Suggestionof Empirical Equationsfor Damping Ratioof Plastican
Suggestionof Empirical Equationsfor Damping Ratioof Plastican
Suggestionof Empirical Equationsfor Damping Ratioof Plastican
Scholars' Mine
International Conferences on Recent Advances in 2001 - Fourth International Conference on Recent
Geotechnical Earthquake Engineering and Soil Advances in Geotechnical Earthquake Engineering
Dynamics and Soil Dynamics
Harry E. Stewart
Cornell University, Ithaca, NY
Recommended Citation
Park, Dugkeun and Stewart, Harry E., "Suggestion of Empirical Equations for Damping Ratio of Plastic and Non-Plastic Soils Based on
the Previous Studies" (2001). International Conferences on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics.
17.
https://scholarsmine.mst.edu/icrageesd/04icrageesd/session01/17
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Suggestion of Empirical Equations for Damping Ratio of Plastic
and Nonplastic Soils based on the Previous Studies
ABSTRACT
Several empirical correlations between damping ratio with various factors are available and many researchers have developed
empirical correlations between damping ratio and normalized shear modulus. Regardless of the appearance of the equations, the
estimation of the damping ratio from the normalized shear modulus at any given shear strain amplitude is possible. In this paper,
thirty-two sets of normalized shear modulus versus damping ratio from ten references are used to find a new empirical equation for
damping ratio of sandy soils. For clayey soils, more than forty sets of normalized shear modulus versus damping ratio from nine
references are used. After compiling and analyzing these previous studies and data, equations of sandy and clayey soils for the
correlation are determined with coefficient of determination (r*) = 0.918 and 0.844, respectively.
INTRODUCTION
D(%)=(50-600c).y(%)0.3 (2)
Several empirical correlations between damping ratio (D) with where Oc is effective confining pressure in psi. They
various factors are available. Hardin (1965) proposed an modified Equation 2 further to include the effects of soil
empirical equation for damping ratio of clean dry sands for gradation and number of cycles as:
shear strain amplitudes (v) in the order of lo-4% to 10-z%. D(%)=((50-60Crc)/38)(73F-53)
For confining pressures (00 ) between 500 psf and 3,000 psf (3)
x (1 .Ol - 0.0461ogN). y (%)“.3
and frequencies less than 600 Hz, the damping ratio (D) were
where N is number of loading cycles, and F is soil gradation
recommendedto be calculated as:
and sphericity factor, which is functions of soil sphericity and
D = 4.5 y”.2 ,o-o.5 coefficient of curvature. Typically, the value of F varies from
1.0 to 2.0 (Saxena and Reddy, 1989).
Sherif et al. (1977) establishedanother empirical equation for
Edil and Luh (1978) observed a significant effect of the
damping ratio based on cyclic torsional shear tests on dry
number of cycles on damping ratio for Ottawa sandand other
Ottawa sandat secondloading cycle as:
where Pa is the atmospheric pressure. The unit of a0 and effect of PI on G/G,, and D separately. However, PI effect
Pa should be the same, and shear strain amplitude (y) is does not have to be included in the formulation for damping
expressed in percentage. ratio when the equation is based on the normalized shear
modulus. This statement is justifiable, since the trend and
shape of damping change with PI are systematically similar to
METHODOLOGY those of modulus reduction change with PI. Figure 2
illustrates the relationship between damping ratio (D) and
As can be observed above, shear strain is the key parameter modulus ratio (G/Grnax), which was a function of inversely
for most empirical equations. After compiling available data proportional at various PI.
for saturated clays from twelve references, Seed and Idriss Many researchers have developed empirical correlations
(1970) suggested typical shear reduction curve and damping between damping ratio and normalized shear modulus.
ratio with shear strain. These relationships may be displayed Hardin and Dmevich (1972) derived a simple relationship
more clearly in 3-D diagram as shown in Fig. 1. It shows between modulus and damping ratio as:
that both properties (G/G,, and D) depend on shear strain,
D = D max Cl- (G/G max )) (7)
and it may be possible to demonstrate a relationship between Tatsuoka et al. (1978) also investigated the relationship
these two properties. Indeed, empirical equations for between shear modulus and damping ratio. They found
damping ratio using G/G,,, seem to be most convenient and
decreasing trend of damping ratio with increasing shear
widely used, since both modulus ratio and damping ratio modulus ratio (G/G,,,) qualitatively. Damping ratio could
depend, in general, on the same parameters. For example, it be expressed as a function of the normalized shear modulus
was found that plasticity index (PI) has significant effects on ratio (G/Grnax) as:
both normalized shear modulus and damping ratio. It was D=f (G/Grnax) (8)
demonstrated that it was not an easy task to formulate the
0.6 15
10
5 ANALYSES
0
0.01 0.1 1 0.01 0.1 1
Shear Strain (%) In this study, thirty-two sets of normalized shear modulus
Fig. 2. Systematic change of damping ratio (0) with versus damping ratio from ten references are used to find a
normalized shear modulus (G/GmaJ for various new empirical equation for damping ratio of sandy soils. For
plastici& indices (modifiedfrom Vucetic and Dobry, clayey soils, forty-eight sets of normalized shear modulus
1991). versus and damping ratio from nine references are used.
Table 1 and 2 summarize the references used for sandy and
Using the idea of Equation 8, several different forms of clayey soils, respectively.
equations were proposed. Uchida et al. (1980) proposed the Figure 3 shows the damping ratio (D) change with normalized
relationship between damping ratio and shear modulus ratio shear modulus. It shows a clear relationship between D and
defined as D,, = 33.3%, which corresponds to very high As can be noticed in Equation 13, Dmax would be about
shear strain levels where the G/G,, ratio is nearly equal to 32.8% when GiGmax is zero and Dmin is calculated as about
zero. Dmax = 33.3% is a representative value from many 0.3% when GiGmax is one.
previous researchers (Hardin and Dmevich, 1972; Sherif et al., Similarly, the damping ratio of clayey soils versus normalized
1977; Tatsuoka et al., 1978) for sands. Zhang (1994) shear modulus is plotted in Fig. 4, and can be represented by
r
Hommoku clay
Puri (1984) loessial silty clay c, d, e * see Table 2
for index
Saada (1985) various clays f, g * 0 a
0 b
Tawfiq (1986) Edger plastic kaolin h 0 c. d. and
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