Abstract
This paper proposes a unified and entirely geometrical methodology for generating 2D and 3D force diagrams for given planar and spatial trusses in static equilibrium within the context of graphic statics. The trusses, regarded as form diagrams, are projections of higher dimensional simply-connected stress functions, can be either self—stressed or loaded with external forces, and need not be tension-only/ compression-only. First, we provide an in-depth overview of fundamental notions of graphic statics and projective geometry related to the construction of reciprocal form and force diagrams. Specifically, we describe a series of polar transformations and discuss them from a geometric and an algebraic standpoint. Moreover, we provide an exegesis and visualization of Maxwell's reciprocal methods while discussing them within the scientific framework of his time. We then develop a contemporary graphic statics framework grounded on projective geometry and higher-dimensional reciprocal stress functions. Within this framework, we provide a general methodology for deriving any type of reciprocal diagrams for the planar and the spatial case. Advantages of this method include: direct control on any of the four reciprocal objects (form diagram, force diagram, corresponding stress functions) and no need for iterative/ procedural reconstruction of any of the diagrams since it is based on direct geometrical transformations. Finally, we highlight how duality, and in particular polarity, can be used for a wide range of applications in structural analysis and design.
Highlights
Graphic statics is a geometrical framework for the design and analysis of 2D and 3D trusses in static equilibrium mostly developed during the 19th century by natural philosophers, scientists, and practitioners of engineering such as (Culmann, 1857; Rankine, 1864; Maxwell, 1864; Maxwell, 1870; Cremona, 1872) among others (Kurrer, 2008; Charlton, 1982)
In its initial widespread implementation, graphic statics consisted of a series of graphical procedures for the construction and transformation of two reciprocal diagrams, the form and force diagrams, the former representing the geometric layout of a loaded truss and the latter the equilibrium of the internal and external forces applied to the nodes of the truss
To set up a consistent theoretical ground for the proposed framework, in Section 2 we summarise a series of fundamental notions related to graphic statics, reciprocal diagrams and Airy stress functions, as well as projective geometry and polar transformations
Summary
Graphic statics is a geometrical framework for the design and analysis of 2D and 3D trusses in static equilibrium mostly developed during the 19th century by natural philosophers, scientists, and practitioners of engineering such as (Culmann, 1857; Rankine, 1864; Maxwell, 1864; Maxwell, 1870; Cremona, 1872) among others (Kurrer, 2008; Charlton, 1982). The possibility to support graphic statics with contemporary computational tools has made it possible to develop visual and intuitive approaches for the design, analysis, and optimisation of structures based on the interplay between the. Most of the current methodologies are generally case specific, work either for 2D or 3D form diagrams and do not make use of the possibilities offered by the underlying fundamental geometrical framework, firstly introduced by Maxwell (1864) and Maxwell (1870), based on projective geometry and reciprocal stress functions
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