采用圆弧分段法对大跨度梁挠度的研究
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摘要
针对挠曲线近似微分方程在计算大跨度梁挠度精度不足的问题,采用近似挠曲线方程和圆弧挠曲线方程分析纯弯曲悬臂梁大跨度的挠度,指出用近似微分方程计算大跨度梁的挠度会出现极大误差。以圆弧挠曲线方程为基础,发明圆弧分段法分析大跨度悬臂梁的挠度,并与挠曲线精确微分方程进行比较,发现采用圆弧分段法计算大跨度悬臂梁的挠度具有收敛快速和精度高的优点,该方法将对大跨度钢构零部件、大跨度桥梁挠曲变形、细长机械零部件的受力变形问题的研究有一定应用价值。
In order to solve the problem that the deflection accuracy of large-span beam is insufficient by using approximate deflection curve equation and circular deflection curve equation,the large-span deflection of pure bending cantilever beam is analyzed.It is pointed out that there will be great error when using approximate differential equation to calculate the deflection of large-span beam.Based on the arc deflection curve equation,the arc sectional method is invented to analyze the deflection of large-span cantilever beams.Compared with the exact differential equation of deflection curve,it is found that the arc sectional method has the advantages of fast convergence and high precision in calculating the deflection of large-span cantilever beams.This method will be applied to large-span steel structure components and large-span bridges.It is of certain application value to study the deflection of beams and the deformation of slender mechanical parts.
In order to solve the problem that the deflection accuracy of large-span beam is insufficient by using approximate deflection curve equation and circular deflection curve equation,the large-span deflection of pure bending cantilever beam is analyzed.It is pointed out that there will be great error when using approximate differential equation to calculate the deflection of large-span beam.Based on the arc deflection curve equation,the arc sectional method is invented to analyze the deflection of large-span cantilever beams.Compared with the exact differential equation of deflection curve,it is found that the arc sectional method has the advantages of fast convergence and high precision in calculating the deflection of large-span cantilever beams.This method will be applied to large-span steel structure components and large-span bridges.It is of certain application value to study the deflection of beams and the deformation of slender mechanical parts.
引文
[1]刘鸿文.材料力学(第五版)[M].北京:高等教育出版社,2010.
[2] James M G,Stephen P.Timoshenko.Mechanics of Materials(2nd Ed)[M].Pacific Grove,CA:Brooks/Cole,1984.
[3] Johnson R P,May I M.Partial-interaction design of composite beams[J].Structure Engineering,1975,8(53):309-311.
[4]蒋秀根,剧锦三,傅向荣.考虑滑移效应的钢-混凝土组合梁弹性应力计算[J].工程力学,2007,24(1):143-146.
[5]丁敏,蒋秀根,孟石平,等.整体-局部弯曲模型及其在简支组合梁中的应用[J].工程力学,2012,29(12):233-240.
[6] Biot M A.Bending of an infinite beam on an elastic foundation[J]. ZeitschriftfiirAngewandte Maihematik und Mechanik,1922,2(3):165-184.
[7] Thurston G A.A numerical solution of the nonlinear equations for axisymmetric bending of shallow spherical shells[J].Journal of Applied Mechanics,1961,28(4):557-562.
[8] Lee S L,Manuel F S,Rossow E C.Large Deflections and Stability of Elastic Frame[J].Journal of the Engineering Mechanics Division,1968,94(2):521-548.
[9] Huang T C.The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions[J].Journal of Applied Mechanics,1961,28(4):579-584.
[10] Oden J T,Childs S B.Finite deflections of a nonlinearly elastic bar[J].Journal of Applied Mechanics,1970,37(1):48-52.
[11] Prathap G,Varadan T K.The inelastic large deformation of beams[J].Journal of Applied Mechanics,1976,43(4):689-690.
[12]曹天捷,杜蓬娟.一次超静定理想弹塑性梁的全过程分析[J].工程力学,1999,16(3):105-112.
[2] James M G,Stephen P.Timoshenko.Mechanics of Materials(2nd Ed)[M].Pacific Grove,CA:Brooks/Cole,1984.
[3] Johnson R P,May I M.Partial-interaction design of composite beams[J].Structure Engineering,1975,8(53):309-311.
[4]蒋秀根,剧锦三,傅向荣.考虑滑移效应的钢-混凝土组合梁弹性应力计算[J].工程力学,2007,24(1):143-146.
[5]丁敏,蒋秀根,孟石平,等.整体-局部弯曲模型及其在简支组合梁中的应用[J].工程力学,2012,29(12):233-240.
[6] Biot M A.Bending of an infinite beam on an elastic foundation[J]. ZeitschriftfiirAngewandte Maihematik und Mechanik,1922,2(3):165-184.
[7] Thurston G A.A numerical solution of the nonlinear equations for axisymmetric bending of shallow spherical shells[J].Journal of Applied Mechanics,1961,28(4):557-562.
[8] Lee S L,Manuel F S,Rossow E C.Large Deflections and Stability of Elastic Frame[J].Journal of the Engineering Mechanics Division,1968,94(2):521-548.
[9] Huang T C.The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions[J].Journal of Applied Mechanics,1961,28(4):579-584.
[10] Oden J T,Childs S B.Finite deflections of a nonlinearly elastic bar[J].Journal of Applied Mechanics,1970,37(1):48-52.
[11] Prathap G,Varadan T K.The inelastic large deformation of beams[J].Journal of Applied Mechanics,1976,43(4):689-690.
[12]曹天捷,杜蓬娟.一次超静定理想弹塑性梁的全过程分析[J].工程力学,1999,16(3):105-112.