This research paper introduces a theoretical framework for the design and analysis of compression-and-tension grid-shells in static equilibrium where the states of self-stress function as design freedoms. This is based on a synthesis of reciprocal discrete Airy stress functions in the context of graphic statics and the Force Density Method (FDM). Specifically, the former is a direct method for generating 2-dimensional global static equilibrium whereas the latter allows for its 3-dimensional implementation. As a result, creative design explorations can take place directly within the equilibrium space without the need for iterative convergence algorithms to obtain equilibrium. The use of reciprocal Airy stress functions in conjunction with the lower bound theorem gives insight and explicit control with regards to the states of self-stress as design and analysis freedoms which can define the structural form and its load path.
LachauerLBlockP. Interactive equilibrium modelling. Int J Space Struct2014; 29: 25–37.
2.
RippmannMLachauerLBlockP. Interactive vault design. Int J Space Struct2012; 27: 219–230.
3.
OhlbrockPOD’AcuntoP. A computer-aided approach to equilibrium design based on graphic statics and combinatorial variations. Comput Des2020; 121: 102802. DOI: 10.1016/j.cad.2019.102802
4.
KurrerKE. The history of the theory of structures, from arch analysis to computational mechanics. Berlin: Ernst & Sohn, 2008.
5.
CharltonTM. A history of theory of structures in the nineteenth century. Cambridge: Cambridge University Press, 1982.
6.
MaxwellJC. XLV. On reciprocal figures and diagrams of forces. Lond Edinb Dubl Philos Mag J Sci1864; 27: 250–261.
7.
MaxwellJC. I. —On reciprocal figures, frames, and diagrams of forces. Trans R Soc Edinb Earth Sci1870; 26: 1–40.
8.
AiryGB. On the strains in the interior of beams. Philos Trans R Soc Lond1862; 153: 49–79.
9.
TimoshenkoSGoodierJN. Theory of elasticity. New York: McGraw-Hill, 1951.
10.
AkbarzadehMVan MeleTBlockP. On the equilibrium of funicular polyhedral frames and convex polyhedral force diagrams. Comput Des2015; 63(Suppl C): 118–128.
11.
LeeJMeleTVBlockP. Disjointed force polyhedra. Comput Des2018; 99: 11–28.
12.
D’AcuntoPJasienskiJPOhlbrockPO, et al. Vector-based 3D graphic statics: a framework for the design of spatial structures based on the relation between form and forces. Int J Solids Struct2019; 167: 58–70.
13.
McRobieABakerWMitchellT, et al. Mechanisms and states of self-stress of planar trusses using graphic statics, part II: applications and extensions. Int J Space Struct2016; 31: 102–111.
14.
MitchellTBakerWMcRobieA, et al. Mechanisms and states of self-stress of planar trusses using graphic statics, part I: the fundamental theorem of linear algebra and the airy stress function. Int J Space Struct2016; 31: 85–101.
15.
McRobieA. Maxwell and Rankine reciprocal diagrams via Minkowski sums for two-dimensional and three-dimensional trusses under load. Int J Space Struct2016; 31: 203–216.
16.
KonstantatouMD’AcuntoPMcRobieA. Polarities in structural analysis and design: n-dimensional graphic statics and structural transformations. Int J Solids Struct2018; 152-153: 272–293.
17.
McRobieA. Correction to ‘The geometry of structural equilibrium’. R Soc Open Sci2017; 4(5): 170338.
18.
FraternaliF. A thrust network approach to the equilibrium problem of unreinforced masonry vaults via polyhedral stress functions. Mech Res Commun2010; 37: 198–204.
19.
MikiMIgarashiTBlockP. Parametric self-supporting surfaces via direct computation of airy stress functions. ACM Trans Graph2015; 34(4): 1–89:12.
20.
MikiMAdielsEBakerW, et al. Form-finding of shells containing both tension and compression using the airy stress function. Preprint2020. DOI: 10.20944/preprints202012.0355.v2.
21.
WilliamsC. The generation of a class of structural forms for vaults and sails. Struct Eng1990; 68(12): 231–235.
22.
VougaEHöbingerMWallnerJ, et al. Design of self-supporting surfaces. ACM Trans Graph2012; 31(4): 1–11.
23.
PellisDPottmannH. Aligning principal stress and curvature directions. In: HesselgrenLKilianAMalekS, et al. (eds) Advances in architectural geometry. Vienna: Klein Publishing GmbH, 2018, pp. 34–53.
24.
LinkwitzKSchekHJ. Einige Bemerkungen zur Berechnung von vorgespannten Seilnetzkonstruktionen. Ing Arch1971; 40: 145–158.
25.
SchekHJ. The force density method for form finding and computation of general networks. Comput Methods Appl Mech Eng1974; 3(1): 115–134.
26.
SouthwellRV. On the calculation of stresses in braced frameworks. Proc R Soc Lond1933; A139: 475–507.
27.
AdriaenssensSBlockPVeenendaalD, et al. Shell structures for architecture: form finding and optimization. New York: Routledge, 2014.
28.
LewisWJ. Tension structures: form and behaviour. London: Thomas Telford, 2003.
29.
LiddellI. Frei Otto and the development of gridshells. Case Stud Struct Eng2015; 4: 39–49.
30.
BlockPOchsendorfJ. Thrust network analysis: a new methodology for three-dimensional equilibrium. J Int Assoc Shell Spatial Struct2007; 48: 167–173.
MikiMKawaguchiK. Extended force density method for form-finding of tension structures. J Int Assoc Shell Spatial Struct2010; 51(4): 291–303.
33.
ZhangJYOhsakiM. Adaptive force density method for form-finding problem of tensegrity structures. Int J Solids Struct2006; 43: 5658–5673.
34.
MalerbaPGPatelliMQuagliaroliM. An extended force density method for the form finding of cable systems with new forms. Struct Eng Mech2012; 42: 191–210.
35.
TachiT. Interactive freeform design of tensegrity. In: HesselgrenLSharmaSWallnerJ, et al. (eds) Advances in architectural geometry. Vienna: Springer, 2012, pp. 269–278.
36.
ZanniGPennockGR. A unified graphical approach to the static analysis of axially loaded structures. Mech Mach Theory2009; 44: 2187–2203.
37.
HeymanJ. The stone skeleton. Cambridge: Cambridge University Press, 1995.
38.
WilliamsC. The analytic and numerical definition of the geometry of the British Museum great Court roof. In: BurryMDattaSDawsonA, et al. (eds) Mathematics & design. Geelong, Victoria: Deakin University, 2001, pp.434–440.
