Abstract
A numerical form-finding procedure, based on Isogeometric Analysis (IgA), is proposed to determine the elevation of a shell having a prescribed covered area so as to carry applied loads in a pure membrane state of stress. The boundary-value problem of a membrane shell is described by Pucher’s equation in terms of external load, Airy potential and shell mid-surface height. Membrane stress components at a point are computed via second derivatives of the Airy potential. Within the IgA framework, the equivalent weak form of the problem is derived and the resulting integral equation is discretized by approximating the relevant fields as a linear combination of control point values and B-Spline basis functions. Once the external load is prescribed, control point values of the Airy potential are computed via a nonlinear optimization routine aimed at minimizing thrusts at edge supports while ensuring a no-traction principal stress state fulfilling boundary conditions. The resulting shell form is then computed in terms of control point heights. Two numerical examples show the effectiveness of the proposed approach.
Supported by the University of Naples Federico II and the Compagnia di San Paolo, FRA grants, CUP E69C21000250005.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Projections are intended to be taken on the (x,y) Cartesian plane.
References
Veenendaal, D., Block, P.: An overview and comparison of structural form finding methods for general networks. Int. J. of Solids Struct. 49, 3741–3753 (2012)
Li, Q., Su, Y., Wu, Y., Borgart, A.: Form-finding of shell structures generated from physical models. Int. J. Space Struct. 32(3), 11–33 (2017)
Chen, L., Nguyen-Than, N., Rabczuk, T., Bordas, S., Limbert, G.: Explicit finite deformation analysis of isogeometric membranes. Comput. Meth. in Appl. Mech. Eng. 277, 104–130 (2014)
Nguyen, N., Rabczuk, T., Nguyen-Xuan, H., Bordas, S.: A smoothed finite element method for shell analysis. Comp. Meth. Appl. Mech. Eng. 198(2), 165–177 (2008)
Nguyen, N., Hien, T., Nguyen-Thoi, T., Lee, J.: A unified adaptive approach for membrane structures: form finding and large deflection isogeometric analysis. Comp. Meth. Appl. Mech. Eng. 369, 113239 (2020)
Schek, H.J.: The force density method for form finding and computation of general networks. Comput. Methods Appl. Mech. Eng. 3(1), 115–134 (1974)
Marmo, F., et al.: On the form of the Musmeci’s bridge over the Basento river. Eng. Struct. 191, 658–673 (2019)
Block, P., Ochsendorf, J.: Thrust network analysis: a new methodology for three-dimensional equilibrium. J. Int. Ass. Shell Spat. Struct. 48(3), 8 (2007)
Marmo, F., Vaiana, N.: Form finding of shell structures by using membrane theory. In: Marmo, F., Sessa, S., Barchiesi, E., Spagnuolo, M. (eds.) Mathematical Applications in Continuum and Structural Mechanics. ASM, vol. 127, pp. 213–237. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-42707-8_11
Pucher, A.: Über den spannungszustand in gekrümmten flächen. Beton Eisen 33, 298 (1934)
Manuello, A.: Multi-body rope approach for grid shells: form-finding and imperfection sensitivity. Eng. Struct. 221, 111029 (2020)
Manuello, A., Riberi, F.: Form-finding of pierced vaults and digital fabrication of scaled prototypes. Curved Layered Struct. 8, 210–224 (2021)
Kilian, A., Ochsendorf, J.: Particle-spring systems for structural form finding. J. Int. Ass. Shell Spat. Struct. 46(148), 77–84 (2005)
Ye, J., Feng, R., Zhuo, S.: The modified dynamic relaxation method for the form-finding of membrane structures. J. Comput. Theoret. Nanosci. 4(8–10), 2845–2853 (2011)
Adriaenssens, S., Rhode-Barbarigos, L.: Form-finding and analysis of bending-active systema using dynamic relaxation. In: VI International Conference on Textile Composites and Inflatable Structures (2013)
Hilmersson, J., Larsson, F., Olsson, J., Ander, M.: Isogeometric analysis and form finding for thin elastic shells. In: Proceedings of IASS Annual Symposia: Contributions in Memory of Mike Barnes, pp. 1-8(8) (2019)
Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric analysis: towards integration of CAD and FEA. Wiley (2009)
Xia, Y., Mantzaflaris, A., Juttler, B., Pan, H., Hu, P., Wang, W.: Design of self-supporting surfaces with isogeometric analysis. Comput. Methods Appl. Mech. Eng. 353, 328–347 (2019)
Miki, M., Igarashi, T.: Parametric self-supporting surfaces via direct computation of airy stress functions. ACM Trans. Graph. 34(4), 89 (2016)
Alic, V., Persson, K.: Form finding with dynamic relaxation and isogeometric membrane elements. Comput. Methods Appl. Mech. Eng. 300, 734–747 (2016)
Milosevic, J., Kovacevic, S.: Form finding of shell structures based on isogeometric analysis. Izgradnja 70, 353–360 (2016)
Funding and Acknowledgments
The present research was supported by the University of Naples Federico II and the Compagnia di San Paolo, which are gratefully acknowledged by the authors, FRA grants, CUP E69C21000250005.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Chianese, C., Marmo, F., Rosati, L. (2024). Form-Finding of Membrane Shells via Isogeometric Analysis. In: Gabriele, S., Manuello Bertetto, A., Marmo, F., Micheletti, A. (eds) Shell and Spatial Structures. IWSS 2023. Lecture Notes in Civil Engineering, vol 437. Springer, Cham. https://doi.org/10.1007/978-3-031-44328-2_5
Download citation
DOI: https://doi.org/10.1007/978-3-031-44328-2_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-44327-5
Online ISBN: 978-3-031-44328-2
eBook Packages: EngineeringEngineering (R0)