Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Prediction of elastic constants of Timoshenko rectangular beams using the first two bending modes

  • Chen, Hung-Liang (Roger) (Department of Civil and Environmental Engineering, West Virginia University) ;
  • Leon, Guadalupe (Department of Civil and Environmental Engineering, West Virginia University)
  • Received : 2021.04.02
  • Accepted : 2021.09.29
  • Published : 2021.12.25
⟨ Previous Next ⟩

Abstract

In this study, a relationship between the resonance frequency ratio and Poisson's ratio was proposed that can be used to directly determine the elastic constants. Using this relationship, the frequency ratio between the 1st bending mode and 2nd bending mode for any rectangular Timoshenko beam can be directly estimated and used to determine the elastic constants efficiently. The exact solution of the Timoshenko beam vibration frequency equation under free-free boundary conditions was determined with an accurate shear shape factor. The highest percent difference for the frequency ratio between the theoretical values and the estimated values for all the beam dimensions studied was less than 0.02%. The proposed equations were used to obtain the elastic constants of beams with different material properties and dimensions using the first two measured transverse bending frequencies. Results show that using the equations proposed in this study, the Young's modulus and Poisson's ratio of rectangular Timoshenko beams can be determined more efficiently and accurately than those obtained from industry standards such as ASTM E1876-15 without the need to test the torsional vibration.

Keywords

Acknowledgement

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

  1. Alfano, M. and Pagnotta, L. (2007), "A non-destructive technique for the elastic characterization of thin isotropic plates", NDT E Int., 40(2), 112-120. https://doi.org/10.1016/j.ndteint.2006.10.002.
  2. Carneiro, V.H., Lopes, D., Puga, H. and Meireles, J. (2021), "Numerical inverse engineering as a route to determine the dynamic mechanical properties of metallic cellular solids", Mater. Sci. Eng.: A, 800, 140428. https://doi.org/10.1016/j.msea.2020.140428.
  3. ASTM C09 Committee (2019), Test Method for Fundamental Transverse, Longitudinal, and Torsional Resonant Frequencies of Concrete Specimens, ASTM International, West Conshohocken, PA.
  4. ASTM E28 Committee (2015), Test Method for Dynamic Youngs Modulus, Shear Modulus, and Poisson Ratio by Impulse Excitation of Vibration, ASTM International, West Conshohocken, PA.
  5. Bosomworth, P. (2010), "Improved frequency equations for calculating the young's modulus of bars of rectangular or circular cross section from their flexural resonant frequencies", J. ASTM Int., 7(8), 1-15. https://doi.org/10.1520/JAI102953.
  6. Brancheriau, L. (2014), "An alternative solution for the determination of elastic parameters in free-free flexural vibration of a Timoshenko beam", Wood Sci. Technol., 48, 1269-1279. https://doi.org/10.1007/s00226-014-0672-x.
  7. Chen, H.L.R. and Kiriakidis, A.C. (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.
  8. Chen, H.L.R. and Leon, G. (2019), "Direct determination of dynamic elastic modulus and Poisson's ratio of rectangular Timoshenko prisms", J. Eng. Mech., 145(9), 04019071. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001643.
  9. 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
  10. Diaz-de-Anda, A., Flores, J., Gutierrez, L., Mendez-Sanchez, R. A., Monsivais, G. and Morales, A. (2012), "Experimental study of the Timoshenko beam theory predictions", J. Sound Vib., 331(26), 5732-5744. https://doi.org/10.1016/j.jsv.2012.07.041.
  11. Goens, E. (1931), "Uber die Bestimmung des Elastizitatsmoduls von Staben mit Hilfe von Biegungsschwingungen", Annalen der Physik, 403(6), 649-678. https://doi.org/10.1002/andp.19314030602
  12. 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.
  13. Hutchinson, J.R. (2000), "Shear coefficients for Timoshenko beam theory", J. Appl. Mech., 68(1), 87-92. https://doi.org/10.1115/1.1349417.
  14. Ip, K.H. and Tse, P.C. (2001), "Determination of dynamic flexural and shear moduli of thick composite beams using natural frequencies", J. Compos. Mater., 35(17), 1553-1569. https://doi.org/10.1106/U0FU-9BR5-JNTG-B57R.
  15. Kolluru, S.V., Popovics, J.S. and Shah, S.P. (2000), "Determining elastic properties of concrete using vibrational resonance frequencies of standard test cylinders", Cement Concrete Aggregat., 22(2), 81-89. https://doi.org/10.1520/CCA10467J.
  16. Larsson, P.O. (1991), "Determination of young's and shear moduli from flexural vibrations of beams", J. Sound Vib., 146(1), 111-123. https://doi.org/10.1016/0022-460X(91)90525-O.
  17. Lee, B.J., Kee, S.H., Oh, T. and Kim, Y.Y. (2017), "Evaluating the dynamic elastic modulus of concrete using shear-wave velocity measurements", Adv. Mater. Sci. Eng., 2017, Article ID 1651753. https://doi.org/10.1155/2017/1651753.
  18. Leon, G. and Chen, H.L.R. (2019), "Direct determination of dynamic elastic modulus and Poisson's ratio of Timoshenko rods", Vib., 2(1), 157-173. https://doi.org/10.3390/vibration2010010.
  19. Medina, R. and Bayon, A. (2010), "Elastic constants of a plate from impact-echo resonance and Rayleigh wave velocity", J. Sound Vib., 329(11), 2114-2126. https://doi.org/10.1016/j.jsv.2009.12.026.
  20. Park, J.Y., Sim, S.H., Yoon, Y.G. and Oh, T.K. (2020), "Prediction of static modulus and compressive strength of concrete from dynamic modulus associated with wave velocity and resonance frequency using machine learning techniques", Mater., 13(13), 2886. https://doi.org/10.3390/ma13132886.
  21. Quaglio, O.A., Da Silva, J.M., Rodovalho, E.C. and Costa, L.V. (2020), "Determination of young's modulus by specific vibration of basalt and diabase", Adv. Mater. Sci. Eng., 2020, Article ID 4706384. https://doi.org/10.1155/2020/4706384.
  22. Rossit, C.A., Bambill, D.V. and Gilardi, G.J. (2018), "Timoshenko theory effect on the vibration of axially functionally graded cantilever beams carrying concentrated masses", Struct. Eng. Mech., 66(6), 703-711. http://doi.org/10.12989/sem.2018.66.6.703.
  23. Safari, M., Mohammadimehr, M. and Ashrafi, H. (2021), "Free vibration of electro-magneto-thermo sandwich Timoshenko beam made of porous core and GPLRC", Adv. Nano Res., 10(2), 115-128. http://doi.org/10.12989/anr.2021.10.2.115.
  24. Timoshenko, S. (1937), Vibration Problems in Engineering, D. Van Nostrand Company, New York.
  25. Wang, J.J., Chang, T.P., Chen, B.T. and Wang, H. (2012), "Determination of Poisson's ratio of solid circular rods by impact-echo method", J. Sound Vib., 331(5), 1059-1067. https://doi.org/10.1016/j.jsv.2011.10.030.