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Determining the Young’s modulus of solid wood by considering the fundamental frequency under the free-free flexural vibration mode

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Abstract

The Young’s modulus in the longitudinal direction of a Sitka spruce specimen was determined by conducting free-free flexural vibration tests based on Euler–Bernoulli’s and Timoshenko’s vibration theories. Comparing the Young’s moduli obtained considering these theories, an equation to calculate the Young’s modulus by considering only the fundamental frequency was formulated by modifying the equation derived from the Euler–Bernoulli’s theory. The modification was conducted with the consideration that the modified equation was applicable to various wood species with a wide range of length/depth ratio. Additionally, the accuracy of the proposed equation was examined using a statistical method for comparing the Young’s moduli obtained considering the aforementioned theories. Using the proposed equation, accurate Young’s moduli could be determined considering only the fundamental frequency, with a reduced influence of the length/depth ratio. The statistical analysis results indicated that the proposed equation can effectively yield accurate Young’s moduli for various wood species with a wide range of length/depth ratio.

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References

  • ASTM (2012) ASTM D6874-12, Standard test methods for nondestructive evaluation of wood-based flexural members using transverse vibration. West Conshohocken, PA, ASTM International. https://doi.org/10.1520/D6874-12

  • ASTM (2016) ASTM D143-14, Standard test methods for small clear specimens of timber. West Conshohocken, PA, ASTM International. https://doi.org/10.1520/D0143-14

  • ASTM (2020) ASTM C1548-02, Standard test method for dynamic Young’s modulus, shear modulus, and Poisson’s ratio of refractory materials by impulse excitation of vibration. West Conshohocken, PA, ASTM International. https://doi.org/10.1520/C1548-02R20

  • Brancheriau L (2006) Influence of cross section dimensions on Timoshenko’s shear factor. Application to wooden beams in free-free flexural vibration. Ann For Sci 63:319–321

    Article  Google Scholar 

  • Brancheriau L, Bailleres H (2002) Natural vibration analysis of clear wooden beams: a theoretical review. Wood Sci Technol 36:347–365

    Article  CAS  Google Scholar 

  • Brancheriau L, Baillères H (2003) Use of the partial least squares method with acoustic vibration spectra as a new grading technique for structural timber. Holzforschung 57:644–652

    Article  CAS  Google Scholar 

  • Chui YH, Smith I (1990) Influence of rotatory inertia, shear deformation and support condition on natural frequencies of wooden beams. Wood Sci Technol 24:233–245

    Article  Google Scholar 

  • Divós F, Tanaka T, Nagao H, Kato H (1998) Determination of shear modulus on construction size timber. Wood Sci Technol 32:393–402

    Article  Google Scholar 

  • Divos F, Denes L, Iñigues G (2005) Effect of cross-sectional change of a board specimen on stress wave velocity determination. Holzforschung 59:230–231

    Article  CAS  Google Scholar 

  • Fisher RA, Yates F (1963) Statistical tables for biological, agricultural and medical research, 6th edn. Oliver and Boyd, Edinburgh

    Google Scholar 

  • Goens E (1931) Über die Bestimmung des Elastizitätsmoduls von Stäben mit Hilfe von Biegungsschwingungen [Determination of Young’s modulus from flexural vibrations]. Ann Phys Ser 7(11):649–678 (in German)

    Article  Google Scholar 

  • Guitard D (1987) Mécanique du matériau bois et composites [Mechanics of wood and composite materials]. Editions Cépaduès, Toulouse, France

  • Haines DW, Leban JM, Herbé C (1996) Determination of Young’s modulus for spruce, fir and isotropic materials by the resonance flexure method with comparisons to static flexure and other dynamic method. Wood Sci Technol 30:253–263

    Article  CAS  Google Scholar 

  • Handbook of wood industry (1982) Handbook of wood industry 3rd edition, Forestry and forest products research institute Japan, Maruzen, Tokyo, Japan

  • Hearmon RFS (1948) Elasticity of wood and plywood. HM Stationary Office, London

    Google Scholar 

  • Hearmon RFS (1958) The influence of shear and rotatory inertia on the free flexural vibration of wooden beams. Brit J Appl Phys 9:381–388

    Article  Google Scholar 

  • Hearmon RFS (1966) Vibration testing of wood. For Prod J 16(8):29–40

    Google Scholar 

  • ISO (1975), ISO 3133:1975, Wood—Determination of ultimate strength in static bending. International Organization for Standardization, Geneva

  • ISO (1975), ISO 3349:1975, Wood—Determination of modulus of elasticity in static bending. International Organization for Standardization, Geneva

  • ISO (2012) ISO 3129:2012, Wood—Sampling methods and general requirements for physical and mechanical testing of small clear wood specimens. International Organization for Standardization, Geneva

  • JIS (1993) JIS Z 2280-1993, Test method for Young’s modulus of metallic materials at elevated temperature. Japanese Standards Association, Tokyo

  • JIS (1995) JIS R 1602-1995, Testing methods for elastic modulus of fine ceramics. Japanese Standards Association, Tokyo

  • JIS (2009) JIS Z 2101-2009, Methods of test for woods. Japanese Standards Association, Tokyo

  • JIS (2010) JIS A 1127-2010, Methods of test for dynamic modulus of elasticity, rigidity and Poisson’s ratio of concrete by resonance vibration. Japanese Standards Association, Tokyo

  • Kollmann FFP, Côté WA (1968) Principles of wood science and technology: I solid wood. Springer, Berlin

    Book  Google Scholar 

  • Matsumoto T (1962) Studies on the dynamic modulus E and the logarithmic decrement of wood by transverse vibration. Bull Kyushu Univ For 36:1–86

    Google Scholar 

  • Murata K, Kanazawa T (2007) Determination of Young’s modulus and shear modulus by means of deflection curves for wood beams obtained in static bending tests. Holzforschung 61:589–594

    Article  CAS  Google Scholar 

  • Ono T (1983) Effect of grain angle on dynamic mechanical properties of wood. J Mater Res Soc Jpn 32:108–113

    Article  Google Scholar 

  • Ono T, Kataoka A (1979a) The frequency dependence of the dynamic Young’s modulus and internal friction of wood used for the soundboards of musical instruments I. Effect of rotatory inertia and shear on the flexural vibration of free-free beams. Mokuzai Gakkaishi 25:461–468

    Google Scholar 

  • Ono T, Kataoka A (1979b) The frequency dependence of the dynamic Young’s modulus and internal friction of wood used for the soundboards of musical instruments II. The dependence of the Young’s modulus and internal friction on frequency, and the mechanical frequency dispertion. Mokuzai Gakkaishi 25:535–542

    Google Scholar 

  • Pickett G (1945) Equations for computing elastic constants from flexural and torsional resonant frequencies of vibration of prisms and cylinders. Proc ASTM 45:846–865

    Google Scholar 

  • Roohnia M (2019) Wood: acoustic properties. In: Hashmi S (ed) Reference module in materials science and materials engineering. Elsevier, Oxford, pp 1–13. ISBN: 978-0-12-803581-8.

  • Roohnia M, Yavari A, Tajdini A (2010) Elastic parameters of poplar wood with end-cracks. Ann for Sci 67:409

    Article  Google Scholar 

  • Roohnia M, Manouchehri N, Tajdini A, Yaghmaeipour A, Bayramzadeh V (2011) Modal frequencies to estimate the defect position in a flexural wooden beam. BioResources 6:3676–3686

    CAS  Google Scholar 

  • Sawada M (1983) Deformation behavior of wood. J Soc Mater Sci Jpn 32:838–847

    Article  Google Scholar 

  • Shamaiirani S, Roohnia M (2016) Dynamic modulus of wood containing water-resistant glue finger joint after severe steaming. BioResources 11:8900–8913

    Article  CAS  Google Scholar 

  • Sobue N (1986) Instantaneous measurement of elastic constants by analysis of the tap tone of wood: application to flexural vibration of beams. Mokuzai Gakkaishi 32:274–279

    Google Scholar 

  • Sohi AMA, Khademi-Eslam H, Hemmasi AH, Roohnia M, Talaiepour M (2011) Nondestructive detection on the effect of drilling on acoustic performance of wood. BioResources 6:2632–2646

    Google Scholar 

  • Spinner S, Tefft WE (1961) A method for determining mechanical resonance frequencies and for calculating elastic moduli from these frequencies. Proc ASTM 61:1221–1238

    Google Scholar 

  • Spinner S, Reichard TW, Tefft WE (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 64A:147–155

    Article  CAS  Google Scholar 

  • Timoshenko SP (1921) On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Phil Mag 41:744–746

    Article  Google Scholar 

  • Yoshihara H (2011) Measurement of the Young’s modulus and shear modulus of in-plane quasi-isotropic medium-density fiberboard by flexural vibration. BioResources 6:4871–4885

    CAS  Google Scholar 

  • Yoshihara H (2012a) Off-axis Young’s modulus and off-axis shear modulus of wood measured by flexural vibration tests. Holzforschung 66:207–213

    CAS  Google Scholar 

  • Yoshihara H (2012b) Examination of the specimen configuration and analysis method in the flexural vibration test of solid wood and wood-based materials. For Prod J 62(3):191–200

    Google Scholar 

  • Yoshihara H (2012c) Influence of the specimen depth to length ratio and lamination construction on Young’s modulus and in-plane shear modulus of plywood measured by flexural vibration. BioResources 7:1337–1351

    CAS  Google Scholar 

  • Yoshihara H, Maruta M (2020) Effect of the specimen configuration on the accuracy in measuring the shear modulus of western hemlock by torsional vibration test. Wood Sci Technol 54:1479–1496

    Article  CAS  Google Scholar 

  • Yoshihara H, Maruta M (2021) Effect of specimen configuration and orthotropy on the Young’s modulus of solid wood obtained from a longitudinal vibration test. Holzforschung 75:428–435. https://doi.org/10.1515/hf-2020-0115

    Article  Google Scholar 

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Number 20K06165.

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Authors and Affiliations

  1. Faculty of Science and Engineering, Shimane University, Matsue, Japan

    Hiroshi Yoshihara

  2. Faculty of Science and Engineering, Shizuoka Institute Science and Technology, Fukuroi, Japan

    Makoto Maruta

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  1. Hiroshi Yoshihara
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  2. Makoto Maruta
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Correspondence to Hiroshi Yoshihara.

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Yoshihara, H., Maruta, M. Determining the Young’s modulus of solid wood by considering the fundamental frequency under the free-free flexural vibration mode. Wood Sci Technol 55, 919–936 (2021). https://doi.org/10.1007/s00226-021-01306-5

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