MINIMUM-WEIGHT DESIGN OF COMPRESSIVELY LOADED COMPOSITE PLATES AND STIFFENED PANELS FOR POSTBUCKLING STRENGTH BY GENETIC ALGORITHM

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1 MINIMUM-WEIGHT DESIGN OF COMPRESSIVELY LOADED COMPOSITE PLATES AND STIFFENED PANELS FOR POSTBUCKLING STRENGTH BY GENETIC ALGORITHM Ji-Ho Kag a, Jae-Hu Lee a, Joo-Hyu Ha a, Chu-Go Kim a, ad Dog-Mi Lee b a Divisio of Aerospace Egieerig, Departmet of Mechaical Egieerig Korea Advaced Istitute of Sciece ad Techology 373-1, Kuseog-dog, Yuseog-gu, Daejeo, 35-71, Korea b Agecy of Defese Developmet, Yuseog P. O. Box 35, Daejeo, Korea SUMMARY: The miimum-weight desig of compressively loaded lamiated plates ad stiffeed paels was aimed uder costraied postbucklig stregth i this paper. A oliear fiite elemet code, COSAP(COmposite Structure Aalysis Program) was applied to aalyze the precise bucklig ad postbucklig behaviour. As a optimizatio techique, Geetic Algorithm was used to fid the optimum poits i the desig space. A covetioal geetic algorithm was modified with two ew techiques. Oe is the use of the pre-calculated result to reduce the calculatio cost, ad the other is the use of variable populatio size i geetic algorithm to fid better optimum poits. The modified Geetic Algorithm code was the parallelized with MPI(Message Passig Iterface) library ad the optimizatio was executed with a parallel-computig supercomputer, CRAY T3E. The optimal desig was performed i two cases; a composite plate ad a composite stiffeed pael. The desig variables were umber of plies ad the ply agle i each ply. I case of the stiffeed paels, the size ad spacig of the stiffeers were also cosidered as the desig variables. The objective fuctio was defied as the product of the odimesioal weight ad stregth. The optimizatio was performed with two chose examples, ad the results showed that the optimal desigs have better performaces tha covetioal desigs. KEYWORDS: weight-miimizatio, optimal desig, geetic algorithm, composite, postbucklig INTRODUCTION Fiber reiforced composite materials have superior characteristics as the variable stackig sequeces

2 ad ply agles whe compared to covetioal materials. Therefore optimal desig ca be achieved by determiig the proper stackig sequeces ad ply agles. However, there are some difficulties i optimal desig like discreteess of the desig values ad complexity of the desig spaces. I 199 s, may researches usig discrete ply agles as desig variables were coducted[1-5]. I the previous researches, it is ievitable to use discrete ply agles such as, 9, or ±45 for desigig realistic composite structures. Weight ad maufacturig efficiecy as well as stiffess or stregth were cosidered. Furthermore the umber of plies ad the shapes should be optimized to reduce weight, ad some researchers cosidered them. Oe more importat issue is cosiderig postbucklig behavior. Bucklig of aerospace structures does ot mea whole structural collapse ad it is reasoable to desig structures allowig bucklig of skis[1,5]. I this paper, the miimum-weight desig of compressively loaded lamiated plates ad stiffeed paels was performed usig a oliear fiite elemet code ad a modified geetic algorithm with parallel computig scheme. NONLINEAR FINITE ELEMENT ANALYSIS FOR COMPOSITE STRUCTURES A oliear fiite elemet aalysis code for composite structures, COSAP(COmposite Structure Aalysis Program) which was developed i the papers[6-8], was used to aalyze bucklig ad postbucklig behavior of composite plates ad stiffeed paels i this study. A brief formulatio procedure of the aalysis is stated below. At a arbitrary (+1)th equilibrium state, the priciple of virtual work without body force terms ca be rewritte i terms of the secod Piola-Kirchhoff stress, S ij ad the Gree strai, ε ij with takig the cofiguratio at the th equilibrium state as the referece oe: V ( ij + Sij) δ( εij) dv ( Ti + Ti) δ( ui) ds = S σ (1) T where σ ij, e ij, T i, u i, ad δ are the Cauchy stress, ifiitesimal strai, surface tractio, displacemet, ad variatio operator respectively. The Gree strai, Δε ij ca be divided ito the liear term, Δe ij ad oliear term, Δη ij. ε = + (2) ij eij ηij By substitutig Δε ij give i eq (2) to eq (1), elimiatig secod-order terms, ad implemetig stress-strai relatio, the equatio ca be obtaied as

3 = S T ( eij) Dijkl ekldv + σ ( u ) u ijδ k, i k V ( Ti + Ti) δ( ui) ds σ ( e )dv V ijδ ij δ, j V dv (3) where D ijkl is the stress-strai relatio matrix i he global coordiate system. The degeerated shell elemet with 8-odes is used for the formulatio. Each ode has 5 DOF s ad the shear deformatio was cosidered from the first-order shear deformatio theory. The strai ad the displacemet ca be expressed with the shape fuctios of the elemet ad the odal DOF vector. { e} = [ B ]{ U } { u, } = [ B ]{ U } L, (4) k NL The fiite elemet equatio ca be obtaied by puttig eq (4) to eq (3) as ([ K ] [ K ]){ U } = { P} L + (5) NL where = (6) [ ] [ ] T L L [ ][ L] K B D B dv V = σ (7) T [ NL] [ NL] [ ][ NL] K B B dv V T { P} = [ B ] { } dv { F } σ (8) V I the previous eqs, {F} is the odal force vector ad { σ }, [ σ ], ad [ σ ] are defied as follows. L { } [ σ σ σ τ τ τ ] T σ = (9) x y z yz xz xy σ x τ xy τ xz [ σ ] = τ xy σ y τ yz (1) τ xz τ yz σ z [ σ] [ σ] = [ σ] [ ] σ (11)

4 I the iteratio process of the fiite elemet equatio, the arc-legth method was used i the loadicremet. To estimate the failure load of the structures, the maximum stress criterio is applied to the average stresses i the pricipal material directios of each layer i each elemet. The stress compoet correspodig to the failure mode is uloaded istataeously. The postbucklig stregth is assumed to be the load at the momet of the first fiber failure. GENETIC ALGORITHM Geetic algorithm[7] was used as the optimizatio method i this study. It simulates the atural evolutio so that multiple desig poits evolve to be coverged to a global optimum. Its calculatio process uses odetermiistic scheme ad has othig to do with differetiability or covexity. The most useful advatage is that it uses discrete desig variables by ature; therefore, it is simple to use the discrete ply agles of composites as desig variables. Parallel Computig Techique Geetic algorithm is the oe that is very suitable for parallel computig scheme because multiple desig poits should be evaluated i a calculatio step. I other words, the algorithm ca be programmed so that multiple desig poits i a geeratio may be divided ito some sub-populatio ad oe processor i a parallel computer calculates oe sub-populatio respectively. The programmig was doe with MPI(Message Passig Iterface) library i this study. Its schematic diagram is show i Fig. 1. The computig system used was CRAY-T3E i the KISTI Supercomputig Ceter i Korea ad 16 processors of the system were implemeted i this study. Modificatio of Geetic Algorithm for Acceleratio I Geetic algorithm process, the fitess of the whole populatio should be evaluated for each geeratio. However, populatio aggregates to a optimum as covergece is accomplished ad some desig poits that are same to the oes i the previous geeratio are re-evaluated, which meas the waste of computig resources ad time. Therefore, it is ecessary to avoid the waste by modifyig the algorithm. There ca be may kids of modificatios possible but we implemeted our ow idea to solve the problem. The procedure ca be explaied briefly as follows: 1. Write all the fitess evaluatio results ito a file. 2. Make ew geeratio cosiderig the fitess of the populatio. 3. I the ew geeratio, fid out which desig poit is same to the previous oe that is i the writte file.

5 4. Read the writte results from the file for the overlapped desig poits that are foud i Step Do the real evaluatio for ewly geerated desig poits oly. 6. Apped the fitess evaluatio results i Step 3 to the file. 7. Go to Step 2 ad repeat the procedure. This method dramatically reduces the umber of the real evaluatio of fitess values. However, the fidig process(step 3) might be a time-cosumig work, so the method should be applied carefully. I this study, we used the oliear fiite elemet aalysis for the fitess evaluatio. The time cosumptio of the fitess evaluatio is eormous ad caot be compared with the method stated above. Therefore, we decided to apply this method ad modify the Geetic Algorithm. Oe more thig was implemeted i the modificatio, the variable populatio size. The populatio size automatically icreases i order to guaratee that the miimum of the umber of the real evaluatio of fitess is equal to or greater tha a particular umber which the desiger desigates. The utility of the variable populatio size makes the algorithm robust eve i the case where the iitial populatio size is relatively small ad too early covergece is iduced, which is udesirable. Root processor Start Other processors Sed iformatio to be shared Radom geeratio of iitial populatio Receive iformatio to be shared Sed cotrol sigal CALC Divide populatio to sub-populatios Sed each sub-populatio to correspodig processor Receive cotrol sigal Yes EXIT? No Receive my sub-populatio from root processor SPBUCK Evaluate fitess of each desig poit i my sub-populatio Evaluate fitess of each desig poit i my sub-populatio SPBUCK Receive each sub-populatio from correspodig processor Sed my sub-populatio to root processor Assemble sub-populatios to make whole populatio Select & reproduce ew populatio Crossover & mutatio Coverged? Yes No Sed cotrol sigal EXIT Ed Fig. 1. Schematic diagram of Geetic Algorithm with parallel computig.

6 OPTIMAL DESIGN OF COMPOSITE PLATES AND STIFFENED PANELS Problem Defiitio The optimal shapes, stackig sequeces, ad ply agles were searched for some composite structures with the modified Geetic Algorithm stated above. The objective of the optimizatio was to fid the miimum weights of the structures that resist desig stregth. The objective fuctio was defied as: f = W W max P 11P fail fail, desig fail 1Ncr + P P fail, desig,, P P fail fail P < P fail, desig fail, desig (12) where W max is the possible heaviest weight ad W is the weight of the desig poit. P fail,desig ad P fail are the desig stregth ad the stregth of the desig poit respectively. Optimal Desig of Composite Plates The shape of the plate, the boudary coditios, ad the loadig coditios are show i Fig. 2. The possible maximum umbers of plies was set to 16 ad oly 8 plies were used as desig variables because all lamiates were assumed to be symmetric. The usable ply agles were set to, ±45, 9 for practical applicatio. The material properties of the composite material are show i Table 1. The desig stregth was fixed 3kN i this example. Fig. 2. The shape, the boudary coditios, ad the loadig coditios of composite plate.

7 Table 1. The material properties of the composite material property value property value E GPa ν E 2 1. GPa ν E 3 1. GPa ν G GPa ν G GPa ν G GPa ν X T 1933 Mpa Y T 51 Mpa X C 151 MPa Y C 141 MPa S 61 MPa ply thickess.125 mm The optimizatio result showed that the optimal stakig sequece ad ply agles were [/9/ 2 /9] S. The optimized plate has the stregth of 34.9kN which is 16% greater tha the desig value ad is 37.5% lighter tha the possible heaviest plate. Fig. 3 shows the load-deflectio curves of three differet plates of same weight; the optimized plate, a quasi-isotropic plate with [/±36/±72] S, ad a uidirectioal plate with [ 1 ] T. It shows that oly the optimized plate resists the desig stregth. Fig. 3. Load-deflectio curves of three differet plates of same weight: the optimized plate, a quasiisotropic plate, ad a uidirectioal plate.

8 Optimal Desig of Composite Stiffeed Paels The shape of the stiffeed pael, the boudary coditios, ad the loadig coditios are show i Fig. 4. The desig variables were selected as show i Table 2. The material properties of the composite material are the same as i Table 1. I order to determie the referece desig stregth, a composite stiffeed pael which has the shape ad the stackig sequece as show i Table 3 was selected ad the postbucklig aalysis was performed for the referece stiffed pael. As the result of the aalysis, the stregth of the pael was 65.5kN ad we set the desig stregth to be a slightly smaller value, 6kN. Table 2. Defiitio of desig variables of composite stiffeed pael desig Ski size L (mm) 25 W (mm) 16 Stiffeer type I Desig failure load, P fail.desig (N) 6 s mi (mm) 25 Stiffeer Locatio, s s max (mm) 56 bits * 5 Flage size, f (mm) 24 (fixed) w mi (mm) 15 Web size, w w max (mm) 78 bits * 6 c mi (mm) 15 Cap size, c c max (mm) 78 bits * 6 Max. umber of the ski plies 8 2 Max. umber of the stiffer plies 8 2 Bits for a degree 3 Total bits 65 Number of possible desigs * : umber of bits for s, w, c. Table 3. Shape ad stackig sequece of the referece composite stiffeed pael Stiffeer locatio, Web height, Cap width, Ski stackig Stiffeer stackig Ultimate failure Weight, W s (mm) w (mm) c (mm) sequece sequece load, P fail (N) (mm 3 ) [/9/±45] S [/9/±45] S

9 The optimizatio result is show i Table 4. The stregth is 72kN which is quite larger tha the referece desig value; however, the weight is smaller tha the result of the pael i Table 3. Fig. 4 shows the compariso of the load-deflectio curves ad the bucklig shapes betwee the referece stiffeed pael i Table 3 ad the optimized result. Table 4. Optimal desig results of composite stiffeed paels Stiffeer locatio, Web height, Cap width, Ski stackig Stiffeer stackig Ultimate failure Weight, W s (mm) w (mm) c (mm) sequece sequece load, P fail (N) (mm 3 ) [-45/9 2 ] S [45//45/ 2 ] S Load, P (N) Referece 2 1 Referece Stiffeed Pael Optimal Result Shear Failure Matrix Failure Fiber Failure Deflectio, u (mm) Optimal Desig Fig. 4. Compariso of the load-deflectio curves ad bucklig modes betwee the referece ad the optimized result. CONCLUSION I this paper, the weight optimizatios of composite structures were studied with cosiderig the postbucklig behavior of composite structures. I order to estimate the postbucklig behavior, a oliear fiite elemet aalysis code was applied ad the modified Geetic Algorithm was used as the optimizatio method. The optimizatio was performed for composite plates ad stiffeed paels ad the result showed that the preset optimal desigs have better performaces tha covetioal desigs.

10 REFERENCES [1] D. K. Shi ad Jr. O. H. Griffi, "Miimum-Weight Desig of Lamiated Composite Plates for Postbucklig Performace," AIAA Paper, preseted at the AIAA/ASME/ASCE/AHS 31th structures, Structural Dyamics, ad Materials Coferece, pp , [2] S. Adali, A. Richter, V. E. Verijeko ad E. B. Summers, "Optimal Desig of Hybrid Lamiates with Discrete Ply Agles for Maximum Bucklig Load ad Miimum Cost," Composite Structures, Vol.32, pp , [3] B. Malott, R. C. Averill, E. D. Goodma ad W. F. Puch, "Use of Geetic Algorithms for Optimal Desig of Lamiated Composite Sadwich Paels with Bedig-Twistig Couplig," AIAA Paper, preseted at the AIAA/ASME/ASCE/AHS 36th structures, Structural Dyamics, ad Materials Coferece, pp , [4] A. S. Fie ad G. S. Spriger, "Desig of Composite Lamiates for Stregth, weight, ad Maufacturig," Joural of Composite Materials, Vol.31, No.23, pp , [5] C. A. Perry. Z. Gürdal ad J. H. Stares JR, "Miimum-Weight Desig of Compressively Loaded Stiffeed Paels for Postbucklig Respose", Egieerig Optimizatio, Vol.28, pp , [6] J. H. Kweo ad C. S. Hog, A Improved Arc-Legth Method for Postbucklig Aalysis of Composite Cylidrical Paels, Computers & Structures, Vol.53, pp , [7] C. W. Kog, I. C. Lee, C. G. Kim, ad C. S. Hog, Postbucklig ad failure of stiffeed composite paels uder axial compressio, Composite Structures, Vol.42, pp.13-21, [8] C. W. Kog, C. S. Hog, ad C. G. Kim, Postbucklig Stregth of Composite Plate with a Hole, Reiforced Plastics ad Composites, Vol.2, pp , 21. [9] D. E. Goldberg, Geetic Algorithms i Search, Optimizatio ad Machie Learig, Addiso Wesley, Bosto, MA, 1989.

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