An insight into time rate of consolidation

Terzaghi's elegant theory of one-dimensional consolidation is dependent upon a number of assumptions which can, at times, severely limit the predictive capabilities of the resulting analytical model. Although other more complex models exist, Terzaghi's one-dimensional model remains popular amongst practicing engineers due to its inherent simplicity and notoriety. The purposes of this study have been to explore key aspects of Terzaghi's consolidation theory, and extend the analytical solution to incorporate a variety of loading scenarios that may give rise to non-uniform distributions of excess pore water pressure. To do this, Terzaghi's consolidation equation was solved within the program MATLAB using a collocation approach to solve for series coefficients, instead of the more traditional orthogonality approach. A novel truncation technique was also employed in cases where discontinuities were present in the initial condition, which would have otherwise elicited Gibbs phenomena, an undesirable trait of series solutions.\ud \ud By varying the initial condition in the MATLAB program, the consolidation behaviour of a soil subjected to a variety of different initial excess pore water pressure (uᵢ) distributions was analysed in terms of excess pore water pressure decay and percentage consolidation settlement. These simulations were conducted for both singly and doubly drained soil layers. In many singly drained cases, the excess pore water pressure within the soil layer decayed in a peculiar fashion, where a 'redistribution' of pore pressure occurred during the early stages of consolidation.\ud \ud When viewing consolidation behaviour in terms of percentage consolidation (U), it was easily shown that any reference to drainage path length (H(d)ᵣ) should be avoided. In fact, continuing to use the traditional expression for time factor (T) in terms of H(d)ᵣ can actually complicate analyses. Instead, T should be expressed in terms of layer thickness (H) only. By adopting this alternative time factor expression, a relationship between the consolidation behaviour due to uniform and non-uniform uᵢ-distributions was developed. This relationship utilised the knowledge that after some short time during consolidation, any skewness attributed to the non-uniform uᵢ-distribution will disappear, and the decay of excess pore water pressure with depth will revert to a sinusoidal or half-sinusoidal shape, for doubly or singly drained cases, respectively. Correction factors were then developed so that the widely available U -T values can be easily adjusted to account for any non-uniform uᵢ-distribution.\ud \ud Currently, some form of Terzaghi's consolidation theory is also used to analyse laboratory time-settlement data so that important consolidation properties such as the coefficient of consolidation (cᵥ) can be back-calculated. The efficacy of some of the more popular curvefitting techniques when applied to different soil types was assessed using a new cᵥ - calculation procedure which steers away from traditional curve-fitting procedures and instead takes advantage of the matrix manipulation capabilities of MATLAB. It was found that this proposed method and Taylor's square-root of time method yielded the most accurate values of cᵥ. Previously restricted to data obtained from a uniform uᵢ-distribution, the Taylor and Casagrande curve-fitting techniques were also generalised to account for a variety of non-uniform uᵢ-distributions. Two of these modified procedures (a singly/doubly drained layer subjected to a sinusoidal uᵢ-distribution) were also experimentally verified.\ud \ud It was also shown that the traditional restrictions associated with consolidation oedometers are not as inflexible as previously assumed. Currently, standard practice requires the height to diameter ratio of a consolidating sample to remain less than 0.4 to avoid any effect of wall friction. However, results suggest that data obtained from a 'tall' oedometer with a height as much as twice its diameter can still be analysed using conventional curve-fitting techniques.\ud \ud Finally, the effects of time-dependent loading were investigated using two approaches; a constant-rate loading approach, and a discretised loading approach, which more closely models the stepped nature of fill application in the field. It was found that for T increments less than 0.0143, the discretised loading approach effectively became a constant-rate loading problem, an inference that was also experimentally verified.\u

A method to determine c(v)under sinusoidal pore pressure distributions

The coefficient of consolidation (cv) is traditionally evaluated by fitting experimental settlement-time data to the theoretical percentage consolidation-time factor curve of a layer subjected to a uniform initial excess pore water pressure (ui) distribution. Over the years, numerous curve-fitting techniques have been developed for this experimental-theoretical correlation, the most popular of which are Casagrande's logarithm-of-time method and Taylor's square-root time method. These classical curve-fitting techniques have recently been generalized to account for a variety of different ui distributions. In this paper, basic consolidation principles have been applied in a novel fashion to both standard and tall oedometer tests on two clays to simulate a sinusoidal ui distribution (operating under either singly or doubly drained conditions) within a laboratory setting. The settlement-time data obtained from these tests were analyzed using the modified curve-fitting procedures previously put forward by the authors to determine appropriate values for cv, which were found to closely align with values of cv obtained using the traditional Taylor and Casagrande methods, when the ui distribution is considered uniform. The most valuable feature of these modified curve-fitting techniques was found to be their ability to analyze traditionally obtained settlement-time data to supplement any conventionally obtained cv values. This is useful in terms of validation when settlement-time data do not exhibit the usual trends and traditionally calculated values of cv require further authentication

An in-depth comparison of cv values determined using common curve-fitting techniques

The coefficient of consolidation (cv) is traditionally determined by fitting observed settlement–time data to the theoretical average degree of consolidation versus time factor relationship developed by Terzaghi. Although it is widely accepted that different curve-fitting methods can produce different values of cv, very few comparisons have been conducted to assess the validity of these methods. In this study, the settlement–time data gathered from conventional oedometer tests conducted on three different clays were analysed using three common curve-fitting techniques: the Casagrande log-time method, Taylor's root-time method, and the Cour inflection point method. A new method proposed by the authors for calculating cv, which abandons the traditional curve-fitting approach in favour of a computational-based approach, was also used to compare these results. To assess the validity of each cv value, the experimental results were compared with the theoretical average degree of consolidation curve and quantified using the root-mean-square (rms) error. The efficacy of the designated curve-fitting method was found to significantly depend upon the "shape" of the settlement–time curve generated during testing. For example, in this study, clay containing a significant fraction of fine sand often resulted in settlement–time curves that exhibited no clear inflection point, which made analysis using the Cour method very difficult. In general, the Taylor method predicted larger values of cv than the Casagrande method, and correspondingly smaller rms errors. The variance method proposed by the authors resulted in values of cv that more closely matched those generated using the Casagrande method. However, smaller rms errors were achieved using the variance method, which suggests that this technique may produce a more realistic estimate of cv than the Casagrande method

Tall oedometer testing: method to account for wall friction

The oedometers used for one-dimensional consolidation tests are proportioned such that their height-to-diameter ratio lies in the range 0.17–0.40. Sometimes, it is desirable to test a soil specimen that has a significantly larger height-to-diameter ratio, where the wall friction must be considered in the analysis. Such tall oedometers can become useful tools in consolidation tests, if the wall friction can be accounted for rationally. The objective of this paper is to develop some theoretical basis for analyzing consolidation test data from tests carried out in tall oedometers. It is shown that the same average degree of consolidation versus time factor charts can be used for height-to-diameter ratios as much as 3, provided the specimen is doubly drained. Consolidation tests carried out in the standard oedometer and tall oedometer in the laboratory gave very similar values of coefficients of consolidation

Aerodynamic Performance of the NREL S826 Airfoil in Icing Conditions

The demand for wind power is rapidly increasing, creating an opportunity for wind farm installations in more challenging climates. Cold climate areas, where ice accretion can be an issue, are often sparsely populated and have high wind energy potential. Icing may lead to severely reduced aerodynamic performance and thereby reduced power output. To reach a greater understanding of how icing affects the aerodynamics of a wind turbine blade, three representative icing cases; rime ice, glaze ice and a mixed ice, were defined and investigated experimentally and computationally. Experiments at Re= 1.0 E5 - 4.0 E5 were conducted in the low-speed wind tunnel at NTNU, determining lift, drag and surface pressure distributions. Computational results, obtained from the Reynolds Averaged Navier-Stokes fluid dynamics code FENSAP, complement the experiments. Measured and predicted data show a reduction in lift for all icing cases. Most severe is the mixed ice case, with a lift reduction of up to 30% in the linear lift area, compared to a clean reference airfoil. Computational results show an under-prediction in maximum lift of 7 - 18% compared to experimental values. Curvature and tendencies for both lift and drag show good agreement

A simple method to account for non-uniform initial excess pore water pressures in settlement computations

The variation in percentage consolidation with time within a clay layer subjected to a non-uniform initial excess pore water pressure distribution can be difficult to evaluate, and as a result, often a uniform initial distribution is assumed in most analyses. However, by utilizing some of the key features of consolidation in terms of excess pore water pressure dissipation, it is possible to simply adjust the uniform case to account for any number of non-uniform initial excess pore pressure distributions. By observing the decay of excess pore water pressure with time resulting from various non-uniform initial distributions, it is clear that any initial asymmetry or skewness is quickly dispersed, and the distribution of excess pore pressure with depth becomes sinusoidal (or half-sinusoidal if singly drained) shortly after consolidation has commenced. In other words, once the pore pressure decay due to a non-uniform initial distribution has become sinusoidal, it will actually decay at the same rate as the uniform case – however, it will be "ahead" or "behind" the uniform case by some constant factor. Once this factor has been determined, it is possible to simply adjust the rate of consolidation resulting from a uniform initial pore pressure distribution (the values of which are widely available in literature) to account for any number of realistic non-uniform initial excess pore pressure distributions

An overview of electrokinetic consolidation of soils

Electrokinetic stabilization is one of the techniques that improve the geotechnical properties of the soils. It was pioneered by Casagrande in late 1940s and has not seen much development since then, especially in large-scale field applications. Some bench scale studies have been carried out during the past two decades and there have been some small scale field studies and limited field applications, mostly on soft soils. Due to lack of understanding of the physiochemical and electrochemical changes in the soil during electrokinetic stabilization, uncertain energy costs, loss of efficiency with time and the corrosion of electrodes, this method is usually considered as a last resort for large-scale practical applications. The objective of this paper is to highlight the critical parameters affecting electrokinetic consolidation, and to discuss their effects on the efficiency of the process. A better understanding of these critical parameters and their effects will enable geotechnical engineers to design the electrokinetic consolidation operation more effectively and make it an economically viable process for many situations

Simple approach to consolidation due to constant rate loading in clays

Terzaghi's one-dimensional consolidation theory is extended to constant rate of loading, where a simple expression relating average degree of consolidation and time factor is derived. This expression is used to assess Terzaghi's approximation that the ramp load can be assumed as an instantaneous load, applied at t₀/2, where t₀ is the duration of loading. It is shown that this assumption overestimates the degree of consolidation by approximately 10%. Terzaghi's approximation can be improved in two simple ways: (1) assume the load acts at t₀/2, estimate the degree of consolidation, and reduce it by 10% or (2) assume the load to act at 2t₀/5 rather than t₀/2 in estimating the degree of consolidation, with no further adjustment. Nevertheless, the single U-T expression developed herein maintains a level of simplicity comparable to Terzaghi's approximation and is adequate for all practical purposes for computing the degree of consolidation during the loading, as well as beyond completion of loading. A numerical example is also given to illustrate the use of the proposed U-T values for constant rate of loading

Time factor in consolidation: critical review

The magnitude of consolidation settlement is often calculated using Terzaghi's expression for average degree of consolidation (U) with respect to time. Developed during a time of limited computing capabilities, Terzaghi's series solution to the one-dimensional consolidation equation was generalized using dimensionless time factor (T), where a single U-T curve is used to describe the consolidation behaviour of both singly and doubly drained strata. As a result, any comparisons between one- and two-way drainage are indirect and confined to discrete values of time. By introducing a modified time factor T* in terms of layer thickness (D) instead of the maximum drainage pat length (Hdr), it is now possible to observe the effect of drainage conditions over a continuous range of time for a variety of asymmetric initial excess pore pressure distributions. although two separate U-T plots are required (for singly and doubly drained cases), the time factor at specific times remains the same for both cases, enabling a direct visual comparison. The importance of a revised time factor is evident when observing the endpoint of consolidation, which occurs as U approaches 100%. This occurs at T*≅0.5 for two-way drainage and at T*≅2 for one-way drainage, an observation not possible using the traditional expression for time factor