scholarly journals Form-Finding of Shells Containing Both Tension and Compression Using the Airy Stress Function

2020 ◽  
Author(s):  
Masaaki Miki ◽  
Emil Adiels ◽  
William Baker ◽  
Toby Mitchell ◽  
Alexander Sehlstrom ◽  
...  
Keyword(s):  
Stress Function ◽  
Mixed Type ◽  
Airy Stress Function ◽  
Form Finding ◽  
Graphic Statics ◽  
Tension And Compression ◽  
Stress Functions ◽  
Complete Set ◽  
Pure Compression ◽  
Asymptotic Lines

Pure-compression shells have been the central topic in the form-finding of shells. This paper studies tension-compression mixed type shells by utilizing a NURBS-based isogeometric form-finding approach that analyzes Airy stress functions to expand the possible plan geometry. A complete set of smooth version graphic statics tools is provided to support the analyses. The method is validated using examples with known solutions, and a further example demonstrates the possible forms of shells that the proposed method permits. Additionally, a guideline to configure a proper set of boundary conditions is presented through the lens of asymptotic lines of the stress functions.

scholarly journals Form-Finding of Shells Containing Both Tension and Compression Using the Airy Stress Function

2020 ◽  
Author(s):  
Masaaki Miki ◽  
Emil Adiels ◽  
William Baker ◽  
Toby Mitchell ◽  
Alexander Sehlstrom ◽  
...  
Keyword(s):  
Stress Function ◽  
Mixed Type ◽  
Airy Stress Function ◽  
Form Finding ◽  
Graphic Statics ◽  
Tension And Compression ◽  
Stress Functions ◽  
Complete Set ◽  
Pure Compression ◽  
Asymptotic Lines

Pure-compression shells have been the central topic in the form-finding of shells. This paper studies tension-compression mixed type shells by utilizing a NURBS-based isogeometric form-finding approach that analyzes Airy stress functions to expand the possible plan geometry. A complete set of smooth version graphic statics tools is provided to support the analyses. The method is validated using examples with known solutions, and a further example demonstrates the possible forms of shells that the proposed method permits. Additionally, a guideline to configure a proper set of boundary conditions is presented through the lens of asymptotic lines of the stress functions.

Design of tension structures and shells using the Airy stress function

2021 ◽  
pp. 095605992110641
Author(s):  
Alexander Sehlström ◽  
Karl-Gunnar Olsson ◽  
Chris JK Williams
Keyword(s):  
Stress Function ◽  
Bending Moment ◽  
Internal Stresses ◽  
Airy Stress Function ◽  
Membrane Action ◽  
Form Finding ◽  
Support Design ◽  
Beam Bending ◽  
Cable Nets ◽  
Curved Membrane

Discontinuities in the Airy stress function for in-plane stress analysis represent forces and moments in connected one-dimensional elements. We expand this representation to curved membrane-action structures, such as shells and cable nets, and graphically visualise the internal stresses and section forces at the boundary necessary for equilibrium. The approach enhances understanding of the interplay between form and forces and can support design decisions related to form-finding and force efficiency. As illustrative examples, the prestressing needed for three existing cable nets is determined, and its influence on the edge-beam bending moment is explored.

Stress Function Interface and Boundary Conditions in Anisotropic Materials

1982 ◽  
Vol 49 (4) ◽  
pp. 787-791 ◽  
Author(s):  
E. E. Gdoutos ◽  
M. Kattis
Keyword(s):  
Boundary Conditions ◽  
Stress Function ◽  
Anisotropic Media ◽  
Anisotropic Materials ◽  
Function Approach ◽  
Airy Stress Function ◽  
Stress Functions ◽  
The Media

The stress and displacement continuity conditions for interfaces between two different anisotropic media were formulated in terms of the Airy stress functions of the media. It was shown that such formulation greatly facilitates the solution of the problems of composite anisotropic materials by the Airy stress function approach. Two examples were given to demonstrate the potentiality of the method.

scholarly journals Analytical graphic statics

2021 ◽  
pp. 095605992110016
Author(s):  
Tamás Baranyai
Keyword(s):  
Stress Function ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Linear Functionals ◽  
The Past ◽  
Force Diagram ◽  
Graphic Statics ◽  
Reciprocal Diagrams ◽  
Force Systems ◽  
Stress Functions

Graphic statics is undergoing a renaissance, with computerized visual representation becoming both easier and more spectacular as time passes. While methods of the past are revived, little emphasis has been placed on studying the mathematics behind these methods. Due to the considerable advances of our mathematical understanding since the birth of graphic statics, we can learn a lot by examining these old methods from a more modern viewpoint. As such, this work shows the mathematical fabric joining different aspects of graphic statics, like dualities, reciprocal diagrams, and discontinuous stress functions. This is done by introducing a new, three dimensional force diagram (containing the old two dimensional force diagram) depicting the three dimensional equilibrium of planar force systems. A corresponding three dimensional “form diagram” (dual diagram) is introduced, in which forces are treated as linear functionals (dual vectors). It is shown that the polyhedral stress function introduced by Maxwell is in fact a linear combination of these functionals; and the projective dualities connecting these three dimensional diagrams are also explained.

The Sector Problem

1957 ◽  
Vol 24 (4) ◽  
pp. 574-581
Author(s):  
G. Horvay ◽  
K. L. Hanson
Keyword(s):  
Variational Method ◽  
Orthogonal Polynomials ◽  
Biharmonic Equation ◽  
Approximate Solutions ◽  
Circular Sector ◽  
Orthonormal Polynomials ◽  
Circular Boundary ◽  
Stress Functions ◽  
Complete Set ◽  
Derivatives Of

Abstract On the basis of the variational method, approximate solutions f k ( r ) h k ( θ ) , f k ( r ) g k ( θ ) , F k ( θ ) H k ( r ) , F k ( θ ) G k ( r ) of the biharmonic equation are established for the circular sector with the following properties: The stress functions fkhk create shear tractions on the radial boundaries; the stress functions fkgk create normal tractions on the radial boundaries; the stress functions FkHk create both shear and normal tractions on the circular boundary, and the stress functions FkGk create normal tractions on the circular boundary. The enumerated tractions are the only tractions which these function sets create on the various boundaries of the sector. The factors fk(r) constitute a complete set of orthonormal polynomials in r into which (more exactly, into the derivatives of which) self-equilibrating normal or shear tractions applied to the radial boundaries of the sector may be expanded; the factors Fk(θ) constitute a complete set of orthonormal polynomials in θ into which shear tractions applied to the circular boundary of the sector may be expanded; and the functions Fk″ + Fk constitute a complete set of non-orthogonal polynomials into which normal tractions applied to the circular boundary of the sector may be expanded. Function tables, to facilitate the use of the stress functions, are also presented.

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