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Moving least squares meshless method for static bending of reinforced folded shells

DOI：

 作者 单位 邮编 陈卫 南华大学 土木工程学院 421001 彭林欣* 广西大学 土木建筑工程学院 530004

可展结构体系中的折板结构凭借着高刚度-重量比、易拼接成型等特点,已被广泛应用于实际结构工程中。相比折板而言,折壳具有几何形状多变、外观优美等特点。此外,加筋板壳结构中的筋条可以采用基于欧拉-伯努利梁(EBT)和铁木辛柯梁(TBT)理论框架下的计算方法。然而,当筋条跨高比达到一定限度时,EBT和TBT则不再满足工程精度需求。本文通过将筋条视为基于3D连续壳理论的长条形板或曲板,提出一种求解加筋折叠壳静力弯曲问题的移动最小二乘无网格法。首先,基于3D连续壳理论,通过映射技术将随动坐标系上的二维无网格节点信息映射到三维壳体中,采用移动最小二乘近似对曲面几何及位移场拟合,根据最小势能原理,推导出壳静力弯曲的无网格控制方程；其次,采用完全转换法对控制方程进行修正；接着,将经过修正后各壳的刚度矩阵和载荷列阵转换到整体坐标系上；最后,通过子结构法叠加得到整个加筋折叠壳结构的控制方程。文末通过折板壳、加筋板、加筋壳四个数值算例,将本文计算结果与ABAQUS有限元解进行对比,验证了本文方法计算加筋折叠壳静力弯曲的收敛性及准确性。

Folded plate structure in deployable structure system has been widely used in practical Structural engineering due to its high stiffness weight ratio and easy splicing. Compared to folded plates, folded shells have the characteristics of variable geometric shapes and beautiful appearance. In addition, the reinforcement bars in reinforced plate and shell structures can be calculated using methods based on the Euler Bernoulli beam (EBT) and Timoshenko beam theoretical (TBT) frameworks. However, when the span-depth ratio of the reinforcement reaches a certain limit, the EBT and TBT no longer meet the engineering accuracy requirements. This article proposes a moving least squares meshless method for solving the static bending problem of reinforced folded shells by treating the ribs as elongated or curved plates based on 3D continuous shell theory. Firstly, based on the 3D continuous shell theory, the two-dimensional meshless node information on the following coordinate system is mapped into the 3D shell using mapping technology. The moving least squares approximation is used to fit the surface geometry and displacement field. Based on the principle of minimum potential energy, the meshless control equation for static bending of the shell is derived; Secondly, the control equation is modified using the complete transformation method; Then, the modified Stiffness matrix and load array of each shell are converted to the overall coordinate system; Finally, the control equation of the entire reinforced folded shell structure is obtained by superimposing the substructure method. At the end of the article, the convergence and accuracy of the method in calculating the static bending of reinforced folded shells were verified by comparing the calculation results of this paper with the ABAQUS finite element solution through four numerical examples: folded plate and shells, reinforced plates, and reinforced shells.
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