ON STATISTICAL INFERENCE OF THE COEFFICIENT OF CONSOLIDATION BY USE OF NUMERICAL OPTIMISATION

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1 57ième CONGRÈS CANADIEN DE GÉOTECHNIQUE 5ième CONGRÈS CONJOINT SCG/AIH-CNN 57TH CANADIAN GEOTECHNICAL CONFERENCE 5TH JOINT CGS/IAH-CNC CONFERENCE ON STATISTICAL INFERENCE OF THE COEFFICIENT OF CONSOLIDATION BY USE OF NUMERICAL OPTIMISATION Bhamisha Ramharry, Faculté ingéniérie, Université e Moncton, Moncton E1A 3E9 Paul Chiasson, Faculté Ingéniérie, Université e Moncton, Moncton, NB, E1A 3E9 chssp@umoncton.ca ABSTRACT A Least Squares metho is evelope to estimate the coefficient of consoliation from oeometetric ata. It involves the comparison of the theoretical an experimental values of the average egree of consoliation base on the Terzaghi one-imensional theory of consoliation. The sum of the squares of resiuals between the two factors are minimise using a Best Linear Unbiase Estimator (BLUE) uner the conition that the sum of the resiuals is zero. Experimental ata was collecte to calculate the coefficient of consoliation by existing methos an the Least Squares metho. The results are foun to be in goo agreement, proving that the propose metho can be an alternative way of etermining the coefficient of consoliation. RÉSUMÉ Une méthoe par moinres carrés est éveloppée afin estimer le coefficient e consoliation es mesures un essai à l oeomètre. Elle consiste à comparer les valeurs théoriques et expérimentales u pourcentage e consoliation moyen qui sont érivées e la théorie uniimensionnelle e Terzaghi. Le principe repose sur l application un estimateur optimal non biaisé. Le coefficient e consoliation estimé minimise la somme es carrées e la ifférence entre les valeurs théoriques et expérimentales, tout en satisfaisant la contrainte que la somme e ces ifférences soit nulle. Des onnées expérimentales ont été recueillies pour calculer le coefficient e consoliation par les ifférentes méthoes isponibles, ainsi que par la méthoe es moinres carrés. On peut éuire que les résultats e la méthoe proposée sont comparables. La méthoe est onc une approche alternative à la étermination u coefficient e consoliation. 1. INTRODUCTION There has been consierable research one in etermining alternative methos for computing the coefficient of consoliation from laboratory oeometer tests. Certainly the most commonly use are the logarithm of time fitting metho (Casagrane an Faum 1940) an the square root of time fitting metho (Taylor 1948). Other methos foun in the literature are the Inflection Point metho (Mesri et al. 1999), the Rectangular Hyperbola metho (Sriharan et al. 1987), an the coefficient of consoliation from the linear segment of the t½ metho (Feng an Lee 2001), which are all iscusse further in this paper. In spite of the existence of all these methos, there is still a nee for a metho which is inepenent of jugement (Casagrane s metho) or of irregularities in the shapes of curves ue to seconary compression (Inflection Point metho). As such, the Least Squares metho has been elaborate to provie optimal unbiase results an structure to be inepenent of the jugement of the interpreter of the experimental ata. This is possible via the use of a Best Linear Unbiase Estimator (BLUE). 2. LEAST SQUARES METHOD The Least Squares metho consists of curve fitting the experimental average egree of consoliation, U avg versus the time factor, T by means of the coefficient of consoliation obtaine from the Casagrane or Taylor methos as see value, an ajusting it with the theoretical Terzaghi T-U avg curve using, as mentione above, a Best Linear Unbiase Estimator (BLUE) which minimises the sum of the squares of resiuals uner the conition that the sum of the resiuals is zero. 2.1 Existing methos for the etermination of C V. Casagrane s metho is base on the ientification of the inflection point, as well as the compression values corresponing to 0% an 100% consoliation, from which the time corresponing to 50% consoliation, t 50 can be euce an use for the etermination of the coefficient of consoliation. C V = 0,197 H r ² [1] t 50 Casagrane s metho, though wiely use, poses two concerns: the use of personal jugement in the ientification of the inflection point from which a tangent is rawn an the possibility of seconary compression occurring before t 100. Taylor (1948) proposes that the en of primary compression occurs at t 90. This value is obtaine by ientifying the linear portion approximately up to U = 60% an rawing a line with the abscissa being 1.15 times that of the line previously obtaine. The time corresponing to Page 9

2 90% consoliation an eformation 0 at t = 0 can thus be etermine. But in this case also, seconary compression might affect the shape of the curve an the results for the value of the coefficient of consoliation. The coefficient of consoliation from the linear segment of the t½ curve (Feng an Lee 2001) requires plotting the curve of U avg versus T 1/2. The initial part of the curve is a straight line up to about 60% consoliation where the curve eviates from the straight line. The coefficient of consoliation can then be calculate using t 60. In the Inflection Point metho (Mesri et al. 1999), the compression versus log time curve is plotte an the inflection point is visually ientifie. This correspons to an average egree of consoliation of 70%. However, this metho is less reliable because of the personal jugement factor an the possibility of having no inflection point ue to the presence of seconary compression. Note that the effect of seconary compression is minimise in the last two methos by having their reference point taken at an average egree of consoliation near 50%. The Rectangular Hyperbola metho (Sriharan et al. 1987) implies that a straight line is obtaine between the average egrees of consoliation points of 60% an 90% on a time/compression versus time graph. This metho oes not involve the ifficulties encountere with the Taylor or Casagrane metho resulting from irregular shapes of curves. 2.2 Propose interpretation metho The Terzaghi theory of one-imensional consoliation efines the relationship between the theoretical time factor, T an the average egree of consoliation, U avg. The solution of this relationship for an initial excess pore water pressure constant with epth is 2 avg 1 exp( M² T) m 0M² U [2] where (2m 1) M [3] 2 The coefficient of consoliation can then be calculate from Equation [4] where t is the consoliation time an H r the rainage istance: Cv t T [4] Hr² With the consoliation time from experimental ata, the time factor T is calculate using Equation 4 with the values of the coefficient of consoliation obtaine from the Casagrane or the Taylor proceure as initial value. The theoretical values of the average egree of consoliation are calculate as follows: For U 60% T U²avg [5a] 4 Rearrange in function of T, this gives 4 T Uavg [5b] For U > 60% T log 10[(1 Uavg)100] Rearrange in terms of T, this gives T ( 2) Uavg [6a] [6b] The theoretical curve of U avg versus T is plotte. The experimental values of the egree of consoliation are calculate using Equation 7 or Equation 8 These two equations were explore following methos of Casagrane an Taylor. Uavg, exp 0 1 [7] Uavg, exp 0.9 [8] 90 0 where is the compression value at time t, 0 is the initial compression, 90 an 100 are the compression values at average egree of consoliation of 90% an 100% respectively. The experimental curve can now be plotte. The fitting between the theoretical an experimental curves is achieve by using a Best Linear Unbiase Estimator (BLUE) which is use to evaluate unknown parameters via iterations from the initial values obtaine from the Casagrane or Taylor proceure: 0, 100 or 90 an C V. The resiual between the estimator of the function U avg (T) an the measure average egree of consoliation U avg, exp is foun from Equation 9. Page 10

3 j = U avg (T) j (U avg, exp ) j [9] The solution for 0, x (x = 90 or 100) an C v is obtaine by minimising the sum of square resiuals n Min j w j [10] j 1 with the unbiase conition of n j w j 0 j 1 [11] where n is the number of measurements an w j is the weight of the resiual. The weight allows control over seconary compression. The analysis of the experimental ata has been performe with a weight threshol of 60% an 90%. If, for example it is ecie that seconary compression affects consoliation as from 60%, the weight of the resiual is set to 1 if the measure average egree of consoliation is below 60% an the eviation of the measure average egree of consoliation from the estimate value is accepte. If the measure average egree of consoliation is above 60%, the weight is set to 0, an the resiual is 0 (i.e. rejecte). The weight of the resiual can also be varie an therefore allows the elimination of suspecte seconary compression in the calculation of the coefficient of consoliation. The complexity of the equation of consoliation (Equation 2) or the approximate Equations 5b, 6b oes not allow the evelopment of an analytical solution to this problem. Numerical optimization techniques are therefore require. These are reaily available in most moern spreasheets. The following summarises the least Squares metho 1. Compute initial values for 0, x ( x = 90 for Equation 8 or 100 for Equation 7) an C v from the metho of Casagrane or Taylor. 2. Compute (U avg,exp ) j from the experimental compression measurements j using Equations 7 or 8 accoringly. 3. Compute T exp from experimental times t j of measurements using Equation 4 an C v of step Compute U avg for T exp using Equation 5b an [6b]. 5. Compute the resiuals using Equation 9, sum of square resiuals using Equation 10 an the sum of resiuals using Equation Repeat steps 2 to 5 until a minimum for Equation 10 is foun that satisfies the constraint of Equation 11. This is one through numerical optimisation. Table 1. Experimental ata use [Data # 1 an 5 Holtz an Kovacs (1991), Data # 2 Robitaille an Tremblay (1997), Data # 3 an # 4 Bowles (1992), Data # 6 Liu an Evett (2003), Data # 7 Wray (1986) Experimental ata #1 #2 #3 #4 #5 #6 #7 Initial height of specimen (H 0 )before aing loa increment (in.) , , , (in.) Page 11

4 2.3 Tests an results A typical curve fitting result for the Least Squares metho is illustrate in Figure 1. It can be seen that the measure T-U avg curve fits the estimate one perfectly until approximately 90%. problems leaing to ifferent solutions. This may be attributable to a more appreciable contribution of seconary compression in the efinition of U avg, exp in Equation 8 than in Equation 7. Average egre of consoliation U (%) Estimate 60 Measure (exp) Factor T Cv, Least Squares Metho (Taylor) (cm²/s) 0,01 0,001 0,0001 0,0001 0,001 0,01 Cv, Taylor (cm²/s) Figure 1. Curve fitting of experimental an estimate average egree of consoliation versus time factor. Figure 2. Values of coefficient of consoliation obtaine from Least Squares metho an Taylor metho. Table 1 shows experimental ata collecte to compare values of coefficient of consoliation obtaine by the methos mentione in this paper. For each set of ata, the coefficient of consoliation has been calculate by each metho an compare with the values obtaine from the Least Squares metho. Some of the results are as shown in Figures 2 an 3. The values of the coefficient of consoliation are in goo agreement with those obtaine from the other methos. Those compute from the Least Squares metho with 0, 90 an C V obtaine from a see value of the Taylor metho vary from 1 to 1.6 times the values obtaine from Taylor s metho (Figure 2). When using 0, 100 an C V obtaine a see value from Casagrane metho, the results are between 1.1 an 2.1 times those obtaine from Taylor s metho (Figure 3). Comparison of the application of the Least Squares metho using the 0, 100 an C V obtaine from a see value of the Casagrane metho an the 0, 90 an C V from a Taylor metho see is illustrate in Figure 4. In this case, the values iffer by a factor of 1.1 to 1.8. The slight ifference can be explaine by the sensibility of the estimator with respect to initial values 0, 90 or 100 an C V as well as the possible existence of more than one local minimum. The numerical estimator can be trappe in a local minimum. Also, since the initial values are use to calculate U avg, exp, the ifference in the efinition between Equation 7 an 8 can in fact efine two slightly ifferent Cv, Least Squares Metho (Casagrane) (cm²/s) 0,01 0,001 0,0001 0,0001 0,001 0,01 Cv, Taylor (cm²/s) Figure 4. Values of coefficient of consoliation obtaine from Least Squares metho an Taylor s metho. Page 12

5 Cv, Least Squares Metho (Casagrane) (cm²/s) 0,01 0,001 0,0001 0,0001 0,001 0,01 Cv, Least squares Metho (T) (cm²/s) Figure 4. Comparison of C V obtaine using Least Squares metho with Casagrane an Taylor starting points. Table 2. Classification of methos in increasing orer of squares of resiuals. Weights are zero when U avg, exp greater than rank 60% 90% 100% 1 Least Squares Least Squares Least Squares (Casagrane) 2 Taylor Taylor Taylor 3 Linear segment of the t½ metho Casagrane Casagrane 4 Casagrane Linear segment of the t½ metho Linear segment of the t½ metho 5 6 Inflection Point Metho Rectangular hyperbola Inflection Point Metho Rectangular hyperbola Inflection Point Metho Rectangular hyperbola For the purpose of comparison on a same scale, the sum of the squares of resiuals between the theoretical an measure average egree of consoliation have been calculate. To achieve this, the C V obtaine by each metho was use to calculate the estimate average egree of consoliation. The experimental values have been calculate using the same proceure as for the Least Squares metho, but using the corresponing values of compression available from each metho, for example 70 can be obtaine from the Inflection Point Metho. In the evaluation of the squares of the resiuals, the en of primary compression has been taken up to 60% an 90%. Table 2 shows the various methos classifie in increasing orer of the sum of the squares of resiuals. For the weight of 100%, the Least Squares metho was compute from initial parameters obtaine from the Casagrane metho. The Least squares metho preominates in terms of the least sum of the squares of resiuals, which infers a most successful fitting of the Terzaghi theoretical T-U avg an experimental curves. The shift between the Casagrane metho an the Linear segment of the t½ metho from an average consoliation of 60% an 90% is ue to the weight of the system. This means that between 60% an 90%, the sum of the squares of the resiuals increase to a greater extent in the linear segment of the t½ metho than in the Casagrane metho because the en of primary compression in the first metho is at 60% compare to 100% in Casagrane s metho. 3. CONCLUSION The coefficient of consoliation is a useful tool in engineering for the preiction of the rate of settlement in soils. In this paper, some methos of fining this coefficient from laboratory oeometer tests have been reviewe an compare to an alternative metho which is the Least Squares metho. Its principle relies on the fitting between theoretical an experimental values of the average egree of consoliation. An unbiase estimator is use to compute values of compression 0 an 90 or 100 as well as the value of C V base on minimising the squares of the resiuals between the average egree of consoliation values. The coefficients etermine from this metho are in goo agreement with the results foun from the other methos an provie an excellent numerical tool for this purpose. It can therefore be conclue that the Least Squares metho is effective in the etermination of the coefficient of consoliation. The metho is also foun to give the best fit to the experimental ata. 4. REFERENCES Bowles, J. E., Engineering Properties of soils an their measurement. 4 th Eition, McGraw Hill. Casagrane, A. & Faum 1940 Notes on soil testing for Engineering Purposes. Harvar Soil Mechanics No 8. Feng, T.W. an Lee, Y.J., Coefficient of Consoliation from the linear segment of the t½ curve. Canaian Geotechnical Journal 2001, 38: Holtz, R.D. an Kovacs, W.D., Introuction à la Géotechnique. Éitions e l école polytechnique. Liu, C. an Evett, J. B Soil properties, Testing, Measurement an their Evaluation. 5 th eition, Prentice Hall. Mesri, G., Feng, T.W., an Shahien M Coefficient of consoliation by Inflection Point metho. Journal of geotechnical an geoenvironmental engineering, 125(8): Page 13

6 Robitaille, V. et Tremblay, D Mécanique es sols, théorie et pratique. Éitions Moulo. Sriharan, A., Murthy, N. S., Prakash, K Rectangular Hyperbola metho of consoliation analysis. Géotechnique 37 (3) : Taylor, D. W Funamentals of soil mechanics. John Wiley an Sons, New York. Wray, W. K Measuring Engineering Properties of Soils. Prentice Hall. Page 14