7 Resonant Frequency
7.2 Resonant Frequency Method
where
E = dynamic modulus of elasticity d = density of the material L = length of the specimen N = fundamental flexural frequency
k = radius of gyration of the section about an axis perpendicular to the plane of bending k = t/12 for rectangular cross section where t = thickness
m = a constant (4.73 for the fundamental mode of vibration)
The dynamic modulus of elasticity can also be computed from the fundamental longitudinal frequency of vibration of a specimen, according to the following equation:2
E = 4L2dN2 (7.3)
Equations 7.1 and 7.3 were obtained by solving the respective differential equations for the motion of a bar vibrating: (1) in flexure in the free-free mode, and (2) in the longitudinal mode.
Thus, the resonant frequency of vibration of a concrete specimen or structure directly relates to its dynamic modulus of elasticity and, hence, its mechanical integrity. The method of determining the dynamic elastic moduli of solid bodies from their resonant frequencies has been in use for the past 55 years. However, until up to the last few years, resonant frequency methods had been used almost exclusively in laboratory studies. In these studies, natural frequencies of vibration are determined on concrete prisms and cylinders in order to calculate the dynamic moduli of elasticity and rigidity, the Poisson’s ratio, and for monitoring the degradation of concrete during durability tests.
7.2 Resonant Frequency Method
This method was first developed by Powers3in the United States in 1938. He determined the resonant
frequency by matching the musical tone created by concrete specimens, usually 51 = 51 = 241-mm prisms, when tapped by a hammer, with the tone created by one of a set of orchestra bells calibrated according
N m k L E d 2 2 2/ E L N d m k 4 2 4 2 4 2 /
to frequency. The error likely to occur in matching the frequency of the concrete specimens to the calibrated bells was of the order of 3%. The shortcomings of this approach, such as the subjective nature of the test, are obvious. But this method laid the groundwork for the subsequent development of more sophisticated methods.
In 1939 Hornibrook4refined the method by using electronic equipment to measure resonant frequency.
Other early investigations on the development of this method included those by Thomson5in 1940, by
Obert and Duvall2in 1941, and by Stanton6in 1944. In all the tests that followed the work of Hornibrook,
the specimens were excited by a vibrating force. Resonance was indicated by the attainment of vibrations having maximum amplitude as the driving frequency was changed. The resonant frequency was read accurately from the graduated scale of the variable driving audio oscillator. The equipment is usually known as a sonometer, and the equipment has been used to measure various dynamic moduli of concrete.7–15
7.2.1 Test Equipment
The testing apparatus required by ASTM C 215-85, entitled “Standard Test Method for Fundamental Transverse, Longitudinal, and Torsional Frequencies of Concrete Specimens,”7is shown schematically in
Figure 7.1. Equipment meeting the ASTM requirements has been designed by various commercial organizations. One of the commercially available sonometers is shown in Figure 7.2. The resonant frequency test equipment presently used in monitoring the long-term deterioration of massive concrete blocks (dimension 305 = 305 = 915-mm) exposed to sea water in Treat Island, Maine, is shown in
Figure 7.3.
The testing apparatus consists primarily of two sections, one generates mechanical vibrations and the other senses these vibrations.8
FIGURE 7.1 Schematic diagram of a typical apparatus for the forced resonance method showing driver and pick- up positions for the three types of vibration. (A) Transverse resonance. (B) Torsional resonance. (C) Longitudinal resonance. (Adapted from ASTM C 215-85, Annual Book of ASTM Standards, Vol. 04.02, 1986, p. 123.)
Oscillator Oscilloscope Meter Amplifier Pickup Circuit Y Y Y Y Y YY Y Y X X X X X X X X X Driving Circuit Amplifier A B C To Driving Unit
7.2.2 Vibration Generating Section
The principal part of this section is an electronic audio-frequency oscillator, which generates electrical audio-frequency voltages. The oscillator output is amplified to a level suitable for producing mechanical vibrations. The relatively undistorted power output of the amplifier is fed to the driver unit for conversion into mechanical vibrations.
7.2.3 Vibration Sensing Section
The mechanical vibrations are sensed by a piezoelectric transducer. The transducer is contained in a separate unit and converts mechanical vibrations to electrical AC voltage of the same frequencies. These voltages are amplified for the operation of a panel-mounted meter which indicates the amplitude of the transducer output. As the frequency of the driver oscillator is varied, maximum deflection of the meter needle indicates when resonance is attained. Visible indications that the specimens are vibrating at their fundamental modes can be obtained easily through the use of an auxiliary cathode-ray oscilloscope, and its use is generally recommended.
FIGURE 7.2 Longitudinal resonance testing of a 76 = 102 = 406-mm concrete prism by a sonometer.
FIGURE 7.3 The resonant frequency test equipment used in monitoring the deterioration of 305 = 305 = 915-mm concrete blocks exposed to sea water in Treat Island, Maine.
7.2.4 Operation of the Sonometer
Some skill and experience are needed to determine the fundamental resonant frequency using a meter- type indicator because several resonant frequencies may be obtained corresponding to different modes of vibration. Specimens having either very small or very large ratios of length to maximum transverse direction are frequently difficult to excite in the fundamental mode of transverse vibration. It has been suggested that the best results are obtained when this ratio is between three and five.
The supports for the specimen under test should be of a material having a fundamental frequency outside the frequency range being investigated and should permit the specimen to vibrate without significant restriction. Ideally, the specimens should be held at the nodal points, but a sheet of soft sponge rubber is quite satisfactory and is preferred if the specimens are being used for freezing and thawing studies.
The fundamental transverse vibration of a specimen has two nodal points, at distances from each end of 0.224 times the length. The vibration amplitude is maximum at the ends, about three fifths of the maximum at the center, and zero at the nodal points. Therefore, movement of the pickup along the length of the specimen and observation of the meter reading will show whether the specimen is vibrating at its fundamental frequency.
For fundamental longitudinal and torsional vibrations, there is a point of zero vibration (node) at the midpoint of the specimen and the maximum amplitude is at the ends.
Sometimes in resonance testing of concrete specimens, two resonant frequencies may appear which are close together. Kesler and Higuchi12 believed this to be caused by a nonsymmetrical shape of the
specimen that causes interference due to vibration of the specimen in some direction other than that intended. Proper choice of specimen size and shape should practically eliminate this problem; for example, in a specimen of rectangular cross section the above problem can be eliminated by vibrating the specimen in the direction parallel to the short side.
In performing resonant frequency tests, it is helpful to have an estimate of the expected fundamental frequency. Table 7.1 shows the approximate ranges of fundamental longitudinal and flexural resonant frequencies of standard concrete specimens given by Jones.13
7.2.5 Calculation of Dynamic Moduli of Elasticity and Rigidity and Poisson’s
Ratio
The dynamic moduli of elasticity and rigidity and the Poisson’s ratio of the concrete can be calculated by equations given in ASTM C 215-02. These are modifications of theoretical equations applicable to specimens that are very long in relation to their cross section, and were developed and verified by Pickett9
and Spinner and Tefft.10The corrections to the theoretical equations in all cases involve Poisson’s ratio
and are considerably greater for transverse resonant frequency than for longitudinal resonant frequency. For example, a standard 102 = 102 = 510-mm prism requires a correction factor of about 27% at the fundamental transverse resonance, as compared with less than 0.5% at the fundamental longitudinal resonance.13,15The longitudinal and flexural modes of vibration give nearly the same value for the dynamic
TABLE 7.1 Approximate Ranges of Resonant Frequencies of Concrete Prism and Cylinder Specimens
Approximate Range of Resonant Frequency, Hz
Size of Specimens (mm) Transverse Longitudinal
152 = 152 = 710-mm prism 550–1150 1800–3200
102 = 102 = 510-mm prism 900–1600 2500–4500
152 = 305-mm cylinder 2500–4500 4000–7500
From Jones, R., Non-Destructive Testing of Concrete, Cambridge University Press, London, 1962. With permission.
modulus of elasticity. The dynamic modulus of elasticity may range from 14.0 GPa, for low quality concretes at early ages, to 48.0 GPa for good quality concrete at later ages.11 The dynamic modulus of
rigidity is about 40% of the modulus of elasticity.14