Notwithstanding the general absence of triangular, sinusoidal or other non-uniform initial excess porewater pressure distributions in the standard incremental loading oedometer tests, the authors have overlooked the inflection point method of fitting the Terzaghi theory of consolidation to compression against time measurements to calculate coefficient of consolidation. This alternative was proposed by Robinson (1997), and confirmed by Mesri et al. (1999) using data from incremental loading as well as unloading oedometer tests on undisturbed specimens of a large number of soft clays and shales. The coefficient of consolidation using field observations of settlement for non-uniform soil profiles, time-dependent loading and non-uniform initial excess pore-water pressure distribution, is best estimated using the Asaoka method (Asaoka, 1978; Jamiolkowski et al., 1985; Mesri & Huvaj-Sarihan, 2009).
The Terzaghi average degree of consolidation U plotted against log time factor T relation, and therefore, compression d against log time t relation, has an inflection point at which the sense of concavity of the curve changes. This is true for all combinations of initial excess pore-water pressure distribution and drainage boundary conditions. Because of secondary compression, however, the actual compression against log time curve may not display an inflection point for small pressure increment ratios, and pressure increments spanning the preconsolidation pressure for certain soils (Mesri & Godlewski, 1977; Mesri et al., 1978, 1999).
The Terzaghi coefficient of consolidation by the inflection point method is computed using
where TvI is the time factor at the inflection point, H is maximum drainage distance and tI is the time elapsed from the application of the pressure increment to the inflection point. The values of TvI and corresponding UI for different combinations of initial and boundary conditions are shown in Table 5. It is significant to note that independent of initial and boundary conditions, the Terzaghi time factor at the inflection point TvI = 0·405. However, the average degree of consolidation at which the inflection point on the U against log T curve is observed is a function of initial and drainage boundary conditions.
Values of TvI and UI for different initial and boundary conditions
| Initial u′ distribution | f(z) | TvI | UI: % |
|---|---|---|---|
 | 0·405 | 70 | |
 | 0·405 | 62 | |
 | 0·405 | 78 | |
 | 0·405 | 68 | |
 | 0·405 | 73 | |
 | 0·405 | 64 | |
 | 4/π2 | 63 | |
Incidentally, the assumption of a sinusoidal distribution of initial excess pore-water pressure combined with drainage only from one boundary is unrealistic because such an initial condition is only encountered for drainage from both boundaries. As consolidation of a uniform layer subjected to uniform initial excess pore-water pressure and free drainage from top and bottom continues, the excess pore-water pressure distribution tends toward a sinusoidal or parabolic shape. Hence, if consolidation analysis begins at a sufficiently long time after the application of a load, the initial excess pore-water pressure distribution will have approximately a sinusoidal or parabolic shape. The initial excess pore-water pressure is zero at the two boundaries and increases sinusoidally or parabolically to a value of at the mid-depth of the compressible layer.
The derivation of (TvI, UI) is illustrated for a sinusoidal distribution of initial excess porewater pressure, for which
In order to find the inflection point (TvI, UI) on the U against log Tv relation, equation (14) is differentiated twice as follows
By setting equation (16) equal to zero, TvI and UI are obtained
TvI = 0·405 and UI = 63%.
In summary, using the inflection point method, the Terzaghi coefficient of consolidation is calculated using
The advantage of the inflection point method is that it does not require the definition of the beginning and end of the Terzaghi hydrodynamic consolidation, which are required for the Casagrande and the Taylor methods. Another advantage of the inflection point method is that the inflection point, at the average degree of consolidation of 62–78%, is within the mid-range of the compression curve and is least affected by initial compression and secondary compression. However, the inflection point on the laboratory d against log t curve must be carefully visually identified for determining tI.
Authors' reply
The authors greatly appreciate the interest shown by Professor Mesri and Mr Funk in the proposed modifications to classical curve-fitting methods used to determine cv. They have concisely adapted the inflection point method to incorporate a variety of non-uniform initial excess pore-water pressure (ui) distributions. The inflection point method is an excellent addition to the current work – in light of the many discrepancies between experimental and theoretical consolidation curves, it appears prudent to consider as many different curve-fitting techniques as possible. However, the authors wish to address some minor points raised by the discussers regarding the efficacy of modifying existing curve-fitting methods to account for non-uniform ui distributions.
The authors did in fact critically review the inflection point method (along with several others) in Lovisa & Sivakugan (2013). However, in Lovisa & Sivakugan (2013), the inflection point method is credited to Cour (1971) rather than Robinson (1997). The standard issues with Cour's inflection point method were encountered in Lovisa & Sivakugan (2013), where in many cases, a clearly identifiable inflection point was not visible on the compression against log time curve. As astutely stated by the discussers, this can be attributed to secondary compression and often occurs for small pressure increment ratios, and pressure increments encompassing the pre-consolidation pressure. In light of this issue and in the interests of brevity, the authors chose not to include the inflection point method in the present study.
The discussers note the general absence of non-uniform initial excess pore water pressure distributions in the standard incremental loading oedometer tests. Furthermore, they also consider the assumption of a sinusoidal ui distribution unrealistic. Although these statements are entirely reasonable, the authors wish to make note of a recently completed study in which non-uniform ui distributions were recreated within a laboratory setting to verify the efficacy of some of the modified Taylor and Casagrande curve-fitting methods proposed in the current study. One of these cases involved a sinusoidal ui distribution with one-way drainage. While the authors acknowledge that a sinusoidal ui distribution with one-way drainage is highly unlikely to exist in the field, it proved to be a useful tool for laboratory-based predictions of cv.
In conclusion, the authors wish to state that the modified curve-fitting techniques for both singly and doubly drained cases where a sinusoidal ui distribution is anticipated were successfully used in the new study to determine realistic and accurate values of cv. The authors look forward to hearing more from the discussers regarding this new work.
