October 2012, Volume 46, Issue 4, pp 477486,
Open Access
This content is freely available online to anyone, anywhere at any time.
Date:
17 Mar 2012
New analytical benchmarks for topology optimization and their implications. Part I: bisymmetric trusses with two point loads between supports
 Tomasz Sokół,
 George I. N. Rozvany
 … show all 2 hide
Abstract
Extending some results by Sokół and Lewiński (Struct Multidisc Optim 42:835–853, 2010), the optimal topology of bisymmetric trusses with two symmetrically placed pointloads is determined analytically, and verified by highly accurate numerical calculations. It is rather remarkable that Michell’s bestknown classical solution had to wait over hundred years for its generalization from one to two point loads. Some implications of these solutions, including properties of socalled Oregions, are also discussed.
 Chan ASL (1960) The design of Michell optimum structures. College of Aeronautics Report No 142, Cranfield
 Chan HSY (1963) Optimum Michell frameworks for three parallel forces. College of Aeronautics Report No 167, Cranfield
 Gilbert, M, Tyas, A (2003) Layout optimization of largescale pinjointed frames. Eng Comput 20: pp. 10441064 CrossRef
 Hemp, WS (1973) Optimum structures. Clarendon, Oxford
 Lewiński, T, Zhou, M, Rozvany, GIN (1994) Extended exact solutions for leastweight truss layouts – Part I: cantilever with a horizontal axis of symmetry. Int J Mech Sci 36: pp. 375398 CrossRef
 Lowe, PG, Melchers, RE (1972) On the theory of optimal constant thickness, fibrereinforced plates I. Int J Mech Sci 14: pp. 311324 CrossRef
 Melchers, R (2005) On extending the range of Michelllike optimal topology structures. Struct Multidisc Optim 29: pp. 8592 CrossRef
 Michell, AGM (1904) The limits of economy of material in frame structures. Philos Mag 8: pp. 589597 CrossRef
 Morley, CT (1966) The minimum reinforcement of concrete slabs. Int J Mech Sci 8: pp. 305319 CrossRef
 Prager, W, Rozvany, GIN Optimization of the structural geometry. In: Bednarek, AR, Cesari, L eds. (1977) Dynamical systems (Proc. Int. Conf. Gainsville, Florida). Academic Press, New York, pp. 265293
 Rozvany, GIN (1972) Grillages of maximum strength and maximum stiffness. Int J Mech Sci 14: pp. 651666 CrossRef
 Rozvany, GIN (2011) On symmetry and nonuniqueness in exact topology optimization. Struct Multidisc Optim 43: pp. 297317 CrossRef
 Rozvany, GIN, Gollub, W, Zhou, M (1997) Exact Michell layouts for various combinations of line supports – part II. Struct Optim 14: pp. 138149 CrossRef
 Rozvany, GIN, Hill, RD (1976) The theory of optimal load transmission by flexure. Adv Appl Mech 16: pp. 183308 CrossRef
 Rozvany, GIN, Sokół, T (2012) Exact truss topology optimization: allowance for support costs and different permissible stresses in tension and compression – extensions of a classical solution by Michell. Struct Multidisc Optim.
 Sokół, T (2011) A 99 line code for discretized Michell truss optimization written in Mathematica. Struct Multidisc Optim 43: pp. 181190 CrossRef
 Sokół T (2011b) Topology optimization of largescale trusses using ground structure approach with selective subsets of active bars. In: Borkowski A, Lewiński T, Dzierżanowski G (eds) 19th int conf on ‘computer methods in mechanics’, CMM 2011, 09–12 May, Warsaw, Book of Abstract, pp 457–458
 Sokół, T, Lewiński, T (2010) On the solution of the three forces problem and its application in optimal designing of a class of symmetric plane frameworks of least weight. Struct Multidisc Optim 42: pp. 835853 CrossRef
 Sokół T, Lewiński T (2011a) On the three forces problem in truss topology optimization. Analytical and numerical solutions. In: 9th world congress on structural and multidisciplinary optimization, WCSMO9, 13–17 June, 2011, Shizuoka, Japan, Book of Abstract, p 76 (Full paper on CD)
 Sokół, T, Lewiński, T (2011) Optimal design of a class of symmetric plane frameworks of least weight. Struct Multidisc Optim 44: pp. 729734 CrossRef
 Title
 New analytical benchmarks for topology optimization and their implications. Part I: bisymmetric trusses with two point loads between supports
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Structural and Multidisciplinary Optimization
Volume 46, Issue 4 , pp 477486
 Cover Date
 20121001
 DOI
 10.1007/s0015801207864
 Print ISSN
 1615147X
 Online ISSN
 16151488
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Exact topology optimization
 Michell trusses
 Optimal regions
 Symmetry
 Stress constraints
 Industry Sectors
 Authors

 Tomasz Sokół ^{(1)}
 George I. N. Rozvany ^{(2)}
 Author Affiliations

 1. Department of Structural Mechanics and Computer Aided Engineering, Faculty of Civil Engineering, Warsaw University of Technology, Al. Armii Ludowej 16, 00637, Warsaw, Poland
 2. Department of Structural Mechanics, Budapest University of Technology and Economics, Müegyetem rkpt. 3, Kmf. 35, 1521, Budapest, Hungary