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Structural and Multidisciplinary Optimization

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08-11-2017 | RESEARCH PAPER

Form-finding of grid-shells using the ground structure and potential energy methods: a comparative study and assessment

Authors: Yang Jiang, Tomás Zegard, William F. Baker, Glaucio H. Paulino

Published in: Structural and Multidisciplinary Optimization | Issue 3/2018

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Abstract

The structural performance of a grid-shell depends directly on the geometry of the design. Form-finding methods, which are typically based on the search for bending-free configurations, aid in achieving structurally efficient geometries. This manuscript proposes two form-finding methods for grid-shells: one method is the potential energy method, which finds the form in equilibrium by minimizing the total potential energy in the system; the second method is based on an augmented version of the ground structure method, in which the load application points become variables of the topology optimization problem. The proposed methods, together with the well-known force density method, are evaluated and compared using numerical examples. The advantages and drawbacks of the methods are reviewed, compared and highlighted.

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Minor concavities can be handled, but are generally not advised since they may introduce structural instabilities in the node–bar network.

Each inner member forms the edge of 2 load panels while each boundary members forms the edge of 1 load panel.

The load could depend on the displacements or the cross-sections, but because the displacements are small, they are assumed constant for the sake of simplicity.

Considering the material’s yield limit, and optionally including considerations for buckling or other criteria.

There are two common versions of the GSM: one based on the plastic formulation, and one based on the elastic formulation. Only the plastic formulation can be formulated as a linear programming problem. The elastic formulation (Bendsøe et al., 1994; Christensen and Klarbring 2009; Ramos and Paulino 2015), despite being computationally more expensive, considers compatibility and can include material and geometric non-linearities to name a few features. The present work uses the plastic formulation in the form-finding process.

Refer to Zegard and Paulino (2014) for details on the effect of the collinear tolerance on the ground structure generation process.

ACI Committee 318, American Concrete Institute (2014) Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary (ACI 318R-14)

Addis B (2007) Building: 3000 years of design, engineering and construction. Phaidon Press, London

Akbarzadeh M, Van Mele T, Block P (2015) Spatial compression-only form finding through subdivision of external force polyhedron. In: Proceedings of the international association for shell and spatial structures (IASS). Symposium, Amsterdam

American Institute of Steel Construction (2011) Steel Construction Manual. 14th Edn

Argyris JH, Angelopoulos T, Bichat B (1974) A general method for the shape finding of lightweight tension structures. Comput Methods Appl Mech Eng 3:135–149CrossRef

Baker WF (1992) Energy-based design of lateral systems. Struct Eng Int 2(2):99–102CrossRef

Baker WF, Beghini LL, Mazurek A, Carrion J, Beghini A (2013) Maxwell’s reciprocal diagrams and discrete Michell frames. Struct Multidiscip Optim 48:267–277MathSciNetCrossRef

Barnes MR (1977) Form-finding and analysis of tension space structures by dynamic relaxation. PhD Thesis, City University London

Barnes MR (1988) Form-finding and analysis of prestressed nets and membranes. Comput Struct 30(3):685–695MathSciNetCrossRef

Barnes MR, Topping BHV, Wakefield DS (1977) Aspects of form-finding by dynamic relaxation. In: International conference on the behaviour of slender structures

Bendsøe MP, Ben-Tal A, Zowe J (1994) Optimization methods for truss geometry and topology design. Struct Multidiscip Optim 7(3):141–159CrossRef

Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods, and applications. Springer, BerlinMATH

Bergós J, Llimargas M (1999) Gaudí: The Man and His Work. Taschen, Köln

Bletzinger K-U, Ramm E (1993) Form finding of shells by structural optimization. Eng Comput 9(1):27–35CrossRef

Bletzinger K-U, Ramm E (1999) A general finite element approach to the form finding of tensile structures by the upyeard reference strategy. Int J Space Struct 14(2):131–145CrossRef

Bletzinger K-U, Ramm E (2001) Structural optimization and form finding of light weight structures. Comput Struct 79(22):2053–2062CrossRef

Bletzinger K-U, Wüchner R, Daoud F, Camprubí N (2005) Computational methods for form finding and optimization of shells and membranes. Comput Methods Appl Mech Eng 194(30):3438–3452MathSciNetCrossRefMATH

Block P (2009) Thrust network analysis: exploring three-dimensional equilibrium. PhD thesis, Massachusetts Institute of Technology, USA

Block P, Ochsendorf J (2007) Thrust network analysis: a new methodology for three-dimensional equilibrium. J Intern Assoc Shell Spatial Struct 48(3):167–173

Chiandussi G, Codegone M, Ferrero S (2009) Topology optimization with optimality criteria and transmissible loads. Comput Math Appl 57(5):772–788MathSciNetCrossRefMATH

Chilton J (2010) Heinz Isler’s infinite spectrum form-finding in design. Archit Des 80(4):64–71

Christensen PW, Klarbring A (2009) An introduction to structural optimization, vol 153 of solid mechanics and its applications. Springer Netherlands, Dordrecht

Coelho RF, Tysmans T, Verwimp E (2014) Form finding & structural optimization. Struct Multidiscip Optim 49(6):1037–1046CrossRef

Collins GR (1977) The Drawings of Antonio gaudí. The drawing center, New York

Darwich W, Gilbert M, Tyas A (2010) Optimum structure to carry a uniform load between pinned supports. Struct Multidiscip Optim 42(1):33–42CrossRef

Dorn W, Gomory R, Greenberg H (1964) Automatic design of optimal structures. J de mécanique 3 (1):25–52

Fuchs MB, Moses E (2000) Optimal structural topologies with transmissible loads. Struct Multidiscip Optim 19(4):263– 273CrossRef

Gallagher RH, Zienkiewicz OC (1973) Optimum structural design: theory and applications. Wiley, LondonMATH

Gilbert M, Darwich W, Tyas A, Shepherd P (2005) Application of large-scale layout optimization techniques in structural engineering practice. In: 6th world congress of structural and multidisciplinary optimization, June 1–10

Haber R, Abel J (1982) Initial equilibrium solution methods for cable reinforced membranes part I-formulations. Comput Methods Appl Mech Eng 30(3):263–284CrossRefMATH

Hemp W (1973) Optimum structures, 1st edn. Oxford University Press, Oxford

International Code Council (2015) 2015 International Building Code

Jiang Y (2015) Free form finding of grid shell structures. Master’s thesis, University of Illinois at Urbana-Champaign, USA

Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4(4):373–395MathSciNetCrossRefMATH

Kilian A, Ochsendorf J (2006) Particle spring systems for structural form finding. J Intern Assoc Shell Spatial Struct 46(2):77– 84

Leon SE, Paulino GH, Pereira A, Menezes IFM, Lages EN (2011) A unified library of nonlinear solution schemes. Appl Mech Rev 64(4):040803CrossRef

Lewiński T, Rozvany GIN, Sokół T, Bołbotowski K (2013) Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains revisited. Struct Multidiscip Optim 47 (6):937–942MathSciNetCrossRefMATH

Lewiński T, Zhou M, Rozvany GIN (1994) Extended exact solutions for least-weight truss layouts—Part I: cantilever with a horizontal axis of symmetry. Int J Mech Sci 36(5):375–398CrossRefMATH

Lewis WJ (2003) Tension structures: form and behaviour. Thomas Telford Publishing, LondonCrossRef

Linkwitz K, Schek H (1971) Einige Bemerkungen zur Berechnung von vorgespannten Seilnetzkonstruktionen. Ingenieur-Archiv 40(3):145–158CrossRef

Martinell C, Collins GR, Rohrer J (1975) Gaudí: his life, his theories, his work. MIT Press, Cambridge

Maxwell JC (1890) In: Niven WD (ed) The scientific papers of James Clerk Maxwell. Library Collection, Cambridge

Michell A (1904) The limits of economy of material in frame structures. Phil Mag 8(47):589–597CrossRefMATH

Miki M, Adriaenssens S, Igarashi T, Kawaguchi K (2014) The geodesic dynamic relaxation method for problems of equilibrium with equality constraint conditions. Int J Numer Methods Eng 99:682–710MathSciNetCrossRefMATH

Mitchell T (2013) A limit of eonomy of material in shell structures. PhD thesis, University of California Berkeley, USA

Nouri-Baranger T (2004) Computational methods for tension-loaded structures. Arch Comput Meth Eng 11 (2003):143– 186CrossRefMATH

Pauletti RMO, Pimenta PM (2008) The natural force density method for the shape finding of taut structures. Comput Methods Appl Mech Eng 197(49):4419–4428CrossRefMATH

Pichugin A, Tyas A, Gilbert M (2012) On the optimality of Hemp’s arch with vertical hangers. Struct Multidiscip Optim 46(1):17– 25MathSciNetCrossRefMATH

Ramos AS, Paulino GH (2015) Convex topology optimization for hyperelastic trusses based on the ground-structure approach. Struct Multidiscip Optim 51(2):287–304MathSciNetCrossRef

Richardson JN, Adriaenssens S, Coelho RF, Bouillard P (2013) Coupled form-finding and grid optimization approach for single layer grid shells. Eng Struct 52:230–239CrossRef

Rozvany GIN (2001) On design-dependent constraints and singular topologies. Struct Multidiscip Optim 21 (2):164–172MathSciNetCrossRef

Rozvany GIN, Gollub W, Zhou M (1997) Exact Michell layouts for various combinations of line supports-Part II. Struct Optim 14(2-3):138–149CrossRef

Rozvany GIN, Prager W (1979) A new class of structural optimization problems: optimal archgrids. Comput Methods Appl Mech Eng 19(1):127–150MathSciNetCrossRefMATH

Rozvany GIN, Sokół T (2013) Validation of numerical methods by analytical benchmarks, and verification of exact solutions by numerical methods. Topology Optimization in Structural and Continuum Mechanics

Rozvany GIN, Wang CM (1983) On plane Prager-structures—I. Int J Mech Sci 25(7):519–527CrossRefMATH

Rozvany GIN, Wang CM, Dow M (1982) Prager-structures: Archgrids and cable networks of optimal layout. Comput Methods Appl Mech Eng 31(1):91–113MathSciNetCrossRefMATH

Sánchez J, Serna MÁ, Morer P (2007) A multi-step force–density method and surface-fitting approach for the preliminary shape design of tensile structures. Eng Struct 29(8):1966– 1976CrossRef

Schek HJ (1974) The force density method for form finding and computation of general networks. Comput Methods Appl Mech Eng 3(1):115–134MathSciNetCrossRef

Siev A, Eidelman J (1964) Stress analysis of prestressed suspended roofs. J Struct Div 90(4):103–122

Sokół T (2011) A 99 line code for discretized Michell truss optimization written in Mathematica. Struct Multidiscip Optim 43(2):181–190CrossRefMATH

Sokół T (2014) Multi-load truss topology optimization using the adaptive ground structure approach. In: Łodygowski T, Rakowski J, Litewka P (eds) Recent advances in computational mechanics. CRC Press, Boca Raton, pp 9–16

Sokół T, Rozvany GIN (2013) On the adaptive ground structure approach for the multi-load truss topology optimization. In: 10th world congress on structural and multidisciplinary optimization, pp 1–10

Tabarrok B, Qin Z (1992) Nonlinear analysis of tension structures. Comput Struct 45(5):973–984CrossRefMATH

Talischi C, Paulino GH, Pereira A, Menezes IFM (2012) Polymesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct Multidiscip Optim 45:309–328MathSciNetCrossRefMATH

Thrall AP, Billington DP, Bréa KL (2012) The Maria Pia Bridge: A major work of structural art. Eng Struct 40:479–486CrossRef

Tyas A, Pichugin AV, Gilbert M (2010) Optimum structure to carry a uniform load between pinned supports: exact analytical solution. Proc R Soc A Math Phys Eng sciences 467(2128):1101–1120MathSciNetCrossRefMATH

Veenendaal D, Block P (2012) An overview and comparison of structural form finding methods for general networks. Int J Solids Struct 49(26):3741–3753CrossRef

Wang CM, Rozvany GIN (1983) On plane prager-structures—II: non-parallel external loads and allowances for selfweight. Int J Mech Sci 25(7):529–541CrossRefMATH

Wright M (2004) The interior-point revolution in optimization: history, recent developments, and lasting consequences. Bull Am Math Soc 42(01):39–56MathSciNetCrossRefMATH

Yang XY, Xie YM, Steven GP (2005) Evolutionary methods for topology optimisation of continuous structures with design dependent loads. Comput Struct 83(12-13):956–963CrossRef

Zalewski W, Allen E (1997) Shaping structures: statics. Wiley, New York

Zegard T (2014) Structural optimization: from continuum and ground structures to additive manufacturing. PhD thesis, University of Illinois at Urbana-Champaign, USA

Zegard T, Paulino GH (2014) GRAND — Ground Structure based topology optimization for arbitrary 2D domains using MATLAB. Struct Multidiscip Optim 50(5):861–882MathSciNetCrossRef

Zegard T, Paulino GH (2015) GRAND3 — Ground Structure based topology optimization for arbitrary 3D domains using MATLAB. Struct Multidiscip Optim 52(6):1161–1184CrossRef

Zhang Y (1996a) Solving large-scale linear programs by interior-point methods under the MATLAB Environment. Technical report, Department of Mathematics and Statistics University of Maryland, Baltimore, USA

Zhang Y (1996b) Solving large-scale linear programs by interior-point methods under the Matlab Environment. Optim Methods Softw 10(1):1–31

Title: Form-finding of grid-shells using the ground structure and potential energy methods: a comparative study and assessment
Authors: Yang Jiang
Tomás Zegard
William F. Baker
Glaucio H. Paulino
Publication date: 08-11-2017
Publisher: Springer Berlin Heidelberg
Published in: Structural and Multidisciplinary Optimization / Issue 3/2018
Print ISSN: 1615-147X
Electronic ISSN: 1615-1488
DOI: https://doi.org/10.1007/s00158-017-1804-3

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