Politecnico di Torino (logo)

Long span roofs : structural stability of partial gridshell with stiffned bounds

Lorenzo Polia

Long span roofs : structural stability of partial gridshell with stiffned bounds.

Rel. Luca Bruno, Fiammetta Venuti. Politecnico di Torino, Corso di laurea magistrale in Architettura Costruzione Città, 2015


Stability failure is one of the main issues concerning gridshell design. Since there are many types of potential failure modes and critical load are in fact influenced by several parameters, the need of an accurate research is fundamental. Member slenderness in steel gridshells is usually very high, and structural collapse is generally caused by buckling instead of material failure. Stability checks are therefore of great importance in a proper design process. Global and local stability of a hybrid single layer gridshell are also caused by geometric nodal imperfection, hard to be measured and controlled in experimental facilities.

Problem statement

Different structural models are adopted and the results are given in terms of two parameters: the Load Factor, that is a ratio of the theoretical design strength to the maximum load expected in service, and the Buckling Shape Length, a continuous measure of the degree of globalness of the instability. Moreover another parameter is introduced for each structural node, named nodal apex, defines global metrics of the geometry.

Some questions immediately follow the state of art:

- Do local maxima or minima exist in the trend of Load Factor versus the imperfection amplitude?

- The mechanical behavior of the imperfect grid shell is influenced by the correlation between the apex of the initial imperfect geometry and the apex of the deformed shape at collapse?

- Which is the influence of location and stiffness of the boundary structures on the stability of grid shells?


The buckling capacity is evaluated for two type of gridshell: the spherical cap, or dome, and for a cut geometry, with stiffed bounds introduced with the nomenclature "partial grid shell". The first objective is to map the existing grid shells with boundary structures and classify these structures according to their most significant parameters. After that, this work investigates with sensitivity analysis the stability of partial gridshells with imperfection to location and stiffness of the boundary structure. Analyzes are performed with numerical experiments through the so called Eigenmode Imperfection Method. The organizational set up for the whole process is presented as well, with in-depth illustration of the parametric tool used to model and analyze the structures.

Keywords: single layer grid shell, buckling instability\ equivalent geometric nodal imperfection, Eigenmode Imperfection Method, stiffened bounds, cut geometry structure, partial gridshell.


The present work can be divided into two main parts.

The first part is a State-of-art about grid shell typology, gives a definition of this type of structure. It gives also a synthetic overview of stability issues in reticulated shell design and analysis and a brief summary of the most popular analysis method adopted for the stability checks of reticulated. The second part includes a categorization of gridshell and an explanation of the parameters used, thus introducing the concept of partial gridshell. The Chapter 5 is focused on the parametric side of the set up for the modelling and on the analysis process. The main objective is to study the buckling behavior of a reticulated domes, involving variable imperfection amplitude.

Relators: Luca Bruno, Fiammetta Venuti
Publication type: Printed
Subjects: D Industrial Design and Applied Arts > DI Digital graphics
S Scienze e Scienze Applicate > SD Computer software
Corso di laurea: Corso di laurea magistrale in Architettura Costruzione Città
Classe di laurea: UNSPECIFIED
Aziende collaboratrici: UNSPECIFIED
URI: http://webthesis.biblio.polito.it/id/eprint/4431


a. problem statement (or open issue, anyway in dubitative form): which is the influence of location and stiffness of the boundary structures on the stability of grid shells? Let's introduce the nomenclature "partial grid shell"

b. objectives

i. map the existing grid shells with boundary structures

ii. sensitivity analysis of a partial grid shell with imperfection to location and stiffness of the boundary structure.

c. Thesis organization Two main parts.

Part I: A State-of-art about grid shell typology (chapter 2) and instability (chapter 3)

Part II: Original contents:

Categorization of the partial grid shells (objective i) (chapter 4) Sensitivity analysis (chapter 5)



a. Definition of gridshell as a doubly curved surface, form-resistant structure

b. Main design parameters (span-to-height ratio, shell thickness, grid type [e.g. tri, quad, hybrid, exa) and length] and some exemple, form (e.g. spherical, parabolic, elliptic caps, free forms)


c. general features of the Stability problem

d. grid shell stability

e. imperfect grid shell stability



a. rationale of the classification

b. examples


a. modelling approach setup description

b. definition of the observables

c. results




- Ansys, 2014 - multiphysics, release 14.0, help system, structural analysis guide (2014)

- David Bushnell - Buckling of shells - pitfall for designers, AIAA Journal, Vol. 19, No. 9, (1981)

- Otto Frei, Natürliche Konstruktionen (L'architettura della natura. Forme e costruzioni della natura e della tecnica e processi della loro formazione), Milano, Saggiatore (1984)

- David Wendland - Model-based formfinding processes: free forms in structural and architectural design (2000)

- Th. Bulenda & J. Knippers - Stability of gridshells - Computers & Structures 79, pp. 1161-1174 (2001)

- Carlos A. Felippa - Nonlinear finite elements methods (August 2001)

- Williams, C. J. K. - The analytic and numerical definition of the geometry of the British Museum Great Court Roof, pages 434-440. Mathematics & design 2001. Deakin University, Geelong, Victoria, Australia. Conference on Mathematics & Design, MD-2001. Deakin University, Geelong, Australia, July 3-5, 2001. 26, 28, 29 (2001)

- Pople, Nicolas - Off the grid: Weald and Downland gridshell. RIBA journal, No.109 (5), pp. 36-44. (2002)

- Helmut Pottmann, Michael Eigensatz, Amir Vaxman, Johannes Wallner-Architectural Geometry (2005)

- C. Douthe, O. Baverel, J.-F. Caron - Form-finding of a grid shell in composite materials -Journal of the international association for shell and spatial structures: IASS (December 2006)

- Hans Schober, Schlaich Ergermann und Partner - Freeform Glass Structures (2006)

- Ouroussoff, N. - A delicate glass roof with links to the past (2007)

- Uemuet Goerguelue - Beam theories, The difference between Euler-Bernoulli and Timoschenko (2010)

- Antony Darby, Tim Ibell and Mark Evernden - Innovative Use and Characterization of Polymers for Timber-Related Construction (2010)

- Feng Fan, Zhenggang Cao, Shizhao Shen - Elasto-plastic stability of single-layer reticulated shells (2010)

- Buro Happold - SMART Structural Solutions (2011)

- Keuning, D. - Sander and ney cover two historic buildings with glazed roofs (2011)

- Nicola Usai - Grandi strutture per il tempo libero. Trasformazione urbana e governance territoriale - edizioni Franco Angeli (2011)

- Matteo Dini - Form finding e ottimizzazione del buckling di grid shells con l'utilizzo di algoritmi genetici (2012)

- Sigrid Adriaenssens, M.ASCE, Laurent Ney Eric Bodarwe and Chris Williams - Finding the Form of an Irregular Meshed Steel and Glass Shell Based on Construction Constraints (2012)

- Kitti Gidófalvy, Levente Katula - Imperfection for the Buckling Analysis of Grid Shells - Second Conference of Junior Researchers in Civil Engineering - pp. 74-80 (2013)

- Carlos A. Felippa - Stability of structures: basic concepts (July 2013)

- Jianguo Cai, Leming Gu, Yixiang Xu, Jian Feng and Jin Zhang - Nonlinear stability analysis of hybrid grid shells (March 2013)

- Lina Bouhaya, Olivier BAVEREL, Jean-Fran_cois Caron - Optimization of gridshell bar orientation using a simplified genetic approach. Structural and Multidisciplinary Optimization, Springer Verlag (Germany), pp. 839-848 (2014).

- Luca Bruno, Mario Sassone, Fiammetta Venuti - Effects of the Equivalent Geometric Nodal Imperfections on the stability of Single Layer Grid Shells (2015)

Modify record (reserved for operators) Modify record (reserved for operators)