基于无网格法的弹性地基加肋斜板频率数值解
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摘要
为了给实际地基工程中的经济效益提供技术指标参考,应用无网格法计算加肋斜板在地基上的自由振动行为,应用移动最小二乘近似MLS和一阶剪切变形原理描述加肋斜板的位移场,分别建立斜板与肋条的势能泛函,使用Winkler弹性地基模型处理加肋板与地基之间的接触势能。通过斜板与肋条的位移协调关系寻找斜板和肋条节点参数转换公式,确立加肋斜板的自由振动控制方程。本文运用了无网格法的优势,即使肋条位置改变也不需重置网格。与有限元解的对比验证了本文方法的有效性和精确性。
Meshfree method was used to calculate the free vibration behaviors of a ribbed skew plate on a foundation for its efficiency and for providing design information in practical foundation engineering.The displacement field of the skew plate was described by the moving-least square approximation and the first-order shear deformation theory,and then the potential energy functional was established.The contact potential energy between the ribbed plate and the foundation was simulated by Winkler elastic foundation model.The nodal parameter transformation equation was established according to the displacement compatibility condition between the plate and the ribs,and the governing equation for free vibration of the stiffened skew plate on the elastic foundation was derived.A position change of ribs would not require resetting the mesh because of the advantage of the proposed method,and the cost of calculation was reduced.The proposed method was verified by comparing its results with those from the finite element method.
Meshfree method was used to calculate the free vibration behaviors of a ribbed skew plate on a foundation for its efficiency and for providing design information in practical foundation engineering.The displacement field of the skew plate was described by the moving-least square approximation and the first-order shear deformation theory,and then the potential energy functional was established.The contact potential energy between the ribbed plate and the foundation was simulated by Winkler elastic foundation model.The nodal parameter transformation equation was established according to the displacement compatibility condition between the plate and the ribs,and the governing equation for free vibration of the stiffened skew plate on the elastic foundation was derived.A position change of ribs would not require resetting the mesh because of the advantage of the proposed method,and the cost of calculation was reduced.The proposed method was verified by comparing its results with those from the finite element method.
引文
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[15] 徐腾飞,邢誉峰.弹性地基上矩形薄板自由振动的精确解[J].工程力学,2014,31(5):43-48.(XU Teng-fei,XING Yu-feng.Exact solutions for free vibrations of rectangular thin plates on the winkler foundation [J].Engineering Mechanics,2014,31(5):43-48.(in Chinese))
[2] 李潇,王宏志,李世萍,等.解析型Winkler弹性地基梁单元构造[J].工程力学,2015,32(3):66-72.(LI Xiao,WANG Hong-zhi,LI Shi-ping,et al.Element for beam on Winkler elastic foundation based on analytical trial functions[J].Engineering Mechanics,2015,32(3):66-72.(in Chinese))
[3] Rad A B,Shariyat M.Thermo-magneto-elasticity analysis of variable thickness annular FGM plates with asymmetric shear and normal loads and non-uniform elastic foundations [J].Archives of Civil and Mechanical Engineering,2016,16(3):448-466.
[4] Liew K M,Xiang Y,Kitipornchai S,et al.Formulation of Mindlin-Engesser model for stiffened plate vibration[J].Computer Methods in Applied Mechanics and Engineering,1995,120(3-4):339-353.
[5] Belytschko T,Lu Y Y,Gu L.Element-free Galerkin methods[J].International Journal for Numerical Methods in Engineering,1994,37(2):229-256.
[6] Liu G R.Mesh Free Methods:Moving Beyond the Finite Element Method[M].CRC Press,2002.
[7] 张雄,刘岩,马上.无网格法的理论及应用[J].力学进展,2009,39(1):1-36.(ZHANG Xiong,LIU Yan,MA Shang.Meshfree methods and their applications[J].Advances in Mechanics,2009,39(1):1-36.(in Chinese))
[8] 张雄,宋康祖,陆明万.无网格法研究进展及其应用[J].计算力学学报,2003,20(6):730-742.(ZHANG Xiong,SONG Kang-zu,LU Ming-wan.Research pro -gress and application of meshless method [J].Chinese Journal of Computational Mechanics,2003,20(6):730-742.(in Chinese))
[9] 顾元通,丁桦.无网格法及其最新进展[J].力学进展,2005,35(3):323-337.(GU Yuan-tong,DING Hua.Recent developments of meshless method [J].Advances in Mechanics,2005,35(3):323-337.(in Chinese))
[10] 吴俊超,邓俊俊,王家睿,等.伽辽金型无网格法的数值积分方法[J].固体力学学报,2016,37(3):208-233.(WU Jun-chao,DENG Jun-jun,WANG Jia-rui,et al.A review of numerical integration approaches for Galerkin meshfree methods [J].Chinese Journal of Solid Mechanics,2016,37(3):208-233.(in Chinese))
[11] Li S,Liu W K.Meshfree and particle methods and their applications[J].Applied Mechanics Reviews,2002,55(1):1-34.
[12] 彭林欣.加肋板自由振动的移动最小二乘无单元分析[J].振动与冲击,2011,30(6):67-73.(PENG Lin-xin.Moving-least-squares free element analysis of free vibration of stiffened plates[J].Journal of Vibration and Shock,2011,30(6):67-73.(in Chinese))
[13] Reddy J N.Theory and Analysis of Elastic Plates and Shells[M].CRC Press,2006.
[14] Chen J S,Pan C,Wu C T,et al.Reproducing Kernel Particle Methods for large deformation analysis of non-linearstructures[J].Computer Methods in Applied Mechanics and Engineering,1996,139(1-4):195-227.
[15] 徐腾飞,邢誉峰.弹性地基上矩形薄板自由振动的精确解[J].工程力学,2014,31(5):43-48.(XU Teng-fei,XING Yu-feng.Exact solutions for free vibrations of rectangular thin plates on the winkler foundation [J].Engineering Mechanics,2014,31(5):43-48.(in Chinese))
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