Optimal design of Schwedler and ribbed domes via hybrid Big Bang–Big Crunch algorithm

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Abstract

An optimum topology design algorithm based on the hybrid Big Bang–Big Crunch optimization (HBB–BC) method is developed for the Schwedler and ribbed domes. A simple procedure is defined to determine the Schwedler and ribbed dome configuration. This procedure includes calculating the joint coordinates and element constructions. The nonlinear response of the dome is considered during the optimization process. The effect of diagonal members on the results is investigated and the optimum results of Schwedler domes obtained by the HBB–BC method demonstrate the efficiency of these domes to cover large areas without intermediate supports.

Introduction

Recently a new optimization method relying on one of the theories of the evolution of the universe, namely the Big Bang and Big Crunch theory, has been introduced by Erol and Eksin [1]. This method has a low computational cost and a high convergence speed. Similarly to the genetic algorithms, the ant colony optimization, the particle swarm optimizer and the harmony search, the Big Bang–Big Crunch (BB–BC) optimization method is a natural evolutionary algorithm. The BB–BC is a heuristic population-based search procedure that incorporates random variation and selection. The random selection and the information obtained in each cycle are used to choose the new points in the subsequent cycles. According to the Big Bang and Big Crunch theory, in the Big Bang phase energy dissipation produces disorder and randomness is the main feature of this phase; whereas, in the Big Crunch phase, randomly distributed particles are drawn into an order. The BB–BC method similarly generates random points in the Big Bang phase and shrinks these points to a single representative point via a center of mass in the Big Crunch phase. After a number of sequential Big Bangs and Big Crunches, where the distribution of randomness within the search space during the Big Bang becomes smaller and smaller about the average point computed during the Big Crunch, the algorithm converges to a solution.

Although BB–BC performs well in the exploitation (the fine search around a local optimum), there are some problems in the exploration (global investigation of the search place) stage. If all of the candidates in the initial Big Bang are collected in a small part of the search space, the BB–BC method may not find the optimum solution and with a high probability, it may be trapped in that subdomain. The authors introduced a Hybrid Big Bang–Big Crunch optimization (HBB–BC) to solve this problem for the space trusses [2]. The HBB–BC method consists of two phases: a Big Bang phase where candidate solutions are randomly distributed over the search space, and a Big Crunch phase working as a convergence operator where the center of mass is generated. Then new solutions are created by using the center of mass, to be used as the next Big Bang. This algorithm uses the Particle Swarm Optimization (PSO) capacities to improve the exploration ability of the BB–BC algorithm. The Particle Swarm Optimization is motivated from the social behavior of bird flocking and fish schooling which has a population of individuals, called particles, that adjust their movements depending on both their own experience and the population’s experience [3]. The HBB–BC algorithm considers the combination of the center of mass, the best position of each particle and the best visited position of all particles, as the average point in the beginning of each Big Bang. The HBB–BC method has been shown to outperform simple BB–BC and some other evolutionary algorithms for many medium- and large-scaled truss examples [2].

On the other hand, covering large areas without intermediate supports has always been an interesting problem for architects and a challenging task for structural engineers. Dome structures are lightweight and elegant structures that provide economical solutions for covering large areas with their splendid aesthetic appearance. The joints of dome structures are considered to be rigidly connected and the members are exposed to both axial forces and bending moments. Therefore, bending moments of members affect the axial stiffness of these elements because of being slender members. Consequently, consideration of geometric nonlinearity in the analysis of these structures becomes important if the real behavior of these structures is intended to be obtained [4]. Furthermore, the instability of domes is also required to be checked during the nonlinear analysis 5., 6.. Some recent researches by Saka have shown that consideration of nonlinear behavior in the optimum design of domes does not only provide more realistic results, it also produces lighter structures 7., 8..

In this paper, optimum topology design algorithm based on the HBB–BC method is developed for the Schwedler and ribbed domes. The algorithm determines the optimum number of rings, the optimum height of crown, and sectional designations for the members of the Schwedler domes under the external loads. Due to the selection of the number of rings as the design variable, a simple procedure is necessary to determine the dome configuration. In order to fulfill this aim, a simple methodology is introduced in here. This procedure consists of calculating the joint coordinates and the element constructions. Diagonal members are considered in the Schwedler domes to stiffen the structure. The effect of these members on the results of the optimization is investigated. The serviceability and the strength requirements are considered in the design problem as specified in LRFD–AISC [9]. The steel pipe sections list of LRFD–AISC is adopted for the cross sections of dome members and the nonlinear response of the dome is considered during the optimization process.

Section snippets

Dome structure optimization problems

Optimal design of Schwedler and ribbed domes consists of finding optimal sections for elements, optimal height for the crown, and the optimum number of rings, under the determined loading conditions. The allowable cross sections are considered as 37 steel pipe sections, as shown in Table 1, where abbreviations ST, EST, and DEST stand for standard weight, extra strong, and double-extra strong, respectively. These sections are taken from LRFD–AISC [9] which is also utilized as the code of

A hybrid Big Bang–Big Crunch algorithm

The BB–BC method developed by Erol and Eksin [1] consists of two phases: a Big Bang phase, and a Big Crunch phase. In the Big Bang phase, candidate solutions are randomly distributed over the search space. The Big Crunch is a convergence operator that has many inputs but only one output, which is named as the center of mass, since the only output has been derived by calculating the center of mass. After the Big Crunch phase, the algorithm creates the new solutions to be used as the Big Bang of

Elastic critical load analysis of spatial structures

The dome structures are rigid structures for which the overall loss of stability might take place when these structures are subjected to equipment loading concentrated at the apex. Therefore, stability check is necessary during the analysis to ensure that the structure does not lose its load carrying capacity due to instability [4] and furthermore, considering the nonlinear behavior in the design of domes is necessary because of the change in geometry under external loads.

Details of the elastic

Configuration of Schwedler and ribbed domes

The configuration of a Schwedler dome is shown in Fig. 1. Schwedler, a German engineer, who introduced this type of dome in 1863, built numerous braced domes during his lifetime. A Schwedler dome, one of the most popular types of braced domes, consists of meridional ribs connected together to a number of horizontal polygonal rings. To stiffen the resulting structure, each trapezium formed by intersecting meridional ribs with horizontal rings is subdivided into two triangles by introducing a

Results and discussion

This section presents the optimum design of the Schwedler and ribbed domes using the HBB–BC algorithm. The modulus of elasticity for the steel is taken as 205 kN/mm2. The limitations imposed on the joint displacements are 28 mm in the z direction and 33 mm in the x and y directions for the 1st, 2nd and 3rd nodes, respectively.

For the proposed algorithm, a population of 50 individuals is used. Using α1=1.0 allows the initial search of the full range of values for each design variable. Previous

Concluding remarks

A Hybrid Big Bang–Big Crunch optimization (HBB–BC) is developed for optimal design of geometrically nonlinear Schwedler and ribbed domes. This method consists of a Big Bang phase where candidate solutions are randomly distributed over the search space, and a Big Crunch phase working as a convergence operator where the center of mass is generated. The Particle Swarm Optimization capacities are added to improve the exploration ability of the algorithm. A simple procedure is developed to determine

Acknowledgement

The first author is grateful to the Iran National Science Foundation for support.

References (12)

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