Direct Determination of Dynamic Elastic Modulus and Poisson’s Ratio of Rectangular Timoshenko Prisms
Publication: Journal of Engineering Mechanics
Volume 145, Issue 9
Abstract
In this paper, the exact solution of the Timoshenko beam vibration frequency equation under free-free boundary conditions is determined with an accurate shear shape factor. The exact solution is compared with a three-dimensional (3D) finite element calculation using ABAQUS, and the difference between the exact solution and the 3D finite-element model are within 0.05% for both the transverse and torsional modes. Furthermore, a relationship between the resonance frequencies and Poisson’s ratio was proposed that can directly determine the elastic modulus and Poisson’s ratio simultaneously, without the need for iteration, unlike the equations provided by an industry standard. The frequency ratio between the first bending and torsional mode for any combination of specimen dimensions can be directly estimated. Rectangular concrete beam specimens with three different mix designs were produced, and the transverse and torsional frequencies of these beams were tested. Results show that using the equations proposed in this study, the Young’s modulus and Poisson’s ratio of the concrete beams can be determined more directly than those obtained from the industry standard and with excellent accuracy.
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Acknowledgments
The authors acknowledge the support provided by the West Virginia Transportation Division of Highways (WVDOH) and FHWA for Research Project WVDOH RP#312. Special thanks are extended to our project monitors, Mike Mance, Donald Williams, and Ryan Arnold of WVDOH.
References
ACI (American Concrete Institute). 2014. Building code requirements for structural concrete (ACI 318-14) and commentary (ACI 318R-15). ACI Committee 318. Farmington Hills, MI: ACI.
ASTM. 2014. Standard test method for fundamental transverse, longitudinal, and torsional resonant frequencies of concrete specimens. ASTM C215. West Conshohocken, PA: ASTM.
Augustyn, E., and M. Kozien. 2015. “Possibility of existence of torsional vibrations of beams in low frequency range.” Tech. Trans. Mech. 3-M (21): 3–11. https://doi.org/10.4467/2353737XCT.15.170.4375.
Chavhan, P., and M. Vyawahare. 2015. “Correlation of static and dynamic modulus of elasticity for different SCC mixes.” Int. J. Recent Innovation Trends Comput. Commun. 3 (7): 4914–4919.
Chen, H.-L. R., and A. C. Kiriakidis. 2000. “Stiffness evaluation and damage detection of ceramic candle filters.” J. Eng. Mech. 126 (3): 308–319. https://doi.org/10.1061/(ASCE)0733-9399(2000)126:3(308).
Chen, H.-L. R., and A. C. Kiriakidis. 2005. “Nondestructive evaluation of ceramic candle filter with various boundary conditions.” J. Nondestr. Eval. 24 (2): 67–81. https://doi.org/10.1007/s10921-005-3483-z.
Cowper, G. R. 1966. “The shear coefficient in Timoshenko’s beam theory.” J. Appl. Mech. 33 (2): 335–340. https://doi.org/10.1115/1.3625046.
Gladwell, G. M. L., and D. K. Vijay. 1975. “Natural frequencies of free finite-length circular cylinders.” J. Sound Vibr. 42 (3): 387–397. https://doi.org/10.1016/0022-460X(75)90252-7.
Goens, E. 1931. “Uber die Bestimmung des Elastizitätsmoduls von Stäben mit hilfe von Biegungsschwingungen.” Ann. Phys. 403 (6): 649–678. https://doi.org/10.1002/andp.19314030602.
Gucunski, N., NRCTR Board, and SSHR Program. 2013. Nondestructive testing to identify concrete bridge deck deterioration. Washington, DC: Transportation Research Board.
Huang, T. C. 1961. “The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions.” J. Appl. Mech. 28 (4): 579–584. https://doi.org/10.1115/1.3641787.
Hutchinson, J. R. 2001. “Shear coefficients for Timoshenko beam theory.” J. Appl. Mech. 68 (1): 87–92. https://doi.org/10.1115/1.1349417.
Lee, B. J., S.-H. Kee, T. Oh, and Y.-Y. Kim. 2015. “Effect of cylinder size on the modulus of elasticity and compressive strength of concrete from static and dynamic tests.” In Advances in materials science and engineering, 12. London: Hindawi Publishing Corporation.
Lee, B. J., S.-H. Kee, T. Oh, and Y.-Y. Kim. 2017. “Evaluating the dynamic elastic modulus of concrete using shear-wave velocity measurements.” In Advances in materials science and engineering, 13. London: Hindawi Publishing Corporation.
Lydon, F. D., and R. V. Balendran. 1986. “Some observations on elastic properties of plain concrete.” Cem. Concr. Res. 16 (3): 314–324. https://doi.org/10.1016/0008-8846(86)90106-7.
Pickett, G. 1945. Equations for computing elastic constants from flexural and torsional resonant frequencies of vibration of prisms and cylinders. Chicago: Portland Cement Association.
Popovics, J. S. 2008. A study of static and dynamic modulus of elasticity of concrete. Urbana, IL: American Concrete Institute-Concrete Research Council.
Powers, T. C. 1938. “Measuring young’s modulus of elasticity by means of sonic vibrations.” In Vol. 38 of Proc., American Society for Testing and Materials, 460. West Conshohocken, PA: ASTM.
Spinner, S., T. W. Reichard, and W. E. Tefft. 1960. “A comparison of experimental and theoretical relations between Young’s modulus and the flexural and longitudinal resonance frequencies of uniform bars.” J. Res. Natl. Bur. Stand. Sect. A 64 (2): 147. https://doi.org/10.6028/jres.064A.014.
Timoshenko, S. 1937. Vibration problems in engineering. New York: D. Van Nostrand Company.
Vet, M. 1962. “Torsional vibration of beams having rectangular cross sections.” J. Acoust. Soc. Am. 34 (10): 1570–1575. https://doi.org/10.1121/1.1909051.
Weiss, J. 2006. “Elastic properties, creep, and relaxation.” Chap. 19 in Significance of tests and properties of concrete and concrete-making materials. Philadelphia: ASTM.
Yıldırım, H., and O. Sengul. 2011. “Modulus of elasticity of substandard and normal concretes.” Constr. Build. Mater. 25 (4): 1645–1652. https://doi.org/10.1016/j.conbuildmat.2010.10.009.
Zhou, Y., J. Gao, Z. Sun, and W. Qu. 2015. “A fundamental study on compressive strength, static and dynamic elastic moduli of young concrete.” Constr. Build. Mater. 98 (Nov): 137–145. https://doi.org/10.1016/j.conbuildmat.2015.08.110.
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©2019 American Society of Civil Engineers.
History
Received: Oct 15, 2018
Accepted: Jan 16, 2019
Published online: Jul 10, 2019
Published in print: Sep 1, 2019
Discussion open until: Dec 10, 2019
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