OPTIMIZATION OF THE BEHAVIOR OF BUCKLING OF THE STRATIFIEES BY GENETIC ALGORITHM

OPTIMIZATION OF THE BEHAVIOR OF BUCKLING OF THE STRATIFIEES BY GENETIC ALGORITHM

Belkheir/F

Department of Materials technology Faculty of Science - USTO MB

Oran, Algeria bel.tiha@gmail.com

Safer /M

Department of Materials technology Faculty of Science - USTO MB

Oran, Algeria mal_rel85@yahoo.com

Abstract—

The laminated composite structures which are largely used nowadays, become unstable when they are prone to loadings of mechanical or thermal nature and flame in the elastic zone.

Consequently, buckling has a very great importance when designing this kind of structure. In this work, we studied the phenomenon of the buckling which was devoted under investigation of the optimization of a composite material laminated by the use of a data-processing method based on a mathematical theory which uses the stochastic statistics it is the theory of the genetic algorithms (GA). The objective of this work is, the maximization of the rigidity of the plate laminated by maximizing the critical load of buckling according to the orientations of the folds.

Keywords— Composite laminated, optimization, buckling, genetic Algorithm.

I. INTRODUCTION

The buckling of a structure corresponds to an instability of behavior. Instability is primarily a property of the structures of extreme geometry, such as for example the compressed elements of important twinge, the thin sections or the cylindrical thin hulls. Generally, buckling intervenes for constraints in material quite lower than the limits with rupture.

To analyze the buckling of a structure amounts determining the conditions for which the balance of the structure ceases being stable. Generally, it is a question of determining the value of loading, one speaks about critical loading, beyond which initial balance becomes unstable and the least disturbance involves the swing towards a new steady balance, correspondent with great displacements in the structure. One usually associates buckling with loadings of compression, but shearing or torsion can lead to this same type of instability.

Other parameters can also start buckling, like the temperature in the case of thermal buckling related to phenomena of opposed dilation.

The laminates consist of successive layers of reinforcements impregnated of resin, directed in an unspecified way the ones compared to the others. (Figure 1), the advantage of the laminates lies in the possibility of adopting and of controlling the orientation of fibres so that the material resists requests determined under better conditions.

Figure 1: designation of a laminate

II. THE CLASSICAL THEORY OF THE LAMINATES The classical theory of laminates (CLT) constitutes the method of meso-macro scaling most largely used in the analytical approaches multi scales published. This simple method makes it possible to estimate, starting from the macroscopic loading applied (membrane flows of efforts and torsion and bending moments), the deformation and stress fields at the level of the fold. The material is thus characterized at the level of fold by its elastic properties:

module of elasticity, Poisson's ratios and modulus of rigidity respectively noted E1, E2, v12 and G12.

The relation between constraints and deformations is written, in the axes of orthotropic fold, noted (1, 2, 3),

Where the no worthless terms of the matrix of rigidity Q are written (eq I):

, , , Eq(I)

For the passage on a macroscopic scale, the CLT is based on the theory of the thin sections of Kirchhoff-Coils, which applies that:

- displacements and deformations remain small (assumption of small disturbances),

- The interfaces between the folds are perfect,

- The section of the plate remains normal with the average plan during the loading, which implies the nullity of the transverse deformations of shearing.

Consequently, the total deflections, expressed in the total reference mark, are supposed to be linear in the thickness of the laminate and are written:

One easily deduces the classical relations from the theory of the laminates in the case of elasticity:

Where the parameters A, B, D are described by the equation

Under this writing, the analysis of the matrix of rigidity assembled of the laminate (matrix ABD) makes it possible to highlight certain elastic behaviors characteristic of the laminates:

- block A corresponds to the behavior of membrane, - the block B corresponds to the behavior of inflection, - the block D corresponds under the terms of coupling

between the phenomena of membrane and inflection.

For a symmetrical laminate, the matrix coupling membrane-inflection [B] is worthless. Moreover, if the laminate are balanced, the A16 coefficients and A26 are also worthless.

III. METHOD OF OPTIMIZATION:

To formulate a problem of optimization returns to identify three ingredients clearly: variables of optimization, objectives of the problems and constraints. Without loss of general information, one places oneself in the continuation within the framework of a problem of maximization.

AGs have the advantage of working with the only values of the functions to optimize, without any assumption of derivability or even of continuity, what returns them particularly adapted for the resolution of problems to the mathematical formulations complex and very indicated for combinative optimization.

The adjustment of the algorithm constitutes the major difficulty for the implementation of the approaches revolutionaries. Indeed, the comprehension of the interactions between the various elements of the algorithm (coding, selection, criterion of stop, etc) requests a certain expert testimony. The adjustment of the corresponding parameters (size of the population, probabilities of crossing, etc) is a exercise delicate, likely to condition the efficiency of the algorithm, , and for which one has generally only one empirical approach. However, one would wish to arrive, in the ideal, with adjustments dynamic, or adaptive, automatically modifying the behavior of the algorithm according to the already got results.

Three elements characterize an GA:

• The maintenance of a set of solutions candidate,

• The selection process of the best solutions according to their quality,

• Their handling by genetic operators, generally of standard crossing and change.

A. Generel architecture of the genetique algorithm

Each GA is determined by its basic cycle, which is represented by architecture follow:

Input: Np (size of the population)

T (maximum number of generations) PC (probability crossing)

pm (probability change)

Outputs: A (a whole of the best solutions)

Stage 1:Initialization:(t=0) to generate an initial population P0 of size Np.

Stage 2:Evaluation:calculation of the function-objective for each individual contained in Pt.

Stage 3:Calculation of the function of quality,for each individual contained in Pt.

Stage 4:Terminationif t > = T or that another criterion of stop is satisfied then to turn over A.

The whole of the best solutions contained in Pt.

Stage 5:Selection:pulling with handing-over of Np solutions of Pt by a given method based on the

value of the function of quality. That is to say P′

temporary population obtained.

Stage 6:Reproduction:to apply the operators of crossing and change to the population P ′ with the

respective probabilities: PC and pm. That is to say Pt+1 resulting population.

To increment the meter of generations (t=t+1) and return at Stage 2.

IV. DESCRIPTION OF THE BEHAVIOR OF THE LAMINATED PLATES

The optimal design of the laminates presents one of the most interesting problems. with the dealt problem consists in maximizing the factor criticizes buckling of a plate rectangular, out of carbon/epoxy.

The equation (2) gives the factor of buckling (λf (m, N)) of a balanced and symmetrical rectangular plate which is simply supported and subjected to Nx loadings and Ny in compression (figure (2.a)). The laminated plate flames then in m and N half sine in directions X and there respectively (see figure 2.b). The critical factor of buckling is used in order to determine the critical loads of buckling Nxfet Nyf (see equation (II.29)) who will cause buckling.

Eq 2

Figure 2.a: Simply supported plate

V. DIGITAL RESULTS AND DISCUSSION

The index properties of Carbon-Epoxy material are given to table (I), they got results are a set of laminates optimal that each one is characterized by the critical factor of buckling

« »,

Table I: Properties of material, data input.

Properties Carbone-

époxy

Longitudinal module E1 127.59 GPa

Longitudinal module E2 13.03 Gpa

Modulus of rigidity in the G12plan 6.41 GPa Poisson's ratio in the v12plan 0.30

Principal deformation maximum e1 0.008 Maximum transverse deformation e2 0.029

Maximum shearing γ_12 0.015

Thickness of the folds h 0.127 mm

The objective of this optimization of the sequences of stacking is to find best stacking for the critical factor of buckling (λcr) for a thickness and proportions given.

The general formulation of this problem is given to tableII.

Dimensions and the loading are given to table II.

Table II: Formulation of problem of optimization To maximize:

While changing:

-

Under the constraints:

-

-

A. The effect of the orientation of the folds on the factor of buckling

To examine the effect of the orientation of the folds of our laminate (Carbon-Epoxy) on the factor of buckling criticizes λcr, we took the data of the sequences of stacking of the literature [4], while keeping the data input fixed of table I. the got results are illustrated in table III. This table gives the best designs obtained by the Rich person and Haftka [1] and Soremekun and collar. [5] (solutions has with F) and best solutions obtained by our program (solutions 1 to 4).

Table III: Comparative table of the sequences of stacking on the factor of buckling.

Sequence of stacking Literatur e ^(a)

Ourresults

A [9010, ±452,902, ±453,902, ±454]s 3976.64 (AG)4104,90 B [±45,9010, ±45,908, ±45,908]s 3976.64 4104,90 C [904, ±452,9016, ±45,906]s 3976.64 4104,90 D [90², ±45,906, ±45,908,

±45,9010]s 3976.64 4104,90

E [90⁸, ±45,902, ±45,902, ±45,902,

±456]s 3976.64 4104,90

F [±45,908, ±45,9018, ±45]s 3976.64 4104,90

1

[-452,90,-452,453,90,- 45,45,90,452,90,452, -

45,90,45,90,-452,904,-45,902,45,- 45]s

--- 4020,56

2 [90,-45,90,-45,90,45,-

45,90,45,902,-45,902,45,-45,902,-

45,45,904,45,90,45,903,452]s --- 4020,56 3 [907,+45,90,-452,903,+452,0,-

45,90,-45,90,02,-45,902,-

45,02,+453]s --- 3976,41

4 [(90,+45)2,(902,-

45)2,903,+45,904,+45,90,0,903,-

45,0,-45,90,02,902]s ---- 3955,49

(a) Ref[1]

It is noticed that this problem counts several optimal solutions. At this time, the optimal designs have a factor of buckling, which is critical for this problem, of 3976.64. The results got by our program gives a factor of buckling of 4104.90, this result confirms the sensitivity of the program screw-have-screw the objective of the problem of optimization.

It is as interesting to notice as there exist several different solutions which have optimal factors of buckling. Table IV.5 details the number of folds of each orientation of the optimal laminates quoted previously. The last column of this table gives the number of distinct stacking, obtained by the program, which contains the same proportion of folds of each orientation. For example, there exist at least two different ways to pile up 18 folds with ± 45° with 14 directed folds with 90°, to form an optimal design (see solutions has and E of Figure 2.b: Chart of m and n

table III). According to table IV, it is interesting to notice that the use of directed folds with 0° is not useful for the attack of the objective.

Table IV: the effect of the orientation of the folds on the factor of buckling.

0°2 ±45° 90°2 Distinct stackings

* 0 18 14 2(a, e)

* 0 6 26 4 (b-c-d-f)

* 0 14 18 1

* 0 20 12 1

* 5 12 15 1

* 4 8 20 1

The results are presented in the form of graph (see Figure3).

15 20 25 30 35 40 45 50

3400 3500 3600 3700 3800 3900 4000

4100 lambda 90°

lambda 45°

Facteur de Flambement

Nombre de plis

Figure 3: variation of buckling according to the angle.

Learned this graph, one notices that the results found on stacking have an impact on calculates critical factor of buckling, because this one depends on certin coefficients of the matrix [D]. One finds a usual result on the level of the mechanical behavior, the folds with +-45 are most effective to be opposed to buckling.

VI. CONCLUSION

The genetic algorithms converge quickly towards a set of good solutions. That could make it possible to find one more a large number of good solutions among the final population.

Also, as it is not necessary to evaluate the same individual twice, it could be interesting to study the possibility of establishing a control mechanism on the individuals already evaluated in order to avoid the useless repetition of the evaluation of this individual. That would make it possible to decrease the number of times that one calls on the function of evaluation (optimization of computing time). However, this way of making can take a storage capacity and an increased time computing.

References

[1] Book «MATERIAUX COMPOSITES, Comportement mécanique et nalyse des structures ». J-M. BERTHELOT.

[2] DOCTORAT DE L’UNIVERSITÉ DE TOULOUSE spécialité : Génie Mécanique, Mécanique des matériaux « Stratégies de calcul pour l'optimisation multiobjectif des structures composites. », soutenue par François-Xavier IRISARRI, Le 23 Janvier 2009

[3] « 2006» Optimisation multicritère de stratifié par algorithmes génétiques Laurent-Blanchard ; Nathalie-Blaszka ; Stéphane-Lantéri ; Jean-Antoine Désidéri (Alcatel Space Industries ; Institut National de recherche en Informatique et en Automatique).

[4] Doctorat de l’université de TOULOUSE spécialité : Génie Mécanique, Mécanique des matériaux « Stratégies de calcul pour l'optimisation multiobjectif des structures composites. », soutenue par François-Xavier IRISARRI, Le 23 Janvier 2009.

[5]. Kosorukoff, A., Goldberg, D.E., “Genetic Algorithms for Social Innovation and Creativity”, IlliGAL Report No. 2001005, 2001.