Statistical evaluation of compression index equations

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  • 1. INTERNATIONAL JOURNAL and Technology (IJCIET), ISSN 0976 – 6308 International Journal of Civil Engineering OF CIVIL ENGINEERING AND (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME TECHNOLOGY (IJCIET)ISSN 0976 – 6308 (Print)ISSN 0976 – 6316(Online)Volume 4, Issue 2, March - April (2013), pp. 104-117 IJCIET© IAEME: www.iaeme.com/ijciet.aspJournal Impact Factor (2013): 5.3277 (Calculated by GISI) © IAEMEwww.jifactor.com STATISTICAL EVALUATION OF COMPRESSION INDEX EQUATIONS 1 2 Ch.Sudha Rani , K.Mallikarjuna Rao 1 Associate Professor, Dept of Civil Engineering, Sri Venkateswara University College of Engineerring, Tirupati, India-517502 2 Professor, Dept of Civil Engineering, Sri Venkateswara University College of Engineerring, Tirupati, India-517502 ABSTRACT Several correlations were developed in practice for predicting Compression Index in terms of either Liquid Limit or Plasticity Index or Dry Density or initial Moisture Content. In this investigation an attempt has been made to quantify statistically the effectiveness of twelve such models statistically by comparing predicted and observed Compression Index values for 180 soils test data obtained from literature. A statistical technique called Analysis of variance (ANOVA) is used to analyse the differences between predicted and observed Compression Index values with and without considering soil classification. One-Factor and Two-Factor ANOVA test results indicate that the influence of soil classification and method of prediction is significant on the deviation between observed and predicted Compression Index values. Certain models were found to have applicability only for some soil classification groups. The best models for prediction of Compression Index of six soil classification groups as well as for all soil types were assessed by conducting statistical Dunnett’s test. Two models were found to have general applicability considering all soil classification groups. KeyWords: Compression Index, Liquid Limit, Plasticity Index, Soil Classification, Soil Type 104
  • 2. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME1. INTRODUCTION Correlations have been a significant part of soil mechanics from its earliest days, asthe soil is known to exhibit greatest degree of variability and uncertainty. This is due to theinherent nature and diversity of geological processes involving stress, time and environmentin soil formation. Correlations using Atterberg limits are fairly common in soil mechanicsliterature from the time Casagrande found that the Atterberg limits provide a much morereliable indication of engineering properties. Virgin Compression of soils is most commonlyexpressed by Compression Index (Cc), determined from the slope of compression curve.Several investigators proposed empirical or semi empirical correlations to predictCompression Index using Liquid Limit (Skempton 1944, Terzaghi&Peck 1967, and Bowles1979) or initial Void Ratio (Nishida 1956, Hough 1957, and Bowles 1979) or initial MoistureContent (Bowles 1979, and Koppula 1981) or in-situ Dry Density (Oswald 1980). Burland(1990), and Nagraj et.al. (1990) expressed Compression Index as a function of generalizedparameters namely Void Index (IV) and e/eL respectively. According to Jian-Han Yin (1999),Sridharan and Nagraj (2001), and Amithnath and DeDelal (2004) Compression Index yieldsgood correlation with Plasticity Index. The engineering properties of soils are known todepend on the composite effect of compositional and environmental factors (Mitchel, 1993). Liquid Limit, Plasticity Index are known to reflect compositional factors while in-situDry Density and natural Moisture Content are the important environmental factors thatinfluence the engineering properties significantly. Review of literature reveals that generallyCc is correlated with any one of the parameters reflecting either composition or environmentof soil excepting the one suggested by Mallikarjuna Rao et.al.(2006). Mallikarjuna Raoet.al., 2006/ Sudha Rani, 2007 developed a regression model for predicting CompressionIndex in terms of four parameters namely, Liquid Limit (WL), Plasticity Index (IP), DryDensity (γd) and initial Moisture Content (mc) which reflect both composition andenvironment of soil. The objective of the present investigation is to quantify statistically theeffectiveness of most popular methods for prediction of Cc by comparing the predicted andobserved Cc values for soils other than those from which the correlations were developed.2. COMPRESSION INDEX EQUATIONS STUDIED From literature it is clear that there are several correlations available for prediction ofCompression Index using one of the parameters namely, Liquid Limit (WL), Plasticity Index(IP), Dry Density (γd), initial Moisture Content (mc), initial Void Ratio (eo) and Porosity (η),which reflect either composition or environment. Some of the most commonly usedcorrelations along with the regions/conditions of applicability are reported by Nagraj &Srinivasa Murthy (1986). The same are shown in Table 1 along with the one suggested byMallikarjuna Rao et.al.(2006) / Sudha Rani(2007). These methods are designated as M1, M2,M3, M4, M5, M6, M7, M8, M9, M10, M11 and M12 for convenience. Regression modelsM2, M6 and M7 correlate Compression Index with the Liquid Limit which is dependent oncomposition of the soil. Models M3, M4, M5, M9 and M10 used environmental factornamely in-situ Void Ratio to predict Compression Index. Model M1 and M8 adopted naturalMoisture Content, while model M11 used in-situ Dry Density for development of regressionmodels. Both natural Moisture Content and in-situ Dry Density are environmental factors.Model M12 accounted for all the environmental factors and compositional factors in thedevelopment of the model. 105
  • 3. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME Table 1 Commonly Used Correlations for Prediction of Compression Index Model Regions / ConditionsS.No. Equation Reference Desig of Applicability nation from Reference 1 M1 Cc = 0.0115 wn Bowles (1979) Organic Silt & Clays 2 M2 Cc = 0.0046(wL -9) Bowles (1979) Brazilian Clays 3 M3 Cc = 0.156 eo+0.0107 Bowles (1979) All Clays 4 M4 Cc= 0.208(eo-0.0083) Bowles (1979) Chicago Clays 5 M5 Cc = 0.75(eo-0.5) Bowles (1979) Soils with Low Plasticity 6 M6 Cc = 0.007(wL -7) Skempton (1944) Remoulded Clays Terzaghi & Peck Normally Consolidated, 7 M7 Cc = 0.009(wL -10) (1967) Moderately Sensitive Clays 8 M8 Cc = 0.01 wn Koppula (1981) Chicago & Alberta Clays 9 M9 Cc = 0.30(eo-0.27) Hough (1957) Inorganic Silty Sandy-Silty Clay 10 M10 Cc = 1.15(eo-0.35) Nishida (1956) All Clays Soil Systems of all Complexities 11 M11 Cc = 0.5(γw/γ2d)1.2 Oswald (1980) and Types Cc = (-0.629+(0.0027* Mallikarjuna 12 M12 WL)+(0.007*mc)+(0.031*γd) + et.al.,(2006)/ All Uncemented Soils (0.002*IP)) Sudha Rani(2007)3. DATABASE USED In order to assess the general applicability of the above mentioned twelve methods,one hundred and seventy eight soils test data was collected from different sources reported inthe literature. Oswald (1980) reported about 100 soils consolidation test data, obtained fromUnited States Army Corps of Engineers (USACE) records covering the offices throughout theContinental United States.Amongst them about eighty soils test data were used for evaluationin this investigation. Other twenty soils data could not be used, as either liquid limit or in-situ void ratio was not reported. Sridharan (1990) reported the e-log p plots of twelveundisturbed samples. Compression Index values were obtained from the e-log p plots and thesame were used for evaluation here. Stalin (1995) conducted a series of consolidation testson about seventy remoulded samples obtained by mixing Bentonite with Kaolinite, fine sand,coarse sand and silt in different proportions. All these tests were conducted on samples withwater content brought out to their respective liquid limit consistency. The same are used herefor evaluation purposes. One dimensional Consolidation tests were conducted on undisturbedsamples by Bayan (2005) for determining compression index on forty two soil samples fromIndian Oil Corporation Limited site in Assam, India and the same are used here for evaluationof methods. Table 2 summarizes test data collected from literature giving the details ofrelevant index properties, soil classification group and Cc values. 106
  • 4. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME Table 2 Typical Soil Data Base Used for Verification of Compression Index Models WP WL mc γd IPS.No. SOIL LOCATION I.S Classification (Cc)a Reference % % % kN/m3 % 1 Thomasville@18ft CH 31.00 87.00 32.70 13.86 56.00 0.13 Oswald 2 Ft.Gordon@d=5ft CH 26.00 51.00 26.80 14.80 25.00 0.31 ” 3 Ft.Stewart@d=19ft CH 23.00 92.00 45.60 11.93 69.00 0.39 ” 4 RobbinsAFB@d=11ft CH 28.00 55.00 30.30 14.32 27.00 0.14 ” 5 Robbins AFB@d=12ft CH 30.00 65.00 28.70 14.27 35.00 0.09 ” 6 Thomasville@d=15ft CH 27.00 60.00 41.70 12.54 33.00 0.34 ” 7 IT1 CH 15.00 53.00 26.10 15.40 38.00 0.17 Sridharan 8 IT2 CH 31.00 50.50 29.00 14.60 19.50 0.12 ” 9 LockandDam@13ft CH 28.00 81.00 44.00 12.34 53.00 0.37 Oswald 10 RedRiver@10ft CH 24.00 55.00 37.30 13.33 31.00 0.21 ”4. STATISTICAL EVALUATION OF COMPRESSION INDEX EQUATIONS The Compression Index of all the 178 soils test data is predicted using the twelvemethods namely M1, M2, M3, M4, M5, M6, M7, M8, M9, M10, M11and M12 presented inTable 1. The observed Cc values are plotted against Cc values predicted by the twelveregression models and the typical plots are shown in Figs 1 to 6. The solid line in the plots isthe line of equality. Careful observation of these plots indicate that the predictability of 6models namely M1, M6, M7, M8, M9 and M12 appear to be fair to good since most of thepoints are falling close to the line of equality. All other models are found to either underpredicting or over predicting, even though the predictions are good for some of the lowcompressible soils. Though the prediction by 6 models namely M1, M6, M7, M8, M9 and M12 appear tobe fair to good based on graphical plots of observed and predicted Cc values, there is a needfurther to quantify the effectiveness of each of these twelve methods in order to identify thebest one. In the context of statistical analysis, if we wish to compare two methods say,Method A with Method B about its superiority, it is customary to proceed on the assumptionthat both the methods are equally good (it is known as Null Hypothesis) and the hypothesis istested through z-test or t-test at 5% or 1% level of significance (α), which implies that thenull hypothesis will be rejected when sampling result has probability of occurrence less thanor equal to the level of significance considered (0.01 for 1% or 0.05 for 5%) and vice-versa.If null hypothesis is true, such groups are identified as samples from same population. If wehappen to examine the significance of the difference between more than twomethods/samples, it necessitates considering all possible combinations of the twomethods/groups of data at a time and that would require a great number of tests before wewould be able to arrive at a decision. In all these situations, ANOVA technique developed bySnedcor and others (Snedcor and Cochran 1973) which permits comparison of all groups ofdata/methods simultaneously is used widely in practice. Analysis of Variance popularlyknown as ANOVA in short is a statistical technique for testing differences between two ormore methods/samples/groups of data. 107
  • 5. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME The basic principle of ANOVA is to test for differences among the means of themethods/groups by examining the amount of variation within each of the groups, relative tothe amount of variation between groups/methods. In ANOVA technique, investigation of anynumber of factors that influence the variable known as dependent variable is possible. Thereare two types of ANOVA tests, based on the number of independent variables considerednamely (i) One-Way ANOVA or One-Factor ANOVA and (ii) Two-Way ANOVA or Two-Factor ANOVA . The analysis for the research situations where single independent variable isconsidered is called One-Way Analysis of Variance and if two factors are investigated at atime, then it is called Two-Way Analysis of Variance.In this investigation, in order to quantify the effectiveness of each of these 12 methods inpredicting Cc, One-Way ANOVA is carried out on predicted Cc values using these 12 methodsfor 178 soils test data that is presented in Table 2. Except Oswald’s method i.e. method M11,none of the methods have used any of these 178 soils test data in the development of the 12models under consideration. About 80 soils test data was actually used in the development ofmodel M11 i.e. Oswald’s method. The analysis is for finding the best method that predictsvalues closer to actual value (from experimental study) among the twelve methods namelyM1, M2, M3, M4, M5, M6, M7, M8, M9, M10, M11and M12 for general applicability.Hence, in One-Way ANOVA, the factor under consideration here is method for prediction ofCompression Index of soils. 6.00 Predicted Cc 5.00 4.00 3.00 2.00 1.00 0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Observed Cc Fig 1 Predicted Vs Observed Cc (Model, M1) 6.00 Predicted Cc 5.00 4.00 3.00 2.00 1.00 0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Observed Cc Fig 2 Predicted Vs Observed Cc (Model, M2) 108
  • 6. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME 6.00 Predicted Cc 5.00 4.00 3.00 2.00 1.00 0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 ObservedCc Fig 3 Predicted Vs Observed Cc (Model, M5) 6.00 Predicted Cc 5.00 4.00 3.00 2.00 1.00 0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Observed Cc Fig 4 Predicted Vs Observed Cc (Model, M6) 6.00 Predicted Cc 5.00 4.00 3.00 2.00 1.00 0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Observed Cc Fig 5 Predicted Vs Observed Cc (Model, M7) 109
  • 7. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME 6.00 Predicted Cc 5.00 4.00 3.00 2.00 1.00 0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Observed Cc Fig 6 Predicted Vs Observed Cc (Model, M12)4.1 ONE-WAY ANOVA TEST The One-Way ANOVA is a statistical testing procedure for comparing the meansof more than two groups of data. Here, we have thirteen groups of data, the first groupdata are the observed Cc values and the data in the twelve groups are predicted values ofCompression Index by the twelve methods M1 to M12. The method begins with the assumption that there is no difference between groupmeans i.e. Ĉc1= Ĉc2 = Ĉc3= Ĉc4 = Ĉc5= Ĉc6 = Ĉc7= Ĉc8 = Ĉc9= Ĉc10 = Ĉc11= Ĉc12 = Ĉc13which is normally known as null hypothesis against the alternative hypothesis that thegroup means are not equal. The variance ratio (‘F-value’/ ‘Fstatic’ / ’F’) is the ratio ofMean Square (MS) between groups and the Mean Square within the groups. F-test isbased on F-distribution and is used to compare the variance of the two-independentsamples. This test is also used in the context of analysis of variance (ANOVA) for judgingthe significance of more than two group/sample means at 5% or 1% level. In this test, F-value (F) evaluated is compared with critical value of variance (‘Fcrit’/ ‘F-limit’), which isthe limiting value for given degrees of freedom and this can be obtained by making use ofthe F-distribution given by Fisher. The method was introduced by Fisher (Snedcor &Cochran 1973). MS-EXCEL and SPSS softwares have a routine to perform this analysis. Table 3 presents the summary of the results obtained by carrying out the One-WayANOVA test. From the ANOVA table, the F-value is found to be 22.41, whereas thecritical F-value at 5% level of significance is 1.76. The P-Value in the table which isequal to 0.00 indicates the probability of acceptance of null hypothesis. Since the F valueis greater than Fcrit, it can be concluded that the means of the groups do differsignificantly. Having concluded that the group means differ significantly, it is nownecessary to determine which method is best among all and to rank all the methods basedon their reliability to predict Cc values. Dunnett’s test, which is a multiple comparisontest, can be used for this purpose. The details of the Dunnett’s test may be found inMontgomery (2005) or any other standard textbook on statistical methods. 110
  • 8. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME Dunnett’s Formula for Critical Difference (CD) is given below CD= dα ((a-1), df) √MSE (1/n1 + 1/n2) … (1) Where CD = Critical Difference α = Significance level at 5% = 0.05 (a-1) = No. of Treatment Means = 12 df = Degrees of Freedom (can be obtained from the ANOVA table) dα = F- distribution value at (a-1) denominator and df numerator =2.69 n1, n2 = No. of samples in actual group and comparing groups =178 MSE = Mean Square Error within the groups (can be obtained from the ANOVA table) Table 3 One-Way ANOVA Summary Sheet Groups n Sum Average Varianc e Mactual 178 97.99 0.551 0.52 M1 178 129.72 0.73 0.54 M2 178 54.60 0.31 0.08 M3 178 48.90 0.27 0.07 M4 178 62.35 0.35 0.13 M5 178 159.18 0.89 1.74 M6 178 85.57 0.48 0.18 M7 178 105.22 0.59 0.29 M8 178 112.79 0.63 0.41 M9 178 75.95 0.43 0.28 M10 178 274.77 1.54 4.08 M11 178 159.91 0.89 2.78 M12 178 103.27 0.58 0.67 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 243.68 12 20.31 22.41 0.00 1.76 Within Groups 2085.48 2301 0.91 Total 2329.16 2313 n - No. of soils considered, SS – Sum of Squares, df – Degrees of freedom, MS – MeanSquare Error F-Value – Probability Value Fcrit - Critical Variance Ratio F - Variance Ratio The critical difference (CD) is calculated using equation (1) and the value is 0.271. SPSS software provides a subroutine for Dunnett’s test and the summary of the results are presented in Table 4. Ranking is assigned to the methods of prediction based on the absolute difference between the mean of each method and the mean of the actual method. If the absolute difference does not exceed critical difference, that difference is considered to be insignificant, indicating that the observed data and the predicted data by the specific prediction method are close to each other and this method can be used for prediction with confidence. From Dunnett’s test results given in Table 4, the absolute difference of the prediction methods M3, M5, M11 and M10 are 0.28, 0.34, 0.35 and 1.54, respectively, which are slightly greater than or greater than the critical difference from Dunnett’s formula (0.271). Hence, these methods may be considered inferior to the other eight methods. 111
  • 9. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME The absolute difference of the other eight methods namely M1, M2, M4, M6, M7,M8, M9 and M12 are 0.18, 0.24, 0.20, 0.07, 0.04, 0.08, 0.12 and 0.03, respectively. Thesevalues are less than the critical difference. Hence, any one of these eight methods can beadopted to predict Cc values with higher confidence. However, the absolute difference ofmeans is also lowest for method M12 being 0.03. Hence it may be concluded that the modelM12, which relates Cc with wL, mc, γd, and IP has more general applicability for predicting Ccthan any other model. Model M7 which relates Cc with wL may also be considered equallygood as the absolute difference is only 0.04 which is very low and very nearer to 0.03. Table 4 Dunnett’s Test Summary (for ALL Soils) Groups Average Abs Diff Rank Mactual 0.55 0 - M12 0.58 0.03 1 M7 0.59 0.04 2 M6 0.48 0.07 3 M8 0.63 0.08 4 M9 0.43 0.12 5 M1 0.73 0.18 6 M4 0.35 0.20 7 M2 0.31 0.24 8 M3 0.27 0.28 9(NA) M5 0.89 0.34 10(NA) M11 0.90 0.35 11(NA) M10 1.54 0.99 12(NA)4.2 TWO-WAY ANOVA TEST Soils are generally not homogenous in nature. Studying engineering behaviour andengineering use of each and every soil in isolation is neither possible nor encouraged. That iswhy soils are generally classified adopting any of the engineering classification systems likeUnified Soil Classification System (Casagrande, 1948), Indian Standard ClassificationSystem (IS: 1498, 1970) and American Society of Testing Materials Classification System(ASTM: D 2487-83, 1983). In these classification systems any given soil is classified usingdual symbol system based on grain size distribution and plasticity characteristics. All the soilsfalling under one classification group are expected to exhibit similar engineering behaviour.Hence, it may be expected that the empirical compression index equations may have abearing on soil classification too. This aspect has not been considered by any of theinvestigators. However, Wesley (2003) suggested that correlations involving Liquid Limit orPlasticity Index on their own are unlikely to be applicable to soils on a general basis. It is theposition of soil occupying on the plasticity chart (involving both IP and wL), that is morelikely to lead to general correlations. An attempt was made here to find out whether there wasany relationship between classification of soil (type of soil) and the applicability of theempirical compression index equations. This objective can be achieved by the statisticaltechnique called Two-Way Analysis of variance test in which two factors are consideredsimultaneously to test equivalence of different methods of prediction of Cc. Two-WayANOVA is performed in this investigation considering type of soil/soil classification as onefactor and the method for prediction of compression index as another factor. SPSS softwarepackage extends facility for Two-Factor ANOVA testing also. The test is performed fordifferent types of soils (soil classification groups) using different methods of prediction 112
  • 10. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME(methods M1 to M12) by including the observed (actual) values of compression index. Thedifference between the actual compression index and the predicted value from the equation istaken as the measure of adequacy. The mean of the predicted values by various methods fordifferent types of soils was obtained. The difference of the means from mean of the observedvalues (control) for particular type of soil should be close to zero if the prediction equation istruly suitable. Deviation from observed compression index could also occur due to type ofsoil accounted. Statistical treatment of the errors can be carried out with the help of twofactor ANOVA with factors as soil type (i.e. soil classification group) and the method used forprediction. The 178 soils test data collected from literature and reported in Table 2 is usedfor carrying out two factor ANOVA test. All the 178 soils are classified based on IndianStandard Soil Classification System (IS 1498, 1970). The classification group so obtained isone of the two factors i.e. soil type in Two-Factor ANOVA. Indian Standard SoilClassification is no different from Unified Soil Classification System excepting that the finegrained soils having wL in the range of 35% - 50% are classified as Intermediate compressiblesoils (i.e. CI and MI). Method of prediction (i.e. M1, M2, M3, M4, M5, M6, M7, M8, M9,M10, M11 and M12) is another factor considered in two factor ANOVA test. The details ofthe test can be found in Montgomery (2003) or in any standard textbook on Statistics. In theanalysis, the Soil type is designated as SOIL_COD, the method code (i.e. M1, M2, M3, M4,M5, M6, M7, M8, M9, M10, M11 and M12) is designated as METHOD_C and the jointeffect of soil type and the method code is denoted as SOIL_COD * METHOD_C. TheANOVA table with means and standard deviation of error (deviation) is shown in Table 5. Thenull hypotheses are:Hypothesis 1: The average error (deviation) between observed and predicted Cc value usingempirical equation/model remains same in all soils (labeled as SOIL_COD in Table 5).Hypothesis 2: The average deviation with respect to each empirical equation/model remainsthe same (labeled as METHOD_C in Table 5).Hypothesis 3: There is no joint effect of soil and the equation on the deviation (labeledSOIL_COD * METHOD_C in Table 5).The ANOVA table gives the components into which the total variation is divided. From Table5 the Fstatic for the three factors SOIL_COD, METHOD_C and SOIL_COD*METHOD_C(read as SOIL_COD by METHOD_C) are 79.130, 8.101and 2.807, respectively. Theprobability of acceptance of the three null hypotheses mentioned above is 0.000 forHypothesis 1 i.e. SOIL_COD, 0.000 for Hypothesis 2 i.e. METHOD_C and 0.000 forHypothesis 3 i.e. SOIL_COD*METHOD_C. The probability being very much less than 0.05(i.e. 5% level of significance), all the three hypotheses are rejected. Rejection of all the threehypotheses indicates that the average deviation between observed and predicted Cc values issignificantly different for different soil types and for different methods of prediction. Furtherthe joint effect of soil type and method of prediction is significant which implies that certainmethods are more suitable for certain soil types. Hence, it may be concluded that there issignificant main effect for the SOIL_COD (soil type) factor, METHOD_C (method) factorand the interaction factor SOIL_COD *METHOD_C (joint effect). Having concluded thatthe effect of soil type and method for prediction of compression index are significant, it isnecessary to determine the best method and the methods applicable to predict Cc values foreach type of soil. Eleven types of soils namely CH, CI, CL, MH, MI, ML, CL-ML, OH, SC,SC-CH and SP-SC are found among the 178 soils test data listed in Table 2. Out of theseCH, CI, CL, MI, OH and SC groups have more than 10 sets of soils test data. For these sevensoil types, an attempt has been made here to identify the best method and methods applicablefor prediction of Cc amongst the twelve methods presented in Table 1 by analyzingstatistically the observed and predicted Cc values. 113
  • 11. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME Table 5 Two-Way ANOVA Summary Sheet Sum of Source df Mean Square F Sig. Squares Corrected Model 940.626 142 6.624 10.802 0.000 Intercept 215.938 1 215.938 352.118 0.000 SOIL_COD 485.268 10 48.527 79.130 0.000 METHOD_C 59.619 12 4.968 8.101 0.000 SOIL_COD * METHOD_C 206.562 120 1.721 2.807 0.000 Error 1331.376 2171 0.613 Total 3181.043 2314 Corrected Total 2272.002 2313 This objective can be met by carrying out statistical Dunnett’s test for each soil typeseparately while comparing the observed and predicted Cc values. Dunnett’s test is carriedfor each type of soil separately to find the critical difference using equation 1. The absolutedifference is the difference between the mean of the actual and the mean of a method. If theabsolute difference is less than the critical difference then that particular method is acceptablefor prediction of Cc for the particular soil type and vice versa. The methods suitable for eachclass of soil are concluded, excluding the methods, which have the absolute difference greaterthan the critical difference. Ranking is given to the suitable methods by sorting the absolutedifference values of these methods, so that the method ranked as one predicts a closer valueof compression index to actual measured value. More details concerning Dunnett’s test canbe found in Montgomery (2003) or any standard textbook on Statistics. The SPSS softwareprovides a subroutine and the same is used in this investigation. Dunnett’s test results for CH soil type are presented in Table 6. The critical differenceaccording to Dunnett’s formulae is 0.58 for this group of soils. The absolute differences ofmeans for all the 12 methods are also shown in Table 6 arranged in ascending order. Theabsolute difference is less than 0.58 for 9 methods namely M12, M7, M8, M6, M9, M1, M4,M2 and M3. Further the absolute difference is increasing from 0.09 to 0.53 in that order for allthese nine methods. Hence it may be concluded that any of these nine methods could be used topredict Cc values with reasonable accuracy. However, the absolute difference being lowest forM12 it may be considered best among all these nine methods. The other three methods namelyM5, M11 and M10 are not applicable for use with CH soils since the absolute difference ismore than 0.58. ‘NA’ under the rank column indicates that the absolute difference for thatmethod is more than the critical difference and the method is not applicable for prediction ofCc. Table 7 summarizes Dunnett’s test results of all the seven soil types along with ALLsoils giving the methods applicable and methods not applicable for each soil type separately.The methods are presented in the order of their ranking. From this table it may be observed thatthe methods M4, M6, M7, M8 and M12 are applicable for almost all soil types whereas eitherM12 or M7 are found to be the best method for any given soil type. Hence, methods M12 andM7 can be adopted to predict Cc values with more confidence, while methods M4, M6 and M8can be also used with reasonable degree of confidence. The Dunnett’s test for all soils ispresented in Table 3 after carrying one-Factor ANOVA test. Here also M12 and M7 were foundto be most suitable methods among all the twelve methods in that order, reinforcing the above-derived conclusion from Two-Factor ANOVA test. Prediction model M12 fails to predict Ccvalues for low compressible clays (i.e., soils falling above A-Line in Plasticity chart withwL<35%) and organic soils of high compressibility. On the other hand the performance ofmethod M7 is not upto the mark for Intermediate compressible fine grained soils (i.e. finegrained soils having wL between 35% and 50%). 114
  • 12. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME Table 6 Dunnett’s Test Summary for CH Soils CH Soils Groups Count Average Abs Diff RANK Mactual 57 0.994 0.000 - M12 57 0.985 0.009 1 M7 57 1.011 0.017 2 M8 57 1.060 0.066 3 M6 57 0.808 0.186 4 M9 57 0.790 0.204 5 M1 57 1.219 0.225 6 M4 57 0.602 0.392 7 M2 57 0.521 0.473 8 M3 57 0.464 0.530 9 M5 57 1.803 0.809 NA M11 57 1.976 0.982 NA M10 57 2.937 1.943 NA Critical Difference = CD=dα(a-1,f) √MSE ((1/n1) + (1/n2)) CD=2.69* √1.787* ((1/57) + (1/57)) (CD)CH = 0.58 NA – Not Applicable Table 7 Summary of Models for Prediction of Compression Index from Two-Factor ANOVA test No. of Soils inSoil Type Methods Applicable Methods Not Applicable the GroupCH M12 , M7 , M8, M6, M9, M1, M4, M2, M3 M5, M10 , M11 57 M1, M3, M6, M7, M8, M9,CI M12 , M11, M2, M4, M5 39 M10 M1, M2, M3, M4, M5,CL M7 , M8, M11, M6 17 M9, M10 , M12MI M12 , M11, M5, M2, M4, M6, M9 M1, M3, M7, M8, M10 20 M2, M3, M4, M5, M9, M10 ,OH M7 , M8, M1, M6 11 M11, M!2SC M7 , M9, M11, M6, M12, M4, M8 M1, M2, M5, M10 14ALL Soils* M12 , M7 , M6, M8, M9, M1, M4 M2, M3, M5, M10, M11 178* From One-Factor ANOVA test 115
  • 13. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME5. CONCLUSIONS The performance of twelve different models for prediction of Compression Index isstatistically evaluated using One-Factor ANOVA and Two-Factor ANOVA by comparing thepredicted and observed values of Cc values for 178 soils test data collected from literature.The statistical analysis reveals that both soil classification (i.e. the position of soil inengineering classification chart) and the method of prediction have bearing on theperformance of models. Most suitable models for each soil type for prediction of Cc areobtained by statistical technique called Dunnett’s test. Two models, one suggested byMallikarjuna Rao et.al. (2006) and the other suggested by Terzaghi & Peck (1967) werefound to have more general applicability considering all soil types.REFERENCESJournal Papers1. Amithnath and DeDalal SS (2004) The Role of Plasticity Index in Predicting Compression Index behaviour of clays. Electronic Journal of Geotechnical Engineering http://www.ejge.com/2004/Per0466/Ppr0466.htm.2. Burland JB (1990) On the Compressibility and Shear Strength of Natural Clays. Geotechnique 40(2): 327-378.3. Jian- Hua Yin (1999) Properties and Behaviour of Hong Kong Marine Deposits with different Clay Contents. Canadian Geotechnical Journal 36: 1085-1095.4. Koppula SD (1981) Statistical Estimation of Compression Index. ASTM Geotechnical Testing Journal 4(2): 68-73.5. Nagraj TS Srinivasa Murthy BR and Vatsala A (1990) Prediction of Soil Behaviour Part I – Development of Generalised Approach. Indian Geotechnical Journal 20: 4.6. Nagraj TS and Srinivasa Murthy BR (1986) A Critical reappraisal of Compression Index equations. Geotechnique Vol 36(1): 27-32.7. Nishida Y (1956) A Brief Note on the Compression Index of Soil. Journal of Soil Mechanics and. Foundation Division, American Society of Civil Engineers 82(3): 1-14.8. Oswald RH (1980) Universal Compression Index Equation. Journal of.Geotechnical Engineering Division, American Society of Civil Engineers 106: 1179-1200.9. Skempton AW (1944) Notes on the Compressibility of Clays. Quaterly Journal of Geotechnical Society. London 100:119-135.10. Sridharan A and Nagraj HB (2001) Compressibility behaviour of remoulded, fine- grained soils and correlation with index properties. Candian Geotechnical Journal 38:1139-1154.11. Wesley LD (2003) Residual Strength of Clays and correlations using Atterberg Limits. Geotechnique 543(7): 669-672.12. Ch. Sudha Rani and K Mallikarjuna Rao, “Compositional and Environmental Factors Role on Compression Index” International Journal of Civil Engineering & Technology (IJCIET), Volume 3, Issue 2, 2012, pp. 392 - 403, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316Books13. ASTM: D 2487-83 (1983) standard test method for classification of soils for engineering purposes, American Society for Testing and Materials, Philadelphia, USA 116
  • 14. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME14. Bowles JW (1979) Physical and Geotechnical Properties of Soils, McGraw Hill, New York.15. Hough BK (1957) Basic Soil Engineering, Ronald, New York.16. IS: 1498 (1970) (Reaffirmed 2002) Classification and Identification of Soils for General Engineering Purposes, Bureau of Indian Standards, New Delhi.17. Mitchell JK (1993) Fundamentals of Soil Behavior, John Wiley and Sons, New York.18. Montgomery CD (2005) Design and Analysis of Experiments, John Wiley & Sons, New York.19. Snedcor GW, Cochran WG (1973) Statistical Methods, Mc Graw Hill New York.20. Terzaghi K and Peck RB (1967) Soil Mechanics in Engineering Practice, Wiley New YorkTheses21. Sreelatha N (2001) Analysis Compressibility and Shear Behaviour of Tropical Residual Soils with Induced Cementation. M.Tech Thesis of Sri Venkateswara University College of Engg, Tirupati, India.22. Stalin VK (1995) Factors and Mechanisms Controlling the Index Properties and Engineering Behaviour of Soil Mixtures. Ph.D Thesis of Indian Institute of Science, Bangalore, India.23. Sudha Rani Ch (2007) A Knowledge Based System for Soil Identification and Assessment of Volume Change Characteristics of Clayey Soils. Ph.D Thesis of Sri Venkateswara University, Tirupati, India.Proceedings Papers24. Bayan GK (2005) Prediction of Historical Loading Condition of Alluvium Soil: Problem and Possible New Solution – A Case Study. Proceedings of National Symposium on Prediction Methods in Geotechnical Engineering GEOPREDICT2005, Indian Institute of Technology, Chennai, 113-120.25. Casagrande A (1948) Classification and Identification of Soils. Transactions of American Society of Civil Engineers 113.26. Mallikarjuna Rao K, Subba Reddy PV and Sudha Rani Ch (2006) Proper Parameters for Prediction of Compression Index. Proceedings National Conference on Corrective Engineering Practices in Troublesome Soils CONCEPTS 2006, JNTU College of Engineering, Kakinada, 35-40.27. Sridharan A (1990) Engineering Behavior of Soils – A Fundamental Approach IGS Lecture. 13th Indian Geotechnical Conference 36(1): 27-32. 117