The Resonant Column Test

Imagine suspending a dumb bell shaped object using a wire. After permitting the wire to relax, the dumb bell could be rotated slightly, and then released. The elasticity of the wire would exert a torque on the dumb bell, causing it to oscillate about the axis of the wire. Using the period of oscillation, inertial moment of the dumb bell, and dimensions of the wire, it is possible to calculate the shear modulus of the wire material, and by observing the decay rate of its oscillations, it is possible to determine its viscosity. This instrument, called a torsional balance oscillator, was originally developed by Coulomb in the late 18th century. Working from his laboratory at the University of Tokyo’s Earthquake Research Institute, Kumizi Iida made several improvements to the device that allowed him to make precise measurements of the shear modulus and viscosity of earth materials such as quartz and sandstone (Iida, 1935). The natural frequency of thicker speciments will generally be much greater than those of wire, so Iida added an electrical torsion motor and optical detector that allowed him to drive the system at much higher frequencies. He could determine the natural frequency of the system from the frequency that produced the maximum

response in his optical detector. In a collaboration, Ishimoto and Iida (1936) modified the device so that it could be used to determine the resonant frequency of soil samples and other unconsolidated media (Figure 1.11).

The resulting resonant column device consisted of an iron disk (base), on which was affixed a cellophane tube containing the soil sample (sample tube). Atop the soil sat another iron disk that compressed the soil sample into the tube (weight). The iron base was oscillated by means of an oscillating magnet. Iidra improved his resonant column device several times. Initially, oscillation of the weight was monitored mechanically, but in later versions, oscillation was monitored using a light and mirror. An advantage of the latter apparatus was that it could be used with moving photographic film to record transient signals. The device could also include an axial excitation mechanism and sensor, so that dilational properties could be measured, as well.

Iidra would vary the excitation frequency until he either obtained a maxima or a null point, calculate the dilational or shear wave velocity necessary for resonance, and then use the velocities so obtained to estimate Young’s modulus, shear modulus, Poisson’s ratio, and Lame’s parameters. By turning-off the instrument, and monitor- ing its transient response, he could calculate both normal and shear viscosity values. Iida’s last paper using a resonant column device, On the Elastic Properties of Soil, Particularly in relation to its Water Content, was published in September 1940 (Iida, 1940)

Interest in the resonant column method resumed in the United States, with re- searchers such as Shannon, Yamane, Dietrich, Hall, Richart, and Hardin publishing many papers on the method and results obtained from it in the 1950s and 1960s

(Richart et al., 1970). Most instruments followed the general pattern set by Iidra; however, researchers often changed the methods for exciting the sample or measur- ing sample response. In most cases, the column was either excited or its response monitored using some sort of magnetic inductor. Such instruments are called free- free instruments, because both their base and weighted top are free (Figure 1.12). A second variant, with a fixed base and free top, was developed by Hall (Hall and Richart, 1963). In 1981, the American Society for the Testing of Materials adopted a standardized test using the free-free instrument, ASTM-D-4015 (1996). The method has been updated twice since that date.

In practice, a specimen is prepared, weighed, and its physical dimensions mea- sured. The resonant frequency is determined, and then machine power is cut-off. Vibrational decay is measured, and then the operator uses uses a series of calcula- tions, curves, or a Fortran program, to determine modulii and the damping ratio, defined as:

DR= ηω

2µ (1.10)

where η is soil viscosity, ω is angular frequency is radians per second, and µ is the shear modulus. Note that we have altered ASTM’s symbology to be consistent with that used throughout the rest of this paper.

One of the most frequent criticisms of ASTM-D-4015 involves Equation 1.10. An influential early practitioner of the resonant column method in the U.S. was Bobby O. Hardin of the University of Kentucky. His paper, The nature of damping in sands, included a thorough treatment of the mathematics and calculations necessary to implement the resonant column method (Hardin, 1965). Using experiments based

Figure 1.12: Resonant Column Apparatus (excerpted from ASTM-D-4015 (92)).

on dry Ottawa sands, he concluded that soil viscosity varies inversely with frequency, such that ηω/G is a constant. This is, essentially, Equation (1.10). The 2 in the denominator of Equation (1.10) aligns the definition of damping ratio with that of a related quantity called the damping capacity. Hardin seems to have been surprised by this finding, and emphasized that this result was for dry sands only, and that this assumption probably only holds near the measurement frequency.

Rearranging Equation 1.10, we arrive at the surprising result that for a given soil sample, its viscosity is inversely proportional to the frequency at which the res- onant column test was conducted: η = 2G(DR)/ω. This very odd result, a material property that is dependent on test conditions, seems to belie a problem with the ASTM-D-4015 methodology.

In response to Hardin’s article, Weissmann (1965) suggested that Hardin’s fre- quency dependent viscosity actually indicates that Hardin used the wrong model.

Indeed, the requirement that the product of viscosity and frequency be constant is exactly what we would expect of a Coulombic damping model. In any event, it is diffi- cult to reconcile a constantηω/Gwith the static viscosity values that were published for the same materials in Hardin’s paper.

We should note that a resonant column instrument is a very crude device for determining frequency-material property relationships. For a given material, we are limited to the column’s resonant frequency and its overtones. Hardin indicated that he used the fundamental and first two overtones to arrive at his conclusion, but he didn’t actually publish any of the frequencies that he used; instead, he published normalized frequencies without giving the normalization factor. In another discussion, about confining pressure relationships, he gave a frequency ofω= 1290 rad/s, which is about 205 Hz. Given this frequency, his first and second overtones would have been about 610 Hz and 1025 Hz, respectively. In a discussion of one of Hardin’s earlier papers (Hardin and Richart, 1963), Rao (1964) asked whether, given Hardin’s caveat about the limited frequency range over which the constantηω/G assumption is valid, measurements taken at frequencies that are two or more orders of magnitude greater than what might be expected in an earthquake are really worthwhile.

Data obtained by Ishimoto and Iida data showed an overwhelming dependence of soil viscosity on moisture conditions, with high moisture content soils (approx 50%) exhibiting viscosities nearly two orders of magnitude less than their dry counterparts. Furthermore, they found that this relationship held across dozens of soil types (Ishi- moto and Iida, 1936). By using columns of several different heights, they were able to take measurements at approximately a dozen different fundamental mode frequencies, and concluded that the relationship between frequency and measured viscosity was

much flatter for dry samples than for moist samples–This is consistent with the idea that a Coulombic model might be appropriate for dry soils.

Other criticisms have been leveled at the resonant column apparatus, itself. H.D. McNiven and C.B. Brown suggested that some of the resonant behavior being ob- served might be due to tube waves (McNiven and Brown, 1963). Viscosity measure- ments typically are made by switching off the power to the device; however, Wang et al. (2003) demonstrated that it could take several seconds for the magnetic field used by the inductor to collapse, and that during that time, the system would create a counter-emf that could explain some or all of the frequency-dependent viscosity effect. In other words, it is quite likely that the amplitude decay measurements used to determine damping ratio are at least partially due to the resonant column device, itself.

A final criticism could be leveled at nearly any laboratory test: It is difficult to know whether or not a laboratory sample is behaving as it would in-situ. Ishimoto and Iida (1936 and 1937), collected soil samples in a carefully designed sampling jar. After making a resonant column with a sample, they would disturb the same soil sample, and repack it into the same sampling jar. They determined that repacking the sample usually effected a 2x to 10x decrease in viscosity.