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Abstract
In this paper optimization of volume fraction distribution in a thick hollow cylinder with finite length made of two-dimensional functionally graded material (2D-FGM) and subjected to steady state thermal and mechanical loadings is considered. The finite element method with graded material properties within each element (graded finite elements) is used to model the structure. Volume fractions of constituent materials on a finite number of design points are taken as design variables and the volume fractions at any arbitrary point in the cylinder are obtained via cubic spline interpolation functions. The objective function selected as having the normalized effective stress equal to one at all points that leads to a uniform stress distribution in the structure. Genetic Algorithm jointed with interior penalty-function method for implementing constraints is effectively employed to find the global solution of the optimization problem. Obtained results indicates that by using the uniform distribution of normalized effective stress as objective function, considerably more efficient usage of materials can be achieved compared with the power law volume fraction distribution. Also considering uniform distribution of safety factor as design criteria instead of minimizing peak effective stress affects remarkably the optimum volume fractions.

Key Words
2D heterogeneous;volume fraction optimization; thermo-mechanical stresses;genetic algorithm

Address
Masoud Asgari: Faculty of Mechanical Engineering, K. N. Toosi University of Technology, Pardis Street, Molla-Sadra Avenue, Vanak Square, P.O. Box 19395-1999, Tehran, Iran

Abstract
Natural frequencies of the structural systems should be far away from the excitation frequency in order to avoid or reduce the destructive effects of dynamic loads on structures. To accomplish this goal, a structural optimization on size and shape has been performed considering frequency constraints. Such an optimization problem has highly nonlinear property. Thus, the quality of the solution is not independent of the optimization technique to be applied. This study presents the performance evaluation of the recently proposed meta-heuristic algorithm called Teaching Learning Based Optimization (TLBO) as an optimization engine in the weight optimization of the truss structures under frequency constraints. Some examples regarding the optimization of trusses on shape and size with frequency constraints are solved. Also, the results obtained are tabulated for comparison. The results demonstrated that the performance of the TLBO is satisfactory. Additionally, TLBO is better than other methods in some cases.

Key Words
teaching learning based optimization; frequency constraints; shape and size optimization; trusses

Address
Tayfun Dede and Vedat Togan: Department of Civil Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey

Abstract
This paper presents the application of a recently developed meta-heuristic algorithm, called Colliding Bodies Optimization (CBO), for size and topology optimization of steel trusses. This method is based on the one-dimensional collisions between two bodies, where each agent solution is considered as a body. The performance of the proposed algorithm is investigated through four benchmark trusses for minimum weight with static and dynamic constraints. A comparison of the numerical results of the CBO with those of other available algorithms indicates that the proposed technique is capable of locating promising solutions using lesser or identical computational effort, with no need for internal parameter tuning.

Key Words
Colliding Bodies optimization; meta-heuristic algorithms; optimum design; size and topology optimization; truss structures

Address
A. Kaveh: Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Tehran, Iran V.R. Mahdavi: School of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran-16, Iran

Abstract
The optimum design of liquid column dampers in seismic vibration control considering system parameter uncertainty is usually performed by minimizing the unconditional response of a structure without any consideration to the variation of damper performance due to uncertainty. However, the system so designed may be sensitive to the variations of input system parameters due to uncertainty. The present study is concerned with robust design optimization (RDO) of liquid column vibration absorber (LCVA) considering random system parameters characterizing the primary structure and ground motion model. The RDO is obtained by minimizing the weighted sum of the mean value of the root mean square displacement of the primary structure as well as its standard deviation. A numerical study elucidates the importance of the RDO procedure for design of LCVA system by comparing the RDO results with the results obtained by the conventional stochastic structural optimization procedure and the unconditional response based optimization.

Key Words
seismic vibration control; liquid column vibration absorber; random system parameters; robust optimization

Address
Rama Debbarma: Department of Civil Engineering, National Institute of Technology, Agartala, India Subrata Chakraborty: Department of Civil Engineering, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India

Abstract
This study aims at comparing the optimum design of two common types open web expanded beams: with hexagonal openings, also called castellated beams and beams with circular openings referred to as cellular beams. The minimum weights of both beams are taken as the objective functions while the design constraints are respectively implemented from The Steel Construction Institute Publication Numbers 5 and 100. The design methods adopted in these publications are consistent with BS5950 parts. The formulation of the design problem considering the limitations of the above mentioned turns out to be a discrete programming problem. Improved harmony search algorithm is suggested to compare the optimum design of mentioned web-expanded beams to analysis the performance of both beams. The design algorithms based on the technique select the optimum Universal Beam sections, dimensional properties of hexagonal and circular holes and total number of openings along the beam as design variables.

Key Words
structural optimization; web-expanded beams; castellated beams; cellular beams; harmony search algorithm

Address
Ferhat Erdal: Department of Civil Engineering, Akdeniz University, 07058, Antalya, Turkey

Abstract
A hybrid approach of Particle Swarm Optimization (PSO) and Swallow Swarm Optimization algorithm (SSO) namely Hybrid Particle Swallow Swarm Optimization algorithm (HPSSO), is presented as a new variant of PSO algorithm for the highly nonlinear dynamic truss shape and size optimization with multiple natural frequency constraints. Experimentally validation of HPSSO on four benchmark trusses results in high performance in comparison to PSO variants and to those of different optimization techniques. The simulation results clearly show a good balance between global and local exploration abilities and consequently results in good optimum solution.

Key Words
hybrid meta-heuristics; Hybrid Particle Swallow Swarm optimization algorithm; truss optimization; frequency constraints

Address
A. Kaveh, T. Bakhshpoori and E. Afshari: Centre of Excellence for Fundamental Studies in Structural Engineering, School of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran-16, Iran

Abstract
This study presents critical buckling load optimization of the axially graded layered uniform columns. In the first place, characteristic equations for the critical buckling loads for all boundary conditions are obtained using the transfer matrix method. Then, for each case, square of this equation is taken as a fitness function together with constraints. Due to explicitly unavailable objective function for the critical buckling loads as a function of segment length and volume fraction of the materials, especially for the column structures with higher segment numbers, initially, prescribed value is assumed for it and then the design variables satisfying constraints are searched using Differential Evolution (DE) optimization method coupled with eigen-value routine. For constraint handling, Exterior Penalty Function formulation is adapted to the optimization cycle. Different boundary conditions are considered. The results reveal that maximum increments in the critical buckling loads are attained about 20% for cantilevered and pinned-pinned end conditions and 18% for clamped-clamped case. Finally, the strongest column structure configurations will be determined. The scientific and statistical results confirmed efficiency, reliability and robustness of the Differential Evolution optimization method and it can be used in the similar problems which especially include transcendental functions.

Key Words
axially graded; uniform column; buckling; optimization; differential evolution

Address
Veysel Alkan: Department of Mechanical Engineering, Pamukkale University, Kinikli, 20070, Denizli, Turkey

Abstract
A novel optimization approach for reinforced concrete (RC) biaxially loaded columns is proposed. Since there are several design constraints and influences, a new computation methodology using iterative analyses for several stages is proposed. In the proposed methodology random iterations are combined with music inspired metaheuristic algorithm called harmony search by modifying the classical rules of the employed algorithm for the problem. Differently from previous approaches, a detailed and practical optimum reinforcement design is done in addition to optimization of dimensions. The main objective of the optimization is the total material cost and the optimization is important for RC members since steel and concrete are very different materials in cost and properties. The methodology was applied for 12 cases of flexural moment combinations. Also, the optimum results are found by using 3 different axial forces for all cases. According to the results, the proposed method is effective to find a detailed optimum result with different number of bars and various sizes which can be only found by 2000 trial of an engineer. Thus, the cost economy is provided by using optimum bars with different sizes.

Key Words
reinforced concrete; biaxially loaded columns; optimization; metaheuristic algorithms; harmony search algorithm

Address
Sinan Melih Nigdeli, Gebrail Bekdaş: Department of Civil Engineering, Istanbul University, 34320 Avcilar, Istanbul, Turkey Sanghun Kim: Department of Civil and Environmental Engineering, Temple University, Philadelphia 19122, USA Zong Woo Geem: Department of Energy IT, Gachon University, Seongnam 461-701, South Korea

Abstract
In this paper, an optimization process using Genetic Algorithm (GA) that mimics biological processes is presented for optimum design of planar frames with semi-rigid connections by selecting suitable standard sections from a specified list taken from American Institute of Steel Construction (AISC). The stress constraints as indicated in AISC-LRFD (American Institute of Steel Construction - Load and Resistance Factor Design), maximum lateral displacement constraints and geometric constraints are considered for optimum design. Two different planar frames with semi-rigid connections taken from the literature are carried out first without considering concrete slab effects in finite element analyses and the results are compared with the ones available in literature. The same optimization procedures are then repeated for full and semi rigid planar frames with composite (steel and concrete) beams. A program is developed in MATLAB for all optimization procedures. Results obtained from this study proved that consideration of the contribution of the concrete on the behavior of the floor beams provides lighter planar frames.

Key Words
genetic algorithm, weight optimization, planar frame, composite beams, semi-rigid connection

Address
Musa Artar: Department of Civil Engineering, Bayburt University, Bayburt 69000, Turkey Ayse T. Daloglu: Department of Civil Engineering, Karadeniz Technical University, Trabzon 61000, Turkey

Abstract
Dolphin echolocation (DE) optimization algorithm is a recently developed meta-heuristic in which echolocation behavior of Dolphins is utilized for seeking a design space. The computational performance of meta-heuristic algorithms is highly dependent to its internal parameters. But the computational time of adjusting these parameters is usually extensive. The DE is an efficient optimization algorithm as it includes few internal parameters compared with other meta-heuristics. In the present paper a modified Dolphin echolocation (MDE) algorithm is proposed for optimization of steel frame structures. In the MDE the step locations are determined using one-dimensional chaotic maps and this improves the convergence behavior of the algorithm. The effectiveness of the proposed MDE algorithm is illustrated in three benchmark steel frame optimization test examples. Results demonstrate the efficiency of the proposed MDE algorithm in finding better solutions compared to standard DE and other existing algorithms.

Key Words
steel structure; optimization; Dolphin echolocation; meta-heuristic

Address
Saeed Gholizadeh and Hamed Poorhoseini: Department of Civil Engineering, Urmia University, Urmia, West Azerbaijan Province, Iran

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