Optimum structure for a uniform load over multiple spans

Abstract

This paper presents a new half-plane Michell structure that transmits a uniformly distributed load of infinite horizontal extent to a series of equally-spaced pinned supports. A full kinematic description of the structure is obtained for the case when the maximum allowable tensile stress is greater than or equal to the allowable compressive stress. Although formal proof of optimality of the solution presented is not yet available, the proposed analytical solution is supported by substantial numerical evidence, involving the solution of problems with in excess of 10 billion potential members. Furthermore, numerical soluThis is an extended version of a paper prepared by tions for various combinations of unequal allowable stresses suggest the existence of a family of related, simple, and practically relevant structures, which range in form from a Hemp-type arch with vertical hangers to a structure which strongly resembles a cable-stayed bridge.

Theoretically optimal forms for very long-span bridges under gravity loading

“… a Obtained by extrapolation, using a power-law extrapolation scheme as used in [ 18 ]. b Analytical solution obtained by Pichugin et al [ 20 ]. …”

“…Quoted span lengths assume a limiting material stress σ 0 = 500 MPa and unit weight ρg = 80 kN m −3 . The quoted volumes for the weightless designs are computed using explicit analytical expressions given in [ 31 , 32 ] (for suspension bridges), [ 31 ] (for cable-stayed bridges) and [ 20 ] (for optimal Michell structures). a Optimal span-to-dip ( L / h ) ratios for weightless designs indicated in parenthesis, where available in the cited sources.…”

“…For example, layout optimization was used to show that the minimum volume structure to carry a uniformly distributed load between pinned supports was not the parabolic arch (or cable) form which had presumed to be optimal since the time of Christiaan Huygens [ 18 ], but instead a more complex form comprising networks of orthogonal tensile and compressive elements near the supports and a central parabolic section; in this case the numerical volume obtained via extrapolation techniques was later shown to be within 0.001% of the exact analytical solution subsequently derived [ 19 ]. Layout optimization has also been used to investigate bridge-like forms, for example being used to show that changing the ratio of the limiting compressive to tensile material strength gives rise to a family of optimal structures which range from arch to cable-stayed forms [ 20 ], prompting follow on analytical studies by practitioners [ 21 ].…”

Fundamentals of exact multi-load topology optimization – stress-based least-volume trusses (generalized Michell structures) – Part I: Plastic design

“…Optimal truss layouts for uniformly distributed load between supports was considered by Hemp (1974), Chan (1975), Darwich et al (2010), Tyas et al (2011) and Pichugin et al (2011Pichugin et al ( , 2012, and for line supports by Rozvany and Gollub (1990) and by Rozvany et al (1997). Layouts for symmetric and unsymmetric cantilevers were studied by Lewiński et al (1994a, b).…”

Topology optimization in structural mechanics

“…The cantilevers in the exterior of polygons were found in [37,39], while the structures within Lshape domains were dealt with in [38] and [50]. The problem of transmitting a uniform load to the successive supports located at equal distances has been solved in [51]. The problem in Fig.…”