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

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 statistical 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 laboratory 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.

• 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. Then, the solution of the com- secondary compression is supposed to begin. The previous equa- plete system of equations can be obtained using a simple 3ϫ3 sys- Ն Ͻ tion is valid only for t t0; for t t0 no secondary compression is tem of linear equations solver coupled to a numerical root finding supposed to occur. method. The proposed regression method can be implemented in It is important to note that t0 may differ substantially from the common spreadsheet programs in a very straightforward way, ei- actual beginning of secondary compression; t0 is the time when the ther through macro-programming or entirely with spreadsheet built linear-log t secondary compression model is assumed to begin. in functions. In the last case, most spreadsheet programs do incor- Therefore, t0 is called the model beginning of secondary compres- porate in their standard configuration a solver for systems of linear sion. equations and also a root finding facility called “goal seek,” which The time of beginning of secondary compression t0 is usually can be used in the procedure. Derivation details for the proposed defined in terms of U͑T͒. Special consolidation tests with pore method are given in the Appendix. BARROS AND PINTO ON OEDOMETER CONSOLIDATION TEST ANALYSIS 3

TABLE 1—Clay samples characteristics.

Sample Description Gs LL PI 1 Bluish-gray clay 2.63 46 % 23 % 2 Commercial bentonite 2.75 237 % 188 % 3 Organic clay 1.95 230 % 46 %

As an alternative, one can use Eq 8 directly as input to a general minimization 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 is required. Moreover, general minimization routines are more ill- convergence prone. That is, the minimization procedure can fail to FIG. 1—Consolidation curve for load increment from 100 kPa to 200 kPa, for converge, or converge to unsatisfactory values. The proposed pro- Specimen 1. cedure is, by contrast, much more stable and robust, because the nonlinear minimization is performed in only one direction. In the After each load increment, settlement measurements at times t in a other three directions the minimization is accomplished by the so- geometric sequence with ratio t /t =ͱ2 were taken. For those lution of a simple system of linear equations. i i−1 measurements, a digital dial gage with 0.001 mm resolution, linked to a data logging program in a computer was used. Each load increment was left on the soil specimen for 24 hours, or until the second- Consolidation Tests ary compression was clearly defined. In order to access the performance of the proposed regression After the tests, the datasets were processed both by hand calcu- method, a series of consolidation tests on different types of clay lation and by the proposed regression method. The hand calculation soils were executed. Three clay samples were used. The first one is was carried out independently by two experienced laboratory tech- a low plasticity, bluish-gray clay from a deposit near the city of nicians; one of them using Taylor’s square root of time method and Santa Gertrudes, in the State of São Paulo, Brazil. The second clay the other using Casagrande’s logarithm of time method. is a commercial bentonite sample, from the State of Paraiba, Brazil. The last sample is a highly organic, black, marine clay, from the city of Cubatão, State of São Paulo, Brazil. The main characteris- Results and Discussion tics of those samples are given in Table 1, where Gs is the specific gravity of soil solids, LL is the liquid limit, and PI is the plasticity The plots in Figs. 1–4 show the measured points along with the index. regression curves obtained with the proposed method, for the load ␴Ј ␴Ј Four remolded soil specimens for the consolidation tests were increment from v =100 kPa to v =200 kPa, for the four speci- prepared with those three samples. The first one was prepared with mens. The agreement between the data points and the regression Sample 1 only.The second and third specimens were prepared from curves is excellent in all cases. The diversity in the curve character- mixtures of Sample 1 and Sample 2. Finally, Specimen 4 was pre- istics for each soils is worth noting. Specimen 1 curve shows small pared with Sample 3 only. The specimens’ composition is shown in deformability and high coefficient of consolidation cv. As the con- Table 2. tent of bentonite in the soil is increased in Specimens 2 and 3, the The specimens, measuring 71.4 mm in diameter and 20 mm in deformability increases and the magnitude of cv decreases. The co- height, were prepared for the tests by putting the soil mixture at a efficient of secondary compression c␣ also increases with the ben- water content in the plastic range and then gently molding by hand tonite content. a small block from which the specimen was cut with the test ring. In this way, very soft, saturated specimens for which the magnitude order of the coefficient of consolidation remained constant for all load 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 drainage on both top and bottom ends of the specimen. The consolidation tests were conducted in the standard way, with load increment ratio LIR=1, with first load q1 =12.5 kPa and last load q8 =1600 kPa.

TABLE 2—Specimens composition and initial data.

Specimen Bluish-gray clay Bentonite Organic Clay w0 e0 1 100 % 0 % 0 % 36.9 % 1.00 2 75 % 25 % 0 % 63.4 % 1.72 3 50 % 50 % 0 % 79.6 % 2.23 4 0 % 0 % 100 % 177.1 % 3.33 FIG. 2—Consolidation curve for load increment from 100 kPa to 200 kPa, for Specimen 2. 4 GEOTECHNICAL TESTING JOURNAL

FIG. 3—Consolidation curve for load increment from 100 kPa to 200 kPa, for Specimen 3.

FIG. 5—Coefficient of consolidation cv for Specimen 1 evaluated by different methods. For the organic clay (Specimen 4), the consolidation curve shows high deformability and also high values for both cv and c␣. The almost absence of the inflexion point in this last curve should whereas the smaller values were obtained for the low plasticity clay. be noted. This fact makes the application of the standard graphical Also, the effect of the bentonite content on c␣ can be clearly ob- methods to the analysis very difficult, specially the Casagrande’s served in the plot, increasing c␣ by a large amount. It should be method. But the regression method is capable of dealing with this type of situation without much trouble. The application of the proposed method to the analysis of the other load increments, not shown here for brevity, presented similar results. Figures 5–8 show the variation of cv as a function of the ␴Ј applied 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 be seen that the results obtained by the three methods are not far from each other, with exception of the cv values obtained for the organic clay. For that soil, the differences between the cv values obtained with the three methods as well as the cv variation with the applied vertical stress is larger than for the other specimens. In all cases, however, the regression method gives results that are in the range of variation of the values obtained by the standard methods. The magnitude of the coefficient of secondary compression c␣ is also obtained with the proposed method. Figure 9 shows the varia- ␴Ј tion of c␣ with the vertical effective stress v for the four speci- FIG. 6—Coefficient of consolidation cv for Specimen 2 evaluated by different mens. methods. The plot in Fig. 9 shows that c␣ is heavily dependent on the soil type, but is less dependent on the applied load magnitude. As ex- pected, the larger values of c␣ were obtained for the organic clay,

FIG. 4—Consolidation curve for load increment from 100 kPa to 200 kPa, for FIG. 7—Coefﬁcient of consolidation cv for Specimen 3 evaluated by different Specimen 4. methods. BARROS AND PINTO ON OEDOMETER CONSOLIDATION TEST ANALYSIS 5

FIG. 8—Coefﬁcient of consolidation cv for Specimen 4 evaluated by different methods.

noted, however, that the obtained c␣ values, specially those values FIG. 10—Variationof cv with the model beginning of secondary compression as ␴Ј for Sample 3, which has the lowest coefficient of consolidation, are a percentage of the primary consolidation (load increment from v =100 kPa to not completely trustful, since they are based on a small number of 200 kPa). readings in the secondary compression range of the consolidation curve. Long-term consolidation tests would be necessary to get tional information on the ongoing secondary compression, even higher confidence on the c␣ values. Anyway, the obtained c␣ values taking the reports of beginning of secondary compression, based on for all soils are in the range reported in the literature for those types experimental observations, at points prior to 75 % of primary con- of soil (Mesri and Godlewski 1977). solidation for peat (Robinson 2003), into account. The linear-log t As stated in the previous section, the model beginning of sec- model used here probably does not apply to those reported, extreme ondary compression should be arbitrarily set in the proposed cases, which require a more refined secondary compression model method. For the numerical results shown hereto, the model begin- to match the experimental observations. Also, those early beginning of secondary compression is set to 95 % of the primary con- ning of secondary compression times were observed in consolida- ͑ ͒ solidation TBoS =1.129 . This setting has some effect on the ob- tion tests with load increment ratio much lower than 1. Thus, in tained values of cv, depending on the amount of secondary order to stay consistent with the standard graphical methods, values compression in the soil behavior. Figure 10 shows the variation of between 90 and 95 % are suggested for the model beginning of sec- ␴Ј the cv values obtained for the load increment from v =100 kPa to ondary compression, for all soil types, when LIR=1 and the linear- 200 kPa with the setting of the beginning of secondary compres- log t model is used, unless more information on the secondary com- sion, for all samples. pression for that soil type is available. The increment of the calculated cv values with the reduction of It is interesting to note, also, that the regression curves resulting t0, shown in Fig. 10, is much larger for the organic clay than for the from different settings of the model beginning of secondary com- other three samples, due to the larger amount of secondary compression are close to each other. Figure 11 shows the regression pression in the organic clay settlement. But the earlier values of t0 curves for Specimen 4, obtained by setting t at 50 %, 75 %, and at used to get the results in Fig. 10 are hardly justified, without addi- 0

FIG. 11—Regression curves obtained with the beginning of secondary compression set at 50 %, 75 %, and 90 % of the primary consolidation (Specimen ␴Ј ␴Ј FIG. 9—Variation of c␣ with v for the four clay specimens. 4, load increment from v =100 kPa to 200 kPa). 6 GEOTECHNICAL TESTING JOURNAL

TABLE 3—Coefficient of consolidation values for original, reduced, and served relative standard deviation magnitude can be considered ␴Ј ␴Ј reduced-with-noise datasets (load increment from v =100 kPa to v low, given the magnitude of the Gaussian noise introduced. As ex- =200 kPa). pected, the relative standard deviation magnitude is higher for the 3 ϫ ͑ 2 ͒ stiffer specimens, specially for Specimen 1, because the noise am- 10 cv mm /s Set with Noise plitude relative to the total settlement is higher for those specimens. It is concluded, from the simulations made, that the proposed re- Specimen Original Set Reduced Set Mean Std. Dev. gression method is very stable and gives consistent results. 1 73.9 75.6 76.6 13.3 ͑17.3 %͒ Additional extensions to the regression method proposed herein 2 5.72 5.80 5.94 0.87 ͑14.6 %͒ can be devised. One can assign different weights to the readings, 3 1.81 1.82 1.79 0.20 ͑10.9 %͒ thus favoring some of them over the others. For example, larger 4 33.2 31.5 32.2 3.97 ͑12.3 %͒ weights can be assigned to the initial readings, which are less affected by secondary compression. In this way the resulting cv will be less affected by the secondary compression. This weighted re- 95 % of the primary consolidation. The plot in Fig. 11 shows the gression method can be implemented by replacing the least square three regression curves almost coincident. Thus, it is not feasible to procedure in Eq 8 with a weighted least square procedure: use a regression method to estimate an optimal point for the model beginning of secondary compression. N It should be noted that the linear-log t secondary compression ͓␦͑ ͒ ␦ ͔2 min ͚ wi ti − i (9) model introduces a small “kink” in the regression curve at t . This ␦ ␦ 0 0, 100,c␣,cv i=1 kink, which is an artifact of the regression model, is apparent only when the coefficient of secondary compression is high, as for soft where wi are the weights assigned to each reading. organic clays. A consequence of this discontinuity is that in some Another possible development is to perform a compound simul- rare cases the numerical procedure may fail to converge. This may taneous regression with datasets from more than one load incre- happen when a test reading point is too close to t0 and the quality of ment at once, sharing the same cv (and possibly the same c␣) the data is poor. As a workaround, one can try small changes in the (Bowen and Jerman 1995). This procedure will result in an average TBoS,sot0 is moved away from that reading point. value for cv (and c␣) that is appropriate for a load range wider than As a final note on this topic, it should pointed out that the sec- one single load increment. ondary compression model can be replaced by another one more appropriate, if it is available, with only minor modifications in the proposed method. Concluding Remarks The test data used in this research were obtained with an equipment setup that may be of higher quality than the equipment avail- A nonlinear regression method based on the least squares proce- able in some laboratories, specially the apparatus setup for settle- dure for the analysis of the oedometer consolidation test data was ment measuring, which employs a high precision electronic dial presented. The method can deal with test data from different types gage coupled to an automatic data logging system. In many cases, of clay soils and calculates the values of the coefficient of consoli- mechanical dial gage extensometers with 0.01 mm precision and dation cv and of secondary compression c␣ without the need of manual data logging are used in practice. In those cases, a much hand drawing the consolidation curve and of other graphical con- reduced number of settlement readings are taken, usually at times structions. The resulting cv and c␣ obtained with this method have 8, 15, and 30 s, 1, 2, 4, 8, 15, and 30 min, 1, 2, 4, 8, and 24 h. statistical significance since the least squares procedure is used. Moreover, the manual data logging can lead to even less precision The regression method can be implemented in spreadsheet com- in the readings, specially for the initial ones, when the dial pointer puter programs that is commonly found in most soil laboratories. is moving faster. This implementation can be done with macro-programming or with In order to test the proposed method behavior in face of those the spreadsheet built in functions. In the last case the “goal seek” adverse conditions, a number of numerical simulations were made. facility found in those computer programs can be used. The simulation process was applied in two steps. First, the extra Numerical values for cv, from consolidation tests executed on readings between those taken at times close to those listed above different types of clays, show that the values obtained by the pro- were deleted from the datasets and the kept settlement values were posed method are in good agreement with those obtained by the rounded to two decimal figures. New values of cv were then ob- standard graphical methods. Also, the values of c␣ obtained with tained with the regression method from these reduced datasets. the proposed method are in the range of values reported in the lit- Next, in the second step, a Gaussian random noise with a pessimis- erature. tic standard deviation of 0.02 mm (two divisions on the mechanical Simulations through variation of data quality showed that the dial gage face) was added to that reduced settlement values and the numerical regression procedure is very stable and leads to consis- regression was applied to the resulting datasets. This second step tent results for most cases. For the rare cases that have convergence was repeated ten times for each reduced dataset. Then, mean and problems, a simple workaround can be applied. standard deviation values of the cv were calculated. Table 3 shows The main limitation of the proposed method is the time of model the results obtained with those simulations. beginning of secondary compression which has to be arbitrarily set The numerical results in Table 3 show that the regression at some degree of the primary consolidation. This limitation is method works very well with the datasets with reduced number of common to all analysis methods based only on settlement measure- points and lower precision, since the cv values obtained with both ments. the original and reduced datasets are very close to each other for all Possible extensions of the proposed method include the use of four specimens. Also, the mean cv values obtained with the reduced different models for the secondary compression, the use of a set with added noise are close to the original cv values. The ob- weighted least squares procedure, and the compound simultaneous BARROS AND PINTO ON OEDOMETER CONSOLIDATION TEST ANALYSIS 7 regression, in which more than one load increment dataset is pro- ͑ ͒ ͑ ͒ ␦ ␦ N U c¯v,ti S t0,ti 0 i cessed at once. ͑¯ ͒ 2͑¯ ͒ ͑¯ ͒ ͑ ͒ ␦ ␦ ͑¯ ͒ ΄U cv,ti U cv,ti U cv,ti S t0,ti ΅Ά 100 · = Ά iU cv,ti · ͑ ͒ ͑ ͒ ͑ ͒ 2͑ ͒ ␦ ͑ ͒ S t0,ti U c¯v,ti S t0,ti S t0,ti c¯B iS t0,ti Acknowledgments (16) The authors want to thank to the laboratory technicians, L. E. with implied summation over i. Meyer and J. B. Cipriano, for the graphical calculations, and to the A fourth equation is obtained by taking the derivative of D rela- technicians, E. Jurgensen and R. B. L. Silva, for the help with the tive to c¯v and equating it to zero: consolidation tests. The second author was partially supported by n Dץ the Conselho Nacional de Desenvolvimento Científico e Tec- F = =2͚ ͓␦ UЈ͑c¯ ,t ͒ + c¯ SЈ͑c¯ ,t ͔͒R = 0 (17) v i B v i i 100 ץ nológico CNPq, Brazil. This research has received partial support c¯v i=1 from Maccaferri do Brasil Ltda. where ϱ 2m +1͒2␲2͑ ͒ ͑ ץ U c¯v,t − c¯ t UЈ͑c¯ ,t͒ = =2t͚ e 4 v (18) ¯cץ Appendix: Algorithm Derivation for the Regres- v sion Method v m=0 and The model assumed for the soil sample settlement in the oedometer, under a stress increment, as a function of the time is: t log maxͩ1, i ͪͬ 0 t Յ tͫץ 10 t 0 ␦ = ␦ + ␦ U͑c¯ ,t͒ + c¯ S͑t ,t͒ (10) SЈ͑c¯ ,t͒ = 0 = 1 (19) Ά Ͼ ץ v B 0 v 100 0 c¯v t t0 ␦ ␦ c¯ ln 10 where 0 is the initial instantaneous displacement, 100 is the total v primary consolidation displacement, The solution of the set of four equations is obtained by an itera- ϫ ϱ tive procedure based on a simple solution for the 3 3 linear sys- ͑2m +1͒2␲2 8 1 − c¯ t tem and on the Newton-Raphson root-finding method for the last U͑c¯ ,t͒ =1− ͚ e 4 v (11) v ␲2 ͑2m +1͒2 equation. m=0 0 First, an initial guess c¯v is estimated. The initial t0 is then calcu- is the percentage of primary consolidation from Terzaghi’s theory, ␦ ␦ lated from Eq 13 and 0, 100, and c¯B are calculated from Eq 16. 2 c¯v =cv /Hd is the modified coefficient of consolidation, Hd is the Next, a better approximation for c¯ can be obtained from: ͑ ͒ v height of drainage, c¯B =cBH=c␣H/ 1+e0 is the modified coefficient of secondary compression, H is the sample height, e is the ͑ ͒ ͑ ͒ F 0 c¯ k+1 = c¯ k − ͯ ͯ (20) v v ͑ ͒ initial void ratio, and FЈ ¯ k cv t ͑ ͒ ͩ ͪ where S t0,t = log10 max 1, (12) t0 n n Fץ ͓␦ ͑ ͒ ͑ ͔͒ ͓␦ ͑ ͒ where t is the time of the model beginning of secondary compres- FЈ = =2͚ 100UЉ c¯v,ti + c¯BSЉ c¯v,ti Ri +2͚ 100UЈ c¯v,ti ¯cץ 0 sion. The parameters for the model above, to be determined from v i=1 i=1 ␦ the measured settlement i at time ti during the consolidation test, + c¯ SЈ͑c¯ ,t ͔͒2 (21) ␦ ␦ B v i are 0, 100, c¯v, and c¯B. The time t0 is supposed to be related to c¯v by: and

TBoS ϱ ␲ ͑2m +1͒2␲2 ͒ ͑ 2ץ (t = (13 0 U c¯v,t 2 2 − c¯ t c¯v UЉ͑c¯ ,t͒ = =− t ͚ ͑2m +1͒ e 4 v (22) 2 ץ v c¯v 2 m=0 where TBoS is the time factor which corresponds to the model beginning of secondary compression, which is supposed to be known. t i ͪͬ Յ ͩ 2ͫץ -The regression algorithm is based on a least-squares ﬁt proce log max 1, 0 t t0 dure. The sum of the squared differences between the model dis- 10 t SЉ͑c¯ ,t͒ = 0 = 1 (23) Ά Ͼ 2 ץ placements at t and the measured displacement d is: v i i c¯v − 2 t t0 c¯v ln 10 N ͑k+1͒ ͑k͒ 2 The procedure is repeated until ͉c¯ −c¯ ͉Յ⑀, where ⑀ is the D = ͚ Ri (14) v v i=1 required precision. ͑ ͒ The initial guess c¯ 0 can be taken as: where v T 0.192 ␦ ␦ ͑ ͒ ͑ ͒ ␦ ͑0͒ 50 Ri = 0 + 100U c¯v,ti + c¯BS t0,ti − i (15) c¯v = = (24) ¯t50 ¯t50 is the residue at the ith point. ␦ ␦ where ¯t is the estimated time for 50 % of the consolidation, which Taking the derivatives of D relative to 0, 100, and c¯B, and 50 equating each one to zero in order to get a minimum, a linear sys- can be taken as the time that corresponds to half of the total settle- ␦¯ tem of equations is obtained. This system, in matricial form, is: ment, excluding the initial settlement, due the load increment, 50: 8 GEOTECHNICAL TESTING JOURNAL

␦ + ␦ Gibson, R. E. and Lo, K. Y., 1961, “A Theory of Consolidation for ␦¯ = 1 N (25) 50 2 Soils Exhibiting Secondary Compression,” Norwegian Geo- technical Institute Publication 41, pp. 1–16. ␦¯ After calculating 50, ¯t50 may be estimated by linear interpola- Mesri, G. and Godlewski, P. M., 1977, “Time- and Stress- ␦¯ Compressibility Interrelationship,” J. Geotech. Engrg. Div., tion from the two available readings closest to 50. ASCE, Vol. 103, No. GT5, pp. 417–430. Robinson, R. G., 2003, “A Study on the Beginning of Secondary References Compression of Soils,” J. Test. Eval., ASTM, Vol. 31, No. 5, pp. 1–10. Bowen, W.P.and Jerman, J. C., 1995, “Nonlinear Regression Using Robinson, R. G. and Allam, M. M., 1996, “Determination of the Spreadsheets,” Trends Pharmacol. Sci., Elsevier, Vol. 16, No. Coefficient of Consolidation from Early Stage of Log t Plot,” 12, pp. 413–417. Geotech. Test. J., ASTM, Vol. 19, No. 3, pp. 316–320. Buisman, A. S. K., 1936, “Results of Long Duration Tests,” Pro- Robinson, R. G. and Allam, M. M., 1998, “Analysis of Consolida- ceedings of the First International Conference on Soil Mechan- tion Data by a Non-Graphical Matching Method,” Geotech.Test. ics and Foundation Engineering, Cambridge, Massachusetts, J., ASTM, Vol. 21, No. 2, pp. 140–143. Vol. 1, pp. 103–105. Sridharan, A. and Prakash, K., 1985, “Improved Rectangular Hy- Casagrande, A. and Fadum, R. E., 1940, “Notes on Soil Testing for Engineering Purposes,” Harvard University Graduate School of perbola Method for Determination of Coefficient of Consolida- Engineering Publication, n. 8. tion,” Geotech. Test. J., ASTM, Vol. 8, No. 1, pp. 37–40. Chan, A. H. C., 2003, “Determination of the Coefficient of Con- Taylor, D. W., 1948, Fundamentals of Soil Mechanics, John Wiley solidation Using a Least Squares Method,” Geotechnique, Vol. and Sons, New York. 53, No. 7, pp. 673–678. Terzaghi, K., 1943, Theoretical Soil Mechanics, John Wiley and Day, R. A. and Morris, P. H., 2006, Discussion on “Determination Sons, New York. of the Coefficient of Consolidation Using a Least Squares Wahls, H. E., 1962, “Analysis of Primary and Secondary Consoli- Method,” by Chan, A. H. C., Geotechnique, Vol. 56, No. 1, dation,” J. Soil Mech. and Found. Div., ASCE, Vol. 88, No. pp. 73–76. SM6, pp. 207–231.