基于四边形面积坐标法的复杂结构极限与安定分析研究

Limit and shakedown analysis of structures is a practical and useful branch of plasticity. Its purpose is to determine the limit and shakedown loads of engineering structures and therefore to provide necessary theoretical basis for engineering design and safety assessment. The modern development direction of limit and shakedown analysis is to search for efficient and feasible computational methods so that the limit and shakedown theories can find their applications in engineering practice. Although the finite element method has been widely used in numerical limit and shakedown analyses, the problem of mesh distortion often hinders the implementation of the finite element analysis. Compared with the traditional isoparametric elements, elements by the quadrilateral area coordinate method can keep their accuracy under severely distorted mesh. Based on taking full advantages of the quadrilateral area coordinate method, this project intends to solve the present difficulties arising from the construction of the self-equilibrium stress field, the linearization of the nonlinear yield surface and the non-differential objective function. Subsequently, numerical methods are established for limit and shakedown analyses (including upper bound and lower bound) of plane structures and plate structures. In addition, the corresponding computational program with high accuracy and efficiency is also developed in this project. The proposed method in this project for limit and shakedown analysis can exhibit good performance in distorted meshes and can be employed to analyze very thin to moderately thick plates. The completeness of this project has important theoretical significance and wide application prospect.

结构极限与安定分析是塑性力学最有实用意义的分支之一，其宗旨是确定结构的极限与安定载荷，为工程设计和安全评估提供必要的理论依据。极限与安定分析的近代发展方向是寻找切实可行的计算方法，使极限安定性理论得以在工程实际中应用。虽然有限元法在数值极限与安定分析中得到了非常广泛的应用，但是网格畸变问题常常制约着有限元法的实施。与传统的等参坐标方法相比，四边形面积坐标单元在网格严重畸变时仍能保持较高的计算精度。本项目拟在充分发挥四边形面积坐标法的基础上，解决目前存在的自平衡应力场构造，非线性屈服面的线性化和目标函数不可微等困难，进而构建平面结构和板结构进行极限安定分析(包括上限和下限)的计算理论和方法，并开发出精确高效的计算程序，达到对网格畸变不敏感，而且在板结构的极限安定分析中无需特殊的数值技巧并可自动彻底地消除剪切闭锁现象之目的。该项目的完成具有重要的理论意义和广泛的应用前景。

结构极限与安定分析是塑性力学最有实用意义的分支之一，其宗旨是确定结构的极限与安定载荷，为工程设计和安全评估提供必要的理论依据。当前极限与安定性研究的一个重要课题是研究应用的策略，寻找切实可行的计算方法，使极限与安定性理论得以在工程实际中应用。到目前为止，绝大多数求解结构极限与安定问题的数值方法都基于有限元方法。但是，网格畸变敏感问题一直是当前有限元法难以解决的问题。发展高精度且抗网格畸变的高性能有限元方法，是计算力学重要的研究课题之一。相对于传统的等参坐标方法，四边形面积坐标单元在网格严重畸变时仍能保持较高的计算精度。此外，在中厚板中难以解决的剪切自锁问题也可以由四边形面积坐标法得到很好的解决。本项目在充分发挥四边形面积坐标法优势的基础上，引入二阶锥规划方法系统地研究了平面结构、薄板结构和中厚板结构的极限安定分析(包括上限和下限)的计算理论和方法。算例分析验证了提出的分析方法的可行性和有效性。项目所提的极限安定分析方法不仅具有很强的抗网格畸变能力，而且在中厚板的极限安定分析中无需特殊的数值技巧并可自动地消除剪切闭锁现象。本项研究是对极限安定分析理论和应用的丰富，能为继续深入研究结构的塑性失效行为提供有益借鉴。

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