The Use of CPTU and DMT Methods to Determine Soil Deformation Moduli—Perspectives and Limitations

Soil deformation moduli play a vital role in the preparation of a geotechnical project for investments, such as road facilities, high-capacity buildings, and wind farms. A very valuable and, at the same time, expected element of the geotechnical design is a complete profile of changes of constrained modulus M₀, Young’s modulus E, and small strain shear modulus G₀ in the subsoil. In situ tests, such as cone penetration test CPTU, seismic cone SCPTU, and flat dilatometer tests DMT, SDMT are highly applicable for obtaining the profile of changes of the abovementioned moduli in the subsoil. The fact that these studies are already commonly used in Poland works in their favor (Młynarek, 2010). The interpretation of CPTU and DMT tests has a good theoretical basis (Lunne et al., 1997, Marchetti, 1980) and numerous empirical relationships have been developed to determine the deformation moduli based on the parameters from these tests (e.g., Mayne, 2006, Młynarek et al., 2013, 2015, Robertson & Cabal, 2012). The measurements of building settlements and the extent to which they comply with the settlements predicted based on the deformation moduli determined from CPTU and DMT are also known. Some of these studies showed high compliance of settlements measured with those calculated on the basis of the moduli determined from the CPTU (Młynarek et al., 2013, Rzeźniczak et al. 2019), as well as from the DMT tests (Monaco et al., 2007). There are also studies that document a significant discrepancy between the predicted and measured settlements and in the case of the moduli from the DMT, predicted settlements are significantly lower than the measured ones. One of the goals of this article is to clarify this interesting issue.

This article uses the results of research carried out in various regions of Poland, differing in structure and geological history of sediments (Fig. 15).

Figure 15:

The largest group of locations is the one in which sediments of the youngest of the Scandinavian glaciations—Weichsel glaciation—were studied. Glacial clays of the Weichsel glaciation from Darłowo, Jarosławiec, Barwic, and Starogard can be divided into two groups of sediments. The first group is glacial clay of an older level, associated with the transgression and regression of the Poznań phase. The second group consists of younger settlements, associated with the transgression and regression of the Pomeranian phase of this glaciation.

The sediments found in Derkacze, Budzyń, Batkowo, Kaźmierz, Chełmno, Rzepin, Poznań, and Lipno are glacial clays and interglacial silts lying in the zone, which the Pomeranian phase of the Weichsel glaciation no longer reached. These sediments are classified as sandy loams and loamy sands because of their texture.

Glacial clays from Jarocin, Krotoszyn, and Koźmin are located in the Riss glacial zone on the outskirts of the Weichsel glacial line. The dominant soils in the profile are sandy loams and loamy sands, which are strongly preconsolidated. A characteristic feature of these clays is the high content of calci carbonate, above 10%, reduced content of the sand fraction, as opposed to the Weichsel glaciation clays, with a simultaneous increase in the content of the silt fraction (up to 40%) (Rząsa & Młynarek, 1968). These clays are also grey–brown in color and are often called grey clays. They are considered to be very good construction subsoil.

Neogene clays were present in the tested profiles in Warsaw and Bydgoszcz. These sediments are strongly preconsolidated as a result of the impact of subsequent Pleistocene glaciations. The consistency of clay is classified as hard or compact, with the exception of top areas in which local yielding of these soils occurs, as a result of the impact of the groundwater deposited on them.

Loess were tested in Łańcut, at the western end of the vast belt of loess covers of the Podolian Upland stretching from Ukraine (Bogucki et al., 2014). The thickness of the loess cover varies in this area from 9 to about 20 meters. These lands, created in the central and upper Pleistocene, rest on older glacial and fluvio-lacial formations associated with the Mindel glaciations. Loess are, in the granulometric sense, silts, sandy silts and sometimes silty clays, i.e. soils corresponding to PN-ISO soils from the range of silt and sand mixtures (saSi–siCl). The individual grain fractions were in the range: 24–33% sand fraction, 55–71% silty fraction, and 7–14% clay fraction. Both Frankowski et al. (2010) and Bogucki et al. (2014) indicate at least a dichotomy of the loess profile of the Podolian Upland. The top zone of the profile (approx. 3 m in depth), is characterized by a greater possibility to collapse. The lower zone of the soil profile, despite its similar genesis and granulometric composition, is characterized by lower porosity, higher degree of humidity, and clearly lower maturity. The presence of carbonate cementation in the top part of sediments, typical for alluvial silty formations, is an additional element influencing the diversification of the loess profile (Stefaniak 2014).

The introduction of the measuring tip into the subsoil in the CPTU and DMT methods generates excess pore shearing in the subsoil. The dissipation effect of pore pressure is closely related to soil texture. In this context, Lunne et al. (1997) proposed the following subsoil subdivision for the interpretation of penetration characteristics in the CPTU method: –

fine-grained soils

This division is particularly justified for the subsoil found in Poland, where there are soils with a significantly different origin and grain size composition. The division adopted in this way is also justified according to equation (2) to construct a partial function, which is the relationship between, e.g. the cone resistance and the constrained modulus with other variables established at a constant level. Such division contains the variables x₁, x₂, x₃. Equation (2) requires the condition that observation pairs for this relationship are determined each time at one stress level σ′_vo in the subsoil.

4.1

Constrained modulus from CPTU–fine-grained soils

The relationship between constrained modulus M, which is determined in an oedometer test, and cone resistance q_c is expressed with the relationship (Lunne et al. 1997) (10) M=αm qc M = {\alpha _m}\;{q_c} This relationship is empirical and in general linear interpretation models are used to determine the α_m coefficient.

In the case of CPTU, the relationship (10) is as follows: (11) M=αi qn=αm(qt−σv0) M = {\alpha _i}\;{q_n} = {\alpha _m}\left( {{q_t} - {\sigma _{v0}}} \right) where: (12) qt=qc+u2(1−α) {q_t} = {q_c} + {u_{\it 2}}\left( {1 - \alpha } \right) u₂ – pore pressure acting behind the cone, α – cone area ratio

In the normally consolidated stress range, Senneset et al. (1989) proposes the relationship α_i – between 4 and 8

A more general relationship suggested Kulhaway & Mayne (1990) (13) M=8.25(qt−σv0) M = 8.25\left( {{q_t} - {\sigma _{v0}}} \right) The Hyson 20Tf static probe from AP van den Berg from the Netherlands was used to carry out detailed tests in order to assess the values of the α_i and coefficients for soils from Poland.

In order to determine the effect of preconsolidation on the relationship recorded by equation (10), the study was conducted in Szczecinek, where the subsoil was characterized by strongly preconsolidated Posnanian phase moraine clays and in the subsurface zone, Pomeranian phase moraine clays (Fig. 16). These subsoil zones differed significantly in the values of overconsolidation ratio—OCR. Oedometer reference tests were performed (Fig. 17) to determine values of the constrained modulus from the CPTU, SDMT, and the profile of changes of this modulus in the subsoil. For laboratory tests, samples were taken with the AP van den Berg MOSTAP probe, and oedometer tests were performed in a Geonor oedometer, according to the CRS oedometer method (Sandbaekken et al., 1986).

Figure 16:

Figure 17:

Fig. 17 shows the assessment of the OCR and its changes in the subsoil obtained from the CPTU and DMT and the reference oedometer test. Wierzbicki’s nomogram was used to determine the value of the OCR from the CPTU (Fig. 18).

Figure 18:

The evaluation of the α_i coefficient values in equation (11) can be obtained by analyzing changes in the oedometric compressibility modulus of M_oedo and the modulus from the CPTU test, with a change in stress σ′_v0 in the subsoil. Constrained modulus M_CPTU was calculated using the values of α_i = 8.25 (equation 13). Fig. 19 well documents the impact of the preconsolidation effect on the value of the coefficient α_i. The conducted research made it possible to formulate a significant statement that has practical recommendations to use the values of the α_i = 8.25 coefficient for preconsolidated clays and α_i = 13.23 for clays normally consolidated (Młynarek et al., 2016, Wierzbicki, 2010).

Figure 19:

4.3

Intermediate soils

Silty and loamy sands qualify for this soil group. It is characterized by heterogenous grain size, as well as origin, and often the effect of cementation. For silty soils, Lunne et al. (1997) recommend the following relationships in order to determine the constrained modulus: (20) qt>25 MPaM0=2qt MPa \matrix{ {{q_t} > 25\;{\rm{MPa}}} & {} & {{M_{\it 0}} = 2{q_t}\;{\rm{MPa}}} \cr } (21) 2.5<qt<5 MPaM0=(4qt−5) MPa \matrix{ {2.5 < {q_t} < 5\;{\rm{MPa}}} & {} & {{M_{\it 0}} = \left( {4{q_t} - 5} \right)\;{\rm{MPa}}} \cr } In-house research has shown that these relationships are very useful to forecast the value of the constrained modulus in the subsoil, where loamy sands and silty sands can be found. As for the silt and clay silt group, greater compliance with the constrained modulus with the oedometer reference tests is obtained using the formula of Mayne (1990)—formula (13).

Alluvial soils, loess and sandy loams, and loamy sands occupy a special position in the intermediate soils group in Poland. In this group of soils there are two previously mentioned effects, i.e. preconsolidation and cementation. Detailed test results for these sediments are presented in the works of Młynarek et al. (2012, 2015). In order to determine the impact of the preconsolidation effect on the value of the constrained modulus in the group of intermediate soils represented by sandy loams and loamy sands, tests were carried out in two locations, namely Poznań and Elbląg. Fig. 20 shows a typical profile of the subsoil in Poznań and parameters from CPTU, while DMT in Fig. 21 documents the classification of soil in the subsoil from Poznań and Elbląg test sites to the intermediate soils group. The variability of the effect of subsoil preconsolidation from the Poznań test site is shown in Fig. 22. The OCR coefficient from the DMT was calculated based on the relationship proposed by Lunne et al. (1990) (22) OCR=0.3KD1,17 OCR = 0.3K_D^{1,17} where K_D = (p₀ – u₀) p₀ = corrected pressure from DMT test, u₀ – hydrostatic pressure on σ′_v0 level for measured parameter p₀.

Figure 20:

Figure 21:

Figure 22:

The OCR for the CPTU test was determined from the nomogram—Fig. 18.

Fig. 22 shows that for the zone of normally consolidated subsoil, a differential assessment of the OCR coefficient values from CPTU and DMT is obtained. The obtained result justifies the comment presented in Section 2.2 regarding the recording of stress σ_h in CPTU and DMT. The influence of the preconsolidation effect on the change of the constrained modulus in the subsoil is shown in Fig. 23.

Figure 23:

A significant relationship between the change in the state of the soil, defined by the liquidity index LI and the constrained modulus M₀ for soils from both locations is illustrated in Fig. 24. The influence of both variables, i.e. the OCR and the liquidity index LI on the change in the constrained modulus can be presented in the following empirical relationship: (23) M0=22.16−1.16LI−0.19 OCR {M_{\it 0}} = 22.16 - 1.16LI - 0.19\;{{OCR}} Equation (23) has a significant statistical value. The OCR for this relationship was adopted according to Wierzbicki (2010).

Figure 24:

The following Marchetti procedure is commonly used to determine the constrained modulus M₀ from the dilatometer test (1980): (24) M0=Rm ED {M_{\it 0}} = {R_m}\;{E_D} (25) ED=34.7(p1−p0) {E_D} = 34.7\left( {{p_{\it 1}} - {p_{\it 0}}} \right) (26) KD=Po−μoσ′νo {K_D} = {{Po - \mu o} \over {\sigma^\prime \nu o}} (27) RM=0.14+2.36logKD if ID≤0.6 {R_M} = 0.14 + 2.36{\rm{\;log\;}}{K_D}\;\;\;{\rm{if}}\;{I_D} \le 0.6 (28) RM=0.5+2logKD if ID≥3.0 {R_M} = 0.5 + 2{\rm{\;log\;}}{K_D}\;\;\;{\rm{if}}\;{I_D} \ge 3.0 (29) RM=Rm.0=(2.5−Rm.0)logKD 0.6<ID<3.0 {R_M} = {R_{m.{\it 0}}} = \left( {2.5 - {R_{m.{\it 0}}}} \right){\rm{\;log\;}}{K_D}\;\;\;0.6 < {I_D} < 3.0 (30) RM=0.14+0.15(ID−0.6) {R_M} = 0.14 + 0.15\left( {{I_D} - 0.6} \right) (31) RM=0.32+2.18logKD if KD>10 {R_M} = 0.32 + 2.18{\rm{\;log\;}}{K_D}\;\;\;{\rm{if}}\;{K_D} > 10 where p₁, p₀—corrected pressure of the dilatometer test.

Owing to different load directions in the CPTU—vertical test and the DMT—horizontal test (Fig. 1), in the case of soils with exposed structure and cementation, the influence of these factors on the CPTU and DMT constrained modulus determined from the tests should be taken into account. An example of this type of sediment are loess. The study of these soils was carried out in the vicinity of Łańcut (Mlynarek et al., 2015). Fig. 25 shows a typical geotechnical profile from the test sites, while Fig. 26 shows the location of these soils in the CPTU classification system, DMT. Two zones can be clearly separated in the test medium. The upper zone, which is characterized by high heterogeneity of the macrostructure due to cementation (Fig. 19) and the effect of preconsolidation (Fig. 26).

Figure 25:

Figure 26:

The lower zone classifies the subsoil as normally consolidated. The influence of the abovementioned factors on the determined values of the constrained modulus from both studies is well illustrated in Figs. 27 and 28. For the lower zone of the subsoil, a relationship was established between the two moduli. This relationship is defined by the empirical relationship: (32) MDMT=0.021 MCPTU2+0.711 MCPTU {M_{DMT}} = 0.021\;M_{CPTU}^2 + 0.711\;{M_{CPTU}} The differential impact of the preconsolidation effect on the value of the constrained modulus from CPTU and DMT tests is also noticeable in moraine clays. This problem is well justified by previously commented research from the vicinity of Poznań. Fig. 29 points out a very interesting observation that in the plastic states of moraine clay, the effect of preconsolidation disappears and the constrained moduli from both studies are very similar. Lechowicz et al. (2011) point out that in order to determine the constrained modulus from DMT in heavily preconsolidated clays, a correction of the R_m coefficient should be made in the formula (24). High values of the constrained modulus obtained from the original formula may lead to underestimate the expected settlement of the structure, as previously mentioned.

Figure 27:

Figure 28:

Figure 29:

The fact that the assessment of constrained moduli from CPTU and DMT for organic subsoil is compatible can be considered an interesting issue. It is also important due to generally known difficulties in obtaining high-quality samples for reference laboratory tests. Thus, CPTU and DMT methods seem to be beneficial for determining the profile of constrained moduli changes in this subsoil. In order to recognize this issue, the research was carried out in three locations (Fig. 15), Exemplary characteristics from CPTU and DMT for the Poznań—Bogdanka River location are shown in Fig. 30.

Figure 30:

In order to determine the constrained modulus from the CPTU, the α coefficient in equation (11) was taken as Mitchel, Gardner (1975) 1.3 for peats, 1.6 for gyttjas, and 8,25 for silty clays. Constrained moduli for the DMT were calculated according to the relationship of Marchetti (1980) and Robarijoely (1999). The results of the research from the Poznan—River Bogdanka location (Fig. 31) prove that the adoption of appropriate values of the α coefficient in the equation (11) or the correction of the Marchetti formula (1980) is a complex problem. Although a better prognosis for the assessment of constrained modulus is obtained from the DMT test (Fig. 32), it is necessary to perform a reference laboratory determination for organic soils for both tests.

Figure 31:

Figure 32:

The cumulative results from the research at the analyzed locations (Fig. 33) confirmed the previously formulated opinion that index “α,” e.g. depends clearly on the type of organic soil. The results of these studies suggest that estimated values of this index amounts to 10.2 for peats, while for organic silts (mud) amounts to 8.5. However, Figs. 33 and 34 indicate that this relationship has a relatively low statistical value.

Figure 33:

Figure 34:

To determine the profile of changes for the shear modulus G₀ in the subsoil, static probing with SCPTU seismic tip and SDMT type dilatometer test are used. These studies are also commonly used in Poland (Godlewski, Szczepański, 2013). The definition of the shear modulus and rigidity index is presented from Mayne (2006) in Fig. 35.

Figure 35:

From the SCPTU and SDMT, the small strain shear modulus is determined from the relationship (Lunne et al., 1997) (33) G0=ρVs2 {G_{\it 0}} = \rho V_s^2 where: ρ - soil density, V_s - shear wave velocity.

Empirical relationships between the measured shear modulus with parameters from tests in in situ CPTU or DMT conditions are searched for due to the cost of SDMT and SCPTU. Such dependencies can be a valuable complement to SCPTU or SDMT research and this can limit their number for the upcoming geotechnical project. The second valuable advantage of these relationships is that they make it possible to determine the profile of shear modulus changes in the subsoil and to construct a 3D model of subsoil stiffness (Młynarek et al., 2007 & 2013). In constructing these empirical relationships, variables that affect the variability of shear modulus G should be taken into account (Hardin, 1979, Lee & Stoke, 1986). Functions that identify the relationship between modulus G and G₀ and variables, which describe parameters of soil medium is expressed in the following form: (34) G/G0=f(σ′vo, e0, OCR, S, C, K, T) G/{G_{\it 0}} = {\rm{f}}\left( {{{\sigma ^\prime }_{vo}},\;{e_{\it 0}},\;OCR,\;S,\;C,\;K,\;T} \right) where: σ′_vo—effective vertical stress, e₀—initial void ratio, OCR—overconsolidation ratio, S—degree of saturation, C—grain characteristics, K—soil structure, T—temperature for noncohesive soils.

Empirical relationships for noncohesive soils between the cone resistance q_t and shear modulus G₀ taking into account some variables from equation (34) were presented by Baldi et al. (1989) in the form of (35) G0=ρ(277 qt0.13 σ′v00.27)2 {G_{\it 0}} = \rho {(277\;{q_t}^{0.13}\;{\sigma ^\prime _{v0}}^{0.27})^2} and Hegazy and Mayne (1995) for cohesive soils (36) G0=ρ(14,13 qt0.359 e−0.479)2 {G_{\it 0}} = \rho {(14,13\;{q_t}^{0.359}\;{e^{ - 0.479}})^2} where: ρ—soil density.

In order to determine the relationship between the cone resistance from the CPTU and the shear modulus G₀ for the subsoil in Greater Poland, where moraine clays of different degree of preconsolidation are found, tests were carried out in the vicinity of Poznań. A typical subsoil profile is shown in Fig. 36. SDMT tests were performed to determine the shear modulus G₀. Based on the replication test, the following correlation was determined: (37) G0=361−323.23 LI−0.323 σ′v0−0.125 σ′p {G_{\it 0}} = 361 - 323.23\;LI - 0.323\;{\sigma ^\prime _{v0}} - 0.125\;{\sigma ^\prime _p} Preconsolidation stress values σ′_p at individual levels σ′_v0, where G₀ modules from SDMT were recorded, were calculated from the relationships given by Wierzbicki (2010). The basis for these calculations were the values of normalized cone resistance Q_t.

Figure 36:

Owing to the significant amount of noncohesive soils of different origins in the subsoil in Poland, in order to determine the relationship between the cone resistance q_c and the constrained modulus M, we carried out tests at six locations (Fig. 15), namely: Darłowo, Derkacze, Gnojewo, Rzepin, Warsaw, and in Norway—Holmen (Młynarek et al., 2021).

The analysis consisted of 238 measurements of G₀ shear modulus values determined from SDMT and SCPTU at various levels σ′_vo in the subsoil. In order to eliminate the fact that the instrument affected the relationship between the cone resistance q_c and shear modulus G₀, the trend of modulus G₀ with depth was analyzed (Fig. 37).

Figure 37:

The analysis showed that the trend of effectiveness does not significantly differ in terms of statistics. The construction of the empirical relationship between the G₀ modulus and the cone resistance q_c was carried out in three stages. The first step was to examine the basic correlation q_c—G₀ for the entire population (fig 38). This relationship is linear, but its statistical significance is not high. In accordance with equation (37), preconsolidation stress σ′_v0 was introduced into the analysis in the second stage. The entire population of soils was divided into normally consolidated NC and preconsolidated OC (Fig. 39). The statistical assessment of this dependence was also not high. The third stage analyzed the relationships G₀ = f(q_c) in individual soil groups with the adopted division into OC and NC soils. Fig. 40 illustrates an example of this relationship for normally consolidated soils. Preconsolidation stress σ′_p was calculated based on the Wierzbicki relationship (2010): (38) σ′p=σ′v0(5.52lnQt−14.97) {\sigma ^\prime _p} = {\sigma ^\prime _{v0}}\left( {5.52{\rm{\;ln\;}}{Q_t} - 14.97} \right) A multivariable dependency model which highly assesses the shear modulus G₀ prognosis based on cone resistance and preconsolidation stress for individual soil groups was adopted in the third stage. For this model, the recommended dependencies are as follows: –

fine sands NC (39) G0=26.197+0.648qc+0.29 σ′pR2=0.85, n=43 \matrix{ {{G_{\it 0}} = 26.197 + 0.648{q_c} + 0.29{{\sigma}_p^\prime}} \cr {{{\rm R}^2} = 0.85,\;\;{\rm{n}} = 43} \cr }

Figure 38:

Figure 39:

Figure 40:

A convenient way to supplement the G₀ profile obtained from the SDMT is the empirical relationship between the G₀ modulus and the parameters from the standard DMT. This type of relationship for three basic groups of soils, i.e. clay, silt, and sand was provided by Marchetti et al. (2008) (Fig. 41). This relationship has been presented in the form of the following equation (45) G0MDMT=A(KD)B {{{G_{\it 0}}} \over {{M_{DMT}}}} = A{({K_D})^B} where: G₀ – initial shear modulus, obtained from the measured shear wave velocity (eq. 33), M_DMT – constrained modulus (eq. 24), K_D – horizontal stress index (eq. 26).

Figure 41:

Soils studied by Marchetti (2008) qualified as normal and slightly preconsolidated soils. To assess the suitability of Marchetti equations for soils from Poland, tests were carried out at five test sites: Kazimierz, Lipno, Jarocin, Kaźmierz, and Łańcut (Fig. 15) The analysis was carried out based on 989 SDMT test results obtained from soil deposits of different origin, macrostructure, and OCR. The procedure for determining correlation relationships was adopted in the same way as for the relationship G₀ = f (q_c).

In the first stage, general relationships between the G_0(m) modules determined based on the measured V_s and the K_D coefficient were forged. Fig. 42 confirms the conclusion formulated by Marchetti et al. (2008) that functional correlation between these variables should be formed for specific groups of soils, which at least distinguish between fine- and coarse-grained soils. The conducted research showed that this correlation is also affected by the preconsolidation effect, which can be defined by σ′_p or OCR. This conclusion is presented well in Fig. 43, where for the strongly preconsolidated clay the value of the calculated G₀ modulus from the Marchetti formula is outside 1:1 line. A high compliance between the G_0(m) modulus and G_0(C) was obtained for less preconsolidated sandy loom and sands. Examples of such a relationship are shown in Fig. 44.

Figure 42:

Figure 43:

Figure 44:

After the calibration process, the obtained coefficients for the overall empirical dependence (45) in individual soil groups are presented in Tab. 2.

Table 2 shows that the determined correlations are of high statistical value and can be recommended for practical use. An empirical relationship, which takes into account the impact of preconsolidation stress σ′_p, can be formed for preconsolidated cohesive soils (sandy loom, loom, clay, and silt) (Fig. 45): (46) σo(m)σ′p=1548KD−1.058R2=0.6703 \matrix{ {{{{\sigma _{o\left( m \right)}}} \over {{{\sigma}_p^\prime}}} = 1548K_D^{ - 1.058}} \hfill & {} \hfill & {{{\rm R}^2} = 0.6703} \hfill \cr } Zones A and B in Fig. 45 related to fissured clays and cemented silts confirm the previously formulated opinion that these soils need separate interpretation.

Figure 45:

In order to prepare the geotechnical design for many investments, including wind farms (Guidelines for Design of Wind Turbines—DNV/Riso 2002) and road facilities—guidelines for soil stabilization with rigid columns, Wydawnictwo Naukowe PWN, parameters such as Young modulus E and rigidity index I_R are necessary. CPTU and SCPTU methods are a convenient way to determine the profile of changes in these parameters on the tested subsoil. There are numerous empirical relationships between the cone resistance q_t and Young modulus E (e.g. Robertson & Cabal, 2012, Mayne, 2001, Lunne et al, 1997). An example of this is the relationship established for uncemented silica sands by Robertson (2012): (47) E=αE(qt−σv0) E = {\alpha _E}\left( {{q_t} - {\sigma _{v0}}} \right) where: α _E = 0.015 [10^{0.55Ic + 1.86}], I_c = [(3.47 – log Q_t)² + (log F_r + 1.22)²]^0.5, F_r—normalized friction ratio

From the solution from elastic theory, the relationship between shear modulus G and Young’s modulus E is written as follows: (48) E=2 G(1+v) E = 2\;G\left( {1 + v} \right) where: ν – Poisson’s ratio, G – shear modulus G assigned to the appropriate strain level (see Fig. 34).

Shear modulus ratio G/G₀ is used to determine the shear modulus G. The numerical value of this ratio depends on shear strain (Mayne 2001).

Robertson (2012) states that for typical engineering structures, the G/G₀ ratio can be assumed in the ranges of 0.30 to 0.38. The instruction “Guidelines for design of wind turbines—DNV/Riso” recommends for deformations of 10⁻⁴ ratio G/G₀ = 0.35. This manual also gives more detailed values of the Poisson ratio (Table 3).

For wind turbine foundation projects, the Instruction for design, calculation, installation, and inspection of wind turbine foundation Rev. Francisse de Geotechnique no 138 (2012) recommends the following values G/G₀ ratio for deformation 10⁻³ to 10⁻⁴: –

Clayey soils compact material – 0.33

This manual also introduces the concept of “static moduli” for deformation 10⁻² and dynamic moduli for deformation about 10⁻⁶.

The presented information proves that the registration of shear modulus G₀ in the SCPTU or SDMT test process at individual stress levels σ_vo in the subsoil will make it possible to determine the Young’s modulus in the profile of changes in the studied area.

Another parameter that determines the rigidity of the subsoil is the rigidity index I_R presented by formula (47) (Lunne et al., 1997) (49) IR=G/Su {I_R} = G/Su where: s_u – undrained shear strength.

Determination of this parameter using CPTU and DMT is given a lot of attention in the literature.

Massarsch (2009) indicates that the α = G/G₀ coefficient depends significantly on the soil type and the value of plasticity index PI in particular.

This coefficient denoted as R_M (Massarsch, 2009) will be determined from the formula: (50) RM=0.043 PI+0.103 {R_M} = 0.043\;PI + 0.103 In order to obtain the profile of changes of the rigidity index in the subsoil, the most advantageous method is the CPTU method. Next, the value of s_u is written by the formula (51) Lunne et al. (1997): (51) su=qt−σv0Nkt {s_u} = {{{q_t} - {\sigma _{v0}}} \over {{N_{kt}}}} A detailed analysis of the factors affecting the values of the rigidity index for soils from Poland is presented in the work by Mlynarek et al. (2018). For this purpose, the study was conducted in 10 locations in northern, western, and southern Poland. CPTU, SCPTU, or DMT and drilling was performed at each location. The oldest studied sediments were preconsolidated Pleistocene clays found in the vicinity of Bydgoszcz and Warsaw. The group of preconsolidated sediments includes Warta glaciation clay found in the area of Derkacze (Fig. 15). The analysis also included glacial formations of the youngest glaciation, preconsolidated sediments of the Poznań phase (Jarocin and Maryszew villages) and normally consolidated soils of the Pomeranian phase from the vicinity of Bartek, Boryszew, and Rzepin. The dominant soils in this group are sandy loams and loamy sands. Another analyzed group of soils were eolic loess deposits from the vicinity of Łańcut. The last analyzed group of sediments were cohesive and organic soils, deposited in the conditions of proglacial ponding, which was found in northern Poland. Such grouping of soils allowed to analyze, as in the case of the analysis of M₀ and G₀ moduli, the impact of texture, the effect of preconsolidation and macrostructure and the effect of cementation on the rigidity index. The analysis also included the liquidity index LI, which, as generally known, has a significant impact on the change of the parameter determining the shear strength s_u and shear modulus G₀. Fig. 46 shows the relationship between the rigidity index I_R and the liquidity index LI, while Fig. 47 shows the relationship between the rigidity index I_R and the preconsolidation stress σ′_p. The total impact of these variables on the rigidity index I_R was determined using multiple linear regression (Draper & Smith, 1981). The coefficient R_M was used according to the equation (50) for the calculation of the value I_R.

Figure 46:

Figure 47:

As a result of this analysis, the following relationships were obtained:

OC clay: (52) IR=−38.16+3.81PI+231.78LIR2=0.64 \matrix{ {{I_R} = - 38.16 + 3.81PI + 231.78LI} \cr {{{\rm{R}}^2} = 0.64} \cr }

Equation (52) does not contain the OCR, because its values in the clay zone changed very little, hence the impact of this variable on the rigidity index was statistically insignificant. The obtained values of the multiple regression coefficient R² for NC sandy loam by 0.26 and 0.51 for organic and alluvial soil confirm that the rigidity index depends on the effect of cementation and anisotropic macrostructure of these sediments. In the case of NC sandy loam, the anisotropy of the macrostructure of these soils is associated with thin interbedding of sands. The need to search for separate relationships between the rigidity index and the parameters from the CPTU for soils from Poland is presented by Fig. 47, which shows the values of the rigidity index determined from the CPTU, equations 52–55 and the formulas by Keaveny & Mitchell (1986)—equation (56), and Krage et al. (2014) —equation (57): (56) IR=0.26(Goσν0) (10.33(0.33Qt)0.75) {I_R} = 0.26\left( {{{Go} \over {{\sigma _{\nu 0}}}}} \right)\;\left( {\matrix{ 1 \cr {0.33{{(0.33{Q_t})}^{0.75}}} \cr } } \right) where: Q_t -normalized cone resistance (57) IR≈exp(137−PI23)1+1n[1+(OCR+1)3.226]0.8 {I_R} \approx {{{\rm{\;exp\;}}\left( {{{137 - PI} \over {23}}} \right)} \over {{1^ + }1n{{\left[ {1 + {{{{(OCR + 1)}^{3.2}}} \over {26}}} \right]}^{0.8}}}} Fig. 48 shows that the results for the established soil groups are well placed on the 1:1 calibration line, while the calculated values of the I_R from the Keaveny and Mitchell formula and Krage et al. are largely outside the calibration line.

Figure 48:

One of the important tasks of the geotechnical project is to separate areas of similar strength and rigidity in the subsoil in the area of planned investment. Continuous measurement of parameters in the CPTU and DMT predisposes these methods to solve this issue. The theoretical basis for the use of the inverse distance weighting (IDW) method, the clustering of data from the CPTU or DMT and then the separation of “homogeneous” zones in the subsoil in the context of constrained modulus M_o or shear modulus G₀ are found in the work of Młynarek (2005), Młynarek et al. (2007 & 2013).

Fig. 49 shows a diagram of data grouping from the CPTU for the construction of a quasi-3D model and 3D model. The grouping of the cone resistance values q_t allows to use the IDW method to create a model of the tested subsoil rigidity based on the value of constrained modulus M₀. Fig. 50 shows a fragment of this model. For the wind farm construction project, a subsoil rigidity model with the shear modulus G₀ (Fig. 51) was also created.

Figure 49:

Figure 50:

Figure 51:

The current state of knowledge justifies the advisability of using the static penetration method and dilatometer test to assess the stiffness of the subsoil very well. Even in complex geological conditions like in Poland, these methods allow to obtain the profile of changes for constrained modulus and initial shear modulus and the rigidity index for the tested subsoil. Undoubtedly, the advantage of these methods is the possibility of constructing a model of subsoil rigidity as one-dimensional, flat, and 3D. Such models allow to distinguish areas of heterogenous or similar rigidity in the area of the planned investment and to select an appropriate foundation system of the facility.

The quality of the obtained constrained modulus, initial shear modulus from these studies is influenced by several factors analyzed in the article. The quality of the CPTU and SCPTU is closely related, as the Norwegian Geotechnical Institute studies have shown, to the level of operator education and equipment quality. For this reason, the obtained moduli based on the parameters from these tests should be verified by laboratory tests of high-quality soil samples. Literary empirical relationships for the determination of the constrained modulus, initial shear modulus and rigidity index require analogous verification.

Local empirical relationships for soils from Poland, which take into account the specificity of these sediments and their genesis, are very helpful for the purpose of this verification. The search for these relationships is still a current research problem.