Introducing the stiffness as a feature of construction soils allows to adopt computational techniques commonly used for materials of elastic properties. Fundamental mechanical parameter of the soil which describes its stiffness, is soil modulus. Research for the mechanical properties of soils conducted over the last 30 years have led to the conclusion that the basic parameter determining the soil stiffness is shear modulus *G* [1]. This parameter is the instantaneous value under the conditions specified by strain level and the stress state, which depends on the load history, its velocity and character (static, dynamic, cyclic), drainage conditions as well as the type of structure that transmits the load. The nonlinear nature of the stiffness variability for the shear modulus *G* can be presented as:
$$\begin{array}{}G\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{G}_{max}\cdot F(\gamma ),\end{array}$$(1)

where:

*G*_{max} – initial shear modulus;

*F*(*γ*) – stiffness degradation function;

*γ* – shear strain in soil.

Initial shear modulus *G*_{max} is widely recognized as a parameter of the state of the soil [1]. It can be assumed that the value of the dynamic shear module for very small shear strain is the same as the *G*_{max} modulus for static problems in both “drained” and “undrained” conditions [2]. For this reason, many of the techniques used in dynamic and seismic applications have been adopted for estimating the *G*_{max} module. In dynamic studies based on the measurement of shear wave propagation velocity in a soild medium, the initial shear modulus is derived from a simple relationship:
$$\begin{array}{}{G}_{dyn}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{G}_{max}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\rho \cdot {V}_{s}{\phantom{\rule{thinmathspace}{0ex}}}^{2},\end{array}$$(2)

where:

*ρ* – volume density of soil;

*V*_{s} – shear wave velocity.

This dependence is used in many uncomplicated research methods, where the main goal is to measure the time of the propagation of generated elastic wave on the separated area of natural subsoil or soil specimen [3, 4]. Determined in this way modulus is a value corresponding to very small shear strain level.

Over the past 50 years, a number of correlation formulas have been developed to estimate the initial *G*_{max} strain modulus based on the results of the velocity measurement of the transverse mechanical wave in soil and the study of other geotechnical characteristics. For example, Hardin and Richart [5] have developed one of the first correlations for uniformly-grained coarse sand (Ottawa sand, No. 2030, *d*_{50} = 0.72 mm, *C*_{U} = 1.2):
$$\begin{array}{}{G}_{max}\phantom{\rule{thinmathspace}{0ex}}[\text{MPa}]=69{\displaystyle \frac{(2.17-e{)}^{2}}{1+e}\sqrt{\frac{{p}^{\prime}}{{p}_{ref}}},}\end{array}$$(3)

where:

*e* – void ratio,

*p*’ – average effective stress [kPa],

*p*_{ref} – reference pressure (*p*_{ref} = 100 kPa).

On the basis of further investigations of natural and artificially prepared from sanded soils, Hardin and Black [6] reported a universal relationship:
$$\begin{array}{}{G}_{max}\phantom{\rule{thinmathspace}{0ex}}[\text{MPa}]=33{\displaystyle \frac{(2.97-e{)}^{2}}{1+e}\sqrt{\frac{{p}^{\prime}}{{p}_{ref}}}.}\end{array}$$(4)

The equations (3) and (4) represent the two specific forms of a so-called Hardin’s formula.

Based on research carried out worldwide in the field, it has been confirmed that the value of the initial shear modulus can be functionally linked to the void ratio and the average effective stress [7]:
$$\begin{array}{}{G}_{max}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}A\cdot f(e)\cdot {\displaystyle (\frac{{p}^{\prime}}{{p}_{ref}}{)}^{m}.}\end{array}$$(5)

where *A* is a constant dependent on soil type, grain properties and distribution, *f*(*e*) is a function of void ratio *e* reflecting the soil density, *m* is the stress exponent reflecting the confining pressure.

A number of studies conducted in the world on noncohesive soils differing in grading from the *d*_{50} diameter and the coefficient of uniformity *C*_{U} have enabled the development of formulas to determine the value of the *G*_{max} modulus. The formulas shown in relate to noncohesive soils with specified grain size distribution characteristics (*d*_{50} and *C*_{U}). Based on the recent research, it has been found that for non-uniformly grained soils, these formulas produce overstated values of *G*_{max} modulus. Research carried out over the last decade, including Wichtmann and Triantafyllidis [8, 9, 10] at the Karlsruhe Institute of Soil Mechanics and Rock Mechanics, have led to develop a universal formula that will take into account the value of the coefficient of uniformity *C*_{U} by function:
$$\begin{array}{}{G}_{max}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}A\cdot {\displaystyle \frac{(a-e{)}^{2}}{1+e}\cdot (\frac{{p}^{\prime}}{{p}_{ref}}{)}^{n}\cdot \phantom{\rule{thinmathspace}{0ex}}{p}_{ref},}\end{array}$$(6)

Table 1 Empirical formulas for modulus *G*_{max} of sand and gravel, which were derived from the Hardin’s formula.

where appropriate constants were proposed as [9]:
$$\begin{array}{}a=1.94\phantom{\rule{thinmathspace}{0ex}}\cdot \phantom{\rule{thinmathspace}{0ex}}{e}^{(-0,066{C}_{U})},\end{array}$$(7)
$$\begin{array}{}n=0.40\phantom{\rule{thinmathspace}{0ex}}\cdot \phantom{\rule{thinmathspace}{0ex}}{C}_{\text{U}}{\phantom{\rule{thinmathspace}{0ex}}}^{0.18},\end{array}$$(8)
$$\begin{array}{}A=1563+3.13\phantom{\rule{thinmathspace}{0ex}}\cdot \phantom{\rule{thinmathspace}{0ex}}{C}_{\text{U}}{\phantom{\rule{thinmathspace}{0ex}}}^{2.98}.\end{array}$$(9)

The selected most used formulas are graphically represented in Figure 1 in relation to the results obtained from equations 6-9.

Figure 1 Comparison of selected correlation formulas for *G*_{max}.

In Figure 2 the relationship between *G*_{max} values and coefficient of uniformity *C*_{u} is given and in Figure 3 between *G*_{max} and average effective stress *p*′, for constant value of the void ratio *e* = 0.37. The value of the *G*_{max} modulus decreases with the increase of the coefficient of graining uniformity *C*_{u} and increses with the increase of the average effective stress *p*’.

Figure 2 Comparison of the empirical relationship between *G*_{max} values and the *C*_{U} for *e* = 0.37.

Figure 3 Comparison of the empirical relationship between *G*_{max} values and the *p*’ for *e* = 0.37.

This article presents selected issues related to the continuation of non-cohesive soils survey conducted at the Institute of Soil Mechanics and Rock Mechanics in Karlsruhe. The purpose of the present study is to verify the correlation between the characteristics of grain size distribution of non-cohesive soils and the value of the dynamic shear modulus. For this purpose, elastic wave propagation studies were conducted on specimens with artificially generated grain size distribution, using piezoelectric bender elements (BE) and triaxial compression cell. The purpose of the research was also to verify and evaluate the different methods of interpreting BE test results.

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