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针对薄板弯曲大变形问题, 运用变分原理, 建立了薄板弯曲大变形问题的高阶非线性偏微分方程. 运用有限差分法和动态设计变量优化算法原理, 以离散坐标点的上未知挠度为设计变量, 以离散坐标点的差分方程组构建目标函数, 提出了薄板弯曲大变形挠度求解的动态设计变量优化算法, 编制了相应的优化求解程序. 分析了具有固定边界、均布载荷下的矩形薄板挠度的典型算例. 通过与有限元的结果对比, 表明了本文求解算法的有效性和精确性, 提供了直接求解实际工程问题的基础.
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关键词:
- 高阶非线性偏微分方程 /
- 薄板弯曲大变形 /
- 动态设计变量优化算法 /
- 程序设计
For a thin plate bending large deformation problem, variational principle is applied, and higher-order nonlinear partial differential equations about thin plate bending large deformation is established. Based on difference method and dynamic design variable optimization method, making unknown deflection of discrete coordinate points as design variables, differential equations sets of the discrete coordinates points as building objective function, a dynamic design variable optimization algorithm for computing thin plate bending deflection is proposed. Universal computing program is designed. Practical example about rectangular thin plate with fixed boundary under uniform load is analyzed. Comparing the program computing result with finite element solution. Effectiveness and feasibility of the method are verified. This method can be used to solve engineering problem.-
Keywords:
- higher-order nonlinear partial differential equations /
- thin plate-bending large deformation /
- dynamic design variables optimization method /
- program design
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[2] Alzheimer W E, Davis R T 1968 J. Eng. Mech. Div. Proc. ASCE 4 905
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[4] Huang Z Y 1957 Acta Phys. Sin. 13 312 (in Chinese) [黄泽言 1957 13 312]
[5] Qian W C, Ye K Y 1954 Acta Phys. Sin. 10 209 (in Chinese) [钱伟长, 叶开沅 1954 10 209]
[6] Hu H C 1955 Acta Phys. Sin. 11 19 (in Chinese) [胡海昌 1955 11 19]
[7] Xie Y X, Tang J S 2004 Acta Phys. Sin. 53 2828 (in Chinese) [谢元喜, 唐驾时 2004 53 2828]
[8] Ma W X, Gu X, Gao L 2009 Adv. Appl. Math. Mech. 1 573
[9] Yang B, Ding H J, Chen W Q 2008 Appl. Math. Mech. 29 9994
[10] Huang J J 2009 Chin. Phys. 18 B 3616
[11] Eerdun Buhe, Temuer Chaolu 2012 Chin. Phys. B 21 035201
[12] Xu H, Chen L H, Mo J Q 2010 Acta Phys. Sin. 60 100201 (in Chinese) [徐惠, 陈丽华, 莫嘉琪 2011 60 100201]
[13] Chen L J, Ma C F 2010 Chin. Phys. B 19 010504
[14] Polyanin A D, Zaitsev V F 2004 Handbook of Nonlinear Partial Differential Equations (Boca Raton, London, New York: Chapman & Hall/CRC) pp1-2
[15] Hou X L, Qian Y, Wu H T 2010 Acta Math. Eng. 27 663 (in Chinese) [侯祥林, 钱颖, 吴海涛 2010 工程数学学报 27 663]
[16] Hou X L, Liu T L, Zhai Z H 2011 Acta Phys. Sin. 60 090202 (in Chinese) [侯祥林, 刘铁林, 翟仲海 2011 60 090202]
[17] Hou X L, Zhai Z H, Zheng L, Liu T L 2012 Acta Phys. Sin. 61 010201 (in Chinese) [侯祥林, 翟中海, 郑莉, 刘铁林 2012 61 010201]
[18] Lao D Z 2007 Variational Method Basis (2nd Ed.) (National Defence Industry Press) pp74-91 (in Chinese) [老大中 2007 变分法基础(第2版) (北京: 国防工业出版社) 第74-91页]
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[1] Timoshenk S, Woinowsky-Krieger S 1959 Theory of Plates and Shells (Now York: McGraw-Hill)
[2] Alzheimer W E, Davis R T 1968 J. Eng. Mech. Div. Proc. ASCE 4 905
[3] Xu Z L 2006 Elasticity (4th Ed.) (Beijing: Higher Education Press) pp149-150 (in Chinese) [徐芝纶 2006 弹性力学(第4版) (北京: 高等教育出版社) 第149-150页)]
[4] Huang Z Y 1957 Acta Phys. Sin. 13 312 (in Chinese) [黄泽言 1957 13 312]
[5] Qian W C, Ye K Y 1954 Acta Phys. Sin. 10 209 (in Chinese) [钱伟长, 叶开沅 1954 10 209]
[6] Hu H C 1955 Acta Phys. Sin. 11 19 (in Chinese) [胡海昌 1955 11 19]
[7] Xie Y X, Tang J S 2004 Acta Phys. Sin. 53 2828 (in Chinese) [谢元喜, 唐驾时 2004 53 2828]
[8] Ma W X, Gu X, Gao L 2009 Adv. Appl. Math. Mech. 1 573
[9] Yang B, Ding H J, Chen W Q 2008 Appl. Math. Mech. 29 9994
[10] Huang J J 2009 Chin. Phys. 18 B 3616
[11] Eerdun Buhe, Temuer Chaolu 2012 Chin. Phys. B 21 035201
[12] Xu H, Chen L H, Mo J Q 2010 Acta Phys. Sin. 60 100201 (in Chinese) [徐惠, 陈丽华, 莫嘉琪 2011 60 100201]
[13] Chen L J, Ma C F 2010 Chin. Phys. B 19 010504
[14] Polyanin A D, Zaitsev V F 2004 Handbook of Nonlinear Partial Differential Equations (Boca Raton, London, New York: Chapman & Hall/CRC) pp1-2
[15] Hou X L, Qian Y, Wu H T 2010 Acta Math. Eng. 27 663 (in Chinese) [侯祥林, 钱颖, 吴海涛 2010 工程数学学报 27 663]
[16] Hou X L, Liu T L, Zhai Z H 2011 Acta Phys. Sin. 60 090202 (in Chinese) [侯祥林, 刘铁林, 翟仲海 2011 60 090202]
[17] Hou X L, Zhai Z H, Zheng L, Liu T L 2012 Acta Phys. Sin. 61 010201 (in Chinese) [侯祥林, 翟中海, 郑莉, 刘铁林 2012 61 010201]
[18] Lao D Z 2007 Variational Method Basis (2nd Ed.) (National Defence Industry Press) pp74-91 (in Chinese) [老大中 2007 变分法基础(第2版) (北京: 国防工业出版社) 第74-91页]
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