McRobie Allan
Cambridge University Engineering Department , Trumpington Street, Cambridge CB2 1PZ, UK.
R Soc Open Sci. 2017 Mar 22;4(3):160759. doi: 10.1098/rsos.160759. eCollection 2017 Mar.
Building on a long tradition from Maxwell, Rankine, Klein and others, this paper puts forward a geometrical description of structural equilibrium which contains a procedure for the graphic analysis of stress resultants within general three-dimensional frames. The method is a natural generalization of Rankine's reciprocal diagrams for three-dimensional trusses. The vertices and edges of dual abstract 4-polytopes are embedded within dual four-dimensional vector spaces, wherein the oriented area of generalized polygons give all six components (axial and shear forces with torsion and bending moments) of the stress resultants. The relevant quantities may be readily calculated using four-dimensional Clifford algebra. As well as giving access to frame analysis and design, the description resolves a number of long-standing problems with the incompleteness of Rankine's description of three-dimensional trusses. Examples are given of how the procedure may be applied to structures of engineering interest, including an outline of a two-stage procedure for addressing the equilibrium of loaded gridshell rooves.
基于麦克斯韦、兰金、克莱因等人的悠久传统，本文提出了一种结构平衡的几何描述，其中包含一种用于一般三维框架内应力合力图形分析的程序。该方法是兰金三维桁架互易图的自然推广。对偶抽象4 - 多面体的顶点和边嵌入在对偶四维向量空间中，其中广义多边形的定向面积给出了应力合力的所有六个分量（轴向力、剪力、扭矩和弯矩）。使用四维克利福德代数可以很容易地计算相关量。除了用于框架分析和设计外，该描述还解决了兰金对三维桁架描述不完整的一些长期存在的问题。文中给出了该程序如何应用于具有工程意义的结构的示例，包括解决加载网壳屋顶平衡的两阶段程序概述。
