On the solution of the three forces problem and its application in optimal designing of a class of symmetric plane frameworks of least weight

Two problems of minimum weight design of plane trusses are dealt with. The first problem concerns construction of the lightest fully stressed truss subject to three self-equilibrated forces applied at three given points. This problem has been solved analytically by H.S.Y. Chan in 1966. This analytical solution is re-derived in the present paper. It compares favourably with new numerical solutions found here by the method developed recently by the first author. The solution to the three forces problem paves the way to half-analytical as well as numerical solutions to the problem of minimum weight design of plane symmetric frameworks transmitting two symmetrically located vertical forces to two fixed supports lying along the line linking the points of application of the forces.

[1]  G. Rozvany,et al.  Extended exact least-weight truss layouts—Part II: Unsymmetric cantilevers , 1994 .

[2]  Cezary Graczykowski,et al.  Michell cantilevers constructed within trapezoidal domains , 2006 .

[3]  Richard E. McConnel.  Least-Weight Frameworks for Loads Across Span , 1974 .

[4]  W. Achtziger.  Topology Optimization of Discrete Structures , 1997 .

[5]  George I. N. Rozvany,et al.  A critical review of established methods of structural topology optimization , 2009 .

[6]  R. Melchers.  On extending the range of Michell-like optimal topology structures , 2005 .

[7]  P. Seppecher,et al.  Locking materials and the topology of optimal shapes , 2001 .

[8]  Cezary Graczykowski,et al.  Michell cantilevers constructed within trapezoidal domains—Part III: force fields , 2006 .

[9]  H. Chan,et al.  Symmetric Plane Frameworks of Least Weight , 1975 .

[10]  Cezary Graczykowski,et al.  Michell cantilevers constructed within trapezoidal domains—Part II: virtual displacement fields , 2006 .

[11]  Tomasz Lewiński,et al.  Force Fields within Michell-Like Cantilevers Transmitting a Point Load to a Straight Support , 2006 .

[12]  Tomasz Sokół,et al.  A 99 line code for discretized Michell truss optimization written in Mathematica , 2011 .

[13]  G. Rozvany.  Author’s reply to a discussion by Gengdong Cheng and Xiaofeng Liu of the review article “On symmetry and non-uniqueness in exact topology optimization” by George I. N. Rozvany (2011, Struct Multidisc Optim 43:297–317) , 2011 .

[14]  G. Rozvany,et al.  Analytical benchmarks for topological optimization IV: square-shaped line support , 2008 .

[15]  W. S. Hemp.  MICHELL FRAMEWORK FOR UNIFORM LOAD BETWEEN FIXED SUPPORTS , 1974 .

[16]  Cezary Graczykowski,et al.  Michell cantilevers constructed within trapezoidal domains—Part I: geometry of Hencky nets , 2006 .

[17]  J. Lógó,et al.  Optimal Topologies in Case of Probabilistic Loading: The Influence of Load Correlation , 2009 .

[18]  M. Gilbert,et al.  Layout optimization of large‐scale pin‐jointed frames , 2003 .

[19]  George I. N. Rozvany,et al.  On symmetry and non-uniqueness in exact topology optimization , 2011 .

[20]  A.S.L. Chan,et al.  The Design of Michell Optimum Structures , 1960 .

[21]  H. Chan,et al.  HALF-PLANE SLIP-LINE FIELDS AND MICHELL STRUCTURES , 1967 .

[22]  C. Carathéodory,et al.  Über die Hencky-Prandtlschen Kurven , 1923 .

[23]  G. Rozvany.  Symmetry and non-uniqueness in exact topology optimization of structures , 2010 .

[24]  G. Rozvany,et al.  Extended exact solutions for least-weight truss layouts—Part I: Cantilever with a horizontal axis of symmetry , 1994 .

[25]  G. Rozvany,et al.  Exact analytical solutions for some popular benchmark problems in topology optimization II: three-sided polygonal supports , 2007 .

[26]  G. Rozvany.  Exact analytical solutions for some popular benchmark problems in topology optimization , 1998 .

[27]  Peter Dewhurst,et al.  Analytical solutions and numerical procedures for minimum-weight Michell structures , 2001 .

[28]  A. Michell.  LVIII. The limits of economy of material in frame-structures , 1904 .