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# examination of the specimen configuration and analysis method in the flexural and longitudinal vibration tests of solid wood and wood-based materials.

by：Teehome     2020-02-28
Elastic modulus and shear modulus of solid wood (
Spruce, western spruce), medium-
Density cardboard (MDF)
And Laurel Wood (Shorea sp. )with five-
Bending and longitudinal vibration tests were performed with different sample depth/length ratios and subsequent finite element analysis was performed (FEA).
The values of Young\'s modulus and shear modulus are calculated by three analytical methods :(1)
Method to solve the differential equation strictly by Timothy enkoPhil. Mag. 41:744-746, 1921), (2)
Iterative program proposed by Hearmon (Brit. J. Appl. Phys. 9:381-388, 1958), and (3)
The Young\'s modulus measured by longitudinal vibration test is replaced with the approximate equation proposed by Goens (Ann. Physik. Ser. 7 11:649-678, 1931).
The results of the finite element analysis show that the analytical method does not affect the value of Young\'s modulus or shear modulus.
However, the results obtained show that the analytical method affects the measurements of these modulus.
Although method 3 is simpler than methods 1 and 2, the effect of depth/span ratio is more obvious when using resonance frequencies below the second bending vibration mode.
However, when using the resonance frequency of bending vibration higher than the third mode, it is promising that the shear modulus can be measured while reducing the effect of the depth/length ratio.
Solid wood and wood-
Basic materials are often used in construction projects.
Therefore, obtaining reliable strength and deformation properties for these materials, including their Young\'s modulus and shear modulus, is essential to ensure efficient and economical construction procedureseffective.
In the experimental method used to determine the elastic modulus and shear modulus of solid wood and wood --
Base materials, such as media-
Density cardboard (MDF)
And plywood, the vibration method is very effective, there are many examples of measuring the Young\'s modulus and shear modulus of these materials by this method.
The longitudinal vibration method is an effective method to measure Young\'s modulus because of its simplicity;
Therefore, there are many examples of measuring the elastic modulus of solid wood and wood
Basic materials obtained by longitudinal vibration method (
Ono and Norimoto 1985, Sobue 1986b, Haines, etc.
1996, Ilic 2003, Yang et al.
2003, Ji Yuan 202c).
In contrast, the torsional vibration method is effective in measuring shear modulus, as this method can induce pure shear stress conditions.
As a result, several studies have used the torsional vibration method to measure shear modulus (
Beeker and Noack 1968, Nakao and others.
1985, middle tail and Okano 1987, Sobue 1999, tonghuki et al. 2010).
However, it is often difficult to measure the shear modulus of orthogonal materials such as wood and wood
The material based on the method of torsional vibration, because the vibration is composed of three-
The dimensions make the resonance frequency of the torsional vibration mode inevitably contain the effect of the two shear modulus in the lateral plane around the torsional axis.
To measure the shear modulus of a single plane with a torsional vibration method, it is necessary to perform multiple vibration tests using samples with different aspect ratios along the cross section to separate the shear modulus from each other (Nakao et al. 1985).
Otherwise, a plate-
The shape of the sample is used to determine the effect of the plane measuring the shear modulus (
Mid-tail and Ono 1987, yoshi original 2009)
Or ignore the difference between shear modulus (Tonosaki et al. 2010). [
Figure 1 slightly]
The bending vibration method is advantageous when comparing longitudinal and torsional vibration methods, because Young\'s modulus and shear modulus can be determined at the same time and can be easily obtained.
In addition, the bending vibration method does not have the above geometric problems, which are related to the method of measuring shear modulus and the torsional vibration of Young\'s modulus, the solid wood bending vibration method based on the Tymoshenko vibration theory (
Hearmon 1958,1966; Nakao 1984; Sobue 1986a;
Cui and Smith 1990; Kubojima et al. 1996, 1997; Divos et al. 1998, 2005;
Brancheriau and Bailleres;
Bran Eucalyptol Limonene and Pinene Enteric Soft Capsules 2006;
Yoshida, Jinze 2007; Tonosaki et al. 2010; Khademi-Eslam et al. 2011; Sohi et al. 2011).
Measuring in-time
However, the effect of the plane shear modulus of the medium fiber board and plywood on the structure of the sample is so significant that
If the depth of the sample is not large enough relative to its length, the plane shear modulus cannot be accurately obtained from the bending vibration test (
Yoshiwara 2011 2012a);
Therefore, the sample configuration should be properly determined to measure
The plane shear modulus is accurately measured by bending vibration test.
In addition to the configuration of the samples used for vibration testing, the method used to analyze the data obtained from vibration testing affects the accuracy of Young\'s modulus and shear modulus measured from solid wood and woodBasic materials.
In the example above (
Hearmon 1958,1966; Nakao 1984; Sobue 1986a;
Cui and Smith 1990; Kubojima et al. 1996, 1997;
Brancheriau and Bailleres; Divds et al. 1998, 2005;
Bran Eucalyptol Limonene and Pinene Enteric Soft Capsules 2006;
Yoshida, Jinze 2007; Tonosaki et al. 2010; Khademi-Eslam et al. 2011; Sohi et al. 2011)
Based on the iterative method proposed by Hearmon, the Young\'s modulus and shear modulus are determined (1946)
Its details are described below.
In previous studies
Yoshiwara 2011 2012a)
, By the strict solution of the Tymoshenko equation derived from Goens, the Young\'s modulus and shear modulus of the medium fiber plate and plywood are obtained by numerical value (1931)
The details are also described below.
As shown in these examples, there are several methods for analyzing the data obtained from the vibration test.
However, a definitive method has not yet been identified, although there is concern that the incorrect analytical approach would result in inaccurate measurement of Young\'s modulus and shear modulus of wood and woodBasic materials.
As mentioned above, in order to ensure the construction procedure of solid wood and wood, it is essential to accurately determine the Young\'s modulus and shear modulus values
High efficiency and low cost of using basic materials
Therefore, an appropriate methodology must be developed for this purpose.
In addition, the method used should be simple when considering the actual measurement of these modulus.
Establish a method for determining the elastic modulus and shear modulus of solid wood and wood
The basic materials are obtained by vibration method, and the comparative study of analytical methods is very important, which has not been carried out at all in previous studies.
In this work, free
Free bending vibration tests were performed on solid wood, medium fiber plate and plywood samples with different depth/length ratios, and Young\'s modulus and shear modulus were obtained through three different analytical methods, this is based on Tymoshenko\'s theory of vibration.
According to the theory of Timothy Schenko, Figure 1a of the bending vibration equation shows a diagram of the bending vibration test.
In 1921, Timothy presents the following bending differential equations considering shear deflection and rotational inertia :[
Mathematical expressions that cannot be reproduced in ASCII](1)where [E. sub. x]
= Young\'s modulus in X direction ,[G. sub. xy]
= Shear modulus in Xy plane, I = moment of secondary rotation, A = Cross
[Section arearho]
= The density of the beam, the middle tail 1984; Sobue 1986a;
Cui and Smith 1990;
Brancheriau and Bailleres 2002;
Bran Eucalyptol Limonene and Pinene Enteric Soft Capsules 2006;
Yoshida, Jinze 2007; Yoshihara 2009; Tonosaki et al. 2010; Khademi-Eslam et al. 2011; Sohi et al. 2011).
Nevertheless ,[E. sub. x]and [G. sub. xy]
The value is not explicitly included in equation 5 and should be calculated through an iterative process with details as described below.
Here, using the Young\'s modulus values obtained from the longitudinal vibration test, another simple method of using Equation 5 is proposed.
As shown in the previous articles (
New House 1983, Garab, etc. 2010)Wood and Wood
The base material is not the ideal positive shot material, so the Young\'s modulus obtained from the bending and longitudinal vibration methods often deviate from each other.
However, when the Young\'s modulus value obtained from the longitudinal vibration method can replace the value obtained from the bending vibration method, the shear modulus can be easily obtained through the following procedure.
At the beginning, longitudinal vibration test was carried out and [E. sub. x]
The value is obtained by replacing the basic frequency of the longitudinal vibration mode [f. sub. L]
Into the following equation :[E. sub. x]= 4[f. sup. 2. sub. L][L. sup. 2][rho](9)Then the [E. sub. x]
The value is replaced with the following equation, which is transformed from Equation 5 :[
Mathematical expressions that cannot be reproduced in ASCII](10)and [
Mathematical expressions that cannot be reproduced in ASCII](11)
Mild and Joannides (1991)
And Kubo and others. (1996, 1997)
[Calculation]G. sub. xy]
By replacing [the value corresponding to each bending vibration mode]E. sub. x]
The values obtained from the longitudinal vibration test are converted into equation 2.
However, as mentioned above, the use of equation 2 is inconvenient due to the need for a mathematical package.
By substitution [E. sub. x]
The value obtained from the longitudinal vibration test to equation 10, however ,[G. sub. xy]
The value can be determined without any program that uses any mathematical software package to solve a differential equation or an iterative process.
In previous studies on the vibration properties of medium fiber plates and plywood, finite element analysis (FEAs)
Reveals the dependence of in-
The plane shear modulus on the sample depth/length ratio (
Yoshiwara 2011 2012a).
Therefore, it is expected that the effect of the analytical method on the measurement of Young\'s modulus and shear modulus can be revealed through finite element analysis. [
Figure 2:Three-dimensional (3D)
Finite element analysis is performed using the finite element program ANSYS 12 independently of the bending vibration test. 0.
Figure 2 shows the finite element mesh of the sample.
In the analysis, solid spruce, medium fiber plate and five-Willow Wood (Shorea sp. )
The model is simulated.
The length of this model is L 300mm.
Depth H changes between 10-60mmmm increments.
The value of width T is 9mm.
Model by eight-
Node block element.
In solid spruce and medium fiber plate models, the finite element mesh is evenly divided and the size of one unit is 5,0.
The length, width and depth directions are 75 and H/20mm, respectively.
By contrast, the elements in the five elements
The dimensions of the Ply Lauan wood model in the direction of length and depth are 5mm and H/20mm, respectively.
However, in the width direction, the component length in the surface and core veneer is 0.
6mm, while in the veneer adjacent to the surface veneer, 0. 675 mm.
Table 1 shows the hypothetical elastic constants of solid Sitka spruce (
, Medium fiber board and Laian veneer.
By performing a bending vibration test, the elastic constant values of spruce and medium fiber plates were obtained from several previous studies (
Ji Yuan 2011, Ji Yuan and Nakano 2011).
In contrast, it is difficult to measure the elastic constants of the Laian veneer composed of the actual plywood.
Therefore, the elastic constants of Lauan veneer in Table 1 are taken from the timber industry Manual (
Forestry and Forestry Research Institute 2004).
The density of spruce, medium fiber board and Laian Plywood models was 380, 650 and 500 kg/m3, respectively.
In the analysis of Lawan plywood, the following two models were simulated with wood grain :(1)
The length direction of the surface veneer is equal to the longitudinal direction, and (2)
The length direction of the surface is equal to the tangent direction.
They are defined as L-and T-
Type the model separately.
According to the classical lamination theory, the Young\'s modulus along the length direction is derived as follows :[
Mathematical expressions that cannot be reproduced in ASCII](12)where [E. sub. L]
Young\'s modulus of = L-
Type model of length direction ,[E. sub. T]
Young\'s modulus of = T-
Type model of length direction ,[T. sub. a]
= Total thickness of core and surface stickers, and [T. sub. b]
= Total thickness close to core veneer. Therefore, [E. sub. L]and [E. sub. T]should be 5. 47 and 7. 94 GPa.
By contrast, the value of In-
The plane shear modulus should be 0.
Both models have 48 GPa.
Modal analysis was carried out, and the resonance frequency of the first to fourth bending vibration modes and the resonance frequency of the first longitudinal vibration mode, the value of Young\'s modulus and shear modulus were extracted ,[E. sub. x]and [G. sub. xy]
Determined separately from the following three procedures. (1)The [E. sub. x]and [G. sub. xy]
The term is calculated using Mathematica 6 from the solution of equation 5. The [G. sub. xy]
Calculate the value corresponding to each vibration mode by changing the value [E. sub. x]
, And coefficient of variation (COV)of the [G. sub. xy]
The value is determined. The [E. sub. x]
The value of the minimum cv that generates [G. sub. xy]
Mean value and [G. sub. xy]
It can be considered the most feasible, as described above (
Yoshiwara 2011 2012a 2012b). (2)The [E. sub. x]and [G. sub. xy]
The term is calculated from the iteration using the resonance of Equation 8 and the first to fourth modes.
Initially, the virtual value [G. sub. xy]
Replaced with the refined value of Y, [of Equation 8]G. sub. xy]
Since replacing sq/p with Y again (Hearmon 1958).
The iteration process is done using the functions contained in Microsoft Excel version 14. 1. 4.
After all the values in the formula change less than 0, the process stops.
Iteration between 001. (3)The [E. sub. x]
Calculate the value by replacing the basic frequency of the longitudinal vibration mode [f. sub. L]
Enter equation 9Then the [G. sub. xy]
By replacing [, the values corresponding to each vibration mode are calculatedE. sub. x]
Obtained from the longitudinal vibration test and the resonance frequency corresponding to each bending vibration mode [f. sub. n]
The average value of the equation 10, and [G. sub. xy]was obtained. The [E. sub. x]
Compare the values predicted by the above three different programs with the input values in the FEA program, the former being 10. 8, 3. 0, 5. 47, and 7.
94 GPa of solid spruce, medium fiber board, L-
Type T plywood
Plywood models, respectively.
Again ,[G. sub. xy]
Compare the values obtained by the three programs with the input modulus: 0. 65, 1. 18, and 0.
The GPa for solid spruce, medium fiber board and two Plywood models is 48 GPa, respectively.
Experimental verification material Sitka spruce (
Wood, medium fiber board and with five-
In this study, the laying structure was used as a sample. The densities ([+ or -]
Standard deviation)
12% the moisture content is 375 [+ or -]5,657 [+ or -]5, and 496 [+ or -]3 kg/[m. sup. 3], respectively.
Sitka spruce samples contain 3 to 4 rings per 10mm kilometers in the radial direction;
These rings are flat enough that their curvature can be ignored.
This wood has no defects such as knots or texture distortions, so samples cut from the wood can be treated as \"small and clear \".
\"The medium fiber board and plywood were manufactured by Ueno Mokuzai Kogyo Co. (Himeji, Japan)
The initial length, width and thickness were 1,820, 910 and 9mm, respectively.
Medium fiber board with Cork (
Typical fiber length from 2 to 4mm)and urea-
Formaldehyde Resin.
The thickness of the table and core veneer is 1.
2mm, and the thickness of the veneer adjacent to the surface veneer is 2. 7 mm.
Spruce wood samples were cut to 1,000, 140 and 100mm in the longitudinal, tangent and radial directions, respectively, while medium fiber plates and plywood samples were cut to 450, 450 and, the length, width and thickness directions are 9mm and respectively.
Samples are stored in a room at a constant temperature of 20 [degrees]
C and 65% relative humidity before the test to achieve a balanced water content of about 12%.
These samples were cut into samples with initial length, depth and width dimensions of 300, 60 and 9mm, respectively.
For spruce samples, these directions are consistent with longitudinal, tangent, and radial directions, respectively.
For the medium fiber plate sample, the length direction is the same as the length direction of the plate.
Plywood specimens whose length direction coincides with the longitudinal direction of the surface veneer are defined as L-
Type samples, while samples whose length direction is consistent with the tangent direction of the surface veneer are defined as T-type specimen.
These definitions are similar to those of FEA.
10 samples were tested under each test condition.
After the bending and longitudinal vibration tests described below were performed, the depth of the sample was reduced, followed by a series of vibration tests using the sample with the depth reduction. The depth (H)
At intervals of 10mm, the number of samples decreased from 60 to 10mm. [
Figure 3 slightly]
Bending and longitudinal vibration test in bending vibration test, the sample is free-
Free resonance vibration mode]f. sub. n]
Excited in depth with a hammer (Fig. l a).
The suspension point is the outermost position of each vibration mode.
In the bending vibration test, it is difficult to measure the resonance frequency above the fifth mode due to the small amplitudeto fourth-
The mode resonance frequency is measured.
In the longitudinal vibration test, the sample is supported by a soft foam at a medium length and is excited along the length direction with a hammer (Fig. lb).
The basic frequency of the longitudinal vibration mode, defined [f. sub. L], was measured.
The resonance frequency was analyzed using a fast Fourier transform analysis program.
In the finite element analysis ,[E. sub. x]and [G. sub. xy]
, Calculate values using three programs with details as described above.
However, when analyzing the mid-fiber plate samples with depth of 10 and 20mm according to equation 5, only the data obtained from the first and second modes and the first mode, respectively, are not used because of the value [G. sub. xy]
Obtained under these conditions is much smaller than obtained from the third and fourth modes, even if [E. sub. x]
Change in value (Yoshihara 2011).
In addition to these special circumstances,G. sub. xy]
The value is obtained from the resonance frequency of the first to fourth modes. [
Figure 4 slightly]
The values of [E. sub. x]and [G. sub. xy]
The data obtained from the three programs are compared and the validity of each method is tested.
Results and discussion the Young\'s modulus and shear modulus obtained by fitting the three analytical methods with the results predicted by FEA figure 3 show the effect of the analytical method on the relationship between Young\'s modulus [E. sub. x]
Obtained by fitting the results predicted by finite element analysis and depth/length ratio H/L
For spruce and medium fiber plate models ,[E. sub. x]
The values obtained from the longitudinal vibration method using Equation 9 increase with the decrease of the H/L ratio, whereas for these two Plywood models the opposite trend of these values is observed.
In contrast, feature trends were not found in the results obtained from the bending vibration method using equations 2 and 8.
However, the change of Young\'s modulus relative to the H/L ratio is less than 5%, which is not more significant than the change of shear modulus, and the details are described below.
Figure 4 shows the effect of the analytical method on the [shear modulus relationship]G. sub. xy]
Fitting the results of finite element analysis and prediction of H/L ratio.
And 【E. sub. x]value, the [G. sub. xy]
Other values, in addition to the spruce model, were significantly dependent on the H/L ratio.
As shown in the previous articles (
Yoshiwara 2011 2012a), the [G. sub. xy]
With the decrease of H/L ratio, the value of the medium fiber plate model tends to increase, while the values of both Plywood models show the opposite trend, with only a few exceptions.
However, it is difficult to identify any significant differences [G. sub. xy]-
H/L relationship obtained by three analytical methods. [
Figure 5 Slightly]
The Young\'s modulus and shear modulus obtained by the above actual vibration test, the finite element analysis results show that the analytical methods examined in this study have no significant effect on the measured Young\'s modulus and shear modulus values.
However, the actual vibration test results are often different from those predicted by the finite element analysis.
Figure 5 shows the value of Young\'s modulus [E. sub. x]
Cho was obtained from the actual vibration test.
Effect of H/L ratio on spruce specimens [j]E. sub. x]
Value is not particularly important.
In addition, there are also small differences [E. sub. x]
Values obtained from bending vibration analysis using equations 2 and 8, longitudinal vibration analysis using Equation 9.
However, for samples of medium fiber plates and plywood ,[E. sub. x]
Values obtained using Equation 9 are generally greater than those obtained using equation 2 and 8.
 Statistical analysis of differenceE. sub. x]
Sample value display of different analysis methods ,[E. sub. x]
When the H/L ratio is less than 0, the value of the medium fiber plate sample obtained using Equation 9 is significantly greater than the value obtained using equation 2 and 8. 166 (
Corresponding depth 50mm)
Because the probability value (P value)
Obtained by comparison with the corresponding [E. sub. x]
Significance level with value less than 0. 01. For the L-and T-
Type plywood samples, statistical analysis shows that the difference is significant when the H/L ratio is less than 0. 133 (
Corresponding depth 40mm)for the L-
The type plywood samples were significantly different when the H/L ratio was 0. 133 and 0. 1 (
The corresponding depth is 40 and 30mm, respectively)for the T-
Plywood model.
Figure 6 shows the shear modulus value [G. sub. xy]
Cho was obtained from the actual vibration test.
Under the action of the bending load, the deflection caused by the shear force is significantly less than that caused by the bending moment.
Therefore, measurement errors are often included in the shear modulus calculated by extracting shear deflection, which is more significant than the measurement error of Young\'s modulus.
This trend is more evident as the depth/length ratio decreases, so the kv value of the shear modulus is greater than the kv value of the Young\'s modulus.
As mentioned above ,[G. sub. xy]
Values derived using equations 2 and 10 are calculated by averaging the values corresponding to each vibration mode.
Similar to the trend observed by elastic modulus, the effect of H/L ratio and analytical methods on [G. sub. xy]
The value of spruce specimens is not particularly significant.
For specimens of medium fiber plates ,【G. sub. xy]
With the decrease of H/L ratio, the value tends to increase.
Nevertheless, the negative value [G. sub. xy]
When the H/L ratio is 0, it is obtained using Equation 8. 033 (
Corresponding depth 10mm).
Negative value []G. sub. xy]
It is because in this case the contribution of the shear deflection to the bending deflection is small (Yoshihara 2011).
By contrast ,[G. sub. xy]
With the decrease of H/L ratio, the value of plywood samples tends to decrease.
[These reduction and increase trends]G. sub. xy]
Values obtained using equations 2 and 8 are similar to those obtained from experiments and numerical analyses conducted in previous studies (
Yoshiwara 2011 2012a).
In the analysis using equations 2 and 8, the H/L ratio of the difference [G. sub. xy]
The values of samples with different H/L ratios did not drop significantly to the range of 0. 133 to 0. 2, 0. 067 to 0. 2, and 0. 1 to 0. 2 (
The corresponding depth is 40 to 60, 20 to 60, 30 to 60mm, respectively)for the MDF, L-
Type T plywood
Plywood model respectively.
Within these H/L ratios, there was no significant difference between the gay values obtained using equation 2 and 8.
Therefore, the shear modulus can be effectively measured by bending vibration test, while reducing the influence of the sample geometry in the H/L range
By contrast ,[G. sub. xy]
When comparing values of the same H/L ratio, the values obtained using equation 10 are generally less than those obtained using equation 2 and 8.
Statistical analysis shows that the effective range of the H/L ratio required to obtain [G. sub. xy]
At the same time, the value of reducing the influence of sample geometry is 0. 2, 0. 167 to 0. 2, and 0. 133 to 0. 2 (
Corresponding to the depth of 60, 50 to 60, 40 to 60mm, respectively)for the MDF, L-
Type T plywood
Plywood model respectively.
These ranges are significantly smaller than those obtained using the above equations 2 and 8. [
Figure 6 slightly]In the low-
The order of bending, relative to the bending deformation, the contribution of the shear deformation is relatively small.
In addition, the frequency of low level
The order mode of bending is often affected by irregularities such as the internal non-uniformity of the sample (
Therefore, the possible inaccurate calculation of shear modulus is \"low\"
Order mode frequency, especially when using equation 10 combined with bending and longitudinal vibration analysis.
In order to reduce this problem, [G. sub. xy]
The values corresponding to each bending vibration mode are evaluated without averaging.
Figure 7 shows the relationship [G. sub. xy]
The values corresponding to each bending vibration mode and H/L ratio.
[Dependence]G. sub. xy]
On H/L, the first and second modes of vibration are more obvious than the third and fourth modes of vibration, where [G. sub. xy]
These values fit well with each other.
When using the third and fourth modes, obtain a valid H/L range [G. sub. xy]
The decrease value of H/L ratio is 0. 133 to 0. 2, 0. 133 to 0. 2, and 0. 1 to 0. 2 (
Corresponding to depth 40 to 60, 30 to 60mm, respectively)for the MDF, L-
Type T plywood
Plywood model respectively.
In the range of these H/L ratios ,[G. sub. xy]
The values obtained from the three analytical methods are not significant.
Therefore, it is recommended to obtain shear modulus by performing bending and longitudinal vibration tests using the third and fourth resonance modes of bending vibration.
This method does not require any mathematical software packages and iterative programs required in the above method, so it can be done very simply and with real use. [
Figure 7 Slightly]
In this study, the effects of sample structure and analytical methods on elastic modulus and shear modulus values of solid wood and wood
Basic materials were investigated.
Nevertheless, there are several other factors in measuring these values using the vibration method.
For example, support effects not considered in this study may affect the measurement of these modulus (
Brancheriau and Baillrres 2002).
In addition, the non-uniformity of several factors such as defects and density distribution may affect the measurement of these modulus (Kubojima et al.
2005. Tonghuki and others. 2010, Khademi-Eslam et al.
2011, Sohi and others. 2011).
In order to improve the accuracy of measuring Young\'s modulus and shear modulus of solid wood and wood ,-
Based on materials, further research is needed to solve the applicability of the method studied here by using various solids for vibration testing
Wood type and Wood
Basic materials under various experimental conditions.
Conclusion The samples of Sitka spruce, medium fiber plate and Lauan plywood were tested to determine their Young\'s modulus and shear modulus values.
The test adopts bending and longitudinal vibration tests, and the results obtained are analyzed by three different methods.
Finite element analysis results show that when the sample depth is properly determined, Young\'s modulus and shear modulus can be effectively obtained from the analytical methods examined in this study.
However, the experimental results show that the analytical method affects the values of Young\'s modulus and shear modulus.
The analytical method based on the strict solution of Tymoshenko differential equation obtained the values of Young\'s modulus and shear modulus (1921)
And the iterative program proposed by Hearmon (1958)
In a wide range of sample depth while reducing the effect of depth/length ratio.
Although the method used to obtain shear modulus by replacing the Young\'s modulus measured by longitudinal vibration test with the approximate equation proposed by Goens (1931)
The experimental results show that the method is simpler than the other two methods, in the use of low
The order of bending in the analysis.
However, when using the resonant frequency of the bending vibration in the third or fourth modes, there is hope to measure the Young\'s modulus and shear modulus by combining the bending and longitudinal vibration methods, at the same time, the influence method of sample depth/length ratio similar to the above is reduced.
Therefore, the last method is recommended for simplicity and practicality.
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The author is a professor of Sci.
University of Engineering
Japan, Shimen, Masu (yosihara@riko. shimane-u. ac. jp).
The document was published in April 2012. Article no. 12-00047.
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