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Simultaneous geometry and cross-section optimization of aluminum trusses
Abstract
Purpose
Trusses constitute a fertile field to demonstrate the application of optimization techniques because of the possibility of several different configurations. Using such techniques allows the search for designs that minimize the use of material to safely comply with the imposed loads. Truss optimization can be classified into three categories: cross-section, shape, and topology. This work presents a numerical and experimental study developed to minimize the weight of aluminum trusses, taking both the cross-sectional dimensions of the elements and the nodal coordinates as design variables.
Design/methodology/approach
Initially, several numerical computer simulations were performed with an optimization program developed by combining the displacement method and a simulated annealing optimization method. Subsequently, two aluminum trusses were selected and built in order to validate the numerical results obtained.
Findings
Experimental tests verified the excellent performance of the optimized model.
Originality/value
In addition, it was concluded that significant savings could be obtained from the application of the proposed formulation.
... In this way, the problem becomes less dependent on the initial solution (initial values assigned to the cross sections and nodal coordinates of the displaceable nodes). The same strategy was adopted successfully by the authors in previous works [7][8][9][10]. ...
The application of optimization to the solution of practical structural engineering problems has been very limited, despite the great development of the techniques. One of the main reasons for this is the complexity of the generated models, which employ nonlinear functions and generate a space of nonconvex solutions. In addition, most design variables, as in the case of steel profiles, can only assume discrete values. Traditional methods of mathematical programming are very limited in solving problems with these characteristics, opening space to the usage of heuristic methods and, more specifically, to metaheuristic methods. The main advantage of this class of methods is the fact that they only involve values of the functions in the optimization process, regardless of the existence of unimodality or even continuity in the derivatives of the functions involved. The present work presents the application of a heuristic method, the simulated annealing method, to the optimization of steel trusses with parallel chords, also called flat trusses. Initially, several usual configurations of trusses were analyzed, aiming to identify which led to the lowest cost. The top chord coordinates were also included as design variables, with the purpose of verifying the relations between span and height normally indicated in the literature as the most economical. Among the results, it was verified that the inclusion of the height of the truss among the design variables can lead to a significant additional cost savings.
This review presents developed models, theory, and numerical methods for structural optimization of trusses with discrete design variables in the period 1968 – 2014. The comprehensive reference list collects, for the first time, the articles in the field presenting deterministic optimization methods and meta heuristics. The field has experienced a shift in focus from deterministic methods to meta heuristics, i.e. stochastic search methods. Based on the reported numerical results it is however not possible to conclude that this shift has improved the competences to solve application relevant problems. This, and other, observations lead to a set of recommended research tasks and objectives to bring the field forward. The development of a publicly available benchmark library is urgently needed to support development and assessment of existing and new heuristics and methods. Combined with this effort, it is recommended that the field begins to use modern methods such as performance profiles for fair and accurate comparison of optimization methods. Finally, theoretical results are rare in this field. This means that most recent methods and heuristics are not supported by mathematical theory. The field should therefore re-focus on theoretical issues such as problem analysis and convergence properties of new methods.
At present, several methods are available for optimization of structures. The application of
such methods to real structural problems, however, has not been as intense as the
development of the techniques themselves. One of the main reasons is that the great
majority of the methods, based on mathematical programming, consider a continuous
search space. This paper presents an application of the Simulated Annealing method to the optimization of trusses considering the cross sections of the members as discrete variables. The constraints imposed to the analysis were the allowable stresses and the displacements on nodes. Some examples are presented in order to demonstrate the effectiveness of the method when compared with other methods found in literature.
The main issue in this paper is engineering optimization of the telecommunication tower using the generalized perturbation-based Stochastic Finite Element Method. This procedure is based on reliability analysis accounting for the admissible eigenfrequencies and obeys an application of the stainless steel as well as the aluminum as the structural materials. Such a reliability-based optimization follows engineering practice, demands of the Eurocodes, cost minimization of the structures and further installations of the new heavier transmission equipment. The reliability analysis is based upon the FORM approach, while computational realization of the generalized SFEM guarantees reliable determination of the first two probabilistic moments of the structural response for symmetric random dispersion in the design parameters. Computational implementation has hybrid character and is based on interoperability of the FEM design system ROBOT and the computer algebra system MAPLE, where all probabilistic tools are programmed.
Over the past few years, swarm intelligence based optimization techniques such as ant colony optimization and particle swarm
optimization have received considerable attention from engineering researchers and practitioners. These algorithms have been
used in the solution of various engineering problems. Recently, a relatively new swarm based optimization algorithm called
the Artificial Bee Colony (ABC) algorithm has begun to attract interest from researchers to solve optimization problems. The
aim of this study is to present an optimization algorithm based on the ABC algorithm for the discrete optimum design of truss
structures. The ABC algorithm is a meta-heuristic optimization technique that mimics the process of food foraging of honeybees.
Originally the ABC algorithm was developed for continuous function optimization problems. This paper describes the modifications
made to the ABC algorithm in order to solve discrete optimization problems and to improve the algorithm’s performance. In
order to demonstrate the effectiveness of the modified algorithm, four structural problems with up to 582 truss members and
29 design variables were solved and the results were compared with those obtained using other well-known meta-heuristic search
techniques. The results demonstrate that the ABC algorithm is very effective and robust for the discrete optimization designs
of truss structural problems.
KeywordsArtificial bee colony–Discrete optimization–Space truss–C# programming
The paper is concerned with the efficient use of simulated annealing (SA) in structural optimization problems of high complexity. A reformulation of the working mechanism of the Boltzmann parameter is introduced to accelerate and enhance the general productivity of SA in terms of convergence reliability. Two general and complementary parameters, referred as weighted Boltzmann parameter and critical Boltzmann parameter, are proposed. Several alternative methodologies are suggested for these two parameters. Numerical applications are presented to test the performance of the proposed techniques, and the results are compared with those that exist in the literature.
In this article, two algorithms are presented for the optimum design of geometrically nonlinear steel space frames that are
based on simulated annealing and genetic algorithm. The design algorithms obtain minimum weight frames by selecting suitable
sections from a standard set of steel sections such as the American Institute of Steel Construction (AISC) wide-flange shapes.
Stress constraints of AISC Load and Resistance Factor Design (LRFD) and AISC Allowable Stress Design (ASD) specifications,
maximum (lateral displacement) and interstorey drift constraints, and also size constraints for columns were imposed on frames.
The algorithms were applied to the optimum design of three space frame structures, which have a very small amount of nonlinearity.
The unconstrained form of objective function was applied in both optimum design algorithms, and constant penalty factors were
used instead of gradually increasing ones. Although genetic algorithm took much less time to converge, the comparisons showed
that the simulated annealing algorithm yielded better designs together with AISC-LRFD code specification.
A new global optimization algorithm for functions of continuous variables is presented, derived from the “Simulated Annealing” algorithm recently introduced in combinatorial optimization. The algorithm is essentially an iterative random search procedure with adaptive moves along the coordinate directions.
It permits uphill moves under the control of a probabilistic criterion, thus tending to avoid the first local minima encountered. The algorithm has been tested against the Nelder and Mead simplex method and against a version of Adaptive Random Search. The test functions were Rosenbrock valleys and multiminima functions in 2,4, and 10 dimensions. The new method proved to be more reliable than the others, being always able to find the optimum, or at least a point very close to it. It is quite costly in term of function evaluations, but its cost can be predicted in advance, depending only slightly on the starting point.
Evolutionary computation techniques have received a great deal of attention regarding their potential as optimization techniques for complex numerical functions. However, they have not produced a significant breakthrough in the area of nonlinear programming due to the fact that they have not addressed the issue of constraints in a systematic way. Only recently have several methods been proposed for handling nonlinear constraints by evolutionary algorithms for numerical optimization problems; however, these methods have several drawbacks, and the experimental results on many test cases have been disappointing.
In this paper we (1) discuss difficulties connected with solving the general nonlinear programming problem; (2) survey several approaches that have emerged in the evolutionary computation community; and (3) provide a set of 11 interesting test cases that may serve as a handy reference for future methods.
There is a deep and useful connection between statistical mechanics (the behavior of systems with many degrees of freedom
in thermal equilibrium at a finite temperature) and multivariate or combinatorial optimization (finding the minimum of a given
function depending on many parameters). A detailed analogy with annealing in solids provides a framework for optimization
of the properties of very large and complex systems. This connection to statistical mechanics exposes new information and
provides an unfamiliar perspective on traditional optimization problems and methods.
This book contains the refereed and edited versions of papers presented at the IUTAM Symposium on Topological Design Optimization of Structures, - chines and Materials: Status and Perspectives, held at Rungstedgaard, near Copenhagen, Denmark, October 26–29, 2005. The symposium was attended by 74 scientists from 19 countries, and the programme was intensive and - cluded also two evenings for presentations and discussions. It is now more than 15 years ago that the so-called homogenization method was proposed as a basis for computational means to optimize the topology and shape of continuum structures. From initially being capable mainly of treating minimum compliance design we now see the basic material distribution idea of the methodology applied to a wide range of structural and mechanical pr- lems as well as to problems that couple structural response to other physical responses. Also, the method has provided insight for micro-mechanical st- ies, meaning that the method has given feedback to the area which provided impetus to the ?eld of topological design optimization in its creation. Finally, topological design is now an integral part of most FEM software systems and it has become a standard industrial tool in some ?elds. The IUTAM Symposium provided a forum for the exchange of ideas for - ture developments in the area of topological design optimization. This enc- passed the application to ?uid-solid interaction problems, acoustics problems, and to problems in biomechanics, as well as to other multiphysics problems.
Evolutionary computation techniques have received a great deal of attention regarding their potential as optimization techniques for complex numerical functions. However, they have not produced a significant breakthrough in the area of nonlinear programming due to the fact that they have not addressed the issue of constraints in a systematic way. Only recently have several methods been proposed for handling nonlinear constraints by evolutionary algorithms for numerical optimization problems; however, these methods have several drawbacks, and the experimental results on many test cases have been disappointing. In this paper we (1) discuss difficulties connected with solving the general nonlinear programming problem; (2) survey several approaches that have emerged in the evolutionary computation community; and (3) provide a set of 11 interesting test cases that may serve as a handy reference for future methods.
In this paper formulation and solution technique using Simulated Annealing for optimizing the moment capacity of steel fiber reinforced concrete beams, with random orientated steel fibers, is presented along with identification of design variables, objective function and constraints. Steel fibers form an expensive constituent of steel fiber concrete and therefore it is important to determine ways and means of using these fibers in a judicial way with care consistent with economy for achieving the desired benefits. The most important factors which influence the ultimate load carrying capacity of FRC are the volume percentage of the fibers, their aspect ratios and bond characteristics. Hence an attempt has been made to analyze the effective contribution of fibers to bending of reinforced fiber concrete beams. Equations are derived to predict the ultimate strength in flexure of SFRC beam with uniformly dispersed and randomly oriented steel fibers. Predicted strengths using the derived expressions have been compared with the experimental data. A reasonable agreement (within the range of ±20 percent!) was evident with different types of steel fibers, aspect ratio, and material characteristics. A computer coding has been developed based on the formulations and the influence of various parameters on the ultimate flexural strength is discussed. A computer algorithm that conducts a random search in the space of four variables-beam width, beam depth, fiber content and aspect ratio-to yield an optimum solution for a given objective function (ultimate moment (M u)) is presented. The outlined methods provide a simple and effective tool to assess the optimum flexural strength of steel fiber reinforced concrete beams. Using the results obtained the influence of various parameters on the ultimate strength are discussed. Particular attentions are given to the construction practice as well as the reduction of searching space. It has been shown that within a reasonable and finite number of searching the developed algorithm is able to yield optimum solutions for the given objective function.
Cold-formed profiles have been largely used in the building industry because they can be easily produced and because they allow for a wide range of sections and thus can be utilized to meet different project requirements. Attainment of maximum performance by structural elements with low use of material is a challenge for engineering projects. This paper presents a numerical study aimed at minimizing the weight of lipped and unlipped cold-formed channel columns, following the AISI 2007 specification. Flexural, torsional and torsional–flexural buckling of columns was considered as constraints. The simulated annealing method was used for optimization. Several numerical simulations are presented and discussed to validate the proposal, in addition to an experimental example that qualifies its implementation. The ratios between lips, web width, and flange width are analyzed. Finally, it may be concluded that the optimization process yields excellent results in terms of cross-sectional area reduction.
The aim of the present paper is to show the application of optimization strategies for the cost of
beams in reinforced concrete buildings and to propose pre-sizing parameters. In order for these goals to be
met, an optimization software program was developed. The program combines the analysis of structures by
the grid model, reinforced concrete sizing, and the simulated annealing optimization heuristic. Sizing is
compliant with the NBR 6118 (2007) Brazilian standard, according to which flexural, shearing, torsion, and
web reinforcements and serviceability limit states (deflection and crack width limitation) are checked.
Besides the dimensions of the situations mentioned above, the influence the cost of each material (steel,
concrete and formwork) has on the overall cost of structures was also determined.
A aplicação da otimização à resolução de problemas práticos de engenharia estrutural tem sido
pouco verificada na literatura, apesar do grande desenvolvimento das técnicas. Uma das principais
razões normalmente apontadas consiste na complexidade do modelo gerado, descrito por funções de
comportamento não-linear e gerando um espaço de soluções não-convexo, sendo ainda comum que
as variáveis de projeto possam assumir apenas valores discretos. Para a resolução de problemas com
estas características, os métodos tradicionais de programação matemática têm se mostrado pouco
eficientes, razão pela qual as heurísticas vêm conquistando um crescente espaço, uma vez que
envolvem apenas valores das funções no processo, não importando se existe unimodalidade ou
mesmo continuidade nas derivadas das funções envolvidas. O presente trabalho apresenta a
aplicação de um método heurístico, o Método do Recozimento Simulado (Simulated Annealing) à
otimização de treliças metálicas compostas por perfís laminados, dimensionados segundo a Norma
Brasileira NBR-8800/86. Inicialmente foram analisadas diversas configurações usuais de treliças
planas de banzos paralelos, buscando-se identificar a que conduzisse ao menor peso. Na seqüência,
as coordenadas do banzo superior foram também incluídas como variáveis de projeto, com o
objetivo de verificar as relações entre vão e altura normalmente indicadas na literatura como as
mais econômicas. Diversas estruturas analisadas são apresentadas, bem como resultados obtidos a
partir das análises efetuadas,
The main objective here is an engineering optimization of the truss-type structures using the generalized perturbation-based Stochastic Finite Element Method. This procedure is based on triple reliability analysis consisting of the limit states for the admissible eigenfrequencies, horizontal displacements and the reduced stresses; it obeys an application of the stainless steel as well as the aluminum as the structural materials. Such a reliability-based optimization follows strictly engineering practice, demands of the Eurocodes, cost minimization of the structures and future installations of the new transmission equipment. The reliability analysis is based on the FORM approach, while computational realization of the generalized SFEM guarantees reliable determination of the first four probabilistic moments of the structural response for any random dispersion in the design parameters. Numerical implementation has hybrid character and is based on interoperability of the FEM design system ROBOT and the computer algebra system MAPLE, where all probabilistic functions are programmed.
A multivariable optimization technique based on the Monte-Carlo method used in statistical mechanics studies of condensed systems is adapted for solving single and multiobjective structural optimization problems. This procedure, known as simulated annealing, draws an analogy between energy minimization in physical systems and objective function minimization in structural systems. The search for a minimum is simulated by a relaxation of the statistical mechanical system where a probabilistic acceptance criterion is used to accept or reject candidate designs. To model the multiple objective functions in the problem formulation, a cooperative game theoretic approach is used. Numerical results obtained using three different annealing strategies for the single and multiobjective design of structures with discrete-continuous variables are presented. The influence of cooling schedule parameters on the optimum solutions obtained is discussed. Simulation results indicate that, in several instances, the optimum solutions obtained using simulated annealing outperform the optimum solutions obtained using some gradient-based and discrete optimization techniques. The results also indicate that simulated annealing has substantial potential for additional applications in optimization, especially for problems with mixed discrete-continuous variables.
Rescaled Simulated Annealing (RSA) has been adapted to solve combinatorial optimization problems in which the available computational
resources are limited. Simulated Annealing (SA) is one of the most popular combinatorial optimization algorithms because of
its convenience of use and because of the good asymptotic results of convergence to optimal solutions. However, SA is too
slow to converge in many problems. RSA was introduced by extending the Metropolis procedure in SA. The extension rescales
the state's energy candidate for a transition before applying the Metropolis criterion. The rescaling process accelerates
convergence to the optimal solutions by reducing transitions from high energy local minima. In this paper, structural optimization
examples using RSA are provided. Truss structures of which design variables are discrete or continuous are optimized with
stress and displacement constraints. The optimization results by RSA are compared with the results from classical SA. The
comparison shows that the numbers of optimization iterations can be effectively reduced using RSA.
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Corresponding author Moacir Kripka can be contacted at: mkripka@upf.br For instructions on how to order reprints of this article, please visit our website
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Applications, McGraw-Hill, New York, NY.
Corresponding author
Moacir Kripka can be contacted at: mkripka@upf.br
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