On Generalized Reciprocal Diagrams for Self-Stressed Frameworks

We present the duality between edge lengths and axial forces in self-stressed frameworks, upon which are based reciprocal diagrams, introduced by Maxwell and Cremona in the nineteenth century. The main concepts and principles are simplified by using a graph theoretic approach. We describe some unusual orthogonality relations, involving lengths, axial forces and their rates of change. Reciprocal diagrams, which exist for frameworks with underlying planar graph, are extended also to the non-planar case by introducing a new criterion. When this criterion can be applied, different reciprocals can be obtained as symmetric frameworks. The same criterion can also be applied to planar cases giving new reciprocals as a result. Although reciprocal diagrams cannot be obtained for all self-stressed frameworks, the presented duality always holds and it provides useful insights for design and form-finding purposes.

[1]  C. Godsil,et al.  Cycles in graphs , 1985 .

[2]  O. Shai.  Deriving structural theorems and methods using Tellegen's theorem and combinatorial representations , 2001 .

[3]  Philippe Block,et al.  THRUST NETWORK ANALYSIS : A NEW METHODOLOGY FOR THREE-DIMENSIONAL EQUILIBRIUM , 2007 .

[4]  J. Maxwell,et al.  I.—On Reciprocal Figures, Frames, and Diagrams of Forces , 1870, Transactions of the Royal Society of Edinburgh.

[5]  Ioannis G. Tollis,et al.  Algorithms for Drawing Graphs: an Annotated Bibliography , 1988, Comput. Geom..

[6]  K. Linkwitz,et al.  Einige Bemerkungen zur Berechnung von vorgespannten Seilnetzkonstruktionen , 1971 .

[7]  H. Schek.  The force density method for form finding and computation of general networks , 1974 .

[8]  Konstantin A. Rybnikov,et al.  On Traces ofd-stresses in the Skeletons of Lower Dimensions of Piecewise-lineard-manifolds , 2001, Eur. J. Comb..

[9]  S. Pellegrino,et al.  First-order infinitesimal mechanisms , 1991 .

[10]  Stud. Kirchhoff.  Ueber den Durchgang eines elektrischen Stromes durch eine Ebene, insbesondere durch eine kreisförmige , 2022 .

[11]  Walter Whiteley,et al.  Plane Self Stresses and projected Polyhedra I: The Basic Pattem , 1993 .

[12]  Norman Biggs.  Algebraic Graph Theory: Index , 1974 .

[13]  Walter,et al.  Spaces of Stresses , Projections and ParallelDrawings for Spherical PolyhedraHenry Crapo , 1994 .

[14]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[15]  John McPhee,et al.  On the use of linear graph theory in multibody system dynamics , 1996 .

[16]  C. Calladine.  Buckminster Fuller's “Tensegrity” structures and Clerk Maxwell's rules for the construction of stiff frames , 1978 .

[17]  Robert H. Bow,et al.  Economics of Construction in Relation to Framed Structures , 2009 .

[18]  S. Pellegrino,et al.  Matrix analysis of statically and kinematically indeterminate frameworks , 1986 .

[19]  Melody Chan.  A survey of the cycle double cover conjecture , 2009 .

[20]  Henry Crapo.  Structural Rigidity , 2003 .