GB2490767A

GB2490767A - Structural geometric framework

Info

Publication number: GB2490767A
Application number: GB201206608A
Authority: GB
Inventors: Alexander Owen David Lorimer
Original assignee: Individual
Current assignee: Individual
Priority date: 2012-04-16
Filing date: 2012-04-16
Publication date: 2012-11-14
Also published as: GB201206608D0

Abstract

A structural support system comprising a series of tetrahedral joints connecting straight beams of the same length, and a hexagonal or orthogonal grid of edge beams that provide a flat face at the perimeters of the structure. The joints between the straight frame members may be formed by lap joints or mortice and tenon joints of the frame members.

Description

Efficient Structural Support Geometry In structural engineering a support system can be said to be efficient if it achieves a high level of strength with a reduced amount of material. Pioneers in lightweight structures such as these include Alexander Graham Bell and Buckminster Fuller who in the last century developed a highly efficient system known as the space frame. Space frames are commonly used in large spanning roofs and occasionally in vehicles, where a high level of strength is required whilst also remaining lightweight; consequentially conserving material.

The strength of the space frame is said to be achieved through the inherent rigidity of the triangle and pyramid (see figures 1 and 2), widely held as the geometry with most structural integrity. Whilst this may be true for a triangle or pyramid in isolation, it is not necessarily so for a sequence of triangles or pyramids packed together in a tessellating fashion, which adequately describes the ubiquitous space frame.

Considering only the triangle, it is well understood that a tessellating cluster of hexagons will outperform a cluster of triangles in terms of structural efficiency as the triangles require significantly more perimeter material to cover a specific area than do the hexagons. The perimeters of the hexagons may therefore be thickened so that they use the same amount of material, subsequently becoming stronger and more resistant to stresses than the cluster of triangles. It is for this reason that economic hexagonal tessellations occur throughout the natural world, from cross-sections of soap bubble clusters, bones and bodily tissues, to the cracks through volcanic rocks and of course the well known bee's honey-comb.

So a tessellation of hexagons is the most efficient pattern for covering the largest area with the least material in two dimensions, but for three dimensional shapes the problem becomes more difficult. In 1887 Lord Kelvin proposed the Kelvin conjecture; he wanted to know what three dimensional shape tessellates perfectly to cover the largest possible volume of space with the least surface area; in some respects the 3D equivalent of the hexagon. He devised the Kelvin Structure, made up of a 14-sided shape with 6 square faces and 8 hexagonal faces, however in order to achieve a higher level of efficiency each face and edge either bows inward or outward slightly (see figure 3). It was only in 1993 that a more efficient solution to the problem was discovered; known as the Weaire-Phelan Structure, it is composed of two different shapes, one 12-sided shape to every three 14-sided shapes (see figure 4). However, again, each face and edge is required to bow either inward or outward to achieve a higher level of efficiency. The first structure in the world to use this geometry as a support framework was the Beijing National Aquatics Centre; the building is reputed to being so strong that it is able to maintain its form even if placed on its end, according to computer simulations.

However the structure used in the building is not the true Weaire-Phelan structure (where each edge bows inward or outward), but has been simplified so that each beam is straight, as a curved beam is obviously less resistant to compressive stresses. While in two dimensions the hexagonal tessellation is the most efficient pattern, with each line only ever meeting two others each separated at 120°, the most efficient angle for edges to meet in a three dimensional tessellation is 109.5°; this is an angle that can only be achieved in the Kelvin and Weaire-Phelan Structure by curving the edges. Therefore the straightening of these beams in the Beijing National Aquatics centre compromises the efficient geometry of the Weaire-Phelan Structure.

The problem with the 109.5° angle is that it belongs to no regular straight edged shape. Each corner in a regular pentagon has 108° but this is the closest angle that can be achieved, as by widening one angle in a pentagon others will automatically tighten. A shape composed of only 109.5° angles falls just short of forming a pentagon (see figure 5).

Another consideration is that the Kelvin Conjecture was originally set as a problem to find the most efficient tessellation for covering the largest volume of space with the smallest surface area of shapes. However when this pattern is translated into a structural framework the faces of the 3D shapes disappear and it is only the edges which become beams, therefore it better suits an architectural cause to find an arrangement that minimises the lengths of edges rather than surface area of faces.

A tetrahedral joint is one that connects four beams separated at the ideal 109.5° angle.

While it is shown in figure 5 that they cannot form a closed geometry with straight edges, it is however possible to form a closed tessellation of hexagons whose corners do not lie on the same plane. This is achieved by rotating every other joint at a 180° angle (see figure 6).

The resulting framework covers the largest volume with the minimum of structural material, and as each beam in the framework follows a geometry that finds the shortest path through space they are inherently resistant to buckling.

The drawings which form part of this specification are as follows; Figure 6 (Rotating the tetrahedral joints to form a closed 3D framework) Figure 7 (Example of possible lap joints in the tetrahedral structure) Figure 8 (Example of framework with corresponding edge beams A) Figure 9 (Example of framework with corresponding edge beams B) Figure 10 (Mortise and tenon joints connecting edge beams) Figure 11 (Example of a connector constraining beams at 109.5° angles) Figure 12 (The tetrahedral structure without complete edge beams from various angles A) Figure 13 (The tetrahedral structure without complete edge beams from various angles B) Figure 14 (Alternative to the space frame using the tetrahedral angles-side view) Figure 15 (Alternative to the space frame using the tetrahedral angles-view from above) This framework could be constructed with various materials using a variety of different jointing mechanisms. Figure 7 is an example of how this structural system could be implemented with wooden beams using a variety of lap joints, while figures 8 and 9 show how these beams are contained by edge beams composed of tessellating hexagons (which form a flat supporting face at the fringes of the main body of the structure). Figure 10 shows how the edge beams can be joined using mortise and tenon joints. Figure 11 shows how steel poles may be joined to form this framework using tetrahedral connectors. Figures 12 and 13 show the tetrahedral structure (without edge beams) from various angles.

This structure achieves maximum strength in compression from the absolute minimum of material. It may also be used as an alternative to space frames which resist a combination of compressive and tensile stresses. Figures 14 and 15 show how a slice through the tetrahedral structure enclosed by orthogonal edge beams can produce an alternative to the common space frame for long span and cantilever applications. Where the structure meets the orthogonal edge beams, joints that connect six beams at 109.5°; 35.25°; and 90° angles are used (see figures 14 and 15). This arrangement covers the same amount of space as the common space frame while using almost 25% less material. The cross-sectional area of the beams in the tetrahedral joint frame may then be increased by 25% to compensate, resulting in a stronger structure while using the same amount of material.

Claims

Claims The embodiments of the structure in which an exclusive property is claimed are defined as follows; 1. A structural geometry which comprises: in its main body a repeating pattern of straight beams of equal length that meet at tetrahedral joints, four at a time at 109.5° angles; and enclosing the main body, a repeating pattern of straight beams forming a hexagonal or orthogonal grid and connecting to the main body providing a flat face.
2. The combination defined in claim 1, wherein a tetrahedral joint connects four beams that at their other ends are connected by another tetrahedral joint which is a rotated version of the originally stated tetrahedral joint; the centre of rotation being at the cross-sectional midpoint of the one of said beams that belongs to the rotated joint, the axis of rotation being parallel to the length of said beam, and the angle of rotation being 180°.
3. The combination defined in claim 1, wherein the main body of the structure is connected, at its fringes, to edge beams that meet each other three at a time to form a tessellating hexagonal arrangement and a flat face to the main body of the structure, to which it is connected via beams that constitute the fringes of the main body projecting outward into the corresponding edge beams, while at their other ends they are joined to the rest of the main body by tetrahedral joints.
4. A structural geometry of claim 1, wherein each tetrahedral joint connects four beams, two of which may be understood as projecting upward and the other two downward, that at their other ends are connected to one other beam, that lies on the same plane as the connecting beam, of the main body of the structure, which also belongs at its other end to another tetrahedral joint from which it projects either upward or downward matching the beam from the other tetrahedral joint that it connects to, at an angle of 109.5 degrees; with the plane of the measured angle being on the same plane as both upward projecting beams if connecting upward beams or both downward projecting beams if connecting downward beams, with the angle being measured from the underside of the beams that may be understood as projecting upward or the topside of the beams that may be understood as projecting downward.
5. The combination defined in claim 4, wherein the joints that connect two beams of the main body at 109.5° angles, also connect said beams to four edge beams separated from themselves at 90° angles, with two of said edge beams running along the same plane as said beams of the main body and separated from them at two 35.25° angles so that the edge beams form a square grid that sandwiches the main body of tetrahedral joints.