Maxwell and Rankine reciprocal diagrams via Minkowski sums for two-dimensional and three-dimensional trusses under load

Abstract

A method is presented for computing and displaying the Maxwell (two-dimensional) or Rankine (three-dimensional) diagram reciprocal to a truss under load. Reciprocals are constructed via the two-dimensional Airy stress function or its three-dimensional analogue. A linear combination of the coordinates of each original node and the coordinates of its reciprocal object leads to the finished diagram. This is related to an underlying Minkowski sum of the polyhedral stress functions of the original and reciprocal diagrams. Some regions of the resulting combined diagram have units of length $L,$ and other regions use units of force $F .$ These regions are connected in a consistent manner, in two dimensions by rectangular areas and in three dimensions by prismatic volumes, each having work units of dimension $F L,$ and which give a visual representation of the $F \cdot L$ terms in Maxwell’s load path theorem. A linear combination weighted in favour of the original leads to a diagram where original bars are augmented by a thickness (in two-dimensional) or a polygonal cross-sectional area (in three-dimensional). The diagrams thus give an appealing visual representation of the material required for a constant stress design. Although the method gives a complete description of loaded trusses in two dimensions, in three dimensions, it is currently only applicable to those trusses which possess a Rankine reciprocal.

Keywords

Airy stress function graphic statics Maxwell reciprocal diagrams Minkowski sum projective geometry Rankine reciprocal diagrams

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