## Schwedler Dome Horizontal Polygonal Rings

FIGURE 24.14 Braced domes.

A Schwedler dome also consists of meridional ribs connected together to a number of horizontal polygonal rings to stiffen the resulting structure so that it will be able to take unsymmetrical loads (Figure 24.14b). Each trapezium formed by intersecting meridional ribs with horizontal rings is subdivided into two triangles by a diagonal member. Sometimes the trapezium may also be subdivided by two cross-diagonal members. This type of dome was introduced by a German engineer J.W. Schwedler in 1863. The great popularity of Schwedler domes is due to the fact that, on the assumption of pin-connected joints, the structure can be analyzed as statically determinate. In practice, in addition to axial forces, all the members are also under the action of bending and torsional moments. Many attempts have been made in the past to simplify their analysis, but precise methods of analysis using computers have finally been applied to find the actual stress distribution.

The construction of three-way grid dome is self-explanatory, which may be imagined as a curved form of three-way double-layer grids (Figure 24.14c). It can also be constructed in single layer for the dome. The Japanese ''diamond dome'' system by Tomoegumi Iron Works belongs to this category. The theoretical analysis of three-way grid domes shows that even under unsymmetrical loading the forces in this configuration are very evenly distributed leading to economy in material consumption.

A lamella dome is formed by intersecting two-way ribs diagonally to form a rhombus-shaped grid pattern. As in lamella braced barrel vault, each lamella element has a length that is twice the length of the side of a diamond. The lamella dome can further be distinguished into parallel and curved domes. For parallel lamella as shown in Figure 24.14d, the circular plan is divided into several sectors (usually six or eight), and each sector is subdivided by parallel ribs into rhombus grids of the same size. This type of lamella dome is very popular in the United States. It is sometimes called a Kiewitt dome, after its developer. For curved lamella as shown in Figure 24.14e, rhombus grids of different sizes, gradually increasing from the center of the dome, are formed by diagonal ribs along the radial lines. Sometimes, for the purpose of establishing purlins for roof decks, concentric rings are introduced, and a triangular network is generated.

The geodesic dome was developed by the American designer Buckminster Fuller, who turned architects' attention to the advantages of braced domes in which the elements forming the framework of the structure are lying on the great circle of a sphere. This is where the name ''geodesic'' came from (Figure 24.14f). The framework of these intersecting elements forms a three-way grid comprising virtually equilateral spherical triangles. In Fuller's original geodesic domes, he used an icosahedron as the basis for the geodesic subdivision of a sphere; then, the spherical surface is divided into 20 equilateral triangles as shown in Figure 24.15a. This is the maximum number of equilateral triangles into which a sphere can be divided. For domes of larger span, each of these triangles can be subdivided into six triangles by drawing medians and bisecting the sides of each triangle. It is therefore possible to form 15 complete great circles regularly arranged on the surface of a sphere (see Figure 24.15b). Practice shows that the primary type of bracing, which is truly geodesic, is not sufficient since it would lead to an excessive length for members in geodesic dome, therefore a secondary bracing has to be introduced. To obtain a more or less regular network of the bracing bars, the edges of the basic triangle are divided modularly. The number of modules into which each edge of the spherical icosahedron is divided depends mainly on the size of the dome, its span, and the type of roof cladding. This subdivision is usually referred to as ''frequency,'' as depicted in Figure 24.15c. It must be pointed out that during such a subdivision the resulting triangles are no longer equilateral. The members forming the skeleton of the dome show slight variation in their length. As the frequency of the subdivision increases, the member length reduces, the number of components as well as the number of types of connecting joints increase. Consequently, this reflects in the increase of the final price of geodesic dome and is one of the reasons why geodesic domes, in spite of their undoubted advantages for smaller spans, do not compare equally well with other types of braced domes for larger span.

The rise of a braced dome can be as flat as one seventh of the diameter or as high as three fourths of the diameter, which will constitute the greater part of a sphere. For diameters larger than 60 m, double-layer grids are recommended. The ratio of the thickness to the diameter of double-layer braced dome is in the range of 35 to 65; for long spans the thickness can be as small as 155 of the diameter.

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