This article contains the analysis of the correlation between the cone resistance _{c}_{0}_{p}_{v0}_{0}^{2}. For more variables, multivariable regression analysis was applied for assessment. Conducted analysis showed that the overall view of the relation between the cone resistance _{c}_{0}_{p}_{vo}

#### Keywords

- shear modulus
- CPTU
- non-cohesive soils

To design and examine the stability of many building structures, the knowledge of small strain shear modulus _{0}_{0}_{c}_{0}_{0}_{0}_{c}_{0}_{c}_{0}

The research was carried out in 5 locations that differed in geological history and allowed the collection of results regarding sediments formed in various depository environments and subjected to various post-sedimentation processes (Fig. 1). Five research sites located in Poland have a complex geological structure. Four of them (Gnojewo, Derkacze, Darłowo and Rzepin) are in the zone of impact of the Weichselian Glaciation, which left moraine deposits covered with fluvioglacial sands.

The grain size distribution of the fluvioglacial sands is very diverse, and it ranges from gravels to fine and silty sands. The layers are generally thin and do not exceed 1.5 m. However, outwash plain ‘sandur’ forms were formed in stages and over hundreds of years, which resulted into a certain variation in the state of these soils and had an impact on the formation of a minor preconsolidation effect in the subsoil. The overconsolidation ratio (

Older sediments, the so-called interglacial sands found below the layer of the youngest moraine clay, were also examined in two locations (Derkacze and Darłowo). These deposits, which developed into the form of medium and fine sands, are characterised by a homogeneous layer structure and high values of

_{c}and shear modulus

_{0}

The empirical correlation between cone resistance and shear modulus G_{0} is built on the functional parameters that describe two different processes. One process is the process of static penetration, the other one is the course and registration of a seismic wave in the subsoil. The static penetration process is expressed with Eq. (1) (Młynarek, 2007; Młynarek et al., 2018):
_{s}_{c}, q_{t}_{p}_{1}_{2}

For non-cohesive soils, parameter _{2}_{1}_{2}_{3}_{4}_{5}_{6}_{7}_{1}_{5}_{8}

The quality of the measured cone resistance values also depends on the measurement uncertainty associated with the used test technique (Młynarek, 2010; Lumb, 1974). The impact of the independent variables recorded in parameter _{1}_{0}_{0}_{1}_{n}

The function that describes the course and registration of a seismic wave and creates the basis for determining shear modulus _{0}_{s}

Equation (3) is supplemented with Eq. (4), which determines the independent variables that affect shear modulus _{0}_{v0}_{0}

Measurement uncertainty, as in CPTU, has an impact on the determined value of _{0}_{0}

During the research, a cone manufactured by AP vd Berg with an seismic module with a single geophone and a Studio Marchetti dilatometer with a seismic module with a pair of geophones located 0.5 m apart were used. To determine the time of arrival of the wave in the case of SCPTU, the pseudo-interval and cross-correlation methods were used. In the case of SDMT, the true interval method and phase shift analysis were applied (ASTM Standard, 2008). Examples of the conducted analysis of the arrival time of the wave are presented in Fig. 2. Determining the time of arrival of the wave allows calculation of the small strain shear modulus _{0}_{c}_{s}_{2}

The seismic measurements have been done every 0.5 m or 1.0 m of profile (dependently on the testing site). The CPTU data were averaged within defined geotechnical layers and were correlated with the seismic measurements carried out within each particular layer and depth. Data groups from this set, which were correlated with the depths from which samples were taken for laboratory analyses, were selected for further examination.

The statistical significance of differences between the designated moduli _{0}_{0}_{0}_{0}

Figure 6 additionally shows the distribution of population of shear moduli _{0}_{c}_{0}_{0}

Several solutions for Eq. (4) are known in the literature. An example of such correlation is Eq. (5) by Jamiolkowski et al. (1995):
_{a}_{0}_{v0}_{h0}

In Eqs. (1) and (2), and (3) and (4), the same independent variables related to the ground are found. This fact is an accurate justification for the purpose of constructing the correlation for non-cohesive soils between cone resistance and shear modulus _{0}_{0}_{p}_{0}_{p}_{v0}

_{c}and shear modulus

_{0}

An important issue for using the correlations between shear modulus _{0}^{2}. For this reason, the analysis of the correlation between modulus _{0}_{c}_{0}

The correlation between modulus _{0}_{c}^{2} was obtained for normally consolidated soils (Fig. 8).

To test the effect of grain roughness, which is defined by variable _{2}_{c} and shear modulus Go in Fig. 9 shows the test results of Norwegian Geotechnical Institute (Lunne et al., 2003). This research covered deposits of fluvial sands from Holmen (Norway). These deposits are normally or lightly overconsolidated from a geological point of view. Despite the similar granulometric and mineral composition and thickness, these soils significantly differ in origin and degree of roughness of grains from those found in Poland.

The third step of the analysis additionally considered the impact on the correlation between the _{0}_{c}

To determine the unit weight of soils in the subsoil, empirical dependencies between the cone resistance _{c}_{0}_{r}

Figure 9 shows that the normally consolidated fine sands from Holmen are located on the diagram in different part of the plot than the normally consolidated fine sands from Poland. It could be because of a different origin and some diferences between them in angularity. Fluvioglacial sediments from Poland are more sharp-edged; however, the quasi preconsolidation effect present in the Holmen test site soils makes them occupy the upper part of the graph in Fig. 9. This fact is confirmed by the impact of this variable expressed in Eq. (2) for the mentioned correlation. In most cases, the used division allowed to obtain the value of the determination coefficient confirming the significant impact of grain size on the analysed correlation.

The first and second stage of the analysis showed that the dominant role in the analysed correlation between shear modulus _{0}_{c}_{0}_{0}_{c}_{p}

Soils with different grain sizes were grouped into normally consolidated and overconsolidated soils in the first part of this analysis. In the case of normally consolidated soils, the analysis includes both the division of soils into individual types and the absence of such a division. The preconsolidation stress _{p}_{t}_{t}_{v0}_{v0}.

The following correlations were obtained as a result of the analysis:

fine sands NC:

R^{2}=0.85, n=43

medium sands NC:

R^{2}=0.71, n=128

silty sands NC:

R^{2}=0.83, n=14

coarse sands and gravels NC:

R^{2}=0.60, n=11

fine sands OC:

R^{2}=0.51, n=17

medium sands OC:

R^{2}=0.68, n=25

_{0}

_{c}

_{p}

It is worth to note that the value of _{p}_{p}_{0}_{v0}

The obtained values of the determination coefficient R^{2} prove that the multivariate dependency model quite well assesses the shear modulus _{0}

The purpose of constructing a multivariate model is also demonstrated by the use of the correlation proposed by Młynarek et al. (2012) for overconsolidated clayey sand from Poland – Eq. (13):

R^{2}=0.42, n=48

Figure 11 shows the location of the shear modulus _{0}_{0}_{0}

_{0}variability

The assessment of the impact of soil physical parameters and stress in the subsoil on modulus _{0}

The impact of effective vertical stress _{v0}_{0}_{0}_{0}_{v0}_{0}_{c}

Another analysed partial function was the correlation between modulus _{0}_{p}_{0}_{0}_{v0}_{0}_{p}_{0}_{p} is quite uncertain because it is based on another empirical correlation.

Another analysed partial function is the correlation between modulus _{0}_{r}_{m0}_{m0}_{h0}_{v0}_{0}= 24.94; C_{1} = 0.46; C_{2}_{r}_{c}_{v0}

The impact of soil origin is less exposed (Fig. 14) in correlations between modulus _{0}_{0}_{p}

fine sands NC and OC:

R^{2} = 0.67

medium sands NC and OC:

R^{2} = 0.71

coarse sands and gravels NC:

R^{2} = 0.52

silty sands NC:

R^{2} = 0.83

The analysis of partial regression coefficients shows that the impact of individual variables on modulus _{0}_{0}_{c}^{2}, which ranges between 0.54 and 0.84 for individual soil groups.

The value of coefficient R^{2} rapidly declines to 0.44 for NC soils and 0.43 for OC soils if we construct the general dependency between _{0}_{r}_{c}_{0}

The conducted tests show that the correlation between cone resistance from the SCPTU method and shear modulus _{0}_{0}_{p}_{0}_{0}

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