Settlement calculation methods

The evaluation of soil settlement is generally based on 1D consolidation properties evaluated from oedometer testing. As previously noted, the introduction of the CRS testing procedure has significantly improved the accuracy of soil deformation properties evaluation. In this section, existing settlement calculation methods are reviewed, including the compressibility index method, tangent modulus method (Janbu, 1967), and CRS Swedish settlement calculation method (Sällfors 1975), hereafter also referred to as “Janbu’s method” and “Sällfors’ method,” respectively.

2.4.2.1 Compression index method

The compression index method is based on two parameters, the compression index (Cc) and the recompression or swelling index (Cr). These parameters are defined based on the linearization of the compressibility curve in the log W–e plot. In particular, Cr describes the variation of the void ratio as a function of the change of effective vertical stress in the OC range, whereas Cc refers to the compressibility behavior in the NC range. Therefore, they can be obtained as follows.

C_୰= െ ^οୣోి

ο୪୭୥஢_౬^ᇲ (2.1)

C_ୡ = െ ^οୣొి

ο୪୭୥஢_౬^ᇲ (2.2)

Similarly, with reference to the log V'v–H plot, the slope parameters are referred to as modified recompression index (ԑ) and modified compression index (O), defined as:

ԑ = ^οகోి

Table 2.2 presents a summary of some existing correlations that are widely used in common practice. It is important to consider several aspects when using the compression index method. First, the parameters Cr and Cc are arbitrary fitting values based on the consolidation test results, which have no physical meaning.

Moreover, in case of low-quality samples, the evaluation of V'p from the stress-strain plot is rather difficult (Fig. 2.5). In relation to this, several interpretation methods have been proposed in the literature to determine the V'p (e.g., Casagrande, 1936;

Sällfors 1975). Finally, the compression index method is characterized by several issues when applied to soft sensitive clays. These clays show highly nonlinear behavior beyond the V'p in the log V'v–e plot. Therefore, the assumption of a constant value Cc would not be representative for the entire NC range. This concept is summarized in Fig. 2.6, which presents the CRS oedometer test result from Perniö clay (Finland). Based on the experimental result, Cc is evaluated considering three different stress ranges (V'p + 10 kPa; V'p + 20 kPa; V'p + 50 kPa). Note that the Cc

value is considerably high just beyond V'p, thus decreasing when the virgin compression line is reached. These aspects should be considered when performing settlement analysis in sensitive clays employing the compression index method.

Table 2.2. Existing correlations for evaluating the compression index Cc. Proposed equation Reference

Cc = 0.007(wL-10) Skempton and Jones (1944) Cc = 0.017(wL-20) Shouka (1964)

Cc = (wL-13)/109 Mayne (1980) Cc = 0.01 w Koppula (1981) Cc = 0.85(w/100)^1.5 Helenelund (1951) Cc = 0.01(w-7.549) Herrero (1983)

Figure 2.5. Schematization of the compressibility of natural clays.

Figure 2.6. Application of the compression index method on CRS consolidation test result from Perniö, Finland (Di Buò et al. 2019c).

2.4.2.2 Tangent modulus calculation method

The tangent modulus method, also known as “Janbu method” (Janbu 1967), is widely used in soft sensitive clays. It was developed by Janbu in the early 1960s to model the stress-strain behavior of cohesive and cohesionless soils in 1D constrained conditions. By observing the behavior of different soils (Fig. 2.7), Janbu developed a method to model the constrained modulus (M) as a function of the effective stress based on two dimensionless parameters: a modulus number (m) and a stress exponent (E, as follows:

where Va is the reference stress, which is equal to 100 kPa. Considering the different behavior in the NC and OC range, the constrained modulus can be expressed as:

M = mଶɐୟቀ^஢ᇲ respectively. Therefore, the Janbu expression for strain can be obtained as:

ɂ = ׬_஢^஢౜ᇲ _୑^ଵdɐ^ᇱ

౬బᇲ , (2.8)

where V'f is the final effective vertical stress reached after loading. Based on this formulation, the vertical strain can be derived by substituting (2.6) and (2.7) into (2.8), thus obtaining

Note that the value of the stress exponent defines the soil type. In particular, the dependency between the constrained modulus and the effective stress can be defined

by the value of E. As an example, by setting E= 1, the constrained modulus is assumed as a constant value:

M = m ɐୟ= m 100. (2.11)

This is the case of overconsolidated clays and rock. Similarly, E= 0 leads to a linear stress-dependent modulus:

M = m ɐ୴ᇱ. (2.12)

Therefore, the Janbu method has the capability to describe several stress-strain relationships. These are summarized in Table 2.3.

Table 2.3. Schematization of the Janbu method for different soil types (according to Canadian Foundation Engineering Manual, 1992).

Soil type Ƣ Strain formula

OC clays and rock 1 ɂ = ^ο஢౬ᇲ

୫ ஢_౗

NC sand and silt 0.5

ɂ =^ଶ

୫ቈට^஢౜ᇲ

஢౗െ ට^஢౬బᇲ

஢౗቉ NC clay, silt, and silty or clayey sand 0 ɂ =_୫^ଵln ቀ_஢^஢౜ᇲ

౬బᇲ ቁ Highly sensitive and quick clays î0.5 ɂ =^ଶ

୫ቈට_஢^஢౗

౬బᇲ െ ට^஢_஢^౗

౜ᇲ቉

Figure 2.7. Stress-strain relationships for different soil types (Janbu 1967).

In particular, considering the behavior of soft sensitive clays, the oedometer modulus can be modeled as a constant value in the OC region while it is stress-dependent in the NC region. Therefore, the stress exponent in the OC region (E2) can be assumed to be equal to 1, thus obtaining two different equations.

M_୒େ=^ப஢ Typical values of the modulus number m2 for different soil types are summarized in Table 2.4. The variability is wide, from 1000 to values close to 1 for very soft clays or peats.

Table 2.4. Values of modulus number for different soil types (according to Canadian Foundation Engineering Manual, 1992).

Soil type Modulus number,m

Very dense till, glacial till 1000–300

Gravel 400–40

2.4.2.3 CRS Swedish settlement calculation method

Similarly to the Janbu method, the CRS Swedish settlement calculation method proposed by Sällfors (1975) is based on a continuous M derived from the CRS consolidation test. As observed from CRS oedometer testing on soft sensitive clays, the constrained modulus can be assumed as a constant value (M0) up to V'p. Once the V'p is reached, the constrained modulus drops to a minimum value (ML), which remains constant until the limit stress V'L. After a further increase in stress, the modulus starts increasing linearly with the stress (M' = 'M/'V). This formulation divides the constrained modulus curve into three parts (Fig. 2.8), defined as follows.

 ൌ ଴ for  < V' < V'p (2.19)

 ൌ _୐ for V'p< V' < V'L (2.20)

 ൌ _୐൅ ^ᇱሺɐ^ᇱെ ɐ_୐^ᇱሻ for V' > V'L (2.21)

Based on the formulation provided, the vertical strain can be derived as follows.

ɂ_୴ൌ^{஢ᇲି஢బᇲ}

୑_బ for  < V' < V'p (2.22)

ɂ୴ൌ^{஢౦ᇲି஢బ}

ᇲ

୑_బ ൅^{஢ᇲି஢౦ᇲ}

୑_ై for V'p< V' < V'L (2.23)

ɂ_୴ൌ^{஢౦ᇲି஢బᇲ}

୑_బ ൅^஢ై

ᇲି஢_౦^ᇲ

୑_ై ൅ ^ଵ

୑^ᇲŽ ቂ^{୑ᇲሺ஢ᇲି஢ైሻ}

୑_ై ൅ ͳቃ for V' > V'L (2.24) The Sällfors method has been mainly adopted in the Swedish geotechnical practice.

However, for the purpose of this thesis, comparisons between the different methods based on the CRS experimental results are made. These aspects are analyzed in detail in chapter 6.

Figure 2.8. Schematization of the CRS Swedish method.

3 FIELD AND LABORATORY INVESTIGATION