Mahran Mohamed, ELsabbagh Adel, Negm Hani
Aerospace Engineering Department, Cairo University, Giza 12613, Egypt.
Asu Sound and Vibration Lab, Design and Production Engineering Department, Ain Shams University, Abbaseya, Cairo 11517, Egypt.
J Adv Res. 2017 Nov;8(6):635-648. doi: 10.1016/j.jare.2017.06.009. Epub 2017 Jul 1.
In the present paper, a comparison between five different shell finite elements, including the Linear Triangular Element, Linear Quadrilateral Element, Linear Quadrilateral Element based on deformation modes, 8-node Quadrilateral Element, and 9-Node Quadrilateral Element was presented. The shape functions and the element equations related to each element were presented through a detailed mathematical formulation. Additionally, the Jacobian matrix for the second order derivatives was simplified and used to derive each element's strain-displacement matrix in bending. The elements were compared using carefully selected elastic and aero-elastic bench mark problems, regarding the number of elements needed to reach convergence, the resulting accuracy, and the needed computation time. The best suitable element for elastic free vibration analysis was found to be the Linear Quadrilateral Element with deformation-based shape functions, whereas the most suitable element for stress analysis was the 8-Node Quadrilateral Element, and the most suitable element for aero-elastic analysis was the 9-Node Quadrilateral Element. Although the linear triangular element was the last choice for modal and stress analyses, it establishes more accurate results in aero-elastic analyses, however, with much longer computation time. Additionally, the nine-node quadrilateral element was found to be the best choice for laminated composite plates analysis.
在本文中，对五种不同的壳有限元进行了比较，包括线性三角形单元、线性四边形单元、基于变形模式的线性四边形单元、八节点四边形单元和九节点四边形单元。通过详细的数学公式给出了与每个单元相关的形函数和单元方程。此外，简化了二阶导数的雅可比矩阵，并用于推导每个单元在弯曲时的应变 - 位移矩阵。使用精心挑选的弹性和气动弹性基准问题对这些单元进行了比较，涉及达到收敛所需的单元数量、所得精度以及所需的计算时间。结果发现，对于弹性自由振动分析，最合适的单元是具有基于变形的形函数的线性四边形单元；对于应力分析，最合适的单元是八节点四边形单元；对于气动弹性分析，最合适的单元是九节点四边形单元。尽管线性三角形单元在模态和应力分析中是最后的选择，但它在气动弹性分析中能得出更准确的结果，不过计算时间要长得多。此外，发现九节点四边形单元是层合复合材料板分析的最佳选择。
