Structural Form ARK-A3001 Design of Structures_Basics Lecture 5. (De)formation Bending. Toni Kotnik

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1 Structural Form ARK-A3001 Design of Structures_Basics Toni Kotnik Professor of Design of Structures Aalto University Department of Architecture Department of Civil Engineering (De)formation Bending AA/ETH Pavilion Stefano-Franscini Plaza Zurich, Switzerland,

2 Cantilever ideal cantilever real cantilever stiffening of cantilever 3 Frank Lloyd Wright Kaufman House Bear Run, USA,

3 Frank Lloyd Wright Kaufman House Bear Run, USA, Frank Lloyd Wright Kaufman House Bear Run, USA,

4 Frank Lloyd Wright Kaufman House Bear Run, USA, Frank Lloyd Wright Kaufman House Bear Run, USA,

5 Frank Lloyd Wright Kaufman House Bear Run, USA, Frank Lloyd Wright Kaufman House Bear Run, USA,

6 Frank Lloyd Wright Kaufman House Bear Run, USA, Frank Lloyd Wright Kaufman House Bear Run, USA,

7 Frank Lloyd Wright Kaufman House Bear Run, USA, Frank Lloyd Wright Kaufman House Bear Run, USA,

8 Frank Lloyd Wright Kaufman House Bear Run, USA, Cantilever 16 8

9 Post-Tensioning of Cantilever Frank Lloyd Wright Kaufman House Bear Run, USA, The resistant virtues of the structures that we make depend on their form. It is through their form that they are stable and not because of an awkward accumulation of materials. There is nothing more noble and elegant from an intellectual point of view than this, resistance through form. Eladio Dieste Sergio Musmeci Basento Viaduct Potenza, Italy,

10 Stabilization by Material Accumulation 19 Stabilization by Form 20 10

11 Form in 3d Gaussian Curvature Κ = Κ 1 Κ 2 Gaussian curvature is the product of the principal curvature and is a measurement of the local behaviour of the surface Κ < 0 hyperbolic point curves in opposite direction Κ = 0 parabolic point at least one curve is flat Κ > 0 elliptic point curves in same direction Κ > 0 elliptic point prototype: sphere Κ = 0 parabolic point prototype: cylinder 21 Κ < 0 hyperbolic point prototype: hyperboloid From Curve to Curvature Borromini: San Carlo alle Quattro Fontane Rome, Italy,

12 From Curve to Curvature Borromini: San Carlo alle Quattro Fontane Rome, Italy, Parametric Design Curves and Surfaces From Curve to Curvature Herzog & de Meuron: University Library Cottbus, Germany, Parametric Design Curves and Surfaces

13 From Curve to Curvature Herzog & de Meuron: University Library Cottbus, Germany, Herzog & de Meuron: University Library, Cottbus, Germany, Parametric Design Curves and Surfaces From Curve to Curvature circles as well-known and easy-to-construct curvy curves with radius as measurement for curvature! 26 Parametric Design Curves and Surfaces

14 Curvature for a circle the curvature к is defined as the invers of the radius r к = 1/r r P c the curvature к is a measure for the roundness of the circle; by means of the limit circle r(t) the local behaviour of a curve at point c(t) can be approximated c к(t) = 1/r(t) P 27 Parametric Design Curves and Surfaces Curvature the limit circle is unique in the case of smooth curves like NURBS and enables the definition of a curvature graph for a curve c 28 Parametric Design Curves and Surfaces

15 Joining Curves different degrees of smoothness of joining two curves are possible dependent on the continuity of the curvature graph c 4 same tangent direction same curvature value same tangent direction different curvature direction c 5 c 3 c 1 different tangent direction different curvature direction same tangent direction different curvature value c 2 29 Parametric Design Curves and Surfaces Frenet-Frame based on the limit circle a point P = c(t) a local coordinate-system at P can be defined bi-normal direction tangent plane tangent direction normal direction orientation by right-hand rule 30 Parametric Design Curves and Surfaces

16 Normal & Principal Curvature tangent plane at P intersection curve of normal plane and surface normal plane to X N X c normal curvature at P into direction X P r limit circle principal curvature at P 31 Parametric Design Curves and Surfaces Principal Curvature Curve c 2 c 1 P 32 Parametric Design Curves and Surfaces

17 Principal Curvature Curve Peter Cook & Colin Fournier: Art Museum Graz, Austria, Parametric Design Curves and Surfaces Principal Curvature Curve Peter Cook & Colin Fournier: Art Museum Graz, Austria, Parametric Design Curves and Surfaces

18 Principal Curvature Curve Peter Cook & Colin Fournier: Art Museum Graz, Austria, Parametric Design Curves and Surfaces Layou of Panels subdivision with respect to coordinate axis results in unresolved corner principal curvature curves for an ellipsoid 36 Parametric Design Curves and Surfaces

19 Form in 3d Gaussian Curvature Κ = Κ 1 Κ 2 Gaussian curvature is the product of the principal curvature and is a measurement of the local behaviour of the surface Κ < 0 hyperbolic point curves in opposite direction Κ = 0 parabolic point at least one curve is flat Κ > 0 elliptic point curves in same direction Κ > 0 elliptic point prototype: sphere Κ > 0 elliptic point collection of arches Κ > 0 elliptic point collection of cables 37 Form in 3d Gaussian Curvature Κ = Κ 1 Κ 2 Gaussian curvature is the product of the principal curvature and is a measurement of the local behaviour of the surface Κ < 0 hyperbolic point curves in opposite direction Κ = 0 parabolic point at least one curve is flat Κ > 0 elliptic point curves in same direction Κ = 0 parabolic point prototype: cylinder Κ = 0 parabolic point collection of arch & beam Κ = 0 parabolic point collection of cable & beam 38 19

20 Form in 3d Gaussian Curvature Κ = Κ 1 Κ 2 Gaussian curvature is the product of the principal curvature and is a measurement of the local behaviour of the surface Κ < 0 hyperbolic point curves in opposite direction Κ = 0 parabolic point at least one curve is flat Κ > 0 elliptic point curves in same direction Κ < 0 hyperbolic point prototype: hyperboloid Κ < 0 hyperbolic point combination of arch & cable 39 Form in 3d Gaussian Curvature bracing effect cable & arch stabilize each other in a hyperbolic point hyperbolic points stiffen a surface Κ < 0 hyperbolic point prototype: hyperboloid 40 20

21 Hyperbolic Surface in Nature Imogen Cunnigham: Calla Lilly, before Hyperbolic Surface in Architecture Hyperbolic paraboloid 42 21

22 Hyperbolic Surface in Architecture Félix Candela: Lomas de Cuernavaca Chapel Morelos, Mexico, Hyperbolic Surface in Architecture Félix Candela: Lomas de Cuernavaca Chapel Morelos, Mexico, 1959 Félix Candela: Lomas de Cuernavaca Chapel, Morelos, Mexico,

23 Hyperbolic Surface in Architecture Félix Candela: Lomas de Cuernavaca Chapel Morelos, Mexico, 1959 Félix Candela: Lomas de Cuernavaca Chapel, Morelos, Mexico, Hyperbolic Paraboloid Modularization 46 23

24 Hyperbolic Paraboloid Modularization 47 Hyperbolic Paraboloid Modularization 48 24

25 Hyperbolic Paraboloid Modularization 49 Hyperbolic Paraboloid Modularization 50 25

26 Hyperbolic Paraboloid Modularization 51 Hyperbolic Paraboloid Modularization Félix Candela: Coyocán Market, Mexico City, Mexico, 1955 Félix Candela: High Life Textile Factory, Mexico City, Mexico,

27 Hyperbolic Paraboloid Transformation Félix Candela: Factory Gate, Lederle Laboratories, Mexico City, Mexico, Hyperbolic Paraboloid Transformation 54 27

28 Hyperbolic Paraboloid Transformation 55 Félix Candela: Church of Our Lady of the Miraculous Medal, Mexico City, Mexico,

29 Félix Candela: Church of Our Lady of the Miraculous Medal, Mexico City, Mexico, Hyperbolic Paraboloid Transformation Félix Candela: Church of Our Lady of the Miraculous Medal Mexico City, Mexico,

30 Félix Candela: Church of Our Lady of the Miraculous Medal, Mexico City, Mexico, Hyperbolic Paraboloid Articulation Pierluigi Nervi & Pietro Belluschi: Cathedral of Saint Mary of the Assumption San Francisco, USA,

31 Hyperbolic Paraboloid Articulation Pierluigi Nervi & Pietro Belluschi: Cathedral of Saint Mary of the Assumption San Francisco, USA, Hyperbolic Paraboloid Articulation Pierluigi Nervi & Pietro Belluschi: Cathedral of Saint Mary of the Assumption San Francisco, USA,

32 Hyperbolic Paraboloid Articulation Pierluigi Nervi & Pietro Belluschi: Cathedral of Saint Mary of the Assumption San Francisco, USA, Form in 3d Gaussian Curvature Κ = Κ 1 Κ 2 Gaussian curvature is the product of the principal curvature and is a measurement of the local behaviour of the surface Κ < 0 hyperbolic point curves in opposite direction Κ = 0 parabolic point at least one curve is flat Κ > 0 elliptic point curves in same direction Κ > 0 elliptic point prototype: sphere Κ = 0 parabolic point prototype: cylinder 64 Κ < 0 hyperbolic point prototype: hyperboloid 32

33 Form in 3d Fabrication Different types of Gaussian curvature result in specific distortion behaviour around a point. For elliptic points the perimeter of a circle of radius r is less than 2πr, for a hyperbolic point its more. Only for parabolic points the perimeter is as expected. This means that a flat material gets compressed at elliptic points and stretched at hyperbolic ones. And that is why parabolic points can be build without distortion (developable surface). hyperbolic elliptic parabolic curvature analysis of surface in Rhino 65 Mapping Mathematically, the problem of making a map is related to projecting figures from a sphere into the flat. The major difficulty arises out of the fact that a sphere cannot be laid out flat without distorting geometric properties like distance, angle, or area. Because of this it is impossible to represent exactly the geometric properties that exist on a sphere. gnomic projection stereographic projection perspective cylindrical projection 66 33

34 Fabrication as Cartographic Problem Frank Gehry: Walt Disney Concert Hall, Los Angeles, USA, Developable Surface The cartographic problem of distortion is immanent to the construction of curvilinear architecture. Most building material is planar. This means only surfaces that can be mapped without any distortion so called developable surfaces - can be fabricated and assembled relatively easy. Frank Gehry: Walt Disney Concert Hall, Los Angeles, USA,

35 Hyperboloid as Ruled Surface Carlo Aiello Design Studio The Parabolic Chair, Hyperboloid as Ruled Surface Le Corbusier & Iannis Xenakis: Philips Pavilion, World s Fair Expo, Brussels,

36 Hyperboloid as Ruled Surface Le Corbusier & Iannis Xenakis: Philips Pavilion, World s Fair Expo, Brussels, Hyperboloid as Ruled Surface Le Corbusier & Iannis Xenakis: Philips Pavilion, World s Fair Expo, Brussels,

37 Hyperboloid as Ruled Surface Edgrad Varèse: Poème électronique, score 73 Thanks! 37