Optimum topological design of geometrically nonlinear single layer latticed domes using coupled genetic algorithm
Introduction
Domes are one of the oldest magnificent structural systems that are used to cover large areas such as exhibition halls, stadium and concert halls. They provide a completely unobstructed inner space and they are economical in terms of materials compare to the more conventional forms of structures [1]. They consist of one or more layers of elements that are arched in all directions. Domes are given different names depending upon the way their surface is formed.
Modeling braced domes as rigidly connected three-dimensional structures yields more realistic representation of their behaviour. Members in a braced dome are subjected to bending moments as well as axial forces. Furthermore, due to the slenderness of the members the bending moments affect the axial stiffness of these members. As a result, the behaviour of braced domes turns out to be nonlinear and it is important that the geometric nonlinearity is considered in their analysis. Furthermore, it is required that instability check also should be investigated through the nonlinear analysis [1], [2], [3], [4], [5]. Some of the algorithms developed for the optimum design of braced domes accommodated the nonlinear elastic behaviour [6], [7], [8]. It is shown that consideration of nonlinear behaviour in the optimum design of structures does not only provide more realistic results, it also produces lighter structures [9], [10], [11], [12]. Recently, an algorithm is presented which determines the optimum height of the crown as well as the optimum tubular cross-sectional designations for members of a nonlinear geodesic dome [13]. However, the algorithm determines the optimum geometry of a single layer latticed domes but falls short of determining the optimum topology of the structure. In this study this algorithm is extended to cover the topological design of the dome which determines the optimum number of members required for the dome, its optimum height and the optimum tubular designation for its members.
Depending upon the form selected the morphology of a single layer latticed dome usually follows a relatively simple geometrical outline. It is possible to obtain the entire topological information of such a dome by just having the values of two parameters. These two parameters are the total number of rings and the height of crown in the dome. By making use of the geometrical properties of the form selected the total number of members and joints as well as member incidences can be determined, if the total number of rings in the dome is known. Furthermore the coordinates of all joints can also be computed by using the geometrical form and the height of the crown. Hence two domes with different number of rings and crown heights have different topologies. The algorithm presented in this paper treats the total number of rings as well as the height of crown in a dome as design variables in addition to cross-sectional designations of dome members. Furthermore the optimum design algorithm takes into account the nonlinear response of the dome due to the slenderness of its member. The presence of bending moments in slender members affects the axial stiffness of members. The interaction between the bending moments and the axial forces in members renders the overall stiffness matrix of these structures into nonlinear. The stability functions for three-dimensional beam-columns are included in the overall stiffness matrix to obtain the nonlinear response of the dome [7], [14]. Serviceability and strength requirements are included in the design problem according to BS 5950 [15]. Genetic algorithm is used to obtain the solution of the design problem.
Section snippets
Morphology of single layer geodesic dome
The geodesic dome shown in Fig. 1 is a commonly used form as a structural system. It has relatively simple geometry. In this dome all the structural data related with the geometry of the dome can be obtained automatically provided that the diameter D, the total number of rings nr and the height of the crown h in the dome are known. It is worthwhile to mention that rings in the dome are arranged in such a way that the distance between them on the meridian line is equal. It can be noticed from
Optimum topology design of geodesic domes
The optimum topology design of a single layer geodesic dome which has the base diameter D requires determination of the optimum number of rings and the height of crown in the dome as well as selection of steel sections such as circular hollow sections or other types for its members from a standard steel sections table. These selections should be carried out in such a way that the dome obtained with its topology and member designations should satisfy the serviceability and strength requirements
Genetic algorithm
The solution of the optimum design problem given in both cases requires the selection of appropriate steel circular hollow sections from a standard list as well as the height of crown and the total number of rings such that the frame with the minimum weight is obtained while the design constraints are satisfied. Since the standard list contains discrete values for the steel sections, the design problem becomes a discrete programming problem. Among the solution techniques available, the genetic
Section classification
It is worth mentioning that BS5950 necessitates the determination of the classification of the cross-section of the steel sections selected for the dome members prior to computation of their load capacities. The capacity expressions depend on whether the cross-section is slender or not. In this study, steel circular hollow sections listed in [16] are adopted for members of the dome. None of these sections given in this table is slender. Consequently, section classification step is omitted from
Elastic instability analysis of braced domes
The computation of the fitness factor of each individual in the population requires the response of the structure under the applied loads. Since latticed domes are slender structures consideration of geometric nonlinearity in their analysis becomes a necessity. They are sometimes subjected to equipment loading concentrated at the crown in addition to uniform gravity loading. This also necessitates to check on the overall loss of stability. The elastic instability analysis of space frames
Optimum topological design algorithm
The optimum topological design algorithm presented for geodesic domes based on genetic algorithm for case I formulation consists of the following steps.
- 1.
Construct the initial population by generating individuals randomly in a binary form.
- 2.
Decode each individual and identify the circular hollow section adopted for each group in the dome from the standard steel sections table, the height selected for the crown and the total number of rings in the dome.
- 3.
Set up the member incidences and joint
Design example
The design algorithm presented is used to determine the optimum number of rings, height of crown and circular steel hollow section designations for members of the geodesic dome shown in Fig. 4. The grade of steel adopted is Grade 43. The modulus of elasticity is taken as 205 kN/mm2. The value of constant C in Eq. (7) is taken as 10 in all the design cases considered. This value is decided after carrying out number of trials. The dome is considered to be subjected to equipment loading of 2000 kN
Conclusions
A new algorithm is presented in this study for the optimum topology of geodesic domes. The algorithm determines the optimum number of members, the optimum height of crown and the optimum steel section designations for the members of the dome. During the topological design process total number of joints and members as well as the height of the crown changes from one design to another. Hence it becomes necessary to automate the construction of member incidences and computation of joint
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