Keywords

1 Introduction

Hardening Soil Model (HS) was formulated in 1999 by Schanz et al. [4]. Its modification, which includes elastic behaviour of soil for the small strain (HSS) was presented by Benz [1]. Both models are among the most popular constitutive models used in the geotechnical practice.

Strain parameters of the model are defined in Fig. 1. The manual [3] and publication [2] listed in the bibliography below, proposed the relationship between Eur and E50 as Eur/E50 = 3. However, Obrzud [2] and Truty [5] noted that Eur/E50 ratio should be higher.

Fig. 1.
figure 1

Evaluation of E0, Eur and E50 based on stress-strain characteristics.

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Below an empirical equation for Eur modulus calculation based on effective stress and plasticity index (PI) is proposed.

2 Test Methodology and Analyses

2.1 Triaxial Tests

The tests were carried out on undisturbed soil samples in GEOTEKO’s laboratory. The triaxial tests included the following stages: back pressure saturation, isotropic consolidation and strain controlled drained shearing along standard stress path i.e. with constant cell pressure and increasing vertical stress.

2.2 Analyses

The analyses were performed using the results of triaxial tests carried out in terms of effective stress of values higher than in situ stress. Below, Fig. 2 shows a relationship between Eur modulus and mean effective stress at the end of consolidation stage (\( {\text{p}}_{\text{c}}^{'} \)) taking into account plasticity index value.

Fig. 2.
figure 2

Eur vs. mean effective stress at the end of consolidation stage.

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The relationship between Eur and \( {\text{p}}_{\text{c}}^{'} \) can be described by the following general formula:

$$ E_{ur} = a \cdot p_{c}^{'n} $$
(1)

Parameter (a) in Eq. (1) changes together with plasticity index variations. Figure 3 shows the relationship between parameter (a) and plasticity index.

Fig. 3.
figure 3

“a” vs. Ip

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Based on the above, Eur moduli values can be calculated from the following formula:

$$ E_{ur} = 21.769 \cdot PI^{ - 0.65} \cdot p_{c}^{'0.65} $$
(2)

Using the proposed empirical formula, Eur moduli were determined and compared with the relevant values resulting from the laboratory tests (Fig. 4). In conclusion one may declare that a significant part of Eur value determined by empirical formula is within ±30% of Eur value determined in the laboratory tests. Thus it seems to form a better approximation than the ratio Eur/E50 = 3 which was mentioned earlier.

Fig. 4.
figure 4

Eur values from empirical formula and from the laboratory tests

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3 Conclusions

The following conclusions can be drawn on the basis of the performed tests and analyses:

  • as soil plasticity index increases, the influence of effective stress on Eur value becomes smaller.

  • the best correlation between Eur modulus and effective stress was obtained for soils of plasticity index smaller than 30%.

  • the proposed empirical relationship allows calculating Eur modulus on the basis of plasticity index (PI) and average effective stress (\( {\text{p}}_{\text{c}}^{'} \)) values. A significant part of Eur value determined from the proposed empirical formula is included within the range of ±30% of Eur value determined in the laboratory tests.

The analyses presented in the paper were carried out within R&D activity of Geoteko Geotechnical Consultants Ltd. The a/m works resulted also in the proposal of other empirical formulas for estimation of geotechnical parameters. They will be presented in the next papers to be published.