BUCKLING OPTIMIZATION OF UNSYMMETRICALLY LAMINATED PLATES UNDER TRANSVERSE LOADS

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1 BUCKLIG OPTIMIZATIO OF USYMMETRICALLY LAMIATED PLATES UDER TRASVERSE LOADS Hsuan-Teh Hu and Zhng-Zhi Chen Department f Civil Engineering, atinal Cheng Kung University Tainan, Taiwan 7, R.O.C. SUMMARY: The itical buckling lads f unsymmetrically laminated rectangular plates with a given material system and subjected t cmbined lateral and inplane lads are maximized against fiber rientatins by using a sequential linear prgramming methd tgether with a simple mve-limit strategy. Significant influence f plate aspect ratis, central circular cututs, lateral lads and end cnditins n the ptimal fiber rientatins and the assciated ptimal buckling lads f unsymmetrically laminated plates has been shwn thrugh this investigatin. KEYWORDS: buckling, ptimizatin, unsymmetrically laminated plates ITRODUCTIO The cmpsite laminate plates in service are cmmnly subjected t cmpressive frces which may cause buckling. Hence, structural instability becmes a majr cncern in safe and reliable design f the cmpsite plates. The buckling resistance f cmpsite laminate plates depends n end cnditins, ply rientatins [-3], and gemetric variables such as aspect ratis, thicknesses and cututs [,4]. Therefre, fr cmpsite plates with a given material system, gemetric shape, thickness and end cnditin, the prper selectin f apprpriate laminatin t realize the maximum buckling resistance f the plates becmes a ucial prblem. Amng varius ptimizatin schemes, the methd f sequential linear prgramming has been successfully applied t many large scale structural prblems [5,6]. In this study, the itical buckling lads f unsymmetrically laminated plates subjected t inplane and transverse lads are calculated by the bifurcatin buckling analysis implemented in the ABAQUS finite element prgram [7]. Then, buckling ptimizatin f unsymmetrically laminated plates with respect t fiber rientatins is perfrmed by using a sequential linear prgramming methd tgether with a simple mve-limit strategy. In the paper, the buckling analysis, the cnstitutive equatins fr fiber-cmpsite laminate and the ptimizatin methd are briefly reviewed first. Then the influence f plate aspect ratis, central circular cututs, lateral lads and end cnditins n the ptimal fiber rientatins and the assciated ptimal buckling lads f unsymmetrically laminated cmpsite plates is presented.

2 BIFURCATIO BUCKLIG AALYSIS In the finite-element analysis, a set f nnlinear equatins results in the inemental frm: [K t ]d{u} = d{p} ( where [K t ] is the tangent stiffness matrix, d{u} the inemental ndal displacement vectr and d{p} the inemental ndal frce vectr. When the structural defrmatin is small, the nnlinear thery leads t the same itical lad as the linear thery. The linearized frmulatin then gives rise t a tangent stiffness matrix in the fllwing expressin: [K t ] = [K L ] + [K σ ] ( where [K L ] is a linear stiffness matrix and [K σ ] a stress stiffness matrix. If a stress stiffness matrix [K σ ] ref is generated accrding t a reference lad {p} ref, fr anther lad level {p} with λ a scalar multiplier, we have {p} = λ{p} ref, [K σ ] = λ[k σ ] ref (3 When buckling ccurs, the external lads d nt change, i.e., d{p} =. Then the slutin fr the linearized buckling prblem may be determined frm the fllwing eigenvalue equatin: ([K L ] + λ [K σ ] ref d{u} = {} (4 where λ is an eigenvalue and d{u} becmes the eigenvectr defining the buckling mde. The itical lad {p} can be btained frm {p} = λ {p} ref. COSTITUTIVE MATRIX FOR FIBER-COMPOSITE LAMIATE The laminate plates are mdeled by eight-nde isparametric laminate shell elements with six degrees f freedm per nde. The frmulatin f the shell element allws transverse shear defrmatin [7]. Fr fiber-cmpsite laminate materials, the stress-strain relatins fr a lamina in the material crdinates (,,3 at an element integratin pint can be written as { σ '} = ' [ Q ]{ ε '}, { '} [ ' τ = Q ]{ '} γ (5 [Q ' ] = E ν E ν ν ν ν ν E E, [Q' ν ν ν ν ] = α G 3 (6 α G 3 G where {σ'} = {σ, σ, τ } T, {τ'} = {τ 3, τ 3 } T, {ε'} = {ε, ε, γ } T, {γ'} = {γ 3, γ 3 } T. The α and α are shear crrectin factrs calculated by assuming that the transverse shear energy thrugh the thickness f laminate is equal t that f the case f unidirectinal bending [7]. The cnstitutive equatins fr the lamina in the element crdinates (x,y,z then becme [ T { σ } = [ Q ]{ ε }, ] [ ] [ ' Q = T Q ][ T ] (7 [ T ' { τ} = [ Q ]{ γ }, Q ] = [ T ] [ Q ][ T ] (8

3 cs sin sin cs cs sin [T ] = sin cs sin cs, [ T ] = (9 sin cs sin cs cs sin sin cs where {σ} = {σ x, σ y, τ xy } T, {τ} = {τ xz, τ yz } T, {ε} = {ε x, ε y, γ xy } T, {γ} = {γ xz, γ yz } T, and fiber angle is measured cunterclckwise frm the element lcal x-axis t the material - axis. Let {ε } = {ε x, ε y, γ xy } T be the in-plane strains at the mid-surface f the laminate sectin, {κ} = {κ x, κ y, κ xy } T the curvatures, and h the ttal thickness f the sectin. If there are n layers in the layup, the stress resultants, {} = { x, y, xy } T, {M} = {M x, M y, M xy } T and {V} = {V x, V y } T, can be defined as { } { M} {} V = h / {} σ z{} σ {} τ dz = h / j= n ( z jt z jb [ Q ] ( z jt z jb [ Q ] T [] ( z ( 3 z jt 3 jt z z [] T jb 3 jb [ Q ] [ Q ] ( z jt [] [] z jb [ Q { ε } {} κ {} ( ] γ where z jt and z jb are the distance frm the mid-surface f the sectin t the tp and the bttm f the j-th layer. The [] is a 3 by matrix with all the cefficients equal t zer. SEQUETIAL LIEAR PROGRAMMIG A general ptimizatin prblem may be defined as the fllwing: Maximize: f(x (.a Subjected t: g i (x, i =,..., r (.b h j (x =, j = r+,..., m (.c p k x k q k, k =,..., n (.d where x = {x, x,..., x n } T is a vectr f design variables, f(x is an bjective functin, g i (x are inequality cnstraints, and h j (x are equality cnstraints. The p k and q k are lwer and upper limits f the variable x k. Fr the ptimizatin prblem f Eqs. (.a-(.d, a linearized prblem may be cnstructed by apprximating the nnlinear functins at a current slutin pint, x = {x, x,..., x n } T, in a first-rder Taylr series expansin as fllws Maximize: f(x f(x + f(x T δx (.a Subjected t: g i (x g i (x + g i (x T δx, i =,..., r (.b h j (x h j (x + h j (x T δx =, j = r+,..., m (.c p k x k q k, k =,..., n (.d where δx = {x -x, x -x,..., x n -x n } T. It is clear that the abve expressins represent a linear prgramming prblem and the slutin can be btained by the simplex methd [8]. After btaining an apprximate slutin fr Eqs. (.a-(.d, say x, we can linearize the riginal prblem at x and slve the new linear prgramming prblem. The prcess is repeated until a precise slutin is achieved. This apprach is referred t as sequential linear prgramming [5,6]. Althugh the prcedure fr sequential linear prgramming is simple, the ptimum slutin fr the apprximated linear prblem may vilate the cnstraint cnditins f the riginal ptimizatin prblem. In additin, if the true ptimum slutin f a nnlinear prblem appears between tw cnstraint intersectins, a straightfrward successive

4 linearizatin may lead t an scillatin f the slutin between the widely separated values. Difficulties in dealing with such prblems may be avided by impsing a mve limit [5,6] n the linear apprximatin, which is a set f bx-like admissible cnstraints placed n the range f δx. The mve limit shuld gradually apprach t zer as the iterative prcess f the sequential linear prgramming cntinues. RESULTS OF THE OPTIMIZATIO AALYSIS Laminate Plates with Simply Supprted Edges In this sectin rectangular cmpsite laminate plates subjected t uniaxial cmpressive frce per unit length applied at the edges nrmal t the x directin as shwn in Fig. (a are analyzed. The width f the plates, b, is cm while the length, a, is varied between 5 cm and 3 cm. The edges f the plates are all simply supprted, which prevents ut f plane displacement w but allws sme inplane u and v mvements. In this regard, all the pints n the right edge f the plates are enfrced t displace the same amunt u in the x directin, while all the pints n the upper edge f the plates are enfrced t displace the same amunt v in the y directin. The thickness f each ply is.5 mm. The laminate lay-ups f the plates are [ ± / 9 / ] (symmetric lay-up and [( ± / 9 / ] /( / 9 / ] ] s! (asymmetric layup. The lamina cnsists f Graphite/Epxy with material cnstitutive prperties taken frm Crawley [9], which are E = 8 GPa, E = GPa, ν =.5, G = G 3 = 4.48 GPa, G 3 =.53 GPa. w=y =z=,same v V y,v u= w= x,u x= z= v= w =y =z = (a simply supprted plates same u w= x= z= y = u=x= w=z= w=x=y =z=,same v b a v= w =x =y =z = (b clamped plates same u x=y = w=z= Fig. : Rectangular cmpsite laminate plates with different edge cnditins Based n the sequential linear prgramming methd, in each iteratin the current linearized ptimizatin prblem fr symmetrically laminated cmpsite plates becmes: Maximize: ( ( + ( = (3.a Subjected t: 9 (3.b -R Q.5 s (- R Q.5 s (3.c where is the itical buckling lad. The is a slutin btained in the previus iteratin. The R and Q in Eq. (3.c are the size and the reductin rate f the mve limit. In the present study, the values f R and Q are selected t be and.9 (M-, where M is a current iteratin number. T cntrl the scillatin f the slutin, a parameter.5 s is intrduced in the mve limit, where s is the number f scillatins f the derivative / that has taken place befre the current iteratin. The value f s ineases by if the sign f / changes. Whenever scillatin f the slutin ccurs, the range f the mve limit is reduced t half f

5 its current value. This expedites the slutin cnvergent rate very rapidly. The / term in Eq. (3.a may be apprximated by using a frward finite-difference methd with the fllwing frm: + ( ( (4 Hence, t determine the value f / numerically, tw bifurcatin buckling analyses t cmpute ( ο and ( ο + are needed in each iteratin. In this study, the value f is selected t be in mst iteratins. The linearized ptimizatin prblem fr asymmetrically laminated plates can be written as: Maximize: (, (, + ( =, = + ( (5.a =, = Subjected t: 9 (5.b 9 (5.c -R Q.5 s ( - R Q.5 s (5.d -R Q.5 s ( - R Q.5 s (5.e where (, is a slutin btained in the previus iteratin. The values f R and R are selected t be and Q and Q are selected t be.9 (M-. Again, s and s are the numbers f scillatins f the derivatives / and / that have taken place befre the current iteratin, respectively. The / and / terms in eq. (5.a may be apprximated by frward finite-difference expressins as fllws: ( +, (, (6.a, + (, ( (6.b Hence, three bifurcatin buckling analyses t cmpute ( +,, (, + and (, are needed in each iteratin. Again, the value f is selected t be in mst iteratins. Fr each case studied, several different initial guesses f fiber angles are selected t make sure that they all cnverge t the same glbal maximum slutin.

6 Op 5 tim al 4 An gle 3 (de gre es (symmetric (unsymmetric (unsymmetric (k/m (a Aspect rati vs. ptimal fiber angle (b Aspect rati vs. itical lad Fig. : Effect f plate aspect ratis n buckling ptimizatin f [ ± Θ/9/] and [( s ± Θ /9/ /(/9/! Θ ] rectangular cmpsite plates with simply supprted edges and subjected t inplane frce Figure shws the ptimal fiber angle and the assciated ptimal buckling lad with respect t plate aspect rati fr [ ± / 9 / ] and [( ± / 9 / ] /( / 9 / ] ] s! rectangular cmpsite plates. Frm the figure we can see that the ptimal fiber angles and ptimal buckling lads attenuate t cnstant values when the plate aspect ratis are large. It is nt surprising t find that the results f ptimizatin fr plates with symmetric and unsymmetric lay-ups are all the same. This is because when buckling ccurs, there is n preference fr plates t buckle up r dwn. Thus, fr cmpsite plates under inplane lading cnditins, there is n benefit t emply the unsymmetric laminate lay-up. a (symmetric (unsymmetric y x P b d q Op 8 tim al 7 An 6 (a cncentrated lad at center (b unifrm line lad arund central cutut Fig. 3: Rectangular plates with cmbined inplane and lateral frces (symmetric (unsymmetric gle5 (unsymmetric (de4 gre 3 es (a Aspect rati vs. ptimal fiber angle (k/m (b Aspect rati vs. itical lad Fig. 4: Effect f plate aspect ratis n buckling ptimizatin f [±/9/] s and [( ± Θ /9/ /(/9/! Θ ] rectangular cmpsite plates with simply supprted edges and subjected t cmbined inplane and lateral frces 4 3 (symmetric (unsymmetric

7 In additin t inplane frces, the cmpsite plates in service may als be subjected t lateral frces. As anther example, cmpsite plates similar t previus nes but with an additinal lateral cncentrated lad P acting at the center f the plates as shwn in Fig. 3(a are analyzed. In the analysis, the rati P/(b =.5 is used. Figure 4 shws the ptimal fiber angle and the assciated ptimal buckling lad with respect t plate aspect rati fr [ ± / 9 / ] s and [( ± / 9 / ] /( / 9 /! ] ] rectangular cmpsite plates. The figure shws that when the plate aspect rati is large (say >, the results f ptimizatin fr plates with unsymmetric lay-ups are very similar t thse fr plates with symmetric lay-ups. In additin, as rati ineases, the ptimal fiber angles and ptimal buckling lads gradually apprach cnstant values. Hwever, when the plate aspect rati is small (say <, the ptimal fiber angles f plates with asymmetric lay-ups are quite different frm thse f plates with symmetric lay-ups. In additin the ptimal buckling lads f the frmer plates are much higher than thse f the latter plates. By cmparing with the plates with symmetric layups, the implementatin f unsymmetric layups in sme cases (say =.7 may inease the ptimal buckling lads f plates by 7%. It is nted [] that when the plate aspect rati is small the ptimal buckling mdes f unsymmetrically laminated plates are quite different frm thse f symmetrically laminated plates. Hwever, when the plate aspect rati is large, the ptimal buckling mdes f unsymmetrically laminated plates are very similar t thse f symmetrically laminated plates. Als, as rati ineases, the ptimal buckling mdes f plates have mre waves in x directin. Laminate Plates with Clamped Edges In rder t find the effect f bundary cnditins n the results f ptimizatin, the cmpsite plates subjected t cmbined inplane and lateral frces in previus sectin are analyzed again, hwever, with all edges clamped as shwn in Fig. (b. Figure 5 shws the ptimal fiber angle and the assciated ptimal buckling lad with respect t plate aspect rati fr [ ± / 9 / ] s and [( ± / 9 / ] /( / 9 /! ] ] rectangular cmpsite plates with clamped edges. Frm the figure we can bserve that except plates with small aspect rati (say arund.5, the results f ptimizatin fr plates with unsymmetric lay-ups are very different frm thse fr plates with symmetric lay-ups. In additin, when, the ptimal buckling lads f plates with unsymmetric lay-ups are higher than thse f plates with symmetric layups by up t %. It is als knwn [] that the ptimal buckling mdes fr symmetrically and unsymmetrically laminated plates with clamped edges and under ptimal fiber rientatin are very similar. Again, as rati ineases, the ptimal buckling mdes f plates have mre waves in x directin. Op 8 tim 7 (sym al An 6 (unsym gle 5 (unsym (de 4 gre 3 es (a Aspect rati vs. ptimal fiber angle (k/m (symmetric (unsymmetric (b Aspect rati vs. itical lad Fig. 5: Effect f plate aspect ratis n buckling ptimizatin f [±/9/] s and [( ± Θ /9/ /(/9/! Θ ] rectangular cmpsite plates with clamped edges and subjected t cmbined inplane and lateral frces

8 Laminate Plates with Varius Central Circular Cututs and with Simply Supprted Edges In this sectin [ ± / 9 / ] and [( ± / 9 / ] /( / 9 / ] ] s! rectangular cmpsite laminate plates with central circular cututs and subjected t cmbined in-plane frce and lateral frce (a unifrm line lad f intensity q as shwn in Fig. 3(b are analyzed. The edges f the plates are all simply supprted. The width f the plates, b, is cm, the length f the plates, a, is 7 cm, the diameter f the hle, d, varies frm t 5 cm, and the rati qπd/(b =.5 is used. Figure 6 shws the ptimal fiber angle and the assciated ptimal buckling lad with respect t the rati d/b fr symmetric and unsymmetric rectangular cmpsite plates. The figure shws that the results f ptimizatin fr plates with unsymmetric lay-ups are very different frm thse fr plates with symmetric layups. In additin, the ptimal buckling lads f the frmer plates are much higher than thse f the latter plates. In sme cases (say d/b =.5, the implementatin f unsymmetric lay-ups may inease the buckling lads f the plates by 3%. Fr plates with symmetric lay-ups, the ptimal buckling lad (say d/b <.4 first deeases with the ineasing f d/b rati then it (say d/b >.4 ineases with the inease f the cutut sizes. Fr plates with unsymmetric lay-ups, the ptimal buckling lad ineases with the inease f the sizes f cututs. This phenmenn is quite different frm ur intuitin that intrducing a large hle int a plate can cause a reductin in the buckling lad f the plate. Hwever, past research did shw (numerically and experimentally that intrducing a hle int an istrpic plate r a cmpsite plate des nt always reduce the buckling lad and, in sme instances, may inease its buckling lad [4,]. This is because that the buckling lad f a plate is nt nly influenced by cutut, but als influenced by material rthtrpy, end cnditin, and plate gemetry. It is als knwn [] that the buckling mdes f plates with unsymmetric lay-ups under ptimal cnditins are quite different frm thse f plates with symmetric lay-ups. Generally, the buckling mdes f plates with unsymmetric lay-ups have mre waves in bth x and y directins. Op 8 tim al 7 An 6 (symmetric gle 5 (unsymmetric (de gre4 (unsymmetric es d/b (a d/b vs. ptimal fiber angle (k/m (symmetric (unsymmetric d/b (b d/b vs. itical lad Fig. 6: Effect f cutut size n buckling ptimizatin f [±/9/] s and [( ± Θ /9/ /(/9/! Θ ] rectangular cmpsite plates with simply supprted edges and subjected t cmbined inplane and lateral frces ( =.7

9 Laminate Plates with Varius Central Circular Cututs and with Clamped Edges In this sectin, the cmpsite laminate plates with central circular cututs similar t thse in previus sectin are analyzed again, hwever, with all edges changed t clamped cnditins and rati changed t.5. The width b f the plates is still cm, the cutut size varies between cm and 8 cm, and the rati qπd/(b =.5 is still kept in the analysis. Figure 7 shws the ptimal fiber angle and the assciated ptimal buckling lad with respect t the rati d/b fr [±/9/] s and [(± /9/ /(/9/± ] rectangular cmpsite plates. Figure 7(a shws that the ptimal fiber angles fr plates with asymmetric lay-ups are very different frm thse fr plates with symmetric lay-ups. Figure 7(b shws that the ptimal buckling lads f the frmer plates are generally higher than thse f the latter plates. In sme cases (say d/b <., the implementatin f unsymmetric lay-ups may inease the buckling lads f the plates by 5%. Fr plates with either symmetric r asymmetric lay-up, the ptimal buckling lad seems t be a secnd-rder functin f the cutut size. It is als knwn [] that when d/b >., the buckling mdes f plates with unsymmetric lay-ups under ptimal cnditins are quite different frm thse f plates with symmetric lay-ups. evertheless, fr bth types f plates, when the cutut sizes are large, the buckling mdes are mre lcally arund the cutut areas. Op 7 tim al 6 (sym An 5 (unsym gle 4 (unsym (de gre3 es d/b (a d/b vs. ptimal fiber angle (k/m (symmetric (unsymmetric d/b (b d/b vs. itical lad Fig. 7: Effect f cutut size n buckling ptimizatin f [±/9/] s and [( ± Θ /9/ /(/9/! Θ ] rectangular cmpsite plates with clamped edges and subjected t cmbined inplane and lateral frces ( =.5 COCLUSIOS In the prcess f sequential linear prgramming, mst ptimal results are btained within 3 iteratins, and the results are all verified by chsing different initial guesses. Hence, as a general cnclusin, the sequential linear prgramming is efficient and stable t slve nnlinear ptimizatin prblems. Fr the ptimal buckling analysis f uniaxially cmpressed symmetric [ ± / 9 / ] and unsymmetric [( ± / 9 / ] /( / 9 / ] ] s! laminated plates with varius plate aspect ratis, circular cututs and end cnditins, the fllwing cnclusins may be drawn:. Fr cmpsite plates under inplane lading cnditins, there is n benefit t be derived frm emplying an unsymmetric laminate layup.. Fr cmpsite plates with simply supprted edges and subjected t cmbined inplane and lateral lads, it is beneficial t emply the unsymmetric laminate lay-ups when the plate

10 aspect rati is small (say <. 3. Fr cmpsite plates with clamped edges and subjected t cmbined inplane and lateral lads, the adptin f unsymmetric laminate design is beneficial when the plate aspect rati is large (say >. 4. Fr cmpsite plates with a central circular cutut and subjected t cmbined inplane and lateral lads, the use f unsymmetric laminate layups is recmmended. The ptimal buckling lads f these plates in sme cases (say plates with clamped edges may inease with the ineasing f cutut sizes. Hence, it is pssible t tailr the cutut size and fiber angle t inease the buckling lads f these plates beynd thse f crrespnding plates withut cututs. In this paper, bifurcatin buckling analysis is carried ut based n the assumptin that the cmpsite laminate material behaves linearly. Fr lw aspect rati plates and fr plates with large cututs, the stresses in the laminates may exceed the elastic range and these laminates are prbably driven by cmpressin strength failure in stead f buckling. In these cases, buckling analyses f cmpsite plates based n nnlinear material prperties are recmmended []. REFERECES.Hiran, Y., Optimum Design f Laminated Plates under Axial Cmpressin, AIAA Jurnal, Vl. 7, 979, pp Leissa, A.W., Buckling f Laminated Cmpsite Plates and Shell Panels. AFWAL-TR , Flight Dynamics Labratry, Air Frce Wright Aernautical Labratries, Wright- Pattersn Air Frce Base, Ohi, Muc, A., Optimal Fibre Orientatin fr Simply-Supprted Angle-Ply Plates under Biaxial Cmpressin, Cmpsite Structures, Vl. 9, 988, pp emeth, M.P., Buckling Behavir f Cmpressin-Laded Symmetrically Laminated Angle-Ply Plates with Hles, AIAA Jurnal, Vl. 6, 988, pp Zienkiewicz, O.C. and Champbell, J.S., Shape Optimizatin and Sequential Linear Prgramming, Optimum Structural Design, Thery and Applicatins, edited by Gallagher, R.H. and Zienkiewicz, O.C.,Wiley, ew Yrk, 973, pp Vanderplaats, G.., umerical Optimizatin Techniques fr Engineering Design with Applicatins, Chapter 6, McGraw-Hill, ew Yrk Hibbitt, Karlssn & Srensen, Inc., ABAQUS User and Thery Manuals Versin 5.7, Prvidence, Rhde Island, Klman, B. and Beck, R.E., Elementary Linear Prgramming with Applicatins, Chapter, Academic Press, Orland, Crawley, E.F., The atural Mdes f Graphite/Eever Plates and Shells, Jurnal f Cmpsite Materials, Vl. 3, 979, pp

11 . Chen, Z.-Z., Buckling Optimizatin f Symmetric and Unsymmetric Laminate Plates Subjected t Cmbined Inplane and Lateral Lads, M.S. Thesis, Department f Civil Engineering, atinal Cheng Kung University, Tainan, Taiwan, R.O.C., Ritchie, D. and Rhades, J., Buckling and Pstbuckling Behavir f Pates with Hles, Aernautical Quarterly, Vl. 6, 975, pp Hu, H.-T., Buckling Analyses f Fiber Cmpsite Laminate Plates with Material nlinearity, Finite Elements in Analysis and Design, Vl. 9, 995, pp