## 2424 Design Parameters

Before any work can proceed on the analysis of a double-layer grid, it is necessary to determine the depth and the module size. The depth is the distance between the top and bottom-layers and the module is the distance between two joints in the layer of the grid (see Figure 24.8). Although these two parameters seem simple enough to determine, yet they will play an important role in the economy of the roof design. There are many factors influencing these parameters, such as the type of double-layer grid, the span between the supports, the roof cladding, and the proprietary system used. In fact, the depth and module size are mutually dependent, which is related by the permissible angle between the center line of web members and the plane of the top and bottom chord members. This should be less than 30°, or the forces in the web members and the length will be relatively excessive, or greater than 60°, or the density of the web members in the grid will become too high. For some of the proprietary systems, the depth and module are standardized.

The depth and module size of double-layer grids are usually determined by practical experience. In some of the papers and handbooks, figures on these parameters are recommended, and one may find that the difference is quite large. For example, the span-depth ratio varies from 12.5 to 25, or even more. It is usually considered that the depth of space frame can be relatively small when compared with more conventional structures. This is generally true because double-layer grids produce smaller deflections under load. However, depths that are small in relation to span will tend to use smaller modules, and hence a heavier structure will result. In the design, almost unlimited possibilities exist in practice for the choice of geometry. It is best to determine these parameters through structural optimization.

Module

Module

### FIGURE 24.8 Depth and module.

Work has been done on the optimum design of double-layer grids supported along perimeters. In an investigation by Lan [4], seven types of double-layer grids were studied. The module dimension and depth of the space frame are chosen as the design variables. The total cost is taken as the objective function, which includes the cost of members and joints as well as of roofing systems and enclosing walls. Such assumption makes the results realistic to a practical design. A series of double-layer grids of different types spanning from 24 to 72 m were analyzed by optimization. It was found that the optimum design parameters were different for different types of roof system. The module number generally increases with the span, and the steel purlin roofing system allows larger module sizes than that of reinforced concrete. The optimum depth is less dependent on the span and a smaller depth can be used for steel purlin roofing system. It should be observed that a smaller member density will lead to a grid with relatively few nodal points and thus the least possible production costs for nodes, erection expense, etc.

Through regression analysis of the calculated values by optimization method where the costs are within the 3% optimum, the following empirical formulas for optimum span-depth ratio are obtained. It was found that the optimum depths are distributed in a belt and all the span-depth ratios within this range will have optimum effect in construction.

For roofing system composed of reinforced concrete slabs where L is the short span and d is the depth of the double-layer grids.

Few data could be obtained from the past works. Regarding the optimum depth for steel purlin roofing systems, Geiger suggested the span-depth ratio to be varied from 10 to 20 with less than 10% variation in cost. Motro recommended a span-depth ratio of 15. Curves for diagonal square pyramid space grids (diagonal on square) were given by Hirata et al., and an optimum ratio of 10 was suggested. In the earlier edition of the Specifications for the Design and Construction of Space Trusses issued in China, the span-depth ratio is specified according to the span. These figures were obtained through the analysis of the parameters used in numerous design projects. A design handbook for double-layer grids also gives graphs for determining upper and lower bounds of module dimension and depth. The relation between depth and span obtained from Equation 24.2 and the relevant source is shown in Figure 24.9. For short and medium spans, the optimum values are in good agreement with those obtained from experience. It is noticeable that the span-depth ratio should decrease with the span, yet an increasing tendency is found from experience, which gives irrational large values for long spans.

In the revised edition of the Specifications for the Design and Construction of Space Trusses issued in China, appropriate values of module size and depth for commonly used double-layer grids simply supported along the perimeters are given. Table 24.2 shows the range of module numbers of the top chord and the span-depth ratios prescribed by the Specifications.

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