Optimum design of laminated composite under axial compressive load

Transcription

1 Sādhanā Vol. 36, Part 1, February 2011, pp Indan Academy of Scences Optmum desgn of lamnated composte under axal compressve load NGRIYENGAR 1, and NILESH VYAS 2 1 Department of Aerospace Engneerng, Jan Unversty, Jan Global Campus, Jakkasandra Post, Bangalore , Inda 2 Hndustan Petroleum Corporaton Lmted (HPCL), Mangalore LPG Import Faclty, Mangalore , Inda e-mal: ngr@tk.ac.n; nleshvyas.nv@gmal.com MS receved 28 Aprl 2010; revsed 27 August 2010; accepted 17 September 2010 Abstract. In the present study optmal desgn of composte lamnates, wth and wthout rectangular cut-out, s carred out for maxmzng the bucklng load. Optmzaton study s carred out for obtanng the maxmum bucklng load wth desgn varables as ply thckness, cut-out sze and orentaton of cut-out wth respect to lamnate. Bucklng load s evaluated usng a smple hgher order shear deformaton theory based on four unknown dsplacements u, v, w b and w s.ac 1 contnuous shear flexble fnte element based on HSDT model s developed usng Hermte cubc polynomal. It s observed that for thck ant-symmetrc lamnates, the non-dmensonal bucklng load decreases wth ncrease n aspect rato and ncrease n fbre orentaton angle. There s a decrease n the non-dmensonal bucklng load of symmetrc lamnate n the presence of cut-out. Keywords. Composte lamnate; genetc algorthm; bucklng; optmum desgn; fnte element. 1. Introducton Bucklng of structural elements lke columns, plates, shells, etc. whether slender or thn s an mportant phenomenon and certanly has to be looked nto desgn phase tself. In aerospace structures generally thn walled members are used from the weght consderaton. Hence they are prone to bucklng under n-plane loads. Cut-outs are generally made n the structures ether to lghten the structures or to carry cables, etc. Two survey artcles by Lessa (1981, 1987) deal extensvely wth bucklng of lamnated composte plates. Results for both the symmetrc and ant-symmetrc modes of bucklng of ant-symmetrc cross-ply plates wth varous aspect ratos have been presented n a graphcal form by Hu (1984). Studes conducted by Ressner (1945), Whtney (1969) and Whtney & Pagano (1970) show that transverse shear effect s qute sgnfcant n layered composte plates due to hgh rato of n-plane elastc modulus to transverse shear modulus. They predct better results than the classcal lamnaton theory. However, the error n For correspondence 73

2 74 N G R Iyengar and Nlesh Vyas the soluton ncreases wth ncrease n plate thckness to sde rato. To account for a better representaton of the shear dstrbuton through the thckness, (Reddy 1984) proposed a hgher order shear deformaton theory. Reddy (1990) presented a general non-lnear thrd order theory for plates wth moderate thckness. Reddy (2006) n hs recent book dscussed the response of composte plates and shells. A smplfed hgher order theory proposed by Lm et al (1988) nvolves only four unknowns nstead of fve by Reddy (1984). Sngh (1993) and Chakraborty (2003) used ths model for the response of lamnated composte beams and plates. Srvatsa & Murthy (1991) presented a parametrc study of the compresson bucklng behavour of stress loaded composte plate wth a central crcular cut-out. Jan & Kumar (2003) analysed the post bucklng response of square lamnates wth a central crcular/ellptcal cut-out. Prabhakara & Datta (1997) studed the vbraton and bucklng behavour of plates wth centrally located cutouts. Anl (2004) analysed lamnated plates for ntal bucklng wth cut-outs under un-axal and baxal compressve and shear n-plane loadng, usng smple HSDT. Genetc algorthm (GA) has become a powerful and robust tool for functon optmzaton (Goldberg 1989, Deb 2001). These algorthms are computatonal smple but powerful n ther search for mprovement n successve generatons (Nagendra et al 1992). GA mmcs some of the natural process observed n natural evoluton. The basc technques of GA are desgned to smulate mechansm of populaton genetcs and natural laws of survval. One of the bg advantages of GA s that t does not requre dfferentablty of ether objectve functon or constrants (Nagendra et al 1992). The constrant handlng capacty of GA s better than classcal optmzaton technques because of populaton based approach (Callahan & Weeks 1992). Nagendra et al (1992) studed the bucklng optmzaton of lamnate sequence wth stran constrants. Callahan & Weeks (1992) studed the optmum desgn of composte lamnates for maxmzng lamnate strength and stffness wth fxed number of ples. They employed tournament selecton scheme n the selecton process. Sngle pont crossover s used wth a crossover probablty of P c = 0.75 and mutaton probablty of 0.1 per cent. Kogso et al (1994) appled GA wth memory for desgn of mnmum thckness composte lamnates subject to strength, bucklng and ply contguty condtons. Muc & Gurba (2001) used GA for layout optmzaton of composte structures wth fnte element optmzaton of objectve functon. Svakumar et al (1999) used GA for optmzng the composte lamnates wth cut-outs undergong large ampltude oscllatons. They compared GA wth many other algorthms and found GA to be better n almost every aspect. It s observed that most researchers have used closed form soluton to solve the bendng, bucklng or vbraton problems. FEM has proved to be a good approxmaton for structures for whch closed form soluton s not possble. In the study presented here, optmzaton of composte lamnates s carred out for maxmzng the bucklng load wth and wthout cut-out usng GA. The effects of varous parameters such as aspect rato, cut-out sze, crossover and mutaton probabltes on the bucklng load are nvestgated. 2. Formulaton 2.1 Consttutve relatons Ths secton deals wth detals of the plate theory proposed by (Lm et al 1988) for sotropc plates and extended to composte lamnates. The dsplacement feld ncludes plate theory and frst order shear deformaton theory as subsets and accounts for parabolc varaton of the transverse shear strans as well as the surface boundary condtons of zero transverse shear stresses at the top and bottom surfaces of the lamnate.

3 Optmum desgn of lamnated composte 75 The coordnate system used s shown n fgure 1. The dsplacements u (x, y, z), v(x, y, z) and w (x, y, z) at any pont n the lamnate s wrtten as u (x,y,z) = u 0 (x,y) z w b x (x,y) z2 φ x (x,y) z 3 ψ x (x,y) v (x,y,z) = v 0 (x,y) z w b (x,y) z2 φ y (x,y) z 3 ψ y (x,y) w (x,y,z) = w b (x,y) w s (x,y), (1) where u 0, v 0, w denote the dsplacements of a pont on the md-plane, and x and y are the warpng of the normal to the md-plane about the Y and X axes respectvely. The transverse dsplacement component w b s such that ts dervatves are numercally equal to the rotaton of the cross secton and w s s the dsplacement due to the effect of transverse shear deformaton of the cross secton. The von-karman type non-lnear stran dsplacement relatons s wrtten as ε x = u 0 x 1 ( wb 2 x w ) 2 s z 2 w b x x 2 ε y = v 0 1 ( wb 2 w ) 2 s z 2 w b 2 ε z = 0 γ xy = u 0 v ( 0 x wb x w s x ( z 3 ψy x ψ ) x γ xz = w s x 2zφ x 3z 2 ψ x φ z2 x x ψ z3 x x φ z2 x ψ z3 x )( wb w ) ( s 2z 2 w b x φy z2 x φ ) x γ yz = w s 2zφ y 3z 2 ψ y. (2) Fgure 1. Coordnate system for the lamnate.

4 76 N G R Iyengar and Nlesh Vyas Incorporatng the condtons that transverse shear stresses vansh at the top and bottom faces of the lamnate, the resultng dsplacement feld s gven as u (x,y,z) = u 0 (x,y) z w b x 4z3 w s 3h 2 x v (x,y,z) = v 0 (x,y) z w b 4z3 w s 3h 2 w (x,y,z) = w b (x,y) w s (x, y). (3) It can be shown that the present formulaton degenerates to frst order shear deformaton theory by sutably droppng the hgher order terms and ncorporatng the approprate shear correcton factors. The classcal lamnate theory can be derved from the present theory by equatng w s to zero. The stress stran relatons for the k th lamna n the materal coordnate axes are gven as (Jones 1975): σ L σ T τ LT τ LT τ TT = Q 11 Q Q 12 Q Q Q Q 55 ε L ε T γ LT γ LT λ TT. (4) Eq. (4) s rewrtten for each lamna dependng on the fbre orentatons. Usng these transformed consttutve equatons and ntegratng over the lamnate thckness, the stress resultants, stress couples, transverse shear resultants and hgher order stress couples per unt length for the lamnate are; {N } {M } {P } {Q } = [ ] [ ] [ ] [ Aj ] [ Bj ] [ Ej ] [ Bj ] [ Dj ] [ Fj ] Ej Fj Hj [0] [0] [0] [0] [G lm ] { } [0] [0] { εoj } { κoj } κlj {γ m } (, j = 1, 2and6), (l, m = 4, 5), where {ε 0 } are the md-surface strans, {κ 0 } are the md-surface bendng and twstng curvatures, {κ l } are the hgher order terms and {γ } are the transverse shear slope (for detals of the dervaton of varous elements of Eq. (5), see Vyas 2005). (5) 2.2 Fnte element formulaton In the present formulaton, there are four varables namely, two n-plane dsplacements and two out-of-plane dsplacements. Four-noded rectangular elements are used to represent the entre doman of the lamnate. A C 1 contnuous shear flexble element s developed usng Hermte nterpolaton formulae as suggested by Bogner et al (1966). A typcal mesh generated over the entre lamnate wth centrally located cut-out s shown n fgure 2. The nodal dsplacement vector {δ} s wrtten n terms of the four unknown dsplacements as {δ} T = { u 0 v 0 w b w s }. (6)

5 Optmum desgn of lamnated composte 77 Fgure 2. Graded mesh for a rectangular doman wth a cut-out. For any element the feld varables can be wrtten n terms of the shape functons and nodal varables as: u 0 (x,y) = v 0 (x,y) = w b (x,y) = w s (x,y) = N u 0 N v 0 N w b N w s N 4 N 4 N 4 N 4 ( ) u0 x ( ) v0 ( wb x ( ws x ) ) N 8 N 8 N 8 N 8 ( ) u0 ( ) v0 ( wb ( ws ) ) N 12 N 12 N 12 N 12 ( 2 ) u 0 x ( 2 ) v 0 x ( 2 ) w b x ( 2 ) w s x. (7) In Eq (7), u 0, v 0, w b, w s, are the sxteen degrees of freedom per node, N s are the element shape functons n local coordnates. Reddy (2004) gves the shape functons n natural coordnates. The boundary condtons appled along the edges of the lamnate are as follows: smply supported along all the edges: w b = w s = 0, clamped along all the edges: w b = w s = w b,n = w s,n = 0. The nvestgatons are confned to the ntal bucklng of the lamnate. Therefore, w b and w s areassumedtobeverysmall.

6 78 N G R Iyengar and Nlesh Vyas The undeformed confguraton of the lamnate s denoted by and ts boundary by. The nfntesmal stran s defned n terms of the Cartesan components of the dsplacements (u, = 1, 2 and 3) ε j = 1 ( ) uj u j, (8) 2 whch s the smplfcaton of the Green Lagrange strans defned by j = ε j 1 ) (u α, u 2 α,j. (9) The smplfcaton s justfed by the assumpton that u α, 1 and hence the product terms are neglgble n relaton to u j. The consttutve equaton s σ j = σ 0 j C jkl ε kl. (10) In whch σj 0 s the pre-exstng stress state ndependent of u and C jkl s the tenor of the elastc modul of the materal. An mportant property of σj 0 s that t s n equlbrum wth the correspondng tractons T 0 = σj 0 n j n the sense (Paul 1998). 1 2 σj 0 ( ) u, j u j, dv = T 0 u da for all u E ( ), (11) where dv and da represent the dfferental volume and dfferental area respectvely. E( ) s the space of knematcally admssble perturbatons. When the reference confguraton s stress free, then the potental energy s defned as (u) = 1 2 C jkl ε j ε kl dv T u da. (12) The exact soluton mnmzes on the set of all knematcally admssble functons denoted by E( ). When the reference confguraton s not stress free, then the work done by the stress σj 0 due to nonlnear stran terms may not be neglgble. The potental energy expresson becomes (u) = 1 2 C jkl ε j ε kl dv 1 2 σj 0 u α, j u α, j dv T u da. (13) The frst term n Eq. (13) represents the nternal stran energy and the second term represents the work done by the ntal stresses due to the nonlnear stran terms. The work done by σj 0 due to the lnear stran terms s cancelled by the work done by T 0 n the sense of Eq. (12). The exact soluton to the problem s obtaned by mnmzng the total potental energy.ths leads to the generalzed fnte element formulaton whch can be expressed as [ [ ]] { } [K ] Kg δ = 0. (14) The elastc stffness and geometrc stffness matrces are obtaned n the usual manner.

7 Optmum desgn of lamnated composte Optmzaton formulaton Genetc algorthm (GA) s used for optmzng the bucklng load. The optmzaton problem s defned n the followng manner. Maxmze λ (bucklng load) Subject to θ ± 60, ± 45, ±30, 0, 90 N h = h h l h h u. The bucklng load λ s obtaned from Eq. (14). 2.3a Evaluaton of ftness: If the constrant s not volated, the ftness s the bucklng load obtaned by solvng the characterstc equaton of Eq. (14). If the constrant s volated, a very hgh penalty factor of the order of 100,000 s mposed. Ths s expressed as f the constrant s not volated, Ftness = bucklng load λ or else Ftness = Bucklng load ( 10 5 Prescrbed thckness Evaluated total thckness ) 2.3b Codng n genetc algorthm: The ply angles and the group ply thckness are coded n strngs. A ten layer lamnate s consdered for the study. The ply angles are coded from 1 to 8. These stand for the stackng sequence, 60, 45, 30,0,30,45,60,90 respectvely. The lamnate thckness s consdered n multples of ten and s generated wthn the specfed bounds on ndvdual ply thckness. 2.3c Convergence crtera: The convergence of the soluton to the optmum value s decded by The constrant value at the optmum pont Repetton of same value wth dfferent startng values Repetton of the populaton n fnal generaton. 2.3d Optmzaton study: Problem 1: For optmzaton study of the lamnate wthout cut-out, the followng data have been consdered. Materal: E L = 130 GPa, (E L /E T ) = 13, (G LT /E T ) = 0.5, (G LT /E T ) = 0.5, (G TT /E T ) = 0.5,ν LT = 0.35 Number of ples: 10 Boundary condton: all edges are smply supported Objectve functon: Maxmze bucklng load.

8 80 N G R Iyengar and Nlesh Vyas Constrants: θ ± 60, ± 45, ± 30, 0, h = 2mm, 0.1 h 0.4 GA Parameters: Maxmum number of generatons = 100 Populaton sze = 50 Probablty of cross-over = 0.80 Probablty of mutaton = 0.02 Optmzed results: Optmal code of the lamnate: [ 45 /30 / 45 /0 ] 2 s Ply thckness = [ 0.1/0.2/0.2/0.3/0.2 ] s. Problem 2: For optmzaton study of the lamnate wth rectangular cut-out, the followng data has been consdered. Materal: E L = 200 GPa, (E L /E T ) = 40, (G LT /E T ) = 0.6, (G LT /E T ) = 0.6, (G TT /E T ) = 0.5,ν LT = 0.25 Number of ples: 8 Boundary condtons: all edges smply supported Objectve functon: Maxmze bucklng load. Constrants: θ ± 60, ± 45, ± 30, 0, 90 8 h = 1.6mm h c/a 0.8. GA parameters: Maxmum number of generatons: 50 Populaton sze = 40 Probablty of cross over = 0.85 Probablty of mutaton = Optmzed results: Optmal code of the lamnate [ 30 /45 / 45 /30 ] s Ply thckness: [ 0.1/0.2/0.2/0.3 ] s Optmum c/a rato: Results and dscusson Both symmetrc and ant-symmetrc lamnates, wth smply supported and clamped edges are consdered for computaton. The valdaton of the genetc algorthm code s carred out on three sets of problems for whch results are avalable. The tests problems are hard globalzaton

9 Optmum desgn of lamnated composte 81 Table 1. Comparson of optmzed results for varable thckness lamnates wth unform thckness lamnates. a/b = 1 and a/h = 100 Unoptmzed results Optmzed results S. No Ply ND Thckness ND Thckness % orentatons Bucklng (mm) Bucklng (mm) Increase load wth load wth constant varable thckness thckness 1 [30/ 30/30/ 30]s [0.2/0.2/0.2/0.2]s [0.1/0.2/0.2/0.3]s [45/ 45/45/ 45]s [0.2/0.2/0.2/0.2]s [0.1/0.2/0.2/0.3]s [60/ 60/60/ 60]s [0.2/0.2/0.2/0.2]s [0.1/0.2/0.2/0.3]s [90/0/30/ 30]s [0.2/0.2/0.2/0.2]s [0.1/0.1/0.3/0.3]s [90/0/45/ 45]s [0.2/0.2/0.2/0.2]s [0.1/0.1/0.3/0.3]s [90/0/60/ 60]s [0.2/0.2/0.2/0.2]s [0.1/0.2/0.3/0.3]s [0/30/0/30]s [0.2/0.2/0.2/0.2]s [0.1/0.2/0.3/0.2]s [0/45/0/45]s [0.2/0.2/0.2/0.2]s [0.1/0.2/0.3/0.2]s [0/60/0/60]s [0.2/0.2/0.2/0.2]s [0.1/0.2/0.3/0.2]s 0.82 problems on whch most conventonal local optmzaton algorthms would fal mserably (Paul 1998). For the response of square lamnate wth a square cut-out, the followng lamnate sequencng was studed ( [ 45 / 45 /45 / 45 ] and [ 0 s /90 /0 /90 ] ) for smply supported s boundary condtons. From the analyss of the results t s observed that for smply supported boundary condtons, the non-dmensonal bucklng load remans farly constant for a cutout rato greater than 0.4. Smlar trend s not observed n the case of clamped boundary condton. Table 1 shows the results obtaned for a unform and varable thckness lamnate along wth unoptmzed and optmzed non-dmensonal bucklng load. It s seen that wth constrant on overall thckness, varable thckness lamnate s better than unform thckness. Some typcal results are presented n ths paper. Fgures 3 and 4 show the varaton of fbre orentaton θ wth optmzed non-dmensonal bucklng load for a square lamnate wthout Fgure 3. Optmzed non-dmensonal bucklng load versus fbre orentaton for a square lamnate wthout cut-out. (a/h = 10).

10 82 N G R Iyengar and Nlesh Vyas Fgure 4. Optmzed non-dmensonal bucklng load versus fbre orentaton for a square lamnate wthout cut-out. (a/h = 100). cut-out for a/h ratos of 10 and 100 respectvely. From fgure 3 t s observed that for a/h = 100, θ crc s approxmately 45 and as the thckness of the lamnate ncreases θ crc reduces as seen n fgure 4 for a/h= 10. Fgures 5 and 6 show the results for a square lamnate wth rectangular cut-out wth a/h = 10 and 100 respectvely. θ crc, remans approxmately the same as for lamnates wthout cut-out. However, the maxmum bucklng load occurs when (c/a = d/b = 0.2). Results have been computed for both thn and thck lamnates and wth and wthout cut-outs for varous aspect ratos. It was observed that the non-dmensonal bucklng load decreases as the lamnates become thcker. Further, the fbre orentaton θ crc (the value of θ at whch the maxmum bucklng load occurs) decreases as the lamnates become thck. Fgures 7 and 8 show the effect of crossover probablty and mutaton probablty on the convergence of the soluton for a lamnate wthout cut-out. From fgure 8, t s observed that hgh mutaton probablty gves better soluton for earler generatons only, but at the same tme t destroys the good soluton whch s already created. At hgher mutaton probablty the ntal convergence s faster. However, as the number of generatons s ncreased the convergence to global optmum s lost. Fgure 9 shows the effect of crossover probablty on the convergence of the soluton for a lamnate wth cut-out. The soluton mproves wth hgher crossover probablty. However, the rate of convergence s slow. Fgure 5. Optmzed non-dmensonal bucklng load versus fbre orentaton for a square lamnate wth cut-out. (a/h = 10, optmum c/a = d/b = 0.2).

11 Optmum desgn of lamnated composte 83 Fgure 6. Optmzed non-dmensonal bucklng load versus fbre orentaton for a square lamnate wth cut-out. (a/h = 100, optmum c/a = d/b = 0.2). Fgure 7. Non-dmensonal bucklng load versus the number of generaton (square plate). Fgure 8. Non-dmensonal bucklng load versus number of generaton (square plate).

12 84 N G R Iyengar and Nlesh Vyas Fgure 9. Non-dmensonal bucklng load versus number of generatons (square plate). 4. Conclusons The followng conclusons can be made on the bass of the lmted study: () For thck ant-symmetrc lamnates, the non-dmensonal optmum bucklng load decreases wth ncrease n aspect rato and ncrease n fbre orentaton angle. () There s a decrease n the non-dmensonal bucklng load of a symmetrc lamnate n the presence of cut-out. There s a crtcal cut-out sze wth respect to lamnate sze, where the bucklng load exceeds that of plate wthout cut-out. () Mutaton s clearly needed as a source of varablty, but too much of t defntely deleterous. (v) For a gven number of varables, populaton sze, there exsts a crtcal probablty of mutaton. (v) Effect of crossover probablty s not sgnfcant on the optmum soluton. References Anl V 2004 Stablty analyss of composte lamnates wth and wthout cut-out. M.Tech Thess, Indan Insttute of Technology, Kanpur, Inda Bogner F K, Fox R L, Schmt L A 1966 The generaton of nter element compatble stffness and mass matrces by the use of a nterpolaton formula. Matrx Methods n Structural Mechancs Callahan J K, Weeks E G 1992 Optmum desgn of composte lamnates usng genetc algorthm. Compos. Eng. 2: Chakraborty A 2003 Stablty analyss of composte lamnates usng smple hgher order shear deformaton theory. M.Tech Thess, Indan Insttute of Technology, Kanpur, Inda Deb K 2001 Mult-objectve optmzaton usng evolutonary algorthms. John Wley and Sons Goldberg D D 1989 Genetc algorthm n search, optmzaton, and machne learnng. Addson Wesley Hu D 1984 Shear bucklng of ant-symmetrc cross-ply rectangular plates. Fbre Sc. Technol. 21: 327 Jan P, Kumar A 2003 Post bucklng response of square lamnates wth a central crcular/ellptcal cutout. Comput. Struct. 65: Jones R M 1975 Mechancs of composte materals. Phladelpha, PA: Taylor & Francs, Second Edton Kogso M, Watson L T, Gurdal L, Hftka R T, Nagendra S 1994 Desgn of composte lamnates by a genetc algorthm wth memory. Mech. Compos. Mater. Struct. 1: Lessa A W 1981 Advances n vbraton, bucklng and post bucklng studes n composte plates, composte structures. Proceedngs I st Internatonal Conference on Composte Structures, London: Appled Scence Publshers, Lessa A W 1987 A revew of lamnated composte plate bucklng. Appl. Mech. Rev. 40: 5

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