When designing foundations for wind farms, a decisive role is played by the regime of subsoil deformations. This is so because foundations for such structures are loaded by considerable torque. In order to ensure foundation stability, despite the relatively low stresses found in the subsoil, a key role is played by the magnitude and the lack of uniformity of the foundation settlement. Advanced computational methods such as the finite element method (FEM), which may be applied in the calculation of settlement and non-uniformity of the foundation slab settlement in a wind farm, are effective if a model of subsoil rigidity is constructed. The model is most frequently constructed based on constrained moduli of soils. This model should show continuous and spatial changes in the constrained modulus of the subsoil under the entire foundation. The piezocone penetration test (CPTU), which is frequently supported by the flat dilatometer test (DMT), proves to be highly advantageous for the construction of such a model. The CPTU may also be used to determine the best location for the wind farm in a specific area. The best location is understood as such a zone in the area where the subsoil has the greatest rigidity (stiffness).
The construction of a model of subsoil rigidity, in which zones with homogeneous values of the constrained modulus are isolated using isolines, is based on the grouping of statistically non-different penetration characteristics recorded using the CPTU. An identical rule is binding for the separation of homogeneous areas in terms of rigidity of soil layers.
These two tasks may be solved using statistical methods of data clustering (e.g. Młynarek and Lunne [1] and Młynarek et al. [2]). Until now, data from CPTU were grouped as discrete values. This article presents a new concept for grouping of penetration characteristics as linear functions, which represent penetration characteristics. Such grouped data constitute the basis for the construction of rigidity models for the dimensioning of foundations for wind farms in the area covered by this study.
Studies were conducted in the northern part of Poland in an area with a characteristic, young glacial geological structure (Figure 1). In an area of several hundred square kilometres, the subsoil to a depth of almost 200 m is composed of alternating beds of boulder clays from successive Pleistocene glaciations, separated by non-cohesive inter-glacial deposits.
Location of the test site in Poland.Figure 1
Nine testing sites selected for analyses were arranged relatively uniformly over an area of approximately 1 km2. Normally consolidated or slightly overconsolidated clays of the last stage of the youngest Wisła (Vistulian) glaciation with variable thicknesses are found within the depth of analyses, i.e. 16 m from the ground surface. These deposits are transformed into more consolidated boulder clays of the older stage of the Vistulian glaciation, occasionally separated from younger clays with sandy inter-beddings. In turn, deposits of the last glaciation are lying on strongly overconsolidated inter-glacial sands, constituting very good building subsoils but, unfortunately, also forming a barrier for static penetrations. Figure 2 presents the characteristic geotechnical and CPTU profiles for the analysed area.
Typical soil and CPTU profiles of the test site.Figure 2
When calculating the volume and non-uniformity of the foundation slab settlement of a wind farm, in a vast majority of cases, secant constrained modulus is applied, which corresponds to the oedometric modulus. Secant moduli and their changes in subsoil with changes in geostatic stresses are determined based on CPTU penetration characteristics using separate formulas for soils with so-called drained conditions (coarse-grained soils) and for undrained (fine-grained soils, e.g. clays) and intermediate soils [3,4,5]. The first step in the construction of a subsoil rigidity model involves the isolation of statistically homogeneous layers in the context of drainage conditions. A convenient method to perform a preliminary qualification of homogeneous groups of soils is provided by the classification system proposed by Robertson [5].
The soil type was also verified by results of the grain size composition of samples, collected from laboratory analyses. The separation of soil layers in the first stage, according to the criterion of drainage conditions, was based on two parameters of the penetration characteristics, i.e.
Discrete values for the three parameters specified herein were grouped using the cluster method [2]. The results of layer grouping and separation are discussed in Section 3. In the second stage of construction of the subsoil rigidity model, the values of the constrained modulus
For soils with drained conditions [6],
where
For soils with undrained conditions [7],
A change in these moduli with a change in the vertical component of geostatic stress for each CPTU test was transformed to a curve, which was analogous to the characteristic
Curves for
The division of a subsoil sample according to the concept of functional data analysis.Figure 3
In view of the indirect relationship of the values measured in the CPTU test (e.g. cone resistance) to the constrained modulus established in the subsoil, first of all, for the constructed algorithm, the value of modulus
In the determination of the scale of similarity between the analysed objects in the first step of functional data analysis, the following measure of distance between them is assumed:
It is a distance in space
Assume that each curve
where {
The most popular basis systems are as follows:
the Fourier basis system (recommended for periodic functions), the splines basis system (recommended for non-periodic functions).
In this study, the splines basis system was used in the calculations. The coefficients
For each
Then, we obtain the following expression:
The degree of smoothness of the function
the Bayesian information criterion (BIC) – applied in this analysis; the Akaike information criterion (AIC); and generalised cross-validation (GCV).
In the case of the
where
Then, we obtain the following expression:
Distances between the objects are calculated from Eq. (9). After they have been established, the final stage is initiated in the construction of the data analysis algorithm, again based on the conventional method of hierarchical cluster analysis [2]. As a result, we obtain the hierarchy of object similarity, which may be presented most comprehensibly in the form of a dendrogram.
In accordance with assumptions presented in Section 3, in the first stage of the analysis, a preliminary division of each analysed CPTU profile into homogeneous layers was conducted in terms of soil behaviour. For this purpose, classification systems were used [5]. The analysis of similarity in terms of soil properties was performed based on two CPTU parameters
where tr(
In the case of the analysed data, the statistical criteria clearly showed the optimal number of isolated geotechnical layers to be nine (Figure 4a). The coefficient of variation criterion is not as clear cut but, in this case also, at the number of 8–9 layers, the first minimum is observed in the coefficient of variation in the layer (Figure 4b). In further analysis, the division of subsoil into nine geotechnical layers was adopted, and their location in the classification system according to Robertson [5] is presented in Figure 5. In the second stage of analysis, the values of the constrained modulus
Indication of optimal cluster number based on the Caliński-Harabasz criterion (a) and the mean weighted coefficient of variation for index Figure 4
Location of the tested soils and the soil behaviour chart, according to Robertson [5].Figure 5
Changes in the constrained modulus Figure 6
The obtained dependencies of the modulus values on depth were adopted as a function of a single variable, and they were clustered in accordance with the assumptions presented in Section 3. In view of the assumed foundation depth for wind farms and the depth of the shallowest CPTU, the analysis was conducted in the depth range from 2 m to 8 m. Due to the varied thickness of the surface, which was a weak layer, apart from the investigation of the entire profile, analyses were additionally conducted separately in two depth ranges: 2–5 m and 5–8 m below the surface. In accordance with the principles of functional data analysis, the first step included smoothing of the function of the modulus depending on the depth, in each of the testing point providing the spline function. Calculations were performed for coefficient
Graphs of the spline funcion in CPTU profiles 1 and 3, smoothing the course of dependencies of modulus Figure 7
In the second step of the functional data analysis, the smoothed spline functions were compared, using a hierarchical algorithm presented in Section 3. As a result, dendrograms were obtained, illustrating the clustering of individual objects depending on their similarity. The Ward method was applied in the clustering of objects. The shorter the distance, the more statistically significant are the similarities between the objects. Results of cluster analysis showed that the value of the adopted smoothing coefficient
Dendrograms of the clustering hierarchy of curves Figure 8
As a result, models of subsoil rigidity were obtained, resulting from the division of subsoil into zones characterised by a similarity of function
Division of testing area in view of the subsoil rigidity model for the depth range of 2–8 m, 2–5 m and 5–8 m.Figure 9
Figures 8 and 9 show that a simultaneous analysis of the entire profile requires adoption of three isolations (A, B and C), while the a priori adoption of two depth ranges simplifies the solution to two separations (A and B). However, it needs to be stressed that these isolations are separate for individual depth ranges. In each of the isolations, the included testing points are characterised by a statistically significant similarity of values of modulus
Models of subsoil rigidity composed on the basis of constrained modulus Figure 10
Assuming the reference to be provided by the subsoil rigidity model based on direct values of modulus
Graphs of mean of the function Figure 11
Mean values obtained for the functions of changes in constrained moduli with depth for isolated areas divide the entire testing area into zones with varied rigidity levels. Such a division has a highly important practical advantage, as it facilitates the selection of the beat location for the planned wind farms.
The proposed new clustering method for CPTU characteristics is a highly effective method to isolate homogeneous subsoil zones in the context of rigidity or strength. This paper presented the application of the functional data analysis method, combined with cluster analysis, in the construction of the subsoil rigidity model in a two-dimensional system of coordinates. This method also makes it possible to isolate, in the analysed area, spatial, three-dimensional areas, which statistically are most similar in terms of constrained deformation moduli (Figure 9). Analysis of rigidity models for subsoil profiles, which were constructed using functional data analysis based on curves describing the function