The Resonant Column Test is a commonly used laboratory test for measuring dynamic soil properties at low to medium strains. The test is performed by vibrating a soil cylinder (solid or hollow) at one of its natural modes. The wave propagation velocity and the shear modulus at the tested strain can then be determined from the resonant frequency.
The dynamic properties of soils are important for the analysis and design of structures required to resist dynamic loads such as earthquake shaking, machinery vibration, traffic loading, etc. Each of these dynamic loading conditions subject the soil-structure system to very different amplitudes and frequencies of loads, and therefore the response of the soil to a wide variety of load amplitudes and frequencies must be determined in order to correctly design for all scenarios.
The mechanical behavior of a soil is determined by effective stress, void ratio, water content, the strain level, stress and strain paths, and several other factors. However, these factors are affected differently depending on whether the soil is loaded statically or dynamically. In a static test, the specimen is subjected to an increasing load in a single direction. In a dynamic test, the specimen is instead subjected to a repetitious load at a set amplitude and frequency.
In the Resonant Column Test, the specimen is tested under dynamic conditions to find the shear modulus of a soil sample at low to medium strains. This test can be performed in conjunction with the Torsional Shear Test to determine the shear modulus at most strain values.
After the soil specimen has been fully prepared and consolidated (where an axial load is applied slowly and incrementally to allow water to be squeezed out of the specimen), a harmonic torsional excitation is applied to the top of the specimen by an electromagnetic loading system or motor. This harmonic load is applied with a constant amplitude over a range of frequencies and the response curve (strain amplitude) is measured. The shear wave velocity is obtained by measuring the first-mode resonant frequency. The shear modulus is calculated from this shear wave velocity and the soil density. Material damping can also be obtained from either the free-vibration decay after the forced vibration is removed or from the width of the frequency response curve assuming viscous damping. The torsional harmonic load amplitude is increased with each test to obtain the shear modulus and damping values for different strain ranges.
Typical specimens used in the resonant column test should have a uniform circular cross section with ends perpendicular to the axis of the specimen. The minimum diameter of the test specimens should be 33 mm (1.3 in.) and should be at least ten times the diameter of the largest particle. However, if the specimen is larger than 70 mm (2.8 in.) in diameter, the largest particle inside the specimen must not be larger than one-sixth of the specimen’s diameter. The test specimen must have a length of at least two times the diameter, but cannot have a length greater than seven times the diameter.
From this test, multiple soil characteristics can be determined. As the loading frequency is gradually changed, a maximum response (strain amplitude) can be found. The lowest frequency at which the strain amplitude is maximized is the fundamental frequency of the soil specimen and the driving system. The fundamental frequency is then a function of the soil stiffness, the specimen geometry, and the characteristics of the resonant column device. This fundamental frequency can then be used to determine the shear wave velocity, which is then used to calculate the shear modulus of the soil at the strain amplidude resulting from the resonant frequency.
Material damping is determined by both the free vibration decay method and the half-power bandwidth method. It is not easy to define true material damping, but it is common practice to express the damping of real materials in terms of its equivalent viscous damping ratio. The free vibration response for a system with a single degree of freedom with viscous damping is typically expressed with the formula (0 = mẍ + cẋ + kx). From relationships between this formula and the damping ratio (D = c / cc), three general solutions can be found. These solutions depend on whether the single degree of freedom system is underdamped, critically damped, or overdamped. The second method to measure material damping in the resonant column test is the half-power bandwidth method. From the forced-vibration test, the logarithmic decrement is calculated by measuring the width of the frequency response curve near resonance.
Damping Properties (D) describe how oscillations in a system decay after a disturbance. Often, a system exhibits oscillatory behavior when an external force disturbs the system from its original position of static equilibrium. For example, imagine a system consisting of a spring attached to a ceiling on one end and attached to a weight on the other end. The weight and spring system is motionless until someone or something pulls or pushes the weight and releases it. The weight will then bounce up and down until eventually returning to its original resting state (static equilibrium). The damping ratio is a measurement of how rapidly the system will return to static equilibrium.
The damping ratio is a dimensionless parameter that is denoted by zeta (ζ) and is equal to the ratio of two coefficients of identical units:
ζ= Actual Damping (c) / Critical Damping (cc)
In this equation, c is the damping coefficient, which is a material property that will indicate whether a material will bounce back or return energy to a system. A low damping coefficient means the material will have a strong bounce back, like that of a rubber bouncing ball. A high damping coefficient, on the other hand, will result in a material having a very weak bounce back. Therefore, when attempting to design an object to withstand an earthquake, which causes large vibrations, it is essential to have a large damping coefficient that will prevent the system from oscillating.
There are four types of damping:
(1) Overdamped: the system returns to static equilibrium without oscillating. ζ > 1
(2) Underdamped: the system oscillates with the amplitude gradually decreasing to zero until equilibrium has been reached. ζ < 1
(3) Undamped: the system oscillates at its natural resonant frequency (ω0). ζ → 0
(4) Critically Damped: the system returns to static equilibrium as quickly as possible. ζ = 1
Critically damped is the case where ζ is equal to 1 and is the border between overdamped and underdamped cases. This damping method is very desirable in many cases of engineering where design of a damped oscillator is necessary (such as that of a door closing mechanism or an earthquake-resistant building).
Keywords: Resonant Column — Elastic Modulus (E*) — Shear Modulus — Damping — Damping Coefficient — Shear Wave Velocity — Material Damping — Critically Damped — Free Vibration Decay — Half-Power Bandwidth — Fundamental Frequency
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