On the optimality of Hemp’s arch with vertical hangers

In 1974 W. S. Hemp constructed a prototype structure to carry a uniformly distributed load between two pinned supports. Although Hemp’s structure had a significantly lower volume than a parabolic arch with vertical hangers, it was shown to fail the Michell optimality criteria, and therefore to be non-optimal. In this paper we demonstrate that if limiting compressive and tensile stresses are unequal then Hemp’s structure is optimal for the half-plane provided the ratio of limiting tensile to compressive stresses falls below a certain threshold. An analytical proof is presented and the finding is confirmed by results from large scale numerical layout optimization simulations.

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

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

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

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

[5]  W. Prager,et al.  A new class of structural optimization problems: Optimal archgrids , 1979 .

[6]  George I. N. Rozvany,et al.  On plane Prager-structures—I☆ , 1983 .

[7]  George I. N. Rozvany,et al.  Layout Optimization of Structures , 1995 .

[8]  G. I. N. Rozvany,et al.  Partial relaxation of the orthogonality requirement for classical Michell trusses , 1997 .

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

[10]  Paul Shepherd,et al.  Application of Large-scale Layout Optimization Techniques in Structural Engineering Practice , 2005 .

[11]  Matthew Gilbert,et al.  Optimum structure to carry a uniform load between pinned supports , 2010 .

[12]  Tomasz Sokół,et al.  On the solution of the three forces problem and its application in optimal designing of a class of symmetric plane frameworks of least weight , 2010 .

[13]  Matthew Gilbert,et al.  Optimum structure to carry a uniform load between pinned supports , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[15]  G. Rozvany,et al.  Exact truss topology optimization: allowance for support costs and different permissible stresses in tension and compression—extensions of a classical solution by Michell , 2012 .