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

Abstract

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.

Optimum structure for a uniform load over multiple spans

“…Michell (1904), Chan (1962), Chan (1967), Hemp (1973), Lewiński et al (1994), and Rozvany (1998). Furthermore, whilst some notable exceptions exist, see Hemp (1974), Chan (1975), Sokół and Lewiński (2010), and Tyas et al (2011), the majority of known Michell structures are designed to support only a single external point load.…”

Application of layout optimization to the design of additively manufactured metallic components

“…Darwich et al 2010, Sokół andLewiński 2010). Although the corresponding truss layouts cannot be manufactured directly, due to the high number of members that are usually involved, they do provide a useful reference volume.…”

“…see Gilbert and Tyas 2003;Sokół and Lewiński 2010;Pichugin et al 2012). These can serve as reference solutions for use in later stages of the design process.…”

New analytical benchmarks for topology optimization and their implications. Part I: bi-symmetric trusses with two point loads between supports

“…Many important properties of functions G n and F n can be found in the paper mentioned before (Lewinski et al 1994). Now, the coordinates of the point D, connecting the horizontal bar with the Michell continuum, can be written as: Equations (8) are obtained by rearranging (2.9-11) of the paper by Sokół and Lewiński (2010). In the latter paper the displacement field of the domain RBDCNR (divided by appropriate sub-regions) was derived in detail.…”

“…The second constraint can be derived from the fact that the upper chord BD should connect smoothly to horizontal bar DS at the point D, see (5.1) by Sokół and Lewiński (2010) and the detailed explanation given there. Now, by (12) the coordinates of point D can be expressed as…”