One of the key parameters essential for conducting numerical analyses of the geotechnical structure or to conduct its design calculations is the deformation modulus of the separated soil layer. The basis for determining the magnitude of the deformation modulus is the stress–strain relation obtained by an empirical study of the appropriately prepared soil sample. Traditionally, the test enabling the determination of the relationship between the states of stress and strain is the triaxial compression test conducted on cylindrical test specimens. In these tests, extremely important are the particular measurement methods used to determine the value of the vertical component of the compressive stress and the strain of the soil sample for the determined conditions (consolidation and drainage conditions, rate of shearing, etc.). Determination of the momentary strain of the soil sample (for a given and recorded state of stress) requires advanced techniques for the precise control of the changing dimensions of the sample.
The accuracy of continuous measurement of changing physical quantities is a very important issue in soil studies due to the non-linearity of the manifested mechanical characteristics, e.g. the non-linear change (degradation) of the deformation modulus with the increase in strain. Many computational methods (to a greater or lesser extent adapting the basis of the theory of elasticity) require knowledge of the exact values of the deformation modulus as the most important material parameter. It should be defined according to the used calculation method (Figure 1). Moreover, knowledge of the value of
Non-linear stress–strain characteristics: Figure 1
In many cases of numerical analysis of geotechnical problems, each stage of the development of the state of stress and strain should be considered, dealing with strains in the range of
where:
The concept of threshold strain, beyond which the soil behaviour depends on the number of load cycles, was introduced by Lo Presti [4] and recorded by Vucetic [5]. According to Tabata and Vucetic [6], the threshold strain value depends on the plasticity index
In many calculation methods for subsequent phases of subsoil loading, it is required to use a tangent modulus, the value of which changes depending on strain and stress. Such an analytical solution was used, among others, in the method of determining the settlement curve of a single pile loaded with axial force [7].
In the modelling of many engineering problems, it is extremely important to determine the expected degradation of the value of the strain modulus, which can be presented in the following generalised form [8]:
where
The authors of this work became interested in determining this function. It should be noted that the initial shear modulus
The proposal of a new method for determining the soil stiffness degradation curve is based on the assumption that the intact soil structure undergoes the most significant changes in the first phase of the cyclic load applied during the torsional test (first quarter of the hysteresis loop). An interesting fragment of the hysteresis loop can be successfully recorded in detail during the test carried out at very low rates of torque changes and high frequency of recording measured physical quantities. The proposed concept eliminates the necessity for repeated soil testing at various levels of applied strain. It allows the determination of the full course of soil stiffness degradation only on the basis of the results of a single test. The article has been divided into the following parts: a synthetic description of the selected
The variability of the G module is most often described as the hyperbolic function of the relationship between the current value of the
where:
Based on the above equation, Hardin and Drnevich [11], introducing an additional parameter controlling the shape of the curve, namely the reference strain
where
This equation can also be presented as a function of the value change of tangential stress:
In the above model of the
The modified form of the hyperbolic
where the constant of the hyperbolic curve is taken in the range of
In 1993, Fahey and Carter [13] proposed a hyperbolic function of variability of the shear modulus in the generalised form:
where
Kuwabara [14] proposed a relationship for the analysis of axial-loaded piles in which a linear characteristic for the change of the
Darendeli [16], in his Ph.D. thesis under the supervision of Professor Stokoe, proposed a modification of the Hardin and Drnevich equation, a lower value of the reference strain (Eq. 10). The defined reference strain corresponds to the value of tangential stress at which
where
A comparison of the selected degradation functions of the modulus is shown in Figure 2.
Comparison of the Figure 2
The problem of describing the function of the strain modulus degradation in the modelling of mechanical phenomena occurring in various geotechnical problems is still being developed. Each formula is verified on the basis of the results of laboratory tests using a triaxial apparatus, a resonant column (RC) or a torsional shear (TS) apparatus. Most of the above-mentioned shear modulus degradation functions were created on the basis of soil analysis studies in the RC. In the research, the dependence of the shape of the degradation curve was determined, among others, on the isotropic stress value and the plasticity index.
The RC/TS apparatus is part of the equipment of the Geotechnical Laboratory in Olsztyn (31-WF8500 model). It is a device that can function as a RC or a TS apparatus. In the RC mode, the apparatus enables dynamic, cyclic excitation of torque in the tested soil sample with a frequency of up to 300 Hz. In the TS mode, the apparatus generates a cyclic load with a frequency in the range of 0.01–50 Hz. The scheme of the device is shown in Figure 3.
RC/TS apparatus scheme.Figure 3
Three types of soil were tested: one cohesive (intact silt) and two non-cohesive (reconstituted medium sands). Reconstitution of the non-cohesive soil specimen consisted of forming a sample by compaction in a cylindrical form and applying constant confining pressure for 24 hours. Detailed characteristics of these samples are summarised in Tables 1 and 2, as well as in Figure 4.
The granulometric composition of the studied soils.Figure 4
Parameters of tested soils.Soil Sandy silt saSi 2.66 0.047 0.11 17.1 0.67 1870 Medium sand MSa (1) 2.65 0.33 - 2.6 0.42 1860 Medium sand MSa (2) 2.65 0.33 - 4.0 0.37 1930
Parameters of tested samples.Sample Soil Diameter, mm Height, mm Cell pressure, kPa Maximum angle, mrad S3057 Sandy silt saSi 70.2 141.4 100–300 17.2 S5919 Medium sand MSa (1) 70.0 143.1 50–200 2.2 S3935 Medium sand MSa (2) 72.8 143.2 50–400 1.3
A synthetic graphical visualisation of the stiffness degradation test results in the RC/TS apparatus is shown in Figures 5–7. Due to the low stiffness values of the S3057 sample, it was possible to obtain a degradation of the
The results of the RC tests for sample S3057.Figure 5
The results of the RC tests for sample S5919.Figure 6
The results of the RC tests for sample S3935.Figure 7
Due to the control of twisting with the torque value, different rigidity of samples results in different strains in all specimens. High stiffness of samples prepared from non-cohesive soils made it impossible to obtain sufficiently large torsion angles to observe the effect of decreasing the rate of
Obtaining the full process of the soil stiffness degradation phenomenon is quite a difficult task, due to the hardware limitations associated with the generation of a correspondingly high value of torque (in the WF8500 apparatus, it is driven by electrical signal). Conducting many different tests on the same sample (e.g. RC and TS at different chamber pressures) leads to false results due to the phenomenon of stiffness degradation. On the other hand, the reproducibility of the results of tests conducted on different samples is too small to be successfully used in each case of the type and condition of the tested soil (unless the tests are carried out on reconstituted specimen). The authors therefore propose alternative methods for the interpretation of the results of soil stiffness degradation studies to address the above-mentioned drawbacks.
The concept proposed is based on the assumption that the intact soil structure undergoes the most important degradation changes in the first phase of applying a cyclic load (the first quarter of the hysteresis loop – see Figure 8), which can be successfully recorded in detail during the TS test conducted at very low torque rates (e.g. 0.01 Hz, see Figure 9) and a high frequency of recording measured physical quantities (e.g. 100 Hz).
The results of the TS test of sample S3935.Figure 8
The results of the TS test of sample S3935.Figure 9
Analysing the differential equation describing the TS-based research [19]:
where:
At very low frequencies
from which the
Having a registered dependence of the shearing angle
where:
Certainly, when the application controlling the device has built-in procedures for converting measurable physical quantities into the value of the shear strain
Despite the simplicity of the formulation, practical determination of the local inclination of the secant and tangent lines to the relationship
Results of TS analysis – sample S3057.Figure 10
Another suggestion of an alternative method of interpreting the results of TS tests is the back analysis of the
Exemplary discretisation of sample S3057.Figure 11
Exemplary results of back analysis of sample S3057.Figure 12
All of the samples listed in Table 2 were analysed by the methods proposed in this study and compared with the results obtained from the RC test, and a graphical summary of the analysis results is shown in Figures 13–15. As can be seen in all cases, the results obtained from the RC tests best correspond to the form of the stiffness degradation function determined by back analysis. Significant scattering of RC results in the case of the S3935 sample is also reflected in an unusual order of degradation curves obtained from back analysis and approximation–classification analysis of the secant
The results of RC/TS analyses of the sample S3057.Figure 13
The results of RC/TS analyses of the sample S5919.Figure 14
The results of RC/TS analyses of the sample S3935.Figure 15
The alternative methods of interpretation of soil stiffness degradation tests presented in this work can be successfully used in practical issues. Certainly, performing a fragmentary TS test, providing information only about the first quarter of the hysteresis loop, is sufficient to estimate the full process of the stiffness degradation function of the tested soil (provided the sample has a sufficiently large final angle of shearing) and significantly shortens the time of the test itself. This avoids the problematic, repeated deformation of the sample, which certainly affects the quality of the obtained image of the mechanical characteristics of the material being tested. Particularly noteworthy is the back analysis method based on optimised simulations of the TS research. In the monographs of Srokosz et al. [19], a modification of this method can be found, which consists in analysing the full load–unload cycle and determining four processes of the stiffness degradation function on this basis. The authors plan to modify their apparatus to allow long-term observation of visco-plastic soil deformations in the TS test after the impact of the torque moment has ceased. Work in this direction is in progress.