Membrane Solution for a Paraboloid under Self-Weight

Mitchell Gohnert, Ryan Bradley
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Abstract

Stress flows in a predictable pattern, and structural optimization is achieved by matching the natural flow of stress with the structural shape. The geometry of the parabolic shape simulates the natural flow of stress, and is therefore highly efficient in the conveyance of stress. However, despite its importance, the membrane solution of a parabolic dome has never been solved. Designers have been reliant on numerical methods, such as finite elements, or older techniques such as graphical solutions. For this reason, a closed-form membrane solution for a parabolic dome is derived. The solution solves for the meridian and hoop stresses, in the vertical and horizontal directions of the dome for the case of uniformly distributed loads, such as the self-weight of a uniformly thick shell. Finite element analysis (FEA) was also used to undertake a full shell analysis (i.e., membrane and bending behavior) to examine the edge effects that are not captured in the membrane solution. From this study, the benefits of the parabolic dome were found to be similar to the catenary dome; i.e., the stresses in the meridian and hoop directions are compressive, boundary effects are largely minimal, andstresses flow primarily in-plane (membrane action).
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自重抛物面的膜溶液
应力以可预测的模式流动,通过将应力的自然流动与结构形状相匹配来实现结构优化。抛物线形的几何形状模拟了应力的自然流动,因此在传递应力方面非常有效。然而,尽管其重要性,抛物线圆顶的膜解从未得到解决。设计师一直依赖于数值方法,如有限元素,或更老的技术,如图形解决方案。为此,导出了抛物线穹顶的闭式膜解。该解决方案解决了子午和环向应力,在垂直和水平方向的穹顶为均匀分布荷载的情况下,如自重均匀厚壳。有限元分析(FEA)也用于进行全壳分析(即膜和弯曲行为),以检查膜溶液中未捕获的边缘效应。从本研究中发现,抛物线穹顶的优点与悬链线穹顶相似;也就是说,在子午和环向的应力是压缩的,边界效应很大程度上是最小的,应力主要在平面内流动(膜作用)。
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CiteScore
1.40
自引率
0.00%
发文量
17
期刊介绍: The Association publishes an international journal, the Journal of the IASS, four times yearly, in print (ISSN 1028-365X) and on-line (ISSN 1996-9015). The months of publication are March, June, September and December. Occasional extra electronic-only issues are included in the on-line version. From this page you can access one or more issues -- a sample issue if you are not logged into the members-only portion of the site, or the current issue and several back issues if you are logged in as a member. For any issue that you can view, you can download articles as .pdf files.
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