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Oedometer Consolidation Test Analysis by Nonlinear Regression

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Oedometer Consolidation Test Analysis by Nonlinear Regression Geotechnical Testing Journal, Vol. 31, No. 1 Paper ID GTJ101007 Available online at: www.astm.org 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 in the regression. This approach allows the resulting regression curve to better fit the experimental data. The method takes the settlement-time readings from the oedometer step-loading consolidation test and calculates automatically the magnitudes of the coefficients of consolidation and of secondary compression. The performance of the proposed method is accessed through consolidation tests executed on four different clay soils, which are analyzed by nonlinear regression and by the usual graphical methods. It is concluded that the proposed method gives results that are close to those obtained by the standard methods of analysis. KEYWORDS: consolidation test, nonlinear regression, least squares, coefficient of consolidation, secondary compression Introduction The initial settlement is easily treated by the various graphical methods and poses no special difficulties to the analysis. On the The oedometer step-loading consolidation test is one of the most other hand, secondary compression interpretation is much more widely used tests in the soil mechanics laboratory. Introduced by challenging for there is no well established theoretical model for it. Terzaghi as an experimental support for his one-dimensional con- Even the point in time when secondary compression may first be solidation theory (Terzaghi 1943), the oedometer test has remained detected is subject to controversy (see, e.g., Robinson 2003). essentially unchanged since then. Added to those discrepancies, the observed consolidation curve The main objective of the consolidation test is to access the con- may also differ from the theoretical model due to the nonlinear be- solidation characteristics of a soil from the measured settlement- havior of the soil compressibility during the load increment and time curve. From the consolidation theory, those characteristics are even due to limitations of the test equipment and instrumentation. expressed by the soil coefficient of consolidation cv and total con- It is worth noting that if the soil behavior during the consolida- ␦ solidation settlement 100. tion followed Terzaghi’s consolidation theory then all those fitting ␦ The evaluation of cv and 100 from the measured consolidation methods would lead to the very same results. But the deviations curve is generally performed by hand-draft curve fitting, being the from theory make the result dependent on the particular aspect of Taylor’s ␦ϫͱt method (Taylor 1948) and the Casagrande’s ␦ the theoretical behavior each fitting method arbitrarily takes into ϫlog t method (Casagrande and Fadum 1940), the most used. account. In this way, a less arbitrary procedure, with a sounder sta- tistical base, like those based on least squares regression, is desir- Those are regarded as standard methods of cv evaluation. Other methods were developed later for this same purpose, such as the able. rectangular hyperbola method (Shidharan and Prakash 1985) and The automatic interpretation of the consolidation curve through the early stage log t method (Robinson and Allam 1996). But all the least squares regression was implemented in recent works (Robin- curve fitting methods above require graphical constructions that in- son and Allam 1998; Chan 2003; Day and Morris 2006). The pro- troduce undesirable subjective interpretations in the process. posed method adds to those the inclusion of secondary compres- The various curve fitting methods lead, in general, to results that sion in the regression model. As a consequence, the resulting are different from each other because each one focuses on different regression curve fits better to the final points of the experimental portions of the consolidation curve. The main cause of these dis- consolidation curve. Thus, the effects of the secondary compres- crepancies is the experimental consolidation curve departure from sion on the resulting cv value can be controlled. Furthermore, the theory, which is mainly caused by: regression formulation developed here can be easily implemented in spreadsheet programs which are available in most soil mechanics ␦ • There is an initial settlement 0 just after the load applica- laboratories. The main benefits of the proposed method are: tion, which is due to incomplete sample saturation, confining • The need for human intervention in the interpretation pro- ring expansion, and deformation of the loading apparatus. cess is kept to a minimum. In this way, manual calculation • The settlement continues after the theoretical end of consoli- errors are avoided. dation in a process known as secondary compression. • The interpretation process takes much less time, so the labo- ratory technicians can focus their attention on the test execu- Manuscript received January 15, 2007; accepted for publication July 16, 2007; published online September 2007. tion procedures. 1Associate Professor, Department of Geotechnics and Transportation, and • The values of the consolidation parameters obtained are es- Graduate Student of Civil Engineering, respectively; State University of Campi- sentially free of subjectivity, which makes correlations be- nas, Brazil. tween 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. pressure measurements (Robinson 2003), showed secondary com- • The secondary compression characteristics of the soil are pression beginning, as indicated by the pore pressure measure- also obtained in the process. ments, from about 65 % of primary consolidation for organic clays • It is possible to couple an automatic data-logging system to to about 95 % for inorganic clays. Unfortunately, it is not feasible the evaluation method, resulting in a completely integrated to identify the beginning of secondary compression from the stan- system for logging, calculating, and interpreting the test dard oedometer test results. Thus, the value of t0 should be arbi- ͑ ͒ results. trarily fixed. If t0 is taken as a function of U T , then a relation between t0 and cv can be established: 2 Regression Model and Evaluation of Parameters TBoSHd t0 = (6) cv The nonlinear regression model considered here comprises three where TBoS is the time factor corresponding to the model beginning parts: of secondary compression. 1. The initial instantaneous settlement ␦ . The choice of t0 (or of TBoS) do have some influence on the re- 0 ␦ 2. The primary consolidation ␦ . sulting cv and 100. Smaller values of t0 lead to larger values of cv p ␦ ␦ and to smaller values of 100. This influence depends on the amount 3. The secondary compression s. Then, the settlement at any given time t is given by: of secondary compression exhibited by the soil. This limitation is common to all curve fitting methods. But for the method proposed ␦͑ ͒ ␦ ␦ ͑ ͒ ␦ ͑ ͒ t = 0 + p t + s t (1) here, since the model beginning of secondary compression is ex- plicitly chosen by the user, the influence of the secondary compres- The primary consolidation is given by Terzaghi’s one- sion on the calculated cv values can be assessed. Furthermore, the dimensional consolidation theory (Terzaghi 1943), expressed by: application of the results to field problems can take this choice into ␦ ͑ ͒ ␦ ͑ ͒ account. p t = 100U T (2) It is worth noting that Taylor’s method implicitly assumes that ␦ ͑ ͒ where 100 is the total primary consolidation settlement and U T is no secondary compression occurs before T90 (the time factor corre- the degree of primary consolidation, given by: sponding to 90 % of primary consolidation), and the graphical con- ϱ struction used for the ␦ evaluation in Casagrande’s method im- ͑2m +1͒2␲2 100 8 1 − T U͑T͒ =1− ͚ e 4 (3) plies that the secondary compression, if present, should be linear ␲2 ͑2m +1͒2 m=0 with log t and should begin at around T95. The complete model resulting from the superposition of the and T is the time factor: three parts is expressed by: c t T = v (4) 2 cvt H t Hd ␦͑ ͒ ␦ ␦ ͩ ͪ ͩ ͪ t = 0 + 100U 2 + c␣ log10 max 1, (7) Hd 1+e0 t0 where cv is the coefficient of consolidation and Hd is the drainage ␦ ␦ height. with model parameters 0, 100, c␣, and cv. The model expressed by ␦ ␦ For the secondary compression part, a number of alternatives Eq 7 is linear with respect to 0, 100, and c␣, and nonlinear only are available. Rheological models (Gibson and Lo 1961; Whals with respect to cv. This fact allows for a specialized nonlinear least 1962), can describe very well the secondary compression, but re- squares regression procedure. quire long-term tests to fully evaluate the model parameters. For the The least squares procedure is expressed by the minimization: standard short-term tests, which takes 24 hours for each load step, empirical models are preferred. Among the empirical models, the N ͓␦͑ ͒ ␦ ͔2 linear-log t model (Buisman 1936) is the older, simpler, and most min ͚ ti − i (8) ␦ ␦ used. 0, 100,c␣,cv i=1 According to the linear-log t model, the secondary compression ͑ ␦ ͒ where ti , i are the time-settlement pairs measured during the can be expressed by: consolidation test and N is the number of readings in the load incre- H t ment stage being analyzed. ␦ ͑ ͒ s t = c␣ log10 (5) Since the model is linear with respect to all but one parameter, 1+e t 0 0 the problem reduces, through the use of derivatives with respect to where c␣ is the coefficient of secondary compression, H is the each parameter, to a system of three linear simultaneous equations sample height, e0 is the initial void ratio, and t0 is the time when the coupled to one nonlinear equation.
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