用切比雪夫多项式研究短梁的弯曲计算
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摘要
考虑剪切效应,利用切比雪夫多项式构造严格满足表面切应力边界条件的轴向位移表达式,建立了短梁弯曲问题的新理论.利用奇异函数把作用在短梁上的复杂外载荷表示为分布载荷,推导出了短梁弯曲时的截面正应力公式及挠曲线表达式.把采用切比雪夫多项式推导出短梁的弯曲计算公式计算结果与弹性理论计算结果进行比较,可知该方法的计算精度较高.研究结果表明:在复杂外载荷作用下,当长高比小于等于6时,剪切变形对梁的弯曲挠度影响较大,而当长高比小于3时,剪切变形对梁的弯曲应力影响较大;因此建议采用切比雪夫多项式方法给出的挠度表达式、弯曲应力进行计算,因为切比雪夫多项式方法不但给出了复杂外载荷作用下梁截面挠度、弯曲应力的计算通式,而且该方法具有计算过程简便、精度高的优点.
Considering the shear effect, Chebyshev polynomial is used to construct the axial displacement expression which strictly meets the boundary condition of the surface shear stress, and new theory about short beam bending problem is established. Singular function is utilized to transform the complex external load into the distributed load on the short beam, and to deduce the normal stress formula of the section and the flexural expression when the short beam is bending. The bending calculation formula based on Chebyshev polynomial is taken to calculate the bending problem of the short beam. Compared to the solutions of elastic theory, the calculation accuracy of this method is higher. The research results show that under the complex external load, there is a greater influence of the shear deformation on the beam bending deflection when the length-to-height ratio is less than or equal to 6, and on the beam bending stress when the length-to-height ratio is less than 3. It is recommended to use the proposed deflection expression and the bending stress by Chebyshev polynomial for short beam. This is because the method based on Chebyshev polynomial not only gives the general section deflection and bending stress of the beam under the complex external load, but also has the advantages of simplified computational process and high precision.
Considering the shear effect, Chebyshev polynomial is used to construct the axial displacement expression which strictly meets the boundary condition of the surface shear stress, and new theory about short beam bending problem is established. Singular function is utilized to transform the complex external load into the distributed load on the short beam, and to deduce the normal stress formula of the section and the flexural expression when the short beam is bending. The bending calculation formula based on Chebyshev polynomial is taken to calculate the bending problem of the short beam. Compared to the solutions of elastic theory, the calculation accuracy of this method is higher. The research results show that under the complex external load, there is a greater influence of the shear deformation on the beam bending deflection when the length-to-height ratio is less than or equal to 6, and on the beam bending stress when the length-to-height ratio is less than 3. It is recommended to use the proposed deflection expression and the bending stress by Chebyshev polynomial for short beam. This is because the method based on Chebyshev polynomial not only gives the general section deflection and bending stress of the beam under the complex external load, but also has the advantages of simplified computational process and high precision.
引文
[1]舒小娟,钟新谷,沈明燕,等.基于剪切变形的矩形梁剪力滞求解方法[J].计算力学学报,2015,32(4):518-522.
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[8]吕红明.边界条件对短梁结构有限元分析影响[J].工程设计学报,2013,20(4):321-325.
[9]杨伯源,巫绪涛,李和平.剪切弯曲下短深梁位移数值计算精度的研究[J].应用力学学报,2003,20(2):145-146.
[10]周旺保,蒋丽保,戚菁菁.考虑剪力滞和剪切变形的薄壁箱梁自振特性分析[J].计算力学学报,2013,30(6):802-806.
[11]罗祖道,李思简.各向异性材料力学[M].上海:上海交通大学出版社,1994,458-459.
[12]吴家龙.弹性力学[M].北京:高等教育出版社,2001,117-129.
[13]刘亮,胡玉茹,张元海.用材料力学公式计算任意截面短梁正应力时的修正项研究[J].计算力学学报,2015,32(3):383-387.
[14]孟宪红,邢依琳,刘双行,等.求解梁的切应力的高阶勒让德模型[J].力学与实践,2015,37(5):626-629.
[2]刘润泉,石勇,朱锡,等.夹层复合材料的弯曲理论分析与计算方法研究[J].玻璃钢/复合材料,2006,6:6-9.
[3]胡海昌.弹性力学的变分原理及其应用[M].北京:科学出版社,1981,140-144.
[4]蒋寿文,马建国.广义梁函数在短梁问题中的应用[J].工程力学,1991,8(1):73-82.
[5]张树祥.短梁计算[J].力学与实践,1985,7(3):22-25.
[6]戴瑛,嵇醒.两端固定受均布载荷的短梁的平面应力解[J].同济大学学报:自然科学报,2008,36(7):890-893.
[7]王正中,朱军祚,谌磊,等.集中力作用下深梁弯剪耦合变形应力计算方法[J].工程力学,2008,25(4):115-120.
[8]吕红明.边界条件对短梁结构有限元分析影响[J].工程设计学报,2013,20(4):321-325.
[9]杨伯源,巫绪涛,李和平.剪切弯曲下短深梁位移数值计算精度的研究[J].应用力学学报,2003,20(2):145-146.
[10]周旺保,蒋丽保,戚菁菁.考虑剪力滞和剪切变形的薄壁箱梁自振特性分析[J].计算力学学报,2013,30(6):802-806.
[11]罗祖道,李思简.各向异性材料力学[M].上海:上海交通大学出版社,1994,458-459.
[12]吴家龙.弹性力学[M].北京:高等教育出版社,2001,117-129.
[13]刘亮,胡玉茹,张元海.用材料力学公式计算任意截面短梁正应力时的修正项研究[J].计算力学学报,2015,32(3):383-387.
[14]孟宪红,邢依琳,刘双行,等.求解梁的切应力的高阶勒让德模型[J].力学与实践,2015,37(5):626-629.