US20100162637A1 - Supporting Structure for Freeform Surfaces in Buildings - Google Patents

Supporting Structure for Freeform Surfaces in Buildings Download PDF

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US20100162637A1
US20100162637A1 US12/308,617 US30861707A US2010162637A1 US 20100162637 A1 US20100162637 A1 US 20100162637A1 US 30861707 A US30861707 A US 30861707A US 2010162637 A1 US2010162637 A1 US 2010162637A1
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gons
network
support elements
supporting structure
planes
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Sigrid Brell-Cokcan
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RFR Sas (50 %)
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    • E—FIXED CONSTRUCTIONS
    • E04—BUILDING
    • E04B—GENERAL BUILDING CONSTRUCTIONS; WALLS, e.g. PARTITIONS; ROOFS; FLOORS; CEILINGS; INSULATION OR OTHER PROTECTION OF BUILDINGS
    • E04B1/00—Constructions in general; Structures which are not restricted either to walls, e.g. partitions, or floors or ceilings or roofs
    • E04B1/32—Arched structures; Vaulted structures; Folded structures
    • E—FIXED CONSTRUCTIONS
    • E04—BUILDING
    • E04B—GENERAL BUILDING CONSTRUCTIONS; WALLS, e.g. PARTITIONS; ROOFS; FLOORS; CEILINGS; INSULATION OR OTHER PROTECTION OF BUILDINGS
    • E04B7/00—Roofs; Roof construction with regard to insulation
    • E04B7/08—Vaulted roofs
    • E04B7/10—Shell structures, e.g. of hyperbolic-parabolic shape; Grid-like formations acting as shell structures; Folded structures
    • E04B7/105—Grid-like structures
    • E—FIXED CONSTRUCTIONS
    • E04—BUILDING
    • E04B—GENERAL BUILDING CONSTRUCTIONS; WALLS, e.g. PARTITIONS; ROOFS; FLOORS; CEILINGS; INSULATION OR OTHER PROTECTION OF BUILDINGS
    • E04B1/00—Constructions in general; Structures which are not restricted either to walls, e.g. partitions, or floors or ceilings or roofs
    • E04B1/32—Arched structures; Vaulted structures; Folded structures
    • E04B2001/3235—Arched structures; Vaulted structures; Folded structures having a grid frame
    • E04B2001/3252—Covering details
    • E—FIXED CONSTRUCTIONS
    • E04—BUILDING
    • E04B—GENERAL BUILDING CONSTRUCTIONS; WALLS, e.g. PARTITIONS; ROOFS; FLOORS; CEILINGS; INSULATION OR OTHER PROTECTION OF BUILDINGS
    • E04B1/00—Constructions in general; Structures which are not restricted either to walls, e.g. partitions, or floors or ceilings or roofs
    • E04B1/32—Arched structures; Vaulted structures; Folded structures
    • E04B2001/3294—Arched structures; Vaulted structures; Folded structures with a faceted surface

Definitions

  • Curved envelope geometries of this kind are used in building construction for realizing freeform surfaces in which the curvature is different in two different spatial directions, such as in the case of domed buildings or also more complex surface shapes.
  • Freeform surfaces of this kind are also known as non-developable surfaces, and are designed at first in the course of architectural planning in the computer model as continuous surfaces.
  • the continuous freeform surfaces are approximated by a plurality of individual surface elements which are held in a supporting structure. It is thus possible for example to also realize complex freeform surfaces with multilayered planar glass elements which are fastened above, between and beneath a supporting structure made of steel for example.
  • Said supporting structure is formed from the individual support elements which are each composed into N-gons, i.e. triangles, quadrilaterals, hexagons, etc. The N-gons span the supporting structure, with the support elements abutting in the node region where they are fastened to each other.
  • a further principal disadvantage in realizing freeform surfaces by means of individual curved surface elements is also given in cases where a multilayered structure of the building shell in order to house necessary building infrastructure such as piping and the like.
  • the user will only notice the optically visible layers, i.e. the inner and outer shell of the building.
  • the optically visible layers i.e. the inner and outer shell of the building.
  • Planar intermediate layers are necessary for this infrastructure which needs to be introduced with much effort between the optically visible, curved freeform surfaces.
  • This concept is also pursued for example in the buildings of Frank O'Gehry, as a result of which his complex building shapes were able to be built in an economically viable way.
  • a further principal disadvantage arises in practice from the considerable quantities of data which need to be processed in the course of planning and which need to be exchanged between architect and specialist planners.
  • In order to describe a freeform surface it is necessary to reproduce the spatial position and the shaping of support and surface elements in a spatial system of coordinates, with there hardly being any possibilities for reducing data due to the often different shape of every single support and surface element.
  • “point clouds”, i.e. individual data points in a large number need to be processed, thus causing problems especially in the use of different CAD or FEM software packages, e.g. in the data transfer between individual specialist planners such as from the architect to the planner for the supporting framework.
  • a freeform surface is realized instead with the help of planar surface elements
  • local systems of coordinates can be defined for the individual surface elements in which the boundary points of the surface element are distinctly definable already by two coordinates such as an x and y coordinate, and a z coordinate can easily be determined by the surface normal on the surface element.
  • These local systems of coordinates are distinctly defined relative to the global system of coordinates of the overall structure. This enables a simple exchange of data with the help of an output file for example which shows the position of the normal of the supporting structure in the node region and the associated local coordinates of the planar surface elements.
  • planar surface elements are unrestricted choice of material for the surface and rod-like elements because no special elastic properties or plastic deformability are required. Moreover, the cutting to size of planar surface elements can be made more easily than in the case of curved surface elements. This reduces the overall construction costs considerably for structural shapes with freeform surfaces.
  • triangles as an elementary basic structure of the supporting structure also comes with disadvantages however. Especially, it will not be possible to find a distribution of the support elements with the help of triangles in which the support elements need not be subjected to any torsion in the geometrical sense, i.e. a twisting of the longitudinal axis in the node region for example, during the mounting between two node regions. Only support elements with a circular cross section can be positioned successively in a “torsion-free” manner, in the geometrical sense. When using non-circular cross sections, torsion (in the geometrical sense) will occur in the supporting structure in the node region. This leads to node regions which are unsatisfactory in regard of aesthetics and statics. Moreover, this also leads to the problem that multilayered structure cannot be realized or only be realized with considerable additional effort. It is therefore necessary to provide a separate support system for each layer, thus increasing the material costs and mounting work several times over.
  • the object of the invention to find a constructional implementation of freeform surfaces which reduces the technical and economic requirements and satisfies aesthetic demands.
  • mounting work and costs shall be kept as low as possible.
  • the supporting structure also offers the possibility of an uncomplicated multilayered configuration for approximation of freeform surfaces, i.e. the mounting in parallel offset of several planar surface elements.
  • the support elements form in this section 4-gons or 6-gons each, and the support elements each have a longitudinal axis which extends in a straight line between two node regions each and runs parallel to the imaginary line of intersection of the surface element planes associated with the same, with the cross section of the support elements normal to their longitudinal axis in each case having a relative twist angle of 0° along the entire longitudinal axis of the support element.
  • the “associated surface element planes” of a support element are the planes of the surface elements which are supported by the respective support element.
  • quadrilaterals or hexagons as an elementary basic shape of supporting structure in contrast the triangular network structure as known in the state of the art is of decisive importance because the applicants have recognized a quadrilateral or hexagonal network structure for the approximation of freeform surfaces has remarkable mathematical properties which are highly advantageous for a constructional implementation of freeform surfaces.
  • a form of geometrical approximation of freeform surfaces can be found for quadrilateral or hexagonal network structures which ensure parallel displaceability of the surface elements, with the respectively parallel displaced surface elements having boundary lines which are parallel to the respective original boundary lines, and thus lead to a continuous overall surface, as will be explained below.
  • offset parallel displaceability
  • their relevance for building engineering have not yet been recognized in the state of the art.
  • construction costs can be reduced with the help of a quadrilateral or hexagonal network structure in comparison with triangular network structures because the cutting to size of triangular surface elements causes more work than the one for quadrilateral surface elements for example.
  • a supporting structure consisting of quadrilateral or hexagonal basic shapes requires less material because the applicants were able to show that in the case of equivalent approximations of freeform surfaces with the help of triangular and quadrilateral network structures the realization by means of quadrilateral network structures requires a lower number of support elements than the one with the triangular basic shapes.
  • the support elements which are support elements which extend between the nodes areas in a straight line and without torsion (in the geometric sense).
  • the support elements each comprise a longitudinal axis which extends in a straight line between two node regions each and runs parallel to the imaginary line of intersection of the surface element planes associated with the same, and the cross section of the support elements normal to their longitudinal axis in each case have a relative twist angle of 0° along the entire longitudinal axis of the support element.
  • the support elements transversally to the plane of the surface element can be provided higher, so that a multilayered configuration in one and the same supporting structure is enabled.
  • the support elements can be arranged in a sufficiently high way in order to create space for the building infrastructure between the boundary outer layers.
  • the installation of the electrical and technical systems and a structural-physical multilayered configuration are facilitated.
  • the use of straight support elements without torsion (in the geometric sense) and bending facilitates mounting, thus reducing the assembly costs.
  • the support elements shall be applied along the common boundary lines, so that the surface elements held in the support elements usually no longer abut physically in order to divide a common boundary line, but are spaced from one another.
  • An imaginary line of intersection can still be formed which is defined by the imaginary extension of the respective surface element planes. This imaginary line of intersection corresponds to the aforementioned common boundary line of the polylines of the geometrical approximation.
  • the point of intersection of the imaginary lines of intersection of four surface element planes adjoining in a node region corresponds to the aforementioned node of four polylines.
  • the supporting structure In the nodes, four angles are obtained between the four adjoining polylines, with the supporting structure having to be arranged in accordance with claim 2 in such a way that the sum total of two mutually opposite angles each is equal.
  • This is a sufficient condition for parallel displaceability of the surface elements held in the supporting structure and the support elements can be arranged without torsion (in the geometric sense), which will be explained below in closer detail.
  • the polygon network underlying such a supporting structure is also known as a “conical network”, as will also be explained below in closer detail.
  • a further possibility for ensuring parallel displaceability of planar surface elements in a supporting structure approximating a freeform surface is according to claim 3 that the angular sum of respectively opposite angles between the surface normals of two adjoining surface element planes of four surface element planes adjoining in a node region is equal.
  • the polygon network underlying such a supporting structure is also known as a “dual-isothermal network”, as will also be explained below in closer detail.
  • the support elements have a rectangular cross-sectional shape or can be inscribed into a rectangular cross-sectional shape.
  • the longitudinal axis of these support elements extends between two node regions each, and the transversal axis stands along the entire longitudinal extension of the support element both normally to the longitudinal axis as well as normally to the imaginary line of intersection of the surface element planes associated with the same.
  • the longitudinal axis need not necessarily be an axis of symmetry of the support element. It is relevant that it extends along the support element in a straight line between two node regions and parallel to the imaginary line of intersection of the surface element planes associated with the same.
  • Rectangular cross sections also have an aesthetic advantage because they appear to be more slender than circular cross sections, and can also be arranged in a more slender way because only the height of the supports normally to the line of intersection of the surface element planes associated with the same is relevant for the bending load as a result of the load by the surface elements acting along the transversal axis.
  • other forms of support elements can be advantageous depending on the technical requirements, such that the support elements can have an I-like cross-sectional shape.
  • At least two surface elements are held on the support elements.
  • the advantage of the features in accordance with the invention is utilized in that a multilayered arrangement by mutually parallel offset surface elements on one and the same supporting structure is easily possible.
  • space for the installations of additional building infrastructure is created such as pipelines or structural-physical layers.
  • the intermediate layer area can fulfill tasks for climatic properties of the structural shape such as the circulation of air masses for rear ventilation and thermal insulation.
  • the approximation of the predetermined curved structural shape occurs with the help of a first continuous network of 4-gons or 6-gons which can be transferred to a further continuous network of 4-gons or 6-gons by parallel displacement in a direction normally to the mesh plane of the respective 4-gon or 6-gon, with two respective adjoining N-gons having a common boundary line which determines the progression of the longitudinal axis of one support element associated with these N-gons, and the dimensions of a support element perpendicular to said boundary line being determined by the distance of the respective boundary line of the first network to that of the further, parallel displaced network.
  • Parallel displacement of a network of 4-gons for example means that each mesh of a 4-gon of the first network is displaced parallel in a direction normal to the mesh plane of the respective 4-gon.
  • the network of 4-gons thus displaced in parallel in this manner must result again in a continuous network of 4-gons with respectively planar mesh planes.
  • This leads to the advantage that the distance of a boundary line of two adjoining 4-gons of the first network to the boundary line of the further, parallel displaced network obtained by parallel displacement of the same can be used for determining the dimensions of a support element perpendicular to said boundary line. Since the progression of the longitudinal axis of this support element is also determined by said boundary line in accordance with the invention, i.e. it extends parallel to the same, straight support elements without torsion (in the geometric sense) are thus obtained by the determining of the support frame in accordance with the invention.
  • Claim 7 relates to the determination of a supporting structure with the help of an underlying conical network, such that the angular sum of respectively opposite angles is equal in their common node between the boundary lines of four adjoining 4-gons.
  • Claim 8 relates to the further possibility of the determination of a support frame with the help of an underlying, dual-isothermal network, such that the angular sum of respectively opposite angles between the surface normals of two adjoining mesh planes of four surface element planes adjoining in a node is equal.
  • Claim 9 aims at a multilayered configuration of the curved structural shape, such that in a section of the supporting structure at least one second continuous network of 4-gons is determined which each define a planar mesh plane, with the 4-gons of the second network being formed by parallel displacement of the 4-gons of the first network in a direction normal to the mesh plane of the respective 4-gon.
  • FIG. 1 shows an illustration of two parallel 4-gon networks with respectively planar mesh planes
  • FIG. 2 shows an illustration of a section of a supporting structure in accordance with the invention in architectonic application
  • FIG. 3 shows the construction of a parallel displaced network N 0 from a base network N and the parallel network p(N) in a two-dimensional view;
  • FIG. 4 shows the construction of a parallel displaced network N 0 from a base network N and the parallel network p(N) in a three-dimensional view
  • FIG. 5 a shows an illustration of a conical node
  • FIG. 5 b shows an illustration of two adjacent conical nodes
  • FIG. 6 a shows an illustration of a Schramm circle packing on the sphere and the isothermal network p(N);
  • FIG. 6 b shows an illustration of a dual-isothermal node
  • FIG. 6 c shows an illustration on the required angular relationship in dual-isothermal networks
  • FIG. 7 shows a 4-gon network (in the background) and a refined 4-gon network with respective planar mesh planes (in the foreground);
  • FIG. 8 shows an example of an architectonic application of the 4-gon network according to FIG. 7 ;
  • FIGS. 9 to 12 show an embodiment for determining support elements on the basis of a 4-gon network generated according to the method in accordance with the invention
  • FIG. 13 shows a schematic, two-dimensional illustration of the rectangular cross-sectional shape of the construction space in the node region
  • FIG. 14 shows a schematic, three-dimensional illustration of the rectangular cross-sectional shape of the construction space in the node region
  • FIGS. 15 a to 15 d show possible architectonic applications of a supporting structure
  • FIG. 16 shows a schematic illustration for explaining the construction of the support elements
  • FIG. 17 shows support elements which converge in a node region of a dual-isothermal network
  • FIG. 18 shows an illustration of a multilayered arrangement on the basis of an example of a dual-isothermal network
  • FIGS. 19 a to 19 b show an illustration of the geometric supporting structure of a hexagonal network with support trapezoids of constant height
  • FIGS. 20 a to 20 c in FIG. 20 b show an offset pair N, N o with constant gon distance and constant surface distance which was obtained from the supporting structure of FIG. 20 a , and of which the associated planar surface support system is shown in FIG. 20 c in the form of a diagram.
  • the following will illustrate how a supporting structure 8 in accordance with the invention can be implemented from the start of planning of a freeform surface up to the realized structural shape with the help of the method in accordance with the invention.
  • the starting point is a computer-generated freeform surface, with the architect having to take into account aesthetic and well-proportioned shaping.
  • Architectonic planning of the freeform surface will also include its structure from individual surface elements 5 and the design of the supporting structure 8 .
  • the freeform surface is made up of the individual surface elements 5 which compile the freeform surface in an uninterrupted manner.
  • the object must be achieved at first in the computer model to subdivide a predetermined continuous freeform surface into N-gons in such a way that the freeform surface is represented in a continuous way.
  • the boundary lines 2 of the N-gon are progressions which delimit a surface content which is designated as a mesh 1 .
  • the mesh plane should be planar and represents the plane of the future surface element 5 such as a glass plate.
  • the term “mesh 1 ” shall be used below in connection with the geometric approximation of a freeform surface by a network made of N-gons, and the term “surface element” 5 shall be used in connection with the physical cover element in constructional implementation which is inserted into the support elements 4 and extends in the respective mesh plane of the geometric model.
  • a mesh network made of the same triangles for example is only possible in a number of few special cases.
  • the manner of subdivision of a sphere for example i.e. a shape that can be described in a comparatively simple way, belongs to the oldest tasks of an engineer.
  • One possible solution for the sphere are the geodesic cupolas of Buckminster Fuller for example, which are an example for a surface division of a sphere with similar hexagons. Even more difficult is the finding of suitable solutions for approximating more complex freeform surfaces, as are demanded in connection with randomly shaped, multiply curved surfaces of contemporary architecture.
  • the object of the invention is approximations of freeform surfaces by quadrilateral or hexagonal networks with planar meshes 1 .
  • Quadrilateral network structures are discussed first below.
  • Such a quadrilateral network is continuously made up of planar quadrilaterals in such a way that precisely two quadrilaterals abut along each inside edge of the network.
  • In an inside vertex of the network generally precisely four quadrilaterals abut.
  • Such a vertex is generally designated below as node X and is called regular, otherwise the node X is known as singular.
  • Only a quadrilateral mesh 1 is possible by edges or boundary lines 2 of boundary polygons. Only one or two meshes 1 abut at regular edges of boundary polygons.
  • a quadrilateral network with planar meshes 1 is always meant.
  • a quadrilateral network N is always imagined as an approximation of a surface F, which is mostly a freeform surface.
  • the network can be refined in such a way that the lateral surfaces will become increasingly smaller and move continually close to F.
  • a curve network K to F is thus obtained. If the planarity of the meshes 1 is obtained in the refining, a so-called conjugated curve network K to F is obtained in the boundary.
  • Parallel networks M, N are such in which the meshes 1 of the network M can be mapped on the meshes of the other network N by maintaining all neighborhood relationships in such a way that the planes of respective meshes are parallel. Since in this transformation or parallel displacement, meshes 1 with a common edge 2 are mapped again on meshes 1 with a common edge 2 , respective edges 2 are parallel in the networks M and N (see FIG. 1 ) due to the parallelism of respective mesh planes. It may occur that all meshes 1 of the one network, e.g. M, are convex, but there are meshes 1 with self-intersections in the parallel network N.
  • FIG. 1 illustrates a construction method which produces the parallel relationship of two polygons (bold).
  • the remainder of N (right illustration in FIG. 1 ) is inevitably obtained by parallel drawing to the respective edges of M (left illustration in FIG. 1 ).
  • Such networks N are relevant for the construction of special networks for which there is a parallel network p(N) which approximates a convex surface S (e.g. a sphere). This regularizes the network N in the sense that too may vertex angles are avoided in the meshes 1 and thus too narrow quadrilaterals. The deeper reason is that the following occurs under refining in the sense stated above and by maintaining the planar meshes 1 of N and p(N): N has a curve network K on a surface F as a boundary position; p(N) has a curve network p(K) on the surface S as a boundary position.
  • K and p(K) relate parallel with respect to each other and therefore K is the network of the relative curvature lines of F with respect to the “relative sphere” S. If S is a sphere, then K is the network of ordinary curvature lines and therefore rectangular.
  • the network of the curvature lines describes the directions of the strongest and weakest normal curvature of a surface. It is suitable to provide the spectator with a good imagination of the shape, thus increasing relevance for architecture.
  • a network N with a parallel network p(N) which approximates a convex surface S is also known as a general curve network.
  • an offset N o to a quadrilateral network N shall be a parallel quadrilateral network, with requirements being placed on the distances of respective meshes 1 , edges 2 or nodes X depending on the application.
  • FIG. 2 shows a sectional view of a supporting structure in accordance with the invention. Two offsets occur in this case which are explained by reference to a network mesh 1 which is shown in FIG. 2 as a surface element 5 .
  • a network mesh 1 which is shown in FIG. 2 as a surface element 5 .
  • the second layer e.g. glass construction
  • a third plane is defined at a slightly larger distance which is indicated by the connecting lines between the end points of the spacer elements 6 .
  • FIG. 2 also shows a typical construction of the support elements 4 in which the glass panes are placed.
  • support elements 4 appearing in the geometrical model at first as two-dimensional support quadrilaterals, which hereinafter shall also be referred to as support trapezoids.
  • Respective (parallel) edges of basic network N and offset network N o are connected by planar quadrilaterals (support quadrilaterals).
  • the distance which connects respective nodes X and X o from base to offset is a common edge n(X) of the four support quadrilaterals converging in this node.
  • the common edge n(X) can be imagined as a counterpart to the surface normal of a smooth surface.
  • the quantity of all support quadrilaterals shall be referred to below as a supporting structure.
  • a quadrilateral network N with planar meshes 1 is related parallel to such a network p(N), with p(N) approximating a convex surface S.
  • a point Z can be chosen which plays the role of a central point and governs the distribution of the distances of the offsets.
  • the convexity of S is not absolutely necessary, but certainly so that a point Z exists from which all surfaces of p(N) are visible.
  • Offsets N o of N with respect to p(N) are built in the following way: Each mesh plane Q of network N is displaced in parallel to a new position Q o within the terms of the given orientation. The distance of Q to Q o must be equal to the distance of Z to p(Q).
  • p(Q) designates the plane of the mesh 1 of the parallel network p(N) which belongs to Q (see FIG. 3 ).
  • all distances can be multiplied with a uniform factor ⁇ (or previously carry out a scaling of p(N)).
  • the parallel network p(N) must have quadrilateral surfaces at a constant distance from Z. That is why p(N) of a sphere S with center Z must be circumscribed in a contacting manner, i.e. all mesh planes of p(N) must make contact with the sphere S. A value of approximately 1 can be assumed for the radius of sphere S.
  • This designation is the result of the following geometric characterization:
  • a conical node X is also referred to here.
  • Conical networks thus have the practical property to have offsets at a constant surface distance. Since p(N) approximates a sphere S, conical networks can also be regarded as approximations of networks of lines of curvature. For calculation, the following optimization method is especially suitable in addition to the construction from p(N).
  • the algorithm works with a numeric optimization which runs iteratively. It occurs by step-by-step displacement of nodes X by maintaining connectivity.
  • the function ftraftraf shall assume small values for smooth and aesthetic networks.
  • the function floom holds the network close to the given reference surface F or also in the input network N during the optimization.
  • the sum total of the distances of respective nodes X of the current and the improved network is used in the simplest of cases.
  • the known tangential distance method has proven to be better however.
  • the weights w 1 and w 2 need to be reduced in the course of optimization in order to increase the influence of the planarity term f planar .
  • a Lagrange Newton iteration is used after the penalty process which minimizes the target function w 1 ftraftraftraffic +w 2 fbury under a quantity of constraints.
  • the algorithm can only be used in a useful way when the input network reflects the geometry of a conjugated curve network, which means in particular that it was obtained from such a one.
  • the proposed optimization method can be combined with a subdivision algorithm which works on quadrilateral networks.
  • the network in the foreground of FIG. 7 was generated with such a method, which is also the base of a specific design of a stop for a means of public transport according to FIG. 8 .
  • a term (w 1 + ⁇ 3 ⁇ 2 ⁇ 4 ) 2 is added to the function f planar per node X.
  • the constraint (K) is added to each node X.
  • the network in the foreground of FIG. 7 was calculated by alternating between Catmull-Clark subdivision and optimization by including the condition (K) per node X.
  • Every quadrilateral mesh of p(N) lies in a plane E whose cutting circle with S touches all sides of mesh 1 . Accordingly, each mesh 1 of p(N) has an incircle lying on sphere S. Adjoining meshes 1 generate contacting incircles. In summary, the quantity of the incircles therefore forms a circle packing on the sphere (see FIG. 6 a ). It is a so-called Schramm circle packing because four common tangents of contingence each go through a common point (node X of p(N)).
  • the network p(N) is known as an isothermal network.
  • Networks N which relate in parallel to such a quadrilateral network p(N), are called dual-isothermal networks because they represent the Laguerre-geometric (and thus to a certain extent dual) counterparts to the isothermal networks. It also follows from this however that only such surfaces can be approximated with such networks in which the Gaussian image of the lines of curvature is an isothermal network. It also follows from this however that only such surfaces can be approximated with such networks in which the Gaussian image of the lines of curvature is an isothermal network. This means a certain limitation in the design.
  • the class of surfaces especially contains the minimal surfaces. Networks with a constant edge distance which approximate minimal surfaces are known in the state of the art without the present property of the offsets and their meaning for architecture having been recognized. Important for the practical implementation is the known construction of the networks p(N) which are also known as Koebe's polyeder.
  • a characterization of dual-isothermal networks can also occur by angular conditions.
  • the edges 2 originating from there lie on a circular cone.
  • the angle occurring along an edge 2 between the surfaces is designated below as dihedral angle along said edge 2 , with ⁇ 1 , ⁇ 2 , ⁇ 3 , ⁇ 4 being the dihedral angles occurring along the edges 2 about a node X, with the following applying (see FIG. 6 b in this connection)
  • FIG. 14 shows a node region 3 of a dual-isothermal network, formed by support elements 4 with a rectangular cross section.
  • the longitudinal axes of the support elements 4 form the same angle with the node axis A (axis of the circular cone mentioned above).
  • the glass panes can be applied directly to the edges of the support elements 4 because these edges themselves delimit planar quadrilateral meshes 1 .
  • Every quadrilateral mesh 1 of p(N) lies in a plane E, the vertices of mesh 1 lie on the intersecting circle of E and S. Every quadrilateral mesh 1 of p(N) is therefore a quadrilateral inscribed in a circle, i.e. it comprises a circumscribed circle. By parallel displacement of the sides of a quadrilateral inscribed in a circle, the sides of a quadrilateral inscribed in a circle are obtained again.
  • each mesh 1 of N is parallel to the respective sides of the respective mesh 1 of p(N)
  • every quadrilateral in N must also be a quadrilateral inscribed in a circle, i.e. it comprises a circumscribed circle. That is why N is also designated as a circular network.
  • p(N) approximates a sphere
  • the circular networks must also be regarded as approximations of networks of lines of curvature.
  • the optimization method as described above is suitable for calculating circular networks.
  • networks N whose offsets have both constant surface distance as well as constant vertex distance. They relate in parallel to the networks p(N) whose surfaces touch a sphere S 1 with center Z and whose vertices lie on a concentric sphere S 2 . This is precisely the case when the quadrilateral meshes 1 of p(N) have circumscribed circles with a fixed radius. Such a circular pattern is constructed from a spherical network of rhombuses.
  • FIG. 20 b shows such an offset pair N, N o with a constant vertex distance and constant surface distance which was obtained from the support structure of FIG. 20 a .
  • FIG. 20 c shows the associated planar surface support system in the form of a diagram.
  • a node X is always conical (because three planes always touch a circular cone) and therefore there are always offsets at constant surface distance. If one wishes to achieve constant edge distance, there must be a parallel-related hexagonal network p(N) whose edges 2 touch a sphere.
  • p(N) whose edges 2 touch a sphere.
  • Such a network can be constructed by means of known algorithms for Koebe's polyeder.
  • the associated supporting structure of N has the property again that the occurring trapezoids have a constant height.
  • the support elements 4 with a constant cross section can be used. The longitudinal axes of the support elements 4 open under the same angles into the nodes axes A and the surface elements 5 can be mounted directly on the inner edges of the supporting structure 8 .
  • FIG. 9 shows such a network schematically, which approximately might concern a conical network.
  • FIGS. 10 to 12 it is now shown in an exemplary way how the progression of the support elements 4 can be determined on the basis of such a network of a geometrical model.
  • the starting point for the construction of the support elements 4 is a quadrilateral network N with planar meshes 1 ( FIG. 9 ).
  • each node point X of the network there is a common straight line (node axis A) on which the respective nodes X of the offset networks are situated ( FIG. 10 ).
  • the node axis A is the axis of the circular cone which is touched by the adjacent mesh planes.
  • the node axis A is the axis of the circular cone which contains the adjacent edges.
  • the quadrilateral network N and an associated offset network N o determine a geometric supporting structure which is made up of planar quadrilaterals ( FIG. 11 ). Each of these quadrilaterals is a trapezoid, bordered by an edge XY of N, the respective parallel edge X o Y o of N o and nodes X of N and N o corresponding to the connecting lines XX o and YY o .
  • the support elements 4 In order to build the supporting structure, the support elements 4 must be arranged along the geometric supporting structure.
  • the support trapezoids can be extruded to both sides by a desired width normal to their planes, thus resulting in cuboid building spaces for the support elements 4 with an oblique cut in the node region 3 ( FIG. 12 ).
  • a support layer can be continuous and the second support layer will abut against the first support layer (also see FIG. 14 ).
  • FIG. 13 shows a schematic, two-dimensional illustration of the rectangular cross-sectional shape of the building space.
  • FIG. 14 shows a respective three-dimensional illustration of the building space in the area of the node region 3
  • FIGS. 15 a to 15 d show possible architectural applications of a supporting structure 8 in steel, wood and concrete (not true to scale).
  • the building space need not have a rectangular cross-sectional shape, but must not exceed the maximum possible building space.
  • FIG. 17 shows a convex node region 3 of a dual-isothermal network, formed by support elements 4 with a rectangular cross section.
  • the longitudinal axes L of the support elements 4 form the same angle with the node axis A (axis of the circular cone as mentioned above) (also see FIG. 14 ).
  • the supporting structure 8 offers the possibility of a multilayer arranged for approximating freeform surfaces, i.e. the parallel offset mounting of several surface elements 5 to the support elements 4 of a single supporting structure 8 .

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  • Physics & Mathematics (AREA)
  • Electromagnetism (AREA)
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  • Structural Engineering (AREA)
  • Aerials With Secondary Devices (AREA)
  • Buildings Adapted To Withstand Abnormal External Influences (AREA)
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US12/308,617 2006-06-21 2007-06-20 Supporting Structure for Freeform Surfaces in Buildings Abandoned US20100162637A1 (en)

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AT0104906A AT503021B1 (de) 2006-06-21 2006-06-21 Tragstruktur für freiformflächen in bauwerken
ATA1049/2006 2006-06-21
PCT/AT2007/000302 WO2007147188A2 (de) 2006-06-21 2007-06-20 Tragstruktur für freiformflächen in bauwerken

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WO (1) WO2007147188A2 (de)

Cited By (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20100071301A1 (en) * 2007-03-15 2010-03-25 Mecal Applied Mechanics B.V. Mast for a Wind Turbine
US20130180184A1 (en) * 2012-01-17 2013-07-18 James L. CHEH Method for forming a double-curved structure and double-curved structure formed using the same
WO2014142763A1 (en) * 2013-03-15 2014-09-18 Singapore University Of Technology And Design Grid structure
CN106759887A (zh) * 2016-12-19 2017-05-31 江苏沪宁钢机股份有限公司 一种用于钢结构建筑中的拉花x节点及其安装工艺
US20170284103A1 (en) * 2016-03-31 2017-10-05 Vkr Holding, A/S Skylight cover with advantageous topography
CN113700199A (zh) * 2021-09-29 2021-11-26 深圳市云光绿建科技有限公司 一种异形天幕系统
CN117253012A (zh) * 2023-09-18 2023-12-19 东南大学 一种还原平面建筑自由曲面网格结构至三维空间的方法

Families Citing this family (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
DE102008017335A1 (de) * 2008-04-04 2009-10-15 Eckard Hofmeister Dachkonstruktion
AT506697B1 (de) * 2008-06-24 2009-11-15 Rfr S A S Tragstruktur für gekrümmte hüllgeometrien
CN102645201B (zh) * 2012-04-19 2014-07-16 浙江东南网架股份有限公司 一种构件侧面展开方法
US11191354B2 (en) 2015-08-27 2021-12-07 Xybix Systems, Inc. Adjustable height desk with acoustical dome

Citations (20)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US1970404A (en) * 1932-04-15 1934-08-14 George D Tesch Portable building
US2705349A (en) * 1951-04-20 1955-04-05 Spheric Structures Inc Structural element for portable buildings
US2944370A (en) * 1958-08-14 1960-07-12 Paul J Malarkey Building construction
US3114176A (en) * 1958-10-14 1963-12-17 Pease Woodwork Company Inc Wood building construction
US3359694A (en) * 1965-08-16 1967-12-26 Walter R Hein Domical building structure
US3462893A (en) * 1965-05-20 1969-08-26 Ickes Braun Glasshouses Inc Dome with connected frame members and frame connecting member
US3857150A (en) * 1972-01-28 1974-12-31 Fresa Ets Method of assembling a framework
US3881284A (en) * 1973-11-01 1975-05-06 Sorelle Frankie Ellipse domed structure
US3921555A (en) * 1973-04-06 1975-11-25 Mitsui Shipbuilding Eng Method of constructing a spherical tank or the like
US3925940A (en) * 1972-07-24 1975-12-16 Donn H Gross Building
US4012549A (en) * 1974-10-10 1977-03-15 General Dynamics Corporation High strength composite structure
US4071985A (en) * 1976-03-17 1978-02-07 Wickwire Chester F Arch and building construction
US4075813A (en) * 1976-07-14 1978-02-28 Nalick David L Dome construction method
US4285174A (en) * 1979-11-23 1981-08-25 Knight Brian V Building structure
US4720947A (en) * 1985-05-07 1988-01-26 Yacaboni Joseph D Dome-shaped building structure
US4784172A (en) * 1987-06-25 1988-11-15 Yacoboni Joseph D Instant emergency shelter
US5377460A (en) * 1993-06-08 1995-01-03 Hicks; Carl Dome building
US20020078635A1 (en) * 2000-10-25 2002-06-27 Monson John A. Polygon-shaped structural panel and constrution method for geodesic domes
US7269926B1 (en) * 2000-10-16 2007-09-18 Stanley S. Milic Domed building structure
US7458186B2 (en) * 2006-01-04 2008-12-02 Carter Philip R Dome-shaped structure

Family Cites Families (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
DE102004022879A1 (de) * 2004-05-06 2005-12-01 Stev Bringmann Gitterschale und deren Herstellung

Patent Citations (21)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US1970404A (en) * 1932-04-15 1934-08-14 George D Tesch Portable building
US2705349A (en) * 1951-04-20 1955-04-05 Spheric Structures Inc Structural element for portable buildings
US2944370A (en) * 1958-08-14 1960-07-12 Paul J Malarkey Building construction
US3114176A (en) * 1958-10-14 1963-12-17 Pease Woodwork Company Inc Wood building construction
US3462893A (en) * 1965-05-20 1969-08-26 Ickes Braun Glasshouses Inc Dome with connected frame members and frame connecting member
US3359694A (en) * 1965-08-16 1967-12-26 Walter R Hein Domical building structure
US3857150A (en) * 1972-01-28 1974-12-31 Fresa Ets Method of assembling a framework
US3925940A (en) * 1972-07-24 1975-12-16 Donn H Gross Building
US3921555A (en) * 1973-04-06 1975-11-25 Mitsui Shipbuilding Eng Method of constructing a spherical tank or the like
US3881284A (en) * 1973-11-01 1975-05-06 Sorelle Frankie Ellipse domed structure
US4012549A (en) * 1974-10-10 1977-03-15 General Dynamics Corporation High strength composite structure
US4071985A (en) * 1976-03-17 1978-02-07 Wickwire Chester F Arch and building construction
US4075813A (en) * 1976-07-14 1978-02-28 Nalick David L Dome construction method
US4285174A (en) * 1979-11-23 1981-08-25 Knight Brian V Building structure
US4720947A (en) * 1985-05-07 1988-01-26 Yacaboni Joseph D Dome-shaped building structure
US4784172A (en) * 1987-06-25 1988-11-15 Yacoboni Joseph D Instant emergency shelter
US5377460A (en) * 1993-06-08 1995-01-03 Hicks; Carl Dome building
US7269926B1 (en) * 2000-10-16 2007-09-18 Stanley S. Milic Domed building structure
US20020078635A1 (en) * 2000-10-25 2002-06-27 Monson John A. Polygon-shaped structural panel and constrution method for geodesic domes
US6658800B2 (en) * 2000-10-25 2003-12-09 John A. Monson Polygon-shaped structural panel and construction method for geodesic domes
US7458186B2 (en) * 2006-01-04 2008-12-02 Carter Philip R Dome-shaped structure

Cited By (11)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20100071301A1 (en) * 2007-03-15 2010-03-25 Mecal Applied Mechanics B.V. Mast for a Wind Turbine
US8720161B2 (en) 2007-03-15 2014-05-13 Postensa Wind Structures S.A. De C.V. Mast for a wind turbine
US20130180184A1 (en) * 2012-01-17 2013-07-18 James L. CHEH Method for forming a double-curved structure and double-curved structure formed using the same
US8789317B2 (en) * 2012-01-17 2014-07-29 James L. CHEH Method for forming a double-curved structure and double-curved structure formed using the same
WO2014142763A1 (en) * 2013-03-15 2014-09-18 Singapore University Of Technology And Design Grid structure
US9458620B2 (en) 2013-03-15 2016-10-04 Singapore University Of Technology And Design Grid structure
US20170284103A1 (en) * 2016-03-31 2017-10-05 Vkr Holding, A/S Skylight cover with advantageous topography
US10889990B2 (en) * 2016-03-31 2021-01-12 Vkr Holding A/S Skylight cover with advantageous topography
CN106759887A (zh) * 2016-12-19 2017-05-31 江苏沪宁钢机股份有限公司 一种用于钢结构建筑中的拉花x节点及其安装工艺
CN113700199A (zh) * 2021-09-29 2021-11-26 深圳市云光绿建科技有限公司 一种异形天幕系统
CN117253012A (zh) * 2023-09-18 2023-12-19 东南大学 一种还原平面建筑自由曲面网格结构至三维空间的方法

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DE502007001749D1 (de) 2009-11-26
EP2029822A2 (de) 2009-03-04
EP2029822B1 (de) 2009-10-14
ATE445744T1 (de) 2009-10-15
WO2007147188A2 (de) 2007-12-27
WO2007147188A8 (de) 2009-04-09
WO2007147188A3 (de) 2008-02-14
AT503021B1 (de) 2007-07-15
AT503021A4 (de) 2007-07-15

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