Keywords: optimum design, optimum geometry design, genetic algorithm, braced domes, lamella dome, network dome, elastic instability, three dimensional beam-column, hollow sections.

Domes are one of the oldest magnificent structural systems. They consist of one or more layers of elements that are arched in all directions. The surface of a dome may be a part of a single surface such as a sphere or a parabolic, or it may consist of a patchwork of different surfaces. Domes are used to cover large areas such as exhibition halls, stadium, and concert halls. They provide a completely unobstructed inner space and economy in terms of materials. They are lighter compared with the more conventional forms of structures.

A great majority of domes of large span have been of steel braced domes. These domes have a distinctive importance in engineering practice and posses enormous stiffness. There are several types of braced domes. A ribbed dome consists of a number of intersecting ribs and rings. A rib is a group of elements that lie along a meridional line and a ring is a group of elements that constitute a horizontal polygon. A ribbed dome will not be structurally stable unless it is designed as a rigidly jointed system.

Schwedler dome is a modified form of a ribbed dome, which is obtained by bracing the quadrilateral panels of the dome. A lamella dome has a diagonal pattern. Grid domes are obtained by projecting plane grid pattern onto a curved surface. There are many other dome patterns that are variation of these basic forms.

Large numbers of braced domes that are built recently are the lamella type. They comprise of a great number of interconnecting steel elements, called lamellas. The stresses are distributed evenly in these domes. They can easily take large concentrated or live load, which are rapidly dispersed through out the framework; generating primarily axial forces in lamella units. This results in substantial saving of material. Due to their great structural rigidity and low cost this system has been selected for the worlds biggest steel dome, which covers 300m spans.

Lattice or network domes consist of parallel rings spaced equidistantly. Triangular networks of bars are then subdividing the annular spaces. Members between two adjacent rings are equal in length. The joints are assumed to be rigid and therefore the members in the dome behave as beam-column.

The skeleton of the dome may be single or double layer. It is a must for a large dome to have double layers to prevent buckling. For members of the dome all types of steel sections are used. Hollow sections, circular or square, with welded joints provides elegant appearance particularly at places where the steelwork is exposed.

The behavior of flexible domes is nonlinear and it is important that the geometric non-linearity is to be considered in their analysis. Furthermore, it is required that instability check should be investigated through the nonlinear analysis.

In this paper, an algorithm is developed for the optimum geometry design of steel braced domes. The height of the apex is taken as design variable in addition to cross-sectional properties of the members. A procedure is developed which calculates the coordinates of the joints in the braced dome automatically for the given value of its height.

The optimum design algorithm takes into account the nonlinear response due to the effect of axial forces in members. Due to the slenderness of members axial forces cause lateral deflection in members, which in turn generates additional bending moments. The presence of bending moments affects the axial stiffness of members due to their apparent shortening caused by the bending deformations. The interaction between bending moment and the axial forces in members renders the overall stiffness matrix of these structures nonlinear. The stability functions for three-dimensional beam-columns are included in the overall stiffness matrix to obtain the nonlinear response of these domes.

The optimum design algorithm considers serviceability requirements as well as combined strength limitations set by BS 5950. The genetic algorithm is used to obtain the solution of the design problem. The proposed technique initiates the optimum design procedure by constructing an initial population consists of individuals that are potential candidate for the design of the dome. The elastic instability analysis is then carried out for each individual until the ultimate load factor is reached. During these iterations checks on the overall stability of the dome is conducted. If the loss of stability takes place during the nonlinear analysis, this individual is taken out of the population and replaced by a new individual, which is generated randomly. This replacement policy is repeated until an individual is found which does not have instability problem. Once the population is established with all the individuals which are stable then the regular genetic operations such as selection crossover and mutation are applied and a new generation is produced. The production of new generations is continued until one individual dominates the population or the maximum number of generations is reached. The design example considered has shown that member grouping has an important effect on the optimum height and weight of geodesic dome. The optimum weight of nonlinear geodesic dome where all members are made out of the same section is 25% heavier than the dome where members are collected in six different groups.

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