The technique of optimal design of bearing structures on the basis of genetic algorithm has been suggested. A design of steel frame with varying 9 parameters using the method of finite elements is considered as an example. The best variant corresponding to the volume minimum of the frame material is revealed

Yuriev, А. G.

Kluev, S. V.

Kluev, А. V.

Electronic archive of Tomsk Polytechnic University

In the figure the result of load influence on the earthsurface is shown, where the curves of motion changesu°x=uxμ/p (m), u°y=uyμ/p (m) and normal stressesσ°yy=σyy/p in transverse plane (η=0) are presented.It follows from the analysis of curve behavior thatextreme deflexions ux of the earth surface and extreme normal stress σyy take place at y=–0,4R, but maximum horizontal displacement uy is at y=0,8R (hereσyy is equal to zero). Increasing |y| displacements andstresses damp quickly, and at |y|=3,2R dynamic influence of load on the earth surface is virtually imperceptible.Bulletin of the Томsк Pоlytеchnic University. 2007. V. 310. № 158REFERENCES1. Pozhuev V.I. Affect of mobile torque load on cylindrical shell in elastic environment // Constructional Mechanics and Building Calculations. – 1984. – № 6. – P. 58–61.2. Aitaliyev Sh.М., Alexeeva L.А., Ukrainets V.N. Influence of freesurface on the tunnel of shallow shaft under the action of mobile loads // Kazakh SSR AS. Proc. 3. Series of physicsmathematics. –1986. – № 5. – P. 75–80.3. Novatskiy V. Elasticity theory. – Moscow: Mir, 1975. – 872 p.4. Ukrainets V.N. Reaction of elastic halfspace to moving along theaxis periodical load // Almaty Mathematical Journal. – 2005. –№ 3. – P. 96–102.Arrived on 05.06.2006In the last few decades in the spheres of engineering,economics and planning there is a trend of transitionfrom the admissible technical and managing solutions tooptimal ones. However, the modern optimization theoryhas not met the requirements of a design engineer because of the fact that its strict mathematical methods donot take into account real conditions of design problems.Besides, modern complicating design practice needs inefficient mathematical means of solving such problems.A distinguishing feature of the new approach is acomplex development making possible to design a whole system, but not its separate parts. Therefore, one ofthe most important scientific and applied problems is todevelop methodology of optimal design of complextechnical systems – the system design.A construction is characterised by a number of criteria: cost, reliability, weight, size, engineering time andetc, that can came in mutual contradiction. The difficulty of the problem solution consists in the lack of apriori information necessary for searching for optimalvariant of construction. Therefore, the design procedure is worthwhile to arrange in such a way that the volume of information on construction would increase atevery subsequent stage. At the same time it is necessaryto exclude inadequate variants revealed in the course ofdesign. Thus, the two tendencies are to combine: generation of variety of modifications and truncation of theobtained variety [1]. A suggested design procedure isconsistent with evolution optimization strategy andgenetic algorithm (GA) in particular [3].Construction design is presented in the form of somesequence of its development levels which are characterised by the degree of its element detail elaboration.Such design technique can be connected with some hierarchical model possessing peculiarity of another class ofevolution model i.e. a sequential decision tree.Genetic algorithms received a wide acceptance inthe middle of the 1960’s owing to J. Holland’s works.They simulate evolutionary process with the stress ongenetic mechanisms, i.e. gene inheritance and recombination. It is made by some number (population) of artificial chromosome (individuals). Each chromosomecontains n genes that correspond to n desired variablesof optimization problem.Genetic algorithms like evolution algorithm in general are applied to search for the function global extremum of many variables. The principle of their operationis based on modelling some mechanisms of populationgenetics, manipulation of chromosome set at forminggenotype of a new biological individual by means ofinheritance of parents’ chromosome parts, accidentalvariations of genotype known as mutation.In fig. 1 genetic algorithm is shown. The main idea ofevolution consists in improving the individual fitness of thefirst population generation until cease criteria are achieved.UDC 624(007.2:57.085)OPTIMIZATION OF BUILDING CONSTRUCTIONS ON THE BASIS OF GENETIC ALGORITHM А.G. Yuriev, S.V. Klyuyev, А.V. KlyuyevThe Belgorod State Technological University named after V.G. ShukhovEmail: Klyuyev@yandex.ruThe technique of optimal design of bearing structures on the basis of genetic algorithm has been suggested. A design of steel frame withvarying 9 parameters using the method of finite elements is considered as an example. The best variant corresponding to the volume minimum of the frame material is revealed.Fig. 1. The genetic algorithm1. Initial population. At first it is necessary to create theinitial population of individuals. As nothing isknown about objective function, let us take the individual genes in the earlier stated area as accidentaland uniformly distributed.2. Individual evaluation. It is necessary to determinefitness for each of newborn individuals on the basisof objective function. Then one can start generationloop to improve individual fitness.3. Selection is the first step to this improvement at whichindividuals are selected by chance or on the basis oftheir previous fitness according to the strategy. Thisselection serves as either genus prolongation or elitestatus transfer. Individual with an elite status can beneither excluded from real generation nor changed.4. Reproduction means genus prolongation of individuals selected for this purpose. The simplest way of reproduction is coping individuals. Besides, GA strategy tries to generate the best individuals by generecombination in chromosome. As a rule, in thiscase parent individuals form couples that exchangegenes with each other by chance and form in thisway two newborns.5. Mutation changes separate genes of gene pool.Owing to it at GA gene pool should be renewed because at reproduction there would be loss gene poolvariety usually after a few number of generations.6. Individual evaluation. After changing individualchromosomes by recombination and mutation it isnecessary for every newborn to determine the fitness.7. Individual replacement. At the end of generation loop it is necessary to find out what individuals shouldbe excluded from the population. Otherwise withoutreplacement the population would grow longer. GAis usually replaced with the majority of parents bynewborns.8. Cease criteria define duration of optimization process and play a decisive role in evaluation of results.The two variants of forming cease criteria are used:1) fitness control, as a result of which the process isceased if the maximum fitness value in population isnot fundamentally improved in the range of specified generation number; 2) determination of generation number.Both variants have disadvantages. In the first variant itcan happen that the maximum fitness value in the population does not change for a long time, but then, e.g. dueto successful mutation there is an improvement. Therefore in the case of early cease only suboptimum is achieved.In the second variant optimum criteria are not themain one at all. Often the solution maintaining enoughthe tolerance set up for varied parameters is chosen. Inthis case it is still necessary to carry out optimizationprocesses with different numbers of generations to estimate the results.By means of selection and exclusion of individualswith relatively poor fitness the population is also revalued. It is clear that in this case the gene pool can lose some good genes. GA tries to reduce these losses as muchas possible [1, 2].Let us consider the design of steel frame as an example. The volume is minimized along with weight andmaterial consumption. The additional conditions connected with stress and stability conditions, general andlocal are specified. As a material steel S345 with the following characteristics is used: density ρ=78,5 kN/m3;the module of longitudinal elasticity E=2,1.105 МPа,lateral deformation coefficient ν=0,3, yield stressRуп=360 МPа, design resistance Ry=300 МPа.The frame sizes are specified: lc=4,8 m, lp=4 m andload intensity on horizontal projection of the beam is8 kH/m. 9 parameters of the system are optimized(fig. 2): b is a half of the width of the flange beam; hрл isthe crosssectional height of the beam from the left; hрпthe crosssectional height of the beam from the right; hсвthe crosssectional height of the support upwardly; hсн –the crosssectional height of the support below; tрп andtрс, tсп and tсс – the thickness of flange and the beam walland the support correspondingly.From the conditions of observing the local stabilityof flange beam crosssection elements the admissible relations of their sizes are stated: for the flange it is 10, forthe support it is 110. The upper and lower bound of thedesired variables are stated in the following way (in sm):5≤b≤15; 10≤hрл≤30; 10≤hрп≤30; 0,5≤tрп≤1,5; 0,5≤tрс≤1,5;10≤hсв≤30; 10≤hсн≤30; 0,5≤tcп≤1,5; 0,5≤tсс≤1,5.Objective function, expressing the volume of the frame material has the following form:   i n i t i a lp o p u l a t i o n   i n d i v i d u a l  e v a l u a t i o nr e p r o d u c t i o ns e l e c t i o nm u t a t i o n   i n d i v i d u a l  e v a l u a t i o n  i n d i v i d u a l        r e p l a c e m e n tn oy e se n dc e a s e ?Mathematics and mechanics. Physics59Fig. 2. Frame profile and crosssections of the beam and supportThe same expression with the opposite sign definesthe fitness function.Using the method of finite elements for calculation,let us break the half of the frame volume into 108 elements – 48 in the beam and 60 in the support, in thiscase in the crosssection there are 6 elements (2 per theflange and the wall).According to the number of optimized parameterswe introduce the chromosome modelThe elements are classified in 4 groups – flange andbeam wall, flange and support wall. Junctions are classified into 26 groups, motion of which corresponds tochromosome genes.Table 1. Sizes of profile elementsTo calculate the frame the following parameters areused: 20 individuals; 300 generations; 0,1 selection pressure; 0,05 mutation standard; 15 replacement number.Mutation deviations have linear character and amount 0,001...0, 0001 for flange and wall thickness, butfor the other elements 0,01...0,001.The obtained results (sizes are in mm) are presentedin the table 1. In the table 2 the frame volume V, m3, thelargest normal stresss σ, MPa, and the relation b/tрп andb/tсп are presented.From the Table 2 it is evident that the relations of b/tрпand b/tсп are exceeded in the second and fifth lines. In thefifth and sixth lines the calculated stress is exceeded.Thus, the three variants of solution should be ignored.Table 2. Geometric characteristics of frame and stressThe wall thicknesses correspond to specified lowerbound. The other parameters are in the range of specifiedbounds. In the rest seven variants of solution the volume isnot sufficiently different (in the range of 3,4 %). The minimal volume corresponds to the fourth variant of solution.Conclusion1. Genetic algorithms are powerful finding means. Thesolution obtained on their basis is suboptimal, but itdoes not prevent from application of algorithms tosearch for global extremums at building construction optimization.2. Genetic algorithms are acceptable for solution ofmultiparametric nonconvex problems in comparisonwith the known analytical methods of optimization.3. The solution can be made more accurate having increased the grid density of the finite elements. Increasing the number of optimized parameters results inincrease of the number of individuals and generations. Besides, machine time consumption increases, that sometimes may serve as an evaluation parameter of using genetic algorithms.V σ b/tрп b/tсп1 0,0150 161 7,4 9,42 0,0265 160 11,7 10,13 0,0150 161 8,5 9,54 0,0147 160 8,4 9,55 0,0224 388 11,5 10,46 0,0153 338 6,8 9,57 0,0149 162 8,6 9,58 0,0149 162 8,7 9,79 0,0152 162 7,9 9,710 0,0149 161 8,2 9,3b hрп hрп tрп tpc hсн hсв tсп tcc1 55 138 143 6,4 5,0 100 105 5,0 5,02 69 209 231 5,7 5,1 298 281 6,6 5,03 50 120 157 5,6 5,0 102 158 5,0 5,04 50 120 194 5,6 5,0 101 108 5,0 5,05 63 191 252 5,2 5,0 191 300 5,8 5,06 50 115 123 7,0 5,0 102 119 5,0 5,07 50 161 162 5,5 5,0 101 121 5,0 5,08 51 128 169 5,6 5,0 103 116 5,0 5,19 51 131 154 6,1 5,1 102 120 5,0 5,010 50 126 161 5,8 5,0 110 117 5,1 5,1b hрп hрп tрп tpc hсн hсв tсп tcccc câ cí cnpc ðë pn pn cn ðn2,4 ( 2 )2 ( 2 ) (19,2 16 ).V t h h tt h h t b t t= + − ++ + − + +Bulletin of the Томsк Pоlytеchnic University. 2007. V. 310. № 160REFERENCES1. Yuriev А.G., Klyuyev S.V. Genetic algorithms of building construction optimization // Education, Science, Industry and Managementin the XXI Century: Materials of International Scientific Conference in 4 volumes. – Stary Oskol: STI MIS&S Edition, 2004. – V. 4. –P. 238–240.2. Yuriev А.G., Klyuyev S.V. Evolution and genetic algorithms of building construction optimization. – Belgorod: BSTU Edition, 2006. –134 p.Arrived on 22.09.2006

Optimization of building constructions on the basis of genetic algorithm

Güthenke, G.

Möller, H.

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Application of price performance analysis in different supplier-buyer relationships

Güthenke, Gunnar

Möller, Henning

RWTH Publications

Optimization of building constructions on the basis of genetic algorithm

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