圆管网格结构的自适应有限元分析方法研究
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摘要
为了判断网格结构有限元模型中梁单元的长度和插值函数是否合理并对此进行调整,首先推导了梁在受拉、受压、纯弯3种情况下的挠曲微分方程,以有限元试算得到的梁单元两端节点力为边界条件,求出了梁单元广义力分布场的解析解;然后,根据Zienkiewicz-Zhu后验误差估计理论,以该解析解为广义力相对精确解,推导了广义力有限元解和广义力相对精确解的能量范数以确定梁单元的相对误差。在试算过程中,如果网格结构中每个梁单元的相对误差都满足精度要求,则终止试算过程,否则调整梁单元的插值函数或长度后再进行试算。以单层球面网壳的自适应有限元静力分析为例验证了该方法的正确性和可行性。
The main objective of the present paper is to judge if the length and shape functions of a beam element in the finite element model of an latticed shell structure are suitable.Firstly the deflection function are derived subject to tension,compression,and pure bending according to the boundary conditions from finite element trial analysis and then the generalized force fields are obtained;Secondly,based on the Zienkiewicz-Zhu error post-processing technique,the generalized force field can be used as the relativly accurate solution,then,the energy norm of the finite element solution and the relative solution are given to determine the raltive error of the beam element.The computation trials proceed after the shape functions and the length of beam elements are adjusted,until the relative error requirement of every beam element in the structure is satisfied.An adaptive static finite element analysis of a single-layer reticulated dome is conducted to illustrate the feasibility and the validity of this method.
The main objective of the present paper is to judge if the length and shape functions of a beam element in the finite element model of an latticed shell structure are suitable.Firstly the deflection function are derived subject to tension,compression,and pure bending according to the boundary conditions from finite element trial analysis and then the generalized force fields are obtained;Secondly,based on the Zienkiewicz-Zhu error post-processing technique,the generalized force field can be used as the relativly accurate solution,then,the energy norm of the finite element solution and the relative solution are given to determine the raltive error of the beam element.The computation trials proceed after the shape functions and the length of beam elements are adjusted,until the relative error requirement of every beam element in the structure is satisfied.An adaptive static finite element analysis of a single-layer reticulated dome is conducted to illustrate the feasibility and the validity of this method.
引文
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[12]杨强,吴浩,周维垣.h-型自适应有限元法分析中的大坝应力取值标准[J].水利学报,2005,36(3):321―327.Yang Qiang,Wu Hao,Zhou Weiyuan.Study ondetermination of dam stress based on h-version adaptiveFEM[J].Journal of Hydraulic Engineering,2005,36(3):321―327.(in Chinese)
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[2]尹越,韩庆华,刘锡良,等.北京2008奥运会老山自行车赛馆网壳结构分析与设计[J].天津大学学报,2008,41(5):522―528.Yin Yue,Han Qinghua,Liu Xiliang,et al.Analysis anddesign of reticulated dome of Laoshan cyclinggymnasium for the Beijing 2008 Olympic Games[J].Journal of Tianjin University,2008,41(5):522―528.(inChinese)
[3]张华刚,马克俭,黄永,等.肋环型球面空腹网壳结构的静力性能分析[J].空间结构,2009,15(6):2―15.Zhang Huagang,Ma Kejian,Huang Yong,et al.Analysison static behavior of ribbed-type open web sphericalreticulated shell.Spatial structures[J].Spatial Structures,2009,15(6):2―15.(in Chinese)
[4]张爱林,刘学春,王冬梅,等.2008奥运会羽毛球馆新型弦支穹顶模型静力试验研究[J].建筑结构学报,2007,28(6):58―67.Zhang Ailin,Liu Xuechun,Wang Dongmei,et al.Staticexperimental study on the model of the suspend-dome ofthe badminton gymnasium for 2008 Olympic Games[J].Journal of Building Structures,2007,28(6):58―67.(inChinese)
[5]童根树.钢结构的平面内稳定[M].北京:中国建筑工业出版社,2007.Tong Genshu.Planar stability of steel structures[M].Beijing China Architecture&Building Press,2007.(inChinese)
[6]史俊亮.K8型单层网壳在撞击荷载作用下的动力响应研究[D].太原:太原理工大学,2005.Shi Junliang.Dynamic response research on K8single-layer reticulated dome under impact[D].Taiyuan:Taiyuan Technogical University,2005.(in Chinese)
[7]Kani I M,McConnel R E.Collapse of shallow latticedomes[J].J.Struct.Engng,ASCE,1987:113(ST8):1806―1819.
[8]Babuska I,Zienkiewicz O C.Accuracy estimates andadaptive refinement in finite element computations[M].John Wiley&Sons,1986.
[9]Zienkiewicz O C,Zhu J Z.A simple error estimator andprocedure for practical engineering analysis[J].Int.J.Numer.Meth.Eng.,1987,24:337―357.
[10]Babuska I,Strouboulis T,Upadhyay C S.A model studyof the quality of a-posteriori error estimators for linearelliptic problems.Error estimations in the interior ofpatchwise uniform grids of triangles[J].Comp MethAppl Mech Eng,1994,114:307―540.
[11]杨令强,马静,陈祖坪.自适应有限元在拱坝破坏追踪中的应用[J].水利发电学报,2008,27(3):48―53.Yang Lingqiang,Ma Jing,Chen Zuping.The applicationof adaptive FEM in cracking trace of arch dams[J].Journal of Hydroelectric Engineering,2008,27(3):48―53.(in Chinese)
[12]杨强,吴浩,周维垣.h-型自适应有限元法分析中的大坝应力取值标准[J].水利学报,2005,36(3):321―327.Yang Qiang,Wu Hao,Zhou Weiyuan.Study ondetermination of dam stress based on h-version adaptiveFEM[J].Journal of Hydraulic Engineering,2005,36(3):321―327.(in Chinese)
[13]刘鸿文.材料力学[M].北京:高等教育出版社,2004.Liu Hongwen.Material mechanics[M].Beijing:HigherEducation Press,2006.(in Chinese)
[14]Ansys 13.0 help[R].Ansys.Inc,2010.
[15]严慧,陈武贞,肖志斌.干煤棚网壳结构极限承载能力分析及其合理设计[J].工业建筑,2004,34(2):50―52.Yan Hui,Chen Wuzhen,Xiao Zhibin.Ultimate bearingcapacity analysis of reticulated shell of dry coal shed andits rational design[J].Industrial Construction,2004,34(2):50―52.(in Chinese)
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