网壳结构稳定性的弱形式求积元分析

The newly issued National Technical Specification for frame structures JGJ7-2010 stipulates the necessity of load-displacement analysis of large-scale and complex reticulated shells incorporating both geometrical and material nonlinearities. Although the nonlinear finite element method has been competent for stability analysis of reticulated shells incorporating both material and geometrical nonlinearities, it is often afflicted by large number of degrees-of-freedom, low computational efficiency and even divergence. The weak form quadrature element method, proposed by the applicant, is a highly efficient numerical tool. It has been shown in some nonlinear analyses of structures that the method is far more efficient than the conventional finite element method. Generally, the number of degrees-of-freedom required in weak form quadrature element analysis can be one order less than that for the conventional finite element method to acquire results of the same accuracy. Like the finite element method, the weak form quadrature element method is based upon variational descriptions of a problem but does not need to define shape functions. The conceptual simplicity and straightforward implementation, easy upfront preparation and post-processing that characterize the method facilitate its extension to various engineering structural problems. The primary aim of the present project is to apply the weak form quadrature element method to stability analysis of reticulated shells. The project will be implemented based on the preliminary work that has been conducted over the past few years in the research group of the applicant. The research activities of the project will be focused on the resolution to problems associated with the weak form quadrature element method during stability analysis of reticulated shells. It is hoped that the outcome of the present project will lay down the basis for stability analysis of large-scale and complex reticulated shells.

我国新近实施的空间网格结构技术规程JGJ7-2010中对大型和复杂网壳结构建议进行几何材料双重非线性荷载-位移全过程稳定性分析。虽然非线性有限元对于完成网壳结构的几何材料双重非线性荷载-位移全过程稳定性分析技术是成熟的，但是对于实际网壳结构存在着计算量庞大以及计算效率低甚至不收敛的问题。弱形式求积元法是申请者近年来提出的一套高效的数值计算方法。已有的结构非线性分析实践表明，该方法比传统有限元模型在保证相同求解精度的前提下自由度可少近一个数量级。与有限元一样，该方法也是基于问题的弱形式描述，但无需先构造形函数。其概念简单，实施步骤直接以及前后处理过程简便等特点有利于在工程结构分析中推广。本项目将在已有的工作基础上，对用弱形式求积元法进行网壳结构的稳定性分析面临的一些问题展开研究。为将此方法应用于解决大型和复杂网壳的稳定性分析奠定基础。

对大型和复杂网壳进行考虑几何和材料非线性荷载位移全过程分析是现行空间网格结构技术规程JGJ7-2010的技术建议要求。虽然一些商业软件的结构非线性分析功能十分强大，但在实际网壳工程结构的分析中这些商业软件会出现难收敛甚至不收敛的情形，更不用说其计算效率低下的问题了。本课题就是要将课题负责人提出的弱形式求积元法应用于考虑几何材料双非线性的网壳结构的荷载位移全过程分析中。本课题开发了可以考虑几何材料双重非线性的基于几何精确理论模型的弱形式求积梁单元，并研究了K6、短程线和Schwedler等典型单层球面网壳，柱面网壳的荷载位移全过程曲线。开展了杆件初弯曲缺陷方向角随机分布对网壳承载力的影响的分析。完成了忽略横向剪切变形的几何精确梁弱形式求积单元的开发。通过与ABAQUS商业软件的对比，验证了所开发的程序的正确性和可靠性。与传统有限元多段直线计算曲梁的方法相比，所开发的求积元模型不需额外增加单元数目，每根杆仍然可以用一个单元模拟，自由度较少，精确度较高，故更适用于处理含杆件初弯曲的网壳稳定性的问题。研究发现，网壳结构承载力降低的比例随杆件初缺陷的挠曲比的增大和材料屈服强度的增大而增大。此外，将材料取为理想弹塑性时用一致缺陷模态法基本给出了各种情况下极限承载力的下界，但计算结果稍偏保守。考虑杆件随机初缺陷分布的弱形式求积元分析结果可以给出各网壳极限承载力的分布范围，计算结果更为准确。. 本课题更有意义的工作是围绕现行规范给出的公式未考虑材料屈服强度和初始缺陷的影响的研究成果。当材料屈服强度较高、初始缺陷弯曲幅值较小时，目前规范给出的计算公式太过保守，建议根据材料屈服强度和初始缺陷幅值进行适当修正。...通过本课题的开展，完善了弱形式求积元法进行几何材料双重非线性分析的能力。特别是一套基于弱形式求积元的网壳稳定性分析计算软件的成型，为弱形式求积元进一步服务于工程奠定了基础。

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