Oedometer Consolidation Test Analysis By Nonlinear Regression
Geotechnical Testing Journal, Vol. 31, No. 1Paper ID GTJ101007Available online at: www.astm.orgPérsio L. A. Barros1 and Paulo R. O. Pinto1Oedometer Consolidation Test Analysis byNonlinear RegressionABSTRACT: A numerical method based on least squares nonlinear regression for the evaluation of the consolidation parameters of soils fromconsolidation tests is presented. A model which includes the initial compression, the primary consolidation, and the secondary compression is usedin the regression. This approach allows the resulting regression curve to better fit the experimental data. The method takes the settlement-timereadings from the oedometer step-loading consolidation test and calculates automatically the magnitudes of the coefficients of consolidation and ofsecondary compression. The performance of the proposed method is accessed through consolidation tests executed on four different clay soils, whichare analyzed by nonlinear regression and by the usual graphical methods. It is concluded that the proposed method gives results that are close to thoseobtained by the standard methods of analysis.KEYWORDS: consolidation test, nonlinear regression, least squares, coefficient of consolidation, secondary compressionIntroductionThe oedometer step-loading consolidation test is one of the mostwidely used tests in the soil mechanics laboratory. Introduced byTerzaghi as an experimental support for his one-dimensional consolidation theory (Terzaghi 1943), the oedometer test has remainedessentially unchanged since then.The main objective of the consolidation test is to access the consolidation characteristics of a soil from the measured settlementtime curve. From the consolidation theory, those characteristics areexpressed by the soil coefficient of consolidation cv and total consolidation settlement 100.The evaluation of cv and 100 from the measured consolidationcurve is generally performed by hand-draft curve fitting, being theTaylor’s 冑t method (Taylor 1948) and the Casagrande’s log t method (Casagrande and Fadum 1940), the most used.Those are regarded as standard methods of cv evaluation. Othermethods were developed later for this same purpose, such as therectangular hyperbola method (Shidharan and Prakash 1985) andthe early stage log t method (Robinson and Allam 1996). But all thecurve fitting methods above require graphical constructions that introduce undesirable subjective interpretations in the process.The various curve fitting methods lead, in general, to results thatare different from each other because each one focuses on differentportions of the consolidation curve. The main cause of these discrepancies is the experimental consolidation curve departure fromtheory, which is mainly caused by: There is an initial settlement 0 just after the load application, which is due to incomplete sample saturation, confiningring expansion, and deformation of the loading apparatus. The settlement continues after the theoretical end of consolidation in a process known as secondary compression.Manuscript received January 15, 2007; accepted for publication July 16,2007; published online September 2007.1Associate Professor, Department of Geotechnics and Transportation, andGraduate Student of Civil Engineering, respectively; State University of Campinas, Brazil.The initial settlement is easily treated by the various graphicalmethods and poses no special difficulties to the analysis. On theother hand, secondary compression interpretation is much morechallenging for there is no well established theoretical model for it.Even the point in time when secondary compression may first bedetected is subject to controversy (see, e.g., Robinson 2003).Added to those discrepancies, the observed consolidation curvemay also differ from the theoretical model due to the nonlinear behavior of the soil compressibility during the load increment andeven due to limitations of the test equipment and instrumentation.It is worth noting that if the soil behavior during the consolidation followed Terzaghi’s consolidation theory then all those fittingmethods would lead to the very same results. But the deviationsfrom theory make the result dependent on the particular aspect ofthe theoretical behavior each fitting method arbitrarily takes intoaccount. In this way, a less arbitrary procedure, with a sounder statistical base, like those based on least squares regression, is desirable.The automatic interpretation of the consolidation curve throughleast squares regression was implemented in recent works (Robinson and Allam 1998; Chan 2003; Day and Morris 2006). The proposed method adds to those the inclusion of secondary compression in the regression model. As a consequence, the resultingregression curve fits better to the final points of the experimentalconsolidation curve. Thus, the effects of the secondary compression on the resulting cv value can be controlled. Furthermore, theregression formulation developed here can be easily implementedin spreadsheet programs which are available in most soil mechanicslaboratories. The main benefits of the proposed method are: The need for human intervention in the interpretation process is kept to a minimum. In this way, manual calculationerrors are avoided. The interpretation process takes much less time, so the laboratory technicians can focus their attention on the test execution procedures. The values of the consolidation parameters obtained are essentially free of subjectivity, which makes correlations between them and other soil parameters more consistent.Copyright 2008 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959.1
2 GEOTECHNICAL TESTING JOURNAL All measured points are considered in the analysis. The secondary compression characteristics of the soil arealso obtained in the process. It is possible to couple an automatic data-logging system tothe evaluation method, resulting in a completely integratedsystem for logging, calculating, and interpreting the testresults.pressure measurements (Robinson 2003), showed secondary compression beginning, as indicated by the pore pressure measurements, from about 65 % of primary consolidation for organic claysto about 95 % for inorganic clays. Unfortunately, it is not feasibleto identify the beginning of secondary compression from the standard oedometer test results. Thus, the value of t0 should be arbitrarily fixed. If t0 is taken as a function of U共T兲, then a relationbetween t0 and cv can be established:Regression Model and Evaluation of ParametersThe nonlinear regression model considered here comprises threeparts:1. The initial instantaneous settlement 0.2. The primary consolidation p.3. The secondary compression s.Then, the settlement at any given time t is given by: 共t兲 0 p共t兲 s共t兲(1)The primary consolidation is given by Terzaghi’s onedimensional consolidation theory (Terzaghi 1943), expressed by: p共t兲 100U共T兲(2)where 100 is the total primary consolidation settlement and U共T兲 isthe degree of primary consolidation, given by: 共2m 1兲2 218 TU共T兲 1 242e m 0 共2m 1兲兺(3)and T is the time factor:T c vtH2d(4)where cv is the coefficient of consolidation and Hd is the drainageheight.For the secondary compression part, a number of alternativesare available. Rheological models (Gibson and Lo 1961; Whals1962), can describe very well the secondary compression, but require long-term tests to fully evaluate the model parameters. For thestandard short-term tests, which takes 24 hours for each load step,empirical models are preferred. Among the empirical models, thelinear-log t model (Buisman 1936) is the older, simpler, and mostused.According to the linear-log t model, the secondary compressioncan be expressed by: s共t兲 c Htlog101 e0t0(5)where c is the coefficient of secondary compression, H is thesample height, e0 is the initial void ratio, and t0 is the time when thesecondary compression is supposed to begin. The previous equation is valid only for t ⱖ t0; for t t0 no secondary compression issupposed to occur.It is important to note that t0 may differ substantially from theactual beginning of secondary compression; t0 is the time when thelinear-log t secondary compression model is assumed to begin.Therefore, t0 is called the model beginning of secondary compression.The time of beginning of secondary compression t0 is usuallydefined in terms of U共T兲. Special consolidation tests with poret0 TBoSH2dcv(6)where TBoS is the time factor corresponding to the model beginningof secondary compression.The choice of t0 (or of TBoS) do have some influence on the resulting cv and 100. Smaller values of t0 lead to larger values of cvand to smaller values of 100. This influence depends on the amountof secondary compression exhibited by the soil. This limitation iscommon to all curve fitting methods. But for the method proposedhere, since the model beginning of secondary compression is explicitly chosen by the user, the influence of the secondary compression on the calculated cv values can be assessed. Furthermore, theapplication of the results to field problems can take this choice intoaccount.It is worth noting that Taylor’s method implicitly assumes thatno secondary compression occurs before T90 (the time factor corresponding to 90 % of primary consolidation), and the graphical construction used for the 100 evaluation in Casagrande’s method implies that the secondary compression, if present, should be linearwith log t and should begin at around T95.The complete model resulting from the superposition of thethree parts is expressed by: 共t兲 0 100U冉 冊冉 冊c vtHtlog10 max 1,2 c 1 e0t0Hd(7)with model parameters 0, 100, c , and cv. The model expressed byEq 7 is linear with respect to 0, 100, and c , and nonlinear onlywith respect to cv. This fact allows for a specialized nonlinear leastsquares regression procedure.The least squares procedure is expressed by the minimization:Nmin兺 关 共ti兲 i兴2 0, 100,c ,cv i 1(8)where 共ti , i兲 are the time-settlement pairs measured during theconsolidation test and N is the number of readings in the load increment stage being analyzed.Since the model is linear with respect to all but one parameter,the problem reduces, through the use of derivatives with respect toeach parameter, to a system of three linear simultaneous equationscoupled to one nonlinear equation. Then, the solution of the complete system of equations can be obtained using a simple 3 3 system of linear equations solver coupled to a numerical root findingmethod. The proposed regression method can be implemented incommon spreadsheet programs in a very straightforward way, either through macro-programming or entirely with spreadsheet builtin functions. In the last case, most spreadsheet programs do incorporate in their standard configuration a solver for systems of linearequations and also a root finding facility called “goal seek,” whichcan be used in the procedure. Derivation details for the proposedmethod are given in the Appendix.
BARROS AND PINTO ON OEDOMETER CONSOLIDATION TEST ANALYSIS3TABLE 1—Clay samples characteristics.Sample123DescriptionBluish-gray clayCommercial bentoniteOrganic clayGs2.632.751.95LL46 %237 %230 %PI23 %188 %46 %As an alternative, one can use Eq 8 directly as input to a generalminimization routine, also found in some spreadsheet programs.This approach has some drawbacks when compared to the procedure used here, though. An initial guess for all four model parameters is required when using a general minimization routine,whereas for the specialized procedure, only an initial guess for cv isrequired. Moreover, general minimization routines are more illconvergence prone. That is, the minimization procedure can fail toconverge, or converge to unsatisfactory values. The proposed procedure is, by contrast, much more stable and robust, because thenonlinear minimization is performed in only one direction. In theother three directions the minimization is accomplished by the solution of a simple system of linear equations.Consolidation TestsIn order to access the performance of the proposed regressionmethod, a series of consolidation tests on different types of claysoils were executed. Three clay samples were used. The first one isa low plasticity, bluish-gray clay from a deposit near the city ofSanta Gertrudes, in the State of São Paulo, Brazil. The second clayis a commercial bentonite sample, from the State of Paraiba, Brazil.The last sample is a highly organic, black, marine clay, from thecity of Cubatão, State of São Paulo, Brazil. The main characteristics of those samples are given in Table 1, where Gs is the specificgravity of soil solids, LL is the liquid limit, and PI is the plasticityindex.Four remolded soil specimens for the consolidation tests wereprepared with those three samples. The first one was prepared withSample 1 only. The second and third specimens were prepared frommixtures of Sample 1 and Sample 2. Finally, Specimen 4 was prepared with Sample 3 only. The specimens’ composition is shown inTable 2.The specimens, measuring 71.4 mm in diameter and 20 mm inheight, were prepared for the tests by putting the soil mixture at awater content in the plastic range and then gently molding by handa small block from which the specimen was cut with the test ring. Inthis way, very soft, saturated specimens for which the magnitudeorder of the coefficient of consolidation remained constant for allload increments could be molded. The specimens initial water content w0 and void ratio e0 are also shown in Table 2.The consolidation cell uses a fixed-type ring setup with drainageon both top and bottom ends of the specimen. The consolidationtests were conducted in the standard way, with load increment ratioLIR 1, with first load q1 12.5 kPa and last load q8 1600 kPa.FIG. 1—Consolidation curve for load increment from 100 kPa to 200 kPa, forSpecimen 1.After each load increment, settlement measurements at times t in ageometric sequence with ratio ti / ti 1 冑2 were taken. For thosemeasurements, a digital dial gage with 0.001 mm resolution, linkedto a data logging program in a computer was used. Each load increment was left on the soil specimen for 24 hours, or until the secondary compression was clearly defined.After the tests, the datasets were processed both by hand calculation and by the proposed regression method. The hand calculationwas carried out independently by two experienced laboratory technicians; one of them using Taylor’s square root of time method andthe other using Casagrande’s logarithm of time method.Results and DiscussionThe plots in Figs. 1–4 show the measured points along with theregression curves obtained with the proposed method, for the loadincrement from v 100 kPa to v 200 kPa, for the four specimens. The agreement between the data points and the regressioncurves is excellent in all cases. The diversity in the curve characteristics for each soils is worth noting. Specimen 1 curve shows smalldeformability and high coefficient of consolidation cv. As the content of bentonite in the soil is increased in Specimens 2 and 3, thedeformability increases and the magnitude of cv decreases. The coefficient of secondary compression c also increases with the bentonite content.TABLE 2—Specimens composition and initial data.Specimen1234Bluish-gray clay100 %75 %50 %0%Bentonite0%25 %50 %0%Organic Clay0%0%0%100 %w036.9 %63.4 %79.6 %177.1 %e01.001.722.233.33FIG. 2—Consolidation curve for load increment from 100 kPa to 200 kPa, forSpecimen 2.
4 GEOTECHNICAL TESTING JOURNALFIG. 3—Consolidation curve for load increment from 100 kPa to 200 kPa, forSpecimen 3.For the organic clay (Specimen 4), the consolidation curveshows high deformability and also high values for both cv and c .The almost absence of the inflexion point in this last curve shouldbe noted. This fact makes the application of the standard graphicalmethods to the analysis very difficult, specially the Casagrande’smethod. But the regression method is capable of dealing with thistype of situation without much trouble.The application of the proposed method to the analysis of theother load increments, not shown here for brevity, presented similarresults. Figures 5–8 show the variation of cv as a function of theapplied vertical effective stress v for the four specimens, calculated for all the load increments by the two standard manual methods and by the automatic least squares regression method. It can beseen that the results obtained by the three methods are not far fromeach other, with exception of the cv values obtained for the organicclay. For that soil, the differences between the cv values obtainedwith the three methods as well as the cv variation with the appliedvertical stress is larger than for the other specimens. In all cases,however, the regression method gives results that are in the range ofvariation of the values obtained by the standard methods.The magnitude of the coefficient of secondary compression c isalso obtained with the proposed method. Figure 9 shows the variation of c with the vertical effective stress v for the four specimens.The plot in Fig. 9 shows that c is heavily dependent on the soiltype, but is less dependent on the applied load magnitude. As expected, the larger values of c were obtained for the organic clay,FIG. 4—Consolidation curve for load increment from 100 kPa to 200 kPa, forSpecimen 4.FIG. 5—Coefficient of consolidation cv for Specimen 1 evaluated by differentmethods.whereas the smaller values were obtained for the low plasticity clay.Also, the effect of the bentonite content on c can be clearly observed in the plot, increasing c by a large amount. It should beFIG. 6—Coefficient of consolidation cv for Specimen 2 evaluated by differentmethods.FIG. 7—Coefficient of consolidation cv for Specimen 3 evaluated by differentmethods.
BARROS AND PINTO ON OEDOMETER CONSOLIDATION TEST ANALYSIS5FIG. 8—Coefficient of consolidation cv for Specimen 4 evaluated by differentmethods.noted, however, that the obtained c values, specially those valuesfor Sample 3, which has the lowest coefficient of consolidation, arenot completely trustful, since they are based on a small number ofreadings in the secondary compression range of the consolidationcurve. Long-term consolidation tests would be necessary to gethigher confidence on the c values. Anyway, the obtained c valuesfor all soils are in the range reported in the literature for those typesof soil (Mesri and Godlewski 1977).As stated in the previous section, the model beginning of secondary compression should be arbitrarily set in the proposedmethod. For the numerical results shown hereto, the model beginning of secondary compression is set to 95 % of the primary consolidation 共TBoS 1.129兲. This setting has some effect on the obtained values of cv, depending on the amount of secondarycompression in the soil behavior. Figure 10 shows the variation ofthe cv values obtained for the load increment from v 100 kPa to200 kPa with the setting of the beginning of secondary compression, for all samples.The increment of the calculated cv values with the reduction oft0, shown in Fig. 10, is much larger for the organic clay than for theother three samples, due to the larger amount of secondary compression in the organic clay settlement. But the earlier values of t0used to get the results in Fig. 10 are hardly justified, without addi-FIG. 9—Variation of c with v for the four clay specimens.FIG. 10—Variation of cv with the model beginning of secondary compression asa percentage of the primary consolidation (load increment from v 100 kPa to200 kPa).tional information on the ongoing secondary compression, eventaking the reports of beginning of secondary compression, based onexperimental observations, at points prior to 75 % of primary consolidation for peat (Robinson 2003), into account. The linear-log tmodel used here proba
Pérsio L. A. Barros1 and Paulo R. O. Pinto1 Oedometer Consolidation Test Analysis by Nonlinear Regression ABSTRACT:A numerical method based on least squares nonlinear regression for the evaluation of the consolidation parameters of soils from consolidation tests is presented.A model which includes the initial compression, the primary consolidation, and the secondary compression is used
CONSOLIDATION TEST 1.Objective Consolidation test is used to determine the rate and magnitude of soil consolidation when the soil is restrained laterally and loaded axially. The Consolidation test is also referred to as Standard Oedometer test or One-dimensional compression test. This test is carried out on saturated soil specimens,
n Stress-Strain in Consolidation n 1-D Consolidation Test . การทดสอบแบบ Oedometer (Consolidation) D e fo r m at i on Time (Log scale) Initial consolidation Primary consolidation �ัวคายน้ํา Secondary consolidation.
CONSOLIDATION TEST To predict consolidation settlement in a soil, . characteristics of soil is known as the consolidation test apparatus (or oedometer) and is illustrated in Fig.2a. The soil sample is encased in a steel cutting ring. Porous discs, saturated with air-free water,
The test is known as a one dimensional consolidation test (also known as an oedometer test). The compressibility of granular soil is usually estimated empirically from in situ (field) tests, such as the Standard Penetration Test. The consolidation test generally consists of placi ng an undisturbed sample in a consolidometer, also known as an .
F2.3 Oedometer Test on Slimes An oedometer test was done on a sample of slurried slimes prepared from the January campaign of the Panel’s field program. The test results are given in Appendix D and repeated here in Table F 2-2. Table F2-2 Slimes consolidation laboratory testing Data Source Pressure (kPa) Void Ratio Cc k (cm/s) Appendix D
Database Consolidation Workbench Plan -Gives consolidation advice by identifying candidate databases for the designated consolidation platform using AWR data Migrate -Implements consolidation plan by migrating databases to new consolidation platform using EM's provisioning features Validate
SNU Geotechnical and Geoenvironmental Engineering Lab. 1-26 1.6.2 Consolidation Test - Simulates 1-dimensional state (flow and deformation) Using a circular metal ring confining the sample. (Possible to measure the pore pressure during consolidation and to perform the permeability tests in the oedometer.)
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