Timoshenko自由梁迭代法计算木材弹性模量和剪切模量的适用性

doi:10.11707/j.1001-7488.20220818

摘要/Abstract

摘要：

目的: 基于试验探究Timoshenko自由梁迭代法计算木材弹性模量和剪切模量的适用性，重点考虑木材弹性模量和剪切模量的测试精度，寻求梁试件合适的长厚比。方法: 通过山毛榉、云杉-松-冷杉(SPF)和单板层积材(LVL)试件截短试验，动态测试不同长厚比试件的频谱，从频谱图上读出自由梁的第一阶和第二阶弯曲频率；应用迭代程序计算木材和木质复合材料的弹性模量和剪切模量；采用自由杆件扭转振动法和自由板扭转振型法验证自由梁迭代法计算剪切模量的有效性。结果: 存在一个与树种有关的试件长厚比下限值，当试件长厚比小于该下限值时, 自由梁迭代法不能计算出正常的弹性模量和剪切模量或根本计算不出弹性模型和剪切模量；自由梁迭代法计算出的弹性模量高于Euler自由梁法计算出的弹性模量，其程度与试件长厚比有关，仅当试件长厚比大于24时其偏高值才在4%以内；自由梁迭代法计算出的剪切模量与矩形截面因子k取值有关，当k取0.833和0.913时，分别高于自由杆件扭转振动法(自由板扭转振型法)剪切模量测试值12.2%和2.3%(19.9%和9.3%)。结论: 为保证自由梁迭代法测试精度，推荐采用长厚比20~24的梁试件动态测试其第一阶和第二阶弯曲频率，并用矩形截面因子取0.913进行迭代计算木材的弹性模量和剪切模量。

关键词: 固有频率, Timoshenko自由梁迭代法, 弹性模量, 剪切模量, 适用性, 测试精度

Abstract:

Objective: When Timoshenko free beam iteration method is used to dynamically test the elastic modulus and shear modulus of wood, its test accuracy is greatly affected by the different length thickness ratios of the specimen. Therefore, this paper was implemented to explore the applicability of calculating wood elastic modulus and shear modulus according to the iterative method of Timoshenko beam, and further focus on the test accuracy. Method: Through the truncated test of beech, SPF(spruce, pine and fir) and LVL(laminated venner lumber) specimens, the spectrums under different length thickness ratios were dynamically measured, and the first and second-order bending frequencies of free beam were read out from the spectrum. At the same time, the elastic modulus and shear modulus of wood and wood composites were calculated by iterative program. The effectiveness of the iterative method to calculate the shear modulus was verified by the free member torsional vibration method and the free plate torsional vibration mode method. Result: There was a lower limit value of length thickness ratio related to tree species. When the length thickness ratio of the specimen was less than the lower limit value, the elastic modulus and shear modulus could not be iterated by the iterative method. The elastic modulus determined by iterative method was higher than the elastic modulus test value of Euler beam. The iterative method also could give a too high calculated value of shear modulus. The elastic modulus calculated by iterative method might be higher than the elastic modulus test value based on Euler beam, and its degree is related to the length thickness ratio of the specimen. Only when the length thickness ratio of the specimen is greater than 24, the high value is less than 4%. The calculated value of shear modulus given by iterative method is related to the value of rectangular section factor k. When the rectangular section factor k is equal to 0.833 and 0.913, the calculated value of the shear modulus obtained by the iterative method is 12.2% and 2.3% (19.9% and 9.3%) higher than the measured value of the free member torsional vibration method (free plate torsional vibration mode method), respectively. Conclusion: In order to ensure the test accuracy, it is recommended to use beam specimens with length thickness ratio of 20-24 to dynamically test the first and second-order bending frequencies, and iteratively calculate the elastic modulus and shear modulus of wood with rectangular section factor of 0.913.

Key words: natural frequency, Timoshenko free beam iterative method, modulus of elasticity, shear modulus, applicability, test accuracy

中图分类号:

S781.2

王正,周宇昊,沈肇雨,何宇航. Timoshenko自由梁迭代法计算木材弹性模量和剪切模量的适用性[J]. 林业科学, 2022, 58(8): 173-181.

Zheng Wang,Yuhao Zhou,Zhaoyu Shen,Yuhang He. Applicability of Timoshenko Beam Iterative Method to Calculate Wood Elastic Modulus and Shear Modulus[J]. Scientia Silvae Sinicae, 2022, 58(8): 173-181.

图/表 18

表1

表2

图1

图2

图3

图4

图5

图6

图7

图8

图9

图10

表3

表4

图11

图12

表5

表6

参考文献 0

	程可, 王正. 基于自由板扭转振形测试剪切模量的一个新方法. 南京工业大学学报(自然科学版), 2015, 37 (5): 61- 66. doi: 10.3969/j.issn.1671-7627.2015.05.010
	Cheng K , Wang Z . New method for testing shear modulus based on torsional vibration shape of free plate. Journal of Nanjing Tech University (Natural Science Edition), 2015, 37 (5): 61- 66. doi: 10.3969/j.issn.1671-7627.2015.05.010
	董浩然, 谈永杰, 杨小军, 等. 木构件裂纹特征及碳纤维浆料修复评价. 森林与环境学报, 2022, 42 (4): 442- 448.
	Dong H R , Tan Y J , Yang X J , et al. Crack characteristics of wood members and repair evaluation of carbon fiber slurry. Journal of Forest and Environment, 2022, 42 (4): 442- 448.
	胡英成. 刨花板的动态剪切弹性模量的无损检测. 东北林业大学学报, 2001, 29 (2): 18- 20.
	Hu Y C . Nondestructive testing of the dynamic shear modulus of elasticity for particleboard. Journal of Northeast Forestry University, 2001, 29 (2): 18- 20.
	胡英成, 王逢瑚, 刘一星, 等. 利用振动法检测胶合板的抗弯弹性模量. 木材工业, 2001, 15 (2): 3- 6.
	Hu Y C , Wang F H , Liu Y X , et al. Study on modulus of elasticity in bending of plywood by vibration method. China Wood Industry, 2001, 15 (2): 3- 6.
	刘鸿文. 材料力学.2版北京: 高等教育出版社, 1983, 287
	Liu H W . Mechanics of materials.2nd editionBeijing: Higher Education Press, 1983, 287
	马功勋. 单向复合材料板弹性常数的动(静)态测定方法. 复合材料学报, 1996, 13 (2): 117- 123.
	Ma G X . Dynamic(static) measurement of the elastic constants in uniaxially-reinforced composite sheet. Material Compositae Sinica, 1996, 13 (2): 117- 123.
	阮锡根, 余观夏. 木材物理学.北京: 中国林业出版社, 2005.
	Ruan X G , Yu G X . Wood physics.Beijing: China Forestry Publishing House, 2005.
	邵蓓珠, 马功勋. 剪力对梁变形影响的分析. 南京化工学院学报, 1988, 10 (3): 78- 84.
	Shan P S , Ma G X . Analysis of effect of shear on deformation of beam. Journal of Nanjing University of Chemical Technology, 1988, 10 (3): 78- 84.
	谭守侠, 周定国. 木材工业手册.北京: 中国林业出版社, 2007: 261- 306.
	Tan S X , Zhou D G . Handbook of wood industry.Beijing: China Forestry Publishing House, 2007: 261- 306.
	王正, 李磊, 杨静, 等. 对麦秸板弹性模量和阻尼比的动、静态法的测定与研究. 林业科技, 2006, 31 (5): 48- 50.
	Wang Z , Li L , Yang J , et al. Measuration and study to elasticity model and damp ratio of the wheat straw board with dynamic and static testing method. Forestry Science & Technology, 2006, 31 (5): 48- 50.
	王正. 两种木质复合材料弹性模量与阻尼比的动态测量. 南京林业大学学报(自然科学版), 2007, 31 (3): 147- 149.
	Wang Z . Dynamic measures of elasticity model and damp ratio to HDF and OSB. Journal of Nanjing Forestry University(Natural Sciences Edition), 2007, 31 (3): 147- 149.
	王正, 饶鑫, 杨小军, 等. 轻型木结构规格材抗弯弹模的两种无损法检测与评级. 林产工业, 2013, 40 (5): 30- 33.
	Wang Z , Rao X , Yang X J , et al. Non-destructive testing and ratings of MOE of SPF stock lumbers by the frequency method and sound velocity method. China Forest Products Industry, 2013, 40 (5): 30- 33.
	王正, 高子震, 顾玲玲, 等. 测试木材剪切模量的自由板扭转振形法. 林业科学, 2014, 50 (11): 122- 128.
	Wang Z , Gao Z Z , Gu L L , et al. Torsional vibration shape method of free plate for testing shear modulus of lumber. Scientia Silvae Sinicae, 2014, 50 (11): 122- 128.
	周海宾, 任海青, 费本华, 等. 木质复合板弯曲、剪切弹性模量动态测试. 建筑材料学报, 2007a, 10 (5): 561- 565.
	Zhou H B , Ren H Q , Fei B H , et al. Dynamical test on flexural and shear modulus of composite wood panels. Journal of Building Materials, 2007a, 10 (5): 561- 565.
	周海宾, 任海青, 殷亚芳, 等. 横向振动评估木结构建筑用规格材弹性性质. 建筑材料学报, 2007b, 10 (3): 271- 275.
	Zhou H B , Ren H Q , Yin Y F , et al. Evaluating static elastic properties of wood structure building dimension lumber usingtransverse vibration. Journal of Building Materials, 2007b, 10 (3): 271- 275.
	社团法人日本コンクリート工学协会(JCI). 日本工业规格JIS A1127-2001《共鸣振动によるコンクリートの动弹性系数、动せん断弹性系数及び动ポアソソ比试验方法》. 2001.
	Chui Y H , Smith I . Influence of rotatory ineria, shear deformation and support condition on natural frequencies of wooden beams. Wood Science and Technology, 1990, 24, 233- 245.
	Chui Y H . Simultaneous evaluation of bending and shear moduli of wood and the influence of knots on these parameters. Wood Science and Technology, 1991, 25, 125- 134.
	Ilic J . Dynamic MOE of 55 species using small wood beams. Holz als Roh-und Werkstoff, 2003, 61 (3): 167- 172.
	Ross R J, Pellerin R F. 1994. Nondestructive testing for assessing wood members in structures: a review. USDA Forest Service Forest Products Laboratory, General Tech. Rep. FPL-GTR-70.
	Timoshenko S . Mechanical vibrational science. Mechanical Industry Press, 1965: 316- 333.
	Turk C, Hunt J F, Marr D J. 2008. Cantilever-beam dynamic modulus for wood composite products. Part 1, apparatus. Research Note FPL-RN-0308, Madison, WI: U. S. Department of Agriculture, Forest Service, Forest Products Laboratory, 5.
	Wang Z , Li L , Gong M . Dynamic modulus of elasticity and damping ratio of wood-based composites using a cantileverbeam vibration technique. Construction & Building Materials, 2012, 28 (1): 831- 834.
	Wang Z , Wang Y L , Cao Y , et al. Measurement of shear modulus of materials based on the torsional mode of cantilever plate. Construction and Building Materials, 2016a., 124, 1059- 1071.
	Wang Z , Wang G G , Wang Y L , et al. Torsional vibration method for free board determining the shear modulus of wood. Journal of Forestry Engineering, 2016b, 1 (4): 10- 17.
	Wang Z , Xie W B , Wang Z H , et al. Strain method for synchronous dynamic measurement of elastic, shear modulus and Poisson's ratio of wood and wood composites. Construction & Building Materials, 2018, 182, 608- 617.
	Wang Z , Wang Y L , Cao Y , et al. Measurements of the shear modulus of materials by the free-plate torsional mode shape method. Journal of Testing and Evaluation, 2019a, 47 (2): 1163- 1181.
	Wang Z , Wang Y L , Cao Y , et al. Measurements of the shear modulus of materials by the free-plate torsional mode shape method. Journal of Testing and Evaluation, 2019b, 47 (2): 1163- 1181.

试件 Specimen	长 Length/mm	宽 Width/mm	厚 Thickness/mm	数量 Quantity	密度 Density/(kg·m^-3)	备注 Remark
山毛榉 Fagus sylvatica	600	28	24	8	708	径向 Radial
山毛榉 Fagus sylvatica	600	24	28	8	708	弦向 Tangential
SPF	578	26	24	6	458	径向 Radial
SPF	578	24	26	6	458	弦向 Tangential
LVL	1 002	46	46	4	573	顺纹 Grain
LVL	960	30	30	4	591	顺纹 Grain
山毛榉 Fagus sylvatica	768	24	24	4	720	弦向、径向 Tangential，radial
SPF	768	24	24	2	481	弦向、径向 Tangential，radial
LVL	760	45	45	4	591	横纹 Horizontal
西加云杉 Picea sitchenrsis	300	30	20	3	369	横向 Transverse

试件 Specimen	长 Length/mm	宽 Width/mm	厚 Thickness/mm	数量 Quantity	密度 Density/(kg·m^-3)	备注 Remark
山毛榉 Fagus sylvatica	552	92	10	4	665	弦向 Tangential
SPF	486	89	10	3	439	弦向 Tangential
LVL	960	240	30	3	580	顺纹 Grain

试件 Specimen	梁长厚比 Length-to-thickness ratio of beam(l/h)
试件 Specimen	32	28	24	20	16	14	12	10	8
SPF	1.030	1.029	1.030	1.041	1.080	1.108	1.139	1.225	2.758
山毛榉 Fagus sylvatica	1.024	1.029	1.039	1.054	1.086	1.679	2.554	－	－
LVL	1.020	1.028	1.037	1.050	1.079	1.103	1.140	2.969	－

试件 Specimen	取向 Direction	梁长厚比 Length-to-thickness ratio of beam(l/h)	弹性模量 Elastic modulus/GPa	剪切模量 Shear modulus/MPa
试件 Specimen	取向 Direction	梁长厚比 Length-to-thickness ratio of beam(l/h)	弹性模量 Elastic modulus/GPa	k = 0.833	k = 0.913
山毛榉 Fagus sylvatica	弦向 Tangential	32~16	16.1(0.4%)	1 114(7.1%)	1 017 (7.1%)
山毛榉 Fagus sylvatica	径向 Radial	32~16	16.4(0.7%)	1 417(4.6%)	1 293 (4.6%)
SPF	弦向 Tangential	32~10	13.5(5.2%)	690(9.9%)	628(9.7%)
SPF	径向 Radial	32~10	13.6(3.5%)	778(12.0%)	719(12.5%)
LVL顺纹 LVL grain	垂直胶层 Perpendicular to adhesive layer	28~12	12.7(0.7%)	957(9.3%)	873(9.3%)
LVL顺纹 LVL grain	平行胶层 Parallel to adhesive layer	28-12	12.3(1.0%)	928(4.5%)	847(9.3%)

试件 Specimen	试件数 Number of specimens	板长宽比 Aspect ratio of plate	取向 Direction	剪切模量 Shear modulus/MPa
试件 Specimen	试件数 Number of specimens	板长宽比 Aspect ratio of plate	取向 Direction	自由板扭转振型法 Free plate torsional mode method	自由杆件扭转振动法 Free member torsional vibration method
山毛榉 Fagus sylvatica	4	3.0~6.0	弦向 Tangential	932(3.9%)	994(3.0%)
SPF	3	3.0~5.5	弦向 Tangential	580(7.2%)	618(5.8%)
LVL	3	4	顺纹 Grain	790(0.1%)	848(0.3%)