297Taiwan J For Sci 22(3): 297-306, 2007

Research paper

1) Department of Natural Resources, National I-Lan Universi ty, 1 Shenlung Rd., Sec. 1, I lan 26047,

Taiwan. 國立宜蘭大學自然資源系，26047宜蘭市神農路一段1號。2) Corresponding author, e-mail:rockcho@seed.net. tw 通訊作者。

Received March 2007, Accepted May 2007. 2007年3月送審 2007年5月通過。

Comparison of Three Methods for Determining Young’s Modulus of Wood

Chih-Lung Cho1, 2)

【Summary】

Values of Young’s modulus of China fir, Taiwan red cypress, Taiwan yellow cypress, Japanese cedar, and camphor wood were investigated in this study. These wood species are most typical of the wood used in historic buildings in Taiwan. Discussions of apparent and true Young’s moduli obtained from longitudinal- and complex-vibration tests and a comparison of these results with tests of the same specimens by the static-bending method are presented. A simultaneous determina-tion of the shear modulus (G) of test specimen was obtained by a complex-vibration bending and twisting method. Results showed that the longitudinal-vibration method yielded the highest values for the apparent Young’s modulus, and those from the static-bending method were the lowest. Values of the percentage differences between values of Young’s modulus obtained from different test methods increased with an increase in Eb/G values. Based on the Timoshenko beam theory and Hearmon approximate solution method, taking rotary inertia and shear deformation into account, it was observed that a very high correlation (R2 = 0.999) existed between the true Young’s modulus obtained from the complex-vibration and static-bending tests. There was a tendency for the dy-namic measurements to be approximately 1.2% on average higher than the static measurements. The accuracies of the apparent Young’s modulus obtained from both the complex-vibration and static-bending tests improved as the E-to-G ratios decreased, and this parameter could be used as a guideline for the expected accuracy of the apparent Young’s modulus corresponding to various test methods.Key words: Young’s modulus, shear modulus, vibration test, static-bending test, Timoshenko beam

theory.Cho CL. 2007. Comparison of three methods for determining Young’s modulus of wood. Taiwan J

For Sci 22(3):297-306.

298 Cho CL―Comparison of three methods for determining Young’s modulus of wood

研究報告

三種測定木材彈性模數方法之比較

卓志隆1,2)

摘 要

本研究以振動方法探討杉木、紅檜、台灣扁柏、柳杉及樟樹等五種常作為台灣歷史建築物之木材

的彈性模數，並與由抗彎試驗得到之結果比較其試驗彈性模數與真實彈性模數之差異。此外，由彎曲

及扭轉複合振動方法可同時計算試材的剪斷模數(G)。

試驗結果顯示由縱向振動方法得到之試驗彈性模數(EL)最高，以抗彎試驗得到之試驗彈性模數(Es)最低。由不同試驗方法所得到之試驗彈性模數間之差異百分比值會隨Eb/G值增加而增加。若依

Timoshenko樑理論，考慮剪斷變形及旋轉慣性對試材彈性模數之影響，則由複合振動試驗所推導出

之真實彈性模數較抗彎試驗所推導出之數值平均大約1.2%，且兩者間具有極高之相關，決定係數為0.999。隨著Eb/G值降低，試驗彈性模數Eb與Es值的準確性會提高，E/G值可作為預測試驗彈性模數準

確性的參考指標。

關鍵詞：彈性模數、剪斷模數、振動試驗、抗彎試驗、Timoshenko樑理論。

卓志隆。2007。三種測定木材彈性模數方法之比較。台灣林業科學22(3):297-306。

INTRODUCTIONYoung’s modulus is one of the most fre-

quently used parameters in structural wood member design and evaluation. Vibration techniques have been used in some non-de-structive evaluation applications for determin-ing Young’s modulus, such as the machine stress rating of lumber (Ross and Pellerin 1991). There are certain advantages of using the vibration approach to determine Young’s modulus over the static-bending method. First, the experimental results are not sensi-tive to geometric imperfections of test speci-mens. Second, the time for a vibration test is shorter than that for the alternative static test (Chui 1989).

Longitudinal and flexural free-free beam vibration techniques have been widely used for estimating Young’s modulus of wood products. The free-support conditions provide the most accurate support conditions achiev-

able. Modern instrumentation enables these methods to easily be applied. Impact-induced free-free resonance flexural vibrations were employed for determining Young’s modulus of small clear specimens by Sobue (1986) and on samples of structural lumber (Perstor-per 1994, Hu and Hsu 1996). However, the classic theory (Euler beam theory) of flexural vibration is inadequate for short, thick beams and for higher-frequency modes. The inad-equacy originates from the fact that rotational motions and shear deformations of the beam elements are neglected. Both of these effects cause the resonance frequencies to be lower than those predicted by the classical model and consequently, Young’s moduli calculated on the basis of Euler beam theory may be sig-nificantly lower than the true values. Chui and Smith (1990) theoretically investigated the ef-fects of rotary inertia, shear deformations, and

299Taiwan J For Sci 22(3): 297-306, 2007

support conditions on the natural frequencies of wooden beams. Hearmon (1958) showed that for clear timber, an approximate method for calculating Young’s modulus according to the Timoshenko beam theory is very accurate for length-to-depth ratios of as low as 10. Hu and Hsu (1996) reported that it is essential to have a span-to-depth ratio exceeding 20 in order to obtain a reliable Young’s modulus for wood-based materials when using the transverse simple-beam vibration technique. For a static-bending test of a simply sup-ported beam specimen subjected to a concen-trated load, the Euler-Bernoulli beam theory (which ignores the effects of shear deforma-tion and rotary inertia) yields an apparent Young’s modulus. The effect of shear stress on the load-deflection relationship is marked when the beam has a small span-to-depth ratio. Bodig and Jayne (1982) indicated that Young’s modulus obtained on the basis of the ASTM D198 method would be 8.9% lower than the theoretical Young’s modulus.

True Young’s modulus is a material con-stant and is independent of a specimens’ size or measurement method. Under some specific conditions, the apparent Young’s modulus can be a good approximation of the true Young’s

modulus. In this study, wood specimens were prepared from 5 species, including 4 soft-woods and 1 hardwood. These wood species are most typical for wood used in historic buildings in Taiwan. Discussions of apparent and true values of Young’s modulus obtained by the vibration tests and a comparison of these results with tests of the same specimens by the static-bending method are presented in this paper.

MATERIALS AND METHODS

MaterialsNinety specimens of 35 (width)×35

(thickness)×600 (length) mm were made from each of 5 wood species: China fir (Cun-ninghamia lanceolata; CF), Japanese cedar (Cryptomeria japonica; JC), Taiwan yellow cypress (Chamaecyparis obtusa var. formo-sana; TYC), Taiwan red cypress (Chamaecy-paris formosensis; TRC), and camphor wood (Cinnamomum camphora; CW). The speci-mens were conditioned at 20℃ and 65% rela-tive humidity for more than a month prior to measuring their properties. The moisture con-tent and density of each wood species under designated conditions are given in Table 1.

Table 1. Mean values of Young’s modulus and shear modulus for 5 wood species Wood Density Moisture Young’s modulus (GPa) Shear Modulusspecies (kgm-3) content (%) Es

2) Eb2) EL

2) (GPa)CF1) 419 (36)3) 13.3 (1.16) 8.42 (1.13) 9.71 (1.46) 11.25 (1.64) 0.65 (0.16)JC1) 508 (65) 14.2 (0.63) 8.25 (1.77) 8.75 (1.82) 9.84 (1.96) 1.03 (0.17)TYC1) 468 (53) 11.9 (0.65) 8.33 (2.19) 8.70 (2.09) 9.54 (2.32) 1.07 (0.17)TRC1) 452 (50) 12.1 (0.45) 7.46 (1.83) 7.60 (1.84) 8.29 (2.04) 1.01 (0.17)CW1) 612 (36) 13.1 (0.59) 7.27 (1.00) 8.02 (1.34) 9.00 (1.43) 0.88 (0.12)1) CF, China fir; JC, Japanese cedar; TYC, Taiwan yellow cypress; TRC, Taiwan red cypress; CW,

camphor wood.2) Es, Eb, and EL are measured values of Young’s modulus obtained by the static-bending test, com-

plex-vibration bending and twisting test, and longitudinal-vibration test, respectively.3) Values in parentheses represent standard deviations.

300 Cho CL―Comparison of three methods for determining Young’s modulus of wood

Longitudinal-vibration testThe longitudinal-vibration test was car-

ried out using impact-induced vibrations in the fiber direction by a small hard rubber hammer striking 1 end of the specimen, held horizontally by 2 knife-edge rubber prisms in the center. The resulting vibrations were de-tected by a miniature accelerometer (Endevco type 2250A-10; San Juan Capistrano, U.S.A.), which was mounted on the specimen with the use of a beeswax layer. A Pulse 3560C system produced by the Brüel and Kjær Co. (Nærum, Denmark) was used for recording transient data and for the subsequent determination of the fundamental frequency by a fast Fourier transformation spectrum analysis. Young’s modulus (EL) was calculated from equation (1):EL = 4ρf1

2L2; (1)where f1 is the fundamental frequency of the longitudinal vibration, EL is Young’s modulus, ρ is the density, and L is the total length of the specimens. Figure 1 illustrates the setup for the longitudinal-vibration test.

Complex-vibration bending and twisting test

The free-free complex-vibration bending and twisting test setup is shown in Fig. 2. The beam specimen was supported horizontally via 2 very soft foam prisms to simulate a free-free condition. The specimen was impacted by a hard rubber hammer at an end along the edge line simultaneously producing com-plex vibrations of bending and twisting. The complex vibrations were measured by a pair of accelerometers located at the other end of the beam. In order to identify resonance fre-quencies of the bending and twisting modes from the complex vibrations, an isolation method of the time-domain signals accord-ing to Sobue’s report (1988) was adopted. Isolated signals were introduced to a fast Fourier transformation spectrum analysis, the resonance frequencies corresponding to each of the bending and twisting modes were in-stantaneously identified, and consequently the values of Young’s modulus (Eb) according to the Euler beam theory and shear modulus (G)

Brüel and KjærPulse 3560C with

FFT analysis

Fig. 1. Schematic diagram of the longitudinal-vibration test.

301Taiwan J For Sci 22(3): 297-306, 2007

were deduced. Young’s modulus and the shear modulus corresponding to the fundamental mode were calculated by equations (2) and (3), respectively:

Eb = ; (2)

where Eb is Young’s modulus, L is the length of the test specimen, fb1 (Hz) is the fundamen-tal frequency of the bending vibration, ρ is the density, β1 = 4.73 is a constant corresponding to the fundamental mode of free-free flexural vibration, and i is the radius of gyration of a cross section.

G = [ ] 2

; (3)

where ft1 (Hz) is the fundamental frequency of the torsion vibration, L is the length of the test specimen, Ip is the polar moment of iner-tia, ρ is the density, and Kt = 0.141bh3 is for a square cross section (b is the width and h is the thickness).

Static-bending testThe static-bending test was performed

in accordance with the CNS 454 procedure.

A Shimadzu model UH-10B testing machine (Kyoto, Japan) was used. The span was 500 mm, and a concentrated load was applied to the center of the span. The ultimate load and deflection were measured from the load-de-flection curve, and the static-bending Young’s modulus (Es) was then calculated.

RESULTS AND DISCUSSION

Comparison of values of static-bending and vibration Young’s moduli

Table 1 shows a comparison of the mean values of Young’s modulus for 5 wood spe-cies obtained from each of the 3 test methods. The mean value of Young’s modulus from the longitudinal-vibration method was greater than those obtained from the complex-vibration method and the static-bending test by about 9~16% and 11~33%, respectively. Values of the mean Young’s modulus ob-tained from the complex-vibration method for these 5 wood species were higher than those from the static-bending test by about 2~15%. These results are similar to those obtained by Hearmon (1958) and Haiens et al. (1996). The

Isolation of timedomain signals

(Brüel and Kjær;Pulse 3560C)

Fig. 2. Schematic diagram of the complex-vibration bending and twisting test.

302 Cho CL―Comparison of three methods for determining Young’s modulus of wood

viscoelasticity is the most likely source of the differences noted. The longitudinal-vibration method employs shorter pulses that are re-lated to waves of higher-frequency content. Kolsky (1963) provided an analysis of a com-plex viscoelasticity model that predicts higher velocities of longitudinal waves at higher fre-quencies. Since Young’s modulus determined by the longitudinal mode is proportional to the square of the longitudinal velocity, these results predict a higher value of the calculated Young’s modulus for wood than we obtained from the longitudinal resonance frequency. Brenndorfer (1972) indicated differences of the same order between Young’s modulus de-duced from longitudinal and flexural modes in the principal directions. When measure-ments were performed in the principal direc-tions, Young’s moduli determined from the longitudinal modes were about 2~16% higher than those measured from the flexural modes.

Underestimation of the static-bending Young’s modulus has been recognized in many studies and is usually attributed to the viscoelastic nature of wood. Table 2 presents an ANOVA analysis of the mean percent-age differences given by Eb vs. Es, EL vs. Es,

and Eb vs. EL for the 5 wood species. Values of percentage differences (∆E (%)) between Young’s moduli obtained from the 3 methods were as follows: ∆E (%) Eb vs. Es = [(Eb - Es)/Es]×100,∆E (%) EL vs. Es = [(EL - Es)/Es]×100, and∆E (%) Eb vs. EL = [(EL - Eb)/Eb]×100.

Variations in ∆E (%) for each wood spe-cies in Table 2 were in the following order: ∆E (%)EL vs. Es > ∆E (%)Eb vs. Es > ∆E (%)Eb vs. EL.

Values of the percentage differences between Young’s moduli of Taiwan yellow cypress and Taiwan red cypress were significantly lower than those of the other wood species. A ten-dency for ∆E to increase with an increase in Eb/G values indicated that shear deformation may be an important factor affecting Young’s modulus values obtained from different test methods.

Comparison of the true Young’s modulusFor flexural vibrations, the Timoshenko

beam theory accounts for both shear defor-mation and rotary inertia. These 2 effects are neglected in the Euler beam theory. Both of these effects cause the resonance frequen-cies to be lower than those predicted by the

Table 2. ANOVA of the values of percentage differences ∆E ( ) between Young’s moduli and Eb-to-G ratios for 5 wood species Values of percentage differences (∆E; %)Wood species between values of Young’s modulus Eb /G Eb vs. EL Eb vs. Es EL vs. Es

CF1) 14.8 (11.7)A2) 16.4 (9.8)A 33.3 (14.0)A 16.0 (4.8)A

JC 8.8 (21.5)B 14.6 (8.6)A 23.8 (20.6)B 8.6 (1.6)BC

TYC 5.7 (12.2)C 9.4 (5.8)B 15.4 (12.4)C 7.9 (1.6)C

TRC 2.1 (4.6)C 9.4 (5.4)B 11.1 (4.5)C 7.8 (1.6)C

CW 10.1 (17.1)B 16.4 (21.0)A 25.2 (14.5)B 9.6 (2.0)B

1) The wood species are defined in the footnotes to Table 1.2) Values in parentheses represent standard deviations. Means within a given column with the same

superscript letter do not significantly (p 0.05) differ as determined by Duncan’s multiple-range tests.

303Taiwan J For Sci 22(3): 297-306, 2007

classical model, and consequently, Young’s modulus calculated on the basis of equation (2) may be significantly lower than the true values.

Timoshenko first discussed the full dif-ferential equation by taking both of these ef-fects into account:

+ - i 2 (1 + ) +

= 0; (5)

where E is Young’s modulus, G is the shear modulus, i is the radius of gyration of a cross section, ρ is the density, y is lateral deflection, x is the distance along the beam, t is the time, and S is the shear deflection coefficient for a rectangular cross section. In the case of a free-free elastic beam, Hearmon (1958) pre-sented an approximate solution:

= 1 + [F1 (αn) + F2 (αn) ] -

; (6)

where Eb-true is the true Young’s modulus, EA is the apparent Young’s modulus, L is the length of the test specimen, values of F1 (αn)

and F2 (αn) for the fundamental mode of a free-free beam are 49.98 and 12.30, respec-tively, S = 1.2 is the shear deflection coeffi-cient for a rectangular cross section, and fo is the measured frequency. Using the resonance frequency, the apparent Young’s modulus and shear modulus obtained by complex vibra-tions of bending and twisting corresponded to the fundamental mode, and mean values of the true Young’s modulus for each of the wood species calculated from equation (6) are shown in Table 3. To assess the comparabil-ity of the complex-vibration technique with the static-bending test, a relationship between the apparent Young’s modulus determined by these methods was established as shown in Fig. 3. The relationship can be represented by the following regression formula:Es = 0.707 Eb + 1.806; R2 = 0.692.

It was found that the correlation between static-bending and dynamic Young’s modulus values, although close, was not perfect; theo-retically the 2 should be identical. To develop a relationship between the true Young’s mod-ulus obtained by the complex-vibration meth-od and by the static-bending method, a proper ratio of the static apparent Young’s modulus

Table 3. Mean values of the true Young’s modulus and ratios of the apparent to the true Young’s modulus for 5 wood species True Young’s modulus (GPa) Ratios of the apparent to the trueWood species Young’s modulus (%) Eb-true

2) Es-true2) Eb/Eb-true Es/Es-true

CF1) 10.42 (1.78)3) 10.31 (1.78) 92.8 (1.6) 82.6 (9.1)JC 9.26 (1.95) 9.11 (1.92) 95.3 (0.6) 90.8 (10.8)TYC 9.16 (2.43) 9.04 (2.04) 95.7 (0.6) 92.7 (9.0)TRC 7.97 (2.01) 7.87 (1.98) 95.7 (0.6) 95.2 (4.5)CW 8.84 (1.74) 8.74 (1.72) 95.1 (0.7) 85.2 (14.5)1) The wood species are defined in the footnotes to Table 1.2) Eb-true and Es-true are the true Young’s modulus determined by the Timoshenko beam theory corre-

sponding to complex-vibration bending and twisting and the static-bending tests, respectively.3) Values in parentheses represent standard deviations.

304 Cho CL―Comparison of three methods for determining Young’s modulus of wood

to the true Young’s modulus must be deter-mined. During static bending of the beam, the load-deflection relation is influenced by the shear stress which occurs in the material. A beam with a simple rectangular cross section under a center-point loading situation with shear deformation can be expressed as fol-lows:

= + (h / Ls)2; (7)

where Es is the static apparent Young’s modu-lus, Es-true is the true Young’s modulus, S is 1.2, G is the shear modulus obtained from the complex-vibration test, h is the thickness, and Ls is the span of the test specimen. The mean of the true Young’s modulus calculated using equation (7) for each wood species is shown in Table 3. By taking the rotary inertia and shear deformation into account, Fig. 4 shows that a very high correlation existed between values of the true Young’s modulus calculated from equations (6) and (7), respectively, with a tendency for the dynamic measurements to be approximately 1.2% higher than the static measurements. The correlation coefficient of 0.999 illustrates the almost-perfect agreement between the 2 methods used. In addition, the

shear modulus determined by the complex-vibration technique is accurate; otherwise, discrepancies in values of the true Young’s modulus between Eb-true and Es-true would be great.

The accuracy of these 2 methods was defined as the ratio of the apparent Young’s modulus to the true Young’s modulus, which is plotted in Fig. 5. As shown in Fig. 5, the accuracy of both techniques was improved as the E-to-G ratio decreased, and this can be used as a guideline for the expected accuracy. For instance, the E-to-G ratio should be less than 9 for specimens with a length-to-depth

Fig. 4. Linear regression relationship between the static-bending true Young’s modulus (Es-true) and the complex-vibration true Young’s modulus (Eb-true) of specimens.

Fig. 3. Linear regression relationship between the static-bending Young’s modulus (Es) and the complex-vibration Young’s modulus (Eb) of specimens.

305Taiwan J For Sci 22(3): 297-306, 2007

Fig. 5. Linear regression relationship between the accuracy (apparent Young’s modulus-to-true Young’s modulus ratios) and the Eb-to-G ratios of specimens.

ratio of 17, if 95% accuracy is expected for Young’s modulus determined from the funda-mental frequency using the complex-vibration technique.

CONCLUSIONS

Based on the experimental results de-termined by the vibration and static-bending methods for 5 wood species and of their sta-tistical evaluation, the following conclusions from this study were obtained.1. The agreement between the dynamic and

static tests was good. The dynamic method may provide a simple, more-rapidly per-formed alternative to the static-bending method for determining Young’s modulus of wood. However, values of the percentage differences between Eb vs. Es and EL vs. Es increased with an increase in Eb/G values.

2. Using the complex vibrations of bending and twisting for a wooden free-free beam enabled the simultaneous determination of Young’s modulus and the shear modulus.

3. Based on the Timoshenko beam theory and Hearmon approximate solution method, it was observed that a very high correlation existed between the true Young’s modulus obtained from the complex-vibration and static-bending tests. There was a tendency for the dynamic measurements to be ap-

proximately 1.2% on average higher than the static measurements.

4. The accuracies of values of Young’s modu-lus obtained from the complex-vibration and static-bending tests were both im-proved as the E-to-G ratios decreased, and this parameter can be used as a guideline for the expected accuracy of Young’s modulus corresponding to the test methods used.

LITERATURE CITED

Bodig J, Jayne BA. 1982. Mechanics of wood and wood composite. New York: Van Nostrand Reihold. p 155-8.Brenndorfer D. 1972. Nondestructive testing of wood with acoustic vibrations. Ind Lemn 6:234-40.Chui YH. 1989. Vibration testing of wood and wooden structures--practical difficulties and possible sources of error. In: Pellerin RF, McDonald KA, editors. Seventh international symposium on nondestructive testing of wood. 27~29 September 1989; Washington State University, Pullman, WA. Forest Product Soc. p 173-88.Chui YH, Smith I. 1990. Influence of rotatory inertia, shear deformation and support condi-tion on natural frequencies of wooden beams. Wood Sci Technol 24(3):233-45.Haiens DW, Leban JM, Herbe C. 1996. De-termination of Young’s modulus for spruce, fir and isotropic materials by the resonance flex-ure method with comparisons to static flexure and other dynamic method. Wood Sci Technol 30(4):253-63.Hearmon RFS. 1958. The influence of shear and rotatory inertia on the free flexural vibra-tion of wooden beam. Br J Appl Phys 9:381-8.Hu LJ, Hsu WE. 1996. Implementation of transverse simple beam vibration technique to determine MOE for wood based materials:

306 Cho CL―Comparison of three methods for determining Young’s modulus of wood

Accuracy, comparability, and limitations. In: Sandoz JL, editor. 10th international sympo-sium on nondestructive testing of wood. 26~28 August 1996; Lausanne, Switzerland. Presses Polytechniques Et Universitaires Romandes. p 227-36.Kolsky H. 1963. Stress wave in solids. New York: Dover Publications. p 121.Perstorper M. 1994. Dynamic modal tests of timber evaluation according to the Euler and Timoshenko theories. In: Pellerin RF, McDon-ald KA, editors. 9th international symposium on nondestructive testing of wood. 22~24 September 1993; Washington State University,

Pullman, WA. Forest Product Soc p 45-54.Ross RJ, Pellerin RF. 1991. Nondestructive testing for assessing wood members in struc-tures – a review. General Technical Report FPL-GTR-70. Madison, WI: USDA Forest Service, Forest Product Laboratory. 40 p.Sobue N. 1986. Instantaneous measurement of elastic constants by analysis of the tape tone of wood. Application to flexural vibration of beams. Mokuzai Gakkaishi 32(4):274-9.Sobue N. 1988. Simultaneous determination of Young’s modulus of structural lumber by complex vibrations of bending and twisting. Mokuzai Gakkaishi 34(8):652-7.