58440

58940

59440

59940

60440

60940

61440

61940

62440

62940

63440

63940

64440

Chainage (m)

CBR (%)

Mansfield

Road Cut

RCR Cut North of

Blackburn Road

RCR Cut South of

Blackburn Road


Australian Geomechanics Vol 44 No1 March 2009 39

Figure 2: Spatial variation in cut 5.

Figure 3: Spatial variation in cut 3.



y = 12.64e-0.22x

R2 = 0.97

Swell > 1%

y = 12.64e-0.29x

R2 = 0.52

All Points

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0 2 4 6 8 10 12

CBR Swell (%)

4 Day soked CBR (%)

3 CALIFORNIA BEARING RATIO SWELL AND COMPACTION The custom of reporting only the CBR value in a test certificate discards much useful data that can be used by the designer. Both strength and movement must be used to characterise a site as failure from movement can occur despite an adequate CBR strength.

Test certificates emphasise the CBR value only in reporting the subgrade condition. The designer should use other useful data from this test, such as the CBR swell, level of compaction, surcharge used and the percentage of material coarser than the 38mm sieve. Material above that size is discarded in the tests, and if sufficient quantity of material is removed the test then becomes inappropriate as a design tool.

At CBR values less than 10%, the swell characteristics need to be considered as the CBR swell value may be at or above 1%. Figure 4 shows the average CBR / swell relationship for soils in Southeast Queensland, as well as the upper bound relationship for the soils tested with swells above 1%. This swell is an indicator of the volume change potential of the soil, as often the design may be adequately designed based on strength, but with the potential still to fail by cracking (a serviceability failure condition). Look (2005) discusses these issues and the relevance of equilibrium moisture conditions. The issue of designing for swell is not discussed further in this paper.

Figure 4: Relationship between CBR strength and swell for southeast Queensland soils.

Additionally there is much confusion with the use of standard and modified testing. Some believe that the modified test with its higher level of compaction and its introduction a few decades after the Proctor “Standard” is an improvement. That belief has led to many errors in the design of subgrades. The modified test was introduced to allow for the “heavier” loadings and larger construction plant of the 1940s. Advocating the modified test as better also suggests we should now be moving on to a modified – modified test as the 1950s “modified” test does not represent the modern day “heavy” equipment.

These tests are reference units only, and the modified test is simply just a different reference unit. While there can be an argument for the use of modified tests in pavements, using such tests on subgrade materials often leads to much confusion and misinterpretation - all at a higher cost of testing. For example, a modified compaction value for a high plasticity, CH clay can often be 10% CBR. No experienced designer would use such a value.

The following case study summary for a project between Robina and Varsity lakes (Gold Coast) illustrates some of the issues that may arise by using the modified compaction test in a subgrade assessment. The specifications for testing by the rail authority required modified compaction; the results were:



• 28% of tests on silty/sandy clays recorded CBR >10%. These values were considered inappropriate for designing a subgrade on clays

• 36% of tests on clays recorded a CBR swell > 5% (not associated with above). These values suggest these materials should not be used due to high swell potential, yet high shrink – swell was not confirmed based on other fundamental tests.

• The above suggests issues with some 66% of the test results and puts doubt on the reliability of the remaining 33% of results.

The end result of this uncertainty was these values were not appropriate for design and $10,000 of laboratory results plus costs of sampling and supervision were consigned to the bin.

With a modified compaction, high CBR swells can result with high densities. Therefore such compaction should not be used in highly reactive clays in a wet environment (Look, 2005). Over-compaction may also lead to crushing and deterioration of some materials. Importantly such levels of compaction that occur in the field assume it is compatible with the underlying material. Hammitt (1970) shows that the laboratory CBR may not be achieved in the field unless the material below also has a minimum strength ratio compared to the layer under consideration.

The relationship with the surrounding soil needs to be considered. Discussions with a commercial laboratory indicate requests for a modification of the surcharge weight used in the soaked CBR test is an infrequent occurrence. Yet that surcharge can have a significant affect on both the CBR value and its swell behaviour for medium to high plasticity clays. The 4.5kg surcharge weight used represents a thin paving layer above. This is not representative of thick pavements and high embankments.

During the investigation stage, the laboratory CBR is often carried out near the Maximum Dry Density (MDD) and Optimum Moisture Content (OMC). This is often incompatible with the construction specifications which would be typically 95% or 90% relative compaction. The corresponding CBR could be approximately 2/3rd and 1/3rd, respectively of the CBR at 100% MDD – but this ratio needs to be checked for each material.

Ervin (1994) provides a comprehensive discussion of various earthwork compaction issues and additional considerations for the designer.

4 CHARACTERISTIC VALUES FOR DESIGN Reliability is often discussed in codes and design procedures. However measurement is in terms of failures. Failures demonstrate a lack of reliability. In geotechnical testing this may be termed the percentage of values less than a given value or the percentage defective in the quality control testing

Reliability design typically allows defects as follows

• 90% Reliability for major roads: i.e. 10% of test values less than the specified value. • 70% Reliability for minor roads i.e. 30% of results defective.

This site (Figure 1) used 20% of values allowed to be less than the design value - an 80% reliability. Obtaining the “reliability” and approaches in obtaining these reliability values will be discussed further in later sections.

The lower characteristic value (LCV) for density quality control is based on the approach in Queensland Main Roads Specifications MRS 11.01 using Table 6. This Table is for a 10% proportion defective (with a 90% probability of acceptance) and applies a k value to account for the number of samples. The LCV is calculated by:

LCV = X – k s

Where X = mean; s = standard deviation; k = acceptance constant dependent on the number of tests

While the above is applicable for statistical quality control of density, some designers also use this as a basis for assessment of the design CBR. The following discussion shows this is inappropriate unless the correct statistical distribution model is also used. The test Coefficient of Variation (COV) for density measurements are typically 3% (homogeneous) or 5% where variability exists for a given material. Strength values may vary by over 40% for a given material.

Alternatively, the Design CBR in Pavement Design Manuals (QDMR, 2007; ARRB 1994) can be simply based on:

Design CBR = X – 1.3s



5.0% 45.0% 50.0%

7.1% 43.9% 49.0%

8.2% 40.8% 51.0%

19.1% 17.6% 63.3%

2.5 8.0

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

All CBR DataChainage 51.4 km - 64.6km

Input

Mean 11.4896

Median 8.0000

Std Dev 10.2426

10% 3.0000

25% 4.5000

Values 96

Pearson5

Mean 12.1411

Median 7.8434

Std Dev 19.9630

10% 2.8938

25% 4.6271

Lognorm

Mean 11.5227

Median 8.1696

Std Dev 11.4850

10% 2.7541

25% 4.6294

Normal

Mean 11.4900

Median 11.4900

Std Dev 10.2960

10% -1.7049

25% 4.5455

This approach assumes a normal distribution of the population, and is for 10% defective. Additionally the value of 1.3 assumes a sample size greater than 20 based on the t – distribution, however that assumption is seldom achieved and thus makes this approach independent of sample size in its application.

When the COV is less than 30% the probability of negative values is negligible when using a normal distribution design model (Booker et al., 2001). However, CBR values have a COV much greater than 30% (Lee et al., 1983). If the normal distribution does not provide a good fit to the data then different statistical models need to be considered.

5 STATISTICAL DISTRIBUTION MODELS For a risk based approach, a statistical model is usually adopted. For rock strength index, the Normal distributions can distort the model, especially at the lower values (Look and Griffiths, 2004). Appropriate distributions can be determined by goodness-of-fit test, which measures how well the sample data fits a hypothesized probability density function. Three types of tests are available:

- Chi-Square test. This is the most common goodness-of-fit test.

- Kolmogorov-Smirnov test. This does not depend on the number of bins, which makes it more powerful than the Chi-Square test. The K-S test does not detect tail discrepancies very well.

- Anderson-Darling test. This is very similar to the Kolmogorov-Smirnov test, but it places more emphasis on tail values. It does not depend on the number of intervals.

The latter test is considered more appropriate in soil and rock data due to these tail variations. Using one of the other goodness of fit tests can change the rank of the 35 available distribution models. Fenton and Griffiths (2008) provide a more comprehensive discussion on choosing of distributions and various statistical models.

Figure 5 shows the distribution of the results for the full 13 km of the site, while Figures 6 to 9 show the corresponding statistical models for the four zonation areas. The graphs show the probability density function (PDF) with CBR (%). The side of the graphs shows the statistics for various distributions with the “best fit “distribution shown after the input (true) data. The tops of the graphs then show the difference in prediction that can be obtained for the three distribution functions shown and comparing with the input data at the 5 percentile and 50 percentile CBR value.

Figure 5: Statistical Distribution Models for the full 13 km length of the site.



5.0% 45.0% 50.0%

4.8% 49.2% 46.0%

4.7% 49.3% 46.0%

16.1% 23.6% 60.3%

2.50 7.00

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CBR (%) DataChainage 51.4 km - 54.9km

Input

Mean 8.6129

Median 7.0000

Std Dev 6.0757

10% 3.5000

25% 4.0000

Values 31

InvGauss

Mean 8.6129

Median 6.5230

Std Dev 6.8054

10% 2.9972

25% 4.1941

Lognorm

Mean 8.8053

Median 6.5313

Std Dev 7.7371

10% 3.0106

25% 4.2197

Normal

Mean 8.6129

Median 8.6129

Std Dev 6.1762

10% 0.6978

25% 4.4471

Mean = 8.6129

Median = 7.0000

-1 SD = 2.5372

+1 SD = 14.6886

-10 0

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20

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70

Mean = 29.5000

Median = 25.0000

-1 SD = 16.0619

+1 SD = 42.9381

Figure 6: Statistical Distribution Models for Chainage 51.4 to 54.9 km.




5.0% 50.0%

11.7% 62.6%

13.1% 63.5%

22.9% 68.8%

2.5 5.0

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0.04

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CBR (%) DataChainage 56.5 km to 59.9 km

Input

Mean 9.8421

Median 5.0000

Std Dev 9.7550

10% 3.0000

25% 4.0000

Values 38

Pearson5

Mean 10.1103

Median 6.4820

Std Dev 16.9338

10% 2.3210

25% 3.7773

Lognorm

Mean 9.6973

Median 6.7458

Std Dev 10.0470

10% 2.1619

25% 3.7371

Normal

Mean 9.8421

Median 9.8421

Std Dev 9.8859

10% -2.8272

25% 3.1742



-10 -5 0 5

10

15

20

25

30

35

40

Mean = 13.5714

Median = 11.0000

-1 SD = 4.2348

+1 SD = 22.9081



6 RESULTS OF ANALYSIS A few of the statistical distribution fits are shown in Figures 5 to 9. These visually illustrate the discrepancies that occur especially at the tail of the distribution models when statistics based on normal distributions are used. In each case the anomalies using the normal distribution as a statistical model would err significantly, including negative values in some instances. Thus some “rationalization” of these anomalies may have to occur.

The PDF presented above in Figure 8 has been transformed in Figure 10 to show the cumulative distribution and its variation from better fitting statistical distribution models. The input line would be produced if a graphical approach was used to evaluate the design CBR. One can then read from the graph directly the percentage of results less than or equal to a given CBR value. Graphical means are appropriate when the data is not large. The input data and three statistical distribution models are compared. The lognormal model appears more closely aligned to the best fit model (Pearson V) in this instance.

Figure 10: Cumulative Distribution for Chainage 56.5 to 59.9 km’

Figures 5 to 9 showed the best fit distribution and compared with the log-normal and normal distributions. The latter is not considered appropriate as a statistical distribution model for the CBR index test. The log-normal distribution provides a reasonable distribution model although it may not necessarily be the optimum best fit statistical model in all situations.

The distribution models illustrated in Figures 5 to 9 are summarised in Table 2 for comparisons purposes. The key observations are:

• a 10% characteristic values requires the correct distribution model to be obtained. • a 25% characteristic value has the normal distribution closely aligned with the more appropriate distribution

models in all cases. • the log normal distribution provides a close fit to the best fit distribution model. • Even after zonation of a site into similar geology and materials, the COV is very high. Phoon and Babu (2007)



discuss that while the COV for concrete is considered satisfactory up to 20%, for geo-materials the COV less than 30% is considered low. They classify a COV of 50% to 70% as high.

Table 2: Results of Distribution models at 10% and 25% risk for the various zones.

Chainage From To

All 51.4 km 64.6 km

51.4 km 54.9 km

54.9 km 56.4 km

56.4 km 59.9 km

59.9 km 64.4 km

No. Of values 96 31 6 38 21 COV 90% 72% 50% 101% 71% 10% LCV Best Fit Log – Normal Normal

2.9 2.8

(-1.7)

3.0 3.0 0.7

15.0 14.8 10.6

2.3 2.2

(-2.8)

4.0 4.0 1.3

25% LCV Best Fit Log – Normal Normal

4.6 4.6 4.5

4.2 4.2 4.4

18.8 18.7 19.6

3.8 3.7 3.2

6.5 6.5 7.1

This detailed statistical curve fitting approach is seldom used for assessment of the design parameters. For comparison, the results using the 2 more popular simplified approaches to determine the LCV as discussed in Section 4 are presented in Table 3. With these methods negative LCV values can still occur at the 10% defective. At the 10% and 25% risk level the MRS derived values are still at variance from the best fit model when the number of samples are accounted for. Thus the approach for construction quality control of density should not be applied to the statistical assessment of design CBR.

Table 3: Lower Characteristic Values (LCV) of CBR using a normal distribution.

Chainage From To

All 51.4 km 64.6 km

51.4 km 54.9 km

54.9 km 56.4 km

56.4 km 59.9 km

59.9 km 64.4 km

No. Of values 96 31 6 38 21 COV Mean

90% 11.5

72% 8.6

50% 29.5

101% 9.8

71% 13.6

Using construction

quality approach

10% LCV K factor

(-0.01) 1.117

2.4 1.006

18.9 0.719

(-0.3) 1.030

4.5 0.952

25% LCV K factor

6.0 0.536

5.9 0.437

27.2 0.155

5.3 0.459

9.9 0.388

Not dependent on sample number

10% LCV X - 1.3 s -2.0 0.6 10.4 -3.0 1.1

25% LCV X – 0.7s

4.3 4.3 19.2 2.9 6.9

These results imply that the issue lies with using the normal distribution as a prediction model when used with a 10% defective risk approach. At the higher risk (25%), the appropriateness of the statistical prediction model is less of an issue. Table 3 shows the statistical compliance scheme approach for density should not be applied to design CBR prediction even for the higher risk level.

For the CBR test the non normality of the test data suggests more appropriate models should be used such as the lognormal method when 10% characteristic values are targeted. Look and Wijeyakulasuriya (2009) found a similar conclusion with respect to using the point load index tests on rock. Note that both the CBR and Point Load test results are an index of strength and not fundamental parameters of the material tested.

7 CONCLUSION The paper outlines the procedures required for assessment of the design CBR along a site. Even after geological zonation has occurred the material needs to be assessed both longitudinally and vertically due to the large coefficient of



variation of even “homogeneous” materials. The CBR laboratory model needs to be appropriate to the field CBR – this may involve comparing soaked with unsoaked CBR, equilibrium CBR and levels of compaction achievable.

Selection of the appropriate CBR design parameter needs to consider the appropriate risk profile and selection of the correct statistical distribution model. Due to the large COV, the normal distribution model is not appropriate as negative values can be predicted using that model, and the appropriate PDF shape is required. For the CBR values modelled herein, the lognormal distribution provides a reasonable PDF shape, although other statistical models may provide a best fit depending on the test data.

8 REFERENCES Australian Road Research Board (1995). Sealed Local Roads Manual.

Booker J.D., Raines M., and Swift K.G. (2001). Designing Capable and Reliable Products. Butterworth – Heinemann Publishers

Ervin, M. (1993). Specifications and control of earthworks. Proceedings of the Conference on Engineered Fills, Newcastle upon Tyne, Edited by BG Clarke, CFP Jones and AIB Moffat, Thomas Telford Publications.

Fenton G.A., and Griffiths D.V. (2008). Risk Assessment in Geotechnical Engineering. John Wiley & Sons Publishers.

Hammitt, G.M. (1970). Thickness requirement for unsurfaced roads and airfields, bare base support. Report S – 705, U.S. Army Engineering Waterways, Experiment Station, Vicksburg

Lee, I.K., White W. and Ingles O.G. (1983). Geotechnical Engineering. Pitman Publishers

Look, B. G. (2005). Equilibrium Moisture Content of volumetrically active clay earthworks in Queensland. Australian Geomechanics Journal, Vol 40, No. 3, pp 55 – 66

Look B G. (2007). Design stiffness of subgrades using the CBR test. 10th Australia New Zealand Conference in Geomechanics, Brisbane, Vol 1, pp 496 – 501.

Look B.G, and Griffiths S. 2004. Characterization of rock strengths in South East Queensland. Proc. 9th Aust. NZ Geomech Conf. Auckland, Vol. 1, pp 187 - 194.

Look, B.G. and Wijeyakulasuriya, V. (2009). The statistical modelling of rock strength for reliability assessment (unpublished to date)

Phoon K.K. and Sivakumar Babu G.L. (2007). Potential Role of Reliability in Foreesic Geoetechnical Engineering. 13th Asian Regional Conference on Soil Mechanics and Geotechnical Engineering, Kolkatta, India.

Queensland Main Roads (2007). Standard Specifications.

spatial and statistical distribution models using the cbr · pdf filethe california bearing...

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